(*  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_generator A B =

     \<lparr> space = space A \<times> space B,

       sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},

       measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"

 definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where

   "A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"

 locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2

   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

 abbreviation (in pair_sigma_algebra)

   "E \<equiv> pair_measure_generator M1 M2"

 abbreviation (in pair_sigma_algebra)

   "P \<equiv> M1 \<Otimes>\<^isub>M M2"

 lemma sigma_algebra_pair_measure:

   "sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"

   by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)

 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P

   using M1.space_closed M2.space_closed

   by (rule sigma_algebra_pair_measure)

 lemma pair_measure_generatorI[intro, simp]:

   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"

   by (auto simp add: pair_measure_generator_def)

 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 add: pair_measure_def)

 lemma space_pair_measure:

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

   by (simp add: pair_measure_def pair_measure_generator_def)

 lemma sets_pair_measure_generator:

   "sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y) ` (sets N \<times> sets M)"

   unfolding pair_measure_generator_def by auto

 lemma pair_measure_generator_sets_into_space:

   assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"

   shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"

   using assms by (auto simp: pair_measure_generator_def)

 lemma pair_measure_generator_Int_snd:

   assumes "sets S1 \<subseteq> Pow (space S1)"

   shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =

          sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"

   (is "?L = ?R")

   apply (auto simp: pair_measure_generator_def image_iff)

   using assms

   apply (rule_tac x="a \<times> xa" in exI)

   apply force

   using assms

   apply (rule_tac x="a" in exI)

   apply (rule_tac x="b \<inter> A" in exI)

   apply auto

   done

 lemma (in pair_sigma_algebra)

   shows measurable_fst[intro!, simp]:

     "fst \<in> measurable P M1" (is ?fst)

   and measurable_snd[intro!, simp]:

     "snd \<in> measurable P M2" (is ?snd)

 proof -

   { fix X assume "X \<in> sets M1"

     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"

       apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])

       using M1.sets_into_space by force+ }

   moreover

   { fix X assume "X \<in> sets M2"

     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"

       apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])

       using M2.sets_into_space by force+ }

   ultimately have "?fst \<and> ?snd"

     by (fastforce simp: measurable_def sets_sigma space_pair_measure

                  intro!: sigma_sets.Basic)

   then show ?fst ?snd by auto

 qed

 lemma (in pair_sigma_algebra) measurable_pair_iff:

   assumes "sigma_algebra M"

   shows "f \<in> measurable M P \<longleftrightarrow>

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

 proof -

   interpret M: sigma_algebra M by fact

   from assms show ?thesis

   proof (safe intro!: measurable_comp[where b=P])

     assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"

     show "f \<in> measurable M P" unfolding pair_measure_def

     proof (rule M.measurable_sigma)

       show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"

         unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto

       show "f \<in> space M \<rightarrow> space E"

         using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)

       fix A assume "A \<in> sets E"

       then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"

         unfolding pair_measure_generator_def by auto

       moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M"

         using f `B \<in> sets M1` unfolding measurable_def by auto

       moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M"

         using s `C \<in> sets M2` unfolding measurable_def by auto

       moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)"

         unfolding `A = B \<times> C` by (auto simp: vimage_Times)

       ultimately show "f -` A \<inter> space M \<in> sets M" by auto

     qed

   qed

 qed

 lemma (in pair_sigma_algebra) measurable_pair:

   assumes "sigma_algebra M"

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

   shows "f \<in> measurable M P"

   unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp

 lemma pair_measure_generatorE:

   assumes "X \<in> sets (pair_measure_generator M1 M2)"

   obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"

   using assms unfolding pair_measure_generator_def by auto

 lemma (in pair_sigma_algebra) pair_measure_generator_swap:

   "(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_measure_generator M2 M1)"

 proof (safe elim!: pair_measure_generatorE)

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

   moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"

     using M1.sets_into_space M2.sets_into_space by auto

   ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)"

     by (auto intro: pair_measure_generatorI)

 next

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

   then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E"

     using M1.sets_into_space M2.sets_into_space

     by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI)

 qed

 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:

   assumes Q: "Q \<in> sets P"

   shows "(\<lambda>(x,y). (y, x)) -` Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")

 proof -

   let ?f = "\<lambda>Q. (\<lambda>(x,y). (y, x)) -` Q \<inter> space M2 \<times> space M1"

   have *: "(\<lambda>(x,y). (y, x)) -` Q = ?f Q"

     using sets_into_space[OF Q] by (auto simp: space_pair_measure)

   have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"

     unfolding pair_measure_def ..

   also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f ` sets E)"

     unfolding sigma_def pair_measure_generator_swap[symmetric]

     by (simp add: pair_measure_generator_def)

   also have "\<dots> = ?f ` sigma_sets (space M1 \<times> space M2) (sets E)"

     using M1.sets_into_space M2.sets_into_space

     by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)

   also have "\<dots> = ?f ` sets P"

     unfolding pair_measure_def pair_measure_generator_def sigma_def by simp

   finally show ?thesis

     using Q by (subst *) auto

 qed

 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:

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

     (is "?f \<in> measurable ?P ?Q")

   unfolding measurable_def

 proof (intro CollectI conjI Pi_I ballI)

   fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"

     unfolding pair_measure_generator_def pair_measure_def by auto

 next

   fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"

   interpret Q: pair_sigma_algebra M2 M1 by default

   from Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets (M2 \<Otimes>\<^isub>M M1)`]

   show "?f -` A \<inter> space ?P \<in> sets ?P" by simp

 qed

 lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:

   assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"

 proof -

   let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"

   let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"

   interpret Q: sigma_algebra ?Q

     proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)

   have "sets E \<subseteq> sets ?Q"

     using M1.sets_into_space M2.sets_into_space

     by (auto simp: pair_measure_generator_def space_pair_measure)

   then have "sets P \<subseteq> sets ?Q"

     apply (subst pair_measure_def, intro Q.sets_sigma_subset)

     by (simp add: pair_measure_def)

   with assms show ?thesis by auto

 qed

 lemma (in pair_sigma_algebra) measurable_cut_snd:

   assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1")

 proof -

   interpret Q: pair_sigma_algebra M2 M1 by default

   from Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]

   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)

 qed

 lemma (in pair_sigma_algebra) measurable_pair_image_snd:

   assumes m: "f \<in> measurable P M" and "x \<in> space M1"

   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"

   unfolding measurable_def

 proof (intro CollectI conjI Pi_I ballI)

   fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1`

   show "f (x, y) \<in> space M"

     unfolding measurable_def pair_measure_generator_def pair_measure_def by auto

 next

   fix A assume "A \<in> sets M"

   then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _")

     using `f \<in> measurable P M`

     by (intro measurable_cut_fst) (auto simp: measurable_def)

   also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2"

     using `x \<in> space M1` by (auto simp: pair_measure_generator_def pair_measure_def)

   finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" .

 qed

 lemma (in pair_sigma_algebra) measurable_pair_image_fst:

   assumes m: "f \<in> measurable P M" and "y \<in> space M2"

   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"

 proof -

   interpret Q: pair_sigma_algebra M2 M1 by default

   from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`,

                                       OF Q.pair_sigma_algebra_swap_measurable m]

   show ?thesis by simp

 qed

 lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"

   unfolding Int_stable_def

 proof (intro ballI)

   fix A B assume "A \<in> sets E" "B \<in> sets E"

   then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"

     "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"

     unfolding pair_measure_generator_def by auto

   then show "A \<inter> B \<in> sets E"

     by (auto simp add: times_Int_times pair_measure_generator_def)

 qed

 lemma finite_measure_cut_measurable:

   fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

   assumes "sigma_finite_measure M1" "finite_measure M2"

   assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"

   shows "(\<lambda>x. measure 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

   interpret pair_sigma_algebra M1 M2 by default

   have [intro]: "sigma_algebra M1" by fact

   have [intro]: "sigma_algebra M2" by fact

   let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"

   note space_pair_measure[simp]

   interpret dynkin_system ?D

   proof (intro dynkin_systemI)

     fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"

       using sets_into_space by simp

   next

     from top show "space ?D \<in> sets ?D"

       by (auto simp add: if_distrib intro!: M1.measurable_If)

   next

     fix A assume "A \<in> sets ?D"

     with sets_into_space have "\<And>x. measure M2 (Pair x -` (space M1 \<times> space M2 - A)) =

         (if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"

       by (auto intro!: M2.measure_compl simp: vimage_Diff)

     with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"

       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_ereal_diff)

   next

     fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"

     moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"

       by (intro M2.measure_countably_additive[symmetric])

          (auto simp: disjoint_family_on_def)

     ultimately show "(\<Union>i. F i) \<in> sets ?D"

       by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)

   qed

   have "sets P = sets ?D" apply (subst pair_measure_def)

   proof (intro dynkin_lemma)

     show "Int_stable E" by (rule Int_stable_pair_measure_generator)

     from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"

       by auto

     then show "sets E \<subseteq> sets ?D"

       by (auto simp: pair_measure_generator_def sets_sigma if_distrib

                intro: sigma_sets.Basic intro!: M1.measurable_If)

   qed (auto simp: pair_measure_def)

   with `Q \<in> sets P` have "Q \<in> sets ?D" by simp

   then show "?s Q \<in> borel_measurable M1" by simp

 qed

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

 locale pair_sigma_finite = pair_sigma_algebra M1 M2 + M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2

   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

 lemma (in pair_sigma_finite) measure_cut_measurable_fst:

   assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")

 proof -

   have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+

   have M1: "sigma_finite_measure M1" by default

   from M2.disjoint_sigma_finite guess F .. note F = this

   then have F_sets: "\<And>i. F i \<in> sets M2" by auto

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

   { fix i

     let ?R = "M2.restricted_space (F i)"

     have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"

       using F M2.sets_into_space by auto

     let ?R2 = "M2.restricted_space (F i)"

     have "(\<lambda>x. measure ?R2 (Pair x -` (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"

     proof (intro finite_measure_cut_measurable[OF M1])

       show "finite_measure ?R2"

         using F by (intro M2.restricted_to_finite_measure) auto

       have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i)) ` sets P"

         using `Q \<in> sets P` by (auto simp: image_iff)

       also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i)) ` sets E)"

         unfolding pair_measure_def pair_measure_generator_def sigma_def

         using `F i \<in> sets M2` M2.sets_into_space

         by (auto intro!: sigma_sets_Int sigma_sets.Basic)

       also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"

         using M1.sets_into_space

         apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def

                     intro!: sigma_sets_subseteq)

         apply (rule_tac x="a" in exI)

         apply (rule_tac x="b \<inter> F i" in exI)

         by auto

       finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .

     qed

     moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"

       using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_measure)

     ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"

       by simp }

   moreover

   { fix x

     have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"

     proof (intro M2.measure_countably_additive)

       show "range (?C x) \<subseteq> sets M2"

         using F `Q \<in> sets P` by (auto intro!: M2.Int)

       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 `Q \<in> sets P`

       by (auto simp: space_pair_measure)

     finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))"

       by simp }

   ultimately show ?thesis using `Q \<in> sets P` F_sets

     by (auto intro!: M1.borel_measurable_psuminf M2.Int)

 qed

 lemma (in pair_sigma_finite) measure_cut_measurable_snd:

   assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"

 proof -

   interpret Q: pair_sigma_finite M2 M1 by default

   note sets_pair_sigma_algebra_swap[OF assms]

   from Q.measure_cut_measurable_fst[OF this]

   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)

 qed

 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:

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

 proof -

   interpret Q: pair_sigma_algebra M2 M1 by default

   have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)

   show ?thesis

     using Q.pair_sigma_algebra_swap_measurable assms

     unfolding * by (rule measurable_comp)

 qed

 lemma (in pair_sigma_finite) pair_measure_alt:

   assumes "A \<in> sets P"

   shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x -` A) \<partial>M1)"

   apply (simp add: pair_measure_def pair_measure_generator_def)

 proof (rule M1.positive_integral_cong)

   fix x assume "x \<in> space M1"

   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: ereal)"

     unfolding indicator_def by auto

   show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)"

     unfolding *

     apply (subst M2.positive_integral_indicator)

     apply (rule measurable_cut_fst[OF assms])

     by simp

 qed

 lemma (in pair_sigma_finite) pair_measure_times:

   assumes A: "A \<in> sets M1" and "B \<in> sets M2"

   shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"

 proof -

   have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"

     using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)

   with assms show ?thesis

     by (simp add: M1.positive_integral_cmult_indicator ac_simps)

 qed

 lemma (in measure_space) measure_not_negative[simp,intro]:

   assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"

   using positive_measure[OF A] by auto

 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:

   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>

     (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"

 proof -

   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where

     F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and

     F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"

     using M1.sigma_finite_up M2.sigma_finite_up by auto

   then 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 unfolding space_pair_measure

   proof (intro exI[of _ ?F] conjI allI)

     show "range ?F \<subseteq> sets E" using F1 F2

       by (fastforce intro!: pair_measure_generatorI)

   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 E"

       using space by (auto simp: space pair_measure_generator_def)

   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 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]

     show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"

       by (simp add: pair_measure_times)

   qed

 qed

 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P

 proof

   show "positive P (measure P)"

     unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def

     by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)

   show "countably_additive P (measure P)"

     unfolding countably_additive_def

   proof (intro allI impI)

     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"

     assume F: "range F \<subseteq> sets P" "disjoint_family F"

     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto

     moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x -` F i)) \<in> borel_measurable M1"

       by (intro measure_cut_measurable_fst) 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. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"

       using F by auto

     ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"

       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]

                     M2.measure_countably_additive

                cong: M1.positive_integral_cong)

   qed

   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 P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"

   proof (rule exI[of _ F], intro conjI)

     show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)

     show "(\<Union>i. F i) = space P"

       using F by (auto simp: pair_measure_def pair_measure_generator_def)

     show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto

   qed

 qed

 lemma (in pair_sigma_algebra) sets_swap:

   assumes "A \<in> sets P"

   shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"

     (is "_ -` A \<inter> space ?Q \<in> sets ?Q")

 proof -

   have *: "(\<lambda>(x, y). (y, x)) -` A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) -` A"

     using `A \<in> sets P` sets_into_space by (auto simp: space_pair_measure)

   show ?thesis

     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)

 qed

 lemma (in pair_sigma_finite) pair_measure_alt2:

   assumes A: "A \<in> sets P"

   shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) -` A) \<partial>M2)"

     (is "_ = ?\<nu> A")

 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

   have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"

     unfolding pair_measure_def by simp

   have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` A \<inter> space Q.P)"

   proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])

     show "measure_space P" "measure_space Q.P" by default

     show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)

     show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"

       using assms unfolding pair_measure_def by auto

     show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"

       using F `A \<in> sets P` by (auto simp: pair_measure_def)

     fix X assume "X \<in> sets E"

     then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"

       unfolding pair_measure_def pair_measure_generator_def by auto

     then have "(\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P = B \<times> A"

       using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)

     then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P)"

       using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)

   qed

   then show ?thesis

     using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]

     by (auto simp add: Q.pair_measure_alt space_pair_measure

              intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])

 qed

 lemma pair_sigma_algebra_sigma:

   assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"

   assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"

   shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"

     (is "sets ?S = sets ?E")

 proof -

   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)

   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)

   have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"

     using E1 E2 by (auto simp add: pair_measure_generator_def)

   interpret E: sigma_algebra ?E unfolding pair_measure_generator_def

     using E1 E2 by (intro sigma_algebra_sigma) auto

   { fix A assume "A \<in> sets E1"

     then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"

       using E1 2 unfolding pair_measure_generator_def by auto

     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto

     also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma

       using 2 `A \<in> sets E1`

       by (intro sigma_sets.Union)

          (force simp: image_subset_iff intro!: sigma_sets.Basic)

     finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }

   moreover

   { fix B assume "B \<in> sets E2"

     then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"

       using E2 1 unfolding pair_measure_generator_def by auto

     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto

     also have "\<dots> \<in> sets ?E"

       using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma

       by (intro sigma_sets.Union)

          (force simp: image_subset_iff intro!: sigma_sets.Basic)

     finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }

   ultimately have proj:

     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"

     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)

                    (auto simp: pair_measure_generator_def sets_sigma)

   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"

     with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"

       unfolding measurable_def by simp_all

     moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"

       using A B M1.sets_into_space M2.sets_into_space

       by (auto simp: pair_measure_generator_def)

     ultimately have "A \<times> B \<in> sets ?E" by auto }

   then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"

     by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)

   then have subset: "sets ?S \<subseteq> sets ?E"

     by (simp add: sets_sigma pair_measure_generator_def)

   show "sets ?S = sets ?E"

   proof (intro set_eqI iffI)

     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"

       unfolding sets_sigma

     proof induct

       case (Basic A) then show ?case

         by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)

     qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)

   next

     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto

   qed

 qed

 section "Fubinis theorem"

 lemma (in pair_sigma_finite) simple_function_cut:

   assumes f: "simple_function P 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 P f"

 proof -

   have f_borel: "f \<in> borel_measurable P"

     using f(1) by (rule borel_measurable_simple_function)

   let ?F = "\<lambda>z. f -` {z} \<inter> space P"

   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 P" 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 (intro borel_measurable_vimage measurable_cut_fst)

     ultimately have "simple_function M2 (\<lambda> y. f (x, y))"

       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])

       apply (rule simple_function_indicator_representation[OF f(1)])

       using `x \<in> space M1` by (auto simp del: space_sigma) }

   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 P. z * M2.\<mu> (?F' x z))"

       unfolding M2.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 P)" by (rule simple_functionD)

     next

       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"

         using `x \<in> space M1` by (auto simp: space_pair_measure)

     next

       fix x' y assume "(x', y) \<in> space P"

         "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) * M2.\<mu> (?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 P"

     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)

   moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"

     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)

   moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"

     using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)

   moreover { fix i assume "i \<in> f`space P"

     with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"

       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 P f" using f(2)

     by (auto simp del: vimage_Int cong: measurable_cong

              intro!: M1.borel_measurable_ereal_setsum setsum_cong

              simp add: M1.positive_integral_setsum simple_integral_def

                        M1.positive_integral_cmult

                        M1.positive_integral_cong[OF eq]

                        positive_integral_eq_simple_integral[OF f]

                        pair_measure_alt[symmetric])

 qed

 lemma (in pair_sigma_finite) positive_integral_fst_measurable:

   assumes f: "f \<in> borel_measurable P"

   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 P 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 P"

     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_pair_image_snd[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 M2.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>P) = (SUP i. integral\<^isup>P P (F i))"

     using F_borel F

     by (intro positive_integral_monotone_convergence_SUP) auto

   also have "(SUP i. integral\<^isup>P P (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 M1.positive_integral_monotone_convergence_SUP[symmetric])

        (auto intro!: M2.positive_integral_mono M2.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!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]

              simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)

   finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"

     using F by (simp add: positive_integral_max_0)

 qed

 lemma (in pair_sigma_finite) measure_preserving_swap:

   "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"

 proof

   interpret Q: pair_sigma_finite M2 M1 by default

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

     using pair_sigma_algebra_swap_measurable .

   fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"

   from measurable_sets[OF * this] this Q.sets_into_space[OF this]

   show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) -` X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"

     by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]

       simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)

 qed

 lemma (in pair_sigma_finite) positive_integral_product_swap:

   assumes f: "f \<in> borel_measurable P"

   shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"

 proof -

   interpret Q: pair_sigma_finite M2 M1 by default

   have "sigma_algebra P" by default

   with f show ?thesis

     by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto

 qed

 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

   assumes f: "f \<in> borel_measurable P"

   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"

 proof -

   interpret Q: pair_sigma_finite M2 M1 by default

   note pair_sigma_algebra_measurable[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>Q.P)"

     by simp

   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"

     unfolding positive_integral_product_swap[OF f, symmetric]

     by (auto intro!: Q.positive_integral_cong)

   finally show ?thesis .

 qed

 lemma (in pair_sigma_finite) Fubini:

   assumes f: "f \<in> borel_measurable P"

   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 P. Q x"

   shows "AE x in M1. (AE y in M2. Q (x, y))"

 proof -

   obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"

     using assms unfolding almost_everywhere_def by auto

   show ?thesis

   proof (rule M1.AE_I)

     from N measure_cut_measurable_fst[OF `N \<in> sets P`]

     show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0"

       by (auto simp: pair_measure_alt M1.positive_integral_0_iff)

     show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1"

       by (intro M1.borel_measurable_ereal_neq_const measure_cut_measurable_fst N)

     { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0"

       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"

       proof (rule M2.AE_I)

         show "M2.\<mu> (Pair x -` N) = 0" by fact

         show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)

         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"

           using N `x \<in> space M1` unfolding space_sigma space_pair_measure by auto

       qed }

     then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0}"

       by auto

   qed

 qed

 lemma (in pair_sigma_algebra) measurable_product_swap:

   "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"

 proof -

   interpret Q: pair_sigma_algebra M2 M1 by default

   show ?thesis

     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]

     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)

 qed

 lemma (in pair_sigma_finite) integrable_product_swap:

   assumes "integrable P 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 *

     using assms unfolding integrable_def

     apply (subst (1 2) positive_integral_product_swap)

     using `integrable P f` unfolding integrable_def

     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])

 qed

 lemma (in pair_sigma_finite) integrable_product_swap_iff:

   "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P 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 "integrable P f"

   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"

 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 lebesgue_integral_def *

     apply (subst (1 2) positive_integral_product_swap)

     using `integrable P f` unfolding integrable_def

     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])

 qed

 lemma (in pair_sigma_finite) integrable_fst_measurable:

   assumes f: "integrable P f"

   shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")

     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")

 proof -

   let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"

   have

     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and

     int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?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!: M1.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: M2.positive_integral_positive)

   from AE_pos show ?AE using assms

     by (simp add: measurable_pair_image_snd 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 M2.positive_integral_positive

       by (intro M1.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 P"

       and int: "integral\<^isup>P P (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"

       and AE: "M1.almost_everywhere (\<lambda>x. (\<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!: M1.borel_measurable_real_of_ereal 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 M2.positive_integral_positive

         by (auto intro!: M1.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 M1.positive_integral_cong_pos)

            (simp add: M2.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 P] lebesgue_integral_def[of M2]

       borel[THEN positive_integral_fst_measurable(2), symmetric]

     using AE[THEN M1.integral_real]

     by simp

 qed

 lemma (in pair_sigma_finite) integrable_snd_measurable:

   assumes f: "integrable P f"

   shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")

     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")

 proof -

   interpret Q: pair_sigma_finite M2 M1 by default

   have Q_int: "integrable Q.P (\<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] by simp

   show ?AE

     using Q.integrable_fst_measurable(1)[OF Q_int]

     by simp

 qed

 lemma (in pair_sigma_finite) Fubini_integral:

   assumes f: "integrable P 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 finite spaces"

 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 \<in> Pow A \<and> b \<in> Pow 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 simp: pair_measure_generator_def intro!: sigma_sets.Basic)

     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

 locale pair_finite_sigma_algebra = pair_sigma_algebra M1 M2 + M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2

 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:

   shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"

 proof -

   show ?thesis

     using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]

     by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)

 qed

 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P

 proof

   show "finite (space P)"

     using M1.finite_space M2.finite_space

     by (subst finite_pair_sigma_algebra) simp

   show "sets P = Pow (space P)"

     by (subst (1 2) finite_pair_sigma_algebra) simp

 qed

 locale pair_finite_space = pair_sigma_finite M1 M2 + pair_finite_sigma_algebra M1 M2 +

   M1: finite_measure_space M1 + M2: finite_measure_space M2 for M1 M2

 lemma (in pair_finite_space) pair_measure_Pair[simp]:

   assumes "a \<in> space M1" "b \<in> space M2"

   shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"

 proof -

   have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"

     using M1.sets_eq_Pow M2.sets_eq_Pow assms

     by (subst pair_measure_times) auto

   then show ?thesis by simp

 qed

 lemma (in pair_finite_space) pair_measure_singleton[simp]:

   assumes "x \<in> space M1 \<times> space M2"

   shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"

   using pair_measure_Pair assms by (cases x) auto

 sublocale pair_finite_space \<subseteq> finite_measure_space P

 proof unfold_locales

   show "measure P (space P) \<noteq> \<infinity>"

     by (subst (2) finite_pair_sigma_algebra)

        (simp add: pair_measure_times)

 qed

 end