Quelle Finite_Product_Measure.thy Sprache: Isabelle

(*  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 Function_Topology
begin

lemma Pi_choice: "(\i\I. \x\F i. P i x) \ (\f\Pi I F. \i\I. P i (f i))"
  by (metis Pi_iff)

lemma PiE_choice: "(\i\I. \x\F i. P i x) \(\f\Pi\<^sub>E I F. \i\I. P i (f i))"
  unfolding Pi_choice by (metis Int_iff PiE_def restrict_PiE restrict_apply)

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

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

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

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

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

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

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

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

lemma restrict_merge[simp]:
  "I \ J = {} \ restrict (merge I J (x, y)) I = restrict x I"
  "I \ J = {} \ restrict (merge I J (x, y)) J = restrict y J"
  "J \ I = {} \ restrict (merge I J (x, y)) I = restrict x I"
  "J \ I = {} \ 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) \ (i \ I \ P (x i)) \ (i \ J - I \ P (y i)) \ (i \ I \ J \ P undefined)"
  unfolding merge_def by auto

lemma PiE_cancel_merge[simp]:
  "I \ J = {} \
    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 \ I \ 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 \ J \ K \ merge I J (x, y) \ 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 \ J)"
  unfolding merge_def by auto

lemma injective_vimage_restrict:
  assumes J: "J \ I"
  and sets: "A \ (\\<^sub>E i\J. S i)" "B \ (\\<^sub>E i\J. S i)" and ne: "(\\<^sub>E i\I. S i) \ {}"
  and eq: "(\x. restrict x J) -` A \ (\\<^sub>E i\I. S i) = (\x. restrict x J) -` B \ (\\<^sub>E i\I. S i)"
  shows "A = B"
proof  (intro set_eqI)
  fix x
  from ne obtain y where y: "\i. i \ I \ y i \ S i" by auto
  have "J \ (I - J) = {}" by auto
  show "x \ A \ x \ B"
  proof cases
    assume x: "x \ (\\<^sub>E i\J. S i)"
    have "x \ A \ merge J (I - J) (x,y) \ (\x. restrict x J) -` A \ (\\<^sub>E i\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 \ A \ x \ 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 (use sets in auto)
qed

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

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

definition\<^marker>\<open>tag important\<close> prod_emb where
  "prod_emb I M K X = (\x. restrict x K) -` X \ (\\<^sub>E i\I. space (M i))"

lemma prod_emb_iff:
  "f \ prod_emb I M K X \ f \ extensional I \ (restrict f K \ X) \ (\i\I. f i \ space (M i))"
  unfolding 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 \ B) = prod_emb M L K A \ prod_emb M L K B"
    and prod_emb_Int: "prod_emb M L K (A \ B) = prod_emb M L K A \ prod_emb M L K B"
    and prod_emb_UN[simp]: "prod_emb M L K (\i\I. F i) = (\i\I. prod_emb M L K (F i))"
    and prod_emb_INT[simp]: "I \ {} \ prod_emb M L K (\i\I. F i) = (\i\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 \ I \ (\i. i \ J \ E i \ space (M i)) \
    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]:
    "(\i. i \ I \ E i \ space (M i)) \ 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 \ K \ K \ L \ 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 \ (\ j\J. sets (M j))" "J \ K"
  shows "prod_emb K M J (Pi\<^sub>E J X) = (\\<^sub>E i\K. if i \ 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 \ (\\<^sub>E i\L. space (M i)) \ prod_emb L M L B = B"
  by (auto simp: prod_emb_def subset_eq extensional_restrict)

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

definition\<^marker>\<open>tag important\<close> PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
  "PiM I M = extend_measure (\\<^sub>E i\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\<^marker>\<open>tag important\<close> prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
  "prod_algebra I M = (\(J, X). prod_emb I M J (\\<^sub>E j\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 \ PiM I M"

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

lemma extend_measure_cong:
  assumes "\ = \'" "I = I'" "G = G'" "\i. i \ I' \ \ i = \' i"
  shows "extend_measure \ I G \ = extend_measure \' I' G' \'"
  unfolding extend_measure_def by (auto simp add: assms)

lemma Pi_cong_sets:
    "\I = J; \x. x \ I \ M x = N x\ \ Pi I M = Pi J N"
  by auto

lemma PiM_cong:
  assumes "I = J" "\x. x \ I \ 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 "\i. space (M i)" "\i. space (N i)"]) simp_all
next
  case 2
  have "\K. K \ J \ (\ j\K. sets (M j)) = (\ j\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 \ Pow (\\<^sub>E i\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 = {(\\<^sub>E i\I. X i) |X. X \ (\ j\I. sets (M j))}" (is "?L = ?R")
proof (intro iffI set_eqI)
  fix A assume "A \ ?L"
  then obtain J E where J: "J \ {} \ I = {}" "finite J" "J \ I" "\i\J. E i \ sets (M i)"
    and A: "A = prod_emb I M J (\\<^sub>E j\J. E j)"
    by (auto simp: prod_algebra_def)
  let ?A = "\\<^sub>E i\I. if i \ 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 \ ?R" unfolding A using J sets.top
    by (intro CollectI exI[of _ "\i. if i \ J then E i else space (M i)"]) simp
next
  fix A assume "A \ ?R"
  then obtain X where A: "A = (\\<^sub>E i\I. X i)" and X: "X \ (\ j\I. sets (M j))" by auto
  then have A: "A = prod_emb I M I (\\<^sub>E i\I. X i)"
    by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff)
  from X I show "A \ ?L" unfolding A
    by (auto simp: prod_algebra_def)
qed

lemma prod_algebraI:
  "finite J \ (J \ {} \ I = {}) \ J \ I \ (\i. i \ J \ E i \ 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 \ (\i\I. E i \ sets (M i)) \ (Pi\<^sub>E I E) \ 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. \i\I. E i \ sets (M i)}"
proof (safe intro!: Int_stableI)
  fix E F assume "\i\I. E i \ sets (M i)" "\i\I. F i \ sets (M i)"
  then show "\G. Pi\<^sub>E J E \ Pi\<^sub>E J F = Pi\<^sub>E J G \ (\i\I. G i \ sets (M i))"
    by (auto intro!: exI[of _ "\i. E i \ F i"] simp: PiE_Int)
qed

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

lemma prod_algebraE_all:
  assumes A: "A \ prod_algebra I M"
  obtains E where "A = Pi\<^sub>E I E" "E \ (\ i\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 \ I" and E: "E \ (\ i\J. sets (M i))"
    by (auto simp: prod_algebra_def)
  from E have "\i. i \ J \ E i \ space (M i)"
    using sets.sets_into_space by auto
  then have "A = (\\<^sub>E i\I. if i\J then E i else space (M i))"
    using A J by (auto simp: prod_emb_PiE)
  moreover have "(\i. if i\J then E i else space (M i)) \ (\ i\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)"
  unfolding Int_stable_def
proof safe
  fix A assume "A \ prod_algebra I M"
  from prod_algebraE[OF this] obtain J E where A:
    "A = prod_emb I M J (Pi\<^sub>E J E)"
    "finite J"
    "J \ {} \ I = {}"
    "J \ I"
    "\i. i \ J \ E i \ sets (M i)"
    by auto
  fix B assume "B \ prod_algebra I M"
  from prod_algebraE[OF this] obtain K F where B:
    "B = prod_emb I M K (Pi\<^sub>E K F)"
    "finite K"
    "K \ {} \ I = {}"
    "K \ I"
    "\i. i \ K \ F i \ sets (M i)"
    by auto
  have "A \ B = prod_emb I M (J \ K) (\\<^sub>E i\J \ K. (if i \ J then E i else space (M i)) \
      (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 "\ \ prod_algebra I M"
    using A B by (auto intro!: prod_algebraI)
  finally show "A \ B \ prod_algebra I M" .
qed

proposition prod_algebra_mono:
  assumes space: "\i. i \ I \ space (E i) = space (F i)"
  assumes sets: "\i. i \ I \ sets (E i) \ sets (F i)"
  shows "prod_algebra I E \ prod_algebra I F"
proof
  fix A assume "A \ prod_algebra I E"
  then obtain J G where J: "J \ {} \ I = {}" "finite J" "J \ I"
    and A: "A = prod_emb I E J (\\<^sub>E i\J. G i)"
    and G: "\i. i \ J \ G i \ sets (E i)"
    by (auto simp: prod_algebra_def)
  moreover
  from space have "(\\<^sub>E i\I. space (E i)) = (\\<^sub>E i\I. space (F i))"
    by (rule PiE_cong)
  with A have "A = prod_emb I F J (\\<^sub>E i\J. G i)"
    by (simp add: prod_emb_def)
  moreover
  from sets G J have "\i. i \ J \ G i \ sets (F i)"
    by auto
  ultimately show "A \ 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 "(\i. i \ I \ sets (M i) = sets (N i))"
  shows "prod_algebra I M = prod_algebra J N"
  by (metis assms prod_algebra_mono sets_eq_imp_space_eq subsetI subset_antisym)

lemma space_in_prod_algebra: "(\\<^sub>E i\I. space (M i)) \ 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 \ {}"
  then obtain i where "i \ I" by auto
  then have "(\\<^sub>E i\I. space (M i)) = prod_emb I M {i} (\\<^sub>E i\{i}. space (M i))"
    by (auto simp: prod_emb_def)
  then show ?thesis
    by (simp add: \<open>i \<in> I\<close> prod_algebraI)
qed

lemma space_PiM: "space (\\<^sub>M i\I. M i) = (\\<^sub>E i\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 \ space (PiM I M)"
  by (auto simp: prod_emb_def space_PiM)

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

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

lemma sets_PiM: "sets (\\<^sub>M i\I. M i) = sigma_sets (\\<^sub>E i\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 ?\ ?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 \ prod_algebra I M"
  from prod_algebraE[OF this] obtain J X where X:
    "A = prod_emb I M J (Pi\<^sub>E J X)"
    "finite J"
    "J \ {} \ I = {}"
    "J \ I"
    "\i. i \ J \ X i \ sets (M i)"
    by auto
  show "A \ sigma_sets ?\ ?R"
  proof cases
    assume "I = {}"
    with X show ?thesis
      by (metis (no_types, lifting) PiE_cong R.top empty_iff prod_emb_PiE subset_eq)
  next
    assume "I \ {}"
    with X have "A = (\j\J. {f\(\\<^sub>E i\I. space (M i)). f j \ X j})"
      by (auto simp: prod_emb_def)
    also have "\ \ sigma_sets ?\ ?R"
      using X \<open>I \<noteq> {}\<close> by (intro R.finite_INT sigma_sets.Basic) auto
    finally show "A \ sigma_sets ?\ ?R" .
  qed
next
  fix A assume "A \ ?R"
  then obtain i B where A: "A = {f\\\<^sub>E i\I. space (M i). f i \ B}" "i \ I" "B \ sets (M i)"
    by auto
  then have "A = prod_emb I M {i} (\\<^sub>E i\{i}. B)"
     by (auto simp: prod_emb_def)
  also have "\ \ sigma_sets ?\ (prod_algebra I M)"
    using A by (intro sigma_sets.Basic prod_algebraI) auto
  finally show "A \ sigma_sets ?\ (prod_algebra I M)" .
qed

lemma sets_PiM_eq_proj:
  assumes "I \ {}"
  shows "sets (PiM I M) = sets (SUP i\I. vimage_algebra (\\<^sub>E i\I. space (M i)) (\x. x i) (M i))"
        (is "?lhs = ?rhs")
proof -
  have "?lhs =
        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)}"
    by (simp add: sets_PiM_single)
  also have "\ = sigma_sets (\\<^sub>E i\I. space (M i))
                (\<Union>x\<in>I. sets (vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>xa. xa x) (M x)))"
    apply (subst arg_cong [of _ _ Sup, OF image_cong, OF refl])
     apply (rule sets_vimage_algebra2)
    by (auto intro!: arg_cong2[where f=sigma_sets])
  also have "... = sigma_sets (\\<^sub>E i\I. space (M i))
     (\<Union> (sets ` (\<lambda>i. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i)) ` I))"
    by simp
  also have "... = ?rhs"
    by (subst sets_Sup_eq[where X="\\<^sub>E i\I. space (M i)"]) (use assms in auto)
  finally show ?thesis .
qed

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

    fix Z assume "Z \ (\i\I. ?F i)"
    then obtain i A where i: "i \ I" "A \ E i" and Z_def: "Z = (\x. x i) -` A \ Pi\<^sub>E I \"
      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: "\i. i \ j \ S i \ E i" "\i. i \ j \ countable (S i)"
      "\i. i \ j \ \ i = \(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: "\n. A' n i = A"
      by simp
    { fix n assume "n \ Pi\<^sub>E (j - {i}) S"
      then have "A' n \ Pi j E"
        unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def \A \ E i\ )
      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 \ A" "x \ Pi\<^sub>E I \"
      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: "\i. i \ j \ s i \ S i" "\i. i \ j \ x i \ 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 = (\n\Pi\<^sub>E (j-{i}) S. {f\(\\<^sub>E i\I. \ i). \i\j. f i \ 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 "\ \ sigma_sets (\\<^sub>E i\I. \ 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 \ sigma_sets (\\<^sub>E i\I. \ i) P" .
  next
    interpret F: sigma_algebra "\\<^sub>E i\I. \ i" "sigma_sets (\\<^sub>E i\I. \ i) (\i\I. ?F i)"
      by (auto intro!: sigma_algebra_sigma_sets)

    fix b assume "b \ P"
    then obtain A j where b: "b = {f\(\\<^sub>E i\I. \ i). \i\j. f i \ A i}" "j \ J" "A \ Pi j E"
      by (auto simp: P_def)
    show "b \ sigma_sets (Pi\<^sub>E I \) (\i\I. ?F i)"
    proof cases
      assume "j = {}"
      with b have "b = (\\<^sub>E i\I. \ i)"
        by auto
      then show ?thesis
        by blast
    next
      assume "j \ {}"
      with J b(2,3) have eq: "b = (\i\j. ((\x. x i) -` A i \ Pi\<^sub>E I \))"
        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 = (\\<^sub>E i\I. space (M i))"
  assumes sets: "\i A. i \ I \ A \ sets (M i) \ {x\space N. x i \ A} \ sets N"
  shows "sets (\\<^sub>M i \ I. M i) \ 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" "\i. i \ J \ 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 \ I" "\i\J. E i \ sets (M i)"
  shows "prod_emb I M J (\\<^sub>E j\J. E j) \ sets (\\<^sub>M i\I. M i)"
proof cases
  assume "J = {}"
  then have "prod_emb I M J (\\<^sub>E j\J. E j) = (\\<^sub>E j\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 \ {}" with assms show ?thesis
    by (force simp add: sets_PiM prod_algebra_def)
qed

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

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

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

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

lemma sets_PiM_I_finite[measurable]:
  assumes "finite I" and sets: "(\i. i \ I \ E i \ sets (M i))"
  shows "(\\<^sub>E j\I. E j) \ sets (\\<^sub>M i\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 \ I" shows "(\x. x i) \ measurable (Pi\<^sub>M I M) (M i)"
proof (unfold measurable_def, intro CollectI conjI ballI)
  fix A assume "A \ sets (M i)"
  then have "(\x. x i) -` A \ space (Pi\<^sub>M I M) = prod_emb I M {i} (\\<^sub>E j\{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 "(\x. x i) -` A \ space (Pi\<^sub>M I M) \ 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 (use \<open>i \<in> I\<close> in \<open>auto simp: space_PiM\<close>)

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

lemma measurable_PiM_component_rev:
  "i \ I \ f \ measurable (M i) N \ (\x. f (x i)) \ measurable (PiM I M) N"
  by simp

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

lemma measurable_case_nat'[measurable (raw)]:
  assumes fg[measurable]: "f \ measurable N M" "g \ measurable N (\\<^sub>M i\UNIV. M)"
  shows "(\x. case_nat (f x) (g x)) \ measurable N (\\<^sub>M i\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]:
  "(\(f, y). f(i := y)) \ measurable (Pi\<^sub>M I M \\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
    (is "?f \ measurable ?P ?I")
proof (rule measurable_PiM_single)
  fix j A assume j: "j \ insert i I" and A: "A \ sets (M j)"
  have "{\ \ space ?P. (\(f, x). fun_upd f i x) \ j \ 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 "\ \ sets ?P"
    using A j
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
  finally show "{\ \ space ?P. case_prod (\f. fun_upd f i) \ j \ A} \ sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_def)

proposition measurable_fun_upd:
  assumes I: "I = J \ {i}"
  assumes f[measurable]: "f \ measurable N (PiM J M)"
  assumes h[measurable]: "h \ measurable N (M i)"
  shows "(\x. (f x) (i := h x)) \ measurable N (PiM I M)"
proof (intro measurable_PiM_single')
  fix j assume "j \ I" then show "(\\. ((f \)(i := h \)) j) \ measurable N (M j)"
    unfolding I by (cases "j = i") auto
next
  show "(\x. (f x)(i := h x)) \ space N \ (\\<^sub>E i\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 \ space (Pi\<^sub>M I M) \ i \ I \ (\v. x(i := v)) \ measurable (M i) (Pi\<^sub>M (insert i I) M)"
  by simp

lemma measurable_merge[measurable]:
  "merge I J \ measurable (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \ J) M)"
    (is "?f \ measurable ?P ?U")
proof (rule measurable_PiM_single)
  fix i A assume A: "A \ sets (M i)" "i \ I \ J"
  then have "{\ \ space ?P. merge I J \ i \ 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 "\ \ sets ?P"
    using A
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
  finally show "{\ \ space ?P. merge I J \ i \ A} \ sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)

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

lemma measurable_abs_UNIV:
  "(\n. (\\. f n \) \ measurable M (N n)) \ (\\ n. f n \) \ measurable M (PiM UNIV N)"
  by (intro measurable_PiM_single) (auto dest: measurable_space)

lemma measurable_restrict_subset: "J \ L \ (\f. restrict f J) \ 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 \ L" "\x. x \ J \ sets (M x) = sets (N x)"
  shows "(\f. restrict f J) \ measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
  by (metis (no_types) assms measurable_cong_sets measurable_restrict_subset sets_PiM_cong)

lemma measurable_prod_emb[intro, simp]:
  "J \ L \ X \ sets (Pi\<^sub>M J M) \ prod_emb L M J X \ 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 \ space (PiM I M)" "x \ X" and X: "X \ sets (Pi\<^sub>M J M)" and "J \ I"
  shows "merge J I (x, y) \ 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 \ I" and ne: "space (PiM I M) \ {}" and X: "X \ sets (Pi\<^sub>M J M)"
    and *: "prod_emb I M J X = {}"
  shows "X = {}"
proof safe
  fix x assume "x \ 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 \ (\j. j \ I \ {x\space (M j). x \ F j} \ sets (M j)) \
  {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 \ f \ measurable N (PiM I M) \
  (\<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\space (PiM I M). x \ extensional I} \ sets (PiM I M)"
  by (smt (verit) PiE_iff mem_Collect_eq sets.top space_PiM subsetI subset_antisym)

lemma sets_in_extensional[measurable (raw)]:
  "f \ measurable N (PiM I M) \ Measurable.pred N (\x. f x \ 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: "\i. i \ I \ E i \ sets (M i)" shows "Pi\<^sub>E I E \ sets (Pi\<^sub>M I M)"
proof cases
  assume "I \ {}"
  then have "Pi\<^sub>E I E = (\i\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 "\ \ sets (PiM I M)"
    using I \<open>I \<noteq> {}\<close> by (simp add: E sets.countable_INT' sets_PiM_I subset_eq)
  finally show ?thesis .
qed (simp add: sets_PiM_empty)

lemma sets_PiM_D_countable:
  assumes A: "A \ PiM I M"
  shows "\J\I. \X\PiM J M. countable J \ A = prod_emb I M J X"
  using A[unfolded sets_PiM_single]
proof induction
  case (Basic A)
  then obtain i X where *: "i \ I" "X \ sets (M i)" and "A = {f \ \\<^sub>E i\I. space (M i). f i \ X}"
    by auto
  then have A: "A = prod_emb I M {i} (\\<^sub>E _\{i}. X)"
    by (auto simp: prod_emb_def)
  then show ?case
    by (intro exI[of _ "{i}"] conjI bexI[of _ "\\<^sub>E _\{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 \ I" "X \ 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: "\i. J i \ I" "\i. countable (J i)" and X: "\i. X i \ sets (Pi\<^sub>M (J i) M)"
    and K: "\i. K i = prod_emb I M (J i) (X i)"
    by (metis Union.IH)
  show ?case
  proof (intro exI bexI conjI)
    show "(\i. J i) \ I" "countable (\i. J i)"
      using J by auto
    with J show "\(K ` UNIV) = prod_emb I M (\i. J i) (\i. prod_emb (\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: "\A. (\i. i \ I \ A i \ sets (M i)) \ P (Pi\<^sub>E I A) = Q (Pi\<^sub>E I A)"
  assumes A: "range A \ prod_algebra I M" "(\i. A i) = space (PiM I M)" "\i::nat. P (A i) \ \"
  shows "P = Q"
proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
  show "range A \ prod_algebra I M" "(\i. A i) = (\\<^sub>E i\I. space (M i))" "\i. P (A i) \ \"
    unfolding space_PiM[symmetric] by fact+
  fix X assume "X \ prod_algebra I M"
  then obtain J E where X: "X = prod_emb I M J (\\<^sub>E j\J. E j)"
    and J: "finite J" "J \ I" "\j. j \ J \ E j \ 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: "\A J. finite J \ J \ I \ (\i. i \ J \ A i \ sets (M i)) \
    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 \ I)"
  have i: "I \ {} \ i \ 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 \ prod_algebra I M"
    using prod_algebraI[of "{}" I "\i. space (M i)" M] by (auto intro!: prod_algebraI i)
  have "\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 "(\i. A i) = (\\<^sub>E i\I. space (M i))" "\i. emeasure P (A i) \ \"
    by (auto simp: space_PiM)
next
  fix X assume X: "X \ prod_algebra I M"
  then obtain J E where X: "X = prod_emb I M J (\\<^sub>E j\J. E j)"
    and J: "finite J" "J \ I" "\j. j \ J \ E j \ 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\<^marker>\<open>tag unimportant\<close> product_sigma_finite =
  fixes M :: "'i \ 'a measure"
  assumes sigma_finite_measures: "\i. sigma_finite_measure (M i)"

sublocale\<^marker>\<open>tag unimportant\<close> product_sigma_finite \<subseteq> M?: sigma_finite_measure "M i" for i
  by (rule sigma_finite_measures)

locale\<^marker>\<open>tag unimportant\<close> 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:
  "\F::'i \ nat \ '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 "\i::'i. \F::nat \ 'a set. range F \ sets (M i) \ incseq F \ (\i. F i) = space (M i) \ (\k. emeasure (M i) (F k) \ \)"
    using M.sigma_finite_incseq by metis
  then obtain F :: "'i \ nat \ 'a set"
    where "\x. range (F x) \ sets (M x) \ incseq (F x) \ \ (range (F x)) = space (M x) \ (\k. emeasure (M x) (F x k) \ \)"
    by metis
  then have F: "\i. range (F i) \ sets (M i)" "\i. incseq (F i)" "\i. (\j. F i j) = space (M i)" "\i k. emeasure (M i) (F i k) \ \"
    by auto
  let ?F = "\k. \\<^sub>E i\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) \ sets (M i)" by fact
  next
    fix i k show "emeasure (M i) (F i k) \ \" by fact
  next
    fix x assume "x \ (\i. ?F i)" with F(1) show "x \ space (PiM I M)"
      by (auto simp: PiE_def dest!: sets.sets_into_space)
  next
    fix f assume "f \ space (PiM I M)"
    with Pi_UN[OF finite_index, of "\k i. F i k"] F
    show "f \ (\i. ?F i)" by (auto simp: incseq_def PiE_def)
  next
    fix i show "?F i \ ?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) {\_. undefined} = 1"
proof -
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ennreal)"
  have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\\<^sub>E i\{}. {})) = 1"
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
    show "positive (PiM {} M) ?\"
      by (auto simp: positive_def)
    show "countably_additive (PiM {} M) ?\"
      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 {} (\\<^sub>E i\{}. {})) = {\_. undefined}"
    by (auto simp: prod_emb_def)
  finally show ?thesis
    by simp
qed

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

lemma (in product_sigma_finite) emeasure_PiM:
  "finite I \ (\i. i\I \ A i \ sets (M i)) \ emeasure (PiM I M) (Pi\<^sub>E I A) = (\i\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 = "(\(f, y). f(i := y))"

  let ?P = "distr (Pi\<^sub>M I M \\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
  let ?\<mu> = "emeasure ?P"
  let ?I = "{j \ insert i I. emeasure (M j) (space (M j)) \ 1}"
  let ?f = "\J E j. if j \ 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 \ {} \ insert i I = {}) \ finite J \ J \ insert i I \ E \ (\ j\J. sets (M j))"
    then have J: "J \ {}" "finite J" "J \ insert i I" and E: "\j\J. E j \ 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}) (\\<^sub>E j\J-{i}. E j)"
    have "?\ ?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 \ space (Pi\<^sub>M I M \\<^sub>M M i) = ?p' \ (if i \ 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 \\<^sub>M (M i)) (?p' \ (if i \ 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' = (\\<^sub>E j\I. if j \ 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) (\\<^sub>E j\I. if j \ 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 "(\j\I. if j \ 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 "\ = (\j\J \ ?I. ?f J E j)"
      using insert(1,2) J E by (intro prod.mono_neutral_right) auto
    finally show "?\ ?p = \" .

    show "prod_emb (insert i I) M J (Pi\<^sub>E J E) \ Pow (\\<^sub>E i\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)) ?\" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\"
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
  next
    show "(insert i I \ {} \ insert i I = {}) \ finite (insert i I) \
      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: "\A. (\i. i \ I \ A i \ sets (M i)) \ P (Pi\<^sub>E I A) = (\i\I. emeasure (M i) (A i))"
  shows "P = PiM I M"
proof -
  interpret finite_product_sigma_finite M I
    by standard fact
  from sigma_finite_pairs obtain C where C:
    "\i\I. range (C i) \ sets (M i)" "\k. \i\I. emeasure (M i) (C i k) \ \"
    "incseq (\k. \\<^sub>E i\I. C i k)" "(\k. \\<^sub>E i\I. C i k) = space (Pi\<^sub>M I M)"
    by auto
  show ?thesis
  proof (rule measure_eqI_PiM_finite[OF I refl P, symmetric])
    show "(\i. i \ I \ A i \ sets (M i)) \ (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 = (\\<^sub>E i\I. C i n)" for n
    with C show "range A \ prod_algebra I M" "\i. emeasure (Pi\<^sub>M I M) (A i) \ \" "(\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: "\j. countable (F j)" "\j f. f \ F j \ f \ sets (M j)"
    "\j f. f \ F j \ emeasure (M j) f \ \" and
    in_space: "\j. space (M j) = \(F j)"
    using sigma_finite_countable by (metis subset_eq)
  moreover have "(\(Pi\<^sub>E I ` Pi\<^sub>E I F)) = space (Pi\<^sub>M I M)"
    using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD1])
  ultimately show "\A. countable A \ A \ sets (Pi\<^sub>M I M) \ \A = space (Pi\<^sub>M I M) \ (\a\A. emeasure (Pi\<^sub>M I M) a \ \)"
    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:
  "(\i. i \ I \ A i \ sets (M i)) \ emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\i\I. emeasure (M i) (A i))"
  using emeasure_PiM[OF finite_index] by auto

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

lemma\<^marker>\<open>tag important\<close> (in product_sigma_finite) distr_merge:
  assumes IJ[simp]: "I \ J = {}" and fin: "finite I" "finite J"
  shows "distr (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \ J) M) (merge I J) = Pi\<^sub>M (I \ 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: "\i. i \ I \ J \ A i \ sets (M i)"
  have *: "(merge I J -` Pi\<^sub>E (I \ J) A \ space (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M)) = Pi\<^sub>E I A \ 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 \\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \ J) M) (merge I J)) (Pi\<^sub>E (I \ 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 (use fin in simp_all)

proposition (in product_sigma_finite) product_nn_integral_fold:
  assumes IJ: "I \ J = {}" "finite I" "finite J"
  and f[measurable]: "f \ borel_measurable (Pi\<^sub>M (I \ J) M)"
shows "integral\<^sup>N (Pi\<^sub>M (I \ J) M) f = (\\<^sup>+ x. (\\<^sup>+ y. f (merge I J (x, y)) \(Pi\<^sub>M J M)) \(Pi\<^sub>M I M))"
        (is "?lhs = ?rhs")
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: "(\x. f (merge I J x)) \ borel_measurable (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M)"
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
  have "?lhs = integral\<^sup>N (distr (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \ J) M) (merge I J)) f"
    by (simp add: I.finite_index J.finite_index assms(1) distr_merge)
  also have "... = \\<^sup>+ x. f (merge I J x) \(Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M)"
    by (simp add: nn_integral_distr)
  also have "... = ?rhs"
    using P.Fubini P.nn_integral_snd by force
  finally show ?thesis .
qed

lemma (in product_sigma_finite) distr_singleton:
  "distr (Pi\<^sub>M {i} M) (M i) (\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 \ sets (M i)"
  then have "(\x. x i) -` A \ space (Pi\<^sub>M {i} M) = (\\<^sub>E i\{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 "\_. A"]
    by (simp add: emeasure_distr measurable_component_singleton)
qed simp

lemma (in product_sigma_finite) product_nn_integral_singleton:
  assumes f: "f \ borel_measurable (M i)"
  shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\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
    by (metis distr_singleton insert_iff measurable_component_singleton nn_integral_distr)
qed

proposition (in product_sigma_finite) product_nn_integral_insert:
  assumes I[simp]: "finite I" "i \ I"
    and f: "f \ borel_measurable (Pi\<^sub>M (insert i I) M)"
  shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\\<^sup>+ x. (\\<^sup>+ y. f (x(i := y)) \(M i)) \(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 \ {i} = {}" and insert: "I \ {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 \ space (Pi\<^sub>M I M)"
    let ?f = "\y. f (x(i := y))"
    show "?f \ borel_measurable (M i)"
      using measurable_comp[OF measurable_component_update f, OF x \<open>i \<notin> I\<close>]
      unfolding comp_def .
    show "(\\<^sup>+ y. f (merge I {i} (x, y)) \Pi\<^sub>M {i} M) = (\\<^sup>+ y. f (x(i := y i)) \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 \ I"
    and [measurable]: "f \ borel_measurable (Pi\<^sub>M (insert i I) M)"
  shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\\<^sup>+ y. (\\<^sup>+ x. f (x(i := y)) \(Pi\<^sub>M I M)) \(M i))"
  apply (subst product_nn_integral_insert[OF assms])
  apply (rule pair_sigma_finite.Fubini')
  apply (simp add: local.sigma_finite pair_sigma_finite.intro sigma_finite_measures)
  apply measurable
  done

lemma (in product_sigma_finite) product_nn_integral_prod:
  assumes "finite I" "\i. i \ I \ f i \ borel_measurable (M i)"
  shows "(\\<^sup>+ x. (\i\I. f i (x i)) \Pi\<^sub>M I M) = (\i\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 *: "\x y. (\j\I. f j (if j = i then y else x j)) = (\j\I. f j (x j))"
    using insert by (auto intro!: prod.cong)
  have prod: "\J. J \ insert i I \ (\x. (\i\J. f i (x i))) \ 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
    using product_nn_integral_insert[OF insert(1,2)]
    by (simp add: insert(2-) * nn_integral_multc nn_integral_cmult)
qed (simp add: space_PiM)

proposition (in product_sigma_finite) product_nn_integral_pair:
  assumes [measurable]: "case_prod f \ borel_measurable (M x \\<^sub>M M y)"
  assumes xy: "x \ y"
  shows "(\\<^sup>+\. f (\ x) (\ y) \PiM {x, y} M) = (\\<^sup>+z. f (fst z) (snd z) \(M x \\<^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 "(\\<^sup>+\. f (\ x) (\ y) \PiM {y, x} M) = (\\<^sup>+y. \\<^sup>+\. f (\ x) y \PiM {x} M \M y)"
    using xy by (subst product_nn_integral_insert_rev) simp_all
  also have "... = (\\<^sup>+y. \\<^sup>+x. f x y \M x \M y)"
    by (intro nn_integral_cong, subst product_nn_integral_singleton) simp_all
  also have "... = (\\<^sup>+z. f (fst z) (snd z) \(M x \\<^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) (\x. \i\{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
proof (intro PiM_eqI)
  fix A assume A: "\ia. ia \ {i} \ A ia \ sets (M ia)"
  then have "(\x. \i\{i}. x) -` Pi\<^sub>E {i} A \ space (M i) = A i"
    by (fastforce dest: sets.sets_into_space)
  with A show "emeasure (distr (M i) (Pi\<^sub>M {i} M) (\x. \i\{i}. x)) (Pi\<^sub>E {i} A) = (\i\{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 \ J = {}" "finite I" "finite J" and A: "A \ sets (Pi\<^sub>M (I \ J) M)"
  shows emeasure_fold_integral:
    "emeasure (Pi\<^sub>M (I \ J) M) A = (\\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\y. merge I J (x, y)) -` A \ space (Pi\<^sub>M J M)) \Pi\<^sub>M I M)"
         (is "?lhs = ?rhs")
    and emeasure_fold_measurable:
    "(\x. emeasure (Pi\<^sub>M J M) ((\y. merge I J (x, y)) -` A \ space (Pi\<^sub>M J M))) \ 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 \ space (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M) \ sets (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M)"
    by (intro measurable_sets[OF _ A] measurable_merge assms)
  have "?lhs = emeasure (distr (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \ J) M) (merge I J)) A"
    by (simp add: I.finite_index J.finite_index assms(1) distr_merge)
  also have "... = emeasure (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M) (merge I J -` A \ space (Pi\<^sub>M I M \\<^sub>M Pi\<^sub>M J M))"
    by (meson A emeasure_distr measurable_merge)
  also have "... = ?rhs"
    apply (subst J.emeasure_pair_measure_alt[OF merge])
    by (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
  finally show "?lhs = ?rhs" .

  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 \ I \ A \ sets (M i) \ { x \ space (Pi\<^sub>M I M). x i \ A } \ 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 \\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \\<^sub>M M2) (\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)"

--> --------------------

--> maximum size reached

--> --------------------

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