--- a/NEWS Mon Apr 23 12:23:23 2012 +0100
+++ b/NEWS Mon Apr 23 12:14:35 2012 +0200
@@ -647,6 +647,180 @@
DERIV_nonneg_imp_nonincreasing ~> DERIV_nonneg_imp_nondecreasing
+* Theory Library/Multiset: Improved code generation of multisets.
+
+* Session HOL-Probability: Introduced the type "'a measure" to represent
+measures, this replaces the records 'a algebra and 'a measure_space. The
+locales based on subset_class now have two locale-parameters the space
+\<Omega> and the set of measurables sets M. The product of probability spaces
+uses now the same constant as the finite product of sigma-finite measure
+spaces "PiM :: ('i => 'a) measure". Most constants are defined now
+outside of locales and gain an additional parameter, like null_sets,
+almost_eventually or \<mu>'. Measure space constructions for distributions
+and densities now got their own constants distr and density. Instead of
+using locales to describe measure spaces with a finite space, the
+measure count_space and point_measure is introduced. INCOMPATIBILITY.
+
+ Renamed constants:
+ measure -> emeasure
+ finite_measure.\<mu>' -> measure
+ product_algebra_generator -> prod_algebra
+ product_prob_space.emb -> prod_emb
+ product_prob_space.infprod_algebra -> PiM
+
+ Removed locales:
+ completeable_measure_space
+ finite_measure_space
+ finite_prob_space
+ finite_product_finite_prob_space
+ finite_product_sigma_algebra
+ finite_sigma_algebra
+ measure_space
+ pair_finite_prob_space
+ pair_finite_sigma_algebra
+ pair_finite_space
+ pair_sigma_algebra
+ product_sigma_algebra
+
+ Removed constants:
+ distribution -> use distr measure, or distributed predicate
+ joint_distribution -> use distr measure, or distributed predicate
+ product_prob_space.infprod_algebra -> use PiM
+ subvimage
+ image_space
+ conditional_space
+ pair_measure_generator
+
+ Replacement theorems:
+ sigma_algebra.measurable_sigma -> measurable_measure_of
+ measure_space.additive -> emeasure_additive
+ measure_space.measure_additive -> plus_emeasure
+ measure_space.measure_mono -> emeasure_mono
+ measure_space.measure_top -> emeasure_space
+ measure_space.measure_compl -> emeasure_compl
+ measure_space.measure_Diff -> emeasure_Diff
+ measure_space.measure_countable_increasing -> emeasure_countable_increasing
+ measure_space.continuity_from_below -> SUP_emeasure_incseq
+ measure_space.measure_incseq -> incseq_emeasure
+ measure_space.continuity_from_below_Lim -> Lim_emeasure_incseq
+ measure_space.measure_decseq -> decseq_emeasure
+ measure_space.continuity_from_above -> INF_emeasure_decseq
+ measure_space.measure_insert -> emeasure_insert
+ measure_space.measure_setsum -> setsum_emeasure
+ measure_space.measure_finite_singleton -> emeasure_eq_setsum_singleton
+ finite_additivity_sufficient -> ring_of_sets.countably_additiveI_finite
+ measure_space.measure_setsum_split -> setsum_emeasure_cover
+ measure_space.measure_subadditive -> subadditive
+ measure_space.measure_subadditive_finite -> emeasure_subadditive_finite
+ measure_space.measure_eq_0 -> emeasure_eq_0
+ measure_space.measure_finitely_subadditive -> emeasure_subadditive_finite
+ measure_space.measure_countably_subadditive -> emeasure_subadditive_countably
+ measure_space.measure_UN_eq_0 -> emeasure_UN_eq_0
+ measure_unique_Int_stable -> measure_eqI_generator_eq
+ measure_space.measure_Diff_null_set -> emeasure_Diff_null_set
+ measure_space.measure_Un_null_set -> emeasure_Un_null_set
+ measure_space.almost_everywhere_def -> eventually_ae_filter
+ measure_space.almost_everywhere_vimage -> AE_distrD
+ measure_space.measure_space_vimage -> emeasure_distr
+ measure_space.AE_iff_null_set -> AE_iff_null
+ measure_space.real_measure_Union -> measure_Union
+ measure_space.real_measure_finite_Union -> measure_finite_Union
+ measure_space.real_measure_Diff -> measure_Diff
+ measure_space.real_measure_UNION -> measure_UNION
+ measure_space.real_measure_subadditive -> measure_subadditive
+ measure_space.real_measure_setsum_singleton -> measure_eq_setsum_singleton
+ measure_space.real_continuity_from_below -> Lim_measure_incseq
+ measure_space.continuity_from_above_Lim -> Lim_emeasure_decseq
+ measure_space.real_continuity_from_above -> Lim_measure_decseq
+ measure_space.real_measure_countably_subadditive -> measure_subadditive_countably
+ finite_measure.finite_measure -> finite_measure.emeasure_finite
+ finite_measure.finite_measure_eq -> finite_measure.emeasure_eq_measure
+ finite_measure.positive_measure' -> measure_nonneg
+ finite_measure.real_measure -> finite_measure.emeasure_real
+ finite_measure.empty_measure -> measure_empty
+ finite_measure.finite_measure_countably_subadditive -> finite_measure.finite_measure_subadditive_countably
+ finite_measure.finite_measure_finite_singleton -> finite_measure.finite_measure_eq_setsum_singleton
+ finite_measure.finite_continuity_from_below -> finite_measure.finite_Lim_measure_incseq
+ finite_measure.finite_continuity_from_above -> finite_measure.finite_Lim_measure_decseq
+ measure_space.simple_integral_vimage -> simple_integral_distr
+ measure_space.integrable_vimage -> integrable_distr
+ measure_space.positive_integral_translated_density -> positive_integral_density
+ measure_space.integral_translated_density -> integral_density
+ measure_space.integral_vimage -> integral_distr
+ measure_space_density -> emeasure_density
+ measure_space.positive_integral_vimage -> positive_integral_distr
+ measure_space.simple_function_vimage -> simple_function_comp
+ measure_space.simple_integral_vimage -> simple_integral_distr
+ pair_sigma_algebra.measurable_cut_fst -> sets_Pair1
+ pair_sigma_algebra.measurable_cut_snd -> sets_Pair2
+ pair_sigma_algebra.measurable_pair_image_fst -> measurable_Pair1
+ pair_sigma_algebra.measurable_pair_image_snd -> measurable_Pair2
+ pair_sigma_algebra.measurable_product_swap -> measurable_pair_swap_iff
+ pair_sigma_finite.measure_cut_measurable_fst -> pair_sigma_finite.measurable_emeasure_Pair1
+ pair_sigma_finite.measure_cut_measurable_snd -> pair_sigma_finite.measurable_emeasure_Pair2
+ measure_space.measure_not_negative -> emeasure_not_MInf
+ pair_sigma_finite.measure_preserving_swap -> pair_sigma_finite.distr_pair_swap
+ pair_sigma_finite.pair_measure_alt -> pair_sigma_finite.emeasure_pair_measure_alt
+ pair_sigma_finite.pair_measure_alt2 -> pair_sigma_finite.emeasure_pair_measure_alt2
+ pair_sigma_finite.pair_measure_times -> pair_sigma_finite.emeasure_pair_measure_Times
+ pair_sigma_algebra.pair_sigma_algebra_measurable -> measurable_pair_swap
+ pair_sigma_algebra.pair_sigma_algebra_swap_measurable -> measurable_pair_swap'
+ pair_sigma_algebra.sets_swap -> sets_pair_swap
+ finite_product_sigma_algebra.in_P -> sets_PiM_I_finite
+ Int_stable_product_algebra_generator -> positive_integral
+ product_sigma_finite.measure_fold -> product_sigma_finite.distr_merge
+ product_sigma_finite.measure_preserving_component_singelton -> product_sigma_finite.distr_singleton
+ product_sigma_finite.measure_preserving_merge -> product_sigma_finite.distr_merge
+ finite_product_sigma_algebra.P_empty -> space_PiM_empty, sets_PiM_empty
+ product_algebra_generator_der -> prod_algebra_eq_finite
+ product_algebra_generator_into_space -> prod_algebra_sets_into_space
+ product_sigma_algebra.product_algebra_into_space -> space_closed
+ product_algebraE -> prod_algebraE_all
+ product_algebraI -> sets_PiM_I_finite
+ product_measure_exists -> product_sigma_finite.sigma_finite
+ sets_product_algebra -> sets_PiM
+ sigma_product_algebra_sigma_eq -> sigma_prod_algebra_sigma_eq
+ space_product_algebra -> space_PiM
+ Int_stable_cuboids -> Int_stable_atLeastAtMost
+ measure_space.density_is_absolutely_continuous -> absolutely_continuousI_density
+ sigma_finite_measure.RN_deriv_vimage -> sigma_finite_measure.RN_deriv_distr
+ prob_space_unique_Int_stable -> measure_eqI_prob_space
+ sigma_finite_measure.disjoint_sigma_finite -> sigma_finite_disjoint
+ prob_space.measure_space_1 -> prob_space.emeasure_space_1
+ prob_space.prob_space_vimage -> prob_space_distr
+ prob_space.random_variable_restrict -> measurable_restrict
+ measure_preserving -> equality "distr M N f = N" "f : measurable M N"
+ measure_unique_Int_stable_vimage -> measure_eqI_generator_eq
+ measure_space.measure_preserving_Int_stable -> measure_eqI_generator_eq
+ product_prob_space.finite_index_eq_finite_product -> product_prob_space.sets_PiM_generator
+ product_prob_space.finite_measure_infprod_emb_Pi -> product_prob_space.measure_PiM_emb
+ finite_product_prob_space.finite_measure_times -> finite_product_prob_space.finite_measure_PiM_emb
+ product_prob_space.infprod_spec -> product_prob_space.emeasure_PiM_emb_not_empty
+ product_prob_space.measurable_component -> measurable_component_singleton
+ product_prob_space.measurable_emb -> measurable_prod_emb
+ product_prob_space.measurable_into_infprod_algebra -> measurable_PiM_single
+ product_prob_space.measurable_singleton_infprod -> measurable_component_singleton
+ product_prob_space.measure_emb -> emeasure_prod_emb
+ sequence_space.measure_infprod -> sequence_space.measure_PiM_countable
+ product_prob_space.measure_preserving_restrict -> product_prob_space.distr_restrict
+ prob_space.indep_distribution_eq_measure -> prob_space.indep_vars_iff_distr_eq_PiM
+ prob_space.indep_var_distributionD -> prob_space.indep_var_distribution_eq
+ conditional_entropy_positive -> conditional_entropy_nonneg_simple
+ conditional_entropy_eq -> conditional_entropy_simple_distributed
+ conditional_mutual_information_eq_mutual_information -> conditional_mutual_information_eq_mutual_information_simple
+ conditional_mutual_information_generic_positive -> conditional_mutual_information_nonneg_simple
+ conditional_mutual_information_positive -> conditional_mutual_information_nonneg_simple
+ entropy_commute -> entropy_commute_simple
+ entropy_eq -> entropy_simple_distributed
+ entropy_generic_eq -> entropy_simple_distributed
+ entropy_positive -> entropy_nonneg_simple
+ entropy_uniform_max -> entropy_uniform
+ KL_eq_0 -> KL_same_eq_0
+ KL_eq_0_imp -> KL_eq_0_iff_eq
+ KL_ge_0 -> KL_nonneg
+ mutual_information_eq -> mutual_information_simple_distributed
+ mutual_information_positive -> mutual_information_nonneg_simple
+
* New "case_product" attribute to generate a case rule doing multiple
case distinctions at the same time. E.g.
@@ -655,8 +829,6 @@
produces a rule which can be used to perform case distinction on both
a list and a nat.
-* Theory Library/Multiset: Improved code generation of multisets.
-
* New Transfer package:
- transfer_rule attribute: Maintains a collection of transfer rules,
--- a/src/HOL/IsaMakefile Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/IsaMakefile Mon Apr 23 12:14:35 2012 +0200
@@ -1198,14 +1198,13 @@
$(OUT)/HOL-Probability: $(OUT)/HOL-Multivariate_Analysis \
Probability/Binary_Product_Measure.thy Probability/Borel_Space.thy \
Probability/Caratheodory.thy Probability/Complete_Measure.thy \
- Probability/Conditional_Probability.thy \
Probability/ex/Dining_Cryptographers.thy \
Probability/ex/Koepf_Duermuth_Countermeasure.thy \
Probability/Finite_Product_Measure.thy \
Probability/Independent_Family.thy \
Probability/Infinite_Product_Measure.thy Probability/Information.thy \
Probability/Lebesgue_Integration.thy Probability/Lebesgue_Measure.thy \
- Probability/Measure.thy Probability/Probability_Measure.thy \
+ Probability/Measure_Space.thy Probability/Probability_Measure.thy \
Probability/Probability.thy Probability/Radon_Nikodym.thy \
Probability/ROOT.ML Probability/Sigma_Algebra.thy \
Library/Countable.thy Library/FuncSet.thy Library/Nat_Bijection.thy
--- a/src/HOL/Probability/Binary_Product_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Binary_Product_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -28,413 +28,317 @@
section "Binary products"
-definition
- "pair_measure_generator A B =
- \<lparr> space = space A \<times> space B,
- sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},
- measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"
-
definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
- "A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"
-
-locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
- for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
-
-abbreviation (in pair_sigma_algebra)
- "E \<equiv> pair_measure_generator M1 M2"
-
-abbreviation (in pair_sigma_algebra)
- "P \<equiv> M1 \<Otimes>\<^isub>M M2"
-
-lemma sigma_algebra_pair_measure:
- "sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"
- by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)
-
-sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
- using M1.space_closed M2.space_closed
- by (rule sigma_algebra_pair_measure)
-
-lemma pair_measure_generatorI[intro, simp]:
- "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"
- by (auto simp add: pair_measure_generator_def)
-
-lemma pair_measureI[intro, simp]:
- "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
- by (auto simp add: pair_measure_def)
+ "A \<Otimes>\<^isub>M B = measure_of (space A \<times> space B)
+ {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
+ (\<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
lemma space_pair_measure:
"space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
- by (simp add: pair_measure_def pair_measure_generator_def)
+ unfolding pair_measure_def using space_closed[of A] space_closed[of B]
+ by (intro space_measure_of) auto
+
+lemma sets_pair_measure:
+ "sets (A \<Otimes>\<^isub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
+ unfolding pair_measure_def using space_closed[of A] space_closed[of B]
+ by (intro sets_measure_of) auto
-lemma sets_pair_measure_generator:
- "sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y) ` (sets N \<times> sets M)"
- unfolding pair_measure_generator_def by auto
+lemma pair_measureI[intro, simp]:
+ "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
+ by (auto simp: sets_pair_measure)
-lemma pair_measure_generator_sets_into_space:
- assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"
- shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"
- using assms by (auto simp: pair_measure_generator_def)
+lemma measurable_pair_measureI:
+ assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
+ assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f -` (A \<times> B) \<inter> space M \<in> sets M"
+ shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
+ unfolding pair_measure_def using 1 2
+ by (intro measurable_measure_of) (auto dest: sets_into_space)
-lemma pair_measure_generator_Int_snd:
- assumes "sets S1 \<subseteq> Pow (space S1)"
- shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =
- sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"
- (is "?L = ?R")
- apply (auto simp: pair_measure_generator_def image_iff)
- using assms
- apply (rule_tac x="a \<times> xa" in exI)
- apply force
- using assms
- apply (rule_tac x="a" in exI)
- apply (rule_tac x="b \<inter> A" in exI)
- apply auto
- done
+lemma measurable_fst[intro!, simp]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1"
+ unfolding measurable_def
+proof safe
+ fix A assume A: "A \<in> sets M1"
+ from this[THEN sets_into_space] have "fst -` A \<inter> space M1 \<times> space M2 = A \<times> space M2" by auto
+ with A show "fst -` A \<inter> space (M1 \<Otimes>\<^isub>M M2) \<in> sets (M1 \<Otimes>\<^isub>M M2)" by (simp add: space_pair_measure)
+qed (simp add: space_pair_measure)
-lemma (in pair_sigma_algebra)
- shows measurable_fst[intro!, simp]:
- "fst \<in> measurable P M1" (is ?fst)
- and measurable_snd[intro!, simp]:
- "snd \<in> measurable P M2" (is ?snd)
-proof -
- { fix X assume "X \<in> sets M1"
- then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
- apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
- using M1.sets_into_space by force+ }
- moreover
- { fix X assume "X \<in> sets M2"
- then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
- apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
- using M2.sets_into_space by force+ }
- ultimately have "?fst \<and> ?snd"
- by (fastforce simp: measurable_def sets_sigma space_pair_measure
- intro!: sigma_sets.Basic)
- then show ?fst ?snd by auto
-qed
+lemma measurable_snd[intro!, simp]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2"
+ unfolding measurable_def
+proof safe
+ fix A assume A: "A \<in> sets M2"
+ from this[THEN sets_into_space] have "snd -` A \<inter> space M1 \<times> space M2 = space M1 \<times> A" by auto
+ with A show "snd -` A \<inter> space (M1 \<Otimes>\<^isub>M M2) \<in> sets (M1 \<Otimes>\<^isub>M M2)" by (simp add: space_pair_measure)
+qed (simp add: space_pair_measure)
+
+lemma measurable_fst': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. fst (f x)) \<in> measurable M N"
+ using measurable_comp[OF _ measurable_fst] by (auto simp: comp_def)
+
+lemma measurable_snd': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. snd (f x)) \<in> measurable M P"
+ using measurable_comp[OF _ measurable_snd] by (auto simp: comp_def)
-lemma (in pair_sigma_algebra) measurable_pair_iff:
- assumes "sigma_algebra M"
- shows "f \<in> measurable M P \<longleftrightarrow>
- (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
-proof -
- interpret M: sigma_algebra M by fact
- from assms show ?thesis
- proof (safe intro!: measurable_comp[where b=P])
- assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
- show "f \<in> measurable M P" unfolding pair_measure_def
- proof (rule M.measurable_sigma)
- show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"
- unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto
- show "f \<in> space M \<rightarrow> space E"
- using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)
- fix A assume "A \<in> sets E"
- then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
- unfolding pair_measure_generator_def by auto
- moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M"
- using f `B \<in> sets M1` unfolding measurable_def by auto
- moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M"
- using s `C \<in> sets M2` unfolding measurable_def by auto
- moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)"
- unfolding `A = B \<times> C` by (auto simp: vimage_Times)
- ultimately show "f -` A \<inter> space M \<in> sets M" by auto
- qed
+lemma measurable_fst'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^isub>M P) N"
+ using measurable_comp[OF measurable_fst _] by (auto simp: comp_def)
+
+lemma measurable_snd'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^isub>M M) N"
+ using measurable_comp[OF measurable_snd _] by (auto simp: comp_def)
+
+lemma measurable_pair_iff:
+ "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
+proof safe
+ assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
+ show "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
+ proof (rule measurable_pair_measureI)
+ show "f \<in> space M \<rightarrow> space M1 \<times> space M2"
+ using f s by (auto simp: mem_Times_iff measurable_def comp_def)
+ fix A B assume "A \<in> sets M1" "B \<in> sets M2"
+ moreover have "(fst \<circ> f) -` A \<inter> space M \<in> sets M" "(snd \<circ> f) -` B \<inter> space M \<in> sets M"
+ using f `A \<in> sets M1` s `B \<in> sets M2` by (auto simp: measurable_sets)
+ moreover have "f -` (A \<times> B) \<inter> space M = ((fst \<circ> f) -` A \<inter> space M) \<inter> ((snd \<circ> f) -` B \<inter> space M)"
+ by (auto simp: vimage_Times)
+ ultimately show "f -` (A \<times> B) \<inter> space M \<in> sets M" by auto
qed
-qed
+qed auto
-lemma (in pair_sigma_algebra) measurable_pair:
- assumes "sigma_algebra M"
- assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
- shows "f \<in> measurable M P"
- unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp
-
-lemma pair_measure_generatorE:
- assumes "X \<in> sets (pair_measure_generator M1 M2)"
- obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
- using assms unfolding pair_measure_generator_def by auto
+lemma measurable_pair:
+ "(fst \<circ> f) \<in> measurable M M1 \<Longrightarrow> (snd \<circ> f) \<in> measurable M M2 \<Longrightarrow> f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
+ unfolding measurable_pair_iff by simp
-lemma (in pair_sigma_algebra) pair_measure_generator_swap:
- "(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_measure_generator M2 M1)"
-proof (safe elim!: pair_measure_generatorE)
- fix A B assume "A \<in> sets M1" "B \<in> sets M2"
- moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
- using M1.sets_into_space M2.sets_into_space by auto
- ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)"
- by (auto intro: pair_measure_generatorI)
-next
- fix A B assume "A \<in> sets M1" "B \<in> sets M2"
- then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E"
- using M1.sets_into_space M2.sets_into_space
- by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI)
+lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
+proof (rule measurable_pair_measureI)
+ fix A B assume "A \<in> sets M2" "B \<in> sets M1"
+ moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space (M1 \<Otimes>\<^isub>M M2) = B \<times> A"
+ by (auto dest: sets_into_space simp: space_pair_measure)
+ ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space (M1 \<Otimes>\<^isub>M M2) \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ by auto
+qed (auto simp add: space_pair_measure)
+
+lemma measurable_pair_swap:
+ assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
+proof -
+ have "(\<lambda>x. f (case x of (x, y) \<Rightarrow> (y, x))) = (\<lambda>(x, y). f (y, x))" by auto
+ then show ?thesis
+ using measurable_comp[OF measurable_pair_swap' f] by (simp add: comp_def)
qed
-lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
- assumes Q: "Q \<in> sets P"
- shows "(\<lambda>(x,y). (y, x)) -` Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")
-proof -
- let ?f = "\<lambda>Q. (\<lambda>(x,y). (y, x)) -` Q \<inter> space M2 \<times> space M1"
- have *: "(\<lambda>(x,y). (y, x)) -` Q = ?f Q"
- using sets_into_space[OF Q] by (auto simp: space_pair_measure)
- have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"
- unfolding pair_measure_def ..
- also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f ` sets E)"
- unfolding sigma_def pair_measure_generator_swap[symmetric]
- by (simp add: pair_measure_generator_def)
- also have "\<dots> = ?f ` sigma_sets (space M1 \<times> space M2) (sets E)"
- using M1.sets_into_space M2.sets_into_space
- by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)
- also have "\<dots> = ?f ` sets P"
- unfolding pair_measure_def pair_measure_generator_def sigma_def by simp
- finally show ?thesis
- using Q by (subst *) auto
-qed
+lemma measurable_pair_swap_iff:
+ "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) M"
+ using measurable_pair_swap[of "\<lambda>(x,y). f (y, x)"]
+ by (auto intro!: measurable_pair_swap)
-lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
- shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)"
- (is "?f \<in> measurable ?P ?Q")
- unfolding measurable_def
-proof (intro CollectI conjI Pi_I ballI)
- fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
- unfolding pair_measure_generator_def pair_measure_def by auto
-next
- fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"
- interpret Q: pair_sigma_algebra M2 M1 by default
- from Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets (M2 \<Otimes>\<^isub>M M1)`]
- show "?f -` A \<inter> space ?P \<in> sets ?P" by simp
-qed
+lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)"
+proof (rule measurable_pair_measureI)
+ fix A B assume "A \<in> sets M1" "B \<in> sets M2"
+ moreover then have "Pair x -` (A \<times> B) \<inter> space M2 = (if x \<in> A then B else {})"
+ by (auto dest: sets_into_space simp: space_pair_measure)
+ ultimately show "Pair x -` (A \<times> B) \<inter> space M2 \<in> sets M2"
+ by auto
+qed (auto simp add: space_pair_measure)
-lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:
- assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"
+lemma sets_Pair1: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "Pair x -` A \<in> sets M2"
proof -
- let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"
- let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
- interpret Q: sigma_algebra ?Q
- proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)
- have "sets E \<subseteq> sets ?Q"
- using M1.sets_into_space M2.sets_into_space
- by (auto simp: pair_measure_generator_def space_pair_measure)
- then have "sets P \<subseteq> sets ?Q"
- apply (subst pair_measure_def, intro Q.sets_sigma_subset)
- by (simp add: pair_measure_def)
- with assms show ?thesis by auto
+ have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
+ using A[THEN sets_into_space] by (auto simp: space_pair_measure)
+ also have "\<dots> \<in> sets M2"
+ using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm)
+ finally show ?thesis .
qed
-lemma (in pair_sigma_algebra) measurable_cut_snd:
- assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1")
-proof -
- interpret Q: pair_sigma_algebra M2 M1 by default
- from Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
- show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
-qed
+lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^isub>M M2)"
+ using measurable_comp[OF measurable_Pair1' measurable_pair_swap', of y M2 M1]
+ by (simp add: comp_def)
-lemma (in pair_sigma_algebra) measurable_pair_image_snd:
- assumes m: "f \<in> measurable P M" and "x \<in> space M1"
- shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
- unfolding measurable_def
-proof (intro CollectI conjI Pi_I ballI)
- fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1`
- show "f (x, y) \<in> space M"
- unfolding measurable_def pair_measure_generator_def pair_measure_def by auto
-next
- fix A assume "A \<in> sets M"
- then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
- using `f \<in> measurable P M`
- by (intro measurable_cut_fst) (auto simp: measurable_def)
- also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2"
- using `x \<in> space M1` by (auto simp: pair_measure_generator_def pair_measure_def)
- finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" .
+lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
+proof -
+ have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
+ using A[THEN sets_into_space] by (auto simp: space_pair_measure)
+ also have "\<dots> \<in> sets M1"
+ using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm)
+ finally show ?thesis .
qed
-lemma (in pair_sigma_algebra) measurable_pair_image_fst:
- assumes m: "f \<in> measurable P M" and "y \<in> space M2"
+lemma measurable_Pair2:
+ assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and x: "x \<in> space M1"
+ shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
+ using measurable_comp[OF measurable_Pair1' f, OF x]
+ by (simp add: comp_def)
+
+lemma measurable_Pair1:
+ assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and y: "y \<in> space M2"
shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
-proof -
- interpret Q: pair_sigma_algebra M2 M1 by default
- from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`,
- OF Q.pair_sigma_algebra_swap_measurable m]
- show ?thesis by simp
-qed
+ using measurable_comp[OF measurable_Pair2' f, OF y]
+ by (simp add: comp_def)
-lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"
+lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
unfolding Int_stable_def
-proof (intro ballI)
- fix A B assume "A \<in> sets E" "B \<in> sets E"
- then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
- "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
- unfolding pair_measure_generator_def by auto
- then show "A \<inter> B \<in> sets E"
- by (auto simp add: times_Int_times pair_measure_generator_def)
-qed
+ by safe (auto simp add: times_Int_times)
lemma finite_measure_cut_measurable:
- fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
assumes "sigma_finite_measure M1" "finite_measure M2"
assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"
- shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1"
+ shows "(\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
(is "?s Q \<in> _")
proof -
interpret M1: sigma_finite_measure M1 by fact
interpret M2: finite_measure M2 by fact
- interpret pair_sigma_algebra M1 M2 by default
- have [intro]: "sigma_algebra M1" by fact
- have [intro]: "sigma_algebra M2" by fact
- let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1} \<rparr>"
+ let ?\<Omega> = "space M1 \<times> space M2" and ?D = "{A\<in>sets (M1 \<Otimes>\<^isub>M M2). ?s A \<in> borel_measurable M1}"
note space_pair_measure[simp]
- interpret dynkin_system ?D
+ interpret dynkin_system ?\<Omega> ?D
proof (intro dynkin_systemI)
- fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
- using sets_into_space by simp
+ fix A assume "A \<in> ?D" then show "A \<subseteq> ?\<Omega>"
+ using sets_into_space[of A "M1 \<Otimes>\<^isub>M M2"] by simp
next
- from top show "space ?D \<in> sets ?D"
- by (auto simp add: if_distrib intro!: M1.measurable_If)
+ from top show "?\<Omega> \<in> ?D"
+ by (auto simp add: if_distrib intro!: measurable_If)
next
- fix A assume "A \<in> sets ?D"
- with sets_into_space have "\<And>x. measure M2 (Pair x -` (space M1 \<times> space M2 - A)) =
- (if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"
- by (auto intro!: M2.measure_compl simp: vimage_Diff)
- with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"
- by (auto intro!: Diff M1.measurable_If M1.borel_measurable_ereal_diff)
+ fix A assume "A \<in> ?D"
+ with sets_into_space have "\<And>x. emeasure M2 (Pair x -` (?\<Omega> - A)) =
+ (if x \<in> space M1 then emeasure M2 (space M2) - ?s A x else 0)"
+ by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
+ with `A \<in> ?D` top show "?\<Omega> - A \<in> ?D"
+ by (auto intro!: measurable_If)
next
- fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
- moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
- by (intro M2.measure_countably_additive[symmetric])
- (auto simp: disjoint_family_on_def)
- ultimately show "(\<Union>i. F i) \<in> sets ?D"
- by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
+ fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> ?D"
+ moreover then have "\<And>x. emeasure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
+ by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)
+ ultimately show "(\<Union>i. F i) \<in> ?D"
+ by (auto simp: vimage_UN intro!: borel_measurable_psuminf)
qed
- have "sets P = sets ?D" apply (subst pair_measure_def)
- proof (intro dynkin_lemma)
- show "Int_stable E" by (rule Int_stable_pair_measure_generator)
- from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
- by auto
- then show "sets E \<subseteq> sets ?D"
- by (auto simp: pair_measure_generator_def sets_sigma if_distrib
- intro: sigma_sets.Basic intro!: M1.measurable_If)
- qed (auto simp: pair_measure_def)
- with `Q \<in> sets P` have "Q \<in> sets ?D" by simp
+ let ?G = "{a \<times> b | a b. a \<in> sets M1 \<and> b \<in> sets M2}"
+ have "sigma_sets ?\<Omega> ?G = ?D"
+ proof (rule dynkin_lemma)
+ show "?G \<subseteq> ?D"
+ by (auto simp: if_distrib Int_def[symmetric] intro!: measurable_If)
+ qed (auto simp: sets_pair_measure Int_stable_pair_measure_generator)
+ with `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)` have "Q \<in> ?D"
+ by (simp add: sets_pair_measure[symmetric])
then show "?s Q \<in> borel_measurable M1" by simp
qed
-subsection {* Binary products of $\sigma$-finite measure spaces *}
+subsection {* Binary products of $\sigma$-finite emeasure spaces *}
-locale pair_sigma_finite = pair_sigma_algebra M1 M2 + M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
- for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
+locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
+ for M1 :: "'a measure" and M2 :: "'b measure"
-lemma (in pair_sigma_finite) measure_cut_measurable_fst:
- assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
+lemma sets_pair_measure_cong[cong]:
+ "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^isub>M M2) = sets (M1' \<Otimes>\<^isub>M M2')"
+ unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
+
+lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
+ assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
proof -
- have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
- have M1: "sigma_finite_measure M1" by default
- from M2.disjoint_sigma_finite guess F .. note F = this
+ from M2.sigma_finite_disjoint guess F . note F = this
then have F_sets: "\<And>i. F i \<in> sets M2" by auto
+ have M1: "sigma_finite_measure M1" ..
let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"
{ fix i
- let ?R = "M2.restricted_space (F i)"
have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
- using F M2.sets_into_space by auto
- let ?R2 = "M2.restricted_space (F i)"
- have "(\<lambda>x. measure ?R2 (Pair x -` (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"
+ using F sets_into_space by auto
+ let ?R = "density M2 (indicator (F i))"
+ have "(\<lambda>x. emeasure ?R (Pair x -` (space M1 \<times> space ?R \<inter> Q))) \<in> borel_measurable M1"
proof (intro finite_measure_cut_measurable[OF M1])
- show "finite_measure ?R2"
- using F by (intro M2.restricted_to_finite_measure) auto
- have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i)) ` sets P"
- using `Q \<in> sets P` by (auto simp: image_iff)
- also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i)) ` sets E)"
- unfolding pair_measure_def pair_measure_generator_def sigma_def
- using `F i \<in> sets M2` M2.sets_into_space
- by (auto intro!: sigma_sets_Int sigma_sets.Basic)
- also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"
- using M1.sets_into_space
- apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def
- intro!: sigma_sets_subseteq)
- apply (rule_tac x="a" in exI)
- apply (rule_tac x="b \<inter> F i" in exI)
- by auto
- finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .
+ show "finite_measure ?R"
+ using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
+ show "(space M1 \<times> space ?R) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R)"
+ using Q by (simp add: Int)
qed
- moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"
- using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
- ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"
+ moreover have "\<And>x. emeasure ?R (Pair x -` (space M1 \<times> space ?R \<inter> Q))
+ = emeasure M2 (F i \<inter> Pair x -` (space M1 \<times> space ?R \<inter> Q))"
+ using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
+ moreover have "\<And>x. F i \<inter> Pair x -` (space M1 \<times> space ?R \<inter> Q) = ?C x i"
+ using sets_into_space[OF Q] by (auto simp: space_pair_measure)
+ ultimately have "(\<lambda>x. emeasure M2 (?C x i)) \<in> borel_measurable M1"
by simp }
moreover
{ fix x
- have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
- proof (intro M2.measure_countably_additive)
+ have "(\<Sum>i. emeasure M2 (?C x i)) = emeasure M2 (\<Union>i. ?C x i)"
+ proof (intro suminf_emeasure)
show "range (?C x) \<subseteq> sets M2"
- using F `Q \<in> sets P` by (auto intro!: M2.Int)
+ using F `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)` by (auto intro!: sets_Pair1)
have "disjoint_family F" using F by auto
show "disjoint_family (?C x)"
by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
qed
also have "(\<Union>i. ?C x i) = Pair x -` Q"
- using F sets_into_space `Q \<in> sets P`
+ using F sets_into_space[OF `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)`]
by (auto simp: space_pair_measure)
- finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))"
+ finally have "emeasure M2 (Pair x -` Q) = (\<Sum>i. emeasure M2 (?C x i))"
by simp }
- ultimately show ?thesis using `Q \<in> sets P` F_sets
- by (auto intro!: M1.borel_measurable_psuminf M2.Int)
+ ultimately show ?thesis using `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)` F_sets
+ by auto
qed
-lemma (in pair_sigma_finite) measure_cut_measurable_snd:
- assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
+lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
+ assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
- note sets_pair_sigma_algebra_swap[OF assms]
- from Q.measure_cut_measurable_fst[OF this]
- show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
-qed
-
-lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
- assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
-proof -
- interpret Q: pair_sigma_algebra M2 M1 by default
- have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
- show ?thesis
- using Q.pair_sigma_algebra_swap_measurable assms
- unfolding * by (rule measurable_comp)
+ have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
+ using Q measurable_pair_swap' by (auto intro: measurable_sets)
+ note Q.measurable_emeasure_Pair1[OF this]
+ moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1)) = (\<lambda>x. (x, y)) -` Q"
+ using Q[THEN sets_into_space] by (auto simp: space_pair_measure)
+ ultimately show ?thesis by simp
qed
-lemma (in pair_sigma_finite) pair_measure_alt:
- assumes "A \<in> sets P"
- shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x -` A) \<partial>M1)"
- apply (simp add: pair_measure_def pair_measure_generator_def)
-proof (rule M1.positive_integral_cong)
- fix x assume "x \<in> space M1"
- have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: ereal)"
- unfolding indicator_def by auto
- show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)"
- unfolding *
- apply (subst M2.positive_integral_indicator)
- apply (rule measurable_cut_fst[OF assms])
- by simp
+lemma (in pair_sigma_finite) emeasure_pair_measure:
+ assumes "X \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ shows "emeasure (M1 \<Otimes>\<^isub>M M2) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M2 \<partial>M1)" (is "_ = ?\<mu> X")
+proof (rule emeasure_measure_of[OF pair_measure_def])
+ show "positive (sets (M1 \<Otimes>\<^isub>M M2)) ?\<mu>"
+ by (auto simp: positive_def positive_integral_positive)
+ have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
+ by (auto simp: indicator_def)
+ show "countably_additive (sets (M1 \<Otimes>\<^isub>M M2)) ?\<mu>"
+ proof (rule countably_additiveI)
+ fix F :: "nat \<Rightarrow> ('a \<times> 'b) set" assume F: "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" "disjoint_family F"
+ from F have *: "\<And>i. F i \<in> sets (M1 \<Otimes>\<^isub>M M2)" "(\<Union>i. F i) \<in> sets (M1 \<Otimes>\<^isub>M M2)" by auto
+ moreover from F have "\<And>i. (\<lambda>x. emeasure M2 (Pair x -` F i)) \<in> borel_measurable M1"
+ by (intro measurable_emeasure_Pair1) auto
+ moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
+ by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
+ moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
+ using F by (auto simp: sets_Pair1)
+ ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
+ by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1
+ intro!: positive_integral_cong positive_integral_indicator[symmetric])
+ qed
+ show "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2} \<subseteq> Pow (space M1 \<times> space M2)"
+ using space_closed[of M1] space_closed[of M2] by auto
+qed fact
+
+lemma (in pair_sigma_finite) emeasure_pair_measure_alt:
+ assumes X: "X \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ shows "emeasure (M1 \<Otimes>\<^isub>M M2) X = (\<integral>\<^isup>+x. emeasure M2 (Pair x -` X) \<partial>M1)"
+proof -
+ have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
+ by (auto simp: indicator_def)
+ show ?thesis
+ using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1)
qed
-lemma (in pair_sigma_finite) pair_measure_times:
- assumes A: "A \<in> sets M1" and "B \<in> sets M2"
- shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"
+lemma (in pair_sigma_finite) emeasure_pair_measure_Times:
+ assumes A: "A \<in> sets M1" and B: "B \<in> sets M2"
+ shows "emeasure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = emeasure M1 A * emeasure M2 B"
proof -
- have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"
- using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
- with assms show ?thesis
- by (simp add: M1.positive_integral_cmult_indicator ac_simps)
+ have "emeasure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M2 B * indicator A x \<partial>M1)"
+ using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)
+ also have "\<dots> = emeasure M2 B * emeasure M1 A"
+ using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)
+ finally show ?thesis
+ by (simp add: ac_simps)
qed
-lemma (in measure_space) measure_not_negative[simp,intro]:
- assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
- using positive_measure[OF A] by auto
-
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
- "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
- (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
+ defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
+ shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and>
+ (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
proof -
- obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
- F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
- F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
- using M1.sigma_finite_up M2.sigma_finite_up by auto
- then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
+ from M1.sigma_finite_incseq guess F1 . note F1 = this
+ from M2.sigma_finite_incseq guess F2 . note F2 = this
+ from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
let ?F = "\<lambda>i. F1 i \<times> F2 i"
- show ?thesis unfolding space_pair_measure
+ show ?thesis
proof (intro exI[of _ ?F] conjI allI)
- show "range ?F \<subseteq> sets E" using F1 F2
- by (fastforce intro!: pair_measure_generatorI)
+ show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
next
have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
proof (intro subsetI)
@@ -448,353 +352,315 @@
by (intro SigmaI) (auto simp add: min_max.sup_commute)
then show "x \<in> (\<Union>i. ?F i)" by auto
qed
- then show "(\<Union>i. ?F i) = space E"
- using space by (auto simp: space pair_measure_generator_def)
+ then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
+ using space by (auto simp: space)
next
fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
next
fix i
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
- with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
- show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
- by (simp add: pair_measure_times)
+ with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]
+ show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
+ by (auto simp add: emeasure_pair_measure_Times)
+ qed
+qed
+
+sublocale pair_sigma_finite \<subseteq> sigma_finite_measure "M1 \<Otimes>\<^isub>M M2"
+proof
+ from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
+ show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
+ proof (rule exI[of _ F], intro conjI)
+ show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def)
+ show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)"
+ using F by (auto simp: space_pair_measure)
+ show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto
qed
qed
-sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
-proof
- show "positive P (measure P)"
- unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
- by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
-
- show "countably_additive P (measure P)"
- unfolding countably_additive_def
- proof (intro allI impI)
- fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
- assume F: "range F \<subseteq> sets P" "disjoint_family F"
- from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
- moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x -` F i)) \<in> borel_measurable M1"
- by (intro measure_cut_measurable_fst) auto
- moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
- by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
- moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
- using F by auto
- ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
- by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
- M2.measure_countably_additive
- cong: M1.positive_integral_cong)
- qed
-
- from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
- show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"
- proof (rule exI[of _ F], intro conjI)
- show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
- show "(\<Union>i. F i) = space P"
- using F by (auto simp: pair_measure_def pair_measure_generator_def)
- show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
- qed
+lemma sigma_finite_pair_measure:
+ assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
+ shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)"
+proof -
+ interpret A: sigma_finite_measure A by fact
+ interpret B: sigma_finite_measure B by fact
+ interpret AB: pair_sigma_finite A B ..
+ show ?thesis ..
qed
-lemma (in pair_sigma_algebra) sets_swap:
- assumes "A \<in> sets P"
+lemma sets_pair_swap:
+ assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
- (is "_ -` A \<inter> space ?Q \<in> sets ?Q")
-proof -
- have *: "(\<lambda>(x, y). (y, x)) -` A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) -` A"
- using `A \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
- show ?thesis
- unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
-qed
+ using measurable_pair_swap' assms by (rule measurable_sets)
-lemma (in pair_sigma_finite) pair_measure_alt2:
- assumes A: "A \<in> sets P"
- shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
- (is "_ = ?\<nu> A")
+lemma (in pair_sigma_finite) distr_pair_swap:
+ "M1 \<Otimes>\<^isub>M M2 = distr (M2 \<Otimes>\<^isub>M M1) (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
proof -
interpret Q: pair_sigma_finite M2 M1 by default
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
- have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"
- unfolding pair_measure_def by simp
-
- have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` A \<inter> space Q.P)"
- proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])
- show "measure_space P" "measure_space Q.P" by default
- show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
- show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
- using assms unfolding pair_measure_def by auto
- show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
- using F `A \<in> sets P` by (auto simp: pair_measure_def)
- fix X assume "X \<in> sets E"
- then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
- unfolding pair_measure_def pair_measure_generator_def by auto
- then have "(\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P = B \<times> A"
- using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)
- then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P)"
- using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)
+ let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
+ show ?thesis
+ proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
+ show "?E \<subseteq> Pow (space ?P)"
+ using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
+ show "sets ?P = sigma_sets (space ?P) ?E"
+ by (simp add: sets_pair_measure space_pair_measure)
+ then show "sets ?D = sigma_sets (space ?P) ?E"
+ by simp
+ next
+ show "range F \<subseteq> ?E" "incseq F" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
+ using F by (auto simp: space_pair_measure)
+ next
+ fix X assume "X \<in> ?E"
+ then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto
+ have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A"
+ using sets_into_space[OF A] sets_into_space[OF B] by (auto simp: space_pair_measure)
+ with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X"
+ by (simp add: emeasure_pair_measure_Times Q.emeasure_pair_measure_Times emeasure_distr
+ measurable_pair_swap' ac_simps)
qed
- then show ?thesis
- using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]
- by (auto simp add: Q.pair_measure_alt space_pair_measure
- intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])
qed
-lemma pair_sigma_algebra_sigma:
- assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
- assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
- shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
- (is "sets ?S = sets ?E")
+lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
+ assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
+ (is "_ = ?\<nu> A")
proof -
- interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
- interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
- have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
- using E1 E2 by (auto simp add: pair_measure_generator_def)
- interpret E: sigma_algebra ?E unfolding pair_measure_generator_def
- using E1 E2 by (intro sigma_algebra_sigma) auto
- { fix A assume "A \<in> sets E1"
- then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
- using E1 2 unfolding pair_measure_generator_def by auto
- also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
- also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
- using 2 `A \<in> sets E1`
- by (intro sigma_sets.Union)
- (force simp: image_subset_iff intro!: sigma_sets.Basic)
- finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
- moreover
- { fix B assume "B \<in> sets E2"
- then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
- using E2 1 unfolding pair_measure_generator_def by auto
- also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
- also have "\<dots> \<in> sets ?E"
- using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma
- by (intro sigma_sets.Union)
- (force simp: image_subset_iff intro!: sigma_sets.Basic)
- finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
- ultimately have proj:
- "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
- using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
- (auto simp: pair_measure_generator_def sets_sigma)
- { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
- with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
- unfolding measurable_def by simp_all
- moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"
- using A B M1.sets_into_space M2.sets_into_space
- by (auto simp: pair_measure_generator_def)
- ultimately have "A \<times> B \<in> sets ?E" by auto }
- then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"
- by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)
- then have subset: "sets ?S \<subseteq> sets ?E"
- by (simp add: sets_sigma pair_measure_generator_def)
- show "sets ?S = sets ?E"
- proof (intro set_eqI iffI)
- fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
- unfolding sets_sigma
- proof induct
- case (Basic A) then show ?case
- by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)
- qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)
- next
- fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
- qed
+ interpret Q: pair_sigma_finite M2 M1 by default
+ have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1))) = (\<lambda>x. (x, y)) -` A"
+ using sets_into_space[OF A] by (auto simp: space_pair_measure)
+ show ?thesis using A
+ by (subst distr_pair_swap)
+ (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
+ Q.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
qed
section "Fubinis theorem"
lemma (in pair_sigma_finite) simple_function_cut:
- assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
+ assumes f: "simple_function (M1 \<Otimes>\<^isub>M M2) f" "\<And>x. 0 \<le> f x"
shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
- and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
+ and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
proof -
- have f_borel: "f \<in> borel_measurable P"
+ have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
using f(1) by (rule borel_measurable_simple_function)
- let ?F = "\<lambda>z. f -` {z} \<inter> space P"
+ let ?F = "\<lambda>z. f -` {z} \<inter> space (M1 \<Otimes>\<^isub>M M2)"
let ?F' = "\<lambda>x z. Pair x -` ?F z"
{ fix x assume "x \<in> space M1"
have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
by (auto simp: indicator_def)
- have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1`
+ have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space (M1 \<Otimes>\<^isub>M M2)" using `x \<in> space M1`
by (simp add: space_pair_measure)
moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
- by (intro borel_measurable_vimage measurable_cut_fst)
+ by (rule sets_Pair1[OF measurable_sets]) auto
ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
- apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
+ apply (rule_tac simple_function_cong[THEN iffD2, OF _])
apply (rule simple_function_indicator_representation[OF f(1)])
- using `x \<in> space M1` by (auto simp del: space_sigma) }
+ using `x \<in> space M1` by auto }
note M2_sf = this
{ fix x assume x: "x \<in> space M1"
- then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space P. z * M2.\<mu> (?F' x z))"
- unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
+ then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space (M1 \<Otimes>\<^isub>M M2). z * emeasure M2 (?F' x z))"
+ unfolding positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
unfolding simple_integral_def
proof (safe intro!: setsum_mono_zero_cong_left)
- from f(1) show "finite (f ` space P)" by (rule simple_functionD)
+ from f(1) show "finite (f ` space (M1 \<Otimes>\<^isub>M M2))" by (rule simple_functionD)
next
- fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
+ fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space (M1 \<Otimes>\<^isub>M M2)"
using `x \<in> space M1` by (auto simp: space_pair_measure)
next
- fix x' y assume "(x', y) \<in> space P"
+ fix x' y assume "(x', y) \<in> space (M1 \<Otimes>\<^isub>M M2)"
"f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
then have *: "?F' x (f (x', y)) = {}"
by (force simp: space_pair_measure)
- show "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"
+ show "f (x', y) * emeasure M2 (?F' x (f (x', y))) = 0"
unfolding * by simp
qed (simp add: vimage_compose[symmetric] comp_def
space_pair_measure) }
note eq = this
- moreover have "\<And>z. ?F z \<in> sets P"
- by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
- moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
- by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
- moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
- using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
- moreover { fix i assume "i \<in> f`space P"
- with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
+ moreover have "\<And>z. ?F z \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ by (auto intro!: f_borel borel_measurable_vimage)
+ moreover then have "\<And>z. (\<lambda>x. emeasure M2 (?F' x z)) \<in> borel_measurable M1"
+ by (auto intro!: measurable_emeasure_Pair1 simp del: vimage_Int)
+ moreover have *: "\<And>i x. 0 \<le> emeasure M2 (Pair x -` (f -` {i} \<inter> space (M1 \<Otimes>\<^isub>M M2)))"
+ using f(1)[THEN simple_functionD(2)] f(2) by (intro emeasure_nonneg)
+ moreover { fix i assume "i \<in> f`space (M1 \<Otimes>\<^isub>M M2)"
+ with * have "\<And>x. 0 \<le> i * emeasure M2 (Pair x -` (f -` {i} \<inter> space (M1 \<Otimes>\<^isub>M M2)))"
using f(2) by auto }
ultimately
show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
- and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
- by (auto simp del: vimage_Int cong: measurable_cong
- intro!: M1.borel_measurable_ereal_setsum setsum_cong
- simp add: M1.positive_integral_setsum simple_integral_def
- M1.positive_integral_cmult
- M1.positive_integral_cong[OF eq]
+ and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f" using f(2)
+ by (auto simp del: vimage_Int cong: measurable_cong intro!: setsum_cong
+ simp add: positive_integral_setsum simple_integral_def
+ positive_integral_cmult
+ positive_integral_cong[OF eq]
positive_integral_eq_simple_integral[OF f]
- pair_measure_alt[symmetric])
+ emeasure_pair_measure_alt[symmetric])
qed
lemma (in pair_sigma_finite) positive_integral_fst_measurable:
- assumes f: "f \<in> borel_measurable P"
+ assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
(is "?C f \<in> borel_measurable M1")
- and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
+ and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
proof -
from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
- then have F_borel: "\<And>i. F i \<in> borel_measurable P"
+ then have F_borel: "\<And>i. F i \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
by (auto intro: borel_measurable_simple_function)
note sf = simple_function_cut[OF F(1,5)]
then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
using F(1) by auto
moreover
{ fix x assume "x \<in> space M1"
- from F measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
+ from F measurable_Pair2[OF F_borel `x \<in> space M1`]
have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
- by (intro M2.positive_integral_monotone_convergence_SUP)
+ by (intro positive_integral_monotone_convergence_SUP)
(auto simp: incseq_Suc_iff le_fun_def)
then have "(SUP i. ?C (F i) x) = ?C f x"
unfolding F(4) positive_integral_max_0 by simp }
note SUPR_C = this
ultimately show "?C f \<in> borel_measurable M1"
by (simp cong: measurable_cong)
- have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
+ have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>(M1 \<Otimes>\<^isub>M M2)) = (SUP i. integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (F i))"
using F_borel F
by (intro positive_integral_monotone_convergence_SUP) auto
- also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
+ also have "(SUP i. integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
unfolding sf(2) by simp
also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
- by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
- (auto intro!: M2.positive_integral_mono M2.positive_integral_positive
- simp: incseq_Suc_iff le_fun_def)
+ by (intro positive_integral_monotone_convergence_SUP[symmetric])
+ (auto intro!: positive_integral_mono positive_integral_positive
+ simp: incseq_Suc_iff le_fun_def)
also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
using F_borel F(2,5)
- by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
- simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
- finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
+ by (auto intro!: positive_integral_cong positive_integral_monotone_convergence_SUP[symmetric] measurable_Pair2
+ simp: incseq_Suc_iff le_fun_def)
+ finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
using F by (simp add: positive_integral_max_0)
qed
-lemma (in pair_sigma_finite) measure_preserving_swap:
- "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
-proof
- interpret Q: pair_sigma_finite M2 M1 by default
- show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
- using pair_sigma_algebra_swap_measurable .
- fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"
- from measurable_sets[OF * this] this Q.sets_into_space[OF this]
- show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) -` X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"
- by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]
- simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)
-qed
-
-lemma (in pair_sigma_finite) positive_integral_product_swap:
- assumes f: "f \<in> borel_measurable P"
- shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"
+lemma (in pair_sigma_finite) positive_integral_snd_measurable:
+ assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
- have "sigma_algebra P" by default
- with f show ?thesis
- by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto
-qed
-
-lemma (in pair_sigma_finite) positive_integral_snd_measurable:
- assumes f: "f \<in> borel_measurable P"
- shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"
-proof -
- interpret Q: pair_sigma_finite M2 M1 by default
- note pair_sigma_algebra_measurable[OF f]
+ note measurable_pair_swap[OF f]
from Q.positive_integral_fst_measurable[OF this]
- have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"
+ have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1))"
by simp
- also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"
- unfolding positive_integral_product_swap[OF f, symmetric]
- by (auto intro!: Q.positive_integral_cong)
+ also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
+ by (subst distr_pair_swap)
+ (auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong)
finally show ?thesis .
qed
lemma (in pair_sigma_finite) Fubini:
- assumes f: "f \<in> borel_measurable P"
+ assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
unfolding positive_integral_snd_measurable[OF assms]
unfolding positive_integral_fst_measurable[OF assms] ..
lemma (in pair_sigma_finite) AE_pair:
- assumes "AE x in P. Q x"
+ assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x"
shows "AE x in M1. (AE y in M2. Q (x, y))"
proof -
- obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
- using assms unfolding almost_everywhere_def by auto
+ obtain N where N: "N \<in> sets (M1 \<Otimes>\<^isub>M M2)" "emeasure (M1 \<Otimes>\<^isub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> Q x} \<subseteq> N"
+ using assms unfolding eventually_ae_filter by auto
show ?thesis
- proof (rule M1.AE_I)
- from N measure_cut_measurable_fst[OF `N \<in> sets P`]
- show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0"
- by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
- show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1"
- by (intro M1.borel_measurable_ereal_neq_const measure_cut_measurable_fst N)
- { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0"
- have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
- proof (rule M2.AE_I)
- show "M2.\<mu> (Pair x -` N) = 0" by fact
- show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)
+ proof (rule AE_I)
+ from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^isub>M M2)`]
+ show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
+ by (auto simp: emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg)
+ show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
+ by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)
+ { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
+ have "AE y in M2. Q (x, y)"
+ proof (rule AE_I)
+ show "emeasure M2 (Pair x -` N) = 0" by fact
+ show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
- using N `x \<in> space M1` unfolding space_sigma space_pair_measure by auto
+ using N `x \<in> space M1` unfolding space_pair_measure by auto
qed }
- then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0}"
+ then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0}"
by auto
qed
qed
-lemma (in pair_sigma_algebra) measurable_product_swap:
- "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
+lemma (in pair_sigma_finite) AE_pair_measure:
+ assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ assumes ae: "AE x in M1. AE y in M2. P (x, y)"
+ shows "AE x in M1 \<Otimes>\<^isub>M M2. P x"
+proof (subst AE_iff_measurable[OF _ refl])
+ show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ by (rule sets_Collect) fact
+ then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} =
+ (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
+ by (simp add: emeasure_pair_measure)
+ also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)"
+ using ae
+ apply (safe intro!: positive_integral_cong_AE)
+ apply (intro AE_I2)
+ apply (safe intro!: positive_integral_cong_AE)
+ apply auto
+ done
+ finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp
+qed
+
+lemma (in pair_sigma_finite) AE_pair_iff:
+ "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow>
+ (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x))"
+ using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
+
+lemma AE_distr_iff:
+ assumes f: "f \<in> measurable M N" and P: "{x \<in> space N. P x} \<in> sets N"
+ shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"
+proof (subst (1 2) AE_iff_measurable[OF _ refl])
+ from P show "{x \<in> space (distr M N f). \<not> P x} \<in> sets (distr M N f)"
+ by (auto intro!: sets_Collect_neg)
+ moreover
+ have "f -` {x \<in> space N. P x} \<inter> space M = {x \<in> space M. P (f x)}"
+ using f by (auto dest: measurable_space)
+ then show "{x \<in> space M. \<not> P (f x)} \<in> sets M"
+ using measurable_sets[OF f P] by (auto intro!: sets_Collect_neg)
+ moreover have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
+ using f by (auto dest: measurable_space)
+ ultimately show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
+ (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
+ using f by (simp add: emeasure_distr)
+qed
+
+lemma (in pair_sigma_finite) AE_commute:
+ assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
+ shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
proof -
- interpret Q: pair_sigma_algebra M2 M1 by default
- show ?thesis
- using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
- by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
+ interpret Q: pair_sigma_finite M2 M1 ..
+ have [simp]: "\<And>x. (fst (case x of (x, y) \<Rightarrow> (y, x))) = snd x" "\<And>x. (snd (case x of (x, y) \<Rightarrow> (y, x))) = fst x"
+ by auto
+ have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} =
+ (\<lambda>(x, y). (y, x)) -` {x \<in> space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^isub>M M1)"
+ by (auto simp: space_pair_measure)
+ also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)"
+ by (intro sets_pair_swap P)
+ finally show ?thesis
+ apply (subst AE_pair_iff[OF P])
+ apply (subst distr_pair_swap)
+ apply (subst AE_distr_iff[OF measurable_pair_swap' P])
+ apply (subst Q.AE_pair_iff)
+ apply simp_all
+ done
qed
lemma (in pair_sigma_finite) integrable_product_swap:
- assumes "integrable P f"
+ assumes "integrable (M1 \<Otimes>\<^isub>M M2) f"
shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
show ?thesis unfolding *
- using assms unfolding integrable_def
- apply (subst (1 2) positive_integral_product_swap)
- using `integrable P f` unfolding integrable_def
- by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
+ by (rule integrable_distr[OF measurable_pair_swap'])
+ (simp add: distr_pair_swap[symmetric] assms)
qed
lemma (in pair_sigma_finite) integrable_product_swap_iff:
- "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"
+ "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^isub>M M2) f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
@@ -802,27 +668,25 @@
qed
lemma (in pair_sigma_finite) integral_product_swap:
- assumes "integrable P f"
- shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"
+ assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f"
proof -
- interpret Q: pair_sigma_finite M2 M1 by default
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
- show ?thesis
- unfolding lebesgue_integral_def *
- apply (subst (1 2) positive_integral_product_swap)
- using `integrable P f` unfolding integrable_def
- by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
+ show ?thesis unfolding *
+ by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
qed
lemma (in pair_sigma_finite) integrable_fst_measurable:
- assumes f: "integrable P f"
- shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
- and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
+ assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
+ shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE")
+ and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
proof -
+ have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ using f by auto
let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
have
- borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
- int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
+ borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and
+ int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?nf \<noteq> \<infinity>" "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?pf \<noteq> \<infinity>"
using assms by auto
have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
"(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
@@ -831,69 +695,92 @@
with borel[THEN positive_integral_fst_measurable(1)]
have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
"AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
- by (auto intro!: M1.positive_integral_PInf_AE )
+ by (auto intro!: positive_integral_PInf_AE )
then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
"AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
- by (auto simp: M2.positive_integral_positive)
+ by (auto simp: positive_integral_positive)
from AE_pos show ?AE using assms
- by (simp add: measurable_pair_image_snd integrable_def)
+ by (simp add: measurable_Pair2[OF f_borel] integrable_def)
{ fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
- using M2.positive_integral_positive
- by (intro M1.positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
+ using positive_integral_positive
+ by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
note this[simp]
- { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable P"
- and int: "integral\<^isup>P P (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
- and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
+ { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
+ and AE: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
proof (intro integrable_def[THEN iffD2] conjI)
show "?f \<in> borel_measurable M1"
- using borel by (auto intro!: M1.borel_measurable_real_of_ereal positive_integral_fst_measurable)
+ using borel by (auto intro!: positive_integral_fst_measurable)
have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1)"
- using AE M2.positive_integral_positive
- by (auto intro!: M1.positive_integral_cong_AE simp: ereal_real)
+ using AE positive_integral_positive[of M2]
+ by (auto intro!: positive_integral_cong_AE simp: ereal_real)
then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
using positive_integral_fst_measurable[OF borel] int by simp
have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
- by (intro M1.positive_integral_cong_pos)
- (simp add: M2.positive_integral_positive real_of_ereal_pos)
+ by (intro positive_integral_cong_pos)
+ (simp add: positive_integral_positive real_of_ereal_pos)
then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
qed }
with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
show ?INT
- unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
+ unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2]
borel[THEN positive_integral_fst_measurable(2), symmetric]
- using AE[THEN M1.integral_real]
+ using AE[THEN integral_real]
by simp
qed
lemma (in pair_sigma_finite) integrable_snd_measurable:
- assumes f: "integrable P f"
- shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")
- and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")
+ assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
+ shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE")
+ and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
proof -
interpret Q: pair_sigma_finite M2 M1 by default
- have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"
+ have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))"
using f unfolding integrable_product_swap_iff .
show ?INT
using Q.integrable_fst_measurable(2)[OF Q_int]
- using integral_product_swap[OF f] by simp
+ using integral_product_swap[of f] f by auto
show ?AE
using Q.integrable_fst_measurable(1)[OF Q_int]
by simp
qed
+lemma (in pair_sigma_finite) positive_integral_fst_measurable':
+ assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ shows "(\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M2) \<in> borel_measurable M1"
+ using positive_integral_fst_measurable(1)[OF f] by simp
+
+lemma (in pair_sigma_finite) integral_fst_measurable:
+ "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M2) \<in> borel_measurable M1"
+ by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_fst_measurable')
+
+lemma (in pair_sigma_finite) positive_integral_snd_measurable':
+ assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ shows "(\<lambda>y. \<integral>\<^isup>+ x. f x y \<partial>M1) \<in> borel_measurable M2"
+proof -
+ interpret Q: pair_sigma_finite M2 M1 ..
+ show ?thesis
+ using measurable_pair_swap[OF f]
+ by (intro Q.positive_integral_fst_measurable') (simp add: split_beta')
+qed
+
+lemma (in pair_sigma_finite) integral_snd_measurable:
+ "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>y. \<integral> x. f x y \<partial>M1) \<in> borel_measurable M2"
+ by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_snd_measurable')
+
lemma (in pair_sigma_finite) Fubini_integral:
- assumes f: "integrable P f"
+ assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
unfolding integrable_snd_measurable[OF assms]
unfolding integrable_fst_measurable[OF assms] ..
-section "Products on finite spaces"
+section {* Products on counting spaces, densities and distributions *}
lemma sigma_sets_pair_measure_generator_finite:
assumes "finite A" and "finite B"
- shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"
+ shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
(is "sigma_sets ?prod ?sets = _")
proof safe
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
@@ -904,8 +791,7 @@
case empty show ?case by (rule sigma_sets.Empty)
next
case (insert a x)
- hence "{a} \<in> sigma_sets ?prod ?sets"
- by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)
+ hence "{a} \<in> sigma_sets ?prod ?sets" by auto
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
qed
@@ -916,48 +802,142 @@
show "a \<in> A" and "b \<in> B" by auto
qed
-locale pair_finite_sigma_algebra = pair_sigma_algebra M1 M2 + M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2
-
-lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:
- shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"
-proof -
- show ?thesis
- using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]
- by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)
-qed
-
-sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
-proof
- show "finite (space P)"
- using M1.finite_space M2.finite_space
- by (subst finite_pair_sigma_algebra) simp
- show "sets P = Pow (space P)"
- by (subst (1 2) finite_pair_sigma_algebra) simp
+lemma pair_measure_count_space:
+ assumes A: "finite A" and B: "finite B"
+ shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
+proof (rule measure_eqI)
+ interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
+ interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
+ interpret P: pair_sigma_finite "count_space A" "count_space B" by default
+ show eq: "sets ?P = sets ?C"
+ by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
+ fix X assume X: "X \<in> sets ?P"
+ with eq have X_subset: "X \<subseteq> A \<times> B" by simp
+ with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
+ by (intro finite_subset[OF _ B]) auto
+ have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
+ show "emeasure ?P X = emeasure ?C X"
+ apply (subst P.emeasure_pair_measure_alt[OF X])
+ apply (subst emeasure_count_space)
+ using X_subset apply auto []
+ apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
+ apply (subst positive_integral_count_space)
+ using A apply simp
+ apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
+ apply (subst card_gt_0_iff)
+ apply (simp add: fin_Pair)
+ apply (subst card_SigmaI[symmetric])
+ using A apply simp
+ using fin_Pair apply simp
+ using X_subset apply (auto intro!: arg_cong[where f=card])
+ done
qed
-locale pair_finite_space = pair_sigma_finite M1 M2 + pair_finite_sigma_algebra M1 M2 +
- M1: finite_measure_space M1 + M2: finite_measure_space M2 for M1 M2
+lemma pair_measure_density:
+ assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
+ assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"
+ assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
+ assumes "sigma_finite_measure (density M1 f)" "sigma_finite_measure (density M2 g)"
+ shows "density M1 f \<Otimes>\<^isub>M density M2 g = density (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")
+proof (rule measure_eqI)
+ interpret M1: sigma_finite_measure M1 by fact
+ interpret M2: sigma_finite_measure M2 by fact
+ interpret D1: sigma_finite_measure "density M1 f" by fact
+ interpret D2: sigma_finite_measure "density M2 g" by fact
+ interpret L: pair_sigma_finite "density M1 f" "density M2 g" ..
+ interpret R: pair_sigma_finite M1 M2 ..
+
+ fix A assume A: "A \<in> sets ?L"
+ then have indicator_eq: "\<And>x y. indicator A (x, y) = indicator (Pair x -` A) y"
+ and Pair_A: "\<And>x. Pair x -` A \<in> sets M2"
+ by (auto simp: indicator_def sets_Pair1)
+ have f_fst: "(\<lambda>p. f (fst p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ using measurable_comp[OF measurable_fst f(1)] by (simp add: comp_def)
+ have g_snd: "(\<lambda>p. g (snd p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
+ using measurable_comp[OF measurable_snd g(1)] by (simp add: comp_def)
+ have "(\<lambda>x. \<integral>\<^isup>+ y. g (snd (x, y)) * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
+ using g_snd Pair_A A by (intro R.positive_integral_fst_measurable) auto
+ then have int_g: "(\<lambda>x. \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
+ by simp
-lemma (in pair_finite_space) pair_measure_Pair[simp]:
- assumes "a \<in> space M1" "b \<in> space M2"
- shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"
+ show "emeasure ?L A = emeasure ?R A"
+ apply (subst L.emeasure_pair_measure[OF A])
+ apply (subst emeasure_density)
+ using f_fst g_snd apply (simp add: split_beta')
+ using A apply simp
+ apply (subst positive_integral_density[OF g])
+ apply (simp add: indicator_eq Pair_A)
+ apply (subst positive_integral_density[OF f])
+ apply (rule int_g)
+ apply (subst R.positive_integral_fst_measurable(2)[symmetric])
+ using f g A Pair_A f_fst g_snd
+ apply (auto intro!: positive_integral_cong_AE R.measurable_emeasure_Pair1
+ simp: positive_integral_cmult indicator_eq split_beta')
+ apply (intro AE_I2 impI)
+ apply (subst mult_assoc)
+ apply (subst positive_integral_cmult)
+ apply auto
+ done
+qed simp
+
+lemma sigma_finite_measure_distr:
+ assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
+ shows "sigma_finite_measure M"
proof -
- have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"
- using M1.sets_eq_Pow M2.sets_eq_Pow assms
- by (subst pair_measure_times) auto
- then show ?thesis by simp
+ interpret sigma_finite_measure "distr M N f" by fact
+ from sigma_finite_disjoint guess A . note A = this
+ show ?thesis
+ proof (unfold_locales, intro conjI exI allI)
+ show "range (\<lambda>i. f -` A i \<inter> space M) \<subseteq> sets M"
+ using A f by (auto intro!: measurable_sets)
+ show "(\<Union>i. f -` A i \<inter> space M) = space M"
+ using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def)
+ fix i show "emeasure M (f -` A i \<inter> space M) \<noteq> \<infinity>"
+ using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq)
+ qed
qed
-lemma (in pair_finite_space) pair_measure_singleton[simp]:
- assumes "x \<in> space M1 \<times> space M2"
- shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"
- using pair_measure_Pair assms by (cases x) auto
+lemma measurable_cong':
+ assumes sets: "sets M = sets M'" "sets N = sets N'"
+ shows "measurable M N = measurable M' N'"
+ using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
-sublocale pair_finite_space \<subseteq> finite_measure_space P
-proof unfold_locales
- show "measure P (space P) \<noteq> \<infinity>"
- by (subst (2) finite_pair_sigma_algebra)
- (simp add: pair_measure_times)
+lemma pair_measure_distr:
+ assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
+ assumes "sigma_finite_measure (distr M S f)" "sigma_finite_measure (distr N T g)"
+ shows "distr M S f \<Otimes>\<^isub>M distr N T g = distr (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")
+proof (rule measure_eqI)
+ show "sets ?P = sets ?D"
+ by simp
+ interpret S: sigma_finite_measure "distr M S f" by fact
+ interpret T: sigma_finite_measure "distr N T g" by fact
+ interpret ST: pair_sigma_finite "distr M S f" "distr N T g" ..
+ interpret M: sigma_finite_measure M by (rule sigma_finite_measure_distr) fact+
+ interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
+ interpret MN: pair_sigma_finite M N ..
+ interpret SN: pair_sigma_finite "distr M S f" N ..
+ have [simp]:
+ "\<And>f g. fst \<circ> (\<lambda>(x, y). (f x, g y)) = f \<circ> fst" "\<And>f g. snd \<circ> (\<lambda>(x, y). (f x, g y)) = g \<circ> snd"
+ by auto
+ then have fg: "(\<lambda>(x, y). (f x, g y)) \<in> measurable (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T)"
+ using measurable_comp[OF measurable_fst f] measurable_comp[OF measurable_snd g]
+ by (auto simp: measurable_pair_iff)
+ fix A assume A: "A \<in> sets ?P"
+ then have "emeasure ?P A = (\<integral>\<^isup>+x. emeasure (distr N T g) (Pair x -` A) \<partial>distr M S f)"
+ by (rule ST.emeasure_pair_measure_alt)
+ also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair x -` A) \<inter> space N) \<partial>distr M S f)"
+ using g A by (simp add: sets_Pair1 emeasure_distr)
+ also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair (f x) -` A) \<inter> space N) \<partial>M)"
+ using f g A ST.measurable_emeasure_Pair1[OF A]
+ by (intro positive_integral_distr) (auto simp add: sets_Pair1 emeasure_distr)
+ also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (Pair x -` ((\<lambda>(x, y). (f x, g y)) -` A \<inter> space (M \<Otimes>\<^isub>M N))) \<partial>M)"
+ by (intro positive_integral_cong arg_cong2[where f=emeasure]) (auto simp: space_pair_measure)
+ also have "\<dots> = emeasure (M \<Otimes>\<^isub>M N) ((\<lambda>(x, y). (f x, g y)) -` A \<inter> space (M \<Otimes>\<^isub>M N))"
+ using fg by (intro MN.emeasure_pair_measure_alt[symmetric] measurable_sets[OF _ A])
+ (auto cong: measurable_cong')
+ also have "\<dots> = emeasure ?D A"
+ using fg A by (subst emeasure_distr) auto
+ finally show "emeasure ?P A = emeasure ?D A" .
qed
end
\ No newline at end of file
--- a/src/HOL/Probability/Borel_Space.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Borel_Space.thy Mon Apr 23 12:14:35 2012 +0200
@@ -11,16 +11,14 @@
section "Generic Borel spaces"
-definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = {S. open S}\<rparr>"
-abbreviation "borel_measurable M \<equiv> measurable M borel"
+definition borel :: "'a::topological_space measure" where
+ "borel = sigma UNIV {S. open S}"
-interpretation borel: sigma_algebra borel
- by (auto simp: borel_def intro!: sigma_algebra_sigma)
+abbreviation "borel_measurable M \<equiv> measurable M borel"
lemma in_borel_measurable:
"f \<in> borel_measurable M \<longleftrightarrow>
- (\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = {S. open S}\<rparr>).
- f -` S \<inter> space M \<in> sets M)"
+ (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
by (auto simp add: measurable_def borel_def)
lemma in_borel_measurable_borel:
@@ -36,7 +34,7 @@
assumes "open A" shows "A \<in> sets borel"
proof -
have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
- thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
+ thus ?thesis unfolding borel_def by auto
qed
lemma borel_closed[simp]:
@@ -48,9 +46,9 @@
qed
lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
- unfolding Compl_eq_Diff_UNIV by (intro borel.Diff) auto
+ unfolding Compl_eq_Diff_UNIV by (intro Diff) auto
-lemma (in sigma_algebra) borel_measurable_vimage:
+lemma borel_measurable_vimage:
fixes f :: "'a \<Rightarrow> 'x::t2_space"
assumes borel: "f \<in> borel_measurable M"
shows "f -` {x} \<inter> space M \<in> sets M"
@@ -65,12 +63,12 @@
thus ?thesis by auto
qed
-lemma (in sigma_algebra) borel_measurableI:
+lemma borel_measurableI:
fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
shows "f \<in> borel_measurable M"
unfolding borel_def
-proof (rule measurable_sigma, simp_all)
+proof (rule measurable_measure_of, simp_all)
fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
using assms[of S] by simp
qed
@@ -78,22 +76,22 @@
lemma borel_singleton[simp, intro]:
fixes x :: "'a::t1_space"
shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
- proof (rule borel.insert_in_sets)
+ proof (rule insert_in_sets)
show "{x} \<in> sets borel"
using closed_singleton[of x] by (rule borel_closed)
qed simp
-lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
+lemma borel_measurable_const[simp, intro]:
"(\<lambda>x. c) \<in> borel_measurable M"
- by (auto intro!: measurable_const)
+ by auto
-lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
+lemma borel_measurable_indicator[simp, intro!]:
assumes A: "A \<in> sets M"
shows "indicator A \<in> borel_measurable M"
unfolding indicator_def [abs_def] using A
- by (auto intro!: measurable_If_set borel_measurable_const)
+ by (auto intro!: measurable_If_set)
-lemma (in sigma_algebra) borel_measurable_indicator_iff:
+lemma borel_measurable_indicator_iff:
"(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
(is "?I \<in> borel_measurable M \<longleftrightarrow> _")
proof
@@ -111,49 +109,7 @@
ultimately show "?I \<in> borel_measurable M" by auto
qed
-lemma (in sigma_algebra) borel_measurable_restricted:
- fixes f :: "'a \<Rightarrow> ereal" assumes "A \<in> sets M"
- shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
- (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
- (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
-proof -
- interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
- have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
- by (auto intro!: measurable_cong)
- show ?thesis unfolding *
- unfolding in_borel_measurable_borel
- proof (simp, safe)
- fix S :: "ereal set" assume "S \<in> sets borel"
- "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
- then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
- then have f: "?f -` S \<inter> A \<in> sets M"
- using `A \<in> sets M` sets_into_space by fastforce
- show "?f -` S \<inter> space M \<in> sets M"
- proof cases
- assume "0 \<in> S"
- then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
- using `A \<in> sets M` sets_into_space by auto
- then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
- next
- assume "0 \<notin> S"
- then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
- using `A \<in> sets M` sets_into_space
- by (auto simp: indicator_def split: split_if_asm)
- then show ?thesis using f by auto
- qed
- next
- fix S :: "ereal set" assume "S \<in> sets borel"
- "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
- then have f: "?f -` S \<inter> space M \<in> sets M" by auto
- then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
- using `A \<in> sets M` sets_into_space
- apply (simp add: image_iff)
- apply (rule bexI[OF _ f])
- by auto
- qed
-qed
-
-lemma (in sigma_algebra) borel_measurable_subalgebra:
+lemma borel_measurable_subalgebra:
assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
shows "f \<in> borel_measurable M"
using assms unfolding measurable_def by auto
@@ -220,7 +176,7 @@
shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel"
by (auto intro!: borel_closed closed_halfspace_component_le)
-lemma (in sigma_algebra) borel_measurable_less[simp, intro]:
+lemma borel_measurable_less[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -233,7 +189,7 @@
by simp (blast intro: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_le[simp, intro]:
+lemma borel_measurable_le[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -245,7 +201,7 @@
by simp blast
qed
-lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:
+lemma borel_measurable_eq[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -257,7 +213,7 @@
thus ?thesis using f g by auto
qed
-lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:
+lemma borel_measurable_neq[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -351,23 +307,70 @@
thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
qed auto
-lemma halfspace_span_open:
- "sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))
- \<subseteq> sets borel"
- by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
- open_halfspace_component_lt)
+lemma borel_sigma_sets_subset:
+ "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
+ using sigma_sets_subset[of A borel] by simp
+
+lemma borel_eq_sigmaI1:
+ fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
+ assumes borel_eq: "borel = sigma UNIV X"
+ assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))"
+ assumes F: "\<And>i. F i \<in> sets borel"
+ shows "borel = sigma UNIV (range F)"
+ unfolding borel_def
+proof (intro sigma_eqI antisym)
+ have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
+ unfolding borel_def by simp
+ also have "\<dots> = sigma_sets UNIV X"
+ unfolding borel_eq by simp
+ also have "\<dots> \<subseteq> sigma_sets UNIV (range F)"
+ using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
+ finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" .
+ show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}"
+ unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
+qed auto
-lemma halfspace_lt_in_halfspace:
- "{x\<Colon>'a. x $$ i < a} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>)"
- by (auto intro!: sigma_sets.Basic simp: sets_sigma)
+lemma borel_eq_sigmaI2:
+ fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
+ and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
+ assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
+ assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
+ assumes F: "\<And>i j. F i j \<in> sets borel"
+ shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
+ using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto
+
+lemma borel_eq_sigmaI3:
+ fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
+ assumes borel_eq: "borel = sigma UNIV X"
+ assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
+ assumes F: "\<And>i j. F i j \<in> sets borel"
+ shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
+ using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
+
+lemma borel_eq_sigmaI4:
+ fixes F :: "'i \<Rightarrow> 'a::topological_space set"
+ and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
+ assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
+ assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))"
+ assumes F: "\<And>i. F i \<in> sets borel"
+ shows "borel = sigma UNIV (range F)"
+ using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto
+
+lemma borel_eq_sigmaI5:
+ fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
+ assumes borel_eq: "borel = sigma UNIV (range G)"
+ assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
+ assumes F: "\<And>i j. F i j \<in> sets borel"
+ shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
+ using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
lemma halfspace_gt_in_halfspace:
- "{x\<Colon>'a. a < x $$ i} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>)"
- (is "?set \<in> sets ?SIGMA")
+ "{x\<Colon>'a. a < x $$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))"
+ (is "?set \<in> ?SIGMA")
proof -
- interpret sigma_algebra "?SIGMA"
- by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
- have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"
+ interpret sigma_algebra UNIV ?SIGMA
+ by (intro sigma_algebra_sigma_sets) simp_all
+ have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"
proof (safe, simp_all add: not_less)
fix x assume "a < x $$ i"
with reals_Archimedean[of "x $$ i - a"]
@@ -381,100 +384,78 @@
also assume "\<dots> \<le> x"
finally show "a < x" .
qed
- show "?set \<in> sets ?SIGMA" unfolding *
- by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
-qed
-
-lemma open_span_halfspace:
- "sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i < a})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
- then interpret sigma_algebra ?SIGMA .
- { fix S :: "'a set" assume "S \<in> {S. open S}"
- then have "open S" unfolding mem_Collect_eq .
- from open_UNION[OF this]
- obtain I where *: "S =
- (\<Union>(a, b)\<in>I.
- (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>
- (\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"
- unfolding greaterThanLessThan_def
- unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
- unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
- by blast
- have "S \<in> sets ?SIGMA"
- unfolding *
- by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
- then show ?thesis unfolding borel_def
- by (intro sets_sigma_subset) auto
+ show "?set \<in> ?SIGMA" unfolding *
+ by (auto intro!: Diff)
qed
-lemma halfspace_span_halfspace_le:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x $$ i \<le> a})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- { fix a i
- have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"
- proof (safe, simp_all)
- fix x::'a assume *: "x$$i < a"
- with reals_Archimedean[of "a - x$$i"]
- obtain n where "x $$ i < a - 1 / (real (Suc n))"
- by (auto simp: field_simps inverse_eq_divide)
- then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"
- by (blast intro: less_imp_le)
- next
- fix x::'a and n
- assume "x$$i \<le> a - 1 / real (Suc n)"
- also have "\<dots> < a" by auto
- finally show "x$$i < a" .
- qed
- have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *
- by (safe intro!: countable_UN)
- (auto simp: sets_sigma intro!: sigma_sets.Basic) }
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
+lemma borel_eq_halfspace_less:
+ "borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x $$ i < a}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI3[OF borel_def])
+ fix S :: "'a set" assume "S \<in> {S. open S}"
+ then have "open S" by simp
+ from open_UNION[OF this]
+ obtain I where *: "S =
+ (\<Union>(a, b)\<in>I.
+ (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>
+ (\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"
+ unfolding greaterThanLessThan_def
+ unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
+ unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
+ by blast
+ show "S \<in> ?SIGMA"
+ unfolding *
+ by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace)
+qed auto
-lemma halfspace_span_halfspace_ge:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x $$ i})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- { fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto
- have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *
- by (safe intro!: Diff)
- (auto simp: sets_sigma intro!: sigma_sets.Basic) }
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
+lemma borel_eq_halfspace_le:
+ "borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i \<le> a}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
+ fix a i
+ have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"
+ proof (safe, simp_all)
+ fix x::'a assume *: "x$$i < a"
+ with reals_Archimedean[of "a - x$$i"]
+ obtain n where "x $$ i < a - 1 / (real (Suc n))"
+ by (auto simp: field_simps inverse_eq_divide)
+ then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"
+ by (blast intro: less_imp_le)
+ next
+ fix x::'a and n
+ assume "x$$i \<le> a - 1 / real (Suc n)"
+ also have "\<dots> < a" by auto
+ finally show "x$$i < a" .
+ qed
+ show "{x. x$$i < a} \<in> ?SIGMA" unfolding *
+ by (safe intro!: countable_UN) auto
+qed auto
-lemma halfspace_le_span_halfspace_gt:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x $$ i})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- { fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
- have "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *
- by (safe intro!: Diff)
- (auto simp: sets_sigma intro!: sigma_sets.Basic) }
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
+lemma borel_eq_halfspace_ge:
+ "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
+ fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto
+ show "{x. x$$i < a} \<in> ?SIGMA" unfolding *
+ by (safe intro!: compl_sets) auto
+qed auto
-lemma halfspace_le_span_atMost:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA"
+lemma borel_eq_halfspace_greater:
+ "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x $$ i}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
+ fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
+ show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *
+ by (safe intro!: compl_sets) auto
+qed auto
+
+lemma borel_eq_atMost:
+ "borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
+ fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"
proof cases
- fix a i assume "i < DIM('a)"
+ assume "i < DIM('a)"
then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
fix x
@@ -484,28 +465,19 @@
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"
by (auto intro!: exI[of _ k])
qed
- show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *
- by (safe intro!: countable_UN)
- (auto simp: sets_sigma intro!: sigma_sets.Basic)
- next
- fix a i assume "\<not> i < DIM('a)"
- then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
- using top by auto
- qed
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
+ show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *
+ by (safe intro!: countable_UN) auto
+ qed (auto intro: sigma_sets_top sigma_sets.Empty)
+qed auto
-lemma halfspace_le_span_greaterThan:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA"
+lemma borel_eq_greaterThan:
+ "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
+ fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"
proof cases
- fix a i assume "i < DIM('a)"
- have "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
+ assume "i < DIM('a)"
+ have "{x::'a. x$$i \<le> a} = UNIV - {x::'a. a < x$$i}" by auto
also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)`
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
fix x
@@ -518,30 +490,22 @@
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"
by (auto intro!: exI[of _ k])
qed
- finally show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
+ finally show "{x. x$$i \<le> a} \<in> ?SIGMA"
apply (simp only:)
apply (safe intro!: countable_UN Diff)
- apply (auto simp: sets_sigma intro!: sigma_sets.Basic)
+ apply (auto intro: sigma_sets_top)
done
- next
- fix a i assume "\<not> i < DIM('a)"
- then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
- using top by auto
- qed
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
+ qed (auto intro: sigma_sets_top sigma_sets.Empty)
+qed auto
-lemma halfspace_le_span_lessThan:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- have "\<And>a i. {x. a \<le> x$$i} \<in> sets ?SIGMA"
+lemma borel_eq_lessThan:
+ "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
+ fix a i show "{x. a \<le> x$$i} \<in> ?SIGMA"
proof cases
fix a i assume "i < DIM('a)"
- have "{x::'a. a \<le> x$$i} = space ?SIGMA - {x::'a. x$$i < a}" by auto
+ have "{x::'a. a \<le> x$$i} = UNIV - {x::'a. x$$i < a}" by auto
also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
fix x
@@ -554,184 +518,70 @@
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k"
by (auto intro!: exI[of _ k])
qed
- finally show "{x. a \<le> x$$i} \<in> sets ?SIGMA"
+ finally show "{x. a \<le> x$$i} \<in> ?SIGMA"
apply (simp only:)
apply (safe intro!: countable_UN Diff)
- apply (auto simp: sets_sigma intro!: sigma_sets.Basic)
+ apply (auto intro: sigma_sets_top)
done
- next
- fix a i assume "\<not> i < DIM('a)"
- then show "{x. a \<le> x$$i} \<in> sets ?SIGMA"
- using top by auto
- qed
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
-
-lemma atMost_span_atLeastAtMost:
- "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
- sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
- (is "_ \<subseteq> sets ?SIGMA")
-proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- { fix a::'a
- have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
- proof (safe, simp_all add: eucl_le[where 'a='a])
- fix x
- from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]
- guess k::nat .. note k = this
- { fix i assume "i < DIM('a)"
- with k have "- x$$i \<le> real k"
- by (subst (asm) Max_le_iff) (auto simp: field_simps)
- then have "- real k \<le> x$$i" by simp }
- then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"
- by (auto intro!: exI[of _ k])
- qed
- have "{..a} \<in> sets ?SIGMA" unfolding *
- by (safe intro!: countable_UN)
- (auto simp: sets_sigma intro!: sigma_sets.Basic) }
- then show ?thesis by (intro sets_sigma_subset) auto
-qed
-
-lemma borel_eq_atMost:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
- by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
+ qed (auto intro: sigma_sets_top sigma_sets.Empty)
qed auto
lemma borel_eq_atLeastAtMost:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using atMost_span_atLeastAtMost halfspace_le_span_atMost
- halfspace_span_halfspace_le open_span_halfspace
- by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
-qed auto
-
-lemma borel_eq_greaterThan:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using halfspace_le_span_greaterThan
- halfspace_span_halfspace_le open_span_halfspace
- by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
-qed auto
-
-lemma borel_eq_lessThan:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using halfspace_le_span_lessThan
- halfspace_span_halfspace_ge open_span_halfspace
- by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
+ "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"
+ (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
+ fix a::'a
+ have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
+ proof (safe, simp_all add: eucl_le[where 'a='a])
+ fix x
+ from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]
+ guess k::nat .. note k = this
+ { fix i assume "i < DIM('a)"
+ with k have "- x$$i \<le> real k"
+ by (subst (asm) Max_le_iff) (auto simp: field_simps)
+ then have "- real k \<le> x$$i" by simp }
+ then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"
+ by (auto intro!: exI[of _ k])
+ qed
+ show "{..a} \<in> ?SIGMA" unfolding *
+ by (safe intro!: countable_UN)
+ (auto intro!: sigma_sets_top)
qed auto
lemma borel_eq_greaterThanLessThan:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
+ "borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))"
(is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
- show "sets borel \<subseteq> sets ?SIGMA"
- proof -
- have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
- then interpret sigma_algebra ?SIGMA .
- { fix M :: "'a set" assume "M \<in> {S. open S}"
- then have "open M" by simp
- have "M \<in> sets ?SIGMA"
- apply (subst open_UNION[OF `open M`])
- apply (safe intro!: countable_UN)
- apply (auto simp add: sigma_def intro!: sigma_sets.Basic)
- done }
- then show ?thesis
- unfolding borel_def by (intro sets_sigma_subset) auto
- qed
+proof (rule borel_eq_sigmaI1[OF borel_def])
+ fix M :: "'a set" assume "M \<in> {S. open S}"
+ then have "open M" by simp
+ show "M \<in> ?SIGMA"
+ apply (subst open_UNION[OF `open M`])
+ apply (safe intro!: countable_UN)
+ apply auto
+ done
qed auto
lemma borel_eq_atLeastLessThan:
- "borel = sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real})\<rparr>" (is "_ = ?S")
-proof (intro algebra.equality antisym)
- interpret sigma_algebra ?S
- by (rule sigma_algebra_sigma) auto
- show "sets borel \<subseteq> sets ?S"
- unfolding borel_eq_lessThan
- proof (intro sets_sigma_subset subsetI)
- have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
- fix A :: "real set" assume "A \<in> sets \<lparr>space = UNIV, sets = range lessThan\<rparr>"
- then obtain x where "A = {..< x}" by auto
- then have "A = (\<Union>i::nat. {-real i ..< x})"
- by (auto simp: move_uminus real_arch_simple)
- then show "A \<in> sets ?S"
- by (auto simp: sets_sigma intro!: sigma_sets.intros)
- qed simp
- show "sets ?S \<subseteq> sets borel"
- by (intro borel.sets_sigma_subset) auto
-qed simp_all
-
-lemma borel_eq_halfspace_le:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i \<le> a})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using open_span_halfspace halfspace_span_halfspace_le by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
+ "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
+proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
+ have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
+ fix x :: real
+ have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
+ by (auto simp: move_uminus real_arch_simple)
+ then show "{..< x} \<in> ?SIGMA"
+ by (auto intro: sigma_sets.intros)
qed auto
-lemma borel_eq_halfspace_less:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using open_span_halfspace .
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
-qed auto
-
-lemma borel_eq_halfspace_gt:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
-qed auto
-
-lemma borel_eq_halfspace_ge:
- "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x$$i})\<rparr>)"
- (is "_ = ?SIGMA")
-proof (intro algebra.equality antisym)
- show "sets borel \<subseteq> sets ?SIGMA"
- using halfspace_span_halfspace_ge open_span_halfspace by auto
- show "sets ?SIGMA \<subseteq> sets borel"
- by (rule borel.sets_sigma_subset) auto
-qed auto
-
-lemma (in sigma_algebra) borel_measurable_halfspacesI:
+lemma borel_measurable_halfspacesI:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
- assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
- and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
- and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
+ assumes F: "borel = sigma UNIV (range F)"
+ and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
+ and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
proof safe
fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
then show "S a i \<in> sets M" unfolding assms
- by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
+ by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1))
next
assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
{ fix a i have "S a i \<in> sets M"
@@ -740,61 +590,58 @@
with a show ?thesis unfolding assms(2) by simp
next
assume "\<not> i < DIM('c)"
- from assms(3)[OF this] show ?thesis .
+ from S[OF this] show ?thesis .
qed }
- then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
- by (auto intro!: measurable_sigma simp: assms(2))
- then show "f \<in> borel_measurable M" unfolding measurable_def
- unfolding assms(1) by simp
+ then show "f \<in> borel_measurable M"
+ by (auto intro!: measurable_measure_of simp: S_eq F)
qed
-lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
+lemma borel_measurable_iff_halfspace_le:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
-lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
+lemma borel_measurable_iff_halfspace_less:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
-lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
+lemma borel_measurable_iff_halfspace_ge:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
-lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
+lemma borel_measurable_iff_halfspace_greater:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)"
- by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
+ by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
-lemma (in sigma_algebra) borel_measurable_iff_le:
+lemma borel_measurable_iff_le:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
using borel_measurable_iff_halfspace_le[where 'c=real] by simp
-lemma (in sigma_algebra) borel_measurable_iff_less:
+lemma borel_measurable_iff_less:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
using borel_measurable_iff_halfspace_less[where 'c=real] by simp
-lemma (in sigma_algebra) borel_measurable_iff_ge:
+lemma borel_measurable_iff_ge:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
-lemma (in sigma_algebra) borel_measurable_iff_greater:
+lemma borel_measurable_iff_greater:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
lemma borel_measurable_euclidean_component:
"(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel"
- unfolding borel_def[where 'a=real]
-proof (rule borel.measurable_sigma, simp_all)
+proof (rule borel_measurableI)
fix S::"real set" assume "open S"
from open_vimage_euclidean_component[OF this]
- show "(\<lambda>x. x $$ i) -` S \<in> sets borel"
+ show "(\<lambda>x. x $$ i) -` S \<inter> space borel \<in> sets borel"
by (auto intro: borel_open)
qed
-lemma (in sigma_algebra) borel_measurable_euclidean_space:
+lemma borel_measurable_euclidean_space:
fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"
proof safe
@@ -810,7 +657,7 @@
subsection "Borel measurable operators"
-lemma (in sigma_algebra) affine_borel_measurable_vector:
+lemma affine_borel_measurable_vector:
fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
@@ -821,8 +668,7 @@
assume "b \<noteq> 0"
with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
by (auto intro!: open_affinity simp: scaleR_add_right)
- hence "?S \<in> sets borel"
- unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
+ hence "?S \<in> sets borel" by auto
moreover
from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
@@ -831,13 +677,13 @@
qed simp
qed
-lemma (in sigma_algebra) affine_borel_measurable:
+lemma affine_borel_measurable:
fixes g :: "'a \<Rightarrow> real"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
-lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
+lemma borel_measurable_add[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -855,7 +701,7 @@
by (simp add: borel_measurable_iff_ge)
qed
-lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
+lemma borel_measurable_setsum[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
@@ -864,7 +710,7 @@
thus ?thesis using assms by induct auto
qed simp
-lemma (in sigma_algebra) borel_measurable_square:
+lemma borel_measurable_square:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
@@ -916,7 +762,7 @@
shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
-lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
+lemma borel_measurable_uminus[simp, intro]:
fixes g :: "'a \<Rightarrow> real"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. - g x) \<in> borel_measurable M"
@@ -928,7 +774,7 @@
finally show ?thesis .
qed
-lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
+lemma borel_measurable_times[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -947,7 +793,7 @@
using 1 2 by simp
qed
-lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
+lemma borel_measurable_setprod[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
@@ -956,14 +802,14 @@
thus ?thesis using assms by induct auto
qed simp
-lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
+lemma borel_measurable_diff[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
unfolding diff_minus using assms by fast
-lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
+lemma borel_measurable_inverse[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
@@ -978,7 +824,7 @@
by (auto intro!: Int Un)
qed
-lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
+lemma borel_measurable_divide[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
and "g \<in> borel_measurable M"
@@ -986,7 +832,7 @@
unfolding field_divide_inverse
by (rule borel_measurable_inverse borel_measurable_times assms)+
-lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
+lemma borel_measurable_max[intro, simp]:
fixes f g :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
@@ -1001,7 +847,7 @@
by (auto intro!: Int)
qed
-lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
+lemma borel_measurable_min[intro, simp]:
fixes f g :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
@@ -1016,7 +862,7 @@
by (auto intro!: Int)
qed
-lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
+lemma borel_measurable_abs[simp, intro]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
proof -
@@ -1033,14 +879,14 @@
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
assumes "continuous_on UNIV f"
shows "f \<in> borel_measurable borel"
- apply(rule borel.borel_measurableI)
+ apply(rule borel_measurableI)
using continuous_open_preimage[OF assms] unfolding vimage_def by auto
lemma borel_measurable_continuous_on:
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
assumes cont: "continuous_on A f" "open A"
shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
-proof (rule borel.borel_measurableI)
+proof (rule borel_measurableI)
fix S :: "'b set" assume "open S"
then have "open {x\<in>A. f x \<in> S}"
by (intro continuous_open_preimage[OF cont]) auto
@@ -1049,11 +895,11 @@
{x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"
by (auto split: split_if_asm)
also have "\<dots> \<in> sets borel"
- using * `open A` by (auto simp del: space_borel intro!: borel.Un)
+ using * `open A` by (auto simp del: space_borel intro!: Un)
finally show "?f -` S \<inter> space borel \<in> sets borel" .
qed
-lemma (in sigma_algebra) convex_measurable:
+lemma convex_measurable:
fixes a b :: real
assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"
assumes q: "convex_on { a <..< b} q"
@@ -1091,7 +937,7 @@
finally show ?thesis .
qed
-lemma (in sigma_algebra) borel_measurable_log[simp,intro]:
+lemma borel_measurable_log[simp,intro]:
assumes f: "f \<in> borel_measurable M" and "1 < b"
shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
@@ -1101,25 +947,21 @@
lemma borel_measurable_ereal_borel:
"ereal \<in> borel_measurable borel"
- unfolding borel_def[where 'a=ereal]
-proof (rule borel.measurable_sigma)
- fix X :: "ereal set" assume "X \<in> sets \<lparr>space = UNIV, sets = {S. open S} \<rparr>"
- then have "open X" by simp
+proof (rule borel_measurableI)
+ fix X :: "ereal set" assume "open X"
then have "open (ereal -` X \<inter> space borel)"
by (simp add: open_ereal_vimage)
then show "ereal -` X \<inter> space borel \<in> sets borel" by auto
-qed auto
+qed
-lemma (in sigma_algebra) borel_measurable_ereal[simp, intro]:
+lemma borel_measurable_ereal[simp, intro]:
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
using measurable_comp[OF f borel_measurable_ereal_borel] unfolding comp_def .
lemma borel_measurable_real_of_ereal_borel:
"(real :: ereal \<Rightarrow> real) \<in> borel_measurable borel"
- unfolding borel_def[where 'a=real]
-proof (rule borel.measurable_sigma)
- fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = {S. open S} \<rparr>"
- then have "open B" by simp
+proof (rule borel_measurableI)
+ fix B :: "real set" assume "open B"
have *: "ereal -` real -` (B - {0}) = B - {0}" by auto
have open_real: "open (real -` (B - {0}) :: ereal set)"
unfolding open_ereal_def * using `open B` by auto
@@ -1137,13 +979,13 @@
then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
using open_real by auto
qed
-qed auto
+qed
-lemma (in sigma_algebra) borel_measurable_real_of_ereal[simp, intro]:
+lemma borel_measurable_real_of_ereal[simp, intro]:
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: ereal)) \<in> borel_measurable M"
using measurable_comp[OF f borel_measurable_real_of_ereal_borel] unfolding comp_def .
-lemma (in sigma_algebra) borel_measurable_ereal_iff:
+lemma borel_measurable_ereal_iff:
shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
proof
assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
@@ -1151,7 +993,7 @@
show "f \<in> borel_measurable M" by auto
qed auto
-lemma (in sigma_algebra) borel_measurable_ereal_iff_real:
+lemma borel_measurable_ereal_iff_real:
fixes f :: "'a \<Rightarrow> ereal"
shows "f \<in> borel_measurable M \<longleftrightarrow>
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
@@ -1165,7 +1007,7 @@
finally show "f \<in> borel_measurable M" .
qed (auto intro: measurable_sets borel_measurable_real_of_ereal)
-lemma (in sigma_algebra) less_eq_ge_measurable:
+lemma less_eq_ge_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
proof
@@ -1178,7 +1020,7 @@
ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
qed
-lemma (in sigma_algebra) greater_eq_le_measurable:
+lemma greater_eq_le_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
proof
@@ -1191,28 +1033,27 @@
ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
qed
-lemma (in sigma_algebra) borel_measurable_uminus_borel_ereal:
+lemma borel_measurable_uminus_borel_ereal:
"(uminus :: ereal \<Rightarrow> ereal) \<in> borel_measurable borel"
-proof (subst borel_def, rule borel.measurable_sigma)
- fix X :: "ereal set" assume "X \<in> sets \<lparr>space = UNIV, sets = {S. open S}\<rparr>"
- then have "open X" by simp
+proof (rule borel_measurableI)
+ fix X :: "ereal set" assume "open X"
have "uminus -` X = uminus ` X" by (force simp: image_iff)
then have "open (uminus -` X)" using `open X` ereal_open_uminus by auto
then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
-qed auto
+qed
-lemma (in sigma_algebra) borel_measurable_uminus_ereal[intro]:
+lemma borel_measurable_uminus_ereal[intro]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
using measurable_comp[OF assms borel_measurable_uminus_borel_ereal] by (simp add: comp_def)
-lemma (in sigma_algebra) borel_measurable_uminus_eq_ereal[simp]:
+lemma borel_measurable_uminus_eq_ereal[simp]:
"(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
proof
assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
qed auto
-lemma (in sigma_algebra) borel_measurable_eq_atMost_ereal:
+lemma borel_measurable_eq_atMost_ereal:
fixes f :: "'a \<Rightarrow> ereal"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
proof (intro iffI allI)
@@ -1244,7 +1085,7 @@
qed
qed (simp add: measurable_sets)
-lemma (in sigma_algebra) borel_measurable_eq_atLeast_ereal:
+lemma borel_measurable_eq_atLeast_ereal:
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
proof
assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
@@ -1255,15 +1096,15 @@
then show "f \<in> borel_measurable M" by simp
qed (simp add: measurable_sets)
-lemma (in sigma_algebra) borel_measurable_ereal_iff_less:
+lemma borel_measurable_ereal_iff_less:
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable ..
-lemma (in sigma_algebra) borel_measurable_ereal_iff_ge:
+lemma borel_measurable_ereal_iff_ge:
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable ..
-lemma (in sigma_algebra) borel_measurable_ereal_eq_const:
+lemma borel_measurable_ereal_eq_const:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x = c} \<in> sets M"
proof -
@@ -1271,7 +1112,7 @@
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_ereal_neq_const:
+lemma borel_measurable_ereal_neq_const:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
proof -
@@ -1279,7 +1120,7 @@
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_ereal_le[intro,simp]:
+lemma borel_measurable_ereal_le[intro,simp]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -1294,7 +1135,7 @@
with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_ereal_less[intro,simp]:
+lemma borel_measurable_ereal_less[intro,simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -1304,7 +1145,7 @@
then show ?thesis using g f by auto
qed
-lemma (in sigma_algebra) borel_measurable_ereal_eq[intro,simp]:
+lemma borel_measurable_ereal_eq[intro,simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -1314,7 +1155,7 @@
then show ?thesis using g f by auto
qed
-lemma (in sigma_algebra) borel_measurable_ereal_neq[intro,simp]:
+lemma borel_measurable_ereal_neq[intro,simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
@@ -1324,12 +1165,12 @@
thus ?thesis using f g by auto
qed
-lemma (in sigma_algebra) split_sets:
+lemma split_sets:
"{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
"{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
by auto
-lemma (in sigma_algebra) borel_measurable_ereal_add[intro, simp]:
+lemma borel_measurable_ereal_add[intro, simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
@@ -1344,7 +1185,7 @@
intro!: Un measurable_If measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_ereal_setsum[simp, intro]:
+lemma borel_measurable_ereal_setsum[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
@@ -1354,7 +1195,7 @@
by induct auto
qed (simp add: borel_measurable_const)
-lemma (in sigma_algebra) borel_measurable_ereal_abs[intro, simp]:
+lemma borel_measurable_ereal_abs[intro, simp]:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
proof -
@@ -1362,7 +1203,7 @@
then show ?thesis using assms by (auto intro!: measurable_If)
qed
-lemma (in sigma_algebra) borel_measurable_ereal_times[intro, simp]:
+lemma borel_measurable_ereal_times[intro, simp]:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
proof -
@@ -1386,7 +1227,7 @@
(auto simp: split_sets intro!: Int)
qed
-lemma (in sigma_algebra) borel_measurable_ereal_setprod[simp, intro]:
+lemma borel_measurable_ereal_setprod[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
@@ -1395,21 +1236,21 @@
thus ?thesis using assms by induct auto
qed simp
-lemma (in sigma_algebra) borel_measurable_ereal_min[simp, intro]:
+lemma borel_measurable_ereal_min[simp, intro]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
using assms unfolding min_def by (auto intro!: measurable_If)
-lemma (in sigma_algebra) borel_measurable_ereal_max[simp, intro]:
+lemma borel_measurable_ereal_max[simp, intro]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M"
and "g \<in> borel_measurable M"
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
using assms unfolding max_def by (auto intro!: measurable_If)
-lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
+lemma borel_measurable_SUP[simp, intro]:
fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
@@ -1422,7 +1263,7 @@
using assms by auto
qed
-lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
+lemma borel_measurable_INF[simp, intro]:
fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
@@ -1435,26 +1276,39 @@
using assms by auto
qed
-lemma (in sigma_algebra) borel_measurable_liminf[simp, intro]:
+lemma borel_measurable_liminf[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. f i \<in> borel_measurable M"
shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
unfolding liminf_SUPR_INFI using assms by auto
-lemma (in sigma_algebra) borel_measurable_limsup[simp, intro]:
+lemma borel_measurable_limsup[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. f i \<in> borel_measurable M"
shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
unfolding limsup_INFI_SUPR using assms by auto
-lemma (in sigma_algebra) borel_measurable_ereal_diff[simp, intro]:
+lemma borel_measurable_ereal_diff[simp, intro]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
unfolding minus_ereal_def using assms by auto
-lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
+lemma borel_measurable_ereal_inverse[simp, intro]:
+ assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
+proof -
+ { fix x have "inverse (f x) = (if f x = 0 then \<infinity> else ereal (inverse (real (f x))))"
+ by (cases "f x") auto }
+ with f show ?thesis
+ by (auto intro!: measurable_If)
+qed
+
+lemma borel_measurable_ereal_divide[simp, intro]:
+ "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. f x / g x :: ereal) \<in> borel_measurable M"
+ unfolding divide_ereal_def by auto
+
+lemma borel_measurable_psuminf[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
@@ -1465,7 +1319,7 @@
section "LIMSEQ is borel measurable"
-lemma (in sigma_algebra) borel_measurable_LIMSEQ:
+lemma borel_measurable_LIMSEQ:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
and u: "\<And>i. u i \<in> borel_measurable M"
--- a/src/HOL/Probability/Caratheodory.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Caratheodory.thy Mon Apr 23 12:14:35 2012 +0200
@@ -6,7 +6,7 @@
header {*Caratheodory Extension Theorem*}
theory Caratheodory
-imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
+ imports Measure_Space
begin
lemma sums_def2:
@@ -53,128 +53,53 @@
subsection {* Measure Spaces *}
-record 'a measure_space = "'a algebra" +
- measure :: "'a set \<Rightarrow> ereal"
-
-definition positive where "positive M f \<longleftrightarrow> f {} = (0::ereal) \<and> (\<forall>A\<in>sets M. 0 \<le> f A)"
-
-definition additive where "additive M f \<longleftrightarrow>
- (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) = f x + f y)"
-
-definition countably_additive :: "('a, 'b) algebra_scheme \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
- "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
- (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
-
-definition increasing where "increasing M f \<longleftrightarrow>
- (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
-
definition subadditive where "subadditive M f \<longleftrightarrow>
- (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
+ (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow>
- (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
+ (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
(f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
-definition lambda_system where "lambda_system M f = {l \<in> sets M.
- \<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x}"
+definition lambda_system where "lambda_system \<Omega> M f = {l \<in> M.
+ \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
definition outer_measure_space where "outer_measure_space M f \<longleftrightarrow>
positive M f \<and> increasing M f \<and> countably_subadditive M f"
definition measure_set where "measure_set M f X = {r.
- \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
-
-locale measure_space = sigma_algebra M for M :: "('a, 'b) measure_space_scheme" +
- assumes measure_positive: "positive M (measure M)"
- and ca: "countably_additive M (measure M)"
-
-abbreviation (in measure_space) "\<mu> \<equiv> measure M"
-
-lemma (in measure_space)
- shows empty_measure[simp, intro]: "\<mu> {} = 0"
- and positive_measure[simp, intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> 0 \<le> \<mu> A"
- using measure_positive unfolding positive_def by auto
-
-lemma increasingD:
- "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
- by (auto simp add: increasing_def)
+ \<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
lemma subadditiveD:
- "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> sets M \<Longrightarrow> y \<in> sets M
- \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
+ "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
by (auto simp add: subadditive_def)
-lemma additiveD:
- "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> sets M \<Longrightarrow> y \<in> sets M
- \<Longrightarrow> f (x \<union> y) = f x + f y"
- by (auto simp add: additive_def)
-
-lemma countably_additiveI:
- assumes "\<And>A x. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M
- \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
- shows "countably_additive M f"
- using assms by (simp add: countably_additive_def)
-
-section "Extend binary sets"
-
-lemma LIMSEQ_binaryset:
- assumes f: "f {} = 0"
- shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
-proof -
- have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
- proof
- fix n
- show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
- by (induct n) (auto simp add: binaryset_def f)
- qed
- moreover
- have "... ----> f A + f B" by (rule tendsto_const)
- ultimately
- have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
- by metis
- hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
- by simp
- thus ?thesis by (rule LIMSEQ_offset [where k=2])
-qed
-
-lemma binaryset_sums:
- assumes f: "f {} = 0"
- shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
- by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
-
-lemma suminf_binaryset_eq:
- fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
- shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
- by (metis binaryset_sums sums_unique)
-
subsection {* Lambda Systems *}
lemma (in algebra) lambda_system_eq:
- shows "lambda_system M f = {l \<in> sets M.
- \<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x}"
+ shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
proof -
- have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
+ have [simp]: "!!l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l"
by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
show ?thesis
by (auto simp add: lambda_system_def) (metis Int_commute)+
qed
lemma (in algebra) lambda_system_empty:
- "positive M f \<Longrightarrow> {} \<in> lambda_system M f"
+ "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
by (auto simp add: positive_def lambda_system_eq)
lemma lambda_system_sets:
- "x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
+ "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
by (simp add: lambda_system_def)
lemma (in algebra) lambda_system_Compl:
fixes f:: "'a set \<Rightarrow> ereal"
- assumes x: "x \<in> lambda_system M f"
- shows "space M - x \<in> lambda_system M f"
+ assumes x: "x \<in> lambda_system \<Omega> M f"
+ shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
proof -
- have "x \<subseteq> space M"
+ have "x \<subseteq> \<Omega>"
by (metis sets_into_space lambda_system_sets x)
- hence "space M - (space M - x) = x"
+ hence "\<Omega> - (\<Omega> - x) = x"
by (metis double_diff equalityE)
with x show ?thesis
by (force simp add: lambda_system_def ac_simps)
@@ -182,16 +107,16 @@
lemma (in algebra) lambda_system_Int:
fixes f:: "'a set \<Rightarrow> ereal"
- assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
- shows "x \<inter> y \<in> lambda_system M f"
+ assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ shows "x \<inter> y \<in> lambda_system \<Omega> M f"
proof -
from xl yl show ?thesis
proof (auto simp add: positive_def lambda_system_eq Int)
fix u
- assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M"
- and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z"
- and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z"
- have "u - x \<inter> y \<in> sets M"
+ assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M"
+ and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z"
+ and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z"
+ have "u - x \<inter> y \<in> M"
by (metis Diff Diff_Int Un u x y)
moreover
have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
@@ -216,20 +141,20 @@
lemma (in algebra) lambda_system_Un:
fixes f:: "'a set \<Rightarrow> ereal"
- assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
- shows "x \<union> y \<in> lambda_system M f"
+ assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ shows "x \<union> y \<in> lambda_system \<Omega> M f"
proof -
- have "(space M - x) \<inter> (space M - y) \<in> sets M"
+ have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M"
by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
moreover
- have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
+ have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - y))"
by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
ultimately show ?thesis
by (metis lambda_system_Compl lambda_system_Int xl yl)
qed
lemma (in algebra) lambda_system_algebra:
- "positive M f \<Longrightarrow> algebra (M\<lparr>sets := lambda_system M f\<rparr>)"
+ "positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)"
apply (auto simp add: algebra_iff_Un)
apply (metis lambda_system_sets set_mp sets_into_space)
apply (metis lambda_system_empty)
@@ -238,117 +163,62 @@
done
lemma (in algebra) lambda_system_strong_additive:
- assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
- and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
+ assumes z: "z \<in> M" and disj: "x \<inter> y = {}"
+ and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
proof -
have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
moreover
have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
moreover
- have "(z \<inter> (x \<union> y)) \<in> sets M"
+ have "(z \<inter> (x \<union> y)) \<in> M"
by (metis Int Un lambda_system_sets xl yl z)
ultimately show ?thesis using xl yl
by (simp add: lambda_system_eq)
qed
-lemma (in algebra) lambda_system_additive:
- "additive (M (|sets := lambda_system M f|)) f"
+lemma (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
proof (auto simp add: additive_def)
fix x and y
assume disj: "x \<inter> y = {}"
- and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
- hence "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
+ and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
+ hence "x \<in> M" "y \<in> M" by (blast intro: lambda_system_sets)+
thus "f (x \<union> y) = f x + f y"
using lambda_system_strong_additive [OF top disj xl yl]
by (simp add: Un)
qed
-lemma (in ring_of_sets) disjointed_additive:
- assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> sets M" "incseq A"
- shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
-proof (induct n)
- case (Suc n)
- then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
- by simp
- also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
- using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
- also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
- using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
- finally show ?case .
-qed simp
-
lemma (in ring_of_sets) countably_subadditive_subadditive:
assumes f: "positive M f" and cs: "countably_subadditive M f"
shows "subadditive M f"
proof (auto simp add: subadditive_def)
fix x y
- assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
+ assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
hence "disjoint_family (binaryset x y)"
by (auto simp add: disjoint_family_on_def binaryset_def)
- hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
- (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
+ hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
+ (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
using cs by (auto simp add: countably_subadditive_def)
- hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
+ hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) \<le> f x + f y" using f x y
by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
qed
-lemma (in ring_of_sets) additive_sum:
- fixes A:: "nat \<Rightarrow> 'a set"
- assumes f: "positive M f" and ad: "additive M f" and "finite S"
- and A: "range A \<subseteq> sets M"
- and disj: "disjoint_family_on A S"
- shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
-using `finite S` disj proof induct
- case empty show ?case using f by (simp add: positive_def)
-next
- case (insert s S)
- then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
- by (auto simp add: disjoint_family_on_def neq_iff)
- moreover
- have "A s \<in> sets M" using A by blast
- moreover have "(\<Union>i\<in>S. A i) \<in> sets M"
- using A `finite S` by auto
- moreover
- ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
- using ad UNION_in_sets A by (auto simp add: additive_def)
- with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
- by (auto simp add: additive_def subset_insertI)
-qed
-
-lemma (in algebra) increasing_additive_bound:
- fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
- assumes f: "positive M f" and ad: "additive M f"
- and inc: "increasing M f"
- and A: "range A \<subseteq> sets M"
- and disj: "disjoint_family A"
- shows "(\<Sum>i. f (A i)) \<le> f (space M)"
-proof (safe intro!: suminf_bound)
- fix N
- note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
- have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
- by (rule additive_sum [OF f ad _ A]) (auto simp: disj_N)
- also have "... \<le> f (space M)" using space_closed A
- by (intro increasingD[OF inc] finite_UN) auto
- finally show "(\<Sum>i<N. f (A i)) \<le> f (space M)" by simp
-qed (insert f A, auto simp: positive_def)
-
lemma lambda_system_increasing:
- "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
+ "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
by (simp add: increasing_def lambda_system_def)
lemma lambda_system_positive:
- "positive M f \<Longrightarrow> positive (M (|sets := lambda_system M f|)) f"
+ "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
by (simp add: positive_def lambda_system_def)
lemma (in algebra) lambda_system_strong_sum:
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
- assumes f: "positive M f" and a: "a \<in> sets M"
- and A: "range A \<subseteq> lambda_system M f"
+ assumes f: "positive M f" and a: "a \<in> M"
+ and A: "range A \<subseteq> lambda_system \<Omega> M f"
and disj: "disjoint_family A"
shows "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
proof (induct n)
@@ -357,11 +227,11 @@
case (Suc n)
have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
by (force simp add: disjoint_family_on_def neq_iff)
- have 3: "A n \<in> lambda_system M f" using A
+ have 3: "A n \<in> lambda_system \<Omega> M f" using A
by blast
- interpret l: algebra "M\<lparr>sets := lambda_system M f\<rparr>"
+ interpret l: algebra \<Omega> "lambda_system \<Omega> M f"
using f by (rule lambda_system_algebra)
- have 4: "UNION {0..<n} A \<in> lambda_system M f"
+ have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> M f"
using A l.UNION_in_sets by simp
from Suc.hyps show ?case
by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
@@ -369,23 +239,23 @@
lemma (in sigma_algebra) lambda_system_caratheodory:
assumes oms: "outer_measure_space M f"
- and A: "range A \<subseteq> lambda_system M f"
+ and A: "range A \<subseteq> lambda_system \<Omega> M f"
and disj: "disjoint_family A"
- shows "(\<Union>i. A i) \<in> lambda_system M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
+ shows "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
proof -
have pos: "positive M f" and inc: "increasing M f"
and csa: "countably_subadditive M f"
by (metis oms outer_measure_space_def)+
have sa: "subadditive M f"
by (metis countably_subadditive_subadditive csa pos)
- have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
- by simp
- interpret ls: algebra "M\<lparr>sets := lambda_system M f\<rparr>"
+ have A': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A
+ by auto
+ interpret ls: algebra \<Omega> "lambda_system \<Omega> M f"
using pos by (rule lambda_system_algebra)
- have A'': "range A \<subseteq> sets M"
+ have A'': "range A \<subseteq> M"
by (metis A image_subset_iff lambda_system_sets)
- have U_in: "(\<Union>i. A i) \<in> sets M"
+ have U_in: "(\<Union>i. A i) \<in> M"
by (metis A'' countable_UN)
have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
proof (rule antisym)
@@ -396,22 +266,22 @@
show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis]
using A''
- by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] allI countable_UN)
+ by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] countable_UN)
qed
{
fix a
- assume a [iff]: "a \<in> sets M"
+ assume a [iff]: "a \<in> M"
have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
proof -
show ?thesis
proof (rule antisym)
- have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
+ have "range (\<lambda>i. a \<inter> A i) \<subseteq> M" using A''
by blast
moreover
have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
by (auto simp add: disjoint_family_on_def)
moreover
- have "a \<inter> (\<Union>i. A i) \<in> sets M"
+ have "a \<inter> (\<Union>i. A i) \<in> M"
by (metis Int U_in a)
ultimately
have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
@@ -424,11 +294,11 @@
have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
proof (intro suminf_bound_add allI)
fix n
- have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
+ have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M"
by (metis A'' UNION_in_sets)
have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
- have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
+ have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system \<Omega> M f"
using ls.UNION_in_sets by (simp add: A)
hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
by (simp add: lambda_system_eq UNION_in)
@@ -437,9 +307,9 @@
thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
next
- have "\<And>i. a \<inter> A i \<in> sets M" using A'' by auto
+ have "\<And>i. a \<inter> A i \<in> M" using A'' by auto
then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
- have "\<And>i. a - (\<Union>i. A i) \<in> sets M" using A'' by auto
+ have "\<And>i. a - (\<Union>i. A i) \<in> M" using A'' by auto
then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
qed
@@ -460,66 +330,29 @@
lemma (in sigma_algebra) caratheodory_lemma:
assumes oms: "outer_measure_space M f"
- shows "measure_space \<lparr> space = space M, sets = lambda_system M f, measure = f \<rparr>"
- (is "measure_space ?M")
+ defines "L \<equiv> lambda_system \<Omega> M f"
+ shows "measure_space \<Omega> L f"
proof -
have pos: "positive M f"
by (metis oms outer_measure_space_def)
- have alg: "algebra ?M"
+ have alg: "algebra \<Omega> L"
using lambda_system_algebra [of f, OF pos]
- by (simp add: algebra_iff_Un)
+ by (simp add: algebra_iff_Un L_def)
then
- have "sigma_algebra ?M"
+ have "sigma_algebra \<Omega> L"
using lambda_system_caratheodory [OF oms]
- by (simp add: sigma_algebra_disjoint_iff)
+ by (simp add: sigma_algebra_disjoint_iff L_def)
moreover
- have "measure_space_axioms ?M"
+ have "countably_additive L f" "positive L f"
using pos lambda_system_caratheodory [OF oms]
- by (simp add: measure_space_axioms_def positive_def lambda_system_sets
- countably_additive_def o_def)
+ by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def)
ultimately
show ?thesis
- by (simp add: measure_space_def)
-qed
-
-lemma (in ring_of_sets) additive_increasing:
- assumes posf: "positive M f" and addf: "additive M f"
- shows "increasing M f"
-proof (auto simp add: increasing_def)
- fix x y
- assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
- then have "y - x \<in> sets M" by auto
- then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
- then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
- also have "... = f (x \<union> (y-x))" using addf
- by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
- also have "... = f y"
- by (metis Un_Diff_cancel Un_absorb1 xy(3))
- finally show "f x \<le> f y" by simp
-qed
-
-lemma (in ring_of_sets) countably_additive_additive:
- assumes posf: "positive M f" and ca: "countably_additive M f"
- shows "additive M f"
-proof (auto simp add: additive_def)
- fix x y
- assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
- hence "disjoint_family (binaryset x y)"
- by (auto simp add: disjoint_family_on_def binaryset_def)
- hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
- (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
- f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
- using ca
- by (simp add: countably_additive_def)
- hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
- f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
- by (simp add: range_binaryset_eq UN_binaryset_eq)
- thus "f (x \<union> y) = f x + f y" using posf x y
- by (auto simp add: Un suminf_binaryset_eq positive_def)
+ using pos by (simp add: measure_space_def)
qed
lemma inf_measure_nonempty:
- assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
+ assumes f: "positive M f" and b: "b \<in> M" and a: "a \<subseteq> b" "{} \<in> M"
shows "f b \<in> measure_set M f a"
proof -
let ?A = "\<lambda>i::nat. (if i = 0 then b else {})"
@@ -534,14 +367,14 @@
lemma (in ring_of_sets) inf_measure_agrees:
assumes posf: "positive M f" and ca: "countably_additive M f"
- and s: "s \<in> sets M"
+ and s: "s \<in> M"
shows "Inf (measure_set M f s) = f s"
unfolding Inf_ereal_def
proof (safe intro!: Greatest_equality)
fix z
assume z: "z \<in> measure_set M f s"
from this obtain A where
- A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
+ A: "range A \<subseteq> M" and disj: "disjoint_family A"
and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z"
by (auto simp add: measure_set_def comp_def)
hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
@@ -549,11 +382,11 @@
by (metis additive_increasing ca countably_additive_additive posf)
have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
proof (rule ca[unfolded countably_additive_def, rule_format])
- show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
+ show "range (\<lambda>n. A n \<inter> s) \<subseteq> M" using A s
by blast
show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
by (auto simp add: disjoint_family_on_def)
- show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
+ show "(\<Union>i. A i \<inter> s) \<in> M" using A s
by (metis UN_extend_simps(4) s seq)
qed
hence "f s = (\<Sum>i. f (A i \<inter> s))"
@@ -562,7 +395,7 @@
proof (rule suminf_le_pos)
fix n show "f (A n \<inter> s) \<le> f (A n)" using A s
by (force intro: increasingD [OF inc])
- fix N have "A N \<inter> s \<in> sets M" using A s by auto
+ fix N have "A N \<inter> s \<in> M" using A s by auto
then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto
qed
also have "... = z" by (rule si)
@@ -578,7 +411,7 @@
assumes posf: "positive M f" "r \<in> measure_set M f X"
shows "0 \<le> r"
proof -
- obtain A where "range A \<subseteq> sets M" and r: "r = (\<Sum>i. f (A i))"
+ obtain A where "range A \<subseteq> M" and r: "r = (\<Sum>i. f (A i))"
using `r \<in> measure_set M f X` unfolding measure_set_def by auto
then show "0 \<le> r" using posf unfolding r positive_def
by (intro suminf_0_le) auto
@@ -593,26 +426,25 @@
qed
lemma inf_measure_empty:
- assumes posf: "positive M f" and "{} \<in> sets M"
+ assumes posf: "positive M f" and "{} \<in> M"
shows "Inf (measure_set M f {}) = 0"
proof (rule antisym)
show "Inf (measure_set M f {}) \<le> 0"
- by (metis complete_lattice_class.Inf_lower `{} \<in> sets M`
+ by (metis complete_lattice_class.Inf_lower `{} \<in> M`
inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
qed (rule inf_measure_pos[OF posf])
lemma (in ring_of_sets) inf_measure_positive:
- assumes p: "positive M f" and "{} \<in> sets M"
+ assumes p: "positive M f" and "{} \<in> M"
shows "positive M (\<lambda>x. Inf (measure_set M f x))"
proof (unfold positive_def, intro conjI ballI)
show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto
- fix A assume "A \<in> sets M"
+ fix A assume "A \<in> M"
qed (rule inf_measure_pos[OF p])
lemma (in ring_of_sets) inf_measure_increasing:
assumes posf: "positive M f"
- shows "increasing \<lparr> space = space M, sets = Pow (space M) \<rparr>
- (\<lambda>x. Inf (measure_set M f x))"
+ shows "increasing (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
apply (clarsimp simp add: increasing_def)
apply (rule complete_lattice_class.Inf_greatest)
apply (rule complete_lattice_class.Inf_lower)
@@ -621,13 +453,13 @@
lemma (in ring_of_sets) inf_measure_le:
assumes posf: "positive M f" and inc: "increasing M f"
- and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
+ and x: "x \<in> {r . \<exists>A. range A \<subseteq> M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
shows "Inf (measure_set M f s) \<le> x"
proof -
- obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
+ obtain A where A: "range A \<subseteq> M" and ss: "s \<subseteq> (\<Union>i. A i)"
and xeq: "(\<Sum>i. f (A i)) = x"
using x by auto
- have dA: "range (disjointed A) \<subseteq> sets M"
+ have dA: "range (disjointed A) \<subseteq> M"
by (metis A range_disjointed_sets)
have "\<forall>n. f (disjointed A n) \<le> f (A n)"
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
@@ -648,8 +480,8 @@
lemma (in ring_of_sets) inf_measure_close:
fixes e :: ereal
- assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)" and "Inf (measure_set M f s) \<noteq> \<infinity>"
- shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
+ assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (\<Omega>)" and "Inf (measure_set M f s) \<noteq> \<infinity>"
+ shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
(\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
proof -
from `Inf (measure_set M f s) \<noteq> \<infinity>` have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>"
@@ -662,22 +494,21 @@
lemma (in ring_of_sets) inf_measure_countably_subadditive:
assumes posf: "positive M f" and inc: "increasing M f"
- shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
- (\<lambda>x. Inf (measure_set M f x))"
+ shows "countably_subadditive (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
proof (simp add: countably_subadditive_def, safe)
fix A :: "nat \<Rightarrow> 'a set"
let ?outer = "\<lambda>B. Inf (measure_set M f B)"
- assume A: "range A \<subseteq> Pow (space M)"
+ assume A: "range A \<subseteq> Pow (\<Omega>)"
and disj: "disjoint_family A"
- and sb: "(\<Union>i. A i) \<subseteq> space M"
+ and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
{ fix e :: ereal assume e: "0 < e" and "\<forall>i. ?outer (A i) \<noteq> \<infinity>"
- hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
+ hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> M \<and> disjoint_family (BB n) \<and>
A n \<subseteq> (\<Union>i. BB n i) \<and> (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)"
apply (safe intro!: choice inf_measure_close [of f, OF posf])
using e sb by (auto simp: ereal_zero_less_0_iff one_ereal_def)
then obtain BB
- where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
+ where BB: "\<And>n. (range (BB n) \<subseteq> M)"
and disjBB: "\<And>n. disjoint_family (BB n)"
and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)"
@@ -697,7 +528,7 @@
finally show ?thesis .
qed
def C \<equiv> "(split BB) o prod_decode"
- have C: "!!n. C n \<in> sets M"
+ have C: "!!n. C n \<in> M"
apply (rule_tac p="prod_decode n" in PairE)
apply (simp add: C_def)
apply (metis BB subsetD rangeI)
@@ -744,9 +575,8 @@
qed
lemma (in ring_of_sets) inf_measure_outer:
- "\<lbrakk> positive M f ; increasing M f \<rbrakk>
- \<Longrightarrow> outer_measure_space \<lparr> space = space M, sets = Pow (space M) \<rparr>
- (\<lambda>x. Inf (measure_set M f x))"
+ "\<lbrakk> positive M f ; increasing M f \<rbrakk> \<Longrightarrow>
+ outer_measure_space (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
using inf_measure_pos[of M f]
by (simp add: outer_measure_space_def inf_measure_empty
inf_measure_increasing inf_measure_countably_subadditive positive_def)
@@ -754,14 +584,13 @@
lemma (in ring_of_sets) algebra_subset_lambda_system:
assumes posf: "positive M f" and inc: "increasing M f"
and add: "additive M f"
- shows "sets M \<subseteq> lambda_system \<lparr> space = space M, sets = Pow (space M) \<rparr>
- (\<lambda>x. Inf (measure_set M f x))"
+ shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
proof (auto dest: sets_into_space
simp add: algebra.lambda_system_eq [OF algebra_Pow])
fix x s
- assume x: "x \<in> sets M"
- and s: "s \<subseteq> space M"
- have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
+ assume x: "x \<in> M"
+ and s: "s \<subseteq> \<Omega>"
+ have [simp]: "!!x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - x) = s-x" using s
by blast
have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
\<le> Inf (measure_set M f s)"
@@ -774,7 +603,7 @@
show ?thesis
proof (rule complete_lattice_class.Inf_greatest)
fix r assume "r \<in> measure_set M f s"
- then obtain A where A: "disjoint_family A" "range A \<subseteq> sets M" "s \<subseteq> (\<Union>i. A i)"
+ then obtain A where A: "disjoint_family A" "range A \<subseteq> M" "s \<subseteq> (\<Union>i. A i)"
and r: "r = (\<Sum>i. f (A i))" unfolding measure_set_def by auto
have "Inf (measure_set M f (s \<inter> x)) \<le> (\<Sum>i. f (A i \<inter> x))"
unfolding measure_set_def
@@ -822,34 +651,31 @@
qed
lemma measure_down:
- "measure_space N \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow> measure N = measure M \<Longrightarrow> measure_space M"
- by (simp add: measure_space_def measure_space_axioms_def positive_def
- countably_additive_def)
+ "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
+ by (simp add: measure_space_def positive_def countably_additive_def)
blast
theorem (in ring_of_sets) caratheodory:
assumes posf: "positive M f" and ca: "countably_additive M f"
- shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
- measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
proof -
have inc: "increasing M f"
by (metis additive_increasing ca countably_additive_additive posf)
let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
- def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
- have mls: "measure_space \<lparr>space = space M, sets = ls, measure = ?infm\<rparr>"
+ def ls \<equiv> "lambda_system \<Omega> (Pow \<Omega>) ?infm"
+ have mls: "measure_space \<Omega> ls ?infm"
using sigma_algebra.caratheodory_lemma
[OF sigma_algebra_Pow inf_measure_outer [OF posf inc]]
by (simp add: ls_def)
- hence sls: "sigma_algebra (|space = space M, sets = ls, measure = ?infm|)"
+ hence sls: "sigma_algebra \<Omega> ls"
by (simp add: measure_space_def)
- have "sets M \<subseteq> ls"
+ have "M \<subseteq> ls"
by (simp add: ls_def)
(metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
- hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls"
- using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"]
+ hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls"
+ using sigma_algebra.sigma_sets_subset [OF sls, of "M"]
by simp
- have "measure_space \<lparr> space = space M, sets = sets (sigma M), measure = ?infm \<rparr>"
- unfolding sigma_def
+ have "measure_space \<Omega> (sigma_sets \<Omega> M) ?infm"
by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
(simp_all add: sgs_sb space_closed)
thus ?thesis using inf_measure_agrees [OF posf ca]
@@ -861,12 +687,12 @@
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
assumes f: "positive M f" "additive M f"
shows "countably_additive M f \<longleftrightarrow>
- (\<forall>A. range A \<subseteq> sets M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))"
+ (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))"
unfolding countably_additive_def
proof safe
- assume count_sum: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> sets M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "incseq A" "(\<Union>i. A i) \<in> sets M"
- then have dA: "range (disjointed A) \<subseteq> sets M" by (auto simp: range_disjointed_sets)
+ assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
+ then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets)
with count_sum[THEN spec, of "disjointed A"] A(3)
have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)"
by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
@@ -880,12 +706,12 @@
using disjointed_additive[OF f A(1,2)] .
ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp
next
- assume cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A" "(\<Union>i. A i) \<in> sets M"
+ assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M"
have *: "(\<Union>n. (\<Union>i\<le>n. A i)) = (\<Union>i. A i)" by auto
have "(\<lambda>n. f (\<Union>i\<le>n. A i)) ----> f (\<Union>i. A i)"
proof (unfold *[symmetric], intro cont[rule_format])
- show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> sets M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> sets M"
+ show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> M"
using A * by auto
qed (force intro!: incseq_SucI)
moreover have "\<And>n. f (\<Union>i\<le>n. A i) = (\<Sum>i\<le>n. f (A i))"
@@ -901,18 +727,18 @@
lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
assumes f: "positive M f" "additive M f"
- shows "(\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> sets M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))
- \<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)"
+ shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))
+ \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)"
proof safe
- assume cont: "(\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> sets M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))"
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
+ assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))"
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0"
using `positive M f`[unfolded positive_def] by auto
next
- assume cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "decseq A" "(\<Inter>i. A i) \<in> sets M" "\<forall>i. f (A i) \<noteq> \<infinity>"
+ assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"
- have f_mono: "\<And>a b. a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"
+ have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"
using additive_increasing[OF f] unfolding increasing_def by simp
have decseq_fA: "decseq (\<lambda>i. f (A i))"
@@ -932,7 +758,7 @@
note f_fin = this
have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0"
proof (intro cont[rule_format, OF _ decseq _ f_fin])
- show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> sets M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"
+ show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"
using A by auto
qed
from INF_Lim_ereal[OF decseq_f this]
@@ -956,17 +782,17 @@
qed
lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def)
-lemma positiveD2: "positive M f \<Longrightarrow> A \<in> sets M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def)
+lemma positiveD2: "positive M f \<Longrightarrow> A \<in> M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def)
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
- assumes f: "positive M f" "additive M f" "\<forall>A\<in>sets M. f A \<noteq> \<infinity>"
- assumes cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
- assumes A: "range A \<subseteq> sets M" "incseq A" "(\<Union>i. A i) \<in> sets M"
+ assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>"
+ assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
+ assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
proof -
- have "\<forall>A\<in>sets M. \<exists>x. f A = ereal x"
+ have "\<forall>A\<in>M. \<exists>x. f A = ereal x"
proof
- fix A assume "A \<in> sets M" with f show "\<exists>x. f A = ereal x"
+ fix A assume "A \<in> M" with f show "\<exists>x. f A = ereal x"
unfolding positive_def by (cases "f A") auto
qed
from bchoice[OF this] guess f' .. note f' = this[rule_format]
@@ -991,20 +817,19 @@
qed
lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
- assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>sets M. f A \<noteq> \<infinity>"
- assumes cont: "\<And>A. range A \<subseteq> sets M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
+ assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>"
+ assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
shows "countably_additive M f"
using countably_additive_iff_continuous_from_below[OF f]
using empty_continuous_imp_continuous_from_below[OF f fin] cont
by blast
lemma (in ring_of_sets) caratheodory_empty_continuous:
- assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> sets M \<Longrightarrow> f A \<noteq> \<infinity>"
- assumes cont: "\<And>A. range A \<subseteq> sets M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
- shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
- measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>"
+ assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
+ assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
proof (intro caratheodory empty_continuous_imp_countably_additive f)
- show "\<forall>A\<in>sets M. f A \<noteq> \<infinity>" using fin by auto
+ show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
qed (rule cont)
end
--- a/src/HOL/Probability/Complete_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Complete_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -6,218 +6,231 @@
imports Lebesgue_Integration
begin
-locale completeable_measure_space = measure_space
-
-definition (in completeable_measure_space)
- "split_completion A p = (\<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and>
- fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets)"
-
-definition (in completeable_measure_space)
- "main_part A = fst (Eps (split_completion A))"
+definition
+ "split_completion M A p = (if A \<in> sets M then p = (A, {}) else
+ \<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets M)"
-definition (in completeable_measure_space)
- "null_part A = snd (Eps (split_completion A))"
-
-abbreviation (in completeable_measure_space) "\<mu>' A \<equiv> \<mu> (main_part A)"
+definition
+ "main_part M A = fst (Eps (split_completion M A))"
-definition (in completeable_measure_space) completion :: "('a, 'b) measure_space_scheme" where
- "completion = \<lparr> space = space M,
- sets = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N' },
- measure = \<mu>',
- \<dots> = more M \<rparr>"
-
+definition
+ "null_part M A = snd (Eps (split_completion M A))"
-lemma (in completeable_measure_space) space_completion[simp]:
- "space completion = space M" unfolding completion_def by simp
-
-lemma (in completeable_measure_space) sets_completionE:
- assumes "A \<in> sets completion"
- obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M"
- using assms unfolding completion_def by auto
+definition completion :: "'a measure \<Rightarrow> 'a measure" where
+ "completion M = measure_of (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }
+ (emeasure M \<circ> main_part M)"
-lemma (in completeable_measure_space) sets_completionI:
- assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M"
- shows "A \<in> sets completion"
- using assms unfolding completion_def by auto
+lemma completion_into_space:
+ "{ S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' } \<subseteq> Pow (space M)"
+ using sets_into_space by auto
-lemma (in completeable_measure_space) sets_completionI_sets[intro]:
- "A \<in> sets M \<Longrightarrow> A \<in> sets completion"
- unfolding completion_def by force
+lemma space_completion[simp]: "space (completion M) = space M"
+ unfolding completion_def using space_measure_of[OF completion_into_space] by simp
-lemma (in completeable_measure_space) null_sets_completion:
- assumes "N' \<in> null_sets" "N \<subseteq> N'" shows "N \<in> sets completion"
- apply(rule sets_completionI[of N "{}" N N'])
+lemma completionI:
+ assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
+ shows "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
using assms by auto
-sublocale completeable_measure_space \<subseteq> completion!: sigma_algebra completion
-proof (unfold sigma_algebra_iff2, safe)
- fix A x assume "A \<in> sets completion" "x \<in> A"
- with sets_into_space show "x \<in> space completion"
- by (auto elim!: sets_completionE)
+lemma completionE:
+ assumes "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
+ obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
+ using assms by auto
+
+lemma sigma_algebra_completion:
+ "sigma_algebra (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
+ (is "sigma_algebra _ ?A")
+ unfolding sigma_algebra_iff2
+proof (intro conjI ballI allI impI)
+ show "?A \<subseteq> Pow (space M)"
+ using sets_into_space by auto
next
- fix A assume "A \<in> sets completion"
- from this[THEN sets_completionE] guess S N N' . note A = this
- let ?C = "space completion"
- show "?C - A \<in> sets completion" using A
- by (intro sets_completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"])
- auto
+ show "{} \<in> ?A" by auto
next
- fix A ::"nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion"
- then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N'"
- unfolding completion_def by (auto simp: image_subset_iff)
+ let ?C = "space M"
+ fix A assume "A \<in> ?A" from completionE[OF this] guess S N N' .
+ then show "space M - A \<in> ?A"
+ by (intro completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"]) auto
+next
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> ?A"
+ then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'"
+ by (auto simp: image_subset_iff)
from choice[OF this] guess S ..
from choice[OF this] guess N ..
from choice[OF this] guess N' ..
- then show "UNION UNIV A \<in> sets completion"
+ then show "UNION UNIV A \<in> ?A"
using null_sets_UN[of N']
- by (intro sets_completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"])
- auto
-qed auto
+ by (intro completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"]) auto
+qed
+
+lemma sets_completion:
+ "sets (completion M) = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
+ using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] by (simp add: completion_def)
+
+lemma sets_completionE:
+ assumes "A \<in> sets (completion M)"
+ obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
+ using assms unfolding sets_completion by auto
+
+lemma sets_completionI:
+ assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
+ shows "A \<in> sets (completion M)"
+ using assms unfolding sets_completion by auto
+
+lemma sets_completionI_sets[intro, simp]:
+ "A \<in> sets M \<Longrightarrow> A \<in> sets (completion M)"
+ unfolding sets_completion by force
-lemma (in completeable_measure_space) split_completion:
- assumes "A \<in> sets completion"
- shows "split_completion A (main_part A, null_part A)"
- unfolding main_part_def null_part_def
-proof (rule someI2_ex)
- from assms[THEN sets_completionE] guess S N N' . note A = this
- let ?P = "(S, N - S)"
- show "\<exists>p. split_completion A p"
- unfolding split_completion_def using A
- proof (intro exI conjI)
- show "A = fst ?P \<union> snd ?P" using A by auto
- show "snd ?P \<subseteq> N'" using A by auto
+lemma null_sets_completion:
+ assumes "N' \<in> null_sets M" "N \<subseteq> N'" shows "N \<in> sets (completion M)"
+ using assms by (intro sets_completionI[of N "{}" N N']) auto
+
+lemma split_completion:
+ assumes "A \<in> sets (completion M)"
+ shows "split_completion M A (main_part M A, null_part M A)"
+proof cases
+ assume "A \<in> sets M" then show ?thesis
+ by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
+next
+ assume nA: "A \<notin> sets M"
+ show ?thesis
+ unfolding main_part_def null_part_def if_not_P[OF nA]
+ proof (rule someI2_ex)
+ from assms[THEN sets_completionE] guess S N N' . note A = this
+ let ?P = "(S, N - S)"
+ show "\<exists>p. split_completion M A p"
+ unfolding split_completion_def if_not_P[OF nA] using A
+ proof (intro exI conjI)
+ show "A = fst ?P \<union> snd ?P" using A by auto
+ show "snd ?P \<subseteq> N'" using A by auto
+ qed auto
qed auto
-qed auto
+qed
-lemma (in completeable_measure_space)
- assumes "S \<in> sets completion"
- shows main_part_sets[intro, simp]: "main_part S \<in> sets M"
- and main_part_null_part_Un[simp]: "main_part S \<union> null_part S = S"
- and main_part_null_part_Int[simp]: "main_part S \<inter> null_part S = {}"
- using split_completion[OF assms] by (auto simp: split_completion_def)
+lemma
+ assumes "S \<in> sets (completion M)"
+ shows main_part_sets[intro, simp]: "main_part M S \<in> sets M"
+ and main_part_null_part_Un[simp]: "main_part M S \<union> null_part M S = S"
+ and main_part_null_part_Int[simp]: "main_part M S \<inter> null_part M S = {}"
+ using split_completion[OF assms]
+ by (auto simp: split_completion_def split: split_if_asm)
-lemma (in completeable_measure_space) null_part:
- assumes "S \<in> sets completion" shows "\<exists>N. N\<in>null_sets \<and> null_part S \<subseteq> N"
- using split_completion[OF assms] by (auto simp: split_completion_def)
+lemma main_part[simp]: "S \<in> sets M \<Longrightarrow> main_part M S = S"
+ using split_completion[of S M]
+ by (auto simp: split_completion_def split: split_if_asm)
-lemma (in completeable_measure_space) null_part_sets[intro, simp]:
- assumes "S \<in> sets M" shows "null_part S \<in> sets M" "\<mu> (null_part S) = 0"
+lemma null_part:
+ assumes "S \<in> sets (completion M)" shows "\<exists>N. N\<in>null_sets M \<and> null_part M S \<subseteq> N"
+ using split_completion[OF assms] by (auto simp: split_completion_def split: split_if_asm)
+
+lemma null_part_sets[intro, simp]:
+ assumes "S \<in> sets M" shows "null_part M S \<in> sets M" "emeasure M (null_part M S) = 0"
proof -
- have S: "S \<in> sets completion" using assms by auto
- have "S - main_part S \<in> sets M" using assms by auto
+ have S: "S \<in> sets (completion M)" using assms by auto
+ have "S - main_part M S \<in> sets M" using assms by auto
moreover
from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
- have "S - main_part S = null_part S" by auto
- ultimately show sets: "null_part S \<in> sets M" by auto
+ have "S - main_part M S = null_part M S" by auto
+ ultimately show sets: "null_part M S \<in> sets M" by auto
from null_part[OF S] guess N ..
- with measure_eq_0[of N "null_part S"] sets
- show "\<mu> (null_part S) = 0" by auto
-qed
-
-lemma (in completeable_measure_space) \<mu>'_set[simp]:
- assumes "S \<in> sets M" shows "\<mu>' S = \<mu> S"
-proof -
- have S: "S \<in> sets completion" using assms by auto
- then have "\<mu> S = \<mu> (main_part S \<union> null_part S)" by simp
- also have "\<dots> = \<mu>' S"
- using S assms measure_additive[of "main_part S" "null_part S"]
- by (auto simp: measure_additive)
- finally show ?thesis by simp
+ with emeasure_eq_0[of N _ "null_part M S"] sets
+ show "emeasure M (null_part M S) = 0" by auto
qed
-lemma (in completeable_measure_space) sets_completionI_sub:
- assumes N: "N' \<in> null_sets" "N \<subseteq> N'"
- shows "N \<in> sets completion"
- using assms by (intro sets_completionI[of _ "{}" N N']) auto
-
-lemma (in completeable_measure_space) \<mu>_main_part_UN:
+lemma emeasure_main_part_UN:
fixes S :: "nat \<Rightarrow> 'a set"
- assumes "range S \<subseteq> sets completion"
- shows "\<mu>' (\<Union>i. (S i)) = \<mu> (\<Union>i. main_part (S i))"
+ assumes "range S \<subseteq> sets (completion M)"
+ shows "emeasure M (main_part M (\<Union>i. (S i))) = emeasure M (\<Union>i. main_part M (S i))"
proof -
- have S: "\<And>i. S i \<in> sets completion" using assms by auto
- then have UN: "(\<Union>i. S i) \<in> sets completion" by auto
- have "\<forall>i. \<exists>N. N \<in> null_sets \<and> null_part (S i) \<subseteq> N"
+ have S: "\<And>i. S i \<in> sets (completion M)" using assms by auto
+ then have UN: "(\<Union>i. S i) \<in> sets (completion M)" by auto
+ have "\<forall>i. \<exists>N. N \<in> null_sets M \<and> null_part M (S i) \<subseteq> N"
using null_part[OF S] by auto
from choice[OF this] guess N .. note N = this
- then have UN_N: "(\<Union>i. N i) \<in> null_sets" by (intro null_sets_UN) auto
- have "(\<Union>i. S i) \<in> sets completion" using S by auto
+ then have UN_N: "(\<Union>i. N i) \<in> null_sets M" by (intro null_sets_UN) auto
+ have "(\<Union>i. S i) \<in> sets (completion M)" using S by auto
from null_part[OF this] guess N' .. note N' = this
let ?N = "(\<Union>i. N i) \<union> N'"
- have null_set: "?N \<in> null_sets" using N' UN_N by (intro nullsets.Un) auto
- have "main_part (\<Union>i. S i) \<union> ?N = (main_part (\<Union>i. S i) \<union> null_part (\<Union>i. S i)) \<union> ?N"
+ have null_set: "?N \<in> null_sets M" using N' UN_N by (intro null_sets.Un) auto
+ have "main_part M (\<Union>i. S i) \<union> ?N = (main_part M (\<Union>i. S i) \<union> null_part M (\<Union>i. S i)) \<union> ?N"
using N' by auto
- also have "\<dots> = (\<Union>i. main_part (S i) \<union> null_part (S i)) \<union> ?N"
+ also have "\<dots> = (\<Union>i. main_part M (S i) \<union> null_part M (S i)) \<union> ?N"
unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
- also have "\<dots> = (\<Union>i. main_part (S i)) \<union> ?N"
+ also have "\<dots> = (\<Union>i. main_part M (S i)) \<union> ?N"
using N by auto
- finally have *: "main_part (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part (S i)) \<union> ?N" .
- have "\<mu> (main_part (\<Union>i. S i)) = \<mu> (main_part (\<Union>i. S i) \<union> ?N)"
- using null_set UN by (intro measure_Un_null_set[symmetric]) auto
- also have "\<dots> = \<mu> ((\<Union>i. main_part (S i)) \<union> ?N)"
+ finally have *: "main_part M (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part M (S i)) \<union> ?N" .
+ have "emeasure M (main_part M (\<Union>i. S i)) = emeasure M (main_part M (\<Union>i. S i) \<union> ?N)"
+ using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
+ also have "\<dots> = emeasure M ((\<Union>i. main_part M (S i)) \<union> ?N)"
unfolding * ..
- also have "\<dots> = \<mu> (\<Union>i. main_part (S i))"
- using null_set S by (intro measure_Un_null_set) auto
+ also have "\<dots> = emeasure M (\<Union>i. main_part M (S i))"
+ using null_set S by (intro emeasure_Un_null_set) auto
finally show ?thesis .
qed
-lemma (in completeable_measure_space) \<mu>_main_part_Un:
- assumes S: "S \<in> sets completion" and T: "T \<in> sets completion"
- shows "\<mu>' (S \<union> T) = \<mu> (main_part S \<union> main_part T)"
-proof -
- have UN: "(\<Union>i. binary (main_part S) (main_part T) i) = (\<Union>i. main_part (binary S T i))"
- unfolding binary_def by (auto split: split_if_asm)
- show ?thesis
- using \<mu>_main_part_UN[of "binary S T"] assms
- unfolding range_binary_eq Un_range_binary UN by auto
+lemma emeasure_completion[simp]:
+ assumes S: "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
+proof (subst emeasure_measure_of[OF completion_def completion_into_space])
+ let ?\<mu> = "emeasure M \<circ> main_part M"
+ show "S \<in> sets (completion M)" "?\<mu> S = emeasure M (main_part M S) " using S by simp_all
+ show "positive (sets (completion M)) ?\<mu>"
+ by (simp add: positive_def emeasure_nonneg)
+ show "countably_additive (sets (completion M)) ?\<mu>"
+ proof (intro countably_additiveI)
+ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets (completion M)" "disjoint_family A"
+ have "disjoint_family (\<lambda>i. main_part M (A i))"
+ proof (intro disjoint_family_on_bisimulation[OF A(2)])
+ fix n m assume "A n \<inter> A m = {}"
+ then have "(main_part M (A n) \<union> null_part M (A n)) \<inter> (main_part M (A m) \<union> null_part M (A m)) = {}"
+ using A by (subst (1 2) main_part_null_part_Un) auto
+ then show "main_part M (A n) \<inter> main_part M (A m) = {}" by auto
+ qed
+ then have "(\<Sum>n. emeasure M (main_part M (A n))) = emeasure M (\<Union>i. main_part M (A i))"
+ using A by (auto intro!: suminf_emeasure)
+ then show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (UNION UNIV A)"
+ by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
+ qed
qed
-sublocale completeable_measure_space \<subseteq> completion!: measure_space completion
- where "measure completion = \<mu>'"
-proof -
- show "measure_space completion"
- proof
- show "positive completion (measure completion)"
- by (auto simp: completion_def positive_def)
- next
- show "countably_additive completion (measure completion)"
- proof (intro countably_additiveI)
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion" "disjoint_family A"
- have "disjoint_family (\<lambda>i. main_part (A i))"
- proof (intro disjoint_family_on_bisimulation[OF A(2)])
- fix n m assume "A n \<inter> A m = {}"
- then have "(main_part (A n) \<union> null_part (A n)) \<inter> (main_part (A m) \<union> null_part (A m)) = {}"
- using A by (subst (1 2) main_part_null_part_Un) auto
- then show "main_part (A n) \<inter> main_part (A m) = {}" by auto
- qed
- then have "(\<Sum>n. measure completion (A n)) = \<mu> (\<Union>i. main_part (A i))"
- unfolding completion_def using A by (auto intro!: measure_countably_additive)
- then show "(\<Sum>n. measure completion (A n)) = measure completion (UNION UNIV A)"
- by (simp add: completion_def \<mu>_main_part_UN[OF A(1)])
- qed
- qed
- show "measure completion = \<mu>'" unfolding completion_def by simp
-qed
+lemma emeasure_completion_UN:
+ "range S \<subseteq> sets (completion M) \<Longrightarrow>
+ emeasure (completion M) (\<Union>i::nat. (S i)) = emeasure M (\<Union>i. main_part M (S i))"
+ by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)
-lemma (in completeable_measure_space) completion_ex_simple_function:
- assumes f: "simple_function completion f"
- shows "\<exists>f'. simple_function M f' \<and> (AE x. f x = f' x)"
+lemma emeasure_completion_Un:
+ assumes S: "S \<in> sets (completion M)" and T: "T \<in> sets (completion M)"
+ shows "emeasure (completion M) (S \<union> T) = emeasure M (main_part M S \<union> main_part M T)"
+proof (subst emeasure_completion)
+ have UN: "(\<Union>i. binary (main_part M S) (main_part M T) i) = (\<Union>i. main_part M (binary S T i))"
+ unfolding binary_def by (auto split: split_if_asm)
+ show "emeasure M (main_part M (S \<union> T)) = emeasure M (main_part M S \<union> main_part M T)"
+ using emeasure_main_part_UN[of "binary S T" M] assms
+ unfolding range_binary_eq Un_range_binary UN by auto
+qed (auto intro: S T)
+
+lemma sets_completionI_sub:
+ assumes N: "N' \<in> null_sets M" "N \<subseteq> N'"
+ shows "N \<in> sets (completion M)"
+ using assms by (intro sets_completionI[of _ "{}" N N']) auto
+
+lemma completion_ex_simple_function:
+ assumes f: "simple_function (completion M) f"
+ shows "\<exists>f'. simple_function M f' \<and> (AE x in M. f x = f' x)"
proof -
let ?F = "\<lambda>x. f -` {x} \<inter> space M"
- have F: "\<And>x. ?F x \<in> sets completion" and fin: "finite (f`space M)"
- using completion.simple_functionD[OF f]
- completion.simple_functionD[OF f] by simp_all
- have "\<forall>x. \<exists>N. N \<in> null_sets \<and> null_part (?F x) \<subseteq> N"
+ have F: "\<And>x. ?F x \<in> sets (completion M)" and fin: "finite (f`space M)"
+ using simple_functionD[OF f] simple_functionD[OF f] by simp_all
+ have "\<forall>x. \<exists>N. N \<in> null_sets M \<and> null_part M (?F x) \<subseteq> N"
using F null_part by auto
from choice[OF this] obtain N where
- N: "\<And>x. null_part (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets" by auto
+ N: "\<And>x. null_part M (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets M" by auto
let ?N = "\<Union>x\<in>f`space M. N x"
let ?f' = "\<lambda>x. if x \<in> ?N then undefined else f x"
- have sets: "?N \<in> null_sets" using N fin by (intro nullsets.finite_UN) auto
+ have sets: "?N \<in> null_sets M" using N fin by (intro null_sets.finite_UN) auto
show ?thesis unfolding simple_function_def
proof (safe intro!: exI[of _ ?f'])
have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto
- from finite_subset[OF this] completion.simple_functionD(1)[OF f]
+ from finite_subset[OF this] simple_functionD(1)[OF f]
show "finite (?f' ` space M)" by auto
next
fix x assume "x \<in> space M"
@@ -225,13 +238,13 @@
(if x \<in> ?N then ?F undefined \<union> ?N
else if f x = undefined then ?F (f x) \<union> ?N
else ?F (f x) - ?N)"
- using N(2) sets_into_space by (auto split: split_if_asm)
+ using N(2) sets_into_space by (auto split: split_if_asm simp: null_sets_def)
moreover { fix y have "?F y \<union> ?N \<in> sets M"
proof cases
assume y: "y \<in> f`space M"
- have "?F y \<union> ?N = (main_part (?F y) \<union> null_part (?F y)) \<union> ?N"
+ have "?F y \<union> ?N = (main_part M (?F y) \<union> null_part M (?F y)) \<union> ?N"
using main_part_null_part_Un[OF F] by auto
- also have "\<dots> = main_part (?F y) \<union> ?N"
+ also have "\<dots> = main_part M (?F y) \<union> ?N"
using y N by auto
finally show ?thesis
using F sets by auto
@@ -240,34 +253,34 @@
then show ?thesis using sets by auto
qed }
moreover {
- have "?F (f x) - ?N = main_part (?F (f x)) \<union> null_part (?F (f x)) - ?N"
+ have "?F (f x) - ?N = main_part M (?F (f x)) \<union> null_part M (?F (f x)) - ?N"
using main_part_null_part_Un[OF F] by auto
- also have "\<dots> = main_part (?F (f x)) - ?N"
+ also have "\<dots> = main_part M (?F (f x)) - ?N"
using N `x \<in> space M` by auto
finally have "?F (f x) - ?N \<in> sets M"
using F sets by auto }
ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
next
- show "AE x. f x = ?f' x"
+ show "AE x in M. f x = ?f' x"
by (rule AE_I', rule sets) auto
qed
qed
-lemma (in completeable_measure_space) completion_ex_borel_measurable_pos:
+lemma completion_ex_borel_measurable_pos:
fixes g :: "'a \<Rightarrow> ereal"
- assumes g: "g \<in> borel_measurable completion" and "\<And>x. 0 \<le> g x"
- shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
+ assumes g: "g \<in> borel_measurable (completion M)" and "\<And>x. 0 \<le> g x"
+ shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
proof -
- from g[THEN completion.borel_measurable_implies_simple_function_sequence'] guess f . note f = this
+ from g[THEN borel_measurable_implies_simple_function_sequence'] guess f . note f = this
from this(1)[THEN completion_ex_simple_function]
- have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x. f i x = f' x)" ..
+ have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x in M. f i x = f' x)" ..
from this[THEN choice] obtain f' where
sf: "\<And>i. simple_function M (f' i)" and
- AE: "\<forall>i. AE x. f i x = f' i x" by auto
+ AE: "\<forall>i. AE x in M. f i x = f' i x" by auto
show ?thesis
proof (intro bexI)
from AE[unfolded AE_all_countable[symmetric]]
- show "AE x. g x = (SUP i. f' i x)" (is "AE x. g x = ?f x")
+ show "AE x in M. g x = (SUP i. f' i x)" (is "AE x in M. g x = ?f x")
proof (elim AE_mp, safe intro!: AE_I2)
fix x assume eq: "\<forall>i. f i x = f' i x"
moreover have "g x = (SUP i. f i x)"
@@ -279,20 +292,20 @@
qed
qed
-lemma (in completeable_measure_space) completion_ex_borel_measurable:
+lemma completion_ex_borel_measurable:
fixes g :: "'a \<Rightarrow> ereal"
- assumes g: "g \<in> borel_measurable completion"
- shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
+ assumes g: "g \<in> borel_measurable (completion M)"
+ shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
proof -
- have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable completion" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
+ have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
moreover
- have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable completion" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
+ have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
ultimately
show ?thesis
proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
- show "AE x. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
+ show "AE x in M. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
proof (intro AE_I2 impI)
fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
show "g x = g_pos x - g_neg x" unfolding g[symmetric]
--- a/src/HOL/Probability/Conditional_Probability.thy Mon Apr 23 12:23:23 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,161 +0,0 @@
-(* Title: HOL/Probability/Conditional_Probability.thy
- Author: Johannes Hölzl, TU München
-*)
-
-header {*Conditional probability*}
-
-theory Conditional_Probability
-imports Probability_Measure Radon_Nikodym
-begin
-
-section "Conditional Expectation and Probability"
-
-definition (in prob_space)
- "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x)
- \<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))"
-
-lemma (in prob_space) conditional_expectation_exists:
- fixes X :: "'a \<Rightarrow> ereal" and N :: "('a, 'b) measure_space_scheme"
- assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
- and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
- shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N.
- (\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))"
-proof -
- note N(4)[simp]
- interpret P: prob_space N
- using prob_space_subalgebra[OF N] .
-
- let ?f = "\<lambda>A x. X x * indicator A x"
- let ?Q = "\<lambda>A. integral\<^isup>P M (?f A)"
-
- from measure_space_density[OF borel]
- have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)"
- apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"])
- using N by (auto intro!: P.sigma_algebra_cong)
- then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" .
-
- have "P.absolutely_continuous ?Q"
- unfolding P.absolutely_continuous_def
- proof safe
- fix A assume "A \<in> sets N" "P.\<mu> A = 0"
- then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
- using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
- then show "?Q A = 0"
- by (auto simp add: positive_integral_0_iff_AE)
- qed
- from P.Radon_Nikodym[OF Q this]
- obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x"
- "\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
- by blast
- with N(2) show ?thesis
- by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
-qed
-
-lemma (in prob_space)
- fixes X :: "'a \<Rightarrow> ereal" and N :: "('a, 'b) measure_space_scheme"
- assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
- and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
- shows borel_measurable_conditional_expectation:
- "conditional_expectation N X \<in> borel_measurable N"
- and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow>
- (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) =
- (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
- (is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C")
-proof -
- note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
- then show "conditional_expectation N X \<in> borel_measurable N"
- unfolding conditional_expectation_def by (rule someI2_ex) blast
-
- from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C"
- unfolding conditional_expectation_def by (rule someI2_ex) blast
-qed
-
-lemma (in sigma_algebra) factorize_measurable_function_pos:
- fixes Z :: "'a \<Rightarrow> ereal" and Y :: "'a \<Rightarrow> 'c"
- assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
- assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
- shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
-proof -
- interpret M': sigma_algebra M' by fact
- have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
- from M'.sigma_algebra_vimage[OF this]
- interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
-
- from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
-
- have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
- proof
- fix i
- from f(1)[of i] have "finite (f i`space M)" and B_ex:
- "\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M"
- unfolding simple_function_def by auto
- from B_ex[THEN bchoice] guess B .. note B = this
-
- let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
-
- show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
- proof (intro exI[of _ ?g] conjI ballI)
- show "simple_function M' ?g" using B by auto
-
- fix x assume "x \<in> space M"
- then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::ereal)"
- unfolding indicator_def using B by auto
- then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
- by (subst va.simple_function_indicator_representation) auto
- qed
- qed
- from choice[OF this] guess g .. note g = this
-
- show ?thesis
- proof (intro ballI bexI)
- show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
- using g by (auto intro: M'.borel_measurable_simple_function)
- fix x assume "x \<in> space M"
- have "max 0 (Z x) = (SUP i. f i x)" using f by simp
- also have "\<dots> = (SUP i. g i (Y x))"
- using g `x \<in> space M` by simp
- finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
- qed
-qed
-
-lemma (in sigma_algebra) factorize_measurable_function:
- fixes Z :: "'a \<Rightarrow> ereal" and Y :: "'a \<Rightarrow> 'c"
- assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
- shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
- \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
-proof safe
- interpret M': sigma_algebra M' by fact
- have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
- from M'.sigma_algebra_vimage[OF this]
- interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
-
- { fix g :: "'c \<Rightarrow> ereal" assume "g \<in> borel_measurable M'"
- with M'.measurable_vimage_algebra[OF Y]
- have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by (rule measurable_comp)
- moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
- then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
- g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by (auto intro!: measurable_cong)
- ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by simp }
-
- assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
- "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by auto
- from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
- from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
- let ?g = "\<lambda>x. p x - n x"
- show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
- proof (intro bexI ballI)
- show "?g \<in> borel_measurable M'" using p n by auto
- fix x assume "x \<in> space M"
- then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
- using p n by auto
- then show "Z x = ?g (Y x)"
- by (auto split: split_max)
- qed
-qed
-
-end
\ No newline at end of file
--- a/src/HOL/Probability/Finite_Product_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Finite_Product_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -8,6 +8,9 @@
imports Binary_Product_Measure
begin
+lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
+ by auto
+
lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
unfolding Pi_def by auto
@@ -34,9 +37,6 @@
notation (xsymbols)
funcset_extensional (infixr "\<rightarrow>\<^isub>E" 60)
-lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
- by safe (auto simp add: extensional_def fun_eq_iff)
-
lemma extensional_insert[intro, simp]:
assumes "a \<in> extensional (insert i I)"
shows "a(i := b) \<in> extensional (insert i I)"
@@ -86,7 +86,7 @@
"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
- by (auto simp: restrict_def intro!: ext)
+ by (auto simp: restrict_def)
lemma extensional_insert_undefined[intro, simp]:
assumes "a \<in> extensional (insert i I)"
@@ -130,16 +130,16 @@
using assms by (auto simp: restrict_Pi_cancel)
lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
- by (auto simp: restrict_def intro!: ext)
+ by (auto simp: restrict_def)
lemma merge_restrict[simp]:
"merge I (restrict x I) J y = merge I x J y"
"merge I x J (restrict y J) = merge I x J y"
- unfolding merge_def by (auto intro!: ext)
+ unfolding merge_def by auto
lemma merge_x_x_eq_restrict[simp]:
"merge I x J x = restrict x (I \<union> J)"
- unfolding merge_def by (auto intro!: ext)
+ unfolding merge_def by auto
lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
apply auto
@@ -233,339 +233,355 @@
section "Products"
-locale product_sigma_algebra =
- fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme"
- assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
+definition prod_emb where
+ "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
+
+lemma prod_emb_iff:
+ "f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))"
+ unfolding prod_emb_def by auto
-locale finite_product_sigma_algebra = product_sigma_algebra M
- for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
- fixes I :: "'i set"
- assumes finite_index[simp, intro]: "finite I"
+lemma
+ shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
+ and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
+ and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
+ and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))"
+ and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
+ and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
+ by (auto simp: prod_emb_def)
-definition
- "product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)),
- sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)),
- measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>"
+lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
+ prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))"
+ by (force simp: prod_emb_def Pi_iff split_if_mem2)
+
+lemma prod_emb_PiE_same_index[simp]: "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E"
+ by (auto simp: prod_emb_def Pi_iff)
-definition product_algebra_def:
- "Pi\<^isub>M I M = sigma (product_algebra_generator I M)
- \<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and>
- (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>"
+definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
+ "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
+ {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
+ (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
+ (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
+
+definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
+ "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
+ {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
+
+abbreviation
+ "Pi\<^isub>M I M \<equiv> PiM I M"
syntax
- "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
- ('i => 'a, 'b) measure_space_scheme" ("(3PIM _:_./ _)" 10)
+ "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIM _:_./ _)" 10)
syntax (xsymbols)
- "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
- ('i => 'a, 'b) measure_space_scheme" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10)
+ "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10)
syntax (HTML output)
- "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
- ('i => 'a, 'b) measure_space_scheme" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10)
+ "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10)
translations
- "PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)"
-
-abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M"
-abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M"
-
-sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
-
-lemma sigma_into_space:
- assumes "sets M \<subseteq> Pow (space M)"
- shows "sets (sigma M) \<subseteq> Pow (space M)"
- using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto
+ "PIM x:I. M" == "CONST PiM I (%x. M)"
-lemma (in product_sigma_algebra) product_algebra_generator_into_space:
- "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
- using M.sets_into_space unfolding product_algebra_generator_def
- by auto blast
-
-lemma (in product_sigma_algebra) product_algebra_into_space:
- "sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))"
- using product_algebra_generator_into_space
- by (auto intro!: sigma_into_space simp add: product_algebra_def)
-
-lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)"
- using product_algebra_generator_into_space unfolding product_algebra_def
- by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all
-
-sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
- using sigma_algebra_product_algebra .
+lemma prod_algebra_sets_into_space:
+ "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ using assms by (auto simp: prod_emb_def prod_algebra_def)
-lemma product_algebraE:
- assumes "A \<in> sets (product_algebra_generator I M)"
- obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
- using assms unfolding product_algebra_generator_def by auto
-
-lemma product_algebra_generatorI[intro]:
- assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
- shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)"
- using assms unfolding product_algebra_generator_def by auto
-
-lemma space_product_algebra_generator[simp]:
- "space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))"
- unfolding product_algebra_generator_def by simp
+lemma prod_algebra_eq_finite:
+ assumes I: "finite I"
+ shows "prod_algebra I M = {(\<Pi>\<^isub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
+proof (intro iffI set_eqI)
+ fix A assume "A \<in> ?L"
+ then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
+ and A: "A = prod_emb I M J (PIE j:J. E j)"
+ by (auto simp: prod_algebra_def)
+ let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
+ have A: "A = ?A"
+ unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
+ show "A \<in> ?R" unfolding A using J top
+ by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
+next
+ fix A assume "A \<in> ?R"
+ then obtain X where "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
+ then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
+ using sets_into_space by (force simp: prod_emb_def Pi_iff)
+ from X I show "A \<in> ?L" unfolding A
+ by (auto simp: prod_algebra_def)
+qed
-lemma space_product_algebra[simp]:
- "space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
- unfolding product_algebra_def product_algebra_generator_def by simp
-
-lemma sets_product_algebra:
- "sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))"
- unfolding product_algebra_def sigma_def by simp
+lemma prod_algebraI:
+ "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
+ \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
+ by (auto simp: prod_algebra_def Pi_iff)
-lemma product_algebra_generator_sets_into_space:
- assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
- shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
- using assms by (auto simp: product_algebra_generator_def) blast
-
-lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
- "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
- by (auto simp: sets_product_algebra)
-
-lemma Int_stable_product_algebra_generator:
- "(\<And>i. i \<in> I \<Longrightarrow> Int_stable (M i)) \<Longrightarrow> Int_stable (product_algebra_generator I M)"
- by (auto simp add: product_algebra_generator_def Int_stable_def PiE_Int Pi_iff)
+lemma prod_algebraE:
+ assumes A: "A \<in> prod_algebra I M"
+ obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
+ "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
+ using A by (auto simp: prod_algebra_def)
-section "Generating set generates also product algebra"
+lemma prod_algebraE_all:
+ assumes A: "A \<in> prod_algebra I M"
+ obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
+proof -
+ from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
+ and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
+ by (auto simp: prod_algebra_def)
+ from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
+ using sets_into_space by auto
+ then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
+ using A J by (auto simp: prod_emb_PiE)
+ moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
+ using top E by auto
+ ultimately show ?thesis using that by auto
+qed
-lemma sigma_product_algebra_sigma_eq:
- assumes "finite I"
- assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
- assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"
- assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
- and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
- shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"
- (is "sets ?S = sets ?E")
-proof cases
- assume "I = {}" then show ?thesis
- by (simp add: product_algebra_def product_algebra_generator_def)
-next
- assume "I \<noteq> {}"
- interpret E: sigma_algebra "sigma (E i)" for i
- using E by (rule sigma_algebra_sigma)
- have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)"
- using E by auto
- interpret G: sigma_algebra ?E
- unfolding product_algebra_def product_algebra_generator_def using E
- by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)
- { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
- then have "(\<lambda>x. x i) -` A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E"
- using mono union unfolding incseq_Suc_iff space_product_algebra
- by (auto dest: Pi_mem)
- also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
- unfolding space_product_algebra
- apply simp
- apply (subst Pi_UN[OF `finite I`])
- using mono[THEN incseqD] apply simp
- apply (simp add: PiE_Int)
- apply (intro PiE_cong)
- using A sets_into by (auto intro!: into_space)
- also have "\<dots> \<in> sets ?E"
- using sets_into `A \<in> sets (E i)`
- unfolding sets_product_algebra sets_sigma
- by (intro sigma_sets.Union)
- (auto simp: image_subset_iff intro!: sigma_sets.Basic)
- finally have "(\<lambda>x. x i) -` A \<inter> space ?E \<in> sets ?E" . }
- then have proj:
- "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
- using E by (subst G.measurable_iff_sigma)
- (auto simp: sets_product_algebra sets_sigma)
- { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
- with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) -` (A i) \<inter> space ?E \<in> sets ?E"
- unfolding measurable_def by simp
- have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) -` (A i) \<inter> space ?E)"
- using A E.sets_into_space `I \<noteq> {}` unfolding product_algebra_def by auto blast
- then have "Pi\<^isub>E I A \<in> sets ?E"
- using G.finite_INT[OF `finite I` `I \<noteq> {}` basic, of "\<lambda>i. i"] by simp }
- then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E"
- by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def)
- then have subset: "sets ?S \<subseteq> sets ?E"
- by (simp add: sets_sigma sets_product_algebra)
- show "sets ?S = sets ?E"
- proof (intro set_eqI iffI)
- fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
- unfolding sets_sigma sets_product_algebra
- proof induct
- case (Basic A) then show ?case
- by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic)
- qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def)
- next
- fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
- qed
+lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
+proof (unfold Int_stable_def, safe)
+ fix A assume "A \<in> prod_algebra I M"
+ from prod_algebraE[OF this] guess J E . note A = this
+ fix B assume "B \<in> prod_algebra I M"
+ from prod_algebraE[OF this] guess K F . note B = this
+ have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^isub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter>
+ (if i \<in> K then F i else space (M i)))"
+ unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space]
+ apply (subst (1 2 3) prod_emb_PiE)
+ apply (simp_all add: subset_eq PiE_Int)
+ apply blast
+ apply (intro PiE_cong)
+ apply auto
+ done
+ also have "\<dots> \<in> prod_algebra I M"
+ using A B by (auto intro!: prod_algebraI)
+ finally show "A \<inter> B \<in> prod_algebra I M" .
+qed
+
+lemma prod_algebra_mono:
+ assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
+ assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
+ shows "prod_algebra I E \<subseteq> prod_algebra I F"
+proof
+ fix A assume "A \<in> prod_algebra I E"
+ then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
+ and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
+ and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
+ by (auto simp: prod_algebra_def)
+ moreover
+ from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
+ by (rule PiE_cong)
+ with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
+ by (simp add: prod_emb_def)
+ moreover
+ from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
+ by auto
+ ultimately show "A \<in> prod_algebra I F"
+ apply (simp add: prod_algebra_def image_iff)
+ apply (intro exI[of _ J] exI[of _ G] conjI)
+ apply auto
+ done
qed
-lemma product_algebraI[intro]:
- "E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M I M)"
- using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI)
+lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
+
+lemma sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)"
+ using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
-lemma (in product_sigma_algebra) measurable_component_update:
- assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
- shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
- unfolding product_algebra_def apply simp
-proof (intro measurable_sigma)
- let ?G = "product_algebra_generator (insert i I) M"
- show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space .
- show "?f \<in> space (M i) \<rightarrow> space ?G"
- using M.sets_into_space assms by auto
- fix A assume "A \<in> sets ?G"
- from product_algebraE[OF this] guess E . note E = this
- then have "?f -` A \<inter> space (M i) = E i \<or> ?f -` A \<inter> space (M i) = {}"
- using M.sets_into_space assms by auto
- then show "?f -` A \<inter> space (M i) \<in> sets (M i)"
- using E by (auto intro!: product_algebraI)
+lemma sets_PiM_single: "sets (PiM I M) =
+ sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
+ (is "_ = sigma_sets ?\<Omega> ?R")
+ unfolding sets_PiM
+proof (rule sigma_sets_eqI)
+ interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
+ fix A assume "A \<in> prod_algebra I M"
+ from prod_algebraE[OF this] guess J X . note X = this
+ show "A \<in> sigma_sets ?\<Omega> ?R"
+ proof cases
+ assume "I = {}"
+ with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
+ with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
+ next
+ assume "I \<noteq> {}"
+ with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})"
+ using sets_into_space[OF X(5)]
+ by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast
+ also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
+ using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
+ finally show "A \<in> sigma_sets ?\<Omega> ?R" .
+ qed
+next
+ fix A assume "A \<in> ?R"
+ then obtain i B where A: "A = {f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
+ by auto
+ then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
+ using sets_into_space[OF A(3)]
+ apply (subst prod_emb_PiE)
+ apply (auto simp: Pi_iff split: split_if_asm)
+ apply blast
+ done
+ also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
+ using A by (intro sigma_sets.Basic prod_algebraI) auto
+ finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
+qed
+
+lemma sets_PiM_I:
+ assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
+ shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
+proof cases
+ assume "J = {}"
+ then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
+ by (auto simp: prod_emb_def)
+ then show ?thesis
+ by (auto simp add: sets_PiM intro!: sigma_sets_top)
+next
+ assume "J \<noteq> {}" with assms show ?thesis
+ by (auto simp add: sets_PiM prod_algebra_def intro!: sigma_sets.Basic)
qed
-lemma (in product_sigma_algebra) measurable_add_dim:
- assumes "i \<notin> I"
- shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
-proof -
- let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (insert i I) M"
- interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M i"
- unfolding pair_sigma_algebra_def
- by (intro sigma_algebra_product_algebra sigma_algebras conjI)
- have "?f \<in> measurable Ii.P (sigma ?G)"
- proof (rule Ii.measurable_sigma)
- show "sets ?G \<subseteq> Pow (space ?G)"
- using product_algebra_generator_into_space .
- show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G"
- by (auto simp: space_pair_measure)
- next
- fix A assume "A \<in> sets ?G"
- then obtain F where "A = Pi\<^isub>E (insert i I) F"
- and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M i)"
- by (auto elim!: product_algebraE)
- then have "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)"
- using sets_into_space `i \<notin> I`
- by (auto simp add: space_pair_measure) blast+
- then show "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)"
- using F by (auto intro!: pair_measureI)
- qed
- then show ?thesis
- by (simp add: product_algebra_def)
+lemma measurable_PiM:
+ assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
+ f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N"
+ shows "f \<in> measurable N (PiM I M)"
+ using sets_PiM prod_algebra_sets_into_space space
+proof (rule measurable_sigma_sets)
+ fix A assume "A \<in> prod_algebra I M"
+ from prod_algebraE[OF this] guess J X .
+ with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
+qed
+
+lemma measurable_PiM_Collect:
+ assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
+ {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
+ shows "f \<in> measurable N (PiM I M)"
+ using sets_PiM prod_algebra_sets_into_space space
+proof (rule measurable_sigma_sets)
+ fix A assume "A \<in> prod_algebra I M"
+ from prod_algebraE[OF this] guess J X . note X = this
+ have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
+ using sets_into_space[OF X(5)] X(2-) space unfolding X(1)
+ by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm)
+ also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
+ finally show "f -` A \<inter> space N \<in> sets N" .
qed
-lemma (in product_sigma_algebra) measurable_merge:
- assumes [simp]: "I \<inter> J = {}"
- shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
-proof -
- let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M J M"
- interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J"
- by (intro sigma_algebra_pair_measure product_algebra_into_space)
- let ?f = "\<lambda>(x, y). merge I x J y"
- let ?G = "product_algebra_generator (I \<union> J) M"
- have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)"
- proof (rule P.measurable_sigma)
- fix A assume "A \<in> sets ?G"
- from product_algebraE[OF this]
- obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .
- then have *: "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
- using sets_into_space `I \<inter> J = {}`
- by (auto simp add: space_pair_measure) fast+
- then show "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"
- using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI
- simp: product_algebra_def)
- qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure)
- then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (Pi\<^isub>M (I \<union> J) M)"
- unfolding product_algebra_def[of "I \<union> J"] by simp
-qed
+lemma measurable_PiM_single:
+ assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N"
+ shows "f \<in> measurable N (PiM I M)"
+ using sets_PiM_single
+proof (rule measurable_sigma_sets)
+ fix A assume "A \<in> {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
+ then obtain B i where "A = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
+ by auto
+ with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
+ also have "\<dots> \<in> sets N" using B by (rule sets)
+ finally show "f -` A \<inter> space N \<in> sets N" .
+qed (auto simp: space)
-lemma (in product_sigma_algebra) measurable_component_singleton:
+lemma sets_PiM_I_finite[simp, intro]:
+ assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
+ shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
+ using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
+
+lemma measurable_component_update:
+ assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
+ shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
+proof (intro measurable_PiM_single)
+ fix j A assume "j \<in> insert i I" "A \<in> sets (M j)"
+ moreover have "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} =
+ (if i = j then space (M i) \<inter> A else if x j \<in> A then space (M i) else {})"
+ by auto
+ ultimately show "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} \<in> sets (M i)"
+ by auto
+qed (insert sets_into_space assms, auto simp: space_PiM)
+
+lemma measurable_component_singleton:
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
proof (unfold measurable_def, intro CollectI conjI ballI)
fix A assume "A \<in> sets (M i)"
- then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
- using M.sets_into_space `i \<in> I` by (fastforce dest: Pi_mem split: split_if_asm)
+ then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)"
+ using sets_into_space `i \<in> I`
+ by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
- using `A \<in> sets (M i)` by (auto intro!: product_algebraI)
-qed (insert `i \<in> I`, auto)
+ using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
+qed (insert `i \<in> I`, auto simp: space_PiM)
+
+lemma measurable_add_dim:
+ assumes "i \<notin> I"
+ shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
+ (is "?f \<in> measurable ?P ?I")
+proof (rule measurable_PiM_single)
+ fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
+ have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
+ (if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
+ using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
+ also have "\<dots> \<in> sets ?P"
+ using A j
+ by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
+ finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
+qed (auto simp: space_pair_measure space_PiM)
-lemma (in sigma_algebra) measurable_restrict:
- assumes I: "finite I"
- assumes "\<And>i. i \<in> I \<Longrightarrow> sets (N i) \<subseteq> Pow (space (N i))"
- assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (N i)"
- shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
- unfolding product_algebra_def
-proof (simp, rule measurable_sigma)
- show "sets (product_algebra_generator I N) \<subseteq> Pow (space (product_algebra_generator I N))"
- by (rule product_algebra_generator_sets_into_space) fact
- show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> space M \<rightarrow> space (product_algebra_generator I N)"
- using X by (auto simp: measurable_def)
- fix E assume "E \<in> sets (product_algebra_generator I N)"
- then obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (N i)"
- by (auto simp: product_algebra_generator_def)
- then have "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M = (\<Inter>i\<in>I. X i -` F i \<inter> space M) \<inter> space M"
- by (auto simp: Pi_iff)
- also have "\<dots> \<in> sets M"
- proof cases
- assume "I = {}" then show ?thesis by simp
- next
- assume "I \<noteq> {}" with X F I show ?thesis
- by (intro finite_INT measurable_sets Int) auto
- qed
- finally show "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M \<in> sets M" .
-qed
+lemma measurable_merge:
+ assumes "I \<inter> J = {}"
+ shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
+ (is "?f \<in> measurable ?P ?U")
+proof (rule measurable_PiM_single)
+ fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
+ then have "{\<omega> \<in> space ?P. prod_case (\<lambda>x. merge I x J) \<omega> i \<in> A} =
+ (if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)"
+ using `I \<inter> J = {}` by auto
+ also have "\<dots> \<in> sets ?P"
+ using A
+ by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
+ finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>x. merge I x J) \<omega> i \<in> A} \<in> sets ?P" .
+qed (insert assms, auto simp: space_pair_measure space_PiM)
-locale product_sigma_finite = product_sigma_algebra M
- for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" +
+lemma measurable_restrict:
+ assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
+ shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
+proof (rule measurable_PiM_single)
+ fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
+ then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
+ by auto
+ then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
+ using A X by (auto intro!: measurable_sets)
+qed (insert X, auto dest: measurable_space)
+
+locale product_sigma_finite =
+ fixes M :: "'i \<Rightarrow> 'a measure"
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
by (rule sigma_finite_measures)
-locale finite_product_sigma_finite = finite_product_sigma_algebra M I + product_sigma_finite M
- for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set"
-
-lemma (in finite_product_sigma_finite) product_algebra_generator_measure:
- assumes "Pi\<^isub>E I F \<in> sets G"
- shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"
-proof cases
- assume ne: "\<forall>i\<in>I. F i \<noteq> {}"
- have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') i = F i"
- by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
- (insert ne, auto simp: Pi_eq_iff)
- then show ?thesis
- unfolding product_algebra_generator_def by simp
-next
- assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"
- then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"
- by (auto simp: setprod_ereal_0 intro!: bexI)
- moreover
- have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"
- by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
- (insert empty, auto simp: Pi_eq_empty_iff[symmetric])
- then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"
- by (auto simp: setprod_ereal_0 intro!: bexI)
- ultimately show ?thesis
- unfolding product_algebra_generator_def by simp
-qed
+locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
+ fixes I :: "'i set"
+ assumes finite_index: "finite I"
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
- (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
- (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"
+ (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
+ (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
proof -
- have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)"
- using M.sigma_finite_up by simp
+ have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)"
+ using M.sigma_finite_incseq by metis
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
- then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>"
+ then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. emeasure (M i) (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
- note space_product_algebra[simp]
+ note space_PiM[simp]
show ?thesis
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
fix i show "range (F i) \<subseteq> sets (M i)" by fact
next
- fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact
+ fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
next
- fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
- using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space
- by (force simp: image_subset_iff)
+ fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)"
+ using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space
+ by auto blast
next
- fix f assume "f \<in> space G"
+ fix f assume "f \<in> space (PiM I M)"
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
next
@@ -574,140 +590,144 @@
qed
qed
-lemma sets_pair_cancel_measure[simp]:
- "sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)"
- "sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)"
- unfolding pair_measure_def pair_measure_generator_def sets_sigma
- by simp_all
-
-lemma measurable_pair_cancel_measure[simp]:
- "measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
- "measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
- "measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)"
- "measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)"
- unfolding measurable_def by (auto simp add: space_pair_measure)
-
-lemma (in product_sigma_finite) product_measure_exists:
+lemma (in product_sigma_finite)
assumes "finite I"
- shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and>
- (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))"
+ shows sigma_finite: "sigma_finite_measure (PiM I M)"
+ and emeasure_PiM:
+ "\<And>A. (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
using `finite I` proof induct
case empty
- interpret finite_product_sigma_finite M "{}" by default simp
- let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> ereal"
- show ?case
- proof (intro exI conjI ballI)
- have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"
- by (rule sigma_algebra_cong) (simp_all add: product_algebra_def)
- then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
- by (rule finite_additivity_sufficient)
- (simp_all add: positive_def additive_def sets_sigma
- product_algebra_generator_def image_constant)
- then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
- by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"]
- simp: sigma_finite_measure_def sigma_finite_measure_axioms_def
- product_algebra_generator_def)
- qed auto
+ let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
+ have "prod_algebra {} M = {{\<lambda>_. undefined}}"
+ by (auto simp: prod_algebra_def prod_emb_def intro!: image_eqI)
+ then have sets_empty: "sets (PiM {} M) = {{}, {\<lambda>_. undefined}}"
+ by (simp add: sets_PiM)
+ have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
+ proof (subst emeasure_extend_measure_Pair[OF PiM_def])
+ have "finite (space (PiM {} M))"
+ by (simp add: space_PiM)
+ moreover show "positive (PiM {} M) ?\<mu>"
+ by (auto simp: positive_def)
+ ultimately show "countably_additive (PiM {} M) ?\<mu>"
+ by (rule countably_additiveI_finite) (auto simp: additive_def space_PiM sets_empty)
+ qed (auto simp: prod_emb_def)
+ also have *: "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
+ by (auto simp: prod_emb_def)
+ finally have emeasure_eq: "emeasure (Pi\<^isub>M {} M) {\<lambda>_. undefined} = 1" .
+
+ interpret finite_measure "PiM {} M"
+ by default (simp add: space_PiM emeasure_eq)
+ case 1 show ?case ..
+
+ case 2 show ?case
+ using emeasure_eq * by simp
next
case (insert i I)
interpret finite_product_sigma_finite M I by default fact
have "finite (insert i I)" using `finite I` by auto
interpret I': finite_product_sigma_finite M "insert i I" by default fact
- from insert obtain \<nu> where
- prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and
- "sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by auto
- then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp
- interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" ..
+ interpret I: sigma_finite_measure "PiM I M" by fact
+ interpret P: pair_sigma_finite "PiM I M" "M i" ..
let ?h = "(\<lambda>(f, y). f(i := y))"
- let ?\<nu> = "\<lambda>A. P.\<mu> (?h -` A \<inter> space P.P)"
- have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)"
- by (rule I'.sigma_algebra_cong) simp_all
- interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>"
- using measurable_add_dim[OF `i \<notin> I`]
- by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def)
- show ?case
- proof (intro exI[of _ ?\<nu>] conjI ballI)
- let ?m = "\<lambda>A. measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h -` A \<inter> space P.P)"
- { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
- then have *: "?h -` Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"
- using `i \<notin> I` M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast
- show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))"
- unfolding * using A
- apply (subst P.pair_measure_times)
- using A apply fastforce
- using A apply fastforce
- using `i \<notin> I` `finite I` prod[of A] A by (auto simp: ac_simps) }
- note product = this
- have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>"
- by (simp add: product_algebra_def)
- show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)"
- proof (unfold *, default, simp)
- from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
- then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
- "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
- "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"
- "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"
- by blast+
- let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k"
- show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
- (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"
- proof (intro exI[of _ ?F] conjI allI)
- show "range ?F \<subseteq> sets I'.P" using F(1) by auto
- next
- from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
- next
- fix j
- have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"
- using F(1) by auto
- with F `finite I` setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"
- by (subst product) auto
- qed
- qed
+
+ let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
+ let ?\<mu> = "emeasure ?P"
+ let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
+ let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
+
+ { case 2
+ have "emeasure (Pi\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>E (insert i I) A)) =
+ (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
+ proof (subst emeasure_extend_measure_Pair[OF PiM_def])
+ fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))"
+ then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto
+ let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
+ let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
+ have "?\<mu> ?p =
+ emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))"
+ by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
+ also have "?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
+ using J E[rule_format, THEN sets_into_space]
+ by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm)
+ also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
+ emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
+ using J E by (intro P.emeasure_pair_measure_Times sets_PiM_I) auto
+ also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
+ using J E[rule_format, THEN sets_into_space]
+ by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+
+ also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
+ (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
+ using E by (subst insert) (auto intro!: setprod_cong)
+ also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
+ emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
+ using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
+ also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
+ using insert(1,2) J E by (intro setprod_mono_one_right) auto
+ finally show "?\<mu> ?p = \<dots>" .
+
+ show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>E i\<in>insert i I. space (M i))"
+ using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff)
+ next
+ show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
+ using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
+ next
+ show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
+ insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
+ using insert(1,2) 2 by auto
+ qed (auto intro!: setprod_cong)
+ with 2[THEN sets_into_space] show ?case by (subst (asm) prod_emb_PiE_same_index) auto }
+ note product = this
+
+ from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
+ then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
+ "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
+ "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space (Pi\<^isub>M (insert i I) M)"
+ "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
+ by blast+
+ let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k"
+
+ case 1 show ?case
+ proof (unfold_locales, intro exI[of _ ?F] conjI allI)
+ show "range ?F \<subseteq> sets (Pi\<^isub>M (insert i I) M)" using F(1) insert(1,2) by auto
+ next
+ from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M (insert i I) M)" by simp
+ next
+ fix j
+ from F `finite I` setprod_PInf[of "insert i I", OF emeasure_nonneg, of M]
+ show "emeasure (\<Pi>\<^isub>M i\<in>insert i I. M i) (?F j) \<noteq> \<infinity>"
+ by (subst product) auto
qed
qed
-sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P
- unfolding product_algebra_def
- using product_measure_exists[OF finite_index]
- by (rule someI2_ex) auto
+sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
+ using sigma_finite[OF finite_index] .
lemma (in finite_product_sigma_finite) measure_times:
- assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
- shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
- unfolding product_algebra_def
- using product_measure_exists[OF finite_index]
- proof (rule someI2_ex, elim conjE)
- fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
- have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
- then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
- also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
- finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
- by (simp add: sigma_def)
- qed
+ "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^isub>M I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
+ using emeasure_PiM[OF finite_index] by auto
lemma (in product_sigma_finite) product_measure_empty[simp]:
- "measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
+ "emeasure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
proof -
interpret finite_product_sigma_finite M "{}" by default auto
from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
qed
-lemma (in finite_product_sigma_algebra) P_empty:
- assumes "I = {}"
- shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }"
- unfolding product_algebra_def product_algebra_generator_def `I = {}`
- by (simp_all add: sigma_def image_constant)
+lemma
+ shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
+ and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
+ by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
lemma (in product_sigma_finite) positive_integral_empty:
assumes pos: "0 \<le> f (\<lambda>k. undefined)"
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
proof -
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
- have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
+ have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
using assms by (subst measure_times) auto
then show ?thesis
- unfolding positive_integral_def simple_function_def simple_integral_def [abs_def]
- proof (simp add: P_empty, intro antisym)
+ unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
+ proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
@@ -715,71 +735,72 @@
qed
qed
-lemma (in product_sigma_finite) measure_fold:
+lemma (in product_sigma_finite) distr_merge:
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
- assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
- shows "measure (Pi\<^isub>M (I \<union> J) M) A =
- measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))"
+ shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (\<lambda>(x,y). merge I x J y) = Pi\<^isub>M (I \<union> J) M"
+ (is "?D = ?P")
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
- interpret P: pair_sigma_finite I.P J.P by default
+ interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
let ?g = "\<lambda>(x,y). merge I x J y"
- let ?X = "\<lambda>S. ?g -` S \<inter> space P.P"
+
from IJ.sigma_finite_pairs obtain F where
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
- "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"
- "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"
+ "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
+ "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
- show "IJ.\<mu> A = P.\<mu> (?X A)"
- proof (rule measure_unique_Int_stable_vimage)
- show "measure_space IJ.P" "measure_space P.P" by default
- show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)"
- using A unfolding product_algebra_def by auto
- next
- show "Int_stable IJ.G"
- by (rule Int_stable_product_algebra_generator)
- (auto simp: Int_stable_def)
- show "range ?F \<subseteq> sets IJ.G" using F
- by (simp add: image_subset_iff product_algebra_def
- product_algebra_generator_def)
- show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+
+
+ show ?thesis
+ proof (rule measure_eqI_generator_eq[symmetric])
+ show "Int_stable (prod_algebra (I \<union> J) M)"
+ by (rule Int_stable_prod_algebra)
+ show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
+ by (rule prod_algebra_sets_into_space)
+ show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
+ by (rule sets_PiM)
+ then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
+ by simp
+
+ show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
+ using fin by (auto simp: prod_algebra_eq_finite)
+ show "incseq ?F" by fact
+ show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))"
+ using F(3) by (simp add: space_PiM)
next
fix k
- have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"
- using F(1) by auto
- with F `finite I` setprod_PInf[of "I \<union> J", OF this]
- show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
+ from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
+ show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
next
- fix A assume "A \<in> sets IJ.G"
- then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"
- and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"
- by (auto simp: product_algebra_generator_def)
- then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
- using sets_into_space by (auto simp: space_pair_measure) blast+
- then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
- using `finite J` `finite I` F
- by (simp add: P.pair_measure_times I.measure_times J.measure_times)
- also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
+ fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
+ with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>I \<union> J. F i \<in> sets (M i)"
+ by (auto simp add: prod_algebra_eq_finite)
+ let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
+ let ?X = "?g -` A \<inter> space ?B"
+ have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)"
+ using F[rule_format, THEN sets_into_space] by (auto simp: space_PiM)
+ then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
+ unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
+ have "emeasure ?D A = emeasure ?B ?X"
+ using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
+ also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
+ using `finite J` `finite I` F X
+ by (simp add: P.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff)
+ also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one)
- also have "\<dots> = IJ.\<mu> A"
+ also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
using `finite J` `finite I` F unfolding A
by (intro IJ.measure_times[symmetric]) auto
- finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)
- qed (rule measurable_merge[OF IJ])
+ finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
+ qed
qed
-lemma (in product_sigma_finite) measure_preserving_merge:
- assumes IJ: "I \<inter> J = {}" and "finite I" "finite J"
- shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
- by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms)
-
lemma (in product_sigma_finite) product_positive_integral_fold:
- assumes IJ[simp]: "I \<inter> J = {}" "finite I" "finite J"
+ assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
@@ -787,42 +808,38 @@
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
- interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp
- have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
+ have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def)
show ?thesis
- unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
- proof (rule P.positive_integral_vimage)
- show "sigma_algebra IJ.P" by default
- show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P"
- using IJ by (rule measure_preserving_merge)
- show "f \<in> borel_measurable IJ.P" using f by simp
- qed
+ apply (subst distr_merge[OF IJ, symmetric])
+ apply (subst positive_integral_distr[OF measurable_merge f, OF IJ(1)])
+ apply (subst P.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
+ apply simp
+ done
qed
-lemma (in product_sigma_finite) measure_preserving_component_singelton:
- "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
-proof (intro measure_preservingI measurable_component_singleton)
+lemma (in product_sigma_finite) distr_singleton:
+ "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
+proof (intro measure_eqI[symmetric])
interpret I: finite_product_sigma_finite M "{i}" by default simp
- fix A let ?P = "(\<lambda>x. x i) -` A \<inter> space I.P"
- assume A: "A \<in> sets (M i)"
- then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto
- show "I.\<mu> ?P = M.\<mu> i A" unfolding *
- using A I.measure_times[of "\<lambda>_. A"] by auto
-qed auto
+ fix A assume A: "A \<in> sets (M i)"
+ moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
+ using sets_into_space by (auto simp: space_PiM)
+ ultimately show "emeasure (M i) A = emeasure ?D A"
+ using A I.measure_times[of "\<lambda>_. A"]
+ by (simp add: emeasure_distr measurable_component_singleton)
+qed simp
lemma (in product_sigma_finite) product_positive_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
proof -
interpret I: finite_product_sigma_finite M "{i}" by default simp
- show ?thesis
- proof (rule I.positive_integral_vimage[symmetric])
- show "sigma_algebra (M i)" by (rule sigma_algebras)
- show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
- by (rule measure_preserving_component_singelton)
- show "f \<in> borel_measurable (M i)" by fact
- qed
+ from f show ?thesis
+ apply (subst distr_singleton[symmetric])
+ apply (subst positive_integral_distr[OF measurable_component_singleton])
+ apply simp_all
+ done
qed
lemma (in product_sigma_finite) product_positive_integral_insert:
@@ -832,19 +849,18 @@
proof -
interpret I: finite_product_sigma_finite M I by default auto
interpret i: finite_product_sigma_finite M "{i}" by default auto
- interpret P: pair_sigma_algebra I.P i.P ..
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
using f by auto
show ?thesis
unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]
- proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)
- fix x assume x: "x \<in> space I.P"
+ proof (rule positive_integral_cong, subst product_positive_integral_singleton)
+ fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
have f'_eq: "\<And>y. ?f y = f (x(i := y))"
- using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
+ using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def space_PiM)
show "?f \<in> borel_measurable (M i)" unfolding f'_eq
- using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
- by (simp add: comp_def)
+ using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
+ unfolding comp_def .
show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"
unfolding f'_eq by simp
qed
@@ -856,10 +872,6 @@
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
using assms proof induct
- case empty
- interpret finite_product_sigma_finite M "{}" by default auto
- show ?case by simp
-next
case (insert i I)
note `finite I`[intro, simp]
interpret I: finite_product_sigma_finite M I by default auto
@@ -867,16 +879,16 @@
using insert by (auto intro!: setprod_cong)
have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
using sets_into_space insert
- by (intro sigma_algebra.borel_measurable_ereal_setprod sigma_algebra_product_algebra
+ by (intro borel_measurable_ereal_setprod
measurable_comp[OF measurable_component_singleton, unfolded comp_def])
auto
then show ?case
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
- apply (simp add: insert * pos borel setprod_ereal_pos M.positive_integral_multc)
- apply (subst I.positive_integral_cmult)
- apply (auto simp add: pos borel insert setprod_ereal_pos positive_integral_positive)
+ apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
+ apply (subst positive_integral_cmult)
+ apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
done
-qed
+qed (simp add: space_PiM)
lemma (in product_sigma_finite) product_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
@@ -899,48 +911,44 @@
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
- interpret P: pair_sigma_finite I.P J.P by default
+ interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
let ?M = "\<lambda>(x, y). merge I x J y"
let ?f = "\<lambda>x. f (?M x)"
+ from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
+ by auto
+ have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
+ using measurable_comp[OF measurable_merge[OF IJ(1)] f_borel] by (simp add: comp_def)
+ have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
+ by (rule integrable_distr[OF measurable_merge[OF IJ]]) (simp add: distr_merge[OF IJ fin] f)
show ?thesis
- proof (subst P.integrable_fst_measurable(2)[of ?f, simplified])
- have 1: "sigma_algebra IJ.P" by default
- have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] .
- have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact
- then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
- by (simp add: integrable_def)
- show "integrable P.P ?f"
- by (rule P.integrable_vimage[where f=f, OF 1 2 3])
- show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f"
- by (rule P.integral_vimage[where f=f, OF 1 2 4])
- qed
+ apply (subst distr_merge[symmetric, OF IJ fin])
+ apply (subst integral_distr[OF measurable_merge[OF IJ] f_borel])
+ apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
+ apply simp
+ done
qed
lemma (in product_sigma_finite) product_integral_insert:
- assumes [simp]: "finite I" "i \<notin> I"
+ assumes I: "finite I" "i \<notin> I"
and f: "integrable (Pi\<^isub>M (insert i I) M) f"
shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
proof -
- interpret I: finite_product_sigma_finite M I by default auto
- interpret I': finite_product_sigma_finite M "insert i I" by default auto
- interpret i: finite_product_sigma_finite M "{i}" by default auto
- interpret P: pair_sigma_finite I.P i.P ..
- have IJ: "I \<inter> {i} = {}" by auto
- show ?thesis
- unfolding product_integral_fold[OF IJ, simplified, OF f]
- proof (rule I.integral_cong, subst product_integral_singleton)
- fix x assume x: "x \<in> space I.P"
- let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
- have f'_eq: "\<And>y. ?f y = f (x(i := y))"
- using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
- have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto
- show "?f \<in> borel_measurable (M i)"
- unfolding measurable_cong[OF f'_eq]
- using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
- by (simp add: comp_def)
- show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))"
- unfolding f'_eq by simp
+ have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
+ by simp
+ also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I x {i} y) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)"
+ using f I by (intro product_integral_fold) auto
+ also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
+ proof (rule integral_cong, subst product_integral_singleton[symmetric])
+ fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
+ have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
+ using f by auto
+ show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
+ using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
+ unfolding comp_def .
+ from x I show "(\<integral> y. f (merge I x {i} y) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)"
+ by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def)
qed
+ finally show ?thesis .
qed
lemma (in product_sigma_finite) product_integrable_setprod:
@@ -951,24 +959,23 @@
interpret finite_product_sigma_finite M I by default fact
have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
using integrable unfolding integrable_def by auto
- then have borel: "?f \<in> borel_measurable P"
- using measurable_comp[OF measurable_component_singleton f]
- by (auto intro!: borel_measurable_setprod simp: comp_def)
+ have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
+ using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
proof (unfold integrable_def, intro conjI)
- show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
+ show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
using borel by auto
- have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>P)"
+ have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)"
by (simp add: setprod_ereal abs_setprod)
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
using f by (subst product_positive_integral_setprod) auto
also have "\<dots> < \<infinity>"
- using integrable[THEN M.integrable_abs]
- by (simp add: setprod_PInf integrable_def M.positive_integral_positive)
- finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto
- have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"
+ using integrable[THEN integrable_abs]
+ by (simp add: setprod_PInf integrable_def positive_integral_positive)
+ finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
+ have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
by (intro positive_integral_cong_pos) auto
- then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp
+ then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
qed
ultimately show ?thesis
by (rule integrable_abs_iff[THEN iffD1])
@@ -992,7 +999,69 @@
using `i \<notin> I` by (auto intro!: setprod_cong)
show ?case
unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
- by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)
+ by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
qed
-end
\ No newline at end of file
+lemma sigma_prod_algebra_sigma_eq:
+ fixes E :: "'i \<Rightarrow> 'a set set"
+ assumes "finite I"
+ assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
+ and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
+ and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
+ assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
+ and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
+ defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
+ shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
+proof
+ let ?P = "sigma (space (Pi\<^isub>M I M)) P"
+ have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
+ using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
+ then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ by (simp add: space_PiM)
+ have "sets (PiM I M) =
+ sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
+ using sets_PiM_single[of I M] by (simp add: space_P)
+ also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
+ proof (safe intro!: sigma_sets_subset)
+ fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
+ have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
+ proof (subst measurable_iff_measure_of)
+ show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
+ from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)"
+ by (auto simp: Pi_iff)
+ show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
+ proof
+ fix A assume A: "A \<in> E i"
+ then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
+ using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
+ also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
+ by (intro PiE_cong) (simp add: S_union)
+ also have "\<dots> = (\<Union>n. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j n)"
+ using S_mono
+ by (subst Pi_UN[symmetric, OF `finite I`]) (auto simp: incseq_def)
+ also have "\<dots> \<in> sets ?P"
+ proof (safe intro!: countable_UN)
+ fix n show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j n) \<in> sets ?P"
+ using A S_in_E
+ by (simp add: P_closed)
+ (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j n"])
+ qed
+ finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
+ using P_closed by simp
+ qed
+ qed
+ from measurable_sets[OF this, of A] A `i \<in> I` E_closed
+ have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
+ by (simp add: E_generates)
+ also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
+ using P_closed by (auto simp: space_PiM)
+ finally show "\<dots> \<in> sets ?P" .
+ qed
+ finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
+ by (simp add: P_closed)
+ show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
+ using `finite I`
+ by (auto intro!: sigma_sets_subset simp: E_generates P_def)
+qed
+
+end
--- a/src/HOL/Probability/Independent_Family.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Independent_Family.thy Mon Apr 23 12:14:35 2012 +0200
@@ -5,7 +5,7 @@
header {* Independent families of events, event sets, and random variables *}
theory Independent_Family
- imports Probability_Measure
+ imports Probability_Measure Infinite_Product_Measure
begin
lemma INT_decseq_offset:
@@ -44,7 +44,7 @@
definition (in prob_space)
"indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV"
-lemma (in prob_space) indep_sets_cong[cong]:
+lemma (in prob_space) indep_sets_cong:
"I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
@@ -135,7 +135,7 @@
lemma (in prob_space) indep_sets_dynkin:
assumes indep: "indep_sets F I"
- shows "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) I"
+ shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I"
(is "indep_sets ?F I")
proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
@@ -193,7 +193,7 @@
qed
qed }
note indep_sets_insert = this
- have "dynkin_system \<lparr> space = space M, sets = ?D \<rparr>"
+ have "dynkin_system (space M) ?D"
proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
show "indep_sets (G(j := {{}})) K"
by (rule indep_sets_insert) auto
@@ -206,7 +206,7 @@
using G by auto
have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
- using A_sets sets_into_space X `J \<noteq> {}`
+ using A_sets sets_into_space[of _ M] X `J \<noteq> {}`
by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
@@ -264,9 +264,8 @@
by (auto dest!: sums_unique)
qed (insert F, auto)
qed (insert sets_into_space, auto)
- then have mono: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq>
- sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>"
- proof (rule dynkin_system.dynkin_subset, simp_all cong del: indep_sets_cong, safe)
+ then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
+ proof (rule dynkin_system.dynkin_subset, safe)
fix X assume "X \<in> G j"
then show "X \<in> events" using G `j \<in> K` by auto
from `indep_sets G K`
@@ -292,20 +291,20 @@
by (intro indep_setsD[OF G(1)]) auto
qed
qed
- then have "indep_sets (G(j:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) K"
+ then have "indep_sets (G(j := dynkin (space M) (G j))) K"
by (rule indep_sets_mono_sets) (insert mono, auto)
then show ?case
by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
qed (insert `indep_sets F K`, simp) }
from this[OF `indep_sets F J` `finite J` subset_refl]
- show "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) J"
+ show "indep_sets ?F J"
by (rule indep_sets_mono_sets) auto
qed
lemma (in prob_space) indep_sets_sigma:
assumes indep: "indep_sets F I"
- assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
- shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) I"
+ assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
+ shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
proof -
from indep_sets_dynkin[OF indep]
show ?thesis
@@ -316,18 +315,12 @@
qed
qed
-lemma (in prob_space) indep_sets_sigma_sets:
- assumes "indep_sets F I"
- assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
- shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
- using indep_sets_sigma[OF assms] by (simp add: sets_sigma)
-
lemma (in prob_space) indep_sets_sigma_sets_iff:
- assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
+ assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
proof
assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
- by (rule indep_sets_sigma_sets) fact
+ by (rule indep_sets_sigma) fact
next
assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
@@ -361,15 +354,14 @@
lemma (in prob_space) indep_set_sigma_sets:
assumes "indep_set A B"
- assumes A: "Int_stable \<lparr> space = space M, sets = A \<rparr>"
- assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>"
+ assumes A: "Int_stable A" and B: "Int_stable B"
shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
proof -
have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
- proof (rule indep_sets_sigma_sets)
+ proof (rule indep_sets_sigma)
show "indep_sets (bool_case A B) UNIV"
by (rule `indep_set A B`[unfolded indep_set_def])
- fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>"
+ fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
using A B by (cases i) auto
qed
then show ?thesis
@@ -380,7 +372,7 @@
lemma (in prob_space) indep_sets_collect_sigma:
fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
- assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable \<lparr>space = space M, sets = E i\<rparr>"
+ assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"
assumes disjoint: "disjoint_family_on I J"
shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
proof -
@@ -389,30 +381,29 @@
from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
unfolding indep_sets_def by auto
{ fix j
- let ?S = "sigma \<lparr> space = space M, sets = (\<Union>i\<in>I j. E i) \<rparr>"
+ let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"
assume "j \<in> J"
- from E[OF this] interpret S: sigma_algebra ?S
- using sets_into_space by (intro sigma_algebra_sigma) auto
+ from E[OF this] interpret S: sigma_algebra "space M" ?S
+ using sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
proof (rule sigma_sets_eqI)
fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
then guess i ..
then show "A \<in> sigma_sets (space M) (?E j)"
- by (auto intro!: sigma_sets.intros exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
+ by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
next
fix A assume "A \<in> ?E j"
then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
and A: "A = (\<Inter>k\<in>K. E' k)"
by auto
- then have "A \<in> sets ?S" unfolding A
- by (safe intro!: S.finite_INT)
- (auto simp: sets_sigma intro!: sigma_sets.Basic)
+ then have "A \<in> ?S" unfolding A
+ by (safe intro!: S.finite_INT) auto
then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
- by (simp add: sets_sigma)
+ by simp
qed }
moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
- proof (rule indep_sets_sigma_sets)
+ proof (rule indep_sets_sigma)
show "indep_sets ?E J"
proof (intro indep_setsI)
fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force intro!: finite_INT)
@@ -460,7 +451,7 @@
qed
next
fix j assume "j \<in> J"
- show "Int_stable \<lparr> space = space M, sets = ?E j \<rparr>"
+ show "Int_stable (?E j)"
proof (rule Int_stableI)
fix a assume "a \<in> ?E j" then obtain Ka Ea
where a: "a = (\<Inter>k\<in>Ka. Ea k)" "finite Ka" "Ka \<noteq> {}" "Ka \<subseteq> I j" "\<And>k. k\<in>Ka \<Longrightarrow> Ea k \<in> E k" by auto
@@ -482,12 +473,12 @@
lemma (in prob_space) terminal_events_sets:
assumes A: "\<And>i. A i \<subseteq> events"
- assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
+ assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
assumes X: "X \<in> terminal_events A"
shows "X \<in> events"
proof -
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
- interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
+ interpret A: sigma_algebra "space M" "A i" for i by fact
from X have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: terminal_events_def)
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
then show "X \<in> events"
@@ -495,12 +486,12 @@
qed
lemma (in prob_space) sigma_algebra_terminal_events:
- assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
- shows "sigma_algebra \<lparr> space = space M, sets = terminal_events A \<rparr>"
+ assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
+ shows "sigma_algebra (space M) (terminal_events A)"
unfolding terminal_events_def
proof (simp add: sigma_algebra_iff2, safe)
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
- interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
+ interpret A: sigma_algebra "space M" "A i" for i by fact
{ fix X x assume "X \<in> ?A" "x \<in> X"
then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
@@ -515,29 +506,29 @@
lemma (in prob_space) kolmogorov_0_1_law:
fixes A :: "nat \<Rightarrow> 'a set set"
assumes A: "\<And>i. A i \<subseteq> events"
- assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
+ assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
assumes indep: "indep_sets A UNIV"
and X: "X \<in> terminal_events A"
shows "prob X = 0 \<or> prob X = 1"
proof -
- let ?D = "\<lparr> space = space M, sets = {D \<in> events. prob (X \<inter> D) = prob X * prob D} \<rparr>"
- interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
- interpret T: sigma_algebra "\<lparr> space = space M, sets = terminal_events A \<rparr>"
+ let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"
+ interpret A: sigma_algebra "space M" "A i" for i by fact
+ interpret T: sigma_algebra "space M" "terminal_events A"
by (rule sigma_algebra_terminal_events) fact
have "X \<subseteq> space M" using T.space_closed X by auto
have X_in: "X \<in> events"
by (rule terminal_events_sets) fact+
- interpret D: dynkin_system ?D
+ interpret D: dynkin_system "space M" ?D
proof (rule dynkin_systemI)
- fix D assume "D \<in> sets ?D" then show "D \<subseteq> space ?D"
+ fix D assume "D \<in> ?D" then show "D \<subseteq> space M"
using sets_into_space by auto
next
- show "space ?D \<in> sets ?D"
+ show "space M \<in> ?D"
using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
next
- fix A assume A: "A \<in> sets ?D"
+ fix A assume A: "A \<in> ?D"
have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
also have "\<dots> = prob X - prob (X \<inter> A)"
@@ -547,10 +538,10 @@
also have "\<dots> = prob X * prob (space M - A)"
using X_in A sets_into_space
by (subst finite_measure_Diff) (auto simp: field_simps)
- finally show "space ?D - A \<in> sets ?D"
+ finally show "space M - A \<in> ?D"
using A `X \<subseteq> space M` by auto
next
- fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> sets ?D"
+ fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
by auto
have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
@@ -566,7 +557,7 @@
by (intro sums_mult finite_measure_UNION F dis)
ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
by (auto dest!: sums_unique)
- with F show "(\<Union>i. F i) \<in> sets ?D"
+ with F show "(\<Union>i. F i) \<in> ?D"
by auto
qed
@@ -579,7 +570,7 @@
show "disjoint_family (bool_case {..n} {Suc n..})"
unfolding disjoint_family_on_def by (auto split: bool.split)
fix m
- show "Int_stable \<lparr>space = space M, sets = A m\<rparr>"
+ show "Int_stable (A m)"
unfolding Int_stable_def using A.Int by auto
qed
also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) =
@@ -588,7 +579,7 @@
finally have indep: "indep_set (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
unfolding indep_set_def by simp
- have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> sets ?D"
+ have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"
proof (simp add: subset_eq, rule)
fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
@@ -597,22 +588,27 @@
show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
by (auto simp add: ac_simps)
qed }
- then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> sets ?D" (is "?A \<subseteq> _")
+ then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
by auto
- have "sigma \<lparr> space = space M, sets = ?A \<rparr> =
- dynkin \<lparr> space = space M, sets = ?A \<rparr>" (is "sigma ?UA = dynkin ?UA")
+ note `X \<in> terminal_events A`
+ also {
+ have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
+ by (intro sigma_sets_subseteq UN_mono) auto
+ then have "terminal_events A \<subseteq> sigma_sets (space M) ?A"
+ unfolding terminal_events_def by auto }
+ also have "sigma_sets (space M) ?A = dynkin (space M) ?A"
proof (rule sigma_eq_dynkin)
{ fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
then have "B \<subseteq> space M"
- by induct (insert A sets_into_space, auto) }
- then show "sets ?UA \<subseteq> Pow (space ?UA)" by auto
- show "Int_stable ?UA"
+ by induct (insert A sets_into_space[of _ M], auto) }
+ then show "?A \<subseteq> Pow (space M)" by auto
+ show "Int_stable ?A"
proof (rule Int_stableI)
fix a assume "a \<in> ?A" then guess n .. note a = this
fix b assume "b \<in> ?A" then guess m .. note b = this
- interpret Amn: sigma_algebra "sigma \<lparr>space = space M, sets = (\<Union>i\<in>{..max m n}. A i)\<rparr>"
- using A sets_into_space by (intro sigma_algebra_sigma) auto
+ interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
+ using A sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
by (intro sigma_sets_subseteq UN_mono) auto
with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
@@ -621,23 +617,13 @@
by (intro sigma_sets_subseteq UN_mono) auto
with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
- using Amn.Int[of a b] by (simp add: sets_sigma)
+ using Amn.Int[of a b] by simp
then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
qed
qed
- moreover have "sets (dynkin ?UA) \<subseteq> sets ?D"
- proof (rule D.dynkin_subset)
- show "sets ?UA \<subseteq> sets ?D" using `?A \<subseteq> sets ?D` by auto
- qed simp
- ultimately have "sets (sigma ?UA) \<subseteq> sets ?D" by simp
- moreover
- have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
- by (intro sigma_sets_subseteq UN_mono) (auto intro: sigma_sets.Basic)
- then have "terminal_events A \<subseteq> sets (sigma ?UA)"
- unfolding sets_sigma terminal_events_def by auto
- moreover note `X \<in> terminal_events A`
- ultimately have "X \<in> sets ?D" by auto
- then show ?thesis by auto
+ also have "dynkin (space M) ?A \<subseteq> ?D"
+ using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)
+ finally show ?thesis by auto
qed
lemma (in prob_space) borel_0_1_law:
@@ -648,14 +634,14 @@
show "\<And>i. sigma_sets (space M) {F i} \<subseteq> events"
using F(1) sets_into_space
by (subst sigma_sets_singleton) auto
- { fix i show "sigma_algebra \<lparr>space = space M, sets = sigma_sets (space M) {F i}\<rparr>"
- using sigma_algebra_sigma[of "\<lparr>space = space M, sets = {F i}\<rparr>"] F sets_into_space
- by (auto simp add: sigma_def) }
+ { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
+ using sigma_algebra_sigma_sets[of "{F i}" "space M"] F sets_into_space
+ by auto }
show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
- proof (rule indep_sets_sigma_sets)
+ proof (rule indep_sets_sigma)
show "indep_sets (\<lambda>i. {F i}) UNIV"
unfolding indep_sets_singleton_iff_indep_events by fact
- fix i show "Int_stable \<lparr>space = space M, sets = {F i}\<rparr>"
+ fix i show "Int_stable {F i}"
unfolding Int_stable_def by simp
qed
let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"
@@ -663,17 +649,17 @@
unfolding terminal_events_def
proof
fix j
- interpret S: sigma_algebra "sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>"
+ interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
using order_trans[OF F(1) space_closed]
- by (intro sigma_algebra_sigma) (simp add: sigma_sets_singleton subset_eq)
+ by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
by (intro decseq_SucI INT_decseq_offset UN_mono) auto
- also have "\<dots> \<in> sets (sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>)"
+ also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
using order_trans[OF F(1) space_closed]
by (safe intro!: S.countable_INT S.countable_UN)
- (auto simp: sets_sigma sigma_sets_singleton intro!: sigma_sets.Basic bexI)
+ (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
- by (simp add: sets_sigma)
+ by simp
qed
qed
@@ -710,84 +696,84 @@
lemma (in prob_space) indep_vars_finite:
fixes I :: "'i set"
assumes I: "I \<noteq> {}" "finite I"
- and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (sigma (M' i)) (X i)"
- and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (M' i)"
- and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> sets (M' i)"
- shows "indep_vars (\<lambda>i. sigma (M' i)) X I \<longleftrightarrow>
- (\<forall>A\<in>(\<Pi> i\<in>I. sets (M' i)). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))"
+ and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"
+ and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
+ and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"
+ and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> E i" and closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M' i))"
+ shows "indep_vars M' X I \<longleftrightarrow>
+ (\<forall>A\<in>(\<Pi> i\<in>I. E i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))"
proof -
from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
unfolding measurable_def by simp
{ fix i assume "i\<in>I"
- have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (sigma (M' i))}
- = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
- unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`]
+ from closed[OF `i \<in> I`]
+ have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
+ = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
+ unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]
by (subst sigma_sets_sigma_sets_eq) auto }
- note this[simp]
+ note sigma_sets_X = this
{ fix i assume "i\<in>I"
- have "Int_stable \<lparr>space = space M, sets = {X i -` A \<inter> space M |A. A \<in> sets (M' i)}\<rparr>"
+ have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"
proof (rule Int_stableI)
- fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
- then obtain A where "a = X i -` A \<inter> space M" "A \<in> sets (M' i)" by auto
+ fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
+ then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto
moreover
- fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
- then obtain B where "b = X i -` B \<inter> space M" "B \<in> sets (M' i)" by auto
+ fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
+ then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
moreover
have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
moreover note Int_stable[OF `i \<in> I`]
ultimately
- show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
+ show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
qed }
- note indep_sets_sigma_sets_iff[OF this, simp]
+ note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
{ fix i assume "i \<in> I"
- { fix A assume "A \<in> sets (M' i)"
- then have "A \<in> sets (sigma (M' i))" by (auto simp: sets_sigma intro: sigma_sets.Basic)
+ { fix A assume "A \<in> E i"
+ with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto
moreover
- from rv[OF `i\<in>I`] have "X i \<in> measurable M (sigma (M' i))" by auto
+ from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto
ultimately
have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
with X[OF `i\<in>I`] space[OF `i\<in>I`]
- have "{X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
- "space M \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
+ have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
+ "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
by (auto intro!: exI[of _ "space (M' i)"]) }
- note indep_sets_finite[OF I this, simp]
+ note indep_sets_finite_X = indep_sets_finite[OF I this]
- have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) =
- (\<forall>A\<in>\<Pi> i\<in>I. sets (M' i). prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"
+ have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) =
+ (\<forall>A\<in>\<Pi> i\<in>I. E i. prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"
(is "?L = ?R")
proof safe
- fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. sets (M' i))"
+ fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
show "prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M))"
by (auto simp add: Pi_iff)
next
- fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> sets (M' i)})"
- from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> sets (M' i)" by auto
+ fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"
+ from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
- "B \<in> (\<Pi> i\<in>I. sets (M' i))" by auto
+ "B \<in> (\<Pi> i\<in>I. E i)" by auto
from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
by simp
qed
then show ?thesis using `I \<noteq> {}`
- by (simp add: rv indep_vars_def)
+ by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
qed
lemma (in prob_space) indep_vars_compose:
assumes "indep_vars M' X I"
- assumes rv:
- "\<And>i. i \<in> I \<Longrightarrow> sigma_algebra (N i)"
- "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
+ assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
unfolding indep_vars_def
proof
from rv `indep_vars M' X I`
show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
- by (auto intro!: measurable_comp simp: indep_vars_def)
+ by (auto simp: indep_vars_def)
have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
using `indep_vars M' X I` by (simp add: indep_vars_def)
@@ -806,7 +792,7 @@
qed
qed
-lemma (in prob_space) indep_varsD:
+lemma (in prob_space) indep_varsD_finite:
assumes X: "indep_vars M' X I"
assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"
shows "prob (\<Inter>i\<in>I. X i -` A i \<inter> space M) = (\<Prod>i\<in>I. prob (X i -` A i \<inter> space M))"
@@ -815,96 +801,134 @@
using X by (auto simp: indep_vars_def)
show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto
show "\<forall>i\<in>I. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
- using I by (auto intro: sigma_sets.Basic)
+ using I by auto
qed
-lemma (in prob_space) indep_distribution_eq_measure:
- assumes I: "I \<noteq> {}" "finite I"
+lemma (in prob_space) indep_varsD:
+ assumes X: "indep_vars M' X I"
+ assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"
+ shows "prob (\<Inter>i\<in>J. X i -` A i \<inter> space M) = (\<Prod>i\<in>J. prob (X i -` A i \<inter> space M))"
+proof (rule indep_setsD)
+ show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
+ using X by (auto simp: indep_vars_def)
+ show "\<forall>i\<in>J. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
+ using I by auto
+qed fact+
+
+lemma prod_algebra_cong:
+ assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
+ shows "prod_algebra I M = prod_algebra J N"
+proof -
+ have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
+ using sets_eq_imp_space_eq[OF sets] by auto
+ with sets show ?thesis unfolding `I = J`
+ by (intro antisym prod_algebra_mono) auto
+qed
+
+lemma space_in_prod_algebra:
+ "(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
+proof cases
+ assume "I = {}" then show ?thesis
+ by (auto simp add: prod_algebra_def image_iff prod_emb_def)
+next
+ assume "I \<noteq> {}"
+ then obtain i where "i \<in> I" by auto
+ then have "(\<Pi>\<^isub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i))"
+ by (auto simp: prod_emb_def Pi_iff)
+ also have "\<dots> \<in> prod_algebra I M"
+ using `i \<in> I` by (intro prod_algebraI) auto
+ finally show ?thesis .
+qed
+
+lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
+ fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
+ assumes "I \<noteq> {}"
assumes rv: "\<And>i. random_variable (M' i) (X i)"
shows "indep_vars M' X I \<longleftrightarrow>
- (\<forall>A\<in>sets (\<Pi>\<^isub>M i\<in>I. (M' i \<lparr> measure := ereal\<circ>distribution (X i) \<rparr>)).
- distribution (\<lambda>x. \<lambda>i\<in>I. X i x) A =
- finite_measure.\<mu>' (\<Pi>\<^isub>M i\<in>I. (M' i \<lparr> measure := ereal\<circ>distribution (X i) \<rparr>)) A)"
- (is "_ \<longleftrightarrow> (\<forall>X\<in>_. distribution ?D X = finite_measure.\<mu>' (Pi\<^isub>M I ?M) X)")
+ distr M (\<Pi>\<^isub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i))"
proof -
- interpret M': prob_space "?M i" for i
- using rv by (rule distribution_prob_space)
- interpret P: finite_product_prob_space ?M I
- proof qed fact
+ let ?P = "\<Pi>\<^isub>M i\<in>I. M' i"
+ let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"
+ let ?D = "distr M ?P ?X"
+ have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
+ interpret D: prob_space ?D by (intro prob_space_distr X)
- let ?D' = "(Pi\<^isub>M I ?M) \<lparr> measure := ereal \<circ> distribution ?D \<rparr>"
- have "random_variable P.P ?D"
- using `finite I` rv by (intro random_variable_restrict) auto
- then interpret D: prob_space ?D'
- by (rule distribution_prob_space)
-
+ let ?D' = "\<lambda>i. distr M (M' i) (X i)"
+ let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)"
+ interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
+ interpret P: product_prob_space ?D' I ..
+
show ?thesis
- proof (intro iffI ballI)
+ proof
assume "indep_vars M' X I"
- fix A assume "A \<in> sets P.P"
- moreover
- have "D.prob A = P.prob A"
- proof (rule prob_space_unique_Int_stable)
- show "prob_space ?D'" by unfold_locales
- show "prob_space (Pi\<^isub>M I ?M)" by unfold_locales
- show "Int_stable P.G" using M'.Int
- by (intro Int_stable_product_algebra_generator) (simp add: Int_stable_def)
- show "space P.G \<in> sets P.G"
- using M'.top by (simp add: product_algebra_generator_def)
- show "space ?D' = space P.G" "sets ?D' = sets (sigma P.G)"
- by (simp_all add: product_algebra_def product_algebra_generator_def sets_sigma)
- show "space P.P = space P.G" "sets P.P = sets (sigma P.G)"
- by (simp_all add: product_algebra_def)
- show "A \<in> sets (sigma P.G)"
- using `A \<in> sets P.P` by (simp add: product_algebra_def)
+ show "?D = ?P'"
+ proof (rule measure_eqI_generator_eq)
+ show "Int_stable (prod_algebra I M')"
+ by (rule Int_stable_prod_algebra)
+ show "prod_algebra I M' \<subseteq> Pow (space ?P)"
+ using prod_algebra_sets_into_space by (simp add: space_PiM)
+ show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
+ by (simp add: sets_PiM space_PiM)
+ show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
+ by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
+ let ?A = "\<lambda>i. \<Pi>\<^isub>E i\<in>I. space (M' i)"
+ show "range ?A \<subseteq> prod_algebra I M'" "incseq ?A" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')"
+ by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
+ { fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }
+ next
+ fix E assume E: "E \<in> prod_algebra I M'"
+ from prod_algebraE[OF E] guess J Y . note J = this
- fix E assume E: "E \<in> sets P.G"
- then have "E \<in> sets P.P"
- by (simp add: sets_sigma sigma_sets.Basic product_algebra_def)
- then have "D.prob E = distribution ?D E"
- unfolding D.\<mu>'_def by simp
- also
- from E obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (M' i)"
- by (auto simp: product_algebra_generator_def)
- with `I \<noteq> {}` have "distribution ?D E = prob (\<Inter>i\<in>I. X i -` F i \<inter> space M)"
- using `I \<noteq> {}` by (auto intro!: arg_cong[where f=prob] simp: Pi_iff distribution_def)
- also have "\<dots> = (\<Prod>i\<in>I. prob (X i -` F i \<inter> space M))"
- using `indep_vars M' X I` I F by (rule indep_varsD)
- also have "\<dots> = P.prob E"
- using F by (simp add: `E = Pi\<^isub>E I F` P.prob_times M'.\<mu>'_def distribution_def)
- finally show "D.prob E = P.prob E" .
+ from E have "E \<in> sets ?P" by (auto simp: sets_PiM)
+ then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
+ by (simp add: emeasure_distr X)
+ also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
+ using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm)
+ also have "emeasure M (\<Inter>i\<in>J. X i -` Y i \<inter> space M) = (\<Prod> i\<in>J. emeasure M (X i -` Y i \<inter> space M))"
+ using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]
+ by (auto simp: emeasure_eq_measure setprod_ereal)
+ also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
+ using rv J by (simp add: emeasure_distr)
+ also have "\<dots> = emeasure ?P' E"
+ using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
+ finally show "emeasure ?D E = emeasure ?P' E" .
qed
- ultimately show "distribution ?D A = P.prob A"
- by (simp add: D.\<mu>'_def)
next
- assume eq: "\<forall>A\<in>sets P.P. distribution ?D A = P.prob A"
- have [simp]: "\<And>i. sigma (M' i) = M' i"
- using rv by (intro sigma_algebra.sigma_eq) simp
- have "indep_vars (\<lambda>i. sigma (M' i)) X I"
- proof (subst indep_vars_finite[OF I])
- fix i assume [simp]: "i \<in> I"
- show "random_variable (sigma (M' i)) (X i)"
- using rv[of i] by simp
- show "Int_stable (M' i)" "space (M' i) \<in> sets (M' i)"
- using M'.Int[of _ i] M'.top by (auto simp: Int_stable_def)
+ assume "?D = ?P'"
+ show "indep_vars M' X I" unfolding indep_vars_def
+ proof (intro conjI indep_setsI ballI rv)
+ fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
+ by (auto intro!: sigma_sets_subset measurable_sets rv)
next
- show "\<forall>A\<in>\<Pi> i\<in>I. sets (M' i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M))"
+ fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
+ assume Y': "\<forall>j\<in>J. Y' j \<in> sigma_sets (space M) {X j -` A \<inter> space M |A. A \<in> sets (M' j)}"
+ have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
proof
- fix A assume A: "A \<in> (\<Pi> i\<in>I. sets (M' i))"
- then have A_in_P: "(Pi\<^isub>E I A) \<in> sets P.P"
- by (auto intro!: product_algebraI)
- have "prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = distribution ?D (Pi\<^isub>E I A)"
- using `I \<noteq> {}`by (auto intro!: arg_cong[where f=prob] simp: Pi_iff distribution_def)
- also have "\<dots> = P.prob (Pi\<^isub>E I A)" using A_in_P eq by simp
- also have "\<dots> = (\<Prod>i\<in>I. M'.prob i (A i))"
- using A by (intro P.prob_times) auto
- also have "\<dots> = (\<Prod>i\<in>I. prob (X i -` A i \<inter> space M))"
- using A by (auto intro!: setprod_cong simp: M'.\<mu>'_def Pi_iff distribution_def)
- finally show "prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M))" .
+ fix j assume "j \<in> J"
+ from Y'[rule_format, OF this] rv[of j]
+ show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
+ by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
+ (auto dest: measurable_space simp: sigma_sets_eq)
qed
+ from bchoice[OF this] obtain Y where
+ Y: "\<And>j. j \<in> J \<Longrightarrow> Y' j = X j -` Y j \<inter> space M" "\<And>j. j \<in> J \<Longrightarrow> Y j \<in> sets (M' j)" by auto
+ let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)"
+ from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
+ using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm)
+ then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
+ by simp
+ also have "\<dots> = emeasure ?D ?E"
+ using Y J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
+ also have "\<dots> = emeasure ?P' ?E"
+ using `?D = ?P'` by simp
+ also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
+ using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
+ also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
+ using rv J Y by (simp add: emeasure_distr)
+ finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .
+ then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"
+ by (auto simp: emeasure_eq_measure setprod_ereal)
qed
- then show "indep_vars M' X I"
- by simp
qed
qed
@@ -936,56 +960,188 @@
unfolding UNIV_bool by auto
qed
-lemma (in prob_space) indep_var_distributionD:
- assumes indep: "indep_var S X T Y"
- defines "P \<equiv> S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- assumes "A \<in> sets P"
- shows "joint_distribution X Y A = finite_measure.\<mu>' P A"
-proof -
- from indep have rvs: "random_variable S X" "random_variable T Y"
- by (blast dest: indep_var_rv1 indep_var_rv2)+
+lemma measurable_bool_case[simp, intro]:
+ "(\<lambda>(x, y). bool_case x y) \<in> measurable (M \<Otimes>\<^isub>M N) (Pi\<^isub>M UNIV (bool_case M N))"
+ (is "?f \<in> measurable ?B ?P")
+proof (rule measurable_PiM_single)
+ show "?f \<in> space ?B \<rightarrow> (\<Pi>\<^isub>E i\<in>UNIV. space (bool_case M N i))"
+ by (auto simp: space_pair_measure extensional_def split: bool.split)
+ fix i A assume "A \<in> sets (case i of True \<Rightarrow> M | False \<Rightarrow> N)"
+ moreover then have "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A}
+ = (case i of True \<Rightarrow> A \<times> space N | False \<Rightarrow> space M \<times> A)"
+ by (auto simp: space_pair_measure split: bool.split dest!: sets_into_space)
+ ultimately show "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A} \<in> sets ?B"
+ by (auto split: bool.split)
+qed
+
+lemma borel_measurable_indicator':
+ "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M"
+ using measurable_comp[OF _ borel_measurable_indicator, of f M N A] by (auto simp add: comp_def)
+
+lemma (in product_sigma_finite) distr_component:
+ "distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P")
+proof (intro measure_eqI[symmetric])
+ interpret I: finite_product_sigma_finite M "{i}" by default simp
+
+ have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
+ by (auto simp: extensional_def restrict_def)
+
+ fix A assume A: "A \<in> sets ?P"
+ then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)"
+ by simp
+ also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) x \<partial>M i)"
+ apply (subst product_positive_integral_singleton[symmetric])
+ apply (force intro!: measurable_restrict measurable_sets A)
+ apply (auto intro!: positive_integral_cong simp: space_PiM indicator_def simp: eq)
+ done
+ also have "\<dots> = emeasure (M i) ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i))"
+ by (force intro!: measurable_restrict measurable_sets A positive_integral_indicator)
+ also have "\<dots> = emeasure ?D A"
+ using A by (auto intro!: emeasure_distr[symmetric] measurable_restrict)
+ finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" .
+qed simp
- let ?S = "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
- let ?T = "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- interpret X: prob_space ?S by (rule distribution_prob_space) fact
- interpret Y: prob_space ?T by (rule distribution_prob_space) fact
- interpret XY: pair_prob_space ?S ?T by default
+lemma pair_measure_eqI:
+ assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
+ assumes sets: "sets (M1 \<Otimes>\<^isub>M M2) = sets M"
+ assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)"
+ shows "M1 \<Otimes>\<^isub>M M2 = M"
+proof -
+ interpret M1: sigma_finite_measure M1 by fact
+ interpret M2: sigma_finite_measure M2 by fact
+ interpret pair_sigma_finite M1 M2 by default
+ from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
+ let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
+ let ?P = "M1 \<Otimes>\<^isub>M M2"
+ show ?thesis
+ proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
+ show "?E \<subseteq> Pow (space ?P)"
+ using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
+ show "sets ?P = sigma_sets (space ?P) ?E"
+ by (simp add: sets_pair_measure space_pair_measure)
+ then show "sets M = sigma_sets (space ?P) ?E"
+ using sets[symmetric] by simp
+ next
+ show "range F \<subseteq> ?E" "incseq F" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
+ using F by (auto simp: space_pair_measure)
+ next
+ fix X assume "X \<in> ?E"
+ then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto
+ then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
+ by (simp add: emeasure_pair_measure_Times)
+ also have "\<dots> = emeasure M (A \<times> B)"
+ using A B emeasure by auto
+ finally show "emeasure ?P X = emeasure M X"
+ by simp
+ qed
+qed
- let ?J = "XY.P\<lparr> measure := ereal \<circ> joint_distribution X Y \<rparr>"
- interpret J: prob_space ?J
- by (rule joint_distribution_prob_space) (simp_all add: rvs)
+lemma pair_measure_eq_distr_PiM:
+ fixes M1 :: "'a measure" and M2 :: "'a measure"
+ assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
+ shows "(M1 \<Otimes>\<^isub>M M2) = distr (Pi\<^isub>M UNIV (bool_case M1 M2)) (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. (x True, x False))"
+ (is "?P = ?D")
+proof (rule pair_measure_eqI[OF assms])
+ interpret B: product_sigma_finite "bool_case M1 M2"
+ unfolding product_sigma_finite_def using assms by (auto split: bool.split)
+ let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)"
- have "finite_measure.\<mu>' (XY.P\<lparr> measure := ereal \<circ> joint_distribution X Y \<rparr>) A = XY.\<mu>' A"
- proof (rule prob_space_unique_Int_stable)
- show "Int_stable (pair_measure_generator ?S ?T)" (is "Int_stable ?P")
- by fact
- show "space ?P \<in> sets ?P"
- unfolding space_pair_measure[simplified pair_measure_def space_sigma]
- using X.top Y.top by (auto intro!: pair_measure_generatorI)
+ have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)"
+ by auto
+ fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
+ have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))"
+ by (simp add: UNIV_bool ac_simps)
+ also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))"
+ using A B by (subst B.emeasure_PiM) (auto split: bool.split)
+ also have "Pi\<^isub>E UNIV (bool_case A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
+ using A[THEN sets_into_space] B[THEN sets_into_space]
+ by (auto simp: Pi_iff all_bool_eq space_PiM split: bool.split)
+ finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
+ using A B
+ measurable_component_singleton[of True UNIV "bool_case M1 M2"]
+ measurable_component_singleton[of False UNIV "bool_case M1 M2"]
+ by (subst emeasure_distr) (auto simp: measurable_pair_iff)
+qed simp
- show "prob_space ?J" by unfold_locales
- show "space ?J = space ?P"
- by (simp add: pair_measure_generator_def space_pair_measure)
- show "sets ?J = sets (sigma ?P)"
- by (simp add: pair_measure_def)
+lemma measurable_Pair:
+ assumes rvs: "X \<in> measurable M S" "Y \<in> measurable M T"
+ shows "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
+proof -
+ have [simp]: "fst \<circ> (\<lambda>x. (X x, Y x)) = (\<lambda>x. X x)" "snd \<circ> (\<lambda>x. (X x, Y x)) = (\<lambda>x. Y x)"
+ by auto
+ show " (\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
+ by (auto simp: measurable_pair_iff rvs)
+qed
+
+lemma (in prob_space) indep_var_distribution_eq:
+ "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>
+ distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^isub>M ?T = ?J")
+proof safe
+ assume "indep_var S X T Y"
+ then show rvs: "random_variable S X" "random_variable T Y"
+ by (blast dest: indep_var_rv1 indep_var_rv2)+
+ then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+ by (rule measurable_Pair)
+
+ interpret X: prob_space ?S by (rule prob_space_distr) fact
+ interpret Y: prob_space ?T by (rule prob_space_distr) fact
+ interpret XY: pair_prob_space ?S ?T ..
+ show "?S \<Otimes>\<^isub>M ?T = ?J"
+ proof (rule pair_measure_eqI)
+ show "sigma_finite_measure ?S" ..
+ show "sigma_finite_measure ?T" ..
- show "prob_space XY.P" by unfold_locales
- show "space XY.P = space ?P" "sets XY.P = sets (sigma ?P)"
- by (simp_all add: pair_measure_generator_def pair_measure_def)
+ fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"
+ have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
+ using A B by (intro emeasure_distr[OF XY]) auto
+ also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
+ using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)
+ also have "\<dots> = emeasure ?S A * emeasure ?T B"
+ using rvs A B by (simp add: emeasure_distr)
+ finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
+ qed simp
+next
+ assume rvs: "random_variable S X" "random_variable T Y"
+ then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+ by (rule measurable_Pair)
- show "A \<in> sets (sigma ?P)"
- using `A \<in> sets P` unfolding P_def pair_measure_def by simp
+ let ?S = "distr M S X" and ?T = "distr M T Y"
+ interpret X: prob_space ?S by (rule prob_space_distr) fact
+ interpret Y: prob_space ?T by (rule prob_space_distr) fact
+ interpret XY: pair_prob_space ?S ?T ..
+
+ assume "?S \<Otimes>\<^isub>M ?T = ?J"
- fix X assume "X \<in> sets ?P"
- then obtain A B where "A \<in> sets S" "B \<in> sets T" "X = A \<times> B"
- by (auto simp: sets_pair_measure_generator)
- then show "J.\<mu>' X = XY.\<mu>' X"
- unfolding J.\<mu>'_def XY.\<mu>'_def using indep
- by (simp add: XY.pair_measure_times)
- (simp add: distribution_def indep_varD)
+ { fix S and X
+ have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"
+ proof (safe intro!: Int_stableI)
+ fix A B assume "A \<in> sets S" "B \<in> sets S"
+ then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
+ by (intro exI[of _ "A \<inter> B"]) auto
+ qed }
+ note Int_stable = this
+
+ show "indep_var S X T Y" unfolding indep_var_eq
+ proof (intro conjI indep_set_sigma_sets Int_stable rvs)
+ show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
+ proof (safe intro!: indep_setI)
+ { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
+ using `X \<in> measurable M S` by (auto intro: measurable_sets) }
+ { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
+ using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
+ next
+ fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
+ then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)"
+ using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"])
+ also have "\<dots> = emeasure (?S \<Otimes>\<^isub>M ?T) (A \<times> B)"
+ unfolding `?S \<Otimes>\<^isub>M ?T = ?J` ..
+ also have "\<dots> = emeasure ?S A * emeasure ?T B"
+ using ab by (simp add: XY.emeasure_pair_measure_Times)
+ finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
+ prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
+ using rvs ab by (simp add: emeasure_eq_measure emeasure_distr)
+ qed
qed
- then show ?thesis
- using `A \<in> sets P` unfolding P_def J.\<mu>'_def XY.\<mu>'_def by simp
qed
end
--- a/src/HOL/Probability/Infinite_Product_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -5,9 +5,49 @@
header {*Infinite Product Measure*}
theory Infinite_Product_Measure
- imports Probability_Measure
+ imports Probability_Measure Caratheodory
begin
+lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
+proof
+ fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
+ by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros)
+qed
+
+lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
+proof
+ fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
+ by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
+qed
+
+lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
+ by (auto intro: sigma_sets.Basic)
+
+lemma (in product_sigma_finite)
+ assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
+ shows emeasure_fold_integral:
+ "emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
+ and emeasure_fold_measurable:
+ "(\<lambda>x. emeasure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
+proof -
+ interpret I: finite_product_sigma_finite M I by default fact
+ interpret J: finite_product_sigma_finite M J by default fact
+ interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
+ have merge: "(\<lambda>(x, y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
+ by (intro measurable_sets[OF _ A] measurable_merge assms)
+
+ show ?I
+ apply (subst distr_merge[symmetric, OF IJ])
+ apply (subst emeasure_distr[OF measurable_merge[OF IJ(1)] A])
+ apply (subst IJ.emeasure_pair_measure_alt[OF merge])
+ apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
+ done
+
+ show ?B
+ using IJ.measurable_emeasure_Pair1[OF merge]
+ by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
+qed
+
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
unfolding restrict_def extensional_def by auto
@@ -41,189 +81,178 @@
qed
qed
-lemma (in product_prob_space) measure_preserving_restrict:
+lemma prod_algebraI_finite:
+ "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M"
+ using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp
+
+lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
+proof (safe intro!: Int_stableI)
+ fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
+ then show "\<exists>G. Pi\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
+ by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"])
+qed
+
+lemma prod_emb_trans[simp]:
+ "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
+ by (auto simp add: Int_absorb1 prod_emb_def)
+
+lemma prod_emb_Pi:
+ assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
+ shows "prod_emb K M J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))"
+ using assms space_closed
+ by (auto simp: prod_emb_def Pi_iff split: split_if_asm) blast+
+
+lemma prod_emb_id:
+ "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
+ by (auto simp: prod_emb_def Pi_iff subset_eq extensional_restrict)
+
+lemma measurable_prod_emb[intro, simp]:
+ "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^isub>M L M)"
+ unfolding prod_emb_def space_PiM[symmetric]
+ by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
+
+lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)"
+ by (intro measurable_restrict measurable_component_singleton) auto
+
+lemma (in product_prob_space) distr_restrict:
assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
- shows "(\<lambda>f. restrict f J) \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" (is "?R \<in> _")
-proof -
- interpret K: finite_product_prob_space M K by default fact
- have J: "J \<noteq> {}" "finite J" using assms by (auto simp add: finite_subset)
- interpret J: finite_product_prob_space M J
- by default (insert J, auto)
- from J.sigma_finite_pairs guess F .. note F = this
- then have [simp,intro]: "\<And>k i. k \<in> J \<Longrightarrow> F k i \<in> sets (M k)"
- by auto
- let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. F k i"
- let ?J = "product_algebra_generator J M \<lparr> measure := measure (Pi\<^isub>M J M) \<rparr>"
- have "?R \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (sigma ?J)"
- proof (rule K.measure_preserving_Int_stable)
- show "Int_stable ?J"
- by (auto simp: Int_stable_def product_algebra_generator_def PiE_Int)
- show "range ?F \<subseteq> sets ?J" "incseq ?F" "(\<Union>i. ?F i) = space ?J"
- using F by auto
- show "\<And>i. measure ?J (?F i) \<noteq> \<infinity>"
- using F by (simp add: J.measure_times setprod_PInf)
- have "measure_space (Pi\<^isub>M J M)" by default
- then show "measure_space (sigma ?J)"
- by (simp add: product_algebra_def sigma_def)
- show "?R \<in> measure_preserving (Pi\<^isub>M K M) ?J"
- proof (simp add: measure_preserving_def measurable_def product_algebra_generator_def del: vimage_Int,
- safe intro!: restrict_extensional)
- fix x k assume "k \<in> J" "x \<in> (\<Pi> i\<in>K. space (M i))"
- then show "x k \<in> space (M k)" using `J \<subseteq> K` by auto
- next
- fix E assume "E \<in> (\<Pi> i\<in>J. sets (M i))"
- then have E: "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" by auto
- then have *: "?R -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i)) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
- (is "?X = Pi\<^isub>E K ?M")
- using `J \<subseteq> K` sets_into_space by (auto simp: Pi_iff split: split_if_asm) blast+
- with E show "?X \<in> sets (Pi\<^isub>M K M)"
- by (auto intro!: product_algebra_generatorI)
- have "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = (\<Prod>i\<in>J. measure (M i) (?M i))"
- using E by (simp add: J.measure_times)
- also have "\<dots> = measure (Pi\<^isub>M K M) ?X"
- unfolding * using E `finite K` `J \<subseteq> K`
- by (auto simp: K.measure_times M.measure_space_1
- cong del: setprod_cong
- intro!: setprod_mono_one_left)
- finally show "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = measure (Pi\<^isub>M K M) ?X" .
- qed
- qed
- then show ?thesis
- by (simp add: product_algebra_def sigma_def)
+ shows "(\<Pi>\<^isub>M i\<in>J. M i) = distr (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D")
+proof (rule measure_eqI_generator_eq)
+ have "finite J" using `J \<subseteq> K` `finite K` by (auto simp add: finite_subset)
+ interpret J: finite_product_prob_space M J proof qed fact
+ interpret K: finite_product_prob_space M K proof qed fact
+
+ let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
+ let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
+ let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
+ show "Int_stable ?J"
+ by (rule Int_stable_PiE)
+ show "range ?F \<subseteq> ?J" "incseq ?F" "(\<Union>i. ?F i) = ?\<Omega>"
+ using `finite J` by (auto intro!: prod_algebraI_finite)
+ { fix i show "emeasure ?P (?F i) \<noteq> \<infinity>" by simp }
+ show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
+ show "sets (\<Pi>\<^isub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J"
+ using `finite J` by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
+
+ fix X assume "X \<in> ?J"
+ then obtain E where [simp]: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
+ with `finite J` have X: "X \<in> sets (Pi\<^isub>M J M)" by auto
+
+ have "emeasure ?P X = (\<Prod> i\<in>J. emeasure (M i) (E i))"
+ using E by (simp add: J.measure_times)
+ also have "\<dots> = (\<Prod> i\<in>J. emeasure (M i) (if i \<in> J then E i else space (M i)))"
+ by simp
+ also have "\<dots> = (\<Prod> i\<in>K. emeasure (M i) (if i \<in> J then E i else space (M i)))"
+ using `finite K` `J \<subseteq> K`
+ by (intro setprod_mono_one_left) (auto simp: M.emeasure_space_1)
+ also have "\<dots> = emeasure (Pi\<^isub>M K M) (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
+ using E by (simp add: K.measure_times)
+ also have "(\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i))"
+ using `J \<subseteq> K` sets_into_space E by (force simp: Pi_iff split: split_if_asm)
+ finally show "emeasure (Pi\<^isub>M J M) X = emeasure ?D X"
+ using X `J \<subseteq> K` apply (subst emeasure_distr)
+ by (auto intro!: measurable_restrict_subset simp: space_PiM)
qed
-lemma (in product_prob_space) measurable_restrict:
- assumes *: "J \<noteq> {}" "J \<subseteq> K" "finite K"
- shows "(\<lambda>f. restrict f J) \<in> measurable (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)"
- using measure_preserving_restrict[OF *]
- by (rule measure_preservingD2)
+abbreviation (in product_prob_space)
+ "emb L K X \<equiv> prod_emb L M K X"
+
+lemma (in product_prob_space) emeasure_prod_emb[simp]:
+ assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)"
+ shows "emeasure (Pi\<^isub>M L M) (emb L J X) = emeasure (Pi\<^isub>M J M) X"
+ by (subst distr_restrict[OF L])
+ (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)
-definition (in product_prob_space)
- "emb J K X = (\<lambda>x. restrict x K) -` X \<inter> space (Pi\<^isub>M J M)"
+lemma (in product_prob_space) prod_emb_injective:
+ assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
+ assumes "prod_emb L M J X = prod_emb L M J Y"
+ shows "X = Y"
+proof (rule injective_vimage_restrict)
+ show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
+ using sets[THEN sets_into_space] by (auto simp: space_PiM)
+ have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
+ using M.not_empty by auto
+ from bchoice[OF this]
+ show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto
+ show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
+ using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
+qed fact
-lemma (in product_prob_space) emb_trans[simp]:
- "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> emb L K (emb K J X) = emb L J X"
- by (auto simp add: Int_absorb1 emb_def)
-
-lemma (in product_prob_space) emb_empty[simp]:
- "emb K J {} = {}"
- by (simp add: emb_def)
+definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) set set" where
+ "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"
-lemma (in product_prob_space) emb_Pi:
- assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
- shows "emb K J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))"
- using assms space_closed
- by (auto simp: emb_def Pi_iff split: split_if_asm) blast+
+lemma (in product_prob_space) generatorI':
+ "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> generator"
+ unfolding generator_def by auto
-lemma (in product_prob_space) emb_injective:
- assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
- assumes "emb L J X = emb L J Y"
- shows "X = Y"
-proof -
- interpret J: finite_product_sigma_finite M J by default fact
- show "X = Y"
- proof (rule injective_vimage_restrict)
- show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
- using J.sets_into_space sets by auto
- have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
- using M.not_empty by auto
- from bchoice[OF this]
- show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto
- show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
- using `emb L J X = emb L J Y` by (simp add: emb_def)
- qed fact
+lemma (in product_prob_space) algebra_generator:
+ assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
+proof
+ let ?G = generator
+ show "?G \<subseteq> Pow ?\<Omega>"
+ by (auto simp: generator_def prod_emb_def)
+ from `I \<noteq> {}` obtain i where "i \<in> I" by auto
+ then show "{} \<in> ?G"
+ by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
+ simp: sigma_sets.Empty generator_def prod_emb_def)
+ from `i \<in> I` show "?\<Omega> \<in> ?G"
+ by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
+ simp: generator_def prod_emb_def)
+ fix A assume "A \<in> ?G"
+ then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
+ by (auto simp: generator_def)
+ fix B assume "B \<in> ?G"
+ then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
+ by (auto simp: generator_def)
+ let ?RA = "emb (JA \<union> JB) JA XA"
+ let ?RB = "emb (JA \<union> JB) JB XB"
+ have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
+ using XA A XB B by auto
+ show "A - B \<in> ?G" "A \<union> B \<in> ?G"
+ unfolding * using XA XB by (safe intro!: generatorI') auto
qed
-lemma (in product_prob_space) emb_id:
- "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> emb L L B = B"
- by (auto simp: emb_def Pi_iff subset_eq extensional_restrict)
-
-lemma (in product_prob_space) emb_simps:
- shows "emb L K (A \<union> B) = emb L K A \<union> emb L K B"
- and "emb L K (A \<inter> B) = emb L K A \<inter> emb L K B"
- and "emb L K (A - B) = emb L K A - emb L K B"
- by (auto simp: emb_def)
-
-lemma (in product_prob_space) measurable_emb[intro,simp]:
- assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
- shows "emb L J X \<in> sets (Pi\<^isub>M L M)"
- using measurable_restrict[THEN measurable_sets, OF *] by (simp add: emb_def)
-
-lemma (in product_prob_space) measure_emb[intro,simp]:
- assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
- shows "measure (Pi\<^isub>M L M) (emb L J X) = measure (Pi\<^isub>M J M) X"
- using measure_preserving_restrict[THEN measure_preservingD, OF *]
- by (simp add: emb_def)
-
-definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) measure_space" where
- "generator = \<lparr>
- space = (\<Pi>\<^isub>E i\<in>I. space (M i)),
- sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)),
- measure = undefined
- \<rparr>"
+lemma (in product_prob_space) sets_PiM_generator:
+ assumes "I \<noteq> {}" shows "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
+proof
+ show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
+ unfolding sets_PiM
+ proof (safe intro!: sigma_sets_subseteq)
+ fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator"
+ by (auto intro!: generatorI' elim!: prod_algebraE)
+ qed
+qed (auto simp: generator_def space_PiM[symmetric] intro!: sigma_sets_subset)
lemma (in product_prob_space) generatorI:
- "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> sets generator"
- unfolding generator_def by auto
-
-lemma (in product_prob_space) generatorI':
- "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> sets generator"
+ "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> generator"
unfolding generator_def by auto
-lemma (in product_sigma_finite)
- assumes "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
- shows measure_fold_integral:
- "measure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. measure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
- and measure_fold_measurable:
- "(\<lambda>x. measure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
-proof -
- interpret I: finite_product_sigma_finite M I by default fact
- interpret J: finite_product_sigma_finite M J by default fact
- interpret IJ: pair_sigma_finite I.P J.P ..
- show ?I
- unfolding measure_fold[OF assms]
- apply (subst IJ.pair_measure_alt)
- apply (intro measurable_sets[OF _ A] measurable_merge assms)
- apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure
- intro!: I.positive_integral_cong)
- done
-
- have "(\<lambda>(x, y). merge I x J y) -` A \<inter> space (I.P \<Otimes>\<^isub>M J.P) \<in> sets (I.P \<Otimes>\<^isub>M J.P)"
- by (intro measurable_sets[OF _ A] measurable_merge assms)
- from IJ.measure_cut_measurable_fst[OF this]
- show ?B
- apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure)
- apply (subst (asm) measurable_cong)
- apply auto
- done
-qed
-
definition (in product_prob_space)
"\<mu>G A =
- (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = measure (Pi\<^isub>M J M) X))"
+ (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (Pi\<^isub>M J M) X))"
lemma (in product_prob_space) \<mu>G_spec:
assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
- shows "\<mu>G A = measure (Pi\<^isub>M J M) X"
+ shows "\<mu>G A = emeasure (Pi\<^isub>M J M) X"
unfolding \<mu>G_def
proof (intro the_equality allI impI ballI)
fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)"
- have "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
+ have "emeasure (Pi\<^isub>M K M) Y = emeasure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
using K J by simp
also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
- using K J by (simp add: emb_injective[of "K \<union> J" I])
- also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X"
+ using K J by (simp add: prod_emb_injective[of "K \<union> J" I])
+ also have "emeasure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = emeasure (Pi\<^isub>M J M) X"
using K J by simp
- finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" ..
+ finally show "emeasure (Pi\<^isub>M J M) X = emeasure (Pi\<^isub>M K M) Y" ..
qed (insert J, force)
lemma (in product_prob_space) \<mu>G_eq:
- "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = measure (Pi\<^isub>M J M) X"
+ "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (Pi\<^isub>M J M) X"
by (intro \<mu>G_spec) auto
lemma (in product_prob_space) generator_Ex:
- assumes *: "A \<in> sets generator"
- shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = measure (Pi\<^isub>M J M) X"
+ assumes *: "A \<in> generator"
+ shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (Pi\<^isub>M J M) X"
proof -
from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
unfolding generator_def by auto
@@ -231,11 +260,11 @@
qed
lemma (in product_prob_space) generatorE:
- assumes A: "A \<in> sets generator"
- obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = measure (Pi\<^isub>M J M) X"
+ assumes A: "A \<in> generator"
+ obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (Pi\<^isub>M J M) X"
proof -
from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A"
- "\<mu>G A = measure (Pi\<^isub>M J M) X" by auto
+ "\<mu>G A = emeasure (Pi\<^isub>M J M) X" by auto
then show thesis by (intro that) auto
qed
@@ -243,11 +272,7 @@
assumes "finite J" "finite K" "J \<inter> K = {}" and A: "A \<in> sets (Pi\<^isub>M (J \<union> K) M)" and x: "x \<in> space (Pi\<^isub>M J M)"
shows "merge J x K -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
proof -
- interpret J: finite_product_sigma_algebra M J by default fact
- interpret K: finite_product_sigma_algebra M K by default fact
- interpret JK: pair_sigma_algebra J.P K.P ..
-
- from JK.measurable_cut_fst[OF
+ from sets_Pair1[OF
measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x
show ?thesis
by (simp add: space_pair_measure comp_def vimage_compose[symmetric])
@@ -266,75 +291,27 @@
have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
have [simp]: "(K - J) \<inter> K = K - J" by auto
from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
- by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto
-qed
-
-definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where
- "infprod_algebra = sigma generator \<lparr> measure :=
- (SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
- prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>"
-
-syntax
- "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3PIP _:_./ _)" 10)
-
-syntax (xsymbols)
- "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10)
-
-syntax (HTML output)
- "_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10)
-
-abbreviation
- "Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I"
-
-translations
- "PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)"
-
-lemma (in product_prob_space) algebra_generator:
- assumes "I \<noteq> {}" shows "algebra generator"
-proof
- let ?G = generator
- show "sets ?G \<subseteq> Pow (space ?G)"
- by (auto simp: generator_def emb_def)
- from `I \<noteq> {}` obtain i where "i \<in> I" by auto
- then show "{} \<in> sets ?G"
- by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
- simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def)
- from `i \<in> I` show "space ?G \<in> sets ?G"
- by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
- simp: generator_def emb_def)
- fix A assume "A \<in> sets ?G"
- then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
- by (auto simp: generator_def)
- fix B assume "B \<in> sets ?G"
- then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
- by (auto simp: generator_def)
- let ?RA = "emb (JA \<union> JB) JA XA"
- let ?RB = "emb (JA \<union> JB) JB XB"
- interpret JAB: finite_product_sigma_algebra M "JA \<union> JB"
- by default (insert XA XB, auto)
- have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
- using XA A XB B by (auto simp: emb_simps)
- then show "A - B \<in> sets ?G" "A \<union> B \<in> sets ?G"
- using XA XB by (auto intro!: generatorI')
+ by (simp split: split_merge add: prod_emb_def Pi_iff extensional_merge_sub set_eq_iff space_PiM)
+ auto
qed
lemma (in product_prob_space) positive_\<mu>G:
assumes "I \<noteq> {}"
shows "positive generator \<mu>G"
proof -
- interpret G!: algebra generator by (rule algebra_generator) fact
+ interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
show ?thesis
proof (intro positive_def[THEN iffD2] conjI ballI)
from generatorE[OF G.empty_sets] guess J X . note this[simp]
interpret J: finite_product_sigma_finite M J by default fact
have "X = {}"
- by (rule emb_injective[of J I]) simp_all
+ by (rule prod_emb_injective[of J I]) simp_all
then show "\<mu>G {} = 0" by simp
next
- fix A assume "A \<in> sets generator"
+ fix A assume "A \<in> generator"
from generatorE[OF this] guess J X . note this[simp]
interpret J: finite_product_sigma_finite M J by default fact
- show "0 \<le> \<mu>G A" by simp
+ show "0 \<le> \<mu>G A" by (simp add: emeasure_nonneg)
qed
qed
@@ -342,102 +319,47 @@
assumes "I \<noteq> {}"
shows "additive generator \<mu>G"
proof -
- interpret G!: algebra generator by (rule algebra_generator) fact
+ interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
show ?thesis
proof (intro additive_def[THEN iffD2] ballI impI)
- fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this
- fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this
+ fix A assume "A \<in> generator" with generatorE guess J X . note J = this
+ fix B assume "B \<in> generator" with generatorE guess K Y . note K = this
assume "A \<inter> B = {}"
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
using J K by auto
interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
- apply (rule emb_injective[of "J \<union> K" I])
+ apply (rule prod_emb_injective[of "J \<union> K" I])
apply (insert `A \<inter> B = {}` JK J K)
- apply (simp_all add: JK.Int emb_simps)
+ apply (simp_all add: Int prod_emb_Int)
done
have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)"
using J K by simp_all
then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))"
- by (simp add: emb_simps)
- also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
- using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un)
+ by simp
+ also have "\<dots> = emeasure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
+ using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un)
also have "\<dots> = \<mu>G A + \<mu>G B"
- using J K JK_disj by (simp add: JK.measure_additive[symmetric])
+ using J K JK_disj by (simp add: plus_emeasure[symmetric])
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
qed
qed
-lemma (in product_prob_space) finite_index_eq_finite_product:
- assumes "finite I"
- shows "sets (sigma generator) = sets (Pi\<^isub>M I M)"
-proof safe
- interpret I: finite_product_sigma_algebra M I by default fact
- have space_generator[simp]: "space generator = space (Pi\<^isub>M I M)"
- by (simp add: generator_def product_algebra_def)
- { fix A assume "A \<in> sets (sigma generator)"
- then show "A \<in> sets I.P" unfolding sets_sigma
- proof induct
- case (Basic A)
- from generatorE[OF this] guess J X . note J = this
- with `finite I` have "emb I J X \<in> sets I.P" by auto
- with `emb I J X = A` show "A \<in> sets I.P" by simp
- qed auto }
- { fix A assume A: "A \<in> sets I.P"
- show "A \<in> sets (sigma generator)"
- proof cases
- assume "I = {}"
- with I.P_empty[OF this] A
- have "A = space generator \<or> A = {}"
- unfolding space_generator by auto
- then show ?thesis
- by (auto simp: sets_sigma simp del: space_generator
- intro: sigma_sets.Empty sigma_sets_top)
- next
- assume "I \<noteq> {}"
- note A this
- moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto
- ultimately show "A \<in> sets (sigma generator)"
- using `finite I` unfolding sets_sigma
- by (intro sigma_sets.Basic generatorI[of I A]) auto
- qed }
-qed
-
-lemma (in product_prob_space) extend_\<mu>G:
- "\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
- prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
+lemma (in product_prob_space) emeasure_PiM_emb_not_empty:
+ assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. X i \<in> sets (M i)"
+ shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)"
proof cases
- assume "finite I"
- interpret I: finite_product_prob_space M I by default fact
- show ?thesis
- proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI)
- fix A assume "A \<in> sets generator"
- from generatorE[OF this] guess J X . note J = this
- from J(1-4) `finite I` show "measure I.P A = \<mu>G A"
- unfolding J(6)
- by (subst J(5)[symmetric]) (simp add: measure_emb)
- next
- have [simp]: "space generator = space (Pi\<^isub>M I M)"
- by (simp add: generator_def product_algebra_def)
- have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr>
- = I.P" (is "?P = _")
- by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`])
- show "prob_space ?P"
- proof
- show "measure_space ?P" using `?P = I.P` by simp default
- show "measure ?P (space ?P) = 1"
- using I.measure_space_1 by simp
- qed
- qed
+ assume "finite I" with X show ?thesis by simp
next
+ let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space (M i)"
let ?G = generator
assume "\<not> finite I"
then have I_not_empty: "I \<noteq> {}" by auto
- interpret G!: algebra generator by (rule algebra_generator) fact
+ interpret G!: algebra ?\<Omega> generator by (rule algebra_generator) fact
note \<mu>G_mono =
G.additive_increasing[OF positive_\<mu>G[OF I_not_empty] additive_\<mu>G[OF I_not_empty], THEN increasingD]
- { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G"
+ { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> ?G"
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
by (metis rev_finite_subset subsetI)
@@ -445,7 +367,7 @@
moreover def K \<equiv> "insert k K'"
moreover def X \<equiv> "emb K K' X'"
ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
- "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X"
+ "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = emeasure (Pi\<^isub>M K M) X"
by (auto simp: subset_insertI)
let ?M = "\<lambda>y. merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
@@ -455,9 +377,9 @@
have **: "?M y \<in> sets (Pi\<^isub>M (K - J) M)"
using J K y by (intro merge_sets) auto
ultimately
- have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G"
+ have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> ?G"
using J K by (intro generatorI) auto
- have "\<mu>G (merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) = measure (Pi\<^isub>M (K - J) M) (?M y)"
+ have "\<mu>G (merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) = emeasure (Pi\<^isub>M (K - J) M) (?M y)"
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto
note * ** *** this }
note merge_in_G = this
@@ -467,16 +389,16 @@
interpret J: finite_product_prob_space M J by default fact+
interpret KmJ: finite_product_prob_space M "K - J" by default fact+
- have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
+ have "\<mu>G Z = emeasure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
using K J by simp
- also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
- using K J by (subst measure_fold_integral) auto
+ also have "\<dots> = (\<integral>\<^isup>+ x. emeasure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
+ using K J by (subst emeasure_fold_integral) auto
also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)"
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)")
- proof (intro J.positive_integral_cong)
+ proof (intro positive_integral_cong)
fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
with K merge_in_G(2)[OF this]
- show "measure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
+ show "emeasure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto
qed
finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" .
@@ -490,21 +412,18 @@
let ?q = "\<lambda>y. \<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M))"
have "?q \<in> borel_measurable (Pi\<^isub>M J M)"
unfolding `Z = emb I K X` using J K merge_in_G(3)
- by (simp add: merge_in_G \<mu>G_eq measure_fold_measurable
- del: space_product_algebra cong: measurable_cong)
+ by (simp add: merge_in_G \<mu>G_eq emeasure_fold_measurable cong: measurable_cong)
note this fold le_1 merge_in_G(3) }
note fold = this
- have "\<exists>\<mu>. (\<forall>s\<in>sets ?G. \<mu> s = \<mu>G s) \<and>
- measure_space \<lparr>space = space ?G, sets = sets (sigma ?G), measure = \<mu>\<rparr>"
- (is "\<exists>\<mu>. _ \<and> measure_space (?ms \<mu>)")
+ have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
- fix A assume "A \<in> sets ?G"
+ fix A assume "A \<in> ?G"
with generatorE guess J X . note JX = this
interpret JK: finite_product_prob_space M J by default fact+
from JX show "\<mu>G A \<noteq> \<infinity>" by simp
next
- fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}"
+ fix A assume A: "range A \<subseteq> ?G" "decseq A" "(\<Inter>i. A i) = {}"
then have "decseq (\<lambda>i. \<mu>G (A i))"
by (auto intro!: \<mu>G_mono simp: decseq_def)
moreover
@@ -515,7 +434,7 @@
using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def)
ultimately have "0 < ?a" by auto
- have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = measure (Pi\<^isub>M J M) X"
+ have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = emeasure (Pi\<^isub>M J M) X"
using A by (intro allI generator_Ex) auto
then obtain J' X' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" "\<And>n. X' n \<in> sets (Pi\<^isub>M (J' n) M)"
and A': "\<And>n. A n = emb I (J' n) (X' n)"
@@ -524,8 +443,8 @@
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. X n \<in> sets (Pi\<^isub>M (J n) M)"
by auto
- with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G"
- unfolding J_def X_def by (subst emb_trans) (insert A, auto)
+ with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> ?G"
+ unfolding J_def X_def by (subst prod_emb_trans) (insert A, auto)
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
unfolding J_def by force
@@ -538,8 +457,8 @@
let ?M = "\<lambda>K Z y. merge K y (I - K) -` Z \<inter> space (Pi\<^isub>M I M)"
- { fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
- then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto
+ { fix Z k assume Z: "range Z \<subseteq> ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
+ then have Z_sets: "\<And>n. Z n \<in> ?G" by auto
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
interpret J': finite_product_prob_space M J' by default fact+
@@ -552,13 +471,13 @@
by (rule measurable_sets) auto }
note Q_sets = this
- have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))"
+ have "?a / 2^(k+1) \<le> (INF n. emeasure (Pi\<^isub>M J' M) (?Q n))"
proof (intro INF_greatest)
fix n
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)"
- unfolding fold(2)[OF J' `Z n \<in> sets ?G`]
- proof (intro J'.positive_integral_mono)
+ unfolding fold(2)[OF J' `Z n \<in> ?G`]
+ proof (intro positive_integral_mono)
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
then have "?q n x \<le> 1 + 0"
using J' Z fold(3) Z_sets by auto
@@ -568,15 +487,15 @@
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
by (auto split: split_indicator simp del: power_Suc)
qed
- also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
- using `0 \<le> ?a` Q_sets J'.measure_space_1
- by (subst J'.positive_integral_add) auto
- finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
- by (cases rule: ereal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"])
+ also have "\<dots> = emeasure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
+ using `0 \<le> ?a` Q_sets J'.emeasure_space_1
+ by (subst positive_integral_add) auto
+ finally show "?a / 2^(k+1) \<le> emeasure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
+ by (cases rule: ereal2_cases[of ?a "emeasure (Pi\<^isub>M J' M) (?Q n)"])
(auto simp: field_simps)
qed
- also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
- proof (intro J'.continuity_from_above)
+ also have "\<dots> = emeasure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
+ proof (intro INF_emeasure_decseq)
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
show "decseq ?Q"
unfolding decseq_def
@@ -587,13 +506,13 @@
also have "?q n x \<le> ?q m x"
proof (rule \<mu>G_mono)
from fold(4)[OF J', OF Z_sets x]
- show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto
+ show "?M J' (Z n) x \<in> ?G" "?M J' (Z m) x \<in> ?G" by auto
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
qed
finally show "?a / 2^(k+1) \<le> ?q m x" .
qed
- qed (intro J'.finite_measure Q_sets)
+ qed simp
finally have "(\<Inter>n. ?Q n) \<noteq> {}"
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
@@ -631,12 +550,12 @@
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)"
using Suc by simp
then show "restrict (w k) (J k) = w k"
- by (simp add: extensional_restrict)
+ by (simp add: extensional_restrict space_PiM)
qed
next
assume "J k \<noteq> J (Suc k)"
with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
- have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G"
+ have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> ?G"
"decseq (\<lambda>n. ?M (J k) (A n) (w k))"
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))"
using `decseq A` fold(4)[OF J(1-3) A_eq(2), of "w k" k] Suc
@@ -651,11 +570,11 @@
by (auto intro!: ext split: split_merge)
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w"
using w'(1) J(3)[of "Suc k"]
- by (auto split: split_merge intro!: extensional_merge_sub) force+
+ by (auto simp: space_PiM split: split_merge intro!: extensional_merge_sub) force+
show ?thesis
apply (rule exI[of _ ?w])
using w' J_mono[of k "Suc k"] wk unfolding *
- apply (auto split: split_merge intro!: extensional_merge_sub ext)
+ apply (auto split: split_merge intro!: extensional_merge_sub ext simp: space_PiM)
apply (force simp: extensional_def)
done
qed
@@ -675,7 +594,7 @@
then have "merge (J k) (w k) (I - J k) x \<in> A k" by auto
then have "\<exists>x\<in>A k. restrict x (J k) = w k"
using `w k \<in> space (Pi\<^isub>M (J k) M)`
- by (intro rev_bexI) (auto intro!: ext simp: extensional_def)
+ by (intro rev_bexI) (auto intro!: ext simp: extensional_def space_PiM)
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)"
"\<exists>x\<in>A k. restrict x (J k) = w k"
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1)"
@@ -707,17 +626,17 @@
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
- using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ }
+ using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq space_PiM)+ }
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
have w': "w' \<in> space (Pi\<^isub>M I M)"
- using w(1) by (auto simp add: Pi_iff extensional_def)
+ using w(1) by (auto simp add: Pi_iff extensional_def space_PiM)
{ fix n
have "restrict w' (J n) = w n" using w(1)
- by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def)
+ by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def space_PiM)
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto
- then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) }
+ then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: prod_emb_def space_PiM) }
then have "w' \<in> (\<Inter>i. A i)" by auto
with `(\<Inter>i. A i) = {}` show False by auto
qed
@@ -726,276 +645,76 @@
qed fact+
then guess \<mu> .. note \<mu> = this
show ?thesis
- proof (intro exI[of _ \<mu>] conjI)
- show "\<forall>S\<in>sets ?G. \<mu> S = \<mu>G S" using \<mu> by simp
- show "prob_space (?ms \<mu>)"
- proof
- show "measure_space (?ms \<mu>)" using \<mu> by simp
- obtain i where "i \<in> I" using I_not_empty by auto
- interpret i: finite_product_sigma_finite M "{i}" by default auto
- let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)"
- have X: "?X \<in> sets (Pi\<^isub>M {i} M)"
- by auto
- with `i \<in> I` have "emb I {i} ?X \<in> sets generator"
- by (intro generatorI') auto
- with \<mu> have "\<mu> (emb I {i} ?X) = \<mu>G (emb I {i} ?X)" by auto
- with \<mu>G_eq[OF _ _ _ X] `i \<in> I`
- have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))"
- by (simp add: i.measure_times)
- also have "emb I {i} ?X = space (Pi\<^isub>P I M)"
- using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def)
- finally show "measure (?ms \<mu>) (space (?ms \<mu>)) = 1"
- using M.measure_space_1 by (simp add: infprod_algebra_def)
- qed
+ proof (subst emeasure_extend_measure_Pair[OF PiM_def, of I M \<mu> J X])
+ from assms show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))"
+ by (simp add: Pi_iff)
+ next
+ fix J X assume J: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))"
+ then show "emb I J (Pi\<^isub>E J X) \<in> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ by (auto simp: Pi_iff prod_emb_def dest: sets_into_space)
+ have "emb I J (Pi\<^isub>E J X) \<in> generator"
+ using J `I \<noteq> {}` by (intro generatorI') auto
+ then have "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))"
+ using \<mu> by simp
+ also have "\<dots> = (\<Prod> j\<in>J. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
+ using J `I \<noteq> {}` by (subst \<mu>G_spec[OF _ _ _ refl]) (auto simp: emeasure_PiM Pi_iff)
+ also have "\<dots> = (\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}.
+ if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
+ using J `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1)
+ finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = \<dots>" .
+ next
+ let ?F = "\<lambda>j. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j))"
+ have "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = (\<Prod>j\<in>J. ?F j)"
+ using X `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1)
+ then show "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) =
+ emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)"
+ using X by (auto simp add: emeasure_PiM)
+ next
+ show "positive (sets (Pi\<^isub>M I M)) \<mu>" "countably_additive (sets (Pi\<^isub>M I M)) \<mu>"
+ using \<mu> unfolding sets_PiM_generator[OF `I \<noteq> {}`] by (auto simp: measure_space_def)
qed
qed
-lemma (in product_prob_space) infprod_spec:
- "(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> prob_space (Pi\<^isub>P I M)"
- (is "?Q infprod_algebra")
- unfolding infprod_algebra_def
- by (rule someI2_ex[OF extend_\<mu>G])
- (auto simp: sigma_def generator_def)
-
-sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M"
- using infprod_spec by simp
-
-lemma (in product_prob_space) measure_infprod_emb:
- assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
- shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X"
-proof -
- have "emb I J X \<in> sets generator"
- using assms by (rule generatorI')
- with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto
-qed
-
-lemma (in product_prob_space) measurable_component:
- assumes "i \<in> I"
- shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
-proof (unfold measurable_def, safe)
- fix x assume "x \<in> space (Pi\<^isub>P I M)"
- then show "x i \<in> space (M i)"
- using `i \<in> I` by (auto simp: infprod_algebra_def generator_def)
-next
- fix A assume "A \<in> sets (M i)"
- with `i \<in> I` have
- "(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)"
- "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)"
- by (auto simp: infprod_algebra_def generator_def emb_def)
- from generatorI[OF _ _ _ this] `i \<in> I`
- show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)"
- unfolding infprod_algebra_def by auto
-qed
-
-lemma (in product_prob_space) emb_in_infprod_algebra[intro]:
- fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)"
- shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)"
-proof cases
- assume "J = {}"
- with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}"
- by (auto simp: emb_def infprod_algebra_def generator_def
- product_algebra_def product_algebra_generator_def image_constant sigma_def)
- then show ?thesis by auto
-next
- assume "J \<noteq> {}"
- show ?thesis unfolding infprod_algebra_def
- by simp (intro in_sigma generatorI' `J \<noteq> {}` J X)
-qed
-
-lemma (in product_prob_space) finite_measure_infprod_emb:
- assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
- shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X"
-proof -
- interpret J: finite_product_prob_space M J by default fact+
- from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto
- with assms show "\<mu>' (emb I J X) = J.\<mu>' X"
- unfolding \<mu>'_def J.\<mu>'_def
- unfolding measure_infprod_emb[OF assms]
- by auto
-qed
-
-lemma (in finite_product_prob_space) finite_measure_times:
- assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
- shows "\<mu>' (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu>' i (A i))"
- using assms
- unfolding \<mu>'_def M.\<mu>'_def
- by (subst measure_times[OF assms])
- (auto simp: finite_measure_eq M.finite_measure_eq setprod_ereal)
-
-lemma (in product_prob_space) finite_measure_infprod_emb_Pi:
- assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)"
- shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))"
-proof cases
- assume "J = {}"
- then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra"
- by (auto simp: infprod_algebra_def generator_def sigma_def emb_def)
- then show ?thesis using `J = {}` P.prob_space
- by simp
-next
- assume "J \<noteq> {}"
- interpret J: finite_product_prob_space M J by default fact+
- have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)"
- using J `J \<noteq> {}` by (subst J.finite_measure_times) auto
- also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))"
- using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto
- finally show ?thesis by simp
-qed
-
-lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
+sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>M I M"
proof
- fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
- by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros)
-qed
-
-lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
-proof
- fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
- by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
+ show "emeasure (Pi\<^isub>M I M) (space (Pi\<^isub>M I M)) = 1"
+ proof cases
+ assume "I = {}" then show ?thesis by (simp add: space_PiM_empty)
+ next
+ assume "I \<noteq> {}"
+ then obtain i where "i \<in> I" by auto
+ moreover then have "emb I {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i)) = (space (Pi\<^isub>M I M))"
+ by (auto simp: prod_emb_def space_PiM)
+ ultimately show ?thesis
+ using emeasure_PiM_emb_not_empty[of "{i}" "\<lambda>i. space (M i)"]
+ by (simp add: emeasure_PiM emeasure_space_1)
+ qed
qed
-lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
- by (auto intro: sigma_sets.Basic)
-
-lemma (in product_prob_space) infprod_algebra_alt:
- "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
- sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i))),
- measure = measure (Pi\<^isub>P I M) \<rparr>"
- (is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>")
-proof (rule measure_space.equality)
- let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)"
- have "sigma_sets ?O ?M = sigma_sets ?O ?G"
- proof (intro equalityI sigma_sets_mono UN_least)
- fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}"
- have "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> emb I J ` sets (Pi\<^isub>M J M)" by auto
- also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper)
- also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_superset_generator)
- finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" .
- have "emb I J ` sets (Pi\<^isub>M J M) = emb I J ` sigma_sets (space (Pi\<^isub>M J M)) (Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
- by (simp add: sets_sigma product_algebra_generator_def product_algebra_def)
- also have "\<dots> = sigma_sets (space (Pi\<^isub>M I M)) (emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
- using J M.sets_into_space
- by (auto simp: emb_def [abs_def] intro!: sigma_sets_vimage[symmetric]) blast
- also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M"
- using J by (intro sigma_sets_mono') auto
- finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M"
- by (simp add: infprod_algebra_def generator_def)
- qed
- then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)"
- by (simp_all add: infprod_algebra_def generator_def sets_sigma)
-qed simp_all
-
-lemma (in product_prob_space) infprod_algebra_alt2:
- "Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
- sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)),
- measure = measure (Pi\<^isub>P I M) \<rparr>"
- (is "_ = ?S")
-proof (rule measure_space.equality)
- let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S
- let ?G = "(\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
- have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G"
- by (subst infprod_algebra_alt) (simp add: sets_sigma)
- also have "\<dots> = sigma_sets ?O ?A"
- proof (intro equalityI sigma_sets_mono subsetI)
- interpret A: sigma_algebra ?S
- by (rule sigma_algebra_sigma) auto
- fix A assume "A \<in> ?G"
- then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)"
- and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)"
- by auto
- then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M))"
- by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
- { fix j assume "j\<in>J"
- with `J \<subseteq> I` have "j \<in> I" by auto
- with `j \<in> J` B have "(\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S"
- by (auto simp: sets_sigma intro: sigma_sets.Basic) }
- with `finite J` `J \<noteq> {}` have "A \<in> sets ?S"
- unfolding A by (intro A.finite_INT) auto
- then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma)
- next
- fix A assume "A \<in> ?A"
- then obtain i B where i: "i \<in> I" "B \<in> sets (M i)"
- and "A = (\<lambda>x. x i) -` B \<inter> space (Pi\<^isub>P I M)"
- by auto
- then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))"
- by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
- with i show "A \<in> sigma_sets ?O ?G"
- by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto
- qed
- also have "\<dots> = sets ?S"
- by (simp add: sets_sigma)
- finally show "sets (Pi\<^isub>P I M) = sets ?S" .
-qed simp_all
-
-lemma (in product_prob_space) measurable_into_infprod_algebra:
- assumes "sigma_algebra N"
- assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
- assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I"
- shows "f \<in> measurable N (Pi\<^isub>P I M)"
-proof -
- interpret N: sigma_algebra N by fact
- have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)"
- using f by (auto simp: measurable_def)
- { fix i A assume i: "i \<in> I" "A \<in> sets (M i)"
- then have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N = (\<lambda>x. f x i) -` A \<inter> space N"
- using f_in ext by (auto simp: infprod_algebra_def generator_def)
- also have "\<dots> \<in> sets N"
- by (rule measurable_sets f i)+
- finally have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N \<in> sets N" . }
- with f_in ext show ?thesis
- by (subst infprod_algebra_alt2)
- (auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def)
+lemma (in product_prob_space) emeasure_PiM_emb:
+ assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
+ shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. emeasure (M i) (X i))"
+proof cases
+ assume "J = {}"
+ moreover have "emb I {} {\<lambda>x. undefined} = space (Pi\<^isub>M I M)"
+ by (auto simp: space_PiM prod_emb_def)
+ ultimately show ?thesis
+ by (simp add: space_PiM_empty P.emeasure_space_1)
+next
+ assume "J \<noteq> {}" with X show ?thesis
+ by (subst emeasure_PiM_emb_not_empty) (auto simp: emeasure_PiM)
qed
-lemma (in product_prob_space) measurable_singleton_infprod:
- assumes "i \<in> I"
- shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
-proof (unfold measurable_def, intro CollectI conjI ballI)
- show "(\<lambda>x. x i) \<in> space (Pi\<^isub>P I M) \<rightarrow> space (M i)"
- using M.sets_into_space `i \<in> I`
- by (auto simp: infprod_algebra_def generator_def)
- fix A assume "A \<in> sets (M i)"
- have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E _\<in>{i}. A)"
- by (auto simp: infprod_algebra_def generator_def emb_def)
- also have "\<dots> \<in> sets (Pi\<^isub>P I M)"
- using `i \<in> I` `A \<in> sets (M i)`
- by (intro emb_in_infprod_algebra product_algebraI) auto
- finally show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" .
-qed
+lemma (in product_prob_space) measure_PiM_emb:
+ assumes "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
+ shows "measure (PiM I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. measure (M i) (X i))"
+ using emeasure_PiM_emb[OF assms]
+ unfolding emeasure_eq_measure M.emeasure_eq_measure by (simp add: setprod_ereal)
-lemma (in product_prob_space) sigma_product_algebra_sigma_eq:
- assumes M: "\<And>i. i \<in> I \<Longrightarrow> M i = sigma (E i)"
- shows "sets (Pi\<^isub>P I M) = sigma_sets (space (Pi\<^isub>P I M)) (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
-proof -
- let ?E = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
- let ?M = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i))"
- { fix i A assume "i\<in>I" "A \<in> sets (E i)"
- then have "A \<in> sets (M i)" using M by auto
- then have "A \<in> Pow (space (M i))" using M.sets_into_space by auto
- then have "A \<in> Pow (space (E i))" using M[OF `i \<in> I`] by auto }
- moreover
- have "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) \<in> space infprod_algebra \<rightarrow> space (E i)"
- by (auto simp: M infprod_algebra_def generator_def Pi_iff)
- ultimately have "sigma_sets (space (Pi\<^isub>P I M)) ?M \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?E"
- apply (intro sigma_sets_mono UN_least)
- apply (simp add: sets_sigma M)
- apply (subst sigma_sets_vimage[symmetric])
- apply (auto intro!: sigma_sets_mono')
- done
- moreover have "sigma_sets (space (Pi\<^isub>P I M)) ?E \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?M"
- by (intro sigma_sets_mono') (auto simp: M)
- ultimately show ?thesis
- by (subst infprod_algebra_alt2) (auto simp: sets_sigma)
-qed
-
-lemma (in product_prob_space) Int_proj_eq_emb:
- assumes "J \<noteq> {}" "J \<subseteq> I"
- shows "(\<Inter>i\<in>J. (\<lambda>x. x i) -` A i \<inter> space (Pi\<^isub>P I M)) = emb I J (Pi\<^isub>E J A)"
- using assms by (auto simp: infprod_algebra_def generator_def emb_def Pi_iff)
-
-lemma (in product_prob_space) emb_insert:
- "i \<notin> J \<Longrightarrow> emb I J (Pi\<^isub>E J f) \<inter> ((\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) =
- emb I (insert i J) (Pi\<^isub>E (insert i J) (f(i := A)))"
- by (auto simp: emb_def Pi_iff infprod_algebra_def generator_def split: split_if_asm)
+lemma (in finite_product_prob_space) finite_measure_PiM_emb:
+ "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> measure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))"
+ using measure_PiM_emb[of I A] finite_index prod_emb_PiE_same_index[OF sets_into_space, of I A M]
+ by auto
subsection {* Sequence space *}
@@ -1003,36 +722,30 @@
lemma (in sequence_space) infprod_in_sets[intro]:
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
- shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)"
+ shows "Pi UNIV E \<in> sets (Pi\<^isub>M UNIV M)"
proof -
have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))"
- using E E[THEN M.sets_into_space]
- by (auto simp: emb_def Pi_iff extensional_def) blast
- with E show ?thesis
- by (auto intro: emb_in_infprod_algebra)
+ using E E[THEN sets_into_space]
+ by (auto simp: prod_emb_def Pi_iff extensional_def) blast
+ with E show ?thesis by auto
qed
-lemma (in sequence_space) measure_infprod:
+lemma (in sequence_space) measure_PiM_countable:
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
- shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) ----> \<mu>' (Pi UNIV E)"
+ shows "(\<lambda>n. \<Prod>i\<le>n. measure (M i) (E i)) ----> measure (Pi\<^isub>M UNIV M) (Pi UNIV E)"
proof -
let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^isub>E {.. n} E)"
- { fix n :: nat
- interpret n: finite_product_prob_space M "{..n}" by default auto
- have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)"
- using E by (subst n.finite_measure_times) auto
- also have "\<dots> = \<mu>' (?E n)"
- using E by (intro finite_measure_infprod_emb[symmetric]) auto
- finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . }
+ have "\<And>n. (\<Prod>i\<le>n. measure (M i) (E i)) = measure (Pi\<^isub>M UNIV M) (?E n)"
+ using E by (simp add: measure_PiM_emb)
moreover have "Pi UNIV E = (\<Inter>n. ?E n)"
- using E E[THEN M.sets_into_space]
- by (auto simp: emb_def extensional_def Pi_iff) blast
- moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)"
+ using E E[THEN sets_into_space]
+ by (auto simp: prod_emb_def extensional_def Pi_iff) blast
+ moreover have "range ?E \<subseteq> sets (Pi\<^isub>M UNIV M)"
using E by auto
moreover have "decseq ?E"
- by (auto simp: emb_def Pi_iff decseq_def)
+ by (auto simp: prod_emb_def Pi_iff decseq_def)
ultimately show ?thesis
- by (simp add: finite_continuity_from_above)
+ by (simp add: finite_Lim_measure_decseq)
qed
end
\ No newline at end of file
--- a/src/HOL/Probability/Information.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Information.thy Mon Apr 23 12:14:35 2012 +0200
@@ -172,147 +172,152 @@
Kullback$-$Leibler distance. *}
definition
- "entropy_density b M \<nu> = log b \<circ> real \<circ> RN_deriv M \<nu>"
+ "entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N"
definition
- "KL_divergence b M \<nu> = integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) (entropy_density b M \<nu>)"
+ "KL_divergence b M N = integral\<^isup>L N (entropy_density b M N)"
lemma (in information_space) measurable_entropy_density:
- assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes ac: "absolutely_continuous \<nu>"
- shows "entropy_density b M \<nu> \<in> borel_measurable M"
+ assumes ac: "absolutely_continuous M N" "sets N = events"
+ shows "entropy_density b M N \<in> borel_measurable M"
proof -
- interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- have "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by fact
- from RN_deriv[OF this ac] b_gt_1 show ?thesis
+ from borel_measurable_RN_deriv[OF ac] b_gt_1 show ?thesis
unfolding entropy_density_def
by (intro measurable_comp) auto
qed
-lemma (in information_space) KL_gt_0:
- assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes ac: "absolutely_continuous \<nu>"
- assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
- assumes A: "A \<in> sets M" "\<nu> A \<noteq> \<mu> A"
- shows "0 < KL_divergence b M \<nu>"
-proof -
- interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
- have fms: "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" by unfold_locales
- note RN = RN_deriv[OF ms ac]
-
- from real_RN_deriv[OF fms ac] guess D . note D = this
- with absolutely_continuous_AE[OF ms] ac
- have D\<nu>: "AE x in M\<lparr>measure := \<nu>\<rparr>. RN_deriv M \<nu> x = ereal (D x)"
- by auto
-
- def f \<equiv> "\<lambda>x. if D x = 0 then 1 else 1 / D x"
- with D have f_borel: "f \<in> borel_measurable M"
- by (auto intro!: measurable_If)
+lemma (in sigma_finite_measure) KL_density:
+ fixes f :: "'a \<Rightarrow> real"
+ assumes "1 < b"
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ shows "KL_divergence b M (density M f) = (\<integral>x. f x * log b (f x) \<partial>M)"
+ unfolding KL_divergence_def
+proof (subst integral_density)
+ show "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M"
+ using f `1 < b`
+ by (auto simp: comp_def entropy_density_def intro!: borel_measurable_log borel_measurable_RN_deriv_density)
+ have "density M (RN_deriv M (density M f)) = density M f"
+ using f by (intro density_RN_deriv_density) auto
+ then have eq: "AE x in M. RN_deriv M (density M f) x = f x"
+ using f
+ by (intro density_unique)
+ (auto intro!: borel_measurable_log borel_measurable_RN_deriv_density simp: RN_deriv_density_nonneg)
+ show "(\<integral>x. f x * entropy_density b M (density M (\<lambda>x. ereal (f x))) x \<partial>M) = (\<integral>x. f x * log b (f x) \<partial>M)"
+ apply (intro integral_cong_AE)
+ using eq
+ apply eventually_elim
+ apply (auto simp: entropy_density_def)
+ done
+qed fact+
- have "KL_divergence b M \<nu> = 1 / ln b * (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
- unfolding KL_divergence_def using int b_gt_1
- by (simp add: integral_cmult)
-
- { fix A assume "A \<in> sets M"
- with RN D have "\<nu>.\<mu> A = (\<integral>\<^isup>+ x. ereal (D x) * indicator A x \<partial>M)"
- by (auto intro!: positive_integral_cong_AE) }
- note D_density = this
+lemma (in sigma_finite_measure) KL_density_density:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes "1 < b"
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+ assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
+ shows "KL_divergence b (density M f) (density M g) = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
+proof -
+ interpret Mf: sigma_finite_measure "density M f"
+ using f by (subst sigma_finite_iff_density_finite) auto
+ have "KL_divergence b (density M f) (density M g) =
+ KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
+ using f g ac by (subst density_density_divide) simp_all
+ also have "\<dots> = (\<integral>x. (g x / f x) * log b (g x / f x) \<partial>density M f)"
+ using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density divide_nonneg_nonneg)
+ also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
+ using ac f g `1 < b` by (subst integral_density) (auto intro!: integral_cong_AE)
+ finally show ?thesis .
+qed
- have ln_entropy: "(\<lambda>x. ln b * entropy_density b M \<nu> x) \<in> borel_measurable M"
- using measurable_entropy_density[OF ps ac] by auto
+lemma (in information_space) KL_gt_0:
+ fixes D :: "'a \<Rightarrow> real"
+ assumes "prob_space (density M D)"
+ assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
+ assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
+ assumes A: "density M D \<noteq> M"
+ shows "0 < KL_divergence b M (density M D)"
+proof -
+ interpret N: prob_space "density M D" by fact
- have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x)"
- using int by auto
- moreover have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x) \<longleftrightarrow>
- integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
- using D D_density ln_entropy
- by (intro integral_translated_density) auto
- ultimately have M_int: "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
- by simp
+ obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A"
+ using measure_eqI[of "density M D" M] `density M D \<noteq> M` by auto
+
+ let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
+ have [simp, intro]: "?D_set \<in> sets M"
+ using D by auto
have D_neg: "(\<integral>\<^isup>+ x. ereal (- D x) \<partial>M) = 0"
using D by (subst positive_integral_0_iff_AE) auto
- have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = \<nu> (space M)"
- using RN D by (auto intro!: positive_integral_cong_AE)
+ have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)"
+ using D by (simp add: emeasure_density cong: positive_integral_cong)
then have D_pos: "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = 1"
- using \<nu>.measure_space_1 by simp
-
- have "integrable M D"
- using D_pos D_neg D by (auto simp: integrable_def)
+ using N.emeasure_space_1 by simp
- have "integral\<^isup>L M D = 1"
- using D_pos D_neg by (auto simp: lebesgue_integral_def)
+ have "integrable M D" "integral\<^isup>L M D = 1"
+ using D D_pos D_neg unfolding integrable_def lebesgue_integral_def by simp_all
- let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
- have [simp, intro]: "?D_set \<in> sets M"
- using D by (auto intro: sets_Collect)
-
- have "0 \<le> 1 - \<mu>' ?D_set"
+ have "0 \<le> 1 - measure M ?D_set"
using prob_le_1 by (auto simp: field_simps)
also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
using `integrable M D` `integral\<^isup>L M D = 1`
- by (simp add: \<mu>'_def)
- also have "\<dots> < (\<integral> x. D x * (ln b * entropy_density b M \<nu> x) \<partial>M)"
+ by (simp add: emeasure_eq_measure)
+ also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)"
proof (rule integral_less_AE)
show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
using `integrable M D`
by (intro integral_diff integral_indicator) auto
next
- show "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
- by fact
+ from integral_cmult(1)[OF int, of "ln b"]
+ show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))"
+ by (simp add: ac_simps)
next
- show "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
+ show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
proof
- assume eq_0: "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
- then have disj: "AE x. D x = 1 \<or> D x = 0"
+ assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
+ then have disj: "AE x in M. D x = 1 \<or> D x = 0"
using D(1) by (auto intro!: AE_I[OF subset_refl] sets_Collect)
- have "\<mu> {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
+ have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
using D(1) by auto
- also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) * indicator {x\<in>space M. D x \<noteq> 0} x \<partial>M)"
+ also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) \<partial>M)"
using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def)
- also have "\<dots> = \<nu> {x\<in>space M. D x \<noteq> 0}"
- using D(1) D_density by auto
- also have "\<dots> = \<nu> (space M)"
- using D_density D(1) by (auto intro!: positive_integral_cong simp: indicator_def)
- finally have "AE x. D x = 1"
- using D(1) \<nu>.measure_space_1 by (intro AE_I_eq_1) auto
+ finally have "AE x in M. D x = 1"
+ using D D_pos by (intro AE_I_eq_1) auto
then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)"
by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
- also have "\<dots> = \<nu> A"
- using `A \<in> sets M` D_density by simp
- finally show False using `A \<in> sets M` `\<nu> A \<noteq> \<mu> A` by simp
+ also have "\<dots> = density M D A"
+ using `A \<in> sets M` D by (simp add: emeasure_density)
+ finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp
qed
show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
- using D(1) by (auto intro: sets_Collect)
+ using D(1) by (auto intro: sets_Collect_conj)
- show "AE t. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
- D t - indicator ?D_set t \<noteq> D t * (ln b * entropy_density b M \<nu> t)"
+ show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
+ D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))"
using D(2)
- proof (elim AE_mp, safe intro!: AE_I2)
- fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0"
- and RN: "RN_deriv M \<nu> t = ereal (D t)"
- and eq: "D t - indicator ?D_set t = D t * (ln b * entropy_density b M \<nu> t)"
+ proof (eventually_elim, safe)
+ fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t"
+ and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))"
have "D t - 1 = D t - indicator ?D_set t"
using Dt by simp
also note eq
- also have "D t * (ln b * entropy_density b M \<nu> t) = - D t * ln (1 / D t)"
- using RN b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
- by (simp add: entropy_density_def log_def ln_div less_le)
+ also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)"
+ using b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
+ by (simp add: log_def ln_div less_le)
finally have "ln (1 / D t) = 1 / D t - 1"
using `D t \<noteq> 0` by (auto simp: field_simps)
from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
show False by auto
qed
- show "AE t. D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
- using D(2)
- proof (elim AE_mp, intro AE_I2 impI)
- fix t assume "t \<in> space M" and RN: "RN_deriv M \<nu> t = ereal (D t)"
- show "D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
+ show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
+ using D(2) AE_space
+ proof eventually_elim
+ fix t assume "t \<in> space M" "0 \<le> D t"
+ show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
proof cases
assume asm: "D t \<noteq> 0"
then have "0 < D t" using `0 \<le> D t` by auto
@@ -321,592 +326,425 @@
using asm `t \<in> space M` by (simp add: field_simps)
also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
- also have "\<dots> = D t * (ln b * entropy_density b M \<nu> t)"
- using `0 < D t` RN b_gt_1
- by (simp_all add: log_def ln_div entropy_density_def)
+ also have "\<dots> = D t * (ln b * log b (D t))"
+ using `0 < D t` b_gt_1
+ by (simp_all add: log_def ln_div)
finally show ?thesis by simp
qed simp
qed
qed
- also have "\<dots> = (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
- using D D_density ln_entropy
- by (intro integral_translated_density[symmetric]) auto
- also have "\<dots> = ln b * (\<integral> x. entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
- using int by (rule \<nu>.integral_cmult)
- finally show "0 < KL_divergence b M \<nu>"
- using b_gt_1 by (auto simp: KL_divergence_def zero_less_mult_iff)
+ also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)"
+ by (simp add: ac_simps)
+ also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)"
+ using int by (rule integral_cmult)
+ finally show ?thesis
+ using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff)
qed
-lemma (in sigma_finite_measure) KL_eq_0:
- assumes eq: "\<forall>A\<in>sets M. \<nu> A = measure M A"
- shows "KL_divergence b M \<nu> = 0"
+lemma (in sigma_finite_measure) KL_same_eq_0: "KL_divergence b M M = 0"
proof -
- have "AE x. 1 = RN_deriv M \<nu> x"
+ have "AE x in M. 1 = RN_deriv M M x"
proof (rule RN_deriv_unique)
- show "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
- using eq by (intro measure_space_cong) auto
- show "absolutely_continuous \<nu>"
- unfolding absolutely_continuous_def using eq by auto
- show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x. 0 \<le> (1 :: ereal)" by auto
- fix A assume "A \<in> sets M"
- with eq show "\<nu> A = \<integral>\<^isup>+ x. 1 * indicator A x \<partial>M" by simp
+ show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto
+ show "density M (\<lambda>x. 1) = M"
+ apply (auto intro!: measure_eqI emeasure_density)
+ apply (subst emeasure_density)
+ apply auto
+ done
qed
- then have "AE x. log b (real (RN_deriv M \<nu> x)) = 0"
+ then have "AE x in M. log b (real (RN_deriv M M x)) = 0"
by (elim AE_mp) simp
from integral_cong_AE[OF this]
- have "integral\<^isup>L M (entropy_density b M \<nu>) = 0"
+ have "integral\<^isup>L M (entropy_density b M M) = 0"
by (simp add: entropy_density_def comp_def)
- with eq show "KL_divergence b M \<nu> = 0"
+ then show "KL_divergence b M M = 0"
unfolding KL_divergence_def
- by (subst integral_cong_measure) auto
+ by auto
qed
-lemma (in information_space) KL_eq_0_imp:
- assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes ac: "absolutely_continuous \<nu>"
- assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
- assumes KL: "KL_divergence b M \<nu> = 0"
- shows "\<forall>A\<in>sets M. \<nu> A = \<mu> A"
- by (metis less_imp_neq KL_gt_0 assms)
+lemma (in information_space) KL_eq_0_iff_eq:
+ fixes D :: "'a \<Rightarrow> real"
+ assumes "prob_space (density M D)"
+ assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
+ assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
+ shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M"
+ using KL_same_eq_0[of b] KL_gt_0[OF assms]
+ by (auto simp: less_le)
-lemma (in information_space) KL_ge_0:
- assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes ac: "absolutely_continuous \<nu>"
- assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
- shows "0 \<le> KL_divergence b M \<nu>"
- using KL_eq_0 KL_gt_0[OF ps ac int]
- by (cases "\<forall>A\<in>sets M. \<nu> A = measure M A") (auto simp: le_less)
-
-
-lemma (in sigma_finite_measure) KL_divergence_vimage:
- assumes T: "T \<in> measure_preserving M M'"
- and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
- and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
- and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
- and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
- and "1 < b"
- shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
+lemma (in information_space) KL_eq_0_iff_eq_ac:
+ fixes D :: "'a \<Rightarrow> real"
+ assumes "prob_space N"
+ assumes ac: "absolutely_continuous M N" "sets N = sets M"
+ assumes int: "integrable N (entropy_density b M N)"
+ shows "KL_divergence b M N = 0 \<longleftrightarrow> N = M"
proof -
- interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
- have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
- by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
- have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
- then have saM': "sigma_algebra M'" by simp
- then interpret M': measure_space M' by (rule measure_space_vimage) fact
- have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
- proof safe
- fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
- then have N': "T' -` N \<inter> space M' \<in> sets M'"
- using T' by (auto simp: measurable_def measure_preserving_def)
- have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
- using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
- then have "measure M' (T' -` N \<inter> space M') = 0"
- using measure_preservingD[OF T N'] N_0 by auto
- with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
- unfolding M'.absolutely_continuous_def measurable_def by auto
- qed
+ interpret N: prob_space N by fact
+ have "finite_measure N" by unfold_locales
+ from real_RN_deriv[OF this ac] guess D . note D = this
+
+ have "N = density M (RN_deriv M N)"
+ using ac by (rule density_RN_deriv[symmetric])
+ also have "\<dots> = density M D"
+ using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong)
+ finally have N: "N = density M D" .
- have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
- have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
- by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
- show ?thesis
- unfolding KL_divergence_def entropy_density_def comp_def
- proof (subst \<nu>'.integral_vimage[OF sa T'])
- show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
- by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
- have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
- (\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
- using inv' by (auto intro!: \<nu>'.integral_cong)
- also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
- using M ac AE
- by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
- (auto elim!: AE_mp)
- finally show "?l = ?r" .
- qed
+ from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density
+ have "integrable N (\<lambda>x. log b (D x))"
+ by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int])
+ (auto simp: N entropy_density_def)
+ with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))"
+ by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def)
+ with `prob_space N` D show ?thesis
+ unfolding N
+ by (intro KL_eq_0_iff_eq) auto
qed
-lemma (in sigma_finite_measure) KL_divergence_cong:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
- assumes [simp]: "sets N = sets M" "space N = space M"
- "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
- "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
- shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
-proof -
- interpret \<nu>: measure_space ?\<nu> by fact
- have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
- by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def entropy_density_def)
- also have "\<dots> = KL_divergence b N \<nu>'"
- by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def entropy_density_def comp_def)
- finally show ?thesis .
-qed
+lemma (in information_space) KL_nonneg:
+ assumes "prob_space (density M D)"
+ assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
+ assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
+ shows "0 \<le> KL_divergence b M (density M D)"
+ using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0)
-lemma (in finite_measure_space) KL_divergence_eq_finite:
- assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes ac: "absolutely_continuous \<nu>"
- shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
-proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v] entropy_density_def)
- interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
- show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
- using RN_deriv_finite_measure[OF ms ac]
- by (auto intro!: setsum_cong simp: field_simps)
-qed
+lemma (in sigma_finite_measure) KL_density_density_nonneg:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes "1 < b"
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)"
+ assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)"
+ assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
+ assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))"
+ shows "0 \<le> KL_divergence b (density M f) (density M g)"
+proof -
+ interpret Mf: prob_space "density M f" by fact
+ interpret Mf: information_space "density M f" b by default fact
+ have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _")
+ using f g ac by (subst density_density_divide) simp_all
-lemma (in finite_prob_space) KL_divergence_positive_finite:
- assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes ac: "absolutely_continuous \<nu>"
- and "1 < b"
- shows "0 \<le> KL_divergence b M \<nu>"
-proof -
- interpret information_space M by default fact
- interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- have ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)" by unfold_locales
- from KL_ge_0[OF this ac v.integral_finite_singleton(1)] show ?thesis .
+ have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
+ proof (rule Mf.KL_nonneg)
+ show "prob_space ?DD" unfolding eq by fact
+ from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)"
+ by auto
+ show "AE x in density M f. 0 \<le> g x / f x"
+ using f g by (auto simp: AE_density divide_nonneg_nonneg)
+ show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))"
+ using `1 < b` f g ac
+ by (subst integral_density)
+ (auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
+ qed
+ also have "\<dots> = KL_divergence b (density M f) (density M g)"
+ using f g ac by (subst density_density_divide) simp_all
+ finally show ?thesis .
qed
subsection {* Mutual Information *}
definition (in prob_space)
"mutual_information b S T X Y =
- KL_divergence b (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>)
- (ereal\<circ>joint_distribution X Y)"
+ KL_divergence b (distr M S X \<Otimes>\<^isub>M distr M T Y) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)))"
-lemma (in information_space)
+lemma (in information_space) mutual_information_indep_vars:
fixes S T X Y
- defines "P \<equiv> S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
+ defines "P \<equiv> distr M S X \<Otimes>\<^isub>M distr M T Y"
+ defines "Q \<equiv> distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
shows "indep_var S X T Y \<longleftrightarrow>
(random_variable S X \<and> random_variable T Y \<and>
- measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y) \<and>
- integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
- (entropy_density b P (ereal\<circ>joint_distribution X Y)) \<and>
- mutual_information b S T X Y = 0)"
+ absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and>
+ mutual_information b S T X Y = 0)"
+ unfolding indep_var_distribution_eq
proof safe
- assume indep: "indep_var S X T Y"
- then have "random_variable S X" "random_variable T Y"
- by (blast dest: indep_var_rv1 indep_var_rv2)+
- then show "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
- by blast+
-
- interpret X: prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
- by (rule distribution_prob_space) fact
- interpret Y: prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- by (rule distribution_prob_space) fact
- interpret XY: pair_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
- interpret XY: information_space XY.P b by default (rule b_gt_1)
-
- let ?J = "XY.P\<lparr> measure := (ereal\<circ>joint_distribution X Y) \<rparr>"
- { fix A assume "A \<in> sets XY.P"
- then have "ereal (joint_distribution X Y A) = XY.\<mu> A"
- using indep_var_distributionD[OF indep]
- by (simp add: XY.P.finite_measure_eq) }
- note j_eq = this
-
- interpret J: prob_space ?J
- using j_eq by (intro XY.prob_space_cong) auto
-
- have ac: "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
- by (simp add: XY.absolutely_continuous_def j_eq)
- then show "measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y)"
- unfolding P_def .
-
- have ed: "entropy_density b XY.P (ereal\<circ>joint_distribution X Y) \<in> borel_measurable XY.P"
- by (rule XY.measurable_entropy_density) (default | fact)+
+ assume rv: "random_variable S X" "random_variable T Y"
- have "AE x in XY.P. 1 = RN_deriv XY.P (ereal\<circ>joint_distribution X Y) x"
- proof (rule XY.RN_deriv_unique[OF _ ac])
- show "measure_space ?J" by default
- fix A assume "A \<in> sets XY.P"
- then show "(ereal\<circ>joint_distribution X Y) A = (\<integral>\<^isup>+ x. 1 * indicator A x \<partial>XY.P)"
- by (simp add: j_eq)
- qed (insert XY.measurable_const[of 1 borel], auto)
- then have ae_XY: "AE x in XY.P. entropy_density b XY.P (ereal\<circ>joint_distribution X Y) x = 0"
- by (elim XY.AE_mp) (simp add: entropy_density_def)
- have ae_J: "AE x in ?J. entropy_density b XY.P (ereal\<circ>joint_distribution X Y) x = 0"
- proof (rule XY.absolutely_continuous_AE)
- show "measure_space ?J" by default
- show "XY.absolutely_continuous (measure ?J)"
- using ac by simp
- qed (insert ae_XY, simp_all)
- then show "integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
- (entropy_density b P (ereal\<circ>joint_distribution X Y))"
- unfolding P_def
- using ed XY.measurable_const[of 0 borel]
- by (subst J.integrable_cong_AE) auto
+ interpret X: prob_space "distr M S X"
+ by (rule prob_space_distr) fact
+ interpret Y: prob_space "distr M T Y"
+ by (rule prob_space_distr) fact
+ interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default
+ interpret P: information_space P b unfolding P_def by default (rule b_gt_1)
- show "mutual_information b S T X Y = 0"
- unfolding mutual_information_def KL_divergence_def P_def
- by (subst J.integral_cong_AE[OF ae_J]) simp
-next
- assume "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
- then have rvs: "random_variable S X" "random_variable T Y" by blast+
+ interpret Q: prob_space Q unfolding Q_def
+ by (rule prob_space_distr) (simp add: comp_def measurable_pair_iff rv)
- interpret X: prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
- by (rule distribution_prob_space) fact
- interpret Y: prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- by (rule distribution_prob_space) fact
- interpret XY: pair_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
- interpret XY: information_space XY.P b by default (rule b_gt_1)
+ { assume "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+ then have [simp]: "Q = P" unfolding Q_def P_def by simp
- let ?J = "XY.P\<lparr> measure := (ereal\<circ>joint_distribution X Y) \<rparr>"
- interpret J: prob_space ?J
- using rvs by (intro joint_distribution_prob_space) auto
+ show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def)
+ then have ed: "entropy_density b P Q \<in> borel_measurable P"
+ by (rule P.measurable_entropy_density) simp
- assume ac: "measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y)"
- assume int: "integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
- (entropy_density b P (ereal\<circ>joint_distribution X Y))"
- assume I_eq_0: "mutual_information b S T X Y = 0"
+ have "AE x in P. 1 = RN_deriv P Q x"
+ proof (rule P.RN_deriv_unique)
+ show "density P (\<lambda>x. 1) = Q"
+ unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density)
+ qed auto
+ then have ae_0: "AE x in P. entropy_density b P Q x = 0"
+ by eventually_elim (auto simp: entropy_density_def)
+ then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)"
+ using ed unfolding `Q = P` by (intro integrable_cong_AE) auto
+ then show "integrable Q (entropy_density b P Q)" by simp
- have eq: "\<forall>A\<in>sets XY.P. (ereal \<circ> joint_distribution X Y) A = XY.\<mu> A"
- proof (rule XY.KL_eq_0_imp)
- show "prob_space ?J" by unfold_locales
- show "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
- using ac by (simp add: P_def)
- show "integrable ?J (entropy_density b XY.P (ereal\<circ>joint_distribution X Y))"
- using int by (simp add: P_def)
- show "KL_divergence b XY.P (ereal\<circ>joint_distribution X Y) = 0"
- using I_eq_0 unfolding mutual_information_def by (simp add: P_def)
- qed
-
- { fix S X assume "sigma_algebra S"
- interpret S: sigma_algebra S by fact
- have "Int_stable \<lparr>space = space M, sets = {X -` A \<inter> space M |A. A \<in> sets S}\<rparr>"
- proof (safe intro!: Int_stableI)
- fix A B assume "A \<in> sets S" "B \<in> sets S"
- then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
- by (intro exI[of _ "A \<inter> B"]) auto
- qed }
- note Int_stable = this
+ show "mutual_information b S T X Y = 0"
+ unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P`
+ using ae_0 by (simp cong: integral_cong_AE) }
- show "indep_var S X T Y" unfolding indep_var_eq
- proof (intro conjI indep_set_sigma_sets Int_stable)
- show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
- proof (safe intro!: indep_setI)
- { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
- using `X \<in> measurable M S` by (auto intro: measurable_sets) }
- { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
- using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
- next
- fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
- have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) =
- ereal (joint_distribution X Y (A \<times> B))"
- unfolding distribution_def
- by (intro arg_cong[where f="\<lambda>C. ereal (prob C)"]) auto
- also have "\<dots> = XY.\<mu> (A \<times> B)"
- using ab eq by (auto simp: XY.finite_measure_eq)
- also have "\<dots> = ereal (distribution X A) * ereal (distribution Y B)"
- using ab by (simp add: XY.pair_measure_times)
- finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
- prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
- unfolding distribution_def by simp
- qed
- qed fact+
-qed
+ { assume ac: "absolutely_continuous P Q"
+ assume int: "integrable Q (entropy_density b P Q)"
+ assume I_eq_0: "mutual_information b S T X Y = 0"
-lemma (in information_space) mutual_information_commute_generic:
- assumes X: "random_variable S X" and Y: "random_variable T Y"
- assumes ac: "measure_space.absolutely_continuous
- (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>) (ereal\<circ>joint_distribution X Y)"
- shows "mutual_information b S T X Y = mutual_information b T S Y X"
-proof -
- let ?S = "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- interpret S: prob_space ?S using X by (rule distribution_prob_space)
- interpret T: prob_space ?T using Y by (rule distribution_prob_space)
- interpret P: pair_prob_space ?S ?T ..
- interpret Q: pair_prob_space ?T ?S ..
- show ?thesis
- unfolding mutual_information_def
- proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
- show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
- (P.P \<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := ereal\<circ>joint_distribution Y X\<rparr>)"
- using X Y unfolding measurable_def
- unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
- by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
- have "prob_space (P.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)"
- using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
- then show "measure_space (P.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)"
- unfolding prob_space_def finite_measure_def sigma_finite_measure_def by simp
- qed auto
+ have eq: "Q = P"
+ proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1])
+ show "prob_space Q" by unfold_locales
+ show "absolutely_continuous P Q" by fact
+ show "integrable Q (entropy_density b P Q)" by fact
+ show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure)
+ show "KL_divergence b P Q = 0"
+ using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def)
+ qed
+ then show "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+ unfolding P_def Q_def .. }
qed
-definition (in prob_space)
- "entropy b s X = mutual_information b s s X X"
-
abbreviation (in information_space)
mutual_information_Pow ("\<I>'(_ ; _')") where
- "\<I>(X ; Y) \<equiv> mutual_information b
- \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
- \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr> X Y"
+ "\<I>(X ; Y) \<equiv> mutual_information b (count_space (X`space M)) (count_space (Y`space M)) X Y"
-lemma (in prob_space) finite_variables_absolutely_continuous:
- assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
- shows "measure_space.absolutely_continuous
- (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>)
- (ereal\<circ>joint_distribution X Y)"
+lemma (in information_space)
+ fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
+ assumes "sigma_finite_measure S" "sigma_finite_measure T"
+ assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
+ shows mutual_information_distr: "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R")
+ and mutual_information_nonneg: "integrable (S \<Otimes>\<^isub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y"
proof -
- interpret X: finite_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
- using X by (rule distribution_finite_prob_space)
- interpret Y: finite_prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- using Y by (rule distribution_finite_prob_space)
- interpret XY: pair_finite_prob_space
- "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr> measure := ereal\<circ>distribution Y\<rparr>" by default
- interpret P: finite_prob_space "XY.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>"
- using assms by (auto intro!: joint_distribution_finite_prob_space)
- note rv = assms[THEN finite_random_variableD]
- show "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
- proof (rule XY.absolutely_continuousI)
- show "finite_measure_space (XY.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)" by unfold_locales
- fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
- then obtain a b where "x = (a, b)"
- and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
- by (cases x) (auto simp: space_pair_measure)
- with finite_distribution_order(5,6)[OF X Y]
- show "(ereal \<circ> joint_distribution X Y) {x} = 0" by auto
+ have X: "random_variable S X"
+ using Px by (auto simp: distributed_def)
+ have Y: "random_variable T Y"
+ using Py by (auto simp: distributed_def)
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret ST: pair_sigma_finite S T ..
+ interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
+ interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
+ interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
+ let ?P = "S \<Otimes>\<^isub>M T"
+ let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"
+
+ { fix A assume "A \<in> sets S"
+ with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
+ by (auto simp: emeasure_distr measurable_Pair measurable_space
+ intro!: arg_cong[where f="emeasure M"]) }
+ note marginal_eq1 = this
+ { fix A assume "A \<in> sets T"
+ with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
+ by (auto simp: emeasure_distr measurable_Pair measurable_space
+ intro!: arg_cong[where f="emeasure M"]) }
+ note marginal_eq2 = this
+
+ have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
+ by auto
+
+ have distr_eq: "distr M S X \<Otimes>\<^isub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
+ unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
+ proof (subst pair_measure_density)
+ show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
+ "AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
+ using Px Py by (auto simp: distributed_def)
+ show "sigma_finite_measure (density S Px)" unfolding Px(1)[THEN distributed_distr_eq_density, symmetric] ..
+ show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
+ qed (fact | simp)+
+
+ have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))"
+ unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..
+
+ from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
+ by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
+ have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
+ proof (rule ST.AE_pair_measure)
+ show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
+ using f by auto
+ show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
+ using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
+ qed
+
+ have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
+ by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
+ moreover
+ have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
+ by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
+ ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
+ by eventually_elim auto
+
+ show "?M = ?R"
+ unfolding M f_def
+ using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
+ by (rule ST.KL_density_density)
+
+ assume int: "integrable (S \<Otimes>\<^isub>M T) f"
+ show "0 \<le> ?M" unfolding M
+ proof (rule ST.KL_density_density_nonneg
+ [OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
+ show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) "
+ unfolding distributed_distr_eq_density[OF Pxy, symmetric]
+ using distributed_measurable[OF Pxy] by (rule prob_space_distr)
+ show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
+ unfolding distr_eq[symmetric] by unfold_locales
qed
qed
lemma (in information_space)
- assumes MX: "finite_random_variable MX X"
- assumes MY: "finite_random_variable MY Y"
- shows mutual_information_generic_eq:
- "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
- joint_distribution X Y {(x,y)} *
- log b (joint_distribution X Y {(x,y)} /
- (distribution X {x} * distribution Y {y})))"
- (is ?sum)
- and mutual_information_positive_generic:
- "0 \<le> mutual_information b MX MY X Y" (is ?positive)
+ fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
+ assumes "sigma_finite_measure S" "sigma_finite_measure T"
+ assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
+ shows mutual_information_eq_0: "mutual_information b S T X Y = 0"
proof -
- interpret X: finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>"
- using MX by (rule distribution_finite_prob_space)
- interpret Y: finite_prob_space "MY\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
- using MY by (rule distribution_finite_prob_space)
- interpret XY: pair_finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>" "MY\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
- interpret P: finite_prob_space "XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>"
- using assms by (auto intro!: joint_distribution_finite_prob_space)
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret ST: pair_sigma_finite S T ..
- have P_ms: "finite_measure_space (XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>)" by unfold_locales
- have P_ps: "finite_prob_space (XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>)" by unfold_locales
-
- show ?sum
- unfolding Let_def mutual_information_def
- by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
- (auto simp add: space_pair_measure setsum_cartesian_product')
-
- show ?positive
- using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
- unfolding mutual_information_def .
+ have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
+ by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
+ moreover
+ have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
+ by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
+ moreover
+ have "AE x in S \<Otimes>\<^isub>M T. Pxy x = Px (fst x) * Py (snd x)"
+ using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy]
+ by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'')
+ ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0"
+ by eventually_elim simp
+ then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
+ by (rule integral_cong_AE)
+ then show ?thesis
+ by (subst mutual_information_distr[OF assms(1-5)]) simp
qed
-lemma (in information_space) mutual_information_commute:
- assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
- shows "mutual_information b S T X Y = mutual_information b T S Y X"
- unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
- unfolding joint_distribution_commute_singleton[of X Y]
- by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
-
-lemma (in information_space) mutual_information_commute_simple:
- assumes X: "simple_function M X" and Y: "simple_function M Y"
- shows "\<I>(X;Y) = \<I>(Y;X)"
- by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
-
-lemma (in information_space) mutual_information_eq:
- assumes "simple_function M X" "simple_function M Y"
- shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
- distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
- (distribution X {x} * distribution Y {y})))"
- using assms by (simp add: mutual_information_generic_eq)
+lemma (in information_space) mutual_information_simple_distributed:
+ assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py"
+ assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+ shows "\<I>(X ; Y) = (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x))`space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
+proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
+ note fin = simple_distributed_joint_finite[OF XY, simp]
+ show "sigma_finite_measure (count_space (X ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ show "sigma_finite_measure (count_space (Y ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
+ let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))"
+ have "\<And>x. ?f x = (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) else 0)"
+ by auto
+ with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M))) =
+ (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
+ by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum_cases split_beta'
+ intro!: setsum_cong)
+qed
-lemma (in information_space) mutual_information_generic_cong:
- assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
- assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
- shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
- unfolding mutual_information_def using X Y
- by (simp cong: distribution_cong)
-
-lemma (in information_space) mutual_information_cong:
- assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
- assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
- shows "\<I>(X; Y) = \<I>(X'; Y')"
- unfolding mutual_information_def using X Y
- by (simp cong: distribution_cong image_cong)
-
-lemma (in information_space) mutual_information_positive:
- assumes "simple_function M X" "simple_function M Y"
- shows "0 \<le> \<I>(X;Y)"
- using assms by (simp add: mutual_information_positive_generic)
+lemma (in information_space)
+ fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
+ assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py"
+ assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+ assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)"
+ shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0"
+proof (subst mutual_information_simple_distributed[OF Px Py Pxy])
+ have "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) =
+ (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)"
+ by (intro setsum_cong) (auto simp: ae)
+ then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M.
+ Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp
+qed
subsection {* Entropy *}
+definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where
+ "entropy b S X = - KL_divergence b S (distr M S X)"
+
abbreviation (in information_space)
entropy_Pow ("\<H>'(_')") where
- "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr> X"
-
-lemma (in information_space) entropy_generic_eq:
- fixes X :: "'a \<Rightarrow> 'c"
- assumes MX: "finite_random_variable MX X"
- shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
-proof -
- interpret MX: finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>"
- using MX by (rule distribution_finite_prob_space)
- let ?X = "\<lambda>x. distribution X {x}"
- let ?XX = "\<lambda>x y. joint_distribution X X {(x, y)}"
+ "\<H>(X) \<equiv> entropy b (count_space (X`space M)) X"
- { fix x y :: 'c
- { assume "x \<noteq> y"
- then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
- then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
- then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
- (if x = y then - ?X y * log b (?X y) else 0)"
- by (auto simp: log_simps zero_less_mult_iff) }
- note remove_XX = this
-
- show ?thesis
- unfolding entropy_def mutual_information_generic_eq[OF MX MX]
- unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
- using MX.finite_space by (auto simp: setsum_cases)
+lemma (in information_space) entropy_distr:
+ fixes X :: "'a \<Rightarrow> 'b"
+ assumes "sigma_finite_measure MX" and X: "distributed M MX X f"
+ shows "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)"
+proof -
+ interpret MX: sigma_finite_measure MX by fact
+ from X show ?thesis
+ unfolding entropy_def X[THEN distributed_distr_eq_density]
+ by (subst MX.KL_density[OF b_gt_1]) (simp_all add: distributed_real_AE distributed_real_measurable)
qed
-lemma (in information_space) entropy_eq:
- assumes "simple_function M X"
- shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
- using assms by (simp add: entropy_generic_eq)
-
-lemma (in information_space) entropy_positive:
- "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
- unfolding entropy_def by (simp add: mutual_information_positive)
+lemma (in information_space) entropy_uniform:
+ assumes "sigma_finite_measure MX"
+ assumes A: "A \<in> sets MX" "emeasure MX A \<noteq> 0" "emeasure MX A \<noteq> \<infinity>"
+ assumes X: "distributed M MX X (\<lambda>x. 1 / measure MX A * indicator A x)"
+ shows "entropy b MX X = log b (measure MX A)"
+proof (subst entropy_distr[OF _ X])
+ let ?f = "\<lambda>x. 1 / measure MX A * indicator A x"
+ have "- (\<integral>x. ?f x * log b (?f x) \<partial>MX) =
+ - (\<integral>x. (log b (1 / measure MX A) / measure MX A) * indicator A x \<partial>MX)"
+ by (auto intro!: integral_cong simp: indicator_def)
+ also have "\<dots> = - log b (inverse (measure MX A))"
+ using A by (subst integral_cmult(2))
+ (simp_all add: measure_def real_of_ereal_eq_0 integral_cmult inverse_eq_divide)
+ also have "\<dots> = log b (measure MX A)"
+ using b_gt_1 A by (subst log_inverse) (auto simp add: measure_def less_le real_of_ereal_eq_0
+ emeasure_nonneg real_of_ereal_pos)
+ finally show "- (\<integral>x. ?f x * log b (?f x) \<partial>MX) = log b (measure MX A)" by simp
+qed fact+
-lemma (in information_space) entropy_certainty_eq_0:
- assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
- shows "\<H>(X) = 0"
-proof -
- let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal\<circ>distribution X\<rparr>"
- note simple_function_imp_finite_random_variable[OF `simple_function M X`]
- from distribution_finite_prob_space[OF this, of "\<lparr> measure = ereal\<circ>distribution X \<rparr>"]
- interpret X: finite_prob_space ?X by simp
- have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
- using X.measure_compl[of "{x}"] assms by auto
- also have "\<dots> = 0" using X.prob_space assms by auto
- finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
- { fix y assume *: "y \<in> X ` space M"
- { assume asm: "y \<noteq> x"
- with * have "{y} \<subseteq> X ` space M - {x}" by auto
- from X.measure_mono[OF this] X0 asm *
- have "distribution X {y} = 0" by (auto intro: antisym) }
- then have "distribution X {y} = (if x = y then 1 else 0)"
- using assms by auto }
- note fi = this
- have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
- show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
+lemma (in information_space) entropy_simple_distributed:
+ fixes X :: "'a \<Rightarrow> 'b"
+ assumes X: "simple_distributed M X f"
+ shows "\<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))"
+proof (subst entropy_distr[OF _ simple_distributed[OF X]])
+ show "sigma_finite_measure (count_space (X ` space M))"
+ using X by (simp add: sigma_finite_measure_count_space_finite simple_distributed_def)
+ show "- (\<integral>x. f x * log b (f x) \<partial>(count_space (X`space M))) = - (\<Sum>x\<in>X ` space M. f x * log b (f x))"
+ using X by (auto simp add: lebesgue_integral_count_space_finite)
qed
lemma (in information_space) entropy_le_card_not_0:
- assumes X: "simple_function M X"
- shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
+ assumes X: "simple_distributed M X f"
+ shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))"
proof -
- let ?p = "\<lambda>x. distribution X {x}"
- have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
- unfolding entropy_eq[OF X] setsum_negf[symmetric]
- by (auto intro!: setsum_cong simp: log_simps)
- also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
- using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
- by (intro log_setsum') (auto simp: simple_function_def)
- also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
+ have "\<H>(X) = (\<Sum>x\<in>X`space M. f x * log b (1 / f x))"
+ unfolding entropy_simple_distributed[OF X] setsum_negf[symmetric]
+ using X by (auto dest: simple_distributed_nonneg intro!: setsum_cong simp: log_simps less_le)
+ also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. f x * (1 / f x))"
+ using not_empty b_gt_1 `simple_distributed M X f`
+ by (intro log_setsum') (auto simp: simple_distributed_nonneg simple_distributed_setsum_space)
+ also have "\<dots> = log b (\<Sum>x\<in>X`space M. if f x \<noteq> 0 then 1 else 0)"
by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
finally show ?thesis
- using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
-qed
-
-lemma (in prob_space) measure'_translate:
- assumes X: "random_variable S X" and A: "A \<in> sets S"
- shows "finite_measure.\<mu>' (S\<lparr> measure := ereal\<circ>distribution X \<rparr>) A = distribution X A"
-proof -
- interpret S: prob_space "S\<lparr> measure := ereal\<circ>distribution X \<rparr>"
- using distribution_prob_space[OF X] .
- from A show "S.\<mu>' A = distribution X A"
- unfolding S.\<mu>'_def by (simp add: distribution_def [abs_def] \<mu>'_def)
-qed
-
-lemma (in information_space) entropy_uniform_max:
- assumes X: "simple_function M X"
- assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
- shows "\<H>(X) = log b (real (card (X ` space M)))"
-proof -
- let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := ereal\<circ>distribution X\<rparr>"
- note frv = simple_function_imp_finite_random_variable[OF X]
- from distribution_finite_prob_space[OF this, of "\<lparr> measure = ereal\<circ>distribution X \<rparr>"]
- interpret X: finite_prob_space ?X by simp
- note rv = finite_random_variableD[OF frv]
- have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
- using `simple_function M X` not_empty by (auto simp: simple_function_def)
- { fix x assume "x \<in> space ?X"
- moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
- proof (rule X.uniform_prob)
- fix x y assume "x \<in> space ?X" "y \<in> space ?X"
- with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
- by (subst (1 2) measure'_translate[OF rv]) auto
- qed
- ultimately have "distribution X {x} = 1 / card (space ?X)"
- by (subst (asm) measure'_translate[OF rv]) auto }
- thus ?thesis
- using not_empty X.finite_space b_gt_1 card_gt0
- by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
+ using `simple_distributed M X f` by (auto simp: setsum_cases real_eq_of_nat)
qed
lemma (in information_space) entropy_le_card:
- assumes "simple_function M X"
+ assumes "simple_distributed M X f"
shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
proof cases
- assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
- then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
+ assume "X ` space M \<inter> {x. f x \<noteq> 0} = {}"
+ then have "\<And>x. x\<in>X`space M \<Longrightarrow> f x = 0" by auto
moreover
have "0 < card (X`space M)"
- using `simple_function M X` not_empty
- by (auto simp: card_gt_0_iff simple_function_def)
+ using `simple_distributed M X f` not_empty by (auto simp: card_gt_0_iff)
then have "log b 1 \<le> log b (real (card (X`space M)))"
using b_gt_1 by (intro log_le) auto
- ultimately show ?thesis using assms by (simp add: entropy_eq)
+ ultimately show ?thesis using assms by (simp add: entropy_simple_distributed)
next
- assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
- have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
- (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
+ assume False: "X ` space M \<inter> {x. f x \<noteq> 0} \<noteq> {}"
+ have "card (X ` space M \<inter> {x. f x \<noteq> 0}) \<le> card (X ` space M)"
+ (is "?A \<le> ?B") using assms not_empty
+ by (auto intro!: card_mono simp: simple_function_def simple_distributed_def)
note entropy_le_card_not_0[OF assms]
also have "log b (real ?A) \<le> log b (real ?B)"
using b_gt_1 False not_empty `?A \<le> ?B` assms
- by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
+ by (auto intro!: log_le simp: card_gt_0_iff simp: simple_distributed_def)
finally show ?thesis .
qed
-lemma (in information_space) entropy_commute:
- assumes "simple_function M X" "simple_function M Y"
- shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
-proof -
- have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
- using assms by (auto intro: simple_function_Pair)
- have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
- by auto
- have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
- by (auto intro!: inj_onI)
- show ?thesis
- unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
- by (simp add: joint_distribution_commute[of Y X] split_beta)
-qed
-
-lemma (in information_space) entropy_eq_cartesian_product:
- assumes "simple_function M X" "simple_function M Y"
- shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
- joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
-proof -
- have sf: "simple_function M (\<lambda>x. (X x, Y x))"
- using assms by (auto intro: simple_function_Pair)
- { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
- then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
- then have "joint_distribution X Y {x} = 0"
- unfolding distribution_def by auto }
- then show ?thesis using sf assms
- unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
- by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
-qed
-
subsection {* Conditional Mutual Information *}
definition (in prob_space)
@@ -917,489 +755,553 @@
abbreviation (in information_space)
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
- \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
- \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr>
- \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = ereal\<circ>distribution Z \<rparr>
- X Y Z"
+ (count_space (X ` space M)) (count_space (Y ` space M)) (count_space (Z ` space M)) X Y Z"
lemma (in information_space) conditional_mutual_information_generic_eq:
- assumes MX: "finite_random_variable MX X"
- and MY: "finite_random_variable MY Y"
- and MZ: "finite_random_variable MZ Z"
- shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
- distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
- log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
- (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
- (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
+ assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
+ assumes Px: "distributed M S X Px"
+ assumes Pz: "distributed M P Z Pz"
+ assumes Pyz: "distributed M (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
+ assumes Pxz: "distributed M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz"
+ assumes Pxyz: "distributed M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
+ assumes I1: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
+ assumes I2: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
+ shows "conditional_mutual_information b S T P X Y Z
+ = (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
proof -
- let ?X = "\<lambda>x. distribution X {x}"
- note finite_var = MX MY MZ
- note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
- note XYZ = finite_random_variable_pairI[OF MX YZ]
- note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
- note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
- note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
- note order1 =
- finite_distribution_order(5,6)[OF finite_var(1) YZ]
- finite_distribution_order(5,6)[OF finite_var(1,3)]
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret P: sigma_finite_measure P by fact
+ interpret TP: pair_sigma_finite T P ..
+ interpret SP: pair_sigma_finite S P ..
+ interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T ..
+ interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" ..
+ have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" ..
+ have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" ..
+ have YZ: "random_variable (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x))"
+ using Pyz by (simp add: distributed_measurable)
+
+ have Pxyz_f: "\<And>M f. f \<in> measurable M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. Pxyz (f x)) \<in> borel_measurable M"
+ using measurable_comp[OF _ Pxyz[THEN distributed_real_measurable]] by (auto simp: comp_def)
+
+ { fix f g h M
+ assume f: "f \<in> measurable M S" and g: "g \<in> measurable M P" and h: "h \<in> measurable M (S \<Otimes>\<^isub>M P)"
+ from measurable_comp[OF h Pxz[THEN distributed_real_measurable]]
+ measurable_comp[OF f Px[THEN distributed_real_measurable]]
+ measurable_comp[OF g Pz[THEN distributed_real_measurable]]
+ have "(\<lambda>x. log b (Pxz (h x) / (Px (f x) * Pz (g x)))) \<in> borel_measurable M"
+ by (simp add: comp_def b_gt_1) }
+ note borel_log = this
+
+ have measurable_cut: "(\<lambda>(x, y, z). (x, z)) \<in> measurable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (S \<Otimes>\<^isub>M P)"
+ by (auto simp add: split_beta' comp_def intro!: measurable_Pair measurable_snd')
+
+ from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) =
+ distr (distr M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). (x, z))"
+ by (subst distr_distr[OF measurable_cut]) (auto dest: distributed_measurable simp: comp_def)
- note random_var = finite_var[THEN finite_random_variableD]
- note finite = finite_var(1) YZ finite_var(3) XZ YZX
-
- have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
- \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
- unfolding joint_distribution_commute_singleton[of X]
- unfolding joint_distribution_assoc_singleton[symmetric]
- using finite_distribution_order(6)[OF finite_var(2) ZX]
- by auto
-
- have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
- (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
- (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
- proof (safe intro!: setsum_cong)
- fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
- show "?L x y z = ?R x y z"
+ have "mutual_information b S P X Z =
+ (\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))"
+ by (rule mutual_information_distr[OF S P Px Pz Pxz])
+ also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
+ using b_gt_1 Pxz Px Pz
+ by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"])
+ (auto simp: split_beta' intro!: measurable_Pair measurable_snd' measurable_snd'' measurable_fst'' borel_measurable_times
+ dest!: distributed_real_measurable)
+ finally have mi_eq:
+ "mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" .
+
+ have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) (auto intro: measurable_snd')
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) (auto intro: measurable_Pair measurable_snd')
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) (auto intro: measurable_Pair)
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)"
+ using Px by (intro STP.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable)
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)"
+ using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))"
+ using Pz Pz[THEN distributed_real_measurable] by (auto intro!: measurable_snd'' TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE)
+ moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
+ using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
+ using measurable_comp[OF measurable_Pair[OF measurable_fst measurable_comp[OF measurable_snd measurable_snd]] Pxz[THEN distributed_real_measurable], of T]
+ using measurable_comp[OF measurable_snd measurable_Pair2[OF Pxz[THEN distributed_real_measurable]], of _ T]
+ by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def)
+ moreover note Pxyz[THEN distributed_real_AE]
+ ultimately have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P.
+ Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
+ Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
+ Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
+ proof eventually_elim
+ case (goal1 x)
+ show ?case
proof cases
- assume "?XYZ x y z \<noteq> 0"
- with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
- using order1 order2 by (auto simp: less_le)
- with b_gt_1 show ?thesis
- by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
+ assume "Pxyz x \<noteq> 0"
+ with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x"
+ by auto
+ then show ?thesis
+ using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
qed simp
qed
- also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
- (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
- by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
- also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
- (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
- unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
- setsum_left_distrib[symmetric]
- unfolding joint_distribution_commute_singleton[of X]
- unfolding joint_distribution_assoc_singleton[symmetric]
- using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
- by (intro setsum_cong refl) (simp add: space_pair_measure)
- also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
- (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
- conditional_mutual_information b MX MY MZ X Y Z"
+ with I1 I2 show ?thesis
unfolding conditional_mutual_information_def
- unfolding mutual_information_generic_eq[OF finite_var(1,3)]
- unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
- by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
- finally show ?thesis by simp
+ apply (subst mi_eq)
+ apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
+ apply (subst integral_diff(2)[symmetric])
+ apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
+ done
qed
lemma (in information_space) conditional_mutual_information_eq:
- assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
- shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
- distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
- log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
- (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
- by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
- simp
+ assumes Pz: "simple_distributed M Z Pz"
+ assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"
+ assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"
+ assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz"
+ shows "\<I>(X ; Y | Z) =
+ (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x))`space M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
+proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _
+ simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz]
+ simple_distributed_joint2[OF Pxyz]])
+ note simple_distributed_joint2_finite[OF Pxyz, simp]
+ show "sigma_finite_measure (count_space (X ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ show "sigma_finite_measure (count_space (Y ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ show "sigma_finite_measure (count_space (Z ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) =
+ count_space (X`space M \<times> Y`space M \<times> Z`space M)"
+ (is "?P = ?C")
+ by (simp add: pair_measure_count_space)
-lemma (in information_space) conditional_mutual_information_eq_mutual_information:
- assumes X: "simple_function M X" and Y: "simple_function M Y"
- shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
-proof -
- have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
- have C: "simple_function M (\<lambda>x. ())" by auto
- show ?thesis
- unfolding conditional_mutual_information_eq[OF X Y C]
- unfolding mutual_information_eq[OF X Y]
- by (simp add: setsum_cartesian_product' distribution_remove_const)
+ let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)"
+ have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^isub>M count_space (Z ` space M))"
+ using simple_distributed_joint[OF Pxz] by (rule distributed_measurable)
+ from measurable_comp[OF this measurable_fst]
+ have "random_variable (count_space (X ` space M)) X"
+ by (simp add: comp_def)
+ then have "simple_function M X"
+ unfolding simple_function_def by auto
+ then have "simple_distributed M X ?Px"
+ by (rule simple_distributedI) auto
+ then show "distributed M (count_space (X ` space M)) X ?Px"
+ by (rule simple_distributed)
+
+ let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)"
+ let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)"
+ let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)"
+ show
+ "integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))"
+ "integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))"
+ by (auto intro!: integrable_count_space simp: pair_measure_count_space)
+ let ?i = "\<lambda>x y z. ?f (x, y, z) * log b (?f (x, y, z) / (?h (x, z) * (?g (y, z) / Pz z)))"
+ let ?j = "\<lambda>x y z. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z)))"
+ have "(\<lambda>(x, y, z). ?i x y z) = (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then ?j (fst x) (fst (snd x)) (snd (snd x)) else 0)"
+ by (auto intro!: ext)
+ then show "(\<integral> (x, y, z). ?i x y z \<partial>?P) = (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x)) ` space M. ?j x y z)"
+ by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum_cases split_beta')
qed
-lemma (in information_space) conditional_mutual_information_generic_positive:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
- shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
-proof (cases "space MX \<times> space MY \<times> space MZ = {}")
- case True show ?thesis
- unfolding conditional_mutual_information_generic_eq[OF assms] True
- by simp
-next
- case False
- let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
- let ?dXZ = "joint_distribution X Z"
- let ?dYZ = "joint_distribution Y Z"
- let ?dX = "distribution X"
- let ?dZ = "distribution Z"
- let ?M = "space MX \<times> space MY \<times> space MZ"
+lemma (in information_space) conditional_mutual_information_nonneg:
+ assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z"
+ shows "0 \<le> \<I>(X ; Y | Z)"
+proof -
+ def Pz \<equiv> "\<lambda>x. if x \<in> Z`space M then measure M (Z -` {x} \<inter> space M) else 0"
+ def Pxz \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x))`space M then measure M ((\<lambda>x. (X x, Z x)) -` {x} \<inter> space M) else 0"
+ def Pyz \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x))`space M then measure M ((\<lambda>x. (Y x, Z x)) -` {x} \<inter> space M) else 0"
+ def Pxyz \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then measure M ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M) else 0"
+ let ?M = "X`space M \<times> Y`space M \<times> Z`space M"
- note YZ = finite_random_variable_pairI[OF Y Z]
- note XZ = finite_random_variable_pairI[OF X Z]
- note ZX = finite_random_variable_pairI[OF Z X]
- note YZ = finite_random_variable_pairI[OF Y Z]
- note XYZ = finite_random_variable_pairI[OF X YZ]
- note finite = Z YZ XZ XYZ
- have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
- \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
- unfolding joint_distribution_commute_singleton[of X]
- unfolding joint_distribution_assoc_singleton[symmetric]
- using finite_distribution_order(6)[OF Y ZX]
- by auto
+ note XZ = simple_function_Pair[OF X Z]
+ note YZ = simple_function_Pair[OF Y Z]
+ note XYZ = simple_function_Pair[OF X simple_function_Pair[OF Y Z]]
+ have Pz: "simple_distributed M Z Pz"
+ using Z by (rule simple_distributedI) (auto simp: Pz_def)
+ have Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"
+ using XZ by (rule simple_distributedI) (auto simp: Pxz_def)
+ have Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"
+ using YZ by (rule simple_distributedI) (auto simp: Pyz_def)
+ have Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz"
+ using XYZ by (rule simple_distributedI) (auto simp: Pxyz_def)
- note order = order
- finite_distribution_order(5,6)[OF X YZ]
- finite_distribution_order(5,6)[OF Y Z]
+ { fix z assume z: "z \<in> Z ` space M" then have "(\<Sum>x\<in>X ` space M. Pxz (x, z)) = Pz z"
+ using distributed_marginal_eq_joint_simple[OF X Pz Pxz z]
+ by (auto intro!: setsum_cong simp: Pxz_def) }
+ note marginal1 = this
- have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
- log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
- unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
- also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
+ { fix z assume z: "z \<in> Z ` space M" then have "(\<Sum>y\<in>Y ` space M. Pyz (y, z)) = Pz z"
+ using distributed_marginal_eq_joint_simple[OF Y Pz Pyz z]
+ by (auto intro!: setsum_cong simp: Pyz_def) }
+ note marginal2 = this
+
+ have "- \<I>(X ; Y | Z) = - (\<Sum>(x, y, z) \<in> ?M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
+ unfolding conditional_mutual_information_eq[OF Pz Pyz Pxz Pxyz]
+ using X Y Z by (auto intro!: setsum_mono_zero_left simp: Pxyz_def simple_functionD)
+ also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. Pxz (x, z) * (Pyz (y,z) / Pz z))"
unfolding split_beta'
proof (rule log_setsum_divide)
- show "?M \<noteq> {}" using False by simp
+ show "?M \<noteq> {}" using not_empty by simp
show "1 < b" using b_gt_1 .
- show "finite ?M" using assms
- unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
-
- show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
- unfolding setsum_cartesian_product'
- unfolding setsum_commute[of _ "space MY"]
- unfolding setsum_commute[of _ "space MZ"]
- by (simp_all add: space_pair_measure
- setsum_joint_distribution_singleton[OF X YZ]
- setsum_joint_distribution_singleton[OF Y Z]
- setsum_distribution[OF Z])
+ show "finite ?M" using X Y Z by (auto simp: simple_functionD)
- fix x assume "x \<in> ?M"
- let ?x = "(fst x, fst (snd x), snd (snd x))"
-
- show "0 \<le> ?dXYZ {?x}"
- "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
- by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
-
- assume *: "0 < ?dXYZ {?x}"
- with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
- by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
+ then show "(\<Sum>x\<in>?M. Pxyz (fst x, fst (snd x), snd (snd x))) = 1"
+ apply (subst Pxyz[THEN simple_distributed_setsum_space, symmetric])
+ apply simp
+ apply (intro setsum_mono_zero_right)
+ apply (auto simp: Pxyz_def)
+ done
+ let ?N = "(\<lambda>x. (X x, Y x, Z x)) ` space M"
+ fix x assume x: "x \<in> ?M"
+ let ?Q = "Pxyz (fst x, fst (snd x), snd (snd x))"
+ let ?P = "Pxz (fst x, snd (snd x)) * (Pyz (fst (snd x), snd (snd x)) / Pz (snd (snd x)))"
+ from x show "0 \<le> ?Q" "0 \<le> ?P"
+ using Pxyz[THEN simple_distributed, THEN distributed_real_AE]
+ using Pxz[THEN simple_distributed, THEN distributed_real_AE]
+ using Pyz[THEN simple_distributed, THEN distributed_real_AE]
+ using Pz[THEN simple_distributed, THEN distributed_real_AE]
+ by (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg simp: AE_count_space Pxyz_def Pxz_def Pyz_def Pz_def)
+ moreover assume "0 < ?Q"
+ moreover have "AE x in count_space ?N. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz[THEN simple_distributed] Pz[THEN simple_distributed]]) (auto intro: measurable_snd')
+ then have "\<And>x. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
+ by (auto simp: Pz_def Pxyz_def AE_count_space)
+ moreover have "AE x in count_space ?N. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz[THEN simple_distributed] Pxz[THEN simple_distributed]]) (auto intro: measurable_Pair measurable_snd')
+ then have "\<And>x. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
+ by (auto simp: Pz_def Pxyz_def AE_count_space)
+ moreover have "AE x in count_space ?N. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
+ by (intro subdensity_real[of snd, OF _ Pxyz[THEN simple_distributed] Pyz[THEN simple_distributed]]) (auto intro: measurable_Pair)
+ then have "\<And>x. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
+ by (auto simp: Pz_def Pxyz_def AE_count_space)
+ ultimately show "0 < ?P" using x by (auto intro!: divide_pos_pos mult_pos_pos simp: less_le)
qed
- also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
+ also have "(\<Sum>(x, y, z) \<in> ?M. Pxz (x, z) * (Pyz (y,z) / Pz z)) = (\<Sum>z\<in>Z`space M. Pz z)"
apply (simp add: setsum_cartesian_product')
apply (subst setsum_commute)
apply (subst (2) setsum_commute)
- by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
- setsum_joint_distribution_singleton[OF X Z]
- setsum_joint_distribution_singleton[OF Y Z]
+ apply (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric] marginal1 marginal2
intro!: setsum_cong)
- also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
- unfolding setsum_distribution[OF Z] by simp
+ done
+ also have "log b (\<Sum>z\<in>Z`space M. Pz z) = 0"
+ using Pz[THEN simple_distributed_setsum_space] by simp
finally show ?thesis by simp
qed
-lemma (in information_space) conditional_mutual_information_positive:
- assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
- shows "0 \<le> \<I>(X;Y|Z)"
- by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
-
subsection {* Conditional Entropy *}
definition (in prob_space)
- "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
+ "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y"
abbreviation (in information_space)
conditional_entropy_Pow ("\<H>'(_ | _')") where
- "\<H>(X | Y) \<equiv> conditional_entropy b
- \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
- \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr> X Y"
-
-lemma (in information_space) conditional_entropy_positive:
- "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
- unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
+ "\<H>(X | Y) \<equiv> conditional_entropy b (count_space (X`space M)) (count_space (Y`space M)) X Y"
lemma (in information_space) conditional_entropy_generic_eq:
- fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
- assumes MX: "finite_random_variable MX X"
- assumes MZ: "finite_random_variable MZ Z"
- shows "conditional_entropy b MX MZ X Z =
- - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
- joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
+ fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
+ assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
+ assumes Px: "distributed M S X Px"
+ assumes Py: "distributed M T Y Py"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ assumes I1: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
+ assumes I2: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
+ shows "conditional_entropy b S T X Y = - (\<integral>(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y) \<partial>(S \<Otimes>\<^isub>M T))"
proof -
- interpret MX: finite_sigma_algebra MX using MX by simp
- interpret MZ: finite_sigma_algebra MZ using MZ by simp
- let ?XXZ = "\<lambda>x y z. joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
- let ?XZ = "\<lambda>x z. joint_distribution X Z {(x, z)}"
- let ?Z = "\<lambda>z. distribution Z {z}"
- let ?f = "\<lambda>x y z. log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
- { fix x z have "?XXZ x x z = ?XZ x z"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
- note this[simp]
- { fix x x' :: 'c and z assume "x' \<noteq> x"
- then have "?XXZ x x' z = 0"
- by (auto simp: distribution_def empty_measure'[symmetric]
- simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
- note this[simp]
- { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
- then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
- = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
- by (auto intro!: setsum_cong)
- also have "\<dots> = ?XZ x z * ?f x x z"
- using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
- also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
- also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
- using finite_distribution_order(6)[OF MX MZ]
- by (auto simp: log_simps field_simps zero_less_mult_iff)
- finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
- note * = this
- show ?thesis
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret ST: pair_sigma_finite S T ..
+ have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" ..
+
+ interpret Pxy: prob_space "density (S \<Otimes>\<^isub>M T) Pxy"
+ unfolding Pxy[THEN distributed_distr_eq_density, symmetric]
+ using Pxy[THEN distributed_measurable] by (rule prob_space_distr)
+
+ from Py Pxy have distr_eq: "distr M T Y =
+ distr (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) T snd"
+ by (subst distr_distr[OF measurable_snd]) (auto dest: distributed_measurable simp: comp_def)
+
+ have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)"
+ by (rule entropy_distr[OF T Py])
+ also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))"
+ using b_gt_1 Py[THEN distributed_real_measurable]
+ by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong)
+ finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" .
+
+ have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
+ by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair)
+ moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
+ by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
+ moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)"
+ using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable)
+ moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)"
+ using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
+ moreover note Pxy[THEN distributed_real_AE]
+ ultimately have pos: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Pxy x \<and> 0 \<le> Px (fst x) \<and> 0 \<le> Py (snd x) \<and>
+ (Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Px (fst x) \<and> 0 < Py (snd x)))"
+ by eventually_elim auto
+
+ from pos have "AE x in S \<Otimes>\<^isub>M T.
+ Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))"
+ by eventually_elim (auto simp: log_simps mult_pos_pos field_simps b_gt_1)
+ with I1 I2 show ?thesis
unfolding conditional_entropy_def
- unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
- by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
- setsum_commute[of _ "space MZ"] *
- intro!: setsum_cong)
+ apply (subst e_eq)
+ apply (subst entropy_distr[OF ST Pxy])
+ unfolding minus_diff_minus
+ apply (subst integral_diff(2)[symmetric])
+ apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
+ done
qed
lemma (in information_space) conditional_entropy_eq:
- assumes "simple_function M X" "simple_function M Z"
- shows "\<H>(X | Z) =
- - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
- joint_distribution X Z {(x, z)} *
- log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
- by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
- simp
+ assumes Y: "simple_distributed M Y Py" and X: "simple_function M X"
+ assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+ shows "\<H>(X | Y) = - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
+proof (subst conditional_entropy_generic_eq[OF _ _
+ simple_distributed[OF simple_distributedI[OF X refl]] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
+ have [simp]: "finite (X`space M)" using X by (simp add: simple_function_def)
+ note Y[THEN simple_distributed_finite, simp]
+ show "sigma_finite_measure (count_space (X ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ show "sigma_finite_measure (count_space (Y ` space M))"
+ by (simp add: sigma_finite_measure_count_space_finite)
+ let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
+ have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
+ (is "?P = ?C")
+ using X Y by (simp add: simple_distributed_finite pair_measure_count_space)
+ with X Y show
+ "integrable ?P (\<lambda>x. ?f x * log b (?f x))"
+ "integrable ?P (\<lambda>x. ?f x * log b (Py (snd x)))"
+ by (auto intro!: integrable_count_space simp: simple_distributed_finite)
+ have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) =
+ (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / Py (snd x)) else 0)"
+ by auto
+ from X Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) =
+ - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
+ by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum_cases split_beta')
+qed
-lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
+lemma (in information_space) conditional_mutual_information_eq_conditional_entropy:
assumes X: "simple_function M X" and Y: "simple_function M Y"
- shows "\<H>(X | Y) =
- -(\<Sum>y\<in>Y`space M. distribution Y {y} *
- (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
- log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
- unfolding conditional_entropy_eq[OF assms]
- using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
- by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib
- intro!: setsum_cong)
+ shows "\<I>(X ; X | Y) = \<H>(X | Y)"
+proof -
+ def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0"
+ def Pxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M) else 0"
+ def Pxxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, X x, Y x))`space M then measure M ((\<lambda>x. (X x, X x, Y x)) -` {x} \<inter> space M) else 0"
+ let ?M = "X`space M \<times> X`space M \<times> Y`space M"
-lemma (in information_space) conditional_entropy_eq_cartesian_product:
- assumes "simple_function M X" "simple_function M Y"
- shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
- joint_distribution X Y {(x,y)} *
- log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
- unfolding conditional_entropy_eq[OF assms]
- by (auto intro!: setsum_cong simp: setsum_cartesian_product')
+ note XY = simple_function_Pair[OF X Y]
+ note XXY = simple_function_Pair[OF X XY]
+ have Py: "simple_distributed M Y Py"
+ using Y by (rule simple_distributedI) (auto simp: Py_def)
+ have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+ using XY by (rule simple_distributedI) (auto simp: Pxy_def)
+ have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy"
+ using XXY by (rule simple_distributedI) (auto simp: Pxxy_def)
+ have eq: "(\<lambda>x. (X x, X x, Y x)) ` space M = (\<lambda>(x, y). (x, x, y)) ` (\<lambda>x. (X x, Y x)) ` space M"
+ by auto
+ have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A"
+ by (auto simp: inj_on_def)
+ have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)"
+ by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob])
+ have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
+ by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair)
+ then show ?thesis
+ apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy])
+ apply (subst conditional_entropy_eq[OF Py X Pxy])
+ apply (auto intro!: setsum_cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum_reindex[OF inj]
+ log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space)
+ using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE]
+ apply (auto simp add: not_le[symmetric] AE_count_space)
+ done
+qed
+
+lemma (in information_space) conditional_entropy_nonneg:
+ assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)"
+ using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y]
+ by simp
subsection {* Equalities *}
-lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
- assumes X: "simple_function M X" and Z: "simple_function M Z"
- shows "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
+lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr:
+ fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
+ assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
+ assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
+ assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
+ assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
+ shows "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
proof -
- let ?XZ = "\<lambda>x z. joint_distribution X Z {(x, z)}"
- let ?Z = "\<lambda>z. distribution Z {z}"
- let ?X = "\<lambda>x. distribution X {x}"
- note fX = X[THEN simple_function_imp_finite_random_variable]
- note fZ = Z[THEN simple_function_imp_finite_random_variable]
- note finite_distribution_order[OF fX fZ, simp]
- { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
- have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
- ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
- by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
- note * = this
+ have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^isub>M T))"
+ using b_gt_1 Px[THEN distributed_real_measurable]
+ apply (subst entropy_distr[OF S Px])
+ apply (subst distributed_transform_integral[OF Pxy Px, where T=fst])
+ apply (auto intro!: integral_cong)
+ done
+
+ have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
+ using b_gt_1 Py[THEN distributed_real_measurable]
+ apply (subst entropy_distr[OF T Py])
+ apply (subst distributed_transform_integral[OF Pxy Py, where T=snd])
+ apply (auto intro!: integral_cong)
+ done
+
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret ST: pair_sigma_finite S T ..
+ have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" ..
+
+ have XY: "entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^isub>M T))"
+ by (subst entropy_distr[OF ST Pxy]) (auto intro!: integral_cong)
+
+ have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
+ by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair)
+ moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
+ by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
+ moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)"
+ using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable)
+ moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)"
+ using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
+ moreover note Pxy[THEN distributed_real_AE]
+ ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)) =
+ Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
+ (is "AE x in _. ?f x = ?g x")
+ proof eventually_elim
+ case (goal1 x)
+ show ?case
+ proof cases
+ assume "Pxy x \<noteq> 0"
+ with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x"
+ by auto
+ then show ?thesis
+ using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
+ qed simp
+ qed
+
+ have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?f"
+ unfolding X Y XY
+ apply (subst integral_diff)
+ apply (intro integral_diff Ixy Ix Iy)+
+ apply (subst integral_diff)
+ apply (intro integral_diff Ixy Ix Iy)+
+ apply (simp add: field_simps)
+ done
+ also have "\<dots> = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?g"
+ using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE)
+ also have "\<dots> = mutual_information b S T X Y"
+ by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric])
+ finally show ?thesis ..
+qed
+
+lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
+ assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
+ shows "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)"
+proof -
+ have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
+ using sf_X by (rule simple_distributedI) auto
+ have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
+ using sf_Y by (rule simple_distributedI) auto
+ have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
+ using sf_X sf_Y by (rule simple_function_Pair)
+ then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
+ by (rule simple_distributedI) auto
+ from simple_distributed_joint_finite[OF this, simp]
+ have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
+ by (simp add: pair_measure_count_space)
+
+ have "\<I>(X ; Y) = \<H>(X) + \<H>(Y) - entropy b (count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x))"
+ using sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]
+ by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space)
+ then show ?thesis
+ unfolding conditional_entropy_def by simp
+qed
+
+lemma (in information_space) mutual_information_nonneg_simple:
+ assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
+ shows "0 \<le> \<I>(X ; Y)"
+proof -
+ have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
+ using sf_X by (rule simple_distributedI) auto
+ have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
+ using sf_Y by (rule simple_distributedI) auto
+
+ have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
+ using sf_X sf_Y by (rule simple_function_Pair)
+ then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
+ by (rule simple_distributedI) auto
+
+ from simple_distributed_joint_finite[OF this, simp]
+ have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
+ by (simp add: pair_measure_count_space)
+
show ?thesis
- unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
- using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
- by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
- setsum_distribution)
+ by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
+ (simp_all add: eq integrable_count_space sigma_finite_measure_count_space_finite)
qed
lemma (in information_space) conditional_entropy_less_eq_entropy:
assumes X: "simple_function M X" and Z: "simple_function M Z"
shows "\<H>(X | Z) \<le> \<H>(X)"
proof -
- have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
- with mutual_information_positive[OF X Z] entropy_positive[OF X]
- show ?thesis by auto
+ have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple)
+ also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
+ finally show ?thesis by auto
qed
lemma (in information_space) entropy_chain_rule:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
proof -
- let ?XY = "\<lambda>x y. joint_distribution X Y {(x, y)}"
- let ?Y = "\<lambda>y. distribution Y {y}"
- let ?X = "\<lambda>x. distribution X {x}"
- note fX = X[THEN simple_function_imp_finite_random_variable]
- note fY = Y[THEN simple_function_imp_finite_random_variable]
- note finite_distribution_order[OF fX fY, simp]
- { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
- have "?XY x y * log b (?XY x y / ?X x) =
- ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
- by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
- note * = this
- show ?thesis
- using setsum_joint_distribution_singleton[OF fY fX]
- unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
- unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
- by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
-qed
-
-section {* Partitioning *}
-
-definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
-
-lemma subvimageI:
- assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
- shows "subvimage A f g"
- using assms unfolding subvimage_def by blast
-
-lemma subvimageE[consumes 1]:
- assumes "subvimage A f g"
- obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
- using assms unfolding subvimage_def by blast
-
-lemma subvimageD:
- "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
- using assms unfolding subvimage_def by blast
-
-lemma subvimage_subset:
- "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
- unfolding subvimage_def by auto
-
-lemma subvimage_idem[intro]: "subvimage A g g"
- by (safe intro!: subvimageI)
-
-lemma subvimage_comp_finer[intro]:
- assumes svi: "subvimage A g h"
- shows "subvimage A g (f \<circ> h)"
-proof (rule subvimageI, simp)
- fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
- from svi[THEN subvimageD, OF this]
- show "f (h x) = f (h y)" by simp
-qed
-
-lemma subvimage_comp_gran:
- assumes svi: "subvimage A g h"
- assumes inj: "inj_on f (g ` A)"
- shows "subvimage A (f \<circ> g) h"
- by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
-
-lemma subvimage_comp:
- assumes svi: "subvimage (f ` A) g h"
- shows "subvimage A (g \<circ> f) (h \<circ> f)"
- by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
-
-lemma subvimage_trans:
- assumes fg: "subvimage A f g"
- assumes gh: "subvimage A g h"
- shows "subvimage A f h"
- by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
-
-lemma subvimage_translator:
- assumes svi: "subvimage A f g"
- shows "\<exists>h. \<forall>x \<in> A. h (f x) = g x"
-proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
- fix x assume "x \<in> A"
- show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
- by (rule theI2[of _ "g x"])
- (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
-qed
-
-lemma subvimage_translator_image:
- assumes svi: "subvimage A f g"
- shows "\<exists>h. h ` f ` A = g ` A"
-proof -
- from subvimage_translator[OF svi]
- obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
- thus ?thesis
- by (auto intro!: exI[of _ h]
- simp: image_compose[symmetric] comp_def cong: image_cong)
-qed
-
-lemma subvimage_finite:
- assumes svi: "subvimage A f g" and fin: "finite (f`A)"
- shows "finite (g`A)"
-proof -
- from subvimage_translator_image[OF svi]
- obtain h where "g`A = h`f`A" by fastforce
- with fin show "finite (g`A)" by simp
-qed
-
-lemma subvimage_disj:
- assumes svi: "subvimage A f g"
- shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
- f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
-proof (rule disjCI)
- assume "\<not> ?dist"
- then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
- thus "?sub" using svi unfolding subvimage_def by auto
-qed
-
-lemma setsum_image_split:
- assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
- shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
- (is "?lhs = ?rhs")
-proof -
- have "f ` A =
- snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
- (is "_ = snd ` ?SIGMA")
- unfolding image_split_eq_Sigma[symmetric]
- by (simp add: image_compose[symmetric] comp_def)
- moreover
- have snd_inj: "inj_on snd ?SIGMA"
- unfolding image_split_eq_Sigma[symmetric]
- by (auto intro!: inj_onI subvimageD[OF svi])
- ultimately
- have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
- by (auto simp: setsum_reindex intro: setsum_cong)
- also have "... = ?rhs"
- using subvimage_finite[OF svi fin] fin
- apply (subst setsum_Sigma[symmetric])
- by (auto intro!: finite_subset[of _ "f`A"])
- finally show ?thesis .
+ note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl]
+ note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl]
+ note simple_distributed_joint_finite[OF this, simp]
+ let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)"
+ let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)"
+ let ?h = "\<lambda>x. if x \<in> (\<lambda>x. (Y x, X x)) ` space M then prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M) else 0"
+ have "\<H>(\<lambda>x. (X x, Y x)) = - (\<Sum>x\<in>(\<lambda>x. (X x, Y x)) ` space M. ?f x * log b (?f x))"
+ using XY by (rule entropy_simple_distributed)
+ also have "\<dots> = - (\<Sum>x\<in>(\<lambda>(x, y). (y, x)) ` (\<lambda>x. (X x, Y x)) ` space M. ?g x * log b (?g x))"
+ by (subst (2) setsum_reindex) (auto simp: inj_on_def intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
+ also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))"
+ by (auto intro!: setsum_cong)
+ also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))"
+ by (subst entropy_distr[OF _ simple_distributed_joint[OF YX]])
+ (auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite
+ cong del: setsum_cong intro!: setsum_mono_zero_left)
+ finally have "\<H>(\<lambda>x. (X x, Y x)) = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" .
+ then show ?thesis
+ unfolding conditional_entropy_def by simp
qed
lemma (in information_space) entropy_partition:
- assumes sf: "simple_function M X" "simple_function M P"
- assumes svi: "subvimage (space M) X P"
- shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
+ assumes X: "simple_function M X"
+ shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)"
proof -
- let ?XP = "\<lambda>x p. joint_distribution X P {(x, p)}"
- let ?X = "\<lambda>x. distribution X {x}"
- let ?P = "\<lambda>p. distribution P {p}"
- note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
- note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
- note finite_distribution_order[OF fX fP, simp]
- have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
- (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
- proof (subst setsum_image_split[OF svi],
- safe intro!: setsum_mono_zero_cong_left imageI)
- show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
- using sf unfolding simple_function_def by auto
- next
- fix p x assume in_space: "p \<in> space M" "x \<in> space M"
- assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
- hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
- with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
- show "x \<in> P -` {P p}" by auto
- next
- fix p x assume in_space: "p \<in> space M" "x \<in> space M"
- assume "P x = P p"
- from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
- have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
- by auto
- hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
- by auto
- thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
- by (auto simp: distribution_def)
- qed
- moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
- ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
- by (auto simp add: log_simps zero_less_mult_iff field_simps)
- ultimately show ?thesis
- unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
- using setsum_joint_distribution_singleton[OF fX fP]
- by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
- setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
+ note fX = simple_function_compose[OF X, of f]
+ have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto
+ have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A"
+ by (auto simp: inj_on_def)
+ show ?thesis
+ apply (subst entropy_chain_rule[symmetric, OF fX X])
+ apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]])
+ apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
+ unfolding eq
+ apply (subst setsum_reindex[OF inj])
+ apply (auto intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
+ done
qed
corollary (in information_space) entropy_data_processing:
assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
proof -
- note X
- moreover have fX: "simple_function M (f \<circ> X)" using X by auto
- moreover have "subvimage (space M) X (f \<circ> X)" by auto
- ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
+ note fX = simple_function_compose[OF X, of f]
+ from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
then show "\<H>(f \<circ> X) \<le> \<H>(X)"
- by (auto intro: conditional_entropy_positive[OF X fX])
+ by (auto intro: conditional_entropy_nonneg[OF X fX])
qed
corollary (in information_space) entropy_of_inj:
@@ -1411,7 +1313,11 @@
have sf: "simple_function M (f \<circ> X)"
using X by auto
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
- by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
+ unfolding o_assoc
+ apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
+ apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_compose[OF X]], where f="\<lambda>x. prob (X -` {x} \<inter> space M)"])
+ apply (auto intro!: setsum_cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def)
+ done
also have "... \<le> \<H>(f \<circ> X)"
using entropy_data_processing[OF sf] .
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
--- a/src/HOL/Probability/Lebesgue_Integration.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Mon Apr 23 12:14:35 2012 +0200
@@ -6,9 +6,13 @@
header {*Lebesgue Integration*}
theory Lebesgue_Integration
- imports Measure Borel_Space
+ imports Measure_Space Borel_Space
begin
+lemma ereal_minus_eq_PInfty_iff:
+ fixes x y :: ereal shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
+ by (cases x y rule: ereal2_cases) simp_all
+
lemma real_ereal_1[simp]: "real (1::ereal) = 1"
unfolding one_ereal_def by simp
@@ -28,17 +32,17 @@
by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
qed
-lemma (in measure_space) measure_Union:
+lemma measure_Union:
assumes "finite S" "S \<subseteq> sets M" "\<And>A B. A \<in> S \<Longrightarrow> B \<in> S \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
- shows "setsum \<mu> S = \<mu> (\<Union>S)"
+ shows "setsum (emeasure M) S = (emeasure M) (\<Union>S)"
proof -
- have "setsum \<mu> S = \<mu> (\<Union>i\<in>S. i)"
- using assms by (intro measure_setsum[OF `finite S`]) (auto simp: disjoint_family_on_def)
- also have "\<dots> = \<mu> (\<Union>S)" by (auto intro!: arg_cong[where f=\<mu>])
+ have "setsum (emeasure M) S = (emeasure M) (\<Union>i\<in>S. i)"
+ using assms by (intro setsum_emeasure[OF _ _ `finite S`]) (auto simp: disjoint_family_on_def)
+ also have "\<dots> = (emeasure M) (\<Union>S)" by (auto intro!: arg_cong[where f="emeasure M"])
finally show ?thesis .
qed
-lemma (in sigma_algebra) measurable_sets2[intro]:
+lemma measurable_sets2[intro]:
assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
and "A \<in> sets M'" "B \<in> sets M''"
shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
@@ -55,7 +59,7 @@
lemma borel_measurable_real_floor:
"(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
- unfolding borel.borel_measurable_iff_ge
+ unfolding borel_measurable_iff_ge
proof (intro allI)
fix a :: real
{ fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
@@ -65,19 +69,7 @@
then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
qed
-lemma measure_preservingD2:
- "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
- unfolding measure_preserving_def by auto
-
-lemma measure_preservingD3:
- "f \<in> measure_preserving A B \<Longrightarrow> f \<in> space A \<rightarrow> space B"
- unfolding measure_preserving_def measurable_def by auto
-
-lemma measure_preservingD:
- "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
- unfolding measure_preserving_def by auto
-
-lemma (in sigma_algebra) borel_measurable_real_natfloor[intro, simp]:
+lemma borel_measurable_real_natfloor[intro, simp]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
proof -
@@ -87,8 +79,8 @@
show ?thesis by (simp add: comp_def)
qed
-lemma (in measure_space) AE_not_in:
- assumes N: "N \<in> null_sets" shows "AE x. x \<notin> N"
+lemma AE_not_in:
+ assumes N: "N \<in> null_sets M" shows "AE x in M. x \<notin> N"
using N by (rule AE_I') auto
lemma sums_If_finite:
@@ -128,7 +120,7 @@
finite (g ` space M) \<and>
(\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
-lemma (in sigma_algebra) simple_functionD:
+lemma simple_functionD:
assumes "simple_function M g"
shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
proof -
@@ -140,7 +132,7 @@
by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
qed
-lemma (in sigma_algebra) simple_function_measurable2[intro]:
+lemma simple_function_measurable2[intro]:
assumes "simple_function M f" "simple_function M g"
shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
proof -
@@ -149,7 +141,7 @@
then show ?thesis using assms[THEN simple_functionD(2)] by auto
qed
-lemma (in sigma_algebra) simple_function_indicator_representation:
+lemma simple_function_indicator_representation:
fixes f ::"'a \<Rightarrow> ereal"
assumes f: "simple_function M f" and x: "x \<in> space M"
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
@@ -164,7 +156,7 @@
finally show ?thesis by auto
qed
-lemma (in measure_space) simple_function_notspace:
+lemma simple_function_notspace:
"simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
proof -
have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
@@ -173,7 +165,7 @@
thus ?thesis unfolding simple_function_def by auto
qed
-lemma (in sigma_algebra) simple_function_cong:
+lemma simple_function_cong:
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "simple_function M f \<longleftrightarrow> simple_function M g"
proof -
@@ -183,12 +175,12 @@
thus ?thesis unfolding simple_function_def using assms by simp
qed
-lemma (in sigma_algebra) simple_function_cong_algebra:
+lemma simple_function_cong_algebra:
assumes "sets N = sets M" "space N = space M"
shows "simple_function M f \<longleftrightarrow> simple_function N f"
unfolding simple_function_def assms ..
-lemma (in sigma_algebra) borel_measurable_simple_function:
+lemma borel_measurable_simple_function:
assumes "simple_function M f"
shows "f \<in> borel_measurable M"
proof (rule borel_measurableI)
@@ -204,24 +196,23 @@
thus "f -` S \<inter> space M \<in> sets M" unfolding * .
qed
-lemma (in sigma_algebra) simple_function_borel_measurable:
+lemma simple_function_borel_measurable:
fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
shows "simple_function M f"
using assms unfolding simple_function_def
by (auto intro: borel_measurable_vimage)
-lemma (in sigma_algebra) simple_function_eq_borel_measurable:
+lemma simple_function_eq_borel_measurable:
fixes f :: "'a \<Rightarrow> ereal"
shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
- using simple_function_borel_measurable[of f]
- borel_measurable_simple_function[of f]
+ using simple_function_borel_measurable[of f] borel_measurable_simple_function[of M f]
by (fastforce simp: simple_function_def)
-lemma (in sigma_algebra) simple_function_const[intro, simp]:
+lemma simple_function_const[intro, simp]:
"simple_function M (\<lambda>x. c)"
by (auto intro: finite_subset simp: simple_function_def)
-lemma (in sigma_algebra) simple_function_compose[intro, simp]:
+lemma simple_function_compose[intro, simp]:
assumes "simple_function M f"
shows "simple_function M (g \<circ> f)"
unfolding simple_function_def
@@ -238,7 +229,7 @@
by (rule_tac finite_UN) (auto intro!: finite_UN)
qed
-lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
+lemma simple_function_indicator[intro, simp]:
assumes "A \<in> sets M"
shows "simple_function M (indicator A)"
proof -
@@ -250,7 +241,7 @@
using assms by (auto simp: indicator_def [abs_def])
qed
-lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
+lemma simple_function_Pair[intro, simp]:
assumes "simple_function M f"
assumes "simple_function M g"
shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
@@ -268,13 +259,13 @@
using assms unfolding simple_function_def by auto
qed
-lemma (in sigma_algebra) simple_function_compose1:
+lemma simple_function_compose1:
assumes "simple_function M f"
shows "simple_function M (\<lambda>x. g (f x))"
using simple_function_compose[OF assms, of g]
by (simp add: comp_def)
-lemma (in sigma_algebra) simple_function_compose2:
+lemma simple_function_compose2:
assumes "simple_function M f" and "simple_function M g"
shows "simple_function M (\<lambda>x. h (f x) (g x))"
proof -
@@ -283,7 +274,7 @@
thus ?thesis by (simp_all add: comp_def)
qed
-lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
+lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
@@ -291,24 +282,24 @@
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
-lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
+lemma simple_function_setsum[intro, simp]:
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
proof cases
assume "finite P" from this assms show ?thesis by induct auto
qed auto
-lemma (in sigma_algebra)
+lemma
fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
shows simple_function_ereal[intro, simp]: "simple_function M (\<lambda>x. ereal (f x))"
by (auto intro!: simple_function_compose1[OF sf])
-lemma (in sigma_algebra)
+lemma
fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
by (auto intro!: simple_function_compose1[OF sf])
-lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
+lemma borel_measurable_implies_simple_function_sequence:
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "u \<in> borel_measurable M"
shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
@@ -416,14 +407,14 @@
qed (auto simp: divide_nonneg_pos)
qed
-lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
+lemma borel_measurable_implies_simple_function_sequence':
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "u \<in> borel_measurable M"
obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
"\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
using borel_measurable_implies_simple_function_sequence[OF u] by auto
-lemma (in sigma_algebra) simple_function_If_set:
+lemma simple_function_If_set:
assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
proof -
@@ -445,7 +436,7 @@
qed
qed
-lemma (in sigma_algebra) simple_function_If:
+lemma simple_function_If:
assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
shows "simple_function M (\<lambda>x. if P x then f x else g x)"
proof -
@@ -453,58 +444,17 @@
with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
qed
-lemma (in measure_space) simple_function_restricted:
- fixes f :: "'a \<Rightarrow> ereal" assumes "A \<in> sets M"
- shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A x)"
- (is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
-proof -
- interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
- have f: "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
- proof cases
- assume "A = space M"
- then have "f`A = ?f`space M" by (fastforce simp: image_iff)
- then show ?thesis by simp
- next
- assume "A \<noteq> space M"
- then obtain x where x: "x \<in> space M" "x \<notin> A"
- using sets_into_space `A \<in> sets M` by auto
- have *: "?f`space M = f`A \<union> {0}"
- proof (auto simp add: image_iff)
- show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
- using x by (auto intro!: bexI[of _ x])
- next
- fix x assume "x \<in> A"
- then show "\<exists>y\<in>space M. f x = f y * indicator A y"
- using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
- next
- fix x
- assume "indicator A x \<noteq> (0::ereal)"
- then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
- moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
- ultimately show "f x = 0" by auto
- qed
- then show ?thesis by auto
- qed
- then show ?thesis
- unfolding simple_function_eq_borel_measurable
- R.simple_function_eq_borel_measurable
- unfolding borel_measurable_restricted[OF `A \<in> sets M`]
- using assms(1)[THEN sets_into_space]
- by (auto simp: indicator_def)
-qed
-
-lemma (in sigma_algebra) simple_function_subalgebra:
+lemma simple_function_subalgebra:
assumes "simple_function N f"
and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
shows "simple_function M f"
using assms unfolding simple_function_def by auto
-lemma (in measure_space) simple_function_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
+lemma simple_function_comp:
+ assumes T: "T \<in> measurable M M'"
and f: "simple_function M' f"
shows "simple_function M (\<lambda>x. f (T x))"
proof (intro simple_function_def[THEN iffD2] conjI ballI)
- interpret T: sigma_algebra M' by fact
have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
using T unfolding measurable_def by auto
then show "finite ((\<lambda>x. f (T x)) ` space M)"
@@ -523,16 +473,16 @@
section "Simple integral"
-definition simple_integral_def:
- "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {x} \<inter> space M))"
+definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^isup>S") where
+ "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
syntax
- "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ereal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
+ "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
translations
- "\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
+ "\<integral>\<^isup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
-lemma (in measure_space) simple_integral_cong:
+lemma simple_integral_cong:
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "integral\<^isup>S M f = integral\<^isup>S M g"
proof -
@@ -542,19 +492,8 @@
thus ?thesis unfolding simple_integral_def by simp
qed
-lemma (in measure_space) simple_integral_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
- and "simple_function M f"
- shows "integral\<^isup>S N f = integral\<^isup>S M f"
-proof -
- interpret v: measure_space N
- by (rule measure_space_cong) fact+
- from simple_functionD[OF `simple_function M f`] assms show ?thesis
- by (auto intro!: setsum_cong simp: simple_integral_def)
-qed
-
-lemma (in measure_space) simple_integral_const[simp]:
- "(\<integral>\<^isup>Sx. c \<partial>M) = c * \<mu> (space M)"
+lemma simple_integral_const[simp]:
+ "(\<integral>\<^isup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
proof (cases "space M = {}")
case True thus ?thesis unfolding simple_integral_def by simp
next
@@ -562,9 +501,9 @@
thus ?thesis unfolding simple_integral_def by simp
qed
-lemma (in measure_space) simple_function_partition:
+lemma simple_function_partition:
assumes f: "simple_function M f" and g: "simple_function M g"
- shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
+ shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * (emeasure M) A)"
(is "_ = setsum _ (?p ` space M)")
proof-
let ?sub = "\<lambda>x. ?p ` (f -` {x} \<inter> space M)"
@@ -586,13 +525,13 @@
{ fix x assume "x \<in> space M"
have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
- with sets have "\<mu> (f -` {f x} \<inter> space M) = setsum \<mu> (?sub (f x))"
+ with sets have "(emeasure M) (f -` {f x} \<inter> space M) = setsum (emeasure M) (?sub (f x))"
by (subst measure_Union) auto }
- hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
+ hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * (emeasure M) A)"
unfolding simple_integral_def using f sets
by (subst setsum_Sigma[symmetric])
(auto intro!: setsum_cong setsum_ereal_right_distrib)
- also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
+ also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * (emeasure M) A)"
proof -
have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
@@ -610,7 +549,7 @@
finally show ?thesis .
qed
-lemma (in measure_space) simple_integral_add[simp]:
+lemma simple_integral_add[simp]:
assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
proof -
@@ -628,7 +567,7 @@
(auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
qed
-lemma (in measure_space) simple_integral_setsum[simp]:
+lemma simple_integral_setsum[simp]:
assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
@@ -638,11 +577,11 @@
by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
qed auto
-lemma (in measure_space) simple_integral_mult[simp]:
+lemma simple_integral_mult[simp]:
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
proof -
- note mult = simple_function_mult[OF simple_function_const[of c] f(1)]
+ note mult = simple_function_mult[OF simple_function_const[of _ c] f(1)]
{ fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
assume "x \<in> space M"
hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
@@ -654,9 +593,9 @@
(auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc)
qed
-lemma (in measure_space) simple_integral_mono_AE:
+lemma simple_integral_mono_AE:
assumes f: "simple_function M f" and g: "simple_function M g"
- and mono: "AE x. f x \<le> g x"
+ and mono: "AE x in M. f x \<le> g x"
shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
proof -
let ?S = "\<lambda>x. (g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
@@ -669,55 +608,55 @@
proof (safe intro!: setsum_mono)
fix x assume "x \<in> space M"
then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
- show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
+ show "the_elem (f`?S x) * (emeasure M) (?S x) \<le> the_elem (g`?S x) * (emeasure M) (?S x)"
proof (cases "f x \<le> g x")
case True then show ?thesis
using * assms(1,2)[THEN simple_functionD(2)]
by (auto intro!: ereal_mult_right_mono)
next
case False
- obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
+ obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "(emeasure M) N = 0"
using mono by (auto elim!: AE_E)
have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
moreover have "?S x \<in> sets M" using assms
by (rule_tac Int) (auto intro!: simple_functionD)
- ultimately have "\<mu> (?S x) \<le> \<mu> N"
- using `N \<in> sets M` by (auto intro!: measure_mono)
- moreover have "0 \<le> \<mu> (?S x)"
+ ultimately have "(emeasure M) (?S x) \<le> (emeasure M) N"
+ using `N \<in> sets M` by (auto intro!: emeasure_mono)
+ moreover have "0 \<le> (emeasure M) (?S x)"
using assms(1,2)[THEN simple_functionD(2)] by auto
- ultimately have "\<mu> (?S x) = 0" using `\<mu> N = 0` by auto
+ ultimately have "(emeasure M) (?S x) = 0" using `(emeasure M) N = 0` by auto
then show ?thesis by simp
qed
qed
qed
-lemma (in measure_space) simple_integral_mono:
+lemma simple_integral_mono:
assumes "simple_function M f" and "simple_function M g"
and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
using assms by (intro simple_integral_mono_AE) auto
-lemma (in measure_space) simple_integral_cong_AE:
+lemma simple_integral_cong_AE:
assumes "simple_function M f" and "simple_function M g"
- and "AE x. f x = g x"
+ and "AE x in M. f x = g x"
shows "integral\<^isup>S M f = integral\<^isup>S M g"
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
-lemma (in measure_space) simple_integral_cong':
+lemma simple_integral_cong':
assumes sf: "simple_function M f" "simple_function M g"
- and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
+ and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
shows "integral\<^isup>S M f = integral\<^isup>S M g"
proof (intro simple_integral_cong_AE sf AE_I)
- show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
+ show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
using sf[THEN borel_measurable_simple_function] by auto
qed simp
-lemma (in measure_space) simple_integral_indicator:
+lemma simple_integral_indicator:
assumes "A \<in> sets M"
assumes "simple_function M f"
shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
- (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
+ (\<Sum>x \<in> f ` space M. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
proof cases
assume "A = space M"
moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f"
@@ -737,7 +676,7 @@
show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
qed
have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
- (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
+ (\<Sum>x \<in> f ` space M \<union> {0}. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
unfolding simple_integral_def I
proof (rule setsum_mono_zero_cong_left)
show "finite (f ` space M \<union> {0})"
@@ -747,118 +686,83 @@
have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
by (auto simp: image_iff)
thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
- i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
+ i * (emeasure M) (f -` {i} \<inter> space M \<inter> A) = 0" by auto
next
fix x assume "x \<in> f`A \<union> {0}"
hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
by (auto simp: indicator_def split: split_if_asm)
- thus "x * \<mu> (?I -` {x} \<inter> space M) =
- x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
+ thus "x * (emeasure M) (?I -` {x} \<inter> space M) =
+ x * (emeasure M) (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
qed
show ?thesis unfolding *
using assms(2) unfolding simple_function_def
by (auto intro!: setsum_mono_zero_cong_right)
qed
-lemma (in measure_space) simple_integral_indicator_only[simp]:
+lemma simple_integral_indicator_only[simp]:
assumes "A \<in> sets M"
- shows "integral\<^isup>S M (indicator A) = \<mu> A"
+ shows "integral\<^isup>S M (indicator A) = emeasure M A"
proof cases
assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
next
assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto
thus ?thesis
- using simple_integral_indicator[OF assms simple_function_const[of 1]]
+ using simple_integral_indicator[OF assms simple_function_const[of _ 1]]
using sets_into_space[OF assms]
- by (auto intro!: arg_cong[where f="\<mu>"])
+ by (auto intro!: arg_cong[where f="(emeasure M)"])
qed
-lemma (in measure_space) simple_integral_null_set:
- assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets"
+lemma simple_integral_null_set:
+ assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
proof -
- have "AE x. indicator N x = (0 :: ereal)"
- using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
+ have "AE x in M. indicator N x = (0 :: ereal)"
+ using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
using assms apply (intro simple_integral_cong_AE) by auto
then show ?thesis by simp
qed
-lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
- assumes sf: "simple_function M f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
+lemma simple_integral_cong_AE_mult_indicator:
+ assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)"
using assms by (intro simple_integral_cong_AE) auto
-lemma (in measure_space) simple_integral_restricted:
- assumes "A \<in> sets M"
- assumes sf: "simple_function M (\<lambda>x. f x * indicator A x)"
- shows "integral\<^isup>S (restricted_space A) f = (\<integral>\<^isup>Sx. f x * indicator A x \<partial>M)"
- (is "_ = integral\<^isup>S M ?f")
+lemma simple_integral_distr:
+ assumes T: "T \<in> measurable M M'"
+ and f: "simple_function M' f"
+ shows "integral\<^isup>S (distr M M' T) f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
unfolding simple_integral_def
-proof (simp, safe intro!: setsum_mono_zero_cong_left)
- from sf show "finite (?f ` space M)"
- unfolding simple_function_def by auto
+proof (intro setsum_mono_zero_cong_right ballI)
+ show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space (distr M M' T)"
+ using T unfolding measurable_def by auto
+ show "finite (f ` space (distr M M' T))"
+ using f unfolding simple_function_def by auto
next
- fix x assume "x \<in> A"
- then show "f x \<in> ?f ` space M"
- using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
-next
- fix x assume "x \<in> space M" "?f x \<notin> f`A"
- then have "x \<notin> A" by (auto simp: image_iff)
- then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
+ fix i assume "i \<in> f ` space (distr M M' T) - (\<lambda>x. f (T x)) ` space M"
+ then have "T -` (f -` {i} \<inter> space (distr M M' T)) \<inter> space M = {}" by (auto simp: image_iff)
+ with f[THEN simple_functionD(2), of "{i}"]
+ show "i * emeasure (distr M M' T) (f -` {i} \<inter> space (distr M M' T)) = 0"
+ using T by (simp add: emeasure_distr)
next
- fix x assume "x \<in> A"
- then have "f x \<noteq> 0 \<Longrightarrow>
- f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
- using `A \<in> sets M` sets_into_space
- by (auto simp: indicator_def split: split_if_asm)
- then show "f x * \<mu> (f -` {f x} \<inter> A) =
- f x * \<mu> (?f -` {f x} \<inter> space M)"
- unfolding ereal_mult_cancel_left by auto
+ fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
+ then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
+ using T unfolding measurable_def by auto
+ with f[THEN simple_functionD(2), of "{i}"] T
+ show "i * emeasure (distr M M' T) (f -` {i} \<inter> space (distr M M' T)) =
+ i * (emeasure M) ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
+ by (auto simp add: emeasure_distr)
qed
-lemma (in measure_space) simple_integral_subalgebra:
- assumes N: "measure_space N" and [simp]: "space N = space M" "measure N = measure M"
- shows "integral\<^isup>S N = integral\<^isup>S M"
- unfolding simple_integral_def [abs_def] by simp
-
-lemma (in measure_space) simple_integral_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
- and f: "simple_function M' f"
- shows "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
-proof -
- interpret T: measure_space M' by (rule measure_space_vimage[OF T])
- show "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
- unfolding simple_integral_def
- proof (intro setsum_mono_zero_cong_right ballI)
- show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
- using T unfolding measurable_def measure_preserving_def by auto
- show "finite (f ` space M')"
- using f unfolding simple_function_def by auto
- next
- fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
- then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
- with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
- show "i * T.\<mu> (f -` {i} \<inter> space M') = 0" by simp
- next
- fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
- then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
- using T unfolding measurable_def measure_preserving_def by auto
- with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
- show "i * T.\<mu> (f -` {i} \<inter> space M') = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
- by auto
- qed
-qed
-
-lemma (in measure_space) simple_integral_cmult_indicator:
+lemma simple_integral_cmult_indicator:
assumes A: "A \<in> sets M"
- shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * \<mu> A"
+ shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * (emeasure M) A"
using simple_integral_mult[OF simple_function_indicator[OF A]]
unfolding simple_integral_indicator_only[OF A] by simp
-lemma (in measure_space) simple_integral_positive:
- assumes f: "simple_function M f" and ae: "AE x. 0 \<le> f x"
+lemma simple_integral_positive:
+ assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
shows "0 \<le> integral\<^isup>S M f"
proof -
have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
@@ -868,29 +772,23 @@
section "Continuous positive integration"
-definition positive_integral_def:
+definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^isup>P") where
"integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
syntax
- "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ereal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
+ "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
translations
- "\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
+ "\<integral>\<^isup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
-lemma (in measure_space) positive_integral_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
- shows "integral\<^isup>P N f = integral\<^isup>P M f"
- unfolding positive_integral_def
- unfolding simple_function_cong_algebra[OF assms(2,3), symmetric]
- using AE_cong_measure[OF assms]
- using simple_integral_cong_measure[OF assms]
- by (auto intro!: SUP_cong)
-
-lemma (in measure_space) positive_integral_positive:
+lemma positive_integral_positive:
"0 \<le> integral\<^isup>P M f"
by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
-lemma (in measure_space) positive_integral_def_finite:
+lemma positive_integral_not_MInfty[simp]: "integral\<^isup>P M f \<noteq> -\<infinity>"
+ using positive_integral_positive[of M f] by auto
+
+lemma positive_integral_def_finite:
"integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
(is "_ = SUPR ?A ?f")
unfolding positive_integral_def
@@ -898,7 +796,7 @@
fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
note gM = g(1)[THEN borel_measurable_simple_function]
- have \<mu>G_pos: "0 \<le> \<mu> ?G" using gM by auto
+ have \<mu>G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
apply (safe intro!: simple_function_max simple_function_If)
@@ -907,21 +805,22 @@
show "integral\<^isup>S M g \<le> SUPR ?A ?f"
proof cases
have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
- assume "\<mu> ?G = 0"
- with gM have "AE x. x \<notin> ?G" by (simp add: AE_iff_null_set)
+ assume "(emeasure M) ?G = 0"
+ with gM have "AE x in M. x \<notin> ?G"
+ by (auto simp add: AE_iff_null intro!: null_setsI)
with gM g show ?thesis
by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
(auto simp: max_def intro!: simple_function_If)
next
- assume \<mu>G: "\<mu> ?G \<noteq> 0"
+ assume \<mu>G: "(emeasure M) ?G \<noteq> 0"
have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
proof (intro SUP_PInfty)
fix n :: nat
- let ?y = "ereal (real n) / (if \<mu> ?G = \<infinity> then 1 else \<mu> ?G)"
+ let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: ereal_divide_eq)
then have "?g ?y \<in> ?A" by (rule g_in_A)
- have "real n \<le> ?y * \<mu> ?G"
- using \<mu>G \<mu>G_pos by (cases "\<mu> ?G") (auto simp: field_simps)
+ have "real n \<le> ?y * (emeasure M) ?G"
+ using \<mu>G \<mu>G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
using `0 \<le> ?y` `?g ?y \<in> ?A` gM
by (subst simple_integral_cmult_indicator) auto
@@ -934,15 +833,15 @@
qed
qed (auto intro: SUP_upper)
-lemma (in measure_space) positive_integral_mono_AE:
- assumes ae: "AE x. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
+lemma positive_integral_mono_AE:
+ assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
unfolding positive_integral_def
proof (safe intro!: SUP_mono)
fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
from ae[THEN AE_E] guess N . note N = this
- then have ae_N: "AE x. x \<notin> N" by (auto intro: AE_not_in)
+ then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
let ?n = "\<lambda>x. n x * indicator (space M - N) x"
- have "AE x. n x \<le> ?n x" "simple_function M ?n"
+ have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
using n N ae_N by auto
moreover
{ fix x have "?n x \<le> max 0 (v x)"
@@ -959,19 +858,19 @@
by force
qed
-lemma (in measure_space) positive_integral_mono:
+lemma positive_integral_mono:
"(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
by (auto intro: positive_integral_mono_AE)
-lemma (in measure_space) positive_integral_cong_AE:
- "AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
+lemma positive_integral_cong_AE:
+ "AE x in M. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
by (auto simp: eq_iff intro!: positive_integral_mono_AE)
-lemma (in measure_space) positive_integral_cong:
+lemma positive_integral_cong:
"(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
by (auto intro: positive_integral_cong_AE)
-lemma (in measure_space) positive_integral_eq_simple_integral:
+lemma positive_integral_eq_simple_integral:
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
proof -
let ?f = "\<lambda>x. f x * indicator (space M) x"
@@ -987,10 +886,10 @@
by (simp cong: positive_integral_cong simple_integral_cong)
qed
-lemma (in measure_space) positive_integral_eq_simple_integral_AE:
- assumes f: "simple_function M f" "AE x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
+lemma positive_integral_eq_simple_integral_AE:
+ assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
proof -
- have "AE x. f x = max 0 (f x)" using f by (auto split: split_max)
+ have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
add: positive_integral_eq_simple_integral)
@@ -998,7 +897,7 @@
by (auto intro!: simple_integral_cong_AE split: split_max)
qed
-lemma (in measure_space) positive_integral_SUP_approx:
+lemma positive_integral_SUP_approx:
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
@@ -1028,7 +927,7 @@
note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
- have measure_conv: "\<And>i. \<mu> (u -` {i} \<inter> space M) = (SUP n. \<mu> (?B' i n))"
+ have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
proof -
fix i
have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
@@ -1051,17 +950,17 @@
thus ?thesis using `x \<in> space M` by auto
qed
qed
- then show "?thesis i" using continuity_from_below[OF 1 2] by simp
+ then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
qed
have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
unfolding simple_integral_indicator[OF B `simple_function M u`]
proof (subst SUPR_ereal_setsum, safe)
fix x n assume "x \<in> space M"
- with u_range show "incseq (\<lambda>i. u x * \<mu> (?B' (u x) i))" "\<And>i. 0 \<le> u x * \<mu> (?B' (u x) i)"
- using B_mono B_u by (auto intro!: measure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
+ with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
+ using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
next
- show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (?B' i n))"
+ show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
using measure_conv u_range B_u unfolding simple_integral_def
by (auto intro!: setsum_cong SUPR_ereal_cmult[symmetric])
qed
@@ -1089,7 +988,7 @@
ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
qed
-lemma (in measure_space) incseq_positive_integral:
+lemma incseq_positive_integral:
assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
proof -
have "\<And>i x. f i x \<le> f (Suc i) x"
@@ -1099,7 +998,7 @@
qed
text {* Beppo-Levi monotone convergence theorem *}
-lemma (in measure_space) positive_integral_monotone_convergence_SUP:
+lemma positive_integral_monotone_convergence_SUP:
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
proof (rule antisym)
@@ -1107,7 +1006,7 @@
by (auto intro!: SUP_least SUP_upper positive_integral_mono)
next
show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
- unfolding positive_integral_def_finite[of "\<lambda>x. SUP i. f i x"]
+ unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"]
proof (safe intro!: SUP_least)
fix g assume g: "simple_function M g"
and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
@@ -1119,15 +1018,15 @@
qed
qed
-lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE:
- assumes f: "\<And>i. AE x. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
+lemma positive_integral_monotone_convergence_SUP_AE:
+ assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
proof -
- from f have "AE x. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
+ from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
by (simp add: AE_all_countable)
from this[THEN AE_E] guess N . note N = this
let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
- have f_eq: "AE x. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ N])
+ have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
by (auto intro!: positive_integral_cong_AE)
also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
@@ -1143,14 +1042,14 @@
finally show ?thesis .
qed
-lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE_incseq:
- assumes f: "incseq f" "\<And>i. AE x. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
+lemma positive_integral_monotone_convergence_SUP_AE_incseq:
+ assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
using f[unfolded incseq_Suc_iff le_fun_def]
by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
auto
-lemma (in measure_space) positive_integral_monotone_convergence_simple:
+lemma positive_integral_monotone_convergence_simple:
assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
@@ -1161,7 +1060,7 @@
"(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
by (simp add: le_fun_def positive_integral_def)
-lemma (in measure_space) positive_integral_cong_pos:
+lemma positive_integral_cong_pos:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
shows "integral\<^isup>P M f = integral\<^isup>P M g"
proof -
@@ -1174,10 +1073,10 @@
then show ?thesis by (simp add: positive_integral_max_0)
qed
-lemma (in measure_space) SUP_simple_integral_sequences:
+lemma SUP_simple_integral_sequences:
assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
- and eq: "AE x. (SUP i. f i x) = (SUP i. g i x)"
+ and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
(is "SUPR _ ?F = SUPR _ ?G")
proof -
@@ -1190,32 +1089,11 @@
finally show ?thesis by simp
qed
-lemma (in measure_space) positive_integral_const[simp]:
- "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
+lemma positive_integral_const[simp]:
+ "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
by (subst positive_integral_eq_simple_integral) auto
-lemma (in measure_space) positive_integral_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
- and f: "f \<in> borel_measurable M'"
- shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
-proof -
- interpret T: measure_space M' by (rule measure_space_vimage[OF T])
- from T.borel_measurable_implies_simple_function_sequence'[OF f]
- guess f' . note f' = this
- let ?f = "\<lambda>i x. f' i (T x)"
- have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
- have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
- using f'(4) .
- have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
- using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)] f'(1)] .
- show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
- using
- T.positive_integral_monotone_convergence_simple[OF f'(2,5,1)]
- positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
- by (simp add: positive_integral_max_0 simple_integral_vimage[OF T f'(1)] f')
-qed
-
-lemma (in measure_space) positive_integral_linear:
+lemma positive_integral_linear:
assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
@@ -1254,7 +1132,7 @@
by (subst SUPR_ereal_cmult[symmetric, OF u(6) `0 \<le> a`])
(auto intro!: SUPR_ereal_add
simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) }
- then show "AE x. (SUP i. l i x) = (SUP i. ?L' i x)"
+ then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
by (intro AE_I2) (auto split: split_max simp add: ereal_add_nonneg_nonneg)
qed
@@ -1268,8 +1146,8 @@
then show ?thesis by (simp add: positive_integral_max_0)
qed
-lemma (in measure_space) positive_integral_cmult:
- assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
+lemma positive_integral_cmult:
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "0 \<le> c"
shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
proof -
have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
@@ -1277,68 +1155,68 @@
have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
by (simp add: positive_integral_max_0)
then show ?thesis
- using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" "\<lambda>x. 0"] f
+ using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
by (auto simp: positive_integral_max_0)
qed
-lemma (in measure_space) positive_integral_multc:
- assumes "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
+lemma positive_integral_multc:
+ assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "0 \<le> c"
shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
-lemma (in measure_space) positive_integral_indicator[simp]:
- "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = \<mu> A"
+lemma positive_integral_indicator[simp]:
+ "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = (emeasure M) A"
by (subst positive_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
-lemma (in measure_space) positive_integral_cmult_indicator:
- "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
+lemma positive_integral_cmult_indicator:
+ "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
by (subst positive_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
-lemma (in measure_space) positive_integral_add:
- assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
- and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
+lemma positive_integral_add:
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
proof -
- have ae: "AE x. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
+ have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
using assms by (auto split: split_max simp: ereal_add_nonneg_nonneg)
have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
by (simp add: positive_integral_max_0)
also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
unfolding ae[THEN positive_integral_cong_AE] ..
also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
- using positive_integral_linear[of "\<lambda>x. max 0 (f x)" 1 "\<lambda>x. max 0 (g x)"] f g
+ using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
by auto
finally show ?thesis
by (simp add: positive_integral_max_0)
qed
-lemma (in measure_space) positive_integral_setsum:
- assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x. 0 \<le> f i x"
+lemma positive_integral_setsum:
+ assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
proof cases
assume f: "finite P"
- from assms have "AE x. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
+ from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
from f this assms(1) show ?thesis
proof induct
case (insert i P)
- then have "f i \<in> borel_measurable M" "AE x. 0 \<le> f i x"
- "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x. 0 \<le> (\<Sum>i\<in>P. f i x)"
+ then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
+ "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)"
by (auto intro!: borel_measurable_ereal_setsum setsum_nonneg)
from positive_integral_add[OF this]
show ?case using insert by auto
qed simp
qed simp
-lemma (in measure_space) positive_integral_Markov_inequality:
- assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
- shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
- (is "\<mu> ?A \<le> _ * ?PI")
+lemma positive_integral_Markov_inequality:
+ assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
+ shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
+ (is "(emeasure M) ?A \<le> _ * ?PI")
proof -
have "?A \<in> sets M"
using `A \<in> sets M` u by auto
- hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
+ hence "(emeasure M) ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
using positive_integral_indicator by simp
also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
by (auto intro!: positive_integral_mono_AE
@@ -1349,17 +1227,17 @@
finally show ?thesis .
qed
-lemma (in measure_space) positive_integral_noteq_infinite:
- assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
+lemma positive_integral_noteq_infinite:
+ assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
and "integral\<^isup>P M g \<noteq> \<infinity>"
- shows "AE x. g x \<noteq> \<infinity>"
+ shows "AE x in M. g x \<noteq> \<infinity>"
proof (rule ccontr)
- assume c: "\<not> (AE x. g x \<noteq> \<infinity>)"
- have "\<mu> {x\<in>space M. g x = \<infinity>} \<noteq> 0"
- using c g by (simp add: AE_iff_null_set)
- moreover have "0 \<le> \<mu> {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
- ultimately have "0 < \<mu> {x\<in>space M. g x = \<infinity>}" by auto
- then have "\<infinity> = \<infinity> * \<mu> {x\<in>space M. g x = \<infinity>}" by auto
+ assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
+ have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
+ using c g by (auto simp add: AE_iff_null)
+ moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
+ ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
+ then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
using g by (subst positive_integral_cmult_indicator) auto
also have "\<dots> \<le> integral\<^isup>P M g"
@@ -1367,34 +1245,34 @@
finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
qed
-lemma (in measure_space) positive_integral_diff:
+lemma positive_integral_diff:
assumes f: "f \<in> borel_measurable M"
- and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
+ and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
- and mono: "AE x. g x \<le> f x"
+ and mono: "AE x in M. g x \<le> f x"
shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
proof -
- have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x. 0 \<le> f x - g x"
+ have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
using assms by (auto intro: ereal_diff_positive)
- have pos_f: "AE x. 0 \<le> f x" using mono g by auto
+ have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
{ fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
by (cases rule: ereal2_cases[of a b]) auto }
note * = this
- then have "AE x. f x = f x - g x + g x"
+ then have "AE x in M. f x = f x - g x + g x"
using mono positive_integral_noteq_infinite[OF g fin] assms by auto
then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
unfolding positive_integral_add[OF diff g, symmetric]
by (rule positive_integral_cong_AE)
show ?thesis unfolding **
- using fin positive_integral_positive[of g]
+ using fin positive_integral_positive[of M g]
by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
qed
-lemma (in measure_space) positive_integral_suminf:
- assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> f i x"
+lemma positive_integral_suminf:
+ assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
proof -
- have all_pos: "AE x. \<forall>i. 0 \<le> f i x"
+ have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
using assms by (auto simp: AE_all_countable)
have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
using positive_integral_positive by (rule suminf_ereal_eq_SUPR)
@@ -1409,12 +1287,12 @@
qed
text {* Fatou's lemma: convergence theorem on limes inferior *}
-lemma (in measure_space) positive_integral_lim_INF:
+lemma positive_integral_lim_INF:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
- assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> u i x"
+ assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
proof -
- have pos: "AE x. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
+ have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
(SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
unfolding liminf_SUPR_INFI using pos u
@@ -1426,120 +1304,31 @@
finally show ?thesis .
qed
-lemma (in measure_space) measure_space_density:
- assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x"
- and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<rparr>)"
- shows "measure_space M'"
-proof -
- interpret M': sigma_algebra M' by (intro sigma_algebra_cong) auto
- show ?thesis
- proof
- have pos: "\<And>A. AE x. 0 \<le> u x * indicator A x"
- using u by (auto simp: ereal_zero_le_0_iff)
- then show "positive M' (measure M')" unfolding M'
- using u(1) by (auto simp: positive_def intro!: positive_integral_positive)
- show "countably_additive M' (measure M')"
- proof (intro countably_additiveI)
- fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
- then have *: "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
- using u by (auto intro: borel_measurable_indicator)
- assume disj: "disjoint_family A"
- have "(\<Sum>n. measure M' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. u x * indicator (A n) x) \<partial>M)"
- unfolding M' using u(1) *
- by (simp add: positive_integral_suminf[OF _ pos, symmetric])
- also have "\<dots> = (\<integral>\<^isup>+ x. u x * (\<Sum>n. indicator (A n) x) \<partial>M)" using u
- by (intro positive_integral_cong_AE)
- (elim AE_mp, auto intro!: AE_I2 suminf_cmult_ereal)
- also have "\<dots> = (\<integral>\<^isup>+ x. u x * indicator (\<Union>n. A n) x \<partial>M)"
- unfolding suminf_indicator[OF disj] ..
- finally show "(\<Sum>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
- unfolding M' by simp
- qed
- qed
-qed
-
-lemma (in measure_space) positive_integral_null_set:
- assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
+lemma positive_integral_null_set:
+ assumes "N \<in> null_sets M" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
proof -
have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
proof (intro positive_integral_cong_AE AE_I)
show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
by (auto simp: indicator_def)
- show "\<mu> N = 0" "N \<in> sets M"
+ show "(emeasure M) N = 0" "N \<in> sets M"
using assms by auto
qed
then show ?thesis by simp
qed
-lemma (in measure_space) positive_integral_translated_density:
- assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
- assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
- and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
- shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
-proof -
- from measure_space_density[OF f M']
- interpret T: measure_space M' .
- have borel[simp]:
- "borel_measurable M' = borel_measurable M"
- "simple_function M' = simple_function M"
- unfolding measurable_def simple_function_def [abs_def] by (auto simp: M')
- from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
- note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
- note G'(2)[simp]
- { fix P have "AE x. P x \<Longrightarrow> AE x in M'. P x"
- using positive_integral_null_set[of _ f]
- unfolding T.almost_everywhere_def almost_everywhere_def
- by (auto simp: M') }
- note ac = this
- from G(4) g(2) have G_M': "AE x in M'. (SUP i. G i x) = g x"
- by (auto intro!: ac split: split_max)
- { fix i
- let ?I = "\<lambda>y x. indicator (G i -` {y} \<inter> space M) x"
- { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
- then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
- from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
- by (subst setsum_ereal_right_distrib) (auto simp: ac_simps)
- also have "\<dots> = f x * G i x"
- by (simp add: indicator_def if_distrib setsum_cases)
- finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
- note to_singleton = this
- have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
- using G T.positive_integral_eq_simple_integral by simp
- also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
- unfolding simple_integral_def M' by simp
- also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
- using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
- also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
- using f G' G by (auto intro!: positive_integral_setsum[symmetric])
- finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
- using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
- note [simp] = this
- have "integral\<^isup>P M' g = (SUP i. integral\<^isup>P M' (G i))" using G'(1) G_M'(1) G
- using T.positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`]
- by (simp cong: T.positive_integral_cong_AE)
- also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
- also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
- using f G' G(2)[THEN incseq_SucD] G
- by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
- (auto simp: ereal_mult_left_mono le_fun_def ereal_zero_le_0_iff)
- also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
- by (intro positive_integral_cong_AE)
- (auto simp add: SUPR_ereal_cmult split: split_max)
- finally show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)" .
-qed
-
-lemma (in measure_space) positive_integral_0_iff:
- assumes u: "u \<in> borel_measurable M" and pos: "AE x. 0 \<le> u x"
- shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
- (is "_ \<longleftrightarrow> \<mu> ?A = 0")
+lemma positive_integral_0_iff:
+ assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
+ shows "integral\<^isup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
+ (is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
proof -
have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
by (auto intro!: positive_integral_cong simp: indicator_def)
show ?thesis
proof
- assume "\<mu> ?A = 0"
- with positive_integral_null_set[of ?A u] u
- show "integral\<^isup>P M u = 0" by (simp add: u_eq)
+ assume "(emeasure M) ?A = 0"
+ with positive_integral_null_set[of ?A M u] u
+ show "integral\<^isup>P M u = 0" by (simp add: u_eq null_sets_def)
next
{ fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
@@ -1547,17 +1336,17 @@
note gt_1 = this
assume *: "integral\<^isup>P M u = 0"
let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
- have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
+ have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
proof -
{ fix n :: nat
from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
- have "\<mu> (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
- moreover have "0 \<le> \<mu> (?M n \<inter> ?A)" using u by auto
- ultimately have "\<mu> (?M n \<inter> ?A) = 0" by auto }
+ have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
+ moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
+ ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
thus ?thesis by simp
qed
- also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
- proof (safe intro!: continuity_from_below)
+ also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
+ proof (safe intro!: SUP_emeasure_incseq)
fix n show "?M n \<inter> ?A \<in> sets M"
using u by (auto intro!: Int)
next
@@ -1570,8 +1359,8 @@
finally show "1 \<le> real (Suc n) * u x" by auto
qed
qed
- also have "\<dots> = \<mu> {x\<in>space M. 0 < u x}"
- proof (safe intro!: arg_cong[where f="\<mu>"] dest!: gt_1)
+ also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
+ proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
proof (cases "u x")
@@ -1582,88 +1371,48 @@
thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
qed (insert `0 < u x`, auto)
qed auto
- finally have "\<mu> {x\<in>space M. 0 < u x} = 0" by simp
+ finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
moreover
- from pos have "AE x. \<not> (u x < 0)" by auto
- then have "\<mu> {x\<in>space M. u x < 0} = 0"
- using AE_iff_null_set u by auto
- moreover have "\<mu> {x\<in>space M. u x \<noteq> 0} = \<mu> {x\<in>space M. u x < 0} + \<mu> {x\<in>space M. 0 < u x}"
- using u by (subst measure_additive) (auto intro!: arg_cong[where f=\<mu>])
- ultimately show "\<mu> ?A = 0" by simp
+ from pos have "AE x in M. \<not> (u x < 0)" by auto
+ then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
+ using AE_iff_null[of M] u by auto
+ moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}"
+ using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
+ ultimately show "(emeasure M) ?A = 0" by simp
qed
qed
-lemma (in measure_space) positive_integral_0_iff_AE:
+lemma positive_integral_0_iff_AE:
assumes u: "u \<in> borel_measurable M"
- shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x \<le> 0)"
+ shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
proof -
have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
using u by auto
from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
- have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. max 0 (u x) = 0)"
+ have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
unfolding positive_integral_max_0
- using AE_iff_null_set[OF sets] u by auto
- also have "\<dots> \<longleftrightarrow> (AE x. u x \<le> 0)" by (auto split: split_max)
+ using AE_iff_null[OF sets] u by auto
+ also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
finally show ?thesis .
qed
-lemma (in measure_space) positive_integral_const_If:
- "(\<integral>\<^isup>+x. a \<partial>M) = (if 0 \<le> a then a * \<mu> (space M) else 0)"
+lemma positive_integral_const_If:
+ "(\<integral>\<^isup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
-lemma (in measure_space) positive_integral_restricted:
- assumes A: "A \<in> sets M"
- shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
- (is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
-proof -
- interpret R: measure_space ?R
- by (rule restricted_measure_space) fact
- let ?I = "\<lambda>g x. g x * indicator A x :: ereal"
- show ?thesis
- unfolding positive_integral_def
- unfolding simple_function_restricted[OF A]
- unfolding AE_restricted[OF A]
- proof (safe intro!: SUPR_eq)
- fix g assume g: "simple_function M (?I g)" and le: "g \<le> max 0 \<circ> f"
- show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> ?I f}.
- integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M j"
- proof (safe intro!: bexI[of _ "?I g"])
- show "integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M (?I g)"
- using g A by (simp add: simple_integral_restricted)
- show "?I g \<le> max 0 \<circ> ?I f"
- using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
- qed fact
- next
- fix g assume g: "simple_function M g" and le: "g \<le> max 0 \<circ> ?I f"
- show "\<exists>i\<in>{g. simple_function M (?I g) \<and> g \<le> max 0 \<circ> f}.
- integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) i"
- proof (safe intro!: bexI[of _ "?I g"])
- show "?I g \<le> max 0 \<circ> f"
- using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
- from le have "\<And>x. g x \<le> ?I (?I g) x"
- by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
- then show "integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) (?I g)"
- using A g by (auto intro!: simple_integral_mono simp: simple_integral_restricted)
- show "simple_function M (?I (?I g))" using g A by auto
- qed
- qed
-qed
-
-lemma (in measure_space) positive_integral_subalgebra:
+lemma positive_integral_subalgebra:
assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f x"
- and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
- and sa: "sigma_algebra N"
+ and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
shows "integral\<^isup>P N f = integral\<^isup>P M f"
proof -
- interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
- from N.borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
+ from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
note sf = simple_function_subalgebra[OF fs(1) N(1,2)]
- from N.positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
- have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
+ from positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
+ have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * emeasure N (fs i -` {x} \<inter> space M))"
unfolding fs(4) positive_integral_max_0
unfolding simple_integral_def `space N = space M` by simp
- also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
- using N N.simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
+ also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * (emeasure M) (fs i -` {x} \<inter> space M))"
+ using N simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
also have "\<dots> = integral\<^isup>P M f"
using positive_integral_monotone_convergence_simple[OF fs(2,5) sf, symmetric]
unfolding fs(4) positive_integral_max_0
@@ -1673,7 +1422,7 @@
section "Lebesgue Integral"
-definition integrable where
+definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
"integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
@@ -1682,55 +1431,44 @@
shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
using assms unfolding integrable_def by auto
-definition lebesgue_integral_def:
+definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^isup>L") where
"integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. ereal (- f x) \<partial>M))"
syntax
- "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
+ "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
translations
- "\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
+ "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)"
-lemma (in measure_space) integrableE:
+lemma integrableE:
assumes "integrable M f"
obtains r q where
"(\<integral>\<^isup>+x. ereal (f x)\<partial>M) = ereal r"
"(\<integral>\<^isup>+x. ereal (-f x)\<partial>M) = ereal q"
"f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
using assms unfolding integrable_def lebesgue_integral_def
- using positive_integral_positive[of "\<lambda>x. ereal (f x)"]
- using positive_integral_positive[of "\<lambda>x. ereal (-f x)"]
+ using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
+ using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
by (cases rule: ereal2_cases[of "(\<integral>\<^isup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. ereal (f x)\<partial>M)"]) auto
-lemma (in measure_space) integral_cong:
+lemma integral_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
shows "integral\<^isup>L M f = integral\<^isup>L M g"
using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
-lemma (in measure_space) integral_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
- shows "integral\<^isup>L N f = integral\<^isup>L M f"
- by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
-
-lemma (in measure_space) integrable_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
- shows "integrable N f \<longleftrightarrow> integrable M f"
- using assms
- by (simp add: positive_integral_cong_measure[OF assms] integrable_def measurable_def)
-
-lemma (in measure_space) integral_cong_AE:
- assumes cong: "AE x. f x = g x"
+lemma integral_cong_AE:
+ assumes cong: "AE x in M. f x = g x"
shows "integral\<^isup>L M f = integral\<^isup>L M g"
proof -
- have *: "AE x. ereal (f x) = ereal (g x)"
- "AE x. ereal (- f x) = ereal (- g x)" using cong by auto
+ have *: "AE x in M. ereal (f x) = ereal (g x)"
+ "AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto
show ?thesis
unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
qed
-lemma (in measure_space) integrable_cong_AE:
+lemma integrable_cong_AE:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- assumes "AE x. f x = g x"
+ assumes "AE x in M. f x = g x"
shows "integrable M f = integrable M g"
proof -
have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (g x) \<partial>M)"
@@ -1740,11 +1478,23 @@
by (auto simp: integrable_def)
qed
-lemma (in measure_space) integrable_cong:
+lemma integrable_cong:
"(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
-lemma (in measure_space) integral_eq_positive_integral:
+lemma positive_integral_eq_integral:
+ assumes f: "integrable M f"
+ assumes nonneg: "AE x in M. 0 \<le> f x"
+ shows "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = integral\<^isup>L M f"
+proof -
+ have "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
+ using nonneg by (intro positive_integral_cong_AE) (auto split: split_max)
+ with f positive_integral_positive show ?thesis
+ by (cases "\<integral>\<^isup>+ x. ereal (f x) \<partial>M")
+ (auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def)
+qed
+
+lemma integral_eq_positive_integral:
assumes f: "\<And>x. 0 \<le> f x"
shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
proof -
@@ -1755,84 +1505,12 @@
unfolding lebesgue_integral_def by simp
qed
-lemma (in measure_space) integral_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
- assumes f: "f \<in> borel_measurable M'"
- shows "integral\<^isup>L M' f = (\<integral>x. f (T x) \<partial>M)"
-proof -
- interpret T: measure_space M' by (rule measure_space_vimage[OF T])
- from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
- have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'"
- and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
- using f by (auto simp: comp_def)
- then show ?thesis
- using f unfolding lebesgue_integral_def integrable_def
- by (auto simp: borel[THEN positive_integral_vimage[OF T]])
-qed
-
-lemma (in measure_space) integrable_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
- assumes f: "integrable M' f"
- shows "integrable M (\<lambda>x. f (T x))"
-proof -
- interpret T: measure_space M' by (rule measure_space_vimage[OF T])
- from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
- have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'"
- and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
- using f by (auto simp: comp_def)
- then show ?thesis
- using f unfolding lebesgue_integral_def integrable_def
- by (auto simp: borel[THEN positive_integral_vimage[OF T]])
-qed
-
-lemma (in measure_space) integral_translated_density:
- assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
- and g: "g \<in> borel_measurable M"
- and N: "space N = space M" "sets N = sets M"
- and density: "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
- (is "\<And>A. _ \<Longrightarrow> _ = ?d A")
- shows "integral\<^isup>L N g = (\<integral> x. f x * g x \<partial>M)" (is ?integral)
- and "integrable N g = integrable M (\<lambda>x. f x * g x)" (is ?integrable)
-proof -
- from f have ms: "measure_space (M\<lparr>measure := ?d\<rparr>)"
- by (intro measure_space_density[where u="\<lambda>x. ereal (f x)"]) auto
-
- from ms density N have "(\<integral>\<^isup>+ x. g x \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (g x)) \<partial>M\<lparr>measure := ?d\<rparr>)"
- unfolding positive_integral_max_0
- by (intro measure_space.positive_integral_cong_measure) auto
- also have "\<dots> = (\<integral>\<^isup>+ x. ereal (f x) * max 0 (ereal (g x)) \<partial>M)"
- using f g by (intro positive_integral_translated_density) auto
- also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (ereal (f x * g x)) \<partial>M)"
- using f by (intro positive_integral_cong_AE)
- (auto simp: ereal_max_0 zero_le_mult_iff split: split_max)
- finally have pos: "(\<integral>\<^isup>+ x. g x \<partial>N) = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
- by (simp add: positive_integral_max_0)
-
- from ms density N have "(\<integral>\<^isup>+ x. - (g x) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (- g x)) \<partial>M\<lparr>measure := ?d\<rparr>)"
- unfolding positive_integral_max_0
- by (intro measure_space.positive_integral_cong_measure) auto
- also have "\<dots> = (\<integral>\<^isup>+ x. ereal (f x) * max 0 (ereal (- g x)) \<partial>M)"
- using f g by (intro positive_integral_translated_density) auto
- also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (ereal (- f x * g x)) \<partial>M)"
- using f by (intro positive_integral_cong_AE)
- (auto simp: ereal_max_0 mult_le_0_iff split: split_max)
- finally have neg: "(\<integral>\<^isup>+ x. - g x \<partial>N) = (\<integral>\<^isup>+ x. - (f x * g x) \<partial>M)"
- by (simp add: positive_integral_max_0)
-
- have g_N: "g \<in> borel_measurable N"
- using g N unfolding measurable_def by simp
-
- show ?integral ?integrable
- unfolding lebesgue_integral_def integrable_def
- using pos neg f g g_N by auto
-qed
-
-lemma (in measure_space) integral_minus[intro, simp]:
+lemma integral_minus[intro, simp]:
assumes "integrable M f"
shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
using assms by (auto simp: integrable_def lebesgue_integral_def)
-lemma (in measure_space) integral_minus_iff[simp]:
+lemma integral_minus_iff[simp]:
"integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
proof
assume "integrable M (\<lambda>x. - f x)"
@@ -1841,7 +1519,7 @@
then show "integrable M f" by simp
qed (rule integral_minus)
-lemma (in measure_space) integral_of_positive_diff:
+lemma integral_of_positive_diff:
assumes integrable: "integrable M u" "integrable M v"
and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
@@ -1851,7 +1529,7 @@
let ?u = "\<lambda>x. max 0 (ereal (u x))"
let ?v = "\<lambda>x. max 0 (ereal (v x))"
- from borel_measurable_diff[of u v] integrable
+ from borel_measurable_diff[of u M v] integrable
have f_borel: "?f \<in> borel_measurable M" and
mf_borel: "?mf \<in> borel_measurable M" and
v_borel: "?v \<in> borel_measurable M" and
@@ -1881,7 +1559,7 @@
using integrable f by (auto elim!: integrableE)
qed
-lemma (in measure_space) integral_linear:
+lemma integral_linear:
assumes "integrable M f" "integrable M g" and "0 \<le> a"
shows "integrable M (\<lambda>t. a * f t + g t)"
and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
@@ -1917,18 +1595,18 @@
by (auto elim!: integrableE simp: field_simps)
qed
-lemma (in measure_space) integral_add[simp, intro]:
+lemma integral_add[simp, intro]:
assumes "integrable M f" "integrable M g"
shows "integrable M (\<lambda>t. f t + g t)"
and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g"
using assms integral_linear[where a=1] by auto
-lemma (in measure_space) integral_zero[simp, intro]:
+lemma integral_zero[simp, intro]:
shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
unfolding integrable_def lebesgue_integral_def
by (auto simp add: borel_measurable_const)
-lemma (in measure_space) integral_cmult[simp, intro]:
+lemma integral_cmult[simp, intro]:
assumes "integrable M f"
shows "integrable M (\<lambda>t. a * f t)" (is ?P)
and "(\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" (is ?I)
@@ -1942,37 +1620,80 @@
assume "a \<le> 0" hence "0 \<le> - a" by auto
have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
- integral_minus(1)[of "\<lambda>t. - a * f t"]
+ integral_minus(1)[of M "\<lambda>t. - a * f t"]
unfolding * integral_zero by simp
qed
thus ?P ?I by auto
qed
-lemma (in measure_space) integral_multc:
+lemma lebesgue_integral_cmult_nonneg:
+ assumes f: "f \<in> borel_measurable M" and "0 \<le> c"
+ shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^isup>L M f"
+proof -
+ { have "c * real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (f x)))) =
+ real (ereal c * integral\<^isup>P M (\<lambda>x. max 0 (ereal (f x))))"
+ by simp
+ also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))"
+ using f `0 \<le> c` by (subst positive_integral_cmult) auto
+ also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (c * f x))))"
+ using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff)
+ finally have "real (integral\<^isup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^isup>P M (\<lambda>x. ereal (f x)))"
+ by (simp add: positive_integral_max_0) }
+ moreover
+ { have "c * real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (- f x)))) =
+ real (ereal c * integral\<^isup>P M (\<lambda>x. max 0 (ereal (- f x))))"
+ by simp
+ also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))"
+ using f `0 \<le> c` by (subst positive_integral_cmult) auto
+ also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (- c * f x))))"
+ using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff)
+ finally have "real (integral\<^isup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^isup>P M (\<lambda>x. ereal (- f x)))"
+ by (simp add: positive_integral_max_0) }
+ ultimately show ?thesis
+ by (simp add: lebesgue_integral_def field_simps)
+qed
+
+lemma lebesgue_integral_uminus:
+ "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
+ unfolding lebesgue_integral_def by simp
+
+lemma lebesgue_integral_cmult:
+ assumes f: "f \<in> borel_measurable M"
+ shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^isup>L M f"
+proof (cases rule: linorder_le_cases)
+ assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg)
+next
+ assume "c \<le> 0"
+ with lebesgue_integral_cmult_nonneg[OF f, of "-c"]
+ show ?thesis
+ by (simp add: lebesgue_integral_def)
+qed
+
+lemma integral_multc:
assumes "integrable M f"
shows "(\<integral> x. f x * c \<partial>M) = integral\<^isup>L M f * c"
unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
-lemma (in measure_space) integral_mono_AE:
+lemma integral_mono_AE:
assumes fg: "integrable M f" "integrable M g"
- and mono: "AE t. f t \<le> g t"
+ and mono: "AE t in M. f t \<le> g t"
shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
proof -
- have "AE x. ereal (f x) \<le> ereal (g x)"
+ have "AE x in M. ereal (f x) \<le> ereal (g x)"
using mono by auto
- moreover have "AE x. ereal (- g x) \<le> ereal (- f x)"
+ moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)"
using mono by auto
ultimately show ?thesis using fg
by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
simp: positive_integral_positive lebesgue_integral_def diff_minus)
qed
-lemma (in measure_space) integral_mono:
+lemma integral_mono:
assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
using assms by (auto intro: integral_mono_AE)
-lemma (in measure_space) integral_diff[simp, intro]:
+lemma integral_diff[simp, intro]:
assumes f: "integrable M f" and g: "integrable M g"
shows "integrable M (\<lambda>t. f t - g t)"
and "(\<integral> t. f t - g t \<partial>M) = integral\<^isup>L M f - integral\<^isup>L M g"
@@ -1980,9 +1701,9 @@
unfolding diff_minus integral_minus(2)[OF g]
by auto
-lemma (in measure_space) integral_indicator[simp, intro]:
- assumes "A \<in> sets M" and "\<mu> A \<noteq> \<infinity>"
- shows "integral\<^isup>L M (indicator A) = real (\<mu> A)" (is ?int)
+lemma integral_indicator[simp, intro]:
+ assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>"
+ shows "integral\<^isup>L M (indicator A) = real ((emeasure M) A)" (is ?int)
and "integrable M (indicator A)" (is ?able)
proof -
from `A \<in> sets M` have *:
@@ -1994,10 +1715,10 @@
by (auto simp: * positive_integral_indicator borel_measurable_indicator)
qed
-lemma (in measure_space) integral_cmul_indicator:
- assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<infinity>"
+lemma integral_cmul_indicator:
+ assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>"
shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
- and "(\<integral>x. c * indicator A x \<partial>M) = c * real (\<mu> A)" (is ?I)
+ and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I)
proof -
show ?P
proof (cases "c = 0")
@@ -2010,7 +1731,7 @@
qed simp
qed
-lemma (in measure_space) integral_setsum[simp, intro]:
+lemma integral_setsum[simp, intro]:
assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^isup>L M (f i))" (is "?int S")
and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
@@ -2023,7 +1744,7 @@
thus "?int S" and "?I S" by auto
qed
-lemma (in measure_space) integrable_abs:
+lemma integrable_abs:
assumes "integrable M f"
shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
proof -
@@ -2034,23 +1755,22 @@
by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
qed
-lemma (in measure_space) integral_subalgebra:
+lemma integral_subalgebra:
assumes borel: "f \<in> borel_measurable N"
- and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" and sa: "sigma_algebra N"
+ and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
proof -
- interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
have "(\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>M)"
"(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)"
- using borel by (auto intro!: positive_integral_subalgebra N sa)
+ using borel by (auto intro!: positive_integral_subalgebra N)
moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
using assms unfolding measurable_def by auto
ultimately show ?P ?I
by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
qed
-lemma (in measure_space) integrable_bound:
+lemma integrable_bound:
assumes "integrable M f"
and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
"\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
@@ -2077,11 +1797,21 @@
unfolding integrable_def by auto
qed
-lemma (in measure_space) integrable_abs_iff:
+lemma lebesgue_integral_nonneg:
+ assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^isup>L M f"
+proof -
+ have "(\<integral>\<^isup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^isup>+x. 0 \<partial>M)"
+ using ae by (intro positive_integral_cong_AE) (auto simp: max_def)
+ then show ?thesis
+ by (auto simp: lebesgue_integral_def positive_integral_max_0
+ intro!: real_of_ereal_pos positive_integral_positive)
+qed
+
+lemma integrable_abs_iff:
"f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
by (auto intro!: integrable_bound[where g=f] integrable_abs)
-lemma (in measure_space) integrable_max:
+lemma integrable_max:
assumes int: "integrable M f" "integrable M g"
shows "integrable M (\<lambda> x. max (f x) (g x))"
proof (rule integrable_bound)
@@ -2095,7 +1825,7 @@
by auto
qed
-lemma (in measure_space) integrable_min:
+lemma integrable_min:
assumes int: "integrable M f" "integrable M g"
shows "integrable M (\<lambda> x. min (f x) (g x))"
proof (rule integrable_bound)
@@ -2109,18 +1839,18 @@
by auto
qed
-lemma (in measure_space) integral_triangle_inequality:
+lemma integral_triangle_inequality:
assumes "integrable M f"
shows "\<bar>integral\<^isup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
proof -
have "\<bar>integral\<^isup>L M f\<bar> = max (integral\<^isup>L M f) (- integral\<^isup>L M f)" by auto
also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
- using assms integral_minus(2)[of f, symmetric]
+ using assms integral_minus(2)[of M f, symmetric]
by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
finally show ?thesis .
qed
-lemma (in measure_space) integral_positive:
+lemma integral_positive:
assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
shows "0 \<le> integral\<^isup>L M f"
proof -
@@ -2130,7 +1860,7 @@
finally show ?thesis .
qed
-lemma (in measure_space) integral_monotone_convergence_pos:
+lemma integral_monotone_convergence_pos:
assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
and pos: "\<And>x i. 0 \<le> f i x"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
@@ -2157,7 +1887,7 @@
using i borel_u pos pos_u by (auto simp: positive_integral_0_iff_AE integrable_def)
have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. ereal (f n x) \<partial>M) = ereal (integral\<^isup>L M (f n))"
- using i positive_integral_positive by (auto simp: ereal_real lebesgue_integral_def integrable_def)
+ using i positive_integral_positive[of M] by (auto simp: ereal_real lebesgue_integral_def integrable_def)
have pos_integral: "\<And>n. 0 \<le> integral\<^isup>L M (f n)"
using pos i by (auto simp: integral_positive)
@@ -2177,7 +1907,7 @@
unfolding integrable_def lebesgue_integral_def by auto
qed
-lemma (in measure_space) integral_monotone_convergence:
+lemma integral_monotone_convergence:
assumes f: "\<And>i. integrable M (f i)" and "mono f"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
@@ -2201,9 +1931,9 @@
by (auto simp: integral_diff)
qed
-lemma (in measure_space) integral_0_iff:
+lemma integral_0_iff:
assumes "integrable M f"
- shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
+ shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0"
proof -
have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
@@ -2212,57 +1942,57 @@
"(\<integral>\<^isup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
from positive_integral_0_iff[OF this(1)] this(2)
show ?thesis unfolding lebesgue_integral_def *
- using positive_integral_positive[of "\<lambda>x. ereal \<bar>f x\<bar>"]
+ using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"]
by (auto simp add: real_of_ereal_eq_0)
qed
-lemma (in measure_space) positive_integral_PInf:
+lemma positive_integral_PInf:
assumes f: "f \<in> borel_measurable M"
and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>"
- shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
+ shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
proof -
- have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
+ have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
also have "\<dots> \<le> integral\<^isup>P M (\<lambda>x. max 0 (f x))"
by (auto intro!: positive_integral_mono simp: indicator_def max_def)
- finally have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
+ finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
by (simp add: positive_integral_max_0)
- moreover have "0 \<le> \<mu> (f -` {\<infinity>} \<inter> space M)"
- using f by (simp add: measurable_sets)
+ moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
+ by (rule emeasure_nonneg)
ultimately show ?thesis
using assms by (auto split: split_if_asm)
qed
-lemma (in measure_space) positive_integral_PInf_AE:
- assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x. f x \<noteq> \<infinity>"
+lemma positive_integral_PInf_AE:
+ assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
proof (rule AE_I)
- show "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
+ show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
by (rule positive_integral_PInf[OF assms])
show "f -` {\<infinity>} \<inter> space M \<in> sets M"
using assms by (auto intro: borel_measurable_vimage)
qed auto
-lemma (in measure_space) simple_integral_PInf:
+lemma simple_integral_PInf:
assumes "simple_function M f" "\<And>x. 0 \<le> f x"
and "integral\<^isup>S M f \<noteq> \<infinity>"
- shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
+ shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
proof (rule positive_integral_PInf)
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
show "integral\<^isup>P M f \<noteq> \<infinity>"
using assms by (simp add: positive_integral_eq_simple_integral)
qed
-lemma (in measure_space) integral_real:
- "AE x. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
+lemma integral_real:
+ "AE x in M. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
using assms unfolding lebesgue_integral_def
by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
lemma (in finite_measure) lebesgue_integral_const[simp]:
shows "integrable M (\<lambda>x. a)"
- and "(\<integral>x. a \<partial>M) = a * \<mu>' (space M)"
+ and "(\<integral>x. a \<partial>M) = a * (measure M) (space M)"
proof -
{ fix a :: real assume "0 \<le> a"
- then have "(\<integral>\<^isup>+ x. ereal a \<partial>M) = ereal a * \<mu> (space M)"
+ then have "(\<integral>\<^isup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)"
by (subst positive_integral_const) auto
moreover
from `0 \<le> a` have "(\<integral>\<^isup>+ x. ereal (-a) \<partial>M) = 0"
@@ -2277,8 +2007,8 @@
then have "0 \<le> -a" by auto
from *[OF this] show ?thesis by simp
qed
- show "(\<integral>x. a \<partial>M) = a * \<mu>' (space M)"
- by (simp add: \<mu>'_def lebesgue_integral_def positive_integral_const_If)
+ show "(\<integral>x. a \<partial>M) = a * measure M (space M)"
+ by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure)
qed
lemma indicator_less[simp]:
@@ -2287,8 +2017,8 @@
lemma (in finite_measure) integral_less_AE:
assumes int: "integrable M X" "integrable M Y"
- assumes A: "\<mu> A \<noteq> 0" "A \<in> sets M" "AE x. x \<in> A \<longrightarrow> X x \<noteq> Y x"
- assumes gt: "AE x. X x \<le> Y x"
+ assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
+ assumes gt: "AE x in M. X x \<le> Y x"
shows "integral\<^isup>L M X < integral\<^isup>L M Y"
proof -
have "integral\<^isup>L M X \<le> integral\<^isup>L M Y"
@@ -2301,26 +2031,27 @@
using gt by (intro integral_cong_AE) auto
also have "\<dots> = 0"
using eq int by simp
- finally have "\<mu> {x \<in> space M. Y x - X x \<noteq> 0} = 0"
+ finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
using int by (simp add: integral_0_iff)
moreover
have "(\<integral>\<^isup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^isup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
using A by (intro positive_integral_mono_AE) auto
- then have "\<mu> A \<le> \<mu> {x \<in> space M. Y x - X x \<noteq> 0}"
+ then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
using int A by (simp add: integrable_def)
- moreover note `\<mu> A \<noteq> 0` positive_measure[OF `A \<in> sets M`]
- ultimately show False by auto
+ ultimately have "emeasure M A = 0"
+ using emeasure_nonneg[of M A] by simp
+ with `(emeasure M) A \<noteq> 0` show False by auto
qed
ultimately show ?thesis by auto
qed
lemma (in finite_measure) integral_less_AE_space:
assumes int: "integrable M X" "integrable M Y"
- assumes gt: "AE x. X x < Y x" "\<mu> (space M) \<noteq> 0"
+ assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0"
shows "integral\<^isup>L M X < integral\<^isup>L M Y"
using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
-lemma (in measure_space) integral_dominated_convergence:
+lemma integral_dominated_convergence:
assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
and w: "integrable M w"
and u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
@@ -2336,7 +2067,7 @@
from u[unfolded integrable_def]
have u'_borel: "u' \<in> borel_measurable M"
- using u' by (blast intro: borel_measurable_LIMSEQ[of u])
+ using u' by (blast intro: borel_measurable_LIMSEQ[of M u])
{ fix x assume x: "x \<in> space M"
then have "0 \<le> \<bar>u 0 x\<bar>" by auto
@@ -2385,7 +2116,7 @@
using diff_less_2w[of _ n] unfolding positive_integral_max_0
by (intro positive_integral_mono) auto
then have "?f n = 0"
- using positive_integral_positive[of ?f'] eq_0 by auto }
+ using positive_integral_positive[of M ?f'] eq_0 by auto }
then show ?thesis by (simp add: Limsup_const)
next
assume neq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
@@ -2411,10 +2142,10 @@
also have "\<dots> = (\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) -
limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
unfolding PI_diff positive_integral_max_0
- using positive_integral_positive[of "\<lambda>x. ereal (2 * w x)"]
+ using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"]
by (subst liminf_ereal_cminus) auto
finally show ?thesis
- using neq_0 I2w_fin positive_integral_positive[of "\<lambda>x. ereal (2 * w x)"] pos
+ using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos
unfolding positive_integral_max_0
by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
auto
@@ -2430,8 +2161,8 @@
ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
using `limsup ?f = 0` by auto
have "\<And>n. (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
- using diff positive_integral_positive
- by (subst integral_eq_positive_integral) (auto simp: ereal_real integrable_def)
+ using diff positive_integral_positive[of M]
+ by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def)
then show ?lim_diff
using ereal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
by (simp add: lim_ereal)
@@ -2456,7 +2187,7 @@
qed
qed
-lemma (in measure_space) integral_sums:
+lemma integral_sums:
assumes borel: "\<And>i. integrable M (f i)"
and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
@@ -2513,18 +2244,18 @@
section "Lebesgue integration on countable spaces"
-lemma (in measure_space) integral_on_countable:
+lemma integral_on_countable:
assumes f: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M)"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
- and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
- and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
+ and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
+ and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
shows "integrable M f"
- and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
+ and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
proof -
let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
- have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral\<^isup>L M (?F r)"
+ have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^isup>L M (?F r)"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
{ fix x assume "x \<in> space M"
@@ -2551,9 +2282,9 @@
{ fix r
have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
by (auto simp: indicator_def intro!: integral_cong)
- also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
+ also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
- finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
+ finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
note int_abs_F = this
@@ -2570,67 +2301,469 @@
show "integrable M f" unfolding int_f by simp
qed
-section "Lebesgue integration on finite space"
+section {* Distributions *}
+
+lemma simple_function_distr[simp]:
+ "simple_function (distr M M' T) f \<longleftrightarrow> simple_function M' (\<lambda>x. f x)"
+ unfolding simple_function_def by simp
+
+lemma positive_integral_distr:
+ assumes T: "T \<in> measurable M M'"
+ and f: "f \<in> borel_measurable M'"
+ shows "integral\<^isup>P (distr M M' T) f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
+proof -
+ from borel_measurable_implies_simple_function_sequence'[OF f]
+ guess f' . note f' = this
+ then have f_distr: "\<And>i. simple_function (distr M M' T) (f' i)"
+ by simp
+ let ?f = "\<lambda>i x. f' i (T x)"
+ have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
+ have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
+ using f'(4) .
+ have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
+ using simple_function_comp[OF T(1) f'(1)] .
+ show "integral\<^isup>P (distr M M' T) f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
+ using
+ positive_integral_monotone_convergence_simple[OF f'(2,5) f_distr]
+ positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
+ by (simp add: positive_integral_max_0 simple_integral_distr[OF T f'(1)] f')
+qed
+
+lemma integral_distr:
+ assumes T: "T \<in> measurable M M'"
+ assumes f: "f \<in> borel_measurable M'"
+ shows "integral\<^isup>L (distr M M' T) f = (\<integral>x. f (T x) \<partial>M)"
+proof -
+ from measurable_comp[OF T, of f borel]
+ have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'"
+ and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
+ using f by (auto simp: comp_def)
+ then show ?thesis
+ using f unfolding lebesgue_integral_def integrable_def
+ by (auto simp: borel[THEN positive_integral_distr[OF T]])
+qed
-lemma (in measure_space) integral_on_finite:
- assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
- and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
- shows "integrable M f"
- and "(\<integral>x. f x \<partial>M) =
- (\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
+lemma integrable_distr:
+ assumes T: "T \<in> measurable M M'" and f: "integrable (distr M M' T) f"
+ shows "integrable M (\<lambda>x. f (T x))"
+proof -
+ from measurable_comp[OF T, of f borel]
+ have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'"
+ and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
+ using f by (auto simp: comp_def)
+ then show ?thesis
+ using f unfolding lebesgue_integral_def integrable_def
+ using borel[THEN positive_integral_distr[OF T]]
+ by (auto simp: borel[THEN positive_integral_distr[OF T]])
+qed
+
+section {* Lebesgue integration on @{const count_space} *}
+
+lemma simple_function_count_space[simp]:
+ "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
+ unfolding simple_function_def by simp
+
+lemma positive_integral_count_space:
+ assumes A: "finite {a\<in>A. 0 < f a}"
+ shows "integral\<^isup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
proof -
- let ?A = "\<lambda>r. f -` {r} \<inter> space M"
- let ?S = "\<lambda>x. \<Sum>r\<in>f`space M. r * indicator (?A r) x"
+ have *: "(\<integral>\<^isup>+x. max 0 (f x) \<partial>count_space A) =
+ (\<integral>\<^isup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
+ by (auto intro!: positive_integral_cong
+ simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less)
+ also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^isup>+ x. f a * indicator {a} x \<partial>count_space A)"
+ by (subst positive_integral_setsum)
+ (simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
+ also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
+ by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric])
+ finally show ?thesis by (simp add: positive_integral_max_0)
+qed
+
+lemma integrable_count_space:
+ "finite X \<Longrightarrow> integrable (count_space X) f"
+ by (auto simp: positive_integral_count_space integrable_def)
+
+lemma positive_integral_count_space_finite:
+ "finite A \<Longrightarrow> (\<integral>\<^isup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
+ by (subst positive_integral_max_0[symmetric])
+ (auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
+
+lemma lebesgue_integral_count_space_finite:
+ "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
+ apply (auto intro!: setsum_mono_zero_left
+ simp: positive_integral_count_space_finite lebesgue_integral_def)
+ apply (subst (1 2) setsum_real_of_ereal[symmetric])
+ apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong)
+ done
+
+section {* Measure spaces with an associated density *}
+
+definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
+ "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
- { fix x assume "x \<in> space M"
- have "f x = (\<Sum>r\<in>f`space M. if x \<in> ?A r then r else 0)"
- using finite `x \<in> space M` by (simp add: setsum_cases)
- also have "\<dots> = ?S x"
- by (auto intro!: setsum_cong)
- finally have "f x = ?S x" . }
- note f_eq = this
+lemma
+ shows sets_density[simp]: "sets (density M f) = sets M"
+ and space_density[simp]: "space (density M f) = space M"
+ by (auto simp: density_def)
+
+lemma
+ shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
+ and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
+ and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
+ unfolding measurable_def simple_function_def by simp_all
+
+lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
+ (AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
+ unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE space_closed)
+
+lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
+proof -
+ have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
+ by (auto simp: indicator_def)
+ then show ?thesis
+ unfolding density_def by (simp add: positive_integral_max_0)
+qed
+
+lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
+ by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
- have f_eq_S: "integrable M f \<longleftrightarrow> integrable M ?S" "integral\<^isup>L M f = integral\<^isup>L M ?S"
- by (auto intro!: integrable_cong integral_cong simp only: f_eq)
+lemma emeasure_density:
+ assumes f: "f \<in> borel_measurable M" and A: "A \<in> sets M"
+ shows "emeasure (density M f) A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
+ (is "_ = ?\<mu> A")
+ unfolding density_def
+proof (rule emeasure_measure_of_sigma)
+ show "sigma_algebra (space M) (sets M)" ..
+ show "positive (sets M) ?\<mu>"
+ using f by (auto simp: positive_def intro!: positive_integral_positive)
+ have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^isup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
+ apply (subst positive_integral_max_0[symmetric])
+ apply (intro ext positive_integral_cong_AE AE_I2)
+ apply (auto simp: indicator_def)
+ done
+ show "countably_additive (sets M) ?\<mu>"
+ unfolding \<mu>_eq
+ proof (intro countably_additiveI)
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
+ then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
+ using f by (auto intro!: borel_measurable_ereal_times)
+ assume disj: "disjoint_family A"
+ have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
+ using f * by (simp add: positive_integral_suminf)
+ also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
+ by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE)
+ also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
+ unfolding suminf_indicator[OF disj] ..
+ finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
+ qed
+qed fact
- show "integrable M f" ?integral using fin f f_eq_S
- by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
+lemma null_sets_density_iff:
+ assumes f: "f \<in> borel_measurable M"
+ shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
+proof -
+ { assume "A \<in> sets M"
+ have eq: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. max 0 (f x) * indicator A x \<partial>M)"
+ apply (subst positive_integral_max_0[symmetric])
+ apply (intro positive_integral_cong)
+ apply (auto simp: indicator_def)
+ done
+ have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow>
+ emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
+ unfolding eq
+ using f `A \<in> sets M`
+ by (intro positive_integral_0_iff) auto
+ also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
+ using f `A \<in> sets M`
+ by (intro AE_iff_measurable[OF _ refl, symmetric])
+ (auto intro!: sets_Collect borel_measurable_ereal_eq)
+ also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
+ by (auto simp add: indicator_def max_def split: split_if_asm)
+ finally have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . }
+ with f show ?thesis
+ by (simp add: null_sets_def emeasure_density cong: conj_cong)
+qed
+
+lemma AE_density:
+ assumes f: "f \<in> borel_measurable M"
+ shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
+proof
+ assume "AE x in density M f. P x"
+ with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0"
+ by (auto simp: eventually_ae_filter null_sets_density_iff)
+ then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
+ with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
+ by (rule eventually_elim2) auto
+next
+ fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
+ then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
+ by (auto simp: eventually_ae_filter)
+ then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
+ "N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
+ using f by (auto simp: subset_eq intro!: sets_Collect_neg AE_not_in)
+ show "AE x in density M f. P x"
+ using ae2
+ unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
+ by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
+ (auto elim: eventually_elim2)
qed
-lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function M f"
- unfolding simple_function_def using finite_space by auto
-
-lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
- by (auto intro: borel_measurable_simple_function)
+lemma positive_integral_density:
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ assumes g': "g' \<in> borel_measurable M"
+ shows "integral\<^isup>P (density M f) g' = (\<integral>\<^isup>+ x. f x * g' x \<partial>M)"
+proof -
+ def g \<equiv> "\<lambda>x. max 0 (g' x)"
+ then have g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+ using g' by auto
+ from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
+ note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
+ note G'(2)[simp]
+ { fix P have "AE x in M. P x \<Longrightarrow> AE x in M. P x"
+ using positive_integral_null_set[of _ _ f]
+ by (auto simp: eventually_ae_filter ) }
+ note ac = this
+ with G(4) f g have G_M': "AE x in density M f. (SUP i. G i x) = g x"
+ by (auto simp add: AE_density[OF f(1)] max_def)
+ { fix i
+ let ?I = "\<lambda>y x. indicator (G i -` {y} \<inter> space M) x"
+ { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
+ then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
+ from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
+ by (subst setsum_ereal_right_distrib) (auto simp: ac_simps)
+ also have "\<dots> = f x * G i x"
+ by (simp add: indicator_def if_distrib setsum_cases)
+ finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
+ note to_singleton = this
+ have "integral\<^isup>P (density M f) (G i) = integral\<^isup>S (density M f) (G i)"
+ using G by (intro positive_integral_eq_simple_integral) simp_all
+ also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
+ using f G(1)
+ by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_density
+ simp: simple_function_def simple_integral_def)
+ also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
+ using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
+ also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
+ using f G' G by (auto intro!: positive_integral_setsum[symmetric])
+ finally have "integral\<^isup>P (density M f) (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
+ using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
+ note [simp] = this
+ have "integral\<^isup>P (density M f) g = (SUP i. integral\<^isup>P (density M f) (G i))" using G'(1) G_M'(1) G
+ using positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`, of "density M f"]
+ by (simp cong: positive_integral_cong_AE)
+ also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
+ also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
+ using f G' G(2)[THEN incseq_SucD] G
+ by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
+ (auto simp: ereal_mult_left_mono le_fun_def ereal_zero_le_0_iff)
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
+ by (intro positive_integral_cong_AE)
+ (auto simp add: SUPR_ereal_cmult split: split_max)
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * g' x \<partial>M)"
+ using f(2)
+ by (subst (2) positive_integral_max_0[symmetric])
+ (auto simp: g_def max_def ereal_zero_le_0_iff intro!: positive_integral_cong_AE)
+ finally show "integral\<^isup>P (density M f) g' = (\<integral>\<^isup>+x. f x * g' x \<partial>M)"
+ unfolding g_def positive_integral_max_0 .
+qed
-lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
- assumes pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
- shows "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
+lemma integral_density:
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ and g: "g \<in> borel_measurable M"
+ shows "integral\<^isup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
+ and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
+ unfolding lebesgue_integral_def integrable_def using f g
+ by (auto simp: positive_integral_density)
+
+lemma emeasure_restricted:
+ assumes S: "S \<in> sets M" and X: "X \<in> sets M"
+ shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
proof -
- have *: "integral\<^isup>P M f = (\<integral>\<^isup>+ x. (\<Sum>y\<in>space M. f y * indicator {y} x) \<partial>M)"
- by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
- show ?thesis unfolding * using borel_measurable_finite[of f] pos
- by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
+ have "emeasure (density M (indicator S)) X = (\<integral>\<^isup>+x. indicator S x * indicator X x \<partial>M)"
+ using S X by (simp add: emeasure_density)
+ also have "\<dots> = (\<integral>\<^isup>+x. indicator (S \<inter> X) x \<partial>M)"
+ by (auto intro!: positive_integral_cong simp: indicator_def)
+ also have "\<dots> = emeasure M (S \<inter> X)"
+ using S X by (simp add: Int)
+ finally show ?thesis .
+qed
+
+lemma measure_restricted:
+ "S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
+ by (simp add: emeasure_restricted measure_def)
+
+lemma (in finite_measure) finite_measure_restricted:
+ "S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
+ by default (simp add: emeasure_restricted)
+
+lemma emeasure_density_const:
+ "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
+ by (auto simp: positive_integral_cmult_indicator emeasure_density)
+
+lemma measure_density_const:
+ "A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
+ by (auto simp: emeasure_density_const measure_def)
+
+lemma density_density_eq:
+ "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
+ density (density M f) g = density M (\<lambda>x. f x * g x)"
+ by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps)
+
+lemma distr_density_distr:
+ assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
+ and inv: "\<forall>x\<in>space M. T' (T x) = x"
+ assumes f: "f \<in> borel_measurable M'"
+ shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
+proof (rule measure_eqI)
+ fix A assume A: "A \<in> sets ?R"
+ { fix x assume "x \<in> space M"
+ with sets_into_space[OF A]
+ have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
+ using T inv by (auto simp: indicator_def measurable_space) }
+ with A T T' f show "emeasure ?R A = emeasure ?L A"
+ by (simp add: measurable_comp emeasure_density emeasure_distr
+ positive_integral_distr measurable_sets cong: positive_integral_cong)
+qed simp
+
+lemma density_density_divide:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+ assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
+ shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
+proof -
+ have "density M g = density M (\<lambda>x. f x * (g x / f x))"
+ using f g ac by (auto intro!: density_cong measurable_If)
+ then show ?thesis
+ using f g by (subst density_density_eq) auto
qed
-lemma (in finite_measure_space) integral_finite_singleton:
- shows "integrable M f"
- and "integral\<^isup>L M f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
+section {* Point measure *}
+
+definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
+ "point_measure A f = density (count_space A) f"
+
+lemma
+ shows space_point_measure: "space (point_measure A f) = A"
+ and sets_point_measure: "sets (point_measure A f) = Pow A"
+ by (auto simp: point_measure_def)
+
+lemma measurable_point_measure_eq1[simp]:
+ "g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
+ unfolding point_measure_def by simp
+
+lemma measurable_point_measure_eq2_finite[simp]:
+ "finite A \<Longrightarrow>
+ g \<in> measurable M (point_measure A f) \<longleftrightarrow>
+ (g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
+ unfolding point_measure_def by simp
+
+lemma simple_function_point_measure[simp]:
+ "simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
+ by (simp add: point_measure_def)
+
+lemma emeasure_point_measure:
+ assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
+ shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
proof -
- have *:
- "(\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (ereal (f x)) * \<mu> {x})"
- "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (ereal (- f x)) * \<mu> {x})"
- by (simp_all add: positive_integral_finite_eq_setsum)
- then show "integrable M f" using finite_space finite_measure
- by (simp add: setsum_Pinfty integrable_def positive_integral_max_0
- split: split_max)
- show ?I using finite_measure *
- apply (simp add: positive_integral_max_0 lebesgue_integral_def)
- apply (subst (1 2) setsum_real_of_ereal[symmetric])
- apply (simp_all split: split_max add: setsum_subtractf[symmetric])
- apply (intro setsum_cong[OF refl])
- apply (simp split: split_max)
- done
+ have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
+ using `X \<subseteq> A` by auto
+ with A show ?thesis
+ by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
+ point_measure_def indicator_def)
qed
+lemma emeasure_point_measure_finite:
+ "finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a|a\<in>X. f a)"
+ by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
+
+lemma null_sets_point_measure_iff:
+ "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
+ by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
+
+lemma AE_point_measure:
+ "(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
+ unfolding point_measure_def
+ by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
+
+lemma positive_integral_point_measure:
+ "finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
+ integral\<^isup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
+ unfolding point_measure_def
+ apply (subst density_max_0)
+ apply (subst positive_integral_density)
+ apply (simp_all add: AE_count_space positive_integral_density)
+ apply (subst positive_integral_count_space )
+ apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff)
+ apply (rule finite_subset)
+ prefer 2
+ apply assumption
+ apply auto
+ done
+
+lemma positive_integral_point_measure_finite:
+ "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow>
+ integral\<^isup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
+ by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
+
+lemma lebesgue_integral_point_measure_finite:
+ "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^isup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
+ by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
+
+lemma integrable_point_measure_finite:
+ "finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g"
+ unfolding point_measure_def
+ apply (subst density_ereal_max_0)
+ apply (subst integral_density)
+ apply (auto simp: AE_count_space integrable_count_space)
+ done
+
+section {* Uniform measure *}
+
+definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
+
+lemma
+ shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M"
+ and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
+ by (auto simp: uniform_measure_def)
+
+lemma emeasure_uniform_measure[simp]:
+ assumes A: "A \<in> sets M" and B: "B \<in> sets M"
+ shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
+proof -
+ from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^isup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
+ by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
+ intro!: positive_integral_cong)
+ also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
+ using A B
+ by (subst positive_integral_cmult_indicator) (simp_all add: Int emeasure_nonneg)
+ finally show ?thesis .
+qed
+
+lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
+ using emeasure_notin_sets[of A M] by blast
+
+lemma measure_uniform_measure[simp]:
+ assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
+ shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
+ using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
+ by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
+
+section {* Uniform count measure *}
+
+definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
+
+lemma
+ shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
+ and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
+ unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
+
+lemma emeasure_uniform_count_measure:
+ "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
+ by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
+
+lemma measure_uniform_count_measure:
+ "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
+ by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
+
end
--- a/src/HOL/Probability/Lebesgue_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -9,6 +9,15 @@
imports Finite_Product_Measure
begin
+lemma borel_measurable_sets:
+ assumes "f \<in> measurable borel M" "A \<in> sets M"
+ shows "f -` A \<in> sets borel"
+ using measurable_sets[OF assms] by simp
+
+lemma measurable_identity[intro,simp]:
+ "(\<lambda>x. x) \<in> measurable M M"
+ unfolding measurable_def by auto
+
subsection {* Standard Cubes *}
definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
@@ -52,17 +61,13 @@
subsection {* Lebesgue measure *}
-definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
- "lebesgue = \<lparr> space = UNIV,
- sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
- measure = \<lambda>A. SUP n. ereal (integral (cube n) (indicator A)) \<rparr>"
+definition lebesgue :: "'a::ordered_euclidean_space measure" where
+ "lebesgue = measure_of UNIV {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n}
+ (\<lambda>A. SUP n. ereal (integral (cube n) (indicator A)))"
lemma space_lebesgue[simp]: "space lebesgue = UNIV"
unfolding lebesgue_def by simp
-lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
- unfolding lebesgue_def by simp
-
lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue"
unfolding lebesgue_def by simp
@@ -86,23 +91,23 @@
"A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
unfolding indicator_def by auto
-interpretation lebesgue: sigma_algebra lebesgue
-proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI)
- fix A n assume A: "A \<in> sets lebesgue"
- have "indicator (space lebesgue - A) = (\<lambda>x. 1 - indicator A x :: real)"
+lemma sigma_algebra_lebesgue:
+ defines "leb \<equiv> {A. \<forall>n. (indicator A :: 'a::ordered_euclidean_space \<Rightarrow> real) integrable_on cube n}"
+ shows "sigma_algebra UNIV leb"
+proof (safe intro!: sigma_algebra_iff2[THEN iffD2])
+ fix A assume A: "A \<in> leb"
+ moreover have "indicator (UNIV - A) = (\<lambda>x. 1 - indicator A x :: real)"
by (auto simp: fun_eq_iff indicator_def)
- then show "(indicator (space lebesgue - A) :: _ \<Rightarrow> real) integrable_on cube n"
- using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def)
+ ultimately show "UNIV - A \<in> leb"
+ using A by (auto intro!: integrable_sub simp: cube_def leb_def)
next
- fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n"
- by (auto simp: cube_def indicator_def [abs_def])
+ fix n show "{} \<in> leb"
+ by (auto simp: cube_def indicator_def[abs_def] leb_def)
next
- fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue"
- then have A: "\<And>i. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
- by (auto dest: lebesgueD)
- show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _")
- proof (intro dominated_convergence[where g="?g"] ballI)
- fix k show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n"
+ fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> leb"
+ have "\<forall>n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "\<forall>n. ?g integrable_on _")
+ proof (intro dominated_convergence[where g="?g"] ballI allI)
+ fix k n show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n"
proof (induct k)
case (Suc k)
have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k"
@@ -111,36 +116,45 @@
indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _")
by (auto simp: fun_eq_iff * indicator_def)
show ?case
- using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: *)
+ using absolutely_integrable_max[of ?f "cube n" ?g] A Suc
+ by (simp add: * leb_def subset_eq)
qed auto
qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
+ then show "(\<Union>i. A i) \<in> leb" by (auto simp: leb_def)
qed simp
-interpretation lebesgue: measure_space lebesgue
-proof
+lemma sets_lebesgue: "sets lebesgue = {A. \<forall>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n}"
+ unfolding lebesgue_def sigma_algebra.sets_measure_of_eq[OF sigma_algebra_lebesgue] ..
+
+lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
+ unfolding sets_lebesgue by simp
+
+lemma emeasure_lebesgue:
+ assumes "A \<in> sets lebesgue"
+ shows "emeasure lebesgue A = (SUP n. ereal (integral (cube n) (indicator A)))"
+ (is "_ = ?\<mu> A")
+proof (rule emeasure_measure_of[OF lebesgue_def])
have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
- show "positive lebesgue (measure lebesgue)"
- proof (unfold positive_def, safe)
- show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
- fix A assume "A \<in> sets lebesgue"
- then show "0 \<le> measure lebesgue A"
- unfolding lebesgue_def
- by (auto intro!: SUP_upper2 integral_nonneg)
+ show "positive (sets lebesgue) ?\<mu>"
+ proof (unfold positive_def, intro conjI ballI)
+ show "?\<mu> {} = 0" by (simp add: integral_0 *)
+ fix A :: "'a set" assume "A \<in> sets lebesgue" then show "0 \<le> ?\<mu> A"
+ by (auto intro!: SUP_upper2 Integration.integral_nonneg simp: sets_lebesgue)
qed
next
- show "countably_additive lebesgue (measure lebesgue)"
+ show "countably_additive (sets lebesgue) ?\<mu>"
proof (intro countably_additive_def[THEN iffD2] allI impI)
- fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
+ fix A :: "nat \<Rightarrow> 'a set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
by (auto dest: lebesgueD)
let ?m = "\<lambda>n i. integral (cube n) (indicator (A i) :: _\<Rightarrow>real)"
let ?M = "\<lambda>n I. integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)"
- have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: integral_nonneg)
+ have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: Integration.integral_nonneg)
assume "(\<Union>i. A i) \<in> sets lebesgue"
then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
- by (auto dest: lebesgueD)
- show "(\<Sum>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
- proof (simp add: lebesgue_def, subst suminf_SUP_eq, safe intro!: incseq_SucI)
+ by (auto simp: sets_lebesgue)
+ show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (\<Union>i. A i)"
+ proof (subst suminf_SUP_eq, safe intro!: incseq_SucI)
fix i n show "ereal (?m n i) \<le> ereal (?m (Suc n) i)"
using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI)
next
@@ -172,14 +186,15 @@
indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)"
by (auto simp: indicator_add lessThan_Suc ac_simps)
ultimately show ?case
- using Suc A by (simp add: integral_add[symmetric])
+ using Suc A by (simp add: Integration.integral_add[symmetric])
qed auto }
ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) ----> ?M n UNIV"
by (simp add: atLeast0LessThan)
qed
qed
qed
-qed
+next
+qed (auto, fact)
lemma has_integral_interval_cube:
fixes a b :: "'a::ordered_euclidean_space"
@@ -202,9 +217,10 @@
fixes s::"'a::ordered_euclidean_space set"
assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
proof -
- let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
- have *:"?S \<subseteq> sets lebesgue"
- proof (safe intro!: lebesgueI)
+ have "s \<in> sigma_sets (space lebesgue) (range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)}))"
+ using assms by (simp add: borel_eq_atLeastAtMost)
+ also have "\<dots> \<subseteq> sets lebesgue"
+ proof (safe intro!: sigma_sets_subset lebesgueI)
fix n :: nat and a b :: 'a
let ?N = "\<chi>\<chi> i. max (- real n) (a $$ i)"
let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)"
@@ -212,11 +228,7 @@
unfolding integrable_on_def
using has_integral_interval_cube[of a b] by auto
qed
- have "s \<in> sigma_sets UNIV ?S" using assms
- unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
- thus ?thesis
- using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
- by (auto simp: sigma_def)
+ finally show ?thesis .
qed
lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
@@ -224,19 +236,21 @@
using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI)
lemma lmeasure_eq_0:
- fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lebesgue.\<mu> S = 0"
+ fixes S :: "'a::ordered_euclidean_space set"
+ assumes "negligible S" shows "emeasure lebesgue S = 0"
proof -
have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0"
unfolding lebesgue_integral_def using assms
by (intro integral_unique some1_equality ex_ex1I)
(auto simp: cube_def negligible_def)
- then show ?thesis by (auto simp: lebesgue_def)
+ then show ?thesis
+ using assms by (simp add: emeasure_lebesgue lebesgueI_negligible)
qed
lemma lmeasure_iff_LIMSEQ:
- assumes "A \<in> sets lebesgue" "0 \<le> m"
- shows "lebesgue.\<mu> A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
-proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
+ assumes A: "A \<in> sets lebesgue" and "0 \<le> m"
+ shows "emeasure lebesgue A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
+proof (subst emeasure_lebesgue[OF A], intro SUP_eq_LIMSEQ)
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
qed
@@ -261,7 +275,7 @@
lemma lmeasure_finite_has_integral:
fixes s :: "'a::ordered_euclidean_space set"
- assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = ereal m" "0 \<le> m"
+ assumes "s \<in> sets lebesgue" "emeasure lebesgue s = ereal m" "0 \<le> m"
shows "(indicator s has_integral m) UNIV"
proof -
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
@@ -275,7 +289,7 @@
(auto dest!: lebesgueD) }
moreover
{ fix n have "0 \<le> integral (cube n) (?I s)"
- using assms by (auto dest!: lebesgueD intro!: integral_nonneg) }
+ using assms by (auto dest!: lebesgueD intro!: Integration.integral_nonneg) }
ultimately
show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}"
unfolding bounded_def
@@ -303,14 +317,13 @@
unfolding m by (intro integrable_integral **)
qed
-lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "lebesgue.\<mu> s \<noteq> \<infinity>"
+lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "emeasure lebesgue s \<noteq> \<infinity>"
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
-proof (cases "lebesgue.\<mu> s")
+proof (cases "emeasure lebesgue s")
case (real m)
- with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
- lebesgue.positive_measure[OF s]
+ with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] emeasure_nonneg[of lebesgue s]
show ?thesis unfolding integrable_on_def by auto
-qed (insert assms lebesgue.positive_measure[OF s], auto)
+qed (insert assms emeasure_nonneg[of lebesgue s], auto)
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
shows "s \<in> sets lebesgue"
@@ -324,7 +337,7 @@
qed
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
- shows "lebesgue.\<mu> s = ereal m"
+ shows "emeasure lebesgue s = ereal m"
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
@@ -349,55 +362,56 @@
qed
lemma has_integral_iff_lmeasure:
- "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m)"
+ "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue A = ereal m)"
proof
assume "(indicator A has_integral m) UNIV"
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
- show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m"
+ show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue A = ereal m"
by (auto intro: has_integral_nonneg)
next
- assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m"
+ assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue A = ereal m"
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
qed
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
- shows "lebesgue.\<mu> s = ereal (integral UNIV (indicator s))"
+ shows "emeasure lebesgue s = ereal (integral UNIV (indicator s))"
using assms unfolding integrable_on_def
proof safe
fix y :: real assume "(indicator s has_integral y) UNIV"
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
- show "lebesgue.\<mu> s = ereal (integral UNIV (indicator s))" by simp
+ show "emeasure lebesgue s = ereal (integral UNIV (indicator s))" by simp
qed
lemma lebesgue_simple_function_indicator:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f:"simple_function lebesgue f"
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
- by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
+ by (rule, subst simple_function_indicator_representation[OF f]) auto
lemma integral_eq_lmeasure:
- "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
+ "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (emeasure lebesgue s)"
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
-lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<infinity>"
+lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "emeasure lebesgue s \<noteq> \<infinity>"
using lmeasure_eq_integral[OF assms] by auto
lemma negligible_iff_lebesgue_null_sets:
- "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
+ "negligible A \<longleftrightarrow> A \<in> null_sets lebesgue"
proof
assume "negligible A"
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
- show "A \<in> lebesgue.null_sets" by auto
+ show "A \<in> null_sets lebesgue" by auto
next
- assume A: "A \<in> lebesgue.null_sets"
- then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto
+ assume A: "A \<in> null_sets lebesgue"
+ then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A]
+ by (auto simp: null_sets_def)
show "negligible A" unfolding negligible_def
proof (intro allI)
fix a b :: 'a
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}"
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *)
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)"
- using * by (auto intro!: integral_subset_le has_integral_integrable)
+ using * by (auto intro!: integral_subset_le)
moreover have "(0::real) \<le> integral {a..b} (indicator A)"
using integrable by (auto intro!: integral_nonneg)
ultimately have "integral {a..b} (indicator A) = (0::real)"
@@ -412,8 +426,8 @@
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
by (rule integral_unique) (rule has_integral_const)
-lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
-proof (simp add: lebesgue_def, intro SUP_PInfty bexI)
+lemma lmeasure_UNIV[intro]: "emeasure lebesgue (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
+proof (simp add: emeasure_lebesgue, intro SUP_PInfty bexI)
fix n :: nat
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto
moreover
@@ -434,7 +448,7 @@
lemma
fixes a b ::"'a::ordered_euclidean_space"
- shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = ereal (content {a..b})"
+ shows lmeasure_atLeastAtMost[simp]: "emeasure lebesgue {a..b} = ereal (content {a..b})"
proof -
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def])
@@ -454,7 +468,7 @@
qed
lemma lmeasure_singleton[simp]:
- fixes a :: "'a::ordered_euclidean_space" shows "lebesgue.\<mu> {a} = 0"
+ fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0"
using lmeasure_atLeastAtMost[of a a] by simp
declare content_real[simp]
@@ -462,82 +476,68 @@
lemma
fixes a b :: real
shows lmeasure_real_greaterThanAtMost[simp]:
- "lebesgue.\<mu> {a <.. b} = ereal (if a \<le> b then b - a else 0)"
+ "emeasure lebesgue {a <.. b} = ereal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
- then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
- by (subst lebesgue.measure_Diff[symmetric])
- (auto intro!: arg_cong[where f=lebesgue.\<mu>])
+ then have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b} - emeasure lebesgue {a}"
+ by (subst emeasure_Diff[symmetric])
+ (auto intro!: arg_cong[where f="emeasure lebesgue"])
then show ?thesis by auto
qed auto
lemma
fixes a b :: real
shows lmeasure_real_atLeastLessThan[simp]:
- "lebesgue.\<mu> {a ..< b} = ereal (if a \<le> b then b - a else 0)"
+ "emeasure lebesgue {a ..< b} = ereal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
- then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
- by (subst lebesgue.measure_Diff[symmetric])
- (auto intro!: arg_cong[where f=lebesgue.\<mu>])
+ then have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b} - emeasure lebesgue {b}"
+ by (subst emeasure_Diff[symmetric])
+ (auto intro!: arg_cong[where f="emeasure lebesgue"])
then show ?thesis by auto
qed auto
lemma
fixes a b :: real
shows lmeasure_real_greaterThanLessThan[simp]:
- "lebesgue.\<mu> {a <..< b} = ereal (if a \<le> b then b - a else 0)"
+ "emeasure lebesgue {a <..< b} = ereal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
- then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
- by (subst lebesgue.measure_Diff[symmetric])
- (auto intro!: arg_cong[where f=lebesgue.\<mu>])
+ then have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a <.. b} - emeasure lebesgue {b}"
+ by (subst emeasure_Diff[symmetric])
+ (auto intro!: arg_cong[where f="emeasure lebesgue"])
then show ?thesis by auto
qed auto
subsection {* Lebesgue-Borel measure *}
-definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>"
+definition "lborel = measure_of UNIV (sets borel) (emeasure lebesgue)"
lemma
shows space_lborel[simp]: "space lborel = UNIV"
and sets_lborel[simp]: "sets lborel = sets borel"
- and measure_lborel[simp]: "measure lborel = lebesgue.\<mu>"
- and measurable_lborel[simp]: "measurable lborel = measurable borel"
- by (simp_all add: measurable_def [abs_def] lborel_def)
+ and measurable_lborel1[simp]: "measurable lborel = measurable borel"
+ and measurable_lborel2[simp]: "measurable A lborel = measurable A borel"
+ using sigma_sets_eq[of borel]
+ by (auto simp add: lborel_def measurable_def[abs_def])
-interpretation lborel: measure_space "lborel :: ('a::ordered_euclidean_space) measure_space"
- where "space lborel = UNIV"
- and "sets lborel = sets borel"
- and "measure lborel = lebesgue.\<mu>"
- and "measurable lborel = measurable borel"
-proof (rule lebesgue.measure_space_subalgebra)
- have "sigma_algebra (lborel::'a measure_space) \<longleftrightarrow> sigma_algebra (borel::'a algebra)"
- unfolding sigma_algebra_iff2 lborel_def by simp
- then show "sigma_algebra (lborel::'a measure_space)" by simp default
-qed auto
+lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A"
+ by (rule emeasure_measure_of[OF lborel_def])
+ (auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure)
interpretation lborel: sigma_finite_measure lborel
- where "space lborel = UNIV"
- and "sets lborel = sets borel"
- and "measure lborel = lebesgue.\<mu>"
- and "measurable lborel = measurable borel"
-proof -
- show "sigma_finite_measure lborel"
- proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
- show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
- { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
- thus "(\<Union>i. cube i) = space lborel" by auto
- show "\<forall>i. measure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
- qed
-qed simp_all
+proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
+ show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
+ { fix x :: 'a have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
+ then show "(\<Union>i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto
+ show "\<forall>i. emeasure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
+qed
interpretation lebesgue: sigma_finite_measure lebesgue
proof
- from lborel.sigma_finite guess A ..
- moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
- ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<infinity>)"
- by auto
+ from lborel.sigma_finite guess A :: "nat \<Rightarrow> 'a set" ..
+ then show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. emeasure lebesgue (A i) \<noteq> \<infinity>)"
+ by (intro exI[of _ A]) (auto simp: subset_eq)
qed
subsection {* Lebesgue integrable implies Gauge integrable *}
@@ -556,11 +556,11 @@
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f:"simple_function lebesgue f"
and f':"range f \<subseteq> {0..<\<infinity>}"
- and om:"\<And>x. x \<in> range f \<Longrightarrow> lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
+ and om:"\<And>x. x \<in> range f \<Longrightarrow> emeasure lebesgue (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
unfolding simple_integral_def space_lebesgue
proof (subst lebesgue_simple_function_indicator)
- let ?M = "\<lambda>x. lebesgue.\<mu> (f -` {x} \<inter> UNIV)"
+ let ?M = "\<lambda>x. emeasure lebesgue (f -` {x} \<inter> UNIV)"
let ?F = "\<lambda>x. indicator (f -` {x})"
{ fix x y assume "y \<in> range f"
from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)"
@@ -571,7 +571,7 @@
have "x * ?M x = real x * real (?M x)"
proof cases
assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto
- with subsetD[OF f' x] f[THEN lebesgue.simple_functionD(2)] show ?thesis
+ with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis
by (cases rule: ereal2_cases[of x "?M x"]) auto
qed simp }
ultimately
@@ -580,11 +580,11 @@
by simp
also have \<dots>
proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral
- real_of_ereal_pos lebesgue.positive_measure ballI)
- show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue" "\<And>y. f -` {y} \<inter> UNIV \<in> sets lebesgue"
- using lebesgue.simple_functionD[OF f] by auto
+ real_of_ereal_pos emeasure_nonneg ballI)
+ show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue"
+ using simple_functionD[OF f] by auto
fix y assume "real y \<noteq> 0" "y \<in> range f"
- with * om[OF this(2)] show "lebesgue.\<mu> (f -` {y}) = ereal (real (?M y))"
+ with * om[OF this(2)] show "emeasure lebesgue (f -` {y}) = ereal (real (?M y))"
by (auto simp: ereal_real)
qed
finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" .
@@ -601,28 +601,28 @@
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
proof -
let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x"
- note f(1)[THEN lebesgue.simple_functionD(2)]
+ note f(1)[THEN simple_functionD(2)]
then have [simp, intro]: "\<And>X. f -` X \<in> sets lebesgue" by auto
have f': "simple_function lebesgue ?f"
- using f by (intro lebesgue.simple_function_If_set) auto
+ using f by (intro simple_function_If_set) auto
have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto
have "AE x in lebesgue. f x = ?f x"
- using lebesgue.simple_integral_PInf[OF f i]
- by (intro lebesgue.AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
+ using simple_integral_PInf[OF f i]
+ by (intro AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f"
- by (rule lebesgue.simple_integral_cong_AE)
+ by (rule simple_integral_cong_AE)
have real_eq: "\<And>x. real (f x) = real (?f x)" by auto
show ?thesis
unfolding eq real_eq
proof (rule simple_function_has_integral[OF f' rng])
- fix x assume x: "x \<in> range ?f" and inf: "lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<infinity>"
- have "x * lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
- using f'[THEN lebesgue.simple_functionD(2)]
- by (simp add: lebesgue.simple_integral_cmult_indicator)
+ fix x assume x: "x \<in> range ?f" and inf: "emeasure lebesgue (?f -` {x} \<inter> UNIV) = \<infinity>"
+ have "x * emeasure lebesgue (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
+ using f'[THEN simple_functionD(2)]
+ by (simp add: simple_integral_cmult_indicator)
also have "\<dots> \<le> integral\<^isup>S lebesgue f"
- using f'[THEN lebesgue.simple_functionD(2)] f
- by (intro lebesgue.simple_integral_mono lebesgue.simple_function_mult lebesgue.simple_function_indicator)
+ using f'[THEN simple_functionD(2)] f
+ by (intro simple_integral_mono simple_function_mult simple_function_indicator)
(auto split: split_indicator)
finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm)
qed
@@ -633,16 +633,16 @@
assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
proof -
- from lebesgue.borel_measurable_implies_simple_function_sequence'[OF f(1)]
+ from borel_measurable_implies_simple_function_sequence'[OF f(1)]
guess u . note u = this
have SUP_eq: "\<And>x. (SUP i. u i x) = f x"
using u(4) f(2)[THEN subsetD] by (auto split: split_max)
let ?u = "\<lambda>i x. real (u i x)"
- note u_eq = lebesgue.positive_integral_eq_simple_integral[OF u(1,5), symmetric]
+ note u_eq = positive_integral_eq_simple_integral[OF u(1,5), symmetric]
{ fix i
note u_eq
also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)"
- by (intro lebesgue.positive_integral_mono) (auto intro: SUP_upper simp: u(4)[symmetric])
+ by (intro positive_integral_mono) (auto intro: SUP_upper simp: u(4)[symmetric])
finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>"
unfolding positive_integral_max_0 using f by auto }
note u_fin = this
@@ -684,10 +684,10 @@
also have "\<dots> = real (integral\<^isup>S lebesgue (u k))"
using u_int[THEN integral_unique] by (simp add: u')
also have "\<dots> = real (integral\<^isup>P lebesgue (u k))"
- using lebesgue.positive_integral_eq_simple_integral[OF u(1,5)] by simp
+ using positive_integral_eq_simple_integral[OF u(1,5)] by simp
also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f
- by (auto intro!: real_of_ereal_positive_mono lebesgue.positive_integral_positive
- lebesgue.positive_integral_mono SUP_upper simp: SUP_eq[symmetric])
+ by (auto intro!: real_of_ereal_positive_mono positive_integral_positive
+ positive_integral_mono SUP_upper simp: SUP_eq[symmetric])
finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" .
qed
qed
@@ -695,21 +695,21 @@
have "integral\<^isup>P lebesgue f = ereal (integral UNIV f')"
proof (rule tendsto_unique[OF trivial_limit_sequentially])
have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> (SUP i. integral\<^isup>P lebesgue (u i))"
- unfolding u_eq by (intro LIMSEQ_ereal_SUPR lebesgue.incseq_positive_integral u)
- also note lebesgue.positive_integral_monotone_convergence_SUP
- [OF u(2) lebesgue.borel_measurable_simple_function[OF u(1)] u(5), symmetric]
+ unfolding u_eq by (intro LIMSEQ_ereal_SUPR incseq_positive_integral u)
+ also note positive_integral_monotone_convergence_SUP
+ [OF u(2) borel_measurable_simple_function[OF u(1)] u(5), symmetric]
finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> integral\<^isup>P lebesgue f"
unfolding SUP_eq .
{ fix k
have "0 \<le> integral\<^isup>S lebesgue (u k)"
- using u by (auto intro!: lebesgue.simple_integral_positive)
+ using u by (auto intro!: simple_integral_positive)
then have "integral\<^isup>S lebesgue (u k) = ereal (real (integral\<^isup>S lebesgue (u k)))"
using u_fin by (auto simp: ereal_real) }
note * = this
show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> ereal (integral UNIV f')"
using convergent using u_int[THEN integral_unique, symmetric]
- by (subst *) (simp add: lim_ereal u')
+ by (subst *) (simp add: u')
qed
then show ?thesis using convergent by (simp add: f' integrable_integral)
qed
@@ -721,8 +721,8 @@
proof -
let ?n = "\<lambda>x. real (ereal (max 0 (- f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))"
have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: ereal_max)
- { fix f have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)"
- by (intro lebesgue.positive_integral_cong_pos) (auto split: split_max) }
+ { fix f :: "'a \<Rightarrow> real" have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)"
+ by (intro positive_integral_cong_pos) (auto split: split_max) }
note eq = this
show ?thesis
unfolding lebesgue_integral_def
@@ -732,7 +732,7 @@
apply (safe intro!: positive_integral_has_integral)
using integrableD[OF f]
by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0 split: split_max
- intro!: lebesgue.measurable_If lebesgue.borel_measurable_ereal)
+ intro!: measurable_If)
qed
lemma lebesgue_positive_integral_eq_borel:
@@ -740,7 +740,7 @@
shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
proof -
from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))"
- by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
+ by (auto intro!: positive_integral_subalgebra[symmetric])
then show ?thesis unfolding positive_integral_max_0 .
qed
@@ -749,9 +749,8 @@
shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P)
and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I)
proof -
- have *: "sigma_algebra lborel" by default
have "sets lborel \<subseteq> sets lebesgue" by auto
- from lebesgue.integral_subalgebra[of f lborel, OF _ this _ _ *] assms
+ from integral_subalgebra[of f lborel, OF _ this _ _] assms
show ?P ?I by auto
qed
@@ -783,152 +782,109 @@
"p2e (e2p x) = (x::'a::ordered_euclidean_space)"
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
-interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure_space"
+interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure"
by default
-interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure_space" "{..<n}" for n :: nat
- where "space lborel = UNIV"
- and "sets lborel = sets borel"
- and "measure lborel = lebesgue.\<mu>"
- and "measurable lborel = measurable borel"
-proof -
- show "finite_product_sigma_finite (\<lambda>x. lborel::real measure_space) {..<n}"
- by default simp
-qed simp_all
+interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure" "{..<n}" for n :: nat
+ by default auto
+
+lemma bchoice_iff: "(\<forall>x\<in>A. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>A. P x (f x))"
+ by metis
lemma sets_product_borel:
- assumes [intro]: "finite I"
- shows "sets (\<Pi>\<^isub>M i\<in>I.
- \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>) =
- sets (\<Pi>\<^isub>M i\<in>I. lborel)" (is "sets ?G = _")
-proof -
- have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
- sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
- by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
- (auto intro!: measurable_sigma_sigma incseq_SucI reals_Archimedean2
- simp: product_algebra_def)
- then show ?thesis
- unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
-qed
+ assumes I: "finite I"
+ shows "sets (\<Pi>\<^isub>M i\<in>I. lborel) = sigma_sets (\<Pi>\<^isub>E i\<in>I. UNIV) { \<Pi>\<^isub>E i\<in>I. {..< x i :: real} | x. True}" (is "_ = ?G")
+proof (subst sigma_prod_algebra_sigma_eq[where S="\<lambda>_ i::nat. {..<real i}" and E="\<lambda>_. range lessThan", OF I])
+ show "sigma_sets (space (Pi\<^isub>M I (\<lambda>i. lborel))) {Pi\<^isub>E I F |F. \<forall>i\<in>I. F i \<in> range lessThan} = ?G"
+ by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff)
+qed (auto simp: borel_eq_lessThan incseq_def reals_Archimedean2 image_iff intro: real_natceiling_ge)
lemma measurable_e2p:
- "e2p \<in> measurable (borel::'a::ordered_euclidean_space algebra)
- (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))"
- (is "_ \<in> measurable ?E ?P")
-proof -
- let ?B = "\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>"
- let ?G = "product_algebra_generator {..<DIM('a)} (\<lambda>_. ?B)"
- have "e2p \<in> measurable ?E (sigma ?G)"
- proof (rule borel.measurable_sigma)
- show "e2p \<in> space ?E \<rightarrow> space ?G" by (auto simp: e2p_def)
- fix A assume "A \<in> sets ?G"
- then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
- and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
- by (auto elim!: product_algebraE simp: )
- then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
- from this[THEN bchoice] guess xs ..
- then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
- using A by auto
- have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
- using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
- euclidean_eq[where 'a='a] eucl_less[where 'a='a])
- then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
- qed (auto simp: product_algebra_generator_def)
- with sets_product_borel[of "{..<DIM('a)}"] show ?thesis
- unfolding measurable_def product_algebra_def by simp
-qed
+ "e2p \<in> measurable (borel::'a::ordered_euclidean_space measure) (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure))"
+proof (rule measurable_sigma_sets[OF sets_product_borel])
+ fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> {\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i} |x. True} "
+ then obtain x where "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i})" by auto
+ then have "e2p -` A = {..< (\<chi>\<chi> i. x i) :: 'a}"
+ using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def
+ euclidean_eq[where 'a='a] eucl_less[where 'a='a])
+ then show "e2p -` A \<inter> space (borel::'a measure) \<in> sets borel" by simp
+qed (auto simp: e2p_def)
lemma measurable_p2e:
- "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
- (borel :: 'a::ordered_euclidean_space algebra)"
+ "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure))
+ (borel :: 'a::ordered_euclidean_space measure)"
(is "p2e \<in> measurable ?P _")
- unfolding borel_eq_lessThan
-proof (intro lborel_space.measurable_sigma)
- let ?E = "\<lparr> space = UNIV :: 'a set, sets = range lessThan \<rparr>"
- show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
- fix A assume "A \<in> sets ?E"
- then obtain x where "A = {..<x}" by auto
- then have "p2e -` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})"
- using DIM_positive
- by (auto simp: Pi_iff set_eq_iff p2e_def
- euclidean_eq[where 'a='a] eucl_less[where 'a='a])
- then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
-qed simp
+proof (safe intro!: borel_measurable_iff_halfspace_le[THEN iffD2])
+ fix x i
+ let ?A = "{w \<in> space ?P. (p2e w :: 'a) $$ i \<le> x}"
+ assume "i < DIM('a)"
+ then have "?A = (\<Pi>\<^isub>E j\<in>{..<DIM('a)}. if i = j then {.. x} else UNIV)"
+ using DIM_positive by (auto simp: space_PiM p2e_def split: split_if_asm)
+ then show "?A \<in> sets ?P"
+ by auto
+qed
+
+lemma Int_stable_atLeastAtMost:
+ fixes x::"'a::ordered_euclidean_space"
+ shows "Int_stable (range (\<lambda>(a, b::'a). {a..b}))"
+ by (auto simp: inter_interval Int_stable_def)
-lemma Int_stable_cuboids:
- fixes x::"'a::ordered_euclidean_space"
- shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
- by (auto simp: inter_interval Int_stable_def)
+lemma lborel_eqI:
+ fixes M :: "'a::ordered_euclidean_space measure"
+ assumes emeasure_eq: "\<And>a b. emeasure M {a .. b} = content {a .. b}"
+ assumes sets_eq: "sets M = sets borel"
+ shows "lborel = M"
+proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost])
+ let ?P = "\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel"
+ let ?E = "range (\<lambda>(a, b). {a..b} :: 'a set)"
+ show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
+ by (simp_all add: borel_eq_atLeastAtMost sets_eq)
+
+ show "range cube \<subseteq> ?E" unfolding cube_def [abs_def] by auto
+ show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
+ { fix x :: 'a have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce }
+ then show "(\<Union>i. cube i :: 'a set) = UNIV" by auto
+
+ { fix i show "emeasure lborel (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
+ { fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
+ by (auto simp: emeasure_eq) }
+qed
lemma lborel_eq_lborel_space:
- fixes A :: "('a::ordered_euclidean_space) set"
- assumes "A \<in> sets borel"
- shows "lborel.\<mu> A = lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a))))"
- (is "_ = measure ?P (?T A)")
-proof (rule measure_unique_Int_stable_vimage)
- show "measure_space ?P" by default
- show "measure_space lborel" by default
-
- let ?E = "\<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
- show "Int_stable ?E" using Int_stable_cuboids .
- show "range cube \<subseteq> sets ?E" unfolding cube_def [abs_def] by auto
- show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
- { fix x have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce }
- then show "(\<Union>i. cube i) = space ?E" by auto
- { fix i show "lborel.\<mu> (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
- show "A \<in> sets (sigma ?E)" "sets (sigma ?E) = sets lborel" "space ?E = space lborel"
- using assms by (simp_all add: borel_eq_atLeastAtMost)
-
- show "p2e \<in> measurable ?P (lborel :: 'a measure_space)"
- using measurable_p2e unfolding measurable_def by simp
- { fix X assume "X \<in> sets ?E"
- then obtain a b where X[simp]: "X = {a .. b}" by auto
- have *: "?T X = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a $$ i .. b $$ i})"
- by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def)
- show "lborel.\<mu> X = measure ?P (?T X)"
- proof cases
- assume "X \<noteq> {}"
- then have "a \<le> b"
- by (simp add: interval_ne_empty eucl_le[where 'a='a])
- then have "lborel.\<mu> X = (\<Prod>x<DIM('a). lborel.\<mu> {a $$ x .. b $$ x})"
- by (auto simp: content_closed_interval eucl_le[where 'a='a]
- intro!: setprod_ereal[symmetric])
- also have "\<dots> = measure ?P (?T X)"
- unfolding * by (subst lborel_space.measure_times) auto
- finally show ?thesis .
- qed simp }
+ "(lborel :: 'a measure) = distr (\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel) lborel p2e"
+ (is "?B = ?D")
+proof (rule lborel_eqI)
+ show "sets ?D = sets borel" by simp
+ let ?P = "(\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel)"
+ fix a b :: 'a
+ have *: "p2e -` {a .. b} \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a $$ i .. b $$ i})"
+ by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def space_PiM)
+ have "emeasure ?P (p2e -` {a..b} \<inter> space ?P) = content {a..b}"
+ proof cases
+ assume "{a..b} \<noteq> {}"
+ then have "a \<le> b"
+ by (simp add: interval_ne_empty eucl_le[where 'a='a])
+ then have "emeasure lborel {a..b} = (\<Prod>x<DIM('a). emeasure lborel {a $$ x .. b $$ x})"
+ by (auto simp: content_closed_interval eucl_le[where 'a='a]
+ intro!: setprod_ereal[symmetric])
+ also have "\<dots> = emeasure ?P (p2e -` {a..b} \<inter> space ?P)"
+ unfolding * by (subst lborel_space.measure_times) auto
+ finally show ?thesis by simp
+ qed simp
+ then show "emeasure ?D {a .. b} = content {a .. b}"
+ by (simp add: emeasure_distr measurable_p2e)
qed
-lemma measure_preserving_p2e:
- "p2e \<in> measure_preserving (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
- (lborel::'a::ordered_euclidean_space measure_space)" (is "_ \<in> measure_preserving ?P ?E")
-proof
- show "p2e \<in> measurable ?P ?E"
- using measurable_p2e by (simp add: measurable_def)
- fix A :: "'a set" assume "A \<in> sets lborel"
- then show "lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a)))) = lborel.\<mu> A"
- by (intro lborel_eq_lborel_space[symmetric]) simp
-qed
-
-lemma lebesgue_eq_lborel_space_in_borel:
- fixes A :: "('a::ordered_euclidean_space) set"
- assumes A: "A \<in> sets borel"
- shows "lebesgue.\<mu> A = lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a))))"
- using lborel_eq_lborel_space[OF A] by simp
-
lemma borel_fubini_positiv_integral:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable borel"
- shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
-proof (rule lborel_space.positive_integral_vimage[OF _ measure_preserving_p2e _])
- show "f \<in> borel_measurable lborel"
- using f by (simp_all add: measurable_def)
-qed default
+ shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel)"
+ by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f)
lemma borel_fubini_integrable:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
shows "integrable lborel f \<longleftrightarrow>
- integrable (lborel_space.P DIM('a)) (\<lambda>x. f (p2e x))"
+ integrable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) (\<lambda>x. f (p2e x))"
(is "_ \<longleftrightarrow> integrable ?B ?f")
proof
assume "integrable lborel f"
@@ -941,9 +897,9 @@
by (simp add: comp_def borel_fubini_positiv_integral integrable_def)
next
assume "integrable ?B ?f"
- moreover then
- have "?f \<circ> e2p \<in> borel_measurable (borel::'a algebra)"
- by (auto intro!: measurable_e2p measurable_comp)
+ moreover
+ then have "?f \<circ> e2p \<in> borel_measurable (borel::'a measure)"
+ by (auto intro!: measurable_e2p)
then have "f \<in> borel_measurable borel"
by (simp cong: measurable_cong)
ultimately show "integrable lborel f"
@@ -953,100 +909,35 @@
lemma borel_fubini:
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
assumes f: "f \<in> borel_measurable borel"
- shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
+ shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))"
using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def)
-
-lemma Int_stable_atLeastAtMost:
- "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a,b). {a::'a::ordered_euclidean_space .. b}) \<rparr>"
-proof (simp add: Int_stable_def image_iff, intro allI)
- fix a1 b1 a2 b2 :: 'a
- have "\<forall>i<DIM('a). \<exists>a b. {a1$$i..b1$$i} \<inter> {a2$$i..b2$$i} = {a..b}" by auto
- then have "\<exists>a b. \<forall>i<DIM('a). {a1$$i..b1$$i} \<inter> {a2$$i..b2$$i} = {a i..b i}"
- unfolding choice_iff' .
- then guess a b by safe
- then have "{a1..b1} \<inter> {a2..b2} = {(\<chi>\<chi> i. a i) .. (\<chi>\<chi> i. b i)}"
- by (simp add: set_eq_iff eucl_le[where 'a='a] all_conj_distrib[symmetric]) blast
- then show "\<exists>a' b'. {a1..b1} \<inter> {a2..b2} = {a'..b'}" by blast
-qed
-
-lemma (in sigma_algebra) borel_measurable_sets:
- assumes "f \<in> measurable borel M" "A \<in> sets M"
- shows "f -` A \<in> sets borel"
- using measurable_sets[OF assms] by simp
-
-lemma (in sigma_algebra) measurable_identity[intro,simp]:
- "(\<lambda>x. x) \<in> measurable M M"
- unfolding measurable_def by auto
+lemma borel_measurable_indicator':
+ "A \<in> sets borel \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M"
+ using measurable_comp[OF _ borel_measurable_indicator, of f M borel A] by (auto simp add: comp_def)
lemma lebesgue_real_affine:
- fixes X :: "real set"
- assumes "X \<in> sets borel" and "c \<noteq> 0"
- shows "measure lebesgue X = ereal \<bar>c\<bar> * measure lebesgue ((\<lambda>x. t + c * x) -` X)"
- (is "_ = ?\<nu> X")
-proof -
- let ?E = "\<lparr>space = UNIV, sets = range (\<lambda>(a,b). {a::real .. b})\<rparr> :: real algebra"
- let ?M = "\<lambda>\<nu>. \<lparr>space = space ?E, sets = sets (sigma ?E), measure = \<nu>\<rparr> :: real measure_space"
- have *: "?M (measure lebesgue) = lborel"
- unfolding borel_eq_atLeastAtMost[symmetric]
- by (simp add: lborel_def lebesgue_def)
- have **: "?M ?\<nu> = lborel \<lparr> measure := ?\<nu> \<rparr>"
- unfolding borel_eq_atLeastAtMost[symmetric]
- by (simp add: lborel_def lebesgue_def)
- show ?thesis
- proof (rule measure_unique_Int_stable[where X=X, OF Int_stable_atLeastAtMost], unfold * **)
- show "X \<in> sets (sigma ?E)"
- unfolding borel_eq_atLeastAtMost[symmetric] by fact
- have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastforce
- then show "(\<Union>i. cube i) = space ?E" by auto
- show "incseq cube" by (intro incseq_SucI cube_subset_Suc)
- show "range cube \<subseteq> sets ?E"
- unfolding cube_def [abs_def] by auto
- show "\<And>i. measure lebesgue (cube i) \<noteq> \<infinity>"
- by (simp add: cube_def)
- show "measure_space lborel" by default
- then interpret sigma_algebra "lborel\<lparr>measure := ?\<nu>\<rparr>"
- by (auto simp add: measure_space_def)
- show "measure_space (lborel\<lparr>measure := ?\<nu>\<rparr>)"
- proof
- show "positive (lborel\<lparr>measure := ?\<nu>\<rparr>) (measure (lborel\<lparr>measure := ?\<nu>\<rparr>))"
- by (auto simp: positive_def intro!: ereal_0_le_mult borel.borel_measurable_sets)
- show "countably_additive (lborel\<lparr>measure := ?\<nu>\<rparr>) (measure (lborel\<lparr>measure := ?\<nu>\<rparr>))"
- proof (simp add: countably_additive_def, safe)
- fix A :: "nat \<Rightarrow> real set" assume A: "range A \<subseteq> sets borel" "disjoint_family A"
- then have Ai: "\<And>i. A i \<in> sets borel" by auto
- have "(\<Sum>n. measure lebesgue ((\<lambda>x. t + c * x) -` A n)) = measure lebesgue (\<Union>n. (\<lambda>x. t + c * x) -` A n)"
- proof (intro lborel.measure_countably_additive)
- { fix n have "(\<lambda>x. t + c * x) -` A n \<inter> space borel \<in> sets borel"
- using A borel.measurable_ident unfolding id_def
- by (intro measurable_sets[where A=borel] borel.borel_measurable_add[OF _ borel.borel_measurable_times]) auto }
- then show "range (\<lambda>i. (\<lambda>x. t + c * x) -` A i) \<subseteq> sets borel" by auto
- from `disjoint_family A`
- show "disjoint_family (\<lambda>i. (\<lambda>x. t + c * x) -` A i)"
- by (rule disjoint_family_on_bisimulation) auto
- qed
- with Ai show "(\<Sum>n. ?\<nu> (A n)) = ?\<nu> (UNION UNIV A)"
- by (subst suminf_cmult_ereal)
- (auto simp: vimage_UN borel.borel_measurable_sets)
- qed
- qed
- fix X assume "X \<in> sets ?E"
- then obtain a b where [simp]: "X = {a .. b}" by auto
- show "measure lebesgue X = ?\<nu> X"
- proof cases
- assume "0 < c"
- then have "(\<lambda>x. t + c * x) -` {a..b} = {(a - t) / c .. (b - t) / c}"
- by (auto simp: field_simps)
- with `0 < c` show ?thesis
- by (cases "a \<le> b") (auto simp: field_simps)
- next
- assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto
- then have *: "(\<lambda>x. t + c * x) -` {a..b} = {(b - t) / c .. (a - t) / c}"
- by (auto simp: field_simps)
- with `c < 0` show ?thesis
- by (cases "a \<le> b") (auto simp: field_simps)
- qed
+ fixes c :: real assumes "c \<noteq> 0"
+ shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D")
+proof (rule lborel_eqI)
+ fix a b show "emeasure ?D {a..b} = content {a .. b}"
+ proof cases
+ assume "0 < c"
+ then have "(\<lambda>x. t + c * x) -` {a..b} = {(a - t) / c .. (b - t) / c}"
+ by (auto simp: field_simps)
+ with `0 < c` show ?thesis
+ by (cases "a \<le> b")
+ (auto simp: field_simps emeasure_density positive_integral_distr positive_integral_cmult
+ borel_measurable_indicator' emeasure_distr)
+ next
+ assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto
+ then have *: "(\<lambda>x. t + c * x) -` {a..b} = {(b - t) / c .. (a - t) / c}"
+ by (auto simp: field_simps)
+ with `c < 0` show ?thesis
+ by (cases "a \<le> b")
+ (auto simp: field_simps emeasure_density positive_integral_distr
+ positive_integral_cmult borel_measurable_indicator' emeasure_distr)
qed
-qed
+qed simp
end
--- a/src/HOL/Probability/Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1452 +0,0 @@
-(* Title: HOL/Probability/Measure.thy
- Author: Lawrence C Paulson
- Author: Johannes Hölzl, TU München
- Author: Armin Heller, TU München
-*)
-
-header {* Properties about measure spaces *}
-
-theory Measure
- imports Caratheodory
-begin
-
-lemma measure_algebra_more[simp]:
- "\<lparr> space = A, sets = B, \<dots> = algebra.more M \<rparr> \<lparr> measure := m \<rparr> =
- \<lparr> space = A, sets = B, \<dots> = algebra.more (M \<lparr> measure := m \<rparr>) \<rparr>"
- by (cases M) simp
-
-lemma measure_algebra_more_eq[simp]:
- "\<And>X. measure \<lparr> space = T, sets = A, \<dots> = algebra.more X \<rparr> = measure X"
- unfolding measure_space.splits by simp
-
-lemma measure_sigma[simp]: "measure (sigma A) = measure A"
- unfolding sigma_def by simp
-
-lemma algebra_measure_update[simp]:
- "algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> algebra M'"
- unfolding algebra_iff_Un by simp
-
-lemma sigma_algebra_measure_update[simp]:
- "sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> sigma_algebra M'"
- unfolding sigma_algebra_def sigma_algebra_axioms_def by simp
-
-lemma finite_sigma_algebra_measure_update[simp]:
- "finite_sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> finite_sigma_algebra M'"
- unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
-
-lemma measurable_cancel_measure[simp]:
- "measurable M1 (M2\<lparr>measure := m2\<rparr>) = measurable M1 M2"
- "measurable (M2\<lparr>measure := m1\<rparr>) M1 = measurable M2 M1"
- unfolding measurable_def by auto
-
-lemma inj_on_image_eq_iff:
- assumes "inj_on f S"
- assumes "A \<subseteq> S" "B \<subseteq> S"
- shows "(f ` A = f ` B) \<longleftrightarrow> (A = B)"
-proof -
- have "inj_on f (A \<union> B)"
- using assms by (auto intro: subset_inj_on)
- from inj_on_Un_image_eq_iff[OF this]
- show ?thesis .
-qed
-
-lemma image_vimage_inter_eq:
- assumes "f ` S = T" "X \<subseteq> T"
- shows "f ` (f -` X \<inter> S) = X"
-proof (intro antisym)
- have "f ` (f -` X \<inter> S) \<subseteq> f ` (f -` X)" by auto
- also have "\<dots> = X \<inter> range f" by simp
- also have "\<dots> = X" using assms by auto
- finally show "f ` (f -` X \<inter> S) \<subseteq> X" by auto
-next
- show "X \<subseteq> f ` (f -` X \<inter> S)"
- proof
- fix x assume "x \<in> X"
- then have "x \<in> T" using `X \<subseteq> T` by auto
- then obtain y where "x = f y" "y \<in> S"
- using assms by auto
- then have "{y} \<subseteq> f -` X \<inter> S" using `x \<in> X` by auto
- moreover have "x \<in> f ` {y}" using `x = f y` by auto
- ultimately show "x \<in> f ` (f -` X \<inter> S)" by auto
- qed
-qed
-
-text {*
- This formalisation of measure theory is based on the work of Hurd/Coble wand
- was later translated by Lawrence Paulson to Isabelle/HOL. Later it was
- modified to use the positive infinite reals and to prove the uniqueness of
- cut stable measures.
-*}
-
-section {* Equations for the measure function @{text \<mu>} *}
-
-lemma (in measure_space) measure_countably_additive:
- assumes "range A \<subseteq> sets M" "disjoint_family A"
- shows "(\<Sum>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
-proof -
- have "(\<Union> i. A i) \<in> sets M" using assms(1) by (rule countable_UN)
- with ca assms show ?thesis by (simp add: countably_additive_def)
-qed
-
-lemma (in sigma_algebra) sigma_algebra_cong:
- assumes "space N = space M" "sets N = sets M"
- shows "sigma_algebra N"
- by default (insert sets_into_space, auto simp: assms)
-
-lemma (in measure_space) measure_space_cong:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
- shows "measure_space N"
-proof -
- interpret N: sigma_algebra N by (intro sigma_algebra_cong assms)
- show ?thesis
- proof
- show "positive N (measure N)" using assms by (auto simp: positive_def)
- show "countably_additive N (measure N)" unfolding countably_additive_def
- proof safe
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets N" "disjoint_family A"
- then have "\<And>i. A i \<in> sets M" "(UNION UNIV A) \<in> sets M" unfolding assms by auto
- from measure_countably_additive[of A] A this[THEN assms(1)]
- show "(\<Sum>n. measure N (A n)) = measure N (UNION UNIV A)"
- unfolding assms by simp
- qed
- qed
-qed
-
-lemma (in measure_space) additive: "additive M \<mu>"
- using ca by (auto intro!: countably_additive_additive simp: positive_def)
-
-lemma (in measure_space) measure_additive:
- "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {}
- \<Longrightarrow> \<mu> a + \<mu> b = \<mu> (a \<union> b)"
- by (metis additiveD additive)
-
-lemma (in measure_space) measure_mono:
- assumes "a \<subseteq> b" "a \<in> sets M" "b \<in> sets M"
- shows "\<mu> a \<le> \<mu> b"
-proof -
- have "b = a \<union> (b - a)" using assms by auto
- moreover have "{} = a \<inter> (b - a)" by auto
- ultimately have "\<mu> b = \<mu> a + \<mu> (b - a)"
- using measure_additive[of a "b - a"] Diff[of b a] assms by auto
- moreover have "\<mu> a + 0 \<le> \<mu> a + \<mu> (b - a)" using assms by (intro add_mono) auto
- ultimately show "\<mu> a \<le> \<mu> b" by auto
-qed
-
-lemma (in measure_space) measure_top:
- "A \<in> sets M \<Longrightarrow> \<mu> A \<le> \<mu> (space M)"
- using sets_into_space[of A] by (auto intro!: measure_mono)
-
-lemma (in measure_space) measure_compl:
- assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<infinity>"
- shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
-proof -
- have s_less_space: "\<mu> s \<le> \<mu> (space M)"
- using s by (auto intro!: measure_mono sets_into_space)
- from s have "0 \<le> \<mu> s" by auto
- have "\<mu> (space M) = \<mu> (s \<union> (space M - s))" using s
- by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
- also have "... = \<mu> s + \<mu> (space M - s)"
- by (rule additiveD [OF additive]) (auto simp add: s)
- finally have "\<mu> (space M) = \<mu> s + \<mu> (space M - s)" .
- then show ?thesis
- using fin `0 \<le> \<mu> s`
- unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
-qed
-
-lemma (in measure_space) measure_Diff:
- assumes finite: "\<mu> B \<noteq> \<infinity>"
- and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
- shows "\<mu> (A - B) = \<mu> A - \<mu> B"
-proof -
- have "0 \<le> \<mu> B" using assms by auto
- have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
- then have "\<mu> A = \<mu> ((A - B) \<union> B)" by simp
- also have "\<dots> = \<mu> (A - B) + \<mu> B"
- using measurable by (subst measure_additive[symmetric]) auto
- finally show "\<mu> (A - B) = \<mu> A - \<mu> B"
- unfolding ereal_eq_minus_iff
- using finite `0 \<le> \<mu> B` by auto
-qed
-
-lemma (in measure_space) measure_countable_increasing:
- assumes A: "range A \<subseteq> sets M"
- and A0: "A 0 = {}"
- and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
- shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
-proof -
- { fix n
- have "\<mu> (A n) = (\<Sum>i<n. \<mu> (A (Suc i) - A i))"
- proof (induct n)
- case 0 thus ?case by (auto simp add: A0)
- next
- case (Suc m)
- have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
- by (metis ASuc Un_Diff_cancel Un_absorb1)
- hence "\<mu> (A (Suc m)) =
- \<mu> (A m) + \<mu> (A (Suc m) - A m)"
- by (subst measure_additive)
- (auto simp add: measure_additive range_subsetD [OF A])
- with Suc show ?case
- by simp
- qed }
- note Meq = this
- have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
- proof (rule UN_finite2_eq [where k=1], simp)
- fix i
- show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
- proof (induct i)
- case 0 thus ?case by (simp add: A0)
- next
- case (Suc i)
- thus ?case
- by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
- qed
- qed
- have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
- by (metis A Diff range_subsetD)
- have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
- by (blast intro: range_subsetD [OF A])
- have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = (\<Sum>i. \<mu> (A (Suc i) - A i))"
- using A by (auto intro!: suminf_ereal_eq_SUPR[symmetric])
- also have "\<dots> = \<mu> (\<Union>i. A (Suc i) - A i)"
- by (rule measure_countably_additive)
- (auto simp add: disjoint_family_Suc ASuc A1 A2)
- also have "... = \<mu> (\<Union>i. A i)"
- by (simp add: Aeq)
- finally have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
- then show ?thesis by (auto simp add: Meq)
-qed
-
-lemma (in measure_space) continuity_from_below:
- assumes A: "range A \<subseteq> sets M" and "incseq A"
- shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
-proof -
- have *: "(SUP n. \<mu> (nat_case {} A (Suc n))) = (SUP n. \<mu> (nat_case {} A n))"
- using A by (auto intro!: SUPR_eq exI split: nat.split)
- have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
- by (auto simp add: split: nat.splits)
- have meq: "\<And>n. \<mu> (A n) = (\<mu> \<circ> nat_case {} A) (Suc n)"
- by simp
- have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. nat_case {} A i)"
- using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
- by (force split: nat.splits intro!: measure_countable_increasing)
- also have "\<mu> (\<Union>i. nat_case {} A i) = \<mu> (\<Union>i. A i)"
- by (simp add: ueq)
- finally have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. A i)" .
- thus ?thesis unfolding meq * comp_def .
-qed
-
-lemma (in measure_space) measure_incseq:
- assumes "range B \<subseteq> sets M" "incseq B"
- shows "incseq (\<lambda>i. \<mu> (B i))"
- using assms by (auto simp: incseq_def intro!: measure_mono)
-
-lemma (in measure_space) continuity_from_below_Lim:
- assumes A: "range A \<subseteq> sets M" "incseq A"
- shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Union>i. A i)"
- using LIMSEQ_ereal_SUPR[OF measure_incseq, OF A]
- continuity_from_below[OF A] by simp
-
-lemma (in measure_space) measure_decseq:
- assumes "range B \<subseteq> sets M" "decseq B"
- shows "decseq (\<lambda>i. \<mu> (B i))"
- using assms by (auto simp: decseq_def intro!: measure_mono)
-
-lemma (in measure_space) continuity_from_above:
- assumes A: "range A \<subseteq> sets M" and "decseq A"
- and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
- shows "(INF n. \<mu> (A n)) = \<mu> (\<Inter>i. A i)"
-proof -
- have le_MI: "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
- using A by (auto intro!: measure_mono)
- hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
-
- have A0: "0 \<le> \<mu> (A 0)" using A by auto
-
- have "\<mu> (A 0) - (INF n. \<mu> (A n)) = \<mu> (A 0) + (SUP n. - \<mu> (A n))"
- by (simp add: ereal_SUPR_uminus minus_ereal_def)
- also have "\<dots> = (SUP n. \<mu> (A 0) - \<mu> (A n))"
- unfolding minus_ereal_def using A0 assms
- by (subst SUPR_ereal_add) (auto simp add: measure_decseq)
- also have "\<dots> = (SUP n. \<mu> (A 0 - A n))"
- using A finite `decseq A`[unfolded decseq_def] by (subst measure_Diff) auto
- also have "\<dots> = \<mu> (\<Union>i. A 0 - A i)"
- proof (rule continuity_from_below)
- show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
- using A by auto
- show "incseq (\<lambda>n. A 0 - A n)"
- using `decseq A` by (auto simp add: incseq_def decseq_def)
- qed
- also have "\<dots> = \<mu> (A 0) - \<mu> (\<Inter>i. A i)"
- using A finite * by (simp, subst measure_Diff) auto
- finally show ?thesis
- unfolding ereal_minus_eq_minus_iff using finite A0 by auto
-qed
-
-lemma (in measure_space) measure_insert:
- assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
- shows "\<mu> (insert x A) = \<mu> {x} + \<mu> A"
-proof -
- have "{x} \<inter> A = {}" using `x \<notin> A` by auto
- from measure_additive[OF sets this] show ?thesis by simp
-qed
-
-lemma (in measure_space) measure_setsum:
- assumes "finite S" and "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
- assumes disj: "disjoint_family_on A S"
- shows "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>i\<in>S. A i)"
-using assms proof induct
- case (insert i S)
- then have "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>a\<in>S. A a)"
- by (auto intro: disjoint_family_on_mono)
- moreover have "A i \<inter> (\<Union>a\<in>S. A a) = {}"
- using `disjoint_family_on A (insert i S)` `i \<notin> S`
- by (auto simp: disjoint_family_on_def)
- ultimately show ?case using insert
- by (auto simp: measure_additive finite_UN)
-qed simp
-
-lemma (in measure_space) measure_finite_singleton:
- assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
- shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
- using measure_setsum[of S "\<lambda>x. {x}", OF assms]
- by (auto simp: disjoint_family_on_def)
-
-lemma finite_additivity_sufficient:
- assumes "sigma_algebra M"
- assumes fin: "finite (space M)" and pos: "positive M (measure M)" and add: "additive M (measure M)"
- shows "measure_space M"
-proof -
- interpret sigma_algebra M by fact
- show ?thesis
- proof
- show [simp]: "positive M (measure M)" using pos by (simp add: positive_def)
- show "countably_additive M (measure M)"
- proof (auto simp add: countably_additive_def)
- fix A :: "nat \<Rightarrow> 'a set"
- assume A: "range A \<subseteq> sets M"
- and disj: "disjoint_family A"
- and UnA: "(\<Union>i. A i) \<in> sets M"
- def I \<equiv> "{i. A i \<noteq> {}}"
- have "Union (A ` I) \<subseteq> space M" using A
- by auto (metis range_subsetD subsetD sets_into_space)
- hence "finite (A ` I)"
- by (metis finite_UnionD finite_subset fin)
- moreover have "inj_on A I" using disj
- by (auto simp add: I_def disjoint_family_on_def inj_on_def)
- ultimately have finI: "finite I"
- by (metis finite_imageD)
- hence "\<exists>N. \<forall>m\<ge>N. A m = {}"
- proof (cases "I = {}")
- case True thus ?thesis by (simp add: I_def)
- next
- case False
- hence "\<forall>i\<in>I. i < Suc(Max I)"
- by (simp add: Max_less_iff [symmetric] finI)
- hence "\<forall>m \<ge> Suc(Max I). A m = {}"
- by (simp add: I_def) (metis less_le_not_le)
- thus ?thesis
- by blast
- qed
- then obtain N where N: "\<forall>m\<ge>N. A m = {}" by blast
- then have "\<forall>m\<ge>N. measure M (A m) = 0" using pos[unfolded positive_def] by simp
- then have "(\<Sum>n. measure M (A n)) = (\<Sum>m<N. measure M (A m))"
- by (simp add: suminf_finite)
- also have "... = measure M (\<Union>i<N. A i)"
- proof (induct N)
- case 0 thus ?case using pos[unfolded positive_def] by simp
- next
- case (Suc n)
- have "measure M (A n \<union> (\<Union> x<n. A x)) = measure M (A n) + measure M (\<Union> i<n. A i)"
- proof (rule Caratheodory.additiveD [OF add])
- show "A n \<inter> (\<Union> x<n. A x) = {}" using disj
- by (auto simp add: disjoint_family_on_def nat_less_le) blast
- show "A n \<in> sets M" using A
- by force
- show "(\<Union>i<n. A i) \<in> sets M"
- proof (induct n)
- case 0 thus ?case by simp
- next
- case (Suc n) thus ?case using A
- by (simp add: lessThan_Suc Un range_subsetD)
- qed
- qed
- thus ?case using Suc
- by (simp add: lessThan_Suc)
- qed
- also have "... = measure M (\<Union>i. A i)"
- proof -
- have "(\<Union> i<N. A i) = (\<Union>i. A i)" using N
- by auto (metis Int_absorb N disjoint_iff_not_equal lessThan_iff not_leE)
- thus ?thesis by simp
- qed
- finally show "(\<Sum>n. measure M (A n)) = measure M (\<Union>i. A i)" .
- qed
- qed
-qed
-
-lemma (in measure_space) measure_setsum_split:
- assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
- assumes "(\<Union>i\<in>S. B i) = space M"
- assumes "disjoint_family_on B S"
- shows "\<mu> A = (\<Sum>i\<in>S. \<mu> (A \<inter> (B i)))"
-proof -
- have *: "\<mu> A = \<mu> (\<Union>i\<in>S. A \<inter> B i)"
- using assms by auto
- show ?thesis unfolding *
- proof (rule measure_setsum[symmetric])
- show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
- using `disjoint_family_on B S`
- unfolding disjoint_family_on_def by auto
- qed (insert assms, auto)
-qed
-
-lemma (in measure_space) measure_subadditive:
- assumes measurable: "A \<in> sets M" "B \<in> sets M"
- shows "\<mu> (A \<union> B) \<le> \<mu> A + \<mu> B"
-proof -
- from measure_additive[of A "B - A"]
- have "\<mu> (A \<union> B) = \<mu> A + \<mu> (B - A)"
- using assms by (simp add: Diff)
- also have "\<dots> \<le> \<mu> A + \<mu> B"
- using assms by (auto intro!: add_left_mono measure_mono)
- finally show ?thesis .
-qed
-
-lemma (in measure_space) measure_subadditive_finite:
- assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
- shows "\<mu> (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu> (A i))"
-using assms proof induct
- case (insert i I)
- then have "\<mu> (\<Union>i\<in>insert i I. A i) = \<mu> (A i \<union> (\<Union>i\<in>I. A i))"
- by simp
- also have "\<dots> \<le> \<mu> (A i) + \<mu> (\<Union>i\<in>I. A i)"
- using insert by (intro measure_subadditive finite_UN) auto
- also have "\<dots> \<le> \<mu> (A i) + (\<Sum>i\<in>I. \<mu> (A i))"
- using insert by (intro add_mono) auto
- also have "\<dots> = (\<Sum>i\<in>insert i I. \<mu> (A i))"
- using insert by auto
- finally show ?case .
-qed simp
-
-lemma (in measure_space) measure_eq_0:
- assumes "N \<in> sets M" and "\<mu> N = 0" and "K \<subseteq> N" and "K \<in> sets M"
- shows "\<mu> K = 0"
- using measure_mono[OF assms(3,4,1)] assms(2) positive_measure[OF assms(4)] by auto
-
-lemma (in measure_space) measure_finitely_subadditive:
- assumes "finite I" "A ` I \<subseteq> sets M"
- shows "\<mu> (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu> (A i))"
-using assms proof induct
- case (insert i I)
- then have "(\<Union>i\<in>I. A i) \<in> sets M" by auto
- then have "\<mu> (\<Union>i\<in>insert i I. A i) \<le> \<mu> (A i) + \<mu> (\<Union>i\<in>I. A i)"
- using insert by (simp add: measure_subadditive)
- also have "\<dots> \<le> (\<Sum>i\<in>insert i I. \<mu> (A i))"
- using insert by (auto intro!: add_left_mono)
- finally show ?case .
-qed simp
-
-lemma (in measure_space) measure_countably_subadditive:
- assumes "range f \<subseteq> sets M"
- shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>i. \<mu> (f i))"
-proof -
- have "\<mu> (\<Union>i. f i) = \<mu> (\<Union>i. disjointed f i)"
- unfolding UN_disjointed_eq ..
- also have "\<dots> = (\<Sum>i. \<mu> (disjointed f i))"
- using range_disjointed_sets[OF assms] measure_countably_additive
- by (simp add: disjoint_family_disjointed comp_def)
- also have "\<dots> \<le> (\<Sum>i. \<mu> (f i))"
- using range_disjointed_sets[OF assms] assms
- by (auto intro!: suminf_le_pos measure_mono disjointed_subset)
- finally show ?thesis .
-qed
-
-lemma (in measure_space) measure_UN_eq_0:
- assumes "\<And>i::nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
- shows "\<mu> (\<Union> i. N i) = 0"
-proof -
- have "0 \<le> \<mu> (\<Union> i. N i)" using assms by auto
- moreover have "\<mu> (\<Union> i. N i) \<le> 0"
- using measure_countably_subadditive[OF assms(2)] assms(1) by simp
- ultimately show ?thesis by simp
-qed
-
-lemma (in measure_space) measure_inter_full_set:
- assumes "S \<in> sets M" "T \<in> sets M" and fin: "\<mu> (T - S) \<noteq> \<infinity>"
- assumes T: "\<mu> T = \<mu> (space M)"
- shows "\<mu> (S \<inter> T) = \<mu> S"
-proof (rule antisym)
- show " \<mu> (S \<inter> T) \<le> \<mu> S"
- using assms by (auto intro!: measure_mono)
-
- have pos: "0 \<le> \<mu> (T - S)" using assms by auto
- show "\<mu> S \<le> \<mu> (S \<inter> T)"
- proof (rule ccontr)
- assume contr: "\<not> ?thesis"
- have "\<mu> (space M) = \<mu> ((T - S) \<union> (S \<inter> T))"
- unfolding T[symmetric] by (auto intro!: arg_cong[where f="\<mu>"])
- also have "\<dots> \<le> \<mu> (T - S) + \<mu> (S \<inter> T)"
- using assms by (auto intro!: measure_subadditive)
- also have "\<dots> < \<mu> (T - S) + \<mu> S"
- using fin contr pos by (intro ereal_less_add) auto
- also have "\<dots> = \<mu> (T \<union> S)"
- using assms by (subst measure_additive) auto
- also have "\<dots> \<le> \<mu> (space M)"
- using assms sets_into_space by (auto intro!: measure_mono)
- finally show False ..
- qed
-qed
-
-lemma measure_unique_Int_stable:
- fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
- assumes "Int_stable E"
- and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E"
- and M: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<mu>\<rparr>" (is "measure_space ?M")
- and N: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<nu>\<rparr>" (is "measure_space ?N")
- and eq: "\<And>X. X \<in> sets E \<Longrightarrow> \<mu> X = \<nu> X"
- and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
- assumes "X \<in> sets (sigma E)"
- shows "\<mu> X = \<nu> X"
-proof -
- let ?D = "\<lambda>F. {D. D \<in> sets (sigma E) \<and> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)}"
- interpret M: measure_space ?M
- where "space ?M = space E" and "sets ?M = sets (sigma E)" and "measure ?M = \<mu>" by (simp_all add: M)
- interpret N: measure_space ?N
- where "space ?N = space E" and "sets ?N = sets (sigma E)" and "measure ?N = \<nu>" by (simp_all add: N)
- { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<infinity>"
- then have [intro]: "F \<in> sets (sigma E)" by auto
- have "\<nu> F \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` `F \<in> sets E` eq by simp
- interpret D: dynkin_system "\<lparr>space=space E, sets=?D F\<rparr>"
- proof (rule dynkin_systemI, simp_all)
- fix A assume "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
- then show "A \<subseteq> space E" using M.sets_into_space by auto
- next
- have "F \<inter> space E = F" using `F \<in> sets E` by auto
- then show "\<mu> (F \<inter> space E) = \<nu> (F \<inter> space E)"
- using `F \<in> sets E` eq by auto
- next
- fix A assume *: "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
- then have **: "F \<inter> (space (sigma E) - A) = F - (F \<inter> A)"
- and [intro]: "F \<inter> A \<in> sets (sigma E)"
- using `F \<in> sets E` M.sets_into_space by auto
- have "\<nu> (F \<inter> A) \<le> \<nu> F" by (auto intro!: N.measure_mono)
- then have "\<nu> (F \<inter> A) \<noteq> \<infinity>" using `\<nu> F \<noteq> \<infinity>` by auto
- have "\<mu> (F \<inter> A) \<le> \<mu> F" by (auto intro!: M.measure_mono)
- then have "\<mu> (F \<inter> A) \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` by auto
- then have "\<mu> (F \<inter> (space (sigma E) - A)) = \<mu> F - \<mu> (F \<inter> A)" unfolding **
- using `F \<inter> A \<in> sets (sigma E)` by (auto intro!: M.measure_Diff)
- also have "\<dots> = \<nu> F - \<nu> (F \<inter> A)" using eq `F \<in> sets E` * by simp
- also have "\<dots> = \<nu> (F \<inter> (space (sigma E) - A))" unfolding **
- using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<infinity>` by (auto intro!: N.measure_Diff[symmetric])
- finally show "space E - A \<in> sets (sigma E) \<and> \<mu> (F \<inter> (space E - A)) = \<nu> (F \<inter> (space E - A))"
- using * by auto
- next
- fix A :: "nat \<Rightarrow> 'a set"
- assume "disjoint_family A" "range A \<subseteq> {X \<in> sets (sigma E). \<mu> (F \<inter> X) = \<nu> (F \<inter> X)}"
- then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sets (sigma E)" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
- "disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. \<mu> (F \<inter> A i) = \<nu> (F \<inter> A i)" "range A \<subseteq> sets (sigma E)"
- by (auto simp: disjoint_family_on_def subset_eq)
- then show "(\<Union>x. A x) \<in> sets (sigma E) \<and> \<mu> (F \<inter> (\<Union>x. A x)) = \<nu> (F \<inter> (\<Union>x. A x))"
- by (auto simp: M.measure_countably_additive[symmetric]
- N.measure_countably_additive[symmetric]
- simp del: UN_simps)
- qed
- have *: "sets (sigma E) = sets \<lparr>space = space E, sets = ?D F\<rparr>"
- using `F \<in> sets E` `Int_stable E`
- by (intro D.dynkin_lemma)
- (auto simp add: sets_sigma Int_stable_def eq intro: sigma_sets.Basic)
- have "\<And>D. D \<in> sets (sigma E) \<Longrightarrow> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)"
- by (subst (asm) *) auto }
- note * = this
- let ?A = "\<lambda>i. A i \<inter> X"
- have A': "range ?A \<subseteq> sets (sigma E)" "incseq ?A"
- using A(1,2) `X \<in> sets (sigma E)` by (auto simp: incseq_def)
- { fix i have "\<mu> (?A i) = \<nu> (?A i)"
- using *[of "A i" X] `X \<in> sets (sigma E)` A finite by auto }
- with M.continuity_from_below[OF A'] N.continuity_from_below[OF A']
- show ?thesis using A(3) `X \<in> sets (sigma E)` by auto
-qed
-
-section "@{text \<mu>}-null sets"
-
-abbreviation (in measure_space) "null_sets \<equiv> {N\<in>sets M. \<mu> N = 0}"
-
-sublocale measure_space \<subseteq> nullsets!: ring_of_sets "\<lparr> space = space M, sets = null_sets \<rparr>"
- where "space \<lparr> space = space M, sets = null_sets \<rparr> = space M"
- and "sets \<lparr> space = space M, sets = null_sets \<rparr> = null_sets"
-proof -
- { fix A B assume sets: "A \<in> sets M" "B \<in> sets M"
- moreover then have "\<mu> (A \<union> B) \<le> \<mu> A + \<mu> B" "\<mu> (A - B) \<le> \<mu> A"
- by (auto intro!: measure_subadditive measure_mono)
- moreover assume "\<mu> B = 0" "\<mu> A = 0"
- ultimately have "\<mu> (A - B) = 0" "\<mu> (A \<union> B) = 0"
- by (auto intro!: antisym) }
- note null = this
- show "ring_of_sets \<lparr> space = space M, sets = null_sets \<rparr>"
- by default (insert sets_into_space null, auto)
-qed simp_all
-
-lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
-proof -
- have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
- unfolding SUP_def image_compose
- unfolding surj_from_nat ..
- then show ?thesis by simp
-qed
-
-lemma (in measure_space) null_sets_UN[intro]:
- assumes "\<And>i::'i::countable. N i \<in> null_sets"
- shows "(\<Union>i. N i) \<in> null_sets"
-proof (intro conjI CollectI)
- show "(\<Union>i. N i) \<in> sets M" using assms by auto
- then have "0 \<le> \<mu> (\<Union>i. N i)" by simp
- moreover have "\<mu> (\<Union>i. N i) \<le> (\<Sum>n. \<mu> (N (Countable.from_nat n)))"
- unfolding UN_from_nat[of N]
- using assms by (intro measure_countably_subadditive) auto
- ultimately show "\<mu> (\<Union>i. N i) = 0" using assms by auto
-qed
-
-lemma (in measure_space) null_set_Int1:
- assumes "B \<in> null_sets" "A \<in> sets M" shows "A \<inter> B \<in> null_sets"
-using assms proof (intro CollectI conjI)
- show "\<mu> (A \<inter> B) = 0" using assms by (intro measure_eq_0[of B "A \<inter> B"]) auto
-qed auto
-
-lemma (in measure_space) null_set_Int2:
- assumes "B \<in> null_sets" "A \<in> sets M" shows "B \<inter> A \<in> null_sets"
- using assms by (subst Int_commute) (rule null_set_Int1)
-
-lemma (in measure_space) measure_Diff_null_set:
- assumes "B \<in> null_sets" "A \<in> sets M"
- shows "\<mu> (A - B) = \<mu> A"
-proof -
- have *: "A - B = (A - (A \<inter> B))" by auto
- have "A \<inter> B \<in> null_sets" using assms by (rule null_set_Int1)
- then show ?thesis
- unfolding * using assms
- by (subst measure_Diff) auto
-qed
-
-lemma (in measure_space) null_set_Diff:
- assumes "B \<in> null_sets" "A \<in> sets M" shows "B - A \<in> null_sets"
-using assms proof (intro CollectI conjI)
- show "\<mu> (B - A) = 0" using assms by (intro measure_eq_0[of B "B - A"]) auto
-qed auto
-
-lemma (in measure_space) measure_Un_null_set:
- assumes "A \<in> sets M" "B \<in> null_sets"
- shows "\<mu> (A \<union> B) = \<mu> A"
-proof -
- have *: "A \<union> B = A \<union> (B - A)" by auto
- have "B - A \<in> null_sets" using assms(2,1) by (rule null_set_Diff)
- then show ?thesis
- unfolding * using assms
- by (subst measure_additive[symmetric]) auto
-qed
-
-section "Formalise almost everywhere"
-
-definition (in measure_space)
- almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
- "almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
-
-syntax
- "_almost_everywhere" :: "pttrn \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
-
-translations
- "AE x in M. P" == "CONST measure_space.almost_everywhere M (%x. P)"
-
-lemma (in measure_space) AE_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
- shows "(AE x in N. P x) \<longleftrightarrow> (AE x. P x)"
-proof -
- interpret N: measure_space N
- by (rule measure_space_cong) fact+
- show ?thesis
- unfolding N.almost_everywhere_def almost_everywhere_def
- by (auto simp: assms)
-qed
-
-lemma (in measure_space) AE_I':
- "N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
- unfolding almost_everywhere_def by auto
-
-lemma (in measure_space) AE_iff_null_set:
- assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
- shows "(AE x. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets"
-proof
- assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
- unfolding almost_everywhere_def by auto
- have "0 \<le> \<mu> ?P" using assms by simp
- moreover have "\<mu> ?P \<le> \<mu> N"
- using assms N(1,2) by (auto intro: measure_mono)
- ultimately have "\<mu> ?P = 0" unfolding `\<mu> N = 0` by auto
- then show "?P \<in> null_sets" using assms by simp
-next
- assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
-qed
-
-lemma (in measure_space) AE_iff_measurable:
- "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x. P x) \<longleftrightarrow> \<mu> N = 0"
- using AE_iff_null_set[of P] by simp
-
-lemma (in measure_space) AE_True[intro, simp]: "AE x. True"
- unfolding almost_everywhere_def by auto
-
-lemma (in measure_space) AE_E[consumes 1]:
- assumes "AE x. P x"
- obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "\<mu> N = 0" "N \<in> sets M"
- using assms unfolding almost_everywhere_def by auto
-
-lemma (in measure_space) AE_E2:
- assumes "AE x. P x" "{x\<in>space M. P x} \<in> sets M"
- shows "\<mu> {x\<in>space M. \<not> P x} = 0" (is "\<mu> ?P = 0")
-proof -
- have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}"
- by auto
- with AE_iff_null_set[of P] assms show ?thesis by auto
-qed
-
-lemma (in measure_space) AE_I:
- assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "\<mu> N = 0" "N \<in> sets M"
- shows "AE x. P x"
- using assms unfolding almost_everywhere_def by auto
-
-lemma (in measure_space) AE_mp[elim!]:
- assumes AE_P: "AE x. P x" and AE_imp: "AE x. P x \<longrightarrow> Q x"
- shows "AE x. Q x"
-proof -
- from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
- and A: "A \<in> sets M" "\<mu> A = 0"
- by (auto elim!: AE_E)
-
- from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
- and B: "B \<in> sets M" "\<mu> B = 0"
- by (auto elim!: AE_E)
-
- show ?thesis
- proof (intro AE_I)
- have "0 \<le> \<mu> (A \<union> B)" using A B by auto
- moreover have "\<mu> (A \<union> B) \<le> 0"
- using measure_subadditive[of A B] A B by auto
- ultimately show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B by auto
- show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
- using P imp by auto
- qed
-qed
-
-lemma (in measure_space)
- shows AE_iffI: "AE x. P x \<Longrightarrow> AE x. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x. Q x"
- and AE_disjI1: "AE x. P x \<Longrightarrow> AE x. P x \<or> Q x"
- and AE_disjI2: "AE x. Q x \<Longrightarrow> AE x. P x \<or> Q x"
- and AE_conjI: "AE x. P x \<Longrightarrow> AE x. Q x \<Longrightarrow> AE x. P x \<and> Q x"
- and AE_conj_iff[simp]: "(AE x. P x \<and> Q x) \<longleftrightarrow> (AE x. P x) \<and> (AE x. Q x)"
- by auto
-
-lemma (in measure_space) AE_measure:
- assumes AE: "AE x. P x" and sets: "{x\<in>space M. P x} \<in> sets M"
- shows "\<mu> {x\<in>space M. P x} = \<mu> (space M)"
-proof -
- from AE_E[OF AE] guess N . note N = this
- with sets have "\<mu> (space M) \<le> \<mu> ({x\<in>space M. P x} \<union> N)"
- by (intro measure_mono) auto
- also have "\<dots> \<le> \<mu> {x\<in>space M. P x} + \<mu> N"
- using sets N by (intro measure_subadditive) auto
- also have "\<dots> = \<mu> {x\<in>space M. P x}" using N by simp
- finally show "\<mu> {x\<in>space M. P x} = \<mu> (space M)"
- using measure_top[OF sets] by auto
-qed
-
-lemma (in measure_space) AE_space: "AE x. x \<in> space M"
- by (rule AE_I[where N="{}"]) auto
-
-lemma (in measure_space) AE_I2[simp, intro]:
- "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x. P x"
- using AE_space by auto
-
-lemma (in measure_space) AE_Ball_mp:
- "\<forall>x\<in>space M. P x \<Longrightarrow> AE x. P x \<longrightarrow> Q x \<Longrightarrow> AE x. Q x"
- by auto
-
-lemma (in measure_space) AE_cong[cong]:
- "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x. P x) \<longleftrightarrow> (AE x. Q x)"
- by auto
-
-lemma (in measure_space) AE_all_countable:
- "(AE x. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x. P i x)"
-proof
- assume "\<forall>i. AE x. P i x"
- from this[unfolded almost_everywhere_def Bex_def, THEN choice]
- obtain N where N: "\<And>i. N i \<in> null_sets" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
- have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
- also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
- finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
- moreover from N have "(\<Union>i. N i) \<in> null_sets"
- by (intro null_sets_UN) auto
- ultimately show "AE x. \<forall>i. P i x"
- unfolding almost_everywhere_def by auto
-qed auto
-
-lemma (in measure_space) AE_finite_all:
- assumes f: "finite S" shows "(AE x. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x. P i x)"
- using f by induct auto
-
-lemma (in measure_space) restricted_measure_space:
- assumes "S \<in> sets M"
- shows "measure_space (restricted_space S)"
- (is "measure_space ?r")
- unfolding measure_space_def measure_space_axioms_def
-proof safe
- show "sigma_algebra ?r" using restricted_sigma_algebra[OF assms] .
- show "positive ?r (measure ?r)" using `S \<in> sets M` by (auto simp: positive_def)
-
- show "countably_additive ?r (measure ?r)"
- unfolding countably_additive_def
- proof safe
- fix A :: "nat \<Rightarrow> 'a set"
- assume *: "range A \<subseteq> sets ?r" and **: "disjoint_family A"
- from restriction_in_sets[OF assms *[simplified]] **
- show "(\<Sum>n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
- using measure_countably_additive by simp
- qed
-qed
-
-lemma (in measure_space) AE_restricted:
- assumes "A \<in> sets M"
- shows "(AE x in restricted_space A. P x) \<longleftrightarrow> (AE x. x \<in> A \<longrightarrow> P x)"
-proof -
- interpret R: measure_space "restricted_space A"
- by (rule restricted_measure_space[OF `A \<in> sets M`])
- show ?thesis
- proof
- assume "AE x in restricted_space A. P x"
- from this[THEN R.AE_E] guess N' .
- then obtain N where "{x \<in> A. \<not> P x} \<subseteq> A \<inter> N" "\<mu> (A \<inter> N) = 0" "N \<in> sets M"
- by auto
- moreover then have "{x \<in> space M. \<not> (x \<in> A \<longrightarrow> P x)} \<subseteq> A \<inter> N"
- using `A \<in> sets M` sets_into_space by auto
- ultimately show "AE x. x \<in> A \<longrightarrow> P x"
- using `A \<in> sets M` by (auto intro!: AE_I[where N="A \<inter> N"])
- next
- assume "AE x. x \<in> A \<longrightarrow> P x"
- from this[THEN AE_E] guess N .
- then show "AE x in restricted_space A. P x"
- using null_set_Int1[OF _ `A \<in> sets M`] `A \<in> sets M`[THEN sets_into_space]
- by (auto intro!: R.AE_I[where N="A \<inter> N"] simp: subset_eq)
- qed
-qed
-
-lemma (in measure_space) measure_space_subalgebra:
- assumes "sigma_algebra N" and "sets N \<subseteq> sets M" "space N = space M"
- and measure[simp]: "\<And>X. X \<in> sets N \<Longrightarrow> measure N X = measure M X"
- shows "measure_space N"
-proof -
- interpret N: sigma_algebra N by fact
- show ?thesis
- proof
- from `sets N \<subseteq> sets M` have "\<And>A. range A \<subseteq> sets N \<Longrightarrow> range A \<subseteq> sets M" by blast
- then show "countably_additive N (measure N)"
- by (auto intro!: measure_countably_additive simp: countably_additive_def subset_eq)
- show "positive N (measure_space.measure N)"
- using assms(2) by (auto simp add: positive_def)
- qed
-qed
-
-lemma (in measure_space) AE_subalgebra:
- assumes ae: "AE x in N. P x"
- and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
- and sa: "sigma_algebra N"
- shows "AE x. P x"
-proof -
- interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
- from ae[THEN N.AE_E] guess N .
- with N show ?thesis unfolding almost_everywhere_def by auto
-qed
-
-section "@{text \<sigma>}-finite Measures"
-
-locale sigma_finite_measure = measure_space +
- assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
-
-lemma (in sigma_finite_measure) restricted_sigma_finite_measure:
- assumes "S \<in> sets M"
- shows "sigma_finite_measure (restricted_space S)"
- (is "sigma_finite_measure ?r")
- unfolding sigma_finite_measure_def sigma_finite_measure_axioms_def
-proof safe
- show "measure_space ?r" using restricted_measure_space[OF assms] .
-next
- obtain A :: "nat \<Rightarrow> 'a set" where
- "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
- using sigma_finite by auto
- show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<infinity>)"
- proof (safe intro!: exI[of _ "\<lambda>i. A i \<inter> S"] del: notI)
- fix i
- show "A i \<inter> S \<in> sets ?r"
- using `range A \<subseteq> sets M` `S \<in> sets M` by auto
- next
- fix x i assume "x \<in> S" thus "x \<in> space ?r" by simp
- next
- fix x assume "x \<in> space ?r" thus "x \<in> (\<Union>i. A i \<inter> S)"
- using `(\<Union>i. A i) = space M` `S \<in> sets M` by auto
- next
- fix i
- have "\<mu> (A i \<inter> S) \<le> \<mu> (A i)"
- using `range A \<subseteq> sets M` `S \<in> sets M` by (auto intro!: measure_mono)
- then show "measure ?r (A i \<inter> S) \<noteq> \<infinity>" using `\<mu> (A i) \<noteq> \<infinity>` by auto
- qed
-qed
-
-lemma (in sigma_finite_measure) sigma_finite_measure_cong:
- assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
- shows "sigma_finite_measure M'"
-proof -
- interpret M': measure_space M' by (intro measure_space_cong cong)
- from sigma_finite guess A .. note A = this
- then have "\<And>i. A i \<in> sets M" by auto
- with A have fin: "\<forall>i. measure M' (A i) \<noteq> \<infinity>" using cong by auto
- show ?thesis
- apply default
- using A fin cong by auto
-qed
-
-lemma (in sigma_finite_measure) disjoint_sigma_finite:
- "\<exists>A::nat\<Rightarrow>'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and>
- (\<forall>i. \<mu> (A i) \<noteq> \<infinity>) \<and> disjoint_family A"
-proof -
- obtain A :: "nat \<Rightarrow> 'a set" where
- range: "range A \<subseteq> sets M" and
- space: "(\<Union>i. A i) = space M" and
- measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
- using sigma_finite by auto
- note range' = range_disjointed_sets[OF range] range
- { fix i
- have "\<mu> (disjointed A i) \<le> \<mu> (A i)"
- using range' disjointed_subset[of A i] by (auto intro!: measure_mono)
- then have "\<mu> (disjointed A i) \<noteq> \<infinity>"
- using measure[of i] by auto }
- with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
- show ?thesis by (auto intro!: exI[of _ "disjointed A"])
-qed
-
-lemma (in sigma_finite_measure) sigma_finite_up:
- "\<exists>F. range F \<subseteq> sets M \<and> incseq F \<and> (\<Union>i. F i) = space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<infinity>)"
-proof -
- obtain F :: "nat \<Rightarrow> 'a set" where
- F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
- using sigma_finite by auto
- then show ?thesis
- proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
- from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
- then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
- using F by fastforce
- next
- fix n
- have "\<mu> (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. \<mu> (F i))" using F
- by (auto intro!: measure_finitely_subadditive)
- also have "\<dots> < \<infinity>"
- using F by (auto simp: setsum_Pinfty)
- finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
- qed (force simp: incseq_def)+
-qed
-
-section {* Measure preserving *}
-
-definition "measure_preserving A B =
- {f \<in> measurable A B. (\<forall>y \<in> sets B. measure B y = measure A (f -` y \<inter> space A))}"
-
-lemma measure_preservingI[intro?]:
- assumes "f \<in> measurable A B"
- and "\<And>y. y \<in> sets B \<Longrightarrow> measure A (f -` y \<inter> space A) = measure B y"
- shows "f \<in> measure_preserving A B"
- unfolding measure_preserving_def using assms by auto
-
-lemma (in measure_space) measure_space_vimage:
- fixes M' :: "('c, 'd) measure_space_scheme"
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
- shows "measure_space M'"
-proof -
- interpret M': sigma_algebra M' by fact
- show ?thesis
- proof
- show "positive M' (measure M')" using T
- by (auto simp: measure_preserving_def positive_def measurable_sets)
-
- show "countably_additive M' (measure M')"
- proof (intro countably_additiveI)
- fix A :: "nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets M'" "disjoint_family A"
- then have A: "\<And>i. A i \<in> sets M'" "(\<Union>i. A i) \<in> sets M'" by auto
- then have *: "range (\<lambda>i. T -` (A i) \<inter> space M) \<subseteq> sets M"
- using T by (auto simp: measurable_def measure_preserving_def)
- moreover have "(\<Union>i. T -` A i \<inter> space M) \<in> sets M"
- using * by blast
- moreover have **: "disjoint_family (\<lambda>i. T -` A i \<inter> space M)"
- using `disjoint_family A` by (auto simp: disjoint_family_on_def)
- ultimately show "(\<Sum>i. measure M' (A i)) = measure M' (\<Union>i. A i)"
- using measure_countably_additive[OF _ **] A T
- by (auto simp: comp_def vimage_UN measure_preserving_def)
- qed
- qed
-qed
-
-lemma (in measure_space) almost_everywhere_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
- and AE: "measure_space.almost_everywhere M' P"
- shows "AE x. P (T x)"
-proof -
- interpret M': measure_space M' using T by (rule measure_space_vimage)
- from AE[THEN M'.AE_E] guess N .
- then show ?thesis
- unfolding almost_everywhere_def M'.almost_everywhere_def
- using T(2) unfolding measurable_def measure_preserving_def
- by (intro bexI[of _ "T -` N \<inter> space M"]) auto
-qed
-
-lemma measure_unique_Int_stable_vimage:
- fixes A :: "nat \<Rightarrow> 'a set"
- assumes E: "Int_stable E"
- and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure M (A i) \<noteq> \<infinity>"
- and N: "measure_space N" "T \<in> measurable N M"
- and M: "measure_space M" "sets (sigma E) = sets M" "space E = space M"
- and eq: "\<And>X. X \<in> sets E \<Longrightarrow> measure M X = measure N (T -` X \<inter> space N)"
- assumes X: "X \<in> sets (sigma E)"
- shows "measure M X = measure N (T -` X \<inter> space N)"
-proof (rule measure_unique_Int_stable[OF E A(1,2,3) _ _ eq _ X])
- interpret M: measure_space M by fact
- interpret N: measure_space N by fact
- let ?T = "\<lambda>X. T -` X \<inter> space N"
- show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = measure M\<rparr>"
- by (rule M.measure_space_cong) (auto simp: M)
- show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<lambda>X. measure N (?T X)\<rparr>" (is "measure_space ?E")
- proof (rule N.measure_space_vimage)
- show "sigma_algebra ?E"
- by (rule M.sigma_algebra_cong) (auto simp: M)
- show "T \<in> measure_preserving N ?E"
- using `T \<in> measurable N M` by (auto simp: M measurable_def measure_preserving_def)
- qed
- show "\<And>i. M.\<mu> (A i) \<noteq> \<infinity>" by fact
-qed
-
-lemma (in measure_space) measure_preserving_Int_stable:
- fixes A :: "nat \<Rightarrow> 'a set"
- assumes E: "Int_stable E" "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure E (A i) \<noteq> \<infinity>"
- and N: "measure_space (sigma E)"
- and T: "T \<in> measure_preserving M E"
- shows "T \<in> measure_preserving M (sigma E)"
-proof
- interpret E: measure_space "sigma E" by fact
- show "T \<in> measurable M (sigma E)"
- using T E.sets_into_space
- by (intro measurable_sigma) (auto simp: measure_preserving_def measurable_def)
- fix X assume "X \<in> sets (sigma E)"
- show "\<mu> (T -` X \<inter> space M) = E.\<mu> X"
- proof (rule measure_unique_Int_stable_vimage[symmetric])
- show "sets (sigma E) = sets (sigma E)" "space E = space (sigma E)"
- "\<And>i. E.\<mu> (A i) \<noteq> \<infinity>" using E by auto
- show "measure_space M" by default
- next
- fix X assume "X \<in> sets E" then show "E.\<mu> X = \<mu> (T -` X \<inter> space M)"
- using T unfolding measure_preserving_def by auto
- qed fact+
-qed
-
-section "Real measure values"
-
-lemma (in measure_space) real_measure_Union:
- assumes finite: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
- and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
- shows "real (\<mu> (A \<union> B)) = real (\<mu> A) + real (\<mu> B)"
- unfolding measure_additive[symmetric, OF measurable]
- using measurable(1,2)[THEN positive_measure]
- using finite by (cases rule: ereal2_cases[of "\<mu> A" "\<mu> B"]) auto
-
-lemma (in measure_space) real_measure_finite_Union:
- assumes measurable:
- "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M" "disjoint_family_on A S"
- assumes finite: "\<And>i. i \<in> S \<Longrightarrow> \<mu> (A i) \<noteq> \<infinity>"
- shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
- using finite measurable(2)[THEN positive_measure]
- by (force intro!: setsum_real_of_ereal[symmetric]
- simp: measure_setsum[OF measurable, symmetric])
-
-lemma (in measure_space) real_measure_Diff:
- assumes finite: "\<mu> A \<noteq> \<infinity>"
- and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
- shows "real (\<mu> (A - B)) = real (\<mu> A) - real (\<mu> B)"
-proof -
- have "\<mu> (A - B) \<le> \<mu> A" "\<mu> B \<le> \<mu> A"
- using measurable by (auto intro!: measure_mono)
- hence "real (\<mu> ((A - B) \<union> B)) = real (\<mu> (A - B)) + real (\<mu> B)"
- using measurable finite by (rule_tac real_measure_Union) auto
- thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
-qed
-
-lemma (in measure_space) real_measure_UNION:
- assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
- assumes finite: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
- shows "(\<lambda>i. real (\<mu> (A i))) sums (real (\<mu> (\<Union>i. A i)))"
-proof -
- have "\<And>i. 0 \<le> \<mu> (A i)" using measurable by auto
- with summable_sums[OF summable_ereal_pos, of "\<lambda>i. \<mu> (A i)"]
- measure_countably_additive[OF measurable]
- have "(\<lambda>i. \<mu> (A i)) sums (\<mu> (\<Union>i. A i))" by simp
- moreover
- { fix i
- have "\<mu> (A i) \<le> \<mu> (\<Union>i. A i)"
- using measurable by (auto intro!: measure_mono)
- moreover have "0 \<le> \<mu> (A i)" using measurable by auto
- ultimately have "\<mu> (A i) = ereal (real (\<mu> (A i)))"
- using finite by (cases "\<mu> (A i)") auto }
- moreover
- have "0 \<le> \<mu> (\<Union>i. A i)" using measurable by auto
- then have "\<mu> (\<Union>i. A i) = ereal (real (\<mu> (\<Union>i. A i)))"
- using finite by (cases "\<mu> (\<Union>i. A i)") auto
- ultimately show ?thesis
- unfolding sums_ereal[symmetric] by simp
-qed
-
-lemma (in measure_space) real_measure_subadditive:
- assumes measurable: "A \<in> sets M" "B \<in> sets M"
- and fin: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
- shows "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
-proof -
- have "0 \<le> \<mu> (A \<union> B)" using measurable by auto
- then show "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
- using measure_subadditive[OF measurable] fin
- by (cases rule: ereal3_cases[of "\<mu> (A \<union> B)" "\<mu> A" "\<mu> B"]) auto
-qed
-
-lemma (in measure_space) real_measure_setsum_singleton:
- assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
- and fin: "\<And>x. x \<in> S \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
- shows "real (\<mu> S) = (\<Sum>x\<in>S. real (\<mu> {x}))"
- using measure_finite_singleton[OF S] fin
- using positive_measure[OF S(2)]
- by (force intro!: setsum_real_of_ereal[symmetric])
-
-lemma (in measure_space) real_continuity_from_below:
- assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
- shows "(\<lambda>i. real (\<mu> (A i))) ----> real (\<mu> (\<Union>i. A i))"
-proof -
- have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
- then have "ereal (real (\<mu> (\<Union>i. A i))) = \<mu> (\<Union>i. A i)"
- using fin by (auto intro: ereal_real')
- then show ?thesis
- using continuity_from_below_Lim[OF A]
- by (intro lim_real_of_ereal) simp
-qed
-
-lemma (in measure_space) continuity_from_above_Lim:
- assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
- shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Inter>i. A i)"
- using LIMSEQ_ereal_INFI[OF measure_decseq, OF A]
- using continuity_from_above[OF A fin] by simp
-
-lemma (in measure_space) real_continuity_from_above:
- assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
- shows "(\<lambda>n. real (\<mu> (A n))) ----> real (\<mu> (\<Inter>i. A i))"
-proof -
- have "0 \<le> \<mu> (\<Inter>i. A i)" using A by auto
- moreover
- have "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
- using A by (auto intro!: measure_mono)
- ultimately have "ereal (real (\<mu> (\<Inter>i. A i))) = \<mu> (\<Inter>i. A i)"
- using fin by (auto intro: ereal_real')
- then show ?thesis
- using continuity_from_above_Lim[OF A fin]
- by (intro lim_real_of_ereal) simp
-qed
-
-lemma (in measure_space) real_measure_countably_subadditive:
- assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. \<mu> (A i)) \<noteq> \<infinity>"
- shows "real (\<mu> (\<Union>i. A i)) \<le> (\<Sum>i. real (\<mu> (A i)))"
-proof -
- { fix i
- have "0 \<le> \<mu> (A i)" using A by auto
- moreover have "\<mu> (A i) \<noteq> \<infinity>" using A by (intro suminf_PInfty[OF _ fin]) auto
- ultimately have "\<bar>\<mu> (A i)\<bar> \<noteq> \<infinity>" by auto }
- moreover have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
- ultimately have "ereal (real (\<mu> (\<Union>i. A i))) \<le> (\<Sum>i. ereal (real (\<mu> (A i))))"
- using measure_countably_subadditive[OF A] by (auto simp: ereal_real)
- also have "\<dots> = ereal (\<Sum>i. real (\<mu> (A i)))"
- using A
- by (auto intro!: sums_unique[symmetric] sums_ereal[THEN iffD2] summable_sums summable_real_of_ereal fin)
- finally show ?thesis by simp
-qed
-
-locale finite_measure = sigma_finite_measure +
- assumes finite_measure_of_space: "\<mu> (space M) \<noteq> \<infinity>"
-
-lemma finite_measureI[Pure.intro!]:
- assumes "measure_space M"
- assumes *: "measure M (space M) \<noteq> \<infinity>"
- shows "finite_measure M"
-proof -
- interpret measure_space M by fact
- show "finite_measure M"
- proof
- show "measure M (space M) \<noteq> \<infinity>" by fact
- show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
- using * by (auto intro!: exI[of _ "\<lambda>x. space M"])
- qed
-qed
-
-lemma (in finite_measure) finite_measure[simp, intro]:
- assumes "A \<in> sets M"
- shows "\<mu> A \<noteq> \<infinity>"
-proof -
- from `A \<in> sets M` have "A \<subseteq> space M"
- using sets_into_space by blast
- then have "\<mu> A \<le> \<mu> (space M)"
- using assms top by (rule measure_mono)
- then show ?thesis
- using finite_measure_of_space by auto
-qed
-
-definition (in finite_measure)
- "\<mu>' A = (if A \<in> sets M then real (\<mu> A) else 0)"
-
-lemma (in finite_measure) finite_measure_eq: "A \<in> sets M \<Longrightarrow> \<mu> A = ereal (\<mu>' A)"
- by (auto simp: \<mu>'_def ereal_real)
-
-lemma (in finite_measure) positive_measure'[simp, intro]: "0 \<le> \<mu>' A"
- unfolding \<mu>'_def by (auto simp: real_of_ereal_pos)
-
-lemma (in finite_measure) real_measure:
- assumes A: "A \<in> sets M" shows "\<exists>r. 0 \<le> r \<and> \<mu> A = ereal r"
- using finite_measure[OF A] positive_measure[OF A] by (cases "\<mu> A") auto
-
-lemma (in finite_measure) bounded_measure: "\<mu>' A \<le> \<mu>' (space M)"
-proof cases
- assume "A \<in> sets M"
- moreover then have "\<mu> A \<le> \<mu> (space M)"
- using sets_into_space by (auto intro!: measure_mono)
- ultimately show ?thesis
- by (auto simp: \<mu>'_def intro!: real_of_ereal_positive_mono)
-qed (simp add: \<mu>'_def real_of_ereal_pos)
-
-lemma (in finite_measure) restricted_finite_measure:
- assumes "S \<in> sets M"
- shows "finite_measure (restricted_space S)"
- (is "finite_measure ?r")
-proof
- show "measure_space ?r" using restricted_measure_space[OF assms] .
- show "measure ?r (space ?r) \<noteq> \<infinity>" using finite_measure[OF `S \<in> sets M`] by auto
-qed
-
-lemma (in measure_space) restricted_to_finite_measure:
- assumes "S \<in> sets M" "\<mu> S \<noteq> \<infinity>"
- shows "finite_measure (restricted_space S)"
-proof
- show "measure_space (restricted_space S)"
- using `S \<in> sets M` by (rule restricted_measure_space)
- show "measure (restricted_space S) (space (restricted_space S)) \<noteq> \<infinity>"
- by simp fact
-qed
-
-lemma (in finite_measure) finite_measure_Diff:
- assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
- shows "\<mu>' (A - B) = \<mu>' A - \<mu>' B"
- using sets[THEN finite_measure_eq]
- using Diff[OF sets, THEN finite_measure_eq]
- using measure_Diff[OF _ assms] by simp
-
-lemma (in finite_measure) finite_measure_Union:
- assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
- shows "\<mu>' (A \<union> B) = \<mu>' A + \<mu>' B"
- using measure_additive[OF assms]
- using sets[THEN finite_measure_eq]
- using Un[OF sets, THEN finite_measure_eq]
- by simp
-
-lemma (in finite_measure) finite_measure_finite_Union:
- assumes S: "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
- and dis: "disjoint_family_on A S"
- shows "\<mu>' (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. \<mu>' (A i))"
- using measure_setsum[OF assms]
- using finite_UN[of S A, OF S, THEN finite_measure_eq]
- using S(2)[THEN finite_measure_eq]
- by simp
-
-lemma (in finite_measure) finite_measure_UNION:
- assumes A: "range A \<subseteq> sets M" "disjoint_family A"
- shows "(\<lambda>i. \<mu>' (A i)) sums (\<mu>' (\<Union>i. A i))"
- using real_measure_UNION[OF A]
- using countable_UN[OF A(1), THEN finite_measure_eq]
- using A(1)[THEN subsetD, THEN finite_measure_eq]
- by auto
-
-lemma (in finite_measure) finite_measure_mono:
- assumes B: "B \<in> sets M" and "A \<subseteq> B" shows "\<mu>' A \<le> \<mu>' B"
-proof cases
- assume "A \<in> sets M"
- from this[THEN finite_measure_eq] B[THEN finite_measure_eq]
- show ?thesis using measure_mono[OF `A \<subseteq> B` `A \<in> sets M` `B \<in> sets M`] by simp
-next
- assume "A \<notin> sets M" then show ?thesis
- using positive_measure'[of B] unfolding \<mu>'_def by auto
-qed
-
-lemma (in finite_measure) finite_measure_subadditive:
- assumes m: "A \<in> sets M" "B \<in> sets M"
- shows "\<mu>' (A \<union> B) \<le> \<mu>' A + \<mu>' B"
- using measure_subadditive[OF m]
- using m[THEN finite_measure_eq] Un[OF m, THEN finite_measure_eq] by simp
-
-lemma (in finite_measure) finite_measure_subadditive_finite:
- assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
- shows "\<mu>' (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu>' (A i))"
- using measure_subadditive_finite[OF assms] assms
- by (simp add: finite_measure_eq finite_UN)
-
-lemma (in finite_measure) finite_measure_countably_subadditive:
- assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. \<mu>' (A i))"
- shows "\<mu>' (\<Union>i. A i) \<le> (\<Sum>i. \<mu>' (A i))"
-proof -
- note A[THEN subsetD, THEN finite_measure_eq, simp]
- note countable_UN[OF A, THEN finite_measure_eq, simp]
- from `summable (\<lambda>i. \<mu>' (A i))`
- have "(\<lambda>i. ereal (\<mu>' (A i))) sums ereal (\<Sum>i. \<mu>' (A i))"
- by (simp add: sums_ereal) (rule summable_sums)
- from sums_unique[OF this, symmetric]
- measure_countably_subadditive[OF A]
- show ?thesis by simp
-qed
-
-lemma (in finite_measure) finite_measure_finite_singleton:
- assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
- shows "\<mu>' S = (\<Sum>x\<in>S. \<mu>' {x})"
- using real_measure_setsum_singleton[OF assms]
- using *[THEN finite_measure_eq]
- using finite_UN[of S "\<lambda>x. {x}", OF assms, THEN finite_measure_eq]
- by simp
-
-lemma (in finite_measure) finite_continuity_from_below:
- assumes A: "range A \<subseteq> sets M" and "incseq A"
- shows "(\<lambda>i. \<mu>' (A i)) ----> \<mu>' (\<Union>i. A i)"
- using real_continuity_from_below[OF A, OF `incseq A` finite_measure] assms
- using A[THEN subsetD, THEN finite_measure_eq]
- using countable_UN[OF A, THEN finite_measure_eq]
- by auto
-
-lemma (in finite_measure) finite_continuity_from_above:
- assumes A: "range A \<subseteq> sets M" and "decseq A"
- shows "(\<lambda>n. \<mu>' (A n)) ----> \<mu>' (\<Inter>i. A i)"
- using real_continuity_from_above[OF A, OF `decseq A` finite_measure] assms
- using A[THEN subsetD, THEN finite_measure_eq]
- using countable_INT[OF A, THEN finite_measure_eq]
- by auto
-
-lemma (in finite_measure) finite_measure_compl:
- assumes S: "S \<in> sets M"
- shows "\<mu>' (space M - S) = \<mu>' (space M) - \<mu>' S"
- using measure_compl[OF S, OF finite_measure, OF S]
- using S[THEN finite_measure_eq]
- using compl_sets[OF S, THEN finite_measure_eq]
- using top[THEN finite_measure_eq]
- by simp
-
-lemma (in finite_measure) finite_measure_inter_full_set:
- assumes S: "S \<in> sets M" "T \<in> sets M"
- assumes T: "\<mu>' T = \<mu>' (space M)"
- shows "\<mu>' (S \<inter> T) = \<mu>' S"
- using measure_inter_full_set[OF S finite_measure]
- using T Diff[OF S(2,1)] Diff[OF S, THEN finite_measure_eq]
- using Int[OF S, THEN finite_measure_eq]
- using S[THEN finite_measure_eq] top[THEN finite_measure_eq]
- by simp
-
-lemma (in finite_measure) empty_measure'[simp]: "\<mu>' {} = 0"
- unfolding \<mu>'_def by simp
-
-section "Finite spaces"
-
-locale finite_measure_space = finite_measure + finite_sigma_algebra
-
-lemma finite_measure_spaceI[Pure.intro!]:
- assumes "finite (space M)"
- assumes sets_Pow: "sets M = Pow (space M)"
- and space: "measure M (space M) \<noteq> \<infinity>"
- and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> measure M {x}"
- and add: "\<And>A. A \<subseteq> space M \<Longrightarrow> measure M A = (\<Sum>x\<in>A. measure M {x})"
- shows "finite_measure_space M"
-proof -
- interpret finite_sigma_algebra M
- proof
- show "finite (space M)" by fact
- qed (auto simp: sets_Pow)
- interpret measure_space M
- proof (rule finite_additivity_sufficient)
- show "sigma_algebra M" by default
- show "finite (space M)" by fact
- show "positive M (measure M)"
- by (auto simp: add positive_def intro!: setsum_nonneg pos)
- show "additive M (measure M)"
- using `finite (space M)`
- by (auto simp add: additive_def add
- intro!: setsum_Un_disjoint dest: finite_subset)
- qed
- interpret finite_measure M
- proof
- show "\<mu> (space M) \<noteq> \<infinity>" by fact
- qed default
- show "finite_measure_space M"
- by default
-qed
-
-lemma (in finite_measure_space) sum_over_space: "(\<Sum>x\<in>space M. \<mu> {x}) = \<mu> (space M)"
- using measure_setsum[of "space M" "\<lambda>i. {i}"]
- by (simp add: disjoint_family_on_def finite_space)
-
-lemma (in finite_measure_space) finite_measure_singleton:
- assumes A: "A \<subseteq> space M" shows "\<mu>' A = (\<Sum>x\<in>A. \<mu>' {x})"
- using A finite_subset[OF A finite_space]
- by (intro finite_measure_finite_singleton) auto
-
-lemma (in finite_measure_space) finite_measure_subadditive_setsum:
- assumes "finite I"
- shows "\<mu>' (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu>' (A i))"
-proof cases
- assume "(\<Union>i\<in>I. A i) \<subseteq> space M"
- then have "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M" by auto
- from finite_measure_subadditive_finite[OF `finite I` this]
- show ?thesis by auto
-next
- assume "\<not> (\<Union>i\<in>I. A i) \<subseteq> space M"
- then have "\<mu>' (\<Union>i\<in>I. A i) = 0"
- by (simp add: \<mu>'_def)
- also have "0 \<le> (\<Sum>i\<in>I. \<mu>' (A i))"
- by (auto intro!: setsum_nonneg)
- finally show ?thesis .
-qed
-
-lemma suminf_cmult_indicator:
- fixes f :: "nat \<Rightarrow> ereal"
- assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
- shows "(\<Sum>n. f n * indicator (A n) x) = f i"
-proof -
- have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
- using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
- then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
- by (auto simp: setsum_cases)
- moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
- proof (rule ereal_SUPI)
- fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
- from this[of "Suc i"] show "f i \<le> y" by auto
- qed (insert assms, simp)
- ultimately show ?thesis using assms
- by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def)
-qed
-
-lemma suminf_indicator:
- assumes "disjoint_family A"
- shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
-proof cases
- assume *: "x \<in> (\<Union>i. A i)"
- then obtain i where "x \<in> A i" by auto
- from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
- show ?thesis using * by simp
-qed simp
-
-end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Measure_Space.thy Mon Apr 23 12:14:35 2012 +0200
@@ -0,0 +1,1457 @@
+(* Title: HOL/Probability/Measure_Space.thy
+ Author: Lawrence C Paulson
+ Author: Johannes Hölzl, TU München
+ Author: Armin Heller, TU München
+*)
+
+header {* Measure spaces and their properties *}
+
+theory Measure_Space
+imports
+ Sigma_Algebra
+ "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
+begin
+
+lemma suminf_eq_setsum:
+ fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, t2_space}"
+ assumes "finite {i. f i \<noteq> 0}" (is "finite ?P")
+ shows "(\<Sum>i. f i) = (\<Sum>i | f i \<noteq> 0. f i)"
+proof cases
+ assume "?P \<noteq> {}"
+ have [dest!]: "\<And>i. Suc (Max ?P) \<le> i \<Longrightarrow> f i = 0"
+ using `finite ?P` `?P \<noteq> {}` by (auto simp: Suc_le_eq)
+ have "(\<Sum>i. f i) = (\<Sum>i<Suc (Max ?P). f i)"
+ by (rule suminf_finite) auto
+ also have "\<dots> = (\<Sum>i | f i \<noteq> 0. f i)"
+ using `finite ?P` `?P \<noteq> {}`
+ by (intro setsum_mono_zero_right) (auto simp: less_Suc_eq_le)
+ finally show ?thesis .
+qed simp
+
+lemma suminf_cmult_indicator:
+ fixes f :: "nat \<Rightarrow> ereal"
+ assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
+ shows "(\<Sum>n. f n * indicator (A n) x) = f i"
+proof -
+ have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
+ using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
+ then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
+ by (auto simp: setsum_cases)
+ moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
+ proof (rule ereal_SUPI)
+ fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
+ from this[of "Suc i"] show "f i \<le> y" by auto
+ qed (insert assms, simp)
+ ultimately show ?thesis using assms
+ by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def)
+qed
+
+lemma suminf_indicator:
+ assumes "disjoint_family A"
+ shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
+proof cases
+ assume *: "x \<in> (\<Union>i. A i)"
+ then obtain i where "x \<in> A i" by auto
+ from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
+ show ?thesis using * by simp
+qed simp
+
+text {*
+ The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to
+ represent sigma algebras (with an arbitrary emeasure).
+*}
+
+section "Extend binary sets"
+
+lemma LIMSEQ_binaryset:
+ assumes f: "f {} = 0"
+ shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
+proof -
+ have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
+ proof
+ fix n
+ show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
+ by (induct n) (auto simp add: binaryset_def f)
+ qed
+ moreover
+ have "... ----> f A + f B" by (rule tendsto_const)
+ ultimately
+ have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
+ by metis
+ hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
+ by simp
+ thus ?thesis by (rule LIMSEQ_offset [where k=2])
+qed
+
+lemma binaryset_sums:
+ assumes f: "f {} = 0"
+ shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
+ by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
+
+lemma suminf_binaryset_eq:
+ fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
+ shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
+ by (metis binaryset_sums sums_unique)
+
+section {* Properties of a premeasure @{term \<mu>} *}
+
+text {*
+ The definitions for @{const positive} and @{const countably_additive} should be here, by they are
+ necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
+*}
+
+definition additive where
+ "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
+
+definition increasing where
+ "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
+
+lemma positiveD_empty:
+ "positive M f \<Longrightarrow> f {} = 0"
+ by (auto simp add: positive_def)
+
+lemma additiveD:
+ "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
+ by (auto simp add: additive_def)
+
+lemma increasingD:
+ "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
+ by (auto simp add: increasing_def)
+
+lemma countably_additiveI:
+ "(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i))
+ \<Longrightarrow> countably_additive M f"
+ by (simp add: countably_additive_def)
+
+lemma (in ring_of_sets) disjointed_additive:
+ assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"
+ shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
+proof (induct n)
+ case (Suc n)
+ then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
+ by simp
+ also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
+ using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
+ also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
+ using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
+ finally show ?case .
+qed simp
+
+lemma (in ring_of_sets) additive_sum:
+ fixes A:: "'i \<Rightarrow> 'a set"
+ assumes f: "positive M f" and ad: "additive M f" and "finite S"
+ and A: "A`S \<subseteq> M"
+ and disj: "disjoint_family_on A S"
+ shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
+using `finite S` disj A proof induct
+ case empty show ?case using f by (simp add: positive_def)
+next
+ case (insert s S)
+ then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
+ by (auto simp add: disjoint_family_on_def neq_iff)
+ moreover
+ have "A s \<in> M" using insert by blast
+ moreover have "(\<Union>i\<in>S. A i) \<in> M"
+ using insert `finite S` by auto
+ moreover
+ ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
+ using ad UNION_in_sets A by (auto simp add: additive_def)
+ with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
+ by (auto simp add: additive_def subset_insertI)
+qed
+
+lemma (in ring_of_sets) additive_increasing:
+ assumes posf: "positive M f" and addf: "additive M f"
+ shows "increasing M f"
+proof (auto simp add: increasing_def)
+ fix x y
+ assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
+ then have "y - x \<in> M" by auto
+ then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
+ then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
+ also have "... = f (x \<union> (y-x))" using addf
+ by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
+ also have "... = f y"
+ by (metis Un_Diff_cancel Un_absorb1 xy(3))
+ finally show "f x \<le> f y" by simp
+qed
+
+lemma (in ring_of_sets) countably_additive_additive:
+ assumes posf: "positive M f" and ca: "countably_additive M f"
+ shows "additive M f"
+proof (auto simp add: additive_def)
+ fix x y
+ assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
+ hence "disjoint_family (binaryset x y)"
+ by (auto simp add: disjoint_family_on_def binaryset_def)
+ hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
+ (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
+ f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
+ using ca
+ by (simp add: countably_additive_def)
+ hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
+ f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
+ by (simp add: range_binaryset_eq UN_binaryset_eq)
+ thus "f (x \<union> y) = f x + f y" using posf x y
+ by (auto simp add: Un suminf_binaryset_eq positive_def)
+qed
+
+lemma (in algebra) increasing_additive_bound:
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
+ assumes f: "positive M f" and ad: "additive M f"
+ and inc: "increasing M f"
+ and A: "range A \<subseteq> M"
+ and disj: "disjoint_family A"
+ shows "(\<Sum>i. f (A i)) \<le> f \<Omega>"
+proof (safe intro!: suminf_bound)
+ fix N
+ note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
+ have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
+ using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
+ also have "... \<le> f \<Omega>" using space_closed A
+ by (intro increasingD[OF inc] finite_UN) auto
+ finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
+qed (insert f A, auto simp: positive_def)
+
+lemma (in ring_of_sets) countably_additiveI_finite:
+ assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
+ shows "countably_additive M \<mu>"
+proof (rule countably_additiveI)
+ fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
+
+ have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
+ from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
+
+ have inj_f: "inj_on f {i. F i \<noteq> {}}"
+ proof (rule inj_onI, simp)
+ fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
+ then have "f i \<in> F i" "f j \<in> F j" using f by force+
+ with disj * show "i = j" by (auto simp: disjoint_family_on_def)
+ qed
+ have "finite (\<Union>i. F i)"
+ by (metis F(2) assms(1) infinite_super sets_into_space)
+
+ have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
+ by (auto simp: positiveD_empty[OF `positive M \<mu>`])
+ moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
+ proof (rule finite_imageD)
+ from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
+ then show "finite (f`{i. F i \<noteq> {}})"
+ by (rule finite_subset) fact
+ qed fact
+ ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
+ by (rule finite_subset)
+
+ have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
+ using disj by (auto simp: disjoint_family_on_def)
+
+ from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
+ by (rule suminf_eq_setsum)
+ also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
+ using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto
+ also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
+ using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
+ also have "\<dots> = \<mu> (\<Union>i. F i)"
+ by (rule arg_cong[where f=\<mu>]) auto
+ finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
+qed
+
+section {* Properties of @{const emeasure} *}
+
+lemma emeasure_positive: "positive (sets M) (emeasure M)"
+ by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
+
+lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
+ using emeasure_positive[of M] by (simp add: positive_def)
+
+lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A"
+ using emeasure_notin_sets[of A M] emeasure_positive[of M]
+ by (cases "A \<in> sets M") (auto simp: positive_def)
+
+lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
+ using emeasure_nonneg[of M A] by auto
+
+lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
+ by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
+
+lemma suminf_emeasure:
+ "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
+ using countable_UN[of A UNIV M] emeasure_countably_additive[of M]
+ by (simp add: countably_additive_def)
+
+lemma emeasure_additive: "additive (sets M) (emeasure M)"
+ by (metis countably_additive_additive emeasure_positive emeasure_countably_additive)
+
+lemma plus_emeasure:
+ "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)"
+ using additiveD[OF emeasure_additive] ..
+
+lemma setsum_emeasure:
+ "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
+ (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
+ by (metis additive_sum emeasure_positive emeasure_additive)
+
+lemma emeasure_mono:
+ "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
+ by (metis additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets
+ emeasure_positive increasingD)
+
+lemma emeasure_space:
+ "emeasure M A \<le> emeasure M (space M)"
+ by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets_into_space top)
+
+lemma emeasure_compl:
+ assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>"
+ shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
+proof -
+ from s have "0 \<le> emeasure M s" by auto
+ have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s
+ by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
+ also have "... = emeasure M s + emeasure M (space M - s)"
+ by (rule plus_emeasure[symmetric]) (auto simp add: s)
+ finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
+ then show ?thesis
+ using fin `0 \<le> emeasure M s`
+ unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
+qed
+
+lemma emeasure_Diff:
+ assumes finite: "emeasure M B \<noteq> \<infinity>"
+ and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
+ shows "emeasure M (A - B) = emeasure M A - emeasure M B"
+proof -
+ have "0 \<le> emeasure M B" using assms by auto
+ have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
+ then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
+ also have "\<dots> = emeasure M (A - B) + emeasure M B"
+ using measurable by (subst plus_emeasure[symmetric]) auto
+ finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
+ unfolding ereal_eq_minus_iff
+ using finite `0 \<le> emeasure M B` by auto
+qed
+
+lemma emeasure_countable_increasing:
+ assumes A: "range A \<subseteq> sets M"
+ and A0: "A 0 = {}"
+ and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
+ shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
+proof -
+ { fix n
+ have "emeasure M (A n) = (\<Sum>i<n. emeasure M (A (Suc i) - A i))"
+ proof (induct n)
+ case 0 thus ?case by (auto simp add: A0)
+ next
+ case (Suc m)
+ have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
+ by (metis ASuc Un_Diff_cancel Un_absorb1)
+ hence "emeasure M (A (Suc m)) =
+ emeasure M (A m) + emeasure M (A (Suc m) - A m)"
+ by (subst plus_emeasure)
+ (auto simp add: emeasure_additive range_subsetD [OF A])
+ with Suc show ?case
+ by simp
+ qed }
+ note Meq = this
+ have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
+ proof (rule UN_finite2_eq [where k=1], simp)
+ fix i
+ show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
+ proof (induct i)
+ case 0 thus ?case by (simp add: A0)
+ next
+ case (Suc i)
+ thus ?case
+ by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
+ qed
+ qed
+ have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
+ by (metis A Diff range_subsetD)
+ have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
+ by (blast intro: range_subsetD [OF A])
+ have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = (\<Sum>i. emeasure M (A (Suc i) - A i))"
+ using A by (auto intro!: suminf_ereal_eq_SUPR[symmetric])
+ also have "\<dots> = emeasure M (\<Union>i. A (Suc i) - A i)"
+ by (rule suminf_emeasure)
+ (auto simp add: disjoint_family_Suc ASuc A1 A2)
+ also have "... = emeasure M (\<Union>i. A i)"
+ by (simp add: Aeq)
+ finally have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = emeasure M (\<Union>i. A i)" .
+ then show ?thesis by (auto simp add: Meq)
+qed
+
+lemma SUP_emeasure_incseq:
+ assumes A: "range A \<subseteq> sets M" and "incseq A"
+ shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
+proof -
+ have *: "(SUP n. emeasure M (nat_case {} A (Suc n))) = (SUP n. emeasure M (nat_case {} A n))"
+ using A by (auto intro!: SUPR_eq exI split: nat.split)
+ have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
+ by (auto simp add: split: nat.splits)
+ have meq: "\<And>n. emeasure M (A n) = (emeasure M \<circ> nat_case {} A) (Suc n)"
+ by simp
+ have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. nat_case {} A i)"
+ using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
+ by (force split: nat.splits intro!: emeasure_countable_increasing)
+ also have "emeasure M (\<Union>i. nat_case {} A i) = emeasure M (\<Union>i. A i)"
+ by (simp add: ueq)
+ finally have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. A i)" .
+ thus ?thesis unfolding meq * comp_def .
+qed
+
+lemma incseq_emeasure:
+ assumes "range B \<subseteq> sets M" "incseq B"
+ shows "incseq (\<lambda>i. emeasure M (B i))"
+ using assms by (auto simp: incseq_def intro!: emeasure_mono)
+
+lemma Lim_emeasure_incseq:
+ assumes A: "range A \<subseteq> sets M" "incseq A"
+ shows "(\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
+ using LIMSEQ_ereal_SUPR[OF incseq_emeasure, OF A]
+ SUP_emeasure_incseq[OF A] by simp
+
+lemma decseq_emeasure:
+ assumes "range B \<subseteq> sets M" "decseq B"
+ shows "decseq (\<lambda>i. emeasure M (B i))"
+ using assms by (auto simp: decseq_def intro!: emeasure_mono)
+
+lemma INF_emeasure_decseq:
+ assumes A: "range A \<subseteq> sets M" and "decseq A"
+ and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
+ shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
+proof -
+ have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
+ using A by (auto intro!: emeasure_mono)
+ hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
+
+ have A0: "0 \<le> emeasure M (A 0)" using A by auto
+
+ have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))"
+ by (simp add: ereal_SUPR_uminus minus_ereal_def)
+ also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))"
+ unfolding minus_ereal_def using A0 assms
+ by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure)
+ also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
+ using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
+ also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
+ proof (rule SUP_emeasure_incseq)
+ show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
+ using A by auto
+ show "incseq (\<lambda>n. A 0 - A n)"
+ using `decseq A` by (auto simp add: incseq_def decseq_def)
+ qed
+ also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
+ using A finite * by (simp, subst emeasure_Diff) auto
+ finally show ?thesis
+ unfolding ereal_minus_eq_minus_iff using finite A0 by auto
+qed
+
+lemma Lim_emeasure_decseq:
+ assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
+ shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
+ using LIMSEQ_ereal_INFI[OF decseq_emeasure, OF A]
+ using INF_emeasure_decseq[OF A fin] by simp
+
+lemma emeasure_subadditive:
+ assumes measurable: "A \<in> sets M" "B \<in> sets M"
+ shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
+proof -
+ from plus_emeasure[of A M "B - A"]
+ have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
+ using assms by (simp add: Diff)
+ also have "\<dots> \<le> emeasure M A + emeasure M B"
+ using assms by (auto intro!: add_left_mono emeasure_mono)
+ finally show ?thesis .
+qed
+
+lemma emeasure_subadditive_finite:
+ assumes "finite I" "A ` I \<subseteq> sets M"
+ shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
+using assms proof induct
+ case (insert i I)
+ then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))"
+ by simp
+ also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)"
+ using insert by (intro emeasure_subadditive finite_UN) auto
+ also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))"
+ using insert by (intro add_mono) auto
+ also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))"
+ using insert by auto
+ finally show ?case .
+qed simp
+
+lemma emeasure_subadditive_countably:
+ assumes "range f \<subseteq> sets M"
+ shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
+proof -
+ have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
+ unfolding UN_disjointed_eq ..
+ also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
+ using range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
+ by (simp add: disjoint_family_disjointed comp_def)
+ also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
+ using range_disjointed_sets[OF assms] assms
+ by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset)
+ finally show ?thesis .
+qed
+
+lemma emeasure_insert:
+ assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
+ shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
+proof -
+ have "{x} \<inter> A = {}" using `x \<notin> A` by auto
+ from plus_emeasure[OF sets this] show ?thesis by simp
+qed
+
+lemma emeasure_eq_setsum_singleton:
+ assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
+ shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
+ using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
+ by (auto simp: disjoint_family_on_def subset_eq)
+
+lemma setsum_emeasure_cover:
+ assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
+ assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
+ assumes disj: "disjoint_family_on B S"
+ shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
+proof -
+ have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
+ proof (rule setsum_emeasure)
+ show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
+ using `disjoint_family_on B S`
+ unfolding disjoint_family_on_def by auto
+ qed (insert assms, auto)
+ also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
+ using A by auto
+ finally show ?thesis by simp
+qed
+
+lemma emeasure_eq_0:
+ "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
+ by (metis emeasure_mono emeasure_nonneg order_eq_iff)
+
+lemma emeasure_UN_eq_0:
+ assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
+ shows "emeasure M (\<Union> i. N i) = 0"
+proof -
+ have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto
+ moreover have "emeasure M (\<Union> i. N i) \<le> 0"
+ using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
+ ultimately show ?thesis by simp
+qed
+
+lemma measure_eqI_finite:
+ assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
+ assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
+ shows "M = N"
+proof (rule measure_eqI)
+ fix X assume "X \<in> sets M"
+ then have X: "X \<subseteq> A" by auto
+ then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
+ using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
+ also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
+ using X eq by (auto intro!: setsum_cong)
+ also have "\<dots> = emeasure N X"
+ using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
+ finally show "emeasure M X = emeasure N X" .
+qed simp
+
+lemma measure_eqI_generator_eq:
+ fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
+ assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
+ and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
+ and M: "sets M = sigma_sets \<Omega> E"
+ and N: "sets N = sigma_sets \<Omega> E"
+ and A: "range A \<subseteq> E" "incseq A" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
+ shows "M = N"
+proof -
+ let ?D = "\<lambda>F. {D. D \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)}"
+ interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
+ { fix F assume "F \<in> E" and "emeasure M F \<noteq> \<infinity>"
+ then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
+ have "emeasure N F \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` `F \<in> E` eq by simp
+ interpret D: dynkin_system \<Omega> "?D F"
+ proof (rule dynkin_systemI, simp_all)
+ fix A assume "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
+ then show "A \<subseteq> \<Omega>" using S.sets_into_space by auto
+ next
+ have "F \<inter> \<Omega> = F" using `F \<in> E` `E \<subseteq> Pow \<Omega>` by auto
+ then show "emeasure M (F \<inter> \<Omega>) = emeasure N (F \<inter> \<Omega>)"
+ using `F \<in> E` eq by (auto intro: sigma_sets_top)
+ next
+ fix A assume *: "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
+ then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
+ and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
+ using `F \<in> E` S.sets_into_space by auto
+ have "emeasure N (F \<inter> A) \<le> emeasure N F" by (auto intro!: emeasure_mono simp: M N)
+ then have "emeasure N (F \<inter> A) \<noteq> \<infinity>" using `emeasure N F \<noteq> \<infinity>` by auto
+ have "emeasure M (F \<inter> A) \<le> emeasure M F" by (auto intro!: emeasure_mono simp: M N)
+ then have "emeasure M (F \<inter> A) \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` by auto
+ then have "emeasure M (F \<inter> (\<Omega> - A)) = emeasure M F - emeasure M (F \<inter> A)" unfolding **
+ using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
+ also have "\<dots> = emeasure N F - emeasure N (F \<inter> A)" using eq `F \<in> E` * by simp
+ also have "\<dots> = emeasure N (F \<inter> (\<Omega> - A))" unfolding **
+ using `F \<inter> A \<in> sigma_sets \<Omega> E` `emeasure N (F \<inter> A) \<noteq> \<infinity>`
+ by (auto intro!: emeasure_Diff[symmetric] simp: M N)
+ finally show "\<Omega> - A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Omega> - A)) = emeasure N (F \<inter> (\<Omega> - A))"
+ using * by auto
+ next
+ fix A :: "nat \<Rightarrow> 'a set"
+ assume "disjoint_family A" "range A \<subseteq> {X \<in> sigma_sets \<Omega> E. emeasure M (F \<inter> X) = emeasure N (F \<inter> X)}"
+ then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sigma_sets \<Omega> E" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
+ "disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. emeasure M (F \<inter> A i) = emeasure N (F \<inter> A i)" "range A \<subseteq> sigma_sets \<Omega> E"
+ by (auto simp: disjoint_family_on_def subset_eq)
+ then show "(\<Union>x. A x) \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Union>x. A x)) = emeasure N (F \<inter> (\<Union>x. A x))"
+ by (auto simp: M N suminf_emeasure[symmetric] simp del: UN_simps)
+ qed
+ have *: "sigma_sets \<Omega> E = ?D F"
+ using `F \<in> E` `Int_stable E`
+ by (intro D.dynkin_lemma) (auto simp add: Int_stable_def eq)
+ have "\<And>D. D \<in> sigma_sets \<Omega> E \<Longrightarrow> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
+ by (subst (asm) *) auto }
+ note * = this
+ show "M = N"
+ proof (rule measure_eqI)
+ show "sets M = sets N"
+ using M N by simp
+ fix X assume "X \<in> sets M"
+ then have "X \<in> sigma_sets \<Omega> E"
+ using M by simp
+ let ?A = "\<lambda>i. A i \<inter> X"
+ have "range ?A \<subseteq> sigma_sets \<Omega> E" "incseq ?A"
+ using A(1,2) `X \<in> sigma_sets \<Omega> E` by (auto simp: incseq_def)
+ moreover
+ { fix i have "emeasure M (?A i) = emeasure N (?A i)"
+ using *[of "A i" X] `X \<in> sigma_sets \<Omega> E` A finite by auto }
+ ultimately show "emeasure M X = emeasure N X"
+ using SUP_emeasure_incseq[of ?A M] SUP_emeasure_incseq[of ?A N] A(3) `X \<in> sigma_sets \<Omega> E`
+ by (auto simp: M N SUP_emeasure_incseq)
+ qed
+qed
+
+lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
+proof (intro measure_eqI emeasure_measure_of_sigma)
+ show "sigma_algebra (space M) (sets M)" ..
+ show "positive (sets M) (emeasure M)"
+ by (simp add: positive_def emeasure_nonneg)
+ show "countably_additive (sets M) (emeasure M)"
+ by (simp add: emeasure_countably_additive)
+qed simp_all
+
+section "@{text \<mu>}-null sets"
+
+definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
+ "null_sets M = {N\<in>sets M. emeasure M N = 0}"
+
+lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
+ by (simp add: null_sets_def)
+
+lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
+ unfolding null_sets_def by simp
+
+lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
+ unfolding null_sets_def by simp
+
+interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
+proof
+ show "null_sets M \<subseteq> Pow (space M)"
+ using sets_into_space by auto
+ show "{} \<in> null_sets M"
+ by auto
+ fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M"
+ then have "A \<in> sets M" "B \<in> sets M"
+ by auto
+ moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
+ "emeasure M (A - B) \<le> emeasure M A"
+ by (auto intro!: emeasure_subadditive emeasure_mono)
+ moreover have "emeasure M B = 0" "emeasure M A = 0"
+ using sets by auto
+ ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
+ by (auto intro!: antisym)
+qed
+
+lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
+proof -
+ have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
+ unfolding SUP_def image_compose
+ unfolding surj_from_nat ..
+ then show ?thesis by simp
+qed
+
+lemma null_sets_UN[intro]:
+ assumes "\<And>i::'i::countable. N i \<in> null_sets M"
+ shows "(\<Union>i. N i) \<in> null_sets M"
+proof (intro conjI CollectI null_setsI)
+ show "(\<Union>i. N i) \<in> sets M" using assms by auto
+ have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg)
+ moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))"
+ unfolding UN_from_nat[of N]
+ using assms by (intro emeasure_subadditive_countably) auto
+ ultimately show "emeasure M (\<Union>i. N i) = 0"
+ using assms by (auto simp: null_setsD1)
+qed
+
+lemma null_set_Int1:
+ assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
+proof (intro CollectI conjI null_setsI)
+ show "emeasure M (A \<inter> B) = 0" using assms
+ by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
+qed (insert assms, auto)
+
+lemma null_set_Int2:
+ assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
+ using assms by (subst Int_commute) (rule null_set_Int1)
+
+lemma emeasure_Diff_null_set:
+ assumes "B \<in> null_sets M" "A \<in> sets M"
+ shows "emeasure M (A - B) = emeasure M A"
+proof -
+ have *: "A - B = (A - (A \<inter> B))" by auto
+ have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
+ then show ?thesis
+ unfolding * using assms
+ by (subst emeasure_Diff) auto
+qed
+
+lemma null_set_Diff:
+ assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
+proof (intro CollectI conjI null_setsI)
+ show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
+qed (insert assms, auto)
+
+lemma emeasure_Un_null_set:
+ assumes "A \<in> sets M" "B \<in> null_sets M"
+ shows "emeasure M (A \<union> B) = emeasure M A"
+proof -
+ have *: "A \<union> B = A \<union> (B - A)" by auto
+ have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
+ then show ?thesis
+ unfolding * using assms
+ by (subst plus_emeasure[symmetric]) auto
+qed
+
+section "Formalize almost everywhere"
+
+definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
+ "ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
+
+abbreviation
+ almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "almost_everywhere M P \<equiv> eventually P (ae_filter M)"
+
+syntax
+ "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
+
+translations
+ "AE x in M. P" == "CONST almost_everywhere M (%x. P)"
+
+lemma eventually_ae_filter:
+ fixes M P
+ defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N"
+ shows "eventually P (ae_filter M) \<longleftrightarrow> F P"
+ unfolding ae_filter_def F_def[symmetric]
+proof (rule eventually_Abs_filter)
+ show "is_filter F"
+ proof
+ fix P Q assume "F P" "F Q"
+ then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N"
+ and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L"
+ by auto
+ then have "L \<union> N \<in> null_sets M" "{x \<in> space M. \<not> (P x \<and> Q x)} \<subseteq> L \<union> N" by auto
+ then show "F (\<lambda>x. P x \<and> Q x)" by auto
+ next
+ fix P Q assume "F P"
+ then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto
+ moreover assume "\<forall>x. P x \<longrightarrow> Q x"
+ ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto
+ then show "F Q" by auto
+ qed auto
+qed
+
+lemma AE_I':
+ "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
+ unfolding eventually_ae_filter by auto
+
+lemma AE_iff_null:
+ assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
+ shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
+proof
+ assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
+ unfolding eventually_ae_filter by auto
+ have "0 \<le> emeasure M ?P" by auto
+ moreover have "emeasure M ?P \<le> emeasure M N"
+ using assms N(1,2) by (auto intro: emeasure_mono)
+ ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
+ then show "?P \<in> null_sets M" using assms by auto
+next
+ assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
+qed
+
+lemma AE_iff_null_sets:
+ "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
+ using Int_absorb1[OF sets_into_space, of N M]
+ by (subst AE_iff_null) (auto simp: Int_def[symmetric])
+
+lemma AE_iff_measurable:
+ "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0"
+ using AE_iff_null[of _ P] by auto
+
+lemma AE_E[consumes 1]:
+ assumes "AE x in M. P x"
+ obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
+ using assms unfolding eventually_ae_filter by auto
+
+lemma AE_E2:
+ assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
+ shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
+proof -
+ have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
+ with AE_iff_null[of M P] assms show ?thesis by auto
+qed
+
+lemma AE_I:
+ assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
+ shows "AE x in M. P x"
+ using assms unfolding eventually_ae_filter by auto
+
+lemma AE_mp[elim!]:
+ assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
+ shows "AE x in M. Q x"
+proof -
+ from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
+ and A: "A \<in> sets M" "emeasure M A = 0"
+ by (auto elim!: AE_E)
+
+ from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
+ and B: "B \<in> sets M" "emeasure M B = 0"
+ by (auto elim!: AE_E)
+
+ show ?thesis
+ proof (intro AE_I)
+ have "0 \<le> emeasure M (A \<union> B)" using A B by auto
+ moreover have "emeasure M (A \<union> B) \<le> 0"
+ using emeasure_subadditive[of A M B] A B by auto
+ ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto
+ show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
+ using P imp by auto
+ qed
+qed
+
+(* depricated replace by laws about eventually *)
+lemma
+ shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
+ and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
+ and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
+ and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x"
+ and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)"
+ by auto
+
+lemma AE_impI:
+ "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
+ by (cases P) auto
+
+lemma AE_measure:
+ assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")
+ shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
+proof -
+ from AE_E[OF AE] guess N . note N = this
+ with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
+ by (intro emeasure_mono) auto
+ also have "\<dots> \<le> emeasure M ?P + emeasure M N"
+ using sets N by (intro emeasure_subadditive) auto
+ also have "\<dots> = emeasure M ?P" using N by simp
+ finally show "emeasure M ?P = emeasure M (space M)"
+ using emeasure_space[of M "?P"] by auto
+qed
+
+lemma AE_space: "AE x in M. x \<in> space M"
+ by (rule AE_I[where N="{}"]) auto
+
+lemma AE_I2[simp, intro]:
+ "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
+ using AE_space by force
+
+lemma AE_Ball_mp:
+ "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
+ by auto
+
+lemma AE_cong[cong]:
+ "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)"
+ by auto
+
+lemma AE_all_countable:
+ "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
+proof
+ assume "\<forall>i. AE x in M. P i x"
+ from this[unfolded eventually_ae_filter Bex_def, THEN choice]
+ obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
+ have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
+ also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
+ finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
+ moreover from N have "(\<Union>i. N i) \<in> null_sets M"
+ by (intro null_sets_UN) auto
+ ultimately show "AE x in M. \<forall>i. P i x"
+ unfolding eventually_ae_filter by auto
+qed auto
+
+lemma AE_finite_all:
+ assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)"
+ using f by induct auto
+
+lemma AE_finite_allI:
+ assumes "finite S"
+ shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"
+ using AE_finite_all[OF `finite S`] by auto
+
+lemma emeasure_mono_AE:
+ assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
+ and B: "B \<in> sets M"
+ shows "emeasure M A \<le> emeasure M B"
+proof cases
+ assume A: "A \<in> sets M"
+ from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M"
+ by (auto simp: eventually_ae_filter)
+ have "emeasure M A = emeasure M (A - N)"
+ using N A by (subst emeasure_Diff_null_set) auto
+ also have "emeasure M (A - N) \<le> emeasure M (B - N)"
+ using N A B sets_into_space by (auto intro!: emeasure_mono)
+ also have "emeasure M (B - N) = emeasure M B"
+ using N B by (subst emeasure_Diff_null_set) auto
+ finally show ?thesis .
+qed (simp add: emeasure_nonneg emeasure_notin_sets)
+
+lemma emeasure_eq_AE:
+ assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
+ assumes A: "A \<in> sets M" and B: "B \<in> sets M"
+ shows "emeasure M A = emeasure M B"
+ using assms by (safe intro!: antisym emeasure_mono_AE) auto
+
+section {* @{text \<sigma>}-finite Measures *}
+
+locale sigma_finite_measure =
+ fixes M :: "'a measure"
+ assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set.
+ range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
+
+lemma (in sigma_finite_measure) sigma_finite_disjoint:
+ obtains A :: "nat \<Rightarrow> 'a set"
+ where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
+proof atomize_elim
+ case goal1
+ obtain A :: "nat \<Rightarrow> 'a set" where
+ range: "range A \<subseteq> sets M" and
+ space: "(\<Union>i. A i) = space M" and
+ measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
+ using sigma_finite by auto
+ note range' = range_disjointed_sets[OF range] range
+ { fix i
+ have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
+ using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono)
+ then have "emeasure M (disjointed A i) \<noteq> \<infinity>"
+ using measure[of i] by auto }
+ with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
+ show ?case by (auto intro!: exI[of _ "disjointed A"])
+qed
+
+lemma (in sigma_finite_measure) sigma_finite_incseq:
+ obtains A :: "nat \<Rightarrow> 'a set"
+ where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
+proof atomize_elim
+ case goal1
+ obtain F :: "nat \<Rightarrow> 'a set" where
+ F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
+ using sigma_finite by auto
+ then show ?case
+ proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
+ from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
+ then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
+ using F by fastforce
+ next
+ fix n
+ have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F
+ by (auto intro!: emeasure_subadditive_finite)
+ also have "\<dots> < \<infinity>"
+ using F by (auto simp: setsum_Pinfty)
+ finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
+ qed (force simp: incseq_def)+
+qed
+
+section {* Measure space induced by distribution of @{const measurable}-functions *}
+
+definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
+ "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
+
+lemma
+ shows sets_distr[simp]: "sets (distr M N f) = sets N"
+ and space_distr[simp]: "space (distr M N f) = space N"
+ by (auto simp: distr_def)
+
+lemma
+ shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
+ and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
+ by (auto simp: measurable_def)
+
+lemma emeasure_distr:
+ fixes f :: "'a \<Rightarrow> 'b"
+ assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
+ shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
+ unfolding distr_def
+proof (rule emeasure_measure_of_sigma)
+ show "positive (sets N) ?\<mu>"
+ by (auto simp: positive_def)
+
+ show "countably_additive (sets N) ?\<mu>"
+ proof (intro countably_additiveI)
+ fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
+ then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
+ then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
+ using f by (auto simp: measurable_def)
+ moreover have "(\<Union>i. f -` A i \<inter> space M) \<in> sets M"
+ using * by blast
+ moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
+ using `disjoint_family A` by (auto simp: disjoint_family_on_def)
+ ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
+ using suminf_emeasure[OF _ **] A f
+ by (auto simp: comp_def vimage_UN)
+ qed
+ show "sigma_algebra (space N) (sets N)" ..
+qed fact
+
+lemma AE_distrD:
+ assumes f: "f \<in> measurable M M'"
+ and AE: "AE x in distr M M' f. P x"
+ shows "AE x in M. P (f x)"
+proof -
+ from AE[THEN AE_E] guess N .
+ with f show ?thesis
+ unfolding eventually_ae_filter
+ by (intro bexI[of _ "f -` N \<inter> space M"])
+ (auto simp: emeasure_distr measurable_def)
+qed
+
+lemma null_sets_distr_iff:
+ "f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N"
+ by (auto simp add: null_sets_def emeasure_distr measurable_sets)
+
+lemma distr_distr:
+ assumes f: "g \<in> measurable N L" and g: "f \<in> measurable M N"
+ shows "distr (distr M N f) L g = distr M L (g \<circ> f)" (is "?L = ?R")
+ using measurable_comp[OF g f] f g
+ by (auto simp add: emeasure_distr measurable_sets measurable_space
+ intro!: arg_cong[where f="emeasure M"] measure_eqI)
+
+section {* Real measure values *}
+
+lemma measure_nonneg: "0 \<le> measure M A"
+ using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
+
+lemma measure_empty[simp]: "measure M {} = 0"
+ unfolding measure_def by simp
+
+lemma emeasure_eq_ereal_measure:
+ "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)"
+ using emeasure_nonneg[of M A]
+ by (cases "emeasure M A") (auto simp: measure_def)
+
+lemma measure_Union:
+ assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
+ and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
+ shows "measure M (A \<union> B) = measure M A + measure M B"
+ unfolding measure_def
+ using plus_emeasure[OF measurable, symmetric] finite
+ by (simp add: emeasure_eq_ereal_measure)
+
+lemma measure_finite_Union:
+ assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
+ assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
+ shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
+ unfolding measure_def
+ using setsum_emeasure[OF measurable, symmetric] finite
+ by (simp add: emeasure_eq_ereal_measure)
+
+lemma measure_Diff:
+ assumes finite: "emeasure M A \<noteq> \<infinity>"
+ and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
+ shows "measure M (A - B) = measure M A - measure M B"
+proof -
+ have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
+ using measurable by (auto intro!: emeasure_mono)
+ hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
+ using measurable finite by (rule_tac measure_Union) auto
+ thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
+qed
+
+lemma measure_UNION:
+ assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
+ assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
+ shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
+proof -
+ from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"]
+ suminf_emeasure[OF measurable] emeasure_nonneg[of M]
+ have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp
+ moreover
+ { fix i
+ have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
+ using measurable by (auto intro!: emeasure_mono)
+ then have "emeasure M (A i) = ereal ((measure M (A i)))"
+ using finite by (intro emeasure_eq_ereal_measure) auto }
+ ultimately show ?thesis using finite
+ unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure)
+qed
+
+lemma measure_subadditive:
+ assumes measurable: "A \<in> sets M" "B \<in> sets M"
+ and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
+ shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
+proof -
+ have "emeasure M (A \<union> B) \<noteq> \<infinity>"
+ using emeasure_subadditive[OF measurable] fin by auto
+ then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
+ using emeasure_subadditive[OF measurable] fin
+ by (auto simp: emeasure_eq_ereal_measure)
+qed
+
+lemma measure_subadditive_finite:
+ assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
+ shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
+proof -
+ { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
+ using emeasure_subadditive_finite[OF A] .
+ also have "\<dots> < \<infinity>"
+ using fin by (simp add: setsum_Pinfty)
+ finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp }
+ then show ?thesis
+ using emeasure_subadditive_finite[OF A] fin
+ unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
+qed
+
+lemma measure_subadditive_countably:
+ assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
+ shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
+proof -
+ from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty)
+ moreover
+ { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
+ using emeasure_subadditive_countably[OF A] .
+ also have "\<dots> < \<infinity>"
+ using fin by simp
+ finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp }
+ ultimately show ?thesis
+ using emeasure_subadditive_countably[OF A] fin
+ unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
+qed
+
+lemma measure_eq_setsum_singleton:
+ assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
+ and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>"
+ shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))"
+ unfolding measure_def
+ using emeasure_eq_setsum_singleton[OF S] fin
+ by simp (simp add: emeasure_eq_ereal_measure)
+
+lemma Lim_measure_incseq:
+ assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
+ shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
+proof -
+ have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
+ using fin by (auto simp: emeasure_eq_ereal_measure)
+ then show ?thesis
+ using Lim_emeasure_incseq[OF A]
+ unfolding measure_def
+ by (intro lim_real_of_ereal) simp
+qed
+
+lemma Lim_measure_decseq:
+ assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
+ shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
+proof -
+ have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
+ using A by (auto intro!: emeasure_mono)
+ also have "\<dots> < \<infinity>"
+ using fin[of 0] by auto
+ finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)"
+ by (auto simp: emeasure_eq_ereal_measure)
+ then show ?thesis
+ unfolding measure_def
+ using Lim_emeasure_decseq[OF A fin]
+ by (intro lim_real_of_ereal) simp
+qed
+
+section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
+
+locale finite_measure = sigma_finite_measure M for M +
+ assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
+
+lemma finite_measureI[Pure.intro!]:
+ assumes *: "emeasure M (space M) \<noteq> \<infinity>"
+ shows "finite_measure M"
+proof
+ show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
+ using * by (auto intro!: exI[of _ "\<lambda>_. space M"])
+qed fact
+
+lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>"
+ using finite_emeasure_space emeasure_space[of M A] by auto
+
+lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)"
+ unfolding measure_def by (simp add: emeasure_eq_ereal_measure)
+
+lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r"
+ using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto
+
+lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
+ using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
+
+lemma (in finite_measure) finite_measure_Diff:
+ assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
+ shows "measure M (A - B) = measure M A - measure M B"
+ using measure_Diff[OF _ assms] by simp
+
+lemma (in finite_measure) finite_measure_Union:
+ assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
+ shows "measure M (A \<union> B) = measure M A + measure M B"
+ using measure_Union[OF _ _ assms] by simp
+
+lemma (in finite_measure) finite_measure_finite_Union:
+ assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
+ shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
+ using measure_finite_Union[OF assms] by simp
+
+lemma (in finite_measure) finite_measure_UNION:
+ assumes A: "range A \<subseteq> sets M" "disjoint_family A"
+ shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
+ using measure_UNION[OF A] by simp
+
+lemma (in finite_measure) finite_measure_mono:
+ assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
+ using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
+
+lemma (in finite_measure) finite_measure_subadditive:
+ assumes m: "A \<in> sets M" "B \<in> sets M"
+ shows "measure M (A \<union> B) \<le> measure M A + measure M B"
+ using measure_subadditive[OF m] by simp
+
+lemma (in finite_measure) finite_measure_subadditive_finite:
+ assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
+ using measure_subadditive_finite[OF assms] by simp
+
+lemma (in finite_measure) finite_measure_subadditive_countably:
+ assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
+ shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
+proof -
+ from `summable (\<lambda>i. measure M (A i))`
+ have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
+ by (simp add: sums_ereal) (rule summable_sums)
+ from sums_unique[OF this, symmetric]
+ measure_subadditive_countably[OF A]
+ show ?thesis by (simp add: emeasure_eq_measure)
+qed
+
+lemma (in finite_measure) finite_measure_eq_setsum_singleton:
+ assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
+ shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
+ using measure_eq_setsum_singleton[OF assms] by simp
+
+lemma (in finite_measure) finite_Lim_measure_incseq:
+ assumes A: "range A \<subseteq> sets M" "incseq A"
+ shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
+ using Lim_measure_incseq[OF A] by simp
+
+lemma (in finite_measure) finite_Lim_measure_decseq:
+ assumes A: "range A \<subseteq> sets M" "decseq A"
+ shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
+ using Lim_measure_decseq[OF A] by simp
+
+lemma (in finite_measure) finite_measure_compl:
+ assumes S: "S \<in> sets M"
+ shows "measure M (space M - S) = measure M (space M) - measure M S"
+ using measure_Diff[OF _ top S sets_into_space] S by simp
+
+lemma (in finite_measure) finite_measure_mono_AE:
+ assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
+ shows "measure M A \<le> measure M B"
+ using assms emeasure_mono_AE[OF imp B]
+ by (simp add: emeasure_eq_measure)
+
+lemma (in finite_measure) finite_measure_eq_AE:
+ assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
+ assumes A: "A \<in> sets M" and B: "B \<in> sets M"
+ shows "measure M A = measure M B"
+ using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
+
+section {* Counting space *}
+
+definition count_space :: "'a set \<Rightarrow> 'a measure" where
+ "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
+
+lemma
+ shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
+ and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
+ using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
+ by (auto simp: count_space_def)
+
+lemma measurable_count_space_eq1[simp]:
+ "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
+ unfolding measurable_def by simp
+
+lemma measurable_count_space_eq2[simp]:
+ assumes "finite A"
+ shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
+proof -
+ { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
+ with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
+ by (auto dest: finite_subset)
+ moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
+ ultimately have "f -` X \<inter> space M \<in> sets M"
+ using `X \<subseteq> A` by (auto intro!: finite_UN simp del: UN_simps) }
+ then show ?thesis
+ unfolding measurable_def by auto
+qed
+
+lemma emeasure_count_space:
+ assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)"
+ (is "_ = ?M X")
+ unfolding count_space_def
+proof (rule emeasure_measure_of_sigma)
+ show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
+
+ show "positive (Pow A) ?M"
+ by (auto simp: positive_def)
+
+ show "countably_additive (Pow A) ?M"
+ proof (unfold countably_additive_def, safe)
+ fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F"
+ show "(\<Sum>i. ?M (F i)) = ?M (\<Union>i. F i)"
+ proof cases
+ assume "\<forall>i. finite (F i)"
+ then have finite_F: "\<And>i. finite (F i)" by auto
+ have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
+ from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
+
+ have inj_f: "inj_on f {i. F i \<noteq> {}}"
+ proof (rule inj_onI, simp)
+ fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
+ then have "f i \<in> F i" "f j \<in> F j" using f by force+
+ with disj * show "i = j" by (auto simp: disjoint_family_on_def)
+ qed
+ have fin_eq: "finite (\<Union>i. F i) \<longleftrightarrow> finite {i. F i \<noteq> {}}"
+ proof
+ assume "finite (\<Union>i. F i)"
+ show "finite {i. F i \<noteq> {}}"
+ proof (rule finite_imageD)
+ from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
+ then show "finite (f`{i. F i \<noteq> {}})"
+ by (rule finite_subset) fact
+ qed fact
+ next
+ assume "finite {i. F i \<noteq> {}}"
+ with finite_F have "finite (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
+ by auto
+ also have "(\<Union>i\<in>{i. F i \<noteq> {}}. F i) = (\<Union>i. F i)"
+ by auto
+ finally show "finite (\<Union>i. F i)" .
+ qed
+
+ show ?thesis
+ proof cases
+ assume *: "finite (\<Union>i. F i)"
+ with finite_F have "finite {i. ?M (F i) \<noteq> 0} "
+ by (simp add: fin_eq)
+ then have "(\<Sum>i. ?M (F i)) = (\<Sum>i | ?M (F i) \<noteq> 0. ?M (F i))"
+ by (rule suminf_eq_setsum)
+ also have "\<dots> = ereal (\<Sum>i | F i \<noteq> {}. card (F i))"
+ using finite_F by simp
+ also have "\<dots> = ereal (card (\<Union>i \<in> {i. F i \<noteq> {}}. F i))"
+ using * finite_F disj
+ by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def fin_eq)
+ also have "\<dots> = ?M (\<Union>i. F i)"
+ using * by (auto intro!: arg_cong[where f=card])
+ finally show ?thesis .
+ next
+ assume inf: "infinite (\<Union>i. F i)"
+ { fix i
+ have "\<exists>N. i \<le> (\<Sum>i<N. card (F i))"
+ proof (induct i)
+ case (Suc j)
+ from Suc obtain N where N: "j \<le> (\<Sum>i<N. card (F i))" by auto
+ have "infinite ({i. F i \<noteq> {}} - {..< N})"
+ using inf by (auto simp: fin_eq)
+ then have "{i. F i \<noteq> {}} - {..< N} \<noteq> {}"
+ by (metis finite.emptyI)
+ then obtain i where i: "F i \<noteq> {}" "N \<le> i"
+ by (auto simp: not_less[symmetric])
+
+ note N
+ also have "(\<Sum>i<N. card (F i)) \<le> (\<Sum>i<i. card (F i))"
+ by (rule setsum_mono2) (auto simp: i)
+ also have "\<dots> < (\<Sum>i<i. card (F i)) + card (F i)"
+ using finite_F `F i \<noteq> {}` by (simp add: card_gt_0_iff)
+ finally have "j < (\<Sum>i<Suc i. card (F i))"
+ by simp
+ then show ?case unfolding Suc_le_eq by blast
+ qed simp }
+ with finite_F inf show ?thesis
+ by (auto simp del: real_of_nat_setsum intro!: SUP_PInfty
+ simp add: suminf_ereal_eq_SUPR real_of_nat_setsum[symmetric])
+ qed
+ next
+ assume "\<not> (\<forall>i. finite (F i))"
+ then obtain j where j: "infinite (F j)" by auto
+ then have "infinite (\<Union>i. F i)"
+ using finite_subset[of "F j" "\<Union>i. F i"] by auto
+ moreover have "\<And>i. 0 \<le> ?M (F i)" by auto
+ ultimately show ?thesis
+ using suminf_PInfty[of "\<lambda>i. ?M (F i)" j] j by auto
+ qed
+ qed
+ show "X \<in> Pow A" using `X \<subseteq> A` by simp
+qed
+
+lemma emeasure_count_space_finite[simp]:
+ "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)"
+ using emeasure_count_space[of X A] by simp
+
+lemma emeasure_count_space_infinite[simp]:
+ "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
+ using emeasure_count_space[of X A] by simp
+
+lemma emeasure_count_space_eq_0:
+ "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
+proof cases
+ assume X: "X \<subseteq> A"
+ then show ?thesis
+ proof (intro iffI impI)
+ assume "emeasure (count_space A) X = 0"
+ with X show "X = {}"
+ by (subst (asm) emeasure_count_space) (auto split: split_if_asm)
+ qed simp
+qed (simp add: emeasure_notin_sets)
+
+lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
+ unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
+
+lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
+ unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
+
+lemma sigma_finite_measure_count_space:
+ fixes A :: "'a::countable set"
+ shows "sigma_finite_measure (count_space A)"
+proof
+ show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (count_space A) \<and> (\<Union>i. F i) = space (count_space A) \<and>
+ (\<forall>i. emeasure (count_space A) (F i) \<noteq> \<infinity>)"
+ using surj_from_nat by (intro exI[of _ "\<lambda>i. {from_nat i} \<inter> A"]) (auto simp del: surj_from_nat)
+qed
+
+lemma finite_measure_count_space:
+ assumes [simp]: "finite A"
+ shows "finite_measure (count_space A)"
+ by rule simp
+
+lemma sigma_finite_measure_count_space_finite:
+ assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
+proof -
+ interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
+ show "sigma_finite_measure (count_space A)" ..
+qed
+
+end
+
--- a/src/HOL/Probability/Probability.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Probability.thy Mon Apr 23 12:14:35 2012 +0200
@@ -4,7 +4,7 @@
Probability_Measure
Infinite_Product_Measure
Independent_Family
- Conditional_Probability
Information
begin
+
end
--- a/src/HOL/Probability/Probability_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Probability_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -6,110 +6,219 @@
header {*Probability measure*}
theory Probability_Measure
-imports Lebesgue_Measure
+ imports Lebesgue_Measure Radon_Nikodym
begin
+lemma funset_eq_UN_fun_upd_I:
+ assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
+ and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
+ and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
+ shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
+proof safe
+ fix f assume f: "f \<in> F (insert a A)"
+ show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
+ proof (rule UN_I[of "f(a := d)"])
+ show "f(a := d) \<in> F A" using *[OF f] .
+ show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
+ proof (rule image_eqI[of _ _ "f a"])
+ show "f a \<in> G (f(a := d))" using **[OF f] .
+ qed simp
+ qed
+next
+ fix f x assume "f \<in> F A" "x \<in> G f"
+ from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
+qed
+
+lemma extensional_funcset_insert_eq[simp]:
+ assumes "a \<notin> A"
+ shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
+ apply (rule funset_eq_UN_fun_upd_I)
+ using assms
+ by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
+
+lemma finite_extensional_funcset[simp, intro]:
+ assumes "finite A" "finite B"
+ shows "finite (extensional A \<inter> (A \<rightarrow> B))"
+ using assms by induct auto
+
+lemma finite_PiE[simp, intro]:
+ assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
+ shows "finite (Pi\<^isub>E A B)"
+proof -
+ have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
+ show ?thesis
+ using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
+qed
+
+
+lemma countably_additiveI[case_names countably]:
+ assumes "\<And>A. \<lbrakk> range A \<subseteq> M ; disjoint_family A ; (\<Union>i. A i) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
+ shows "countably_additive M \<mu>"
+ using assms unfolding countably_additive_def by auto
+
+lemma convex_le_Inf_differential:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex_on I f"
+ assumes "x \<in> interior I" "y \<in> I"
+ shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
+ (is "_ \<ge> _ + Inf (?F x) * (y - x)")
+proof -
+ show ?thesis
+ proof (cases rule: linorder_cases)
+ assume "x < y"
+ moreover
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e . note e = this
+ moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
+ ultimately have "x < t" "t < y" "t \<in> ball x e"
+ by (auto simp: dist_real_def field_simps split: split_min)
+ with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e .
+ moreover def K \<equiv> "x - e / 2"
+ with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: dist_real_def)
+ ultimately have "K \<in> I" "K < x" "x \<in> I"
+ using interior_subset[of I] `x \<in> interior I` by auto
+
+ have "Inf (?F x) \<le> (f x - f y) / (x - y)"
+ proof (rule Inf_lower2)
+ show "(f x - f t) / (x - t) \<in> ?F x"
+ using `t \<in> I` `x < t` by auto
+ show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+ using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
+ next
+ fix y assume "y \<in> ?F x"
+ with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
+ show "(f K - f x) / (K - x) \<le> y" by auto
+ qed
+ then show ?thesis
+ using `x < y` by (simp add: field_simps)
+ next
+ assume "y < x"
+ moreover
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e . note e = this
+ moreover def t \<equiv> "x + e / 2"
+ ultimately have "x < t" "t \<in> ball x e"
+ by (auto simp: dist_real_def field_simps)
+ with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+ have "(f x - f y) / (x - y) \<le> Inf (?F x)"
+ proof (rule Inf_greatest)
+ have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
+ using `y < x` by (auto simp: field_simps)
+ also
+ fix z assume "z \<in> ?F x"
+ with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
+ have "(f y - f x) / (y - x) \<le> z" by auto
+ finally show "(f x - f y) / (x - y) \<le> z" .
+ next
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e . note e = this
+ then have "x + e / 2 \<in> ball x e" by (auto simp: dist_real_def)
+ with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
+ then show "?F x \<noteq> {}" by blast
+ qed
+ then show ?thesis
+ using `y < x` by (simp add: field_simps)
+ qed simp
+qed
+
+lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
+ by (rule measure_eqI) (auto simp: emeasure_distr)
+
locale prob_space = finite_measure +
- assumes measure_space_1: "measure M (space M) = 1"
+ assumes emeasure_space_1: "emeasure M (space M) = 1"
lemma prob_spaceI[Pure.intro!]:
- assumes "measure_space M"
- assumes *: "measure M (space M) = 1"
+ assumes *: "emeasure M (space M) = 1"
shows "prob_space M"
proof -
interpret finite_measure M
proof
- show "measure_space M" by fact
- show "measure M (space M) \<noteq> \<infinity>" using * by simp
+ show "emeasure M (space M) \<noteq> \<infinity>" using * by simp
qed
show "prob_space M" by default fact
qed
abbreviation (in prob_space) "events \<equiv> sets M"
-abbreviation (in prob_space) "prob \<equiv> \<mu>'"
-abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
+abbreviation (in prob_space) "prob \<equiv> measure M"
+abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
-definition (in prob_space)
- "distribution X A = \<mu>' (X -` A \<inter> space M)"
-
-abbreviation (in prob_space)
- "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
-
-lemma (in prob_space) prob_space_cong:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
- shows "prob_space N"
-proof
- show "measure_space N" by (intro measure_space_cong assms)
- show "measure N (space N) = 1"
- using measure_space_1 assms by simp
+lemma (in prob_space) prob_space_distr:
+ assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
+proof (rule prob_spaceI)
+ have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
+ with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
+ by (auto simp: emeasure_distr emeasure_space_1)
qed
-lemma (in prob_space) distribution_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
- shows "distribution X = distribution Y"
- unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) joint_distribution_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
- assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
- shows "joint_distribution X Y = joint_distribution X' Y'"
- unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) distribution_id[simp]:
- "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
- by (auto simp: distribution_def intro!: arg_cong[where f=prob])
-
lemma (in prob_space) prob_space: "prob (space M) = 1"
- using measure_space_1 unfolding \<mu>'_def by (simp add: one_ereal_def)
+ using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
using bounded_measure[of A] by (simp add: prob_space)
-lemma (in prob_space) distribution_positive[simp, intro]:
- "0 \<le> distribution X A" unfolding distribution_def by auto
-
-lemma (in prob_space) not_zero_less_distribution[simp]:
- "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
- using distribution_positive[of X A] by arith
-
-lemma (in prob_space) joint_distribution_remove[simp]:
- "joint_distribution X X {(x, x)} = distribution X {x}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+lemma (in prob_space) not_empty: "space M \<noteq> {}"
+ using prob_space by auto
-lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
- unfolding distribution_def using prob_space by auto
-
-lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) not_empty: "space M \<noteq> {}"
- using prob_space empty_measure' by auto
-
-lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
- unfolding measure_space_1[symmetric]
- using sets_into_space
- by (intro measure_mono) auto
+lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
+ using emeasure_space[of M X] by (simp add: emeasure_space_1)
lemma (in prob_space) AE_I_eq_1:
- assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
- shows "AE x. P x"
+ assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
+ shows "AE x in M. P x"
proof (rule AE_I)
- show "\<mu> (space M - {x \<in> space M. P x}) = 0"
- using assms measure_space_1 by (simp add: measure_compl)
+ show "emeasure M (space M - {x \<in> space M. P x}) = 0"
+ using assms emeasure_space_1 by (simp add: emeasure_compl)
qed (insert assms, auto)
-lemma (in prob_space) distribution_1:
- "distribution X A \<le> 1"
- unfolding distribution_def by simp
-
lemma (in prob_space) prob_compl:
assumes A: "A \<in> events"
shows "prob (space M - A) = 1 - prob A"
using finite_measure_compl[OF A] by (simp add: prob_space)
+lemma (in prob_space) AE_in_set_eq_1:
+ assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
+proof
+ assume ae: "AE x in M. x \<in> A"
+ have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
+ using `A \<in> events`[THEN sets_into_space] by auto
+ with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
+ by (simp add: emeasure_compl emeasure_space_1)
+ then show "prob A = 1"
+ using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
+next
+ assume prob: "prob A = 1"
+ show "AE x in M. x \<in> A"
+ proof (rule AE_I)
+ show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
+ show "emeasure M (space M - A) = 0"
+ using `A \<in> events` prob
+ by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
+ show "space M - A \<in> events"
+ using `A \<in> events` by auto
+ qed
+qed
+
+lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
+proof
+ assume "AE x in M. False"
+ then have "AE x in M. x \<in> {}" by simp
+ then show False
+ by (subst (asm) AE_in_set_eq_1) auto
+qed simp
+
+lemma (in prob_space) AE_prob_1:
+ assumes "prob A = 1" shows "AE x in M. x \<in> A"
+proof -
+ from `prob A = 1` have "A \<in> events"
+ by (metis measure_notin_sets zero_neq_one)
+ with AE_in_set_eq_1 assms show ?thesis by simp
+qed
+
lemma (in prob_space) prob_space_increasing: "increasing M prob"
by (auto intro!: finite_measure_mono simp: increasing_def)
@@ -164,9 +273,8 @@
shows "prob (\<Union> i :: nat. c i) = 0"
proof (rule antisym)
show "prob (\<Union> i :: nat. c i) \<le> 0"
- using finite_measure_countably_subadditive[OF assms(1)]
- by (simp add: assms(2) suminf_zero summable_zero)
-qed simp
+ using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
+qed (simp add: measure_nonneg)
lemma (in prob_space) prob_equiprobable_finite_unions:
assumes "s \<in> events"
@@ -178,7 +286,7 @@
from someI_ex[OF this] assms
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
have "prob s = (\<Sum> x \<in> s. prob {x})"
- using finite_measure_finite_singleton[OF s_finite] by simp
+ using finite_measure_eq_setsum_singleton[OF s_finite] by simp
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
using setsum_constant assms by (simp add: real_eq_of_nat)
@@ -199,96 +307,20 @@
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
proof (rule finite_measure_finite_Union)
show "finite s" by fact
- show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
+ show "(\<lambda>i. e \<inter> f i)`s \<subseteq> events" using assms(2) by auto
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
using disjoint by (auto simp: disjoint_family_on_def)
qed
finally show ?thesis .
qed
-lemma (in prob_space) prob_space_vimage:
- assumes S: "sigma_algebra S"
- assumes T: "T \<in> measure_preserving M S"
- shows "prob_space S"
-proof
- interpret S: measure_space S
- using S and T by (rule measure_space_vimage)
- show "measure_space S" ..
-
- from T[THEN measure_preservingD2]
- have "T -` space S \<inter> space M = space M"
- by (auto simp: measurable_def)
- with T[THEN measure_preservingD, of "space S", symmetric]
- show "measure S (space S) = 1"
- using measure_space_1 by simp
-qed
-
-lemma prob_space_unique_Int_stable:
- fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
- assumes E: "Int_stable E" "space E \<in> sets E"
- and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)"
- and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)"
- and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
- assumes "X \<in> sets (sigma E)"
- shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
-proof -
- interpret M!: prob_space M by fact
- interpret N!: prob_space N by fact
- have "measure M X = measure N X"
- proof (rule measure_unique_Int_stable[OF `Int_stable E`])
- show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E"
- using E M N by auto
- show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>"
- using M.measure_space_1 by simp
- show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>"
- using E M N by (auto intro!: M.measure_space_cong)
- show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>"
- using E M N by (auto intro!: N.measure_space_cong)
- { fix X assume "X \<in> sets E"
- then have "X \<in> sets (sigma E)"
- by (auto simp: sets_sigma sigma_sets.Basic)
- with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X"
- by (simp add: M.finite_measure_eq N.finite_measure_eq) }
- qed fact
- with `X \<in> sets (sigma E)` M N show ?thesis
- by (simp add: M.finite_measure_eq N.finite_measure_eq)
-qed
-
-lemma (in prob_space) distribution_prob_space:
- assumes X: "random_variable S X"
- shows "prob_space (S\<lparr>measure := ereal \<circ> distribution X\<rparr>)" (is "prob_space ?S")
-proof (rule prob_space_vimage)
- show "X \<in> measure_preserving M ?S"
- using X
- unfolding measure_preserving_def distribution_def [abs_def]
- by (auto simp: finite_measure_eq measurable_sets)
- show "sigma_algebra ?S" using X by simp
-qed
-
-lemma (in prob_space) AE_distribution:
- assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := ereal \<circ> distribution X\<rparr>. Q x"
- shows "AE x. Q (X x)"
-proof -
- interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
- obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
- using assms unfolding X.almost_everywhere_def by auto
- from X[unfolded measurable_def] N show "AE x. Q (X x)"
- by (intro AE_I'[where N="X -` N \<inter> space M"])
- (auto simp: finite_measure_eq distribution_def measurable_sets)
-qed
-
-lemma (in prob_space) distribution_eq_integral:
- "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
- using finite_measure_eq[of "X -` A \<inter> space M"]
- by (auto simp: measurable_sets distribution_def)
-
lemma (in prob_space) expectation_less:
assumes [simp]: "integrable M X"
assumes gt: "\<forall>x\<in>space M. X x < b"
shows "expectation X < b"
proof -
have "expectation X < expectation (\<lambda>x. b)"
- using gt measure_space_1
+ using gt emeasure_space_1
by (intro integral_less_AE_space) auto
then show ?thesis using prob_space by simp
qed
@@ -299,80 +331,11 @@
shows "a < expectation X"
proof -
have "expectation (\<lambda>x. a) < expectation X"
- using gt measure_space_1
+ using gt emeasure_space_1
by (intro integral_less_AE_space) auto
then show ?thesis using prob_space by simp
qed
-lemma convex_le_Inf_differential:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_on I f"
- assumes "x \<in> interior I" "y \<in> I"
- shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
- (is "_ \<ge> _ + Inf (?F x) * (y - x)")
-proof -
- show ?thesis
- proof (cases rule: linorder_cases)
- assume "x < y"
- moreover
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e . note e = this
- moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
- ultimately have "x < t" "t < y" "t \<in> ball x e"
- by (auto simp: mem_ball dist_real_def field_simps split: split_min)
- with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e .
- moreover def K \<equiv> "x - e / 2"
- with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def)
- ultimately have "K \<in> I" "K < x" "x \<in> I"
- using interior_subset[of I] `x \<in> interior I` by auto
-
- have "Inf (?F x) \<le> (f x - f y) / (x - y)"
- proof (rule Inf_lower2)
- show "(f x - f t) / (x - t) \<in> ?F x"
- using `t \<in> I` `x < t` by auto
- show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
- next
- fix y assume "y \<in> ?F x"
- with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
- show "(f K - f x) / (K - x) \<le> y" by auto
- qed
- then show ?thesis
- using `x < y` by (simp add: field_simps)
- next
- assume "y < x"
- moreover
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e . note e = this
- moreover def t \<equiv> "x + e / 2"
- ultimately have "x < t" "t \<in> ball x e"
- by (auto simp: mem_ball dist_real_def field_simps)
- with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
- have "(f x - f y) / (x - y) \<le> Inf (?F x)"
- proof (rule Inf_greatest)
- have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
- using `y < x` by (auto simp: field_simps)
- also
- fix z assume "z \<in> ?F x"
- with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
- have "(f y - f x) / (y - x) \<le> z" by auto
- finally show "(f x - f y) / (x - y) \<le> z" .
- next
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e . note e = this
- then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def)
- with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
- then show "?F x \<noteq> {}" by blast
- qed
- then show ?thesis
- using `y < x` by (simp add: field_simps)
- qed simp
-qed
-
lemma (in prob_space) jensens_inequality:
fixes a b :: real
assumes X: "integrable M X" "X ` space M \<subseteq> I"
@@ -410,8 +373,7 @@
fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
then guess x .. note x = this
have "q x + ?F x * (expectation X - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
- using prob_space
- by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X)
+ using prob_space by (simp add: X)
also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
using `x \<in> I` `open I` X(2)
by (intro integral_mono integral_add integral_cmult integral_diff
@@ -422,31 +384,6 @@
finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
qed
-lemma (in prob_space) distribution_eq_translated_integral:
- assumes "random_variable S X" "A \<in> sets S"
- shows "distribution X A = integral\<^isup>P (S\<lparr>measure := ereal \<circ> distribution X\<rparr>) (indicator A)"
-proof -
- interpret S: prob_space "S\<lparr>measure := ereal \<circ> distribution X\<rparr>"
- using assms(1) by (rule distribution_prob_space)
- show ?thesis
- using S.positive_integral_indicator(1)[of A] assms by simp
-qed
-
-lemma (in prob_space) finite_expectation1:
- assumes f: "finite (X`space M)" and rv: "random_variable borel X"
- shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
-proof (subst integral_on_finite)
- show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
- show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
- "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
- using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
-qed
-
-lemma (in prob_space) finite_expectation:
- assumes "finite (X`space M)" "random_variable borel X"
- shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
- using assms unfolding distribution_def using finite_expectation1 by auto
-
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
assumes "{x} \<in> events"
assumes "prob {x} = 1"
@@ -455,119 +392,25 @@
shows "prob {y} = 0"
using prob_one_inter[of "{y}" "{x}"] assms by auto
-lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
- unfolding distribution_def by simp
-
-lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1"
-proof -
- have "X -` X ` space M \<inter> space M = space M" by auto
- thus ?thesis unfolding distribution_def by (simp add: prob_space)
-qed
-
-lemma (in prob_space) distribution_one:
- assumes "random_variable M' X" and "A \<in> sets M'"
- shows "distribution X A \<le> 1"
-proof -
- have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def
- using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono)
- thus ?thesis by (simp add: prob_space)
-qed
-
-lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0:
- assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
- (is "random_variable ?S X")
- assumes "distribution X {x} = 1"
- assumes "y \<noteq> x"
- shows "distribution X {y} = 0"
-proof cases
- { fix x have "X -` {x} \<inter> space M \<in> sets M"
- proof cases
- assume "x \<in> X`space M" with X show ?thesis
- by (auto simp: measurable_def image_iff)
- next
- assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
- then show ?thesis by auto
- qed } note single = this
- have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
- "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
- using `y \<noteq> x` by auto
- with finite_measure_inter_full_set[OF single single, of x y] assms(2)
- show ?thesis by (auto simp: distribution_def prob_space)
-next
- assume "{y} \<notin> sets ?S"
- then have "X -` {y} \<inter> space M = {}" by auto
- thus "distribution X {y} = 0" unfolding distribution_def by auto
-qed
-
lemma (in prob_space) joint_distribution_Times_le_fst:
- assumes X: "random_variable MX X" and Y: "random_variable MY Y"
- and A: "A \<in> sets MX" and B: "B \<in> sets MY"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
- unfolding distribution_def
-proof (intro finite_measure_mono)
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
- show "X -` A \<inter> space M \<in> events"
- using X A unfolding measurable_def by simp
- have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
- (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_commute:
- "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+ "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
+ \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MX X) A"
+ by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
lemma (in prob_space) joint_distribution_Times_le_snd:
- assumes X: "random_variable MX X" and Y: "random_variable MY Y"
- and A: "A \<in> sets MX" and B: "B \<in> sets MY"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
- using assms
- by (subst joint_distribution_commute)
- (simp add: swap_product joint_distribution_Times_le_fst)
-
-lemma (in prob_space) random_variable_pairI:
- assumes "random_variable MX X"
- assumes "random_variable MY Y"
- shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
- interpret MX: sigma_algebra MX using assms by simp
- interpret MY: sigma_algebra MY using assms by simp
- interpret P: pair_sigma_algebra MX MY by default
- show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
- have sa: "sigma_algebra M" by default
- show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
- unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) joint_distribution_commute_singleton:
- "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) joint_distribution_assoc_singleton:
- "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
- joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+ "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
+ \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MY Y) B"
+ by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
-sublocale pair_prob_space \<subseteq> P: prob_space P
+sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
proof
- show "measure_space P" ..
- show "measure P (space P) = 1"
- by (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
+ show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
+ by (simp add: emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
qed
-lemma countably_additiveI[case_names countably]:
- assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
- (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
- shows "countably_additive M \<mu>"
- using assms unfolding countably_additive_def by auto
-
-lemma (in prob_space) joint_distribution_prob_space:
- assumes "random_variable MX X" "random_variable MY Y"
- shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
- using random_variable_pairI[OF assms] by (rule distribution_prob_space)
-
-locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
+locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
fixes I :: "'i set"
assumes prob_space: "\<And>i. prob_space (M i)"
@@ -578,648 +421,401 @@
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
proof
- show "measure_space P" ..
- show "measure P (space P) = 1"
- by (simp add: measure_times M.measure_space_1 setprod_1)
+ show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
+ by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
qed
lemma (in finite_product_prob_space) prob_times:
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
proof -
- have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
- using X by (intro finite_measure_eq[symmetric] in_P) auto
- also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
+ have "ereal (measure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)) = emeasure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)"
+ using X by (simp add: emeasure_eq_measure)
+ also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
using measure_times X by simp
- also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
- using X by (simp add: M.finite_measure_eq setprod_ereal)
- finally show ?thesis by simp
-qed
-
-lemma (in prob_space) random_variable_restrict:
- assumes I: "finite I"
- assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)"
- shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)"
-proof
- { fix i assume "i \<in> I"
- with X interpret N: sigma_algebra "N i" by simp
- have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) }
- note N_closed = this
- then show "sigma_algebra (Pi\<^isub>M I N)"
- by (simp add: product_algebra_def)
- (intro sigma_algebra_sigma product_algebra_generator_sets_into_space)
- show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
- using X by (intro measurable_restrict[OF I N_closed]) auto
-qed
-
-section "Probability spaces on finite sets"
-
-locale finite_prob_space = prob_space + finite_measure_space
-
-abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
-
-lemma (in prob_space) finite_random_variableD:
- assumes "finite_random_variable M' X" shows "random_variable M' X"
-proof -
- interpret M': finite_sigma_algebra M' using assms by simp
- show "random_variable M' X" using assms by simp default
-qed
-
-lemma (in prob_space) distribution_finite_prob_space:
- assumes "finite_random_variable MX X"
- shows "finite_prob_space (MX\<lparr>measure := ereal \<circ> distribution X\<rparr>)"
-proof -
- interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>"
- using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
- interpret MX: finite_sigma_algebra MX
- using assms by auto
- show ?thesis by default (simp_all add: MX.finite_space)
-qed
-
-lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
- assumes "simple_function M X"
- shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
- (is "finite_random_variable ?X _")
-proof (intro conjI)
- have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp
- interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
- show "finite_sigma_algebra ?X"
- by default auto
- show "X \<in> measurable M ?X"
- proof (unfold measurable_def, clarsimp)
- fix A assume A: "A \<subseteq> X`space M"
- then have "finite A" by (rule finite_subset) simp
- then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
- unfolding vimage_UN UN_extend_simps
- apply (rule finite_UN)
- using A assms unfolding simple_function_def by auto
- then show "X -` A \<inter> space M \<in> events" by simp
- qed
-qed
-
-lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
- assumes "simple_function M X"
- shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
- using simple_function_imp_finite_random_variable[OF assms, of ext]
- by (auto dest!: finite_random_variableD)
-
-lemma (in prob_space) sum_over_space_real_distribution:
- "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric]
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def simple_function_def
- intro!: arg_cong[where f=prob])
-
-lemma (in prob_space) finite_random_variable_pairI:
- assumes "finite_random_variable MX X"
- assumes "finite_random_variable MY Y"
- shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
- interpret MX: finite_sigma_algebra MX using assms by simp
- interpret MY: finite_sigma_algebra MY using assms by simp
- interpret P: pair_finite_sigma_algebra MX MY by default
- show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" ..
- have sa: "sigma_algebra M" by default
- show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
- unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) finite_random_variable_imp_sets:
- "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
- unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
-
-lemma (in prob_space) finite_random_variable_measurable:
- assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
-proof -
- interpret X: finite_sigma_algebra MX using X by simp
- from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
- "X \<in> space M \<rightarrow> space MX"
- by (auto simp: measurable_def)
- then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
- by auto
- show "X -` A \<inter> space M \<in> events"
- unfolding * by (intro vimage) auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_fst:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
- unfolding distribution_def
-proof (intro finite_measure_mono)
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
- show "X -` A \<inter> space M \<in> events"
- using finite_random_variable_measurable[OF X] .
- have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
- (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_snd:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
- using assms
- by (subst joint_distribution_commute)
- (simp add: swap_product joint_distribution_finite_Times_le_fst)
-
-lemma (in prob_space) finite_distribution_order:
- fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
- assumes "finite_random_variable MX X" "finite_random_variable MY Y"
- shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
- and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
- using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
- by (auto intro: antisym)
-
-lemma (in prob_space) setsum_joint_distribution:
- assumes X: "finite_random_variable MX X"
- assumes Y: "random_variable MY Y" "B \<in> sets MY"
- shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
- unfolding distribution_def
-proof (subst finite_measure_finite_Union[symmetric])
- interpret MX: finite_sigma_algebra MX using X by auto
- show "finite (space MX)" using MX.finite_space .
- let ?d = "\<lambda>i. (\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
- { fix i assume "i \<in> space MX"
- moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
- ultimately show "?d i \<in> events"
- using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
- using MX.sets_eq_Pow by auto }
- show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
- show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
- using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
-qed
-
-lemma (in prob_space) setsum_joint_distribution_singleton:
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
- shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
- using setsum_joint_distribution[OF X
- finite_random_variableD[OF Y(1)]
- finite_random_variable_imp_sets[OF Y]] by simp
-
-lemma (in prob_space) setsum_distribution:
- assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
- using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
- using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
-
-locale pair_finite_prob_space = pair_prob_space M1 M2 + pair_finite_space M1 M2 + M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
-
-sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
-
-lemma funset_eq_UN_fun_upd_I:
- assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
- and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
- and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
- shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
-proof safe
- fix f assume f: "f \<in> F (insert a A)"
- show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
- proof (rule UN_I[of "f(a := d)"])
- show "f(a := d) \<in> F A" using *[OF f] .
- show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
- proof (rule image_eqI[of _ _ "f a"])
- show "f a \<in> G (f(a := d))" using **[OF f] .
- qed simp
- qed
-next
- fix f x assume "f \<in> F A" "x \<in> G f"
- from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
-qed
-
-lemma extensional_funcset_insert_eq[simp]:
- assumes "a \<notin> A"
- shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
- apply (rule funset_eq_UN_fun_upd_I)
- using assms
- by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
-
-lemma finite_extensional_funcset[simp, intro]:
- assumes "finite A" "finite B"
- shows "finite (extensional A \<inter> (A \<rightarrow> B))"
- using assms by induct (auto simp: extensional_funcset_insert_eq)
-
-lemma finite_PiE[simp, intro]:
- assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
- shows "finite (Pi\<^isub>E A B)"
-proof -
- have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
- show ?thesis
- using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
-qed
-
-locale finite_product_finite_prob_space = finite_product_prob_space M I for M I +
- assumes finite_space: "\<And>i. finite_prob_space (M i)"
-
-sublocale finite_product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space .
-
-lemma (in finite_product_finite_prob_space) singleton_eq_product:
- assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})"
-proof (safe intro!: ext[of _ x])
- fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I"
- with x show "y i = x i"
- by (cases "i \<in> I") (auto simp: extensional_def)
-qed (insert x, auto)
-
-sublocale finite_product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M"
-proof
- show "finite (space P)"
- using finite_index M.finite_space by auto
-
- { fix x assume "x \<in> space P"
- with this[THEN singleton_eq_product]
- have "{x} \<in> sets P"
- by (auto intro!: in_P) }
- note x_in_P = this
-
- have "Pow (space P) \<subseteq> sets P"
- proof
- fix X assume "X \<in> Pow (space P)"
- moreover then have "finite X"
- using `finite (space P)` by (blast intro: finite_subset)
- ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P"
- by (intro finite_UN x_in_P) auto
- then show "X \<in> sets P" by simp
- qed
- with space_closed show [simp]: "sets P = Pow (space P)" ..
-qed
-
-lemma (in finite_product_finite_prob_space) measure_finite_times:
- "(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))"
- by (rule measure_times) simp
-
-lemma (in finite_product_finite_prob_space) measure_singleton_times:
- assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})"
- unfolding singleton_eq_product[OF x] using x
- by (intro measure_finite_times) auto
-
-lemma (in finite_product_finite_prob_space) prob_finite_times:
- assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)"
- shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
-proof -
- have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
- using X by (intro finite_measure_eq[symmetric] in_P) auto
- also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
- using measure_finite_times X by simp
- also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
- using X by (simp add: M.finite_measure_eq setprod_ereal)
+ also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
+ using X by (simp add: M.emeasure_eq_measure setprod_ereal)
finally show ?thesis by simp
qed
-lemma (in finite_product_finite_prob_space) prob_singleton_times:
- assumes x: "x \<in> space P"
- shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})"
- unfolding singleton_eq_product[OF x] using x
- by (intro prob_finite_times) auto
-
-lemma (in finite_product_finite_prob_space) prob_finite_product:
- "A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})"
- by (auto simp add: finite_measure_singleton prob_singleton_times
- simp del: space_product_algebra
- intro!: setsum_cong prob_singleton_times)
+section {* Distributions *}
-lemma (in prob_space) joint_distribution_finite_prob_space:
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y"
- shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
- by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
-
-lemma finite_prob_space_eq:
- "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1"
- unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
- by auto
-
-lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
- using measure_space_1 sum_over_space by simp
+definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and>
+ f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
-lemma (in finite_prob_space) joint_distribution_restriction_fst:
- "joint_distribution X Y A \<le> distribution X (fst ` A)"
- unfolding distribution_def
-proof (safe intro!: finite_measure_mono)
- fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
- show "x \<in> X -` fst ` A"
- by (auto intro!: image_eqI[OF _ *])
-qed (simp_all add: sets_eq_Pow)
-
-lemma (in finite_prob_space) joint_distribution_restriction_snd:
- "joint_distribution X Y A \<le> distribution Y (snd ` A)"
- unfolding distribution_def
-proof (safe intro!: finite_measure_mono)
- fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
- show "x \<in> Y -` snd ` A"
- by (auto intro!: image_eqI[OF _ *])
-qed (simp_all add: sets_eq_Pow)
-
-lemma (in finite_prob_space) distribution_order:
- shows "0 \<le> distribution X x'"
- and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
- and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- using
- joint_distribution_restriction_fst[of X Y "{(x, y)}"]
- joint_distribution_restriction_snd[of X Y "{(x, y)}"]
- by (auto intro: antisym)
-
-lemma (in finite_prob_space) distribution_mono:
- assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
- shows "distribution X x \<le> distribution Y y"
- unfolding distribution_def
- using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
+lemma
+ shows distributed_distr_eq_density: "distributed M N X f \<Longrightarrow> distr M N X = density N f"
+ and distributed_measurable: "distributed M N X f \<Longrightarrow> X \<in> measurable M N"
+ and distributed_borel_measurable: "distributed M N X f \<Longrightarrow> f \<in> borel_measurable N"
+ and distributed_AE: "distributed M N X f \<Longrightarrow> (AE x in N. 0 \<le> f x)"
+ by (simp_all add: distributed_def)
-lemma (in finite_prob_space) distribution_mono_gt_0:
- assumes gt_0: "0 < distribution X x"
- assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
- shows "0 < distribution Y y"
- by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
-
-lemma (in finite_prob_space) sum_over_space_distrib:
- "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow
- intro!: arg_cong[where f=\<mu>'])
-
-lemma (in finite_prob_space) finite_sum_over_space_eq_1:
- "(\<Sum>x\<in>space M. prob {x}) = 1"
- using prob_space finite_space
- by (subst (asm) finite_measure_finite_singleton) auto
-
-lemma (in prob_space) distribution_remove_const:
- shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
- and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
- and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
- and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
- and "distribution (\<lambda>x. ()) {()} = 1"
- by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
+lemma
+ shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
+ and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
+ by (simp_all add: distributed_def borel_measurable_ereal_iff)
-lemma (in finite_prob_space) setsum_distribution_gen:
- assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
- and "inj_on f (X`space M)"
- shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
- unfolding distribution_def assms
- using finite_space assms
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
- intro!: arg_cong[where f=prob])
-
-lemma (in finite_prob_space) setsum_distribution_cut:
- "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
- "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
- "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
- "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
- "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
- by (auto intro!: inj_onI setsum_distribution_gen)
-
-lemma (in finite_prob_space) uniform_prob:
- assumes "x \<in> space M"
- assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
- shows "prob {x} = 1 / card (space M)"
+lemma distributed_count_space:
+ assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
+ shows "P a = emeasure M (X -` {a} \<inter> space M)"
proof -
- have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
- using assms(2)[OF _ `x \<in> space M`] by blast
- have "1 = prob (space M)"
- using prob_space by auto
- also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
- using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
- sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
- finite_space unfolding disjoint_family_on_def prob_space[symmetric]
- by (auto simp add:setsum_restrict_set)
- also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
- using prob_x by auto
- also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
- finally have one: "1 = real (card (space M)) * prob {x}"
- using real_eq_of_nat by auto
- hence two: "real (card (space M)) \<noteq> 0" by fastforce
- from one have three: "prob {x} \<noteq> 0" by fastforce
- thus ?thesis using one two three divide_cancel_right
- by (auto simp:field_simps)
+ have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
+ using X a A by (simp add: distributed_measurable emeasure_distr)
+ also have "\<dots> = emeasure (density (count_space A) P) {a}"
+ using X by (simp add: distributed_distr_eq_density)
+ also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
+ using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
+ also have "\<dots> = P a"
+ using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
+ finally show ?thesis ..
qed
-lemma (in prob_space) prob_space_subalgebra:
- assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M"
- and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
- shows "prob_space N"
-proof
- interpret N: measure_space N
- by (rule measure_space_subalgebra[OF assms])
- show "measure_space N" ..
- show "measure N (space N) = 1"
- using assms(4)[OF N.top] by (simp add: assms measure_space_1)
+lemma distributed_cong_density:
+ "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
+ distributed M N X f \<longleftrightarrow> distributed M N X g"
+ by (auto simp: distributed_def intro!: density_cong)
+
+lemma subdensity:
+ assumes T: "T \<in> measurable P Q"
+ assumes f: "distributed M P X f"
+ assumes g: "distributed M Q Y g"
+ assumes Y: "Y = T \<circ> X"
+ shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
+proof -
+ have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
+ using g Y by (auto simp: null_sets_density_iff distributed_def)
+ also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
+ using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
+ finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
+ using T by (subst (asm) null_sets_distr_iff) auto
+ also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
+ using T by (auto dest: measurable_space)
+ finally show ?thesis
+ using f g by (auto simp add: null_sets_density_iff distributed_def)
qed
-lemma (in prob_space) prob_space_of_restricted_space:
- assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
- shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
- (is "prob_space ?P")
+lemma subdensity_real:
+ fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
+ assumes T: "T \<in> measurable P Q"
+ assumes f: "distributed M P X f"
+ assumes g: "distributed M Q Y g"
+ assumes Y: "Y = T \<circ> X"
+ shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
+ using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
+
+lemma distributed_integral:
+ "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
+ by (auto simp: distributed_real_measurable distributed_real_AE distributed_measurable
+ distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
+
+lemma distributed_transform_integral:
+ assumes Px: "distributed M N X Px"
+ assumes "distributed M P Y Py"
+ assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
+ shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
proof -
- interpret A: measure_space "restricted_space A"
- using `A \<in> sets M` by (rule restricted_measure_space)
- interpret A': sigma_algebra ?P
- by (rule A.sigma_algebra_cong) auto
- show "prob_space ?P"
- proof
- show "measure_space ?P"
- proof
- show "positive ?P (measure ?P)"
- proof (simp add: positive_def, safe)
- fix B assume "B \<in> events"
- with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
- show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
- qed
- show "countably_additive ?P (measure ?P)"
- proof (simp add: countably_additive_def, safe)
- fix B and F :: "nat \<Rightarrow> 'a set"
- assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
- { fix i
- from F have "F i \<in> op \<inter> A ` events" by auto
- with `A \<in> events` have "F i \<in> events" by auto }
- moreover then have "range F \<subseteq> events" by auto
- moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
- by (simp add: mult_commute divide_ereal_def)
- moreover have "0 \<le> inverse (\<mu> A)"
- using real_measure[OF `A \<in> events`] by auto
- ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
- using measure_countably_additive[of F] F
- by (auto simp: suminf_cmult_ereal)
- qed
- qed
- show "measure ?P (space ?P) = 1"
- using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
- qed
-qed
-
-lemma finite_prob_spaceI:
- assumes "finite (space M)" "sets M = Pow(space M)"
- and 1: "measure M (space M) = 1" and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> measure M {x}"
- and add: "\<And>A B. A \<subseteq> space M \<Longrightarrow> measure M A = (\<Sum>x\<in>A. measure M {x})"
- shows "finite_prob_space M"
-proof -
- interpret finite_measure_space M
- proof
- show "measure M (space M) \<noteq> \<infinity>" using 1 by simp
- qed fact+
- show ?thesis by default fact
-qed
-
-lemma (in finite_prob_space) distribution_eq_setsum:
- "distribution X A = (\<Sum>x\<in>A \<inter> X ` space M. distribution X {x})"
-proof -
- have *: "X -` A \<inter> space M = (\<Union>x\<in>A \<inter> X ` space M. X -` {x} \<inter> space M)"
- by auto
- then show "distribution X A = (\<Sum>x\<in>A \<inter> X ` space M. distribution X {x})"
- using finite_space unfolding distribution_def *
- by (intro finite_measure_finite_Union)
- (auto simp: disjoint_family_on_def)
-qed
-
-lemma (in finite_prob_space) distribution_eq_setsum_finite:
- assumes "finite A"
- shows "distribution X A = (\<Sum>x\<in>A. distribution X {x})"
-proof -
- note distribution_eq_setsum[of X A]
- also have "(\<Sum>x\<in>A \<inter> X ` space M. distribution X {x}) = (\<Sum>x\<in>A. distribution X {x})"
- proof (intro setsum_mono_zero_cong_left ballI)
- fix i assume "i\<in>A - A \<inter> X ` space M"
- then have "X -` {i} \<inter> space M = {}" by auto
- then show "distribution X {i} = 0"
- by (simp add: distribution_def)
- next
- show "finite A" by fact
- qed simp_all
+ have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
+ by (rule distributed_integral) fact+
+ also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
+ using Y by simp
+ also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
+ using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
finally show ?thesis .
qed
-lemma (in finite_prob_space) finite_measure_space:
- fixes X :: "'a \<Rightarrow> 'x"
- shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X\<rparr>"
- (is "finite_measure_space ?S")
-proof (rule finite_measure_spaceI, simp_all)
- show "finite (X ` space M)" using finite_space by simp
-next
- fix A assume "A \<subseteq> X ` space M"
- then show "distribution X A = (\<Sum>x\<in>A. distribution X {x})"
- by (subst distribution_eq_setsum) (simp add: Int_absorb2)
+lemma distributed_marginal_eq_joint:
+ assumes T: "sigma_finite_measure T"
+ assumes S: "sigma_finite_measure S"
+ assumes Px: "distributed M S X Px"
+ assumes Py: "distributed M T Y Py"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
+proof (rule sigma_finite_measure.density_unique[OF T])
+ interpret ST: pair_sigma_finite S T using S T unfolding pair_sigma_finite_def by simp
+ show "Py \<in> borel_measurable T" "AE y in T. 0 \<le> Py y"
+ "(\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S) \<in> borel_measurable T" "AE y in T. 0 \<le> \<integral>\<^isup>+ x. Pxy (x, y) \<partial>S"
+ using Pxy[THEN distributed_borel_measurable]
+ by (auto intro!: Py[THEN distributed_borel_measurable] Py[THEN distributed_AE]
+ ST.positive_integral_snd_measurable' positive_integral_positive)
+
+ show "density T Py = density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)"
+ proof (rule measure_eqI)
+ fix A assume A: "A \<in> sets (density T Py)"
+ have *: "\<And>x y. x \<in> space S \<Longrightarrow> indicator (space S \<times> A) (x, y) = indicator A y"
+ by (auto simp: indicator_def)
+ have "emeasure (density T Py) A = emeasure (distr M T Y) A"
+ unfolding Py[THEN distributed_distr_eq_density] ..
+ also have "\<dots> = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (space S \<times> A)"
+ using A Px Py Pxy
+ by (subst (1 2) emeasure_distr)
+ (auto dest: measurable_space distributed_measurable intro!: arg_cong[where f="emeasure M"])
+ also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (space S \<times> A)"
+ unfolding Pxy[THEN distributed_distr_eq_density] ..
+ also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator (space S \<times> A) x \<partial>(S \<Otimes>\<^isub>M T))"
+ using A Pxy by (simp add: emeasure_density distributed_borel_measurable)
+ also have "\<dots> = (\<integral>\<^isup>+y. \<integral>\<^isup>+x. Pxy (x, y) * indicator (space S \<times> A) (x, y) \<partial>S \<partial>T)"
+ using A Pxy
+ by (subst ST.positive_integral_snd_measurable) (simp_all add: emeasure_density distributed_borel_measurable)
+ also have "\<dots> = (\<integral>\<^isup>+y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S) * indicator A y \<partial>T)"
+ using measurable_comp[OF measurable_Pair1[OF measurable_identity] distributed_borel_measurable[OF Pxy]]
+ using distributed_borel_measurable[OF Pxy] distributed_AE[OF Pxy, THEN ST.AE_pair]
+ by (subst (asm) ST.AE_commute) (auto intro!: positive_integral_cong_AE positive_integral_multc cong: positive_integral_cong simp: * comp_def)
+ also have "\<dots> = emeasure (density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)) A"
+ using A by (intro emeasure_density[symmetric]) (auto intro!: ST.positive_integral_snd_measurable' Pxy[THEN distributed_borel_measurable])
+ finally show "emeasure (density T Py) A = emeasure (density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)) A" .
+ qed simp
qed
-lemma (in finite_prob_space) finite_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X \<rparr>"
- by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_ereal_def)
+lemma (in prob_space) distr_marginal1:
+ fixes Pxy :: "('b \<times> 'c) \<Rightarrow> real"
+ assumes "sigma_finite_measure S" "sigma_finite_measure T"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ defines "Px \<equiv> \<lambda>x. real (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
+ shows "distributed M S X Px"
+ unfolding distributed_def
+proof safe
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret ST: pair_sigma_finite S T by default
+
+ have XY: "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
+ using Pxy by (rule distributed_measurable)
+ then show X: "X \<in> measurable M S"
+ unfolding measurable_pair_iff by (simp add: comp_def)
+ from XY have Y: "Y \<in> measurable M T"
+ unfolding measurable_pair_iff by (simp add: comp_def)
+
+ from Pxy show borel: "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S"
+ by (auto intro!: ST.positive_integral_fst_measurable borel_measurable_real_of_ereal dest!: distributed_real_measurable simp: Px_def)
-lemma (in finite_prob_space) finite_product_measure_space:
- fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
- assumes "finite s1" "finite s2"
- shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = ereal \<circ> joint_distribution X Y\<rparr>"
- (is "finite_measure_space ?M")
-proof (rule finite_measure_spaceI, simp_all)
- show "finite (s1 \<times> s2)"
- using assms by auto
-next
- fix A assume "A \<subseteq> (s1 \<times> s2)"
- with assms show "joint_distribution X Y A = (\<Sum>x\<in>A. joint_distribution X Y {x})"
- by (intro distribution_eq_setsum_finite) (auto dest: finite_subset)
-qed
-
-lemma (in finite_prob_space) finite_product_measure_space_of_images:
- shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
- sets = Pow (X ` space M \<times> Y ` space M),
- measure = ereal \<circ> joint_distribution X Y \<rparr>"
- using finite_space by (auto intro!: finite_product_measure_space)
-
-lemma (in finite_prob_space) finite_product_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M),
- measure = ereal \<circ> joint_distribution X Y \<rparr>"
- (is "finite_prob_space ?S")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_ereal_def)
- have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
- thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
- by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
+ interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+ using XY by (rule prob_space_distr)
+ have "(\<integral>\<^isup>+ x. max 0 (ereal (- Pxy x)) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
+ using Pxy
+ by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_real_measurable distributed_real_AE)
+ then have Pxy_integrable: "integrable (S \<Otimes>\<^isub>M T) Pxy"
+ using Pxy Pxy.emeasure_space_1
+ by (simp add: integrable_def emeasure_density positive_integral_max_0 distributed_def borel_measurable_ereal_iff cong: positive_integral_cong)
+
+ show "distr M S X = density S Px"
+ proof (rule measure_eqI)
+ fix A assume A: "A \<in> sets (distr M S X)"
+ with X Y XY have "emeasure (distr M S X) A = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (A \<times> space T)"
+ by (auto simp add: emeasure_distr
+ intro!: arg_cong[where f="emeasure M"] dest: measurable_space)
+ also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
+ using Pxy by (simp add: distributed_def)
+ also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
+ using A borel Pxy
+ by (simp add: emeasure_density ST.positive_integral_fst_measurable(2)[symmetric] distributed_def)
+ also have "\<dots> = \<integral>\<^isup>+ x. ereal (Px x) * indicator A x \<partial>S"
+ apply (rule positive_integral_cong_AE)
+ using Pxy[THEN distributed_real_AE, THEN ST.AE_pair] ST.integrable_fst_measurable(1)[OF Pxy_integrable] AE_space
+ proof eventually_elim
+ fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)" and i: "integrable T (\<lambda>y. Pxy (x, y))"
+ moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
+ by (auto simp: indicator_def)
+ ultimately have "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T) =
+ (\<integral>\<^isup>+ y. ereal (Pxy (x, y)) \<partial>T) * indicator A x"
+ using Pxy[THEN distributed_real_measurable] by (simp add: eq positive_integral_multc measurable_Pair2 cong: positive_integral_cong)
+ also have "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) \<partial>T) = Px x"
+ using i by (simp add: Px_def ereal_real integrable_def positive_integral_positive)
+ finally show "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T) = ereal (Px x) * indicator A x" .
+ qed
+ finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
+ using A borel Pxy by (simp add: emeasure_density)
+ qed simp
+
+ show "AE x in S. 0 \<le> ereal (Px x)"
+ by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
qed
-subsection "Borel Measure on {0 ..< 1}"
-
-definition pborel :: "real measure_space" where
- "pborel = lborel.restricted_space {0 ..< 1}"
-
-lemma space_pborel[simp]:
- "space pborel = {0 ..< 1}"
- unfolding pborel_def by auto
-
-lemma sets_pborel:
- "A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 ..< 1}"
- unfolding pborel_def by auto
+definition
+ "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
+ finite (X`space M)"
-lemma in_pborel[intro, simp]:
- "A \<subseteq> {0 ..< 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel"
- unfolding pborel_def by auto
-
-interpretation pborel: measure_space pborel
- using lborel.restricted_measure_space[of "{0 ..< 1}"]
- by (simp add: pborel_def)
+lemma simple_distributed:
+ "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
+ unfolding simple_distributed_def by auto
-interpretation pborel: prob_space pborel
-proof
- show "measure pborel (space pborel) = 1"
- by (simp add: one_ereal_def pborel_def)
-qed default
-
-lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 ..< 1} then real (lborel.\<mu> A) else 0)"
- unfolding pborel.\<mu>'_def by (auto simp: pborel_def)
+lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
+ by (simp add: simple_distributed_def)
-lemma pborel_singelton[simp]: "pborel.prob {a} = 0"
- by (auto simp: pborel_prob)
-
-lemma
- shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
- and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
- and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
- and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
- unfolding pborel_prob
- by (auto simp: atLeastAtMost_subseteq_atLeastLessThan_iff
- greaterThanAtMost_subseteq_atLeastLessThan_iff greaterThanLessThan_subseteq_atLeastLessThan_iff)
-
-lemma pborel_lebesgue_measure:
- "A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)"
- by (simp add: sets_pborel pborel_prob)
+lemma (in prob_space) distributed_simple_function_superset:
+ assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
+ assumes A: "X`space M \<subseteq> A" "finite A"
+ defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
+ shows "distributed M S X P'"
+ unfolding distributed_def
+proof safe
+ show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
+ show "AE x in S. 0 \<le> ereal (P' x)"
+ using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
+ show "distr M S X = density S P'"
+ proof (rule measure_eqI_finite)
+ show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
+ using A unfolding S_def by auto
+ show "finite A" by fact
+ fix a assume a: "a \<in> A"
+ then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
+ with A a X have "emeasure (distr M S X) {a} = P' a"
+ by (subst emeasure_distr)
+ (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure
+ intro!: arg_cong[where f=prob])
+ also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
+ using A X a
+ by (subst positive_integral_cmult_indicator)
+ (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
+ also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
+ by (auto simp: indicator_def intro!: positive_integral_cong)
+ also have "\<dots> = emeasure (density S P') {a}"
+ using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
+ finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
+ qed
+ show "random_variable S X"
+ using X(1) A by (auto simp: measurable_def simple_functionD S_def)
+qed
-lemma pborel_alt:
- "pborel = sigma \<lparr>
- space = {0..<1},
- sets = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1}),
- measure = measure lborel \<rparr>" (is "_ = ?R")
+lemma (in prob_space) simple_distributedI:
+ assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
+ shows "simple_distributed M X P"
+ unfolding simple_distributed_def
+proof
+ have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
+ (is "?A")
+ using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
+ also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
+ by (rule distributed_cong_density) auto
+ finally show "\<dots>" .
+qed (rule simple_functionD[OF X(1)])
+
+lemma simple_distributed_joint_finite:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
+ shows "finite (X ` space M)" "finite (Y ` space M)"
proof -
- have *: "{0..<1::real} \<in> sets borel" by auto
- have **: "op \<inter> {0..<1::real} ` range (\<lambda>(x, y). {x..<y}) = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1})"
- unfolding image_image by (intro arg_cong[where f=range]) auto
- have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real}),
- measure = measure pborel\<rparr>) {0 ..< 1}"
- by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastLessThan lborel_def)
- also have "\<dots> = ?R"
- by (subst restricted_sigma)
- (simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"])
- finally show ?thesis .
+ have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
+ using X by (auto simp: simple_distributed_def simple_functionD)
+ then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
+ by auto
+ then show fin: "finite (X ` space M)" "finite (Y ` space M)"
+ by (auto simp: image_image)
+qed
+
+lemma simple_distributed_joint2_finite:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
+ shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
+proof -
+ have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ using X by (auto simp: simple_distributed_def simple_functionD)
+ then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ by auto
+ then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
+ by (auto simp: image_image)
qed
-subsection "Bernoulli space"
+lemma simple_distributed_simple_function:
+ "simple_distributed M X Px \<Longrightarrow> simple_function M X"
+ unfolding simple_distributed_def distributed_def
+ by (auto simp: simple_function_def)
+
+lemma simple_distributed_measure:
+ "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
+ using distributed_count_space[of M "X`space M" X P a, symmetric]
+ by (auto simp: simple_distributed_def measure_def)
+
+lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
+ by (auto simp: simple_distributed_measure measure_nonneg)
-definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV,
- measure = ereal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>"
+lemma (in prob_space) simple_distributed_joint:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
+ defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
+ defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
+ shows "distributed M S (\<lambda>x. (X x, Y x)) P"
+proof -
+ from simple_distributed_joint_finite[OF X, simp]
+ have S_eq: "S = count_space (X`space M \<times> Y`space M)"
+ by (simp add: S_def pair_measure_count_space)
+ show ?thesis
+ unfolding S_eq P_def
+ proof (rule distributed_simple_function_superset)
+ show "simple_function M (\<lambda>x. (X x, Y x))"
+ using X by (rule simple_distributed_simple_function)
+ fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
+ from simple_distributed_measure[OF X this]
+ show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
+ qed auto
+qed
-interpretation bernoulli: finite_prob_space "bernoulli_space p" for p
- by (rule finite_prob_spaceI)
- (auto simp: bernoulli_space_def UNIV_bool one_ereal_def setsum_Un_disjoint intro!: setsum_nonneg)
+lemma (in prob_space) simple_distributed_joint2:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
+ defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
+ defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
+ shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
+proof -
+ from simple_distributed_joint2_finite[OF X, simp]
+ have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
+ by (simp add: S_def pair_measure_count_space)
+ show ?thesis
+ unfolding S_eq P_def
+ proof (rule distributed_simple_function_superset)
+ show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
+ using X by (rule simple_distributed_simple_function)
+ fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
+ from simple_distributed_measure[OF X this]
+ show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
+ qed auto
+qed
+
+lemma (in prob_space) simple_distributed_setsum_space:
+ assumes X: "simple_distributed M X f"
+ shows "setsum f (X`space M) = 1"
+proof -
+ from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
+ by (subst finite_measure_finite_Union)
+ (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
+ intro!: setsum_cong arg_cong[where f="prob"])
+ also have "\<dots> = prob (space M)"
+ by (auto intro!: arg_cong[where f=prob])
+ finally show ?thesis
+ using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
+qed
-lemma bernoulli_measure:
- "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)"
- unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong)
+lemma (in prob_space) distributed_marginal_eq_joint_simple:
+ assumes Px: "simple_function M X"
+ assumes Py: "simple_distributed M Y Py"
+ assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+ assumes y: "y \<in> Y`space M"
+ shows "Py y = (\<Sum>x\<in>X`space M. if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
+proof -
+ note Px = simple_distributedI[OF Px refl]
+ have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
+ by (simp add: setsum_ereal[symmetric] zero_ereal_def)
+ from distributed_marginal_eq_joint[OF sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite
+ simple_distributed[OF Px] simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
+ OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
+ y Px[THEN simple_distributed_finite] Py[THEN simple_distributed_finite]
+ Pxy[THEN simple_distributed, THEN distributed_real_AE]
+ show ?thesis
+ unfolding AE_count_space
+ apply (elim ballE[where x=y])
+ apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
+ done
+qed
-lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p"
- and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1 - p"
- unfolding bernoulli_measure by simp_all
+
+lemma prob_space_uniform_measure:
+ assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
+ shows "prob_space (uniform_measure M A)"
+proof
+ show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
+ using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
+ using sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
+ by (simp add: Int_absorb2 emeasure_nonneg)
+qed
+
+lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
+ by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
end
--- a/src/HOL/Probability/Radon_Nikodym.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy Mon Apr 23 12:14:35 2012 +0200
@@ -8,45 +8,79 @@
imports Lebesgue_Integration
begin
+definition "diff_measure M N =
+ measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)"
+
+lemma
+ shows space_diff_measure[simp]: "space (diff_measure M N) = space M"
+ and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M"
+ by (auto simp: diff_measure_def)
+
+lemma emeasure_diff_measure:
+ assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N"
+ assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M"
+ shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A")
+ unfolding diff_measure_def
+proof (rule emeasure_measure_of_sigma)
+ show "sigma_algebra (space M) (sets M)" ..
+ show "positive (sets M) ?\<mu>"
+ using pos by (simp add: positive_def ereal_diff_positive)
+ show "countably_additive (sets M) ?\<mu>"
+ proof (rule countably_additiveI)
+ fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> sets M" and "disjoint_family A"
+ then have suminf:
+ "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
+ "(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)"
+ by (simp_all add: suminf_emeasure sets_eq)
+ with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
+ (\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))"
+ using fin
+ by (intro suminf_ereal_minus pos emeasure_nonneg)
+ (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure)
+ then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
+ emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) "
+ by (simp add: suminf)
+ qed
+qed fact
+
lemma (in sigma_finite_measure) Ex_finite_integrable_function:
shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)"
proof -
obtain A :: "nat \<Rightarrow> 'a set" where
range: "range A \<subseteq> sets M" and
space: "(\<Union>i. A i) = space M" and
- measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>" and
+ measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and
disjoint: "disjoint_family A"
- using disjoint_sigma_finite by auto
- let ?B = "\<lambda>i. 2^Suc i * \<mu> (A i)"
+ using sigma_finite_disjoint by auto
+ let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)"
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
proof
- fix i have Ai: "A i \<in> sets M" using range by auto
- from measure positive_measure[OF this]
- show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
- by (auto intro!: ereal_dense simp: ereal_0_gt_inverse)
+ fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
+ using measure[of i] emeasure_nonneg[of M "A i"]
+ by (auto intro!: ereal_dense simp: ereal_0_gt_inverse ereal_zero_le_0_iff)
qed
from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
- "\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto
+ "\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto
{ fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x"
show ?thesis
proof (safe intro!: bexI[of _ ?h] del: notI)
have "\<And>i. A i \<in> sets M"
using range by fastforce+
- then have "integral\<^isup>P M ?h = (\<Sum>i. n i * \<mu> (A i))" using pos
+ then have "integral\<^isup>P M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos
by (simp add: positive_integral_suminf positive_integral_cmult_indicator)
also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)"
proof (rule suminf_le_pos)
fix N
- have "n N * \<mu> (A N) \<le> inverse (2^Suc N * \<mu> (A N)) * \<mu> (A N)"
- using positive_measure[OF `A N \<in> sets M`] n[of N]
+ have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)"
+ using n[of N]
by (intro ereal_mult_right_mono) auto
also have "\<dots> \<le> (1 / 2) ^ Suc N"
using measure[of N] n[of N]
- by (cases rule: ereal2_cases[of "n N" "\<mu> (A N)"])
+ by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"])
(simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
- finally show "n N * \<mu> (A N) \<le> (1 / 2) ^ Suc N" .
- show "0 \<le> n N * \<mu> (A N)" using n[of N] `A N \<in> sets M` by simp
+ finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" .
+ show "0 \<le> n N * emeasure M (A N)" using n[of N] `A N \<in> sets M` by (simp add: emeasure_nonneg)
qed
finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
next
@@ -71,68 +105,45 @@
subsection "Absolutely continuous"
-definition (in measure_space)
- "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: ereal))"
+definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
+ "absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N"
+
+lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M"
+ unfolding absolutely_continuous_def by (auto simp: null_sets_count_space)
-lemma (in measure_space) absolutely_continuous_AE:
- assumes "measure_space M'" and [simp]: "sets M' = sets M" "space M' = space M"
- and "absolutely_continuous (measure M')" "AE x. P x"
+lemma absolutely_continuousI_density:
+ "f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)"
+ by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in)
+
+lemma absolutely_continuousI_point_measure_finite:
+ "(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)"
+ unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff)
+
+lemma absolutely_continuous_AE:
+ assumes sets_eq: "sets M' = sets M"
+ and "absolutely_continuous M M'" "AE x in M. P x"
shows "AE x in M'. P x"
proof -
- interpret \<nu>: measure_space M' by fact
- from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
- unfolding almost_everywhere_def by auto
+ from `AE x in M. P x` obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
+ unfolding eventually_ae_filter by auto
show "AE x in M'. P x"
- proof (rule \<nu>.AE_I')
- show "{x\<in>space M'. \<not> P x} \<subseteq> N" by simp fact
- from `absolutely_continuous (measure M')` show "N \<in> \<nu>.null_sets"
- using N unfolding absolutely_continuous_def by auto
+ proof (rule AE_I')
+ show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
+ from `absolutely_continuous M M'` show "N \<in> null_sets M'"
+ using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
qed
qed
-lemma (in finite_measure_space) absolutely_continuousI:
- assumes "finite_measure_space (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure_space ?\<nu>")
- assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
- shows "absolutely_continuous \<nu>"
-proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
- fix N assume "\<mu> N = 0" "N \<subseteq> space M"
- interpret v: finite_measure_space ?\<nu> by fact
- have "\<nu> N = measure ?\<nu> (\<Union>x\<in>N. {x})" by simp
- also have "\<dots> = (\<Sum>x\<in>N. measure ?\<nu> {x})"
- proof (rule v.measure_setsum[symmetric])
- show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
- show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
- fix x assume "x \<in> N" thus "{x} \<in> sets ?\<nu>" using `N \<subseteq> space M` sets_eq_Pow by auto
- qed
- also have "\<dots> = 0"
- proof (safe intro!: setsum_0')
- fix x assume "x \<in> N"
- hence "\<mu> {x} \<le> \<mu> N" "0 \<le> \<mu> {x}"
- using sets_eq_Pow `N \<subseteq> space M` positive_measure[of "{x}"]
- by (auto intro!: measure_mono)
- then have "\<mu> {x} = 0" using `\<mu> N = 0` by simp
- thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
- qed
- finally show "\<nu> N = 0" by simp
-qed
-
-lemma (in measure_space) density_is_absolutely_continuous:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
- shows "absolutely_continuous \<nu>"
- using assms unfolding absolutely_continuous_def
- by (simp add: positive_integral_null_set)
-
subsection "Existence of the Radon-Nikodym derivative"
lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
fixes e :: real assumes "0 < e"
- assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
- shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
- \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
- (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
+ assumes "finite_measure N" and sets_eq: "sets N = sets M"
+ shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le> measure M A - measure N A \<and>
+ (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < measure M B - measure N B)"
proof -
- interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
- let ?d = "\<lambda>A. \<mu>' A - M'.\<mu>' A"
+ interpret M': finite_measure N by fact
+ let ?d = "\<lambda>A. measure M A - measure N A"
let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
then {}
else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
@@ -159,7 +170,7 @@
fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
hence "?d (A n \<union> B) = ?d (A n) + ?d B"
- using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by simp
+ using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq)
also have "\<dots> \<le> ?d (A n) - e" using dB by simp
finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
qed }
@@ -187,8 +198,8 @@
proof (induct n)
fix n assume "?d (space M) \<le> ?d (space M - A n)"
also have "\<dots> \<le> ?d (space M - A (Suc n))"
- using A_in_sets sets_into_space dA_mono[of n]
- by (simp del: A_simps add: finite_measure_Diff M'.finite_measure_Diff)
+ using A_in_sets sets_into_space dA_mono[of n] finite_measure_compl M'.finite_measure_compl
+ by (simp del: A_simps add: sets_eq sets_eq_imp_space_eq[OF sets_eq])
finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
qed simp
qed
@@ -199,17 +210,17 @@
proof (induct n)
case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
next
- case 0 with M'.empty_measure show ?case by (simp add: zero_ereal_def)
+ case 0 with measure_empty show ?case by (simp add: zero_ereal_def)
qed } note dA_less = this
have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
proof (rule incseq_SucI)
fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
qed
have A: "incseq A" by (auto intro!: incseq_SucI)
- from finite_continuity_from_below[OF _ A] `range A \<subseteq> sets M`
- M'.finite_continuity_from_below[OF _ A]
+ from finite_Lim_measure_incseq[OF _ A] `range A \<subseteq> sets M`
+ M'.finite_Lim_measure_incseq[OF _ A]
have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
- by (auto intro!: tendsto_diff)
+ by (auto intro!: tendsto_diff simp: sets_eq)
obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
@@ -217,57 +228,24 @@
qed
qed
-lemma (in finite_measure) restricted_measure_subset:
- assumes S: "S \<in> sets M" and X: "X \<subseteq> S"
- shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
-proof cases
- note r = restricted_finite_measure[OF S]
- { assume "X \<in> sets M" with S X show ?thesis
- unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def by auto }
- { assume "X \<notin> sets M"
- moreover then have "S \<inter> X \<notin> sets M"
- using X by (simp add: Int_absorb1)
- ultimately show ?thesis
- unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def using S by auto }
-qed
-
-lemma (in finite_measure) restricted_measure:
- assumes X: "S \<in> sets M" "X \<in> sets (restricted_space S)"
- shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
+lemma (in finite_measure) Radon_Nikodym_aux:
+ assumes "finite_measure N" and sets_eq: "sets N = sets M"
+ shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le>
+ measure M A - measure N A \<and>
+ (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> measure M B - measure N B)"
proof -
- from X have "S \<in> sets M" "X \<subseteq> S" by auto
- from restricted_measure_subset[OF this] show ?thesis .
-qed
-
-lemma (in finite_measure) Radon_Nikodym_aux:
- assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
- shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
- \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
- (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
-proof -
- interpret M': finite_measure ?M' where
- "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
- let ?d = "\<lambda>A. \<mu>' A - M'.\<mu>' A"
+ interpret N: finite_measure N by fact
+ let ?d = "\<lambda>A. measure M A - measure N A"
let ?P = "\<lambda>A B n. A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
let ?r = "\<lambda>S. restricted_space S"
{ fix S n assume S: "S \<in> sets M"
- note r = M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
- then have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
- "finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
+ then have "finite_measure (density M (indicator S))" "0 < 1 / real (Suc n)"
+ "finite_measure (density N (indicator S))" "sets (density N (indicator S)) = sets (density M (indicator S))"
+ by (auto simp: finite_measure_restricted N.finite_measure_restricted sets_eq)
from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
- have "?P X S n"
- proof (intro conjI ballI impI)
- show "X \<in> sets M" "X \<subseteq> S" using X(1) `S \<in> sets M` by auto
- have "S \<in> op \<inter> S ` sets M" using `S \<in> sets M` by auto
- then show "?d S \<le> ?d X"
- using S X restricted_measure[OF S] M'.restricted_measure[OF S] by simp
- fix C assume "C \<in> sets M" "C \<subseteq> X"
- then have "C \<in> sets (restricted_space S)" "C \<subseteq> X"
- using `S \<in> sets M` `X \<subseteq> S` by auto
- with X(2) show "- 1 / real (Suc n) < ?d C"
- using S X restricted_measure[OF S] M'.restricted_measure[OF S] by auto
- qed
- hence "\<exists>A. ?P A S n" by auto }
+ with S have "?P (S \<inter> X) S n"
+ by (simp add: measure_restricted sets_eq Int) (metis inf_absorb2)
+ hence "\<exists>A. ?P A S n" .. }
note Ex_P = this
def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
@@ -292,12 +270,11 @@
proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
have A: "decseq A" using A_mono by (auto intro!: decseq_SucI)
- from
- finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
- M'.finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
- have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (intro tendsto_diff)
+ from `range A \<subseteq> sets M`
+ finite_Lim_measure_decseq[OF _ A] N.finite_Lim_measure_decseq[OF _ A]
+ have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq)
thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
- by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
+ by (rule_tac LIMSEQ_le_const) auto
next
fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
show "0 \<le> ?d B"
@@ -315,14 +292,12 @@
qed
lemma (in finite_measure) Radon_Nikodym_finite_measure:
- assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
- assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ assumes "finite_measure N" and sets_eq: "sets N = sets M"
+ assumes "absolutely_continuous M N"
+ shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N"
proof -
- interpret M': finite_measure ?M'
- where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
- using assms(1) by auto
- def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A)}"
+ interpret N: finite_measure N by fact
+ def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A)}"
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
hence "G \<noteq> {}" by auto
{ fix f g assume f: "f \<in> G" and g: "g \<in> G"
@@ -333,6 +308,7 @@
have "?A \<in> sets M" using f g unfolding G_def by auto
fix A assume "A \<in> sets M"
hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
+ hence sets': "?A \<inter> A \<in> sets N" "(space M - ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq)
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
using sets_into_space[OF `A \<in> sets M`] by auto
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
@@ -343,11 +319,11 @@
(\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
using f g sets unfolding G_def
by (auto cong: positive_integral_cong intro!: positive_integral_add)
- also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
+ also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)"
using f g sets unfolding G_def by (auto intro!: add_mono)
- also have "\<dots> = \<nu> A"
- using M'.measure_additive[OF sets] union by auto
- finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
+ also have "\<dots> = N A"
+ using plus_emeasure[OF sets'] union by auto
+ finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" .
next
fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max)
qed }
@@ -368,15 +344,15 @@
using `incseq f` f `A \<in> sets M`
by (intro positive_integral_monotone_convergence_SUP)
(auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
- finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> \<nu> A"
+ finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
qed }
note SUP_in_G = this
let ?y = "SUP g : G. integral\<^isup>P M g"
- have "?y \<le> \<nu> (space M)" unfolding G_def
+ have y_le: "?y \<le> N (space M)" unfolding G_def
proof (safe intro!: SUP_least)
- fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A"
- from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
+ fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A"
+ from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> N (space M)"
by (simp cong: positive_integral_cong)
qed
from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this
@@ -411,8 +387,7 @@
also have "\<dots> = ?y"
proof (rule antisym)
show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
- using g_in_G
- by (auto intro!: exI Sup_mono simp: SUP_def)
+ using g_in_G by (auto intro: Sup_mono simp: SUP_def)
show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
by (auto intro!: SUP_mono positive_integral_mono Max_ge)
qed
@@ -420,107 +395,68 @@
have "\<And>x. 0 \<le> f x"
unfolding f_def using `\<And>i. gs i \<in> G`
by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
- let ?t = "\<lambda>A. \<nu> A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
- let ?M = "M\<lparr> measure := ?t\<rparr>"
- interpret M: sigma_algebra ?M
- by (intro sigma_algebra_cong) auto
- have f_le_\<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> \<nu> A"
+ let ?t = "\<lambda>A. N A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
+ let ?M = "diff_measure N (density M f)"
+ have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A"
using `f \<in> G` unfolding G_def by auto
- have fmM: "finite_measure ?M"
+ have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A"
+ proof (subst emeasure_diff_measure)
+ from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)"
+ by (auto intro!: finite_measureI simp: emeasure_density cong: positive_integral_cong)
+ next
+ fix B assume "B \<in> sets N" with f_le_N[of B] show "emeasure (density M f) B \<le> emeasure N B"
+ by (auto simp: sets_eq emeasure_density cong: positive_integral_cong)
+ qed (auto simp: sets_eq emeasure_density)
+ from emeasure_M[of "space M"] N.finite_emeasure_space positive_integral_positive[of M "?F (space M)"]
+ interpret M': finite_measure ?M
+ by (auto intro!: finite_measureI simp: sets_eq_imp_space_eq[OF sets_eq] N.emeasure_eq_measure ereal_minus_eq_PInfty_iff)
+
+ have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def
proof
- show "measure_space ?M"
- proof (default, simp_all add: countably_additive_def positive_def, safe del: notI)
- fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A"
- have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. (\<Sum>n. ?F (A n) x) \<partial>M)"
- using `range A \<subseteq> sets M` `\<And>x. 0 \<le> f x`
- by (intro positive_integral_suminf[symmetric]) auto
- also have "\<dots> = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)"
- using `\<And>x. 0 \<le> f x`
- by (intro positive_integral_cong) (simp add: suminf_cmult_ereal suminf_indicator[OF `disjoint_family A`])
- finally have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)" .
- moreover have "(\<Sum>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
- using M'.measure_countably_additive A by (simp add: comp_def)
- moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<infinity>" using M'.finite_measure A by (simp add: countable_UN)
- moreover {
- have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
- using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
- also have "\<nu> (\<Union>i. A i) < \<infinity>" using v_fin by simp
- finally have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<noteq> \<infinity>" by simp }
- moreover have "\<And>i. (\<integral>\<^isup>+x. ?F (A i) x \<partial>M) \<le> \<nu> (A i)"
- using A by (intro f_le_\<nu>) auto
- ultimately
- show "(\<Sum>n. ?t (A n)) = ?t (\<Union>i. A i)"
- by (subst suminf_ereal_minus) (simp_all add: positive_integral_positive)
- next
- fix A assume A: "A \<in> sets M" show "0 \<le> \<nu> A - \<integral>\<^isup>+ x. ?F A x \<partial>M"
- using f_le_\<nu>[OF A] `f \<in> G` M'.finite_measure[OF A] by (auto simp: G_def ereal_le_minus_iff)
- qed
- next
- show "measure ?M (space ?M) \<noteq> \<infinity>"
- using positive_integral_positive[of "?F (space M)"]
- by (cases rule: ereal2_cases[of "\<nu> (space M)" "\<integral>\<^isup>+ x. ?F (space M) x \<partial>M"]) auto
+ fix A assume A: "A \<in> null_sets M"
+ with `absolutely_continuous M N` have "A \<in> null_sets N"
+ unfolding absolutely_continuous_def by auto
+ moreover with A have "(\<integral>\<^isup>+ x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` by (auto simp: G_def)
+ ultimately have "N A - (\<integral>\<^isup>+ x. ?F A x \<partial>M) = 0"
+ using positive_integral_positive[of M] by (auto intro!: antisym)
+ then show "A \<in> null_sets ?M"
+ using A by (simp add: emeasure_M null_sets_def sets_eq)
qed
- then interpret M: finite_measure ?M
- where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
- by (simp_all add: fmM)
- have ac: "absolutely_continuous ?t" unfolding absolutely_continuous_def
- proof
- fix N assume N: "N \<in> null_sets"
- with `absolutely_continuous \<nu>` have "\<nu> N = 0" unfolding absolutely_continuous_def by auto
- moreover with N have "(\<integral>\<^isup>+ x. ?F N x \<partial>M) \<le> \<nu> N" using `f \<in> G` by (auto simp: G_def)
- ultimately show "\<nu> N - (\<integral>\<^isup>+ x. ?F N x \<partial>M) = 0"
- using positive_integral_positive by (auto intro!: antisym)
- qed
- have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
+ have upper_bound: "\<forall>A\<in>sets M. ?M A \<le> 0"
proof (rule ccontr)
assume "\<not> ?thesis"
- then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A"
+ then obtain A where A: "A \<in> sets M" and pos: "0 < ?M A"
by (auto simp: not_le)
note pos
- also have "?t A \<le> ?t (space M)"
- using M.measure_mono[of A "space M"] A sets_into_space by simp
- finally have pos_t: "0 < ?t (space M)" by simp
+ also have "?M A \<le> ?M (space M)"
+ using emeasure_space[of ?M A] by (simp add: sets_eq[THEN sets_eq_imp_space_eq])
+ finally have pos_t: "0 < ?M (space M)" by simp
moreover
- then have "\<mu> (space M) \<noteq> 0"
- using ac unfolding absolutely_continuous_def by auto
- then have pos_M: "0 < \<mu> (space M)"
- using positive_measure[OF top] by (simp add: le_less)
+ then have "emeasure M (space M) \<noteq> 0"
+ using ac unfolding absolutely_continuous_def by (auto simp: null_sets_def)
+ then have pos_M: "0 < emeasure M (space M)"
+ using emeasure_nonneg[of M "space M"] by (simp add: le_less)
moreover
- have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
+ have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)"
using `f \<in> G` unfolding G_def by auto
hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
- using M'.finite_measure_of_space by auto
+ using M'.finite_emeasure_space by auto
moreover
- def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
+ def b \<equiv> "?M (space M) / emeasure M (space M) / 2"
ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>"
- using M'.finite_measure_of_space positive_integral_positive[of "?F (space M)"]
- by (cases rule: ereal3_cases[of "integral\<^isup>P M (?F (space M))" "\<nu> (space M)" "\<mu> (space M)"])
- (simp_all add: field_simps)
+ by (auto simp: ereal_divide_eq)
then have b: "b \<noteq> 0" "0 \<le> b" "0 < b" "b \<noteq> \<infinity>" by auto
- let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
- interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
- have Mb: "finite_measure ?Mb"
- proof
- show "measure_space ?Mb"
- proof
- show "positive ?Mb (measure ?Mb)"
- using `0 \<le> b` by (auto simp: positive_def)
- show "countably_additive ?Mb (measure ?Mb)"
- using `0 \<le> b` measure_countably_additive
- by (auto simp: countably_additive_def suminf_cmult_ereal subset_eq)
- qed
- show "measure ?Mb (space ?Mb) \<noteq> \<infinity>"
- using b by auto
- qed
- from M.Radon_Nikodym_aux[OF this]
- obtain A0 where "A0 \<in> sets M" and
- space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
- *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)"
- unfolding M.\<mu>'_def finite_measure.\<mu>'_def[OF Mb] by auto
+ let ?Mb = "density M (\<lambda>_. b)"
+ have Mb: "finite_measure ?Mb" "sets ?Mb = sets ?M"
+ using b by (auto simp: emeasure_density_const sets_eq intro!: finite_measureI)
+ from M'.Radon_Nikodym_aux[OF this] guess A0 ..
+ then have "A0 \<in> sets M"
+ and space_less_A0: "measure ?M (space M) - real b * measure M (space M) \<le> measure ?M A0 - real b * measure M A0"
+ and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B - real b * measure M B"
+ using b by (simp_all add: measure_density_const sets_eq_imp_space_eq[OF sets_eq] sets_eq)
{ fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
- with *[OF this] have "b * \<mu> B \<le> ?t B"
- using M'.finite_measure b finite_measure M.positive_measure[OF B(1)]
- by (cases rule: ereal2_cases[of "?t B" "b * \<mu> B"]) auto }
+ with *[OF this] have "b * emeasure M B \<le> ?M B"
+ using b unfolding M'.emeasure_eq_measure emeasure_eq_measure by (cases b) auto }
note bM_le_t = this
let ?f0 = "\<lambda>x. f x + b * indicator A0 x"
{ fix A assume A: "A \<in> sets M"
@@ -529,71 +465,47 @@
(\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
by (auto intro!: positive_integral_cong split: split_indicator)
hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
- (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
+ (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)"
using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
by (simp add: G_def positive_integral_add positive_integral_cmult_indicator) }
note f0_eq = this
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
- have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
- using `f \<in> G` A unfolding G_def by auto
+ have f_le_v: "(\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` A unfolding G_def by auto
note f0_eq[OF A]
- also have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0) \<le>
- (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t (A \<inter> A0)"
+ also have "(\<integral>\<^isup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^isup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)"
using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
by (auto intro!: add_left_mono)
- also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t A"
- using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
- by (auto intro!: add_left_mono)
- also have "\<dots> \<le> \<nu> A"
- using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`] positive_integral_positive[of "?F A"]
- by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "\<nu> A") auto
- finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
+ also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?M A"
+ using emeasure_mono[of "A \<inter> A0" A ?M] `A \<in> sets M` `A0 \<in> sets M`
+ by (auto intro!: add_left_mono simp: sets_eq)
+ also have "\<dots> \<le> N A"
+ unfolding emeasure_M[OF `A \<in> sets M`]
+ using f_le_v N.emeasure_eq_measure[of A] positive_integral_positive[of M "?F A"]
+ by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "N A") auto
+ finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . }
hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` unfolding G_def
by (auto intro!: ereal_add_nonneg_nonneg)
- have real: "?t (space M) \<noteq> \<infinity>" "?t A0 \<noteq> \<infinity>"
- "b * \<mu> (space M) \<noteq> \<infinity>" "b * \<mu> A0 \<noteq> \<infinity>"
- using `A0 \<in> sets M` b
- finite_measure[of A0] M.finite_measure[of A0]
- finite_measure_of_space M.finite_measure_of_space
- by auto
have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>"
- using M'.finite_measure_of_space pos_t unfolding ereal_less_minus_iff
- by (auto cong: positive_integral_cong)
- have "0 < ?t (space M) - b * \<mu> (space M)" unfolding b_def
- using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
- using positive_integral_positive[of "?F (space M)"]
- by (cases rule: ereal3_cases[of "\<mu> (space M)" "\<nu> (space M)" "integral\<^isup>P M (?F (space M))"])
- (auto simp: field_simps mult_less_cancel_left)
- also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
- using space_less_A0 b
- using
- `A0 \<in> sets M`[THEN M.real_measure]
- top[THEN M.real_measure]
- apply safe
- apply simp
- using
- `A0 \<in> sets M`[THEN real_measure]
- `A0 \<in> sets M`[THEN M'.real_measure]
- top[THEN real_measure]
- top[THEN M'.real_measure]
- by (cases b) auto
- finally have 1: "b * \<mu> A0 < ?t A0"
- using
- `A0 \<in> sets M`[THEN M.real_measure]
- apply safe
- apply simp
- using
- `A0 \<in> sets M`[THEN real_measure]
- `A0 \<in> sets M`[THEN M'.real_measure]
- by (cases b) auto
- have "0 < ?t A0"
- using b `A0 \<in> sets M` by (auto intro!: le_less_trans[OF _ 1])
- then have "\<mu> A0 \<noteq> 0" using ac unfolding absolutely_continuous_def
- using `A0 \<in> sets M` by auto
- then have "0 < \<mu> A0" using positive_measure[OF `A0 \<in> sets M`] by auto
- hence "0 < b * \<mu> A0" using b by (auto simp: ereal_zero_less_0_iff)
- with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
+ by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le)
+ have "0 < ?M (space M) - emeasure ?Mb (space M)"
+ using pos_t
+ by (simp add: b emeasure_density_const)
+ (simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def)
+ also have "\<dots> \<le> ?M A0 - b * emeasure M A0"
+ using space_less_A0 `A0 \<in> sets M` b
+ by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure)
+ finally have 1: "b * emeasure M A0 < ?M A0"
+ by (metis M'.emeasure_real `A0 \<in> sets M` bM_le_t diff_self ereal_less(1) ereal_minus(1)
+ less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def)
+ with b have "0 < ?M A0"
+ by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times
+ ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def)
+ then have "emeasure M A0 \<noteq> 0" using ac `A0 \<in> sets M`
+ by (auto simp: absolutely_continuous_def null_sets_def)
+ then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto
+ hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff)
+ with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * emeasure M A0" unfolding int_f_eq_y
using `f \<in> G`
by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive)
also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
@@ -602,75 +514,67 @@
moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
ultimately show False by auto
qed
+ let ?f = "\<lambda>x. max 0 (f x)"
show ?thesis
- proof (safe intro!: bexI[of _ f])
- fix A assume A: "A\<in>sets M"
- show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
- proof (rule antisym)
- show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
- using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
- show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
- using upper_bound[THEN bspec, OF `A \<in> sets M`]
- using M'.real_measure[OF A]
- by (cases "integral\<^isup>P M (?F A)") auto
- qed
- qed simp
+ proof (intro bexI[of _ ?f] measure_eqI conjI)
+ show "sets (density M ?f) = sets N"
+ by (simp add: sets_eq)
+ fix A assume A: "A\<in>sets (density M ?f)"
+ then show "emeasure (density M ?f) A = emeasure N A"
+ using `f \<in> G` A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A]
+ by (cases "integral\<^isup>P M (?F A)")
+ (auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric])
+ qed auto
qed
lemma (in finite_measure) split_space_into_finite_sets_and_rest:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
- assumes ac: "absolutely_continuous \<nu>"
+ assumes ac: "absolutely_continuous M N" and sets_eq: "sets N = sets M"
shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M - (\<Union>i. B i) \<and>
- (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<infinity>)) \<and>
- (\<forall>i. \<nu> (B i) \<noteq> \<infinity>)"
+ (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>)) \<and>
+ (\<forall>i. N (B i) \<noteq> \<infinity>)"
proof -
- interpret v: measure_space ?N
- where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
- by fact auto
- let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<infinity>}"
- let ?a = "SUP Q:?Q. \<mu> Q"
- have "{} \<in> ?Q" using v.empty_measure by auto
+ let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}"
+ let ?a = "SUP Q:?Q. emeasure M Q"
+ have "{} \<in> ?Q" by auto
then have Q_not_empty: "?Q \<noteq> {}" by blast
- have "?a \<le> \<mu> (space M)" using sets_into_space
- by (auto intro!: SUP_least measure_mono)
- then have "?a \<noteq> \<infinity>" using finite_measure_of_space
+ have "?a \<le> emeasure M (space M)" using sets_into_space
+ by (auto intro!: SUP_least emeasure_mono)
+ then have "?a \<noteq> \<infinity>" using finite_emeasure_space
by auto
- from SUPR_countable_SUPR[OF Q_not_empty, of \<mu>]
- obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
+ from SUPR_countable_SUPR[OF Q_not_empty, of "emeasure M"]
+ obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
by auto
- then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
- from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q"
+ then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto
+ from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q"
by auto
- then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp
+ then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp
let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
- have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
- proof (rule continuity_from_below[of ?O])
- show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
+ have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)"
+ proof (rule SUP_emeasure_incseq[of ?O])
+ show "range ?O \<subseteq> sets M" using Q' by auto
show "incseq ?O" by (fastforce intro!: incseq_SucI)
qed
have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
- have O_sets: "\<And>i. ?O i \<in> sets M"
- using Q' by (auto intro!: finite_UN Un)
+ have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto
then have O_in_G: "\<And>i. ?O i \<in> ?Q"
proof (safe del: notI)
- fix i have "Q' ` {..i} \<subseteq> sets M"
- using Q' by (auto intro: finite_UN)
- with v.measure_finitely_subadditive[of "{.. i}" Q']
- have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
+ fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto
+ then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))"
+ by (simp add: sets_eq emeasure_subadditive_finite)
also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty)
- finally show "\<nu> (?O i) \<noteq> \<infinity>" by simp
+ finally show "N (?O i) \<noteq> \<infinity>" by simp
qed auto
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
- have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
+ have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric]
proof (rule antisym)
- show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
- using Q' by (auto intro!: SUP_mono measure_mono finite_UN)
- show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUP_def
+ show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim
+ using Q' by (auto intro!: SUP_mono emeasure_mono)
+ show "(SUP i. emeasure M (?O i)) \<le> ?a" unfolding SUP_def
proof (safe intro!: Sup_mono, unfold bex_simps)
fix i
have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
- then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<infinity>) \<and>
- \<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
+ then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and>
+ emeasure M (\<Union>Q' ` {..i}) \<le> emeasure M x"
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
qed
qed
@@ -687,51 +591,50 @@
show "range Q \<subseteq> sets M"
using Q_sets by auto
{ fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
- show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
+ show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
proof (rule disjCI, simp)
- assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<infinity>"
- show "\<mu> A = 0 \<and> \<nu> A = 0"
+ assume *: "0 < emeasure M A \<longrightarrow> N A \<noteq> \<infinity>"
+ show "emeasure M A = 0 \<and> N A = 0"
proof cases
- assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
+ assume "emeasure M A = 0" moreover with ac A have "N A = 0"
unfolding absolutely_continuous_def by auto
ultimately show ?thesis by simp
next
- assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<infinity>" using positive_measure[OF A(1)] by auto
- with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
- using Q' by (auto intro!: measure_additive countable_UN)
- also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
- proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
+ assume "emeasure M A \<noteq> 0" with * have "N A \<noteq> \<infinity>" using emeasure_nonneg[of M A] by auto
+ with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)"
+ using Q' by (auto intro!: plus_emeasure countable_UN)
+ also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
+ proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
- using `\<nu> A \<noteq> \<infinity>` O_sets A by auto
+ using `N A \<noteq> \<infinity>` O_sets A by auto
qed (fastforce intro!: incseq_SucI)
also have "\<dots> \<le> ?a"
proof (safe intro!: SUP_least)
fix i have "?O i \<union> A \<in> ?Q"
proof (safe del: notI)
show "?O i \<union> A \<in> sets M" using O_sets A by auto
- from O_in_G[of i] have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
- using v.measure_subadditive[of "?O i" A] A O_sets by auto
- with O_in_G[of i] show "\<nu> (?O i \<union> A) \<noteq> \<infinity>"
- using `\<nu> A \<noteq> \<infinity>` by auto
+ from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
+ using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq)
+ with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
+ using `N A \<noteq> \<infinity>` by auto
qed
- then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
+ then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
qed
- finally have "\<mu> A = 0"
- unfolding a_eq using real_measure[OF `?O_0 \<in> sets M`] real_measure[OF A(1)] by auto
- with `\<mu> A \<noteq> 0` show ?thesis by auto
+ finally have "emeasure M A = 0"
+ unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
+ with `emeasure M A \<noteq> 0` show ?thesis by auto
qed
qed }
- { fix i show "\<nu> (Q i) \<noteq> \<infinity>"
+ { fix i show "N (Q i) \<noteq> \<infinity>"
proof (cases i)
case 0 then show ?thesis
unfolding Q_def using Q'[of 0] by simp
next
case (Suc n)
- then show ?thesis unfolding Q_def
- using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
- using v.measure_mono[OF O_mono, of n] v.positive_measure[of "?O n"] v.positive_measure[of "?O (Suc n)"]
- using v.measure_Diff[of "?O n" "?O (Suc n)", OF _ _ _ O_mono]
- by (cases rule: ereal2_cases[of "\<nu> (\<Union> x\<le>Suc n. Q' x)" "\<nu> (\<Union> i\<le>n. Q' i)"]) auto
+ with `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
+ emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union> x\<le>n. Q' x)"]
+ show ?thesis
+ by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def)
qed }
show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
{ fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
@@ -751,96 +654,93 @@
qed
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
- assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
+ assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M"
+ shows "\<exists>f\<in>borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N"
proof -
- interpret v: measure_space ?N
- where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
- by fact auto
from split_space_into_finite_sets_and_rest[OF assms]
obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
- and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
- and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<infinity>" by force
+ and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
+ and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
- have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
- \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
- proof
+ let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))"
+ have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). (\<forall>x. 0 \<le> f x) \<and> density (?M i) f = ?N i"
+ proof (intro allI finite_measure.Radon_Nikodym_finite_measure)
fix i
- have indicator_eq: "\<And>f x A. (f x :: ereal) * indicator (Q i \<inter> A) x * indicator (Q i) x
- = (f x * indicator (Q i) x) * indicator A x"
- unfolding indicator_def by auto
- have fm: "finite_measure (restricted_space (Q i))"
- (is "finite_measure ?R") by (rule restricted_finite_measure[OF Q_sets[of i]])
- then interpret R: finite_measure ?R .
- have fmv: "finite_measure (restricted_space (Q i) \<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?Q")
- proof
- show "measure_space ?Q"
- using v.restricted_measure_space Q_sets[of i] by auto
- show "measure ?Q (space ?Q) \<noteq> \<infinity>" using Q_fin by simp
- qed
- have "R.absolutely_continuous \<nu>"
- using `absolutely_continuous \<nu>` `Q i \<in> sets M`
- by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
- from R.Radon_Nikodym_finite_measure[OF fmv this]
- obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
- and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x \<partial>M)"
- unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
- positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
- then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
- \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
- by (auto intro!: exI[of _ "\<lambda>x. max 0 (f x * indicator (Q i) x)"] positive_integral_cong_pos
- split: split_indicator split_if_asm simp: max_def)
+ from Q show "finite_measure (?M i)"
+ by (auto intro!: finite_measureI cong: positive_integral_cong
+ simp add: emeasure_density subset_eq sets_eq)
+ from Q have "emeasure (?N i) (space N) = emeasure N (Q i)"
+ by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: positive_integral_cong)
+ with Q_fin show "finite_measure (?N i)"
+ by (auto intro!: finite_measureI)
+ show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
+ show "absolutely_continuous (?M i) (?N i)"
+ using `absolutely_continuous M N` `Q i \<in> sets M`
+ by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
+ intro!: absolutely_continuous_AE[OF sets_eq])
qed
- from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
- and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
- \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
+ from choice[OF this[unfolded Bex_def]]
+ obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
+ and f_density: "\<And>i. density (?M i) (f i) = ?N i"
by auto
+ { fix A i assume A: "A \<in> sets M"
+ with Q borel have "(\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A"
+ by (auto simp add: emeasure_density positive_integral_density subset_eq
+ intro!: positive_integral_cong split: split_indicator)
+ also have "\<dots> = emeasure N (Q i \<inter> A)"
+ using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq)
+ finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. }
+ note integral_eq = this
let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x"
show ?thesis
proof (safe intro!: bexI[of _ ?f])
show "?f \<in> borel_measurable M" using Q0 borel Q_sets
by (auto intro!: measurable_If)
show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def)
- fix A assume "A \<in> sets M"
- have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
- have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
- "\<And>i. AE x. 0 \<le> f i x * indicator (Q i \<inter> A) x"
- using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
- have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
- using borel by (intro positive_integral_cong) (auto simp: indicator_def)
- also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * \<mu> (Q0 \<inter> A)"
- using borel Qi Q0(1) `A \<in> sets M`
- by (subst positive_integral_add) (auto simp del: ereal_infty_mult
- simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
- also have "\<dots> = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)"
- by (subst f[OF `A \<in> sets M`], subst positive_integral_suminf) auto
- finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)" .
- moreover have "(\<Sum>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
- using Q Q_sets `A \<in> sets M`
- by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
- (auto simp: disjoint_family_on_def)
- moreover have "\<infinity> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
- proof -
- have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
- from in_Q0[OF this] show ?thesis by auto
- qed
- moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
- using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
- moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
- using `A \<in> sets M` sets_into_space Q0 by auto
- ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
- using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
- by simp
+ show "density M ?f = N"
+ proof (rule measure_eqI)
+ fix A assume "A \<in> sets (density M ?f)"
+ then have "A \<in> sets M" by simp
+ have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
+ have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
+ "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
+ using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
+ have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
+ using borel by (intro positive_integral_cong) (auto simp: indicator_def)
+ also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)"
+ using borel Qi Q0(1) `A \<in> sets M`
+ by (subst positive_integral_add) (auto simp del: ereal_infty_mult
+ simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
+ also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)"
+ by (subst integral_eq[OF `A \<in> sets M`], subst positive_integral_suminf) auto
+ finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" .
+ moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
+ using Q Q_sets `A \<in> sets M`
+ by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
+ moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)"
+ proof -
+ have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
+ from in_Q0[OF this] show ?thesis by auto
+ qed
+ moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
+ using Q_sets `A \<in> sets M` Q0(1) by auto
+ moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
+ using `A \<in> sets M` sets_into_space Q0 by auto
+ ultimately have "N A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
+ using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq)
+ with `A \<in> sets M` borel Q Q0(1) show "emeasure (density M ?f) A = N A"
+ by (subst emeasure_density)
+ (auto intro!: borel_measurable_ereal_add borel_measurable_psuminf measurable_If
+ simp: subset_eq)
+ qed (simp add: sets_eq)
qed
qed
lemma (in sigma_finite_measure) Radon_Nikodym:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
- assumes ac: "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
+ assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M"
+ shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N"
proof -
from Ex_finite_integrable_function
obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and
@@ -849,47 +749,38 @@
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
"\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
let ?T = "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
- let ?MT = "M\<lparr> measure := ?T \<rparr>"
- interpret T: finite_measure ?MT
- where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
- using borel finite nn
- by (auto intro!: measure_space_density finite_measureI cong: positive_integral_cong)
- have "T.absolutely_continuous \<nu>"
- proof (unfold T.absolutely_continuous_def, safe)
- fix N assume "N \<in> sets M" "(\<integral>\<^isup>+x. h x * indicator N x \<partial>M) = 0"
- with borel ac pos have "AE x. x \<notin> N"
- by (subst (asm) positive_integral_0_iff_AE) (auto split: split_indicator simp: not_le)
- then have "N \<in> null_sets" using `N \<in> sets M` sets_into_space
- by (subst (asm) AE_iff_measurable[OF `N \<in> sets M`]) auto
- then show "\<nu> N = 0" using ac by (auto simp: absolutely_continuous_def)
- qed
- from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
- obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and
- fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
- by (auto simp: measurable_def)
- show ?thesis
- proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
- show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
- using borel f_borel by auto
- show "\<And>x. 0 \<le> h x * f x" using nn f_borel by auto
- fix A assume "A \<in> sets M"
- then show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
- unfolding fT[OF `A \<in> sets M`] mult_assoc using nn borel f_borel
- by (intro positive_integral_translated_density) auto
- qed
+ let ?MT = "density M h"
+ from borel finite nn interpret T: finite_measure ?MT
+ by (auto intro!: finite_measureI cong: positive_integral_cong simp: emeasure_density)
+ have "absolutely_continuous ?MT N" "sets N = sets ?MT"
+ proof (unfold absolutely_continuous_def, safe)
+ fix A assume "A \<in> null_sets ?MT"
+ with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0"
+ by (auto simp add: null_sets_density_iff)
+ with pos sets_into_space have "AE x in M. x \<notin> A"
+ by (elim eventually_elim1) (auto simp: not_le[symmetric])
+ then have "A \<in> null_sets M"
+ using `A \<in> sets M` by (simp add: AE_iff_null_sets)
+ with ac show "A \<in> null_sets N"
+ by (auto simp: absolutely_continuous_def)
+ qed (auto simp add: sets_eq)
+ from T.Radon_Nikodym_finite_measure_infinite[OF this]
+ obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density ?MT f = N" by auto
+ with nn borel show ?thesis
+ by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
qed
section "Uniqueness of densities"
-lemma (in measure_space) finite_density_unique:
+lemma finite_density_unique:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
+ assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
and fin: "integral\<^isup>P M f \<noteq> \<infinity>"
shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. g x * indicator A x \<partial>M))
- \<longleftrightarrow> (AE x. f x = g x)"
+ \<longleftrightarrow> (AE x in M. f x = g x)"
(is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
proof (intro iffI ballI)
- fix A assume eq: "AE x. f x = g x"
+ fix A assume eq: "AE x in M. f x = g x"
then show "?P f A = ?P g A"
by (auto intro: positive_integral_cong_AE)
next
@@ -897,7 +788,7 @@
from this[THEN bspec, OF top] fin
have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- and pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
+ and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
let ?N = "{x\<in>space M. g x < f x}"
have N: "?N \<in> sets M" using borel by simp
@@ -910,167 +801,163 @@
proof (rule positive_integral_diff)
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
using borel N by auto
- show "AE x. g x * indicator ?N x \<le> f x * indicator ?N x"
- "AE x. 0 \<le> g x * indicator ?N x"
+ show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x"
+ "AE x in M. 0 \<le> g x * indicator ?N x"
using pos by (auto split: split_indicator)
qed fact
also have "\<dots> = 0"
- unfolding eq[THEN bspec, OF N] using positive_integral_positive Pg_fin by auto
- finally have "AE x. f x \<le> g x"
+ unfolding eq[THEN bspec, OF N] using positive_integral_positive[of M] Pg_fin by auto
+ finally have "AE x in M. f x \<le> g x"
using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
by (subst (asm) positive_integral_0_iff_AE)
(auto split: split_indicator simp: not_less ereal_minus_le_iff) }
from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
- show "AE x. f x = g x" by auto
+ show "AE x in M. f x = g x" by auto
qed
lemma (in finite_measure) density_unique_finite_measure:
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
- assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> f' x"
+ assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x"
assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
- shows "AE x. f x = f' x"
+ shows "AE x in M. f x = f' x"
proof -
- let ?\<nu> = "\<lambda>A. ?P f A" and ?\<nu>' = "\<lambda>A. ?P f' A"
+ let ?D = "\<lambda>f. density M f"
+ let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A"
let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
- interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
- using borel(1) pos(1) by (rule measure_space_density) simp
- have ac: "absolutely_continuous ?\<nu>"
- using f by (rule density_is_absolutely_continuous)
- from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
+
+ have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
+ using borel by (auto intro!: absolutely_continuousI_density)
+ from split_space_into_finite_sets_and_rest[OF this]
obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
- and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<infinity>"
- and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<infinity>" by force
+ and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>"
+ and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force
+ with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>"
+ and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq)
+
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
- let ?N = "{x\<in>space M. f x \<noteq> f' x}"
- have "?N \<in> sets M" using borel by auto
+ let ?D = "{x\<in>space M. f x \<noteq> f' x}"
+ have "?D \<in> sets M" using borel by auto
have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
unfolding indicator_def by auto
- have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
+ have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
by (intro finite_density_unique[THEN iffD1] allI)
(auto intro!: borel_measurable_ereal_times f Int simp: *)
- moreover have "AE x. ?f Q0 x = ?f' Q0 x"
+ moreover have "AE x in M. ?f Q0 x = ?f' Q0 x"
proof (rule AE_I')
{ fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}"
- have "(\<Union>i. ?A i) \<in> null_sets"
+ have "(\<Union>i. ?A i) \<in> null_sets M"
proof (rule null_sets_UN)
fix i ::nat have "?A i \<in> sets M"
using borel Q0(1) by auto
- have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
+ have "?N (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
unfolding eq[OF `?A i \<in> sets M`]
by (auto intro!: positive_integral_mono simp: indicator_def)
- also have "\<dots> = i * \<mu> (?A i)"
+ also have "\<dots> = i * emeasure M (?A i)"
using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
- also have "\<dots> < \<infinity>"
- using `?A i \<in> sets M`[THEN finite_measure] by auto
- finally have "?\<nu> (?A i) \<noteq> \<infinity>" by simp
- then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
+ also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp
+ finally have "?N (?A i) \<noteq> \<infinity>" by simp
+ then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
qed
also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
- finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" by simp }
+ finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp }
from this[OF borel(1) refl] this[OF borel(2) f]
- have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets" by simp_all
- then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets" by (rule nullsets.Un)
+ have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all
+ then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un)
show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
qed
- moreover have "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
+ moreover have "AE x in M. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
?f (space M) x = ?f' (space M) x"
by (auto simp: indicator_def Q0)
- ultimately have "AE x. ?f (space M) x = ?f' (space M) x"
- by (auto simp: AE_all_countable[symmetric])
- then show "AE x. f x = f' x" by auto
+ ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
+ unfolding AE_all_countable[symmetric]
+ by eventually_elim (auto intro!: AE_I2 split: split_if_asm simp: indicator_def)
+ then show "AE x in M. f x = f' x" by auto
qed
lemma (in sigma_finite_measure) density_unique:
- assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
- assumes f': "f' \<in> borel_measurable M" "AE x. 0 \<le> f' x"
- assumes eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
- (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
- shows "AE x. f x = f' x"
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ assumes f': "f' \<in> borel_measurable M" "AE x in M. 0 \<le> f' x"
+ assumes density_eq: "density M f = density M f'"
+ shows "AE x in M. f x = f' x"
proof -
obtain h where h_borel: "h \<in> borel_measurable M"
and fin: "integral\<^isup>P M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
using Ex_finite_integrable_function by auto
- then have h_nn: "AE x. 0 \<le> h x" by auto
- let ?H = "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
- have H: "measure_space ?H"
- using h_borel h_nn by (rule measure_space_density) simp
- then interpret h: measure_space ?H .
- interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
- by (intro H finite_measureI) (simp cong: positive_integral_cong add: fin)
- let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
- interpret f: measure_space ?fM
- using f by (rule measure_space_density) simp
- let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
- interpret f': measure_space ?f'M
- using f' by (rule measure_space_density) simp
+ then have h_nn: "AE x in M. 0 \<le> h x" by auto
+ let ?H = "density M h"
+ interpret h: finite_measure ?H
+ using fin h_borel pos
+ by (intro finite_measureI) (simp cong: positive_integral_cong emeasure_density add: fin)
+ let ?fM = "density M f"
+ let ?f'M = "density M f'"
{ fix A assume "A \<in> sets M"
then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
using pos(1) sets_into_space by (force simp: indicator_def)
- then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets"
+ then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
note h_null_sets = this
{ fix A assume "A \<in> sets M"
have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
using `A \<in> sets M` h_borel h_nn f f'
- by (intro positive_integral_translated_density[symmetric]) auto
+ by (intro positive_integral_density[symmetric]) auto
also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
- by (rule f'.positive_integral_cong_measure) (simp_all add: eq)
+ by (simp_all add: density_eq)
also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
using `A \<in> sets M` h_borel h_nn f f'
- by (intro positive_integral_translated_density) auto
+ by (intro positive_integral_density) auto
finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)"
by (simp add: ac_simps)
then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
using `A \<in> sets M` h_borel h_nn f f'
- by (subst (asm) (1 2) positive_integral_translated_density[symmetric]) auto }
+ by (subst (asm) (1 2) positive_integral_density[symmetric]) auto }
then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
- by (intro h.density_unique_finite_measure absolutely_continuous_AE[OF H] density_is_absolutely_continuous)
- simp_all
- then show "AE x. f x = f' x"
- unfolding h.almost_everywhere_def almost_everywhere_def
- by (auto simp add: h_null_sets)
+ by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M])
+ (auto simp add: AE_density)
+ then show "AE x in M. f x = f' x"
+ unfolding eventually_ae_filter using h_borel pos
+ by (auto simp add: h_null_sets null_sets_density_iff not_less[symmetric]
+ AE_iff_null_sets[symmetric])
+ blast
qed
+lemma (in sigma_finite_measure) density_unique_iff:
+ assumes f: "f \<in> borel_measurable M" and "AE x in M. 0 \<le> f x"
+ assumes f': "f' \<in> borel_measurable M" and "AE x in M. 0 \<le> f' x"
+ shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)"
+ using density_unique[OF assms] density_cong[OF f f'] by auto
+
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
- assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
- and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
- shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<infinity>)"
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
+ (is "sigma_finite_measure ?N \<longleftrightarrow> _")
proof
assume "sigma_finite_measure ?N"
- then interpret \<nu>: sigma_finite_measure ?N
- where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
- and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
- from \<nu>.Ex_finite_integrable_function obtain h where
+ then interpret N: sigma_finite_measure ?N .
+ from N.Ex_finite_integrable_function obtain h where
h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
h_nn: "\<And>x. 0 \<le> h x" and
fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
- have "AE x. f x * h x \<noteq> \<infinity>"
+ have "AE x in M. f x * h x \<noteq> \<infinity>"
proof (rule AE_I')
have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
- by (subst \<nu>.positive_integral_cong_measure[symmetric,
- of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
- (auto intro!: positive_integral_translated_density simp: eq)
+ by (auto intro!: positive_integral_density)
then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
using h(2) by simp
- then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets"
+ then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M"
using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
qed auto
- then show "AE x. f x \<noteq> \<infinity>"
+ then show "AE x in M. f x \<noteq> \<infinity>"
using fin by (auto elim!: AE_Ball_mp)
next
- assume AE: "AE x. f x \<noteq> \<infinity>"
+ assume AE: "AE x in M. f x \<noteq> \<infinity>"
from sigma_finite guess Q .. note Q = this
- interpret \<nu>: measure_space ?N
- where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
- and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M"
{ fix i j have "A i \<inter> Q j \<in> sets M"
unfolding A_def using f Q
@@ -1113,7 +1000,7 @@
have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
proof (cases i)
case 0
- have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
+ have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
using AE by (auto simp: A_def `i = 0`)
from positive_integral_cong_AE[OF this] show ?thesis by simp
next
@@ -1121,271 +1008,238 @@
then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
(\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat)
- also have "\<dots> = Suc n * \<mu> (Q j)"
+ also have "\<dots> = Suc n * emeasure M (Q j)"
using Q by (auto intro!: positive_integral_cmult_indicator)
also have "\<dots> < \<infinity>"
using Q by (auto simp: real_eq_of_nat[symmetric])
finally show ?thesis by simp
qed
- then show "measure ?N (?A n) \<noteq> \<infinity>"
- using A_in_sets Q eq by auto
+ then show "emeasure ?N (?A n) \<noteq> \<infinity>"
+ using A_in_sets Q f by (auto simp: emeasure_density)
qed
qed
section "Radon-Nikodym derivative"
definition
- "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and>
- (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
+ "RN_deriv M N \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N"
-lemma (in sigma_finite_measure) RN_deriv_cong:
- assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
- and \<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
- shows "RN_deriv M' \<nu>' x = RN_deriv M \<nu> x"
+lemma
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ shows borel_measurable_RN_deriv_density: "RN_deriv M (density M f) \<in> borel_measurable M" (is ?borel)
+ and density_RN_deriv_density: "density M (RN_deriv M (density M f)) = density M f" (is ?density)
+ and RN_deriv_density_nonneg: "0 \<le> RN_deriv M (density M f) x" (is ?pos)
proof -
- interpret \<mu>': sigma_finite_measure M'
- using cong by (rule sigma_finite_measure_cong)
- show ?thesis
- unfolding RN_deriv_def
- by (simp add: cong \<nu> positive_integral_cong_measure[OF cong] measurable_def)
+ let ?f = "\<lambda>x. max 0 (f x)"
+ let ?P = "\<lambda>g. g \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> g x) \<and> density M g = density M f"
+ from f have "?P ?f" using f by (auto intro!: density_cong simp: split: split_max)
+ then have "?P (RN_deriv M (density M f))"
+ unfolding RN_deriv_def by (rule someI[where P="?P"])
+ then show ?borel ?density ?pos by auto
qed
lemma (in sigma_finite_measure) RN_deriv:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
- assumes "absolutely_continuous \<nu>"
- shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
- and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
- (is "\<And>A. _ \<Longrightarrow> ?int A")
- and "0 \<le> RN_deriv M \<nu> x"
+ assumes "absolutely_continuous M N" "sets N = sets M"
+ shows borel_measurable_RN_deriv: "RN_deriv M N \<in> borel_measurable M" (is ?borel)
+ and density_RN_deriv: "density M (RN_deriv M N) = N" (is ?density)
+ and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?pos)
proof -
note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
- then show ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
- from Ex show "0 \<le> RN_deriv M \<nu> x" unfolding RN_deriv_def
- by (rule someI2_ex) simp
- fix A assume "A \<in> sets M"
- from Ex show "?int A" unfolding RN_deriv_def
- by (rule someI2_ex) (simp add: `A \<in> sets M`)
+ from Ex show ?borel unfolding RN_deriv_def by (rule someI2_ex) simp
+ from Ex show ?density unfolding RN_deriv_def by (rule someI2_ex) simp
+ from Ex show ?pos unfolding RN_deriv_def by (rule someI2_ex) simp
qed
lemma (in sigma_finite_measure) RN_deriv_positive_integral:
- assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
+ assumes N: "absolutely_continuous M N" "sets N = sets M"
and f: "f \<in> borel_measurable M"
- shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
+ shows "integral\<^isup>P N f = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)"
proof -
- interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- note RN = RN_deriv[OF \<nu>]
- have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>)"
- unfolding positive_integral_max_0 ..
- also have "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>) =
- (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>)"
- by (intro \<nu>.positive_integral_cong_measure[symmetric]) (simp_all add: RN(2))
- also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * max 0 (f x) \<partial>M)"
- by (intro positive_integral_translated_density) (auto simp add: RN f)
- also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
- using RN_deriv(3)[OF \<nu>]
- by (auto intro!: positive_integral_cong_pos split: split_if_asm
- simp: max_def ereal_mult_le_0_iff)
- finally show ?thesis .
-qed
-
-lemma (in sigma_finite_measure) RN_deriv_unique:
- assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
- and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
- shows "AE x. f x = RN_deriv M \<nu> x"
-proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
- show "AE x. 0 \<le> RN_deriv M \<nu> x" using RN_deriv[OF \<nu>] by auto
- fix A assume A: "A \<in> sets M"
- show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
- unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
+ have "integral\<^isup>P N f = integral\<^isup>P (density M (RN_deriv M N)) f"
+ using N by (simp add: density_RN_deriv)
+ also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)"
+ using RN_deriv(1,3)[OF N] f by (simp add: positive_integral_density)
+ finally show ?thesis by simp
qed
-lemma (in sigma_finite_measure) RN_deriv_vimage:
- assumes T: "T \<in> measure_preserving M M'"
- and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
- and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
- and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
- shows "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
+lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
+ using AE_iff_null_sets[of N M] by auto
+
+lemma (in sigma_finite_measure) RN_deriv_unique:
+ assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+ and eq: "density M f = N"
+ shows "AE x in M. f x = RN_deriv M N x"
+proof (rule density_unique)
+ have ac: "absolutely_continuous M N"
+ using f(1) unfolding eq[symmetric] by (rule absolutely_continuousI_density)
+ have eq2: "sets N = sets M"
+ unfolding eq[symmetric] by simp
+ show "RN_deriv M N \<in> borel_measurable M" "AE x in M. 0 \<le> RN_deriv M N x"
+ "density M f = density M (RN_deriv M N)"
+ using RN_deriv[OF ac eq2] eq by auto
+qed fact+
+
+lemma (in sigma_finite_measure) RN_deriv_distr:
+ fixes T :: "'a \<Rightarrow> 'b"
+ assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
+ and inv: "\<forall>x\<in>space M. T' (T x) = x"
+ and ac: "absolutely_continuous (distr M M' T) (distr N M' T)"
+ and N: "sets N = sets M"
+ shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
proof (rule RN_deriv_unique)
- interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
- show "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
- by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
- interpret M': measure_space M'
- proof (rule measure_space_vimage)
- have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
- then show "sigma_algebra M'" by simp
- qed fact
- show "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
- proof safe
- fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
- then have N': "T' -` N \<inter> space M' \<in> sets M'"
- using T' by (auto simp: measurable_def measure_preserving_def)
- have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
- using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
- then have "measure M' (T' -` N \<inter> space M') = 0"
- using measure_preservingD[OF T N'] N_0 by auto
- with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
- unfolding M'.absolutely_continuous_def measurable_def by auto
- qed
- interpret M': sigma_finite_measure M'
+ have [simp]: "sets N = sets M" by fact
+ note sets_eq_imp_space_eq[OF N, simp]
+ have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
+ { fix A assume "A \<in> sets M"
+ with inv T T' sets_into_space[OF this]
+ have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
+ by (auto simp: measurable_def) }
+ note eq = this[simp]
+ { fix A assume "A \<in> sets M"
+ with inv T T' sets_into_space[OF this]
+ have "(T' \<circ> T) -` A \<inter> space M = A"
+ by (auto simp: measurable_def) }
+ note eq2 = this[simp]
+ let ?M' = "distr M M' T" and ?N' = "distr N M' T"
+ interpret M': sigma_finite_measure ?M'
proof
from sigma_finite guess F .. note F = this
- show "\<exists>A::nat \<Rightarrow> 'c set. range A \<subseteq> sets M' \<and> (\<Union>i. A i) = space M' \<and> (\<forall>i. M'.\<mu> (A i) \<noteq> \<infinity>)"
+ show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets ?M' \<and> (\<Union>i. A i) = space ?M' \<and> (\<forall>i. emeasure ?M' (A i) \<noteq> \<infinity>)"
proof (intro exI conjI allI)
- show *: "range (\<lambda>i. T' -` F i \<inter> space M') \<subseteq> sets M'"
- using F T' by (auto simp: measurable_def measure_preserving_def)
- show "(\<Union>i. T' -` F i \<inter> space M') = space M'"
- using F T' by (force simp: measurable_def measure_preserving_def)
+ show *: "range (\<lambda>i. T' -` F i \<inter> space ?M') \<subseteq> sets ?M'"
+ using F T' by (auto simp: measurable_def)
+ show "(\<Union>i. T' -` F i \<inter> space ?M') = space ?M'"
+ using F T' by (force simp: measurable_def)
fix i
have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto
- note measure_preservingD[OF T this, symmetric]
moreover
have Fi: "F i \<in> sets M" using F by auto
- then have "T -` (T' -` F i \<inter> space M') \<inter> space M = F i"
- using T inv sets_into_space[OF Fi]
- by (auto simp: measurable_def measure_preserving_def)
- ultimately show "measure M' (T' -` F i \<inter> space M') \<noteq> \<infinity>"
- using F by simp
+ ultimately show "emeasure ?M' (T' -` F i \<inter> space ?M') \<noteq> \<infinity>"
+ using F T T' by (simp add: emeasure_distr)
qed
qed
- have "(RN_deriv M' \<nu>') \<circ> T \<in> borel_measurable M"
- by (intro measurable_comp[where b=M'] M'.RN_deriv(1) measure_preservingD2[OF T]) fact+
- then show "(\<lambda>x. RN_deriv M' \<nu>' (T x)) \<in> borel_measurable M"
+ have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
+ using T ac by (intro measurable_comp[where b="?M'"] M'.borel_measurable_RN_deriv) simp_all
+ then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
by (simp add: comp_def)
- show "AE x. 0 \<le> RN_deriv M' \<nu>' (T x)" using M'.RN_deriv(3)[OF \<nu>'] by auto
- fix A let ?A = "T' -` A \<inter> space M'"
- assume A: "A \<in> sets M"
- then have A': "?A \<in> sets M'" using T' unfolding measurable_def measure_preserving_def
- by auto
- from A have "\<nu> A = \<nu>' ?A" using T'[THEN measure_preservingD] by simp
- also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' x * indicator ?A x \<partial>M'"
- using A' by (rule M'.RN_deriv(2)[OF \<nu>'])
- also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator ?A (T x) \<partial>M"
- proof (rule positive_integral_vimage)
- show "sigma_algebra M'" by default
- show "(\<lambda>x. RN_deriv M' \<nu>' x * indicator (T' -` A \<inter> space M') x) \<in> borel_measurable M'"
- by (auto intro!: A' M'.RN_deriv(1)[OF \<nu>'])
- qed fact
- also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M"
- using T inv by (auto intro!: positive_integral_cong simp: measure_preserving_def measurable_def indicator_def)
- finally show "\<nu> A = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M" .
+ show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" using M'.RN_deriv_nonneg[OF ac] by auto
+
+ have "N = distr N M (T' \<circ> T)"
+ by (subst measure_of_of_measure[of N, symmetric])
+ (auto simp add: distr_def sigma_sets_eq intro!: measure_of_eq space_closed)
+ also have "\<dots> = distr (distr N M' T) M T'"
+ using T T' by (simp add: distr_distr)
+ also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
+ using ac by (simp add: M'.density_RN_deriv)
+ also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
+ using M'.borel_measurable_RN_deriv[OF ac] by (simp add: distr_density_distr[OF T T', OF inv])
+ finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
+ by (simp add: comp_def)
qed
lemma (in sigma_finite_measure) RN_deriv_finite:
- assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
- shows "AE x. RN_deriv M \<nu> x \<noteq> \<infinity>"
+ assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
+ shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>"
proof -
- interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
- have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
- from sfm show ?thesis
- using sigma_finite_iff_density_finite[OF \<nu> RN_deriv(1)[OF \<nu> ac]] RN_deriv(2,3)[OF \<nu> ac] by simp
+ interpret N: sigma_finite_measure N by fact
+ from N show ?thesis
+ using sigma_finite_iff_density_finite[OF RN_deriv(1)[OF ac]] RN_deriv(2,3)[OF ac] by simp
qed
lemma (in sigma_finite_measure)
- assumes \<nu>: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
+ assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
and f: "f \<in> borel_measurable M"
- shows RN_deriv_integrable: "integrable (M\<lparr>measure := \<nu>\<rparr>) f \<longleftrightarrow>
- integrable M (\<lambda>x. real (RN_deriv M \<nu> x) * f x)" (is ?integrable)
- and RN_deriv_integral: "integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) f =
- (\<integral>x. real (RN_deriv M \<nu> x) * f x \<partial>M)" (is ?integral)
+ shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
+ integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable)
+ and RN_deriv_integral: "integral\<^isup>L N f =
+ (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
proof -
- interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
- have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
+ note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
+ interpret N: sigma_finite_measure N by fact
have minus_cong: "\<And>A B A' B'::ereal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
- have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f by simp
+ have Nf: "f \<in> borel_measurable N" using f by (simp add: measurable_def)
{ fix f :: "'a \<Rightarrow> real"
- { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<infinity>"
- have "ereal (real (RN_deriv M \<nu> x)) * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
+ { fix x assume *: "RN_deriv M N x \<noteq> \<infinity>"
+ have "ereal (real (RN_deriv M N x)) * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
by (simp add: mult_le_0_iff)
- then have "RN_deriv M \<nu> x * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
- using RN_deriv(3)[OF ms \<nu>(2)] * by (auto simp add: ereal_real split: split_if_asm) }
- then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M \<nu> x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (f x) \<partial>M)"
- "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M \<nu> x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (- f x) \<partial>M)"
- using RN_deriv_finite[OF \<nu>] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
+ then have "RN_deriv M N x * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
+ using RN_deriv(3)[OF ac] * by (auto simp add: ereal_real split: split_if_asm) }
+ then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M N x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M N x * ereal (f x) \<partial>M)"
+ "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M N x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M N x * ereal (- f x) \<partial>M)"
+ using RN_deriv_finite[OF N ac] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
by (auto intro!: positive_integral_cong_AE) }
note * = this
show ?integral ?integrable
unfolding lebesgue_integral_def integrable_def *
- using f RN_deriv(1)[OF ms \<nu>(2)]
- by (auto simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
+ using Nf f RN_deriv(1)[OF ac]
+ by (auto simp: RN_deriv_positive_integral[OF ac])
qed
lemma (in sigma_finite_measure) real_RN_deriv:
- assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?\<nu>")
- assumes ac: "absolutely_continuous \<nu>"
+ assumes "finite_measure N"
+ assumes ac: "absolutely_continuous M N" "sets N = sets M"
obtains D where "D \<in> borel_measurable M"
- and "AE x. RN_deriv M \<nu> x = ereal (D x)"
- and "AE x in M\<lparr>measure := \<nu>\<rparr>. 0 < D x"
+ and "AE x in M. RN_deriv M N x = ereal (D x)"
+ and "AE x in N. 0 < D x"
and "\<And>x. 0 \<le> D x"
proof
- interpret \<nu>: finite_measure ?\<nu> by fact
- have ms: "measure_space ?\<nu>" by default
- note RN = RN_deriv[OF ms ac]
+ interpret N: finite_measure N by fact
+
+ note RN = RN_deriv[OF ac]
- let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M \<nu> x = t}"
+ let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
- show "(\<lambda>x. real (RN_deriv M \<nu> x)) \<in> borel_measurable M"
+ show "(\<lambda>x. real (RN_deriv M N x)) \<in> borel_measurable M"
using RN by auto
- have "\<nu> (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN \<infinity>) x \<partial>M)"
- using RN by auto
+ have "N (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
+ using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
by (intro positive_integral_cong) (auto simp: indicator_def)
- also have "\<dots> = \<infinity> * \<mu> (?RN \<infinity>)"
+ also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
using RN by (intro positive_integral_cmult_indicator) auto
- finally have eq: "\<nu> (?RN \<infinity>) = \<infinity> * \<mu> (?RN \<infinity>)" .
+ finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" .
moreover
- have "\<mu> (?RN \<infinity>) = 0"
+ have "emeasure M (?RN \<infinity>) = 0"
proof (rule ccontr)
- assume "\<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>} \<noteq> 0"
- moreover from RN have "0 \<le> \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
- ultimately have "0 < \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
- with eq have "\<nu> (?RN \<infinity>) = \<infinity>" by simp
- with \<nu>.finite_measure[of "?RN \<infinity>"] RN show False by auto
+ assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0"
+ moreover from RN have "0 \<le> emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto
+ ultimately have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto
+ with eq have "N (?RN \<infinity>) = \<infinity>" by simp
+ with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto
qed
- ultimately have "AE x. RN_deriv M \<nu> x < \<infinity>"
+ ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
using RN by (intro AE_iff_measurable[THEN iffD2]) auto
- then show "AE x. RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
+ then show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))"
using RN(3) by (auto simp: ereal_real)
- then have eq: "AE x in (M\<lparr>measure := \<nu>\<rparr>) . RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
- using ac absolutely_continuous_AE[OF ms] by auto
+ then have eq: "AE x in N. RN_deriv M N x = ereal (real (RN_deriv M N x))"
+ using ac absolutely_continuous_AE by auto
- show "\<And>x. 0 \<le> real (RN_deriv M \<nu> x)"
+ show "\<And>x. 0 \<le> real (RN_deriv M N x)"
using RN by (auto intro: real_of_ereal_pos)
- have "\<nu> (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN 0) x \<partial>M)"
- using RN by auto
+ have "N (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
+ using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)"
by (intro positive_integral_cong) (auto simp: indicator_def)
- finally have "AE x in (M\<lparr>measure := \<nu>\<rparr>). RN_deriv M \<nu> x \<noteq> 0"
- using RN by (intro \<nu>.AE_iff_measurable[THEN iffD2]) auto
- with RN(3) eq show "AE x in (M\<lparr>measure := \<nu>\<rparr>). 0 < real (RN_deriv M \<nu> x)"
+ finally have "AE x in N. RN_deriv M N x \<noteq> 0"
+ using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
+ with RN(3) eq show "AE x in N. 0 < real (RN_deriv M N x)"
by (auto simp: zero_less_real_of_ereal le_less)
qed
lemma (in sigma_finite_measure) RN_deriv_singleton:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
- and ac: "absolutely_continuous \<nu>"
- and "{x} \<in> sets M"
- shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
+ assumes ac: "absolutely_continuous M N" "sets N = sets M"
+ and x: "{x} \<in> sets M"
+ shows "N {x} = RN_deriv M N x * emeasure M {x}"
proof -
- note deriv = RN_deriv[OF assms(1, 2)]
- from deriv(2)[OF `{x} \<in> sets M`]
- have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
- by (auto simp: indicator_def intro!: positive_integral_cong)
- thus ?thesis using positive_integral_cmult_indicator[OF _ `{x} \<in> sets M`] deriv(3)
- by auto
-qed
-
-theorem (in finite_measure_space) RN_deriv_finite_measure:
- assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
- and ac: "absolutely_continuous \<nu>"
- and "x \<in> space M"
- shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
-proof -
- have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
- from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .
+ note deriv = RN_deriv[OF ac]
+ from deriv(1,3) `{x} \<in> sets M`
+ have "density M (RN_deriv M N) {x} = (\<integral>\<^isup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
+ by (auto simp: indicator_def emeasure_density intro!: positive_integral_cong)
+ with x deriv show ?thesis
+ by (auto simp: positive_integral_cmult_indicator)
qed
end
--- a/src/HOL/Probability/Sigma_Algebra.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Sigma_Algebra.thy Mon Apr 23 12:14:35 2012 +0200
@@ -13,6 +13,7 @@
"~~/src/HOL/Library/Countable"
"~~/src/HOL/Library/FuncSet"
"~~/src/HOL/Library/Indicator_Function"
+ "~~/src/HOL/Library/Extended_Real"
begin
text {* Sigma algebras are an elementary concept in measure
@@ -25,203 +26,198 @@
subsection {* Algebras *}
-record 'a algebra =
- space :: "'a set"
- sets :: "'a set set"
+locale subset_class =
+ fixes \<Omega> :: "'a set" and M :: "'a set set"
+ assumes space_closed: "M \<subseteq> Pow \<Omega>"
-locale subset_class =
- fixes M :: "('a, 'b) algebra_scheme"
- assumes space_closed: "sets M \<subseteq> Pow (space M)"
-
-lemma (in subset_class) sets_into_space: "x \<in> sets M \<Longrightarrow> x \<subseteq> space M"
+lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
by (metis PowD contra_subsetD space_closed)
locale ring_of_sets = subset_class +
- assumes empty_sets [iff]: "{} \<in> sets M"
- and Diff [intro]: "\<And>a b. a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a - b \<in> sets M"
- and Un [intro]: "\<And>a b. a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<union> b \<in> sets M"
+ assumes empty_sets [iff]: "{} \<in> M"
+ and Diff [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
+ and Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
lemma (in ring_of_sets) Int [intro]:
- assumes a: "a \<in> sets M" and b: "b \<in> sets M" shows "a \<inter> b \<in> sets M"
+ assumes a: "a \<in> M" and b: "b \<in> M" shows "a \<inter> b \<in> M"
proof -
have "a \<inter> b = a - (a - b)"
by auto
- then show "a \<inter> b \<in> sets M"
+ then show "a \<inter> b \<in> M"
using a b by auto
qed
lemma (in ring_of_sets) finite_Union [intro]:
- "finite X \<Longrightarrow> X \<subseteq> sets M \<Longrightarrow> Union X \<in> sets M"
+ "finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M"
by (induct set: finite) (auto simp add: Un)
lemma (in ring_of_sets) finite_UN[intro]:
- assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
- shows "(\<Union>i\<in>I. A i) \<in> sets M"
+ assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
+ shows "(\<Union>i\<in>I. A i) \<in> M"
using assms by induct auto
lemma (in ring_of_sets) finite_INT[intro]:
- assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
- shows "(\<Inter>i\<in>I. A i) \<in> sets M"
+ assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
+ shows "(\<Inter>i\<in>I. A i) \<in> M"
using assms by (induct rule: finite_ne_induct) auto
lemma (in ring_of_sets) insert_in_sets:
- assumes "{x} \<in> sets M" "A \<in> sets M" shows "insert x A \<in> sets M"
+ assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
proof -
- have "{x} \<union> A \<in> sets M" using assms by (rule Un)
+ have "{x} \<union> A \<in> M" using assms by (rule Un)
thus ?thesis by auto
qed
-lemma (in ring_of_sets) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"
+lemma (in ring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
by (metis Int_absorb1 sets_into_space)
-lemma (in ring_of_sets) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
+lemma (in ring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
by (metis Int_absorb2 sets_into_space)
lemma (in ring_of_sets) sets_Collect_conj:
- assumes "{x\<in>space M. P x} \<in> sets M" "{x\<in>space M. Q x} \<in> sets M"
- shows "{x\<in>space M. Q x \<and> P x} \<in> sets M"
+ assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
+ shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
proof -
- have "{x\<in>space M. Q x \<and> P x} = {x\<in>space M. Q x} \<inter> {x\<in>space M. P x}"
+ have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
by auto
with assms show ?thesis by auto
qed
lemma (in ring_of_sets) sets_Collect_disj:
- assumes "{x\<in>space M. P x} \<in> sets M" "{x\<in>space M. Q x} \<in> sets M"
- shows "{x\<in>space M. Q x \<or> P x} \<in> sets M"
+ assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
+ shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
proof -
- have "{x\<in>space M. Q x \<or> P x} = {x\<in>space M. Q x} \<union> {x\<in>space M. P x}"
+ have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
by auto
with assms show ?thesis by auto
qed
lemma (in ring_of_sets) sets_Collect_finite_All:
- assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>space M. P i x} \<in> sets M" "finite S" "S \<noteq> {}"
- shows "{x\<in>space M. \<forall>i\<in>S. P i x} \<in> sets M"
+ assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
+ shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
proof -
- have "{x\<in>space M. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>space M. P i x})"
+ have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
using `S \<noteq> {}` by auto
with assms show ?thesis by auto
qed
lemma (in ring_of_sets) sets_Collect_finite_Ex:
- assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>space M. P i x} \<in> sets M" "finite S"
- shows "{x\<in>space M. \<exists>i\<in>S. P i x} \<in> sets M"
+ assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
+ shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
proof -
- have "{x\<in>space M. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>space M. P i x})"
+ have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
by auto
with assms show ?thesis by auto
qed
locale algebra = ring_of_sets +
- assumes top [iff]: "space M \<in> sets M"
+ assumes top [iff]: "\<Omega> \<in> M"
lemma (in algebra) compl_sets [intro]:
- "a \<in> sets M \<Longrightarrow> space M - a \<in> sets M"
+ "a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
by auto
lemma algebra_iff_Un:
- "algebra M \<longleftrightarrow>
- sets M \<subseteq> Pow (space M) &
- {} \<in> sets M &
- (\<forall>a \<in> sets M. space M - a \<in> sets M) &
- (\<forall>a \<in> sets M. \<forall> b \<in> sets M. a \<union> b \<in> sets M)" (is "_ \<longleftrightarrow> ?Un")
+ "algebra \<Omega> M \<longleftrightarrow>
+ M \<subseteq> Pow \<Omega> \<and>
+ {} \<in> M \<and>
+ (\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
+ (\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
proof
- assume "algebra M"
- then interpret algebra M .
+ assume "algebra \<Omega> M"
+ then interpret algebra \<Omega> M .
show ?Un using sets_into_space by auto
next
assume ?Un
- show "algebra M"
+ show "algebra \<Omega> M"
proof
- show space: "sets M \<subseteq> Pow (space M)" "{} \<in> sets M" "space M \<in> sets M"
+ show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M" "\<Omega> \<in> M"
using `?Un` by auto
- fix a b assume a: "a \<in> sets M" and b: "b \<in> sets M"
- then show "a \<union> b \<in> sets M" using `?Un` by auto
- have "a - b = space M - ((space M - a) \<union> b)"
- using space a b by auto
- then show "a - b \<in> sets M"
+ fix a b assume a: "a \<in> M" and b: "b \<in> M"
+ then show "a \<union> b \<in> M" using `?Un` by auto
+ have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
+ using \<Omega> a b by auto
+ then show "a - b \<in> M"
using a b `?Un` by auto
qed
qed
lemma algebra_iff_Int:
- "algebra M \<longleftrightarrow>
- sets M \<subseteq> Pow (space M) & {} \<in> sets M &
- (\<forall>a \<in> sets M. space M - a \<in> sets M) &
- (\<forall>a \<in> sets M. \<forall> b \<in> sets M. a \<inter> b \<in> sets M)" (is "_ \<longleftrightarrow> ?Int")
+ "algebra \<Omega> M \<longleftrightarrow>
+ M \<subseteq> Pow \<Omega> & {} \<in> M &
+ (\<forall>a \<in> M. \<Omega> - a \<in> M) &
+ (\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
proof
- assume "algebra M"
- then interpret algebra M .
+ assume "algebra \<Omega> M"
+ then interpret algebra \<Omega> M .
show ?Int using sets_into_space by auto
next
assume ?Int
- show "algebra M"
+ show "algebra \<Omega> M"
proof (unfold algebra_iff_Un, intro conjI ballI)
- show space: "sets M \<subseteq> Pow (space M)" "{} \<in> sets M"
+ show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
using `?Int` by auto
- from `?Int` show "\<And>a. a \<in> sets M \<Longrightarrow> space M - a \<in> sets M" by auto
- fix a b assume sets: "a \<in> sets M" "b \<in> sets M"
- hence "a \<union> b = space M - ((space M - a) \<inter> (space M - b))"
- using space by blast
- also have "... \<in> sets M"
- using sets `?Int` by auto
- finally show "a \<union> b \<in> sets M" .
+ from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
+ fix a b assume M: "a \<in> M" "b \<in> M"
+ hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
+ using \<Omega> by blast
+ also have "... \<in> M"
+ using M `?Int` by auto
+ finally show "a \<union> b \<in> M" .
qed
qed
lemma (in algebra) sets_Collect_neg:
- assumes "{x\<in>space M. P x} \<in> sets M"
- shows "{x\<in>space M. \<not> P x} \<in> sets M"
+ assumes "{x\<in>\<Omega>. P x} \<in> M"
+ shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
proof -
- have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
+ have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
with assms show ?thesis by auto
qed
lemma (in algebra) sets_Collect_imp:
- "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> {x\<in>space M. Q x} \<in> sets M \<Longrightarrow> {x\<in>space M. Q x \<longrightarrow> P x} \<in> sets M"
+ "{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M"
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)
lemma (in algebra) sets_Collect_const:
- "{x\<in>space M. P} \<in> sets M"
+ "{x\<in>\<Omega>. P} \<in> M"
by (cases P) auto
lemma algebra_single_set:
assumes "X \<subseteq> S"
- shows "algebra \<lparr> space = S, sets = { {}, X, S - X, S }\<rparr>"
+ shows "algebra S { {}, X, S - X, S }"
by default (insert `X \<subseteq> S`, auto)
section {* Restricted algebras *}
abbreviation (in algebra)
- "restricted_space A \<equiv> \<lparr> space = A, sets = (\<lambda>S. (A \<inter> S)) ` sets M, \<dots> = more M \<rparr>"
+ "restricted_space A \<equiv> (op \<inter> A) ` M"
lemma (in algebra) restricted_algebra:
- assumes "A \<in> sets M" shows "algebra (restricted_space A)"
+ assumes "A \<in> M" shows "algebra A (restricted_space A)"
using assms by unfold_locales auto
subsection {* Sigma Algebras *}
locale sigma_algebra = algebra +
- assumes countable_nat_UN [intro]:
- "!!A. range A \<subseteq> sets M \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
+ assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
lemma (in algebra) is_sigma_algebra:
- assumes "finite (sets M)"
- shows "sigma_algebra M"
+ assumes "finite M"
+ shows "sigma_algebra \<Omega> M"
proof
- fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
- then have "(\<Union>i. A i) = (\<Union>s\<in>sets M \<inter> range A. s)"
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
+ then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
by auto
- also have "(\<Union>s\<in>sets M \<inter> range A. s) \<in> sets M"
- using `finite (sets M)` by auto
- finally show "(\<Union>i. A i) \<in> sets M" .
+ also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
+ using `finite M` by auto
+ finally show "(\<Union>i. A i) \<in> M" .
qed
lemma countable_UN_eq:
fixes A :: "'i::countable \<Rightarrow> 'a set"
- shows "(range A \<subseteq> sets M \<longrightarrow> (\<Union>i. A i) \<in> sets M) \<longleftrightarrow>
- (range (A \<circ> from_nat) \<subseteq> sets M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> sets M)"
+ shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
+ (range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
proof -
let ?A' = "A \<circ> from_nat"
have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
@@ -240,60 +236,57 @@
lemma (in sigma_algebra) countable_UN[intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
- assumes "A`X \<subseteq> sets M"
- shows "(\<Union>x\<in>X. A x) \<in> sets M"
+ assumes "A`X \<subseteq> M"
+ shows "(\<Union>x\<in>X. A x) \<in> M"
proof -
let ?A = "\<lambda>i. if i \<in> X then A i else {}"
- from assms have "range ?A \<subseteq> sets M" by auto
+ from assms have "range ?A \<subseteq> M" by auto
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
- have "(\<Union>x. ?A x) \<in> sets M" by auto
+ have "(\<Union>x. ?A x) \<in> M" by auto
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
ultimately show ?thesis by simp
qed
lemma (in sigma_algebra) countable_INT [intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
- assumes A: "A`X \<subseteq> sets M" "X \<noteq> {}"
- shows "(\<Inter>i\<in>X. A i) \<in> sets M"
+ assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
+ shows "(\<Inter>i\<in>X. A i) \<in> M"
proof -
- from A have "\<forall>i\<in>X. A i \<in> sets M" by fast
- hence "space M - (\<Union>i\<in>X. space M - A i) \<in> sets M" by blast
+ from A have "\<forall>i\<in>X. A i \<in> M" by fast
+ hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
moreover
- have "(\<Inter>i\<in>X. A i) = space M - (\<Union>i\<in>X. space M - A i)" using space_closed A
+ have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
by blast
ultimately show ?thesis by metis
qed
-lemma ring_of_sets_Pow:
- "ring_of_sets \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
+lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
by default auto
-lemma algebra_Pow:
- "algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
+lemma algebra_Pow: "algebra sp (Pow sp)"
by default auto
-lemma sigma_algebra_Pow:
- "sigma_algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
+lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
by default auto
lemma sigma_algebra_iff:
- "sigma_algebra M \<longleftrightarrow>
- algebra M \<and> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
+ "sigma_algebra \<Omega> M \<longleftrightarrow>
+ algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
lemma (in sigma_algebra) sets_Collect_countable_All:
- assumes "\<And>i. {x\<in>space M. P i x} \<in> sets M"
- shows "{x\<in>space M. \<forall>i::'i::countable. P i x} \<in> sets M"
+ assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
+ shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
proof -
- have "{x\<in>space M. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>space M. P i x})" by auto
+ have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis by auto
qed
lemma (in sigma_algebra) sets_Collect_countable_Ex:
- assumes "\<And>i. {x\<in>space M. P i x} \<in> sets M"
- shows "{x\<in>space M. \<exists>i::'i::countable. P i x} \<in> sets M"
+ assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
+ shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
proof -
- have "{x\<in>space M. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>space M. P i x})" by auto
+ have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis by auto
qed
@@ -301,9 +294,19 @@
sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
+lemma (in sigma_algebra) sets_Collect_countable_Ball:
+ assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
+ shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
+ unfolding Ball_def by (intro sets_Collect assms)
+
+lemma (in sigma_algebra) sets_Collect_countable_Bex:
+ assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
+ shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
+ unfolding Bex_def by (intro sets_Collect assms)
+
lemma sigma_algebra_single_set:
assumes "X \<subseteq> S"
- shows "sigma_algebra \<lparr> space = S, sets = { {}, X, S - X, S }\<rparr>"
+ shows "sigma_algebra S { {}, X, S - X, S }"
using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
subsection {* Binary Unions *}
@@ -321,37 +324,34 @@
by (simp add: INF_def range_binary_eq)
lemma sigma_algebra_iff2:
- "sigma_algebra M \<longleftrightarrow>
- sets M \<subseteq> Pow (space M) \<and>
- {} \<in> sets M \<and> (\<forall>s \<in> sets M. space M - s \<in> sets M) \<and>
- (\<forall>A. range A \<subseteq> sets M \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
+ "sigma_algebra \<Omega> M \<longleftrightarrow>
+ M \<subseteq> Pow \<Omega> \<and>
+ {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
+ (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
algebra_iff_Un Un_range_binary)
subsection {* Initial Sigma Algebra *}
text {*Sigma algebras can naturally be created as the closure of any set of
- sets with regard to the properties just postulated. *}
+ M with regard to the properties just postulated. *}
inductive_set
sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
for sp :: "'a set" and A :: "'a set set"
where
- Basic: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
+ Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
| Empty: "{} \<in> sigma_sets sp A"
| Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
| Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
-definition
- "sigma M = \<lparr> space = space M, sets = sigma_sets (space M) (sets M), \<dots> = more M \<rparr>"
-
lemma (in sigma_algebra) sigma_sets_subset:
- assumes a: "a \<subseteq> sets M"
- shows "sigma_sets (space M) a \<subseteq> sets M"
+ assumes a: "a \<subseteq> M"
+ shows "sigma_sets \<Omega> a \<subseteq> M"
proof
fix x
- assume "x \<in> sigma_sets (space M) a"
- from this show "x \<in> sets M"
+ assume "x \<in> sigma_sets \<Omega> a"
+ from this show "x \<in> M"
by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
qed
@@ -359,35 +359,28 @@
by (erule sigma_sets.induct, auto)
lemma sigma_algebra_sigma_sets:
- "a \<subseteq> Pow (space M) \<Longrightarrow> sets M = sigma_sets (space M) a \<Longrightarrow> sigma_algebra M"
+ "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
lemma sigma_sets_least_sigma_algebra:
assumes "A \<subseteq> Pow S"
- shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra \<lparr>space = S, sets = B\<rparr>}"
+ shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
proof safe
- fix B X assume "A \<subseteq> B" and sa: "sigma_algebra \<lparr> space = S, sets = B \<rparr>"
+ fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
and X: "X \<in> sigma_sets S A"
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
show "X \<in> B" by auto
next
- fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra \<lparr>space = S, sets = B\<rparr>}"
- then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra \<lparr>space = S, sets = B\<rparr> \<Longrightarrow> X \<in> B"
+ fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
+ then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
by simp
- have "A \<subseteq> sigma_sets S A" using assms
- by (auto intro!: sigma_sets.Basic)
- moreover have "sigma_algebra \<lparr>space = S, sets = sigma_sets S A\<rparr>"
+ have "A \<subseteq> sigma_sets S A" using assms by auto
+ moreover have "sigma_algebra S (sigma_sets S A)"
using assms by (intro sigma_algebra_sigma_sets[of A]) auto
ultimately show "X \<in> sigma_sets S A" by auto
qed
-lemma sets_sigma: "sets (sigma M) = sigma_sets (space M) (sets M)"
- unfolding sigma_def by simp
-
-lemma space_sigma [simp]: "space (sigma M) = space M"
- by (simp add: sigma_def)
-
lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
@@ -421,7 +414,7 @@
shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
proof -
from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
- by (simp add: sigma_sets.intros sigma_sets_top)
+ by (simp add: sigma_sets.intros(2-) sigma_sets_top)
hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
by (rule sigma_sets_Inter [OF Asb])
also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
@@ -430,12 +423,12 @@
qed
lemma (in sigma_algebra) sigma_sets_eq:
- "sigma_sets (space M) (sets M) = sets M"
+ "sigma_sets \<Omega> M = M"
proof
- show "sets M \<subseteq> sigma_sets (space M) (sets M)"
+ show "M \<subseteq> sigma_sets \<Omega> M"
by (metis Set.subsetI sigma_sets.Basic)
next
- show "sigma_sets (space M) (sets M) \<subseteq> sets M"
+ show "sigma_sets \<Omega> M \<subseteq> M"
by (metis sigma_sets_subset subset_refl)
qed
@@ -446,52 +439,42 @@
proof (intro set_eqI iffI)
fix a assume "a \<in> sigma_sets M A"
from this A show "a \<in> sigma_sets M B"
- by induct (auto intro!: sigma_sets.intros del: sigma_sets.Basic)
+ by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
next
fix b assume "b \<in> sigma_sets M B"
from this B show "b \<in> sigma_sets M A"
- by induct (auto intro!: sigma_sets.intros del: sigma_sets.Basic)
+ by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
qed
-lemma sigma_algebra_sigma:
- "sets M \<subseteq> Pow (space M) \<Longrightarrow> sigma_algebra (sigma M)"
- apply (rule sigma_algebra_sigma_sets)
- apply (auto simp add: sigma_def)
- done
-
-lemma (in sigma_algebra) sigma_subset:
- "sets N \<subseteq> sets M \<Longrightarrow> space N = space M \<Longrightarrow> sets (sigma N) \<subseteq> (sets M)"
- by (simp add: sigma_def sigma_sets_subset)
-
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
- by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
+ by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
qed
lemma (in sigma_algebra) restriction_in_sets:
fixes A :: "nat \<Rightarrow> 'a set"
- assumes "S \<in> sets M"
- and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` sets M" (is "_ \<subseteq> ?r")
- shows "range A \<subseteq> sets M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` sets M"
+ assumes "S \<in> M"
+ and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
+ shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
proof -
{ fix i have "A i \<in> ?r" using * by auto
- hence "\<exists>B. A i = B \<inter> S \<and> B \<in> sets M" by auto
- hence "A i \<subseteq> S" "A i \<in> sets M" using `S \<in> sets M` by auto }
- thus "range A \<subseteq> sets M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` sets M"
+ hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
+ hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
+ thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
qed
lemma (in sigma_algebra) restricted_sigma_algebra:
- assumes "S \<in> sets M"
- shows "sigma_algebra (restricted_space S)"
+ assumes "S \<in> M"
+ shows "sigma_algebra S (restricted_space S)"
unfolding sigma_algebra_def sigma_algebra_axioms_def
proof safe
- show "algebra (restricted_space S)" using restricted_algebra[OF assms] .
+ show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
next
- fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets (restricted_space S)"
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
from restriction_in_sets[OF assms this[simplified]]
- show "(\<Union>i. A i) \<in> sets (restricted_space S)" by simp
+ show "(\<Union>i. A i) \<in> restricted_space S" by simp
qed
lemma sigma_sets_Int:
@@ -510,7 +493,7 @@
then show ?case
by (auto intro!: sigma_sets.Union
simp add: UN_extend_simps simp del: UN_simps)
- qed (auto intro!: sigma_sets.intros)
+ qed (auto intro!: sigma_sets.intros(2-))
then show "x \<in> sigma_sets A (op \<inter> A ` st)"
using `A \<subseteq> sp` by (simp add: Int_absorb2)
next
@@ -529,93 +512,75 @@
then show ?case
by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
simp add: image_iff)
- qed (auto intro!: sigma_sets.intros)
+ qed (auto intro!: sigma_sets.intros(2-))
qed
-lemma sigma_sets_single[simp]: "sigma_sets {X} {{X}} = {{}, {X}}"
+lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
proof (intro set_eqI iffI)
- fix x assume "x \<in> sigma_sets {X} {{X}}"
- from sigma_sets_into_sp[OF _ this]
- show "x \<in> {{}, {X}}" by auto
+ fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
+ by induct blast+
+qed (auto intro: sigma_sets.Empty sigma_sets_top)
+
+lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
+proof (intro set_eqI iffI)
+ fix x assume "x \<in> sigma_sets A {A}"
+ then show "x \<in> {{}, A}"
+ by induct blast+
next
- fix x assume "x \<in> {{}, {X}}"
- then show "x \<in> sigma_sets {X} {{X}}"
+ fix x assume "x \<in> {{}, A}"
+ then show "x \<in> sigma_sets A {A}"
by (auto intro: sigma_sets.Empty sigma_sets_top)
qed
-lemma (in sigma_algebra) sets_sigma_subset:
- assumes "space N = space M"
- assumes "sets N \<subseteq> sets M"
- shows "sets (sigma N) \<subseteq> sets M"
- by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms)
-
-lemma in_sigma[intro, simp]: "A \<in> sets M \<Longrightarrow> A \<in> sets (sigma M)"
- unfolding sigma_def by (auto intro!: sigma_sets.Basic)
-
-lemma (in sigma_algebra) sigma_eq[simp]: "sigma M = M"
- unfolding sigma_def sigma_sets_eq by simp
-
-lemma sigma_sigma_eq:
- assumes "sets M \<subseteq> Pow (space M)"
- shows "sigma (sigma M) = sigma M"
- using sigma_algebra.sigma_eq[OF sigma_algebra_sigma, OF assms] .
-
lemma sigma_sets_sigma_sets_eq:
"M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
- using sigma_sigma_eq[of "\<lparr> space = S, sets = M \<rparr>"]
- by (simp add: sigma_def)
+ by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
lemma sigma_sets_singleton:
assumes "X \<subseteq> S"
shows "sigma_sets S { X } = { {}, X, S - X, S }"
proof -
- interpret sigma_algebra "\<lparr> space = S, sets = { {}, X, S - X, S }\<rparr>"
+ interpret sigma_algebra S "{ {}, X, S - X, S }"
by (rule sigma_algebra_single_set) fact
have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
by (rule sigma_sets_subseteq) simp
moreover have "\<dots> = { {}, X, S - X, S }"
- using sigma_eq unfolding sigma_def by simp
+ using sigma_sets_eq by simp
moreover
{ fix A assume "A \<in> { {}, X, S - X, S }"
then have "A \<in> sigma_sets S { X }"
- by (auto intro: sigma_sets.intros sigma_sets_top) }
+ by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
by (intro antisym) auto
- with sigma_eq show ?thesis
- unfolding sigma_def by simp
+ with sigma_sets_eq show ?thesis by simp
qed
lemma restricted_sigma:
- assumes S: "S \<in> sets (sigma M)" and M: "sets M \<subseteq> Pow (space M)"
- shows "algebra.restricted_space (sigma M) S = sigma (algebra.restricted_space M S)"
+ assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
+ shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
+ sigma_sets S (algebra.restricted_space M S)"
proof -
from S sigma_sets_into_sp[OF M]
- have "S \<in> sigma_sets (space M) (sets M)" "S \<subseteq> space M"
- by (auto simp: sigma_def)
+ have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
from sigma_sets_Int[OF this]
- show ?thesis
- by (simp add: sigma_def)
+ show ?thesis by simp
qed
lemma sigma_sets_vimage_commute:
- assumes X: "X \<in> space M \<rightarrow> space M'"
- shows "{X -` A \<inter> space M |A. A \<in> sets (sigma M')}
- = sigma_sets (space M) {X -` A \<inter> space M |A. A \<in> sets M'}" (is "?L = ?R")
+ assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
+ shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
+ = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
proof
show "?L \<subseteq> ?R"
proof clarify
- fix A assume "A \<in> sets (sigma M')"
- then have "A \<in> sigma_sets (space M') (sets M')" by (simp add: sets_sigma)
- then show "X -` A \<inter> space M \<in> ?R"
+ fix A assume "A \<in> sigma_sets \<Omega>' M'"
+ then show "X -` A \<inter> \<Omega> \<in> ?R"
proof induct
- case (Basic B) then show ?case
- by (auto intro!: sigma_sets.Basic)
- next
case Empty then show ?case
by (auto intro!: sigma_sets.Empty)
next
case (Compl B)
- have [simp]: "X -` (space M' - B) \<inter> space M = space M - (X -` B \<inter> space M)"
+ have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
by (auto simp add: funcset_mem [OF X])
with Compl show ?case
by (auto intro!: sigma_sets.Compl)
@@ -624,194 +589,34 @@
then show ?case
by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
intro!: sigma_sets.Union)
- qed
+ qed auto
qed
show "?R \<subseteq> ?L"
proof clarify
fix A assume "A \<in> ?R"
- then show "\<exists>B. A = X -` B \<inter> space M \<and> B \<in> sets (sigma M')"
+ then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
proof induct
case (Basic B) then show ?case by auto
next
case Empty then show ?case
- by (auto simp: sets_sigma intro!: sigma_sets.Empty exI[of _ "{}"])
+ by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
next
case (Compl B)
- then obtain A where A: "B = X -` A \<inter> space M" "A \<in> sets (sigma M')" by auto
- then have [simp]: "space M - B = X -` (space M' - A) \<inter> space M"
+ then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
+ then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
by (auto simp add: funcset_mem [OF X])
with A(2) show ?case
- by (auto simp: sets_sigma intro: sigma_sets.Compl)
+ by (auto intro: sigma_sets.Compl)
next
case (Union F)
- then have "\<forall>i. \<exists>B. F i = X -` B \<inter> space M \<and> B \<in> sets (sigma M')" by auto
+ then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
from choice[OF this] guess A .. note A = this
with A show ?case
- by (auto simp: sets_sigma vimage_UN[symmetric] intro: sigma_sets.Union)
+ by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
qed
qed
qed
-section {* Measurable functions *}
-
-definition
- "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
-
-lemma (in sigma_algebra) measurable_sigma:
- assumes B: "sets N \<subseteq> Pow (space N)"
- and f: "f \<in> space M -> space N"
- and ba: "\<And>y. y \<in> sets N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
- shows "f \<in> measurable M (sigma N)"
-proof -
- have "sigma_sets (space N) (sets N) \<subseteq> {y . (f -` y) \<inter> space M \<in> sets M & y \<subseteq> space N}"
- proof clarify
- fix x
- assume "x \<in> sigma_sets (space N) (sets N)"
- thus "f -` x \<inter> space M \<in> sets M \<and> x \<subseteq> space N"
- proof induct
- case (Basic a)
- thus ?case
- by (auto simp add: ba) (metis B subsetD PowD)
- next
- case Empty
- thus ?case
- by auto
- next
- case (Compl a)
- have [simp]: "f -` space N \<inter> space M = space M"
- by (auto simp add: funcset_mem [OF f])
- thus ?case
- by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl)
- next
- case (Union a)
- thus ?case
- by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
- qed
- qed
- thus ?thesis
- by (simp add: measurable_def sigma_algebra_axioms sigma_algebra_sigma B f)
- (auto simp add: sigma_def)
-qed
-
-lemma measurable_cong:
- assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
- shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
- unfolding measurable_def using assms
- by (simp cong: vimage_inter_cong Pi_cong)
-
-lemma measurable_space:
- "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
- unfolding measurable_def by auto
-
-lemma measurable_sets:
- "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
- unfolding measurable_def by auto
-
-lemma (in sigma_algebra) measurable_subset:
- "(\<And>S. S \<in> sets A \<Longrightarrow> S \<subseteq> space A) \<Longrightarrow> measurable M A \<subseteq> measurable M (sigma A)"
- by (auto intro: measurable_sigma measurable_sets measurable_space)
-
-lemma measurable_eqI:
- "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
- sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
- \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
- by (simp add: measurable_def sigma_algebra_iff2)
-
-lemma (in sigma_algebra) measurable_const[intro, simp]:
- assumes "c \<in> space M'"
- shows "(\<lambda>x. c) \<in> measurable M M'"
- using assms by (auto simp add: measurable_def)
-
-lemma (in sigma_algebra) measurable_If:
- assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
- assumes P: "{x\<in>space M. P x} \<in> sets M"
- shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
- unfolding measurable_def
-proof safe
- fix x assume "x \<in> space M"
- thus "(if P x then f x else g x) \<in> space M'"
- using measure unfolding measurable_def by auto
-next
- fix A assume "A \<in> sets M'"
- hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
- ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
- ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
- using measure unfolding measurable_def by (auto split: split_if_asm)
- show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
- using `A \<in> sets M'` measure P unfolding * measurable_def
- by (auto intro!: Un)
-qed
-
-lemma (in sigma_algebra) measurable_If_set:
- assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
- assumes P: "A \<in> sets M"
- shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
-proof (rule measurable_If[OF measure])
- have "{x \<in> space M. x \<in> A} = A" using `A \<in> sets M` sets_into_space by auto
- thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<in> sets M` by auto
-qed
-
-lemma (in ring_of_sets) measurable_ident[intro, simp]: "id \<in> measurable M M"
- by (auto simp add: measurable_def)
-
-lemma measurable_comp[intro]:
- fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
- shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c"
- apply (auto simp add: measurable_def vimage_compose)
- apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a")
- apply force+
- done
-
-lemma measurable_strong:
- fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
- assumes f: "f \<in> measurable a b" and g: "g \<in> (space b -> space c)"
- and a: "sigma_algebra a" and b: "sigma_algebra b" and c: "sigma_algebra c"
- and t: "f ` (space a) \<subseteq> t"
- and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
- shows "(g o f) \<in> measurable a c"
-proof -
- have fab: "f \<in> (space a -> space b)"
- and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
- by (auto simp add: measurable_def)
- have eq: "f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
- by force
- show ?thesis
- apply (auto simp add: measurable_def vimage_compose a c)
- apply (metis funcset_mem fab g)
- apply (subst eq, metis ba cb)
- done
-qed
-
-lemma measurable_mono1:
- "a \<subseteq> b \<Longrightarrow> sigma_algebra \<lparr>space = X, sets = b\<rparr>
- \<Longrightarrow> measurable \<lparr>space = X, sets = a\<rparr> c \<subseteq> measurable \<lparr>space = X, sets = b\<rparr> c"
- by (auto simp add: measurable_def)
-
-lemma measurable_up_sigma:
- "measurable A M \<subseteq> measurable (sigma A) M"
- unfolding measurable_def
- by (auto simp: sigma_def intro: sigma_sets.Basic)
-
-lemma (in sigma_algebra) measurable_range_reduce:
- "\<lbrakk> f \<in> measurable M \<lparr>space = s, sets = Pow s\<rparr> ; s \<noteq> {} \<rbrakk>
- \<Longrightarrow> f \<in> measurable M \<lparr>space = s \<inter> (f ` space M), sets = Pow (s \<inter> (f ` space M))\<rparr>"
- by (simp add: measurable_def sigma_algebra_Pow del: Pow_Int_eq) blast
-
-lemma (in sigma_algebra) measurable_Pow_to_Pow:
- "(sets M = Pow (space M)) \<Longrightarrow> f \<in> measurable M \<lparr>space = UNIV, sets = Pow UNIV\<rparr>"
- by (auto simp add: measurable_def sigma_algebra_def sigma_algebra_axioms_def algebra_def)
-
-lemma (in sigma_algebra) measurable_Pow_to_Pow_image:
- "sets M = Pow (space M)
- \<Longrightarrow> f \<in> measurable M \<lparr>space = f ` space M, sets = Pow (f ` space M)\<rparr>"
- by (simp add: measurable_def sigma_algebra_Pow) intro_locales
-
-lemma (in sigma_algebra) measurable_iff_sigma:
- assumes "sets E \<subseteq> Pow (space E)" and "f \<in> space M \<rightarrow> space E"
- shows "f \<in> measurable M (sigma E) \<longleftrightarrow> (\<forall>A\<in>sets E. f -` A \<inter> space M \<in> sets M)"
- using measurable_sigma[OF assms]
- by (fastforce simp: measurable_def sets_sigma intro: sigma_sets.intros)
-
section "Disjoint families"
definition
@@ -906,8 +711,8 @@
lemma (in ring_of_sets) UNION_in_sets:
fixes A:: "nat \<Rightarrow> 'a set"
- assumes A: "range A \<subseteq> sets M "
- shows "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
+ assumes A: "range A \<subseteq> M"
+ shows "(\<Union>i\<in>{0..<n}. A i) \<in> M"
proof (induct n)
case 0 show ?case by simp
next
@@ -917,16 +722,16 @@
qed
lemma (in ring_of_sets) range_disjointed_sets:
- assumes A: "range A \<subseteq> sets M "
- shows "range (disjointed A) \<subseteq> sets M"
+ assumes A: "range A \<subseteq> M"
+ shows "range (disjointed A) \<subseteq> M"
proof (auto simp add: disjointed_def)
fix n
- show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> sets M" using UNION_in_sets
+ show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
by (metis A Diff UNIV_I image_subset_iff)
qed
lemma (in algebra) range_disjointed_sets':
- "range A \<subseteq> sets M \<Longrightarrow> range (disjointed A) \<subseteq> sets M"
+ "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
using range_disjointed_sets .
lemma disjointed_0[simp]: "disjointed A 0 = A 0"
@@ -942,81 +747,518 @@
by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
lemma sigma_algebra_disjoint_iff:
- "sigma_algebra M \<longleftrightarrow>
- algebra M &
- (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow>
- (\<Union>i::nat. A i) \<in> sets M)"
+ "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
+ (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
proof (auto simp add: sigma_algebra_iff)
fix A :: "nat \<Rightarrow> 'a set"
- assume M: "algebra M"
- and A: "range A \<subseteq> sets M"
- and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>
- disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
- hence "range (disjointed A) \<subseteq> sets M \<longrightarrow>
+ assume M: "algebra \<Omega> M"
+ and A: "range A \<subseteq> M"
+ and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
+ hence "range (disjointed A) \<subseteq> M \<longrightarrow>
disjoint_family (disjointed A) \<longrightarrow>
- (\<Union>i. disjointed A i) \<in> sets M" by blast
- hence "(\<Union>i. disjointed A i) \<in> sets M"
- by (simp add: algebra.range_disjointed_sets' M A disjoint_family_disjointed)
- thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
+ (\<Union>i. disjointed A i) \<in> M" by blast
+ hence "(\<Union>i. disjointed A i) \<in> M"
+ by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
+ thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
+qed
+
+section {* Measure type *}
+
+definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+ "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
+
+definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+ "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
+ (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
+
+definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+ "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
+
+typedef (open) 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
+proof
+ have "sigma_algebra UNIV {{}, UNIV}"
+ proof qed auto
+ then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
+ by (auto simp: measure_space_def positive_def countably_additive_def)
+qed
+
+definition space :: "'a measure \<Rightarrow> 'a set" where
+ "space M = fst (Rep_measure M)"
+
+definition sets :: "'a measure \<Rightarrow> 'a set set" where
+ "sets M = fst (snd (Rep_measure M))"
+
+definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
+ "emeasure M = snd (snd (Rep_measure M))"
+
+definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
+ "measure M A = real (emeasure M A)"
+
+declare [[coercion sets]]
+
+declare [[coercion measure]]
+
+declare [[coercion emeasure]]
+
+lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
+ by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
+
+interpretation sigma_algebra "space M" "sets M" for M :: "'a measure"
+ using measure_space[of M] by (auto simp: measure_space_def)
+
+definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
+ "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, sigma_sets \<Omega> A,
+ \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
+
+abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
+
+lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
+ unfolding measure_space_def
+ by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
+
+lemma (in ring_of_sets) positive_cong_eq:
+ "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
+ by (auto simp add: positive_def)
+
+lemma (in sigma_algebra) countably_additive_eq:
+ "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
+ unfolding countably_additive_def
+ by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
+
+lemma measure_space_eq:
+ assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
+ shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
+proof -
+ interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
+ from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
+ by (auto simp: measure_space_def)
+qed
+
+lemma measure_of_eq:
+ assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
+ shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
+proof -
+ have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
+ using assms by (rule measure_space_eq)
+ with eq show ?thesis
+ by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
+qed
+
+lemma
+ assumes A: "A \<subseteq> Pow \<Omega>"
+ shows sets_measure_of[simp]: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" (is ?sets)
+ and space_measure_of[simp]: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
+proof -
+ have "?sets \<and> ?space"
+ proof cases
+ assume "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
+ moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
+ (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
+ using A by (rule measure_space_eq) auto
+ ultimately show "?sets \<and> ?space"
+ by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def)
+ next
+ assume "\<not> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
+ with A show "?sets \<and> ?space"
+ by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def measure_space_0)
+ qed
+ then show ?sets ?space by auto
+qed
+
+lemma (in sigma_algebra) sets_measure_of_eq[simp]:
+ "sets (measure_of \<Omega> M \<mu>) = M"
+ using space_closed by (auto intro!: sigma_sets_eq)
+
+lemma (in sigma_algebra) space_measure_of_eq[simp]:
+ "space (measure_of \<Omega> M \<mu>) = \<Omega>"
+ using space_closed by (auto intro!: sigma_sets_eq)
+
+lemma measure_of_subset:
+ "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
+ by (auto intro!: sigma_sets_subseteq)
+
+lemma in_extended_measure[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
+ by auto
+
+section {* Constructing simple @{typ "'a measure"} *}
+
+lemma emeasure_measure_of:
+ assumes M: "M = measure_of \<Omega> A \<mu>"
+ assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
+ assumes X: "X \<in> sets M"
+ shows "emeasure M X = \<mu> X"
+proof -
+ interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
+ have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
+ using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
+ moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)
+ = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
+ using ms(1) by (rule measure_space_eq) auto
+ moreover have "X \<in> sigma_sets \<Omega> A"
+ using X M ms by simp
+ ultimately show ?thesis
+ unfolding emeasure_def measure_of_def M
+ by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
+qed
+
+lemma emeasure_measure_of_sigma:
+ assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
+ assumes A: "A \<in> M"
+ shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
+proof -
+ interpret sigma_algebra \<Omega> M by fact
+ have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+ using ms sigma_sets_eq by (simp add: measure_space_def)
+ moreover have "measure_space \<Omega> (sigma_sets \<Omega> M) (\<lambda>a. if a \<in> sigma_sets \<Omega> M then \<mu> a else 0)
+ = measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+ using space_closed by (rule measure_space_eq) auto
+ ultimately show ?thesis using A
+ unfolding emeasure_def measure_of_def
+ by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
+qed
+
+lemma measure_cases[cases type: measure]:
+ obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
+ by atomize_elim (cases x, auto)
+
+lemma sets_eq_imp_space_eq:
+ "sets M = sets M' \<Longrightarrow> space M = space M'"
+ using top[of M] top[of M'] space_closed[of M] space_closed[of M']
+ by blast
+
+lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
+ by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
+
+lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
+ by (simp add: measure_def emeasure_notin_sets)
+
+lemma measure_eqI:
+ fixes M N :: "'a measure"
+ assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
+ shows "M = N"
+proof (cases M N rule: measure_cases[case_product measure_cases])
+ case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
+ interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
+ interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
+ have "A = sets M" "A' = sets N"
+ using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
+ with `sets M = sets N` have "A = A'" by simp
+ moreover with M.top N.top M.space_closed N.space_closed have "\<Omega> = \<Omega>'" by auto
+ moreover { fix B have "\<mu> B = \<mu>' B"
+ proof cases
+ assume "B \<in> A"
+ with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
+ with measure_measure show "\<mu> B = \<mu>' B"
+ by (simp add: emeasure_def Abs_measure_inverse)
+ next
+ assume "B \<notin> A"
+ with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
+ by auto
+ then have "emeasure M B = 0" "emeasure N B = 0"
+ by (simp_all add: emeasure_notin_sets)
+ with measure_measure show "\<mu> B = \<mu>' B"
+ by (simp add: emeasure_def Abs_measure_inverse)
+ qed }
+ then have "\<mu> = \<mu>'" by auto
+ ultimately show "M = N"
+ by (simp add: measure_measure)
qed
+lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
+ using measure_space_0[of A \<Omega>]
+ by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
+
+lemma sigma_eqI:
+ assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
+ shows "sigma \<Omega> M = sigma \<Omega> N"
+ by (rule measure_eqI) (simp_all add: emeasure_sigma)
+
+section {* Measurable functions *}
+
+definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
+ "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
+
+lemma measurable_space:
+ "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
+ unfolding measurable_def by auto
+
+lemma measurable_sets:
+ "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
+ unfolding measurable_def by auto
+
+lemma measurable_sigma_sets:
+ assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
+ and f: "f \<in> space M \<rightarrow> \<Omega>"
+ and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
+ shows "f \<in> measurable M N"
+proof -
+ interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
+ from B top[of N] A.top space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
+
+ { fix X assume "X \<in> sigma_sets \<Omega> A"
+ then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
+ proof induct
+ case (Basic a) then show ?case
+ by (auto simp add: ba) (metis B(2) subsetD PowD)
+ next
+ case (Compl a)
+ have [simp]: "f -` \<Omega> \<inter> space M = space M"
+ by (auto simp add: funcset_mem [OF f])
+ then show ?case
+ by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl)
+ next
+ case (Union a)
+ then show ?case
+ by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
+ qed auto }
+ with f show ?thesis
+ by (auto simp add: measurable_def B \<Omega>)
+qed
+
+lemma measurable_measure_of:
+ assumes B: "N \<subseteq> Pow \<Omega>"
+ and f: "f \<in> space M \<rightarrow> \<Omega>"
+ and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
+ shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
+proof -
+ have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
+ using B by (rule sets_measure_of)
+ from this assms show ?thesis by (rule measurable_sigma_sets)
+qed
+
+lemma measurable_iff_measure_of:
+ assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
+ shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
+ by (metis assms in_extended_measure measurable_measure_of assms measurable_sets)
+
+lemma measurable_cong:
+ assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
+ shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
+ unfolding measurable_def using assms
+ by (simp cong: vimage_inter_cong Pi_cong)
+
+lemma measurable_eqI:
+ "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
+ sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
+ \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
+ by (simp add: measurable_def sigma_algebra_iff2)
+
+lemma measurable_const[intro, simp]:
+ "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
+ by (auto simp add: measurable_def)
+
+lemma measurable_If:
+ assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
+ assumes P: "{x\<in>space M. P x} \<in> sets M"
+ shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
+ unfolding measurable_def
+proof safe
+ fix x assume "x \<in> space M"
+ thus "(if P x then f x else g x) \<in> space M'"
+ using measure unfolding measurable_def by auto
+next
+ fix A assume "A \<in> sets M'"
+ hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
+ ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
+ ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
+ using measure unfolding measurable_def by (auto split: split_if_asm)
+ show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
+ using `A \<in> sets M'` measure P unfolding * measurable_def
+ by (auto intro!: Un)
+qed
+
+lemma measurable_If_set:
+ assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
+ assumes P: "A \<in> sets M"
+ shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
+proof (rule measurable_If[OF measure])
+ have "{x \<in> space M. x \<in> A} = A" using `A \<in> sets M` sets_into_space by auto
+ thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<in> sets M` by auto
+qed
+
+lemma measurable_ident[intro, simp]: "id \<in> measurable M M"
+ by (auto simp add: measurable_def)
+
+lemma measurable_comp[intro]:
+ fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
+ shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c"
+ apply (auto simp add: measurable_def vimage_compose)
+ apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a")
+ apply force+
+ done
+
+lemma measurable_Least:
+ assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
+ shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
+proof -
+ { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
+ proof cases
+ assume i: "(LEAST j. False) = i"
+ have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
+ {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
+ by (simp add: set_eq_iff, safe)
+ (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
+ with meas show ?thesis
+ by (auto intro!: Int)
+ next
+ assume i: "(LEAST j. False) \<noteq> i"
+ then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
+ {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
+ proof (simp add: set_eq_iff, safe)
+ fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
+ have "\<exists>j. P j x"
+ by (rule ccontr) (insert neq, auto)
+ then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
+ qed (auto dest: Least_le intro!: Least_equality)
+ with meas show ?thesis
+ by auto
+ qed }
+ then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
+ by (intro countable_UN) auto
+ moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
+ (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma measurable_strong:
+ fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
+ assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
+ and t: "f ` (space a) \<subseteq> t"
+ and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
+ shows "(g o f) \<in> measurable a c"
+proof -
+ have fab: "f \<in> (space a -> space b)"
+ and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
+ by (auto simp add: measurable_def)
+ have eq: "\<And>y. f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
+ by force
+ show ?thesis
+ apply (auto simp add: measurable_def vimage_compose)
+ apply (metis funcset_mem fab g)
+ apply (subst eq, metis ba cb)
+ done
+qed
+
+lemma measurable_mono1:
+ "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
+ measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
+ using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
+
+subsection {* Extend measure *}
+
+definition "extend_measure \<Omega> I G \<mu> =
+ (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
+ then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>')
+ else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
+
+lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
+ unfolding extend_measure_def by simp
+
+lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
+ unfolding extend_measure_def by simp
+
+lemma emeasure_extend_measure:
+ assumes M: "M = extend_measure \<Omega> I G \<mu>"
+ and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
+ and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
+ and "i \<in> I"
+ shows "emeasure M (G i) = \<mu> i"
+proof cases
+ assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
+ with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
+ by (simp add: extend_measure_def)
+ from measure_space_0[OF ms(1)] ms `i\<in>I`
+ have "emeasure M (G i) = 0"
+ by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
+ with `i\<in>I` * show ?thesis
+ by simp
+next
+ def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
+ assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
+ moreover
+ have "measure_space (space M) (sets M) \<mu>'"
+ using ms unfolding measure_space_def by auto default
+ with ms eq have "\<exists>\<mu>'. P \<mu>'"
+ unfolding P_def
+ by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
+ ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
+ by (simp add: M extend_measure_def P_def[symmetric])
+
+ from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
+ show "emeasure M (G i) = \<mu> i"
+ proof (subst emeasure_measure_of[OF M_eq])
+ have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
+ using M_eq ms by (auto simp: sets_extend_measure)
+ then show "G i \<in> sets M" using `i \<in> I` by auto
+ show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
+ using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
+ qed fact
+qed
+
+lemma emeasure_extend_measure_Pair:
+ assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
+ and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
+ and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
+ and "I i j"
+ shows "emeasure M (G i j) = \<mu> i j"
+ using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
+ by (auto simp: subset_eq)
+
subsection {* Sigma algebra generated by function preimages *}
-definition (in sigma_algebra)
- "vimage_algebra S f = \<lparr> space = S, sets = (\<lambda>A. f -` A \<inter> S) ` sets M, \<dots> = more M \<rparr>"
-
-lemma (in sigma_algebra) in_vimage_algebra[simp]:
- "A \<in> sets (vimage_algebra S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
- by (simp add: vimage_algebra_def image_iff)
+definition
+ "vimage_algebra M S f = sigma S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
-lemma (in sigma_algebra) space_vimage_algebra[simp]:
- "space (vimage_algebra S f) = S"
- by (simp add: vimage_algebra_def)
-
-lemma (in sigma_algebra) sigma_algebra_preimages:
+lemma sigma_algebra_preimages:
fixes f :: "'x \<Rightarrow> 'a"
- assumes "f \<in> A \<rightarrow> space M"
- shows "sigma_algebra \<lparr> space = A, sets = (\<lambda>M. f -` M \<inter> A) ` sets M \<rparr>"
- (is "sigma_algebra \<lparr> space = _, sets = ?F ` sets M \<rparr>")
+ assumes "f \<in> S \<rightarrow> space M"
+ shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
+ (is "sigma_algebra _ (?F ` sets M)")
proof (simp add: sigma_algebra_iff2, safe)
show "{} \<in> ?F ` sets M" by blast
next
- fix S assume "S \<in> sets M"
- moreover have "A - ?F S = ?F (space M - S)"
+ fix A assume "A \<in> sets M"
+ moreover have "S - ?F A = ?F (space M - A)"
using assms by auto
- ultimately show "A - ?F S \<in> ?F ` sets M"
+ ultimately show "S - ?F A \<in> ?F ` sets M"
by blast
next
- fix S :: "nat \<Rightarrow> 'x set" assume *: "range S \<subseteq> ?F ` sets M"
- have "\<forall>i. \<exists>b. b \<in> sets M \<and> S i = ?F b"
+ fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
+ have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
proof safe
fix i
- have "S i \<in> ?F ` sets M" using * by auto
- then show "\<exists>b. b \<in> sets M \<and> S i = ?F b" by auto
+ have "A i \<in> ?F ` M" using * by auto
+ then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
qed
- from choice[OF this] obtain b where b: "range b \<subseteq> sets M" "\<And>i. S i = ?F (b i)"
+ from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
by auto
- then have "(\<Union>i. S i) = ?F (\<Union>i. b i)" by auto
- then show "(\<Union>i. S i) \<in> ?F ` sets M" using b(1) by blast
+ then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
+ then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
qed
-lemma (in sigma_algebra) sigma_algebra_vimage:
+lemma sets_vimage_algebra[simp]:
+ "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
+ using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]
+ by (simp add: vimage_algebra_def)
+
+lemma space_vimage_algebra[simp]:
+ "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
+ using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]
+ by (simp add: vimage_algebra_def)
+
+lemma in_vimage_algebra[simp]:
+ "f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
+ by (simp add: image_iff)
+
+lemma measurable_vimage_algebra:
fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
- shows "sigma_algebra (vimage_algebra S f)"
-proof -
- from sigma_algebra_preimages[OF assms]
- show ?thesis unfolding vimage_algebra_def by (auto simp: sigma_algebra_iff2)
-qed
+ shows "f \<in> measurable (vimage_algebra M S f) M"
+ unfolding measurable_def using assms by force
-lemma (in sigma_algebra) measurable_vimage_algebra:
- fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
- shows "f \<in> measurable (vimage_algebra S f) M"
- unfolding measurable_def using assms by force
-
-lemma (in sigma_algebra) measurable_vimage:
+lemma measurable_vimage:
fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
- shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra S f) M2"
+ shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
proof -
note measurable_vimage_algebra[OF assms(2)]
from measurable_comp[OF this assms(1)]
@@ -1033,7 +1275,7 @@
proof induct
case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
by auto
- then show ?case by (auto intro!: sigma_sets.Basic)
+ then show ?case by auto
next
case Empty then show ?case
by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
@@ -1060,8 +1302,6 @@
then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
then show "X \<in> sigma_sets S' (?F ` A)"
proof (induct arbitrary: X)
- case (Basic X') then show ?case by (auto intro: sigma_sets.Basic)
- next
case Empty then show ?case by (auto intro: sigma_sets.Empty)
next
case (Compl X')
@@ -1078,23 +1318,7 @@
also have "(\<Union>i. f -` F i \<inter> S') = X"
using assms Union by auto
finally show ?case .
- qed
-qed
-
-section {* Conditional space *}
-
-definition (in algebra)
- "image_space X = \<lparr> space = X`space M, sets = (\<lambda>S. X`S) ` sets M, \<dots> = more M \<rparr>"
-
-definition (in algebra)
- "conditional_space X A = algebra.image_space (restricted_space A) X"
-
-lemma (in algebra) space_conditional_space:
- assumes "A \<in> sets M" shows "space (conditional_space X A) = X`A"
-proof -
- interpret r: algebra "restricted_space A" using assms by (rule restricted_algebra)
- show ?thesis unfolding conditional_space_def r.image_space_def
- by simp
+ qed auto
qed
subsection {* A Two-Element Series *}
@@ -1113,80 +1337,64 @@
section {* Closed CDI *}
-definition
- closed_cdi where
- "closed_cdi M \<longleftrightarrow>
- sets M \<subseteq> Pow (space M) &
- (\<forall>s \<in> sets M. space M - s \<in> sets M) &
- (\<forall>A. (range A \<subseteq> sets M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
- (\<Union>i. A i) \<in> sets M) &
- (\<forall>A. (range A \<subseteq> sets M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
+definition closed_cdi where
+ "closed_cdi \<Omega> M \<longleftrightarrow>
+ M \<subseteq> Pow \<Omega> &
+ (\<forall>s \<in> M. \<Omega> - s \<in> M) &
+ (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
+ (\<Union>i. A i) \<in> M) &
+ (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
inductive_set
- smallest_ccdi_sets :: "('a, 'b) algebra_scheme \<Rightarrow> 'a set set"
- for M
+ smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
+ for \<Omega> M
where
Basic [intro]:
- "a \<in> sets M \<Longrightarrow> a \<in> smallest_ccdi_sets M"
+ "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
| Compl [intro]:
- "a \<in> smallest_ccdi_sets M \<Longrightarrow> space M - a \<in> smallest_ccdi_sets M"
+ "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
| Inc:
- "range A \<in> Pow(smallest_ccdi_sets M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
- \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets M"
+ "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
+ \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
| Disj:
- "range A \<in> Pow(smallest_ccdi_sets M) \<Longrightarrow> disjoint_family A
- \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets M"
-
+ "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
+ \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
-definition
- smallest_closed_cdi where
- "smallest_closed_cdi M = (|space = space M, sets = smallest_ccdi_sets M|)"
+lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
+ by auto
-lemma space_smallest_closed_cdi [simp]:
- "space (smallest_closed_cdi M) = space M"
- by (simp add: smallest_closed_cdi_def)
-
-lemma (in algebra) smallest_closed_cdi1: "sets M \<subseteq> sets (smallest_closed_cdi M)"
- by (auto simp add: smallest_closed_cdi_def)
-
-lemma (in algebra) smallest_ccdi_sets:
- "smallest_ccdi_sets M \<subseteq> Pow (space M)"
+lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
apply (rule subsetI)
apply (erule smallest_ccdi_sets.induct)
apply (auto intro: range_subsetD dest: sets_into_space)
done
-lemma (in algebra) smallest_closed_cdi2: "closed_cdi (smallest_closed_cdi M)"
- apply (auto simp add: closed_cdi_def smallest_closed_cdi_def smallest_ccdi_sets)
+lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
+ apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
done
-lemma (in algebra) smallest_closed_cdi3:
- "sets (smallest_closed_cdi M) \<subseteq> Pow (space M)"
- by (simp add: smallest_closed_cdi_def smallest_ccdi_sets)
-
-lemma closed_cdi_subset: "closed_cdi M \<Longrightarrow> sets M \<subseteq> Pow (space M)"
+lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
by (simp add: closed_cdi_def)
-lemma closed_cdi_Compl: "closed_cdi M \<Longrightarrow> s \<in> sets M \<Longrightarrow> space M - s \<in> sets M"
+lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Inc:
- "closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow>
- (\<Union>i. A i) \<in> sets M"
+ "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Disj:
- "closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
+ "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Un:
- assumes cdi: "closed_cdi M" and empty: "{} \<in> sets M"
- and A: "A \<in> sets M" and B: "B \<in> sets M"
+ assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
+ and A: "A \<in> M" and B: "B \<in> M"
and disj: "A \<inter> B = {}"
- shows "A \<union> B \<in> sets M"
+ shows "A \<union> B \<in> M"
proof -
- have ra: "range (binaryset A B) \<subseteq> sets M"
+ have ra: "range (binaryset A B) \<subseteq> M"
by (simp add: range_binaryset_eq empty A B)
have di: "disjoint_family (binaryset A B)" using disj
by (simp add: disjoint_family_on_def binaryset_def Int_commute)
@@ -1196,11 +1404,11 @@
qed
lemma (in algebra) smallest_ccdi_sets_Un:
- assumes A: "A \<in> smallest_ccdi_sets M" and B: "B \<in> smallest_ccdi_sets M"
+ assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
and disj: "A \<inter> B = {}"
- shows "A \<union> B \<in> smallest_ccdi_sets M"
+ shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
proof -
- have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets M)"
+ have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic)
have di: "disjoint_family (binaryset A B)" using disj
by (simp add: disjoint_family_on_def binaryset_def Int_commute)
@@ -1210,33 +1418,32 @@
qed
lemma (in algebra) smallest_ccdi_sets_Int1:
- assumes a: "a \<in> sets M"
- shows "b \<in> smallest_ccdi_sets M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets M"
+ assumes a: "a \<in> M"
+ shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis a Int smallest_ccdi_sets.Basic)
next
case (Compl x)
- have "a \<inter> (space M - x) = space M - ((space M - a) \<union> (a \<inter> x))"
+ have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
by blast
- also have "... \<in> smallest_ccdi_sets M"
+ also have "... \<in> smallest_ccdi_sets \<Omega> M"
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
- Diff_disjoint Int_Diff Int_empty_right Un_commute
- smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl
- smallest_ccdi_sets_Un)
+ Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
+ smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
finally show ?case .
next
case (Inc A)
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
by blast
- have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets M)" using Inc
+ have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
by blast
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
by (simp add: Inc)
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
by blast
- ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets M"
+ ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
@@ -1244,11 +1451,11 @@
case (Disj A)
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
by blast
- have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets M)" using Disj
+ have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
by blast
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
by (auto simp add: disjoint_family_on_def)
- ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets M"
+ ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
@@ -1256,17 +1463,17 @@
lemma (in algebra) smallest_ccdi_sets_Int:
- assumes b: "b \<in> smallest_ccdi_sets M"
- shows "a \<in> smallest_ccdi_sets M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets M"
+ assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
+ shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis b smallest_ccdi_sets_Int1)
next
case (Compl x)
- have "(space M - x) \<inter> b = space M - (x \<inter> b \<union> (space M - b))"
+ have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
by blast
- also have "... \<in> smallest_ccdi_sets M"
+ also have "... \<in> smallest_ccdi_sets \<Omega> M"
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
finally show ?case .
@@ -1274,13 +1481,13 @@
case (Inc A)
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
by blast
- have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets M)" using Inc
+ have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
by blast
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
by (simp add: Inc)
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
by blast
- ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets M"
+ ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
@@ -1288,39 +1495,34 @@
case (Disj A)
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
by blast
- have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets M)" using Disj
+ have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
by blast
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
by (auto simp add: disjoint_family_on_def)
- ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets M"
+ ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed
-lemma (in algebra) sets_smallest_closed_cdi_Int:
- "a \<in> sets (smallest_closed_cdi M) \<Longrightarrow> b \<in> sets (smallest_closed_cdi M)
- \<Longrightarrow> a \<inter> b \<in> sets (smallest_closed_cdi M)"
- by (simp add: smallest_ccdi_sets_Int smallest_closed_cdi_def)
-
lemma (in algebra) sigma_property_disjoint_lemma:
- assumes sbC: "sets M \<subseteq> C"
- and ccdi: "closed_cdi (|space = space M, sets = C|)"
- shows "sigma_sets (space M) (sets M) \<subseteq> C"
+ assumes sbC: "M \<subseteq> C"
+ and ccdi: "closed_cdi \<Omega> C"
+ shows "sigma_sets \<Omega> M \<subseteq> C"
proof -
- have "smallest_ccdi_sets M \<in> {B . sets M \<subseteq> B \<and> sigma_algebra (|space = space M, sets = B|)}"
+ have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
smallest_ccdi_sets_Int)
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Disj)
done
- hence "sigma_sets (space M) (sets M) \<subseteq> smallest_ccdi_sets M"
+ hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
by clarsimp
- (drule sigma_algebra.sigma_sets_subset [where a="sets M"], auto)
+ (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
also have "... \<subseteq> C"
proof
fix x
- assume x: "x \<in> smallest_ccdi_sets M"
+ assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
thus "x \<in> C"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
@@ -1344,21 +1546,21 @@
qed
lemma (in algebra) sigma_property_disjoint:
- assumes sbC: "sets M \<subseteq> C"
- and compl: "!!s. s \<in> C \<inter> sigma_sets (space M) (sets M) \<Longrightarrow> space M - s \<in> C"
- and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (space M) (sets M)
+ assumes sbC: "M \<subseteq> C"
+ and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
+ and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
\<Longrightarrow> (\<Union>i. A i) \<in> C"
- and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (space M) (sets M)
+ and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
- shows "sigma_sets (space M) (sets M) \<subseteq> C"
+ shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
proof -
- have "sigma_sets (space M) (sets M) \<subseteq> C \<inter> sigma_sets (space M) (sets M)"
+ have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
proof (rule sigma_property_disjoint_lemma)
- show "sets M \<subseteq> C \<inter> sigma_sets (space M) (sets M)"
+ show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
next
- show "closed_cdi \<lparr>space = space M, sets = C \<inter> sigma_sets (space M) (sets M)\<rparr>"
+ show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
by (simp add: closed_cdi_def compl inc disj)
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
@@ -1370,97 +1572,97 @@
section {* Dynkin systems *}
locale dynkin_system = subset_class +
- assumes space: "space M \<in> sets M"
- and compl[intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
- and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
- \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
+ assumes space: "\<Omega> \<in> M"
+ and compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
+ and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
+ \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
-lemma (in dynkin_system) empty[intro, simp]: "{} \<in> sets M"
- using space compl[of "space M"] by simp
+lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
+ using space compl[of "\<Omega>"] by simp
lemma (in dynkin_system) diff:
- assumes sets: "D \<in> sets M" "E \<in> sets M" and "D \<subseteq> E"
- shows "E - D \<in> sets M"
+ assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
+ shows "E - D \<in> M"
proof -
- let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then space M - E else {}"
- have "range ?f = {D, space M - E, {}}"
+ let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
+ have "range ?f = {D, \<Omega> - E, {}}"
by (auto simp: image_iff)
- moreover have "D \<union> (space M - E) = (\<Union>i. ?f i)"
+ moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
by (auto simp: image_iff split: split_if_asm)
moreover
then have "disjoint_family ?f" unfolding disjoint_family_on_def
- using `D \<in> sets M`[THEN sets_into_space] `D \<subseteq> E` by auto
- ultimately have "space M - (D \<union> (space M - E)) \<in> sets M"
+ using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
+ ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
using sets by auto
- also have "space M - (D \<union> (space M - E)) = E - D"
+ also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
using assms sets_into_space by auto
finally show ?thesis .
qed
lemma dynkin_systemI:
- assumes "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M" "space M \<in> sets M"
- assumes "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
- assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
- \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
- shows "dynkin_system M"
+ assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
+ assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
+ assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
+ \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
+ shows "dynkin_system \<Omega> M"
using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
lemma dynkin_systemI':
- assumes 1: "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
- assumes empty: "{} \<in> sets M"
- assumes Diff: "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
- assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
- \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
- shows "dynkin_system M"
+ assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
+ assumes empty: "{} \<in> M"
+ assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
+ assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
+ \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
+ shows "dynkin_system \<Omega> M"
proof -
- from Diff[OF empty] have "space M \<in> sets M" by auto
+ from Diff[OF empty] have "\<Omega> \<in> M" by auto
from 1 this Diff 2 show ?thesis
by (intro dynkin_systemI) auto
qed
lemma dynkin_system_trivial:
- shows "dynkin_system \<lparr> space = A, sets = Pow A \<rparr>"
+ shows "dynkin_system A (Pow A)"
by (rule dynkin_systemI) auto
lemma sigma_algebra_imp_dynkin_system:
- assumes "sigma_algebra M" shows "dynkin_system M"
+ assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
proof -
- interpret sigma_algebra M by fact
+ interpret sigma_algebra \<Omega> M by fact
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
qed
subsection "Intersection stable algebras"
-definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> sets M. \<forall> b \<in> sets M. a \<inter> b \<in> sets M)"
+definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
lemma (in algebra) Int_stable: "Int_stable M"
unfolding Int_stable_def by auto
lemma Int_stableI:
- "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable \<lparr> space = \<Omega>, sets = A \<rparr>"
+ "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
unfolding Int_stable_def by auto
lemma Int_stableD:
- "Int_stable M \<Longrightarrow> a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b \<in> sets M"
+ "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
unfolding Int_stable_def by auto
lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
- "sigma_algebra M \<longleftrightarrow> Int_stable M"
+ "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
proof
- assume "sigma_algebra M" then show "Int_stable M"
+ assume "sigma_algebra \<Omega> M" then show "Int_stable M"
unfolding sigma_algebra_def using algebra.Int_stable by auto
next
assume "Int_stable M"
- show "sigma_algebra M"
+ show "sigma_algebra \<Omega> M"
unfolding sigma_algebra_disjoint_iff algebra_iff_Un
proof (intro conjI ballI allI impI)
- show "sets M \<subseteq> Pow (space M)" using sets_into_space by auto
+ show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
next
- fix A B assume "A \<in> sets M" "B \<in> sets M"
- then have "A \<union> B = space M - ((space M - A) \<inter> (space M - B))"
- "space M - A \<in> sets M" "space M - B \<in> sets M"
+ fix A B assume "A \<in> M" "B \<in> M"
+ then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
+ "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
using sets_into_space by auto
- then show "A \<union> B \<in> sets M"
+ then show "A \<union> B \<in> M"
using `Int_stable M` unfolding Int_stable_def by auto
qed auto
qed
@@ -1468,217 +1670,148 @@
subsection "Smallest Dynkin systems"
definition dynkin where
- "dynkin M = \<lparr> space = space M,
- sets = \<Inter>{D. dynkin_system \<lparr> space = space M, sets = D \<rparr> \<and> sets M \<subseteq> D},
- \<dots> = more M \<rparr>"
+ "dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
lemma dynkin_system_dynkin:
- assumes "sets M \<subseteq> Pow (space M)"
- shows "dynkin_system (dynkin M)"
+ assumes "M \<subseteq> Pow (\<Omega>)"
+ shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
proof (rule dynkin_systemI)
- fix A assume "A \<in> sets (dynkin M)"
+ fix A assume "A \<in> dynkin \<Omega> M"
moreover
- { fix D assume "A \<in> D" and d: "dynkin_system \<lparr> space = space M, sets = D \<rparr>"
- then have "A \<subseteq> space M" by (auto simp: dynkin_system_def subset_class_def) }
- moreover have "{D. dynkin_system \<lparr> space = space M, sets = D\<rparr> \<and> sets M \<subseteq> D} \<noteq> {}"
+ { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
+ then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
+ moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
using assms dynkin_system_trivial by fastforce
- ultimately show "A \<subseteq> space (dynkin M)"
+ ultimately show "A \<subseteq> \<Omega>"
unfolding dynkin_def using assms
- by simp (metis dynkin_system_def subset_class_def in_mono)
+ by auto
next
- show "space (dynkin M) \<in> sets (dynkin M)"
+ show "\<Omega> \<in> dynkin \<Omega> M"
unfolding dynkin_def using dynkin_system.space by fastforce
next
- fix A assume "A \<in> sets (dynkin M)"
- then show "space (dynkin M) - A \<in> sets (dynkin M)"
+ fix A assume "A \<in> dynkin \<Omega> M"
+ then show "\<Omega> - A \<in> dynkin \<Omega> M"
unfolding dynkin_def using dynkin_system.compl by force
next
fix A :: "nat \<Rightarrow> 'a set"
- assume A: "disjoint_family A" "range A \<subseteq> sets (dynkin M)"
- show "(\<Union>i. A i) \<in> sets (dynkin M)" unfolding dynkin_def
+ assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
+ show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
proof (simp, safe)
- fix D assume "dynkin_system \<lparr>space = space M, sets = D\<rparr>" "sets M \<subseteq> D"
- with A have "(\<Union>i. A i) \<in> sets \<lparr>space = space M, sets = D\<rparr>"
+ fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
+ with A have "(\<Union>i. A i) \<in> D"
by (intro dynkin_system.UN) (auto simp: dynkin_def)
then show "(\<Union>i. A i) \<in> D" by auto
qed
qed
-lemma dynkin_Basic[intro]:
- "A \<in> sets M \<Longrightarrow> A \<in> sets (dynkin M)"
- unfolding dynkin_def by auto
-
-lemma dynkin_space[simp]:
- "space (dynkin M) = space M"
+lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
unfolding dynkin_def by auto
lemma (in dynkin_system) restricted_dynkin_system:
- assumes "D \<in> sets M"
- shows "dynkin_system \<lparr> space = space M,
- sets = {Q. Q \<subseteq> space M \<and> Q \<inter> D \<in> sets M} \<rparr>"
+ assumes "D \<in> M"
+ shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
proof (rule dynkin_systemI, simp_all)
- have "space M \<inter> D = D"
- using `D \<in> sets M` sets_into_space by auto
- then show "space M \<inter> D \<in> sets M"
- using `D \<in> sets M` by auto
+ have "\<Omega> \<inter> D = D"
+ using `D \<in> M` sets_into_space by auto
+ then show "\<Omega> \<inter> D \<in> M"
+ using `D \<in> M` by auto
next
- fix A assume "A \<subseteq> space M \<and> A \<inter> D \<in> sets M"
- moreover have "(space M - A) \<inter> D = (space M - (A \<inter> D)) - (space M - D)"
+ fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
+ moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
by auto
- ultimately show "space M - A \<subseteq> space M \<and> (space M - A) \<inter> D \<in> sets M"
- using `D \<in> sets M` by (auto intro: diff)
+ ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
+ using `D \<in> M` by (auto intro: diff)
next
fix A :: "nat \<Rightarrow> 'a set"
- assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> space M \<and> Q \<inter> D \<in> sets M}"
- then have "\<And>i. A i \<subseteq> space M" "disjoint_family (\<lambda>i. A i \<inter> D)"
- "range (\<lambda>i. A i \<inter> D) \<subseteq> sets M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
+ assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
+ then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
+ "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
by ((fastforce simp: disjoint_family_on_def)+)
- then show "(\<Union>x. A x) \<subseteq> space M \<and> (\<Union>x. A x) \<inter> D \<in> sets M"
+ then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
by (auto simp del: UN_simps)
qed
lemma (in dynkin_system) dynkin_subset:
- assumes "sets N \<subseteq> sets M"
- assumes "space N = space M"
- shows "sets (dynkin N) \<subseteq> sets M"
+ assumes "N \<subseteq> M"
+ shows "dynkin \<Omega> N \<subseteq> M"
proof -
- have "dynkin_system M" by default
- then have "dynkin_system \<lparr>space = space N, sets = sets M \<rparr>"
+ have "dynkin_system \<Omega> M" by default
+ then have "dynkin_system \<Omega> M"
using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
- with `sets N \<subseteq> sets M` show ?thesis by (auto simp add: dynkin_def)
+ with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
qed
lemma sigma_eq_dynkin:
- assumes sets: "sets M \<subseteq> Pow (space M)"
+ assumes sets: "M \<subseteq> Pow \<Omega>"
assumes "Int_stable M"
- shows "sigma M = dynkin M"
+ shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
proof -
- have "sets (dynkin M) \<subseteq> sigma_sets (space M) (sets M)"
+ have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
using sigma_algebra_imp_dynkin_system
- unfolding dynkin_def sigma_def sigma_sets_least_sigma_algebra[OF sets] by auto
+ unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
moreover
- interpret dynkin_system "dynkin M"
+ interpret dynkin_system \<Omega> "dynkin \<Omega> M"
using dynkin_system_dynkin[OF sets] .
- have "sigma_algebra (dynkin M)"
+ have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
unfolding sigma_algebra_eq_Int_stable Int_stable_def
proof (intro ballI)
- fix A B assume "A \<in> sets (dynkin M)" "B \<in> sets (dynkin M)"
- let ?D = "\<lambda>E. \<lparr> space = space M,
- sets = {Q. Q \<subseteq> space M \<and> Q \<inter> E \<in> sets (dynkin M)} \<rparr>"
- have "sets M \<subseteq> sets (?D B)"
+ fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
+ let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
+ have "M \<subseteq> ?D B"
proof
- fix E assume "E \<in> sets M"
- then have "sets M \<subseteq> sets (?D E)" "E \<in> sets (dynkin M)"
+ fix E assume "E \<in> M"
+ then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
- then have "sets (dynkin M) \<subseteq> sets (?D E)"
- using restricted_dynkin_system `E \<in> sets (dynkin M)`
+ then have "dynkin \<Omega> M \<subseteq> ?D E"
+ using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
by (intro dynkin_system.dynkin_subset) simp_all
- then have "B \<in> sets (?D E)"
- using `B \<in> sets (dynkin M)` by auto
- then have "E \<inter> B \<in> sets (dynkin M)"
+ then have "B \<in> ?D E"
+ using `B \<in> dynkin \<Omega> M` by auto
+ then have "E \<inter> B \<in> dynkin \<Omega> M"
by (subst Int_commute) simp
- then show "E \<in> sets (?D B)"
- using sets `E \<in> sets M` by auto
+ then show "E \<in> ?D B"
+ using sets `E \<in> M` by auto
qed
- then have "sets (dynkin M) \<subseteq> sets (?D B)"
- using restricted_dynkin_system `B \<in> sets (dynkin M)`
+ then have "dynkin \<Omega> M \<subseteq> ?D B"
+ using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
by (intro dynkin_system.dynkin_subset) simp_all
- then show "A \<inter> B \<in> sets (dynkin M)"
- using `A \<in> sets (dynkin M)` sets_into_space by auto
+ then show "A \<inter> B \<in> dynkin \<Omega> M"
+ using `A \<in> dynkin \<Omega> M` sets_into_space by auto
qed
- from sigma_algebra.sigma_sets_subset[OF this, of "sets M"]
- have "sigma_sets (space M) (sets M) \<subseteq> sets (dynkin M)" by auto
- ultimately have "sigma_sets (space M) (sets M) = sets (dynkin M)" by auto
+ from sigma_algebra.sigma_sets_subset[OF this, of "M"]
+ have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
+ ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
then show ?thesis
- by (auto intro!: algebra.equality simp: sigma_def dynkin_def)
+ by (auto simp: dynkin_def)
qed
lemma (in dynkin_system) dynkin_idem:
- "dynkin M = M"
+ "dynkin \<Omega> M = M"
proof -
- have "sets (dynkin M) = sets M"
+ have "dynkin \<Omega> M = M"
proof
- show "sets M \<subseteq> sets (dynkin M)"
+ show "M \<subseteq> dynkin \<Omega> M"
using dynkin_Basic by auto
- show "sets (dynkin M) \<subseteq> sets M"
+ show "dynkin \<Omega> M \<subseteq> M"
by (intro dynkin_subset) auto
qed
then show ?thesis
- by (auto intro!: algebra.equality simp: dynkin_def)
+ by (auto simp: dynkin_def)
qed
lemma (in dynkin_system) dynkin_lemma:
assumes "Int_stable E"
- and E: "sets E \<subseteq> sets M" "space E = space M" "sets M \<subseteq> sets (sigma E)"
- shows "sets (sigma E) = sets M"
+ and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
+ shows "sigma_sets \<Omega> E = M"
proof -
- have "sets E \<subseteq> Pow (space E)"
+ have "E \<subseteq> Pow \<Omega>"
using E sets_into_space by force
- then have "sigma E = dynkin E"
+ then have "sigma_sets \<Omega> E = dynkin \<Omega> E"
using `Int_stable E` by (rule sigma_eq_dynkin)
- moreover then have "sets (dynkin E) = sets M"
- using assms dynkin_subset[OF E(1,2)] by simp
+ moreover then have "dynkin \<Omega> E = M"
+ using assms dynkin_subset[OF E(1)] by simp
ultimately show ?thesis
- using assms by (auto intro!: algebra.equality simp: dynkin_def)
-qed
-
-subsection "Sigma algebras on finite sets"
-
-locale finite_sigma_algebra = sigma_algebra +
- assumes finite_space: "finite (space M)"
- and sets_eq_Pow[simp]: "sets M = Pow (space M)"
-
-lemma (in finite_sigma_algebra) sets_image_space_eq_Pow:
- "sets (image_space X) = Pow (space (image_space X))"
-proof safe
- fix x S assume "S \<in> sets (image_space X)" "x \<in> S"
- then show "x \<in> space (image_space X)"
- using sets_into_space by (auto intro!: imageI simp: image_space_def)
-next
- fix S assume "S \<subseteq> space (image_space X)"
- then obtain S' where "S = X`S'" "S'\<in>sets M"
- by (auto simp: subset_image_iff image_space_def)
- then show "S \<in> sets (image_space X)"
- by (auto simp: image_space_def)
-qed
-
-lemma measurable_sigma_sigma:
- assumes M: "sets M \<subseteq> Pow (space M)" and N: "sets N \<subseteq> Pow (space N)"
- shows "f \<in> measurable M N \<Longrightarrow> f \<in> measurable (sigma M) (sigma N)"
- using sigma_algebra.measurable_subset[OF sigma_algebra_sigma[OF M], of N]
- using measurable_up_sigma[of M N] N by auto
-
-lemma (in sigma_algebra) measurable_Least:
- assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> sets M"
- shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
-proof -
- { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
- proof cases
- assume i: "(LEAST j. False) = i"
- have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
- {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
- by (simp add: set_eq_iff, safe)
- (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
- with meas show ?thesis
- by (auto intro!: Int)
- next
- assume i: "(LEAST j. False) \<noteq> i"
- then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
- {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
- proof (simp add: set_eq_iff, safe)
- fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
- have "\<exists>j. P j x"
- by (rule ccontr) (insert neq, auto)
- then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
- qed (auto dest: Least_le intro!: Least_equality)
- with meas show ?thesis
- by (auto intro!: Int)
- qed }
- then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
- by (intro countable_UN) auto
- moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
- (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
- ultimately show ?thesis by auto
+ using assms by (auto simp: dynkin_def)
qed
end
--- a/src/HOL/Probability/ex/Dining_Cryptographers.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/ex/Dining_Cryptographers.thy Mon Apr 23 12:14:35 2012 +0200
@@ -8,29 +8,6 @@
lemma Ex1_eq: "\<exists>! x. P x \<Longrightarrow> P x \<Longrightarrow> P y \<Longrightarrow> x = y"
by auto
-locale finite_space =
- fixes S :: "'a set"
- assumes finite[simp]: "finite S"
- and not_empty[simp]: "S \<noteq> {}"
-
-definition (in finite_space) "M = \<lparr> space = S, sets = Pow S,
- measure = \<lambda>A. ereal (card A / card S) \<rparr>"
-
-lemma (in finite_space)
- shows space_M[simp]: "space M = S"
- and sets_M[simp]: "sets M = Pow S"
- by (simp_all add: M_def)
-
-sublocale finite_space \<subseteq> finite_measure_space M
- by (rule finite_measure_spaceI)
- (simp_all add: M_def real_of_nat_def)
-
-sublocale finite_space \<subseteq> information_space M 2
- by default (simp_all add: M_def one_ereal_def)
-
-lemma (in finite_space) \<mu>'_eq[simp]: "\<mu>' A = (if A \<subseteq> S then card A / card S else 0)"
- unfolding \<mu>'_def by (auto simp: M_def)
-
section "Define the state space"
text {*
@@ -68,7 +45,7 @@
definition "dining_cryptographers =
({None} \<union> Some ` {0..<n}) \<times> {xs :: bool list. length xs = n}"
definition "payer dc = fst dc"
-definition coin :: "(nat option \<times> bool list) => nat \<Rightarrow> bool" where
+definition coin :: "(nat option \<times> bool list) \<Rightarrow> nat \<Rightarrow> bool" where
"coin dc c = snd dc ! (c mod n)"
definition "inversion dc =
map (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) [0..<n]"
@@ -417,15 +394,12 @@
end
+sublocale dining_cryptographers_space \<subseteq> prob_space "uniform_count_measure dc_crypto"
+ by (rule prob_space_uniform_count_measure[OF finite_dc_crypto])
+ (insert n_gt_3, auto simp: dc_crypto intro: exI[of _ "replicate n True"])
-sublocale
- dining_cryptographers_space \<subseteq> finite_space "dc_crypto"
-proof
- show "finite dc_crypto" using finite_dc_crypto .
- show "dc_crypto \<noteq> {}"
- unfolding dc_crypto
- using n_gt_3 by (auto intro: exI[of _ "replicate n True"])
-qed
+sublocale dining_cryptographers_space \<subseteq> information_space "uniform_count_measure dc_crypto" 2
+ by default auto
notation (in dining_cryptographers_space)
mutual_information_Pow ("\<I>'( _ ; _ ')")
@@ -438,71 +412,64 @@
theorem (in dining_cryptographers_space)
"\<I>( inversion ; payer ) = 0"
-proof -
+proof (rule mutual_information_eq_0_simple)
have n: "0 < n" using n_gt_3 by auto
- have lists: "{xs. length xs = n} \<noteq> {}" using Ex_list_of_length by auto
have card_image_inversion:
"real (card (inversion ` dc_crypto)) = 2^n / 2"
unfolding card_image_inversion using `0 < n` by (cases n) auto
- let ?dIP = "distribution (\<lambda>x. (inversion x, payer x))"
- let ?dP = "distribution payer"
- let ?dI = "distribution inversion"
+ show inversion: "simple_distributed (uniform_count_measure dc_crypto) inversion (\<lambda>x. 2 / 2^n)"
+ proof (rule simple_distributedI)
+ show "simple_function (uniform_count_measure dc_crypto) inversion"
+ using finite_dc_crypto
+ by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
+ fix x assume "x \<in> inversion ` space (uniform_count_measure dc_crypto)"
+ moreover have "inversion -` {x} \<inter> dc_crypto = {dc \<in> dc_crypto. inversion dc = x}" by auto
+ ultimately show "2 / 2^n = prob (inversion -` {x} \<inter> space (uniform_count_measure dc_crypto))"
+ using `0 < n`
+ by (simp add: card_inversion card_dc_crypto finite_dc_crypto
+ subset_eq space_uniform_count_measure measure_uniform_count_measure)
+ qed
- { have "\<H>(inversion|payer) =
- - (\<Sum>x\<in>inversion`dc_crypto. (\<Sum>z\<in>Some ` {0..<n}. ?dIP {(x, z)} * log 2 (?dIP {(x, z)} / ?dP {z})))"
- unfolding conditional_entropy_eq[OF simple_function_finite simple_function_finite]
- by (simp add: image_payer_dc_crypto setsum_Sigma)
- also have "... =
- - (\<Sum>x\<in>inversion`dc_crypto. (\<Sum>z\<in>Some ` {0..<n}. 2 / (real n * 2^n) * (1 - real n)))"
- unfolding neg_equal_iff_equal
- proof (rule setsum_cong[OF refl], rule setsum_cong[OF refl])
- fix x z assume x: "x \<in> inversion`dc_crypto" and z: "z \<in> Some ` {0..<n}"
- hence "(\<lambda>x. (inversion x, payer x)) -` {(x, z)} \<inter> dc_crypto =
- {dc \<in> dc_crypto. payer dc = Some (the z) \<and> inversion dc = x}"
- by (auto simp add: payer_def)
- moreover from x z obtain i where "z = Some i" and "i < n" by auto
- moreover from x have "length x = n" by (auto simp: inversion_def [abs_def] dc_crypto)
- ultimately
- have "?dIP {(x, z)} = 2 / (real n * 2^n)" using x
- by (auto simp add: card_dc_crypto card_payer_and_inversion distribution_def)
- moreover
- from z have "payer -` {z} \<inter> dc_crypto = {z} \<times> {xs. length xs = n}"
- by (auto simp: dc_crypto payer_def)
- hence "card (payer -` {z} \<inter> dc_crypto) = 2^n"
- using card_lists_length_eq[where A="UNIV::bool set"]
- by (simp add: card_cartesian_product_singleton)
- hence "?dP {z} = 1 / real n"
- by (simp add: distribution_def card_dc_crypto)
- ultimately
- show "?dIP {(x,z)} * log 2 (?dIP {(x,z)} / ?dP {z}) =
- 2 / (real n * 2^n) * (1 - real n)"
- by (simp add: log_divide log_nat_power[of 2])
- qed
- also have "... = real n - 1"
- using n finite_space
- by (simp add: card_image_inversion card_image[OF inj_Some] field_simps real_eq_of_nat[symmetric])
- finally have "\<H>(inversion|payer) = real n - 1" . }
- moreover
- { have "\<H>(inversion) = - (\<Sum>x \<in> inversion`dc_crypto. ?dI {x} * log 2 (?dI {x}))"
- unfolding entropy_eq[OF simple_function_finite] by simp
- also have "... = - (\<Sum>x \<in> inversion`dc_crypto. 2 * (1 - real n) / 2^n)"
- unfolding neg_equal_iff_equal
- proof (rule setsum_cong[OF refl])
- fix x assume x_inv: "x \<in> inversion ` dc_crypto"
- hence "length x = n" by (auto simp: inversion_def [abs_def] dc_crypto)
- moreover have "inversion -` {x} \<inter> dc_crypto = {dc \<in> dc_crypto. inversion dc = x}" by auto
- ultimately have "?dI {x} = 2 / 2^n" using `0 < n`
- by (auto simp: card_inversion[OF x_inv] card_dc_crypto distribution_def)
- thus "?dI {x} * log 2 (?dI {x}) = 2 * (1 - real n) / 2^n"
- by (simp add: log_divide log_nat_power power_le_zero_eq inverse_eq_divide)
- qed
- also have "... = real n - 1"
- by (simp add: card_image_inversion real_of_nat_def[symmetric] field_simps)
- finally have "\<H>(inversion) = real n - 1" .
- }
- ultimately show ?thesis
- unfolding mutual_information_eq_entropy_conditional_entropy[OF simple_function_finite simple_function_finite]
+ show "simple_distributed (uniform_count_measure dc_crypto) payer (\<lambda>x. 1 / real n)"
+ proof (rule simple_distributedI)
+ show "simple_function (uniform_count_measure dc_crypto) payer"
+ using finite_dc_crypto
+ by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
+ fix z assume "z \<in> payer ` space (uniform_count_measure dc_crypto)"
+ then have "payer -` {z} \<inter> dc_crypto = {z} \<times> {xs. length xs = n}"
+ by (auto simp: dc_crypto payer_def space_uniform_count_measure)
+ hence "card (payer -` {z} \<inter> dc_crypto) = 2^n"
+ using card_lists_length_eq[where A="UNIV::bool set"]
+ by (simp add: card_cartesian_product_singleton)
+ then show "1 / real n = prob (payer -` {z} \<inter> space (uniform_count_measure dc_crypto))"
+ using finite_dc_crypto
+ by (subst measure_uniform_count_measure) (auto simp add: card_dc_crypto space_uniform_count_measure)
+ qed
+
+ show "simple_distributed (uniform_count_measure dc_crypto) (\<lambda>x. (inversion x, payer x)) (\<lambda>x. 2 / (real n *2^n))"
+ proof (rule simple_distributedI)
+ show "simple_function (uniform_count_measure dc_crypto) (\<lambda>x. (inversion x, payer x))"
+ using finite_dc_crypto
+ by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
+ fix x assume "x \<in> (\<lambda>x. (inversion x, payer x)) ` space (uniform_count_measure dc_crypto)"
+ then obtain i xs where x: "x = (inversion (Some i, xs), payer (Some i, xs))"
+ and "i < n" "length xs = n"
+ by (simp add: image_iff space_uniform_count_measure dc_crypto Bex_def) blast
+ then have xs: "inversion (Some i, xs) \<in> inversion`dc_crypto" and i: "Some i \<in> Some ` {0..<n}"
+ and x: "x = (inversion (Some i, xs), Some i)" by (simp_all add: payer_def dc_crypto)
+ moreover def ys \<equiv> "inversion (Some i, xs)"
+ ultimately have ys: "ys \<in> inversion`dc_crypto"
+ and "Some i \<in> Some ` {0..<n}" "x = (ys, Some i)" by simp_all
+ then have "(\<lambda>x. (inversion x, payer x)) -` {x} \<inter> space (uniform_count_measure dc_crypto) =
+ {dc \<in> dc_crypto. payer dc = Some (the (Some i)) \<and> inversion dc = ys}"
+ by (auto simp add: payer_def space_uniform_count_measure)
+ then show "2 / (real n * 2 ^ n) = prob ((\<lambda>x. (inversion x, payer x)) -` {x} \<inter> space (uniform_count_measure dc_crypto))"
+ using `i < n` ys
+ by (simp add: measure_uniform_count_measure card_payer_and_inversion finite_dc_crypto subset_eq card_dc_crypto)
+ qed
+
+ show "\<forall>x\<in>space (uniform_count_measure dc_crypto). 2 / (real n * 2 ^ n) = 2 / 2 ^ n * (1 / real n) "
by simp
qed
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -103,49 +103,14 @@
lessThan_Suc_eq_insert_0 setprod_reindex setsum_left_distrib[symmetric] setsum_right_distrib[symmetric])
qed
-lemma ex_ordered_bij_betw_nat_finite:
- fixes order :: "nat \<Rightarrow> 'a\<Colon>linorder"
- assumes "finite S"
- shows "\<exists>f. bij_betw f {0..<card S} S \<and> (\<forall>i<card S. \<forall>j<card S. i \<le> j \<longrightarrow> order (f i) \<le> order (f j))"
-proof -
- from ex_bij_betw_nat_finite [OF `finite S`] guess f .. note f = this
- let ?xs = "sort_key order (map f [0 ..< card S])"
+declare space_point_measure[simp]
- have "?xs <~~> map f [0 ..< card S]"
- unfolding multiset_of_eq_perm[symmetric] by (rule multiset_of_sort)
- from permutation_Ex_bij [OF this]
- obtain g where g: "bij_betw g {0..<card S} {0..<card S}" and
- map: "\<And>i. i<card S \<Longrightarrow> ?xs ! i = map f [0 ..< card S] ! g i"
- by (auto simp: atLeast0LessThan)
-
- { fix i assume "i < card S"
- then have "g i < card S" using g by (auto simp: bij_betw_def)
- with map [OF `i < card S`] have "f (g i) = ?xs ! i" by simp }
- note this[simp]
+declare sets_point_measure[simp]
- show ?thesis
- proof (intro exI allI conjI impI)
- show "bij_betw (f\<circ>g) {0..<card S} S"
- using g f by (rule bij_betw_trans)
- fix i j assume [simp]: "i < card S" "j < card S" "i \<le> j"
- from sorted_nth_mono[of "map order ?xs" i j]
- show "order ((f\<circ>g) i) \<le> order ((f\<circ>g) j)" by simp
- qed
-qed
-
-definition (in prob_space)
- "ordered_variable_partition X = (SOME f. bij_betw f {0..<card (X`space M)} (X`space M) \<and>
- (\<forall>i<card (X`space M). \<forall>j<card (X`space M). i \<le> j \<longrightarrow> distribution X {f i} \<le> distribution X {f j}))"
-
-definition (in prob_space)
- "order_distribution X i = real (distribution X {ordered_variable_partition X i})"
-
-definition (in prob_space)
- "guessing_entropy b X = (\<Sum>i<card(X`space M). real i * log b (order_distribution X i))"
-
-abbreviation (in information_space)
- finite_guessing_entropy ("\<G>'(_')") where
- "\<G>(X) \<equiv> guessing_entropy b X"
+lemma measure_point_measure:
+ "finite \<Omega> \<Longrightarrow> A \<subseteq> \<Omega> \<Longrightarrow> (\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> p x) \<Longrightarrow>
+ measure (point_measure \<Omega> (\<lambda>x. ereal (p x))) A = (\<Sum>i\<in>A. p i)"
+ unfolding measure_def by (subst emeasure_point_measure_finite) auto
locale finite_information =
fixes \<Omega> :: "'a set"
@@ -159,17 +124,14 @@
lemma (in finite_information) positive_p_sum[simp]: "0 \<le> setsum p X"
by (auto intro!: setsum_nonneg)
-sublocale finite_information \<subseteq> finite_measure_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. ereal (setsum p S)\<rparr>"
- by (rule finite_measure_spaceI) (simp_all add: setsum_Un_disjoint finite_subset)
+sublocale finite_information \<subseteq> prob_space "point_measure \<Omega> p"
+ by default (simp add: one_ereal_def emeasure_point_measure_finite)
-sublocale finite_information \<subseteq> finite_prob_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. ereal (setsum p S)\<rparr>"
- by default (simp add: one_ereal_def)
-
-sublocale finite_information \<subseteq> information_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. ereal (setsum p S)\<rparr>" b
+sublocale finite_information \<subseteq> information_space "point_measure \<Omega> p" b
by default simp
-lemma (in finite_information) \<mu>'_eq: "A \<subseteq> \<Omega> \<Longrightarrow> \<mu>' A = setsum p A"
- unfolding \<mu>'_def by auto
+lemma (in finite_information) \<mu>'_eq: "A \<subseteq> \<Omega> \<Longrightarrow> prob A = setsum p A"
+ by (auto simp: measure_point_measure)
locale koepf_duermuth = K: finite_information keys K b + M: finite_information messages M b
for b :: real
@@ -247,14 +209,19 @@
abbreviation
"p A \<equiv> setsum P A"
+abbreviation
+ "\<mu> \<equiv> point_measure msgs P"
+
abbreviation probability ("\<P>'(_') _") where
- "\<P>(X) x \<equiv> distribution X x"
+ "\<P>(X) x \<equiv> measure \<mu> (X -` x \<inter> msgs)"
-abbreviation joint_probability ("\<P>'(_, _') _") where
- "\<P>(X, Y) x \<equiv> joint_distribution X Y x"
+abbreviation joint_probability ("\<P>'(_ ; _') _") where
+ "\<P>(X ; Y) x \<equiv> \<P>(\<lambda>x. (X x, Y x)) x"
-abbreviation conditional_probability ("\<P>'(_|_') _") where
- "\<P>(X|Y) x \<equiv> \<P>(X, Y) x / \<P>(Y) (snd`x)"
+no_notation disj (infixr "|" 30)
+
+abbreviation conditional_probability ("\<P>'(_ | _') _") where
+ "\<P>(X | Y) x \<equiv> (\<P>(X ; Y) x) / \<P>(Y) (snd`x)"
notation
entropy_Pow ("\<H>'( _ ')")
@@ -280,8 +247,8 @@
from assms have *:
"fst -` {k} \<inter> msgs = {k}\<times>{ms. set ms \<subseteq> messages \<and> length ms = n}"
unfolding msgs_def by auto
- show "\<P>(fst) {k} = K k"
- apply (simp add: \<mu>'_eq distribution_def)
+ show "(\<P>(fst) {k}) = K k"
+ apply (simp add: \<mu>'_eq)
apply (simp add: * P_def)
apply (simp add: setsum_cartesian_product')
using setprod_setsum_distrib_lists[OF M.finite_space, of M n "\<lambda>x x. True"] `k \<in> keys`
@@ -297,6 +264,67 @@
unfolding msgs_def fst_image_times if_not_P[OF *] by simp
qed
+lemma setsum_distribution_cut:
+ "\<P>(X) {x} = (\<Sum>y \<in> Y`space \<mu>. \<P>(X ; Y) {(x, y)})"
+ by (subst finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def inj_on_def
+ intro!: arg_cong[where f=prob])
+
+lemma prob_conj_imp1:
+ "prob ({x. Q x} \<inter> msgs) = 0 \<Longrightarrow> prob ({x. Pr x \<and> Q x} \<inter> msgs) = 0"
+ using finite_measure_mono[of "{x. Pr x \<and> Q x} \<inter> msgs" "{x. Q x} \<inter> msgs"]
+ using measure_nonneg[of \<mu> "{x. Pr x \<and> Q x} \<inter> msgs"]
+ by (simp add: subset_eq)
+
+lemma prob_conj_imp2:
+ "prob ({x. Pr x} \<inter> msgs) = 0 \<Longrightarrow> prob ({x. Pr x \<and> Q x} \<inter> msgs) = 0"
+ using finite_measure_mono[of "{x. Pr x \<and> Q x} \<inter> msgs" "{x. Pr x} \<inter> msgs"]
+ using measure_nonneg[of \<mu> "{x. Pr x \<and> Q x} \<inter> msgs"]
+ by (simp add: subset_eq)
+
+lemma simple_function_finite: "simple_function \<mu> f"
+ by (simp add: simple_function_def)
+
+lemma entropy_commute: "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
+ apply (subst (1 2) entropy_simple_distributed[OF simple_distributedI[OF simple_function_finite refl]])
+ unfolding space_point_measure
+proof -
+ have eq: "(\<lambda>x. (X x, Y x)) ` msgs = (\<lambda>(x, y). (y, x)) ` (\<lambda>x. (Y x, X x)) ` msgs"
+ by auto
+ show "- (\<Sum>x\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {x}) * log b (\<P>(X ; Y) {x})) =
+ - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` msgs. (\<P>(Y ; X) {x}) * log b (\<P>(Y ; X) {x}))"
+ unfolding eq
+ apply (subst setsum_reindex)
+ apply (auto simp: vimage_def inj_on_def intro!: setsum_cong arg_cong[where f="\<lambda>x. prob x * log b (prob x)"])
+ done
+qed
+
+lemma (in -) measure_eq_0I: "A = {} \<Longrightarrow> measure M A = 0" by simp
+
+lemma conditional_entropy_eq_ce_with_hypothesis:
+ "\<H>(X | Y) = -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) *
+ log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))"
+ apply (subst conditional_entropy_eq[OF
+ simple_distributedI[OF simple_function_finite refl]
+ simple_function_finite
+ simple_distributedI[OF simple_function_finite refl]])
+ unfolding space_point_measure
+proof -
+ have "- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))) =
+ - (\<Sum>x\<in>X`msgs. (\<Sum>y\<in>Y`msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))))"
+ unfolding setsum_cartesian_product
+ apply (intro arg_cong[where f=uminus] setsum_mono_zero_left)
+ apply (auto simp: vimage_def image_iff intro!: measure_eq_0I)
+ apply metis
+ done
+ also have "\<dots> = - (\<Sum>y\<in>Y`msgs. (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))))"
+ by (subst setsum_commute) rule
+ also have "\<dots> = -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) * log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))"
+ by (auto simp add: setsum_right_distrib vimage_def intro!: setsum_cong prob_conj_imp1)
+ finally show "- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))) =
+ -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) * log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))" .
+qed
+
lemma ce_OB_eq_ce_t: "\<H>(fst | OB) = \<H>(fst | t\<circ>OB)"
proof -
txt {* Lemma 2 *}
@@ -314,22 +342,22 @@
then have **: "\<And>ms. length ms = n \<Longrightarrow> OB (k, ms) = obs \<longleftrightarrow> (\<forall>i<n. observe k (ms!i) = obs ! i)"
unfolding OB_def msgs_def by (simp add: image_iff list_eq_iff_nth_eq)
- have "\<P>(OB, fst) {(obs, k)} / K k =
+ have "(\<P>(OB ; fst) {(obs, k)}) / K k =
p ({k}\<times>{ms. (k,ms) \<in> msgs \<and> OB (k,ms) = obs}) / K k"
- apply (simp add: distribution_def \<mu>'_eq) by (auto intro!: arg_cong[where f=p])
+ apply (simp add: \<mu>'_eq) by (auto intro!: arg_cong[where f=p])
also have "\<dots> =
(\<Prod>i<n. \<Sum>m\<in>{m\<in>messages. observe k m = obs ! i}. M m)"
unfolding P_def using `K k \<noteq> 0` `k \<in> keys`
apply (simp add: setsum_cartesian_product' setsum_divide_distrib msgs_def ** cong: conj_cong)
apply (subst setprod_setsum_distrib_lists[OF M.finite_space]) ..
- finally have "\<P>(OB, fst) {(obs, k)} / K k =
+ finally have "(\<P>(OB ; fst) {(obs, k)}) / K k =
(\<Prod>i<n. \<Sum>m\<in>{m\<in>messages. observe k m = obs ! i}. M m)" . }
note * = this
- have "\<P>(OB, fst) {(obs, k)} / K k = \<P>(OB, fst) {(obs', k)} / K k"
+ have "(\<P>(OB ; fst) {(obs, k)}) / K k = (\<P>(OB ; fst) {(obs', k)}) / K k"
unfolding *[OF obs] *[OF obs']
using t_f(1) obs_t_f by (subst (2) t_f(2)) (simp add: setprod_reindex)
- then have "\<P>(OB, fst) {(obs, k)} = \<P>(OB, fst) {(obs', k)}"
+ then have "(\<P>(OB ; fst) {(obs, k)}) = (\<P>(OB ; fst) {(obs', k)})"
using `K k \<noteq> 0` by auto }
note t_eq_imp = this
@@ -339,13 +367,13 @@
(\<Union>obs'\<in>?S obs. ((\<lambda>x. (OB x, fst x)) -` {(obs', k)} \<inter> msgs))" by auto
have df: "disjoint_family_on (\<lambda>obs'. (\<lambda>x. (OB x, fst x)) -` {(obs', k)} \<inter> msgs) (?S obs)"
unfolding disjoint_family_on_def by auto
- have "\<P>(t\<circ>OB, fst) {(t obs, k)} = (\<Sum>obs'\<in>?S obs. \<P>(OB, fst) {(obs', k)})"
- unfolding distribution_def comp_def
- using finite_measure_finite_Union[OF _ _ df]
- by (force simp add: * intro!: setsum_nonneg)
- also have "(\<Sum>obs'\<in>?S obs. \<P>(OB, fst) {(obs', k)}) = real (card (?S obs)) * \<P>(OB, fst) {(obs, k)}"
+ have "\<P>(t\<circ>OB ; fst) {(t obs, k)} = (\<Sum>obs'\<in>?S obs. \<P>(OB ; fst) {(obs', k)})"
+ unfolding comp_def
+ using finite_measure_finite_Union[OF _ df]
+ by (auto simp add: * intro!: setsum_nonneg)
+ also have "(\<Sum>obs'\<in>?S obs. \<P>(OB ; fst) {(obs', k)}) = real (card (?S obs)) * \<P>(OB ; fst) {(obs, k)}"
by (simp add: t_eq_imp[OF `k \<in> keys` `K k \<noteq> 0` obs] real_eq_of_nat)
- finally have "\<P>(t\<circ>OB, fst) {(t obs, k)} = real (card (?S obs)) * \<P>(OB, fst) {(obs, k)}" .}
+ finally have "\<P>(t\<circ>OB ; fst) {(t obs, k)} = real (card (?S obs)) * \<P>(OB ; fst) {(obs, k)}" .}
note P_t_eq_P_OB = this
{ fix k obs assume "k \<in> keys" and obs: "obs \<in> OB`msgs"
@@ -359,11 +387,15 @@
then have "real (card ?S) \<noteq> 0" by auto
have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / \<P>(t\<circ>OB) {t obs}"
- using distribution_order(7,8)[where X=fst and x=k and Y="t\<circ>OB" and y="t obs"]
- by (subst joint_distribution_commute) auto
- also have "\<P>(t\<circ>OB) {t obs} = (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'})"
- using setsum_distribution_cut(2)[of "t\<circ>OB" fst "t obs", symmetric]
- by (auto intro!: setsum_cong distribution_order(8))
+ using finite_measure_mono[of "{x. fst x = k \<and> t (OB x) = t obs} \<inter> msgs" "{x. fst x = k} \<inter> msgs"]
+ using measure_nonneg[of \<mu> "{x. fst x = k \<and> t (OB x) = t obs} \<inter> msgs"]
+ by (auto simp add: vimage_def conj_commute subset_eq)
+ also have "(\<P>(t\<circ>OB) {t obs}) = (\<Sum>k'\<in>keys. (\<P>(t\<circ>OB|fst) {(t obs, k')}) * (\<P>(fst) {k'}))"
+ using finite_measure_mono[of "{x. t (OB x) = t obs \<and> fst x = k} \<inter> msgs" "{x. fst x = k} \<inter> msgs"]
+ using measure_nonneg[of \<mu> "{x. fst x = k \<and> t (OB x) = t obs} \<inter> msgs"]
+ apply (simp add: setsum_distribution_cut[of "t\<circ>OB" "t obs" fst])
+ apply (auto intro!: setsum_cong simp: subset_eq vimage_def prob_conj_imp1)
+ done
also have "\<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'}) =
\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'})"
using CP_t_K[OF `k\<in>keys` obs] CP_t_K[OF _ obs] `real (card ?S) \<noteq> 0`
@@ -371,10 +403,10 @@
mult_divide_mult_cancel_left[OF `real (card ?S) \<noteq> 0`]
cong: setsum_cong)
also have "(\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'}) = \<P>(OB) {obs}"
- using setsum_distribution_cut(2)[of OB fst obs, symmetric]
- by (auto intro!: setsum_cong distribution_order(8))
+ using setsum_distribution_cut[of OB obs fst]
+ by (auto intro!: setsum_cong simp: prob_conj_imp1 vimage_def)
also have "\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / \<P>(OB) {obs} = \<P>(fst | OB) {(k, obs)}"
- by (subst joint_distribution_commute) (auto intro!: distribution_order(8))
+ by (auto simp: vimage_def conj_commute prob_conj_imp2)
finally have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(fst | OB) {(k, obs)}" . }
note CP_T_eq_CP_O = this
@@ -396,15 +428,15 @@
have df: "disjoint_family_on (\<lambda>obs. OB -` {obs} \<inter> msgs) (?S (OB x))"
unfolding disjoint_family_on_def by auto
have "\<P>(t\<circ>OB) {t (OB x)} = (\<Sum>obs\<in>?S (OB x). \<P>(OB) {obs})"
- unfolding distribution_def comp_def
- using finite_measure_finite_Union[OF _ _ df]
+ unfolding comp_def
+ using finite_measure_finite_Union[OF _ df]
by (force simp add: * intro!: setsum_nonneg) }
note P_t_sum_P_O = this
txt {* Lemma 3 *}
+ txt {* Lemma 3 *}
have "\<H>(fst | OB) = -(\<Sum>obs\<in>OB`msgs. \<P>(OB) {obs} * ?Ht (t obs))"
- unfolding conditional_entropy_eq_ce_with_hypothesis[OF
- simple_function_finite simple_function_finite] using * by simp
+ unfolding conditional_entropy_eq_ce_with_hypothesis using * by simp
also have "\<dots> = -(\<Sum>obs\<in>t`OB`msgs. \<P>(t\<circ>OB) {obs} * ?Ht obs)"
apply (subst SIGMA_image_vimage[symmetric, of "OB`msgs" t])
apply (subst setsum_reindex)
@@ -418,8 +450,7 @@
by (simp add: setsum_divide_distrib[symmetric] field_simps **
setsum_right_distrib[symmetric] setsum_left_distrib[symmetric])
also have "\<dots> = \<H>(fst | t\<circ>OB)"
- unfolding conditional_entropy_eq_ce_with_hypothesis[OF
- simple_function_finite simple_function_finite]
+ unfolding conditional_entropy_eq_ce_with_hypothesis
by (simp add: comp_def image_image[symmetric])
finally show ?thesis .
qed
@@ -433,11 +464,11 @@
unfolding ce_OB_eq_ce_t ..
also have "\<dots> = \<H>(t\<circ>OB) - \<H>(t\<circ>OB | fst)"
unfolding entropy_chain_rule[symmetric, OF simple_function_finite simple_function_finite] sign_simps
- by (subst entropy_commute[OF simple_function_finite simple_function_finite]) simp
+ by (subst entropy_commute) simp
also have "\<dots> \<le> \<H>(t\<circ>OB)"
- using conditional_entropy_positive[of "t\<circ>OB" fst] by simp
+ using conditional_entropy_nonneg[of "t\<circ>OB" fst] by simp
also have "\<dots> \<le> log b (real (card ((t\<circ>OB)`msgs)))"
- using entropy_le_card[of "t\<circ>OB"] by simp
+ using entropy_le_card[of "t\<circ>OB", OF simple_distributedI[OF simple_function_finite refl]] by simp
also have "\<dots> \<le> log b (real (n + 1)^card observations)"
using card_T_bound not_empty
by (auto intro!: log_le simp: card_gt_0_iff power_real_of_nat simp del: real_of_nat_power)