src/HOL/Probability/Binary_Product_Measure.thy
author hoelzl
Fri Nov 02 14:00:39 2012 +0100 (2012-11-02)
changeset 49999 dfb63b9b8908
parent 49825 bb5db3d1d6dd
child 50002 ce0d316b5b44
permissions -rw-r--r--
for the product measure it is enough if only one measure is sigma-finite
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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 pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"
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  using space_closed[of A] space_closed[of B] by auto
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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 pair_measure_closed[of A B]
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  by (rule space_measure_of)
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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 pair_measure_closed[of A B]
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  by (rule sets_measure_of)
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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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  using Int_stable_pair_measure_generator pair_measure_closed assms
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  unfolding sets_pair_measure
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proof (induct rule: sigma_sets_induct_disjoint)
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  case (compl A)
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  with sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) =
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      (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"
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    unfolding sets_pair_measure[symmetric]
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    by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
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  with compl top show ?case
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    by (auto intro!: measurable_If simp: space_pair_measure)
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next
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  case (union F)
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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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    unfolding sets_pair_measure[symmetric]
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    by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)
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  ultimately show ?case
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    by (auto simp: vimage_UN)
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qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
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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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proof (rule emeasure_measure_of[OF pair_measure_def])
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  show "positive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"
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    by (auto simp: positive_def positive_integral_positive)
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  have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
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    by (auto simp: indicator_def)
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  show "countably_additive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"
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  proof (rule countably_additiveI)
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    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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    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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    moreover from F have "\<And>i. (\<lambda>x. emeasure M (Pair x -` F i)) \<in> borel_measurable N"
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   240
      by (intro measurable_emeasure_Pair) auto
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   241
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
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   242
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@49776
   243
    moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M"
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   244
      using F by (auto simp: sets_Pair1)
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   245
    ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
hoelzl@49776
   246
      by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1
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   247
               intro!: positive_integral_cong positive_integral_indicator[symmetric])
hoelzl@49776
   248
  qed
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   249
  show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
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   250
    using space_closed[of N] space_closed[of M] by auto
hoelzl@49776
   251
qed fact
hoelzl@49776
   252
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   253
lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
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   254
  assumes X: "X \<in> sets (N \<Otimes>\<^isub>M M)"
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   255
  shows "emeasure (N  \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+x. emeasure M (Pair x -` X) \<partial>N)"
hoelzl@49776
   256
proof -
hoelzl@49776
   257
  have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
hoelzl@49776
   258
    by (auto simp: indicator_def)
hoelzl@49776
   259
  show ?thesis
hoelzl@49776
   260
    using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1)
hoelzl@49776
   261
qed
hoelzl@49776
   262
hoelzl@49776
   263
lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
hoelzl@49776
   264
  assumes A: "A \<in> sets N" and B: "B \<in> sets M"
hoelzl@49776
   265
  shows "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = emeasure N A * emeasure M B"
hoelzl@49776
   266
proof -
hoelzl@49776
   267
  have "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M B * indicator A x \<partial>N)"
hoelzl@49776
   268
    using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)
hoelzl@49776
   269
  also have "\<dots> = emeasure M B * emeasure N A"
hoelzl@49776
   270
    using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)
hoelzl@49776
   271
  finally show ?thesis
hoelzl@49776
   272
    by (simp add: ac_simps)
hoelzl@40859
   273
qed
hoelzl@40859
   274
hoelzl@47694
   275
subsection {* Binary products of $\sigma$-finite emeasure spaces *}
hoelzl@40859
   276
hoelzl@47694
   277
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
hoelzl@47694
   278
  for M1 :: "'a measure" and M2 :: "'b measure"
hoelzl@40859
   279
hoelzl@47694
   280
lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
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   281
  "Q \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
hoelzl@49776
   282
  using M2.measurable_emeasure_Pair .
hoelzl@40859
   283
hoelzl@47694
   284
lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
hoelzl@47694
   285
  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
   286
proof -
hoelzl@47694
   287
  have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
hoelzl@47694
   288
    using Q measurable_pair_swap' by (auto intro: measurable_sets)
hoelzl@49776
   289
  note M1.measurable_emeasure_Pair[OF this]
hoelzl@47694
   290
  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
   291
    using Q[THEN sets_into_space] by (auto simp: space_pair_measure)
hoelzl@47694
   292
  ultimately show ?thesis by simp
hoelzl@39088
   293
qed
hoelzl@39088
   294
hoelzl@41689
   295
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
hoelzl@47694
   296
  defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
hoelzl@47694
   297
  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
   298
    (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
hoelzl@40859
   299
proof -
hoelzl@47694
   300
  from M1.sigma_finite_incseq guess F1 . note F1 = this
hoelzl@47694
   301
  from M2.sigma_finite_incseq guess F2 . note F2 = this
hoelzl@47694
   302
  from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
hoelzl@40859
   303
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
hoelzl@47694
   304
  show ?thesis
hoelzl@40859
   305
  proof (intro exI[of _ ?F] conjI allI)
hoelzl@47694
   306
    show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
hoelzl@40859
   307
  next
hoelzl@40859
   308
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
hoelzl@40859
   309
    proof (intro subsetI)
hoelzl@40859
   310
      fix x assume "x \<in> space M1 \<times> space M2"
hoelzl@40859
   311
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
hoelzl@40859
   312
        by (auto simp: space)
hoelzl@40859
   313
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
hoelzl@41981
   314
        using `incseq F1` `incseq F2` unfolding incseq_def
hoelzl@41981
   315
        by (force split: split_max)+
hoelzl@40859
   316
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
hoelzl@40859
   317
        by (intro SigmaI) (auto simp add: min_max.sup_commute)
hoelzl@40859
   318
      then show "x \<in> (\<Union>i. ?F i)" by auto
hoelzl@40859
   319
    qed
hoelzl@47694
   320
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
hoelzl@47694
   321
      using space by (auto simp: space)
hoelzl@40859
   322
  next
hoelzl@41981
   323
    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
hoelzl@41981
   324
      using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
hoelzl@40859
   325
  next
hoelzl@40859
   326
    fix i
hoelzl@40859
   327
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
hoelzl@47694
   328
    with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]
hoelzl@47694
   329
    show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
hoelzl@47694
   330
      by (auto simp add: emeasure_pair_measure_Times)
hoelzl@47694
   331
  qed
hoelzl@47694
   332
qed
hoelzl@47694
   333
hoelzl@49800
   334
sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^isub>M M2"
hoelzl@47694
   335
proof
hoelzl@47694
   336
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
   337
  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
   338
  proof (rule exI[of _ F], intro conjI)
hoelzl@47694
   339
    show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def)
hoelzl@47694
   340
    show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   341
      using F by (auto simp: space_pair_measure)
hoelzl@47694
   342
    show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto
hoelzl@40859
   343
  qed
hoelzl@40859
   344
qed
hoelzl@40859
   345
hoelzl@47694
   346
lemma sigma_finite_pair_measure:
hoelzl@47694
   347
  assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
hoelzl@47694
   348
  shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)"
hoelzl@47694
   349
proof -
hoelzl@47694
   350
  interpret A: sigma_finite_measure A by fact
hoelzl@47694
   351
  interpret B: sigma_finite_measure B by fact
hoelzl@47694
   352
  interpret AB: pair_sigma_finite A  B ..
hoelzl@47694
   353
  show ?thesis ..
hoelzl@40859
   354
qed
hoelzl@39088
   355
hoelzl@47694
   356
lemma sets_pair_swap:
hoelzl@47694
   357
  assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41689
   358
  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
hoelzl@47694
   359
  using measurable_pair_swap' assms by (rule measurable_sets)
hoelzl@41661
   360
hoelzl@47694
   361
lemma (in pair_sigma_finite) distr_pair_swap:
hoelzl@47694
   362
  "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
   363
proof -
hoelzl@41689
   364
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
   365
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
hoelzl@47694
   366
  show ?thesis
hoelzl@47694
   367
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@47694
   368
    show "?E \<subseteq> Pow (space ?P)"
hoelzl@47694
   369
      using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@47694
   370
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@47694
   371
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@47694
   372
    then show "sets ?D = sigma_sets (space ?P) ?E"
hoelzl@47694
   373
      by simp
hoelzl@47694
   374
  next
hoelzl@49784
   375
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@47694
   376
      using F by (auto simp: space_pair_measure)
hoelzl@47694
   377
  next
hoelzl@47694
   378
    fix X assume "X \<in> ?E"
hoelzl@47694
   379
    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
   380
    have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A"
hoelzl@47694
   381
      using sets_into_space[OF A] sets_into_space[OF B] by (auto simp: space_pair_measure)
hoelzl@47694
   382
    with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X"
hoelzl@49776
   383
      by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
hoelzl@47694
   384
                    measurable_pair_swap' ac_simps)
hoelzl@41689
   385
  qed
hoelzl@41689
   386
qed
hoelzl@41689
   387
hoelzl@47694
   388
lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
hoelzl@47694
   389
  assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   390
  shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
hoelzl@47694
   391
    (is "_ = ?\<nu> A")
hoelzl@41689
   392
proof -
hoelzl@47694
   393
  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
   394
    using sets_into_space[OF A] by (auto simp: space_pair_measure)
hoelzl@47694
   395
  show ?thesis using A
hoelzl@47694
   396
    by (subst distr_pair_swap)
hoelzl@47694
   397
       (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
hoelzl@49776
   398
                 M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
hoelzl@49776
   399
qed
hoelzl@49776
   400
hoelzl@49776
   401
lemma (in pair_sigma_finite) AE_pair:
hoelzl@49776
   402
  assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x"
hoelzl@49776
   403
  shows "AE x in M1. (AE y in M2. Q (x, y))"
hoelzl@49776
   404
proof -
hoelzl@49776
   405
  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
   406
    using assms unfolding eventually_ae_filter by auto
hoelzl@49776
   407
  show ?thesis
hoelzl@49776
   408
  proof (rule AE_I)
hoelzl@49776
   409
    from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^isub>M M2)`]
hoelzl@49776
   410
    show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
hoelzl@49776
   411
      by (auto simp: M2.emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg)
hoelzl@49776
   412
    show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
hoelzl@49776
   413
      by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)
hoelzl@49776
   414
    { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
hoelzl@49776
   415
      have "AE y in M2. Q (x, y)"
hoelzl@49776
   416
      proof (rule AE_I)
hoelzl@49776
   417
        show "emeasure M2 (Pair x -` N) = 0" by fact
hoelzl@49776
   418
        show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
hoelzl@49776
   419
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
hoelzl@49776
   420
          using N `x \<in> space M1` unfolding space_pair_measure by auto
hoelzl@49776
   421
      qed }
hoelzl@49776
   422
    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
   423
      by auto
hoelzl@49776
   424
  qed
hoelzl@49776
   425
qed
hoelzl@49776
   426
hoelzl@49776
   427
lemma (in pair_sigma_finite) AE_pair_measure:
hoelzl@49776
   428
  assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@49776
   429
  assumes ae: "AE x in M1. AE y in M2. P (x, y)"
hoelzl@49776
   430
  shows "AE x in M1 \<Otimes>\<^isub>M M2. P x"
hoelzl@49776
   431
proof (subst AE_iff_measurable[OF _ refl])
hoelzl@49776
   432
  show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@49776
   433
    by (rule sets_Collect) fact
hoelzl@49776
   434
  then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} =
hoelzl@49776
   435
      (\<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
   436
    by (simp add: M2.emeasure_pair_measure)
hoelzl@49776
   437
  also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)"
hoelzl@49776
   438
    using ae
hoelzl@49776
   439
    apply (safe intro!: positive_integral_cong_AE)
hoelzl@49776
   440
    apply (intro AE_I2)
hoelzl@49776
   441
    apply (safe intro!: positive_integral_cong_AE)
hoelzl@49776
   442
    apply auto
hoelzl@49776
   443
    done
hoelzl@49776
   444
  finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp
hoelzl@49776
   445
qed
hoelzl@49776
   446
hoelzl@49776
   447
lemma (in pair_sigma_finite) AE_pair_iff:
hoelzl@49776
   448
  "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow>
hoelzl@49776
   449
    (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
   450
  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
   451
hoelzl@49776
   452
lemma (in pair_sigma_finite) AE_commute:
hoelzl@49776
   453
  assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@49776
   454
  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
   455
proof -
hoelzl@49776
   456
  interpret Q: pair_sigma_finite M2 M1 ..
hoelzl@49776
   457
  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
   458
    by auto
hoelzl@49776
   459
  have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} =
hoelzl@49776
   460
    (\<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
   461
    by (auto simp: space_pair_measure)
hoelzl@49776
   462
  also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)"
hoelzl@49776
   463
    by (intro sets_pair_swap P)
hoelzl@49776
   464
  finally show ?thesis
hoelzl@49776
   465
    apply (subst AE_pair_iff[OF P])
hoelzl@49776
   466
    apply (subst distr_pair_swap)
hoelzl@49776
   467
    apply (subst AE_distr_iff[OF measurable_pair_swap' P])
hoelzl@49776
   468
    apply (subst Q.AE_pair_iff)
hoelzl@49776
   469
    apply simp_all
hoelzl@49776
   470
    done
hoelzl@40859
   471
qed
hoelzl@40859
   472
hoelzl@40859
   473
section "Fubinis theorem"
hoelzl@40859
   474
hoelzl@49800
   475
lemma measurable_compose_Pair1:
hoelzl@49800
   476
  "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^isub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
hoelzl@49800
   477
  by (rule measurable_compose[OF measurable_Pair]) auto
hoelzl@49800
   478
hoelzl@49999
   479
lemma (in sigma_finite_measure) borel_measurable_positive_integral_fst':
hoelzl@49999
   480
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x"
hoelzl@49999
   481
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
hoelzl@49800
   482
using f proof induct
hoelzl@49800
   483
  case (cong u v)
hoelzl@49999
   484
  then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
hoelzl@49800
   485
    by (auto simp: space_pair_measure)
hoelzl@49800
   486
  show ?case
hoelzl@49800
   487
    apply (subst measurable_cong)
hoelzl@49800
   488
    apply (rule positive_integral_cong)
hoelzl@49800
   489
    apply fact+
hoelzl@49800
   490
    done
hoelzl@49800
   491
next
hoelzl@49800
   492
  case (set Q)
hoelzl@49800
   493
  have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y"
hoelzl@49800
   494
    by (auto simp: indicator_def)
hoelzl@49999
   495
  have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^isup>+ y. indicator Q (x, y) \<partial>M"
hoelzl@49800
   496
    by (simp add: sets_Pair1[OF set])
hoelzl@49999
   497
  from this measurable_emeasure_Pair[OF set] show ?case
hoelzl@49800
   498
    by (rule measurable_cong[THEN iffD1])
hoelzl@49800
   499
qed (simp_all add: positive_integral_add positive_integral_cmult measurable_compose_Pair1
hoelzl@49800
   500
                   positive_integral_monotone_convergence_SUP incseq_def le_fun_def
hoelzl@49800
   501
              cong: measurable_cong)
hoelzl@49800
   502
hoelzl@49999
   503
lemma (in sigma_finite_measure) positive_integral_fst:
hoelzl@49999
   504
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x"
hoelzl@49999
   505
  shows "(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f" (is "?I f = _")
hoelzl@49800
   506
using f proof induct
hoelzl@49800
   507
  case (cong u v)
hoelzl@49800
   508
  moreover then have "?I u = ?I v"
hoelzl@49800
   509
    by (intro positive_integral_cong) (auto simp: space_pair_measure)
hoelzl@49800
   510
  ultimately show ?case
hoelzl@49800
   511
    by (simp cong: positive_integral_cong)
hoelzl@49999
   512
qed (simp_all add: emeasure_pair_measure positive_integral_cmult positive_integral_add
hoelzl@49800
   513
                   positive_integral_monotone_convergence_SUP
hoelzl@49800
   514
                   measurable_compose_Pair1 positive_integral_positive
hoelzl@49825
   515
                   borel_measurable_positive_integral_fst' positive_integral_mono incseq_def le_fun_def
hoelzl@49800
   516
              cong: positive_integral_cong)
hoelzl@40859
   517
hoelzl@49999
   518
lemma (in sigma_finite_measure) positive_integral_fst_measurable:
hoelzl@49999
   519
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)"
hoelzl@49999
   520
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
hoelzl@40859
   521
      (is "?C f \<in> borel_measurable M1")
hoelzl@49999
   522
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f"
hoelzl@49800
   523
  using f
hoelzl@49825
   524
    borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (f x)"]
hoelzl@49800
   525
    positive_integral_fst[of "\<lambda>x. max 0 (f x)"]
hoelzl@49800
   526
  unfolding positive_integral_max_0 by auto
hoelzl@40859
   527
hoelzl@49999
   528
lemma (in sigma_finite_measure) borel_measurable_positive_integral:
hoelzl@49999
   529
  "(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M) \<in> borel_measurable M1"
hoelzl@49999
   530
  using positive_integral_fst_measurable(1)[of "split f" M1] by simp
hoelzl@49825
   531
hoelzl@47694
   532
lemma (in pair_sigma_finite) positive_integral_snd_measurable:
hoelzl@47694
   533
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   534
  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
   535
proof -
hoelzl@47694
   536
  note measurable_pair_swap[OF f]
hoelzl@49999
   537
  from M1.positive_integral_fst_measurable[OF this]
hoelzl@47694
   538
  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
   539
    by simp
hoelzl@47694
   540
  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
   541
    by (subst distr_pair_swap)
hoelzl@47694
   542
       (auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong)
hoelzl@40859
   543
  finally show ?thesis .
hoelzl@40859
   544
qed
hoelzl@40859
   545
hoelzl@40859
   546
lemma (in pair_sigma_finite) Fubini:
hoelzl@47694
   547
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41689
   548
  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
   549
  unfolding positive_integral_snd_measurable[OF assms]
hoelzl@49999
   550
  unfolding M2.positive_integral_fst_measurable[OF assms] ..
hoelzl@40859
   551
hoelzl@41026
   552
lemma (in pair_sigma_finite) integrable_product_swap:
hoelzl@47694
   553
  assumes "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41689
   554
  shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
hoelzl@41026
   555
proof -
hoelzl@41689
   556
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@41661
   557
  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
   558
  show ?thesis unfolding *
hoelzl@47694
   559
    by (rule integrable_distr[OF measurable_pair_swap'])
hoelzl@47694
   560
       (simp add: distr_pair_swap[symmetric] assms)
hoelzl@41661
   561
qed
hoelzl@41661
   562
hoelzl@41661
   563
lemma (in pair_sigma_finite) integrable_product_swap_iff:
hoelzl@47694
   564
  "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41661
   565
proof -
hoelzl@41689
   566
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@41661
   567
  from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
hoelzl@41661
   568
  show ?thesis by auto
hoelzl@41026
   569
qed
hoelzl@41026
   570
hoelzl@41026
   571
lemma (in pair_sigma_finite) integral_product_swap:
hoelzl@47694
   572
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   573
  shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41026
   574
proof -
hoelzl@41661
   575
  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
   576
  show ?thesis unfolding *
hoelzl@47694
   577
    by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
hoelzl@41026
   578
qed
hoelzl@41026
   579
hoelzl@41026
   580
lemma (in pair_sigma_finite) integrable_fst_measurable:
hoelzl@47694
   581
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@47694
   582
  shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE")
hoelzl@47694
   583
    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
   584
proof -
hoelzl@47694
   585
  have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   586
    using f by auto
wenzelm@46731
   587
  let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
hoelzl@41026
   588
  have
hoelzl@47694
   589
    borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and
hoelzl@47694
   590
    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
   591
    using assms by auto
hoelzl@43920
   592
  have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
hoelzl@43920
   593
     "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
hoelzl@49999
   594
    using borel[THEN M2.positive_integral_fst_measurable(1)] int
hoelzl@49999
   595
    unfolding borel[THEN M2.positive_integral_fst_measurable(2)] by simp_all
hoelzl@49999
   596
  with borel[THEN M2.positive_integral_fst_measurable(1)]
hoelzl@43920
   597
  have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
hoelzl@43920
   598
    "AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
hoelzl@47694
   599
    by (auto intro!: positive_integral_PInf_AE )
hoelzl@43920
   600
  then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
hoelzl@43920
   601
    "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
hoelzl@47694
   602
    by (auto simp: positive_integral_positive)
hoelzl@41981
   603
  from AE_pos show ?AE using assms
hoelzl@47694
   604
    by (simp add: measurable_Pair2[OF f_borel] integrable_def)
hoelzl@43920
   605
  { 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
   606
      using positive_integral_positive
hoelzl@47694
   607
      by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
hoelzl@43920
   608
    then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
hoelzl@41981
   609
  note this[simp]
hoelzl@47694
   610
  { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   611
      and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
hoelzl@47694
   612
      and AE: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
hoelzl@43920
   613
    have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
hoelzl@41705
   614
    proof (intro integrable_def[THEN iffD2] conjI)
hoelzl@41705
   615
      show "?f \<in> borel_measurable M1"
hoelzl@49999
   616
        using borel by (auto intro!: M2.positive_integral_fst_measurable)
hoelzl@43920
   617
      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
   618
        using AE positive_integral_positive[of M2]
hoelzl@47694
   619
        by (auto intro!: positive_integral_cong_AE simp: ereal_real)
hoelzl@43920
   620
      then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
hoelzl@49999
   621
        using M2.positive_integral_fst_measurable[OF borel] int by simp
hoelzl@43920
   622
      have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
hoelzl@47694
   623
        by (intro positive_integral_cong_pos)
hoelzl@47694
   624
           (simp add: positive_integral_positive real_of_ereal_pos)
hoelzl@43920
   625
      then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
hoelzl@41705
   626
    qed }
hoelzl@41981
   627
  with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
hoelzl@41705
   628
  show ?INT
hoelzl@47694
   629
    unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2]
hoelzl@49999
   630
      borel[THEN M2.positive_integral_fst_measurable(2), symmetric]
hoelzl@47694
   631
    using AE[THEN integral_real]
hoelzl@41981
   632
    by simp
hoelzl@41026
   633
qed
hoelzl@41026
   634
hoelzl@41026
   635
lemma (in pair_sigma_finite) integrable_snd_measurable:
hoelzl@47694
   636
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@47694
   637
  shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE")
hoelzl@47694
   638
    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
   639
proof -
hoelzl@41689
   640
  interpret Q: pair_sigma_finite M2 M1 by default
hoelzl@47694
   641
  have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))"
hoelzl@41661
   642
    using f unfolding integrable_product_swap_iff .
hoelzl@41026
   643
  show ?INT
hoelzl@41026
   644
    using Q.integrable_fst_measurable(2)[OF Q_int]
hoelzl@47694
   645
    using integral_product_swap[of f] f by auto
hoelzl@41026
   646
  show ?AE
hoelzl@41026
   647
    using Q.integrable_fst_measurable(1)[OF Q_int]
hoelzl@41026
   648
    by simp
hoelzl@41026
   649
qed
hoelzl@41026
   650
hoelzl@47694
   651
lemma (in pair_sigma_finite) positive_integral_fst_measurable':
hoelzl@47694
   652
  assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   653
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M2) \<in> borel_measurable M1"
hoelzl@49999
   654
  using M2.positive_integral_fst_measurable(1)[OF f] by simp
hoelzl@47694
   655
hoelzl@47694
   656
lemma (in pair_sigma_finite) integral_fst_measurable:
hoelzl@47694
   657
  "(\<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
   658
  by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_fst_measurable')
hoelzl@47694
   659
hoelzl@47694
   660
lemma (in pair_sigma_finite) positive_integral_snd_measurable':
hoelzl@47694
   661
  assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   662
  shows "(\<lambda>y. \<integral>\<^isup>+ x. f x y \<partial>M1) \<in> borel_measurable M2"
hoelzl@47694
   663
proof -
hoelzl@47694
   664
  interpret Q: pair_sigma_finite M2 M1 ..
hoelzl@47694
   665
  show ?thesis
hoelzl@47694
   666
    using measurable_pair_swap[OF f]
hoelzl@47694
   667
    by (intro Q.positive_integral_fst_measurable') (simp add: split_beta')
hoelzl@47694
   668
qed
hoelzl@47694
   669
hoelzl@47694
   670
lemma (in pair_sigma_finite) integral_snd_measurable:
hoelzl@47694
   671
  "(\<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
   672
  by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_snd_measurable')
hoelzl@47694
   673
hoelzl@41026
   674
lemma (in pair_sigma_finite) Fubini_integral:
hoelzl@47694
   675
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
hoelzl@41689
   676
  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
   677
  unfolding integrable_snd_measurable[OF assms]
hoelzl@41026
   678
  unfolding integrable_fst_measurable[OF assms] ..
hoelzl@41026
   679
hoelzl@47694
   680
section {* Products on counting spaces, densities and distributions *}
hoelzl@40859
   681
hoelzl@41689
   682
lemma sigma_sets_pair_measure_generator_finite:
hoelzl@38656
   683
  assumes "finite A" and "finite B"
hoelzl@47694
   684
  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
hoelzl@40859
   685
  (is "sigma_sets ?prod ?sets = _")
hoelzl@38656
   686
proof safe
hoelzl@38656
   687
  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
hoelzl@38656
   688
  fix x assume subset: "x \<subseteq> A \<times> B"
hoelzl@38656
   689
  hence "finite x" using fin by (rule finite_subset)
hoelzl@40859
   690
  from this subset show "x \<in> sigma_sets ?prod ?sets"
hoelzl@38656
   691
  proof (induct x)
hoelzl@38656
   692
    case empty show ?case by (rule sigma_sets.Empty)
hoelzl@38656
   693
  next
hoelzl@38656
   694
    case (insert a x)
hoelzl@47694
   695
    hence "{a} \<in> sigma_sets ?prod ?sets" by auto
hoelzl@38656
   696
    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
hoelzl@38656
   697
    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
hoelzl@38656
   698
  qed
hoelzl@38656
   699
next
hoelzl@38656
   700
  fix x a b
hoelzl@40859
   701
  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
hoelzl@38656
   702
  from sigma_sets_into_sp[OF _ this(1)] this(2)
hoelzl@40859
   703
  show "a \<in> A" and "b \<in> B" by auto
hoelzl@35833
   704
qed
hoelzl@35833
   705
hoelzl@47694
   706
lemma pair_measure_count_space:
hoelzl@47694
   707
  assumes A: "finite A" and B: "finite B"
hoelzl@47694
   708
  shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
hoelzl@47694
   709
proof (rule measure_eqI)
hoelzl@47694
   710
  interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
hoelzl@47694
   711
  interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
hoelzl@47694
   712
  interpret P: pair_sigma_finite "count_space A" "count_space B" by default
hoelzl@47694
   713
  show eq: "sets ?P = sets ?C"
hoelzl@47694
   714
    by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
hoelzl@47694
   715
  fix X assume X: "X \<in> sets ?P"
hoelzl@47694
   716
  with eq have X_subset: "X \<subseteq> A \<times> B" by simp
hoelzl@47694
   717
  with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
hoelzl@47694
   718
    by (intro finite_subset[OF _ B]) auto
hoelzl@47694
   719
  have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
hoelzl@47694
   720
  show "emeasure ?P X = emeasure ?C X"
hoelzl@49776
   721
    apply (subst B.emeasure_pair_measure_alt[OF X])
hoelzl@47694
   722
    apply (subst emeasure_count_space)
hoelzl@47694
   723
    using X_subset apply auto []
hoelzl@47694
   724
    apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
hoelzl@47694
   725
    apply (subst positive_integral_count_space)
hoelzl@47694
   726
    using A apply simp
hoelzl@47694
   727
    apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
hoelzl@47694
   728
    apply (subst card_gt_0_iff)
hoelzl@47694
   729
    apply (simp add: fin_Pair)
hoelzl@47694
   730
    apply (subst card_SigmaI[symmetric])
hoelzl@47694
   731
    using A apply simp
hoelzl@47694
   732
    using fin_Pair apply simp
hoelzl@47694
   733
    using X_subset apply (auto intro!: arg_cong[where f=card])
hoelzl@47694
   734
    done
hoelzl@45777
   735
qed
hoelzl@35833
   736
hoelzl@47694
   737
lemma pair_measure_density:
hoelzl@47694
   738
  assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
hoelzl@47694
   739
  assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"
hoelzl@47694
   740
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
hoelzl@47694
   741
  assumes "sigma_finite_measure (density M1 f)" "sigma_finite_measure (density M2 g)"
hoelzl@47694
   742
  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
   743
proof (rule measure_eqI)
hoelzl@47694
   744
  interpret M1: sigma_finite_measure M1 by fact
hoelzl@47694
   745
  interpret M2: sigma_finite_measure M2 by fact
hoelzl@47694
   746
  interpret D1: sigma_finite_measure "density M1 f" by fact
hoelzl@47694
   747
  interpret D2: sigma_finite_measure "density M2 g" by fact
hoelzl@47694
   748
  interpret L: pair_sigma_finite "density M1 f" "density M2 g" ..
hoelzl@47694
   749
  interpret R: pair_sigma_finite M1 M2 ..
hoelzl@47694
   750
hoelzl@47694
   751
  fix A assume A: "A \<in> sets ?L"
hoelzl@47694
   752
  then have indicator_eq: "\<And>x y. indicator A (x, y) = indicator (Pair x -` A) y"
hoelzl@47694
   753
   and Pair_A: "\<And>x. Pair x -` A \<in> sets M2"
hoelzl@47694
   754
    by (auto simp: indicator_def sets_Pair1)
hoelzl@47694
   755
  have f_fst: "(\<lambda>p. f (fst p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   756
    using measurable_comp[OF measurable_fst f(1)] by (simp add: comp_def)
hoelzl@47694
   757
  have g_snd: "(\<lambda>p. g (snd p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
hoelzl@47694
   758
    using measurable_comp[OF measurable_snd g(1)] by (simp add: comp_def)
hoelzl@47694
   759
  have "(\<lambda>x. \<integral>\<^isup>+ y. g (snd (x, y)) * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@49999
   760
    using g_snd Pair_A A by (intro M2.positive_integral_fst_measurable) auto
hoelzl@47694
   761
  then have int_g: "(\<lambda>x. \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
hoelzl@47694
   762
    by simp
hoelzl@38656
   763
hoelzl@47694
   764
  show "emeasure ?L A = emeasure ?R A"
hoelzl@49776
   765
    apply (subst D2.emeasure_pair_measure[OF A])
hoelzl@47694
   766
    apply (subst emeasure_density)
hoelzl@47694
   767
        using f_fst g_snd apply (simp add: split_beta')
hoelzl@47694
   768
      using A apply simp
hoelzl@47694
   769
    apply (subst positive_integral_density[OF g])
hoelzl@47694
   770
      apply (simp add: indicator_eq Pair_A)
hoelzl@47694
   771
    apply (subst positive_integral_density[OF f])
hoelzl@47694
   772
      apply (rule int_g)
hoelzl@49999
   773
    apply (subst M2.positive_integral_fst_measurable(2)[symmetric])
hoelzl@47694
   774
      using f g A Pair_A f_fst g_snd
hoelzl@47694
   775
      apply (auto intro!: positive_integral_cong_AE R.measurable_emeasure_Pair1
hoelzl@47694
   776
                  simp: positive_integral_cmult indicator_eq split_beta')
hoelzl@47694
   777
    apply (intro AE_I2 impI)
hoelzl@47694
   778
    apply (subst mult_assoc)
hoelzl@47694
   779
    apply (subst positive_integral_cmult)
hoelzl@47694
   780
          apply auto
hoelzl@47694
   781
    done
hoelzl@47694
   782
qed simp
hoelzl@47694
   783
hoelzl@47694
   784
lemma sigma_finite_measure_distr:
hoelzl@47694
   785
  assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
hoelzl@47694
   786
  shows "sigma_finite_measure M"
hoelzl@40859
   787
proof -
hoelzl@47694
   788
  interpret sigma_finite_measure "distr M N f" by fact
hoelzl@47694
   789
  from sigma_finite_disjoint guess A . note A = this
hoelzl@47694
   790
  show ?thesis
hoelzl@47694
   791
  proof (unfold_locales, intro conjI exI allI)
hoelzl@47694
   792
    show "range (\<lambda>i. f -` A i \<inter> space M) \<subseteq> sets M"
hoelzl@47694
   793
      using A f by (auto intro!: measurable_sets)
hoelzl@47694
   794
    show "(\<Union>i. f -` A i \<inter> space M) = space M"
hoelzl@47694
   795
      using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def)
hoelzl@47694
   796
    fix i show "emeasure M (f -` A i \<inter> space M) \<noteq> \<infinity>"
hoelzl@47694
   797
      using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq)
hoelzl@47694
   798
  qed
hoelzl@38656
   799
qed
hoelzl@38656
   800
hoelzl@47694
   801
lemma measurable_cong':
hoelzl@47694
   802
  assumes sets: "sets M = sets M'" "sets N = sets N'"
hoelzl@47694
   803
  shows "measurable M N = measurable M' N'"
hoelzl@47694
   804
  using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
hoelzl@38656
   805
hoelzl@47694
   806
lemma pair_measure_distr:
hoelzl@47694
   807
  assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
hoelzl@47694
   808
  assumes "sigma_finite_measure (distr M S f)" "sigma_finite_measure (distr N T g)"
hoelzl@47694
   809
  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
   810
proof (rule measure_eqI)
hoelzl@47694
   811
  show "sets ?P = sets ?D"
hoelzl@47694
   812
    by simp
hoelzl@47694
   813
  interpret S: sigma_finite_measure "distr M S f" by fact
hoelzl@47694
   814
  interpret T: sigma_finite_measure "distr N T g" by fact
hoelzl@47694
   815
  interpret ST: pair_sigma_finite "distr M S f"  "distr N T g" ..
hoelzl@47694
   816
  interpret M: sigma_finite_measure M by (rule sigma_finite_measure_distr) fact+
hoelzl@47694
   817
  interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
hoelzl@47694
   818
  interpret MN: pair_sigma_finite M N ..
hoelzl@47694
   819
  interpret SN: pair_sigma_finite "distr M S f" N ..
hoelzl@47694
   820
  have [simp]: 
hoelzl@47694
   821
    "\<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
   822
    by auto
hoelzl@47694
   823
  then have fg: "(\<lambda>(x, y). (f x, g y)) \<in> measurable (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T)"
hoelzl@47694
   824
    using measurable_comp[OF measurable_fst f] measurable_comp[OF measurable_snd g]
hoelzl@47694
   825
    by (auto simp: measurable_pair_iff)
hoelzl@47694
   826
  fix A assume A: "A \<in> sets ?P"
hoelzl@47694
   827
  then have "emeasure ?P A = (\<integral>\<^isup>+x. emeasure (distr N T g) (Pair x -` A) \<partial>distr M S f)"
hoelzl@49776
   828
    by (rule T.emeasure_pair_measure_alt)
hoelzl@47694
   829
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair x -` A) \<inter> space N) \<partial>distr M S f)"
hoelzl@47694
   830
    using g A by (simp add: sets_Pair1 emeasure_distr)
hoelzl@47694
   831
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair (f x) -` A) \<inter> space N) \<partial>M)"
hoelzl@47694
   832
    using f g A ST.measurable_emeasure_Pair1[OF A]
hoelzl@47694
   833
    by (intro positive_integral_distr) (auto simp add: sets_Pair1 emeasure_distr)
hoelzl@47694
   834
  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
   835
    by (intro positive_integral_cong arg_cong2[where f=emeasure]) (auto simp: space_pair_measure)
hoelzl@47694
   836
  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
   837
    using fg by (intro N.emeasure_pair_measure_alt[symmetric] measurable_sets[OF _ A])
hoelzl@47694
   838
                (auto cong: measurable_cong')
hoelzl@47694
   839
  also have "\<dots> = emeasure ?D A"
hoelzl@47694
   840
    using fg A by (subst emeasure_distr) auto
hoelzl@47694
   841
  finally show "emeasure ?P A = emeasure ?D A" .
hoelzl@45777
   842
qed
hoelzl@39097
   843
hoelzl@40859
   844
end