src/HOL/Probability/Binary_Product_Measure.thy
author hoelzl
Wed Oct 10 12:12:18 2012 +0200 (2012-10-10)
changeset 49776 199d1d5bb17e
parent 47694 05663f75964c
child 49784 5e5b2da42a69
permissions -rw-r--r--
tuned product measurability
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(*  Title:      HOL/Probability/Binary_Product_Measure.thy
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    Author:     Johannes Hölzl, TU München
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*)
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header {*Binary product measures*}
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theory Binary_Product_Measure
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imports Lebesgue_Integration
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begin
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lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
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  by auto
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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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section "Binary products"
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definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
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  "A \<Otimes>\<^isub>M B = measure_of (space A \<times> space B)
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      {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
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      (\<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
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lemma space_pair_measure:
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  "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
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  unfolding pair_measure_def using space_closed[of A] space_closed[of B]
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  by (intro space_measure_of) auto
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lemma sets_pair_measure:
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  "sets (A \<Otimes>\<^isub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
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  unfolding pair_measure_def using space_closed[of A] space_closed[of B]
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  by (intro sets_measure_of) auto
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lemma sets_pair_measure_cong[cong]:
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  "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^isub>M M2) = sets (M1' \<Otimes>\<^isub>M M2')"
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  unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
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lemma pair_measureI[intro, simp]:
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  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
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  by (auto simp: sets_pair_measure)
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lemma measurable_pair_measureI:
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  assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
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  assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f -` (A \<times> B) \<inter> space M \<in> sets M"
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  shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
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  unfolding pair_measure_def using 1 2
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  by (intro measurable_measure_of) (auto dest: sets_into_space)
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lemma measurable_Pair:
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  assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
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  shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
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proof (rule measurable_pair_measureI)
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  show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
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    using f g by (auto simp: measurable_def)
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  fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
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  have "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
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    by auto
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  also have "\<dots> \<in> sets M"
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    by (rule Int) (auto intro!: measurable_sets * f g)
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  finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .
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qed
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lemma measurable_pair:
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  assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
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  shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
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  using measurable_Pair[OF assms] by simp
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lemma measurable_fst[intro!, simp]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1"
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  by (auto simp: fst_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)
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lemma measurable_snd[intro!, simp]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2"
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  by (auto simp: snd_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)
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lemma measurable_fst': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. fst (f x)) \<in> measurable M N"
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  using measurable_comp[OF _ measurable_fst] by (auto simp: comp_def)
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lemma measurable_snd': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. snd (f x)) \<in> measurable M P"
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    using measurable_comp[OF _ measurable_snd] by (auto simp: comp_def)
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lemma measurable_fst'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^isub>M P) N"
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  using measurable_comp[OF measurable_fst _] by (auto simp: comp_def)
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lemma measurable_snd'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^isub>M M) N"
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  using measurable_comp[OF measurable_snd _] by (auto simp: comp_def)
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lemma measurable_pair_iff:
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  "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
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  using measurable_pair[of f M M1 M2] by auto
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lemma measurable_split_conv:
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  "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
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  by (intro arg_cong2[where f="op \<in>"]) auto
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lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
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  by (auto intro!: measurable_Pair simp: measurable_split_conv)
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lemma measurable_pair_swap:
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  assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
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  using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
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lemma measurable_pair_swap_iff:
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  "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) M"
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  using measurable_pair_swap[of "\<lambda>(x,y). f (y, x)"]
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  by (auto intro!: measurable_pair_swap)
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lemma measurable_ident[intro, simp]: "(\<lambda>x. x) \<in> measurable M M"
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  unfolding measurable_def by auto
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lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)"
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  by (auto intro!: measurable_Pair)
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lemma sets_Pair1: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "Pair x -` A \<in> sets M2"
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proof -
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  have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
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    using A[THEN sets_into_space] by (auto simp: space_pair_measure)
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  also have "\<dots> \<in> sets M2"
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    using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm)
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  finally show ?thesis .
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qed
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lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^isub>M M2)"
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  by (auto intro!: measurable_Pair)
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lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
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proof -
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  have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
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    using A[THEN sets_into_space] by (auto simp: space_pair_measure)
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  also have "\<dots> \<in> sets M1"
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    using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm)
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  finally show ?thesis .
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qed
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lemma measurable_Pair2:
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  assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and x: "x \<in> space M1"
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  shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
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  using measurable_comp[OF measurable_Pair1' f, OF x]
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  by (simp add: comp_def)
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lemma measurable_Pair1:
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  assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and y: "y \<in> space M2"
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  shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
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  using measurable_comp[OF measurable_Pair2' f, OF y]
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  by (simp add: comp_def)
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lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
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  unfolding Int_stable_def
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  by safe (auto simp add: times_Int_times)
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lemma (in finite_measure) finite_measure_cut_measurable:
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  assumes "Q \<in> sets (N \<Otimes>\<^isub>M M)"
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  shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
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    (is "?s Q \<in> _")
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proof -
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  let ?\<Omega> = "space N \<times> space M" and ?D = "{A\<in>sets (N \<Otimes>\<^isub>M M). ?s A \<in> borel_measurable N}"
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  note space_pair_measure[simp]
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  interpret dynkin_system ?\<Omega> ?D
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  proof (intro dynkin_systemI)
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    fix A assume "A \<in> ?D" then show "A \<subseteq> ?\<Omega>"
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      using sets_into_space[of A "N \<Otimes>\<^isub>M M"] by simp
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  next
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    from top show "?\<Omega> \<in> ?D"
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      by (auto simp add: if_distrib intro!: measurable_If)
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  next
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    fix A assume "A \<in> ?D"
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    with sets_into_space have "\<And>x. emeasure M (Pair x -` (?\<Omega> - A)) =
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        (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"
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      by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
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    with `A \<in> ?D` top show "?\<Omega> - A \<in> ?D"
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      by (auto intro!: measurable_If)
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  next
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    fix F :: "nat \<Rightarrow> ('b\<times>'a) set" assume "disjoint_family F" "range F \<subseteq> ?D"
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    moreover then have "\<And>x. emeasure M (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
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      by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)
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    ultimately show "(\<Union>i. F i) \<in> ?D"
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      by (auto simp: vimage_UN intro!: borel_measurable_psuminf)
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  qed
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  let ?G = "{a \<times> b | a b. a \<in> sets N \<and> b \<in> sets M}"
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  have "sigma_sets ?\<Omega> ?G = ?D"
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  proof (rule dynkin_lemma)
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    show "?G \<subseteq> ?D"
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      by (auto simp: if_distrib Int_def[symmetric] intro!: measurable_If)
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  qed (auto simp: sets_pair_measure  Int_stable_pair_measure_generator)
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  with `Q \<in> sets (N \<Otimes>\<^isub>M M)` have "Q \<in> ?D"
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    by (simp add: sets_pair_measure[symmetric])
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  then show "?s Q \<in> borel_measurable N" by simp
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qed
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lemma (in sigma_finite_measure) measurable_emeasure_Pair:
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  assumes Q: "Q \<in> sets (N \<Otimes>\<^isub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" (is "?s Q \<in> _")
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proof -
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  from sigma_finite_disjoint guess F . note F = this
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  then have F_sets: "\<And>i. F i \<in> sets M" by auto
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  let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"
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  { fix i
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    have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"
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      using F sets_into_space by auto
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    let ?R = "density M (indicator (F i))"
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    have "finite_measure ?R"
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      using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
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    then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"
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     by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)
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    moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))
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        = emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))"
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      using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
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    moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i"
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      using sets_into_space[OF Q] by (auto simp: space_pair_measure)
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    ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"
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      by simp }
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  moreover
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  { fix x
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    have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
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    proof (intro suminf_emeasure)
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      show "range (?C x) \<subseteq> sets M"
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        using F `Q \<in> sets (N \<Otimes>\<^isub>M M)` by (auto intro!: sets_Pair1)
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      have "disjoint_family F" using F by auto
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      show "disjoint_family (?C x)"
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        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
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    qed
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    also have "(\<Union>i. ?C x i) = Pair x -` Q"
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      using F sets_into_space[OF `Q \<in> sets (N \<Otimes>\<^isub>M M)`]
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      by (auto simp: space_pair_measure)
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    finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"
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      by simp }
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  ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^isub>M M)` F_sets
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    by auto
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qed
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lemma (in sigma_finite_measure) emeasure_pair_measure:
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  assumes "X \<in> sets (N \<Otimes>\<^isub>M M)"
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  shows "emeasure (N \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X")
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   243
proof (rule emeasure_measure_of[OF pair_measure_def])
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   244
  show "positive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"
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   245
    by (auto simp: positive_def positive_integral_positive)
hoelzl@49776
   246
  have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
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   247
    by (auto simp: indicator_def)
hoelzl@49776
   248
  show "countably_additive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"
hoelzl@49776
   249
  proof (rule countably_additiveI)
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   250
    fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^isub>M M)" "disjoint_family F"
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   251
    from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^isub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^isub>M M)" by auto
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   252
    moreover from F have "\<And>i. (\<lambda>x. emeasure M (Pair x -` F i)) \<in> borel_measurable N"
hoelzl@49776
   253
      by (intro measurable_emeasure_Pair) auto
hoelzl@49776
   254
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
hoelzl@49776
   255
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@49776
   256
    moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M"
hoelzl@49776
   257
      using F by (auto simp: sets_Pair1)
hoelzl@49776
   258
    ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
hoelzl@49776
   259
      by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1
hoelzl@49776
   260
               intro!: positive_integral_cong positive_integral_indicator[symmetric])
hoelzl@49776
   261
  qed
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   262
  show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
hoelzl@49776
   263
    using space_closed[of N] space_closed[of M] by auto
hoelzl@49776
   264
qed fact
hoelzl@49776
   265
hoelzl@49776
   266
lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
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   267
  assumes X: "X \<in> sets (N \<Otimes>\<^isub>M M)"
hoelzl@49776
   268
  shows "emeasure (N  \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+x. emeasure M (Pair x -` X) \<partial>N)"
hoelzl@49776
   269
proof -
hoelzl@49776
   270
  have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
hoelzl@49776
   271
    by (auto simp: indicator_def)
hoelzl@49776
   272
  show ?thesis
hoelzl@49776
   273
    using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1)
hoelzl@49776
   274
qed
hoelzl@49776
   275
hoelzl@49776
   276
lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
hoelzl@49776
   277
  assumes A: "A \<in> sets N" and B: "B \<in> sets M"
hoelzl@49776
   278
  shows "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = emeasure N A * emeasure M B"
hoelzl@49776
   279
proof -
hoelzl@49776
   280
  have "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M B * indicator A x \<partial>N)"
hoelzl@49776
   281
    using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)
hoelzl@49776
   282
  also have "\<dots> = emeasure M B * emeasure N A"
hoelzl@49776
   283
    using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)
hoelzl@49776
   284
  finally show ?thesis
hoelzl@49776
   285
    by (simp add: ac_simps)
hoelzl@40859
   286
qed
hoelzl@40859
   287
hoelzl@47694
   288
subsection {* Binary products of $\sigma$-finite emeasure spaces *}
hoelzl@40859
   289
hoelzl@47694
   290
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
hoelzl@47694
   291
  for M1 :: "'a measure" and M2 :: "'b measure"
hoelzl@40859
   292
hoelzl@47694
   293
lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
hoelzl@49776
   294
  "Q \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
hoelzl@49776
   295
  using M2.measurable_emeasure_Pair .
hoelzl@40859
   296
hoelzl@47694
   297
lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
hoelzl@47694
   298
  assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
hoelzl@40859
   299
proof -
hoelzl@47694
   300
  have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
hoelzl@47694
   301
    using Q measurable_pair_swap' by (auto intro: measurable_sets)
hoelzl@49776
   302
  note M1.measurable_emeasure_Pair[OF this]
hoelzl@47694
   303
  moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1)) = (\<lambda>x. (x, y)) -` Q"
hoelzl@47694
   304
    using Q[THEN sets_into_space] by (auto simp: space_pair_measure)
hoelzl@47694
   305
  ultimately show ?thesis by simp
hoelzl@39088
   306
qed
hoelzl@39088
   307
hoelzl@41689
   308
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
hoelzl@47694
   309
  defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
hoelzl@47694
   310
  shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and>
hoelzl@47694
   311
    (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
hoelzl@40859
   312
proof -
hoelzl@47694
   313
  from M1.sigma_finite_incseq guess F1 . note F1 = this
hoelzl@47694
   314
  from M2.sigma_finite_incseq guess F2 . note F2 = this
hoelzl@47694
   315
  from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
hoelzl@40859
   316
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
hoelzl@47694
   317
  show ?thesis
hoelzl@40859
   318
  proof (intro exI[of _ ?F] conjI allI)
hoelzl@47694
   319
    show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
hoelzl@40859
   320
  next
hoelzl@40859
   321
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
hoelzl@40859
   322
    proof (intro subsetI)
hoelzl@40859
   323
      fix x assume "x \<in> space M1 \<times> space M2"
hoelzl@40859
   324
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
hoelzl@40859
   325
        by (auto simp: space)
hoelzl@40859
   326
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
hoelzl@41981
   327
        using `incseq F1` `incseq F2` unfolding incseq_def
hoelzl@41981
   328
        by (force split: split_max)+
hoelzl@40859
   329
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
hoelzl@40859
   330
        by (intro SigmaI) (auto simp add: min_max.sup_commute)
hoelzl@40859
   331
      then show "x \<in> (\<Union>i. ?F i)" by auto
hoelzl@40859
   332
    qed
hoelzl@47694
   333
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
hoelzl@47694
   334
      using space by (auto simp: space)
hoelzl@40859
   335
  next
hoelzl@41981
   336
    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
hoelzl@41981
   337
      using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
hoelzl@40859
   338
  next
hoelzl@40859
   339
    fix i
hoelzl@40859
   340
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
hoelzl@47694
   341
    with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]
hoelzl@47694
   342
    show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
hoelzl@47694
   343
      by (auto simp add: emeasure_pair_measure_Times)
hoelzl@47694
   344
  qed
hoelzl@47694
   345
qed
hoelzl@47694
   346
hoelzl@47694
   347
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure "M1 \<Otimes>\<^isub>M M2"
hoelzl@47694
   348
proof
hoelzl@47694
   349
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
   350
  show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
hoelzl@47694
   351
  proof (rule exI[of _ F], intro conjI)
hoelzl@47694
   352
    show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def)
hoelzl@47694
   353
    show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   354
      using F by (auto simp: space_pair_measure)
hoelzl@47694
   355
    show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto
hoelzl@40859
   356
  qed
hoelzl@40859
   357
qed
hoelzl@40859
   358
hoelzl@47694
   359
lemma sigma_finite_pair_measure:
hoelzl@47694
   360
  assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
hoelzl@47694
   361
  shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)"
hoelzl@47694
   362
proof -
hoelzl@47694
   363
  interpret A: sigma_finite_measure A by fact
hoelzl@47694
   364
  interpret B: sigma_finite_measure B by fact
hoelzl@47694
   365
  interpret AB: pair_sigma_finite A  B ..
hoelzl@47694
   366
  show ?thesis ..
hoelzl@40859
   367
qed
hoelzl@39088
   368
hoelzl@47694
   369
lemma sets_pair_swap:
hoelzl@47694
   370
  assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41689
   371
  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
hoelzl@47694
   372
  using measurable_pair_swap' assms by (rule measurable_sets)
hoelzl@41661
   373
hoelzl@47694
   374
lemma (in pair_sigma_finite) distr_pair_swap:
hoelzl@47694
   375
  "M1 \<Otimes>\<^isub>M M2 = distr (M2 \<Otimes>\<^isub>M M1) (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
hoelzl@40859
   376
proof -
hoelzl@41689
   377
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
   378
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
hoelzl@47694
   379
  show ?thesis
hoelzl@47694
   380
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@47694
   381
    show "?E \<subseteq> Pow (space ?P)"
hoelzl@47694
   382
      using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@47694
   383
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@47694
   384
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@47694
   385
    then show "sets ?D = sigma_sets (space ?P) ?E"
hoelzl@47694
   386
      by simp
hoelzl@47694
   387
  next
hoelzl@47694
   388
    show "range F \<subseteq> ?E" "incseq F" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@47694
   389
      using F by (auto simp: space_pair_measure)
hoelzl@47694
   390
  next
hoelzl@47694
   391
    fix X assume "X \<in> ?E"
hoelzl@47694
   392
    then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto
hoelzl@47694
   393
    have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A"
hoelzl@47694
   394
      using sets_into_space[OF A] sets_into_space[OF B] by (auto simp: space_pair_measure)
hoelzl@47694
   395
    with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X"
hoelzl@49776
   396
      by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
hoelzl@47694
   397
                    measurable_pair_swap' ac_simps)
hoelzl@41689
   398
  qed
hoelzl@41689
   399
qed
hoelzl@41689
   400
hoelzl@47694
   401
lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
hoelzl@47694
   402
  assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   403
  shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
hoelzl@47694
   404
    (is "_ = ?\<nu> A")
hoelzl@41689
   405
proof -
hoelzl@47694
   406
  have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1))) = (\<lambda>x. (x, y)) -` A"
hoelzl@47694
   407
    using sets_into_space[OF A] by (auto simp: space_pair_measure)
hoelzl@47694
   408
  show ?thesis using A
hoelzl@47694
   409
    by (subst distr_pair_swap)
hoelzl@47694
   410
       (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
hoelzl@49776
   411
                 M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
hoelzl@49776
   412
qed
hoelzl@49776
   413
hoelzl@49776
   414
lemma (in pair_sigma_finite) AE_pair:
hoelzl@49776
   415
  assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x"
hoelzl@49776
   416
  shows "AE x in M1. (AE y in M2. Q (x, y))"
hoelzl@49776
   417
proof -
hoelzl@49776
   418
  obtain N where N: "N \<in> sets (M1 \<Otimes>\<^isub>M M2)" "emeasure (M1 \<Otimes>\<^isub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> Q x} \<subseteq> N"
hoelzl@49776
   419
    using assms unfolding eventually_ae_filter by auto
hoelzl@49776
   420
  show ?thesis
hoelzl@49776
   421
  proof (rule AE_I)
hoelzl@49776
   422
    from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^isub>M M2)`]
hoelzl@49776
   423
    show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
hoelzl@49776
   424
      by (auto simp: M2.emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg)
hoelzl@49776
   425
    show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
hoelzl@49776
   426
      by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)
hoelzl@49776
   427
    { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
hoelzl@49776
   428
      have "AE y in M2. Q (x, y)"
hoelzl@49776
   429
      proof (rule AE_I)
hoelzl@49776
   430
        show "emeasure M2 (Pair x -` N) = 0" by fact
hoelzl@49776
   431
        show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
hoelzl@49776
   432
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
hoelzl@49776
   433
          using N `x \<in> space M1` unfolding space_pair_measure by auto
hoelzl@49776
   434
      qed }
hoelzl@49776
   435
    then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0}"
hoelzl@49776
   436
      by auto
hoelzl@49776
   437
  qed
hoelzl@49776
   438
qed
hoelzl@49776
   439
hoelzl@49776
   440
lemma (in pair_sigma_finite) AE_pair_measure:
hoelzl@49776
   441
  assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@49776
   442
  assumes ae: "AE x in M1. AE y in M2. P (x, y)"
hoelzl@49776
   443
  shows "AE x in M1 \<Otimes>\<^isub>M M2. P x"
hoelzl@49776
   444
proof (subst AE_iff_measurable[OF _ refl])
hoelzl@49776
   445
  show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@49776
   446
    by (rule sets_Collect) fact
hoelzl@49776
   447
  then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} =
hoelzl@49776
   448
      (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
hoelzl@49776
   449
    by (simp add: M2.emeasure_pair_measure)
hoelzl@49776
   450
  also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)"
hoelzl@49776
   451
    using ae
hoelzl@49776
   452
    apply (safe intro!: positive_integral_cong_AE)
hoelzl@49776
   453
    apply (intro AE_I2)
hoelzl@49776
   454
    apply (safe intro!: positive_integral_cong_AE)
hoelzl@49776
   455
    apply auto
hoelzl@49776
   456
    done
hoelzl@49776
   457
  finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp
hoelzl@49776
   458
qed
hoelzl@49776
   459
hoelzl@49776
   460
lemma (in pair_sigma_finite) AE_pair_iff:
hoelzl@49776
   461
  "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow>
hoelzl@49776
   462
    (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x))"
hoelzl@49776
   463
  using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
hoelzl@49776
   464
hoelzl@49776
   465
lemma (in pair_sigma_finite) AE_commute:
hoelzl@49776
   466
  assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@49776
   467
  shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
hoelzl@49776
   468
proof -
hoelzl@49776
   469
  interpret Q: pair_sigma_finite M2 M1 ..
hoelzl@49776
   470
  have [simp]: "\<And>x. (fst (case x of (x, y) \<Rightarrow> (y, x))) = snd x" "\<And>x. (snd (case x of (x, y) \<Rightarrow> (y, x))) = fst x"
hoelzl@49776
   471
    by auto
hoelzl@49776
   472
  have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} =
hoelzl@49776
   473
    (\<lambda>(x, y). (y, x)) -` {x \<in> space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^isub>M M1)"
hoelzl@49776
   474
    by (auto simp: space_pair_measure)
hoelzl@49776
   475
  also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)"
hoelzl@49776
   476
    by (intro sets_pair_swap P)
hoelzl@49776
   477
  finally show ?thesis
hoelzl@49776
   478
    apply (subst AE_pair_iff[OF P])
hoelzl@49776
   479
    apply (subst distr_pair_swap)
hoelzl@49776
   480
    apply (subst AE_distr_iff[OF measurable_pair_swap' P])
hoelzl@49776
   481
    apply (subst Q.AE_pair_iff)
hoelzl@49776
   482
    apply simp_all
hoelzl@49776
   483
    done
hoelzl@40859
   484
qed
hoelzl@40859
   485
hoelzl@40859
   486
section "Fubinis theorem"
hoelzl@40859
   487
hoelzl@40859
   488
lemma (in pair_sigma_finite) simple_function_cut:
hoelzl@47694
   489
  assumes f: "simple_function (M1 \<Otimes>\<^isub>M M2) f" "\<And>x. 0 \<le> f x"
hoelzl@41689
   490
  shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   491
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@40859
   492
proof -
hoelzl@47694
   493
  have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41981
   494
    using f(1) by (rule borel_measurable_simple_function)
hoelzl@47694
   495
  let ?F = "\<lambda>z. f -` {z} \<inter> space (M1 \<Otimes>\<^isub>M M2)"
wenzelm@46731
   496
  let ?F' = "\<lambda>x z. Pair x -` ?F z"
hoelzl@40859
   497
  { fix x assume "x \<in> space M1"
hoelzl@40859
   498
    have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
hoelzl@40859
   499
      by (auto simp: indicator_def)
hoelzl@47694
   500
    have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space (M1 \<Otimes>\<^isub>M M2)" using `x \<in> space M1`
hoelzl@41689
   501
      by (simp add: space_pair_measure)
hoelzl@40859
   502
    moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
hoelzl@47694
   503
      by (rule sets_Pair1[OF measurable_sets]) auto
hoelzl@41689
   504
    ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
hoelzl@47694
   505
      apply (rule_tac simple_function_cong[THEN iffD2, OF _])
hoelzl@41981
   506
      apply (rule simple_function_indicator_representation[OF f(1)])
hoelzl@47694
   507
      using `x \<in> space M1` by auto }
hoelzl@40859
   508
  note M2_sf = this
hoelzl@40859
   509
  { fix x assume x: "x \<in> space M1"
hoelzl@47694
   510
    then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space (M1 \<Otimes>\<^isub>M M2). z * emeasure M2 (?F' x z))"
hoelzl@47694
   511
      unfolding positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
hoelzl@41689
   512
      unfolding simple_integral_def
hoelzl@40859
   513
    proof (safe intro!: setsum_mono_zero_cong_left)
hoelzl@47694
   514
      from f(1) show "finite (f ` space (M1 \<Otimes>\<^isub>M M2))" by (rule simple_functionD)
hoelzl@40859
   515
    next
hoelzl@47694
   516
      fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41689
   517
        using `x \<in> space M1` by (auto simp: space_pair_measure)
hoelzl@40859
   518
    next
hoelzl@47694
   519
      fix x' y assume "(x', y) \<in> space (M1 \<Otimes>\<^isub>M M2)"
hoelzl@40859
   520
        "f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
hoelzl@40859
   521
      then have *: "?F' x (f (x', y)) = {}"
hoelzl@41689
   522
        by (force simp: space_pair_measure)
hoelzl@47694
   523
      show  "f (x', y) * emeasure M2 (?F' x (f (x', y))) = 0"
hoelzl@40859
   524
        unfolding * by simp
hoelzl@40859
   525
    qed (simp add: vimage_compose[symmetric] comp_def
hoelzl@41689
   526
                   space_pair_measure) }
hoelzl@40859
   527
  note eq = this
hoelzl@47694
   528
  moreover have "\<And>z. ?F z \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   529
    by (auto intro!: f_borel borel_measurable_vimage)
hoelzl@47694
   530
  moreover then have "\<And>z. (\<lambda>x. emeasure M2 (?F' x z)) \<in> borel_measurable M1"
hoelzl@47694
   531
    by (auto intro!: measurable_emeasure_Pair1 simp del: vimage_Int)
hoelzl@47694
   532
  moreover have *: "\<And>i x. 0 \<le> emeasure M2 (Pair x -` (f -` {i} \<inter> space (M1 \<Otimes>\<^isub>M M2)))"
hoelzl@47694
   533
    using f(1)[THEN simple_functionD(2)] f(2) by (intro emeasure_nonneg)
hoelzl@47694
   534
  moreover { fix i assume "i \<in> f`space (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   535
    with * have "\<And>x. 0 \<le> i * emeasure M2 (Pair x -` (f -` {i} \<inter> space (M1 \<Otimes>\<^isub>M M2)))"
hoelzl@41981
   536
      using f(2) by auto }
hoelzl@40859
   537
  ultimately
hoelzl@41689
   538
  show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   539
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f" using f(2)
hoelzl@47694
   540
    by (auto simp del: vimage_Int cong: measurable_cong intro!: setsum_cong
hoelzl@47694
   541
             simp add: positive_integral_setsum simple_integral_def
hoelzl@47694
   542
                       positive_integral_cmult
hoelzl@47694
   543
                       positive_integral_cong[OF eq]
hoelzl@40859
   544
                       positive_integral_eq_simple_integral[OF f]
hoelzl@49776
   545
                       M2.emeasure_pair_measure_alt[symmetric])
hoelzl@40859
   546
qed
hoelzl@40859
   547
hoelzl@40859
   548
lemma (in pair_sigma_finite) positive_integral_fst_measurable:
hoelzl@47694
   549
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41689
   550
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@40859
   551
      (is "?C f \<in> borel_measurable M1")
hoelzl@47694
   552
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@40859
   553
proof -
hoelzl@41981
   554
  from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
hoelzl@47694
   555
  then have F_borel: "\<And>i. F i \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@40859
   556
    by (auto intro: borel_measurable_simple_function)
hoelzl@41981
   557
  note sf = simple_function_cut[OF F(1,5)]
hoelzl@41097
   558
  then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
hoelzl@41097
   559
    using F(1) by auto
hoelzl@40859
   560
  moreover
hoelzl@40859
   561
  { fix x assume "x \<in> space M1"
hoelzl@47694
   562
    from F measurable_Pair2[OF F_borel `x \<in> space M1`]
hoelzl@41981
   563
    have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
hoelzl@47694
   564
      by (intro positive_integral_monotone_convergence_SUP)
hoelzl@41981
   565
         (auto simp: incseq_Suc_iff le_fun_def)
hoelzl@41981
   566
    then have "(SUP i. ?C (F i) x) = ?C f x"
hoelzl@41981
   567
      unfolding F(4) positive_integral_max_0 by simp }
hoelzl@40859
   568
  note SUPR_C = this
hoelzl@40859
   569
  ultimately show "?C f \<in> borel_measurable M1"
hoelzl@41097
   570
    by (simp cong: measurable_cong)
hoelzl@47694
   571
  have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>(M1 \<Otimes>\<^isub>M M2)) = (SUP i. integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (F i))"
hoelzl@41981
   572
    using F_borel F
hoelzl@41981
   573
    by (intro positive_integral_monotone_convergence_SUP) auto
hoelzl@47694
   574
  also have "(SUP i. integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
hoelzl@40859
   575
    unfolding sf(2) by simp
hoelzl@41981
   576
  also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
hoelzl@47694
   577
    by (intro positive_integral_monotone_convergence_SUP[symmetric])
hoelzl@47694
   578
       (auto intro!: positive_integral_mono positive_integral_positive
hoelzl@47694
   579
             simp: incseq_Suc_iff le_fun_def)
hoelzl@41689
   580
  also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
hoelzl@41981
   581
    using F_borel F(2,5)
hoelzl@47694
   582
    by (auto intro!: positive_integral_cong positive_integral_monotone_convergence_SUP[symmetric] measurable_Pair2
hoelzl@47694
   583
             simp: incseq_Suc_iff le_fun_def)
hoelzl@47694
   584
  finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41981
   585
    using F by (simp add: positive_integral_max_0)
hoelzl@40859
   586
qed
hoelzl@40859
   587
hoelzl@47694
   588
lemma (in pair_sigma_finite) positive_integral_snd_measurable:
hoelzl@47694
   589
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   590
  shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41661
   591
proof -
hoelzl@41689
   592
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@47694
   593
  note measurable_pair_swap[OF f]
hoelzl@40859
   594
  from Q.positive_integral_fst_measurable[OF this]
hoelzl@47694
   595
  have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1))"
hoelzl@40859
   596
    by simp
hoelzl@47694
   597
  also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@47694
   598
    by (subst distr_pair_swap)
hoelzl@47694
   599
       (auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong)
hoelzl@40859
   600
  finally show ?thesis .
hoelzl@40859
   601
qed
hoelzl@40859
   602
hoelzl@40859
   603
lemma (in pair_sigma_finite) Fubini:
hoelzl@47694
   604
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41689
   605
  shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
hoelzl@40859
   606
  unfolding positive_integral_snd_measurable[OF assms]
hoelzl@40859
   607
  unfolding positive_integral_fst_measurable[OF assms] ..
hoelzl@40859
   608
hoelzl@41026
   609
lemma (in pair_sigma_finite) integrable_product_swap:
hoelzl@47694
   610
  assumes "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41689
   611
  shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
hoelzl@41026
   612
proof -
hoelzl@41689
   613
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@41661
   614
  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
hoelzl@41661
   615
  show ?thesis unfolding *
hoelzl@47694
   616
    by (rule integrable_distr[OF measurable_pair_swap'])
hoelzl@47694
   617
       (simp add: distr_pair_swap[symmetric] assms)
hoelzl@41661
   618
qed
hoelzl@41661
   619
hoelzl@41661
   620
lemma (in pair_sigma_finite) integrable_product_swap_iff:
hoelzl@47694
   621
  "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41661
   622
proof -
hoelzl@41689
   623
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@41661
   624
  from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
hoelzl@41661
   625
  show ?thesis by auto
hoelzl@41026
   626
qed
hoelzl@41026
   627
hoelzl@41026
   628
lemma (in pair_sigma_finite) integral_product_swap:
hoelzl@47694
   629
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   630
  shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41026
   631
proof -
hoelzl@41661
   632
  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
hoelzl@47694
   633
  show ?thesis unfolding *
hoelzl@47694
   634
    by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
hoelzl@41026
   635
qed
hoelzl@41026
   636
hoelzl@41026
   637
lemma (in pair_sigma_finite) integrable_fst_measurable:
hoelzl@47694
   638
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@47694
   639
  shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE")
hoelzl@47694
   640
    and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
hoelzl@41026
   641
proof -
hoelzl@47694
   642
  have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   643
    using f by auto
wenzelm@46731
   644
  let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
hoelzl@41026
   645
  have
hoelzl@47694
   646
    borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and
hoelzl@47694
   647
    int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?nf \<noteq> \<infinity>" "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?pf \<noteq> \<infinity>"
hoelzl@41026
   648
    using assms by auto
hoelzl@43920
   649
  have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
hoelzl@43920
   650
     "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
hoelzl@41026
   651
    using borel[THEN positive_integral_fst_measurable(1)] int
hoelzl@41026
   652
    unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
hoelzl@41026
   653
  with borel[THEN positive_integral_fst_measurable(1)]
hoelzl@43920
   654
  have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
hoelzl@43920
   655
    "AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
hoelzl@47694
   656
    by (auto intro!: positive_integral_PInf_AE )
hoelzl@43920
   657
  then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
hoelzl@43920
   658
    "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
hoelzl@47694
   659
    by (auto simp: positive_integral_positive)
hoelzl@41981
   660
  from AE_pos show ?AE using assms
hoelzl@47694
   661
    by (simp add: measurable_Pair2[OF f_borel] integrable_def)
hoelzl@43920
   662
  { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
hoelzl@47694
   663
      using positive_integral_positive
hoelzl@47694
   664
      by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
hoelzl@43920
   665
    then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
hoelzl@41981
   666
  note this[simp]
hoelzl@47694
   667
  { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   668
      and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
hoelzl@47694
   669
      and AE: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
hoelzl@43920
   670
    have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
hoelzl@41705
   671
    proof (intro integrable_def[THEN iffD2] conjI)
hoelzl@41705
   672
      show "?f \<in> borel_measurable M1"
hoelzl@47694
   673
        using borel by (auto intro!: positive_integral_fst_measurable)
hoelzl@43920
   674
      have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y))  \<partial>M2) \<partial>M1)"
hoelzl@47694
   675
        using AE positive_integral_positive[of M2]
hoelzl@47694
   676
        by (auto intro!: positive_integral_cong_AE simp: ereal_real)
hoelzl@43920
   677
      then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
hoelzl@41705
   678
        using positive_integral_fst_measurable[OF borel] int by simp
hoelzl@43920
   679
      have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
hoelzl@47694
   680
        by (intro positive_integral_cong_pos)
hoelzl@47694
   681
           (simp add: positive_integral_positive real_of_ereal_pos)
hoelzl@43920
   682
      then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
hoelzl@41705
   683
    qed }
hoelzl@41981
   684
  with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
hoelzl@41705
   685
  show ?INT
hoelzl@47694
   686
    unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2]
hoelzl@41026
   687
      borel[THEN positive_integral_fst_measurable(2), symmetric]
hoelzl@47694
   688
    using AE[THEN integral_real]
hoelzl@41981
   689
    by simp
hoelzl@41026
   690
qed
hoelzl@41026
   691
hoelzl@41026
   692
lemma (in pair_sigma_finite) integrable_snd_measurable:
hoelzl@47694
   693
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@47694
   694
  shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE")
hoelzl@47694
   695
    and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
hoelzl@41026
   696
proof -
hoelzl@41689
   697
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@47694
   698
  have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))"
hoelzl@41661
   699
    using f unfolding integrable_product_swap_iff .
hoelzl@41026
   700
  show ?INT
hoelzl@41026
   701
    using Q.integrable_fst_measurable(2)[OF Q_int]
hoelzl@47694
   702
    using integral_product_swap[of f] f by auto
hoelzl@41026
   703
  show ?AE
hoelzl@41026
   704
    using Q.integrable_fst_measurable(1)[OF Q_int]
hoelzl@41026
   705
    by simp
hoelzl@41026
   706
qed
hoelzl@41026
   707
hoelzl@47694
   708
lemma (in pair_sigma_finite) positive_integral_fst_measurable':
hoelzl@47694
   709
  assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   710
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   711
  using positive_integral_fst_measurable(1)[OF f] by simp
hoelzl@47694
   712
hoelzl@47694
   713
lemma (in pair_sigma_finite) integral_fst_measurable:
hoelzl@47694
   714
  "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   715
  by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_fst_measurable')
hoelzl@47694
   716
hoelzl@47694
   717
lemma (in pair_sigma_finite) positive_integral_snd_measurable':
hoelzl@47694
   718
  assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   719
  shows "(\<lambda>y. \<integral>\<^isup>+ x. f x y \<partial>M1) \<in> borel_measurable M2"
hoelzl@47694
   720
proof -
hoelzl@47694
   721
  interpret Q: pair_sigma_finite M2 M1 ..
hoelzl@47694
   722
  show ?thesis
hoelzl@47694
   723
    using measurable_pair_swap[OF f]
hoelzl@47694
   724
    by (intro Q.positive_integral_fst_measurable') (simp add: split_beta')
hoelzl@47694
   725
qed
hoelzl@47694
   726
hoelzl@47694
   727
lemma (in pair_sigma_finite) integral_snd_measurable:
hoelzl@47694
   728
  "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>y. \<integral> x. f x y \<partial>M1) \<in> borel_measurable M2"
hoelzl@47694
   729
  by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_snd_measurable')
hoelzl@47694
   730
hoelzl@41026
   731
lemma (in pair_sigma_finite) Fubini_integral:
hoelzl@47694
   732
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41689
   733
  shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
hoelzl@41026
   734
  unfolding integrable_snd_measurable[OF assms]
hoelzl@41026
   735
  unfolding integrable_fst_measurable[OF assms] ..
hoelzl@41026
   736
hoelzl@47694
   737
section {* Products on counting spaces, densities and distributions *}
hoelzl@40859
   738
hoelzl@41689
   739
lemma sigma_sets_pair_measure_generator_finite:
hoelzl@38656
   740
  assumes "finite A" and "finite B"
hoelzl@47694
   741
  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
hoelzl@40859
   742
  (is "sigma_sets ?prod ?sets = _")
hoelzl@38656
   743
proof safe
hoelzl@38656
   744
  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
hoelzl@38656
   745
  fix x assume subset: "x \<subseteq> A \<times> B"
hoelzl@38656
   746
  hence "finite x" using fin by (rule finite_subset)
hoelzl@40859
   747
  from this subset show "x \<in> sigma_sets ?prod ?sets"
hoelzl@38656
   748
  proof (induct x)
hoelzl@38656
   749
    case empty show ?case by (rule sigma_sets.Empty)
hoelzl@38656
   750
  next
hoelzl@38656
   751
    case (insert a x)
hoelzl@47694
   752
    hence "{a} \<in> sigma_sets ?prod ?sets" by auto
hoelzl@38656
   753
    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
hoelzl@38656
   754
    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
hoelzl@38656
   755
  qed
hoelzl@38656
   756
next
hoelzl@38656
   757
  fix x a b
hoelzl@40859
   758
  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
hoelzl@38656
   759
  from sigma_sets_into_sp[OF _ this(1)] this(2)
hoelzl@40859
   760
  show "a \<in> A" and "b \<in> B" by auto
hoelzl@35833
   761
qed
hoelzl@35833
   762
hoelzl@47694
   763
lemma pair_measure_count_space:
hoelzl@47694
   764
  assumes A: "finite A" and B: "finite B"
hoelzl@47694
   765
  shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
hoelzl@47694
   766
proof (rule measure_eqI)
hoelzl@47694
   767
  interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
hoelzl@47694
   768
  interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
hoelzl@47694
   769
  interpret P: pair_sigma_finite "count_space A" "count_space B" by default
hoelzl@47694
   770
  show eq: "sets ?P = sets ?C"
hoelzl@47694
   771
    by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
hoelzl@47694
   772
  fix X assume X: "X \<in> sets ?P"
hoelzl@47694
   773
  with eq have X_subset: "X \<subseteq> A \<times> B" by simp
hoelzl@47694
   774
  with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
hoelzl@47694
   775
    by (intro finite_subset[OF _ B]) auto
hoelzl@47694
   776
  have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
hoelzl@47694
   777
  show "emeasure ?P X = emeasure ?C X"
hoelzl@49776
   778
    apply (subst B.emeasure_pair_measure_alt[OF X])
hoelzl@47694
   779
    apply (subst emeasure_count_space)
hoelzl@47694
   780
    using X_subset apply auto []
hoelzl@47694
   781
    apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
hoelzl@47694
   782
    apply (subst positive_integral_count_space)
hoelzl@47694
   783
    using A apply simp
hoelzl@47694
   784
    apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
hoelzl@47694
   785
    apply (subst card_gt_0_iff)
hoelzl@47694
   786
    apply (simp add: fin_Pair)
hoelzl@47694
   787
    apply (subst card_SigmaI[symmetric])
hoelzl@47694
   788
    using A apply simp
hoelzl@47694
   789
    using fin_Pair apply simp
hoelzl@47694
   790
    using X_subset apply (auto intro!: arg_cong[where f=card])
hoelzl@47694
   791
    done
hoelzl@45777
   792
qed
hoelzl@35833
   793
hoelzl@47694
   794
lemma pair_measure_density:
hoelzl@47694
   795
  assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
hoelzl@47694
   796
  assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"
hoelzl@47694
   797
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
hoelzl@47694
   798
  assumes "sigma_finite_measure (density M1 f)" "sigma_finite_measure (density M2 g)"
hoelzl@47694
   799
  shows "density M1 f \<Otimes>\<^isub>M density M2 g = density (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")
hoelzl@47694
   800
proof (rule measure_eqI)
hoelzl@47694
   801
  interpret M1: sigma_finite_measure M1 by fact
hoelzl@47694
   802
  interpret M2: sigma_finite_measure M2 by fact
hoelzl@47694
   803
  interpret D1: sigma_finite_measure "density M1 f" by fact
hoelzl@47694
   804
  interpret D2: sigma_finite_measure "density M2 g" by fact
hoelzl@47694
   805
  interpret L: pair_sigma_finite "density M1 f" "density M2 g" ..
hoelzl@47694
   806
  interpret R: pair_sigma_finite M1 M2 ..
hoelzl@47694
   807
hoelzl@47694
   808
  fix A assume A: "A \<in> sets ?L"
hoelzl@47694
   809
  then have indicator_eq: "\<And>x y. indicator A (x, y) = indicator (Pair x -` A) y"
hoelzl@47694
   810
   and Pair_A: "\<And>x. Pair x -` A \<in> sets M2"
hoelzl@47694
   811
    by (auto simp: indicator_def sets_Pair1)
hoelzl@47694
   812
  have f_fst: "(\<lambda>p. f (fst p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   813
    using measurable_comp[OF measurable_fst f(1)] by (simp add: comp_def)
hoelzl@47694
   814
  have g_snd: "(\<lambda>p. g (snd p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   815
    using measurable_comp[OF measurable_snd g(1)] by (simp add: comp_def)
hoelzl@47694
   816
  have "(\<lambda>x. \<integral>\<^isup>+ y. g (snd (x, y)) * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   817
    using g_snd Pair_A A by (intro R.positive_integral_fst_measurable) auto
hoelzl@47694
   818
  then have int_g: "(\<lambda>x. \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   819
    by simp
hoelzl@38656
   820
hoelzl@47694
   821
  show "emeasure ?L A = emeasure ?R A"
hoelzl@49776
   822
    apply (subst D2.emeasure_pair_measure[OF A])
hoelzl@47694
   823
    apply (subst emeasure_density)
hoelzl@47694
   824
        using f_fst g_snd apply (simp add: split_beta')
hoelzl@47694
   825
      using A apply simp
hoelzl@47694
   826
    apply (subst positive_integral_density[OF g])
hoelzl@47694
   827
      apply (simp add: indicator_eq Pair_A)
hoelzl@47694
   828
    apply (subst positive_integral_density[OF f])
hoelzl@47694
   829
      apply (rule int_g)
hoelzl@47694
   830
    apply (subst R.positive_integral_fst_measurable(2)[symmetric])
hoelzl@47694
   831
      using f g A Pair_A f_fst g_snd
hoelzl@47694
   832
      apply (auto intro!: positive_integral_cong_AE R.measurable_emeasure_Pair1
hoelzl@47694
   833
                  simp: positive_integral_cmult indicator_eq split_beta')
hoelzl@47694
   834
    apply (intro AE_I2 impI)
hoelzl@47694
   835
    apply (subst mult_assoc)
hoelzl@47694
   836
    apply (subst positive_integral_cmult)
hoelzl@47694
   837
          apply auto
hoelzl@47694
   838
    done
hoelzl@47694
   839
qed simp
hoelzl@47694
   840
hoelzl@47694
   841
lemma sigma_finite_measure_distr:
hoelzl@47694
   842
  assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
hoelzl@47694
   843
  shows "sigma_finite_measure M"
hoelzl@40859
   844
proof -
hoelzl@47694
   845
  interpret sigma_finite_measure "distr M N f" by fact
hoelzl@47694
   846
  from sigma_finite_disjoint guess A . note A = this
hoelzl@47694
   847
  show ?thesis
hoelzl@47694
   848
  proof (unfold_locales, intro conjI exI allI)
hoelzl@47694
   849
    show "range (\<lambda>i. f -` A i \<inter> space M) \<subseteq> sets M"
hoelzl@47694
   850
      using A f by (auto intro!: measurable_sets)
hoelzl@47694
   851
    show "(\<Union>i. f -` A i \<inter> space M) = space M"
hoelzl@47694
   852
      using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def)
hoelzl@47694
   853
    fix i show "emeasure M (f -` A i \<inter> space M) \<noteq> \<infinity>"
hoelzl@47694
   854
      using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq)
hoelzl@47694
   855
  qed
hoelzl@38656
   856
qed
hoelzl@38656
   857
hoelzl@47694
   858
lemma measurable_cong':
hoelzl@47694
   859
  assumes sets: "sets M = sets M'" "sets N = sets N'"
hoelzl@47694
   860
  shows "measurable M N = measurable M' N'"
hoelzl@47694
   861
  using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
hoelzl@38656
   862
hoelzl@47694
   863
lemma pair_measure_distr:
hoelzl@47694
   864
  assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
hoelzl@47694
   865
  assumes "sigma_finite_measure (distr M S f)" "sigma_finite_measure (distr N T g)"
hoelzl@47694
   866
  shows "distr M S f \<Otimes>\<^isub>M distr N T g = distr (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")
hoelzl@47694
   867
proof (rule measure_eqI)
hoelzl@47694
   868
  show "sets ?P = sets ?D"
hoelzl@47694
   869
    by simp
hoelzl@47694
   870
  interpret S: sigma_finite_measure "distr M S f" by fact
hoelzl@47694
   871
  interpret T: sigma_finite_measure "distr N T g" by fact
hoelzl@47694
   872
  interpret ST: pair_sigma_finite "distr M S f"  "distr N T g" ..
hoelzl@47694
   873
  interpret M: sigma_finite_measure M by (rule sigma_finite_measure_distr) fact+
hoelzl@47694
   874
  interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
hoelzl@47694
   875
  interpret MN: pair_sigma_finite M N ..
hoelzl@47694
   876
  interpret SN: pair_sigma_finite "distr M S f" N ..
hoelzl@47694
   877
  have [simp]: 
hoelzl@47694
   878
    "\<And>f g. fst \<circ> (\<lambda>(x, y). (f x, g y)) = f \<circ> fst" "\<And>f g. snd \<circ> (\<lambda>(x, y). (f x, g y)) = g \<circ> snd"
hoelzl@47694
   879
    by auto
hoelzl@47694
   880
  then have fg: "(\<lambda>(x, y). (f x, g y)) \<in> measurable (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T)"
hoelzl@47694
   881
    using measurable_comp[OF measurable_fst f] measurable_comp[OF measurable_snd g]
hoelzl@47694
   882
    by (auto simp: measurable_pair_iff)
hoelzl@47694
   883
  fix A assume A: "A \<in> sets ?P"
hoelzl@47694
   884
  then have "emeasure ?P A = (\<integral>\<^isup>+x. emeasure (distr N T g) (Pair x -` A) \<partial>distr M S f)"
hoelzl@49776
   885
    by (rule T.emeasure_pair_measure_alt)
hoelzl@47694
   886
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair x -` A) \<inter> space N) \<partial>distr M S f)"
hoelzl@47694
   887
    using g A by (simp add: sets_Pair1 emeasure_distr)
hoelzl@47694
   888
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair (f x) -` A) \<inter> space N) \<partial>M)"
hoelzl@47694
   889
    using f g A ST.measurable_emeasure_Pair1[OF A]
hoelzl@47694
   890
    by (intro positive_integral_distr) (auto simp add: sets_Pair1 emeasure_distr)
hoelzl@47694
   891
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (Pair x -` ((\<lambda>(x, y). (f x, g y)) -` A \<inter> space (M \<Otimes>\<^isub>M N))) \<partial>M)"
hoelzl@47694
   892
    by (intro positive_integral_cong arg_cong2[where f=emeasure]) (auto simp: space_pair_measure)
hoelzl@47694
   893
  also have "\<dots> = emeasure (M \<Otimes>\<^isub>M N) ((\<lambda>(x, y). (f x, g y)) -` A \<inter> space (M \<Otimes>\<^isub>M N))"
hoelzl@49776
   894
    using fg by (intro N.emeasure_pair_measure_alt[symmetric] measurable_sets[OF _ A])
hoelzl@47694
   895
                (auto cong: measurable_cong')
hoelzl@47694
   896
  also have "\<dots> = emeasure ?D A"
hoelzl@47694
   897
    using fg A by (subst emeasure_distr) auto
hoelzl@47694
   898
  finally show "emeasure ?P A = emeasure ?D A" .
hoelzl@45777
   899
qed
hoelzl@39097
   900
hoelzl@40859
   901
end