src/HOL/Probability/Binary_Product_Measure.thy

added helper -- cf. SET616^5

(*  Title:      HOL/Probability/Binary_Product_Measure.thy
    Author:     Johannes Hölzl, TU München
*)

header {*Binary product measures*}

theory Binary_Product_Measure
imports Lebesgue_Integration
begin

lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
  by auto

lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
  by auto

lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
  by auto

lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
  by auto

lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
  by (cases x) simp

lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
  by (auto simp: fun_eq_iff)

section "Binary products"

definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
  "A \<Otimes>\<^isub>M B = measure_of (space A \<times> space B)
      {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
      (\<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A)"

lemma space_pair_measure:
  "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
  unfolding pair_measure_def using space_closed[of A] space_closed[of B]
  by (intro space_measure_of) auto

lemma sets_pair_measure:
  "sets (A \<Otimes>\<^isub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
  unfolding pair_measure_def using space_closed[of A] space_closed[of B]
  by (intro sets_measure_of) auto

lemma pair_measureI[intro, simp]:
  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
  by (auto simp: sets_pair_measure)

lemma measurable_pair_measureI:
  assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
  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"
  shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
  unfolding pair_measure_def using 1 2
  by (intro measurable_measure_of) (auto dest: sets_into_space)

lemma measurable_fst[intro!, simp]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1"
  unfolding measurable_def
proof safe
  fix A assume A: "A \<in> sets M1"
  from this[THEN sets_into_space] have "fst -` A \<inter> space M1 \<times> space M2 = A \<times> space M2" by auto
  with A show "fst -` A \<inter> space (M1 \<Otimes>\<^isub>M M2) \<in> sets (M1 \<Otimes>\<^isub>M M2)" by (simp add: space_pair_measure)
qed (simp add: space_pair_measure)

lemma measurable_snd[intro!, simp]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2"
  unfolding measurable_def
proof safe
  fix A assume A: "A \<in> sets M2"
  from this[THEN sets_into_space] have "snd -` A \<inter> space M1 \<times> space M2 = space M1 \<times> A" by auto
  with A show "snd -` A \<inter> space (M1 \<Otimes>\<^isub>M M2) \<in> sets (M1 \<Otimes>\<^isub>M M2)" by (simp add: space_pair_measure)
qed (simp add: space_pair_measure)

lemma measurable_fst': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. fst (f x)) \<in> measurable M N"
  using measurable_comp[OF _ measurable_fst] by (auto simp: comp_def)

lemma measurable_snd': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. snd (f x)) \<in> measurable M P"
    using measurable_comp[OF _ measurable_snd] by (auto simp: comp_def)

lemma measurable_fst'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^isub>M P) N"
  using measurable_comp[OF measurable_fst _] by (auto simp: comp_def)

lemma measurable_snd'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^isub>M M) N"
  using measurable_comp[OF measurable_snd _] by (auto simp: comp_def)

lemma measurable_pair_iff:
  "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
proof safe
  assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
  show "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
  proof (rule measurable_pair_measureI)
    show "f \<in> space M \<rightarrow> space M1 \<times> space M2"
      using f s by (auto simp: mem_Times_iff measurable_def comp_def)
    fix A B assume "A \<in> sets M1" "B \<in> sets M2"
    moreover have "(fst \<circ> f) -` A \<inter> space M \<in> sets M" "(snd \<circ> f) -` B \<inter> space M \<in> sets M"
      using f `A \<in> sets M1` s `B \<in> sets M2` by (auto simp: measurable_sets)
    moreover have "f -` (A \<times> B) \<inter> space M = ((fst \<circ> f) -` A \<inter> space M) \<inter> ((snd \<circ> f) -` B \<inter> space M)"
      by (auto simp: vimage_Times)
    ultimately show "f -` (A \<times> B) \<inter> space M \<in> sets M" by auto
  qed
qed auto

lemma measurable_pair:
  "(fst \<circ> f) \<in> measurable M M1 \<Longrightarrow> (snd \<circ> f) \<in> measurable M M2 \<Longrightarrow> f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"
  unfolding measurable_pair_iff by simp

lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
proof (rule measurable_pair_measureI)
  fix A B assume "A \<in> sets M2" "B \<in> sets M1"
  moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space (M1 \<Otimes>\<^isub>M M2) = B \<times> A"
    by (auto dest: sets_into_space simp: space_pair_measure)
  ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space (M1 \<Otimes>\<^isub>M M2) \<in> sets (M1 \<Otimes>\<^isub>M M2)"
    by auto
qed (auto simp add: space_pair_measure)

lemma measurable_pair_swap:
  assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
proof -
  have "(\<lambda>x. f (case x of (x, y) \<Rightarrow> (y, x))) = (\<lambda>(x, y). f (y, x))" by auto
  then show ?thesis
    using measurable_comp[OF measurable_pair_swap' f] by (simp add: comp_def)
qed

lemma measurable_pair_swap_iff:
  "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) M"
  using measurable_pair_swap[of "\<lambda>(x,y). f (y, x)"]
  by (auto intro!: measurable_pair_swap)

lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)"
proof (rule measurable_pair_measureI)
  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
  moreover then have "Pair x -` (A \<times> B) \<inter> space M2 = (if x \<in> A then B else {})"
    by (auto dest: sets_into_space simp: space_pair_measure)
  ultimately show "Pair x -` (A \<times> B) \<inter> space M2 \<in> sets M2"
    by auto
qed (auto simp add: space_pair_measure)

lemma sets_Pair1: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "Pair x -` A \<in> sets M2"
proof -
  have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
    using A[THEN sets_into_space] by (auto simp: space_pair_measure)
  also have "\<dots> \<in> sets M2"
    using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm)
  finally show ?thesis .
qed

lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^isub>M M2)"
  using measurable_comp[OF measurable_Pair1' measurable_pair_swap', of y M2 M1]
  by (simp add: comp_def)

lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
proof -
  have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
    using A[THEN sets_into_space] by (auto simp: space_pair_measure)
  also have "\<dots> \<in> sets M1"
    using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm)
  finally show ?thesis .
qed

lemma measurable_Pair2:
  assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and x: "x \<in> space M1"
  shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
  using measurable_comp[OF measurable_Pair1' f, OF x]
  by (simp add: comp_def)
  
lemma measurable_Pair1:
  assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and y: "y \<in> space M2"
  shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
  using measurable_comp[OF measurable_Pair2' f, OF y]
  by (simp add: comp_def)

lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
  unfolding Int_stable_def
  by safe (auto simp add: times_Int_times)

lemma finite_measure_cut_measurable:
  assumes "sigma_finite_measure M1" "finite_measure M2"
  assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
    (is "?s Q \<in> _")
proof -
  interpret M1: sigma_finite_measure M1 by fact
  interpret M2: finite_measure M2 by fact
  let ?\<Omega> = "space M1 \<times> space M2" and ?D = "{A\<in>sets (M1 \<Otimes>\<^isub>M M2). ?s A \<in> borel_measurable M1}"
  note space_pair_measure[simp]
  interpret dynkin_system ?\<Omega> ?D
  proof (intro dynkin_systemI)
    fix A assume "A \<in> ?D" then show "A \<subseteq> ?\<Omega>"
      using sets_into_space[of A "M1 \<Otimes>\<^isub>M M2"] by simp
  next
    from top show "?\<Omega> \<in> ?D"
      by (auto simp add: if_distrib intro!: measurable_If)
  next
    fix A assume "A \<in> ?D"
    with sets_into_space have "\<And>x. emeasure M2 (Pair x -` (?\<Omega> - A)) =
        (if x \<in> space M1 then emeasure M2 (space M2) - ?s A x else 0)"
      by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
    with `A \<in> ?D` top show "?\<Omega> - A \<in> ?D"
      by (auto intro!: measurable_If)
  next
    fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> ?D"
    moreover then have "\<And>x. emeasure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
      by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)
    ultimately show "(\<Union>i. F i) \<in> ?D"
      by (auto simp: vimage_UN intro!: borel_measurable_psuminf)
  qed
  let ?G = "{a \<times> b | a b. a \<in> sets M1 \<and> b \<in> sets M2}"
  have "sigma_sets ?\<Omega> ?G = ?D"
  proof (rule dynkin_lemma)
    show "?G \<subseteq> ?D"
      by (auto simp: if_distrib Int_def[symmetric] intro!: measurable_If)
  qed (auto simp: sets_pair_measure  Int_stable_pair_measure_generator)
  with `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)` have "Q \<in> ?D"
    by (simp add: sets_pair_measure[symmetric])
  then show "?s Q \<in> borel_measurable M1" by simp
qed

subsection {* Binary products of $\sigma$-finite emeasure spaces *}

locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
  for M1 :: "'a measure" and M2 :: "'b measure"

lemma sets_pair_measure_cong[cong]:
  "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^isub>M M2) = sets (M1' \<Otimes>\<^isub>M M2')"
  unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)

lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
  assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
proof -
  from M2.sigma_finite_disjoint guess F . note F = this
  then have F_sets: "\<And>i. F i \<in> sets M2" by auto
  have M1: "sigma_finite_measure M1" ..
  let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"
  { fix i
    have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
      using F sets_into_space by auto
    let ?R = "density M2 (indicator (F i))"
    have "(\<lambda>x. emeasure ?R (Pair x -` (space M1 \<times> space ?R \<inter> Q))) \<in> borel_measurable M1"
    proof (intro finite_measure_cut_measurable[OF M1])
      show "finite_measure ?R"
        using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
      show "(space M1 \<times> space ?R) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R)"
        using Q by (simp add: Int)
    qed
    moreover have "\<And>x. emeasure ?R (Pair x -` (space M1 \<times> space ?R \<inter> Q))
        = emeasure M2 (F i \<inter> Pair x -` (space M1 \<times> space ?R \<inter> Q))"
      using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
    moreover have "\<And>x. F i \<inter> Pair x -` (space M1 \<times> space ?R \<inter> Q) = ?C x i"
      using sets_into_space[OF Q] by (auto simp: space_pair_measure)
    ultimately have "(\<lambda>x. emeasure M2 (?C x i)) \<in> borel_measurable M1"
      by simp }
  moreover
  { fix x
    have "(\<Sum>i. emeasure M2 (?C x i)) = emeasure M2 (\<Union>i. ?C x i)"
    proof (intro suminf_emeasure)
      show "range (?C x) \<subseteq> sets M2"
        using F `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)` by (auto intro!: sets_Pair1)
      have "disjoint_family F" using F by auto
      show "disjoint_family (?C x)"
        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
    qed
    also have "(\<Union>i. ?C x i) = Pair x -` Q"
      using F sets_into_space[OF `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)`]
      by (auto simp: space_pair_measure)
    finally have "emeasure M2 (Pair x -` Q) = (\<Sum>i. emeasure M2 (?C x i))"
      by simp }
  ultimately show ?thesis using `Q \<in> sets (M1 \<Otimes>\<^isub>M M2)` F_sets
    by auto
qed

lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
  assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
    using Q measurable_pair_swap' by (auto intro: measurable_sets)
  note Q.measurable_emeasure_Pair1[OF this]
  moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^isub>M M1)) = (\<lambda>x. (x, y)) -` Q"
    using Q[THEN sets_into_space] by (auto simp: space_pair_measure)
  ultimately show ?thesis by simp
qed

lemma (in pair_sigma_finite) emeasure_pair_measure:
  assumes "X \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  shows "emeasure (M1 \<Otimes>\<^isub>M M2) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M2 \<partial>M1)" (is "_ = ?\<mu> X")
proof (rule emeasure_measure_of[OF pair_measure_def])
  show "positive (sets (M1 \<Otimes>\<^isub>M M2)) ?\<mu>"
    by (auto simp: positive_def positive_integral_positive)
  have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
    by (auto simp: indicator_def)
  show "countably_additive (sets (M1 \<Otimes>\<^isub>M M2)) ?\<mu>"
  proof (rule countably_additiveI)
    fix F :: "nat \<Rightarrow> ('a \<times> 'b) set" assume F: "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" "disjoint_family F"
    from F have *: "\<And>i. F i \<in> sets (M1 \<Otimes>\<^isub>M M2)" "(\<Union>i. F i) \<in> sets (M1 \<Otimes>\<^isub>M M2)" by auto
    moreover from F have "\<And>i. (\<lambda>x. emeasure M2 (Pair x -` F i)) \<in> borel_measurable M1"
      by (intro measurable_emeasure_Pair1) auto
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
    moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
      using F by (auto simp: sets_Pair1)
    ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
      by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1
               intro!: positive_integral_cong positive_integral_indicator[symmetric])
  qed
  show "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2} \<subseteq> Pow (space M1 \<times> space M2)"
    using space_closed[of M1] space_closed[of M2] by auto
qed fact

lemma (in pair_sigma_finite) emeasure_pair_measure_alt:
  assumes X: "X \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  shows "emeasure (M1  \<Otimes>\<^isub>M M2) X = (\<integral>\<^isup>+x. emeasure M2 (Pair x -` X) \<partial>M1)"
proof -
  have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
    by (auto simp: indicator_def)
  show ?thesis
    using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1)
qed

lemma (in pair_sigma_finite) emeasure_pair_measure_Times:
  assumes A: "A \<in> sets M1" and B: "B \<in> sets M2"
  shows "emeasure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = emeasure M1 A * emeasure M2 B"
proof -
  have "emeasure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M2 B * indicator A x \<partial>M1)"
    using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)
  also have "\<dots> = emeasure M2 B * emeasure M1 A"
    using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)
  finally show ?thesis
    by (simp add: ac_simps)
qed

lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
  defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
  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>
    (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
proof -
  from M1.sigma_finite_incseq guess F1 . note F1 = this
  from M2.sigma_finite_incseq guess F2 . note F2 = this
  from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
  show ?thesis
  proof (intro exI[of _ ?F] conjI allI)
    show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
  next
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
    proof (intro subsetI)
      fix x assume "x \<in> space M1 \<times> space M2"
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
        by (auto simp: space)
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
        using `incseq F1` `incseq F2` unfolding incseq_def
        by (force split: split_max)+
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
        by (intro SigmaI) (auto simp add: min_max.sup_commute)
      then show "x \<in> (\<Union>i. ?F i)" by auto
    qed
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
      using space by (auto simp: space)
  next
    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
      using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
  next
    fix i
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
    with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]
    show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
      by (auto simp add: emeasure_pair_measure_Times)
  qed
qed

sublocale pair_sigma_finite \<subseteq> sigma_finite_measure "M1 \<Otimes>\<^isub>M M2"
proof
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
  show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
  proof (rule exI[of _ F], intro conjI)
    show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def)
    show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)"
      using F by (auto simp: space_pair_measure)
    show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto
  qed
qed

lemma sigma_finite_pair_measure:
  assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
  shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)"
proof -
  interpret A: sigma_finite_measure A by fact
  interpret B: sigma_finite_measure B by fact
  interpret AB: pair_sigma_finite A  B ..
  show ?thesis ..
qed

lemma sets_pair_swap:
  assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
  using measurable_pair_swap' assms by (rule measurable_sets)

lemma (in pair_sigma_finite) distr_pair_swap:
  "M1 \<Otimes>\<^isub>M M2 = distr (M2 \<Otimes>\<^isub>M M1) (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
  show ?thesis
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
    show "?E \<subseteq> Pow (space ?P)"
      using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
    show "sets ?P = sigma_sets (space ?P) ?E"
      by (simp add: sets_pair_measure space_pair_measure)
    then show "sets ?D = sigma_sets (space ?P) ?E"
      by simp
  next
    show "range F \<subseteq> ?E" "incseq F" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
      using F by (auto simp: space_pair_measure)
  next
    fix X assume "X \<in> ?E"
    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
    have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A"
      using sets_into_space[OF A] sets_into_space[OF B] by (auto simp: space_pair_measure)
    with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X"
      by (simp add: emeasure_pair_measure_Times Q.emeasure_pair_measure_Times emeasure_distr
                    measurable_pair_swap' ac_simps)
  qed
qed

lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
  assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
    (is "_ = ?\<nu> A")
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1))) = (\<lambda>x. (x, y)) -` A"
    using sets_into_space[OF A] by (auto simp: space_pair_measure)
  show ?thesis using A
    by (subst distr_pair_swap)
       (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
                 Q.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
qed

section "Fubinis theorem"

lemma (in pair_sigma_finite) simple_function_cut:
  assumes f: "simple_function (M1 \<Otimes>\<^isub>M M2) f" "\<And>x. 0 \<le> f x"
  shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
proof -
  have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
    using f(1) by (rule borel_measurable_simple_function)
  let ?F = "\<lambda>z. f -` {z} \<inter> space (M1 \<Otimes>\<^isub>M M2)"
  let ?F' = "\<lambda>x z. Pair x -` ?F z"
  { fix x assume "x \<in> space M1"
    have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
      by (auto simp: indicator_def)
    have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space (M1 \<Otimes>\<^isub>M M2)" using `x \<in> space M1`
      by (simp add: space_pair_measure)
    moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
      by (rule sets_Pair1[OF measurable_sets]) auto
    ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
      apply (rule_tac simple_function_cong[THEN iffD2, OF _])
      apply (rule simple_function_indicator_representation[OF f(1)])
      using `x \<in> space M1` by auto }
  note M2_sf = this
  { fix x assume x: "x \<in> space M1"
    then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space (M1 \<Otimes>\<^isub>M M2). z * emeasure M2 (?F' x z))"
      unfolding positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
      unfolding simple_integral_def
    proof (safe intro!: setsum_mono_zero_cong_left)
      from f(1) show "finite (f ` space (M1 \<Otimes>\<^isub>M M2))" by (rule simple_functionD)
    next
      fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space (M1 \<Otimes>\<^isub>M M2)"
        using `x \<in> space M1` by (auto simp: space_pair_measure)
    next
      fix x' y assume "(x', y) \<in> space (M1 \<Otimes>\<^isub>M M2)"
        "f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
      then have *: "?F' x (f (x', y)) = {}"
        by (force simp: space_pair_measure)
      show  "f (x', y) * emeasure M2 (?F' x (f (x', y))) = 0"
        unfolding * by simp
    qed (simp add: vimage_compose[symmetric] comp_def
                   space_pair_measure) }
  note eq = this
  moreover have "\<And>z. ?F z \<in> sets (M1 \<Otimes>\<^isub>M M2)"
    by (auto intro!: f_borel borel_measurable_vimage)
  moreover then have "\<And>z. (\<lambda>x. emeasure M2 (?F' x z)) \<in> borel_measurable M1"
    by (auto intro!: measurable_emeasure_Pair1 simp del: vimage_Int)
  moreover have *: "\<And>i x. 0 \<le> emeasure M2 (Pair x -` (f -` {i} \<inter> space (M1 \<Otimes>\<^isub>M M2)))"
    using f(1)[THEN simple_functionD(2)] f(2) by (intro emeasure_nonneg)
  moreover { fix i assume "i \<in> f`space (M1 \<Otimes>\<^isub>M M2)"
    with * have "\<And>x. 0 \<le> i * emeasure M2 (Pair x -` (f -` {i} \<inter> space (M1 \<Otimes>\<^isub>M M2)))"
      using f(2) by auto }
  ultimately
  show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f" using f(2)
    by (auto simp del: vimage_Int cong: measurable_cong intro!: setsum_cong
             simp add: positive_integral_setsum simple_integral_def
                       positive_integral_cmult
                       positive_integral_cong[OF eq]
                       positive_integral_eq_simple_integral[OF f]
                       emeasure_pair_measure_alt[symmetric])
qed

lemma (in pair_sigma_finite) positive_integral_fst_measurable:
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
      (is "?C f \<in> borel_measurable M1")
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
proof -
  from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
  then have F_borel: "\<And>i. F i \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
    by (auto intro: borel_measurable_simple_function)
  note sf = simple_function_cut[OF F(1,5)]
  then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
    using F(1) by auto
  moreover
  { fix x assume "x \<in> space M1"
    from F measurable_Pair2[OF F_borel `x \<in> space M1`]
    have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
      by (intro positive_integral_monotone_convergence_SUP)
         (auto simp: incseq_Suc_iff le_fun_def)
    then have "(SUP i. ?C (F i) x) = ?C f x"
      unfolding F(4) positive_integral_max_0 by simp }
  note SUPR_C = this
  ultimately show "?C f \<in> borel_measurable M1"
    by (simp cong: measurable_cong)
  have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>(M1 \<Otimes>\<^isub>M M2)) = (SUP i. integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (F i))"
    using F_borel F
    by (intro positive_integral_monotone_convergence_SUP) auto
  also have "(SUP i. integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
    unfolding sf(2) by simp
  also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
    by (intro positive_integral_monotone_convergence_SUP[symmetric])
       (auto intro!: positive_integral_mono positive_integral_positive
             simp: incseq_Suc_iff le_fun_def)
  also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
    using F_borel F(2,5)
    by (auto intro!: positive_integral_cong positive_integral_monotone_convergence_SUP[symmetric] measurable_Pair2
             simp: incseq_Suc_iff le_fun_def)
  finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
    using F by (simp add: positive_integral_max_0)
qed

lemma (in pair_sigma_finite) positive_integral_snd_measurable:
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  note measurable_pair_swap[OF f]
  from Q.positive_integral_fst_measurable[OF this]
  have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1))"
    by simp
  also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"
    by (subst distr_pair_swap)
       (auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong)
  finally show ?thesis .
qed

lemma (in pair_sigma_finite) Fubini:
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
  unfolding positive_integral_snd_measurable[OF assms]
  unfolding positive_integral_fst_measurable[OF assms] ..

lemma (in pair_sigma_finite) AE_pair:
  assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x"
  shows "AE x in M1. (AE y in M2. Q (x, y))"
proof -
  obtain N where N: "N \<in> sets (M1 \<Otimes>\<^isub>M M2)" "emeasure (M1 \<Otimes>\<^isub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> Q x} \<subseteq> N"
    using assms unfolding eventually_ae_filter by auto
  show ?thesis
  proof (rule AE_I)
    from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^isub>M M2)`]
    show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
      by (auto simp: emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg)
    show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
      by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)
    { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
      have "AE y in M2. Q (x, y)"
      proof (rule AE_I)
        show "emeasure M2 (Pair x -` N) = 0" by fact
        show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
          using N `x \<in> space M1` unfolding space_pair_measure by auto
      qed }
    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}"
      by auto
  qed
qed

lemma (in pair_sigma_finite) AE_pair_measure:
  assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  assumes ae: "AE x in M1. AE y in M2. P (x, y)"
  shows "AE x in M1 \<Otimes>\<^isub>M M2. P x"
proof (subst AE_iff_measurable[OF _ refl])
  show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
    by (rule sets_Collect) fact
  then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} =
      (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
    by (simp add: emeasure_pair_measure)
  also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)"
    using ae
    apply (safe intro!: positive_integral_cong_AE)
    apply (intro AE_I2)
    apply (safe intro!: positive_integral_cong_AE)
    apply auto
    done
  finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp
qed

lemma (in pair_sigma_finite) AE_pair_iff:
  "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow>
    (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x))"
  using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto

lemma AE_distr_iff:
  assumes f: "f \<in> measurable M N" and P: "{x \<in> space N. P x} \<in> sets N"
  shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"
proof (subst (1 2) AE_iff_measurable[OF _ refl])
  from P show "{x \<in> space (distr M N f). \<not> P x} \<in> sets (distr M N f)"
    by (auto intro!: sets_Collect_neg)
  moreover
  have "f -` {x \<in> space N. P x} \<inter> space M = {x \<in> space M. P (f x)}"
    using f by (auto dest: measurable_space)
  then show "{x \<in> space M. \<not> P (f x)} \<in> sets M"
    using measurable_sets[OF f P] by (auto intro!: sets_Collect_neg)
  moreover have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
    using f by (auto dest: measurable_space)
  ultimately show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
    (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
    using f by (simp add: emeasure_distr)
qed

lemma (in pair_sigma_finite) AE_commute:
  assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)"
  shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
proof -
  interpret Q: pair_sigma_finite M2 M1 ..
  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"
    by auto
  have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} =
    (\<lambda>(x, y). (y, x)) -` {x \<in> space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^isub>M M1)"
    by (auto simp: space_pair_measure)
  also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)"
    by (intro sets_pair_swap P)
  finally show ?thesis
    apply (subst AE_pair_iff[OF P])
    apply (subst distr_pair_swap)
    apply (subst AE_distr_iff[OF measurable_pair_swap' P])
    apply (subst Q.AE_pair_iff)
    apply simp_all
    done
qed

lemma (in pair_sigma_finite) integrable_product_swap:
  assumes "integrable (M1 \<Otimes>\<^isub>M M2) f"
  shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
  show ?thesis unfolding *
    by (rule integrable_distr[OF measurable_pair_swap'])
       (simp add: distr_pair_swap[symmetric] assms)
qed

lemma (in pair_sigma_finite) integrable_product_swap_iff:
  "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^isub>M M2) f"
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
  show ?thesis by auto
qed

lemma (in pair_sigma_finite) integral_product_swap:
  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f"
proof -
  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
  show ?thesis unfolding *
    by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
qed

lemma (in pair_sigma_finite) integrable_fst_measurable:
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
  shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE")
    and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
proof -
  have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
    using f by auto
  let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
  have
    borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and
    int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?nf \<noteq> \<infinity>" "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?pf \<noteq> \<infinity>"
    using assms by auto
  have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
     "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
    using borel[THEN positive_integral_fst_measurable(1)] int
    unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
  with borel[THEN positive_integral_fst_measurable(1)]
  have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
    "AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
    by (auto intro!: positive_integral_PInf_AE )
  then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
    "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
    by (auto simp: positive_integral_positive)
  from AE_pos show ?AE using assms
    by (simp add: measurable_Pair2[OF f_borel] integrable_def)
  { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
      using positive_integral_positive
      by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
    then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
  note this[simp]
  { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
      and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
      and AE: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
    have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
    proof (intro integrable_def[THEN iffD2] conjI)
      show "?f \<in> borel_measurable M1"
        using borel by (auto intro!: positive_integral_fst_measurable)
      have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y))  \<partial>M2) \<partial>M1)"
        using AE positive_integral_positive[of M2]
        by (auto intro!: positive_integral_cong_AE simp: ereal_real)
      then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
        using positive_integral_fst_measurable[OF borel] int by simp
      have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
        by (intro positive_integral_cong_pos)
           (simp add: positive_integral_positive real_of_ereal_pos)
      then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
    qed }
  with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
  show ?INT
    unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2]
      borel[THEN positive_integral_fst_measurable(2), symmetric]
    using AE[THEN integral_real]
    by simp
qed

lemma (in pair_sigma_finite) integrable_snd_measurable:
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
  shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE")
    and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")
proof -
  interpret Q: pair_sigma_finite M2 M1 by default
  have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))"
    using f unfolding integrable_product_swap_iff .
  show ?INT
    using Q.integrable_fst_measurable(2)[OF Q_int]
    using integral_product_swap[of f] f by auto
  show ?AE
    using Q.integrable_fst_measurable(1)[OF Q_int]
    by simp
qed

lemma (in pair_sigma_finite) positive_integral_fst_measurable':
  assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M2) \<in> borel_measurable M1"
  using positive_integral_fst_measurable(1)[OF f] by simp

lemma (in pair_sigma_finite) integral_fst_measurable:
  "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M2) \<in> borel_measurable M1"
  by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_fst_measurable')

lemma (in pair_sigma_finite) positive_integral_snd_measurable':
  assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
  shows "(\<lambda>y. \<integral>\<^isup>+ x. f x y \<partial>M1) \<in> borel_measurable M2"
proof -
  interpret Q: pair_sigma_finite M2 M1 ..
  show ?thesis
    using measurable_pair_swap[OF f]
    by (intro Q.positive_integral_fst_measurable') (simp add: split_beta')
qed

lemma (in pair_sigma_finite) integral_snd_measurable:
  "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>y. \<integral> x. f x y \<partial>M1) \<in> borel_measurable M2"
  by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_snd_measurable')

lemma (in pair_sigma_finite) Fubini_integral:
  assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"
  shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
  unfolding integrable_snd_measurable[OF assms]
  unfolding integrable_fst_measurable[OF assms] ..

section {* Products on counting spaces, densities and distributions *}

lemma sigma_sets_pair_measure_generator_finite:
  assumes "finite A" and "finite B"
  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
  (is "sigma_sets ?prod ?sets = _")
proof safe
  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
  fix x assume subset: "x \<subseteq> A \<times> B"
  hence "finite x" using fin by (rule finite_subset)
  from this subset show "x \<in> sigma_sets ?prod ?sets"
  proof (induct x)
    case empty show ?case by (rule sigma_sets.Empty)
  next
    case (insert a x)
    hence "{a} \<in> sigma_sets ?prod ?sets" by auto
    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
  qed
next
  fix x a b
  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
  from sigma_sets_into_sp[OF _ this(1)] this(2)
  show "a \<in> A" and "b \<in> B" by auto
qed

lemma pair_measure_count_space:
  assumes A: "finite A" and B: "finite B"
  shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
proof (rule measure_eqI)
  interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
  interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
  interpret P: pair_sigma_finite "count_space A" "count_space B" by default
  show eq: "sets ?P = sets ?C"
    by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
  fix X assume X: "X \<in> sets ?P"
  with eq have X_subset: "X \<subseteq> A \<times> B" by simp
  with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
    by (intro finite_subset[OF _ B]) auto
  have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
  show "emeasure ?P X = emeasure ?C X"
    apply (subst P.emeasure_pair_measure_alt[OF X])
    apply (subst emeasure_count_space)
    using X_subset apply auto []
    apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
    apply (subst positive_integral_count_space)
    using A apply simp
    apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
    apply (subst card_gt_0_iff)
    apply (simp add: fin_Pair)
    apply (subst card_SigmaI[symmetric])
    using A apply simp
    using fin_Pair apply simp
    using X_subset apply (auto intro!: arg_cong[where f=card])
    done
qed

lemma pair_measure_density:
  assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
  assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
  assumes "sigma_finite_measure (density M1 f)" "sigma_finite_measure (density M2 g)"
  shows "density M1 f \<Otimes>\<^isub>M density M2 g = density (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")
proof (rule measure_eqI)
  interpret M1: sigma_finite_measure M1 by fact
  interpret M2: sigma_finite_measure M2 by fact
  interpret D1: sigma_finite_measure "density M1 f" by fact
  interpret D2: sigma_finite_measure "density M2 g" by fact
  interpret L: pair_sigma_finite "density M1 f" "density M2 g" ..
  interpret R: pair_sigma_finite M1 M2 ..

  fix A assume A: "A \<in> sets ?L"
  then have indicator_eq: "\<And>x y. indicator A (x, y) = indicator (Pair x -` A) y"
   and Pair_A: "\<And>x. Pair x -` A \<in> sets M2"
    by (auto simp: indicator_def sets_Pair1)
  have f_fst: "(\<lambda>p. f (fst p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
    using measurable_comp[OF measurable_fst f(1)] by (simp add: comp_def)
  have g_snd: "(\<lambda>p. g (snd p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"
    using measurable_comp[OF measurable_snd g(1)] by (simp add: comp_def)
  have "(\<lambda>x. \<integral>\<^isup>+ y. g (snd (x, y)) * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
    using g_snd Pair_A A by (intro R.positive_integral_fst_measurable) auto
  then have int_g: "(\<lambda>x. \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"
    by simp

  show "emeasure ?L A = emeasure ?R A"
    apply (subst L.emeasure_pair_measure[OF A])
    apply (subst emeasure_density)
        using f_fst g_snd apply (simp add: split_beta')
      using A apply simp
    apply (subst positive_integral_density[OF g])
      apply (simp add: indicator_eq Pair_A)
    apply (subst positive_integral_density[OF f])
      apply (rule int_g)
    apply (subst R.positive_integral_fst_measurable(2)[symmetric])
      using f g A Pair_A f_fst g_snd
      apply (auto intro!: positive_integral_cong_AE R.measurable_emeasure_Pair1
                  simp: positive_integral_cmult indicator_eq split_beta')
    apply (intro AE_I2 impI)
    apply (subst mult_assoc)
    apply (subst positive_integral_cmult)
          apply auto
    done
qed simp

lemma sigma_finite_measure_distr:
  assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
  shows "sigma_finite_measure M"
proof -
  interpret sigma_finite_measure "distr M N f" by fact
  from sigma_finite_disjoint guess A . note A = this
  show ?thesis
  proof (unfold_locales, intro conjI exI allI)
    show "range (\<lambda>i. f -` A i \<inter> space M) \<subseteq> sets M"
      using A f by (auto intro!: measurable_sets)
    show "(\<Union>i. f -` A i \<inter> space M) = space M"
      using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def)
    fix i show "emeasure M (f -` A i \<inter> space M) \<noteq> \<infinity>"
      using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq)
  qed
qed

lemma measurable_cong':
  assumes sets: "sets M = sets M'" "sets N = sets N'"
  shows "measurable M N = measurable M' N'"
  using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

lemma pair_measure_distr:
  assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
  assumes "sigma_finite_measure (distr M S f)" "sigma_finite_measure (distr N T g)"
  shows "distr M S f \<Otimes>\<^isub>M distr N T g = distr (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")
proof (rule measure_eqI)
  show "sets ?P = sets ?D"
    by simp
  interpret S: sigma_finite_measure "distr M S f" by fact
  interpret T: sigma_finite_measure "distr N T g" by fact
  interpret ST: pair_sigma_finite "distr M S f"  "distr N T g" ..
  interpret M: sigma_finite_measure M by (rule sigma_finite_measure_distr) fact+
  interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
  interpret MN: pair_sigma_finite M N ..
  interpret SN: pair_sigma_finite "distr M S f" N ..
  have [simp]: 
    "\<And>f g. fst \<circ> (\<lambda>(x, y). (f x, g y)) = f \<circ> fst" "\<And>f g. snd \<circ> (\<lambda>(x, y). (f x, g y)) = g \<circ> snd"
    by auto
  then have fg: "(\<lambda>(x, y). (f x, g y)) \<in> measurable (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T)"
    using measurable_comp[OF measurable_fst f] measurable_comp[OF measurable_snd g]
    by (auto simp: measurable_pair_iff)
  fix A assume A: "A \<in> sets ?P"
  then have "emeasure ?P A = (\<integral>\<^isup>+x. emeasure (distr N T g) (Pair x -` A) \<partial>distr M S f)"
    by (rule ST.emeasure_pair_measure_alt)
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair x -` A) \<inter> space N) \<partial>distr M S f)"
    using g A by (simp add: sets_Pair1 emeasure_distr)
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g -` (Pair (f x) -` A) \<inter> space N) \<partial>M)"
    using f g A ST.measurable_emeasure_Pair1[OF A]
    by (intro positive_integral_distr) (auto simp add: sets_Pair1 emeasure_distr)
  also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (Pair x -` ((\<lambda>(x, y). (f x, g y)) -` A \<inter> space (M \<Otimes>\<^isub>M N))) \<partial>M)"
    by (intro positive_integral_cong arg_cong2[where f=emeasure]) (auto simp: space_pair_measure)
  also have "\<dots> = emeasure (M \<Otimes>\<^isub>M N) ((\<lambda>(x, y). (f x, g y)) -` A \<inter> space (M \<Otimes>\<^isub>M N))"
    using fg by (intro MN.emeasure_pair_measure_alt[symmetric] measurable_sets[OF _ A])
                (auto cong: measurable_cong')
  also have "\<dots> = emeasure ?D A"
    using fg A by (subst emeasure_distr) auto
  finally show "emeasure ?P A = emeasure ?D A" .
qed

end