(*  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