--- a/src/HOL/Probability/Sigma_Algebra.thy Mon May 19 12:04:45 2014 +0200
+++ b/src/HOL/Probability/Sigma_Algebra.thy Mon May 19 13:44:13 2014 +0200
@@ -5,7 +5,7 @@
translated by Lawrence Paulson.
*)
-header {* Sigma Algebras *}
+header {* Describing measurable sets *}
theory Sigma_Algebra
imports
@@ -33,9 +33,7 @@
lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
by (metis PowD contra_subsetD space_closed)
-subsection {* Semiring of sets *}
-
-subsubsection {* Disjoint sets *}
+subsubsection {* Semiring of sets *}
definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
@@ -255,7 +253,7 @@
"X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
by (auto simp: algebra_iff_Int)
-subsection {* Restricted algebras *}
+subsubsection {* Restricted algebras *}
abbreviation (in algebra)
"restricted_space A \<equiv> (op \<inter> A) ` M"
@@ -264,7 +262,7 @@
assumes "A \<in> M" shows "algebra A (restricted_space A)"
using assms by (auto simp: algebra_iff_Int)
-subsection {* Sigma Algebras *}
+subsubsection {* Sigma Algebras *}
locale sigma_algebra = algebra +
assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
@@ -446,7 +444,7 @@
shows "sigma_algebra S { {}, X, S - X, S }"
using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
-subsection {* Binary Unions *}
+subsubsection {* Binary Unions *}
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
where "binary a b = (\<lambda>x. b)(0 := a)"
@@ -468,7 +466,7 @@
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
algebra_iff_Un Un_range_binary)
-subsection {* Initial Sigma Algebra *}
+subsubsection {* Initial Sigma Algebra *}
text {*Sigma algebras can naturally be created as the closure of any set of
M with regard to the properties just postulated. *}
@@ -775,7 +773,7 @@
qed
qed
-subsection "Disjoint families"
+subsubsection "Disjoint families"
definition
disjoint_family_on where
@@ -934,7 +932,7 @@
by (intro disjointD[OF d]) auto
qed
-subsection {* Ring generated by a semiring *}
+subsubsection {* Ring generated by a semiring *}
definition (in semiring_of_sets)
"generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
@@ -1064,759 +1062,7 @@
by (blast intro!: sigma_sets_mono elim: generated_ringE)
qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
-subsection {* Measure type *}
-
-definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
- "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
-
-definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
- "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
- (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
-
-definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
- "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
-
-typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
-proof
- have "sigma_algebra UNIV {{}, UNIV}"
- by (auto simp: sigma_algebra_iff2)
- then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
- by (auto simp: measure_space_def positive_def countably_additive_def)
-qed
-
-definition space :: "'a measure \<Rightarrow> 'a set" where
- "space M = fst (Rep_measure M)"
-
-definition sets :: "'a measure \<Rightarrow> 'a set set" where
- "sets M = fst (snd (Rep_measure M))"
-
-definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
- "emeasure M = snd (snd (Rep_measure M))"
-
-definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
- "measure M A = real (emeasure M A)"
-
-declare [[coercion sets]]
-
-declare [[coercion measure]]
-
-declare [[coercion emeasure]]
-
-lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
- by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
-
-interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
- using measure_space[of M] by (auto simp: measure_space_def)
-
-definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
- "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
- \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
-
-abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
-
-lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
- unfolding measure_space_def
- by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
-
-lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
-by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
-
-lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
-by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
-
-lemma measure_space_closed:
- assumes "measure_space \<Omega> M \<mu>"
- shows "M \<subseteq> Pow \<Omega>"
-proof -
- interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
- show ?thesis by(rule space_closed)
-qed
-
-lemma (in ring_of_sets) positive_cong_eq:
- "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
- by (auto simp add: positive_def)
-
-lemma (in sigma_algebra) countably_additive_eq:
- "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
- unfolding countably_additive_def
- by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
-
-lemma measure_space_eq:
- assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
- shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
-proof -
- interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
- from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
- by (auto simp: measure_space_def)
-qed
-
-lemma measure_of_eq:
- assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
- shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
-proof -
- have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
- using assms by (rule measure_space_eq)
- with eq show ?thesis
- by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
-qed
-
-lemma
- shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
- and sets_measure_of_conv:
- "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
- and emeasure_measure_of_conv:
- "emeasure (measure_of \<Omega> A \<mu>) =
- (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)" (is ?emeasure)
-proof -
- have "?space \<and> ?sets \<and> ?emeasure"
- proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
- case True
- from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
- have "A \<subseteq> Pow \<Omega>" by simp
- hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
- (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
- by(rule measure_space_eq) auto
- with True `A \<subseteq> Pow \<Omega>` show ?thesis
- by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
- next
- case False thus ?thesis
- by(cases "A \<subseteq> Pow \<Omega>")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0')
- qed
- thus ?space ?sets ?emeasure by simp_all
-qed
-
-lemma [simp]:
- assumes A: "A \<subseteq> Pow \<Omega>"
- shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
- and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
-using assms
-by(simp_all add: sets_measure_of_conv space_measure_of_conv)
-
-lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
- using space_closed by (auto intro!: sigma_sets_eq)
-
-lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
- by (rule space_measure_of_conv)
-
-lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
- by (auto intro!: sigma_sets_subseteq)
-
-lemma sigma_sets_mono'':
- assumes "A \<in> sigma_sets C D"
- assumes "B \<subseteq> D"
- assumes "D \<subseteq> Pow C"
- shows "sigma_sets A B \<subseteq> sigma_sets C D"
-proof
- fix x assume "x \<in> sigma_sets A B"
- thus "x \<in> sigma_sets C D"
- proof induct
- case (Basic a) with assms have "a \<in> D" by auto
- thus ?case ..
- next
- case Empty show ?case by (rule sigma_sets.Empty)
- next
- from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
- moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
- ultimately have "A - a \<in> sets (sigma C D)" ..
- thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
- next
- case (Union a)
- thus ?case by (intro sigma_sets.Union)
- qed
-qed
-
-lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
- by auto
-
-subsection {* Constructing simple @{typ "'a measure"} *}
-
-lemma emeasure_measure_of:
- assumes M: "M = measure_of \<Omega> A \<mu>"
- assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
- assumes X: "X \<in> sets M"
- shows "emeasure M X = \<mu> X"
-proof -
- interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
- have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
- using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
- thus ?thesis using X ms
- by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
-qed
-
-lemma emeasure_measure_of_sigma:
- assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
- assumes A: "A \<in> M"
- shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
-proof -
- interpret sigma_algebra \<Omega> M by fact
- have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
- using ms sigma_sets_eq by (simp add: measure_space_def)
- thus ?thesis by(simp add: emeasure_measure_of_conv A)
-qed
-
-lemma measure_cases[cases type: measure]:
- obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
- by atomize_elim (cases x, auto)
-
-lemma sets_eq_imp_space_eq:
- "sets M = sets M' \<Longrightarrow> space M = space M'"
- using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
- by blast
-
-lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
- by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
-
-lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
- using emeasure_notin_sets[of A M] by blast
-
-lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
- by (simp add: measure_def emeasure_notin_sets)
-
-lemma measure_eqI:
- fixes M N :: "'a measure"
- assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
- shows "M = N"
-proof (cases M N rule: measure_cases[case_product measure_cases])
- case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
- interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
- interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
- have "A = sets M" "A' = sets N"
- using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
- with `sets M = sets N` have AA': "A = A'" by simp
- moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
- moreover { fix B have "\<mu> B = \<mu>' B"
- proof cases
- assume "B \<in> A"
- with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
- with measure_measure show "\<mu> B = \<mu>' B"
- by (simp add: emeasure_def Abs_measure_inverse)
- next
- assume "B \<notin> A"
- with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
- by auto
- then have "emeasure M B = 0" "emeasure N B = 0"
- by (simp_all add: emeasure_notin_sets)
- with measure_measure show "\<mu> B = \<mu>' B"
- by (simp add: emeasure_def Abs_measure_inverse)
- qed }
- then have "\<mu> = \<mu>'" by auto
- ultimately show "M = N"
- by (simp add: measure_measure)
-qed
-
-lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
- using measure_space_0[of A \<Omega>]
- by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
-
-lemma sigma_eqI:
- assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
- shows "sigma \<Omega> M = sigma \<Omega> N"
- by (rule measure_eqI) (simp_all add: emeasure_sigma)
-
-subsection {* Measurable functions *}
-
-definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
- "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
-
-lemma measurable_space:
- "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
- unfolding measurable_def by auto
-
-lemma measurable_sets:
- "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
- unfolding measurable_def by auto
-
-lemma measurable_sets_Collect:
- assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M"
-proof -
- have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
- using measurable_space[OF f] by auto
- with measurable_sets[OF f P] show ?thesis
- by simp
-qed
-
-lemma measurable_sigma_sets:
- assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
- and f: "f \<in> space M \<rightarrow> \<Omega>"
- and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
- shows "f \<in> measurable M N"
-proof -
- interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
- from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
-
- { fix X assume "X \<in> sigma_sets \<Omega> A"
- then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
- proof induct
- case (Basic a) then show ?case
- by (auto simp add: ba) (metis B(2) subsetD PowD)
- next
- case (Compl a)
- have [simp]: "f -` \<Omega> \<inter> space M = space M"
- by (auto simp add: funcset_mem [OF f])
- then show ?case
- by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
- next
- case (Union a)
- then show ?case
- by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
- qed auto }
- with f show ?thesis
- by (auto simp add: measurable_def B \<Omega>)
-qed
-
-lemma measurable_measure_of:
- assumes B: "N \<subseteq> Pow \<Omega>"
- and f: "f \<in> space M \<rightarrow> \<Omega>"
- and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
- shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
-proof -
- have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
- using B by (rule sets_measure_of)
- from this assms show ?thesis by (rule measurable_sigma_sets)
-qed
-
-lemma measurable_iff_measure_of:
- assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
- shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
- by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
-
-lemma measurable_cong_sets:
- assumes sets: "sets M = sets M'" "sets N = sets N'"
- shows "measurable M N = measurable M' N'"
- using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
-
-lemma measurable_cong:
- assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
- shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
- unfolding measurable_def using assms
- by (simp cong: vimage_inter_cong Pi_cong)
-
-lemma measurable_cong_strong:
- "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
- f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
- by (metis measurable_cong)
-
-lemma measurable_eqI:
- "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
- sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
- \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
- by (simp add: measurable_def sigma_algebra_iff2)
-
-lemma measurable_compose:
- assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
- shows "(\<lambda>x. g (f x)) \<in> measurable M L"
-proof -
- have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
- using measurable_space[OF f] by auto
- with measurable_space[OF f] measurable_space[OF g] show ?thesis
- by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
- simp del: vimage_Int simp add: measurable_def)
-qed
-
-lemma measurable_comp:
- "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
- using measurable_compose[of f M N g L] by (simp add: comp_def)
-
-lemma measurable_const:
- "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
- by (auto simp add: measurable_def)
-
-lemma measurable_If:
- assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
- assumes P: "{x\<in>space M. P x} \<in> sets M"
- shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
- unfolding measurable_def
-proof safe
- fix x assume "x \<in> space M"
- thus "(if P x then f x else g x) \<in> space M'"
- using measure unfolding measurable_def by auto
-next
- fix A assume "A \<in> sets M'"
- hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
- ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
- ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
- using measure unfolding measurable_def by (auto split: split_if_asm)
- show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
- using `A \<in> sets M'` measure P unfolding * measurable_def
- by (auto intro!: sets.Un)
-qed
-
-lemma measurable_If_set:
- assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
- assumes P: "A \<inter> space M \<in> sets M"
- shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
-proof (rule measurable_If[OF measure])
- have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
- thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
-qed
-
-lemma measurable_ident: "id \<in> measurable M M"
- by (auto simp add: measurable_def)
-
-lemma measurable_ident_sets:
- assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
- using measurable_ident[of M]
- unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
-
-lemma sets_Least:
- assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
- shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
-proof -
- { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
- proof cases
- assume i: "(LEAST j. False) = i"
- have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
- {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
- by (simp add: set_eq_iff, safe)
- (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
- with meas show ?thesis
- by (auto intro!: sets.Int)
- next
- assume i: "(LEAST j. False) \<noteq> i"
- then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
- {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
- proof (simp add: set_eq_iff, safe)
- fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
- have "\<exists>j. P j x"
- by (rule ccontr) (insert neq, auto)
- then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
- qed (auto dest: Least_le intro!: Least_equality)
- with meas show ?thesis
- by auto
- qed }
- then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
- by (intro sets.countable_UN) auto
- moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
- (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
- ultimately show ?thesis by auto
-qed
-
-lemma measurable_strong:
- fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
- assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
- and t: "f ` (space a) \<subseteq> t"
- and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
- shows "(g o f) \<in> measurable a c"
-proof -
- have fab: "f \<in> (space a -> space b)"
- and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
- by (auto simp add: measurable_def)
- have eq: "\<And>y. (g \<circ> f) -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
- by force
- show ?thesis
- apply (auto simp add: measurable_def vimage_comp)
- apply (metis funcset_mem fab g)
- apply (subst eq)
- apply (metis ba cb)
- done
-qed
-
-lemma measurable_mono1:
- "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
- measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
- using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
-
-subsection {* Counting space *}
-
-definition count_space :: "'a set \<Rightarrow> 'a measure" where
- "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
-
-lemma
- shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
- and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
- using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
- by (auto simp: count_space_def)
-
-lemma measurable_count_space_eq1[simp]:
- "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
- unfolding measurable_def by simp
-
-lemma measurable_count_space_eq2:
- assumes "finite A"
- shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
-proof -
- { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
- with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
- by (auto dest: finite_subset)
- moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
- ultimately have "f -` X \<inter> space M \<in> sets M"
- using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) }
- then show ?thesis
- unfolding measurable_def by auto
-qed
-
-lemma measurable_count_space_eq2_countable:
- fixes f :: "'a => 'c::countable"
- shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
-proof -
- { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
- assume *: "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
- have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
- by auto
- also have "\<dots> \<in> sets M"
- using * `X \<subseteq> A` by (intro sets.countable_UN) auto
- finally have "f -` X \<inter> space M \<in> sets M" . }
- then show ?thesis
- unfolding measurable_def by auto
-qed
-
-lemma measurable_compose_countable:
- assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
- shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
- unfolding measurable_def
-proof safe
- fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
- using f[THEN measurable_space] g[THEN measurable_space] by auto
-next
- fix A assume A: "A \<in> sets N"
- have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))"
- by auto
- also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]
- by (auto intro!: sets.countable_UN measurable_sets)
- finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
-qed
-
-lemma measurable_count_space_const:
- "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
- by (simp add: measurable_const)
-
-lemma measurable_count_space:
- "f \<in> measurable (count_space A) (count_space UNIV)"
- by simp
-
-lemma measurable_compose_rev:
- assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
- shows "(\<lambda>x. f (g x)) \<in> measurable M N"
- using measurable_compose[OF g f] .
-
-lemma measurable_count_space_eq_countable:
- assumes "countable A"
- shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
-proof -
- { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
- with `countable A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
- by (auto dest: countable_subset)
- moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
- ultimately have "f -` X \<inter> space M \<in> sets M"
- using `X \<subseteq> A` by (auto intro!: sets.countable_UN' simp del: UN_simps) }
- then show ?thesis
- unfolding measurable_def by auto
-qed
-
-subsection {* Extend measure *}
-
-definition "extend_measure \<Omega> I G \<mu> =
- (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
- then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>')
- else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
-
-lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
- unfolding extend_measure_def by simp
-
-lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
- unfolding extend_measure_def by simp
-
-lemma emeasure_extend_measure:
- assumes M: "M = extend_measure \<Omega> I G \<mu>"
- and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
- and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
- and "i \<in> I"
- shows "emeasure M (G i) = \<mu> i"
-proof cases
- assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
- with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
- by (simp add: extend_measure_def)
- from measure_space_0[OF ms(1)] ms `i\<in>I`
- have "emeasure M (G i) = 0"
- by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
- with `i\<in>I` * show ?thesis
- by simp
-next
- def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
- assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
- moreover
- have "measure_space (space M) (sets M) \<mu>'"
- using ms unfolding measure_space_def by auto default
- with ms eq have "\<exists>\<mu>'. P \<mu>'"
- unfolding P_def
- by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
- ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
- by (simp add: M extend_measure_def P_def[symmetric])
-
- from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
- show "emeasure M (G i) = \<mu> i"
- proof (subst emeasure_measure_of[OF M_eq])
- have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
- using M_eq ms by (auto simp: sets_extend_measure)
- then show "G i \<in> sets M" using `i \<in> I` by auto
- show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
- using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
- qed fact
-qed
-
-lemma emeasure_extend_measure_Pair:
- assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
- and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
- and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
- and "I i j"
- shows "emeasure M (G i j) = \<mu> i j"
- using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
- by (auto simp: subset_eq)
-
-subsection {* Sigma algebra generated by function preimages *}
-
-definition
- "vimage_algebra M S X = sigma S ((\<lambda>A. X -` A \<inter> S) ` sets M)"
-
-lemma sigma_algebra_preimages:
- fixes f :: "'x \<Rightarrow> 'a"
- assumes "f \<in> S \<rightarrow> space M"
- shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
- (is "sigma_algebra _ (?F ` sets M)")
-proof (simp add: sigma_algebra_iff2, safe)
- show "{} \<in> ?F ` sets M" by blast
-next
- fix A assume "A \<in> sets M"
- moreover have "S - ?F A = ?F (space M - A)"
- using assms by auto
- ultimately show "S - ?F A \<in> ?F ` sets M"
- by blast
-next
- fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
- have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
- proof safe
- fix i
- have "A i \<in> ?F ` M" using * by auto
- then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
- qed
- from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
- by auto
- then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
- then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
-qed
-
-lemma sets_vimage_algebra[simp]:
- "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
- using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]
- by (simp add: vimage_algebra_def)
-
-lemma space_vimage_algebra[simp]:
- "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
- using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]
- by (simp add: vimage_algebra_def)
-
-lemma in_vimage_algebra[simp]:
- "f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
- by (simp add: image_iff)
-
-lemma measurable_vimage_algebra:
- fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
- shows "f \<in> measurable (vimage_algebra M S f) M"
- unfolding measurable_def using assms by force
-
-lemma measurable_vimage:
- fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
- assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
- shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
-proof -
- note measurable_vimage_algebra[OF assms(2)]
- from measurable_comp[OF this assms(1)]
- show ?thesis by (simp add: comp_def)
-qed
-
-lemma sigma_sets_vimage:
- assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
- shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
-proof (intro set_eqI iffI)
- let ?F = "\<lambda>X. f -` X \<inter> S'"
- fix X assume "X \<in> sigma_sets S' (?F ` A)"
- then show "X \<in> ?F ` sigma_sets S A"
- proof induct
- case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
- by auto
- then show ?case by auto
- next
- case Empty then show ?case
- by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
- next
- case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
- by auto
- then have "S - X' \<in> sigma_sets S A"
- by (auto intro!: sigma_sets.Compl)
- then show ?case
- using X assms by (auto intro!: image_eqI[where x="S - X'"])
- next
- case (Union F)
- then have "\<forall>i. \<exists>F'. F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
- by (auto simp: image_iff Bex_def)
- from choice[OF this] obtain F' where
- "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
- by auto
- then show ?case
- by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
- qed
-next
- let ?F = "\<lambda>X. f -` X \<inter> S'"
- fix X assume "X \<in> ?F ` sigma_sets S A"
- then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
- then show "X \<in> sigma_sets S' (?F ` A)"
- proof (induct arbitrary: X)
- case Empty then show ?case by (auto intro: sigma_sets.Empty)
- next
- case (Compl X')
- have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
- apply (rule sigma_sets.Compl)
- using assms by (auto intro!: Compl.hyps simp: Compl.prems)
- also have "S' - (S' - X) = X"
- using assms Compl by auto
- finally show ?case .
- next
- case (Union F)
- have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
- by (intro sigma_sets.Union Union.hyps) simp
- also have "(\<Union>i. f -` F i \<inter> S') = X"
- using assms Union by auto
- finally show ?case .
- qed auto
-qed
-
-subsection {* Restricted Space Sigma Algebra *}
-
-definition "restrict_space M \<Omega> = measure_of \<Omega> ((op \<inter> \<Omega>) ` sets M) (\<lambda>A. emeasure M (A \<inter> \<Omega>))"
-
-lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega>"
- unfolding restrict_space_def by (subst space_measure_of) auto
-
-lemma sets_restrict_space: "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
- using sigma_sets_vimage[of "\<lambda>x. x" \<Omega> "space M" "sets M"]
- unfolding restrict_space_def
- by (subst sets_measure_of) (auto simp: sets.sigma_sets_eq Int_commute[of _ \<Omega>] sets.space_closed)
-
-lemma sets_restrict_space_iff:
- "\<Omega> \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
- by (subst sets_restrict_space) (auto dest: sets.sets_into_space)
-
-lemma measurable_restrict_space1:
- assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> measurable M N" shows "f \<in> measurable (restrict_space M \<Omega>) N"
- unfolding measurable_def
-proof (intro CollectI conjI ballI)
- show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"
- using measurable_space[OF f] sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
-
- fix A assume "A \<in> sets N"
- have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> \<Omega>"
- using sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
- also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
- unfolding sets_restrict_space_iff[OF \<Omega>]
- using measurable_sets[OF f `A \<in> sets N`] \<Omega> by blast
- finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
-qed
-
-lemma measurable_restrict_space2:
- "\<Omega> \<in> sets N \<Longrightarrow> f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"
- by (simp add: measurable_def space_restrict_space sets_restrict_space_iff)
-
-subsection {* A Two-Element Series *}
+subsubsection {* A Two-Element Series *}
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
@@ -1830,7 +1076,7 @@
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
by (simp add: SUP_def range_binaryset_eq)
-section {* Closed CDI *}
+subsubsection {* Closed CDI *}
definition closed_cdi where
"closed_cdi \<Omega> M \<longleftrightarrow>
@@ -2064,7 +1310,7 @@
by blast
qed
-subsection {* Dynkin systems *}
+subsubsection {* Dynkin systems *}
locale dynkin_system = subset_class +
assumes space: "\<Omega> \<in> M"
@@ -2126,7 +1372,7 @@
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
qed
-subsection "Intersection stable algebras"
+subsubsection "Intersection sets systems"
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
@@ -2162,7 +1408,7 @@
qed auto
qed
-subsection "Smallest Dynkin systems"
+subsubsection "Smallest Dynkin systems"
definition dynkin where
"dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
@@ -2309,6 +1555,11 @@
using assms by (auto simp: dynkin_def)
qed
+subsubsection {* Induction rule for intersection-stable generators *}
+
+text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
+generated by a generator closed under intersection. *}
+
lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
assumes "Int_stable G"
and closed: "G \<subseteq> Pow \<Omega>"
@@ -2330,4 +1581,756 @@
with A show ?thesis by auto
qed
+subsection {* Measure type *}
+
+definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+ "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
+
+definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+ "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
+ (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
+
+definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+ "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
+
+typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
+proof
+ have "sigma_algebra UNIV {{}, UNIV}"
+ by (auto simp: sigma_algebra_iff2)
+ then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
+ by (auto simp: measure_space_def positive_def countably_additive_def)
+qed
+
+definition space :: "'a measure \<Rightarrow> 'a set" where
+ "space M = fst (Rep_measure M)"
+
+definition sets :: "'a measure \<Rightarrow> 'a set set" where
+ "sets M = fst (snd (Rep_measure M))"
+
+definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
+ "emeasure M = snd (snd (Rep_measure M))"
+
+definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
+ "measure M A = real (emeasure M A)"
+
+declare [[coercion sets]]
+
+declare [[coercion measure]]
+
+declare [[coercion emeasure]]
+
+lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
+ by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
+
+interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
+ using measure_space[of M] by (auto simp: measure_space_def)
+
+definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
+ "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
+ \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
+
+abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
+
+lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
+ unfolding measure_space_def
+ by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
+
+lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
+by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
+
+lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
+by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
+
+lemma measure_space_closed:
+ assumes "measure_space \<Omega> M \<mu>"
+ shows "M \<subseteq> Pow \<Omega>"
+proof -
+ interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
+ show ?thesis by(rule space_closed)
+qed
+
+lemma (in ring_of_sets) positive_cong_eq:
+ "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
+ by (auto simp add: positive_def)
+
+lemma (in sigma_algebra) countably_additive_eq:
+ "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
+ unfolding countably_additive_def
+ by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
+
+lemma measure_space_eq:
+ assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
+ shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
+proof -
+ interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
+ from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
+ by (auto simp: measure_space_def)
+qed
+
+lemma measure_of_eq:
+ assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
+ shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
+proof -
+ have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
+ using assms by (rule measure_space_eq)
+ with eq show ?thesis
+ by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
+qed
+
+lemma
+ shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
+ and sets_measure_of_conv:
+ "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
+ and emeasure_measure_of_conv:
+ "emeasure (measure_of \<Omega> A \<mu>) =
+ (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)" (is ?emeasure)
+proof -
+ have "?space \<and> ?sets \<and> ?emeasure"
+ proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
+ case True
+ from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
+ have "A \<subseteq> Pow \<Omega>" by simp
+ hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
+ (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
+ by(rule measure_space_eq) auto
+ with True `A \<subseteq> Pow \<Omega>` show ?thesis
+ by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
+ next
+ case False thus ?thesis
+ by(cases "A \<subseteq> Pow \<Omega>")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0')
+ qed
+ thus ?space ?sets ?emeasure by simp_all
+qed
+
+lemma [simp]:
+ assumes A: "A \<subseteq> Pow \<Omega>"
+ shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
+ and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
+using assms
+by(simp_all add: sets_measure_of_conv space_measure_of_conv)
+
+lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
+ using space_closed by (auto intro!: sigma_sets_eq)
+
+lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
+ by (rule space_measure_of_conv)
+
+lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
+ by (auto intro!: sigma_sets_subseteq)
+
+lemma sigma_sets_mono'':
+ assumes "A \<in> sigma_sets C D"
+ assumes "B \<subseteq> D"
+ assumes "D \<subseteq> Pow C"
+ shows "sigma_sets A B \<subseteq> sigma_sets C D"
+proof
+ fix x assume "x \<in> sigma_sets A B"
+ thus "x \<in> sigma_sets C D"
+ proof induct
+ case (Basic a) with assms have "a \<in> D" by auto
+ thus ?case ..
+ next
+ case Empty show ?case by (rule sigma_sets.Empty)
+ next
+ from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
+ moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
+ ultimately have "A - a \<in> sets (sigma C D)" ..
+ thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
+ next
+ case (Union a)
+ thus ?case by (intro sigma_sets.Union)
+ qed
+qed
+
+lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
+ by auto
+
+subsubsection {* Constructing simple @{typ "'a measure"} *}
+
+lemma emeasure_measure_of:
+ assumes M: "M = measure_of \<Omega> A \<mu>"
+ assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
+ assumes X: "X \<in> sets M"
+ shows "emeasure M X = \<mu> X"
+proof -
+ interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
+ have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
+ using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
+ thus ?thesis using X ms
+ by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
+qed
+
+lemma emeasure_measure_of_sigma:
+ assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
+ assumes A: "A \<in> M"
+ shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
+proof -
+ interpret sigma_algebra \<Omega> M by fact
+ have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+ using ms sigma_sets_eq by (simp add: measure_space_def)
+ thus ?thesis by(simp add: emeasure_measure_of_conv A)
+qed
+
+lemma measure_cases[cases type: measure]:
+ obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
+ by atomize_elim (cases x, auto)
+
+lemma sets_eq_imp_space_eq:
+ "sets M = sets M' \<Longrightarrow> space M = space M'"
+ using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
+ by blast
+
+lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
+ by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
+
+lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
+ using emeasure_notin_sets[of A M] by blast
+
+lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
+ by (simp add: measure_def emeasure_notin_sets)
+
+lemma measure_eqI:
+ fixes M N :: "'a measure"
+ assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
+ shows "M = N"
+proof (cases M N rule: measure_cases[case_product measure_cases])
+ case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
+ interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
+ interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
+ have "A = sets M" "A' = sets N"
+ using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
+ with `sets M = sets N` have AA': "A = A'" by simp
+ moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
+ moreover { fix B have "\<mu> B = \<mu>' B"
+ proof cases
+ assume "B \<in> A"
+ with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
+ with measure_measure show "\<mu> B = \<mu>' B"
+ by (simp add: emeasure_def Abs_measure_inverse)
+ next
+ assume "B \<notin> A"
+ with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
+ by auto
+ then have "emeasure M B = 0" "emeasure N B = 0"
+ by (simp_all add: emeasure_notin_sets)
+ with measure_measure show "\<mu> B = \<mu>' B"
+ by (simp add: emeasure_def Abs_measure_inverse)
+ qed }
+ then have "\<mu> = \<mu>'" by auto
+ ultimately show "M = N"
+ by (simp add: measure_measure)
+qed
+
+lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
+ using measure_space_0[of A \<Omega>]
+ by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
+
+lemma sigma_eqI:
+ assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
+ shows "sigma \<Omega> M = sigma \<Omega> N"
+ by (rule measure_eqI) (simp_all add: emeasure_sigma)
+
+subsubsection {* Measurable functions *}
+
+definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
+ "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
+
+lemma measurable_space:
+ "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
+ unfolding measurable_def by auto
+
+lemma measurable_sets:
+ "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
+ unfolding measurable_def by auto
+
+lemma measurable_sets_Collect:
+ assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M"
+proof -
+ have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
+ using measurable_space[OF f] by auto
+ with measurable_sets[OF f P] show ?thesis
+ by simp
+qed
+
+lemma measurable_sigma_sets:
+ assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
+ and f: "f \<in> space M \<rightarrow> \<Omega>"
+ and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
+ shows "f \<in> measurable M N"
+proof -
+ interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
+ from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
+
+ { fix X assume "X \<in> sigma_sets \<Omega> A"
+ then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
+ proof induct
+ case (Basic a) then show ?case
+ by (auto simp add: ba) (metis B(2) subsetD PowD)
+ next
+ case (Compl a)
+ have [simp]: "f -` \<Omega> \<inter> space M = space M"
+ by (auto simp add: funcset_mem [OF f])
+ then show ?case
+ by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
+ next
+ case (Union a)
+ then show ?case
+ by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
+ qed auto }
+ with f show ?thesis
+ by (auto simp add: measurable_def B \<Omega>)
+qed
+
+lemma measurable_measure_of:
+ assumes B: "N \<subseteq> Pow \<Omega>"
+ and f: "f \<in> space M \<rightarrow> \<Omega>"
+ and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
+ shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
+proof -
+ have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
+ using B by (rule sets_measure_of)
+ from this assms show ?thesis by (rule measurable_sigma_sets)
+qed
+
+lemma measurable_iff_measure_of:
+ assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
+ shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
+ by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
+
+lemma measurable_cong_sets:
+ assumes sets: "sets M = sets M'" "sets N = sets N'"
+ shows "measurable M N = measurable M' N'"
+ using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
+
+lemma measurable_cong:
+ assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
+ shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
+ unfolding measurable_def using assms
+ by (simp cong: vimage_inter_cong Pi_cong)
+
+lemma measurable_cong_strong:
+ "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
+ f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
+ by (metis measurable_cong)
+
+lemma measurable_eqI:
+ "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
+ sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
+ \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
+ by (simp add: measurable_def sigma_algebra_iff2)
+
+lemma measurable_compose:
+ assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
+ shows "(\<lambda>x. g (f x)) \<in> measurable M L"
+proof -
+ have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
+ using measurable_space[OF f] by auto
+ with measurable_space[OF f] measurable_space[OF g] show ?thesis
+ by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
+ simp del: vimage_Int simp add: measurable_def)
+qed
+
+lemma measurable_comp:
+ "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
+ using measurable_compose[of f M N g L] by (simp add: comp_def)
+
+lemma measurable_const:
+ "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
+ by (auto simp add: measurable_def)
+
+lemma measurable_If:
+ assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
+ assumes P: "{x\<in>space M. P x} \<in> sets M"
+ shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
+ unfolding measurable_def
+proof safe
+ fix x assume "x \<in> space M"
+ thus "(if P x then f x else g x) \<in> space M'"
+ using measure unfolding measurable_def by auto
+next
+ fix A assume "A \<in> sets M'"
+ hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
+ ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
+ ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
+ using measure unfolding measurable_def by (auto split: split_if_asm)
+ show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
+ using `A \<in> sets M'` measure P unfolding * measurable_def
+ by (auto intro!: sets.Un)
+qed
+
+lemma measurable_If_set:
+ assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
+ assumes P: "A \<inter> space M \<in> sets M"
+ shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
+proof (rule measurable_If[OF measure])
+ have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
+ thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
+qed
+
+lemma measurable_ident: "id \<in> measurable M M"
+ by (auto simp add: measurable_def)
+
+lemma measurable_ident_sets:
+ assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
+ using measurable_ident[of M]
+ unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
+
+lemma sets_Least:
+ assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
+ shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
+proof -
+ { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
+ proof cases
+ assume i: "(LEAST j. False) = i"
+ have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
+ {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
+ by (simp add: set_eq_iff, safe)
+ (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
+ with meas show ?thesis
+ by (auto intro!: sets.Int)
+ next
+ assume i: "(LEAST j. False) \<noteq> i"
+ then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
+ {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
+ proof (simp add: set_eq_iff, safe)
+ fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
+ have "\<exists>j. P j x"
+ by (rule ccontr) (insert neq, auto)
+ then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
+ qed (auto dest: Least_le intro!: Least_equality)
+ with meas show ?thesis
+ by auto
+ qed }
+ then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
+ by (intro sets.countable_UN) auto
+ moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
+ (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma measurable_strong:
+ fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
+ assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
+ and t: "f ` (space a) \<subseteq> t"
+ and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
+ shows "(g o f) \<in> measurable a c"
+proof -
+ have fab: "f \<in> (space a -> space b)"
+ and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
+ by (auto simp add: measurable_def)
+ have eq: "\<And>y. (g \<circ> f) -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
+ by force
+ show ?thesis
+ apply (auto simp add: measurable_def vimage_comp)
+ apply (metis funcset_mem fab g)
+ apply (subst eq)
+ apply (metis ba cb)
+ done
+qed
+
+lemma measurable_mono1:
+ "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
+ measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
+ using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
+
+subsubsection {* Counting space *}
+
+definition count_space :: "'a set \<Rightarrow> 'a measure" where
+ "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
+
+lemma
+ shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
+ and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
+ using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
+ by (auto simp: count_space_def)
+
+lemma measurable_count_space_eq1[simp]:
+ "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
+ unfolding measurable_def by simp
+
+lemma measurable_count_space_eq2:
+ assumes "finite A"
+ shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
+proof -
+ { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
+ with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
+ by (auto dest: finite_subset)
+ moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
+ ultimately have "f -` X \<inter> space M \<in> sets M"
+ using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) }
+ then show ?thesis
+ unfolding measurable_def by auto
+qed
+
+lemma measurable_count_space_eq2_countable:
+ fixes f :: "'a => 'c::countable"
+ shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
+proof -
+ { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
+ assume *: "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
+ have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
+ by auto
+ also have "\<dots> \<in> sets M"
+ using * `X \<subseteq> A` by (intro sets.countable_UN) auto
+ finally have "f -` X \<inter> space M \<in> sets M" . }
+ then show ?thesis
+ unfolding measurable_def by auto
+qed
+
+lemma measurable_compose_countable:
+ assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
+ shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
+ unfolding measurable_def
+proof safe
+ fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
+ using f[THEN measurable_space] g[THEN measurable_space] by auto
+next
+ fix A assume A: "A \<in> sets N"
+ have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))"
+ by auto
+ also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]
+ by (auto intro!: sets.countable_UN measurable_sets)
+ finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
+qed
+
+lemma measurable_count_space_const:
+ "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
+ by (simp add: measurable_const)
+
+lemma measurable_count_space:
+ "f \<in> measurable (count_space A) (count_space UNIV)"
+ by simp
+
+lemma measurable_compose_rev:
+ assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
+ shows "(\<lambda>x. f (g x)) \<in> measurable M N"
+ using measurable_compose[OF g f] .
+
+lemma measurable_count_space_eq_countable:
+ assumes "countable A"
+ shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
+proof -
+ { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
+ with `countable A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
+ by (auto dest: countable_subset)
+ moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
+ ultimately have "f -` X \<inter> space M \<in> sets M"
+ using `X \<subseteq> A` by (auto intro!: sets.countable_UN' simp del: UN_simps) }
+ then show ?thesis
+ unfolding measurable_def by auto
+qed
+
+subsubsection {* Extend measure *}
+
+definition "extend_measure \<Omega> I G \<mu> =
+ (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
+ then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>')
+ else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
+
+lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
+ unfolding extend_measure_def by simp
+
+lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
+ unfolding extend_measure_def by simp
+
+lemma emeasure_extend_measure:
+ assumes M: "M = extend_measure \<Omega> I G \<mu>"
+ and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
+ and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
+ and "i \<in> I"
+ shows "emeasure M (G i) = \<mu> i"
+proof cases
+ assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
+ with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
+ by (simp add: extend_measure_def)
+ from measure_space_0[OF ms(1)] ms `i\<in>I`
+ have "emeasure M (G i) = 0"
+ by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
+ with `i\<in>I` * show ?thesis
+ by simp
+next
+ def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
+ assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
+ moreover
+ have "measure_space (space M) (sets M) \<mu>'"
+ using ms unfolding measure_space_def by auto default
+ with ms eq have "\<exists>\<mu>'. P \<mu>'"
+ unfolding P_def
+ by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
+ ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
+ by (simp add: M extend_measure_def P_def[symmetric])
+
+ from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
+ show "emeasure M (G i) = \<mu> i"
+ proof (subst emeasure_measure_of[OF M_eq])
+ have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
+ using M_eq ms by (auto simp: sets_extend_measure)
+ then show "G i \<in> sets M" using `i \<in> I` by auto
+ show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
+ using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
+ qed fact
+qed
+
+lemma emeasure_extend_measure_Pair:
+ assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
+ and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
+ and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
+ and "I i j"
+ shows "emeasure M (G i j) = \<mu> i j"
+ using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
+ by (auto simp: subset_eq)
+
+subsubsection {* Sigma algebra generated by function preimages *}
+
+definition
+ "vimage_algebra M S X = sigma S ((\<lambda>A. X -` A \<inter> S) ` sets M)"
+
+lemma sigma_algebra_preimages:
+ fixes f :: "'x \<Rightarrow> 'a"
+ assumes "f \<in> S \<rightarrow> space M"
+ shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
+ (is "sigma_algebra _ (?F ` sets M)")
+proof (simp add: sigma_algebra_iff2, safe)
+ show "{} \<in> ?F ` sets M" by blast
+next
+ fix A assume "A \<in> sets M"
+ moreover have "S - ?F A = ?F (space M - A)"
+ using assms by auto
+ ultimately show "S - ?F A \<in> ?F ` sets M"
+ by blast
+next
+ fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
+ have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
+ proof safe
+ fix i
+ have "A i \<in> ?F ` M" using * by auto
+ then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
+ qed
+ from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
+ by auto
+ then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
+ then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
+qed
+
+lemma sets_vimage_algebra[simp]:
+ "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
+ using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]
+ by (simp add: vimage_algebra_def)
+
+lemma space_vimage_algebra[simp]:
+ "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
+ using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]
+ by (simp add: vimage_algebra_def)
+
+lemma in_vimage_algebra[simp]:
+ "f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
+ by (simp add: image_iff)
+
+lemma measurable_vimage_algebra:
+ fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
+ shows "f \<in> measurable (vimage_algebra M S f) M"
+ unfolding measurable_def using assms by force
+
+lemma measurable_vimage:
+ fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
+ assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
+ shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
+proof -
+ note measurable_vimage_algebra[OF assms(2)]
+ from measurable_comp[OF this assms(1)]
+ show ?thesis by (simp add: comp_def)
+qed
+
+lemma sigma_sets_vimage:
+ assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
+ shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
+proof (intro set_eqI iffI)
+ let ?F = "\<lambda>X. f -` X \<inter> S'"
+ fix X assume "X \<in> sigma_sets S' (?F ` A)"
+ then show "X \<in> ?F ` sigma_sets S A"
+ proof induct
+ case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
+ by auto
+ then show ?case by auto
+ next
+ case Empty then show ?case
+ by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
+ next
+ case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
+ by auto
+ then have "S - X' \<in> sigma_sets S A"
+ by (auto intro!: sigma_sets.Compl)
+ then show ?case
+ using X assms by (auto intro!: image_eqI[where x="S - X'"])
+ next
+ case (Union F)
+ then have "\<forall>i. \<exists>F'. F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
+ by (auto simp: image_iff Bex_def)
+ from choice[OF this] obtain F' where
+ "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
+ by auto
+ then show ?case
+ by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
+ qed
+next
+ let ?F = "\<lambda>X. f -` X \<inter> S'"
+ fix X assume "X \<in> ?F ` sigma_sets S A"
+ then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
+ then show "X \<in> sigma_sets S' (?F ` A)"
+ proof (induct arbitrary: X)
+ case Empty then show ?case by (auto intro: sigma_sets.Empty)
+ next
+ case (Compl X')
+ have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
+ apply (rule sigma_sets.Compl)
+ using assms by (auto intro!: Compl.hyps simp: Compl.prems)
+ also have "S' - (S' - X) = X"
+ using assms Compl by auto
+ finally show ?case .
+ next
+ case (Union F)
+ have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
+ by (intro sigma_sets.Union Union.hyps) simp
+ also have "(\<Union>i. f -` F i \<inter> S') = X"
+ using assms Union by auto
+ finally show ?case .
+ qed auto
+qed
+
+subsubsection {* Restricted Space Sigma Algebra *}
+
+definition "restrict_space M \<Omega> = measure_of \<Omega> ((op \<inter> \<Omega>) ` sets M) (\<lambda>A. emeasure M (A \<inter> \<Omega>))"
+
+lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega>"
+ unfolding restrict_space_def by (subst space_measure_of) auto
+
+lemma sets_restrict_space: "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
+ using sigma_sets_vimage[of "\<lambda>x. x" \<Omega> "space M" "sets M"]
+ unfolding restrict_space_def
+ by (subst sets_measure_of) (auto simp: sets.sigma_sets_eq Int_commute[of _ \<Omega>] sets.space_closed)
+
+lemma sets_restrict_space_iff:
+ "\<Omega> \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
+ by (subst sets_restrict_space) (auto dest: sets.sets_into_space)
+
+lemma measurable_restrict_space1:
+ assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> measurable M N" shows "f \<in> measurable (restrict_space M \<Omega>) N"
+ unfolding measurable_def
+proof (intro CollectI conjI ballI)
+ show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"
+ using measurable_space[OF f] sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
+
+ fix A assume "A \<in> sets N"
+ have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> \<Omega>"
+ using sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
+ also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
+ unfolding sets_restrict_space_iff[OF \<Omega>]
+ using measurable_sets[OF f `A \<in> sets N`] \<Omega> by blast
+ finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
+qed
+
+lemma measurable_restrict_space2:
+ "\<Omega> \<in> sets N \<Longrightarrow> f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"
+ by (simp add: measurable_def space_restrict_space sets_restrict_space_iff)
+
end