author | hoelzl |
Wed, 01 Dec 2010 19:20:30 +0100 | |
changeset 40859 | de0b30e6c2d2 |
parent 39098 | 21e9bd6cf0a8 |
child 40871 | 688f6ff859e1 |
permissions | -rw-r--r-- |
35833 | 1 |
theory Product_Measure |
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imports Lebesgue_Integration |
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begin |
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lemma in_sigma[intro, simp]: "A \<in> sets M \<Longrightarrow> A \<in> sets (sigma M)" |
6 |
unfolding sigma_def by (auto intro!: sigma_sets.Basic) |
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lemma (in sigma_algebra) sigma_eq[simp]: "sigma M = M" |
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unfolding sigma_def sigma_sets_eq by simp |
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changed definition of dynkin; replaces proofs by metis calles
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parents:
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lemma vimage_algebra_sigma: |
12 |
assumes E: "sets E \<subseteq> Pow (space E)" |
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and f: "f \<in> space F \<rightarrow> space E" |
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and "\<And>A. A \<in> sets F \<Longrightarrow> A \<in> (\<lambda>X. f -` X \<inter> space F) ` sets E" |
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and "\<And>A. A \<in> sets E \<Longrightarrow> f -` A \<inter> space F \<in> sets F" |
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shows "sigma_algebra.vimage_algebra (sigma E) (space F) f = sigma F" |
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proof - |
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interpret sigma_algebra "sigma E" |
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using assms by (intro sigma_algebra_sigma) auto |
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have eq: "sets F = (\<lambda>X. f -` X \<inter> space F) ` sets E" |
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using assms by auto |
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show "vimage_algebra (space F) f = sigma F" |
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unfolding vimage_algebra_def using assms |
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by (simp add: sigma_def eq sigma_sets_vimage) |
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qed |
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lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)" |
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by auto |
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lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x -` (A \<times> B) = (if x \<in> A then B else {})" |
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by auto |
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lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
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by auto |
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lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))" |
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by (cases x) simp |
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abbreviation |
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"Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A" |
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abbreviation |
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funcset_extensional :: "['a set, 'b set] => ('a => 'b) set" |
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(infixr "->\<^isub>E" 60) where |
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"A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)" |
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notation (xsymbols) |
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funcset_extensional (infixr "\<rightarrow>\<^isub>E" 60) |
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lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}" |
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by safe (auto simp add: extensional_def fun_eq_iff) |
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lemma extensional_insert[intro, simp]: |
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assumes "a \<in> extensional (insert i I)" |
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shows "a(i := b) \<in> extensional (insert i I)" |
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using assms unfolding extensional_def by auto |
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lemma extensional_Int[simp]: |
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"extensional I \<inter> extensional I' = extensional (I \<inter> I')" |
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unfolding extensional_def by auto |
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definition |
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"merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)" |
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lemma merge_apply[simp]: |
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"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i" |
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"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i" |
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"J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i" |
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"J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i" |
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"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined" |
|
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unfolding merge_def by auto |
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lemma merge_commute: |
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"I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x" |
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by (auto simp: merge_def intro!: ext) |
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lemma Pi_cancel_merge_range[simp]: |
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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A" |
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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A" |
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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A" |
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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A" |
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by (auto simp: Pi_def) |
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lemma Pi_cancel_merge[simp]: |
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"I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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"J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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"I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
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"J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
|
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by (auto simp: Pi_def) |
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lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)" |
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by (auto simp: extensional_def) |
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93 |
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lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A" |
|
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by (auto simp: restrict_def Pi_def) |
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lemma restrict_merge[simp]: |
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"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I" |
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"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J" |
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"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I" |
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"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J" |
|
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by (auto simp: restrict_def intro!: ext) |
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lemma extensional_insert_undefined[intro, simp]: |
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assumes "a \<in> extensional (insert i I)" |
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shows "a(i := undefined) \<in> extensional I" |
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using assms unfolding extensional_def by auto |
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lemma extensional_insert_cancel[intro, simp]: |
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assumes "a \<in> extensional I" |
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shows "a \<in> extensional (insert i I)" |
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using assms unfolding extensional_def by auto |
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lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)" |
|
115 |
by auto |
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lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B" |
|
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by (auto simp: Pi_def) |
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lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i" |
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by (auto simp: Pi_def) |
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39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
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lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X" |
124 |
by (auto simp: Pi_def) |
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lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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by (auto simp: Pi_def) |
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section "Binary products" |
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definition |
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"pair_algebra A B = \<lparr> space = space A \<times> space B, |
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sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<rparr>" |
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locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2 |
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for M1 M2 |
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137 |
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abbreviation (in pair_sigma_algebra) |
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"E \<equiv> pair_algebra M1 M2" |
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abbreviation (in pair_sigma_algebra) |
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"P \<equiv> sigma E" |
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143 |
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sublocale pair_sigma_algebra \<subseteq> sigma_algebra P |
|
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using M1.sets_into_space M2.sets_into_space |
|
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by (force simp: pair_algebra_def intro!: sigma_algebra_sigma) |
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lemma pair_algebraI[intro, simp]: |
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"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_algebra A B)" |
|
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by (auto simp add: pair_algebra_def) |
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151 |
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152 |
lemma space_pair_algebra: |
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"space (pair_algebra A B) = space A \<times> space B" |
|
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by (simp add: pair_algebra_def) |
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156 |
lemma pair_algebra_Int_snd: |
|
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assumes "sets S1 \<subseteq> Pow (space S1)" |
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shows "pair_algebra S1 (algebra.restricted_space S2 A) = |
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algebra.restricted_space (pair_algebra S1 S2) (space S1 \<times> A)" |
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(is "?L = ?R") |
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proof (intro algebra.equality set_eqI iffI) |
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fix X assume "X \<in> sets ?L" |
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then obtain A1 A2 where X: "X = A1 \<times> (A \<inter> A2)" and "A1 \<in> sets S1" "A2 \<in> sets S2" |
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by (auto simp: pair_algebra_def) |
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then show "X \<in> sets ?R" unfolding pair_algebra_def |
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using assms apply simp by (intro image_eqI[of _ _ "A1 \<times> A2"]) auto |
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next |
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fix X assume "X \<in> sets ?R" |
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then obtain A1 A2 where "X = space S1 \<times> A \<inter> A1 \<times> A2" "A1 \<in> sets S1" "A2 \<in> sets S2" |
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by (auto simp: pair_algebra_def) |
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moreover then have "X = A1 \<times> (A \<inter> A2)" |
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using assms by auto |
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ultimately show "X \<in> sets ?L" |
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unfolding pair_algebra_def by auto |
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qed (auto simp add: pair_algebra_def) |
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lemma (in pair_sigma_algebra) |
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shows measurable_fst[intro!, simp]: |
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"fst \<in> measurable P M1" (is ?fst) |
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and measurable_snd[intro!, simp]: |
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"snd \<in> measurable P M2" (is ?snd) |
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39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
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proof - |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
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{ fix X assume "X \<in> sets M1" |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
184 |
then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2" |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
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apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"]) |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
186 |
using M1.sets_into_space by force+ } |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
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187 |
moreover |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
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188 |
{ fix X assume "X \<in> sets M2" |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
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then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2" |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
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190 |
apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X]) |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
191 |
using M2.sets_into_space by force+ } |
40859 | 192 |
ultimately have "?fst \<and> ?snd" |
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by (fastsimp simp: measurable_def sets_sigma space_pair_algebra |
|
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intro!: sigma_sets.Basic) |
|
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then show ?fst ?snd by auto |
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qed |
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lemma (in pair_sigma_algebra) measurable_pair: |
|
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assumes "sigma_algebra M" |
|
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shows "f \<in> measurable M P \<longleftrightarrow> |
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(fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2" |
|
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proof - |
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interpret M: sigma_algebra M by fact |
|
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from assms show ?thesis |
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proof (safe intro!: measurable_comp[where b=P]) |
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assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2" |
|
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show "f \<in> measurable M P" |
|
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proof (rule M.measurable_sigma) |
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show "sets (pair_algebra M1 M2) \<subseteq> Pow (space E)" |
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unfolding pair_algebra_def using M1.sets_into_space M2.sets_into_space by auto |
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show "f \<in> space M \<rightarrow> space E" |
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using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma space_pair_algebra) |
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fix A assume "A \<in> sets E" |
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then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C" |
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unfolding pair_algebra_def by auto |
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moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M" |
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using f `B \<in> sets M1` unfolding measurable_def by auto |
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moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M" |
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using s `C \<in> sets M2` unfolding measurable_def by auto |
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moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)" |
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unfolding `A = B \<times> C` by (auto simp: vimage_Times) |
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ultimately show "f -` A \<inter> space M \<in> sets M" by auto |
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qed |
|
224 |
qed |
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qed |
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lemma (in pair_sigma_algebra) measurable_prod_sigma: |
|
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assumes "sigma_algebra M" |
|
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assumes 1: "(fst \<circ> f) \<in> measurable M M1" and 2: "(snd \<circ> f) \<in> measurable M M2" |
|
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shows "f \<in> measurable M P" |
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proof - |
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interpret M: sigma_algebra M by fact |
|
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from 1 have fn1: "fst \<circ> f \<in> space M \<rightarrow> space M1" |
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and q1: "\<forall>y\<in>sets M1. (fst \<circ> f) -` y \<inter> space M \<in> sets M" |
|
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by (auto simp add: measurable_def) |
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from 2 have fn2: "snd \<circ> f \<in> space M \<rightarrow> space M2" |
|
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and q2: "\<forall>y\<in>sets M2. (snd \<circ> f) -` y \<inter> space M \<in> sets M" |
|
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by (auto simp add: measurable_def) |
|
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show ?thesis |
|
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proof (rule M.measurable_sigma) |
|
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show "sets E \<subseteq> Pow (space E)" |
|
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using M1.space_closed M2.space_closed |
|
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by (auto simp add: sigma_algebra_iff pair_algebra_def) |
|
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next |
|
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show "f \<in> space M \<rightarrow> space E" |
|
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by (simp add: space_pair_algebra) (rule prod_final [OF fn1 fn2]) |
|
247 |
next |
|
248 |
fix z |
|
249 |
assume z: "z \<in> sets E" |
|
250 |
thus "f -` z \<inter> space M \<in> sets M" |
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proof (auto simp add: pair_algebra_def vimage_Times) |
|
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fix x y |
|
253 |
assume x: "x \<in> sets M1" and y: "y \<in> sets M2" |
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254 |
have "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M = |
|
255 |
((fst \<circ> f) -` x \<inter> space M) \<inter> ((snd \<circ> f) -` y \<inter> space M)" |
|
256 |
by blast |
|
257 |
also have "... \<in> sets M" using x y q1 q2 |
|
258 |
by blast |
|
259 |
finally show "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M \<in> sets M" . |
|
260 |
qed |
|
261 |
qed |
|
262 |
qed |
|
263 |
||
264 |
lemma pair_algebraE: |
|
265 |
assumes "X \<in> sets (pair_algebra M1 M2)" |
|
266 |
obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2" |
|
267 |
using assms unfolding pair_algebra_def by auto |
|
268 |
||
269 |
lemma (in pair_sigma_algebra) pair_algebra_swap: |
|
270 |
"(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_algebra M2 M1)" |
|
271 |
proof (safe elim!: pair_algebraE) |
|
272 |
fix A B assume "A \<in> sets M1" "B \<in> sets M2" |
|
273 |
moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A" |
|
274 |
using M1.sets_into_space M2.sets_into_space by auto |
|
275 |
ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_algebra M2 M1)" |
|
276 |
by (auto intro: pair_algebraI) |
|
277 |
next |
|
278 |
fix A B assume "A \<in> sets M1" "B \<in> sets M2" |
|
279 |
then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E" |
|
280 |
using M1.sets_into_space M2.sets_into_space |
|
281 |
by (auto intro!: image_eqI[where x="A \<times> B"] pair_algebraI) |
|
282 |
qed |
|
283 |
||
284 |
lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap: |
|
285 |
assumes Q: "Q \<in> sets P" |
|
286 |
shows "(\<lambda>(x,y). (y, x)) ` Q \<in> sets (sigma (pair_algebra M2 M1))" (is "_ \<in> sets ?Q") |
|
287 |
proof - |
|
288 |
have *: "(\<lambda>(x,y). (y, x)) \<in> space M2 \<times> space M1 \<rightarrow> (space M1 \<times> space M2)" |
|
289 |
"sets (pair_algebra M1 M2) \<subseteq> Pow (space M1 \<times> space M2)" |
|
290 |
using M1.sets_into_space M2.sets_into_space by (auto elim!: pair_algebraE) |
|
291 |
from Q sets_into_space show ?thesis |
|
292 |
by (auto intro!: image_eqI[where x=Q] |
|
293 |
simp: pair_algebra_swap[symmetric] sets_sigma |
|
294 |
sigma_sets_vimage[OF *] space_pair_algebra) |
|
295 |
qed |
|
296 |
||
297 |
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable: |
|
298 |
shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (sigma (pair_algebra M2 M1))" |
|
299 |
(is "?f \<in> measurable ?P ?Q") |
|
300 |
unfolding measurable_def |
|
301 |
proof (intro CollectI conjI Pi_I ballI) |
|
302 |
fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q" |
|
303 |
unfolding pair_algebra_def by auto |
|
304 |
next |
|
305 |
fix A assume "A \<in> sets ?Q" |
|
306 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
307 |
have "?f -` A \<inter> space ?P = (\<lambda>(x,y). (y, x)) ` A" |
|
308 |
using Q.sets_into_space `A \<in> sets ?Q` by (auto simp: pair_algebra_def) |
|
309 |
with Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets ?Q`] |
|
310 |
show "?f -` A \<inter> space ?P \<in> sets ?P" by simp |
|
311 |
qed |
|
312 |
||
313 |
lemma (in pair_sigma_algebra) measurable_cut_fst: |
|
314 |
assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2" |
|
315 |
proof - |
|
316 |
let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}" |
|
317 |
let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>" |
|
318 |
interpret Q: sigma_algebra ?Q |
|
319 |
proof qed (auto simp: vimage_UN vimage_Diff space_pair_algebra) |
|
320 |
have "sets E \<subseteq> sets ?Q" |
|
321 |
using M1.sets_into_space M2.sets_into_space |
|
322 |
by (auto simp: pair_algebra_def space_pair_algebra) |
|
323 |
then have "sets P \<subseteq> sets ?Q" |
|
324 |
by (subst pair_algebra_def, intro Q.sets_sigma_subset) |
|
325 |
(simp_all add: pair_algebra_def) |
|
326 |
with assms show ?thesis by auto |
|
327 |
qed |
|
328 |
||
329 |
lemma (in pair_sigma_algebra) measurable_cut_snd: |
|
330 |
assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1") |
|
331 |
proof - |
|
332 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
333 |
have "Pair y -` (\<lambda>(x, y). (y, x)) ` Q = (\<lambda>x. (x, y)) -` Q" by auto |
|
334 |
with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y] |
|
335 |
show ?thesis by simp |
|
336 |
qed |
|
337 |
||
338 |
lemma (in pair_sigma_algebra) measurable_pair_image_snd: |
|
339 |
assumes m: "f \<in> measurable P M" and "x \<in> space M1" |
|
340 |
shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M" |
|
341 |
unfolding measurable_def |
|
342 |
proof (intro CollectI conjI Pi_I ballI) |
|
343 |
fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1` |
|
344 |
show "f (x, y) \<in> space M" unfolding measurable_def pair_algebra_def by auto |
|
345 |
next |
|
346 |
fix A assume "A \<in> sets M" |
|
347 |
then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _") |
|
348 |
using `f \<in> measurable P M` |
|
349 |
by (intro measurable_cut_fst) (auto simp: measurable_def) |
|
350 |
also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2" |
|
351 |
using `x \<in> space M1` by (auto simp: pair_algebra_def) |
|
352 |
finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" . |
|
353 |
qed |
|
354 |
||
355 |
lemma (in pair_sigma_algebra) measurable_pair_image_fst: |
|
356 |
assumes m: "f \<in> measurable P M" and "y \<in> space M2" |
|
357 |
shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M" |
|
358 |
proof - |
|
359 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
360 |
from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`, |
|
361 |
OF Q.pair_sigma_algebra_swap_measurable m] |
|
362 |
show ?thesis by simp |
|
363 |
qed |
|
364 |
||
365 |
lemma (in pair_sigma_algebra) Int_stable_pair_algebra: "Int_stable E" |
|
366 |
unfolding Int_stable_def |
|
367 |
proof (intro ballI) |
|
368 |
fix A B assume "A \<in> sets E" "B \<in> sets E" |
|
369 |
then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2" |
|
370 |
"A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2" |
|
371 |
unfolding pair_algebra_def by auto |
|
372 |
then show "A \<inter> B \<in> sets E" |
|
373 |
by (auto simp add: times_Int_times pair_algebra_def) |
|
374 |
qed |
|
375 |
||
376 |
lemma finite_measure_cut_measurable: |
|
377 |
fixes M1 :: "'a algebra" and M2 :: "'b algebra" |
|
378 |
assumes "sigma_finite_measure M1 \<mu>1" "finite_measure M2 \<mu>2" |
|
379 |
assumes "Q \<in> sets (sigma (pair_algebra M1 M2))" |
|
380 |
shows "(\<lambda>x. \<mu>2 (Pair x -` Q)) \<in> borel_measurable M1" |
|
381 |
(is "?s Q \<in> _") |
|
382 |
proof - |
|
383 |
interpret M1: sigma_finite_measure M1 \<mu>1 by fact |
|
384 |
interpret M2: finite_measure M2 \<mu>2 by fact |
|
385 |
interpret pair_sigma_algebra M1 M2 by default |
|
386 |
have [intro]: "sigma_algebra M1" by fact |
|
387 |
have [intro]: "sigma_algebra M2" by fact |
|
388 |
let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1} \<rparr>" |
|
389 |
note space_pair_algebra[simp] |
|
390 |
interpret dynkin_system ?D |
|
391 |
proof (intro dynkin_systemI) |
|
392 |
fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D" |
|
393 |
using sets_into_space by simp |
|
394 |
next |
|
395 |
from top show "space ?D \<in> sets ?D" |
|
396 |
by (auto simp add: if_distrib intro!: M1.measurable_If) |
|
397 |
next |
|
398 |
fix A assume "A \<in> sets ?D" |
|
399 |
with sets_into_space have "\<And>x. \<mu>2 (Pair x -` (space M1 \<times> space M2 - A)) = |
|
400 |
(if x \<in> space M1 then \<mu>2 (space M2) - ?s A x else 0)" |
|
401 |
by (auto intro!: M2.finite_measure_compl measurable_cut_fst |
|
402 |
simp: vimage_Diff) |
|
403 |
with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D" |
|
404 |
by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pinfreal_diff) |
|
405 |
next |
|
406 |
fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D" |
|
407 |
moreover then have "\<And>x. \<mu>2 (\<Union>i. Pair x -` F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)" |
|
408 |
by (intro M2.measure_countably_additive[symmetric]) |
|
409 |
(auto intro!: measurable_cut_fst simp: disjoint_family_on_def) |
|
410 |
ultimately show "(\<Union>i. F i) \<in> sets ?D" |
|
411 |
by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf) |
|
412 |
qed |
|
413 |
have "P = ?D" |
|
414 |
proof (intro dynkin_lemma) |
|
415 |
show "Int_stable E" by (rule Int_stable_pair_algebra) |
|
416 |
from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A" |
|
417 |
by auto |
|
418 |
then show "sets E \<subseteq> sets ?D" |
|
419 |
by (auto simp: pair_algebra_def sets_sigma if_distrib |
|
420 |
intro: sigma_sets.Basic intro!: M1.measurable_If) |
|
421 |
qed auto |
|
422 |
with `Q \<in> sets P` have "Q \<in> sets ?D" by simp |
|
423 |
then show "?s Q \<in> borel_measurable M1" by simp |
|
424 |
qed |
|
425 |
||
426 |
subsection {* Binary products of $\sigma$-finite measure spaces *} |
|
427 |
||
428 |
locale pair_sigma_finite = M1: sigma_finite_measure M1 \<mu>1 + M2: sigma_finite_measure M2 \<mu>2 |
|
429 |
for M1 \<mu>1 M2 \<mu>2 |
|
430 |
||
431 |
sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2 |
|
432 |
by default |
|
433 |
||
434 |
lemma (in pair_sigma_finite) measure_cut_measurable_fst: |
|
435 |
assumes "Q \<in> sets P" shows "(\<lambda>x. \<mu>2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _") |
|
436 |
proof - |
|
437 |
have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+ |
|
438 |
have M1: "sigma_finite_measure M1 \<mu>1" by default |
|
439 |
||
440 |
from M2.disjoint_sigma_finite guess F .. note F = this |
|
441 |
let "?C x i" = "F i \<inter> Pair x -` Q" |
|
442 |
{ fix i |
|
443 |
let ?R = "M2.restricted_space (F i)" |
|
444 |
have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i" |
|
445 |
using F M2.sets_into_space by auto |
|
446 |
have "(\<lambda>x. \<mu>2 (Pair x -` (space M1 \<times> F i \<inter> Q))) \<in> borel_measurable M1" |
|
447 |
proof (intro finite_measure_cut_measurable[OF M1]) |
|
448 |
show "finite_measure (M2.restricted_space (F i)) \<mu>2" |
|
449 |
using F by (intro M2.restricted_to_finite_measure) auto |
|
450 |
have "space M1 \<times> F i \<in> sets P" |
|
451 |
using M1.top F by blast |
|
452 |
from sigma_sets_Int[symmetric, |
|
453 |
OF this[unfolded pair_sigma_algebra_def sets_sigma]] |
|
454 |
show "(space M1 \<times> F i) \<inter> Q \<in> sets (sigma (pair_algebra M1 ?R))" |
|
455 |
using `Q \<in> sets P` |
|
456 |
using pair_algebra_Int_snd[OF M1.space_closed, of "F i" M2] |
|
457 |
by (auto simp: pair_algebra_def sets_sigma) |
|
458 |
qed |
|
459 |
moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i" |
|
460 |
using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_algebra) |
|
461 |
ultimately have "(\<lambda>x. \<mu>2 (?C x i)) \<in> borel_measurable M1" |
|
462 |
by simp } |
|
463 |
moreover |
|
464 |
{ fix x |
|
465 |
have "(\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i)) = \<mu>2 (\<Union>i. ?C x i)" |
|
466 |
proof (intro M2.measure_countably_additive) |
|
467 |
show "range (?C x) \<subseteq> sets M2" |
|
468 |
using F `Q \<in> sets P` by (auto intro!: M2.Int measurable_cut_fst) |
|
469 |
have "disjoint_family F" using F by auto |
|
470 |
show "disjoint_family (?C x)" |
|
471 |
by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto |
|
472 |
qed |
|
473 |
also have "(\<Union>i. ?C x i) = Pair x -` Q" |
|
474 |
using F sets_into_space `Q \<in> sets P` |
|
475 |
by (auto simp: space_pair_algebra) |
|
476 |
finally have "\<mu>2 (Pair x -` Q) = (\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i))" |
|
477 |
by simp } |
|
478 |
ultimately show ?thesis |
|
479 |
by (auto intro!: M1.borel_measurable_psuminf) |
|
480 |
qed |
|
481 |
||
482 |
lemma (in pair_sigma_finite) measure_cut_measurable_snd: |
|
483 |
assumes "Q \<in> sets P" shows "(\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2" |
|
484 |
proof - |
|
485 |
interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default |
|
486 |
have [simp]: "\<And>y. (Pair y -` (\<lambda>(x, y). (y, x)) ` Q) = (\<lambda>x. (x, y)) -` Q" |
|
487 |
by auto |
|
488 |
note sets_pair_sigma_algebra_swap[OF assms] |
|
489 |
from Q.measure_cut_measurable_fst[OF this] |
|
490 |
show ?thesis by simp |
|
491 |
qed |
|
492 |
||
493 |
lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable: |
|
494 |
assumes "f \<in> measurable P M" |
|
495 |
shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (sigma (pair_algebra M2 M1)) M" |
|
496 |
proof - |
|
497 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
498 |
have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff) |
|
499 |
show ?thesis |
|
500 |
using Q.pair_sigma_algebra_swap_measurable assms |
|
501 |
unfolding * by (rule measurable_comp) |
|
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
502 |
qed |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
503 |
|
40859 | 504 |
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap: |
505 |
"sigma (pair_algebra M2 M1) = (vimage_algebra (space M2 \<times> space M1) (\<lambda>(x, y). (y, x)))" |
|
506 |
unfolding vimage_algebra_def |
|
507 |
apply (simp add: sets_sigma) |
|
508 |
apply (subst sigma_sets_vimage[symmetric]) |
|
509 |
apply (fastsimp simp: pair_algebra_def) |
|
510 |
using M1.sets_into_space M2.sets_into_space apply (fastsimp simp: pair_algebra_def) |
|
511 |
proof - |
|
512 |
have "(\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E |
|
513 |
= sets (pair_algebra M2 M1)" (is "?S = _") |
|
514 |
by (rule pair_algebra_swap) |
|
515 |
then show "sigma (pair_algebra M2 M1) = \<lparr>space = space M2 \<times> space M1, |
|
516 |
sets = sigma_sets (space M2 \<times> space M1) ?S\<rparr>" |
|
517 |
by (simp add: pair_algebra_def sigma_def) |
|
518 |
qed |
|
519 |
||
520 |
definition (in pair_sigma_finite) |
|
521 |
"pair_measure A = M1.positive_integral (\<lambda>x. |
|
522 |
M2.positive_integral (\<lambda>y. indicator A (x, y)))" |
|
523 |
||
524 |
lemma (in pair_sigma_finite) pair_measure_alt: |
|
525 |
assumes "A \<in> sets P" |
|
526 |
shows "pair_measure A = M1.positive_integral (\<lambda>x. \<mu>2 (Pair x -` A))" |
|
527 |
unfolding pair_measure_def |
|
528 |
proof (rule M1.positive_integral_cong) |
|
529 |
fix x assume "x \<in> space M1" |
|
530 |
have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: pinfreal)" |
|
531 |
unfolding indicator_def by auto |
|
532 |
show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x -` A)" |
|
533 |
unfolding * |
|
534 |
apply (subst M2.positive_integral_indicator) |
|
535 |
apply (rule measurable_cut_fst[OF assms]) |
|
536 |
by simp |
|
537 |
qed |
|
538 |
||
539 |
lemma (in pair_sigma_finite) pair_measure_times: |
|
540 |
assumes A: "A \<in> sets M1" and "B \<in> sets M2" |
|
541 |
shows "pair_measure (A \<times> B) = \<mu>1 A * \<mu>2 B" |
|
542 |
proof - |
|
543 |
from assms have "pair_measure (A \<times> B) = |
|
544 |
M1.positive_integral (\<lambda>x. \<mu>2 B * indicator A x)" |
|
545 |
by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt) |
|
546 |
with assms show ?thesis |
|
547 |
by (simp add: M1.positive_integral_cmult_indicator ac_simps) |
|
548 |
qed |
|
549 |
||
550 |
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra: |
|
551 |
"\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and> |
|
552 |
(\<forall>i. pair_measure (F i) \<noteq> \<omega>)" |
|
553 |
proof - |
|
554 |
obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where |
|
555 |
F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. \<mu>1 (F1 i) \<noteq> \<omega>" and |
|
556 |
F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. \<mu>2 (F2 i) \<noteq> \<omega>" |
|
557 |
using M1.sigma_finite_up M2.sigma_finite_up by auto |
|
558 |
then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" |
|
559 |
unfolding isoton_def by auto |
|
560 |
let ?F = "\<lambda>i. F1 i \<times> F2 i" |
|
561 |
show ?thesis unfolding isoton_def space_pair_algebra |
|
562 |
proof (intro exI[of _ ?F] conjI allI) |
|
563 |
show "range ?F \<subseteq> sets E" using F1 F2 |
|
564 |
by (fastsimp intro!: pair_algebraI) |
|
565 |
next |
|
566 |
have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)" |
|
567 |
proof (intro subsetI) |
|
568 |
fix x assume "x \<in> space M1 \<times> space M2" |
|
569 |
then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j" |
|
570 |
by (auto simp: space) |
|
571 |
then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)" |
|
572 |
using `F1 \<up> space M1` `F2 \<up> space M2` |
|
573 |
by (auto simp: max_def dest: isoton_mono_le) |
|
574 |
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" |
|
575 |
by (intro SigmaI) (auto simp add: min_max.sup_commute) |
|
576 |
then show "x \<in> (\<Union>i. ?F i)" by auto |
|
577 |
qed |
|
578 |
then show "(\<Union>i. ?F i) = space M1 \<times> space M2" |
|
579 |
using space by (auto simp: space) |
|
580 |
next |
|
581 |
fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)" |
|
582 |
using `F1 \<up> space M1` `F2 \<up> space M2` unfolding isoton_def |
|
583 |
by auto |
|
584 |
next |
|
585 |
fix i |
|
586 |
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto |
|
587 |
with F1 F2 show "pair_measure (F1 i \<times> F2 i) \<noteq> \<omega>" |
|
588 |
by (simp add: pair_measure_times) |
|
589 |
qed |
|
590 |
qed |
|
591 |
||
592 |
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure |
|
593 |
proof |
|
594 |
show "pair_measure {} = 0" |
|
595 |
unfolding pair_measure_def by auto |
|
596 |
||
597 |
show "countably_additive P pair_measure" |
|
598 |
unfolding countably_additive_def |
|
599 |
proof (intro allI impI) |
|
600 |
fix F :: "nat \<Rightarrow> ('a \<times> 'b) set" |
|
601 |
assume F: "range F \<subseteq> sets P" "disjoint_family F" |
|
602 |
from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto |
|
603 |
moreover from F have "\<And>i. (\<lambda>x. \<mu>2 (Pair x -` F i)) \<in> borel_measurable M1" |
|
604 |
by (intro measure_cut_measurable_fst) auto |
|
605 |
moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)" |
|
606 |
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto |
|
607 |
moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2" |
|
608 |
using F by (auto intro!: measurable_cut_fst) |
|
609 |
ultimately show "(\<Sum>\<^isub>\<infinity>n. pair_measure (F n)) = pair_measure (\<Union>i. F i)" |
|
610 |
by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric] |
|
611 |
M2.measure_countably_additive |
|
612 |
cong: M1.positive_integral_cong) |
|
613 |
qed |
|
614 |
||
615 |
from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this |
|
616 |
show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. pair_measure (F i) \<noteq> \<omega>)" |
|
617 |
proof (rule exI[of _ F], intro conjI) |
|
618 |
show "range F \<subseteq> sets P" using F by auto |
|
619 |
show "(\<Union>i. F i) = space P" |
|
620 |
using F by (auto simp: space_pair_algebra isoton_def) |
|
621 |
show "\<forall>i. pair_measure (F i) \<noteq> \<omega>" using F by auto |
|
622 |
qed |
|
623 |
qed |
|
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
624 |
|
40859 | 625 |
lemma (in pair_sigma_finite) pair_measure_alt2: |
626 |
assumes "A \<in> sets P" |
|
627 |
shows "pair_measure A = M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` A))" |
|
628 |
(is "_ = ?\<nu> A") |
|
629 |
proof - |
|
630 |
from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this |
|
631 |
show ?thesis |
|
632 |
proof (rule measure_unique_Int_stable[where \<nu>="?\<nu>", OF Int_stable_pair_algebra], |
|
633 |
simp_all add: pair_sigma_algebra_def[symmetric]) |
|
634 |
show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. pair_measure (F i) \<noteq> \<omega>" |
|
635 |
using F by auto |
|
636 |
show "measure_space P pair_measure" by default |
|
637 |
next |
|
638 |
show "measure_space P ?\<nu>" |
|
639 |
proof |
|
640 |
show "?\<nu> {} = 0" by auto |
|
641 |
show "countably_additive P ?\<nu>" |
|
642 |
unfolding countably_additive_def |
|
643 |
proof (intro allI impI) |
|
644 |
fix F :: "nat \<Rightarrow> ('a \<times> 'b) set" |
|
645 |
assume F: "range F \<subseteq> sets P" "disjoint_family F" |
|
646 |
from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto |
|
647 |
moreover from F have "\<And>i. (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` F i)) \<in> borel_measurable M2" |
|
648 |
by (intro measure_cut_measurable_snd) auto |
|
649 |
moreover have "\<And>y. disjoint_family (\<lambda>i. (\<lambda>x. (x, y)) -` F i)" |
|
650 |
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto |
|
651 |
moreover have "\<And>y. y \<in> space M2 \<Longrightarrow> range (\<lambda>i. (\<lambda>x. (x, y)) -` F i) \<subseteq> sets M1" |
|
652 |
using F by (auto intro!: measurable_cut_snd) |
|
653 |
ultimately show "(\<Sum>\<^isub>\<infinity>n. ?\<nu> (F n)) = ?\<nu> (\<Union>i. F i)" |
|
654 |
by (simp add: vimage_UN M2.positive_integral_psuminf[symmetric] |
|
655 |
M1.measure_countably_additive |
|
656 |
cong: M2.positive_integral_cong) |
|
657 |
qed |
|
658 |
qed |
|
659 |
next |
|
660 |
fix X assume "X \<in> sets E" |
|
661 |
then obtain A B where X: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2" |
|
662 |
unfolding pair_algebra_def by auto |
|
663 |
show "pair_measure X = ?\<nu> X" |
|
664 |
proof - |
|
665 |
from AB have "?\<nu> (A \<times> B) = |
|
666 |
M2.positive_integral (\<lambda>y. \<mu>1 A * indicator B y)" |
|
667 |
by (auto intro!: M2.positive_integral_cong) |
|
668 |
with AB show ?thesis |
|
669 |
unfolding pair_measure_times[OF AB] X |
|
670 |
by (simp add: M2.positive_integral_cmult_indicator ac_simps) |
|
671 |
qed |
|
672 |
qed fact |
|
673 |
qed |
|
674 |
||
675 |
section "Fubinis theorem" |
|
676 |
||
677 |
lemma (in pair_sigma_finite) simple_function_cut: |
|
678 |
assumes f: "simple_function f" |
|
679 |
shows "(\<lambda>x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1" |
|
680 |
and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y))) |
|
681 |
= positive_integral f" |
|
682 |
proof - |
|
683 |
have f_borel: "f \<in> borel_measurable P" |
|
684 |
using f by (rule borel_measurable_simple_function) |
|
685 |
let "?F z" = "f -` {z} \<inter> space P" |
|
686 |
let "?F' x z" = "Pair x -` ?F z" |
|
687 |
{ fix x assume "x \<in> space M1" |
|
688 |
have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y" |
|
689 |
by (auto simp: indicator_def) |
|
690 |
have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1` |
|
691 |
by (simp add: space_pair_algebra) |
|
692 |
moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel |
|
693 |
by (intro borel_measurable_vimage measurable_cut_fst) |
|
694 |
ultimately have "M2.simple_function (\<lambda> y. f (x, y))" |
|
695 |
apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _]) |
|
696 |
apply (rule simple_function_indicator_representation[OF f]) |
|
697 |
using `x \<in> space M1` by (auto simp del: space_sigma) } |
|
698 |
note M2_sf = this |
|
699 |
{ fix x assume x: "x \<in> space M1" |
|
700 |
then have "M2.positive_integral (\<lambda> y. f (x, y)) = |
|
701 |
(\<Sum>z\<in>f ` space P. z * \<mu>2 (?F' x z))" |
|
702 |
unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]] |
|
703 |
unfolding M2.simple_integral_def |
|
704 |
proof (safe intro!: setsum_mono_zero_cong_left) |
|
705 |
from f show "finite (f ` space P)" by (rule simple_functionD) |
|
706 |
next |
|
707 |
fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P" |
|
708 |
using `x \<in> space M1` by (auto simp: space_pair_algebra) |
|
709 |
next |
|
710 |
fix x' y assume "(x', y) \<in> space P" |
|
711 |
"f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2" |
|
712 |
then have *: "?F' x (f (x', y)) = {}" |
|
713 |
by (force simp: space_pair_algebra) |
|
714 |
show "f (x', y) * \<mu>2 (?F' x (f (x', y))) = 0" |
|
715 |
unfolding * by simp |
|
716 |
qed (simp add: vimage_compose[symmetric] comp_def |
|
717 |
space_pair_algebra) } |
|
718 |
note eq = this |
|
719 |
moreover have "\<And>z. ?F z \<in> sets P" |
|
720 |
by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma) |
|
721 |
moreover then have "\<And>z. (\<lambda>x. \<mu>2 (?F' x z)) \<in> borel_measurable M1" |
|
722 |
by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int) |
|
723 |
ultimately |
|
724 |
show "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1" |
|
725 |
and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y))) |
|
726 |
= positive_integral f" |
|
727 |
by (auto simp del: vimage_Int cong: measurable_cong |
|
728 |
intro!: M1.borel_measurable_pinfreal_setsum |
|
729 |
simp add: M1.positive_integral_setsum simple_integral_def |
|
730 |
M1.positive_integral_cmult |
|
731 |
M1.positive_integral_cong[OF eq] |
|
732 |
positive_integral_eq_simple_integral[OF f] |
|
733 |
pair_measure_alt[symmetric]) |
|
734 |
qed |
|
735 |
||
736 |
lemma (in pair_sigma_finite) positive_integral_fst_measurable: |
|
737 |
assumes f: "f \<in> borel_measurable P" |
|
738 |
shows "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1" |
|
739 |
(is "?C f \<in> borel_measurable M1") |
|
740 |
and "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) = |
|
741 |
positive_integral f" |
|
742 |
proof - |
|
743 |
from borel_measurable_implies_simple_function_sequence[OF f] |
|
744 |
obtain F where F: "\<And>i. simple_function (F i)" "F \<up> f" by auto |
|
745 |
then have F_borel: "\<And>i. F i \<in> borel_measurable P" |
|
746 |
and F_mono: "\<And>i x. F i x \<le> F (Suc i) x" |
|
747 |
and F_SUPR: "\<And>x. (SUP i. F i x) = f x" |
|
748 |
unfolding isoton_def le_fun_def SUPR_fun_expand |
|
749 |
by (auto intro: borel_measurable_simple_function) |
|
750 |
note sf = simple_function_cut[OF F(1)] |
|
751 |
then have "(SUP i. ?C (F i)) \<in> borel_measurable M1" |
|
752 |
using F(1) by (auto intro!: M1.borel_measurable_SUP) |
|
753 |
moreover |
|
754 |
{ fix x assume "x \<in> space M1" |
|
755 |
have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))" |
|
756 |
using `F \<up> f` unfolding isoton_fun_expand |
|
757 |
by (auto simp: isoton_def) |
|
758 |
note measurable_pair_image_snd[OF F_borel`x \<in> space M1`] |
|
759 |
from M2.positive_integral_isoton[OF isotone this] |
|
760 |
have "(SUP i. ?C (F i) x) = ?C f x" |
|
761 |
by (simp add: isoton_def) } |
|
762 |
note SUPR_C = this |
|
763 |
ultimately show "?C f \<in> borel_measurable M1" |
|
764 |
unfolding SUPR_fun_expand by (simp cong: measurable_cong) |
|
765 |
have "positive_integral (\<lambda>x. SUP i. F i x) = (SUP i. positive_integral (F i))" |
|
766 |
using F_borel F_mono |
|
767 |
by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric]) |
|
768 |
also have "(SUP i. positive_integral (F i)) = |
|
769 |
(SUP i. M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. F i (x, y))))" |
|
770 |
unfolding sf(2) by simp |
|
771 |
also have "\<dots> = M1.positive_integral (\<lambda>x. SUP i. M2.positive_integral (\<lambda>y. F i (x, y)))" |
|
772 |
by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)] |
|
773 |
M2.positive_integral_mono F_mono) |
|
774 |
also have "\<dots> = M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. SUP i. F i (x, y)))" |
|
775 |
using F_borel F_mono |
|
776 |
by (auto intro!: M2.positive_integral_monotone_convergence_SUP |
|
777 |
M1.positive_integral_cong measurable_pair_image_snd) |
|
778 |
finally show "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) = |
|
779 |
positive_integral f" |
|
780 |
unfolding F_SUPR by simp |
|
781 |
qed |
|
782 |
||
783 |
lemma (in pair_sigma_finite) positive_integral_snd_measurable: |
|
784 |
assumes f: "f \<in> borel_measurable P" |
|
785 |
shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) = |
|
786 |
positive_integral f" |
|
787 |
proof - |
|
788 |
interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default |
|
789 |
||
790 |
have s: "\<And>x y. (case (x, y) of (x, y) \<Rightarrow> f (y, x)) = f (y, x)" by simp |
|
791 |
have t: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> (y, x))) = (\<lambda>(x, y). f (y, x))" by (auto simp: fun_eq_iff) |
|
792 |
||
793 |
have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)" |
|
794 |
by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def) |
|
795 |
||
796 |
note pair_sigma_algebra_measurable[OF f] |
|
797 |
from Q.positive_integral_fst_measurable[OF this] |
|
798 |
have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) = |
|
799 |
Q.positive_integral (\<lambda>(x, y). f (y, x))" |
|
800 |
by simp |
|
801 |
also have "\<dots> = positive_integral f" |
|
802 |
unfolding positive_integral_vimage[OF bij, of f] t |
|
803 |
unfolding pair_sigma_algebra_swap[symmetric] |
|
804 |
proof (rule Q.positive_integral_cong_measure[symmetric]) |
|
805 |
fix A assume "A \<in> sets Q.P" |
|
806 |
from this Q.sets_pair_sigma_algebra_swap[OF this] |
|
807 |
show "pair_measure ((\<lambda>(x, y). (y, x)) ` A) = Q.pair_measure A" |
|
808 |
by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2] |
|
809 |
simp: pair_measure_alt Q.pair_measure_alt2) |
|
810 |
qed |
|
811 |
finally show ?thesis . |
|
812 |
qed |
|
813 |
||
814 |
lemma (in pair_sigma_finite) Fubini: |
|
815 |
assumes f: "f \<in> borel_measurable P" |
|
816 |
shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) = |
|
817 |
M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))" |
|
818 |
unfolding positive_integral_snd_measurable[OF assms] |
|
819 |
unfolding positive_integral_fst_measurable[OF assms] .. |
|
820 |
||
821 |
lemma (in pair_sigma_finite) AE_pair: |
|
822 |
assumes "almost_everywhere (\<lambda>x. Q x)" |
|
823 |
shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))" |
|
824 |
proof - |
|
825 |
obtain N where N: "N \<in> sets P" "pair_measure N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N" |
|
826 |
using assms unfolding almost_everywhere_def by auto |
|
827 |
show ?thesis |
|
828 |
proof (rule M1.AE_I) |
|
829 |
from N measure_cut_measurable_fst[OF `N \<in> sets P`] |
|
830 |
show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x -` N) \<noteq> 0} = 0" |
|
831 |
by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def) |
|
832 |
show "{x \<in> space M1. \<mu>2 (Pair x -` N) \<noteq> 0} \<in> sets M1" |
|
833 |
by (intro M1.borel_measurable_pinfreal_neq_const measure_cut_measurable_fst N) |
|
834 |
{ fix x assume "x \<in> space M1" "\<mu>2 (Pair x -` N) = 0" |
|
835 |
have "M2.almost_everywhere (\<lambda>y. Q (x, y))" |
|
836 |
proof (rule M2.AE_I) |
|
837 |
show "\<mu>2 (Pair x -` N) = 0" by fact |
|
838 |
show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N) |
|
839 |
show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N" |
|
840 |
using N `x \<in> space M1` unfolding space_sigma space_pair_algebra by auto |
|
841 |
qed } |
|
842 |
then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. \<mu>2 (Pair x -` N) \<noteq> 0}" |
|
843 |
by auto |
|
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
844 |
qed |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
845 |
qed |
35833 | 846 |
|
40859 | 847 |
section "Finite product spaces" |
848 |
||
849 |
section "Products" |
|
850 |
||
851 |
locale product_sigma_algebra = |
|
852 |
fixes M :: "'i \<Rightarrow> 'a algebra" |
|
853 |
assumes sigma_algebras: "\<And>i. sigma_algebra (M i)" |
|
854 |
||
855 |
locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" + |
|
856 |
fixes I :: "'i set" |
|
857 |
assumes finite_index: "finite I" |
|
858 |
||
859 |
syntax |
|
860 |
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PIE _:_./ _)" 10) |
|
861 |
||
862 |
syntax (xsymbols) |
|
863 |
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) |
|
864 |
||
865 |
syntax (HTML output) |
|
866 |
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) |
|
867 |
||
868 |
translations |
|
869 |
"PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)" |
|
870 |
||
35833 | 871 |
definition |
40859 | 872 |
"product_algebra M I = \<lparr> space = (\<Pi>\<^isub>E i\<in>I. space (M i)), sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)) \<rparr>" |
873 |
||
874 |
abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I" |
|
875 |
abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G" |
|
876 |
||
877 |
sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras) |
|
878 |
||
879 |
lemma (in finite_product_sigma_algebra) product_algebra_into_space: |
|
880 |
"sets G \<subseteq> Pow (space G)" |
|
881 |
using M.sets_into_space unfolding product_algebra_def |
|
882 |
by auto blast |
|
883 |
||
884 |
sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P |
|
885 |
using product_algebra_into_space by (rule sigma_algebra_sigma) |
|
886 |
||
887 |
lemma space_product_algebra[simp]: |
|
888 |
"space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))" |
|
889 |
unfolding product_algebra_def by simp |
|
890 |
||
891 |
lemma (in finite_product_sigma_algebra) P_empty: |
|
892 |
"I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>" |
|
893 |
unfolding product_algebra_def by (simp add: sigma_def) |
|
894 |
||
895 |
lemma (in finite_product_sigma_algebra) in_P[simp, intro]: |
|
896 |
"\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P" |
|
897 |
by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic) |
|
898 |
||
899 |
lemma bij_betw_prod_fold: |
|
900 |
assumes "i \<notin> I" |
|
901 |
shows "bij_betw (\<lambda>x. (x(i:=undefined), x i)) (\<Pi>\<^isub>E j\<in>insert i I. space (M j)) ((\<Pi>\<^isub>E j\<in>I. space (M j)) \<times> space (M i))" |
|
902 |
(is "bij_betw ?f ?P ?F") |
|
903 |
using `i \<notin> I` |
|
904 |
proof (unfold bij_betw_def, intro conjI set_eqI iffI) |
|
905 |
show "inj_on ?f ?P" |
|
906 |
proof (safe intro!: inj_onI ext) |
|
907 |
fix a b x assume "a(i:=undefined) = b(i:=undefined)" "a i = b i" |
|
908 |
then show "a x = b x" |
|
909 |
by (cases "x = i") (auto simp: fun_eq_iff split: split_if_asm) |
|
910 |
qed |
|
911 |
next |
|
912 |
fix X assume *: "X \<in> ?F" show "X \<in> ?f ` ?P" |
|
913 |
proof (cases X) |
|
914 |
case (Pair a b) with * `i \<notin> I` show ?thesis |
|
915 |
by (auto intro!: image_eqI[where x="a (i := b)"] ext simp: extensional_def) |
|
916 |
qed |
|
917 |
qed auto |
|
918 |
||
919 |
section "Generating set generates also product algebra" |
|
920 |
||
921 |
lemma pair_sigma_algebra_sigma: |
|
922 |
assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)" |
|
923 |
assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)" |
|
924 |
shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)" |
|
925 |
(is "?S = ?E") |
|
926 |
proof - |
|
927 |
interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma) |
|
928 |
interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma) |
|
929 |
have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)" |
|
930 |
using E1 E2 by (auto simp add: pair_algebra_def) |
|
931 |
interpret E: sigma_algebra ?E unfolding pair_algebra_def |
|
932 |
using E1 E2 by (intro sigma_algebra_sigma) auto |
|
933 |
||
934 |
{ fix A assume "A \<in> sets E1" |
|
935 |
then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)" |
|
936 |
using E1 2 unfolding isoton_def pair_algebra_def by auto |
|
937 |
also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto |
|
938 |
also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma |
|
939 |
using 2 `A \<in> sets E1` |
|
940 |
by (intro sigma_sets.Union) |
|
941 |
(auto simp: image_subset_iff intro!: sigma_sets.Basic) |
|
942 |
finally have "fst -` A \<inter> space ?E \<in> sets ?E" . } |
|
943 |
moreover |
|
944 |
{ fix B assume "B \<in> sets E2" |
|
945 |
then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B" |
|
946 |
using E2 1 unfolding isoton_def pair_algebra_def by auto |
|
947 |
also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto |
|
948 |
also have "\<dots> \<in> sets ?E" |
|
949 |
using 1 `B \<in> sets E2` unfolding pair_algebra_def sets_sigma |
|
950 |
by (intro sigma_sets.Union) |
|
951 |
(auto simp: image_subset_iff intro!: sigma_sets.Basic) |
|
952 |
finally have "snd -` B \<inter> space ?E \<in> sets ?E" . } |
|
953 |
ultimately have proj: |
|
954 |
"fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)" |
|
955 |
using E1 E2 by (subst (1 2) E.measurable_iff_sigma) |
|
956 |
(auto simp: pair_algebra_def sets_sigma) |
|
957 |
||
958 |
{ fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)" |
|
959 |
with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E" |
|
960 |
unfolding measurable_def by simp_all |
|
961 |
moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)" |
|
962 |
using A B M1.sets_into_space M2.sets_into_space |
|
963 |
by (auto simp: pair_algebra_def) |
|
964 |
ultimately have "A \<times> B \<in> sets ?E" by auto } |
|
965 |
then have "sigma_sets (space ?E) (sets (pair_algebra (sigma E1) (sigma E2))) \<subseteq> sets ?E" |
|
966 |
by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma) |
|
967 |
then have subset: "sets ?S \<subseteq> sets ?E" |
|
968 |
by (simp add: sets_sigma pair_algebra_def) |
|
969 |
||
970 |
have "sets ?S = sets ?E" |
|
971 |
proof (intro set_eqI iffI) |
|
972 |
fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S" |
|
973 |
unfolding sets_sigma |
|
974 |
proof induct |
|
975 |
case (Basic A) then show ?case |
|
976 |
by (auto simp: pair_algebra_def sets_sigma intro: sigma_sets.Basic) |
|
977 |
qed (auto intro: sigma_sets.intros simp: pair_algebra_def) |
|
978 |
next |
|
979 |
fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto |
|
980 |
qed |
|
981 |
then show ?thesis |
|
982 |
by (simp add: pair_algebra_def sigma_def) |
|
983 |
qed |
|
984 |
||
985 |
lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A" |
|
986 |
apply auto |
|
987 |
apply (drule_tac x=x in Pi_mem) |
|
988 |
apply (simp_all split: split_if_asm) |
|
989 |
apply (drule_tac x=i in Pi_mem) |
|
990 |
apply (auto dest!: Pi_mem) |
|
991 |
done |
|
992 |
||
993 |
lemma Pi_UN: |
|
994 |
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" |
|
995 |
assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" |
|
996 |
shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)" |
|
997 |
proof (intro set_eqI iffI) |
|
998 |
fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)" |
|
999 |
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto |
|
1000 |
from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto |
|
1001 |
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" |
|
1002 |
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto |
|
1003 |
have "f \<in> Pi I (A k)" |
|
1004 |
proof (intro Pi_I) |
|
1005 |
fix i assume "i \<in> I" |
|
1006 |
from mono[OF this, of "n i" k] k[OF this] n[OF this] |
|
1007 |
show "f i \<in> A k i" by auto |
|
1008 |
qed |
|
1009 |
then show "f \<in> (\<Union>n. Pi I (A n))" by auto |
|
1010 |
qed auto |
|
1011 |
||
1012 |
lemma PiE_cong: |
|
1013 |
assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i" |
|
1014 |
shows "Pi\<^isub>E I A = Pi\<^isub>E I B" |
|
1015 |
using assms by (auto intro!: Pi_cong) |
|
1016 |
||
1017 |
lemma sigma_product_algebra_sigma_eq: |
|
1018 |
assumes "finite I" |
|
1019 |
assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))" |
|
1020 |
assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)" |
|
1021 |
and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))" |
|
1022 |
shows "sigma (product_algebra (\<lambda>i. sigma (E i)) I) = sigma (product_algebra E I)" |
|
1023 |
(is "?S = ?E") |
|
1024 |
proof cases |
|
1025 |
assume "I = {}" then show ?thesis by (simp add: product_algebra_def) |
|
1026 |
next |
|
1027 |
assume "I \<noteq> {}" |
|
1028 |
interpret E: sigma_algebra "sigma (E i)" for i |
|
1029 |
using E by (rule sigma_algebra_sigma) |
|
1030 |
||
1031 |
have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)" |
|
1032 |
using E by auto |
|
1033 |
||
1034 |
interpret G: sigma_algebra ?E |
|
1035 |
unfolding product_algebra_def using E |
|
1036 |
by (intro sigma_algebra_sigma) (auto dest: Pi_mem) |
|
1037 |
||
1038 |
{ fix A i assume "i \<in> I" and A: "A \<in> sets (E i)" |
|
1039 |
then have "(\<lambda>x. x i) -` A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E" |
|
1040 |
using isotone unfolding isoton_def product_algebra_def by (auto dest: Pi_mem) |
|
1041 |
also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))" |
|
1042 |
unfolding product_algebra_def |
|
1043 |
apply simp |
|
1044 |
apply (subst Pi_UN[OF `finite I`]) |
|
1045 |
using isotone[THEN isoton_mono_le] apply simp |
|
1046 |
apply (simp add: PiE_Int) |
|
1047 |
apply (intro PiE_cong) |
|
1048 |
using A sets_into by (auto intro!: into_space) |
|
1049 |
also have "\<dots> \<in> sets ?E" unfolding product_algebra_def sets_sigma |
|
1050 |
using sets_into `A \<in> sets (E i)` |
|
1051 |
by (intro sigma_sets.Union) |
|
1052 |
(auto simp: image_subset_iff intro!: sigma_sets.Basic) |
|
1053 |
finally have "(\<lambda>x. x i) -` A \<inter> space ?E \<in> sets ?E" . } |
|
1054 |
then have proj: |
|
1055 |
"\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))" |
|
1056 |
using E by (subst G.measurable_iff_sigma) |
|
1057 |
(auto simp: product_algebra_def sets_sigma) |
|
1058 |
||
1059 |
{ fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))" |
|
1060 |
with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) -` (A i) \<inter> space ?E \<in> sets ?E" |
|
1061 |
unfolding measurable_def by simp |
|
1062 |
have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) -` (A i) \<inter> space ?E)" |
|
1063 |
using A E.sets_into_space `I \<noteq> {}` unfolding product_algebra_def by auto blast |
|
1064 |
then have "Pi\<^isub>E I A \<in> sets ?E" |
|
1065 |
using G.finite_INT[OF `finite I` `I \<noteq> {}` basic, of "\<lambda>i. i"] by simp } |
|
1066 |
then have "sigma_sets (space ?E) (sets (product_algebra (\<lambda>i. sigma (E i)) I)) \<subseteq> sets ?E" |
|
1067 |
by (intro G.sigma_sets_subset) (auto simp add: sets_sigma product_algebra_def) |
|
1068 |
then have subset: "sets ?S \<subseteq> sets ?E" |
|
1069 |
by (simp add: sets_sigma product_algebra_def) |
|
1070 |
||
1071 |
have "sets ?S = sets ?E" |
|
1072 |
proof (intro set_eqI iffI) |
|
1073 |
fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S" |
|
1074 |
unfolding sets_sigma |
|
1075 |
proof induct |
|
1076 |
case (Basic A) then show ?case |
|
1077 |
by (auto simp: sets_sigma product_algebra_def intro: sigma_sets.Basic) |
|
1078 |
qed (auto intro: sigma_sets.intros simp: product_algebra_def) |
|
1079 |
next |
|
1080 |
fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto |
|
1081 |
qed |
|
1082 |
then show ?thesis |
|
1083 |
by (simp add: product_algebra_def sigma_def) |
|
1084 |
qed |
|
1085 |
||
1086 |
lemma (in finite_product_sigma_algebra) pair_sigma_algebra_finite_product_space: |
|
1087 |
"sigma (pair_algebra P (M i)) = sigma (pair_algebra G (M i))" |
|
1088 |
proof - |
|
1089 |
have "sigma (pair_algebra P (M i)) = sigma (pair_algebra P (sigma (M i)))" by simp |
|
1090 |
also have "\<dots> = sigma (pair_algebra G (M i))" |
|
1091 |
proof (rule pair_sigma_algebra_sigma) |
|
1092 |
show "(\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)) \<up> space G" |
|
1093 |
"(\<lambda>_. space (M i)) \<up> space (M i)" |
|
1094 |
by (simp_all add: isoton_const) |
|
1095 |
show "range (\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)) \<subseteq> sets G" "range (\<lambda>_. space (M i)) \<subseteq> sets (M i)" |
|
1096 |
by (auto intro!: image_eqI[where x="\<lambda>i\<in>I. space (M i)"] dest: Pi_mem |
|
1097 |
simp: product_algebra_def) |
|
1098 |
show "sets G \<subseteq> Pow (space G)" "sets (M i) \<subseteq> Pow (space (M i))" |
|
1099 |
using product_algebra_into_space M.sets_into_space by auto |
|
1100 |
qed |
|
1101 |
finally show ?thesis . |
|
1102 |
qed |
|
1103 |
||
1104 |
lemma sets_pair_algebra: "sets (pair_algebra N M) = (\<lambda>(x, y). x \<times> y) ` (sets N \<times> sets M)" |
|
1105 |
unfolding pair_algebra_def by auto |
|
1106 |
||
1107 |
lemma (in finite_product_sigma_algebra) sigma_pair_algebra_sigma_eq: |
|
1108 |
"sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))) = |
|
1109 |
sigma (pair_algebra (product_algebra M I) (product_algebra M J))" |
|
1110 |
using M.sets_into_space |
|
1111 |
by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)", of _ "\<lambda>_. \<Pi>\<^isub>E i\<in>J. space (M i)"]) |
|
1112 |
(auto simp: isoton_const product_algebra_def, blast+) |
|
1113 |
||
1114 |
lemma (in product_sigma_algebra) product_product_vimage_algebra: |
|
1115 |
assumes [simp]: "I \<inter> J = {}" and "finite I" "finite J" |
|
1116 |
shows "sigma_algebra.vimage_algebra |
|
1117 |
(sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J)))) |
|
1118 |
(space (product_algebra M (I \<union> J))) (\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))) = |
|
1119 |
sigma (product_algebra M (I \<union> J))" |
|
1120 |
(is "sigma_algebra.vimage_algebra _ (space ?IJ) ?f = sigma ?IJ") |
|
1121 |
proof - |
|
1122 |
have "finite (I \<union> J)" using assms by auto |
|
1123 |
interpret I: finite_product_sigma_algebra M I by default fact |
|
1124 |
interpret J: finite_product_sigma_algebra M J by default fact |
|
1125 |
interpret IJ: finite_product_sigma_algebra M "I \<union> J" by default fact |
|
1126 |
interpret pair_sigma_algebra I.P J.P by default |
|
1127 |
||
1128 |
show "vimage_algebra (space ?IJ) ?f = sigma ?IJ" |
|
1129 |
unfolding I.sigma_pair_algebra_sigma_eq |
|
1130 |
proof (rule vimage_algebra_sigma) |
|
1131 |
from M.sets_into_space |
|
1132 |
show "sets (pair_algebra I.G J.G) \<subseteq> Pow (space (pair_algebra I.G J.G))" |
|
1133 |
by (auto simp: sets_pair_algebra space_pair_algebra product_algebra_def) blast+ |
|
1134 |
show "?f \<in> space IJ.G \<rightarrow> space (pair_algebra I.G J.G)" |
|
1135 |
by (auto simp: space_pair_algebra product_algebra_def) |
|
1136 |
let ?F = "\<lambda>A. ?f -` A \<inter> (space IJ.G)" |
|
1137 |
let ?s = "\<lambda>I. Pi\<^isub>E I ` (\<Pi> i\<in>I. sets (M i))" |
|
1138 |
{ fix A assume "A \<in> sets IJ.G" |
|
1139 |
then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F" "F \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> (\<Pi> i\<in>J. sets (M i))" |
|
1140 |
by (auto simp: product_algebra_def) |
|
1141 |
show "A \<in> ?F ` sets (pair_algebra I.G J.G)" |
|
1142 |
using A M.sets_into_space |
|
1143 |
by (auto simp: restrict_Pi_cancel product_algebra_def |
|
1144 |
intro!: image_eqI[where x="Pi\<^isub>E I F \<times> Pi\<^isub>E J F"]) blast+ } |
|
1145 |
{ fix A assume "A \<in> sets (pair_algebra I.G J.G)" |
|
1146 |
then obtain E F where A: "A = Pi\<^isub>E I E \<times> Pi\<^isub>E J F" "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> (\<Pi> i\<in>J. sets (M i))" |
|
1147 |
by (auto simp: product_algebra_def sets_pair_algebra) |
|
1148 |
then show "?F A \<in> sets IJ.G" |
|
1149 |
using A M.sets_into_space |
|
1150 |
by (auto simp: restrict_Pi_cancel product_algebra_def |
|
1151 |
intro!: image_eqI[where x="merge I E J F"]) blast+ } |
|
1152 |
qed |
|
1153 |
qed |
|
1154 |
||
1155 |
lemma (in finite_product_sigma_algebra) sigma_pair_algebra_sigma_M_eq: |
|
1156 |
"sigma (pair_algebra P (M i)) = sigma (pair_algebra G (M i))" |
|
1157 |
proof - |
|
1158 |
have "sigma (pair_algebra P (sigma (M i))) = sigma (pair_algebra G (M i))" |
|
1159 |
using M.sets_into_space |
|
1160 |
by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)", of _ "\<lambda>_. space (M i)"]) |
|
1161 |
(auto simp: isoton_const product_algebra_def, blast+) |
|
1162 |
then show ?thesis by simp |
|
1163 |
qed |
|
1164 |
||
1165 |
lemma (in product_sigma_algebra) product_singleton_vimage_algebra_eq: |
|
1166 |
assumes [simp]: "i \<notin> I" "finite I" |
|
1167 |
shows "sigma_algebra.vimage_algebra |
|
1168 |
(sigma (pair_algebra (sigma (product_algebra M I)) (M i))) |
|
1169 |
(space (product_algebra M (insert i I))) (\<lambda>x. ((\<lambda>i\<in>I. x i), x i)) = |
|
1170 |
sigma (product_algebra M (insert i I))" |
|
1171 |
(is "sigma_algebra.vimage_algebra _ (space ?I') ?f = sigma ?I'") |
|
1172 |
proof - |
|
1173 |
have "finite (insert i I)" using assms by auto |
|
1174 |
interpret I: finite_product_sigma_algebra M I by default fact |
|
1175 |
interpret I': finite_product_sigma_algebra M "insert i I" by default fact |
|
1176 |
interpret pair_sigma_algebra I.P "M i" by default |
|
1177 |
show "vimage_algebra (space ?I') ?f = sigma ?I'" |
|
1178 |
unfolding I.sigma_pair_algebra_sigma_M_eq |
|
1179 |
proof (rule vimage_algebra_sigma) |
|
1180 |
from M.sets_into_space |
|
1181 |
show "sets (pair_algebra I.G (M i)) \<subseteq> Pow (space (pair_algebra I.G (M i)))" |
|
1182 |
by (auto simp: sets_pair_algebra space_pair_algebra product_algebra_def) blast |
|
1183 |
show "?f \<in> space I'.G \<rightarrow> space (pair_algebra I.G (M i))" |
|
1184 |
by (auto simp: space_pair_algebra product_algebra_def) |
|
1185 |
let ?F = "\<lambda>A. ?f -` A \<inter> (space I'.G)" |
|
1186 |
{ fix A assume "A \<in> sets I'.G" |
|
1187 |
then obtain F where A: "A = Pi\<^isub>E (insert i I) F" "F \<in> (\<Pi> i\<in>I. sets (M i))" "F i \<in> sets (M i)" |
|
1188 |
by (auto simp: product_algebra_def) |
|
1189 |
show "A \<in> ?F ` sets (pair_algebra I.G (M i))" |
|
1190 |
using A M.sets_into_space |
|
1191 |
by (auto simp: restrict_Pi_cancel product_algebra_def |
|
1192 |
intro!: image_eqI[where x="Pi\<^isub>E I F \<times> F i"]) blast+ } |
|
1193 |
{ fix A assume "A \<in> sets (pair_algebra I.G (M i))" |
|
1194 |
then obtain E F where A: "A = Pi\<^isub>E I E \<times> F" "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> sets (M i)" |
|
1195 |
by (auto simp: product_algebra_def sets_pair_algebra) |
|
1196 |
then show "?F A \<in> sets I'.G" |
|
1197 |
using A M.sets_into_space |
|
1198 |
by (auto simp: restrict_Pi_cancel product_algebra_def |
|
1199 |
intro!: image_eqI[where x="E(i:= F)"]) blast+ } |
|
1200 |
qed |
|
1201 |
qed |
|
38656 | 1202 |
|
40859 | 1203 |
lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I" |
1204 |
by (auto simp: restrict_def intro!: ext) |
|
1205 |
||
1206 |
lemma bij_betw_restrict_I_i: |
|
1207 |
"i \<notin> I \<Longrightarrow> bij_betw (\<lambda>x. (restrict x I, x i)) |
|
1208 |
(space (product_algebra M (insert i I))) |
|
1209 |
(space (pair_algebra (sigma (product_algebra M I)) (M i)))" |
|
1210 |
by (intro bij_betwI[where g="(\<lambda>(x,y). x(i:=y))"]) |
|
1211 |
(auto simp: space_pair_algebra extensional_def intro!: ext) |
|
1212 |
||
1213 |
lemma (in product_sigma_algebra) product_singleton_vimage_algebra_inv_eq: |
|
1214 |
assumes [simp]: "i \<notin> I" "finite I" |
|
1215 |
shows "sigma_algebra.vimage_algebra |
|
1216 |
(sigma (product_algebra M (insert i I))) |
|
1217 |
(space (pair_algebra (sigma (product_algebra M I)) (M i))) (\<lambda>(x,y). x(i:=y)) = |
|
1218 |
sigma (pair_algebra (sigma (product_algebra M I)) (M i))" |
|
1219 |
proof - |
|
1220 |
have "finite (insert i I)" using `finite I` by auto |
|
1221 |
interpret I: finite_product_sigma_algebra M I by default fact |
|
1222 |
interpret I': finite_product_sigma_algebra M "insert i I" by default fact |
|
1223 |
interpret pair_sigma_algebra I.P "M i" by default |
|
1224 |
show ?thesis |
|
1225 |
unfolding product_singleton_vimage_algebra_eq[OF assms, symmetric] |
|
1226 |
using bij_betw_restrict_I_i[OF `i \<notin> I`, of M] |
|
1227 |
by (rule vimage_vimage_inv[unfolded space_sigma]) |
|
1228 |
(auto simp: space_pair_algebra product_algebra_def dest: extensional_restrict) |
|
1229 |
qed |
|
1230 |
||
1231 |
locale product_sigma_finite = |
|
1232 |
fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pinfreal" |
|
1233 |
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i) (\<mu> i)" |
|
1234 |
||
1235 |
locale finite_product_sigma_finite = product_sigma_finite M \<mu> for M :: "'i \<Rightarrow> 'a algebra" and \<mu> + |
|
1236 |
fixes I :: "'i set" assumes finite_index': "finite I" |
|
1237 |
||
1238 |
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" "\<mu> i" for i |
|
1239 |
by (rule sigma_finite_measures) |
|
1240 |
||
1241 |
sublocale product_sigma_finite \<subseteq> product_sigma_algebra |
|
1242 |
by default |
|
1243 |
||
1244 |
sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra |
|
1245 |
by default (fact finite_index') |
|
1246 |
||
1247 |
lemma (in finite_product_sigma_finite) sigma_finite_pairs: |
|
1248 |
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. |
|
1249 |
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> |
|
1250 |
(\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<omega>) \<and> |
|
1251 |
(\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<up> space G" |
|
1252 |
proof - |
|
1253 |
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> F \<up> space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<omega>)" |
|
1254 |
using M.sigma_finite_up by simp |
|
1255 |
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
|
1256 |
then have "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. F i \<up> space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<omega>" |
|
1257 |
by auto |
|
1258 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k" |
|
1259 |
note space_product_algebra[simp] |
|
1260 |
show ?thesis |
|
1261 |
proof (intro exI[of _ F] conjI allI isotoneI set_eqI iffI ballI) |
|
1262 |
fix i show "range (F i) \<subseteq> sets (M i)" by fact |
|
1263 |
next |
|
1264 |
fix i k show "\<mu> i (F i k) \<noteq> \<omega>" by fact |
|
1265 |
next |
|
1266 |
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G" |
|
1267 |
using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space by auto blast |
|
1268 |
next |
|
1269 |
fix f assume "f \<in> space G" |
|
1270 |
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] |
|
1271 |
`\<And>i. F i \<up> space (M i)`[THEN isotonD(2)] |
|
1272 |
`\<And>i. F i \<up> space (M i)`[THEN isoton_mono_le] |
|
1273 |
show "f \<in> (\<Union>i. ?F i)" by auto |
|
1274 |
next |
|
1275 |
fix i show "?F i \<subseteq> ?F (Suc i)" |
|
1276 |
using `\<And>i. F i \<up> space (M i)`[THEN isotonD(1)] by auto |
|
1277 |
qed |
|
1278 |
qed |
|
1279 |
||
1280 |
lemma (in product_sigma_finite) product_measure_exists: |
|
1281 |
assumes "finite I" |
|
1282 |
shows "\<exists>\<nu>. (\<forall>A\<in>(\<Pi> i\<in>I. sets (M i)). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and> |
|
1283 |
sigma_finite_measure (sigma (product_algebra M I)) \<nu>" |
|
1284 |
using `finite I` proof induct |
|
1285 |
case empty then show ?case unfolding product_algebra_def |
|
1286 |
by (auto intro!: exI[of _ "\<lambda>A. if A = {} then 0 else 1"] sigma_algebra_sigma |
|
1287 |
sigma_algebra.finite_additivity_sufficient |
|
1288 |
simp add: positive_def additive_def sets_sigma sigma_finite_measure_def |
|
1289 |
sigma_finite_measure_axioms_def) |
|
1290 |
next |
|
1291 |
case (insert i I) |
|
1292 |
interpret finite_product_sigma_finite M \<mu> I by default fact |
|
1293 |
have "finite (insert i I)" using `finite I` by auto |
|
1294 |
interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default fact |
|
1295 |
from insert obtain \<nu> where |
|
1296 |
prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" and |
|
1297 |
"sigma_finite_measure P \<nu>" by auto |
|
1298 |
interpret I: sigma_finite_measure P \<nu> by fact |
|
1299 |
interpret P: pair_sigma_finite P \<nu> "M i" "\<mu> i" .. |
|
1300 |
||
1301 |
let ?h = "\<lambda>x. (restrict x I, x i)" |
|
1302 |
let ?\<nu> = "\<lambda>A. P.pair_measure (?h ` A)" |
|
1303 |
interpret I': measure_space "sigma (product_algebra M (insert i I))" ?\<nu> |
|
1304 |
unfolding product_singleton_vimage_algebra_eq[OF `i \<notin> I` `finite I`, symmetric] |
|
1305 |
using bij_betw_restrict_I_i[OF `i \<notin> I`, of M] |
|
1306 |
by (intro P.measure_space_isomorphic) auto |
|
1307 |
||
1308 |
show ?case |
|
1309 |
proof (intro exI[of _ ?\<nu>] conjI ballI) |
|
1310 |
{ fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))" |
|
1311 |
moreover then have "A \<in> (\<Pi> i\<in>I. sets (M i))" by auto |
|
1312 |
moreover have "(\<lambda>x. (restrict x I, x i)) ` Pi\<^isub>E (insert i I) A = Pi\<^isub>E I A \<times> A i" |
|
1313 |
using `i \<notin> I` |
|
1314 |
apply auto |
|
1315 |
apply (rule_tac x="a(i:=b)" in image_eqI) |
|
1316 |
apply (auto simp: extensional_def fun_eq_iff) |
|
1317 |
done |
|
1318 |
ultimately show "?\<nu> (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. \<mu> i (A i))" |
|
1319 |
apply simp |
|
1320 |
apply (subst P.pair_measure_times) |
|
1321 |
apply fastsimp |
|
1322 |
apply fastsimp |
|
1323 |
using `i \<notin> I` `finite I` prod[of A] by (auto simp: ac_simps) } |
|
1324 |
note product = this |
|
1325 |
||
1326 |
show "sigma_finite_measure I'.P ?\<nu>" |
|
1327 |
proof |
|
1328 |
from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
|
1329 |
then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)" |
|
1330 |
"(\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) \<up> space I'.G" |
|
1331 |
"\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<omega>" |
|
1332 |
by blast+ |
|
1333 |
let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k" |
|
1334 |
show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and> |
|
1335 |
(\<Union>i. F i) = space I'.P \<and> (\<forall>i. ?\<nu> (F i) \<noteq> \<omega>)" |
|
1336 |
proof (intro exI[of _ ?F] conjI allI) |
|
1337 |
show "range ?F \<subseteq> sets I'.P" using F(1) by auto |
|
1338 |
next |
|
1339 |
from F(2)[THEN isotonD(2)] |
|
1340 |
show "(\<Union>i. ?F i) = space I'.P" by simp |
|
1341 |
next |
|
1342 |
fix j |
|
1343 |
show "?\<nu> (?F j) \<noteq> \<omega>" |
|
1344 |
using F `finite I` |
|
1345 |
by (subst product) (auto simp: setprod_\<omega>) |
|
1346 |
qed |
|
1347 |
qed |
|
1348 |
qed |
|
1349 |
qed |
|
1350 |
||
1351 |
definition (in finite_product_sigma_finite) |
|
1352 |
measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pinfreal" where |
|
1353 |
"measure = (SOME \<nu>. |
|
1354 |
(\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and> |
|
1355 |
sigma_finite_measure P \<nu>)" |
|
1356 |
||
1357 |
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P measure |
|
1358 |
proof - |
|
1359 |
show "sigma_finite_measure P measure" |
|
1360 |
unfolding measure_def |
|
1361 |
by (rule someI2_ex[OF product_measure_exists[OF finite_index]]) auto |
|
1362 |
qed |
|
1363 |
||
1364 |
lemma (in finite_product_sigma_finite) measure_times: |
|
1365 |
assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" |
|
1366 |
shows "measure (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" |
|
1367 |
proof - |
|
1368 |
note ex = product_measure_exists[OF finite_index] |
|
1369 |
show ?thesis |
|
1370 |
unfolding measure_def |
|
1371 |
proof (rule someI2_ex[OF ex], elim conjE) |
|
1372 |
fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" |
|
1373 |
have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem) |
|
1374 |
then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp |
|
1375 |
also have "\<dots> = (\<Prod>i\<in>I. \<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto |
|
1376 |
finally show "\<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" by simp |
|
1377 |
qed |
|
1378 |
qed |
|
35833 | 1379 |
|
40859 | 1380 |
abbreviation (in product_sigma_finite) |
1381 |
"product_measure I \<equiv> finite_product_sigma_finite.measure M \<mu> I" |
|
1382 |
||
1383 |
abbreviation (in product_sigma_finite) |
|
1384 |
"product_positive_integral I \<equiv> |
|
1385 |
measure_space.positive_integral (sigma (product_algebra M I)) (product_measure I)" |
|
1386 |
||
1387 |
abbreviation (in product_sigma_finite) |
|
1388 |
"product_integral I \<equiv> |
|
1389 |
measure_space.integral (sigma (product_algebra M I)) (product_measure I)" |
|
1390 |
||
1391 |
lemma (in product_sigma_finite) positive_integral_empty: |
|
1392 |
"product_positive_integral {} f = f (\<lambda>k. undefined)" |
|
1393 |
proof - |
|
1394 |
interpret finite_product_sigma_finite M \<mu> "{}" by default (fact finite.emptyI) |
|
1395 |
have "\<And>A. measure (Pi\<^isub>E {} A) = 1" |
|
1396 |
using assms by (subst measure_times) auto |
|
1397 |
then show ?thesis |
|
1398 |
unfolding positive_integral_alt simple_function_def simple_integral_def_raw |
|
1399 |
proof (simp add: P_empty, intro antisym) |
|
1400 |
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))" |
|
1401 |
by (intro le_SUPI) auto |
|
1402 |
show "(SUP f:{g. g \<le> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" |
|
1403 |
by (intro SUP_leI) (auto simp: le_fun_def) |
|
1404 |
qed |
|
1405 |
qed |
|
1406 |
||
1407 |
lemma merge_restrict[simp]: |
|
1408 |
"merge I (restrict x I) J y = merge I x J y" |
|
1409 |
"merge I x J (restrict y J) = merge I x J y" |
|
1410 |
unfolding merge_def by (auto intro!: ext) |
|
1411 |
||
1412 |
lemma merge_x_x_eq_restrict[simp]: |
|
1413 |
"merge I x J x = restrict x (I \<union> J)" |
|
1414 |
unfolding merge_def by (auto intro!: ext) |
|
1415 |
||
1416 |
lemma bij_betw_restrict_I_J: |
|
1417 |
"I \<inter> J = {} \<Longrightarrow> bij_betw (\<lambda>x. (restrict x I, restrict x J)) |
|
1418 |
(space (product_algebra M (I \<union> J))) |
|
1419 |
(space (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))" |
|
1420 |
by (intro bij_betwI[where g="\<lambda>(x,y). merge I x J y"]) |
|
1421 |
(auto dest: extensional_restrict simp: space_pair_algebra) |
|
1422 |
||
1423 |
lemma (in product_sigma_algebra) product_product_vimage_algebra_eq: |
|
1424 |
assumes [simp]: "I \<inter> J = {}" and "finite I" "finite J" |
|
1425 |
shows "sigma_algebra.vimage_algebra |
|
1426 |
(sigma (product_algebra M (I \<union> J))) |
|
1427 |
(space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) |
|
1428 |
(\<lambda>(x, y). merge I x J y) = |
|
1429 |
sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J)))" |
|
1430 |
(is "sigma_algebra.vimage_algebra ?IJ ?S ?m = ?P") |
|
1431 |
proof - |
|
1432 |
interpret I: finite_product_sigma_algebra M I by default fact |
|
1433 |
interpret J: finite_product_sigma_algebra M J by default fact |
|
1434 |
have "finite (I \<union> J)" using assms by auto |
|
1435 |
interpret IJ: finite_product_sigma_algebra M "I \<union> J" by default fact |
|
1436 |
interpret P: pair_sigma_algebra I.P J.P by default |
|
1437 |
||
1438 |
let ?g = "\<lambda>x. (restrict x I, restrict x J)" |
|
1439 |
let ?f = "\<lambda>(x, y). merge I x J y" |
|
1440 |
show "IJ.vimage_algebra (space P.P) ?f = P.P" |
|
1441 |
using bij_betw_restrict_I_J[OF `I \<inter> J = {}`] |
|
1442 |
by (subst P.vimage_vimage_inv[of ?g "space IJ.G" ?f, |
|
1443 |
unfolded product_product_vimage_algebra[OF assms]]) |
|
1444 |
(auto simp: pair_algebra_def dest: extensional_restrict) |
|
1445 |
qed |
|
1446 |
||
1447 |
lemma (in product_sigma_finite) measure_fold_left: |
|
1448 |
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" |
|
1449 |
and f: "f \<in> borel_measurable (sigma (product_algebra M (I \<union> J)))" |
|
1450 |
shows "product_positive_integral (I \<union> J) f = |
|
1451 |
product_positive_integral I (\<lambda>x. product_positive_integral J (\<lambda>y. f (merge I x J y)))" |
|
1452 |
proof - |
|
1453 |
interpret I: finite_product_sigma_finite M \<mu> I by default fact |
|
1454 |
interpret J: finite_product_sigma_finite M \<mu> J by default fact |
|
1455 |
have "finite (I \<union> J)" using fin by auto |
|
1456 |
interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact |
|
1457 |
interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default |
|
1458 |
||
1459 |
let ?f = "\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))" |
|
1460 |
||
1461 |
have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P" |
|
1462 |
by (subst product_product_vimage_algebra_eq[OF IJ fin, symmetric]) |
|
1463 |
(auto simp: space_pair_algebra intro!: IJ.measurable_vimage f) |
|
1464 |
||
1465 |
have split_f_image[simp]: "\<And>F. ?f ` (Pi\<^isub>E (I \<union> J) F) = (Pi\<^isub>E I F) \<times> (Pi\<^isub>E J F)" |
|
1466 |
apply auto apply (rule_tac x="merge I a J b" in image_eqI) |
|
1467 |
by (auto dest: extensional_restrict) |
|
1468 |
||
1469 |
have "IJ.positive_integral f = IJ.positive_integral (\<lambda>x. f (restrict x (I \<union> J)))" |
|
1470 |
by (auto intro!: IJ.positive_integral_cong arg_cong[where f=f] dest!: extensional_restrict) |
|
1471 |
also have "\<dots> = I.positive_integral (\<lambda>x. J.positive_integral (\<lambda>y. f (merge I x J y)))" |
|
1472 |
unfolding P.positive_integral_fst_measurable[OF P_borel, simplified] |
|
1473 |
unfolding P.positive_integral_vimage[unfolded space_sigma, OF bij_betw_restrict_I_J[OF IJ]] |
|
1474 |
unfolding product_product_vimage_algebra[OF IJ fin] |
|
1475 |
proof (simp, rule IJ.positive_integral_cong_measure[symmetric]) |
|
1476 |
fix A assume *: "A \<in> sets IJ.P" |
|
1477 |
from IJ.sigma_finite_pairs obtain F where |
|
1478 |
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)" |
|
1479 |
"(\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) \<up> space IJ.G" |
|
1480 |
"\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<omega>" |
|
1481 |
by auto |
|
1482 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k" |
|
1483 |
show "P.pair_measure (?f ` A) = IJ.measure A" |
|
1484 |
proof (rule measure_unique_Int_stable[OF _ _ _ _ _ _ _ _ *]) |
|
1485 |
show "Int_stable IJ.G" by (simp add: PiE_Int Int_stable_def product_algebra_def) auto |
|
1486 |
show "range ?F \<subseteq> sets IJ.G" using F by (simp add: image_subset_iff product_algebra_def) |
|
1487 |
show "?F \<up> space IJ.G " using F(2) by simp |
|
1488 |
show "measure_space IJ.P (\<lambda>A. P.pair_measure (?f ` A))" |
|
1489 |
unfolding product_product_vimage_algebra[OF IJ fin, symmetric] |
|
1490 |
using bij_betw_restrict_I_J[OF IJ, of M] |
|
1491 |
by (auto intro!: P.measure_space_isomorphic) |
|
1492 |
show "measure_space IJ.P IJ.measure" by fact |
|
1493 |
next |
|
1494 |
fix A assume "A \<in> sets IJ.G" |
|
1495 |
then obtain F where A[simp]: "A = Pi\<^isub>E (I \<union> J) F" "F \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" |
|
1496 |
by (auto simp: product_algebra_def) |
|
1497 |
then have F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (M i)" "\<And>i. i \<in> J \<Longrightarrow> F i \<in> sets (M i)" |
|
1498 |
by auto |
|
1499 |
have "P.pair_measure (?f ` A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))" |
|
1500 |
using `finite J` `finite I` F |
|
1501 |
by (simp add: P.pair_measure_times I.measure_times J.measure_times) |
|
1502 |
also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))" |
|
1503 |
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one) |
|
1504 |
also have "\<dots> = IJ.measure A" |
|
1505 |
using `finite J` `finite I` F unfolding A |
|
1506 |
by (intro IJ.measure_times[symmetric]) auto |
|
1507 |
finally show "P.pair_measure (?f ` A) = IJ.measure A" . |
|
1508 |
next |
|
1509 |
fix k |
|
1510 |
have "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i k \<in> sets (M i)" using F by auto |
|
1511 |
then have "P.pair_measure (?f ` ?F k) = (\<Prod>i\<in>I. \<mu> i (F i k)) * (\<Prod>i\<in>J. \<mu> i (F i k))" |
|
1512 |
by (simp add: P.pair_measure_times I.measure_times J.measure_times) |
|
1513 |
then show "P.pair_measure (?f ` ?F k) \<noteq> \<omega>" |
|
1514 |
using `finite I` F by (simp add: setprod_\<omega>) |
|
1515 |
qed simp |
|
1516 |
qed |
|
1517 |
finally show ?thesis . |
|
1518 |
qed |
|
1519 |
||
1520 |
lemma (in product_sigma_finite) finite_pair_measure_singleton: |
|
1521 |
assumes f: "f \<in> borel_measurable (M i)" |
|
1522 |
shows "product_positive_integral {i} (\<lambda>x. f (x i)) = M.positive_integral i f" |
|
1523 |
proof - |
|
1524 |
interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp |
|
1525 |
have bij: "bij_betw (\<lambda>x. \<lambda>j\<in>{i}. x) (space (M i)) (space I.P)" |
|
1526 |
by (auto intro!: bij_betwI ext simp: extensional_def) |
|
1527 |
have *: "(\<lambda>x. (\<lambda>x. \<lambda>j\<in>{i}. x) -` Pi\<^isub>E {i} x \<inter> space (M i)) ` (\<Pi> i\<in>{i}. sets (M i)) = sets (M i)" |
|
1528 |
proof (subst image_cong, rule refl) |
|
1529 |
fix x assume "x \<in> (\<Pi> i\<in>{i}. sets (M i))" |
|
1530 |
then show "(\<lambda>x. \<lambda>j\<in>{i}. x) -` Pi\<^isub>E {i} x \<inter> space (M i) = x i" |
|
1531 |
using sets_into_space by auto |
|
1532 |
qed auto |
|
1533 |
have vimage: "I.vimage_algebra (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i" |
|
1534 |
unfolding I.vimage_algebra_def |
|
1535 |
unfolding product_sigma_algebra_def sets_sigma |
|
1536 |
apply (subst sigma_sets_vimage[symmetric]) |
|
1537 |
apply (simp_all add: image_image sigma_sets_eq product_algebra_def * del: vimage_Int) |
|
1538 |
using sets_into_space by blast |
|
1539 |
show "I.positive_integral (\<lambda>x. f (x i)) = M.positive_integral i f" |
|
1540 |
unfolding I.positive_integral_vimage[OF bij] |
|
1541 |
unfolding vimage |
|
1542 |
apply (subst positive_integral_cong_measure) |
|
1543 |
proof - |
|
1544 |
fix A assume A: "A \<in> sets (M i)" |
|
1545 |
have "(\<lambda>x. \<lambda>j\<in>{i}. x) ` A = (\<Pi>\<^isub>E i\<in>{i}. A)" |
|
1546 |
by (auto intro!: image_eqI ext[where 'b='a] simp: extensional_def) |
|
1547 |
with A show "product_measure {i} ((\<lambda>x. \<lambda>j\<in>{i}. x) ` A) = \<mu> i A" |
|
1548 |
using I.measure_times[of "\<lambda>i. A"] by simp |
|
1549 |
qed simp |
|
1550 |
qed |
|
1551 |
||
1552 |
section "Products on finite spaces" |
|
1553 |
||
1554 |
lemma sigma_sets_pair_algebra_finite: |
|
38656 | 1555 |
assumes "finite A" and "finite B" |
40859 | 1556 |
shows "sigma_sets (A \<times> B) ((\<lambda>(x,y). x \<times> y) ` (Pow A \<times> Pow B)) = Pow (A \<times> B)" |
1557 |
(is "sigma_sets ?prod ?sets = _") |
|
38656 | 1558 |
proof safe |
1559 |
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) |
|
1560 |
fix x assume subset: "x \<subseteq> A \<times> B" |
|
1561 |
hence "finite x" using fin by (rule finite_subset) |
|
40859 | 1562 |
from this subset show "x \<in> sigma_sets ?prod ?sets" |
38656 | 1563 |
proof (induct x) |
1564 |
case empty show ?case by (rule sigma_sets.Empty) |
|
1565 |
next |
|
1566 |
case (insert a x) |
|
40859 | 1567 |
hence "{a} \<in> sigma_sets ?prod ?sets" |
1568 |
by (auto simp: pair_algebra_def intro!: sigma_sets.Basic) |
|
38656 | 1569 |
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto |
1570 |
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) |
|
1571 |
qed |
|
1572 |
next |
|
1573 |
fix x a b |
|
40859 | 1574 |
assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x" |
38656 | 1575 |
from sigma_sets_into_sp[OF _ this(1)] this(2) |
40859 | 1576 |
show "a \<in> A" and "b \<in> B" by auto |
35833 | 1577 |
qed |
1578 |
||
40859 | 1579 |
locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2 |
1580 |
||
1581 |
sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default |
|
1582 |
||
1583 |
lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra[simp]: |
|
1584 |
shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)" |
|
35977 | 1585 |
proof - |
40859 | 1586 |
show ?thesis using M1.finite_space M2.finite_space |
1587 |
by (simp add: sigma_def space_pair_algebra sets_pair_algebra |
|
1588 |
sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow) |
|
1589 |
qed |
|
1590 |
||
1591 |
sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P |
|
1592 |
proof |
|
1593 |
show "finite (space P)" "sets P = Pow (space P)" |
|
1594 |
using M1.finite_space M2.finite_space by auto |
|
35977 | 1595 |
qed |
35833 | 1596 |
|
40859 | 1597 |
locale pair_finite_space = M1: finite_measure_space M1 \<mu>1 + M2: finite_measure_space M2 \<mu>2 |
1598 |
for M1 \<mu>1 M2 \<mu>2 |
|
1599 |
||
1600 |
sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra |
|
1601 |
by default |
|
35833 | 1602 |
|
40859 | 1603 |
sublocale pair_finite_space \<subseteq> pair_sigma_finite |
1604 |
by default |
|
38656 | 1605 |
|
40859 | 1606 |
lemma (in pair_finite_space) finite_pair_sigma_algebra[simp]: |
1607 |
shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)" |
|
38656 | 1608 |
proof - |
40859 | 1609 |
show ?thesis using M1.finite_space M2.finite_space |
1610 |
by (simp add: sigma_def space_pair_algebra sets_pair_algebra |
|
1611 |
sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow) |
|
35833 | 1612 |
qed |
1613 |
||
40859 | 1614 |
lemma (in pair_finite_space) pair_measure_Pair[simp]: |
1615 |
assumes "a \<in> space M1" "b \<in> space M2" |
|
1616 |
shows "pair_measure {(a, b)} = \<mu>1 {a} * \<mu>2 {b}" |
|
1617 |
proof - |
|
1618 |
have "pair_measure ({a}\<times>{b}) = \<mu>1 {a} * \<mu>2 {b}" |
|
1619 |
using M1.sets_eq_Pow M2.sets_eq_Pow assms |
|
1620 |
by (subst pair_measure_times) auto |
|
1621 |
then show ?thesis by simp |
|
38656 | 1622 |
qed |
1623 |
||
40859 | 1624 |
lemma (in pair_finite_space) pair_measure_singleton[simp]: |
1625 |
assumes "x \<in> space M1 \<times> space M2" |
|
1626 |
shows "pair_measure {x} = \<mu>1 {fst x} * \<mu>2 {snd x}" |
|
1627 |
using pair_measure_Pair assms by (cases x) auto |
|
38656 | 1628 |
|
40859 | 1629 |
sublocale pair_finite_space \<subseteq> finite_measure_space P pair_measure |
1630 |
by default auto |
|
39097 | 1631 |
|
40859 | 1632 |
lemma (in pair_finite_space) finite_measure_space_finite_prod_measure_alterantive: |
1633 |
"finite_measure_space \<lparr>space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2)\<rparr> pair_measure" |
|
1634 |
unfolding finite_pair_sigma_algebra[symmetric] |
|
1635 |
by default |
|
39097 | 1636 |
|
40859 | 1637 |
end |