src/HOL/Probability/Binary_Product_Measure.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 60066 14efa7f4ee7b
child 60727 53697011b03a
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
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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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section {*Binary product measures*}
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theory Binary_Product_Measure
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imports Nonnegative_Lebesgue_Integration
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begin
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lemma Pair_vimage_times[simp]: "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]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
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  by auto
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subsection "Binary products"
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definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
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  "A \<Otimes>\<^sub>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>\<^sup>+x. (\<integral>\<^sup>+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 sets.space_closed[of A] sets.space_closed[of B] by auto
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lemma space_pair_measure:
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  "space (A \<Otimes>\<^sub>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 SIGMA_Collect_eq: "(SIGMA x:space M. {y\<in>space N. P x y}) = {x\<in>space (M \<Otimes>\<^sub>M N). P (fst x) (snd x)}"
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  by (auto simp: space_pair_measure)
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lemma sets_pair_measure:
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  "sets (A \<Otimes>\<^sub>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_in_sets:
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  assumes N: "space A \<times> space B = space N"
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  assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"
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  shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
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  using assms by (auto intro!: sets.sigma_sets_subset simp: sets_pair_measure N)
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lemma sets_pair_measure_cong[measurable_cong, cong]:
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  "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>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, measurable]:
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  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
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  by (auto simp: sets_pair_measure)
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lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
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  using pair_measureI[of "{x}" M1 "{y}" M2] by simp
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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>\<^sub>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.sets_into_space)
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lemma measurable_split_replace[measurable (raw)]:
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  "(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. split (f x) (g x)) \<in> measurable M N"
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  unfolding split_beta' .
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lemma measurable_Pair[measurable (raw)]:
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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>\<^sub>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 sets.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_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
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  by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
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    measurable_def)
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lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
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  by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
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    measurable_def)
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lemma measurable_Pair_compose_split[measurable_dest]:
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  assumes f: "split f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N"
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  assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"
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  shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"
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  using measurable_compose[OF measurable_Pair f, OF g h] by simp
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lemma measurable_Pair1_compose[measurable_dest]:
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  assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
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  assumes [measurable]: "h \<in> measurable N M"
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  shows "(\<lambda>x. f (h x)) \<in> measurable N M1"
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  using measurable_compose[OF f measurable_fst] by simp
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lemma measurable_Pair2_compose[measurable_dest]:
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  assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
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  assumes [measurable]: "h \<in> measurable N M"
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  shows "(\<lambda>x. g (h x)) \<in> measurable N M2"
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  using measurable_compose[OF f measurable_snd] by simp
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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>\<^sub>M M2)"
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  using measurable_Pair[OF assms] by simp
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lemma 
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  assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)" 
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  shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"
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    and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"
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  by simp_all
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lemma
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  assumes f[measurable]: "f \<in> measurable M N"
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  shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N"
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    and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N"
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  by simp_all
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lemma sets_pair_eq_sets_fst_snd:
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  "sets (A \<Otimes>\<^sub>M B) = sets (Sup_sigma {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})"
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    (is "?P = sets (Sup_sigma {?fst, ?snd})")
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proof -
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  { fix a b assume ab: "a \<in> sets A" "b \<in> sets B"
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    then have "a \<times> b = (fst -` a \<inter> (space A \<times> space B)) \<inter> (snd -` b \<inter> (space A \<times> space B))"
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      by (auto dest: sets.sets_into_space)
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    also have "\<dots> \<in> sets (Sup_sigma {?fst, ?snd})"
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      using ab by (auto intro: in_Sup_sigma in_vimage_algebra)
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    finally have "a \<times> b \<in> sets (Sup_sigma {?fst, ?snd})" . }
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  moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)"
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    by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric])
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  moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)"  
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    by (rule sets_image_in_sets) (auto simp: space_pair_measure)
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  ultimately show ?thesis
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    by (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets )
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       (auto simp add: space_Sup_sigma space_pair_measure)
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qed
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lemma measurable_pair_iff:
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  "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
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  by (auto intro: measurable_pair[of f M M1 M2]) 
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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>\<^sub>M M2) (M2 \<Otimes>\<^sub>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>\<^sub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^sub>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>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M"
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  by (auto dest: measurable_pair_swap)
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lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
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  by simp
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lemma sets_Pair1[measurable (raw)]:
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  assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>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.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>\<^sub>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>\<^sub>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.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>\<^sub>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>\<^sub>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 disjoint_family_vimageI: "disjoint_family F \<Longrightarrow> disjoint_family (\<lambda>i. f -` F i)"
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  by (auto simp: disjoint_family_on_def)
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lemma (in finite_measure) finite_measure_cut_measurable:
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  assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>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.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 sets.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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  then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)"
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    by (simp add: suminf_emeasure disjoint_family_vimageI subset_eq vimage_UN sets_pair_measure[symmetric])
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  with union show ?case
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    unfolding sets_pair_measure[symmetric] by simp
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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>\<^sub>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.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))"
hoelzl@49776
   245
      using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
hoelzl@49776
   246
    moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i"
immler@50244
   247
      using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure)
hoelzl@49776
   248
    ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"
hoelzl@49776
   249
      by simp }
hoelzl@49776
   250
  moreover
hoelzl@49776
   251
  { fix x
hoelzl@49776
   252
    have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
hoelzl@49776
   253
    proof (intro suminf_emeasure)
hoelzl@49776
   254
      show "range (?C x) \<subseteq> sets M"
wenzelm@53015
   255
        using F `Q \<in> sets (N \<Otimes>\<^sub>M M)` by (auto intro!: sets_Pair1)
hoelzl@49776
   256
      have "disjoint_family F" using F by auto
hoelzl@49776
   257
      show "disjoint_family (?C x)"
hoelzl@49776
   258
        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
hoelzl@49776
   259
    qed
hoelzl@49776
   260
    also have "(\<Union>i. ?C x i) = Pair x -` Q"
wenzelm@53015
   261
      using F sets.sets_into_space[OF `Q \<in> sets (N \<Otimes>\<^sub>M M)`]
hoelzl@49776
   262
      by (auto simp: space_pair_measure)
hoelzl@49776
   263
    finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"
hoelzl@49776
   264
      by simp }
wenzelm@53015
   265
  ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^sub>M M)` F_sets
hoelzl@49776
   266
    by auto
hoelzl@49776
   267
qed
hoelzl@49776
   268
hoelzl@50003
   269
lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
hoelzl@50003
   270
  assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
wenzelm@53015
   271
  assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)"
hoelzl@50003
   272
  shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
hoelzl@50003
   273
proof -
wenzelm@53015
   274
  from space have "\<And>x. x \<in> space N \<Longrightarrow> Pair x -` {x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} = A x"
hoelzl@50003
   275
    by (auto simp: space_pair_measure)
hoelzl@50003
   276
  with measurable_emeasure_Pair[OF A] show ?thesis
hoelzl@50003
   277
    by (auto cong: measurable_cong)
hoelzl@50003
   278
qed
hoelzl@50003
   279
hoelzl@49776
   280
lemma (in sigma_finite_measure) emeasure_pair_measure:
wenzelm@53015
   281
  assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
wenzelm@53015
   282
  shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X")
hoelzl@49776
   283
proof (rule emeasure_measure_of[OF pair_measure_def])
wenzelm@53015
   284
  show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
hoelzl@56996
   285
    by (auto simp: positive_def nn_integral_nonneg)
hoelzl@49776
   286
  have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
hoelzl@49776
   287
    by (auto simp: indicator_def)
wenzelm@53015
   288
  show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
hoelzl@49776
   289
  proof (rule countably_additiveI)
wenzelm@53015
   290
    fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F"
hoelzl@59353
   291
    from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto
hoelzl@49776
   292
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
hoelzl@49776
   293
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@49776
   294
    moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M"
hoelzl@49776
   295
      using F by (auto simp: sets_Pair1)
hoelzl@49776
   296
    ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
hoelzl@59353
   297
      by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure emeasure_nonneg
hoelzl@56996
   298
               intro!: nn_integral_cong nn_integral_indicator[symmetric])
hoelzl@49776
   299
  qed
hoelzl@49776
   300
  show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
immler@50244
   301
    using sets.space_closed[of N] sets.space_closed[of M] by auto
hoelzl@49776
   302
qed fact
hoelzl@49776
   303
hoelzl@49776
   304
lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
wenzelm@53015
   305
  assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
wenzelm@53015
   306
  shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
hoelzl@49776
   307
proof -
hoelzl@49776
   308
  have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
hoelzl@49776
   309
    by (auto simp: indicator_def)
hoelzl@49776
   310
  show ?thesis
hoelzl@56996
   311
    using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
hoelzl@49776
   312
qed
hoelzl@49776
   313
hoelzl@49776
   314
lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
hoelzl@49776
   315
  assumes A: "A \<in> sets N" and B: "B \<in> sets M"
wenzelm@53015
   316
  shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
hoelzl@49776
   317
proof -
wenzelm@53015
   318
  have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
hoelzl@56996
   319
    using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
hoelzl@49776
   320
  also have "\<dots> = emeasure M B * emeasure N A"
hoelzl@56996
   321
    using A by (simp add: emeasure_nonneg nn_integral_cmult_indicator)
hoelzl@49776
   322
  finally show ?thesis
hoelzl@49776
   323
    by (simp add: ac_simps)
hoelzl@40859
   324
qed
hoelzl@40859
   325
hoelzl@47694
   326
subsection {* Binary products of $\sigma$-finite emeasure spaces *}
hoelzl@40859
   327
hoelzl@47694
   328
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
hoelzl@47694
   329
  for M1 :: "'a measure" and M2 :: "'b measure"
hoelzl@40859
   330
hoelzl@47694
   331
lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
wenzelm@53015
   332
  "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
hoelzl@49776
   333
  using M2.measurable_emeasure_Pair .
hoelzl@40859
   334
hoelzl@47694
   335
lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
wenzelm@53015
   336
  assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
hoelzl@40859
   337
proof -
wenzelm@53015
   338
  have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
hoelzl@47694
   339
    using Q measurable_pair_swap' by (auto intro: measurable_sets)
hoelzl@49776
   340
  note M1.measurable_emeasure_Pair[OF this]
wenzelm@53015
   341
  moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1)) = (\<lambda>x. (x, y)) -` Q"
immler@50244
   342
    using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
hoelzl@47694
   343
  ultimately show ?thesis by simp
hoelzl@39088
   344
qed
hoelzl@39088
   345
hoelzl@41689
   346
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
hoelzl@47694
   347
  defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
hoelzl@47694
   348
  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>
wenzelm@53015
   349
    (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
hoelzl@40859
   350
proof -
hoelzl@47694
   351
  from M1.sigma_finite_incseq guess F1 . note F1 = this
hoelzl@47694
   352
  from M2.sigma_finite_incseq guess F2 . note F2 = this
hoelzl@47694
   353
  from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
hoelzl@40859
   354
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
hoelzl@47694
   355
  show ?thesis
hoelzl@40859
   356
  proof (intro exI[of _ ?F] conjI allI)
hoelzl@47694
   357
    show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
hoelzl@40859
   358
  next
hoelzl@40859
   359
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
hoelzl@40859
   360
    proof (intro subsetI)
hoelzl@40859
   361
      fix x assume "x \<in> space M1 \<times> space M2"
hoelzl@40859
   362
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
hoelzl@40859
   363
        by (auto simp: space)
hoelzl@40859
   364
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
hoelzl@41981
   365
        using `incseq F1` `incseq F2` unfolding incseq_def
hoelzl@41981
   366
        by (force split: split_max)+
hoelzl@40859
   367
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
haftmann@54863
   368
        by (intro SigmaI) (auto simp add: max.commute)
hoelzl@40859
   369
      then show "x \<in> (\<Union>i. ?F i)" by auto
hoelzl@40859
   370
    qed
hoelzl@47694
   371
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
hoelzl@47694
   372
      using space by (auto simp: space)
hoelzl@40859
   373
  next
hoelzl@41981
   374
    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
hoelzl@41981
   375
      using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
hoelzl@40859
   376
  next
hoelzl@40859
   377
    fix i
hoelzl@40859
   378
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
hoelzl@47694
   379
    with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]
wenzelm@53015
   380
    show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
hoelzl@47694
   381
      by (auto simp add: emeasure_pair_measure_Times)
hoelzl@47694
   382
  qed
hoelzl@47694
   383
qed
hoelzl@47694
   384
wenzelm@53015
   385
sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
hoelzl@47694
   386
proof
hoelzl@57447
   387
  from M1.sigma_finite_countable guess F1 ..
hoelzl@57447
   388
  moreover from M2.sigma_finite_countable guess F2 ..
hoelzl@57447
   389
  ultimately show
hoelzl@57447
   390
    "\<exists>A. countable A \<and> A \<subseteq> sets (M1 \<Otimes>\<^sub>M M2) \<and> \<Union>A = space (M1 \<Otimes>\<^sub>M M2) \<and> (\<forall>a\<in>A. emeasure (M1 \<Otimes>\<^sub>M M2) a \<noteq> \<infinity>)"
hoelzl@57447
   391
    by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI)
hoelzl@57447
   392
       (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq
hoelzl@57447
   393
             dest: sets.sets_into_space)
hoelzl@40859
   394
qed
hoelzl@40859
   395
hoelzl@47694
   396
lemma sigma_finite_pair_measure:
hoelzl@47694
   397
  assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
wenzelm@53015
   398
  shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
hoelzl@47694
   399
proof -
hoelzl@47694
   400
  interpret A: sigma_finite_measure A by fact
hoelzl@47694
   401
  interpret B: sigma_finite_measure B by fact
hoelzl@47694
   402
  interpret AB: pair_sigma_finite A  B ..
hoelzl@47694
   403
  show ?thesis ..
hoelzl@40859
   404
qed
hoelzl@39088
   405
hoelzl@47694
   406
lemma sets_pair_swap:
wenzelm@53015
   407
  assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
wenzelm@53015
   408
  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
hoelzl@47694
   409
  using measurable_pair_swap' assms by (rule measurable_sets)
hoelzl@41661
   410
hoelzl@47694
   411
lemma (in pair_sigma_finite) distr_pair_swap:
wenzelm@53015
   412
  "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
hoelzl@40859
   413
proof -
hoelzl@41689
   414
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
   415
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
hoelzl@47694
   416
  show ?thesis
hoelzl@47694
   417
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@47694
   418
    show "?E \<subseteq> Pow (space ?P)"
immler@50244
   419
      using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@47694
   420
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@47694
   421
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@47694
   422
    then show "sets ?D = sigma_sets (space ?P) ?E"
hoelzl@47694
   423
      by simp
hoelzl@47694
   424
  next
hoelzl@49784
   425
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@47694
   426
      using F by (auto simp: space_pair_measure)
hoelzl@47694
   427
  next
hoelzl@47694
   428
    fix X assume "X \<in> ?E"
hoelzl@47694
   429
    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
wenzelm@53015
   430
    have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A"
immler@50244
   431
      using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure)
wenzelm@53015
   432
    with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X"
hoelzl@49776
   433
      by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
hoelzl@47694
   434
                    measurable_pair_swap' ac_simps)
hoelzl@41689
   435
  qed
hoelzl@41689
   436
qed
hoelzl@41689
   437
hoelzl@47694
   438
lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
wenzelm@53015
   439
  assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
wenzelm@53015
   440
  shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
hoelzl@47694
   441
    (is "_ = ?\<nu> A")
hoelzl@41689
   442
proof -
wenzelm@53015
   443
  have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A"
immler@50244
   444
    using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
hoelzl@47694
   445
  show ?thesis using A
hoelzl@47694
   446
    by (subst distr_pair_swap)
hoelzl@47694
   447
       (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
hoelzl@49776
   448
                 M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
hoelzl@49776
   449
qed
hoelzl@49776
   450
hoelzl@49776
   451
lemma (in pair_sigma_finite) AE_pair:
wenzelm@53015
   452
  assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
hoelzl@49776
   453
  shows "AE x in M1. (AE y in M2. Q (x, y))"
hoelzl@49776
   454
proof -
wenzelm@53015
   455
  obtain N where N: "N \<in> sets (M1 \<Otimes>\<^sub>M M2)" "emeasure (M1 \<Otimes>\<^sub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> Q x} \<subseteq> N"
hoelzl@49776
   456
    using assms unfolding eventually_ae_filter by auto
hoelzl@49776
   457
  show ?thesis
hoelzl@49776
   458
  proof (rule AE_I)
wenzelm@53015
   459
    from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^sub>M M2)`]
hoelzl@49776
   460
    show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
hoelzl@56996
   461
      by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff emeasure_nonneg)
hoelzl@49776
   462
    show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
hoelzl@49776
   463
      by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)
hoelzl@49776
   464
    { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
hoelzl@49776
   465
      have "AE y in M2. Q (x, y)"
hoelzl@49776
   466
      proof (rule AE_I)
hoelzl@49776
   467
        show "emeasure M2 (Pair x -` N) = 0" by fact
hoelzl@49776
   468
        show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
hoelzl@49776
   469
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
hoelzl@49776
   470
          using N `x \<in> space M1` unfolding space_pair_measure by auto
hoelzl@49776
   471
      qed }
hoelzl@49776
   472
    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
   473
      by auto
hoelzl@49776
   474
  qed
hoelzl@49776
   475
qed
hoelzl@49776
   476
hoelzl@49776
   477
lemma (in pair_sigma_finite) AE_pair_measure:
wenzelm@53015
   478
  assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
hoelzl@49776
   479
  assumes ae: "AE x in M1. AE y in M2. P (x, y)"
wenzelm@53015
   480
  shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
hoelzl@49776
   481
proof (subst AE_iff_measurable[OF _ refl])
wenzelm@53015
   482
  show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
immler@50244
   483
    by (rule sets.sets_Collect) fact
wenzelm@53015
   484
  then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
wenzelm@53015
   485
      (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
hoelzl@49776
   486
    by (simp add: M2.emeasure_pair_measure)
wenzelm@53015
   487
  also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)"
hoelzl@49776
   488
    using ae
hoelzl@56996
   489
    apply (safe intro!: nn_integral_cong_AE)
hoelzl@49776
   490
    apply (intro AE_I2)
hoelzl@56996
   491
    apply (safe intro!: nn_integral_cong_AE)
hoelzl@49776
   492
    apply auto
hoelzl@49776
   493
    done
wenzelm@53015
   494
  finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
hoelzl@49776
   495
qed
hoelzl@49776
   496
hoelzl@49776
   497
lemma (in pair_sigma_finite) AE_pair_iff:
wenzelm@53015
   498
  "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
wenzelm@53015
   499
    (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x))"
hoelzl@49776
   500
  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
   501
hoelzl@49776
   502
lemma (in pair_sigma_finite) AE_commute:
wenzelm@53015
   503
  assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
hoelzl@49776
   504
  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
   505
proof -
hoelzl@49776
   506
  interpret Q: pair_sigma_finite M2 M1 ..
hoelzl@49776
   507
  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
   508
    by auto
wenzelm@53015
   509
  have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} =
wenzelm@53015
   510
    (\<lambda>(x, y). (y, x)) -` {x \<in> space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^sub>M M1)"
hoelzl@49776
   511
    by (auto simp: space_pair_measure)
wenzelm@53015
   512
  also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)"
hoelzl@49776
   513
    by (intro sets_pair_swap P)
hoelzl@49776
   514
  finally show ?thesis
hoelzl@49776
   515
    apply (subst AE_pair_iff[OF P])
hoelzl@49776
   516
    apply (subst distr_pair_swap)
hoelzl@49776
   517
    apply (subst AE_distr_iff[OF measurable_pair_swap' P])
hoelzl@49776
   518
    apply (subst Q.AE_pair_iff)
hoelzl@49776
   519
    apply simp_all
hoelzl@49776
   520
    done
hoelzl@40859
   521
qed
hoelzl@40859
   522
hoelzl@56994
   523
subsection "Fubinis theorem"
hoelzl@40859
   524
hoelzl@49800
   525
lemma measurable_compose_Pair1:
wenzelm@53015
   526
  "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
hoelzl@50003
   527
  by simp
hoelzl@49800
   528
hoelzl@56996
   529
lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst':
wenzelm@53015
   530
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x"
wenzelm@53015
   531
  shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
hoelzl@49800
   532
using f proof induct
hoelzl@49800
   533
  case (cong u v)
hoelzl@49999
   534
  then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
hoelzl@49800
   535
    by (auto simp: space_pair_measure)
hoelzl@49800
   536
  show ?case
hoelzl@49800
   537
    apply (subst measurable_cong)
hoelzl@56996
   538
    apply (rule nn_integral_cong)
hoelzl@49800
   539
    apply fact+
hoelzl@49800
   540
    done
hoelzl@49800
   541
next
hoelzl@49800
   542
  case (set Q)
hoelzl@49800
   543
  have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y"
hoelzl@49800
   544
    by (auto simp: indicator_def)
wenzelm@53015
   545
  have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M"
hoelzl@49800
   546
    by (simp add: sets_Pair1[OF set])
hoelzl@49999
   547
  from this measurable_emeasure_Pair[OF set] show ?case
hoelzl@49800
   548
    by (rule measurable_cong[THEN iffD1])
hoelzl@56996
   549
qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1
hoelzl@56996
   550
                   nn_integral_monotone_convergence_SUP incseq_def le_fun_def
hoelzl@49800
   551
              cong: measurable_cong)
hoelzl@49800
   552
hoelzl@56996
   553
lemma (in sigma_finite_measure) nn_integral_fst':
wenzelm@53015
   554
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x"
hoelzl@56996
   555
  shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f" (is "?I f = _")
hoelzl@49800
   556
using f proof induct
hoelzl@49800
   557
  case (cong u v)
wenzelm@53374
   558
  then have "?I u = ?I v"
hoelzl@56996
   559
    by (intro nn_integral_cong) (auto simp: space_pair_measure)
wenzelm@53374
   560
  with cong show ?case
hoelzl@56996
   561
    by (simp cong: nn_integral_cong)
hoelzl@56996
   562
qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add
hoelzl@56996
   563
                   nn_integral_monotone_convergence_SUP
hoelzl@56996
   564
                   measurable_compose_Pair1 nn_integral_nonneg
hoelzl@56996
   565
                   borel_measurable_nn_integral_fst' nn_integral_mono incseq_def le_fun_def
hoelzl@56996
   566
              cong: nn_integral_cong)
hoelzl@40859
   567
hoelzl@56996
   568
lemma (in sigma_finite_measure) nn_integral_fst:
wenzelm@53015
   569
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
hoelzl@56996
   570
  shows "(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f"
hoelzl@49800
   571
  using f
hoelzl@56996
   572
    borel_measurable_nn_integral_fst'[of "\<lambda>x. max 0 (f x)"]
hoelzl@56996
   573
    nn_integral_fst'[of "\<lambda>x. max 0 (f x)"]
hoelzl@56996
   574
  unfolding nn_integral_max_0 by auto
hoelzl@40859
   575
hoelzl@56996
   576
lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
wenzelm@53015
   577
  "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N"
hoelzl@56996
   578
  using borel_measurable_nn_integral_fst'[of "\<lambda>x. max 0 (split f x)" N]
hoelzl@56996
   579
  by (simp add: nn_integral_max_0)
hoelzl@50003
   580
hoelzl@56996
   581
lemma (in pair_sigma_finite) nn_integral_snd:
wenzelm@53015
   582
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
hoelzl@56996
   583
  shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
hoelzl@41661
   584
proof -
hoelzl@47694
   585
  note measurable_pair_swap[OF f]
hoelzl@56996
   586
  from M1.nn_integral_fst[OF this]
wenzelm@53015
   587
  have "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1))"
hoelzl@40859
   588
    by simp
hoelzl@56996
   589
  also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
hoelzl@47694
   590
    by (subst distr_pair_swap)
hoelzl@56996
   591
       (auto simp: nn_integral_distr[OF measurable_pair_swap' f] intro!: nn_integral_cong)
hoelzl@40859
   592
  finally show ?thesis .
hoelzl@40859
   593
qed
hoelzl@40859
   594
hoelzl@40859
   595
lemma (in pair_sigma_finite) Fubini:
wenzelm@53015
   596
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
wenzelm@53015
   597
  shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
hoelzl@56996
   598
  unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..
hoelzl@41026
   599
hoelzl@57235
   600
lemma (in pair_sigma_finite) Fubini':
hoelzl@57235
   601
  assumes f: "split f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
hoelzl@57235
   602
  shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f x y \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f x y \<partial>M2) \<partial>M1)"
hoelzl@57235
   603
  using Fubini[OF f] by simp
hoelzl@57235
   604
hoelzl@56994
   605
subsection {* Products on counting spaces, densities and distributions *}
hoelzl@40859
   606
hoelzl@59088
   607
lemma sigma_prod:
hoelzl@59088
   608
  assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"
hoelzl@59088
   609
  assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y"
hoelzl@59088
   610
  shows "sigma X A \<Otimes>\<^sub>M sigma Y B = sigma (X \<times> Y) {a \<times> b | a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   611
    (is "?P = ?S")
hoelzl@59088
   612
proof (rule measure_eqI)
hoelzl@59088
   613
  have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X"
hoelzl@59088
   614
    by auto
hoelzl@59088
   615
  let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}"
hoelzl@59088
   616
  have "sets ?P = 
hoelzl@59088
   617
    sets (\<Squnion>\<^sub>\<sigma> xy\<in>?XY. sigma (X \<times> Y) xy)"
hoelzl@59088
   618
    by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)
hoelzl@59088
   619
  also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))"
hoelzl@59088
   620
    by (intro Sup_sigma_sigma arg_cong[where f=sets]) auto
hoelzl@59088
   621
  also have "\<dots> = sets ?S"
hoelzl@59088
   622
  proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI) 
hoelzl@59088
   623
    show "\<Union>?XY \<subseteq> Pow (X \<times> Y)" "{a \<times> b |a b. a \<in> A \<and> b \<in> B} \<subseteq> Pow (X \<times> Y)"
hoelzl@59088
   624
      using A B by auto
hoelzl@59088
   625
  next
hoelzl@59088
   626
    interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   627
      using A B by (intro sigma_algebra_sigma_sets) auto
hoelzl@59088
   628
    fix Z assume "Z \<in> \<Union>?XY"
hoelzl@59088
   629
    then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   630
    proof safe
hoelzl@59088
   631
      fix a assume "a \<in> A"
hoelzl@59088
   632
      from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E"
hoelzl@59088
   633
        by auto
hoelzl@59088
   634
      with `a \<in> A` A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"
hoelzl@59088
   635
        by auto
hoelzl@59088
   636
      show "fst -` a \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   637
        using `a \<in> A` E unfolding eq by (auto intro!: XY.countable_UN')
hoelzl@59088
   638
    next
hoelzl@59088
   639
      fix b assume "b \<in> B"
hoelzl@59088
   640
      from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E"
hoelzl@59088
   641
        by auto
hoelzl@59088
   642
      with `b \<in> B` B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"
hoelzl@59088
   643
        by auto
hoelzl@59088
   644
      show "snd -` b \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   645
        using `b \<in> B` E unfolding eq by (auto intro!: XY.countable_UN')
hoelzl@59088
   646
    qed
hoelzl@59088
   647
  next
hoelzl@59088
   648
    fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   649
    then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B"
hoelzl@59088
   650
      by auto
hoelzl@59088
   651
    then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)"
hoelzl@59088
   652
      using A B by auto
hoelzl@59088
   653
    interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)"
hoelzl@59088
   654
      by (intro sigma_algebra_sigma_sets) auto
hoelzl@59088
   655
    show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)"
hoelzl@59088
   656
      unfolding Z by (rule XY.Int) (blast intro: ab)+
hoelzl@59088
   657
  qed
hoelzl@59088
   658
  finally show "sets ?P = sets ?S" .
hoelzl@59088
   659
next
hoelzl@59088
   660
  interpret finite_measure "sigma X A" for X A
hoelzl@59088
   661
    proof qed (simp add: emeasure_sigma)
hoelzl@59088
   662
  fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A"
hoelzl@59088
   663
    by (simp add: emeasure_pair_measure_alt emeasure_sigma)
hoelzl@59088
   664
qed
hoelzl@59088
   665
hoelzl@41689
   666
lemma sigma_sets_pair_measure_generator_finite:
hoelzl@38656
   667
  assumes "finite A" and "finite B"
hoelzl@47694
   668
  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
hoelzl@40859
   669
  (is "sigma_sets ?prod ?sets = _")
hoelzl@38656
   670
proof safe
hoelzl@38656
   671
  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
hoelzl@38656
   672
  fix x assume subset: "x \<subseteq> A \<times> B"
hoelzl@38656
   673
  hence "finite x" using fin by (rule finite_subset)
hoelzl@40859
   674
  from this subset show "x \<in> sigma_sets ?prod ?sets"
hoelzl@38656
   675
  proof (induct x)
hoelzl@38656
   676
    case empty show ?case by (rule sigma_sets.Empty)
hoelzl@38656
   677
  next
hoelzl@38656
   678
    case (insert a x)
hoelzl@47694
   679
    hence "{a} \<in> sigma_sets ?prod ?sets" by auto
hoelzl@38656
   680
    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
hoelzl@38656
   681
    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
hoelzl@38656
   682
  qed
hoelzl@38656
   683
next
hoelzl@38656
   684
  fix x a b
hoelzl@40859
   685
  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
hoelzl@38656
   686
  from sigma_sets_into_sp[OF _ this(1)] this(2)
hoelzl@40859
   687
  show "a \<in> A" and "b \<in> B" by auto
hoelzl@35833
   688
qed
hoelzl@35833
   689
hoelzl@59088
   690
lemma borel_prod:
hoelzl@59088
   691
  "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
hoelzl@59088
   692
  (is "?P = ?B")
hoelzl@59088
   693
proof -
hoelzl@59088
   694
  have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
hoelzl@59088
   695
    by (rule second_countable_borel_measurable[OF open_prod_generated])
hoelzl@59088
   696
  also have "\<dots> = ?P"
hoelzl@59088
   697
    unfolding borel_def
hoelzl@59088
   698
    by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])
hoelzl@59088
   699
  finally show ?thesis ..
hoelzl@59088
   700
qed
hoelzl@59088
   701
hoelzl@47694
   702
lemma pair_measure_count_space:
hoelzl@47694
   703
  assumes A: "finite A" and B: "finite B"
wenzelm@53015
   704
  shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
hoelzl@47694
   705
proof (rule measure_eqI)
hoelzl@47694
   706
  interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
hoelzl@47694
   707
  interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
hoelzl@47694
   708
  interpret P: pair_sigma_finite "count_space A" "count_space B" by default
hoelzl@47694
   709
  show eq: "sets ?P = sets ?C"
hoelzl@47694
   710
    by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
hoelzl@47694
   711
  fix X assume X: "X \<in> sets ?P"
hoelzl@47694
   712
  with eq have X_subset: "X \<subseteq> A \<times> B" by simp
hoelzl@47694
   713
  with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
hoelzl@47694
   714
    by (intro finite_subset[OF _ B]) auto
hoelzl@47694
   715
  have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
hoelzl@47694
   716
  show "emeasure ?P X = emeasure ?C X"
hoelzl@49776
   717
    apply (subst B.emeasure_pair_measure_alt[OF X])
hoelzl@47694
   718
    apply (subst emeasure_count_space)
hoelzl@47694
   719
    using X_subset apply auto []
hoelzl@47694
   720
    apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
hoelzl@56996
   721
    apply (subst nn_integral_count_space)
hoelzl@47694
   722
    using A apply simp
hoelzl@47694
   723
    apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
hoelzl@47694
   724
    apply (subst card_gt_0_iff)
hoelzl@47694
   725
    apply (simp add: fin_Pair)
hoelzl@47694
   726
    apply (subst card_SigmaI[symmetric])
hoelzl@47694
   727
    using A apply simp
hoelzl@47694
   728
    using fin_Pair apply simp
hoelzl@47694
   729
    using X_subset apply (auto intro!: arg_cong[where f=card])
hoelzl@47694
   730
    done
hoelzl@45777
   731
qed
hoelzl@35833
   732
hoelzl@59426
   733
hoelzl@59426
   734
lemma emeasure_prod_count_space:
hoelzl@59426
   735
  assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
hoelzl@59426
   736
  shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
hoelzl@59426
   737
  by (rule emeasure_measure_of[OF pair_measure_def])
hoelzl@59426
   738
     (auto simp: countably_additive_def positive_def suminf_indicator nn_integral_nonneg A
hoelzl@59426
   739
                 nn_integral_suminf[symmetric] dest: sets.sets_into_space)
hoelzl@59426
   740
hoelzl@59426
   741
lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
hoelzl@59426
   742
proof -
hoelzl@59426
   743
  have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ereal)"
hoelzl@59426
   744
    by (auto split: split_indicator)
hoelzl@59426
   745
  show ?thesis
hoelzl@59426
   746
    by (cases x)
hoelzl@59426
   747
       (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair nn_integral_max_0 one_ereal_def[symmetric])
hoelzl@59426
   748
qed
hoelzl@59426
   749
hoelzl@59426
   750
lemma emeasure_count_space_prod_eq:
hoelzl@59426
   751
  fixes A :: "('a \<times> 'b) set"
hoelzl@59426
   752
  assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
hoelzl@59426
   753
  shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
hoelzl@59426
   754
proof -
hoelzl@59426
   755
  { fix A :: "('a \<times> 'b) set" assume "countable A"
hoelzl@59426
   756
    then have "emeasure (?A \<Otimes>\<^sub>M ?B) (\<Union>a\<in>A. {a}) = (\<integral>\<^sup>+a. emeasure (?A \<Otimes>\<^sub>M ?B) {a} \<partial>count_space A)"
hoelzl@59426
   757
      by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
hoelzl@59426
   758
    also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)"
hoelzl@59426
   759
      by (subst nn_integral_count_space_indicator) auto
hoelzl@59426
   760
    finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
hoelzl@59426
   761
      by simp }
hoelzl@59426
   762
  note * = this
hoelzl@59426
   763
hoelzl@59426
   764
  show ?thesis
hoelzl@59426
   765
  proof cases
hoelzl@59426
   766
    assume "finite A" then show ?thesis
hoelzl@59426
   767
      by (intro * countable_finite)
hoelzl@59426
   768
  next
hoelzl@59426
   769
    assume "infinite A"
hoelzl@59426
   770
    then obtain C where "countable C" and "infinite C" and "C \<subseteq> A"
hoelzl@59426
   771
      by (auto dest: infinite_countable_subset')
hoelzl@59426
   772
    with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A"
hoelzl@59426
   773
      by (intro emeasure_mono) auto
hoelzl@59426
   774
    also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C"
hoelzl@59426
   775
      using `countable C` by (rule *)
hoelzl@59426
   776
    finally show ?thesis
hoelzl@59426
   777
      using `infinite C` `infinite A` by simp
hoelzl@59426
   778
  qed
hoelzl@59426
   779
qed
hoelzl@59426
   780
hoelzl@59426
   781
lemma nn_intergal_count_space_prod_eq':
hoelzl@59426
   782
  assumes [simp]: "\<And>x. 0 \<le> f x"
hoelzl@59426
   783
  shows "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
hoelzl@59426
   784
    (is "nn_integral ?P f = _")
hoelzl@59426
   785
proof cases
hoelzl@59426
   786
  assume cntbl: "countable {x. f x \<noteq> 0}"
hoelzl@59426
   787
  have [simp]: "\<And>x. ereal (real (card ({x} \<inter> {x. f x \<noteq> 0}))) = indicator {x. f x \<noteq> 0} x"
hoelzl@59426
   788
    by (auto split: split_indicator)
hoelzl@59426
   789
  have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P"
hoelzl@59426
   790
    by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y])
hoelzl@59426
   791
       (auto intro: sets_Pair)
hoelzl@59426
   792
hoelzl@59426
   793
  have "(\<integral>\<^sup>+x. f x \<partial>?P) = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x * indicator {x} x' \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)"
hoelzl@59426
   794
    by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator)
hoelzl@59426
   795
  also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)"
hoelzl@59426
   796
    by (auto intro!: nn_integral_cong split: split_indicator)
hoelzl@59426
   797
  also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})"
hoelzl@59426
   798
    by (intro nn_integral_count_space_nn_integral cntbl) auto
hoelzl@59426
   799
  also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})"
hoelzl@59426
   800
    by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair)
hoelzl@59426
   801
  finally show ?thesis
hoelzl@59426
   802
    by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
hoelzl@59426
   803
next
hoelzl@59426
   804
  { fix x assume "f x \<noteq> 0"
hoelzl@59426
   805
    with `0 \<le> f x` have "(\<exists>r. 0 < r \<and> f x = ereal r) \<or> f x = \<infinity>"
hoelzl@59426
   806
      by (cases "f x") (auto simp: less_le)
hoelzl@59426
   807
    then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f x"
hoelzl@59426
   808
      by (auto elim!: nat_approx_posE intro!: less_imp_le) }
hoelzl@59426
   809
  note * = this
hoelzl@59426
   810
hoelzl@59426
   811
  assume cntbl: "uncountable {x. f x \<noteq> 0}"
hoelzl@59426
   812
  also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})"
hoelzl@59426
   813
    using * by auto
hoelzl@59426
   814
  finally obtain n where "infinite {x. 1/Suc n \<le> f x}"
hoelzl@59426
   815
    by (meson countableI_type countable_UN uncountable_infinite)
hoelzl@59426
   816
  then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C"
hoelzl@59426
   817
    by (metis infinite_countable_subset')
hoelzl@59426
   818
hoelzl@59426
   819
  have [measurable]: "C \<in> sets ?P"
hoelzl@59426
   820
    using sets.countable[OF _ `countable C`, of ?P] by (auto simp: sets_Pair)
hoelzl@59426
   821
hoelzl@59426
   822
  have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f"
hoelzl@59426
   823
    using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
hoelzl@59426
   824
  moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>"
hoelzl@59426
   825
    using `infinite C` by (simp add: nn_integral_cmult emeasure_count_space_prod_eq)
hoelzl@59426
   826
  moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f"
hoelzl@59426
   827
    using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
hoelzl@59426
   828
  moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>"
hoelzl@59426
   829
    using `infinite C` by (simp add: nn_integral_cmult)
hoelzl@59426
   830
  ultimately show ?thesis
hoelzl@59426
   831
    by simp
hoelzl@59426
   832
qed
hoelzl@59426
   833
hoelzl@59426
   834
lemma nn_intergal_count_space_prod_eq:
hoelzl@59426
   835
  "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
hoelzl@59426
   836
  by (subst (1 2) nn_integral_max_0[symmetric]) (auto intro!: nn_intergal_count_space_prod_eq')
hoelzl@59426
   837
hoelzl@47694
   838
lemma pair_measure_density:
hoelzl@47694
   839
  assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
hoelzl@47694
   840
  assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"
hoelzl@50003
   841
  assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
wenzelm@53015
   842
  shows "density M1 f \<Otimes>\<^sub>M density M2 g = density (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")
hoelzl@47694
   843
proof (rule measure_eqI)
hoelzl@47694
   844
  interpret M2: sigma_finite_measure M2 by fact
hoelzl@47694
   845
  interpret D2: sigma_finite_measure "density M2 g" by fact
hoelzl@47694
   846
hoelzl@47694
   847
  fix A assume A: "A \<in> sets ?L"
wenzelm@53015
   848
  with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) =
wenzelm@53015
   849
    (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)"
hoelzl@56996
   850
    by (intro nn_integral_cong_AE)
hoelzl@56996
   851
       (auto simp add: nn_integral_cmult[symmetric] ac_simps)
hoelzl@50003
   852
  with A f g show "emeasure ?L A = emeasure ?R A"
hoelzl@56996
   853
    by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density
hoelzl@56996
   854
                  M2.nn_integral_fst[symmetric]
hoelzl@56996
   855
             cong: nn_integral_cong)
hoelzl@47694
   856
qed simp
hoelzl@47694
   857
hoelzl@47694
   858
lemma sigma_finite_measure_distr:
hoelzl@47694
   859
  assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
hoelzl@47694
   860
  shows "sigma_finite_measure M"
hoelzl@40859
   861
proof -
hoelzl@47694
   862
  interpret sigma_finite_measure "distr M N f" by fact
hoelzl@57447
   863
  from sigma_finite_countable guess A .. note A = this
hoelzl@47694
   864
  show ?thesis
hoelzl@57447
   865
  proof
hoelzl@57447
   866
    show "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union>A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
hoelzl@57447
   867
      using A f
hoelzl@57447
   868
      by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"])
hoelzl@57447
   869
         (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)
hoelzl@47694
   870
  qed
hoelzl@38656
   871
qed
hoelzl@38656
   872
hoelzl@47694
   873
lemma pair_measure_distr:
hoelzl@47694
   874
  assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
hoelzl@50003
   875
  assumes "sigma_finite_measure (distr N T g)"
wenzelm@53015
   876
  shows "distr M S f \<Otimes>\<^sub>M distr N T g = distr (M \<Otimes>\<^sub>M N) (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")
hoelzl@47694
   877
proof (rule measure_eqI)
hoelzl@47694
   878
  interpret T: sigma_finite_measure "distr N T g" by fact
hoelzl@47694
   879
  interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
hoelzl@50003
   880
hoelzl@47694
   881
  fix A assume A: "A \<in> sets ?P"
hoelzl@50003
   882
  with f g show "emeasure ?P A = emeasure ?D A"
hoelzl@50003
   883
    by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr
hoelzl@56996
   884
                       T.emeasure_pair_measure_alt nn_integral_distr
hoelzl@56996
   885
             intro!: nn_integral_cong arg_cong[where f="emeasure N"])
hoelzl@50003
   886
qed simp
hoelzl@39097
   887
hoelzl@50104
   888
lemma pair_measure_eqI:
hoelzl@50104
   889
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
wenzelm@53015
   890
  assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
hoelzl@50104
   891
  assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)"
wenzelm@53015
   892
  shows "M1 \<Otimes>\<^sub>M M2 = M"
hoelzl@50104
   893
proof -
hoelzl@50104
   894
  interpret M1: sigma_finite_measure M1 by fact
hoelzl@50104
   895
  interpret M2: sigma_finite_measure M2 by fact
hoelzl@50104
   896
  interpret pair_sigma_finite M1 M2 by default
hoelzl@50104
   897
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@50104
   898
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
wenzelm@53015
   899
  let ?P = "M1 \<Otimes>\<^sub>M M2"
hoelzl@50104
   900
  show ?thesis
hoelzl@50104
   901
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@50104
   902
    show "?E \<subseteq> Pow (space ?P)"
immler@50244
   903
      using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@50104
   904
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@50104
   905
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@50104
   906
    then show "sets M = sigma_sets (space ?P) ?E"
hoelzl@50104
   907
      using sets[symmetric] by simp
hoelzl@50104
   908
  next
hoelzl@50104
   909
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@50104
   910
      using F by (auto simp: space_pair_measure)
hoelzl@50104
   911
  next
hoelzl@50104
   912
    fix X assume "X \<in> ?E"
hoelzl@50104
   913
    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@50104
   914
    then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
hoelzl@50104
   915
       by (simp add: M2.emeasure_pair_measure_Times)
hoelzl@50104
   916
    also have "\<dots> = emeasure M (A \<times> B)"
hoelzl@50104
   917
      using A B emeasure by auto
hoelzl@50104
   918
    finally show "emeasure ?P X = emeasure M X"
hoelzl@50104
   919
      by simp
hoelzl@50104
   920
  qed
hoelzl@50104
   921
qed
hoelzl@57025
   922
  
hoelzl@57025
   923
lemma sets_pair_countable:
hoelzl@57025
   924
  assumes "countable S1" "countable S2"
hoelzl@57025
   925
  assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
hoelzl@57025
   926
  shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
hoelzl@57025
   927
proof auto
hoelzl@57025
   928
  fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
hoelzl@57025
   929
  from sets.sets_into_space[OF x(1)] x(2)
hoelzl@57025
   930
    sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
hoelzl@57025
   931
  show "a \<in> S1" "b \<in> S2"
hoelzl@57025
   932
    by (auto simp: space_pair_measure)
hoelzl@57025
   933
next
hoelzl@57025
   934
  fix X assume X: "X \<subseteq> S1 \<times> S2"
hoelzl@57025
   935
  then have "countable X"
hoelzl@57025
   936
    by (metis countable_subset `countable S1` `countable S2` countable_SIGMA)
hoelzl@57025
   937
  have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto
hoelzl@57025
   938
  also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)"
hoelzl@57025
   939
    using X
hoelzl@57025
   940
    by (safe intro!: sets.countable_UN' `countable X` subsetI pair_measureI) (auto simp: M N)
hoelzl@57025
   941
  finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
hoelzl@57025
   942
qed
hoelzl@57025
   943
hoelzl@57025
   944
lemma pair_measure_countable:
hoelzl@57025
   945
  assumes "countable S1" "countable S2"
hoelzl@57025
   946
  shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
hoelzl@57025
   947
proof (rule pair_measure_eqI)
hoelzl@57025
   948
  show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
hoelzl@57025
   949
    using assms by (auto intro!: sigma_finite_measure_count_space_countable)
hoelzl@57025
   950
  show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
hoelzl@57025
   951
    by (subst sets_pair_countable[OF assms]) auto
hoelzl@57025
   952
next
hoelzl@57025
   953
  fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)"
hoelzl@57025
   954
  then show "emeasure (count_space S1) A * emeasure (count_space S2) B = 
hoelzl@57025
   955
    emeasure (count_space (S1 \<times> S2)) (A \<times> B)"
hoelzl@57025
   956
    by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff)
hoelzl@57025
   957
qed
hoelzl@50104
   958
Andreas@59489
   959
lemma nn_integral_fst_count_space':
Andreas@59489
   960
  assumes nonneg: "\<And>xy. 0 \<le> f xy"
Andreas@59489
   961
  shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
Andreas@59489
   962
  (is "?lhs = ?rhs")
Andreas@59489
   963
proof(cases)
Andreas@59489
   964
  assume *: "countable {xy. f xy \<noteq> 0}"
Andreas@59489
   965
  let ?A = "fst ` {xy. f xy \<noteq> 0}"
Andreas@59489
   966
  let ?B = "snd ` {xy. f xy \<noteq> 0}"
Andreas@59489
   967
  from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+
Andreas@59489
   968
  from nonneg have f_neq_0: "\<And>xy. f xy \<noteq> 0 \<longleftrightarrow> f xy > 0"
Andreas@59489
   969
    by(auto simp add: order.order_iff_strict)
Andreas@59489
   970
  have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)"
Andreas@59489
   971
    by(rule nn_integral_count_space_eq)
Andreas@59489
   972
      (auto simp add: f_neq_0 nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI)
Andreas@59489
   973
  also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)"
Andreas@59489
   974
    by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI)
Andreas@59489
   975
  also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))"
Andreas@59489
   976
    by(subst sigma_finite_measure.nn_integral_fst)
Andreas@59489
   977
      (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable)
Andreas@59489
   978
  also have "\<dots> = ?rhs"
Andreas@59489
   979
    by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI)
Andreas@59489
   980
  finally show ?thesis .
Andreas@59489
   981
next
Andreas@59489
   982
  { fix xy assume "f xy \<noteq> 0"
Andreas@59489
   983
    with `0 \<le> f xy` have "(\<exists>r. 0 < r \<and> f xy = ereal r) \<or> f xy = \<infinity>"
Andreas@59489
   984
      by (cases "f xy") (auto simp: less_le)
Andreas@59489
   985
    then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f xy"
Andreas@59489
   986
      by (auto elim!: nat_approx_posE intro!: less_imp_le) }
Andreas@59489
   987
  note * = this
Andreas@59489
   988
Andreas@59489
   989
  assume cntbl: "uncountable {xy. f xy \<noteq> 0}"
Andreas@59489
   990
  also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})"
Andreas@59489
   991
    using * by auto
Andreas@59489
   992
  finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}"
Andreas@59489
   993
    by (meson countableI_type countable_UN uncountable_infinite)
Andreas@59489
   994
  then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C"
Andreas@59489
   995
    by (metis infinite_countable_subset')
Andreas@59489
   996
Andreas@59489
   997
  have "\<infinity> = (\<integral>\<^sup>+ xy. ereal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)"
Andreas@59489
   998
    using \<open>infinite C\<close> by(simp add: nn_integral_cmult)
Andreas@59489
   999
  also have "\<dots> \<le> ?rhs" using C
Andreas@59489
  1000
    by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg)
Andreas@59489
  1001
  finally have "?rhs = \<infinity>" by simp
Andreas@59489
  1002
  moreover have "?lhs = \<infinity>"
Andreas@59489
  1003
  proof(cases "finite (fst ` C)")
Andreas@59489
  1004
    case True
Andreas@59489
  1005
    then obtain x C' where x: "x \<in> fst ` C" 
Andreas@59489
  1006
      and C': "C' = fst -` {x} \<inter> C"
Andreas@59489
  1007
      and "infinite C'"
Andreas@59489
  1008
      using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE')
Andreas@59489
  1009
    from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto
Andreas@59489
  1010
Andreas@59489
  1011
    from C' \<open>infinite C'\<close> have "infinite (snd ` C')"
Andreas@59489
  1012
      by(auto dest!: finite_imageD simp add: inj_on_def)
Andreas@59489
  1013
    then have "\<infinity> = (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)"
Andreas@59489
  1014
      by(simp add: nn_integral_cmult)
Andreas@59489
  1015
    also have "\<dots> = (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)"
Andreas@59489
  1016
      by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C')
Andreas@59489
  1017
    also have "\<dots> = (\<integral>\<^sup>+ x'. (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV) * indicator {x} x' \<partial>count_space UNIV)"
Andreas@59489
  1018
      by(simp add: one_ereal_def[symmetric] nn_integral_nonneg nn_integral_cmult_indicator)
Andreas@59489
  1019
    also have "\<dots> \<le> (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV \<partial>count_space UNIV)"
Andreas@59489
  1020
      by(rule nn_integral_mono)(simp split: split_indicator add: nn_integral_nonneg)
Andreas@59489
  1021
    also have "\<dots> \<le> ?lhs" using **
Andreas@59489
  1022
      by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg)
Andreas@59489
  1023
    finally show ?thesis by simp
Andreas@59489
  1024
  next
Andreas@59489
  1025
    case False
Andreas@59489
  1026
    def C' \<equiv> "fst ` C"
Andreas@59489
  1027
    have "\<infinity> = \<integral>\<^sup>+ x. ereal (1 / Suc n) * indicator C' x \<partial>count_space UNIV"
Andreas@59489
  1028
      using C'_def False by(simp add: nn_integral_cmult)
Andreas@59489
  1029
    also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) \<in> C} y \<partial>count_space UNIV \<partial>count_space UNIV"
Andreas@59489
  1030
      by(auto simp add: one_ereal_def[symmetric] nn_integral_cmult_indicator intro: nn_integral_cong)
Andreas@59489
  1031
    also have "\<dots> \<le> \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C (x, y) \<partial>count_space UNIV \<partial>count_space UNIV"
Andreas@59489
  1032
      by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI)
Andreas@59489
  1033
    also have "\<dots> \<le> ?lhs" using C
Andreas@59489
  1034
      by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg)
Andreas@59489
  1035
    finally show ?thesis by simp
Andreas@59489
  1036
  qed
Andreas@59489
  1037
  ultimately show ?thesis by simp
Andreas@59489
  1038
qed
Andreas@59489
  1039
Andreas@59489
  1040
lemma nn_integral_fst_count_space:
Andreas@59489
  1041
  "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
Andreas@59489
  1042
by(subst (2 3) nn_integral_max_0[symmetric])(rule nn_integral_fst_count_space', simp)
Andreas@59489
  1043
Andreas@59491
  1044
lemma nn_integral_snd_count_space:
Andreas@59491
  1045
  "(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
Andreas@59491
  1046
  (is "?lhs = ?rhs")
Andreas@59491
  1047
proof -
Andreas@59491
  1048
  have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)"
Andreas@59491
  1049
    by(simp)
Andreas@59491
  1050
  also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV"
Andreas@59491
  1051
    by(rule nn_integral_fst_count_space)
Andreas@59491
  1052
  also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)"
Andreas@59491
  1053
    by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
Andreas@59491
  1054
      (simp_all add: inj_on_def split_def)
Andreas@59491
  1055
  also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto
Andreas@59491
  1056
  finally show ?thesis .
Andreas@59491
  1057
qed
Andreas@59491
  1058
Andreas@60066
  1059
lemma measurable_pair_measure_countable1:
Andreas@60066
  1060
  assumes "countable A"
Andreas@60066
  1061
  and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
Andreas@60066
  1062
  shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
Andreas@60066
  1063
using _ _ assms(1)
Andreas@60066
  1064
by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
Andreas@60066
  1065
hoelzl@57235
  1066
subsection {* Product of Borel spaces *}
hoelzl@57235
  1067
hoelzl@57235
  1068
lemma borel_Times:
hoelzl@57235
  1069
  fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
hoelzl@57235
  1070
  assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
hoelzl@57235
  1071
  shows "A \<times> B \<in> sets borel"
hoelzl@57235
  1072
proof -
hoelzl@57235
  1073
  have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
hoelzl@57235
  1074
    by auto
hoelzl@57235
  1075
  moreover
hoelzl@57235
  1076
  { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)
hoelzl@57235
  1077
    then have "A\<times>UNIV \<in> sets borel"
hoelzl@57235
  1078
    proof (induct A)
hoelzl@57235
  1079
      case (Basic S) then show ?case
hoelzl@57235
  1080
        by (auto intro!: borel_open open_Times)
hoelzl@57235
  1081
    next
hoelzl@57235
  1082
      case (Compl A)
hoelzl@57235
  1083
      moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)"
hoelzl@57235
  1084
        by auto
hoelzl@57235
  1085
      ultimately show ?case
hoelzl@57235
  1086
        unfolding * by auto
hoelzl@57235
  1087
    next
hoelzl@57235
  1088
      case (Union A)
hoelzl@57235
  1089
      moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)"
hoelzl@57235
  1090
        by auto
hoelzl@57235
  1091
      ultimately show ?case
hoelzl@57235
  1092
        unfolding * by auto
hoelzl@57235
  1093
    qed simp }
hoelzl@57235
  1094
  moreover
hoelzl@57235
  1095
  { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)
hoelzl@57235
  1096
    then have "UNIV\<times>B \<in> sets borel"
hoelzl@57235
  1097
    proof (induct B)
hoelzl@57235
  1098
      case (Basic S) then show ?case
hoelzl@57235
  1099
        by (auto intro!: borel_open open_Times)
hoelzl@57235
  1100
    next
hoelzl@57235
  1101
      case (Compl B)
hoelzl@57235
  1102
      moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)"
hoelzl@57235
  1103
        by auto
hoelzl@57235
  1104
      ultimately show ?case
hoelzl@57235
  1105
        unfolding * by auto
hoelzl@57235
  1106
    next
hoelzl@57235
  1107
      case (Union B)
hoelzl@57235
  1108
      moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)"
hoelzl@57235
  1109
        by auto
hoelzl@57235
  1110
      ultimately show ?case
hoelzl@57235
  1111
        unfolding * by auto
hoelzl@57235
  1112
    qed simp }
hoelzl@57235
  1113
  ultimately show ?thesis
hoelzl@57235
  1114
    by auto
hoelzl@57235
  1115
qed
hoelzl@57235
  1116
hoelzl@57235
  1117
lemma finite_measure_pair_measure:
hoelzl@57235
  1118
  assumes "finite_measure M" "finite_measure N"
hoelzl@57235
  1119
  shows "finite_measure (N  \<Otimes>\<^sub>M M)"
hoelzl@57235
  1120
proof (rule finite_measureI)
hoelzl@57235
  1121
  interpret M: finite_measure M by fact
hoelzl@57235
  1122
  interpret N: finite_measure N by fact
hoelzl@57235
  1123
  show "emeasure (N  \<Otimes>\<^sub>M M) (space (N  \<Otimes>\<^sub>M M)) \<noteq> \<infinity>"
hoelzl@57235
  1124
    by (auto simp: space_pair_measure M.emeasure_pair_measure_Times)
hoelzl@57235
  1125
qed
hoelzl@57235
  1126
hoelzl@40859
  1127
end