--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Thu Sep 02 20:44:33 2010 +0200
@@ -5027,7 +5027,7 @@
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$i \<le> x$$i \<and> x$$i \<le> b$$i) \<or> (b$$i \<le> x$$i \<and> x$$i \<le> a$$i))) \<longrightarrow> x \<in> s)"
lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
- "is_interval {a<..<b}" (is ?th2) proof -
+ "is_interval {a<..<b}" (is ?th2) proof -
have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
by(meson order_trans le_less_trans less_le_trans *)+ qed
@@ -5051,6 +5051,9 @@
lemma continuous_at_inner: "continuous (at x) (inner a)"
unfolding continuous_at by (intro tendsto_intros)
+lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
+ unfolding euclidean_component_def by (rule continuous_at_inner)
+
lemma continuous_on_inner:
fixes s :: "'a::real_inner set"
shows "continuous_on s (inner a)"
@@ -5159,6 +5162,9 @@
by (simp add: closed_def open_halfspace_component_lt)
qed
+lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
+ by (auto intro!: continuous_open_vimage)
+
text{* This gives a simple derivation of limit component bounds. *}
lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
--- a/src/HOL/Probability/Borel.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Borel.thy Thu Sep 02 20:44:33 2010 +0200
@@ -6,6 +6,10 @@
imports Sigma_Algebra Positive_Infinite_Real Multivariate_Analysis
begin
+lemma LIMSEQ_max:
+ "u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
+ by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
+
section "Generic Borel spaces"
definition "borel_space = sigma (UNIV::'a::topological_space set) open"
@@ -81,7 +85,7 @@
"(\<lambda>x. c) \<in> borel_measurable M"
by (auto intro!: measurable_const)
-lemma (in sigma_algebra) borel_measurable_indicator:
+lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
assumes A: "A \<in> sets M"
shows "indicator A \<in> borel_measurable M"
unfolding indicator_def_raw using A
@@ -105,6 +109,53 @@
qed (auto simp add: vimage_UN)
qed
+lemma (in sigma_algebra) borel_measurable_restricted:
+ fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" 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_space
+ proof (simp, safe)
+ fix S :: "'x set" assume "S \<in> sets borel_space"
+ "\<forall>S\<in>sets borel_space. ?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 fastsimp
+ 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 :: "'x set" assume "S \<in> sets borel_space"
+ "\<forall>S\<in>sets borel_space. ?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:
+ assumes "N \<subseteq> sets M" "f \<in> borel_measurable (M\<lparr>sets:=N\<rparr>)"
+ shows "f \<in> borel_measurable M"
+ using assms unfolding measurable_def by auto
+
section "Borel spaces on euclidean spaces"
lemma lessThan_borel[simp, intro]:
@@ -658,6 +709,30 @@
"(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_measureable_euclidean_component:
+ "(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel_space"
+ unfolding borel_space_def[where 'a=real]
+proof (rule borel_space.measurable_sigma)
+ fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
+ from open_vimage_euclidean_component[OF this]
+ show "(\<lambda>x. x $$ i) -` S \<inter> space borel_space \<in> sets borel_space"
+ by (auto intro: borel_space_open)
+qed auto
+
+lemma (in sigma_algebra) borel_measureable_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 $$ i) \<in> borel_measurable M)"
+proof safe
+ fix i assume "f \<in> borel_measurable M"
+ then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"
+ using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def]
+ by (auto intro: borel_measureable_euclidean_component)
+next
+ assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M"
+ then show "f \<in> borel_measurable M"
+ unfolding borel_measurable_iff_halfspace_le by auto
+qed
+
subsection "Borel measurable operators"
lemma (in sigma_algebra) affine_borel_measurable_vector:
@@ -1270,4 +1345,46 @@
using assms by auto
qed
+lemma (in sigma_algebra) borel_measurable_psuminf:
+ assumes "\<And>i. f i \<in> borel_measurable M"
+ shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
+ using assms unfolding psuminf_def
+ by (auto intro!: borel_measurable_SUP[unfolded SUPR_fun_expand])
+
+section "LIMSEQ is borel measurable"
+
+lemma (in sigma_algebra) 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"
+ shows "u' \<in> borel_measurable M"
+proof -
+ let "?pu x i" = "max (u i x) 0"
+ let "?nu x i" = "max (- u i x) 0"
+
+ { fix x assume x: "x \<in> space M"
+ have "(?pu x) ----> max (u' x) 0"
+ "(?nu x) ----> max (- u' x) 0"
+ using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
+ from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
+ have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
+ "(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
+ by (simp_all add: Real_max'[symmetric]) }
+ note eq = this
+
+ have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
+ by auto
+
+ have "(SUP n. INF m. (\<lambda>x. Real (u (n + m) x))) \<in> borel_measurable M"
+ "(SUP n. INF m. (\<lambda>x. Real (- u (n + m) x))) \<in> borel_measurable M"
+ using u by (auto intro: borel_measurable_SUP borel_measurable_INF borel_measurable_Real)
+ with eq[THEN measurable_cong, of M "\<lambda>x. x" borel_space]
+ have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
+ "(\<lambda>x. Real (- u' x)) \<in> borel_measurable M"
+ unfolding SUPR_fun_expand INFI_fun_expand by auto
+ note this[THEN borel_measurable_real]
+ from borel_measurable_diff[OF this]
+ show ?thesis unfolding * .
+qed
+
end
--- a/src/HOL/Probability/Caratheodory.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Caratheodory.thy Thu Sep 02 20:44:33 2010 +0200
@@ -445,21 +445,6 @@
by intro_locales (auto simp add: sigma_algebra_def)
qed
-
-lemma (in algebra) inf_measure_nonempty:
- assumes f: "positive f" and b: "b \<in> sets M" and a: "a \<subseteq> b"
- shows "f b \<in> measure_set M f a"
-proof -
- have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
- by (rule psuminf_finite) (simp add: f[unfolded positive_def])
- also have "... = f b"
- by simp
- finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
- thus ?thesis using a b
- by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
- simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
-qed
-
lemma (in algebra) additive_increasing:
assumes posf: "positive f" and addf: "additive M f"
shows "increasing M f"
@@ -494,6 +479,20 @@
by (auto simp add: Un binaryset_psuminf positive_def)
qed
+lemma inf_measure_nonempty:
+ assumes f: "positive f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
+ shows "f b \<in> measure_set M f a"
+proof -
+ have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
+ by (rule psuminf_finite) (simp add: f[unfolded positive_def])
+ also have "... = f b"
+ by simp
+ finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
+ thus ?thesis using assms
+ by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
+ simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
+qed
+
lemma (in algebra) inf_measure_agrees:
assumes posf: "positive f" and ca: "countably_additive M f"
and s: "s \<in> sets M"
@@ -535,11 +534,11 @@
qed
lemma (in algebra) inf_measure_empty:
- assumes posf: "positive f"
+ assumes posf: "positive f" "{} \<in> sets 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 empty_sets inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
+ by (metis complete_lattice_class.Inf_lower `{} \<in> sets M` inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
qed simp
lemma (in algebra) inf_measure_positive:
@@ -597,7 +596,7 @@
next
case True
have "measure_set M f s \<noteq> {}"
- by (metis emptyE ss inf_measure_nonempty [of f, OF posf top])
+ by (metis emptyE ss inf_measure_nonempty [of f, OF posf top _ empty_sets])
then obtain l where "l \<in> measure_set M f s" by auto
moreover from True have "l \<le> Inf (measure_set M f s) + e" by simp
ultimately show ?thesis
--- a/src/HOL/Probability/Information.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Information.thy Thu Sep 02 20:44:33 2010 +0200
@@ -2,11 +2,53 @@
imports Probability_Space Product_Measure Convex Radon_Nikodym
begin
+lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
+ by (subst log_le_cancel_iff) auto
+
+lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
+ by (subst log_less_cancel_iff) auto
+
+lemma setsum_cartesian_product':
+ "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
+ unfolding setsum_cartesian_product by simp
+
lemma real_of_pinfreal_inverse[simp]:
fixes X :: pinfreal
shows "real (inverse X) = 1 / real X"
by (cases X) (auto simp: inverse_eq_divide)
+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)\<rparr>
+ (joint_distribution X Y)"
+ (is "finite_prob_space ?S _")
+proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images)
+ 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)
+qed
+
+lemma (in finite_prob_space) finite_measure_space_prod:
+ assumes X: "finite_measure_space MX (distribution X)"
+ assumes Y: "finite_measure_space MY (distribution Y)"
+ shows "finite_measure_space (prod_measure_space MX MY) (joint_distribution X Y)"
+ (is "finite_measure_space ?M ?D")
+proof (intro finite_measure_spaceI)
+ interpret X: finite_measure_space MX "distribution X" by fact
+ interpret Y: finite_measure_space MY "distribution Y" by fact
+ note finite_measure_space.finite_prod_measure_space[OF X Y, simp]
+ show "finite (space ?M)" using X.finite_space Y.finite_space by auto
+ show "joint_distribution X Y {} = 0" by simp
+ show "sets ?M = Pow (space ?M)" by simp
+ { fix x show "?D (space ?M) \<noteq> \<omega>" by (rule distribution_finite) }
+ { fix A B assume "A \<subseteq> space ?M" "B \<subseteq> space ?M" "A \<inter> B = {}"
+ have *: "(\<lambda>t. (X t, Y t)) -` (A \<union> B) \<inter> space M =
+ (\<lambda>t. (X t, Y t)) -` A \<inter> space M \<union> (\<lambda>t. (X t, Y t)) -` B \<inter> space M"
+ by auto
+ show "?D (A \<union> B) = ?D A + ?D B" unfolding distribution_def *
+ apply (rule measure_additive[symmetric])
+ using `A \<inter> B = {}` by (auto simp: sets_eq_Pow) }
+qed
+
section "Convex theory"
lemma log_setsum:
@@ -105,51 +147,19 @@
finally show ?thesis .
qed
-lemma (in finite_prob_space) sum_over_space_distrib:
- "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def measure_space_1[symmetric] using finite_space
- by (subst measure_finitely_additive'')
- (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=\<mu>])
-
-lemma (in finite_prob_space) sum_over_space_real_distribution:
- "(\<Sum>x\<in>X`space M. real (distribution X {x})) = 1"
- unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst real_finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+lemma split_pairs:
+ shows
+ "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
+ "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
section "Information theory"
-definition
- "KL_divergence b M \<mu> \<nu> =
- measure_space.integral M \<mu> (\<lambda>x. log b (real (sigma_finite_measure.RN_deriv M \<nu> \<mu> x)))"
-
locale finite_information_space = finite_prob_space +
fixes b :: real assumes b_gt_1: "1 < b"
-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!: measure_mono)
-
-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"
- unfolding measure_space_1[symmetric]
- by (auto intro!: arg_cong[where f="\<mu>"] simp: distribution_def)
-
context finite_information_space
begin
-lemma 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
assumes "0 \<le> A" and pos: "0 < A \<Longrightarrow> 0 < B" "0 < A \<Longrightarrow> 0 < C"
shows mult_log_mult: "A * log b (B * C) = A * log b B + A * log b C" (is "?mult")
@@ -165,41 +175,6 @@
thus ?mult and ?div by auto
qed
-lemma split_pairs:
- shows
- "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
- "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
-
-lemma (in finite_information_space) distribution_finite:
- "distribution X A \<noteq> \<omega>"
- using measure_finite[of "X -` A \<inter> space M"]
- unfolding distribution_def sets_eq_Pow by auto
-
-lemma (in finite_information_space) real_distribution_gt_0[simp]:
- "0 < real (distribution Y y) \<longleftrightarrow> 0 < distribution Y y"
- using assms by (auto intro!: real_pinfreal_pos distribution_finite)
-
-lemma real_distribution_mult_pos_pos:
- assumes "0 < distribution Y y"
- and "0 < distribution X x"
- shows "0 < real (distribution Y y * distribution X x)"
- unfolding real_of_pinfreal_mult[symmetric]
- using assms by (auto intro!: mult_pos_pos)
-
-lemma real_distribution_divide_pos_pos:
- assumes "0 < distribution Y y"
- and "0 < distribution X x"
- shows "0 < real (distribution Y y / distribution X x)"
- unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
- using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
-
-lemma real_distribution_mult_inverse_pos_pos:
- assumes "0 < distribution Y y"
- and "0 < distribution X x"
- shows "0 < real (distribution Y y * inverse (distribution X x))"
- unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
- using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
-
ML {*
(* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X - W * log b (Y * Z)"}
@@ -252,31 +227,14 @@
end
-lemma (in finite_measure_space) absolutely_continuousI:
- assumes "finite_measure_space M \<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 M \<nu> by fact
+subsection "Kullback$-$Leibler divergence"
- have "\<nu> N = \<nu> (\<Union>x\<in>N. {x})" by simp
- also have "\<dots> = (\<Sum>x\<in>N. \<nu> {x})"
- proof (rule v.measure_finitely_additive''[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 M" 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" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
- hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
- thus "\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
- qed
- finally show "\<nu> N = 0" .
-qed
+text {* The Kullback$-$Leibler divergence is also known as relative entropy or
+Kullback$-$Leibler distance. *}
+
+definition
+ "KL_divergence b M \<mu> \<nu> =
+ measure_space.integral M \<mu> (\<lambda>x. log b (real (sigma_finite_measure.RN_deriv M \<nu> \<mu> x)))"
lemma (in finite_measure_space) KL_divergence_eq_finite:
assumes v: "finite_measure_space M \<nu>"
@@ -285,19 +243,13 @@
proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v])
interpret v: finite_measure_space M \<nu> by fact
have ms: "measure_space M \<nu>" by fact
-
have ac: "absolutely_continuous \<nu>"
using ac by (auto intro!: absolutely_continuousI[OF v])
-
show "(\<Sum>x \<in> space M. log b (real (RN_deriv \<nu> x)) * real (\<nu> {x})) = ?sum"
using RN_deriv_finite_measure[OF ms ac]
by (auto intro!: setsum_cong simp: field_simps real_of_pinfreal_mult[symmetric])
qed
-lemma (in finite_prob_space) finite_sum_over_space_eq_1:
- "(\<Sum>x\<in>space M. real (\<mu> {x})) = 1"
- using sum_over_space_eq_1 finite_measure by (simp add: real_of_pinfreal_setsum sets_eq_Pow)
-
lemma (in finite_prob_space) KL_divergence_positive_finite:
assumes v: "finite_prob_space M \<nu>"
assumes ac: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
@@ -322,13 +274,15 @@
fix x assume x: "x \<in> space M"
{ assume "0 < real (\<nu> {x})"
hence "\<mu> {x} \<noteq> 0" using ac[OF x] by auto
- thus "0 < prob {x}" using measure_finite[of "{x}"] sets_eq_Pow x
+ thus "0 < prob {x}" using finite_measure[of "{x}"] sets_eq_Pow x
by (cases "\<mu> {x}") simp_all }
qed auto
qed
thus "0 \<le> KL_divergence b M \<nu> \<mu>" using finite_sum_over_space_eq_1 by simp
qed
+subsection {* Mutual Information *}
+
definition (in prob_space)
"mutual_information b S T X Y =
KL_divergence b (prod_measure_space S T)
@@ -341,24 +295,48 @@
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
-lemma prod_measure_times_finite:
- assumes fms: "finite_measure_space M \<mu>" "finite_measure_space N \<nu>" and a: "a \<in> space M \<times> space N"
- shows "prod_measure M \<mu> N \<nu> {a} = \<mu> {fst a} * \<nu> {snd a}"
-proof (cases a)
- case (Pair b c)
- hence a_eq: "{a} = {b} \<times> {c}" by simp
-
- interpret M: finite_measure_space M \<mu> by fact
- interpret N: finite_measure_space N \<nu> by fact
-
- from finite_measure_space.finite_prod_measure_times[OF fms, of "{b}" "{c}"] M.sets_eq_Pow N.sets_eq_Pow a Pair
- show ?thesis unfolding a_eq by simp
+lemma (in finite_information_space) mutual_information_generic_eq:
+ assumes MX: "finite_measure_space MX (distribution X)"
+ assumes MY: "finite_measure_space MY (distribution Y)"
+ shows "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
+ real (joint_distribution X Y {(x,y)}) *
+ log b (real (joint_distribution X Y {(x,y)}) /
+ (real (distribution X {x}) * real (distribution Y {y}))))"
+proof -
+ let ?P = "prod_measure_space MX MY"
+ let ?\<mu> = "prod_measure MX (distribution X) MY (distribution Y)"
+ let ?\<nu> = "joint_distribution X Y"
+ interpret X: finite_measure_space MX "distribution X" by fact
+ moreover interpret Y: finite_measure_space MY "distribution Y" by fact
+ have fms: "finite_measure_space MX (distribution X)"
+ "finite_measure_space MY (distribution Y)" by fact+
+ have fms_P: "finite_measure_space ?P ?\<mu>"
+ by (rule X.finite_measure_space_finite_prod_measure) fact
+ then interpret P: finite_measure_space ?P ?\<mu> .
+ have fms_P': "finite_measure_space ?P ?\<nu>"
+ using finite_product_measure_space[of "space MX" "space MY"]
+ X.finite_space Y.finite_space sigma_prod_sets_finite[OF X.finite_space Y.finite_space]
+ X.sets_eq_Pow Y.sets_eq_Pow
+ by (simp add: prod_measure_space_def sigma_def)
+ then interpret P': finite_measure_space ?P ?\<nu> .
+ { fix x assume "x \<in> space ?P"
+ hence in_MX: "{fst x} \<in> sets MX" "{snd x} \<in> sets MY" using X.sets_eq_Pow Y.sets_eq_Pow
+ by (auto simp: prod_measure_space_def)
+ assume "?\<mu> {x} = 0"
+ with X.finite_prod_measure_times[OF fms(2), of "{fst x}" "{snd x}"] in_MX
+ have "distribution X {fst x} = 0 \<or> distribution Y {snd x} = 0"
+ by (simp add: prod_measure_space_def)
+ hence "joint_distribution X Y {x} = 0"
+ by (cases x) (auto simp: distribution_order) }
+ note measure_0 = this
+ show ?thesis
+ unfolding Let_def mutual_information_def
+ using measure_0 fms_P fms_P' MX MY P.absolutely_continuous_def
+ by (subst P.KL_divergence_eq_finite)
+ (auto simp add: prod_measure_space_def prod_measure_times_finite
+ finite_prob_space_eq setsum_cartesian_product' real_of_pinfreal_mult[symmetric])
qed
-lemma setsum_cartesian_product':
- "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
- unfolding setsum_cartesian_product by simp
-
lemma (in finite_information_space)
assumes MX: "finite_prob_space MX (distribution X)"
assumes MY: "finite_prob_space MY (distribution Y)"
@@ -436,9 +414,26 @@
(real (distribution X {x}) * real (distribution Y {y}))))"
by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images)
+lemma (in finite_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')"
+proof -
+ have "X ` space M = X' ` space M" using X by (auto intro!: image_eqI)
+ moreover have "Y ` space M = Y' ` space M" using Y by (auto intro!: image_eqI)
+ ultimately show ?thesis
+ unfolding mutual_information_eq
+ using
+ assms[THEN distribution_cong]
+ joint_distribution_cong[OF assms]
+ by (auto intro!: setsum_cong)
+qed
+
lemma (in finite_information_space) mutual_information_positive: "0 \<le> \<I>(X;Y)"
by (subst mutual_information_positive_generic) (simp_all add: finite_prob_space_of_images)
+subsection {* Entropy *}
+
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
@@ -446,32 +441,146 @@
finite_entropy ("\<H>'(_')") where
"\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
-lemma (in finite_information_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 finite_information_space) entropy_generic_eq:
+ assumes MX: "finite_measure_space MX (distribution X)"
+ shows "entropy b MX X = -(\<Sum> x \<in> space MX. real (distribution X {x}) * log b (real (distribution X {x})))"
+proof -
+ let "?X x" = "real (distribution X {x})"
+ let "?XX x y" = "real (joint_distribution X X {(x, y)})"
+ interpret MX: finite_measure_space MX "distribution X" by fact
+ { fix x y
+ have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
+ 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)"
+ unfolding distribution_def by (auto simp: mult_log_divide) }
+ 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
+ by (auto simp: setsum_cases MX.finite_space)
+qed
lemma (in finite_information_space) entropy_eq:
"\<H>(X) = -(\<Sum> x \<in> X ` space M. real (distribution X {x}) * log b (real (distribution X {x})))"
-proof -
- { fix f
- { fix x y
- have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
- hence "real (distribution (\<lambda>x. (X x, X x)) {(x,y)}) * f x y =
- (if x = y then real (distribution X {x}) * f x y else 0)"
- unfolding distribution_def by auto }
- hence "(\<Sum>(x, y) \<in> X ` space M \<times> X ` space M. real (joint_distribution X X {(x, y)}) * f x y) =
- (\<Sum>x \<in> X ` space M. real (distribution X {x}) * f x x)"
- unfolding setsum_cartesian_product' by (simp add: setsum_cases finite_space) }
- note remove_cartesian_product = this
-
- show ?thesis
- unfolding entropy_def mutual_information_eq setsum_negf[symmetric] remove_cartesian_product
- by (auto intro!: setsum_cong)
-qed
+ by (simp add: finite_measure_space entropy_generic_eq)
lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
unfolding entropy_def using mutual_information_positive .
+lemma (in finite_information_space) entropy_certainty_eq_0:
+ assumes "x \<in> X ` space M" and "distribution X {x} = 1"
+ shows "\<H>(X) = 0"
+proof -
+ interpret X: finite_prob_space "\<lparr> space = X ` space M, sets = Pow (X ` space M) \<rparr>" "distribution X"
+ by (rule finite_prob_space_of_images)
+
+ 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 asm: "y \<noteq> x" "y \<in> X ` space M"
+ hence "{y} \<subseteq> X ` space M - {x}" by auto
+ from X.measure_mono[OF this] X0 asm
+ have "distribution X {y} = 0" by auto }
+
+ hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> real (distribution X {y}) = (if x = y then 1 else 0)"
+ using assms by auto
+
+ 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 by (auto simp: y fi)
+qed
+
+lemma (in finite_information_space) entropy_le_card_not_0:
+ "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
+proof -
+ let "?d x" = "distribution X {x}"
+ let "?p x" = "real (?d x)"
+ have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
+ by (auto intro!: setsum_cong simp: entropy_eq setsum_negf[symmetric])
+ also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
+ apply (rule log_setsum')
+ using not_empty b_gt_1 finite_space sum_over_space_real_distribution
+ by auto
+ also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?d x \<noteq> 0 then 1 else 0)"
+ apply (rule arg_cong[where f="\<lambda>f. log b (\<Sum>x\<in>X`space M. f x)"])
+ using distribution_finite[of X] by (auto simp: expand_fun_eq real_of_pinfreal_eq_0)
+ finally show ?thesis
+ using finite_space by (auto simp: setsum_cases real_eq_of_nat)
+qed
+
+lemma (in finite_information_space) entropy_uniform_max:
+ 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 -
+ note uniform =
+ finite_prob_space_of_images[of X, THEN finite_prob_space.uniform_prob, simplified]
+
+ have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
+ using finite_space not_empty by auto
+
+ { fix x assume "x \<in> X ` space M"
+ hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
+ proof (rule uniform)
+ fix x y assume "x \<in> X`space M" "y \<in> X`space M"
+ from assms[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
+ qed }
+ thus ?thesis
+ using not_empty finite_space b_gt_1 card_gt0
+ by (simp add: entropy_eq real_eq_of_nat[symmetric] log_divide)
+qed
+
+lemma (in finite_information_space) entropy_le_card:
+ "\<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
+ moreover
+ have "0 < card (X`space M)"
+ using finite_space not_empty unfolding card_gt_0_iff by auto
+ 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 unfolding entropy_eq by simp
+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 finite_space not_empty by (auto intro!: card_mono)
+ note entropy_le_card_not_0
+ also have "log b (real ?A) \<le> log b (real ?B)"
+ using b_gt_1 False finite_space not_empty `?A \<le> ?B`
+ by (auto intro!: log_le simp: card_gt_0_iff)
+ finally show ?thesis .
+qed
+
+lemma (in finite_information_space) entropy_commute:
+ "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
+proof -
+ 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 entropy_eq unfolding * setsum_reindex[OF inj]
+ by (simp add: joint_distribution_commute[of Y X] split_beta)
+qed
+
+lemma (in finite_information_space) entropy_eq_cartesian_sum:
+ "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
+ real (joint_distribution X Y {(x,y)}) *
+ log b (real (joint_distribution X Y {(x,y)})))"
+proof -
+ { 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 finite_space
+ unfolding entropy_eq neg_equal_iff_equal setsum_cartesian_product
+ by (auto intro!: setsum_mono_zero_cong_left)
+qed
+
+subsection {* Conditional Mutual Information *}
+
definition (in prob_space)
"conditional_mutual_information b M1 M2 M3 X Y Z \<equiv>
mutual_information b M1 (prod_measure_space M2 M3) X (\<lambda>x. (Y x, Z x)) -
@@ -485,87 +594,32 @@
\<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr>
X Y Z"
-lemma (in finite_information_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 measure_finitely_additive'')
- (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
- intro!: arg_cong[where f=prob])
-
-lemma (in finite_information_space) setsum_distribution:
- "(\<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_information_space) setsum_real_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. real (distribution Y {f x})) = real (distribution Z {c})"
- unfolding distribution_def assms
- using finite_space assms
- by (subst real_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_information_space) setsum_real_distribution:
- "(\<Sum>x \<in> X`space M. real (joint_distribution X Y {(x, y)})) = real (distribution Y {y})"
- "(\<Sum>y \<in> Y`space M. real (joint_distribution X Y {(x, y)})) = real (distribution X {x})"
- "(\<Sum>x \<in> X`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution Y Z {(y, z)})"
- "(\<Sum>y \<in> Y`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Z {(x, z)})"
- "(\<Sum>z \<in> Z`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Y {(x, y)})"
- by (auto intro!: inj_onI setsum_real_distribution_gen)
+lemma (in finite_information_space) conditional_mutual_information_generic_eq:
+ assumes MX: "finite_measure_space MX (distribution X)"
+ assumes MY: "finite_measure_space MY (distribution Y)"
+ assumes MZ: "finite_measure_space MZ (distribution 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.
+ real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) *
+ log b (real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) /
+ (real (distribution X {x}) * real (joint_distribution Y Z {(y, z)})))) -
+ (\<Sum>(x, y)\<in>space MX \<times> space MZ.
+ real (joint_distribution X Z {(x, y)}) *
+ log b (real (joint_distribution X Z {(x, y)}) / (real (distribution X {x}) * real (distribution Z {y}))))"
+ using assms finite_measure_space_prod[OF MY MZ]
+ unfolding conditional_mutual_information_def
+ by (subst (1 2) mutual_information_generic_eq)
+ (simp_all add: setsum_cartesian_product' finite_measure_space.finite_prod_measure_space)
-lemma (in finite_information_space) conditional_mutual_information_eq_sum:
- "\<I>(X ; Y | Z) =
- (\<Sum>(x, y, z)\<in>X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M.
- real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
- log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)})/
- real (distribution (\<lambda>x. (Y x, Z x)) {(y, z)}))) -
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
- real (distribution (\<lambda>x. (X x, Z x)) {(x,z)}) * log b (real (distribution (\<lambda>x. (X x, Z x)) {(x,z)}) / real (distribution Z {z})))"
- (is "_ = ?rhs")
-proof -
- have setsum_product:
- "\<And>f x. (\<Sum>v\<in>(\<lambda>x. (Y x, Z x)) ` space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)}) * f v)
- = (\<Sum>v\<in>Y ` space M \<times> Z ` space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)}) * f v)"
- proof (safe intro!: setsum_mono_zero_cong_left imageI)
- fix x y z f
- assume *: "(Y y, Z z) \<notin> (\<lambda>x. (Y x, Z x)) ` space M" and "y \<in> space M" "z \<in> space M"
- hence "(\<lambda>x. (X x, Y x, Z x)) -` {(x, Y y, Z z)} \<inter> space M = {}"
- proof safe
- fix x' assume x': "x' \<in> space M" and eq: "Y x' = Y y" "Z x' = Z z"
- have "(Y y, Z z) \<in> (\<lambda>x. (Y x, Z x)) ` space M" using eq[symmetric] x' by auto
- thus "x' \<in> {}" using * by auto
- qed
- thus "real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, Y y, Z z)}) * f (Y y) (Z z) = 0"
- unfolding distribution_def by simp
- qed (simp add: finite_space)
-
- thus ?thesis
- unfolding conditional_mutual_information_def Let_def mutual_information_eq
- by (subst mutual_information_eq_generic)
- (auto simp: prod_measure_space_def sigma_prod_sets_finite finite_space sigma_def
- finite_prob_space_of_images finite_product_prob_space_of_images
- setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
- setsum_left_distrib[symmetric] setsum_real_distribution
- cong: setsum_cong)
-qed
lemma (in finite_information_space) conditional_mutual_information_eq:
"\<I>(X ; Y | Z) = (\<Sum>(x, y, z) \<in> X ` space M \<times> Y ` space M \<times> Z ` space M.
real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
(real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
- unfolding conditional_mutual_information_def Let_def mutual_information_eq
- by (subst mutual_information_eq_generic)
+ by (subst conditional_mutual_information_generic_eq)
(auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
- finite_prob_space_of_images finite_product_prob_space_of_images sigma_def
+ finite_measure_space finite_product_prob_space_of_images sigma_def
setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
setsum_left_distrib[symmetric] setsum_real_distribution setsum_commute[where A="Y`space M"]
real_of_pinfreal_mult[symmetric]
@@ -581,22 +635,6 @@
by (simp add: setsum_cartesian_product' distribution_remove_const)
qed
-lemma (in finite_prob_space) distribution_finite:
- "distribution X A \<noteq> \<omega>"
- by (auto simp: sets_eq_Pow distribution_def intro!: measure_finite)
-
-lemma (in finite_prob_space) real_distribution_order:
- shows "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution X {x})"
- and "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution Y {y})"
- and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution X {x})"
- and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution Y {y})"
- and "distribution X {x} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
- using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
- by auto
-
lemma (in finite_information_space) conditional_mutual_information_positive:
"0 \<le> \<I>(X ; Y | Z)"
proof -
@@ -640,6 +678,8 @@
by (simp add: real_of_pinfreal_mult[symmetric])
qed
+subsection {* Conditional Entropy *}
+
definition (in prob_space)
"conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
@@ -652,19 +692,69 @@
lemma (in finite_information_space) conditional_entropy_positive:
"0 \<le> \<H>(X | Y)" unfolding conditional_entropy_def using conditional_mutual_information_positive .
+lemma (in finite_information_space) conditional_entropy_generic_eq:
+ assumes MX: "finite_measure_space MX (distribution X)"
+ assumes MY: "finite_measure_space MZ (distribution Z)"
+ shows "conditional_entropy b MX MZ X Z =
+ - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
+ real (joint_distribution X Z {(x, z)}) *
+ log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
+ unfolding conditional_entropy_def using assms
+ apply (simp add: conditional_mutual_information_generic_eq
+ setsum_cartesian_product' setsum_commute[of _ "space MZ"]
+ setsum_negf[symmetric] setsum_subtractf[symmetric])
+proof (safe intro!: setsum_cong, simp)
+ fix z x assume "z \<in> space MZ" "x \<in> space MX"
+ let "?XXZ x'" = "real (joint_distribution X (\<lambda>x. (X x, Z x)) {(x, x', z)})"
+ let "?XZ x'" = "real (joint_distribution X Z {(x', z)})"
+ let "?X" = "real (distribution X {x})"
+ interpret MX: finite_measure_space MX "distribution X" by fact
+ have *: "\<And>A. A = {} \<Longrightarrow> prob A = 0" by simp
+ have XXZ: "\<And>x'. ?XXZ x' = (if x' = x then ?XZ x else 0)"
+ by (auto simp: distribution_def intro!: arg_cong[where f=prob] *)
+ have "(\<Sum>x'\<in>space MX. ?XXZ x' * log b (?XXZ x') - (?XXZ x' * log b ?X + ?XXZ x' * log b (?XZ x'))) =
+ (\<Sum>x'\<in>{x}. ?XZ x' * log b (?XZ x') - (?XZ x' * log b ?X + ?XZ x' * log b (?XZ x')))"
+ using `x \<in> space MX` MX.finite_space
+ by (safe intro!: setsum_mono_zero_cong_right)
+ (auto split: split_if_asm simp: XXZ)
+ then show "(\<Sum>x'\<in>space MX. ?XXZ x' * log b (?XXZ x') - (?XXZ x' * log b ?X + ?XXZ x' * log b (?XZ x'))) +
+ ?XZ x * log b ?X = 0" by simp
+qed
+
lemma (in finite_information_space) conditional_entropy_eq:
"\<H>(X | Z) =
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
real (joint_distribution X Z {(x, z)}) *
log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
+ by (simp add: finite_measure_space conditional_entropy_generic_eq)
+
+lemma (in finite_information_space) conditional_entropy_eq_ce_with_hypothesis:
+ "\<H>(X | Y) =
+ -(\<Sum>y\<in>Y`space M. real (distribution Y {y}) *
+ (\<Sum>x\<in>X`space M. real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}) *
+ log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}))))"
+ unfolding conditional_entropy_eq neg_equal_iff_equal
+ apply (simp add: setsum_commute[of _ "Y`space M"] setsum_cartesian_product' setsum_divide_distrib[symmetric])
+ apply (safe intro!: setsum_cong)
+ using real_distribution_order'[of Y _ X _]
+ by (auto simp add: setsum_subtractf[of _ _ "X`space M"])
+
+lemma (in finite_information_space) conditional_entropy_eq_cartesian_sum:
+ "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
+ real (joint_distribution X Y {(x,y)}) *
+ log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {y})))"
proof -
- have *: "\<And>x y z. (\<lambda>x. (X x, X x, Z x)) -` {(x, y, z)} = (if x = y then (\<lambda>x. (X x, Z x)) -` {(x, z)} else {})" by auto
- show ?thesis
- unfolding conditional_mutual_information_eq_sum
- conditional_entropy_def distribution_def *
- by (auto intro!: setsum_0')
+ { 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 finite_space
+ unfolding conditional_entropy_eq neg_equal_iff_equal setsum_cartesian_product
+ by (auto intro!: setsum_mono_zero_cong_left)
qed
+subsection {* Equalities *}
+
lemma (in finite_information_space) mutual_information_eq_entropy_conditional_entropy:
"\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
unfolding mutual_information_eq entropy_eq conditional_entropy_eq
@@ -680,109 +770,15 @@
show ?thesis by auto
qed
-(* -------------Entropy of a RV with a certain event is zero---------------- *)
-
-lemma (in finite_information_space) finite_entropy_certainty_eq_0:
- assumes "x \<in> X ` space M" and "distribution X {x} = 1"
- shows "\<H>(X) = 0"
-proof -
- interpret X: finite_prob_space "\<lparr> space = X ` space M, sets = Pow (X ` space M) \<rparr>" "distribution X"
- by (rule finite_prob_space_of_images)
-
- 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 asm: "y \<noteq> x" "y \<in> X ` space M"
- hence "{y} \<subseteq> X ` space M - {x}" by auto
- from X.measure_mono[OF this] X0 asm
- have "distribution X {y} = 0" by auto }
-
- hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> real (distribution X {y}) = (if x = y then 1 else 0)"
- using assms by auto
-
- 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 by (auto simp: y fi)
-qed
-(* --------------- upper bound on entropy for a rv ------------------------- *)
-
-lemma (in finite_prob_space) distribution_1:
- "distribution X A \<le> 1"
- unfolding distribution_def measure_space_1[symmetric]
- by (auto intro!: measure_mono simp: sets_eq_Pow)
-
-lemma (in finite_prob_space) real_distribution_1:
- "real (distribution X A) \<le> 1"
- unfolding real_pinfreal_1[symmetric]
- by (rule real_of_pinfreal_mono[OF _ distribution_1]) simp
+lemma (in finite_information_space) entropy_chain_rule:
+ "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
+ unfolding entropy_eq[of X] entropy_eq_cartesian_sum conditional_entropy_eq_cartesian_sum
+ unfolding setsum_commute[of _ "X`space M"] setsum_negf[symmetric] setsum_addf[symmetric]
+ by (rule setsum_cong)
+ (simp_all add: setsum_negf setsum_addf setsum_subtractf setsum_real_distribution
+ setsum_left_distrib[symmetric] joint_distribution_commute[of X Y])
-lemma (in finite_information_space) finite_entropy_le_card:
- "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
-proof -
- let "?d x" = "distribution X {x}"
- let "?p x" = "real (?d x)"
- have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
- by (auto intro!: setsum_cong simp: entropy_eq setsum_negf[symmetric])
- also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
- apply (rule log_setsum')
- using not_empty b_gt_1 finite_space sum_over_space_real_distribution
- by auto
- also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?d x \<noteq> 0 then 1 else 0)"
- apply (rule arg_cong[where f="\<lambda>f. log b (\<Sum>x\<in>X`space M. f x)"])
- using distribution_finite[of X] by (auto simp: expand_fun_eq real_of_pinfreal_eq_0)
- finally show ?thesis
- using finite_space by (auto simp: setsum_cases real_eq_of_nat)
-qed
-
-(* --------------- entropy is maximal for a uniform rv --------------------- *)
-
-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 / real (card (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 real_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 fastsimp
- from one have three: "prob {x} \<noteq> 0" by fastsimp
- thus ?thesis using one two three divide_cancel_right
- by (auto simp:field_simps)
-qed
-
-lemma (in finite_information_space) finite_entropy_uniform_max:
- 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 -
- note uniform =
- finite_prob_space_of_images[of X, THEN finite_prob_space.uniform_prob, simplified]
-
- have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
- using finite_space not_empty by auto
-
- { fix x assume "x \<in> X ` space M"
- hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
- proof (rule uniform)
- fix x y assume "x \<in> X`space M" "y \<in> X`space M"
- from assms[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
- qed }
- thus ?thesis
- using not_empty finite_space b_gt_1 card_gt0
- by (simp add: entropy_eq real_eq_of_nat[symmetric] log_divide)
-qed
+section {* Partitioning *}
definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
@@ -934,38 +930,6 @@
"\<H>(f \<circ> X) \<le> \<H>(X)"
by (subst (2) entropy_partition[of _ "f \<circ> X"]) (auto intro: conditional_entropy_positive)
-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 expand_fun_eq
- 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 expand_fun_eq
- using assms by (auto intro!: arg_cong[where f="\<mu>"])
-
-lemma image_cong:
- "\<lbrakk> \<And>x. x \<in> S \<Longrightarrow> X x = X' x \<rbrakk> \<Longrightarrow> X ` S = X' ` S"
- by (auto intro!: image_eqI)
-
-lemma (in finite_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')"
-proof -
- have "X ` space M = X' ` space M" using X by (rule image_cong)
- moreover have "Y ` space M = Y' ` space M" using Y by (rule image_cong)
- ultimately show ?thesis
- unfolding mutual_information_eq
- using
- assms[THEN distribution_cong]
- joint_distribution_cong[OF assms]
- by (auto intro!: setsum_cong)
-qed
-
corollary (in finite_information_space) entropy_of_inj:
assumes "inj_on f (X`space M)"
shows "\<H>(f \<circ> X) = \<H>(X)"
--- a/src/HOL/Probability/Lebesgue_Integration.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Thu Sep 02 20:44:33 2010 +0200
@@ -209,19 +209,6 @@
by (auto intro!: **)
qed
-lemma setsum_indicator_disjoint_family:
- fixes f :: "'d \<Rightarrow> 'e::semiring_1"
- assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
- shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
-proof -
- have "P \<inter> {i. x \<in> A i} = {j}"
- using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
- by auto
- thus ?thesis
- unfolding indicator_def
- by (simp add: if_distrib setsum_cases[OF `finite P`])
-qed
-
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
fixes u :: "'a \<Rightarrow> pinfreal"
assumes u: "u \<in> borel_measurable M"
@@ -426,6 +413,62 @@
with x show thesis by (auto intro!: that[of f])
qed
+lemma (in sigma_algebra) simple_function_eq_borel_measurable:
+ fixes f :: "'a \<Rightarrow> pinfreal"
+ shows "simple_function 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]
+ by (fastsimp simp: simple_function_def)
+
+lemma (in measure_space) simple_function_restricted:
+ fixes f :: "'a \<Rightarrow> pinfreal" assumes "A \<in> sets M"
+ shows "sigma_algebra.simple_function (restricted_space A) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
+ (is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
+proof -
+ interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
+ have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
+ proof cases
+ assume "A = space M"
+ then have "f`A = ?f`space M" by (fastsimp 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::pinfreal)"
+ 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`]
+ by auto
+qed
+
+lemma (in sigma_algebra) simple_function_subalgebra:
+ assumes "sigma_algebra.simple_function (M\<lparr>sets:=N\<rparr>) f"
+ and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets:=N\<rparr>)"
+ shows "simple_function f"
+ using assms
+ unfolding simple_function_def
+ unfolding sigma_algebra.simple_function_def[OF N_subalgebra(2)]
+ by auto
+
section "Simple integral"
definition (in measure_space)
@@ -668,6 +711,41 @@
qed
qed
+lemma (in measure_space) simple_integral_restricted:
+ assumes "A \<in> sets M"
+ assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
+ shows "measure_space.simple_integral (restricted_space A) \<mu> f = simple_integral (\<lambda>x. f x * indicator A x)"
+ (is "_ = simple_integral ?f")
+ unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets 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
+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
+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 pinfreal_mult_cancel_left by auto
+qed
+
+lemma (in measure_space) simple_integral_subalgebra[simp]:
+ assumes "measure_space (M\<lparr>sets := N\<rparr>) \<mu>"
+ shows "measure_space.simple_integral (M\<lparr>sets := N\<rparr>) \<mu> = simple_integral"
+ unfolding simple_integral_def_raw
+ unfolding measure_space.simple_integral_def_raw[OF assms] by simp
+
section "Continuous posititve integration"
definition (in measure_space)
@@ -1077,6 +1155,43 @@
qed
qed
+lemma (in measure_space) positive_integral_translated_density:
+ assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+ shows "measure_space.positive_integral M (\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)) g =
+ positive_integral (\<lambda>x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
+proof -
+ from measure_space_density[OF assms(1)]
+ interpret T: measure_space M ?T .
+ from borel_measurable_implies_simple_function_sequence[OF assms(2)]
+ obtain G where G: "\<And>i. simple_function (G i)" "G \<up> g" by blast
+ note G_borel = borel_measurable_simple_function[OF this(1)]
+ from T.positive_integral_isoton[OF `G \<up> g` G_borel]
+ have *: "(\<lambda>i. T.positive_integral (G i)) \<up> T.positive_integral g" .
+ { fix i
+ have [simp]: "finite (G i ` space M)"
+ using G(1) unfolding simple_function_def by auto
+ have "T.positive_integral (G i) = T.simple_integral (G i)"
+ using G T.positive_integral_eq_simple_integral by simp
+ also have "\<dots> = positive_integral (\<lambda>x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
+ apply (simp add: T.simple_integral_def)
+ apply (subst positive_integral_cmult[symmetric])
+ using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
+ apply (subst positive_integral_setsum[symmetric])
+ using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
+ by (simp add: setsum_right_distrib field_simps)
+ also have "\<dots> = positive_integral (\<lambda>x. f x * G i x)"
+ by (auto intro!: positive_integral_cong
+ simp: indicator_def if_distrib setsum_cases)
+ finally have "T.positive_integral (G i) = positive_integral (\<lambda>x. f x * G i x)" . }
+ with * have eq_Tg: "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> T.positive_integral g" by simp
+ from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
+ unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
+ then have "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> positive_integral (\<lambda>x. f x * g x)"
+ using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pinfreal_times)
+ with eq_Tg show "T.positive_integral g = positive_integral (\<lambda>x. f x * g x)"
+ unfolding isoton_def by simp
+qed
+
lemma (in measure_space) positive_integral_null_set:
assumes borel: "u \<in> borel_measurable M" and "N \<in> null_sets"
shows "positive_integral (\<lambda>x. u x * indicator N x) = 0" (is "?I = 0")
@@ -1222,6 +1337,58 @@
finally show ?thesis by simp
qed
+lemma (in measure_space) positive_integral_restricted:
+ assumes "A \<in> sets M"
+ shows "measure_space.positive_integral (restricted_space A) \<mu> f = positive_integral (\<lambda>x. f x * indicator A x)"
+ (is "measure_space.positive_integral ?R \<mu> f = positive_integral ?f")
+proof -
+ have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
+ then interpret R: measure_space ?R \<mu> .
+ have saR: "sigma_algebra ?R" by fact
+ have *: "R.positive_integral f = R.positive_integral ?f"
+ by (auto intro!: R.positive_integral_cong)
+ show ?thesis
+ unfolding * R.positive_integral_def positive_integral_def
+ unfolding simple_function_restricted[OF `A \<in> sets M`]
+ apply (simp add: SUPR_def)
+ apply (rule arg_cong[where f=Sup])
+ proof (auto simp: image_iff simple_integral_restricted[OF `A \<in> sets M`])
+ fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
+ "g \<le> f" "\<forall>x\<in>A. \<omega> \<noteq> g x"
+ then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and> (\<forall>y\<in>space M. \<omega> \<noteq> x y) \<and>
+ simple_integral (\<lambda>x. g x * indicator A x) = simple_integral x"
+ apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
+ by (auto simp: indicator_def le_fun_def)
+ next
+ fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
+ "\<forall>x\<in>space M. \<omega> \<noteq> g x"
+ then have *: "(\<lambda>x. g x * indicator A x) = g"
+ "\<And>x. g x * indicator A x = g x"
+ "\<And>x. g x \<le> f x"
+ by (auto simp: le_fun_def expand_fun_eq indicator_def split: split_if_asm)
+ from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and> (\<forall>xa\<in>A. \<omega> \<noteq> x xa) \<and>
+ simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
+ using `A \<in> sets M`[THEN sets_into_space]
+ apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
+ by (fastsimp simp: le_fun_def *)
+ qed
+qed
+
+lemma (in measure_space) positive_integral_subalgebra[simp]:
+ assumes borel: "f \<in> borel_measurable (M\<lparr>sets := N\<rparr>)"
+ and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets := N\<rparr>)"
+ shows "measure_space.positive_integral (M\<lparr>sets := N\<rparr>) \<mu> f = positive_integral f"
+proof -
+ note msN = measure_space_subalgebra[OF N_subalgebra]
+ then interpret N: measure_space "M\<lparr>sets:=N\<rparr>" \<mu> .
+ from N.borel_measurable_implies_simple_function_sequence[OF borel]
+ obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast
+ then have sf: "\<And>i. simple_function (fs i)"
+ using simple_function_subalgebra[OF _ N_subalgebra] by blast
+ from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
+ show ?thesis unfolding simple_integral_subalgebra[OF msN] isoton_def by simp
+qed
+
section "Lebesgue Integral"
definition (in measure_space) integrable where
@@ -1629,44 +1796,6 @@
by (simp add: real_of_pinfreal_eq_0)
qed
-lemma LIMSEQ_max:
- "u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
- by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
-
-lemma (in sigma_algebra) 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"
- shows "u' \<in> borel_measurable M"
-proof -
- let "?pu x i" = "max (u i x) 0"
- let "?nu x i" = "max (- u i x) 0"
-
- { fix x assume x: "x \<in> space M"
- have "(?pu x) ----> max (u' x) 0"
- "(?nu x) ----> max (- u' x) 0"
- using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
- from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
- have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
- "(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
- by (simp_all add: Real_max'[symmetric]) }
- note eq = this
-
- have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
- by auto
-
- have "(SUP n. INF m. (\<lambda>x. Real (u (n + m) x))) \<in> borel_measurable M"
- "(SUP n. INF m. (\<lambda>x. Real (- u (n + m) x))) \<in> borel_measurable M"
- using u by (auto intro: borel_measurable_SUP borel_measurable_INF borel_measurable_Real)
- with eq[THEN measurable_cong, of M "\<lambda>x. x" borel_space]
- have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
- "(\<lambda>x. Real (- u' x)) \<in> borel_measurable M"
- unfolding SUPR_fun_expand INFI_fun_expand by auto
- note this[THEN borel_measurable_real]
- from borel_measurable_diff[OF this]
- show ?thesis unfolding * .
-qed
-
lemma (in measure_space) integral_dominated_convergence:
assumes u: "\<And>i. integrable (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
and w: "integrable w" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> w x"
@@ -1926,41 +2055,11 @@
by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
qed
-lemma sigma_algebra_cong:
- fixes M :: "('a, 'b) algebra_scheme" and M' :: "('a, 'c) algebra_scheme"
- assumes *: "sigma_algebra M"
- and cong: "space M = space M'" "sets M = sets M'"
- shows "sigma_algebra M'"
-using * unfolding sigma_algebra_def algebra_def sigma_algebra_axioms_def unfolding cong .
-
-lemma finite_Pow_additivity_sufficient:
- assumes "finite (space M)" and "sets M = Pow (space M)"
- and "positive \<mu>" and "additive M \<mu>"
- and "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<omega>"
- shows "finite_measure_space M \<mu>"
-proof -
- have "sigma_algebra M"
- using assms by (auto intro!: sigma_algebra_cong[OF sigma_algebra_Pow])
-
- have "measure_space M \<mu>"
- by (rule sigma_algebra.finite_additivity_sufficient) (fact+)
- thus ?thesis
- unfolding finite_measure_space_def finite_measure_space_axioms_def
- using assms by simp
-qed
-
-lemma finite_measure_spaceI:
- assumes "measure_space M \<mu>" and "finite (space M)" and "sets M = Pow (space M)"
- and "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<omega>"
- shows "finite_measure_space M \<mu>"
- unfolding finite_measure_space_def finite_measure_space_axioms_def
- using assms by simp
+lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function f"
+ unfolding simple_function_def sets_eq_Pow using finite_space by auto
lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
- unfolding measurable_def sets_eq_Pow by auto
-
-lemma (in finite_measure_space) simple_function_finite: "simple_function f"
- unfolding simple_function_def sets_eq_Pow using finite_space by auto
+ by (auto intro: borel_measurable_simple_function)
lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
"positive_integral f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
@@ -1979,10 +2078,8 @@
"positive_integral (\<lambda>x. Real (f x)) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
"positive_integral (\<lambda>x. Real (- f x)) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
unfolding positive_integral_finite_eq_setsum by auto
-
show "integrable f" using finite_space finite_measure
by (simp add: setsum_\<omega> integrable_def sets_eq_Pow)
-
show ?I using finite_measure
apply (simp add: integral_def sets_eq_Pow real_of_pinfreal_setsum[symmetric]
real_of_pinfreal_mult[symmetric] setsum_subtractf[symmetric])
--- a/src/HOL/Probability/Measure.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Measure.thy Thu Sep 02 20:44:33 2010 +0200
@@ -414,6 +414,19 @@
finally show ?thesis .
qed
+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 intro: finite_UN)
+ 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) measurable_countably_subadditive:
assumes "range f \<subseteq> sets M"
shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>\<^isub>\<infinity> i. \<mu> (f i))"
@@ -432,9 +445,34 @@
finally show ?thesis .
qed
+lemma (in measure_space) measure_inter_full_set:
+ assumes "S \<in> sets M" "T \<in> sets M" and not_\<omega>: "\<mu> (T - S) \<noteq> \<omega>"
+ 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)
+
+ 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"
+ by (rule pinfreal_less_add[OF not_\<omega>]) (insert contr, 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 (in measure_space) restricted_measure_space:
assumes "S \<in> sets M"
- shows "measure_space (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>) \<mu>"
+ shows "measure_space (restricted_space S) \<mu>"
(is "measure_space ?r \<mu>")
unfolding measure_space_def measure_space_axioms_def
proof safe
@@ -451,6 +489,46 @@
qed
qed
+lemma (in measure_space) measure_space_vimage:
+ assumes "f \<in> measurable M M'"
+ and "sigma_algebra M'"
+ shows "measure_space M' (\<lambda>A. \<mu> (f -` A \<inter> space M))" (is "measure_space M' ?T")
+proof -
+ interpret M': sigma_algebra M' by fact
+
+ show ?thesis
+ proof
+ show "?T {} = 0" by simp
+
+ show "countably_additive M' ?T"
+ proof (unfold countably_additive_def, safe)
+ fix A :: "nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets M'" "disjoint_family A"
+ hence *: "\<And>i. f -` (A i) \<inter> space M \<in> sets M"
+ using `f \<in> measurable M M'` 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>\<^isub>\<infinity> i. ?T (A i)) = ?T (\<Union>i. A i)"
+ using measure_countably_additive[OF _ **] by (auto simp: comp_def vimage_UN)
+ qed
+ qed
+qed
+
+lemma (in measure_space) measure_space_subalgebra:
+ assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
+ shows "measure_space (M\<lparr> sets := N \<rparr>) \<mu>"
+proof -
+ interpret N: sigma_algebra "M\<lparr> sets := N \<rparr>" by fact
+ show ?thesis
+ proof
+ show "countably_additive (M\<lparr>sets := N\<rparr>) \<mu>"
+ using `N \<subseteq> sets M`
+ by (auto simp add: countably_additive_def
+ intro!: measure_countably_additive)
+ qed simp
+qed
+
section "@{text \<sigma>}-finite Measures"
locale sigma_finite_measure = measure_space +
@@ -458,7 +536,7 @@
lemma (in sigma_finite_measure) restricted_sigma_finite_measure:
assumes "S \<in> sets M"
- shows "sigma_finite_measure (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>) \<mu>"
+ shows "sigma_finite_measure (restricted_space S) \<mu>"
(is "sigma_finite_measure ?r _")
unfolding sigma_finite_measure_def sigma_finite_measure_axioms_def
proof safe
@@ -486,6 +564,25 @@
qed
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> \<omega>) \<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> \<omega>"
+ 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> \<omega>"
+ 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
+
section "Real measure values"
lemma (in measure_space) real_measure_Union:
@@ -604,7 +701,7 @@
using finite_measure_of_space by (auto intro!: exI[of _ "\<lambda>x. space M"])
qed
-lemma (in finite_measure) finite_measure:
+lemma (in finite_measure) finite_measure[simp, intro]:
assumes "A \<in> sets M"
shows "\<mu> A \<noteq> \<omega>"
proof -
@@ -619,7 +716,7 @@
lemma (in finite_measure) restricted_finite_measure:
assumes "S \<in> sets M"
- shows "finite_measure (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>) \<mu>"
+ shows "finite_measure (restricted_space S) \<mu>"
(is "finite_measure ?r _")
unfolding finite_measure_def finite_measure_axioms_def
proof (safe del: notI)
@@ -707,6 +804,13 @@
shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
using measure_compl[OF s, OF finite_measure, OF s] .
+lemma (in finite_measure) finite_measure_inter_full_set:
+ assumes "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 assms(1,2) finite_measure assms(3)] assms
+ by auto
+
section {* Measure preserving *}
definition "measure_preserving A \<mu> B \<nu> =
@@ -817,10 +921,51 @@
and sets_eq_Pow: "sets M = Pow (space M)"
and finite_single_measure: "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<omega>"
+lemma (in finite_measure_space) 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 sets_eq_Pow image_space_def)
+ then show "S \<in> sets (image_space X)"
+ by (auto simp: image_space_def)
+qed
+
lemma (in finite_measure_space) sum_over_space: "(\<Sum>x\<in>space M. \<mu> {x}) = \<mu> (space M)"
using measure_finitely_additive''[of "space M" "\<lambda>i. {i}"]
by (simp add: sets_eq_Pow disjoint_family_on_def finite_space)
+lemma finite_measure_spaceI:
+ assumes "finite (space M)" "sets M = Pow(space M)" and space: "\<mu> (space M) \<noteq> \<omega>"
+ and add: "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> \<mu> (A \<union> B) = \<mu> A + \<mu> B"
+ and "\<mu> {} = 0"
+ shows "finite_measure_space M \<mu>"
+ unfolding finite_measure_space_def finite_measure_space_axioms_def
+proof (safe del: notI)
+ show "measure_space M \<mu>"
+ proof (rule sigma_algebra.finite_additivity_sufficient)
+ show "sigma_algebra M"
+ apply (rule sigma_algebra_cong)
+ apply (rule sigma_algebra_Pow[of "space M"])
+ using assms by simp_all
+ show "finite (space M)" by fact
+ show "positive \<mu>" unfolding positive_def by fact
+ show "additive M \<mu>" unfolding additive_def using assms by simp
+ qed
+ show "finite (space M)" by fact
+ { fix A x assume "A \<in> sets M" "x \<in> A" then show "x \<in> space M"
+ using assms by auto }
+ { fix A assume "A \<subseteq> space M" then show "A \<in> sets M"
+ using assms by auto }
+ { fix x assume *: "x \<in> space M"
+ with add[of "{x}" "space M - {x}"] space
+ show "\<mu> {x} \<noteq> \<omega>" by (auto simp: insert_absorb[OF *] Diff_subset) }
+qed
+
sublocale finite_measure_space < finite_measure
proof
show "\<mu> (space M) \<noteq> \<omega>"
@@ -828,6 +973,22 @@
using finite_space finite_single_measure by auto
qed
+lemma finite_measure_space_iff:
+ "finite_measure_space M \<mu> \<longleftrightarrow>
+ finite (space M) \<and> sets M = Pow(space M) \<and> \<mu> (space M) \<noteq> \<omega> \<and> \<mu> {} = 0 \<and>
+ (\<forall>A\<subseteq>space M. \<forall>B\<subseteq>space M. A \<inter> B = {} \<longrightarrow> \<mu> (A \<union> B) = \<mu> A + \<mu> B)"
+ (is "_ = ?rhs")
+proof (intro iffI)
+ assume "finite_measure_space M \<mu>"
+ then interpret finite_measure_space M \<mu> .
+ show ?rhs
+ using finite_space sets_eq_Pow measure_additive empty_measure finite_measure
+ by auto
+next
+ assume ?rhs then show "finite_measure_space M \<mu>"
+ by (auto intro!: finite_measure_spaceI)
+qed
+
lemma psuminf_cmult_indicator:
assumes "disjoint_family A" "x \<in> A i"
shows "(\<Sum>\<^isub>\<infinity> n. f n * indicator (A n) x) = f i"
--- a/src/HOL/Probability/Positive_Infinite_Real.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Positive_Infinite_Real.thy Thu Sep 02 20:44:33 2010 +0200
@@ -411,6 +411,10 @@
lemma pinfreal_less_\<omega>: "x < \<omega> \<longleftrightarrow> x \<noteq> \<omega>"
by (cases x) auto
+lemma pinfreal_0_less_mult_iff[simp]:
+ fixes x y :: pinfreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y"
+ by (cases x, cases y) (auto simp: zero_less_mult_iff)
+
subsection {* @{text "x - y"} on @{typ pinfreal} *}
instantiation pinfreal :: minus
--- a/src/HOL/Probability/Probability_Space.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Probability_Space.thy Thu Sep 02 20:44:33 2010 +0200
@@ -1,39 +1,7 @@
theory Probability_Space
-imports Lebesgue_Integration
+imports Lebesgue_Integration Radon_Nikodym
begin
-lemma (in measure_space) measure_inter_full_set:
- assumes "S \<in> sets M" "T \<in> sets M" and not_\<omega>: "\<mu> (T - S) \<noteq> \<omega>"
- 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)
-
- 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"
- by (rule pinfreal_less_add[OF not_\<omega>]) (insert contr, 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 (in finite_measure) finite_measure_inter_full_set:
- assumes "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 assms(1,2) finite_measure assms(3)] assms
- by auto
-
locale prob_space = measure_space +
assumes measure_space_1: "\<mu> (space M) = 1"
@@ -63,6 +31,19 @@
abbreviation
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
+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 expand_fun_eq
+ 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 expand_fun_eq
+ using assms by (auto intro!: arg_cong[where f="\<mu>"])
+
lemma prob_space: "prob (space M) = 1"
unfolding measure_space_1 by simp
@@ -75,10 +56,6 @@
finally show ?thesis .
qed
-lemma measure_finite[simp, intro]:
- assumes "A \<in> events" shows "\<mu> A \<noteq> \<omega>"
- using measure_le_1[OF assms] by auto
-
lemma prob_compl:
assumes "A \<in> events"
shows "prob (space M - A) = 1 - prob A"
@@ -197,35 +174,17 @@
qed
lemma distribution_prob_space:
- fixes S :: "('c, 'd) algebra_scheme"
- assumes "sigma_algebra S" "random_variable S X"
+ assumes S: "sigma_algebra S" "random_variable S X"
shows "prob_space S (distribution X)"
proof -
- interpret S: sigma_algebra S by fact
+ interpret S: measure_space S "distribution X"
+ using measure_space_vimage[OF S(2,1)] unfolding distribution_def .
show ?thesis
proof
- show "distribution X {} = 0" unfolding distribution_def by simp
have "X -` space S \<inter> space M = space M"
using `random_variable S X` by (auto simp: measurable_def)
- then show "distribution X (space S) = 1" using measure_space_1 by (simp add: distribution_def)
-
- show "countably_additive S (distribution X)"
- proof (unfold countably_additive_def, safe)
- fix A :: "nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets S" "disjoint_family A"
- hence *: "\<And>i. X -` A i \<inter> space M \<in> sets M"
- using `random_variable S X` by (auto simp: measurable_def)
- moreover hence "\<And>i. \<mu> (X -` A i \<inter> space M) \<noteq> \<omega>"
- using finite_measure by auto
- moreover have "(\<Union>i. X -` A i \<inter> space M) \<in> sets M"
- using * by blast
- moreover hence "\<mu> (\<Union>i. X -` A i \<inter> space M) \<noteq> \<omega>"
- using finite_measure by auto
- moreover have **: "disjoint_family (\<lambda>i. X -` A i \<inter> space M)"
- using `disjoint_family A` by (auto simp: disjoint_family_on_def)
- ultimately show "(\<Sum>\<^isub>\<infinity> i. distribution X (A i)) = distribution X (\<Union>i. A i)"
- using measure_countably_additive[OF _ **]
- by (auto simp: distribution_def Real_real comp_def vimage_UN)
- qed
+ then show "distribution X (space S) = 1"
+ using measure_space_1 by (simp add: distribution_def)
qed
qed
@@ -379,39 +338,246 @@
joint_distribution_restriction_snd[of X Y "{(x, y)}"]
by auto
+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!: measure_mono)
+
+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 measure_space_1[symmetric] using finite_space
+ by (subst measure_finitely_additive'')
+ (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=\<mu>])
+
+lemma (in finite_prob_space) sum_over_space_real_distribution:
+ "(\<Sum>x\<in>X`space M. real (distribution X {x})) = 1"
+ unfolding distribution_def prob_space[symmetric] using finite_space
+ by (subst real_finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+
+lemma (in finite_prob_space) finite_sum_over_space_eq_1:
+ "(\<Sum>x\<in>space M. real (\<mu> {x})) = 1"
+ using sum_over_space_eq_1 finite_measure by (simp add: real_of_pinfreal_setsum sets_eq_Pow)
+
+lemma (in finite_prob_space) distribution_finite:
+ "distribution X A \<noteq> \<omega>"
+ using finite_measure[of "X -` A \<inter> space M"]
+ unfolding distribution_def sets_eq_Pow by auto
+
+lemma (in finite_prob_space) real_distribution_gt_0[simp]:
+ "0 < real (distribution Y y) \<longleftrightarrow> 0 < distribution Y y"
+ using assms by (auto intro!: real_pinfreal_pos distribution_finite)
+
+lemma (in finite_prob_space) real_distribution_mult_pos_pos:
+ assumes "0 < distribution Y y"
+ and "0 < distribution X x"
+ shows "0 < real (distribution Y y * distribution X x)"
+ unfolding real_of_pinfreal_mult[symmetric]
+ using assms by (auto intro!: mult_pos_pos)
+
+lemma (in finite_prob_space) real_distribution_divide_pos_pos:
+ assumes "0 < distribution Y y"
+ and "0 < distribution X x"
+ shows "0 < real (distribution Y y / distribution X x)"
+ unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
+ using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
+
+lemma (in finite_prob_space) real_distribution_mult_inverse_pos_pos:
+ assumes "0 < distribution Y y"
+ and "0 < distribution X x"
+ shows "0 < real (distribution Y y * inverse (distribution X x))"
+ unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
+ using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
+
+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"
+ unfolding measure_space_1[symmetric]
+ by (auto intro!: arg_cong[where f="\<mu>"] simp: distribution_def)
+
+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 measure_finitely_additive'')
+ (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:
+ "(\<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) setsum_real_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. real (distribution Y {f x})) = real (distribution Z {c})"
+ unfolding distribution_def assms
+ using finite_space assms
+ by (subst real_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_real_distribution:
+ "(\<Sum>x \<in> X`space M. real (joint_distribution X Y {(x, y)})) = real (distribution Y {y})"
+ "(\<Sum>y \<in> Y`space M. real (joint_distribution X Y {(x, y)})) = real (distribution X {x})"
+ "(\<Sum>x \<in> X`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution Y Z {(y, z)})"
+ "(\<Sum>y \<in> Y`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Z {(x, z)})"
+ "(\<Sum>z \<in> Z`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Y {(x, y)})"
+ by (auto intro!: inj_onI setsum_real_distribution_gen)
+
+lemma (in finite_prob_space) real_distribution_order:
+ shows "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution X {x})"
+ and "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution Y {y})"
+ and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution X {x})"
+ and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution Y {y})"
+ and "distribution X {x} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ and "distribution Y {y} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
+ using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
+ by auto
+
+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 finite_prob_space) distribution_1:
+ "distribution X A \<le> 1"
+ unfolding distribution_def measure_space_1[symmetric]
+ by (auto intro!: measure_mono simp: sets_eq_Pow)
+
+lemma (in finite_prob_space) real_distribution_1:
+ "real (distribution X A) \<le> 1"
+ unfolding real_pinfreal_1[symmetric]
+ by (rule real_of_pinfreal_mono[OF _ distribution_1]) simp
+
+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 / real (card (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 real_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 fastsimp
+ from one have three: "prob {x} \<noteq> 0" by fastsimp
+ thus ?thesis using one two three divide_cancel_right
+ by (auto simp:field_simps)
+qed
+
+lemma (in prob_space) prob_space_subalgebra:
+ assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
+ shows "prob_space (M\<lparr> sets := N \<rparr>) \<mu>"
+proof -
+ interpret N: measure_space "M\<lparr> sets := N \<rparr>" \<mu>
+ using measure_space_subalgebra[OF assms] .
+ show ?thesis
+ proof qed (simp add: measure_space_1)
+qed
+
+lemma (in prob_space) prob_space_of_restricted_space:
+ assumes "\<mu> A \<noteq> 0" "\<mu> A \<noteq> \<omega>" "A \<in> sets M"
+ shows "prob_space (restricted_space A) (\<lambda>S. \<mu> S / \<mu> A)"
+ unfolding prob_space_def prob_space_axioms_def
+proof
+ show "\<mu> (space (restricted_space A)) / \<mu> A = 1"
+ using `\<mu> A \<noteq> 0` `\<mu> A \<noteq> \<omega>` by (auto simp: pinfreal_noteq_omega_Ex)
+ have *: "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" by (simp add: mult_commute)
+ interpret A: measure_space "restricted_space A" \<mu>
+ using `A \<in> sets M` by (rule restricted_measure_space)
+ show "measure_space (restricted_space A) (\<lambda>S. \<mu> S / \<mu> A)"
+ proof
+ show "\<mu> {} / \<mu> A = 0" by auto
+ show "countably_additive (restricted_space A) (\<lambda>S. \<mu> S / \<mu> A)"
+ unfolding countably_additive_def psuminf_cmult_right *
+ using A.measure_countably_additive by auto
+ qed
+qed
+
+lemma finite_prob_spaceI:
+ assumes "finite (space M)" "sets M = Pow(space M)" "\<mu> (space M) = 1" "\<mu> {} = 0"
+ and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> \<mu> (A \<union> B) = \<mu> A + \<mu> B"
+ shows "finite_prob_space M \<mu>"
+ unfolding finite_prob_space_eq
+proof
+ show "finite_measure_space M \<mu>" using assms
+ by (auto intro!: finite_measure_spaceI)
+ show "\<mu> (space M) = 1" by fact
+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)\<rparr> (distribution X)"
+ (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 B :: "'x set" assume "A \<inter> B = {}"
+ then show "distribution X (A \<union> B) = distribution X A + distribution X B"
+ unfolding distribution_def
+ by (subst measure_additive)
+ (auto intro!: arg_cong[where f=\<mu>] simp: sets_eq_Pow)
+qed
+
+lemma (in finite_prob_space) finite_prob_space_of_images:
+ "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
+ by (simp add: finite_prob_space_eq finite_measure_space)
+
+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>])
+
+lemma (in finite_prob_space) real_distribution_order':
+ shows "real (distribution X {x}) = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ and "real (distribution Y {y}) = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
+ using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
+ by auto
+
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)\<rparr> (joint_distribution X Y)"
(is "finite_measure_space ?M ?D")
-proof (rule finite_Pow_additivity_sufficient)
- show "positive ?D"
- unfolding positive_def using assms sets_eq_Pow
- by (simp add: distribution_def)
-
- show "additive ?M ?D" unfolding additive_def
- proof safe
- fix x y
- have A: "((\<lambda>x. (X x, Y x)) -` x) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto
- have B: "((\<lambda>x. (X x, Y x)) -` y) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto
- assume "x \<inter> y = {}"
- hence "(\<lambda>x. (X x, Y x)) -` x \<inter> space M \<inter> ((\<lambda>x. (X x, Y x)) -` y \<inter> space M) = {}"
- by auto
- from additive[unfolded additive_def, rule_format, OF A B] this
- finite_measure[OF A] finite_measure[OF B]
- show "?D (x \<union> y) = ?D x + ?D y"
- apply (simp add: distribution_def)
- apply (subst Int_Un_distrib2)
- by (auto simp: real_of_pinfreal_add)
- qed
-
- show "finite (space ?M)"
+proof (rule finite_measure_spaceI, simp_all)
+ show "finite (s1 \<times> s2)"
using assms by auto
-
- show "sets ?M = Pow (space ?M)"
- by simp
-
- { fix x assume "x \<in> space ?M" thus "?D {x} \<noteq> \<omega>"
- unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) }
+ show "joint_distribution X Y (s1\<times>s2) \<noteq> \<omega>"
+ using distribution_finite .
+next
+ fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
+ then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
+ unfolding distribution_def
+ by (subst measure_additive)
+ (auto intro!: arg_cong[where f=\<mu>] simp: sets_eq_Pow)
qed
lemma (in finite_prob_space) finite_product_measure_space_of_images:
@@ -420,47 +586,133 @@
(joint_distribution X Y)"
using finite_space by (auto intro!: finite_product_measure_space)
-lemma (in finite_prob_space) finite_measure_space:
- shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
- (is "finite_measure_space ?S _")
-proof (rule finite_Pow_additivity_sufficient, simp_all)
- show "finite (X ` space M)" using finite_space by simp
+section "Conditional Expectation and Probability"
- show "positive (distribution X)"
- unfolding distribution_def positive_def using sets_eq_Pow by auto
+lemma (in prob_space) conditional_expectation_exists:
+ fixes X :: "'a \<Rightarrow> pinfreal"
+ assumes borel: "X \<in> borel_measurable M"
+ and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
+ shows "\<exists>Y\<in>borel_measurable (M\<lparr> sets := N \<rparr>). \<forall>C\<in>N.
+ positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)"
+proof -
+ interpret P: prob_space "M\<lparr> sets := N \<rparr>" \<mu>
+ using prob_space_subalgebra[OF N_subalgebra] .
+
+ let "?f A" = "\<lambda>x. X x * indicator A x"
+ let "?Q A" = "positive_integral (?f A)"
+
+ from measure_space_density[OF borel]
+ have Q: "measure_space (M\<lparr> sets := N \<rparr>) ?Q"
+ by (rule measure_space.measure_space_subalgebra[OF _ N_subalgebra])
+ then interpret Q: measure_space "M\<lparr> sets := N \<rparr>" ?Q .
- show "additive ?S (distribution X)" unfolding additive_def distribution_def
- proof (simp, safe)
- fix x y
- have x: "(X -` x) \<inter> space M \<in> sets M"
- and y: "(X -` y) \<inter> space M \<in> sets M" using sets_eq_Pow by auto
- assume "x \<inter> y = {}"
- hence "X -` x \<inter> space M \<inter> (X -` y \<inter> space M) = {}" by auto
- from additive[unfolded additive_def, rule_format, OF x y] this
- finite_measure[OF x] finite_measure[OF y]
- have "\<mu> (((X -` x) \<union> (X -` y)) \<inter> space M) =
- \<mu> ((X -` x) \<inter> space M) + \<mu> ((X -` y) \<inter> space M)"
- by (subst Int_Un_distrib2) auto
- thus "\<mu> ((X -` x \<union> X -` y) \<inter> space M) = \<mu> (X -` x \<inter> space M) + \<mu> (X -` y \<inter> space M)"
- by auto
+ have "P.absolutely_continuous ?Q"
+ unfolding P.absolutely_continuous_def
+ proof (safe, simp)
+ fix A assume "A \<in> N" "\<mu> A = 0"
+ moreover then have f_borel: "?f A \<in> borel_measurable M"
+ using borel N_subalgebra by (auto intro: borel_measurable_indicator)
+ moreover have "{x\<in>space M. ?f A x \<noteq> 0} = (?f A -` {0<..} \<inter> space M) \<inter> A"
+ by (auto simp: indicator_def)
+ moreover have "\<mu> \<dots> \<le> \<mu> A"
+ using `A \<in> N` N_subalgebra f_borel
+ by (auto intro!: measure_mono Int[of _ A] measurable_sets)
+ ultimately show "?Q A = 0"
+ by (simp add: positive_integral_0_iff)
qed
-
- { fix x assume "x \<in> X ` space M" thus "distribution X {x} \<noteq> \<omega>"
- unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) }
+ from P.Radon_Nikodym[OF Q this]
+ obtain Y where Y: "Y \<in> borel_measurable (M\<lparr>sets := N\<rparr>)"
+ "\<And>A. A \<in> sets (M\<lparr>sets:=N\<rparr>) \<Longrightarrow> ?Q A = P.positive_integral (\<lambda>x. Y x * indicator A x)"
+ by blast
+ with N_subalgebra show ?thesis
+ by (auto intro!: bexI[OF _ Y(1)])
qed
-lemma (in finite_prob_space) finite_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
- by (simp add: finite_prob_space_eq finite_measure_space)
+definition (in prob_space)
+ "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable (M\<lparr>sets:=N\<rparr>)
+ \<and> (\<forall>C\<in>N. positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)))"
+
+abbreviation (in prob_space)
+ "conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
+
+lemma (in prob_space)
+ fixes X :: "'a \<Rightarrow> pinfreal"
+ assumes borel: "X \<in> borel_measurable M"
+ and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
+ shows borel_measurable_conditional_expectation:
+ "conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)"
+ and conditional_expectation: "\<And>C. C \<in> N \<Longrightarrow>
+ positive_integral (\<lambda>x. conditional_expectation N X x * indicator C x) =
+ positive_integral (\<lambda>x. X x * indicator C x)"
+ (is "\<And>C. C \<in> N \<Longrightarrow> ?eq C")
+proof -
+ note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
+ then show "conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)"
+ unfolding conditional_expectation_def by (rule someI2_ex) blast
+
+ from CE show "\<And>C. C\<in>N \<Longrightarrow> ?eq C"
+ unfolding conditional_expectation_def by (rule someI2_ex) blast
+qed
+
+lemma (in sigma_algebra) factorize_measurable_function:
+ fixes Z :: "'a \<Rightarrow> pinfreal" 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" .
-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)\<rparr>
- (joint_distribution X Y)"
- (is "finite_prob_space ?S _")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images)
- 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)
+ { fix g :: "'c \<Rightarrow> pinfreal" 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 \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ from va.borel_measurable_implies_simple_function_sequence[OF this]
+ obtain f where f: "\<And>i. va.simple_function (f i)" and "f \<up> Z" by blast
+
+ have "\<forall>i. \<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+ proof
+ fix i
+ from f[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 va.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. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+ proof (intro exI[of _ ?g] conjI ballI)
+ show "M'.simple_function ?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::pinfreal)"
+ unfolding indicator_def using B by auto
+ then show "f i x = ?g (Y x)" using `x \<in> space M` f[of i]
+ by (subst va.simple_function_indicator_representation) auto
+ qed
+ qed
+ from choice[OF this] guess g .. note g = this
+
+ show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
+ proof (intro ballI bexI)
+ show "(SUP i. g i) \<in> borel_measurable M'"
+ using g by (auto intro: M'.borel_measurable_simple_function)
+ fix x assume "x \<in> space M"
+ have "Z x = (SUP i. f i) x" using `f \<up> Z` unfolding isoton_def by simp
+ also have "\<dots> = (SUP i. g i) (Y x)" unfolding SUPR_fun_expand
+ using g `x \<in> space M` by simp
+ finally show "Z x = (SUP i. g i) (Y x)" .
+ qed
qed
end
--- a/src/HOL/Probability/Product_Measure.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Product_Measure.thy Thu Sep 02 20:44:33 2010 +0200
@@ -2,14 +2,412 @@
imports Lebesgue_Integration
begin
+definition "dynkin M \<longleftrightarrow>
+ space M \<in> sets M \<and>
+ (\<forall> A \<in> sets M. A \<subseteq> space M) \<and>
+ (\<forall> a \<in> sets M. \<forall> b \<in> sets M. a \<subseteq> b \<longrightarrow> b - a \<in> sets M) \<and>
+ (\<forall>A. disjoint_family A \<and> range A \<subseteq> sets M \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
+
+lemma dynkinI:
+ assumes "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
+ assumes "space M \<in> sets M" and "\<forall> b \<in> sets M. \<forall> a \<in> sets M. a \<subseteq> b \<longrightarrow> b - a \<in> sets M"
+ assumes "\<And> a. (\<And> i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {})
+ \<Longrightarrow> (\<And> i :: nat. a i \<in> sets M) \<Longrightarrow> UNION UNIV a \<in> sets M"
+ shows "dynkin M"
+using assms by (auto simp: dynkin_def disjoint_family_on_def image_subset_iff)
+
+lemma dynkin_subset:
+ assumes "dynkin M"
+ shows "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
+using assms unfolding dynkin_def by auto
+
+lemma dynkin_space:
+ assumes "dynkin M"
+ shows "space M \<in> sets M"
+using assms unfolding dynkin_def by auto
+
+lemma dynkin_diff:
+ assumes "dynkin M"
+ shows "\<And> a b. \<lbrakk> a \<in> sets M ; b \<in> sets M ; a \<subseteq> b \<rbrakk> \<Longrightarrow> b - a \<in> sets M"
+using assms unfolding dynkin_def by auto
+
+lemma dynkin_empty:
+ assumes "dynkin M"
+ shows "{} \<in> sets M"
+using dynkin_diff[OF assms dynkin_space[OF assms] dynkin_space[OF assms]] by auto
+
+lemma dynkin_UN:
+ assumes "dynkin M"
+ assumes "\<And> i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}"
+ assumes "\<And> i :: nat. a i \<in> sets M"
+ shows "UNION UNIV a \<in> sets M"
+using assms by (auto simp: dynkin_def disjoint_family_on_def image_subset_iff)
+
+definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> sets M. \<forall> b \<in> sets M. a \<inter> b \<in> sets M)"
+
+lemma dynkin_trivial:
+ shows "dynkin \<lparr> space = A, sets = Pow A \<rparr>"
+by (rule dynkinI) auto
+
+lemma dynkin_lemma:
+ fixes D :: "'a algebra"
+ assumes stab: "Int_stable E"
+ and spac: "space E = space D"
+ and subsED: "sets E \<subseteq> sets D"
+ and subsDE: "sets D \<subseteq> sigma_sets (space E) (sets E)"
+ and dyn: "dynkin D"
+ shows "sigma (space E) (sets E) = D"
+proof -
+ def sets_\<delta>E == "\<Inter> {sets d | d :: 'a algebra. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d}"
+ def \<delta>E == "\<lparr> space = space E, sets = sets_\<delta>E \<rparr>"
+ have "\<lparr> space = space E, sets = Pow (space E) \<rparr> \<in> {d | d. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d}"
+ using dynkin_trivial spac subsED dynkin_subset[OF dyn] by fastsimp
+ hence not_empty: "{sets (d :: 'a algebra) | d. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d} \<noteq> {}"
+ using exI[of "\<lambda> x. space x = space E \<and> dynkin x \<and> sets E \<subseteq> sets x" "\<lparr> space = space E, sets = Pow (space E) \<rparr>", simplified]
+ by auto
+ have \<delta>E_D: "sets_\<delta>E \<subseteq> sets D"
+ unfolding sets_\<delta>E_def using assms by auto
+ have \<delta>ynkin: "dynkin \<delta>E"
+ proof (rule dynkinI, safe)
+ fix A x assume asm: "A \<in> sets \<delta>E" "x \<in> A"
+ { fix d :: "('a, 'b) algebra_scheme" assume "A \<in> sets d" "dynkin d \<and> space d = space E"
+ hence "A \<subseteq> space d" using dynkin_subset by auto }
+ show "x \<in> space \<delta>E" using asm unfolding \<delta>E_def sets_\<delta>E_def using not_empty
+ by simp (metis dynkin_subset in_mono mem_def)
+ next
+ show "space \<delta>E \<in> sets \<delta>E"
+ unfolding \<delta>E_def sets_\<delta>E_def
+ using dynkin_space by fastsimp
+ next
+ fix a b assume "a \<in> sets \<delta>E" "b \<in> sets \<delta>E" "a \<subseteq> b"
+ thus "b - a \<in> sets \<delta>E"
+ unfolding \<delta>E_def sets_\<delta>E_def by (auto intro:dynkin_diff)
+ next
+ fix a assume asm: "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}" "\<And>i. a i \<in> sets \<delta>E"
+ thus "UNION UNIV a \<in> sets \<delta>E"
+ unfolding \<delta>E_def sets_\<delta>E_def apply (auto intro!:dynkin_UN[OF _ asm(1)])
+ by blast
+ qed
+
+ def Dy == "\<lambda> d. {A | A. A \<in> sets_\<delta>E \<and> A \<inter> d \<in> sets_\<delta>E}"
+ { fix d assume dasm: "d \<in> sets_\<delta>E"
+ have "dynkin \<lparr> space = space E, sets = Dy d \<rparr>"
+ proof (rule dynkinI, safe, simp_all)
+ fix A x assume "A \<in> Dy d" "x \<in> A"
+ thus "x \<in> space E" unfolding Dy_def sets_\<delta>E_def using not_empty
+ by simp (metis dynkin_subset in_mono mem_def)
+ next
+ show "space E \<in> Dy d"
+ unfolding Dy_def \<delta>E_def sets_\<delta>E_def
+ proof auto
+ fix d assume asm: "dynkin d" "space d = space E" "sets E \<subseteq> sets d"
+ hence "space d \<in> sets d" using dynkin_space[OF asm(1)] by auto
+ thus "space E \<in> sets d" using asm by auto
+ next
+ fix da :: "'a algebra" assume asm: "dynkin da" "space da = space E" "sets E \<subseteq> sets da"
+ have d: "d = space E \<inter> d"
+ using dasm dynkin_subset[OF asm(1)] asm(2) dynkin_subset[OF \<delta>ynkin]
+ unfolding \<delta>E_def by auto
+ hence "space E \<inter> d \<in> sets \<delta>E" unfolding \<delta>E_def
+ using dasm by auto
+ have "sets \<delta>E \<subseteq> sets da" unfolding \<delta>E_def sets_\<delta>E_def using asm
+ by auto
+ thus "space E \<inter> d \<in> sets da" using dasm asm d dynkin_subset[OF \<delta>ynkin]
+ unfolding \<delta>E_def by auto
+ qed
+ next
+ fix a b assume absm: "a \<in> Dy d" "b \<in> Dy d" "a \<subseteq> b"
+ hence "a \<in> sets \<delta>E" "b \<in> sets \<delta>E"
+ unfolding Dy_def \<delta>E_def by auto
+ hence *: "b - a \<in> sets \<delta>E"
+ using dynkin_diff[OF \<delta>ynkin] `a \<subseteq> b` by auto
+ have "a \<inter> d \<in> sets \<delta>E" "b \<inter> d \<in> sets \<delta>E"
+ using absm unfolding Dy_def \<delta>E_def by auto
+ hence "(b \<inter> d) - (a \<inter> d) \<in> sets \<delta>E"
+ using dynkin_diff[OF \<delta>ynkin] `a \<subseteq> b` by auto
+ hence **: "(b - a) \<inter> d \<in> sets \<delta>E" by (auto simp add:Diff_Int_distrib2)
+ thus "b - a \<in> Dy d"
+ using * ** unfolding Dy_def \<delta>E_def by auto
+ next
+ fix a assume aasm: "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}" "\<And>i. a i \<in> Dy d"
+ hence "\<And> i. a i \<in> sets \<delta>E"
+ unfolding Dy_def \<delta>E_def by auto
+ from dynkin_UN[OF \<delta>ynkin aasm(1) this]
+ have *: "UNION UNIV a \<in> sets \<delta>E" by auto
+ from aasm
+ have aE: "\<forall> i. a i \<inter> d \<in> sets \<delta>E"
+ unfolding Dy_def \<delta>E_def by auto
+ from aasm
+ have "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> (a i \<inter> d) \<inter> (a j \<inter> d) = {}" by auto
+ from dynkin_UN[OF \<delta>ynkin this]
+ have "UNION UNIV (\<lambda> i. a i \<inter> d) \<in> sets \<delta>E"
+ using aE by auto
+ hence **: "UNION UNIV a \<inter> d \<in> sets \<delta>E" by auto
+ from * ** show "UNION UNIV a \<in> Dy d" unfolding Dy_def \<delta>E_def by auto
+ qed } note Dy_nkin = this
+ have E_\<delta>E: "sets E \<subseteq> sets \<delta>E"
+ unfolding \<delta>E_def sets_\<delta>E_def by auto
+ { fix d assume dasm: "d \<in> sets \<delta>E"
+ { fix e assume easm: "e \<in> sets E"
+ hence deasm: "e \<in> sets \<delta>E"
+ unfolding \<delta>E_def sets_\<delta>E_def by auto
+ have subset: "Dy e \<subseteq> sets \<delta>E"
+ unfolding Dy_def \<delta>E_def by auto
+ { fix e' assume e'asm: "e' \<in> sets E"
+ have "e' \<inter> e \<in> sets E"
+ using easm e'asm stab unfolding Int_stable_def by auto
+ hence "e' \<inter> e \<in> sets \<delta>E"
+ unfolding \<delta>E_def sets_\<delta>E_def by auto
+ hence "e' \<in> Dy e" using e'asm unfolding Dy_def \<delta>E_def sets_\<delta>E_def by auto }
+ hence E_Dy: "sets E \<subseteq> Dy e" by auto
+ have "\<lparr> space = space E, sets = Dy e \<rparr> \<in> {d | d. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d}"
+ using Dy_nkin[OF deasm[unfolded \<delta>E_def, simplified]] E_\<delta>E E_Dy by auto
+ hence "sets_\<delta>E \<subseteq> Dy e"
+ unfolding sets_\<delta>E_def by auto (metis E_Dy simps(1) simps(2) spac)
+ hence "sets \<delta>E = Dy e" using subset unfolding \<delta>E_def by auto
+ hence "d \<inter> e \<in> sets \<delta>E"
+ using dasm easm deasm unfolding Dy_def \<delta>E_def by auto
+ hence "e \<in> Dy d" using deasm
+ unfolding Dy_def \<delta>E_def
+ by (auto simp add:Int_commute) }
+ hence "sets E \<subseteq> Dy d" by auto
+ hence "sets \<delta>E \<subseteq> Dy d" using Dy_nkin[OF dasm[unfolded \<delta>E_def, simplified]]
+ unfolding \<delta>E_def sets_\<delta>E_def
+ by auto (metis `sets E <= Dy d` simps(1) simps(2) spac)
+ hence *: "sets \<delta>E = Dy d"
+ unfolding Dy_def \<delta>E_def by auto
+ fix a assume aasm: "a \<in> sets \<delta>E"
+ hence "a \<inter> d \<in> sets \<delta>E"
+ using * dasm unfolding Dy_def \<delta>E_def by auto } note \<delta>E_stab = this
+ { fix A :: "nat \<Rightarrow> 'a set" assume Asm: "range A \<subseteq> sets \<delta>E" "\<And>A. A \<in> sets \<delta>E \<Longrightarrow> A \<subseteq> space \<delta>E"
+ "\<And>a. a \<in> sets \<delta>E \<Longrightarrow> space \<delta>E - a \<in> sets \<delta>E"
+ "{} \<in> sets \<delta>E" "space \<delta>E \<in> sets \<delta>E"
+ let "?A i" = "A i \<inter> (\<Inter> j \<in> {..< i}. space \<delta>E - A j)"
+ { fix i :: nat
+ have *: "(\<Inter> j \<in> {..< i}. space \<delta>E - A j) \<inter> space \<delta>E \<in> sets \<delta>E"
+ apply (induct i)
+ using lessThan_Suc Asm \<delta>E_stab apply fastsimp
+ apply (subst lessThan_Suc)
+ apply (subst INT_insert)
+ apply (subst Int_assoc)
+ apply (subst \<delta>E_stab)
+ using lessThan_Suc Asm \<delta>E_stab Asm
+ apply (fastsimp simp add:Int_assoc dynkin_diff[OF \<delta>ynkin])
+ prefer 2 apply simp
+ apply (rule dynkin_diff[OF \<delta>ynkin, of _ "space \<delta>E", OF _ dynkin_space[OF \<delta>ynkin]])
+ using Asm by auto
+ have **: "\<And> i. A i \<subseteq> space \<delta>E" using Asm by auto
+ have "(\<Inter> j \<in> {..< i}. space \<delta>E - A j) \<subseteq> space \<delta>E \<or> (\<Inter> j \<in> {..< i}. A j) = UNIV \<and> i = 0"
+ apply (cases i)
+ using Asm ** dynkin_subset[OF \<delta>ynkin, of "A (i - 1)"]
+ by auto
+ hence Aisets: "?A i \<in> sets \<delta>E"
+ apply (cases i)
+ using Asm * apply fastsimp
+ apply (rule \<delta>E_stab)
+ using Asm * **
+ by (auto simp add:Int_absorb2)
+ have "?A i = disjointed A i" unfolding disjointed_def
+ atLeast0LessThan using Asm by auto
+ hence "?A i = disjointed A i" "?A i \<in> sets \<delta>E"
+ using Aisets by auto
+ } note Ai = this
+ from dynkin_UN[OF \<delta>ynkin _ this(2)] this disjoint_family_disjointed[of A]
+ have "(\<Union> i. ?A i) \<in> sets \<delta>E"
+ by (auto simp add:disjoint_family_on_def disjointed_def)
+ hence "(\<Union> i. A i) \<in> sets \<delta>E"
+ using Ai(1) UN_disjointed_eq[of A] by auto } note \<delta>E_UN = this
+ { fix a b assume asm: "a \<in> sets \<delta>E" "b \<in> sets \<delta>E"
+ let ?ab = "\<lambda> i. if (i::nat) = 0 then a else if i = 1 then b else {}"
+ have *: "(\<Union> i. ?ab i) \<in> sets \<delta>E"
+ apply (rule \<delta>E_UN)
+ using asm \<delta>E_UN dynkin_empty[OF \<delta>ynkin]
+ dynkin_subset[OF \<delta>ynkin]
+ dynkin_space[OF \<delta>ynkin]
+ dynkin_diff[OF \<delta>ynkin] by auto
+ have "(\<Union> i. ?ab i) = a \<union> b" apply auto
+ apply (case_tac "i = 0")
+ apply auto
+ apply (case_tac "i = 1")
+ by auto
+ hence "a \<union> b \<in> sets \<delta>E" using * by auto} note \<delta>E_Un = this
+ have "sigma_algebra \<delta>E"
+ apply unfold_locales
+ using dynkin_subset[OF \<delta>ynkin]
+ using dynkin_diff[OF \<delta>ynkin, of _ "space \<delta>E", OF _ dynkin_space[OF \<delta>ynkin]]
+ using dynkin_diff[OF \<delta>ynkin, of "space \<delta>E" "space \<delta>E", OF dynkin_space[OF \<delta>ynkin] dynkin_space[OF \<delta>ynkin]]
+ using dynkin_space[OF \<delta>ynkin]
+ using \<delta>E_UN \<delta>E_Un
+ by auto
+ from sigma_algebra.sigma_subset[OF this E_\<delta>E] \<delta>E_D subsDE spac
+ show ?thesis by (auto simp add:\<delta>E_def sigma_def)
+qed
+
+lemma measure_eq:
+ assumes fin: "\<mu> (space M) = \<nu> (space M)" "\<nu> (space M) < \<omega>"
+ assumes E: "M = sigma (space E) (sets E)" "Int_stable E"
+ assumes eq: "\<And> e. e \<in> sets E \<Longrightarrow> \<mu> e = \<nu> e"
+ assumes ms: "measure_space M \<mu>" "measure_space M \<nu>"
+ assumes A: "A \<in> sets M"
+ shows "\<mu> A = \<nu> A"
+proof -
+ interpret M: measure_space M \<mu>
+ using ms by simp
+ interpret M': measure_space M \<nu>
+ using ms by simp
+
+ let ?D_sets = "{A. A \<in> sets M \<and> \<mu> A = \<nu> A}"
+ have \<delta>: "dynkin \<lparr> space = space M , sets = ?D_sets \<rparr>"
+ proof (rule dynkinI, safe, simp_all)
+ fix A x assume "A \<in> sets M \<and> \<mu> A = \<nu> A" "x \<in> A"
+ thus "x \<in> space M" using assms M.sets_into_space by auto
+ next
+ show "\<mu> (space M) = \<nu> (space M)"
+ using fin by auto
+ next
+ fix a b
+ assume asm: "a \<in> sets M \<and> \<mu> a = \<nu> a"
+ "b \<in> sets M \<and> \<mu> b = \<nu> b" "a \<subseteq> b"
+ hence "a \<subseteq> space M"
+ using M.sets_into_space by auto
+ from M.measure_mono[OF this]
+ have "\<mu> a \<le> \<mu> (space M)"
+ using asm by auto
+ hence afin: "\<mu> a < \<omega>"
+ using fin by auto
+ have *: "b = b - a \<union> a" using asm by auto
+ have **: "(b - a) \<inter> a = {}" using asm by auto
+ have iv: "\<mu> (b - a) + \<mu> a = \<mu> b"
+ using M.measure_additive[of "b - a" a]
+ conjunct1[OF asm(1)] conjunct1[OF asm(2)] * **
+ by auto
+ have v: "\<nu> (b - a) + \<nu> a = \<nu> b"
+ using M'.measure_additive[of "b - a" a]
+ conjunct1[OF asm(1)] conjunct1[OF asm(2)] * **
+ by auto
+ from iv v have "\<mu> (b - a) = \<nu> (b - a)" using asm afin
+ pinfreal_add_cancel_right[of "\<mu> (b - a)" "\<nu> a" "\<nu> (b - a)"]
+ by auto
+ thus "b - a \<in> sets M \<and> \<mu> (b - a) = \<nu> (b - a)"
+ using asm by auto
+ next
+ fix a assume "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}"
+ "\<And>i. a i \<in> sets M \<and> \<mu> (a i) = \<nu> (a i)"
+ thus "(\<Union>x. a x) \<in> sets M \<and> \<mu> (\<Union>x. a x) = \<nu> (\<Union>x. a x)"
+ using M.measure_countably_additive
+ M'.measure_countably_additive
+ M.countable_UN
+ apply (auto simp add:disjoint_family_on_def image_def)
+ apply (subst M.measure_countably_additive[symmetric])
+ apply (auto simp add:disjoint_family_on_def)
+ apply (subst M'.measure_countably_additive[symmetric])
+ by (auto simp add:disjoint_family_on_def)
+ qed
+ have *: "sets E \<subseteq> ?D_sets"
+ using eq E sigma_sets.Basic[of _ "sets E"]
+ by (auto simp add:sigma_def)
+ have **: "?D_sets \<subseteq> sets M" by auto
+ have "M = \<lparr> space = space M , sets = ?D_sets \<rparr>"
+ unfolding E(1)
+ apply (rule dynkin_lemma[OF E(2)])
+ using eq E space_sigma \<delta> sigma_sets.Basic
+ by (auto simp add:sigma_def)
+ from subst[OF this, of "\<lambda> M. A \<in> sets M", OF A]
+ show ?thesis by auto
+qed
+(*
+lemma
+ assumes sfin: "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And> i :: nat. \<nu> (A i) < \<omega>"
+ assumes A: "\<And> i. \<mu> (A i) = \<nu> (A i)" "\<And> i. A i \<subseteq> A (Suc i)"
+ assumes E: "M = sigma (space E) (sets E)" "Int_stable E"
+ assumes eq: "\<And> e. e \<in> sets E \<Longrightarrow> \<mu> e = \<nu> e"
+ assumes ms: "measure_space (M :: 'a algebra) \<mu>" "measure_space M \<nu>"
+ assumes B: "B \<in> sets M"
+ shows "\<mu> B = \<nu> B"
+proof -
+ interpret M: measure_space M \<mu> by (rule ms)
+ interpret M': measure_space M \<nu> by (rule ms)
+ have *: "M = \<lparr> space = space M, sets = sets M \<rparr>" by auto
+ { fix i :: nat
+ have **: "M\<lparr> space := A i, sets := op \<inter> (A i) ` sets M \<rparr> =
+ \<lparr> space = A i, sets = op \<inter> (A i) ` sets M \<rparr>"
+ by auto
+ have mu_i: "measure_space \<lparr> space = A i, sets = op \<inter> (A i) ` sets M \<rparr> \<mu>"
+ using M.restricted_measure_space[of "A i", simplified **]
+ sfin by auto
+ have nu_i: "measure_space \<lparr> space = A i, sets = op \<inter> (A i) ` sets M \<rparr> \<nu>"
+ using M'.restricted_measure_space[of "A i", simplified **]
+ sfin by auto
+ let ?M = "\<lparr> space = A i, sets = op \<inter> (A i) ` sets M \<rparr>"
+ have "\<mu> (A i \<inter> B) = \<nu> (A i \<inter> B)"
+ apply (rule measure_eq[of \<mu> ?M \<nu> "\<lparr> space = space E \<inter> A i, sets = op \<inter> (A i) ` sets E\<rparr>" "A i \<inter> B", simplified])
+ using assms nu_i mu_i
+ apply (auto simp add:image_def) (* TODO *) sorry
+ show ?thesis sorry
+qed
+*)
definition prod_sets where
"prod_sets A B = {z. \<exists>x \<in> A. \<exists>y \<in> B. z = x \<times> y}"
definition
- "prod_measure M \<mu> N \<nu> = (\<lambda>A. measure_space.positive_integral M \<mu> (\<lambda>s0. \<nu> ((\<lambda>s1. (s0, s1)) -` A)))"
+ "prod_measure_space M1 M2 = sigma (space M1 \<times> space M2) (prod_sets (sets M1) (sets M2))"
+
+lemma
+ fixes M1 :: "'a algebra" and M2 :: "'b algebra"
+ assumes "algebra M1" "algebra M2"
+ shows measureable_fst[intro!, simp]:
+ "fst \<in> measurable (prod_measure_space M1 M2) M1" (is ?fst)
+ and measureable_snd[intro!, simp]:
+ "snd \<in> measurable (prod_measure_space M1 M2) M2" (is ?snd)
+proof -
+ interpret M1: algebra M1 by fact
+ interpret M2: algebra M2 by fact
+
+ { 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 show ?fst ?snd
+ by (force intro!: sigma_sets.Basic
+ simp: measurable_def prod_measure_space_def prod_sets_def sets_sigma)+
+qed
+
+lemma (in sigma_algebra) measureable_prod:
+ fixes M1 :: "'a algebra" and M2 :: "'b algebra"
+ assumes "algebra M1" "algebra M2"
+ shows "f \<in> measurable M (prod_measure_space M1 M2) \<longleftrightarrow>
+ (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
+using assms proof (safe intro!: measurable_comp[where b="prod_measure_space M1 M2"])
+ interpret M1: algebra M1 by fact
+ interpret M2: algebra M2 by fact
+ assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
+
+ show "f \<in> measurable M (prod_measure_space M1 M2)" unfolding prod_measure_space_def
+ proof (rule measurable_sigma)
+ show "prod_sets (sets M1) (sets M2) \<subseteq> Pow (space M1 \<times> space M2)"
+ unfolding prod_sets_def using M1.sets_into_space M2.sets_into_space by auto
+ 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 assume "A \<in> prod_sets (sets M1) (sets M2)"
+ then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
+ unfolding prod_sets_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
+qed
definition
- "prod_measure_space M1 M2 = sigma (space M1 \<times> space M2) (prod_sets (sets M1) (sets M2))"
+ "prod_measure M \<mu> N \<nu> = (\<lambda>A. measure_space.positive_integral M \<mu> (\<lambda>s0. \<nu> ((\<lambda>s1. (s0, s1)) -` A)))"
lemma prod_setsI: "x \<in> A \<Longrightarrow> y \<in> B \<Longrightarrow> (x \<times> y) \<in> prod_sets A B"
by (auto simp add: prod_sets_def)
@@ -114,36 +512,25 @@
qed
lemma (in finite_measure_space) finite_measure_space_finite_prod_measure:
- assumes "finite_measure_space N \<nu>"
+ fixes N :: "('c, 'd) algebra_scheme"
+ assumes N: "finite_measure_space N \<nu>"
shows "finite_measure_space (prod_measure_space M N) (prod_measure M \<mu> N \<nu>)"
unfolding finite_prod_measure_space[OF assms]
-proof (rule finite_measure_spaceI)
+proof (rule finite_measure_spaceI, simp_all)
interpret N: finite_measure_space N \<nu> by fact
-
- let ?P = "\<lparr>space = space M \<times> space N, sets = Pow (space M \<times> space N)\<rparr>"
- show "measure_space ?P (prod_measure M \<mu> N \<nu>)"
- proof (rule sigma_algebra.finite_additivity_sufficient)
- show "sigma_algebra ?P" by (rule sigma_algebra_Pow)
- show "finite (space ?P)" using finite_space N.finite_space by auto
- from finite_prod_measure_times[OF assms, of "{}" "{}"]
- show "positive (prod_measure M \<mu> N \<nu>)"
- unfolding positive_def by simp
+ show "finite (space M \<times> space N)" using finite_space N.finite_space by auto
+ show "prod_measure M \<mu> N \<nu> (space M \<times> space N) \<noteq> \<omega>"
+ using finite_prod_measure_times[OF N top N.top] by simp
+ show "prod_measure M \<mu> N \<nu> {} = 0"
+ using finite_prod_measure_times[OF N empty_sets N.empty_sets] by simp
- show "additive ?P (prod_measure M \<mu> N \<nu>)"
- unfolding additive_def
- apply (auto simp add: sets_eq_Pow prod_measure_def positive_integral_add[symmetric]
- intro!: positive_integral_cong)
- apply (subst N.measure_additive[symmetric])
- by (auto simp: N.sets_eq_Pow sets_eq_Pow)
- qed
- show "finite (space ?P)" using finite_space N.finite_space by auto
- show "sets ?P = Pow (space ?P)" by simp
-
- fix x assume "x \<in> space ?P"
- with finite_prod_measure_times[OF assms, of "{fst x}" "{snd x}"]
- finite_measure[of "{fst x}"] N.finite_measure[of "{snd x}"]
- show "prod_measure M \<mu> N \<nu> {x} \<noteq> \<omega>"
- by (auto simp add: sets_eq_Pow N.sets_eq_Pow elim!: SigmaE)
+ fix A B :: "('a * 'c) set" assume "A \<inter> B = {}" "A \<subseteq> space M \<times> space N" "B \<subseteq> space M \<times> space N"
+ then show "prod_measure M \<mu> N \<nu> (A \<union> B) = prod_measure M \<mu> N \<nu> A + prod_measure M \<mu> N \<nu> B"
+ apply (auto simp add: sets_eq_Pow prod_measure_def positive_integral_add[symmetric]
+ intro!: positive_integral_cong)
+ apply (subst N.measure_additive)
+ apply (auto intro!: arg_cong[where f=\<mu>] simp: N.sets_eq_Pow sets_eq_Pow)
+ done
qed
lemma (in finite_measure_space) finite_measure_space_finite_prod_measure_alterantive:
@@ -153,4 +540,18 @@
unfolding finite_prod_measure_space[OF N, symmetric]
using finite_measure_space_finite_prod_measure[OF N] .
-end
\ No newline at end of file
+lemma prod_measure_times_finite:
+ assumes fms: "finite_measure_space M \<mu>" "finite_measure_space N \<nu>" and a: "a \<in> space M \<times> space N"
+ shows "prod_measure M \<mu> N \<nu> {a} = \<mu> {fst a} * \<nu> {snd a}"
+proof (cases a)
+ case (Pair b c)
+ hence a_eq: "{a} = {b} \<times> {c}" by simp
+
+ interpret M: finite_measure_space M \<mu> by fact
+ interpret N: finite_measure_space N \<nu> by fact
+
+ from finite_measure_space.finite_prod_measure_times[OF fms, of "{b}" "{c}"] M.sets_eq_Pow N.sets_eq_Pow a Pair
+ show ?thesis unfolding a_eq by simp
+qed
+
+end
--- a/src/HOL/Probability/Radon_Nikodym.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Radon_Nikodym.thy Thu Sep 02 20:44:33 2010 +0200
@@ -2,201 +2,6 @@
imports Lebesgue_Integration
begin
-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 intro: finite_UN)
- 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 sigma_algebra) borel_measurable_restricted:
- fixes f :: "'a \<Rightarrow> pinfreal" assumes "A \<in> sets M"
- shows "f \<in> borel_measurable (M\<lparr> space := A, sets := op \<inter> A ` sets M \<rparr>) \<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_space
- proof (simp, safe)
- fix S :: "pinfreal set" assume "S \<in> sets borel_space"
- "\<forall>S\<in>sets borel_space. ?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 fastsimp
- 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 :: "pinfreal set" assume "S \<in> sets borel_space"
- "\<forall>S\<in>sets borel_space. ?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) simple_function_eq_borel_measurable:
- fixes f :: "'a \<Rightarrow> pinfreal"
- shows "simple_function 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]
- by (fastsimp simp: simple_function_def)
-
-lemma (in measure_space) simple_function_restricted:
- fixes f :: "'a \<Rightarrow> pinfreal" assumes "A \<in> sets M"
- shows "sigma_algebra.simple_function (M\<lparr> space := A, sets := op \<inter> A ` sets M \<rparr>) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
- (is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
-proof -
- interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
- have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
- proof cases
- assume "A = space M"
- then have "f`A = ?f`space M" by (fastsimp 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::pinfreal)"
- 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`]
- by auto
-qed
-
-lemma (in measure_space) simple_integral_restricted:
- assumes "A \<in> sets M"
- assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
- shows "measure_space.simple_integral (M\<lparr> space := A, sets := op \<inter> A ` sets M \<rparr>) \<mu> f = simple_integral (\<lambda>x. f x * indicator A x)"
- (is "_ = simple_integral ?f")
- unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets 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
-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
-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 pinfreal_mult_cancel_left by auto
-qed
-
-lemma (in measure_space) positive_integral_restricted:
- assumes "A \<in> sets M"
- shows "measure_space.positive_integral (M\<lparr> space := A, sets := op \<inter> A ` sets M \<rparr>) \<mu> f = positive_integral (\<lambda>x. f x * indicator A x)"
- (is "measure_space.positive_integral ?R \<mu> f = positive_integral ?f")
-proof -
- have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
- then interpret R: measure_space ?R \<mu> .
- have saR: "sigma_algebra ?R" by fact
- have *: "R.positive_integral f = R.positive_integral ?f"
- by (auto intro!: R.positive_integral_cong)
- show ?thesis
- unfolding * R.positive_integral_def positive_integral_def
- unfolding simple_function_restricted[OF `A \<in> sets M`]
- apply (simp add: SUPR_def)
- apply (rule arg_cong[where f=Sup])
- proof (auto simp: image_iff simple_integral_restricted[OF `A \<in> sets M`])
- fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
- "g \<le> f" "\<forall>x\<in>A. \<omega> \<noteq> g x"
- then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and> (\<forall>y\<in>space M. \<omega> \<noteq> x y) \<and>
- simple_integral (\<lambda>x. g x * indicator A x) = simple_integral x"
- apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
- by (auto simp: indicator_def le_fun_def)
- next
- fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
- "\<forall>x\<in>space M. \<omega> \<noteq> g x"
- then have *: "(\<lambda>x. g x * indicator A x) = g"
- "\<And>x. g x * indicator A x = g x"
- "\<And>x. g x \<le> f x"
- by (auto simp: le_fun_def expand_fun_eq indicator_def split: split_if_asm)
- from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and> (\<forall>xa\<in>A. \<omega> \<noteq> x xa) \<and>
- simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
- using `A \<in> sets M`[THEN sets_into_space]
- apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
- by (fastsimp simp: le_fun_def *)
- qed
-qed
-
-lemma (in sigma_algebra) borel_measurable_psuminf:
- assumes "\<And>i. f i \<in> borel_measurable M"
- shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
- using assms unfolding psuminf_def
- by (auto intro!: borel_measurable_SUP[unfolded SUPR_fun_expand])
-
-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> \<omega>) \<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> \<omega>"
- 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> \<omega>"
- 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) Ex_finite_integrable_function:
shows "\<exists>h\<in>borel_measurable M. positive_integral h \<noteq> \<omega> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>)"
proof -
@@ -206,7 +11,6 @@
measure: "\<And>i. \<mu> (A i) \<noteq> \<omega>" and
disjoint: "disjoint_family A"
using disjoint_sigma_finite by auto
-
let "?B i" = "2^Suc i * \<mu> (A i)"
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
proof
@@ -225,20 +29,22 @@
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
-
let "?h x" = "\<Sum>\<^isub>\<infinity> i. n i * indicator (A i) x"
show ?thesis
proof (safe intro!: bexI[of _ ?h] del: notI)
- have "positive_integral ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))"
- apply (subst positive_integral_psuminf)
- using range apply (fastsimp intro!: borel_measurable_pinfreal_times borel_measurable_const borel_measurable_indicator)
- apply (subst positive_integral_cmult_indicator)
- using range by auto
+ have "\<And>i. A i \<in> sets M"
+ using range by fastsimp+
+ then have "positive_integral ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))"
+ by (simp add: positive_integral_psuminf positive_integral_cmult_indicator)
also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. Real ((1 / 2)^Suc i))"
proof (rule psuminf_le)
fix N show "n N * \<mu> (A N) \<le> Real ((1 / 2) ^ Suc N)"
using measure[of N] n[of N]
- by (cases "n N") (auto simp: pinfreal_noteq_omega_Ex field_simps zero_le_mult_iff mult_le_0_iff mult_less_0_iff power_less_zero_eq power_le_zero_eq inverse_eq_divide less_divide_eq power_divide split: split_if_asm)
+ by (cases "n N")
+ (auto simp: pinfreal_noteq_omega_Ex field_simps zero_le_mult_iff
+ mult_le_0_iff mult_less_0_iff power_less_zero_eq
+ power_le_zero_eq inverse_eq_divide less_divide_eq
+ power_divide split: split_if_asm)
qed
also have "\<dots> = Real 1"
by (rule suminf_imp_psuminf, rule power_half_series, auto)
@@ -251,13 +57,37 @@
then show "0 < ?h x" and "?h x < \<omega>" using n[of i] by auto
next
show "?h \<in> borel_measurable M" using range
- by (auto intro!: borel_measurable_psuminf borel_measurable_pinfreal_times borel_measurable_indicator)
+ by (auto intro!: borel_measurable_psuminf borel_measurable_pinfreal_times)
qed
qed
definition (in measure_space)
"absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pinfreal))"
+lemma (in finite_measure_space) absolutely_continuousI:
+ assumes "finite_measure_space M \<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 M \<nu> by fact
+ have "\<nu> N = \<nu> (\<Union>x\<in>N. {x})" by simp
+ also have "\<dots> = (\<Sum>x\<in>N. \<nu> {x})"
+ proof (rule v.measure_finitely_additive''[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 M" 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" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
+ hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
+ thus "\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
+ qed
+ finally show "\<nu> N = 0" .
+qed
+
lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
fixes e :: real assumes "0 < e"
assumes "finite_measure M s"
@@ -370,7 +200,7 @@
interpret M': finite_measure M s by fact
- let "?r S" = "M\<lparr> space := S, sets := (\<lambda>C. S \<inter> C)`sets M\<rparr>"
+ let "?r S" = "restricted_space S"
{ fix S n
assume S: "S \<in> sets M"
@@ -838,7 +668,7 @@
= (f x * indicator (Q i) x) * indicator A x"
unfolding indicator_def by auto
- have fm: "finite_measure (M\<lparr>space := Q i, sets := op \<inter> (Q i) ` sets M\<rparr>) \<mu>"
+ have fm: "finite_measure (restricted_space (Q i)) \<mu>"
(is "finite_measure ?R \<mu>") by (rule restricted_finite_measure[OF Q_sets[of i]])
then interpret R: finite_measure ?R .
have fmv: "finite_measure ?R \<nu>"
@@ -935,47 +765,6 @@
qed
qed
-lemma (in measure_space) positive_integral_translated_density:
- assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- shows "measure_space.positive_integral M (\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)) g =
- positive_integral (\<lambda>x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
-proof -
- from measure_space_density[OF assms(1)]
- interpret T: measure_space M ?T .
-
- from borel_measurable_implies_simple_function_sequence[OF assms(2)]
- obtain G where G: "\<And>i. simple_function (G i)" "G \<up> g" by blast
- note G_borel = borel_measurable_simple_function[OF this(1)]
-
- from T.positive_integral_isoton[OF `G \<up> g` G_borel]
- have *: "(\<lambda>i. T.positive_integral (G i)) \<up> T.positive_integral g" .
-
- { fix i
- have [simp]: "finite (G i ` space M)"
- using G(1) unfolding simple_function_def by auto
- have "T.positive_integral (G i) = T.simple_integral (G i)"
- using G T.positive_integral_eq_simple_integral by simp
- also have "\<dots> = positive_integral (\<lambda>x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
- apply (simp add: T.simple_integral_def)
- apply (subst positive_integral_cmult[symmetric])
- using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
- apply (subst positive_integral_setsum[symmetric])
- using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
- by (simp add: setsum_right_distrib field_simps)
- also have "\<dots> = positive_integral (\<lambda>x. f x * G i x)"
- by (auto intro!: positive_integral_cong
- simp: indicator_def if_distrib setsum_cases)
- finally have "T.positive_integral (G i) = positive_integral (\<lambda>x. f x * G i x)" . }
- with * have eq_Tg: "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> T.positive_integral g" by simp
-
- from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
- unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
- then have "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> positive_integral (\<lambda>x. f x * g x)"
- using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pinfreal_times)
- with eq_Tg show "T.positive_integral g = positive_integral (\<lambda>x. f x * g x)"
- unfolding isoton_def by simp
-qed
-
lemma (in sigma_finite_measure) Radon_Nikodym:
assumes "measure_space M \<nu>"
assumes "absolutely_continuous \<nu>"
--- a/src/HOL/Probability/Sigma_Algebra.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/Sigma_Algebra.thy Thu Sep 02 20:44:33 2010 +0200
@@ -6,7 +6,7 @@
header {* Sigma Algebras *}
-theory Sigma_Algebra imports Main Countable FuncSet begin
+theory Sigma_Algebra imports Main Countable FuncSet Indicator_Function begin
text {* Sigma algebras are an elementary concept in measure
theory. To measure --- that is to integrate --- functions, we first have
@@ -95,10 +95,13 @@
lemma (in algebra) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
by (metis Int_absorb2 sets_into_space)
+section {* Restricted algebras *}
+
+abbreviation (in algebra)
+ "restricted_space A \<equiv> \<lparr> space = A, sets = (\<lambda>S. (A \<inter> S)) ` sets M \<rparr>"
+
lemma (in algebra) restricted_algebra:
- assumes "S \<in> sets M"
- shows "algebra (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>)"
- (is "algebra ?r")
+ assumes "A \<in> sets M" shows "algebra (restricted_space A)"
using assms by unfold_locales auto
subsection {* Sigma Algebras *}
@@ -107,6 +110,13 @@
assumes countable_nat_UN [intro]:
"!!A. range A \<subseteq> sets M \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
+lemma sigma_algebra_cong:
+ fixes M :: "('a, 'b) algebra_scheme" and M' :: "('a, 'c) algebra_scheme"
+ assumes *: "sigma_algebra M"
+ and cong: "space M = space M'" "sets M = sets M'"
+ shows "sigma_algebra M'"
+using * unfolding sigma_algebra_def algebra_def sigma_algebra_axioms_def unfolding cong .
+
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>
@@ -320,15 +330,14 @@
lemma (in sigma_algebra) restricted_sigma_algebra:
assumes "S \<in> sets M"
- shows "sigma_algebra (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>)"
- (is "sigma_algebra ?r")
+ shows "sigma_algebra (restricted_space S)"
unfolding sigma_algebra_def sigma_algebra_axioms_def
proof safe
- show "algebra ?r" using restricted_algebra[OF assms] .
+ show "algebra (restricted_space S)" using restricted_algebra[OF assms] .
next
- fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets ?r"
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets (restricted_space S)"
from restriction_in_sets[OF assms this[simplified]]
- show "(\<Union>i. A i) \<in> sets ?r" by simp
+ show "(\<Union>i. A i) \<in> sets (restricted_space S)" by simp
qed
section {* Measurable functions *}
@@ -560,6 +569,19 @@
(metis insert_absorb insert_subset le_SucE le_antisym not_leE)
qed
+lemma setsum_indicator_disjoint_family:
+ fixes f :: "'d \<Rightarrow> 'e::semiring_1"
+ assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
+ shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
+proof -
+ have "P \<inter> {i. x \<in> A i} = {j}"
+ using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
+ by auto
+ thus ?thesis
+ unfolding indicator_def
+ by (simp add: if_distrib setsum_cases[OF `finite P`])
+qed
+
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
@@ -626,6 +648,67 @@
thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
qed
+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 \<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)
+
+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_vimage:
+ fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
+ shows "sigma_algebra (vimage_algebra S f)"
+proof
+ fix A assume "A \<in> sets (vimage_algebra S f)"
+ then guess B unfolding in_vimage_algebra ..
+ then show "space (vimage_algebra S f) - A \<in> sets (vimage_algebra S f)"
+ using sets_into_space assms
+ by (auto intro!: bexI[of _ "space M - B"])
+next
+ fix A assume "A \<in> sets (vimage_algebra S f)"
+ then guess A' unfolding in_vimage_algebra .. note A' = this
+ fix B assume "B \<in> sets (vimage_algebra S f)"
+ then guess B' unfolding in_vimage_algebra .. note B' = this
+ then show "A \<union> B \<in> sets (vimage_algebra S f)"
+ using sets_into_space assms A' B'
+ by (auto intro!: bexI[of _ "A' \<union> B'"])
+next
+ fix A::"nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets (vimage_algebra S f)"
+ then have "\<forall>i. \<exists>B. A i = f -` B \<inter> S \<and> B \<in> sets M"
+ by (simp add: subset_eq) blast
+ from this[THEN choice] obtain B
+ where B: "\<And>i. A i = f -` B i \<inter> S" "range B \<subseteq> sets M" by auto
+ show "(\<Union>i. A i) \<in> sets (vimage_algebra S f)"
+ using B by (auto intro!: bexI[of _ "\<Union>i. B i"])
+qed auto
+
+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
+
+section {* Conditional space *}
+
+definition (in algebra)
+ "image_space X = \<lparr> space = X`space M, sets = (\<lambda>S. X`S) ` sets 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
+
subsection {* A Two-Element Series *}
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
--- a/src/HOL/Probability/ex/Dining_Cryptographers.thy Thu Sep 02 18:45:23 2010 +0200
+++ b/src/HOL/Probability/ex/Dining_Cryptographers.thy Thu Sep 02 20:44:33 2010 +0200
@@ -26,14 +26,13 @@
let ?measure = "\<lambda>s::'a set. real (card s) / real (card S)"
show "finite_measure_space M \<mu>"
- proof (rule finite_Pow_additivity_sufficient, simp_all)
- show "positive \<mu>" by (simp add: positive_def)
-
- show "additive M \<mu>"
- by (simp add: additive_def inverse_eq_divide field_simps Real_real
+ proof (rule finite_measure_spaceI)
+ fix A B :: "'a set" assume "A \<inter> B = {}" "A \<subseteq> space M" "B \<subseteq> space M"
+ then show "\<mu> (A \<union> B) = \<mu> A + \<mu> B"
+ by (simp add: inverse_eq_divide field_simps Real_real
divide_le_0_iff zero_le_divide_iff
card_Un_disjoint finite_subset[OF _ finite])
- qed
+ qed auto
qed simp_all
lemma set_of_list_extend: