src/HOL/Analysis/Finite_Product_Measure.thy

too agressive

(*  Title:      HOL/Analysis/Finite_Product_Measure.thy
    Author:     Johannes Hölzl, TU München
*)

section \<open>Finite Product Measure\<close>

theory Finite_Product_Measure
imports Binary_Product_Measure
begin

lemma PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
  by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1])
     (force intro: exI[of _ "restrict f I" for f])

lemma case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
  by auto

subsection%unimportant \<open>More about Function restricted by \<^const>\<open>extensional\<close>\<close>

definition
  "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"

lemma merge_apply[simp]:
  "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
  "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
  "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
  "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
  "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined"
  unfolding merge_def by auto

lemma merge_commute:
  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)"
  by (force simp: merge_def)

lemma Pi_cancel_merge_range[simp]:
  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
  by (auto simp: Pi_def)

lemma Pi_cancel_merge[simp]:
  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
  "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
  "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
  by (auto simp: Pi_def)

lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)"
  by (auto simp: extensional_def)

lemma restrict_merge[simp]:
  "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
  "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
  "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
  "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
  by (auto simp: restrict_def)

lemma split_merge: "P (merge I J (x,y) i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J - I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)"
  unfolding merge_def by auto

lemma PiE_cancel_merge[simp]:
  "I \<inter> J = {} \<Longrightarrow>
    merge I J (x, y) \<in> Pi\<^sub>E (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B"
  by (auto simp: PiE_def restrict_Pi_cancel)

lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)"
  unfolding merge_def by (auto simp: fun_eq_iff)

lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K"
  unfolding merge_def extensional_def by auto

lemma merge_restrict[simp]:
  "merge I J (restrict x I, y) = merge I J (x, y)"
  "merge I J (x, restrict y J) = merge I J (x, y)"
  unfolding merge_def by auto

lemma merge_x_x_eq_restrict[simp]:
  "merge I J (x, x) = restrict x (I \<union> J)"
  unfolding merge_def by auto

lemma injective_vimage_restrict:
  assumes J: "J \<subseteq> I"
  and sets: "A \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" and ne: "(\<Pi>\<^sub>E i\<in>I. S i) \<noteq> {}"
  and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
  shows "A = B"
proof  (intro set_eqI)
  fix x
  from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
  have "J \<inter> (I - J) = {}" by auto
  show "x \<in> A \<longleftrightarrow> x \<in> B"
  proof cases
    assume x: "x \<in> (\<Pi>\<^sub>E i\<in>J. S i)"
    have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
      using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S]
      by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
    then show "x \<in> A \<longleftrightarrow> x \<in> B"
      using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S]
      by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq)
  qed (insert sets, auto)
qed

lemma restrict_vimage:
  "I \<inter> J = {} \<Longrightarrow>
    (\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^sub>E I E \<times> Pi\<^sub>E J F) = Pi (I \<union> J) (merge I J (E, F))"
  by (auto simp: restrict_Pi_cancel PiE_def)

lemma merge_vimage:
  "I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^sub>E (I \<union> J) E = Pi I E \<times> Pi J E"
  by (auto simp: restrict_Pi_cancel PiE_def)

subsection \<open>Finite product spaces\<close>

definition%important prod_emb where
  "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"

lemma prod_emb_iff:
  "f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))"
  unfolding%unimportant prod_emb_def PiE_def by auto

lemma
  shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
    and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
    and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
    and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))"
    and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
    and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
  by (auto simp: prod_emb_def)

lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
    prod_emb I M J (\<Pi>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
  by (force simp: prod_emb_def PiE_iff if_split_mem2)

lemma prod_emb_PiE_same_index[simp]:
    "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E"
  by (auto simp: prod_emb_def PiE_iff)

lemma prod_emb_trans[simp]:
  "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
  by (auto simp add: Int_absorb1 prod_emb_def PiE_def)

lemma prod_emb_Pi:
  assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
  shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
  using assms sets.space_closed
  by (auto simp: prod_emb_def PiE_iff split: if_split_asm) blast+

lemma prod_emb_id:
  "B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
  by (auto simp: prod_emb_def subset_eq extensional_restrict)

lemma prod_emb_mono:
  "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
  by (auto simp: prod_emb_def)

definition%important PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
  "PiM I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i))
    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j))
    (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"

definition%important prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
  "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) `
    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"

abbreviation
  "Pi\<^sub>M I M \<equiv> PiM I M"

syntax
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^sub>M _\<in>_./ _)"  10)
translations
  "\<Pi>\<^sub>M x\<in>I. M" == "CONST PiM I (%x. M)"

lemma extend_measure_cong:
  assumes "\<Omega> = \<Omega>'" "I = I'" "G = G'" "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i"
  shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'"
  unfolding extend_measure_def by (auto simp add: assms)

lemma Pi_cong_sets:
    "\<lbrakk>I = J; \<And>x. x \<in> I \<Longrightarrow> M x = N x\<rbrakk> \<Longrightarrow> Pi I M = Pi J N"
  unfolding Pi_def by auto

lemma PiM_cong:
  assumes "I = J" "\<And>x. x \<in> I \<Longrightarrow> M x = N x"
  shows "PiM I M = PiM J N"
  unfolding PiM_def
proof (rule extend_measure_cong, goal_cases)
  case 1
  show ?case using assms
    by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all
next
  case 2
  have "\<And>K. K \<subseteq> J \<Longrightarrow> (\<Pi> j\<in>K. sets (M j)) = (\<Pi> j\<in>K. sets (N j))"
    using assms by (intro Pi_cong_sets) auto
  thus ?case by (auto simp: assms)
next
  case 3
  show ?case using assms
    by (intro ext) (auto simp: prod_emb_def dest: PiE_mem)
next
  case (4 x)
  thus ?case using assms
    by (auto intro!: prod.cong split: if_split_asm)
qed


lemma prod_algebra_sets_into_space:
  "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
  by (auto simp: prod_emb_def prod_algebra_def)

lemma prod_algebra_eq_finite:
  assumes I: "finite I"
  shows "prod_algebra I M = {(\<Pi>\<^sub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
proof (intro iffI set_eqI)
  fix A assume "A \<in> ?L"
  then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
    and A: "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
    by (auto simp: prod_algebra_def)
  let ?A = "\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i)"
  have A: "A = ?A"
    unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto
  show "A \<in> ?R" unfolding A using J sets.top
    by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
next
  fix A assume "A \<in> ?R"
  then obtain X where A: "A = (\<Pi>\<^sub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
  then have A: "A = prod_emb I M I (\<Pi>\<^sub>E i\<in>I. X i)"
    by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff)
  from X I show "A \<in> ?L" unfolding A
    by (auto simp: prod_algebra_def)
qed

lemma prod_algebraI:
  "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
    \<Longrightarrow> prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> prod_algebra I M"
  by (auto simp: prod_algebra_def)

lemma prod_algebraI_finite:
  "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M"
  using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp

lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
proof (safe intro!: Int_stableI)
  fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
  then show "\<exists>G. Pi\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
    by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
qed

lemma prod_algebraE:
  assumes A: "A \<in> prod_algebra I M"
  obtains J E where "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
    "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
  using A by (auto simp: prod_algebra_def)

lemma prod_algebraE_all:
  assumes A: "A \<in> prod_algebra I M"
  obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
proof -
  from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)"
    and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
    by (auto simp: prod_algebra_def)
  from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
    using sets.sets_into_space by auto
  then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))"
    using A J by (auto simp: prod_emb_PiE)
  moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
    using sets.top E by auto
  ultimately show ?thesis using that by auto
qed

lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
proof (unfold Int_stable_def, safe)
  fix A assume "A \<in> prod_algebra I M"
  from prod_algebraE[OF this] guess J E . note A = this
  fix B assume "B \<in> prod_algebra I M"
  from prod_algebraE[OF this] guess K F . note B = this
  have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^sub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter>
      (if i \<in> K then F i else space (M i)))"
    unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4)
      B(5)[THEN sets.sets_into_space]
    apply (subst (1 2 3) prod_emb_PiE)
    apply (simp_all add: subset_eq PiE_Int)
    apply blast
    apply (intro PiE_cong)
    apply auto
    done
  also have "\<dots> \<in> prod_algebra I M"
    using A B by (auto intro!: prod_algebraI)
  finally show "A \<inter> B \<in> prod_algebra I M" .
qed

proposition prod_algebra_mono:
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
proof
  fix A assume "A \<in> prod_algebra I E"
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
    and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
    by (auto simp: prod_algebra_def)
  moreover
  from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
    by (rule PiE_cong)
  with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
    by (simp add: prod_emb_def)
  moreover
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
    by auto
  ultimately show "A \<in> prod_algebra I F"
    apply (simp add: prod_algebra_def image_iff)
    apply (intro exI[of _ J] exI[of _ G] conjI)
    apply auto
    done
qed
proposition prod_algebra_cong:
  assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
  shows "prod_algebra I M = prod_algebra J N"
proof -
  have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
    using sets_eq_imp_space_eq[OF sets] by auto
  with sets show ?thesis unfolding \<open>I = J\<close>
    by (intro antisym prod_algebra_mono) auto
qed

lemma space_in_prod_algebra:
  "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
proof cases
  assume "I = {}" then show ?thesis
    by (auto simp add: prod_algebra_def image_iff prod_emb_def)
next
  assume "I \<noteq> {}"
  then obtain i where "i \<in> I" by auto
  then have "(\<Pi>\<^sub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i))"
    by (auto simp: prod_emb_def)
  also have "\<dots> \<in> prod_algebra I M"
    using \<open>i \<in> I\<close> by (intro prod_algebraI) auto
  finally show ?thesis .
qed

lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp

lemma prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)"
  by (auto simp: prod_emb_def space_PiM)

lemma space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow>  (\<exists>i\<in>I. space (M i) = {})"
  by (auto simp: space_PiM PiE_eq_empty_iff)

lemma undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)"
  by (auto simp: space_PiM)

lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp

proposition sets_PiM_single: "sets (PiM I M) =
    sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
    (is "_ = sigma_sets ?\<Omega> ?R")
  unfolding sets_PiM
proof (rule sigma_sets_eqI)
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
  fix A assume "A \<in> prod_algebra I M"
  from prod_algebraE[OF this] guess J X . note X = this
  show "A \<in> sigma_sets ?\<Omega> ?R"
  proof cases
    assume "I = {}"
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
    with \<open>I = {}\<close> show ?thesis by (auto intro!: sigma_sets_top)
  next
    assume "I \<noteq> {}"
    with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
      by (auto simp: prod_emb_def)
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
      using X \<open>I \<noteq> {}\<close> by (intro R.finite_INT sigma_sets.Basic) auto
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
  qed
next
  fix A assume "A \<in> ?R"
  then obtain i B where A: "A = {f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
    by auto
  then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
     by (auto simp: prod_emb_def)
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
    using A by (intro sigma_sets.Basic prod_algebraI) auto
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
qed

lemma sets_PiM_eq_proj:
  "I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
  apply (simp add: sets_PiM_single)
  apply (subst sets_Sup_eq[where X="\<Pi>\<^sub>E i\<in>I. space (M i)"])
  apply auto []
  apply auto []
  apply simp
  apply (subst arg_cong [of _ _ Sup, OF image_cong, OF refl])
  apply (rule sets_vimage_algebra2)
  apply (auto intro!: arg_cong2[where f=sigma_sets])
  done

lemma
  shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
    and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)

proposition sets_PiM_sigma:
  assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S"
  assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)"
  assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I"
  defines "P \<equiv> {{f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i} | A j. j \<in> J \<and> A \<in> Pi j E}"
  shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
proof cases
  assume "I = {}"
  with \<open>\<Union>J = I\<close> have "P = {{\<lambda>_. undefined}} \<or> P = {}"
    by (auto simp: P_def)
  with \<open>I = {}\<close> show ?thesis
    by (auto simp add: sets_PiM_empty sigma_sets_empty_eq)
next
  let ?F = "\<lambda>i. {(\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega> |A. A \<in> E i}"
  assume "I \<noteq> {}"
  then have "sets (Pi\<^sub>M I (\<lambda>i. sigma (\<Omega> i) (E i))) =
      sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<lambda>x. x i) (sigma (\<Omega> i) (E i)))"
    by (subst sets_PiM_eq_proj) (auto simp: space_measure_of_conv)
  also have "\<dots> = sets (SUP i\<in>I. sigma (Pi\<^sub>E I \<Omega>) (?F i))"
    using E by (intro sets_SUP_cong arg_cong[where f=sets] vimage_algebra_sigma) auto
  also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i))"
    using \<open>I \<noteq> {}\<close> by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto
  also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) P)"
  proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
    show "(\<Union>i\<in>I. ?F i) \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" "P \<subseteq> Pow (Pi\<^sub>E I \<Omega>)"
      by (auto simp: P_def)
  next
    interpret P: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
      by (auto intro!: sigma_algebra_sigma_sets simp: P_def)

    fix Z assume "Z \<in> (\<Union>i\<in>I. ?F i)"
    then obtain i A where i: "i \<in> I" "A \<in> E i" and Z_def: "Z = (\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega>"
      by auto
    from \<open>i \<in> I\<close> J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j"
      by auto
    obtain S where S: "\<And>i. i \<in> j \<Longrightarrow> S i \<subseteq> E i" "\<And>i. i \<in> j \<Longrightarrow> countable (S i)"
      "\<And>i. i \<in> j \<Longrightarrow> \<Omega> i = \<Union>(S i)"
      by (metis subset_eq \<Omega>_cover \<open>j \<subseteq> I\<close>)
    define A' where "A' n = n(i := A)" for n
    then have A'_i: "\<And>n. A' n i = A"
      by simp
    { fix n assume "n \<in> Pi\<^sub>E (j - {i}) S"
      then have "A' n \<in> Pi j E"
        unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def \<open>A \<in> E i\<close> )
      with \<open>j \<in> J\<close> have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P"
        by (auto simp: P_def) }
    note A'_in_P = this

    { fix x assume "x i \<in> A" "x \<in> Pi\<^sub>E I \<Omega>"
      with S(3) \<open>j \<subseteq> I\<close> have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s"
        by (auto simp: PiE_def Pi_def)
      then obtain s where s: "\<And>i. i \<in> j \<Longrightarrow> s i \<in> S i" "\<And>i. i \<in> j \<Longrightarrow> x i \<in> s i"
        by metis
      with \<open>x i \<in> A\<close> have "\<exists>n\<in>Pi\<^sub>E (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i"
        by (intro bexI[of _ "restrict (s(i := A)) (j-{i})"]) (auto simp: A'_def split: if_splits) }
    then have "Z = (\<Union>n\<in>Pi\<^sub>E (j-{i}) S. {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A' n i})"
      unfolding Z_def
      by (auto simp add: set_eq_iff ball_conj_distrib \<open>i\<in>j\<close> A'_i dest: bspec[OF _ \<open>i\<in>j\<close>]
               cong: conj_cong)
    also have "\<dots> \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
      using \<open>finite j\<close> S(2)
      by (intro P.countable_UN' countable_PiE) (simp_all add: image_subset_iff A'_in_P)
    finally show "Z \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" .
  next
    interpret F: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<Union>i\<in>I. ?F i)"
      by (auto intro!: sigma_algebra_sigma_sets)

    fix b assume "b \<in> P"
    then obtain A j where b: "b = {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i}" "j \<in> J" "A \<in> Pi j E"
      by (auto simp: P_def)
    show "b \<in> sigma_sets (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i)"
    proof cases
      assume "j = {}"
      with b have "b = (\<Pi>\<^sub>E i\<in>I. \<Omega> i)"
        by auto
      then show ?thesis
        by blast
    next
      assume "j \<noteq> {}"
      with J b(2,3) have eq: "b = (\<Inter>i\<in>j. ((\<lambda>x. x i) -` A i \<inter> Pi\<^sub>E I \<Omega>))"
        unfolding b(1)
        by (auto simp: PiE_def Pi_def)
      show ?thesis
        unfolding eq using \<open>A \<in> Pi j E\<close> \<open>j \<in> J\<close> J(2)
        by (intro F.finite_INT J \<open>j \<in> J\<close> \<open>j \<noteq> {}\<close> sigma_sets.Basic) blast
    qed
  qed
  finally show "?thesis" .
qed

lemma sets_PiM_in_sets:
  assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
  assumes sets: "\<And>i A. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {x\<in>space N. x i \<in> A} \<in> sets N"
  shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N"
  unfolding sets_PiM_single space[symmetric]
  by (intro sets.sigma_sets_subset subsetI) (auto intro: sets)

lemma sets_PiM_cong[measurable_cong]:
  assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)"
  using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong)

lemma sets_PiM_I:
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
  shows "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
proof cases
  assume "J = {}"
  then have "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) = (\<Pi>\<^sub>E j\<in>I. space (M j))"
    by (auto simp: prod_emb_def)
  then show ?thesis
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
next
  assume "J \<noteq> {}" with assms show ?thesis
    by (force simp add: sets_PiM prod_algebra_def)
qed

proposition measurable_PiM:
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
  assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
    f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
  shows "f \<in> measurable N (PiM I M)"
  using sets_PiM prod_algebra_sets_into_space space
proof (rule measurable_sigma_sets)
  fix A assume "A \<in> prod_algebra I M"
  from prod_algebraE[OF this] guess J X .
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
qed

lemma measurable_PiM_Collect:
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
  assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
  shows "f \<in> measurable N (PiM I M)"
  using sets_PiM prod_algebra_sets_into_space space
proof (rule measurable_sigma_sets)
  fix A assume "A \<in> prod_algebra I M"
  from prod_algebraE[OF this] guess J X . note X = this
  then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
    using space by (auto simp: prod_emb_def del: PiE_I)
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
  finally show "f -` A \<inter> space N \<in> sets N" .
qed

lemma measurable_PiM_single:
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
  assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N"
  shows "f \<in> measurable N (PiM I M)"
  using sets_PiM_single
proof (rule measurable_sigma_sets)
  fix A assume "A \<in> {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
  then obtain B i where "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
    by auto
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
  also have "\<dots> \<in> sets N" using B by (rule sets)
  finally show "f -` A \<inter> space N \<in> sets N" .
qed (auto simp: space)

lemma measurable_PiM_single':
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
    and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
  shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
proof (rule measurable_PiM_single)
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
  then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
    by auto
  then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
    using A f by (auto intro!: measurable_sets)
qed fact

lemma sets_PiM_I_finite[measurable]:
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
  shows "(\<Pi>\<^sub>E j\<in>I. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
  using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] \<open>finite I\<close> sets by auto

lemma measurable_component_singleton[measurable (raw)]:
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
proof (unfold measurable_def, intro CollectI conjI ballI)
  fix A assume "A \<in> sets (M i)"
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. A)"
    using sets.sets_into_space \<open>i \<in> I\<close>
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: if_split_asm)
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
    using \<open>A \<in> sets (M i)\<close> \<open>i \<in> I\<close> by (auto intro!: sets_PiM_I)
qed (insert \<open>i \<in> I\<close>, auto simp: space_PiM)

lemma measurable_component_singleton'[measurable_dest]:
  assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
  assumes g: "g \<in> measurable L N"
  assumes i: "i \<in> I"
  shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)"
  using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] .

lemma measurable_PiM_component_rev:
  "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
  by simp

lemma measurable_case_nat[measurable (raw)]:
  assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
    "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
  shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
  by (cases i) simp_all

lemma measurable_case_nat'[measurable (raw)]:
  assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
  shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
  using fg[THEN measurable_space]
  by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)

lemma measurable_add_dim[measurable]:
  "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
    (is "?f \<in> measurable ?P ?I")
proof (rule measurable_PiM_single)
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
    (if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
    using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
  also have "\<dots> \<in> sets ?P"
    using A j
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
  finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_def)

proposition measurable_fun_upd:
  assumes I: "I = J \<union> {i}"
  assumes f[measurable]: "f \<in> measurable N (PiM J M)"
  assumes h[measurable]: "h \<in> measurable N (M i)"
  shows "(\<lambda>x. (f x) (i := h x)) \<in> measurable N (PiM I M)"
proof (intro measurable_PiM_single')
  fix j assume "j \<in> I" then show "(\<lambda>\<omega>. ((f \<omega>)(i := h \<omega>)) j) \<in> measurable N (M j)"
    unfolding I by (cases "j = i") auto
next
  show "(\<lambda>x. (f x)(i := h x)) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
    using I f[THEN measurable_space] h[THEN measurable_space]
    by (auto simp: space_PiM PiE_iff extensional_def)
qed

lemma measurable_component_update:
  "x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)"
  by simp

lemma measurable_merge[measurable]:
  "merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)"
    (is "?f \<in> measurable ?P ?U")
proof (rule measurable_PiM_single)
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
    (if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)"
    by (auto simp: merge_def)
  also have "\<dots> \<in> sets ?P"
    using A
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)

lemma measurable_restrict[measurable (raw)]:
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
proof (rule measurable_PiM_single)
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
    by auto
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
    using A X by (auto intro!: measurable_sets)
qed (insert X, auto simp add: PiE_def dest: measurable_space)

lemma measurable_abs_UNIV:
  "(\<And>n. (\<lambda>\<omega>. f n \<omega>) \<in> measurable M (N n)) \<Longrightarrow> (\<lambda>\<omega> n. f n \<omega>) \<in> measurable M (PiM UNIV N)"
  by (intro measurable_PiM_single) (auto dest: measurable_space)

lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
  by (intro measurable_restrict measurable_component_singleton) auto

lemma measurable_restrict_subset':
  assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)"
  shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
proof-
  from assms(1) have "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
    by (rule measurable_restrict_subset)
  also from assms(2) have "measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M) = measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
    by (intro sets_PiM_cong measurable_cong_sets) simp_all
  finally show ?thesis .
qed

lemma measurable_prod_emb[intro, simp]:
  "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
  unfolding prod_emb_def space_PiM[symmetric]
  by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)

lemma merge_in_prod_emb:
  assumes "y \<in> space (PiM I M)" "x \<in> X" and X: "X \<in> sets (Pi\<^sub>M J M)" and "J \<subseteq> I"
  shows "merge J I (x, y) \<in> prod_emb I M J X"
  using assms sets.sets_into_space[OF X]
  by (simp add: merge_def prod_emb_def subset_eq space_PiM PiE_def extensional_restrict Pi_iff
           cong: if_cong restrict_cong)
     (simp add: extensional_def)

lemma prod_emb_eq_emptyD:
  assumes J: "J \<subseteq> I" and ne: "space (PiM I M) \<noteq> {}" and X: "X \<in> sets (Pi\<^sub>M J M)"
    and *: "prod_emb I M J X = {}"
  shows "X = {}"
proof safe
  fix x assume "x \<in> X"
  obtain \<omega> where "\<omega> \<in> space (PiM I M)"
    using ne by blast
  from merge_in_prod_emb[OF this \<open>x\<in>X\<close> X J] * show "x \<in> {}" by auto
qed

lemma sets_in_Pi_aux:
  "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
  {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
  by (simp add: subset_eq Pi_iff)

lemma sets_in_Pi[measurable (raw)]:
  "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
  (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
  Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
  unfolding pred_def
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto

lemma sets_in_extensional_aux:
  "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
proof -
  have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
    by (auto simp add: extensional_def space_PiM)
  then show ?thesis by simp
qed

lemma sets_in_extensional[measurable (raw)]:
  "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
  unfolding pred_def
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto

lemma sets_PiM_I_countable:
  assumes I: "countable I" and E: "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
proof cases
  assume "I \<noteq> {}"
  then have "Pi\<^sub>E I E = (\<Inter>i\<in>I. prod_emb I M {i} (Pi\<^sub>E {i} E))"
    using E[THEN sets.sets_into_space] by (auto simp: PiE_iff prod_emb_def fun_eq_iff)
  also have "\<dots> \<in> sets (PiM I M)"
    using I \<open>I \<noteq> {}\<close> by (safe intro!: sets.countable_INT' measurable_prod_emb sets_PiM_I_finite E)
  finally show ?thesis .
qed (simp add: sets_PiM_empty)

lemma sets_PiM_D_countable:
  assumes A: "A \<in> PiM I M"
  shows "\<exists>J\<subseteq>I. \<exists>X\<in>PiM J M. countable J \<and> A = prod_emb I M J X"
  using A[unfolded sets_PiM_single]
proof induction
  case (Basic A)
  then obtain i X where *: "i \<in> I" "X \<in> sets (M i)" and "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> X}"
    by auto
  then have A: "A = prod_emb I M {i} (\<Pi>\<^sub>E _\<in>{i}. X)"
    by (auto simp: prod_emb_def)
  then show ?case
    by (intro exI[of _ "{i}"] conjI bexI[of _ "\<Pi>\<^sub>E _\<in>{i}. X"])
       (auto intro: countable_finite * sets_PiM_I_finite)
next
  case Empty then show ?case
    by (intro exI[of _ "{}"] conjI bexI[of _ "{}"]) auto
next
  case (Compl A)
  then obtain J X where "J \<subseteq> I" "X \<in> sets (Pi\<^sub>M J M)" "countable J" "A = prod_emb I M J X"
    by auto
  then show ?case
    by (intro exI[of _ J] bexI[of _ "space (PiM J M) - X"] conjI)
       (auto simp add: space_PiM prod_emb_PiE intro!: sets_PiM_I_countable)
next
  case (Union K)
  obtain J X where J: "\<And>i. J i \<subseteq> I" "\<And>i. countable (J i)" and X: "\<And>i. X i \<in> sets (Pi\<^sub>M (J i) M)"
    and K: "\<And>i. K i = prod_emb I M (J i) (X i)"
    by (metis Union.IH)
  show ?case
  proof (intro exI[of _ "\<Union>i. J i"] bexI[of _ "\<Union>i. prod_emb (\<Union>i. J i) M (J i) (X i)"] conjI)
    show "(\<Union>i. J i) \<subseteq> I" "countable (\<Union>i. J i)" using J by auto
    with J show "\<Union>(K ` UNIV) = prod_emb I M (\<Union>i. J i) (\<Union>i. prod_emb (\<Union>i. J i) M (J i) (X i))"
      by (simp add: K[abs_def] SUP_upper)
  qed(auto intro: X)
qed

proposition measure_eqI_PiM_finite:
  assumes [simp]: "finite I" "sets P = PiM I M" "sets Q = PiM I M"
  assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = Q (Pi\<^sub>E I A)"
  assumes A: "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = space (PiM I M)" "\<And>i::nat. P (A i) \<noteq> \<infinity>"
  shows "P = Q"
proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
  show "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. P (A i) \<noteq> \<infinity>"
    unfolding space_PiM[symmetric] by fact+
  fix X assume "X \<in> prod_algebra I M"
  then obtain J E where X: "X = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
    and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)"
    by (force elim!: prod_algebraE)
  then show "emeasure P X = emeasure Q X"
    unfolding X by (subst (1 2) prod_emb_Pi) (auto simp: eq)
qed (simp_all add: sets_PiM)

proposition measure_eqI_PiM_infinite:
  assumes [simp]: "sets P = PiM I M" "sets Q = PiM I M"
  assumes eq: "\<And>A J. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow>
    P (prod_emb I M J (Pi\<^sub>E J A)) = Q (prod_emb I M J (Pi\<^sub>E J A))"
  assumes A: "finite_measure P"
  shows "P = Q"
proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
  interpret finite_measure P by fact
  define i where "i = (SOME i. i \<in> I)"
  have i: "I \<noteq> {} \<Longrightarrow> i \<in> I"
    unfolding i_def by (rule someI_ex) auto
  define A where "A n =
    (if I = {} then prod_emb I M {} (\<Pi>\<^sub>E i\<in>{}. {}) else prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i)))"
    for n :: nat
  then show "range A \<subseteq> prod_algebra I M"
    using prod_algebraI[of "{}" I "\<lambda>i. space (M i)" M] by (auto intro!: prod_algebraI i)
  have "\<And>i. A i = space (PiM I M)"
    by (auto simp: prod_emb_def space_PiM PiE_iff A_def i ex_in_conv[symmetric] exI)
  then show "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. emeasure P (A i) \<noteq> \<infinity>"
    by (auto simp: space_PiM)
next
  fix X assume X: "X \<in> prod_algebra I M"
  then obtain J E where X: "X = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
    and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)"
    by (force elim!: prod_algebraE)
  then show "emeasure P X = emeasure Q X"
    by (auto intro!: eq)
qed (auto simp: sets_PiM)

locale%unimportant product_sigma_finite =
  fixes M :: "'i \<Rightarrow> 'a measure"
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"

sublocale%unimportant product_sigma_finite \<subseteq> M?: sigma_finite_measure "M i" for i
  by (rule sigma_finite_measures)

locale%unimportant finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
  fixes I :: "'i set"
  assumes finite_index: "finite I"

proposition (in finite_product_sigma_finite) sigma_finite_pairs:
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
    (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and>
    (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
proof -
  have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)"
    using M.sigma_finite_incseq by metis
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
  then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. emeasure (M i) (F i k) \<noteq> \<infinity>"
    by auto
  let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
  note space_PiM[simp]
  show ?thesis
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
  next
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
  next
    fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
      by (auto simp: PiE_def dest!: sets.sets_into_space)
  next
    fix f assume "f \<in> space (PiM I M)"
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
  next
    fix i show "?F i \<subseteq> ?F (Suc i)"
      using \<open>\<And>i. incseq (F i)\<close>[THEN incseq_SucD] by auto
  qed
qed

lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
proof -
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ennreal)"
  have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
    show "positive (PiM {} M) ?\<mu>"
      by (auto simp: positive_def)
    show "countably_additive (PiM {} M) ?\<mu>"
      by (rule sets.countably_additiveI_finite)
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
  qed (auto simp: prod_emb_def)
  also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
    by (auto simp: prod_emb_def)
  finally show ?thesis
    by simp
qed

lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
  by (rule measure_eqI) (auto simp add: sets_PiM_empty)

lemma (in product_sigma_finite) emeasure_PiM:
  "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
proof (induct I arbitrary: A rule: finite_induct)
  case (insert i I)
  interpret finite_product_sigma_finite M I by standard fact
  have "finite (insert i I)" using \<open>finite I\<close> by auto
  interpret I': finite_product_sigma_finite M "insert i I" by standard fact
  let ?h = "(\<lambda>(f, y). f(i := y))"

  let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
  let ?\<mu> = "emeasure ?P"
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
  let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"

  have "emeasure (Pi\<^sub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^sub>E (insert i I) A)) =
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
    fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))"
    then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto
    let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
    have "?\<mu> ?p =
      emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i))"
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
    also have "?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
      using J E[rule_format, THEN sets.sets_into_space]
      by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: if_split_asm)
    also have "emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
      emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
    also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
      using J E[rule_format, THEN sets.sets_into_space]
      by (auto simp: prod_emb_iff PiE_def Pi_iff split: if_split_asm) blast+
    also have "emeasure (Pi\<^sub>M I M) (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
      using E by (subst insert) (auto intro!: prod.cong)
    also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
       emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
      using insert by (auto simp: mult.commute intro!: arg_cong2[where f="(*)"] prod.cong)
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
      using insert(1,2) J E by (intro prod.mono_neutral_right) auto
    finally show "?\<mu> ?p = \<dots>" .

    show "prod_emb (insert i I) M J (Pi\<^sub>E J E) \<in> Pow (\<Pi>\<^sub>E i\<in>insert i I. space (M i))"
      using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
  next
    show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
  next
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
      using insert by auto
  qed (auto intro!: prod.cong)
  with insert show ?case
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
qed simp

lemma (in product_sigma_finite) PiM_eqI:
  assumes I[simp]: "finite I" and P: "sets P = PiM I M"
  assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
  shows "P = PiM I M"
proof -
  interpret finite_product_sigma_finite M I
    proof qed fact
  from sigma_finite_pairs guess C .. note C = this
  show ?thesis
  proof (rule measure_eqI_PiM_finite[OF I refl P, symmetric])
    show "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>M I M) (Pi\<^sub>E I A) = P (Pi\<^sub>E I A)" for A
      by (simp add: eq emeasure_PiM)
    define A where "A n = (\<Pi>\<^sub>E i\<in>I. C i n)" for n
    with C show "range A \<subseteq> prod_algebra I M" "\<And>i. emeasure (Pi\<^sub>M I M) (A i) \<noteq> \<infinity>" "(\<Union>i. A i) = space (PiM I M)"
      by (auto intro!: prod_algebraI_finite simp: emeasure_PiM subset_eq ennreal_prod_eq_top)
  qed
qed

lemma (in product_sigma_finite) sigma_finite:
  assumes "finite I"
  shows "sigma_finite_measure (PiM I M)"
proof
  interpret finite_product_sigma_finite M I by standard fact

  obtain F where F: "\<And>j. countable (F j)" "\<And>j f. f \<in> F j \<Longrightarrow> f \<in> sets (M j)"
    "\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and
    in_space: "\<And>j. space (M j) = \<Union>(F j)"
    using sigma_finite_countable by (metis subset_eq)
  moreover have "(\<Union>(Pi\<^sub>E I ` Pi\<^sub>E I F)) = space (Pi\<^sub>M I M)"
    using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD2])
  ultimately show "\<exists>A. countable A \<and> A \<subseteq> sets (Pi\<^sub>M I M) \<and> \<Union>A = space (Pi\<^sub>M I M) \<and> (\<forall>a\<in>A. emeasure (Pi\<^sub>M I M) a \<noteq> \<infinity>)"
    by (intro exI[of _ "Pi\<^sub>E I ` Pi\<^sub>E I F"])
       (auto intro!: countable_PiE sets_PiM_I_finite
             simp: PiE_iff emeasure_PiM finite_index ennreal_prod_eq_top)
qed

sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
  using sigma_finite[OF finite_index] .

lemma (in finite_product_sigma_finite) measure_times:
  "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
  using emeasure_PiM[OF finite_index] by auto

lemma (in product_sigma_finite) nn_integral_empty:
  "0 \<le> f (\<lambda>k. undefined) \<Longrightarrow> integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
  by (simp add: PiM_empty nn_integral_count_space_finite max.absorb2)

lemma%important (in product_sigma_finite) distr_merge:
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
  shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
   (is "?D = ?P")
proof (rule PiM_eqI)
  interpret I: finite_product_sigma_finite M I by standard fact
  interpret J: finite_product_sigma_finite M J by standard fact
  fix A assume A: "\<And>i. i \<in> I \<union> J \<Longrightarrow> A i \<in> sets (M i)"
  have *: "(merge I J -` Pi\<^sub>E (I \<union> J) A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)) = Pi\<^sub>E I A \<times> Pi\<^sub>E J A"
    using A[THEN sets.sets_into_space] by (auto simp: space_PiM space_pair_measure)
  from A fin show "emeasure (distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J)) (Pi\<^sub>E (I \<union> J) A) =
      (\<Prod>i\<in>I \<union> J. emeasure (M i) (A i))"
    by (subst emeasure_distr)
       (auto simp: * J.emeasure_pair_measure_Times I.measure_times J.measure_times prod.union_disjoint)
qed (insert fin, simp_all)

proposition (in product_sigma_finite) product_nn_integral_fold:
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
  and f[measurable]: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
  shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
    (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
proof -
  interpret I: finite_product_sigma_finite M I by standard fact
  interpret J: finite_product_sigma_finite M J by standard fact
  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by standard
  have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
  show ?thesis
    apply (subst distr_merge[OF IJ, symmetric])
    apply (subst nn_integral_distr[OF measurable_merge])
    apply measurable []
    apply (subst J.nn_integral_fst[symmetric, OF P_borel])
    apply simp
    done
qed

lemma (in product_sigma_finite) distr_singleton:
  "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
proof (intro measure_eqI[symmetric])
  interpret I: finite_product_sigma_finite M "{i}" by standard simp
  fix A assume A: "A \<in> sets (M i)"
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
    using sets.sets_into_space by (auto simp: space_PiM)
  then show "emeasure (M i) A = emeasure ?D A"
    using A I.measure_times[of "\<lambda>_. A"]
    by (simp add: emeasure_distr measurable_component_singleton)
qed simp

lemma (in product_sigma_finite) product_nn_integral_singleton:
  assumes f: "f \<in> borel_measurable (M i)"
  shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f"
proof -
  interpret I: finite_product_sigma_finite M "{i}" by standard simp
  from f show ?thesis
    apply (subst distr_singleton[symmetric])
    apply (subst nn_integral_distr[OF measurable_component_singleton])
    apply simp_all
    done
qed

proposition (in product_sigma_finite) product_nn_integral_insert:
  assumes I[simp]: "finite I" "i \<notin> I"
    and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
  shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
proof -
  interpret I: finite_product_sigma_finite M I by standard auto
  interpret i: finite_product_sigma_finite M "{i}" by standard auto
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
    using f by auto
  show ?thesis
    unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
  proof (rule nn_integral_cong, subst product_nn_integral_singleton[symmetric])
    fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
    let ?f = "\<lambda>y. f (x(i := y))"
    show "?f \<in> borel_measurable (M i)"
      using measurable_comp[OF measurable_component_update f, OF x \<open>i \<notin> I\<close>]
      unfolding comp_def .
    show "(\<integral>\<^sup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^sub>M {i} M) = (\<integral>\<^sup>+ y. f (x(i := y i)) \<partial>Pi\<^sub>M {i} M)"
      using x
      by (auto intro!: nn_integral_cong arg_cong[where f=f]
               simp add: space_PiM extensional_def PiE_def)
  qed
qed

lemma (in product_sigma_finite) product_nn_integral_insert_rev:
  assumes I[simp]: "finite I" "i \<notin> I"
    and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
  shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x(i := y)) \<partial>(Pi\<^sub>M I M)) \<partial>(M i))"
  apply (subst product_nn_integral_insert[OF assms])
  apply (rule pair_sigma_finite.Fubini')
  apply intro_locales []
  apply (rule sigma_finite[OF I(1)])
  apply measurable
  done

lemma (in product_sigma_finite) product_nn_integral_prod:
  assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
  shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
using assms proof (induction I)
  case (insert i I)
  note insert.prems[measurable]
  note \<open>finite I\<close>[intro, simp]
  interpret I: finite_product_sigma_finite M I by standard auto
  have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
    using insert by (auto intro!: prod.cong)
  have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^sub>M J M)"
    using sets.sets_into_space insert
    by (intro borel_measurable_prod_ennreal
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
       auto
  then show ?case
    apply (simp add: product_nn_integral_insert[OF insert(1,2)])
    apply (simp add: insert(2-) * nn_integral_multc)
    apply (subst nn_integral_cmult)
    apply (auto simp add: insert(2-))
    done
qed (simp add: space_PiM)

proposition (in product_sigma_finite) product_nn_integral_pair:
  assumes [measurable]: "case_prod f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
  assumes xy: "x \<noteq> y"
  shows "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {x, y} M) = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
proof -
  interpret psm: pair_sigma_finite "M x" "M y"
    unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all
  have "{x, y} = {y, x}" by auto
  also have "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {y, x} M) = (\<integral>\<^sup>+y. \<integral>\<^sup>+\<sigma>. f (\<sigma> x) y \<partial>PiM {x} M \<partial>M y)"
    using xy by (subst product_nn_integral_insert_rev) simp_all
  also have "... = (\<integral>\<^sup>+y. \<integral>\<^sup>+x. f x y \<partial>M x \<partial>M y)"
    by (intro nn_integral_cong, subst product_nn_integral_singleton) simp_all
  also have "... = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
    by (subst psm.nn_integral_snd[symmetric]) simp_all
  finally show ?thesis .
qed

lemma (in product_sigma_finite) distr_component:
  "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
proof (intro PiM_eqI)
  fix A assume A: "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)"
  then have "(\<lambda>x. \<lambda>i\<in>{i}. x) -` Pi\<^sub>E {i} A \<inter> space (M i) = A i"
    by (fastforce dest: sets.sets_into_space)
  with A show "emeasure (distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x)) (Pi\<^sub>E {i} A) = (\<Prod>i\<in>{i}. emeasure (M i) (A i))"
    by (subst emeasure_distr) (auto intro!: sets_PiM_I_finite measurable_restrict)
qed simp_all

lemma (in product_sigma_finite)
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
  shows emeasure_fold_integral:
    "emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I)
    and emeasure_fold_measurable:
    "(\<lambda>x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M))) \<in> borel_measurable (Pi\<^sub>M I M)" (is ?B)
proof -
  interpret I: finite_product_sigma_finite M I by standard fact
  interpret J: finite_product_sigma_finite M J by standard fact
  interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
  have merge: "merge I J -` A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
    by (intro measurable_sets[OF _ A] measurable_merge assms)

  show ?I
    apply (subst distr_merge[symmetric, OF IJ])
    apply (subst emeasure_distr[OF measurable_merge A])
    apply (subst J.emeasure_pair_measure_alt[OF merge])
    apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
    done

  show ?B
    using IJ.measurable_emeasure_Pair1[OF merge]
    by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
qed

lemma sets_Collect_single:
  "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
  by simp

lemma pair_measure_eq_distr_PiM:
  fixes M1 :: "'a measure" and M2 :: "'a measure"
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
  shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))"
    (is "?P = ?D")
proof (rule pair_measure_eqI[OF assms])
  interpret B: product_sigma_finite "case_bool M1 M2"
    unfolding product_sigma_finite_def using assms by (auto split: bool.split)
  let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"

  have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)"
    by auto
  fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
  have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
    by (simp add: UNIV_bool ac_simps)
  also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
    using A B by (subst B.emeasure_PiM) (auto split: bool.split)
  also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
    using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
    by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
  finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
    using A B
      measurable_component_singleton[of True UNIV "case_bool M1 M2"]
      measurable_component_singleton[of False UNIV "case_bool M1 M2"]
    by (subst emeasure_distr) (auto simp: measurable_pair_iff)
qed simp

lemma infprod_in_sets[intro]:
  fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
  shows "Pi UNIV E \<in> sets (\<Pi>\<^sub>M i\<in>UNIV::nat set. M i)"
proof -
  have "Pi UNIV E = (\<Inter>i. prod_emb UNIV M {..i} (\<Pi>\<^sub>E j\<in>{..i}. E j))"
    using E E[THEN sets.sets_into_space]
    by (auto simp: prod_emb_def Pi_iff extensional_def)
  with E show ?thesis by auto
qed

end