src/HOL/Probability/Product_Measure.thy
author hoelzl
Mon Dec 06 19:54:56 2010 +0100 (2010-12-06)
changeset 41026 bea75746dc9d
parent 41023 9118eb4eb8dc
child 41095 c335d880ff82
permissions -rw-r--r--
folding on arbitrary Lebesgue integrable functions
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theory Product_Measure
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imports Lebesgue_Integration
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begin
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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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lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
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  by (auto simp: fun_eq_iff)
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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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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)"
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  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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lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
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  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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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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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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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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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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proof -
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  { fix X assume "X \<in> sets M1"
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    then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
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      apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
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      using M1.sets_into_space by force+ }
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  moreover
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  { fix X assume "X \<in> sets M2"
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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"
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      apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
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      using M2.sets_into_space by force+ }
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  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
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  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])
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  next
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    fix z
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    assume z: "z \<in> sets E"
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    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
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      assume x: "x \<in> sets M1" and y: "y \<in> sets M2"
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      have "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M =
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            ((fst \<circ> f) -` x \<inter> space M) \<inter> ((snd \<circ> f) -` y \<inter> space M)"
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        by blast
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      also have "...  \<in> sets M" using x y q1 q2
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        by blast
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      finally show "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M \<in> sets M" .
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    qed
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  qed
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qed
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lemma pair_algebraE:
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  assumes "X \<in> sets (pair_algebra M1 M2)"
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  obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
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  using assms unfolding pair_algebra_def by auto
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lemma (in pair_sigma_algebra) pair_algebra_swap:
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  "(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_algebra M2 M1)"
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proof (safe elim!: pair_algebraE)
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  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
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  moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
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    using M1.sets_into_space M2.sets_into_space by auto
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  ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_algebra M2 M1)"
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    by (auto intro: pair_algebraI)
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next
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  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
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  then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E"
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    using M1.sets_into_space M2.sets_into_space
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    by (auto intro!: image_eqI[where x="A \<times> B"] pair_algebraI)
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qed
hoelzl@40859
   264
hoelzl@40859
   265
lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
hoelzl@40859
   266
  assumes Q: "Q \<in> sets P"
hoelzl@40859
   267
  shows "(\<lambda>(x,y). (y, x)) ` Q \<in> sets (sigma (pair_algebra M2 M1))" (is "_ \<in> sets ?Q")
hoelzl@40859
   268
proof -
hoelzl@40859
   269
  have *: "(\<lambda>(x,y). (y, x)) \<in> space M2 \<times> space M1 \<rightarrow> (space M1 \<times> space M2)"
hoelzl@40859
   270
       "sets (pair_algebra M1 M2) \<subseteq> Pow (space M1 \<times> space M2)"
hoelzl@40859
   271
    using M1.sets_into_space M2.sets_into_space by (auto elim!: pair_algebraE)
hoelzl@40859
   272
  from Q sets_into_space show ?thesis
hoelzl@40859
   273
    by (auto intro!: image_eqI[where x=Q]
hoelzl@40859
   274
             simp: pair_algebra_swap[symmetric] sets_sigma
hoelzl@40859
   275
                   sigma_sets_vimage[OF *] space_pair_algebra)
hoelzl@40859
   276
qed
hoelzl@40859
   277
hoelzl@40859
   278
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
hoelzl@40859
   279
  shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (sigma (pair_algebra M2 M1))"
hoelzl@40859
   280
    (is "?f \<in> measurable ?P ?Q")
hoelzl@40859
   281
  unfolding measurable_def
hoelzl@40859
   282
proof (intro CollectI conjI Pi_I ballI)
hoelzl@40859
   283
  fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
hoelzl@40859
   284
    unfolding pair_algebra_def by auto
hoelzl@40859
   285
next
hoelzl@40859
   286
  fix A assume "A \<in> sets ?Q"
hoelzl@40859
   287
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   288
  have "?f -` A \<inter> space ?P = (\<lambda>(x,y). (y, x)) ` A"
hoelzl@40859
   289
    using Q.sets_into_space `A \<in> sets ?Q` by (auto simp: pair_algebra_def)
hoelzl@40859
   290
  with Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets ?Q`]
hoelzl@40859
   291
  show "?f -` A \<inter> space ?P \<in> sets ?P" by simp
hoelzl@40859
   292
qed
hoelzl@40859
   293
hoelzl@40859
   294
lemma (in pair_sigma_algebra) measurable_cut_fst:
hoelzl@40859
   295
  assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"
hoelzl@40859
   296
proof -
hoelzl@40859
   297
  let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"
hoelzl@40859
   298
  let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
hoelzl@40859
   299
  interpret Q: sigma_algebra ?Q
hoelzl@40859
   300
    proof qed (auto simp: vimage_UN vimage_Diff space_pair_algebra)
hoelzl@40859
   301
  have "sets E \<subseteq> sets ?Q"
hoelzl@40859
   302
    using M1.sets_into_space M2.sets_into_space
hoelzl@40859
   303
    by (auto simp: pair_algebra_def space_pair_algebra)
hoelzl@40859
   304
  then have "sets P \<subseteq> sets ?Q"
hoelzl@40859
   305
    by (subst pair_algebra_def, intro Q.sets_sigma_subset)
hoelzl@40859
   306
       (simp_all add: pair_algebra_def)
hoelzl@40859
   307
  with assms show ?thesis by auto
hoelzl@40859
   308
qed
hoelzl@40859
   309
hoelzl@40859
   310
lemma (in pair_sigma_algebra) measurable_cut_snd:
hoelzl@40859
   311
  assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1")
hoelzl@40859
   312
proof -
hoelzl@40859
   313
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   314
  have "Pair y -` (\<lambda>(x, y). (y, x)) ` Q = (\<lambda>x. (x, y)) -` Q" by auto
hoelzl@40859
   315
  with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
hoelzl@40859
   316
  show ?thesis by simp
hoelzl@40859
   317
qed
hoelzl@40859
   318
hoelzl@40859
   319
lemma (in pair_sigma_algebra) measurable_pair_image_snd:
hoelzl@40859
   320
  assumes m: "f \<in> measurable P M" and "x \<in> space M1"
hoelzl@40859
   321
  shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
hoelzl@40859
   322
  unfolding measurable_def
hoelzl@40859
   323
proof (intro CollectI conjI Pi_I ballI)
hoelzl@40859
   324
  fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1`
hoelzl@40859
   325
  show "f (x, y) \<in> space M" unfolding measurable_def pair_algebra_def by auto
hoelzl@40859
   326
next
hoelzl@40859
   327
  fix A assume "A \<in> sets M"
hoelzl@40859
   328
  then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
hoelzl@40859
   329
    using `f \<in> measurable P M`
hoelzl@40859
   330
    by (intro measurable_cut_fst) (auto simp: measurable_def)
hoelzl@40859
   331
  also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2"
hoelzl@40859
   332
    using `x \<in> space M1` by (auto simp: pair_algebra_def)
hoelzl@40859
   333
  finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" .
hoelzl@40859
   334
qed
hoelzl@40859
   335
hoelzl@40859
   336
lemma (in pair_sigma_algebra) measurable_pair_image_fst:
hoelzl@40859
   337
  assumes m: "f \<in> measurable P M" and "y \<in> space M2"
hoelzl@40859
   338
  shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
hoelzl@40859
   339
proof -
hoelzl@40859
   340
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   341
  from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`,
hoelzl@40859
   342
                                      OF Q.pair_sigma_algebra_swap_measurable m]
hoelzl@40859
   343
  show ?thesis by simp
hoelzl@40859
   344
qed
hoelzl@40859
   345
hoelzl@40859
   346
lemma (in pair_sigma_algebra) Int_stable_pair_algebra: "Int_stable E"
hoelzl@40859
   347
  unfolding Int_stable_def
hoelzl@40859
   348
proof (intro ballI)
hoelzl@40859
   349
  fix A B assume "A \<in> sets E" "B \<in> sets E"
hoelzl@40859
   350
  then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
hoelzl@40859
   351
    "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
hoelzl@40859
   352
    unfolding pair_algebra_def by auto
hoelzl@40859
   353
  then show "A \<inter> B \<in> sets E"
hoelzl@40859
   354
    by (auto simp add: times_Int_times pair_algebra_def)
hoelzl@40859
   355
qed
hoelzl@40859
   356
hoelzl@40859
   357
lemma finite_measure_cut_measurable:
hoelzl@40859
   358
  fixes M1 :: "'a algebra" and M2 :: "'b algebra"
hoelzl@40859
   359
  assumes "sigma_finite_measure M1 \<mu>1" "finite_measure M2 \<mu>2"
hoelzl@40859
   360
  assumes "Q \<in> sets (sigma (pair_algebra M1 M2))"
hoelzl@40859
   361
  shows "(\<lambda>x. \<mu>2 (Pair x -` Q)) \<in> borel_measurable M1"
hoelzl@40859
   362
    (is "?s Q \<in> _")
hoelzl@40859
   363
proof -
hoelzl@40859
   364
  interpret M1: sigma_finite_measure M1 \<mu>1 by fact
hoelzl@40859
   365
  interpret M2: finite_measure M2 \<mu>2 by fact
hoelzl@40859
   366
  interpret pair_sigma_algebra M1 M2 by default
hoelzl@40859
   367
  have [intro]: "sigma_algebra M1" by fact
hoelzl@40859
   368
  have [intro]: "sigma_algebra M2" by fact
hoelzl@40859
   369
  let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"
hoelzl@40859
   370
  note space_pair_algebra[simp]
hoelzl@40859
   371
  interpret dynkin_system ?D
hoelzl@40859
   372
  proof (intro dynkin_systemI)
hoelzl@40859
   373
    fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
hoelzl@40859
   374
      using sets_into_space by simp
hoelzl@40859
   375
  next
hoelzl@40859
   376
    from top show "space ?D \<in> sets ?D"
hoelzl@40859
   377
      by (auto simp add: if_distrib intro!: M1.measurable_If)
hoelzl@40859
   378
  next
hoelzl@40859
   379
    fix A assume "A \<in> sets ?D"
hoelzl@40859
   380
    with sets_into_space have "\<And>x. \<mu>2 (Pair x -` (space M1 \<times> space M2 - A)) =
hoelzl@40859
   381
        (if x \<in> space M1 then \<mu>2 (space M2) - ?s A x else 0)"
hoelzl@40859
   382
      by (auto intro!: M2.finite_measure_compl measurable_cut_fst
hoelzl@40859
   383
               simp: vimage_Diff)
hoelzl@40859
   384
    with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"
hoelzl@41023
   385
      by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pextreal_diff)
hoelzl@40859
   386
  next
hoelzl@40859
   387
    fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
hoelzl@40859
   388
    moreover then have "\<And>x. \<mu>2 (\<Union>i. Pair x -` F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)"
hoelzl@40859
   389
      by (intro M2.measure_countably_additive[symmetric])
hoelzl@40859
   390
         (auto intro!: measurable_cut_fst simp: disjoint_family_on_def)
hoelzl@40859
   391
    ultimately show "(\<Union>i. F i) \<in> sets ?D"
hoelzl@40859
   392
      by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
hoelzl@40859
   393
  qed
hoelzl@40859
   394
  have "P = ?D"
hoelzl@40859
   395
  proof (intro dynkin_lemma)
hoelzl@40859
   396
    show "Int_stable E" by (rule Int_stable_pair_algebra)
hoelzl@40859
   397
    from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
hoelzl@40859
   398
      by auto
hoelzl@40859
   399
    then show "sets E \<subseteq> sets ?D"
hoelzl@40859
   400
      by (auto simp: pair_algebra_def sets_sigma if_distrib
hoelzl@40859
   401
               intro: sigma_sets.Basic intro!: M1.measurable_If)
hoelzl@40859
   402
  qed auto
hoelzl@40859
   403
  with `Q \<in> sets P` have "Q \<in> sets ?D" by simp
hoelzl@40859
   404
  then show "?s Q \<in> borel_measurable M1" by simp
hoelzl@40859
   405
qed
hoelzl@40859
   406
hoelzl@40859
   407
subsection {* Binary products of $\sigma$-finite measure spaces *}
hoelzl@40859
   408
hoelzl@40859
   409
locale pair_sigma_finite = M1: sigma_finite_measure M1 \<mu>1 + M2: sigma_finite_measure M2 \<mu>2
hoelzl@40859
   410
  for M1 \<mu>1 M2 \<mu>2
hoelzl@40859
   411
hoelzl@40859
   412
sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
hoelzl@40859
   413
  by default
hoelzl@40859
   414
hoelzl@40859
   415
lemma (in pair_sigma_finite) measure_cut_measurable_fst:
hoelzl@40859
   416
  assumes "Q \<in> sets P" shows "(\<lambda>x. \<mu>2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
hoelzl@40859
   417
proof -
hoelzl@40859
   418
  have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
hoelzl@40859
   419
  have M1: "sigma_finite_measure M1 \<mu>1" by default
hoelzl@40859
   420
hoelzl@40859
   421
  from M2.disjoint_sigma_finite guess F .. note F = this
hoelzl@40859
   422
  let "?C x i" = "F i \<inter> Pair x -` Q"
hoelzl@40859
   423
  { fix i
hoelzl@40859
   424
    let ?R = "M2.restricted_space (F i)"
hoelzl@40859
   425
    have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
hoelzl@40859
   426
      using F M2.sets_into_space by auto
hoelzl@40859
   427
    have "(\<lambda>x. \<mu>2 (Pair x -` (space M1 \<times> F i \<inter> Q))) \<in> borel_measurable M1"
hoelzl@40859
   428
    proof (intro finite_measure_cut_measurable[OF M1])
hoelzl@40859
   429
      show "finite_measure (M2.restricted_space (F i)) \<mu>2"
hoelzl@40859
   430
        using F by (intro M2.restricted_to_finite_measure) auto
hoelzl@40859
   431
      have "space M1 \<times> F i \<in> sets P"
hoelzl@40859
   432
        using M1.top F by blast
hoelzl@40859
   433
      from sigma_sets_Int[symmetric,
hoelzl@40859
   434
        OF this[unfolded pair_sigma_algebra_def sets_sigma]]
hoelzl@40859
   435
      show "(space M1 \<times> F i) \<inter> Q \<in> sets (sigma (pair_algebra M1 ?R))"
hoelzl@40859
   436
        using `Q \<in> sets P`
hoelzl@40859
   437
        using pair_algebra_Int_snd[OF M1.space_closed, of "F i" M2]
hoelzl@40859
   438
        by (auto simp: pair_algebra_def sets_sigma)
hoelzl@40859
   439
    qed
hoelzl@40859
   440
    moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"
hoelzl@40859
   441
      using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_algebra)
hoelzl@40859
   442
    ultimately have "(\<lambda>x. \<mu>2 (?C x i)) \<in> borel_measurable M1"
hoelzl@40859
   443
      by simp }
hoelzl@40859
   444
  moreover
hoelzl@40859
   445
  { fix x
hoelzl@40859
   446
    have "(\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i)) = \<mu>2 (\<Union>i. ?C x i)"
hoelzl@40859
   447
    proof (intro M2.measure_countably_additive)
hoelzl@40859
   448
      show "range (?C x) \<subseteq> sets M2"
hoelzl@40859
   449
        using F `Q \<in> sets P` by (auto intro!: M2.Int measurable_cut_fst)
hoelzl@40859
   450
      have "disjoint_family F" using F by auto
hoelzl@40859
   451
      show "disjoint_family (?C x)"
hoelzl@40859
   452
        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
hoelzl@40859
   453
    qed
hoelzl@40859
   454
    also have "(\<Union>i. ?C x i) = Pair x -` Q"
hoelzl@40859
   455
      using F sets_into_space `Q \<in> sets P`
hoelzl@40859
   456
      by (auto simp: space_pair_algebra)
hoelzl@40859
   457
    finally have "\<mu>2 (Pair x -` Q) = (\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i))"
hoelzl@40859
   458
      by simp }
hoelzl@40859
   459
  ultimately show ?thesis
hoelzl@40859
   460
    by (auto intro!: M1.borel_measurable_psuminf)
hoelzl@40859
   461
qed
hoelzl@40859
   462
hoelzl@40859
   463
lemma (in pair_sigma_finite) measure_cut_measurable_snd:
hoelzl@40859
   464
  assumes "Q \<in> sets P" shows "(\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
hoelzl@40859
   465
proof -
hoelzl@40859
   466
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@40859
   467
  have [simp]: "\<And>y. (Pair y -` (\<lambda>(x, y). (y, x)) ` Q) = (\<lambda>x. (x, y)) -` Q"
hoelzl@40859
   468
    by auto
hoelzl@40859
   469
  note sets_pair_sigma_algebra_swap[OF assms]
hoelzl@40859
   470
  from Q.measure_cut_measurable_fst[OF this]
hoelzl@40859
   471
  show ?thesis by simp
hoelzl@40859
   472
qed
hoelzl@40859
   473
hoelzl@40859
   474
lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
hoelzl@40859
   475
  assumes "f \<in> measurable P M"
hoelzl@40859
   476
  shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (sigma (pair_algebra M2 M1)) M"
hoelzl@40859
   477
proof -
hoelzl@40859
   478
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   479
  have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
hoelzl@40859
   480
  show ?thesis
hoelzl@40859
   481
    using Q.pair_sigma_algebra_swap_measurable assms
hoelzl@40859
   482
    unfolding * by (rule measurable_comp)
hoelzl@39088
   483
qed
hoelzl@39088
   484
hoelzl@40859
   485
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap:
hoelzl@40859
   486
  "sigma (pair_algebra M2 M1) = (vimage_algebra (space M2 \<times> space M1) (\<lambda>(x, y). (y, x)))"
hoelzl@40859
   487
  unfolding vimage_algebra_def
hoelzl@40859
   488
  apply (simp add: sets_sigma)
hoelzl@40859
   489
  apply (subst sigma_sets_vimage[symmetric])
hoelzl@40859
   490
  apply (fastsimp simp: pair_algebra_def)
hoelzl@40859
   491
  using M1.sets_into_space M2.sets_into_space apply (fastsimp simp: pair_algebra_def)
hoelzl@40859
   492
proof -
hoelzl@40859
   493
  have "(\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E
hoelzl@40859
   494
    = sets (pair_algebra M2 M1)" (is "?S = _")
hoelzl@40859
   495
    by (rule pair_algebra_swap)
hoelzl@40859
   496
  then show "sigma (pair_algebra M2 M1) = \<lparr>space = space M2 \<times> space M1,
hoelzl@40859
   497
       sets = sigma_sets (space M2 \<times> space M1) ?S\<rparr>"
hoelzl@40859
   498
    by (simp add: pair_algebra_def sigma_def)
hoelzl@40859
   499
qed
hoelzl@40859
   500
hoelzl@40859
   501
definition (in pair_sigma_finite)
hoelzl@40859
   502
  "pair_measure A = M1.positive_integral (\<lambda>x.
hoelzl@40859
   503
    M2.positive_integral (\<lambda>y. indicator A (x, y)))"
hoelzl@40859
   504
hoelzl@40859
   505
lemma (in pair_sigma_finite) pair_measure_alt:
hoelzl@40859
   506
  assumes "A \<in> sets P"
hoelzl@40859
   507
  shows "pair_measure A = M1.positive_integral (\<lambda>x. \<mu>2 (Pair x -` A))"
hoelzl@40859
   508
  unfolding pair_measure_def
hoelzl@40859
   509
proof (rule M1.positive_integral_cong)
hoelzl@40859
   510
  fix x assume "x \<in> space M1"
hoelzl@41023
   511
  have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: pextreal)"
hoelzl@40859
   512
    unfolding indicator_def by auto
hoelzl@40859
   513
  show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x -` A)"
hoelzl@40859
   514
    unfolding *
hoelzl@40859
   515
    apply (subst M2.positive_integral_indicator)
hoelzl@40859
   516
    apply (rule measurable_cut_fst[OF assms])
hoelzl@40859
   517
    by simp
hoelzl@40859
   518
qed
hoelzl@40859
   519
hoelzl@40859
   520
lemma (in pair_sigma_finite) pair_measure_times:
hoelzl@40859
   521
  assumes A: "A \<in> sets M1" and "B \<in> sets M2"
hoelzl@40859
   522
  shows "pair_measure (A \<times> B) = \<mu>1 A * \<mu>2 B"
hoelzl@40859
   523
proof -
hoelzl@40859
   524
  from assms have "pair_measure (A \<times> B) =
hoelzl@40859
   525
      M1.positive_integral (\<lambda>x. \<mu>2 B * indicator A x)"
hoelzl@40859
   526
    by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
hoelzl@40859
   527
  with assms show ?thesis
hoelzl@40859
   528
    by (simp add: M1.positive_integral_cmult_indicator ac_simps)
hoelzl@40859
   529
qed
hoelzl@40859
   530
hoelzl@40859
   531
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra:
hoelzl@40859
   532
  "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and>
hoelzl@40859
   533
    (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"
hoelzl@40859
   534
proof -
hoelzl@40859
   535
  obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
hoelzl@40859
   536
    F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. \<mu>1 (F1 i) \<noteq> \<omega>" and
hoelzl@40859
   537
    F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. \<mu>2 (F2 i) \<noteq> \<omega>"
hoelzl@40859
   538
    using M1.sigma_finite_up M2.sigma_finite_up by auto
hoelzl@40859
   539
  then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)"
hoelzl@40859
   540
    unfolding isoton_def by auto
hoelzl@40859
   541
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
hoelzl@40859
   542
  show ?thesis unfolding isoton_def space_pair_algebra
hoelzl@40859
   543
  proof (intro exI[of _ ?F] conjI allI)
hoelzl@40859
   544
    show "range ?F \<subseteq> sets E" using F1 F2
hoelzl@40859
   545
      by (fastsimp intro!: pair_algebraI)
hoelzl@40859
   546
  next
hoelzl@40859
   547
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
hoelzl@40859
   548
    proof (intro subsetI)
hoelzl@40859
   549
      fix x assume "x \<in> space M1 \<times> space M2"
hoelzl@40859
   550
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
hoelzl@40859
   551
        by (auto simp: space)
hoelzl@40859
   552
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
hoelzl@40859
   553
        using `F1 \<up> space M1` `F2 \<up> space M2`
hoelzl@40859
   554
        by (auto simp: max_def dest: isoton_mono_le)
hoelzl@40859
   555
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
hoelzl@40859
   556
        by (intro SigmaI) (auto simp add: min_max.sup_commute)
hoelzl@40859
   557
      then show "x \<in> (\<Union>i. ?F i)" by auto
hoelzl@40859
   558
    qed
hoelzl@40859
   559
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
hoelzl@40859
   560
      using space by (auto simp: space)
hoelzl@40859
   561
  next
hoelzl@40859
   562
    fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)"
hoelzl@40859
   563
      using `F1 \<up> space M1` `F2 \<up> space M2` unfolding isoton_def
hoelzl@40859
   564
      by auto
hoelzl@40859
   565
  next
hoelzl@40859
   566
    fix i
hoelzl@40859
   567
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
hoelzl@40859
   568
    with F1 F2 show "pair_measure (F1 i \<times> F2 i) \<noteq> \<omega>"
hoelzl@40859
   569
      by (simp add: pair_measure_times)
hoelzl@40859
   570
  qed
hoelzl@40859
   571
qed
hoelzl@40859
   572
hoelzl@40859
   573
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure
hoelzl@40859
   574
proof
hoelzl@40859
   575
  show "pair_measure {} = 0"
hoelzl@40859
   576
    unfolding pair_measure_def by auto
hoelzl@40859
   577
hoelzl@40859
   578
  show "countably_additive P pair_measure"
hoelzl@40859
   579
    unfolding countably_additive_def
hoelzl@40859
   580
  proof (intro allI impI)
hoelzl@40859
   581
    fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
hoelzl@40859
   582
    assume F: "range F \<subseteq> sets P" "disjoint_family F"
hoelzl@40859
   583
    from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
hoelzl@40859
   584
    moreover from F have "\<And>i. (\<lambda>x. \<mu>2 (Pair x -` F i)) \<in> borel_measurable M1"
hoelzl@40859
   585
      by (intro measure_cut_measurable_fst) auto
hoelzl@40859
   586
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
hoelzl@40859
   587
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@40859
   588
    moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
hoelzl@40859
   589
      using F by (auto intro!: measurable_cut_fst)
hoelzl@40859
   590
    ultimately show "(\<Sum>\<^isub>\<infinity>n. pair_measure (F n)) = pair_measure (\<Union>i. F i)"
hoelzl@40859
   591
      by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric]
hoelzl@40859
   592
                    M2.measure_countably_additive
hoelzl@40859
   593
               cong: M1.positive_integral_cong)
hoelzl@40859
   594
  qed
hoelzl@40859
   595
hoelzl@40859
   596
  from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
hoelzl@40859
   597
  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>)"
hoelzl@40859
   598
  proof (rule exI[of _ F], intro conjI)
hoelzl@40859
   599
    show "range F \<subseteq> sets P" using F by auto
hoelzl@40859
   600
    show "(\<Union>i. F i) = space P"
hoelzl@40859
   601
      using F by (auto simp: space_pair_algebra isoton_def)
hoelzl@40859
   602
    show "\<forall>i. pair_measure (F i) \<noteq> \<omega>" using F by auto
hoelzl@40859
   603
  qed
hoelzl@40859
   604
qed
hoelzl@39088
   605
hoelzl@40859
   606
lemma (in pair_sigma_finite) pair_measure_alt2:
hoelzl@40859
   607
  assumes "A \<in> sets P"
hoelzl@40859
   608
  shows "pair_measure A = M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` A))"
hoelzl@40859
   609
    (is "_ = ?\<nu> A")
hoelzl@40859
   610
proof -
hoelzl@40859
   611
  from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
hoelzl@40859
   612
  show ?thesis
hoelzl@40859
   613
  proof (rule measure_unique_Int_stable[where \<nu>="?\<nu>", OF Int_stable_pair_algebra],
hoelzl@40859
   614
         simp_all add: pair_sigma_algebra_def[symmetric])
hoelzl@40859
   615
    show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. pair_measure (F i) \<noteq> \<omega>"
hoelzl@40859
   616
      using F by auto
hoelzl@40859
   617
    show "measure_space P pair_measure" by default
hoelzl@40859
   618
  next
hoelzl@40859
   619
    show "measure_space P ?\<nu>"
hoelzl@40859
   620
    proof
hoelzl@40859
   621
      show "?\<nu> {} = 0" by auto
hoelzl@40859
   622
      show "countably_additive P ?\<nu>"
hoelzl@40859
   623
        unfolding countably_additive_def
hoelzl@40859
   624
      proof (intro allI impI)
hoelzl@40859
   625
        fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
hoelzl@40859
   626
        assume F: "range F \<subseteq> sets P" "disjoint_family F"
hoelzl@40859
   627
        from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
hoelzl@40859
   628
        moreover from F have "\<And>i. (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` F i)) \<in> borel_measurable M2"
hoelzl@40859
   629
          by (intro measure_cut_measurable_snd) auto
hoelzl@40859
   630
        moreover have "\<And>y. disjoint_family (\<lambda>i. (\<lambda>x. (x, y)) -` F i)"
hoelzl@40859
   631
          by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@40859
   632
        moreover have "\<And>y. y \<in> space M2 \<Longrightarrow> range (\<lambda>i. (\<lambda>x. (x, y)) -` F i) \<subseteq> sets M1"
hoelzl@40859
   633
          using F by (auto intro!: measurable_cut_snd)
hoelzl@40859
   634
        ultimately show "(\<Sum>\<^isub>\<infinity>n. ?\<nu> (F n)) = ?\<nu> (\<Union>i. F i)"
hoelzl@40859
   635
          by (simp add: vimage_UN M2.positive_integral_psuminf[symmetric]
hoelzl@40859
   636
                        M1.measure_countably_additive
hoelzl@40859
   637
                   cong: M2.positive_integral_cong)
hoelzl@40859
   638
      qed
hoelzl@40859
   639
    qed
hoelzl@40859
   640
  next
hoelzl@40859
   641
    fix X assume "X \<in> sets E"
hoelzl@40859
   642
    then obtain A B where X: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
hoelzl@40859
   643
      unfolding pair_algebra_def by auto
hoelzl@40859
   644
    show "pair_measure X = ?\<nu> X"
hoelzl@40859
   645
    proof -
hoelzl@40859
   646
      from AB have "?\<nu> (A \<times> B) =
hoelzl@40859
   647
          M2.positive_integral (\<lambda>y. \<mu>1 A * indicator B y)"
hoelzl@40859
   648
        by (auto intro!: M2.positive_integral_cong)
hoelzl@40859
   649
      with AB show ?thesis
hoelzl@40859
   650
        unfolding pair_measure_times[OF AB] X
hoelzl@40859
   651
        by (simp add: M2.positive_integral_cmult_indicator ac_simps)
hoelzl@40859
   652
    qed
hoelzl@40859
   653
  qed fact
hoelzl@40859
   654
qed
hoelzl@40859
   655
hoelzl@40859
   656
section "Fubinis theorem"
hoelzl@40859
   657
hoelzl@40859
   658
lemma (in pair_sigma_finite) simple_function_cut:
hoelzl@40859
   659
  assumes f: "simple_function f"
hoelzl@40859
   660
  shows "(\<lambda>x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
hoelzl@40859
   661
    and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
hoelzl@40859
   662
      = positive_integral f"
hoelzl@40859
   663
proof -
hoelzl@40859
   664
  have f_borel: "f \<in> borel_measurable P"
hoelzl@40859
   665
    using f by (rule borel_measurable_simple_function)
hoelzl@40859
   666
  let "?F z" = "f -` {z} \<inter> space P"
hoelzl@40859
   667
  let "?F' x z" = "Pair x -` ?F z"
hoelzl@40859
   668
  { fix x assume "x \<in> space M1"
hoelzl@40859
   669
    have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
hoelzl@40859
   670
      by (auto simp: indicator_def)
hoelzl@40859
   671
    have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1`
hoelzl@40859
   672
      by (simp add: space_pair_algebra)
hoelzl@40859
   673
    moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
hoelzl@40859
   674
      by (intro borel_measurable_vimage measurable_cut_fst)
hoelzl@40859
   675
    ultimately have "M2.simple_function (\<lambda> y. f (x, y))"
hoelzl@40859
   676
      apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
hoelzl@40859
   677
      apply (rule simple_function_indicator_representation[OF f])
hoelzl@40859
   678
      using `x \<in> space M1` by (auto simp del: space_sigma) }
hoelzl@40859
   679
  note M2_sf = this
hoelzl@40859
   680
  { fix x assume x: "x \<in> space M1"
hoelzl@40859
   681
    then have "M2.positive_integral (\<lambda> y. f (x, y)) =
hoelzl@40859
   682
        (\<Sum>z\<in>f ` space P. z * \<mu>2 (?F' x z))"
hoelzl@40859
   683
      unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]]
hoelzl@40859
   684
      unfolding M2.simple_integral_def
hoelzl@40859
   685
    proof (safe intro!: setsum_mono_zero_cong_left)
hoelzl@40859
   686
      from f show "finite (f ` space P)" by (rule simple_functionD)
hoelzl@40859
   687
    next
hoelzl@40859
   688
      fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
hoelzl@40859
   689
        using `x \<in> space M1` by (auto simp: space_pair_algebra)
hoelzl@40859
   690
    next
hoelzl@40859
   691
      fix x' y assume "(x', y) \<in> space P"
hoelzl@40859
   692
        "f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
hoelzl@40859
   693
      then have *: "?F' x (f (x', y)) = {}"
hoelzl@40859
   694
        by (force simp: space_pair_algebra)
hoelzl@40859
   695
      show  "f (x', y) * \<mu>2 (?F' x (f (x', y))) = 0"
hoelzl@40859
   696
        unfolding * by simp
hoelzl@40859
   697
    qed (simp add: vimage_compose[symmetric] comp_def
hoelzl@40859
   698
                   space_pair_algebra) }
hoelzl@40859
   699
  note eq = this
hoelzl@40859
   700
  moreover have "\<And>z. ?F z \<in> sets P"
hoelzl@40859
   701
    by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
hoelzl@40859
   702
  moreover then have "\<And>z. (\<lambda>x. \<mu>2 (?F' x z)) \<in> borel_measurable M1"
hoelzl@40859
   703
    by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
hoelzl@40859
   704
  ultimately
hoelzl@40859
   705
  show "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
hoelzl@40859
   706
    and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
hoelzl@40859
   707
    = positive_integral f"
hoelzl@40859
   708
    by (auto simp del: vimage_Int cong: measurable_cong
hoelzl@41023
   709
             intro!: M1.borel_measurable_pextreal_setsum
hoelzl@40859
   710
             simp add: M1.positive_integral_setsum simple_integral_def
hoelzl@40859
   711
                       M1.positive_integral_cmult
hoelzl@40859
   712
                       M1.positive_integral_cong[OF eq]
hoelzl@40859
   713
                       positive_integral_eq_simple_integral[OF f]
hoelzl@40859
   714
                       pair_measure_alt[symmetric])
hoelzl@40859
   715
qed
hoelzl@40859
   716
hoelzl@40859
   717
lemma (in pair_sigma_finite) positive_integral_fst_measurable:
hoelzl@40859
   718
  assumes f: "f \<in> borel_measurable P"
hoelzl@40859
   719
  shows "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
hoelzl@40859
   720
      (is "?C f \<in> borel_measurable M1")
hoelzl@40859
   721
    and "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
hoelzl@40859
   722
      positive_integral f"
hoelzl@40859
   723
proof -
hoelzl@40859
   724
  from borel_measurable_implies_simple_function_sequence[OF f]
hoelzl@40859
   725
  obtain F where F: "\<And>i. simple_function (F i)" "F \<up> f" by auto
hoelzl@40859
   726
  then have F_borel: "\<And>i. F i \<in> borel_measurable P"
hoelzl@40859
   727
    and F_mono: "\<And>i x. F i x \<le> F (Suc i) x"
hoelzl@40859
   728
    and F_SUPR: "\<And>x. (SUP i. F i x) = f x"
hoelzl@40859
   729
    unfolding isoton_def le_fun_def SUPR_fun_expand
hoelzl@40859
   730
    by (auto intro: borel_measurable_simple_function)
hoelzl@40859
   731
  note sf = simple_function_cut[OF F(1)]
hoelzl@40859
   732
  then have "(SUP i. ?C (F i)) \<in> borel_measurable M1"
hoelzl@40859
   733
    using F(1) by (auto intro!: M1.borel_measurable_SUP)
hoelzl@40859
   734
  moreover
hoelzl@40859
   735
  { fix x assume "x \<in> space M1"
hoelzl@40859
   736
    have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))"
hoelzl@40859
   737
      using `F \<up> f` unfolding isoton_fun_expand
hoelzl@40859
   738
      by (auto simp: isoton_def)
hoelzl@40859
   739
    note measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
hoelzl@40859
   740
    from M2.positive_integral_isoton[OF isotone this]
hoelzl@40859
   741
    have "(SUP i. ?C (F i) x) = ?C f x"
hoelzl@40859
   742
      by (simp add: isoton_def) }
hoelzl@40859
   743
  note SUPR_C = this
hoelzl@40859
   744
  ultimately show "?C f \<in> borel_measurable M1"
hoelzl@40859
   745
    unfolding SUPR_fun_expand by (simp cong: measurable_cong)
hoelzl@40859
   746
  have "positive_integral (\<lambda>x. SUP i. F i x) = (SUP i. positive_integral (F i))"
hoelzl@40859
   747
    using F_borel F_mono
hoelzl@40859
   748
    by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])
hoelzl@40859
   749
  also have "(SUP i. positive_integral (F i)) =
hoelzl@40859
   750
    (SUP i. M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. F i (x, y))))"
hoelzl@40859
   751
    unfolding sf(2) by simp
hoelzl@40859
   752
  also have "\<dots> = M1.positive_integral (\<lambda>x. SUP i. M2.positive_integral (\<lambda>y. F i (x, y)))"
hoelzl@40859
   753
    by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)]
hoelzl@40859
   754
                     M2.positive_integral_mono F_mono)
hoelzl@40859
   755
  also have "\<dots> = M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. SUP i. F i (x, y)))"
hoelzl@40859
   756
    using F_borel F_mono
hoelzl@40859
   757
    by (auto intro!: M2.positive_integral_monotone_convergence_SUP
hoelzl@40859
   758
                     M1.positive_integral_cong measurable_pair_image_snd)
hoelzl@40859
   759
  finally show "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
hoelzl@40859
   760
      positive_integral f"
hoelzl@40859
   761
    unfolding F_SUPR by simp
hoelzl@40859
   762
qed
hoelzl@40859
   763
hoelzl@40859
   764
lemma (in pair_sigma_finite) positive_integral_snd_measurable:
hoelzl@40859
   765
  assumes f: "f \<in> borel_measurable P"
hoelzl@40859
   766
  shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
hoelzl@40859
   767
      positive_integral f"
hoelzl@40859
   768
proof -
hoelzl@40859
   769
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@40859
   770
  have s: "\<And>x y. (case (x, y) of (x, y) \<Rightarrow> f (y, x)) = f (y, x)" by simp
hoelzl@40859
   771
  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)
hoelzl@40859
   772
  have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"
hoelzl@40859
   773
    by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
hoelzl@40859
   774
  note pair_sigma_algebra_measurable[OF f]
hoelzl@40859
   775
  from Q.positive_integral_fst_measurable[OF this]
hoelzl@40859
   776
  have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
hoelzl@40859
   777
    Q.positive_integral (\<lambda>(x, y). f (y, x))"
hoelzl@40859
   778
    by simp
hoelzl@40859
   779
  also have "\<dots> = positive_integral f"
hoelzl@40859
   780
    unfolding positive_integral_vimage[OF bij, of f] t
hoelzl@40859
   781
    unfolding pair_sigma_algebra_swap[symmetric]
hoelzl@40859
   782
  proof (rule Q.positive_integral_cong_measure[symmetric])
hoelzl@40859
   783
    fix A assume "A \<in> sets Q.P"
hoelzl@40859
   784
    from this Q.sets_pair_sigma_algebra_swap[OF this]
hoelzl@40859
   785
    show "pair_measure ((\<lambda>(x, y). (y, x)) ` A) = Q.pair_measure A"
hoelzl@40859
   786
      by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]
hoelzl@40859
   787
               simp: pair_measure_alt Q.pair_measure_alt2)
hoelzl@40859
   788
  qed
hoelzl@40859
   789
  finally show ?thesis .
hoelzl@40859
   790
qed
hoelzl@40859
   791
hoelzl@40859
   792
lemma (in pair_sigma_finite) Fubini:
hoelzl@40859
   793
  assumes f: "f \<in> borel_measurable P"
hoelzl@40859
   794
  shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
hoelzl@40859
   795
      M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))"
hoelzl@40859
   796
  unfolding positive_integral_snd_measurable[OF assms]
hoelzl@40859
   797
  unfolding positive_integral_fst_measurable[OF assms] ..
hoelzl@40859
   798
hoelzl@40859
   799
lemma (in pair_sigma_finite) AE_pair:
hoelzl@40859
   800
  assumes "almost_everywhere (\<lambda>x. Q x)"
hoelzl@40859
   801
  shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))"
hoelzl@40859
   802
proof -
hoelzl@40859
   803
  obtain N where N: "N \<in> sets P" "pair_measure N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
hoelzl@40859
   804
    using assms unfolding almost_everywhere_def by auto
hoelzl@40859
   805
  show ?thesis
hoelzl@40859
   806
  proof (rule M1.AE_I)
hoelzl@40859
   807
    from N measure_cut_measurable_fst[OF `N \<in> sets P`]
hoelzl@40859
   808
    show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x -` N) \<noteq> 0} = 0"
hoelzl@40859
   809
      by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)
hoelzl@40859
   810
    show "{x \<in> space M1. \<mu>2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
hoelzl@41023
   811
      by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)
hoelzl@40859
   812
    { fix x assume "x \<in> space M1" "\<mu>2 (Pair x -` N) = 0"
hoelzl@40859
   813
      have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
hoelzl@40859
   814
      proof (rule M2.AE_I)
hoelzl@40859
   815
        show "\<mu>2 (Pair x -` N) = 0" by fact
hoelzl@40859
   816
        show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)
hoelzl@40859
   817
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
hoelzl@40859
   818
          using N `x \<in> space M1` unfolding space_sigma space_pair_algebra by auto
hoelzl@40859
   819
      qed }
hoelzl@40859
   820
    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}"
hoelzl@40859
   821
      by auto
hoelzl@39088
   822
  qed
hoelzl@39088
   823
qed
hoelzl@35833
   824
hoelzl@41026
   825
lemma (in pair_sigma_finite) positive_integral_product_swap:
hoelzl@41026
   826
  "measure_space.positive_integral
hoelzl@41026
   827
    (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f =
hoelzl@41026
   828
  positive_integral (\<lambda>(x,y). f (y,x))"
hoelzl@41026
   829
proof -
hoelzl@41026
   830
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@41026
   831
  have t: "(\<lambda>y. case case y of (y, x) \<Rightarrow> (x, y) of (x, y) \<Rightarrow> f (y, x)) = f"
hoelzl@41026
   832
    by (auto simp: fun_eq_iff)
hoelzl@41026
   833
  have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"
hoelzl@41026
   834
    by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
hoelzl@41026
   835
  show ?thesis
hoelzl@41026
   836
    unfolding positive_integral_vimage[OF bij, of "\<lambda>(y,x). f (x,y)"]
hoelzl@41026
   837
    unfolding pair_sigma_algebra_swap[symmetric] t
hoelzl@41026
   838
  proof (rule Q.positive_integral_cong_measure[symmetric])
hoelzl@41026
   839
    fix A assume "A \<in> sets Q.P"
hoelzl@41026
   840
    from this Q.sets_pair_sigma_algebra_swap[OF this]
hoelzl@41026
   841
    show "pair_measure ((\<lambda>(x, y). (y, x)) ` A) = Q.pair_measure A"
hoelzl@41026
   842
      by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]
hoelzl@41026
   843
               simp: pair_measure_alt Q.pair_measure_alt2)
hoelzl@41026
   844
  qed
hoelzl@41026
   845
qed
hoelzl@41026
   846
hoelzl@41026
   847
lemma (in pair_sigma_algebra) measurable_product_swap:
hoelzl@41026
   848
  "f \<in> measurable (sigma (pair_algebra M2 M1)) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
hoelzl@41026
   849
proof -
hoelzl@41026
   850
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@41026
   851
  show ?thesis
hoelzl@41026
   852
    using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
hoelzl@41026
   853
    by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
hoelzl@41026
   854
qed
hoelzl@41026
   855
hoelzl@41026
   856
lemma (in pair_sigma_finite) integrable_product_swap:
hoelzl@41026
   857
  "measure_space.integrable
hoelzl@41026
   858
    (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f \<longleftrightarrow>
hoelzl@41026
   859
  integrable (\<lambda>(x,y). f (y,x))"
hoelzl@41026
   860
proof -
hoelzl@41026
   861
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@41026
   862
  show ?thesis
hoelzl@41026
   863
    unfolding Q.integrable_def integrable_def
hoelzl@41026
   864
    unfolding positive_integral_product_swap
hoelzl@41026
   865
    unfolding measurable_product_swap
hoelzl@41026
   866
    by (simp add: case_prod_distrib)
hoelzl@41026
   867
qed
hoelzl@41026
   868
hoelzl@41026
   869
lemma (in pair_sigma_finite) integral_product_swap:
hoelzl@41026
   870
  "measure_space.integral
hoelzl@41026
   871
    (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f =
hoelzl@41026
   872
  integral (\<lambda>(x,y). f (y,x))"
hoelzl@41026
   873
proof -
hoelzl@41026
   874
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@41026
   875
  show ?thesis
hoelzl@41026
   876
    unfolding integral_def Q.integral_def positive_integral_product_swap
hoelzl@41026
   877
    by (simp add: case_prod_distrib)
hoelzl@41026
   878
qed
hoelzl@41026
   879
hoelzl@41026
   880
lemma (in pair_sigma_finite) integrable_fst_measurable:
hoelzl@41026
   881
  assumes f: "integrable f"
hoelzl@41026
   882
  shows "M1.almost_everywhere (\<lambda>x. M2.integrable (\<lambda> y. f (x, y)))" (is "?AE")
hoelzl@41026
   883
    and "M1.integral (\<lambda> x. M2.integral (\<lambda> y. f (x, y))) = integral f" (is "?INT")
hoelzl@41026
   884
proof -
hoelzl@41026
   885
  let "?pf x" = "Real (f x)" and "?nf x" = "Real (- f x)"
hoelzl@41026
   886
  have
hoelzl@41026
   887
    borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
hoelzl@41026
   888
    int: "positive_integral ?nf \<noteq> \<omega>" "positive_integral ?pf \<noteq> \<omega>"
hoelzl@41026
   889
    using assms by auto
hoelzl@41026
   890
  have "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y)))) \<noteq> \<omega>"
hoelzl@41026
   891
     "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y)))) \<noteq> \<omega>"
hoelzl@41026
   892
    using borel[THEN positive_integral_fst_measurable(1)] int
hoelzl@41026
   893
    unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
hoelzl@41026
   894
  with borel[THEN positive_integral_fst_measurable(1)]
hoelzl@41026
   895
  have AE: "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y))) \<noteq> \<omega>)"
hoelzl@41026
   896
    "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y))) \<noteq> \<omega>)"
hoelzl@41026
   897
    by (auto intro!: M1.positive_integral_omega_AE)
hoelzl@41026
   898
  then show ?AE
hoelzl@41026
   899
    apply (rule M1.AE_mp[OF _ M1.AE_mp])
hoelzl@41026
   900
    apply (rule M1.AE_cong)
hoelzl@41026
   901
    using assms unfolding M2.integrable_def
hoelzl@41026
   902
    by (auto intro!: measurable_pair_image_snd)
hoelzl@41026
   903
  have "M1.integrable
hoelzl@41026
   904
     (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (f (x, xa)))))" (is "M1.integrable ?f")
hoelzl@41026
   905
  proof (unfold M1.integrable_def, intro conjI)
hoelzl@41026
   906
    show "?f \<in> borel_measurable M1"
hoelzl@41026
   907
      using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
hoelzl@41026
   908
    have "M1.positive_integral (\<lambda>x. Real (?f x)) =
hoelzl@41026
   909
        M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (f (x, xa))))"
hoelzl@41026
   910
      apply (rule M1.positive_integral_cong_AE)
hoelzl@41026
   911
      apply (rule M1.AE_mp[OF AE(1)])
hoelzl@41026
   912
      apply (rule M1.AE_cong)
hoelzl@41026
   913
      by (auto simp: Real_real)
hoelzl@41026
   914
    then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"
hoelzl@41026
   915
      using positive_integral_fst_measurable[OF borel(2)] int by simp
hoelzl@41026
   916
    have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"
hoelzl@41026
   917
      by (intro M1.positive_integral_cong) simp
hoelzl@41026
   918
    then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp
hoelzl@41026
   919
  qed
hoelzl@41026
   920
  moreover have "M1.integrable
hoelzl@41026
   921
     (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (- f (x, xa)))))" (is "M1.integrable ?f")
hoelzl@41026
   922
  proof (unfold M1.integrable_def, intro conjI)
hoelzl@41026
   923
    show "?f \<in> borel_measurable M1"
hoelzl@41026
   924
      using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
hoelzl@41026
   925
    have "M1.positive_integral (\<lambda>x. Real (?f x)) =
hoelzl@41026
   926
        M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (- f (x, xa))))"
hoelzl@41026
   927
      apply (rule M1.positive_integral_cong_AE)
hoelzl@41026
   928
      apply (rule M1.AE_mp[OF AE(2)])
hoelzl@41026
   929
      apply (rule M1.AE_cong)
hoelzl@41026
   930
      by (auto simp: Real_real)
hoelzl@41026
   931
    then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"
hoelzl@41026
   932
      using positive_integral_fst_measurable[OF borel(1)] int by simp
hoelzl@41026
   933
    have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"
hoelzl@41026
   934
      by (intro M1.positive_integral_cong) simp
hoelzl@41026
   935
    then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp
hoelzl@41026
   936
  qed
hoelzl@41026
   937
  ultimately show ?INT
hoelzl@41026
   938
    unfolding M2.integral_def integral_def
hoelzl@41026
   939
      borel[THEN positive_integral_fst_measurable(2), symmetric]
hoelzl@41026
   940
    by (simp add: M1.integral_real[OF AE(1)] M1.integral_real[OF AE(2)])
hoelzl@41026
   941
qed
hoelzl@41026
   942
hoelzl@41026
   943
lemma (in pair_sigma_finite) integrable_snd_measurable:
hoelzl@41026
   944
  assumes f: "integrable f"
hoelzl@41026
   945
  shows "M2.almost_everywhere (\<lambda>y. M1.integrable (\<lambda>x. f (x, y)))" (is "?AE")
hoelzl@41026
   946
    and "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) = integral f" (is "?INT")
hoelzl@41026
   947
proof -
hoelzl@41026
   948
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@41026
   949
  have Q_int: "Q.integrable (\<lambda>(x, y). f (y, x))"
hoelzl@41026
   950
    using f unfolding integrable_product_swap by simp
hoelzl@41026
   951
  show ?INT
hoelzl@41026
   952
    using Q.integrable_fst_measurable(2)[OF Q_int]
hoelzl@41026
   953
    unfolding integral_product_swap by simp
hoelzl@41026
   954
  show ?AE
hoelzl@41026
   955
    using Q.integrable_fst_measurable(1)[OF Q_int]
hoelzl@41026
   956
    by simp
hoelzl@41026
   957
qed
hoelzl@41026
   958
hoelzl@41026
   959
lemma (in pair_sigma_finite) Fubini_integral:
hoelzl@41026
   960
  assumes f: "integrable f"
hoelzl@41026
   961
  shows "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) =
hoelzl@41026
   962
      M1.integral (\<lambda>x. M2.integral (\<lambda>y. f (x, y)))"
hoelzl@41026
   963
  unfolding integrable_snd_measurable[OF assms]
hoelzl@41026
   964
  unfolding integrable_fst_measurable[OF assms] ..
hoelzl@41026
   965
hoelzl@40859
   966
section "Finite product spaces"
hoelzl@40859
   967
hoelzl@40859
   968
section "Products"
hoelzl@40859
   969
hoelzl@40859
   970
locale product_sigma_algebra =
hoelzl@40859
   971
  fixes M :: "'i \<Rightarrow> 'a algebra"
hoelzl@40859
   972
  assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
hoelzl@40859
   973
hoelzl@40859
   974
locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" +
hoelzl@40859
   975
  fixes I :: "'i set"
hoelzl@40859
   976
  assumes finite_index: "finite I"
hoelzl@40859
   977
hoelzl@40859
   978
syntax
hoelzl@40859
   979
  "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
hoelzl@40859
   980
hoelzl@40859
   981
syntax (xsymbols)
hoelzl@40859
   982
  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
hoelzl@40859
   983
hoelzl@40859
   984
syntax (HTML output)
hoelzl@40859
   985
  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
hoelzl@40859
   986
hoelzl@40859
   987
translations
hoelzl@40859
   988
  "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
hoelzl@40859
   989
hoelzl@35833
   990
definition
hoelzl@40859
   991
  "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>"
hoelzl@40859
   992
hoelzl@40859
   993
abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I"
hoelzl@40859
   994
abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G"
hoelzl@40859
   995
hoelzl@40859
   996
sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
hoelzl@40859
   997
hoelzl@40859
   998
lemma (in finite_product_sigma_algebra) product_algebra_into_space:
hoelzl@40859
   999
  "sets G \<subseteq> Pow (space G)"
hoelzl@40859
  1000
  using M.sets_into_space unfolding product_algebra_def
hoelzl@40859
  1001
  by auto blast
hoelzl@40859
  1002
hoelzl@40859
  1003
sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
hoelzl@40859
  1004
  using product_algebra_into_space by (rule sigma_algebra_sigma)
hoelzl@40859
  1005
hoelzl@40859
  1006
lemma space_product_algebra[simp]:
hoelzl@40859
  1007
  "space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))"
hoelzl@40859
  1008
  unfolding product_algebra_def by simp
hoelzl@40859
  1009
hoelzl@40859
  1010
lemma (in finite_product_sigma_algebra) P_empty:
hoelzl@40859
  1011
  "I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"
hoelzl@40872
  1012
  unfolding product_algebra_def by (simp add: sigma_def image_constant)
hoelzl@40859
  1013
hoelzl@40859
  1014
lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
hoelzl@40859
  1015
  "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
hoelzl@40859
  1016
  by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic)
hoelzl@40859
  1017
hoelzl@40859
  1018
lemma bij_betw_prod_fold:
hoelzl@40859
  1019
  assumes "i \<notin> I"
hoelzl@40859
  1020
  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))"
hoelzl@40859
  1021
    (is "bij_betw ?f ?P ?F")
hoelzl@40859
  1022
    using `i \<notin> I`
hoelzl@40859
  1023
proof (unfold bij_betw_def, intro conjI set_eqI iffI)
hoelzl@40859
  1024
  show "inj_on ?f ?P"
hoelzl@40859
  1025
  proof (safe intro!: inj_onI ext)
hoelzl@40859
  1026
    fix a b x assume "a(i:=undefined) = b(i:=undefined)" "a i = b i"
hoelzl@40859
  1027
    then show "a x = b x"
hoelzl@40859
  1028
      by (cases "x = i") (auto simp: fun_eq_iff split: split_if_asm)
hoelzl@40859
  1029
  qed
hoelzl@40859
  1030
next
hoelzl@40859
  1031
  fix X assume *: "X \<in> ?F" show "X \<in> ?f ` ?P"
hoelzl@40859
  1032
  proof (cases X)
hoelzl@40859
  1033
    case (Pair a b) with * `i \<notin> I` show ?thesis
hoelzl@40859
  1034
      by (auto intro!: image_eqI[where x="a (i := b)"] ext simp: extensional_def)
hoelzl@40859
  1035
  qed
hoelzl@40859
  1036
qed auto
hoelzl@40859
  1037
hoelzl@41026
  1038
lemma borel_measurable_product_component:
hoelzl@41026
  1039
  assumes "i \<in> I"
hoelzl@41026
  1040
  shows "(\<lambda>x. x i) \<in> borel_measurable (sigma (product_algebra (\<lambda>x. borel) I))"
hoelzl@41026
  1041
proof -
hoelzl@41026
  1042
  have *: "\<And>A. (\<lambda>x. x i) -` A \<inter> extensional I = (\<Pi>\<^isub>E j\<in>I. if j = i then A else UNIV)"
hoelzl@41026
  1043
    using `i \<in> I` by (auto elim!: PiE)
hoelzl@41026
  1044
  show ?thesis
hoelzl@41026
  1045
    by (auto simp: * measurable_def product_algebra_def)
hoelzl@41026
  1046
qed
hoelzl@41026
  1047
hoelzl@40859
  1048
section "Generating set generates also product algebra"
hoelzl@40859
  1049
hoelzl@40859
  1050
lemma pair_sigma_algebra_sigma:
hoelzl@40859
  1051
  assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
hoelzl@40859
  1052
  assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
hoelzl@40859
  1053
  shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)"
hoelzl@40859
  1054
    (is "?S = ?E")
hoelzl@40859
  1055
proof -
hoelzl@40859
  1056
  interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
hoelzl@40859
  1057
  interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
hoelzl@40859
  1058
  have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
hoelzl@40859
  1059
    using E1 E2 by (auto simp add: pair_algebra_def)
hoelzl@40859
  1060
  interpret E: sigma_algebra ?E unfolding pair_algebra_def
hoelzl@40859
  1061
    using E1 E2 by (intro sigma_algebra_sigma) auto
hoelzl@40859
  1062
  { fix A assume "A \<in> sets E1"
hoelzl@40859
  1063
    then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
hoelzl@40859
  1064
      using E1 2 unfolding isoton_def pair_algebra_def by auto
hoelzl@40859
  1065
    also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
hoelzl@40859
  1066
    also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma
hoelzl@40859
  1067
      using 2 `A \<in> sets E1`
hoelzl@40859
  1068
      by (intro sigma_sets.Union)
hoelzl@40859
  1069
         (auto simp: image_subset_iff intro!: sigma_sets.Basic)
hoelzl@40859
  1070
    finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
hoelzl@40859
  1071
  moreover
hoelzl@40859
  1072
  { fix B assume "B \<in> sets E2"
hoelzl@40859
  1073
    then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
hoelzl@40859
  1074
      using E2 1 unfolding isoton_def pair_algebra_def by auto
hoelzl@40859
  1075
    also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
hoelzl@40859
  1076
    also have "\<dots> \<in> sets ?E"
hoelzl@40859
  1077
      using 1 `B \<in> sets E2` unfolding pair_algebra_def sets_sigma
hoelzl@40859
  1078
      by (intro sigma_sets.Union)
hoelzl@40859
  1079
         (auto simp: image_subset_iff intro!: sigma_sets.Basic)
hoelzl@40859
  1080
    finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
hoelzl@40859
  1081
  ultimately have proj:
hoelzl@40859
  1082
    "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
hoelzl@40859
  1083
    using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
hoelzl@40859
  1084
                   (auto simp: pair_algebra_def sets_sigma)
hoelzl@40859
  1085
  { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
hoelzl@40859
  1086
    with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
hoelzl@40859
  1087
      unfolding measurable_def by simp_all
hoelzl@40859
  1088
    moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"
hoelzl@40859
  1089
      using A B M1.sets_into_space M2.sets_into_space
hoelzl@40859
  1090
      by (auto simp: pair_algebra_def)
hoelzl@40859
  1091
    ultimately have "A \<times> B \<in> sets ?E" by auto }