src/HOL/Analysis/Binary_Product_Measure.thy
author nipkow
Sat Dec 29 15:43:53 2018 +0100 (6 months ago)
changeset 69529 4ab9657b3257
parent 69517 dc20f278e8f3
child 69566 c41954ee87cf
permissions -rw-r--r--
capitalize proper names in lemma names
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(*  Title:      HOL/Analysis/Binary_Product_Measure.thy
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    Author:     Johannes Hölzl, TU München
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*)
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section%important \<open>Binary Product Measure\<close>
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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%unimportant 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%unimportant 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%important "Binary products"
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definition%important 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%important 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%unimportant sets.space_closed[of A] sets.space_closed[of B] by auto
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lemma%important 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%unimportant (rule space_measure_of)
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lemma%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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. case_prod (f x) (g x)) \<in> measurable M N"
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  unfolding split_beta' .
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lemma%important 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%unimportant (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%unimportant 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%unimportant 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%unimportant measurable_Pair_compose_split[measurable_dest]:
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  assumes f: "case_prod 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%unimportant 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%unimportant 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%unimportant 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%unimportant (*FIX ME needs a name *)
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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%unimportant (*FIX ME needs a name *)
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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%unimportant sets_pair_in_sets:
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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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  unfolding sets_pair_measure
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  by (intro sets.sigma_sets_subset') (auto intro!: assms)
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lemma%important  sets_pair_eq_sets_fst_snd:
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  "sets (A \<Otimes>\<^sub>M B) = sets (Sup {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 {?fst, ?snd})")
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proof%unimportant -
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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 {?fst, ?snd})"
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      apply (rule sets.Int)
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      apply (rule in_sets_Sup)
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      apply auto []
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      apply (rule insertI1)
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      apply (auto intro: ab in_vimage_algebra) []
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      apply (rule in_sets_Sup)
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      apply auto []
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      apply (rule insertI2)
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      apply (auto intro: ab in_vimage_algebra)
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      done
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    finally have "a \<times> b \<in> sets (Sup {?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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    apply (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets)
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    apply simp
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    apply simp
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    apply simp
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    apply (elim disjE)
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    apply (simp add: space_pair_measure)
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    apply (simp add: space_pair_measure)
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    apply (auto simp add: space_pair_measure)
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    done
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qed
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lemma%unimportant 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%unimportant  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="(\<in>)"]) auto
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lemma%unimportant 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%unimportant  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%unimportant 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%unimportant 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%unimportant 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: if_split_asm)
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  finally show ?thesis .
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qed
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lemma%unimportant 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%unimportant 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: if_split_asm)
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  finally show ?thesis .
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qed
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lemma%unimportant 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%unimportant 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%unimportant 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%unimportant (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_on_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%unimportant (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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   249
    have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"
immler@50244
   250
      using F sets.sets_into_space by auto
hoelzl@49776
   251
    let ?R = "density M (indicator (F i))"
hoelzl@49776
   252
    have "finite_measure ?R"
hoelzl@49776
   253
      using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
hoelzl@49776
   254
    then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"
hoelzl@49776
   255
     by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)
hoelzl@49776
   256
    moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))
hoelzl@49776
   257
        = emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))"
hoelzl@49776
   258
      using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
hoelzl@49776
   259
    moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i"
immler@50244
   260
      using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure)
hoelzl@49776
   261
    ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"
hoelzl@49776
   262
      by simp }
hoelzl@49776
   263
  moreover
hoelzl@49776
   264
  { fix x
hoelzl@49776
   265
    have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
hoelzl@49776
   266
    proof (intro suminf_emeasure)
hoelzl@49776
   267
      show "range (?C x) \<subseteq> sets M"
wenzelm@61808
   268
        using F \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> by (auto intro!: sets_Pair1)
hoelzl@49776
   269
      have "disjoint_family F" using F by auto
hoelzl@49776
   270
      show "disjoint_family (?C x)"
wenzelm@61808
   271
        by (rule disjoint_family_on_bisimulation[OF \<open>disjoint_family F\<close>]) auto
hoelzl@49776
   272
    qed
hoelzl@49776
   273
    also have "(\<Union>i. ?C x i) = Pair x -` Q"
wenzelm@61808
   274
      using F sets.sets_into_space[OF \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close>]
hoelzl@49776
   275
      by (auto simp: space_pair_measure)
hoelzl@49776
   276
    finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"
hoelzl@49776
   277
      by simp }
wenzelm@61808
   278
  ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets
hoelzl@49776
   279
    by auto
hoelzl@49776
   280
qed
hoelzl@49776
   281
ak2110@68833
   282
lemma%unimportant (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
hoelzl@50003
   283
  assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
wenzelm@53015
   284
  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
   285
  shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
hoelzl@50003
   286
proof -
wenzelm@53015
   287
  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
   288
    by (auto simp: space_pair_measure)
hoelzl@50003
   289
  with measurable_emeasure_Pair[OF A] show ?thesis
hoelzl@50003
   290
    by (auto cong: measurable_cong)
hoelzl@50003
   291
qed
hoelzl@50003
   292
ak2110@68833
   293
lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure:
wenzelm@53015
   294
  assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
wenzelm@53015
   295
  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
   296
proof (rule emeasure_measure_of[OF pair_measure_def])
wenzelm@53015
   297
  show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
hoelzl@62975
   298
    by (auto simp: positive_def)
hoelzl@49776
   299
  have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
hoelzl@49776
   300
    by (auto simp: indicator_def)
wenzelm@53015
   301
  show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
hoelzl@49776
   302
  proof (rule countably_additiveI)
wenzelm@53015
   303
    fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F"
hoelzl@59353
   304
    from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto
hoelzl@49776
   305
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
hoelzl@49776
   306
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@49776
   307
    moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M"
hoelzl@49776
   308
      using F by (auto simp: sets_Pair1)
hoelzl@49776
   309
    ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
hoelzl@62975
   310
      by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure
hoelzl@56996
   311
               intro!: nn_integral_cong nn_integral_indicator[symmetric])
hoelzl@49776
   312
  qed
hoelzl@49776
   313
  show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
immler@50244
   314
    using sets.space_closed[of N] sets.space_closed[of M] by auto
hoelzl@49776
   315
qed fact
hoelzl@49776
   316
ak2110@68833
   317
lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure_alt:
wenzelm@53015
   318
  assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
wenzelm@53015
   319
  shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
hoelzl@49776
   320
proof -
hoelzl@49776
   321
  have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
hoelzl@49776
   322
    by (auto simp: indicator_def)
hoelzl@49776
   323
  show ?thesis
hoelzl@56996
   324
    using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
hoelzl@49776
   325
qed
hoelzl@49776
   326
ak2110@68833
   327
lemma%important (in sigma_finite_measure) emeasure_pair_measure_Times:
hoelzl@49776
   328
  assumes A: "A \<in> sets N" and B: "B \<in> sets M"
wenzelm@53015
   329
  shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
ak2110@68833
   330
proof%unimportant -
wenzelm@53015
   331
  have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
hoelzl@56996
   332
    using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
hoelzl@49776
   333
  also have "\<dots> = emeasure M B * emeasure N A"
hoelzl@62975
   334
    using A by (simp add: nn_integral_cmult_indicator)
hoelzl@49776
   335
  finally show ?thesis
hoelzl@49776
   336
    by (simp add: ac_simps)
hoelzl@40859
   337
qed
hoelzl@40859
   338
ak2110@68833
   339
subsection%important \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
hoelzl@40859
   340
ak2110@68833
   341
locale%important pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
hoelzl@47694
   342
  for M1 :: "'a measure" and M2 :: "'b measure"
hoelzl@40859
   343
ak2110@68833
   344
lemma%unimportant (in pair_sigma_finite) measurable_emeasure_Pair1:
wenzelm@53015
   345
  "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
hoelzl@49776
   346
  using M2.measurable_emeasure_Pair .
hoelzl@40859
   347
ak2110@68833
   348
lemma%important (in pair_sigma_finite) measurable_emeasure_Pair2:
wenzelm@53015
   349
  assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
ak2110@68833
   350
proof%unimportant -
wenzelm@53015
   351
  have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
hoelzl@47694
   352
    using Q measurable_pair_swap' by (auto intro: measurable_sets)
hoelzl@49776
   353
  note M1.measurable_emeasure_Pair[OF this]
wenzelm@53015
   354
  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
   355
    using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
hoelzl@47694
   356
  ultimately show ?thesis by simp
hoelzl@39088
   357
qed
hoelzl@39088
   358
ak2110@68833
   359
lemma%important (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
hoelzl@47694
   360
  defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
hoelzl@47694
   361
  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
   362
    (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
ak2110@68833
   363
proof%unimportant -
hoelzl@47694
   364
  from M1.sigma_finite_incseq guess F1 . note F1 = this
hoelzl@47694
   365
  from M2.sigma_finite_incseq guess F2 . note F2 = this
hoelzl@47694
   366
  from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
hoelzl@40859
   367
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
hoelzl@47694
   368
  show ?thesis
hoelzl@40859
   369
  proof (intro exI[of _ ?F] conjI allI)
hoelzl@47694
   370
    show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
hoelzl@40859
   371
  next
hoelzl@40859
   372
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
hoelzl@40859
   373
    proof (intro subsetI)
hoelzl@40859
   374
      fix x assume "x \<in> space M1 \<times> space M2"
hoelzl@40859
   375
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
hoelzl@40859
   376
        by (auto simp: space)
hoelzl@40859
   377
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
wenzelm@61808
   378
        using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_def
hoelzl@41981
   379
        by (force split: split_max)+
hoelzl@40859
   380
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
haftmann@54863
   381
        by (intro SigmaI) (auto simp add: max.commute)
hoelzl@40859
   382
      then show "x \<in> (\<Union>i. ?F i)" by auto
hoelzl@40859
   383
    qed
hoelzl@47694
   384
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
hoelzl@47694
   385
      using space by (auto simp: space)
hoelzl@40859
   386
  next
hoelzl@41981
   387
    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
wenzelm@61808
   388
      using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_Suc_iff by auto
hoelzl@40859
   389
  next
hoelzl@40859
   390
    fix i
hoelzl@40859
   391
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
hoelzl@62975
   392
    with F1 F2 show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
hoelzl@62975
   393
      by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
hoelzl@47694
   394
  qed
hoelzl@47694
   395
qed
hoelzl@47694
   396
ak2110@68833
   397
sublocale%important pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
hoelzl@47694
   398
proof
hoelzl@57447
   399
  from M1.sigma_finite_countable guess F1 ..
hoelzl@57447
   400
  moreover from M2.sigma_finite_countable guess F2 ..
hoelzl@57447
   401
  ultimately show
hoelzl@57447
   402
    "\<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
   403
    by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI)
hoelzl@62975
   404
       (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
hoelzl@40859
   405
qed
hoelzl@40859
   406
ak2110@68833
   407
lemma%unimportant sigma_finite_pair_measure:
hoelzl@47694
   408
  assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
wenzelm@53015
   409
  shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
hoelzl@47694
   410
proof -
hoelzl@47694
   411
  interpret A: sigma_finite_measure A by fact
hoelzl@47694
   412
  interpret B: sigma_finite_measure B by fact
hoelzl@47694
   413
  interpret AB: pair_sigma_finite A  B ..
hoelzl@47694
   414
  show ?thesis ..
hoelzl@40859
   415
qed
hoelzl@39088
   416
ak2110@68833
   417
lemma%unimportant sets_pair_swap:
wenzelm@53015
   418
  assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
wenzelm@53015
   419
  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
hoelzl@47694
   420
  using measurable_pair_swap' assms by (rule measurable_sets)
hoelzl@41661
   421
ak2110@68833
   422
lemma%important (in pair_sigma_finite) distr_pair_swap:
wenzelm@53015
   423
  "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
ak2110@68833
   424
proof%unimportant -
hoelzl@41689
   425
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
   426
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
hoelzl@47694
   427
  show ?thesis
hoelzl@47694
   428
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@47694
   429
    show "?E \<subseteq> Pow (space ?P)"
immler@50244
   430
      using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@47694
   431
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@47694
   432
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@47694
   433
    then show "sets ?D = sigma_sets (space ?P) ?E"
hoelzl@47694
   434
      by simp
hoelzl@47694
   435
  next
hoelzl@49784
   436
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@47694
   437
      using F by (auto simp: space_pair_measure)
hoelzl@47694
   438
  next
hoelzl@47694
   439
    fix X assume "X \<in> ?E"
hoelzl@47694
   440
    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
   441
    have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A"
immler@50244
   442
      using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure)
wenzelm@53015
   443
    with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X"
hoelzl@49776
   444
      by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
hoelzl@47694
   445
                    measurable_pair_swap' ac_simps)
hoelzl@41689
   446
  qed
hoelzl@41689
   447
qed
hoelzl@41689
   448
ak2110@68833
   449
lemma%unimportant (in pair_sigma_finite) emeasure_pair_measure_alt2:
wenzelm@53015
   450
  assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
wenzelm@53015
   451
  shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
hoelzl@47694
   452
    (is "_ = ?\<nu> A")
ak2110@68833
   453
proof%unimportant -
wenzelm@53015
   454
  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
   455
    using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
hoelzl@47694
   456
  show ?thesis using A
hoelzl@47694
   457
    by (subst distr_pair_swap)
hoelzl@47694
   458
       (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
hoelzl@49776
   459
                 M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
hoelzl@49776
   460
qed
hoelzl@49776
   461
ak2110@68833
   462
lemma%unimportant (in pair_sigma_finite) AE_pair:
wenzelm@53015
   463
  assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
hoelzl@49776
   464
  shows "AE x in M1. (AE y in M2. Q (x, y))"
hoelzl@49776
   465
proof -
wenzelm@53015
   466
  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
   467
    using assms unfolding eventually_ae_filter by auto
hoelzl@49776
   468
  show ?thesis
hoelzl@49776
   469
  proof (rule AE_I)
wenzelm@61808
   470
    from N measurable_emeasure_Pair1[OF \<open>N \<in> sets (M1 \<Otimes>\<^sub>M M2)\<close>]
hoelzl@49776
   471
    show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
hoelzl@62975
   472
      by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff)
hoelzl@49776
   473
    show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
hoelzl@62975
   474
      by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp
hoelzl@49776
   475
    { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
hoelzl@49776
   476
      have "AE y in M2. Q (x, y)"
hoelzl@49776
   477
      proof (rule AE_I)
hoelzl@49776
   478
        show "emeasure M2 (Pair x -` N) = 0" by fact
hoelzl@49776
   479
        show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
hoelzl@49776
   480
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
wenzelm@61808
   481
          using N \<open>x \<in> space M1\<close> unfolding space_pair_measure by auto
hoelzl@49776
   482
      qed }
hoelzl@49776
   483
    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
   484
      by auto
hoelzl@49776
   485
  qed
hoelzl@49776
   486
qed
hoelzl@49776
   487
ak2110@68833
   488
lemma%important (in pair_sigma_finite) AE_pair_measure:
wenzelm@53015
   489
  assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
hoelzl@49776
   490
  assumes ae: "AE x in M1. AE y in M2. P (x, y)"
wenzelm@53015
   491
  shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
ak2110@68833
   492
proof%unimportant (subst AE_iff_measurable[OF _ refl])
wenzelm@53015
   493
  show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
immler@50244
   494
    by (rule sets.sets_Collect) fact
wenzelm@53015
   495
  then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
wenzelm@53015
   496
      (\<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
   497
    by (simp add: M2.emeasure_pair_measure)
wenzelm@53015
   498
  also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)"
hoelzl@49776
   499
    using ae
hoelzl@56996
   500
    apply (safe intro!: nn_integral_cong_AE)
hoelzl@49776
   501
    apply (intro AE_I2)
hoelzl@56996
   502
    apply (safe intro!: nn_integral_cong_AE)
hoelzl@49776
   503
    apply auto
hoelzl@49776
   504
    done
wenzelm@53015
   505
  finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
hoelzl@49776
   506
qed
hoelzl@49776
   507
ak2110@68833
   508
lemma%unimportant (in pair_sigma_finite) AE_pair_iff:
wenzelm@53015
   509
  "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
wenzelm@53015
   510
    (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
   511
  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
   512
ak2110@68833
   513
lemma%unimportant (in pair_sigma_finite) AE_commute:
wenzelm@53015
   514
  assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
hoelzl@49776
   515
  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
   516
proof -
hoelzl@49776
   517
  interpret Q: pair_sigma_finite M2 M1 ..
hoelzl@49776
   518
  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
   519
    by auto
wenzelm@53015
   520
  have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} =
wenzelm@53015
   521
    (\<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
   522
    by (auto simp: space_pair_measure)
wenzelm@53015
   523
  also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)"
hoelzl@49776
   524
    by (intro sets_pair_swap P)
hoelzl@49776
   525
  finally show ?thesis
hoelzl@49776
   526
    apply (subst AE_pair_iff[OF P])
hoelzl@49776
   527
    apply (subst distr_pair_swap)
hoelzl@49776
   528
    apply (subst AE_distr_iff[OF measurable_pair_swap' P])
hoelzl@49776
   529
    apply (subst Q.AE_pair_iff)
hoelzl@49776
   530
    apply simp_all
hoelzl@49776
   531
    done
hoelzl@40859
   532
qed
hoelzl@40859
   533
ak2110@68833
   534
subsection%important "Fubinis theorem"
hoelzl@40859
   535
ak2110@68833
   536
lemma%unimportant measurable_compose_Pair1:
wenzelm@53015
   537
  "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
   538
  by simp
hoelzl@49800
   539
ak2110@68833
   540
lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral_fst:
hoelzl@62975
   541
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
wenzelm@53015
   542
  shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
ak2110@68833
   543
using f proof%unimportant induct
hoelzl@49800
   544
  case (cong u v)
hoelzl@49999
   545
  then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
hoelzl@49800
   546
    by (auto simp: space_pair_measure)
hoelzl@49800
   547
  show ?case
hoelzl@49800
   548
    apply (subst measurable_cong)
hoelzl@56996
   549
    apply (rule nn_integral_cong)
hoelzl@49800
   550
    apply fact+
hoelzl@49800
   551
    done
hoelzl@49800
   552
next
hoelzl@49800
   553
  case (set Q)
hoelzl@49800
   554
  have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y"
hoelzl@49800
   555
    by (auto simp: indicator_def)
wenzelm@53015
   556
  have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M"
hoelzl@49800
   557
    by (simp add: sets_Pair1[OF set])
hoelzl@49999
   558
  from this measurable_emeasure_Pair[OF set] show ?case
hoelzl@49800
   559
    by (rule measurable_cong[THEN iffD1])
hoelzl@56996
   560
qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1
hoelzl@56996
   561
                   nn_integral_monotone_convergence_SUP incseq_def le_fun_def
hoelzl@49800
   562
              cong: measurable_cong)
hoelzl@49800
   563
ak2110@68833
   564
lemma%unimportant (in sigma_finite_measure) nn_integral_fst:
hoelzl@62975
   565
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
hoelzl@56996
   566
  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 = _")
ak2110@68833
   567
  using f proof induct
hoelzl@49800
   568
  case (cong u v)
wenzelm@53374
   569
  then have "?I u = ?I v"
hoelzl@56996
   570
    by (intro nn_integral_cong) (auto simp: space_pair_measure)
wenzelm@53374
   571
  with cong show ?case
hoelzl@56996
   572
    by (simp cong: nn_integral_cong)
hoelzl@56996
   573
qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add
hoelzl@62975
   574
                   nn_integral_monotone_convergence_SUP measurable_compose_Pair1
hoelzl@62975
   575
                   borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def
hoelzl@56996
   576
              cong: nn_integral_cong)
hoelzl@40859
   577
ak2110@68833
   578
lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
haftmann@61424
   579
  "case_prod 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@62975
   580
  using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp
hoelzl@50003
   581
ak2110@68833
   582
lemma%important (in pair_sigma_finite) nn_integral_snd:
hoelzl@62975
   583
  assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
hoelzl@56996
   584
  shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
ak2110@68833
   585
proof%unimportant -
hoelzl@47694
   586
  note measurable_pair_swap[OF f]
hoelzl@56996
   587
  from M1.nn_integral_fst[OF this]
wenzelm@53015
   588
  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
   589
    by simp
hoelzl@56996
   590
  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@62975
   591
    by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong)
hoelzl@40859
   592
  finally show ?thesis .
hoelzl@40859
   593
qed
hoelzl@40859
   594
ak2110@68833
   595
lemma%important (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
ak2110@68833
   600
lemma%important (in pair_sigma_finite) Fubini':
haftmann@61424
   601
  assumes f: "case_prod 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
ak2110@68833
   605
subsection%important \<open>Products on counting spaces, densities and distributions\<close>
hoelzl@40859
   606
ak2110@68833
   607
lemma%important 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")
ak2110@68833
   612
proof%unimportant (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}}"
haftmann@69260
   616
  have "sets ?P = sets (SUP xy\<in>?XY. sigma (X \<times> Y) xy)"
hoelzl@59088
   617
    by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)
hoelzl@59088
   618
  also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))"
hoelzl@63333
   619
    by (intro Sup_sigma arg_cong[where f=sets]) auto
hoelzl@59088
   620
  also have "\<dots> = sets ?S"
hoelzl@62975
   621
  proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
hoelzl@59088
   622
    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
   623
      using A B by auto
hoelzl@59088
   624
  next
hoelzl@59088
   625
    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
   626
      using A B by (intro sigma_algebra_sigma_sets) auto
hoelzl@59088
   627
    fix Z assume "Z \<in> \<Union>?XY"
hoelzl@59088
   628
    then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   629
    proof safe
hoelzl@59088
   630
      fix a assume "a \<in> A"
hoelzl@59088
   631
      from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E"
hoelzl@59088
   632
        by auto
wenzelm@61808
   633
      with \<open>a \<in> A\<close> A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"
hoelzl@59088
   634
        by auto
hoelzl@59088
   635
      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}"
wenzelm@61808
   636
        using \<open>a \<in> A\<close> E unfolding eq by (auto intro!: XY.countable_UN')
hoelzl@59088
   637
    next
hoelzl@59088
   638
      fix b assume "b \<in> B"
hoelzl@59088
   639
      from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E"
hoelzl@59088
   640
        by auto
wenzelm@61808
   641
      with \<open>b \<in> B\<close> B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"
hoelzl@59088
   642
        by auto
hoelzl@59088
   643
      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}"
wenzelm@61808
   644
        using \<open>b \<in> B\<close> E unfolding eq by (auto intro!: XY.countable_UN')
hoelzl@59088
   645
    qed
hoelzl@59088
   646
  next
hoelzl@59088
   647
    fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
hoelzl@59088
   648
    then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B"
hoelzl@59088
   649
      by auto
hoelzl@59088
   650
    then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)"
hoelzl@59088
   651
      using A B by auto
hoelzl@59088
   652
    interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)"
hoelzl@59088
   653
      by (intro sigma_algebra_sigma_sets) auto
hoelzl@59088
   654
    show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)"
hoelzl@59088
   655
      unfolding Z by (rule XY.Int) (blast intro: ab)+
hoelzl@59088
   656
  qed
hoelzl@59088
   657
  finally show "sets ?P = sets ?S" .
hoelzl@59088
   658
next
hoelzl@59088
   659
  interpret finite_measure "sigma X A" for X A
hoelzl@59088
   660
    proof qed (simp add: emeasure_sigma)
hoelzl@59088
   661
  fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A"
hoelzl@59088
   662
    by (simp add: emeasure_pair_measure_alt emeasure_sigma)
hoelzl@59088
   663
qed
hoelzl@59088
   664
ak2110@68833
   665
lemma%unimportant sigma_sets_pair_measure_generator_finite:
hoelzl@38656
   666
  assumes "finite A" and "finite B"
hoelzl@47694
   667
  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
hoelzl@40859
   668
  (is "sigma_sets ?prod ?sets = _")
hoelzl@38656
   669
proof safe
hoelzl@38656
   670
  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
hoelzl@38656
   671
  fix x assume subset: "x \<subseteq> A \<times> B"
hoelzl@38656
   672
  hence "finite x" using fin by (rule finite_subset)
hoelzl@40859
   673
  from this subset show "x \<in> sigma_sets ?prod ?sets"
hoelzl@38656
   674
  proof (induct x)
hoelzl@38656
   675
    case empty show ?case by (rule sigma_sets.Empty)
hoelzl@38656
   676
  next
hoelzl@38656
   677
    case (insert a x)
hoelzl@47694
   678
    hence "{a} \<in> sigma_sets ?prod ?sets" by auto
hoelzl@38656
   679
    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
hoelzl@38656
   680
    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
hoelzl@38656
   681
  qed
hoelzl@38656
   682
next
hoelzl@38656
   683
  fix x a b
hoelzl@40859
   684
  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
hoelzl@38656
   685
  from sigma_sets_into_sp[OF _ this(1)] this(2)
hoelzl@40859
   686
  show "a \<in> A" and "b \<in> B" by auto
hoelzl@35833
   687
qed
hoelzl@35833
   688
ak2110@68833
   689
lemma%important sets_pair_eq:
hoelzl@64008
   690
  assumes Ea: "Ea \<subseteq> Pow (space A)" "sets A = sigma_sets (space A) Ea"
hoelzl@64008
   691
    and Ca: "countable Ca" "Ca \<subseteq> Ea" "\<Union>Ca = space A"
hoelzl@64008
   692
    and Eb: "Eb \<subseteq> Pow (space B)" "sets B = sigma_sets (space B) Eb"
hoelzl@64008
   693
    and Cb: "countable Cb" "Cb \<subseteq> Eb" "\<Union>Cb = space B"
hoelzl@64008
   694
  shows "sets (A \<Otimes>\<^sub>M B) = sets (sigma (space A \<times> space B) { a \<times> b | a b. a \<in> Ea \<and> b \<in> Eb })"
hoelzl@64008
   695
    (is "_ = sets (sigma ?\<Omega> ?E)")
ak2110@68833
   696
proof%unimportant
hoelzl@64008
   697
  show "sets (sigma ?\<Omega> ?E) \<subseteq> sets (A \<Otimes>\<^sub>M B)"
hoelzl@64008
   698
    using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
hoelzl@64008
   699
  have "?E \<subseteq> Pow ?\<Omega>"
hoelzl@64008
   700
    using Ea(1) Eb(1) by auto
hoelzl@64008
   701
  then have E: "a \<in> Ea \<Longrightarrow> b \<in> Eb \<Longrightarrow> a \<times> b \<in> sets (sigma ?\<Omega> ?E)" for a b
hoelzl@64008
   702
    by auto
hoelzl@64008
   703
  have "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (Sup {vimage_algebra ?\<Omega> fst A, vimage_algebra ?\<Omega> snd B})"
hoelzl@64008
   704
    unfolding sets_pair_eq_sets_fst_snd ..
hoelzl@64008
   705
  also have "vimage_algebra ?\<Omega> fst A = vimage_algebra ?\<Omega> fst (sigma (space A) Ea)"
hoelzl@64008
   706
    by (intro vimage_algebra_cong[OF refl refl]) (simp add: Ea)
hoelzl@64008
   707
  also have "\<dots> = sigma ?\<Omega> {fst -` A \<inter> ?\<Omega> |A. A \<in> Ea}"
hoelzl@64008
   708
    by (intro Ea vimage_algebra_sigma) auto
hoelzl@64008
   709
  also have "vimage_algebra ?\<Omega> snd B = vimage_algebra ?\<Omega> snd (sigma (space B) Eb)"
hoelzl@64008
   710
    by (intro vimage_algebra_cong[OF refl refl]) (simp add: Eb)
hoelzl@64008
   711
  also have "\<dots> = sigma ?\<Omega> {snd -` A \<inter> ?\<Omega> |A. A \<in> Eb}"
hoelzl@64008
   712
    by (intro Eb vimage_algebra_sigma) auto
hoelzl@64008
   713
  also have "{sigma ?\<Omega> {fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, sigma ?\<Omega> {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}} =
hoelzl@64008
   714
    sigma ?\<Omega> ` {{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}"
hoelzl@64008
   715
    by auto
haftmann@69260
   716
  also have "sets (SUP S\<in>{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}. sigma ?\<Omega> S) =
hoelzl@64008
   717
    sets (sigma ?\<Omega> (\<Union>{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}))"
hoelzl@64008
   718
    using Ea(1) Eb(1) by (intro sets_Sup_sigma) auto
hoelzl@64008
   719
  also have "\<dots> \<subseteq> sets (sigma ?\<Omega> ?E)"
hoelzl@64008
   720
  proof (subst sigma_le_sets, safe intro!: space_in_measure_of)
hoelzl@64008
   721
    fix a assume "a \<in> Ea"
hoelzl@64008
   722
    then have "fst -` a \<inter> ?\<Omega> = (\<Union>b\<in>Cb. a \<times> b)"
hoelzl@64008
   723
      using Cb(3)[symmetric] Ea(1) by auto
hoelzl@64008
   724
    then show "fst -` a \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)"
hoelzl@64008
   725
      using Cb \<open>a \<in> Ea\<close> by (auto intro!: sets.countable_UN' E)
hoelzl@64008
   726
  next
hoelzl@64008
   727
    fix b assume "b \<in> Eb"
hoelzl@64008
   728
    then have "snd -` b \<inter> ?\<Omega> = (\<Union>a\<in>Ca. a \<times> b)"
hoelzl@64008
   729
      using Ca(3)[symmetric] Eb(1) by auto
hoelzl@64008
   730
    then show "snd -` b \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)"
hoelzl@64008
   731
      using Ca \<open>b \<in> Eb\<close> by (auto intro!: sets.countable_UN' E)
hoelzl@64008
   732
  qed
hoelzl@64008
   733
  finally show "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (sigma ?\<Omega> ?E)" .
hoelzl@64008
   734
qed
hoelzl@64008
   735
ak2110@68833
   736
lemma%important borel_prod:
hoelzl@59088
   737
  "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
hoelzl@59088
   738
  (is "?P = ?B")
ak2110@68833
   739
proof%unimportant -
hoelzl@59088
   740
  have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
hoelzl@59088
   741
    by (rule second_countable_borel_measurable[OF open_prod_generated])
hoelzl@59088
   742
  also have "\<dots> = ?P"
hoelzl@59088
   743
    unfolding borel_def
hoelzl@59088
   744
    by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])
hoelzl@59088
   745
  finally show ?thesis ..
hoelzl@59088
   746
qed
hoelzl@59088
   747
ak2110@68833
   748
lemma%important pair_measure_count_space:
hoelzl@47694
   749
  assumes A: "finite A" and B: "finite B"
wenzelm@53015
   750
  shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
ak2110@68833
   751
proof%unimportant (rule measure_eqI)
hoelzl@47694
   752
  interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
hoelzl@47694
   753
  interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
wenzelm@61169
   754
  interpret P: pair_sigma_finite "count_space A" "count_space B" ..
hoelzl@47694
   755
  show eq: "sets ?P = sets ?C"
hoelzl@47694
   756
    by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
hoelzl@47694
   757
  fix X assume X: "X \<in> sets ?P"
hoelzl@47694
   758
  with eq have X_subset: "X \<subseteq> A \<times> B" by simp
hoelzl@47694
   759
  with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
hoelzl@47694
   760
    by (intro finite_subset[OF _ B]) auto
hoelzl@47694
   761
  have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
lp15@67693
   762
  have card: "0 < card (Pair a -` X)" if "(a, b) \<in> X" for a b
lp15@67693
   763
    using card_gt_0_iff fin_Pair that by auto
lp15@67693
   764
  then have "emeasure ?P X = \<integral>\<^sup>+ x. emeasure (count_space B) (Pair x -` X)
lp15@67693
   765
            \<partial>count_space A"
lp15@67693
   766
    by (simp add: B.emeasure_pair_measure_alt X)
lp15@67693
   767
  also have "... = emeasure ?C X"
hoelzl@47694
   768
    apply (subst emeasure_count_space)
lp15@67693
   769
    using card X_subset A fin_Pair fin_X
lp15@67693
   770
    apply (auto simp add: nn_integral_count_space
lp15@67693
   771
                           of_nat_sum[symmetric] card_SigmaI[symmetric]
lp15@67693
   772
                simp del:  card_SigmaI
hoelzl@62975
   773
                intro!: arg_cong[where f=card])
hoelzl@47694
   774
    done
lp15@67693
   775
  finally show "emeasure ?P X = emeasure ?C X" .
hoelzl@45777
   776
qed
hoelzl@35833
   777
hoelzl@59426
   778
ak2110@68833
   779
lemma%unimportant emeasure_prod_count_space:
hoelzl@59426
   780
  assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
hoelzl@59426
   781
  shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
hoelzl@59426
   782
  by (rule emeasure_measure_of[OF pair_measure_def])
hoelzl@62975
   783
     (auto simp: countably_additive_def positive_def suminf_indicator A
hoelzl@59426
   784
                 nn_integral_suminf[symmetric] dest: sets.sets_into_space)
hoelzl@59426
   785
ak2110@68833
   786
lemma%unimportant emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
hoelzl@59426
   787
proof -
hoelzl@62975
   788
  have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
hoelzl@59426
   789
    by (auto split: split_indicator)
hoelzl@59426
   790
  show ?thesis
hoelzl@62975
   791
    by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
hoelzl@59426
   792
qed
hoelzl@59426
   793
ak2110@68833
   794
lemma%important emeasure_count_space_prod_eq:
hoelzl@59426
   795
  fixes A :: "('a \<times> 'b) set"
hoelzl@59426
   796
  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
   797
  shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
ak2110@68833
   798
proof%unimportant -
hoelzl@59426
   799
  { fix A :: "('a \<times> 'b) set" assume "countable A"
hoelzl@59426
   800
    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
   801
      by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
hoelzl@59426
   802
    also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)"
hoelzl@59426
   803
      by (subst nn_integral_count_space_indicator) auto
hoelzl@59426
   804
    finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
hoelzl@59426
   805
      by simp }
hoelzl@59426
   806
  note * = this
hoelzl@59426
   807
hoelzl@59426
   808
  show ?thesis
hoelzl@59426
   809
  proof cases
hoelzl@59426
   810
    assume "finite A" then show ?thesis
hoelzl@59426
   811
      by (intro * countable_finite)
hoelzl@59426
   812
  next
hoelzl@59426
   813
    assume "infinite A"
hoelzl@59426
   814
    then obtain C where "countable C" and "infinite C" and "C \<subseteq> A"
hoelzl@59426
   815
      by (auto dest: infinite_countable_subset')
hoelzl@59426
   816
    with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A"
hoelzl@59426
   817
      by (intro emeasure_mono) auto
hoelzl@59426
   818
    also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C"
wenzelm@61808
   819
      using \<open>countable C\<close> by (rule *)
hoelzl@59426
   820
    finally show ?thesis
hoelzl@62975
   821
      using \<open>infinite C\<close> \<open>infinite A\<close> by (simp add: top_unique)
hoelzl@59426
   822
  qed
hoelzl@59426
   823
qed
hoelzl@59426
   824
ak2110@68833
   825
lemma%unimportant nn_integral_count_space_prod_eq:
hoelzl@62975
   826
  "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
hoelzl@59426
   827
    (is "nn_integral ?P f = _")
hoelzl@59426
   828
proof cases
hoelzl@59426
   829
  assume cntbl: "countable {x. f x \<noteq> 0}"
hoelzl@62975
   830
  have [simp]: "\<And>x. card ({x} \<inter> {x. f x \<noteq> 0}) = (indicator {x. f x \<noteq> 0} x::ennreal)"
hoelzl@59426
   831
    by (auto split: split_indicator)
hoelzl@59426
   832
  have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P"
hoelzl@59426
   833
    by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y])
hoelzl@59426
   834
       (auto intro: sets_Pair)
hoelzl@59426
   835
hoelzl@59426
   836
  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
   837
    by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator)
hoelzl@59426
   838
  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
   839
    by (auto intro!: nn_integral_cong split: split_indicator)
hoelzl@59426
   840
  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
   841
    by (intro nn_integral_count_space_nn_integral cntbl) auto
hoelzl@59426
   842
  also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})"
hoelzl@59426
   843
    by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair)
hoelzl@59426
   844
  finally show ?thesis
hoelzl@59426
   845
    by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
hoelzl@59426
   846
next
hoelzl@59426
   847
  { fix x assume "f x \<noteq> 0"
hoelzl@62975
   848
    then have "(\<exists>r\<ge>0. 0 < r \<and> f x = ennreal r) \<or> f x = \<infinity>"
hoelzl@62975
   849
      by (cases "f x" rule: ennreal_cases) (auto simp: less_le)
hoelzl@62975
   850
    then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f x"
hoelzl@59426
   851
      by (auto elim!: nat_approx_posE intro!: less_imp_le) }
hoelzl@59426
   852
  note * = this
hoelzl@59426
   853
hoelzl@59426
   854
  assume cntbl: "uncountable {x. f x \<noteq> 0}"
hoelzl@59426
   855
  also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})"
hoelzl@59426
   856
    using * by auto
hoelzl@59426
   857
  finally obtain n where "infinite {x. 1/Suc n \<le> f x}"
hoelzl@59426
   858
    by (meson countableI_type countable_UN uncountable_infinite)
hoelzl@59426
   859
  then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C"
hoelzl@59426
   860
    by (metis infinite_countable_subset')
hoelzl@59426
   861
hoelzl@59426
   862
  have [measurable]: "C \<in> sets ?P"
wenzelm@61808
   863
    using sets.countable[OF _ \<open>countable C\<close>, of ?P] by (auto simp: sets_Pair)
hoelzl@59426
   864
hoelzl@62975
   865
  have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f"
hoelzl@59426
   866
    using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
hoelzl@62975
   867
  moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>"
hoelzl@62975
   868
    using \<open>infinite C\<close> by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top)
hoelzl@62975
   869
  moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f"
hoelzl@59426
   870
    using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
hoelzl@62975
   871
  moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>"
hoelzl@62975
   872
    using \<open>infinite C\<close> by (simp add: nn_integral_cmult ennreal_mult_top)
hoelzl@59426
   873
  ultimately show ?thesis
hoelzl@62975
   874
    by (simp add: top_unique)
hoelzl@59426
   875
qed
hoelzl@59426
   876
ak2110@68833
   877
lemma%important pair_measure_density:
hoelzl@62975
   878
  assumes f: "f \<in> borel_measurable M1"
hoelzl@62975
   879
  assumes g: "g \<in> borel_measurable M2"
hoelzl@50003
   880
  assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
wenzelm@53015
   881
  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")
ak2110@68833
   882
proof%unimportant (rule measure_eqI)
hoelzl@47694
   883
  interpret M2: sigma_finite_measure M2 by fact
hoelzl@47694
   884
  interpret D2: sigma_finite_measure "density M2 g" by fact
hoelzl@47694
   885
hoelzl@47694
   886
  fix A assume A: "A \<in> sets ?L"
wenzelm@53015
   887
  with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) =
wenzelm@53015
   888
    (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)"
hoelzl@56996
   889
    by (intro nn_integral_cong_AE)
hoelzl@56996
   890
       (auto simp add: nn_integral_cmult[symmetric] ac_simps)
hoelzl@50003
   891
  with A f g show "emeasure ?L A = emeasure ?R A"
hoelzl@56996
   892
    by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density
hoelzl@56996
   893
                  M2.nn_integral_fst[symmetric]
hoelzl@56996
   894
             cong: nn_integral_cong)
hoelzl@47694
   895
qed simp
hoelzl@47694
   896
ak2110@68833
   897
lemma%unimportant sigma_finite_measure_distr:
hoelzl@47694
   898
  assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
hoelzl@47694
   899
  shows "sigma_finite_measure M"
hoelzl@40859
   900
proof -
hoelzl@47694
   901
  interpret sigma_finite_measure "distr M N f" by fact
hoelzl@57447
   902
  from sigma_finite_countable guess A .. note A = this
hoelzl@47694
   903
  show ?thesis
hoelzl@57447
   904
  proof
hoelzl@57447
   905
    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
   906
      using A f
hoelzl@57447
   907
      by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"])
hoelzl@57447
   908
         (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)
hoelzl@47694
   909
  qed
hoelzl@38656
   910
qed
hoelzl@38656
   911
ak2110@68833
   912
lemma%unimportant pair_measure_distr:
hoelzl@47694
   913
  assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
hoelzl@50003
   914
  assumes "sigma_finite_measure (distr N T g)"
wenzelm@53015
   915
  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
   916
proof (rule measure_eqI)
hoelzl@47694
   917
  interpret T: sigma_finite_measure "distr N T g" by fact
hoelzl@47694
   918
  interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
hoelzl@50003
   919
hoelzl@47694
   920
  fix A assume A: "A \<in> sets ?P"
hoelzl@50003
   921
  with f g show "emeasure ?P A = emeasure ?D A"
hoelzl@50003
   922
    by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr
hoelzl@56996
   923
                       T.emeasure_pair_measure_alt nn_integral_distr
hoelzl@56996
   924
             intro!: nn_integral_cong arg_cong[where f="emeasure N"])
hoelzl@50003
   925
qed simp
hoelzl@39097
   926
ak2110@68833
   927
lemma%important pair_measure_eqI:
hoelzl@50104
   928
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
wenzelm@53015
   929
  assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
hoelzl@50104
   930
  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
   931
  shows "M1 \<Otimes>\<^sub>M M2 = M"
ak2110@68833
   932
proof%unimportant -
hoelzl@50104
   933
  interpret M1: sigma_finite_measure M1 by fact
hoelzl@50104
   934
  interpret M2: sigma_finite_measure M2 by fact
wenzelm@61169
   935
  interpret pair_sigma_finite M1 M2 ..
hoelzl@50104
   936
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@50104
   937
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
wenzelm@53015
   938
  let ?P = "M1 \<Otimes>\<^sub>M M2"
hoelzl@50104
   939
  show ?thesis
hoelzl@50104
   940
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@50104
   941
    show "?E \<subseteq> Pow (space ?P)"
immler@50244
   942
      using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@50104
   943
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@50104
   944
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@50104
   945
    then show "sets M = sigma_sets (space ?P) ?E"
hoelzl@50104
   946
      using sets[symmetric] by simp
hoelzl@50104
   947
  next
hoelzl@50104
   948
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@50104
   949
      using F by (auto simp: space_pair_measure)
hoelzl@50104
   950
  next
hoelzl@50104
   951
    fix X assume "X \<in> ?E"
hoelzl@50104
   952
    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
   953
    then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
hoelzl@50104
   954
       by (simp add: M2.emeasure_pair_measure_Times)
hoelzl@50104
   955
    also have "\<dots> = emeasure M (A \<times> B)"
hoelzl@50104
   956
      using A B emeasure by auto
hoelzl@50104
   957
    finally show "emeasure ?P X = emeasure M X"
hoelzl@50104
   958
      by simp
hoelzl@50104
   959
  qed
hoelzl@50104
   960
qed
hoelzl@62975
   961
ak2110@68833
   962
lemma%important sets_pair_countable:
hoelzl@57025
   963
  assumes "countable S1" "countable S2"
hoelzl@57025
   964
  assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
hoelzl@57025
   965
  shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
ak2110@68833
   966
proof%unimportant auto
hoelzl@57025
   967
  fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
hoelzl@57025
   968
  from sets.sets_into_space[OF x(1)] x(2)
hoelzl@57025
   969
    sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
hoelzl@57025
   970
  show "a \<in> S1" "b \<in> S2"
hoelzl@57025
   971
    by (auto simp: space_pair_measure)
hoelzl@57025
   972
next
hoelzl@57025
   973
  fix X assume X: "X \<subseteq> S1 \<times> S2"
hoelzl@57025
   974
  then have "countable X"
wenzelm@61808
   975
    by (metis countable_subset \<open>countable S1\<close> \<open>countable S2\<close> countable_SIGMA)
hoelzl@57025
   976
  have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto
hoelzl@57025
   977
  also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)"
hoelzl@57025
   978
    using X
wenzelm@61808
   979
    by (safe intro!: sets.countable_UN' \<open>countable X\<close> subsetI pair_measureI) (auto simp: M N)
hoelzl@57025
   980
  finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
hoelzl@57025
   981
qed
hoelzl@57025
   982
ak2110@68833
   983
lemma%important pair_measure_countable:
hoelzl@57025
   984
  assumes "countable S1" "countable S2"
hoelzl@57025
   985
  shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
ak2110@68833
   986
proof%unimportant (rule pair_measure_eqI)
hoelzl@57025
   987
  show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
hoelzl@57025
   988
    using assms by (auto intro!: sigma_finite_measure_count_space_countable)
hoelzl@57025
   989
  show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
hoelzl@57025
   990
    by (subst sets_pair_countable[OF assms]) auto
hoelzl@57025
   991
next
hoelzl@57025
   992
  fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)"
hoelzl@62975
   993
  then show "emeasure (count_space S1) A * emeasure (count_space S2) B =
hoelzl@57025
   994
    emeasure (count_space (S1 \<times> S2)) (A \<times> B)"
hoelzl@62975
   995
    by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
hoelzl@57025
   996
qed
hoelzl@50104
   997
ak2110@68833
   998
lemma%important nn_integral_fst_count_space:
hoelzl@62975
   999
  "(\<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
  1000
  (is "?lhs = ?rhs")
ak2110@68833
  1001
proof%unimportant(cases)
Andreas@59489
  1002
  assume *: "countable {xy. f xy \<noteq> 0}"
Andreas@59489
  1003
  let ?A = "fst ` {xy. f xy \<noteq> 0}"
Andreas@59489
  1004
  let ?B = "snd ` {xy. f xy \<noteq> 0}"
Andreas@59489
  1005
  from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+
Andreas@59489
  1006
  have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)"
Andreas@59489
  1007
    by(rule nn_integral_count_space_eq)
hoelzl@62975
  1008
      (auto simp add: nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI)
Andreas@59489
  1009
  also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)"
Andreas@59489
  1010
    by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI)
Andreas@59489
  1011
  also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))"
Andreas@59489
  1012
    by(subst sigma_finite_measure.nn_integral_fst)
Andreas@59489
  1013
      (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable)
Andreas@59489
  1014
  also have "\<dots> = ?rhs"
Andreas@59489
  1015
    by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI)
Andreas@59489
  1016
  finally show ?thesis .
Andreas@59489
  1017
next
Andreas@59489
  1018
  { fix xy assume "f xy \<noteq> 0"
hoelzl@62975
  1019
    then have "(\<exists>r\<ge>0. 0 < r \<and> f xy = ennreal r) \<or> f xy = \<infinity>"
hoelzl@62975
  1020
      by (cases "f xy" rule: ennreal_cases) (auto simp: less_le)
hoelzl@62975
  1021
    then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f xy"
Andreas@59489
  1022
      by (auto elim!: nat_approx_posE intro!: less_imp_le) }
Andreas@59489
  1023
  note * = this
Andreas@59489
  1024
Andreas@59489
  1025
  assume cntbl: "uncountable {xy. f xy \<noteq> 0}"
Andreas@59489
  1026
  also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})"
Andreas@59489
  1027
    using * by auto
Andreas@59489
  1028
  finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}"
Andreas@59489
  1029
    by (meson countableI_type countable_UN uncountable_infinite)
Andreas@59489
  1030
  then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C"
Andreas@59489
  1031
    by (metis infinite_countable_subset')
Andreas@59489
  1032
hoelzl@62975
  1033
  have "\<infinity> = (\<integral>\<^sup>+ xy. ennreal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)"
hoelzl@62975
  1034
    using \<open>infinite C\<close> by(simp add: nn_integral_cmult ennreal_mult_top)
Andreas@59489
  1035
  also have "\<dots> \<le> ?rhs" using C
hoelzl@62975
  1036
    by(intro nn_integral_mono)(auto split: split_indicator)
hoelzl@62975
  1037
  finally have "?rhs = \<infinity>" by (simp add: top_unique)
Andreas@59489
  1038
  moreover have "?lhs = \<infinity>"
Andreas@59489
  1039
  proof(cases "finite (fst ` C)")
Andreas@59489
  1040
    case True
hoelzl@62975
  1041
    then obtain x C' where x: "x \<in> fst ` C"
Andreas@59489
  1042
      and C': "C' = fst -` {x} \<inter> C"
Andreas@59489
  1043
      and "infinite C'"
Andreas@59489
  1044
      using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE')
Andreas@59489
  1045
    from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto
Andreas@59489
  1046
Andreas@59489
  1047
    from C' \<open>infinite C'\<close> have "infinite (snd ` C')"
Andreas@59489
  1048
      by(auto dest!: finite_imageD simp add: inj_on_def)
hoelzl@62975
  1049
    then have "\<infinity> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)"
hoelzl@62975
  1050
      by(simp add: nn_integral_cmult ennreal_mult_top)
hoelzl@62975
  1051
    also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)"
Andreas@59489
  1052
      by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C')
hoelzl@62975
  1053
    also have "\<dots> = (\<integral>\<^sup>+ x'. (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV) * indicator {x} x' \<partial>count_space UNIV)"
hoelzl@62975
  1054
      by(simp add: one_ereal_def[symmetric])
hoelzl@62975
  1055
    also have "\<dots> \<le> (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV \<partial>count_space UNIV)"
hoelzl@62975
  1056
      by(rule nn_integral_mono)(simp split: split_indicator)
Andreas@59489
  1057
    also have "\<dots> \<le> ?lhs" using **
hoelzl@62975
  1058
      by(intro nn_integral_mono)(auto split: split_indicator)
hoelzl@62975
  1059
    finally show ?thesis by (simp add: top_unique)
Andreas@59489
  1060
  next
Andreas@59489
  1061
    case False
wenzelm@63040
  1062
    define C' where "C' = fst ` C"
hoelzl@62975
  1063
    have "\<infinity> = \<integral>\<^sup>+ x. ennreal (1 / Suc n) * indicator C' x \<partial>count_space UNIV"
hoelzl@62975
  1064
      using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top)
hoelzl@62975
  1065
    also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) \<in> C} y \<partial>count_space UNIV \<partial>count_space UNIV"
hoelzl@62083
  1066
      by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong)
hoelzl@62975
  1067
    also have "\<dots> \<le> \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C (x, y) \<partial>count_space UNIV \<partial>count_space UNIV"
Andreas@59489
  1068
      by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI)
Andreas@59489
  1069
    also have "\<dots> \<le> ?lhs" using C
hoelzl@62975
  1070
      by(intro nn_integral_mono)(auto split: split_indicator)
hoelzl@62975
  1071
    finally show ?thesis by (simp add: top_unique)
Andreas@59489
  1072
  qed
Andreas@59489
  1073
  ultimately show ?thesis by simp
Andreas@59489
  1074
qed
Andreas@59489
  1075
Andreas@59491
  1076
lemma nn_integral_snd_count_space:
Andreas@59491
  1077
  "(\<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
  1078
  (is "?lhs = ?rhs")
Andreas@59491
  1079
proof -
Andreas@59491
  1080
  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
  1081
    by(simp)
Andreas@59491
  1082
  also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV"
Andreas@59491
  1083
    by(rule nn_integral_fst_count_space)
Andreas@59491
  1084
  also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)"
Andreas@59491
  1085
    by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
Andreas@59491
  1086
      (simp_all add: inj_on_def split_def)
Andreas@59491
  1087
  also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto
Andreas@59491
  1088
  finally show ?thesis .
Andreas@59491
  1089
qed
Andreas@59491
  1090
ak2110@68833
  1091
lemma%unimportant measurable_pair_measure_countable1:
Andreas@60066
  1092
  assumes "countable A"
Andreas@60066
  1093
  and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
Andreas@60066
  1094
  shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
Andreas@60066
  1095
using _ _ assms(1)
Andreas@60066
  1096
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
  1097
ak2110@68833
  1098
subsection%important \<open>Product of Borel spaces\<close>
hoelzl@57235
  1099
ak2110@68833
  1100
lemma%important borel_Times:
hoelzl@57235
  1101
  fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
hoelzl@57235
  1102
  assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
hoelzl@57235
  1103
  shows "A \<times> B \<in> sets borel"
ak2110@68833
  1104
proof%unimportant -
hoelzl@57235
  1105
  have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
hoelzl@57235
  1106
    by auto
hoelzl@57235
  1107
  moreover
hoelzl@57235
  1108
  { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)
hoelzl@57235
  1109
    then have "A\<times>UNIV \<in> sets borel"
hoelzl@57235
  1110
    proof (induct A)
hoelzl@57235
  1111
      case (Basic S) then show ?case
hoelzl@57235
  1112
        by (auto intro!: borel_open open_Times)
hoelzl@57235
  1113
    next
hoelzl@57235
  1114
      case (Compl A)
hoelzl@57235
  1115
      moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)"
hoelzl@57235
  1116
        by auto
hoelzl@57235
  1117
      ultimately show ?case
hoelzl@57235
  1118
        unfolding * by auto
hoelzl@57235
  1119
    next
hoelzl@57235
  1120
      case (Union A)
haftmann@69313
  1121
      moreover have *: "(\<Union>(A ` UNIV)) \<times> UNIV = \<Union>((\<lambda>i. A i \<times> UNIV) ` UNIV)"
hoelzl@57235
  1122
        by auto
hoelzl@57235
  1123
      ultimately show ?case
hoelzl@57235
  1124
        unfolding * by auto
hoelzl@57235
  1125
    qed simp }
hoelzl@57235
  1126
  moreover
hoelzl@57235
  1127
  { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)
hoelzl@57235
  1128
    then have "UNIV\<times>B \<in> sets borel"
hoelzl@57235
  1129
    proof (induct B)
hoelzl@57235
  1130
      case (Basic S) then show ?case
hoelzl@57235
  1131
        by (auto intro!: borel_open open_Times)
hoelzl@57235
  1132
    next
hoelzl@57235
  1133
      case (Compl B)
hoelzl@57235
  1134
      moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)"
hoelzl@57235
  1135
        by auto
hoelzl@57235
  1136
      ultimately show ?case
hoelzl@57235
  1137
        unfolding * by auto
hoelzl@57235
  1138
    next
hoelzl@57235
  1139
      case (Union B)
haftmann@69313
  1140
      moreover have *: "UNIV \<times> (\<Union>(B ` UNIV)) = \<Union>((\<lambda>i. UNIV \<times> B i) ` UNIV)"
hoelzl@57235
  1141
        by auto
hoelzl@57235
  1142
      ultimately show ?case
hoelzl@57235
  1143
        unfolding * by auto
hoelzl@57235
  1144
    qed simp }
hoelzl@57235
  1145
  ultimately show ?thesis
hoelzl@57235
  1146
    by auto
hoelzl@57235
  1147
qed
hoelzl@57235
  1148
ak2110@68833
  1149
lemma%unimportant finite_measure_pair_measure:
hoelzl@57235
  1150
  assumes "finite_measure M" "finite_measure N"
hoelzl@57235
  1151
  shows "finite_measure (N  \<Otimes>\<^sub>M M)"
hoelzl@57235
  1152
proof (rule finite_measureI)
hoelzl@57235
  1153
  interpret M: finite_measure M by fact
hoelzl@57235
  1154
  interpret N: finite_measure N by fact
hoelzl@57235
  1155
  show "emeasure (N  \<Otimes>\<^sub>M M) (space (N  \<Otimes>\<^sub>M M)) \<noteq> \<infinity>"
hoelzl@62975
  1156
    by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
hoelzl@57235
  1157
qed
hoelzl@57235
  1158
hoelzl@62083
  1159
end