(*  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 pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"

   using space_closed[of A] space_closed[of B] by auto

 lemma space_pair_measure:

   "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"

   unfolding pair_measure_def using pair_measure_closed[of A B]

   by (rule space_measure_of)

 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 pair_measure_closed[of A B]

   by (rule sets_measure_of)

 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 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_Pair:

   assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"

   shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"

 proof (rule measurable_pair_measureI)

   show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"

     using f g by (auto simp: measurable_def)

   fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"

   have "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"

     by auto

   also have "\<dots> \<in> sets M"

     by (rule Int) (auto intro!: measurable_sets * f g)

   finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .

 qed

 lemma measurable_pair:

   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"

   shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"

   using measurable_Pair[OF assms] by simp

 lemma measurable_fst[intro!, simp]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1"

   by (auto simp: fst_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)

 lemma measurable_snd[intro!, simp]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2"

   by (auto simp: snd_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)

 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"

   using measurable_pair[of f M M1 M2] by auto

 lemma measurable_split_conv:

   "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"

   by (intro arg_cong2[where f="op \<in>"]) auto

 lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"

   by (auto intro!: measurable_Pair simp: measurable_split_conv)

 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"

   using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)

 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_ident[intro, simp]: "(\<lambda>x. x) \<in> measurable M M"

   unfolding measurable_def by auto

 lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)"

   by (auto intro!: measurable_Pair)

 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)"

   by (auto intro!: measurable_Pair)

 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 (in finite_measure) finite_measure_cut_measurable:

   assumes "Q \<in> sets (N \<Otimes>\<^isub>M M)"

   shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"

     (is "?s Q \<in> _")

   using Int_stable_pair_measure_generator pair_measure_closed assms

   unfolding sets_pair_measure

 proof (induct rule: sigma_sets_induct_disjoint)

   case (compl A)

   with sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) =

       (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"

     unfolding sets_pair_measure[symmetric]

     by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)

   with compl top show ?case

     by (auto intro!: measurable_If simp: space_pair_measure)

 next

   case (union F)

   moreover then have "\<And>x. emeasure M (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"

     unfolding sets_pair_measure[symmetric]

     by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)

   ultimately show ?case

     by (auto simp: vimage_UN)

 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)

 lemma (in sigma_finite_measure) measurable_emeasure_Pair:

   assumes Q: "Q \<in> sets (N \<Otimes>\<^isub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" (is "?s Q \<in> _")

 proof -

   from sigma_finite_disjoint guess F . note F = this

   then have F_sets: "\<And>i. F i \<in> sets M" by auto

   let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"

   { fix i

     have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"

       using F sets_into_space by auto

     let ?R = "density M (indicator (F i))"

     have "finite_measure ?R"

       using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)

     then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"

      by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)

     moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))

         = emeasure M (F i \<inter> Pair x -` (space N \<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 N \<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 M (?C x i)) \<in> borel_measurable N"

       by simp }

   moreover

   { fix x

     have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"

     proof (intro suminf_emeasure)

       show "range (?C x) \<subseteq> sets M"

         using F `Q \<in> sets (N \<Otimes>\<^isub>M M)` 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 (N \<Otimes>\<^isub>M M)`]

       by (auto simp: space_pair_measure)

     finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"

       by simp }

   ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^isub>M M)` F_sets

     by auto

 qed

 lemma (in sigma_finite_measure) emeasure_pair_measure:

   assumes "X \<in> sets (N \<Otimes>\<^isub>M M)"

   shows "emeasure (N \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X")

 proof (rule emeasure_measure_of[OF pair_measure_def])

   show "positive (sets (N \<Otimes>\<^isub>M M)) ?\<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 (N \<Otimes>\<^isub>M M)) ?\<mu>"

   proof (rule countably_additiveI)

     fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^isub>M M)" "disjoint_family F"

     from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^isub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^isub>M M)" by auto

     moreover from F have "\<And>i. (\<lambda>x. emeasure M (Pair x -` F i)) \<in> borel_measurable N"

       by (intro measurable_emeasure_Pair) 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 M"

       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 N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"

     using space_closed[of N] space_closed[of M] by auto

 qed fact

 lemma (in sigma_finite_measure) emeasure_pair_measure_alt:

   assumes X: "X \<in> sets (N \<Otimes>\<^isub>M M)"

   shows "emeasure (N  \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+x. emeasure M (Pair x -` X) \<partial>N)"

 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 sigma_finite_measure) emeasure_pair_measure_Times:

   assumes A: "A \<in> sets N" and B: "B \<in> sets M"

   shows "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = emeasure N A * emeasure M B"

 proof -

   have "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M B * indicator A x \<partial>N)"

     using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)

   also have "\<dots> = emeasure M B * emeasure N A"

     using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)

   finally show ?thesis

     by (simp add: ac_simps)

 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 (in pair_sigma_finite) measurable_emeasure_Pair1:

   "Q \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"

   using M2.measurable_emeasure_Pair .

 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 -

   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 M1.measurable_emeasure_Pair[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) 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> P: 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 -

   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" "(\<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: M2.emeasure_pair_measure_Times M1.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 -

   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']

                  M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])

 qed

 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: M2.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: M2.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 (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

 section "Fubinis theorem"

 lemma measurable_compose_Pair1:

   "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^isub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"

   by (rule measurable_compose[OF measurable_Pair]) auto

 lemma (in pair_sigma_finite) borel_measurable_positive_integral_fst:

   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" "\<And>x. 0 \<le> f x"

   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"

 using f proof induct

   case (cong u v)

   then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M2 \<Longrightarrow> u (w, x) = v (w, x)"

     by (auto simp: space_pair_measure)

   show ?case

     apply (subst measurable_cong)

     apply (rule positive_integral_cong)

     apply fact+

     done

 next

   case (set Q)

   have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y"

     by (auto simp: indicator_def)

   have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M2 (Pair x -` Q) = \<integral>\<^isup>+ y. indicator Q (x, y) \<partial>M2"

     by (simp add: sets_Pair1[OF set])

   from this M2.measurable_emeasure_Pair[OF set] show ?case

     by (rule measurable_cong[THEN iffD1])

 qed (simp_all add: positive_integral_add positive_integral_cmult measurable_compose_Pair1

                    positive_integral_monotone_convergence_SUP incseq_def le_fun_def

               cong: measurable_cong)

 lemma (in pair_sigma_finite) positive_integral_fst:

   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" "\<And>x. 0 \<le> f x"

   shows "(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2 \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f" (is "?I f = _")

 using f proof induct

   case (cong u v)

   moreover then have "?I u = ?I v"

     by (intro positive_integral_cong) (auto simp: space_pair_measure)

   ultimately show ?case

     by (simp cong: positive_integral_cong)

 qed (simp_all add: M2.emeasure_pair_measure positive_integral_cmult positive_integral_add

                    positive_integral_monotone_convergence_SUP

                    measurable_compose_Pair1 positive_integral_positive

                    borel_measurable_positive_integral_fst positive_integral_mono incseq_def le_fun_def

               cong: positive_integral_cong)

 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"

   using f

     borel_measurable_positive_integral_fst[of "\<lambda>x. max 0 (f x)"]

     positive_integral_fst[of "\<lambda>x. max 0 (f x)"]

   unfolding positive_integral_max_0 by auto

 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) 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 B.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 D2.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 T.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 N.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