src/HOL/Probability/Finite_Product_Measure.thy
 author wenzelm Thu, 18 Apr 2013 17:07:01 +0200 changeset 51717 9e7d1c139569 parent 50387 3d8863c41fe8 child 53015 a1119cf551e8 permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
```
(*  Title:      HOL/Probability/Finite_Product_Measure.thy
Author:     Johannes Hölzl, TU München
*)

theory Finite_Product_Measure
imports Binary_Product_Measure
begin

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

subsubsection {* Merge two extensional functions *}

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> PiE (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>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}"
and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^isub>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>\<^isub>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>\<^isub>E i\<in>I. S i)"
using y x `J \<subseteq> I` 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 `J \<subseteq> I` 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\<^isub>E I E \<times> Pi\<^isub>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\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
by (auto simp: restrict_Pi_cancel PiE_def)

section "Finite product spaces"

section "Products"

definition prod_emb where
"prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i: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 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>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))"
by (force simp: prod_emb_def PiE_iff split_if_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\<^isub>E I E) = Pi\<^isub>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\<^isub>E J X) = (\<Pi>\<^isub>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: split_if_asm) blast+

lemma prod_emb_id:
"B \<subseteq> (\<Pi>\<^isub>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 PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
"PiM I M = extend_measure (\<Pi>\<^isub>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>\<^isub>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 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>\<^isub>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\<^isub>M I M \<equiv> PiM I M"

syntax
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)

syntax (xsymbols)
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)

syntax (HTML output)
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)

translations
"PIM x:I. M" == "CONST PiM I (%x. M)"

lemma prod_algebra_sets_into_space:
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>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>\<^isub>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 (PIE j:J. E j)"
by (auto simp: prod_algebra_def)
let ?A = "\<Pi>\<^isub>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>\<^isub>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>\<^isub>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 (PIE j: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\<^isub>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\<^isub>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\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>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 (PIE j: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\<^isub>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\<^isub>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>\<^isub>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 then 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>\<^isub>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 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

lemma 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>\<^isub>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>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
by (rule PiE_cong)
with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
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 (intro exI[of _ J] exI[of _ G] conjI)
apply auto
done
qed

lemma 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 `I = J`
by (intro antisym prod_algebra_mono) auto
qed

lemma space_in_prod_algebra:
"(\<Pi>\<^isub>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>\<^isub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i))"
by (auto simp: prod_emb_def)
also have "\<dots> \<in> prod_algebra I M"
using `i \<in> I` by (intro prod_algebraI) auto
finally show ?thesis .
qed

lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>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 sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>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

lemma sets_PiM_single: "sets (PiM I M) =
sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>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 `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
next
assume "I \<noteq> {}"
with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>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 `I \<noteq> {}` 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>\<^isub>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>\<^isub>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_I:
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
proof cases
assume "J = {}"
then have "prod_emb I M J (PIE j:J. E j) = (PIE 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 \<noteq> {}" with assms show ?thesis
by (force simp add: sets_PiM prod_algebra_def)
qed

lemma measurable_PiM:
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>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\<^isub>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>\<^isub>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>\<^isub>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>\<^isub>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>\<^isub>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>\<^isub>E i\<in>I. space (M i))"
shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^isub>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 "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto

lemma measurable_component_singleton:
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>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\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)"
using sets.sets_into_space `i \<in> I`
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
qed (insert `i \<in> I`, auto simp: space_PiM)

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

lemma measurable_PiM_component_rev[measurable (raw)]:
"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_nat_case[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. nat_case (f x) (g x) i) \<in> measurable M N"
by (cases i) simp_all

lemma measurable_nat_case'[measurable (raw)]:
assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
shows "(\<lambda>x. nat_case (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>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)

"(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>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\<^isub>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. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_def)

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

lemma measurable_merge[measurable]:
"merge I J \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>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\<^isub>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_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)"
by (intro measurable_restrict measurable_component_singleton) auto

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

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)"

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

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

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

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

lemma (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>\<^isub>E i\<in>I. F i k) \<and>
(\<Union>k. \<Pi>\<^isub>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>\<^isub>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 `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
qed
qed

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

lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
proof -
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>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>"
(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>\<^isub>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 one_ereal_def)

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\<^isub>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 default fact
have "finite (insert i I)" using `finite I` by auto
interpret I': finite_product_sigma_finite M "insert i I" by default fact
let ?h = "(\<lambda>(f, y). f(i := y))"

let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>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\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>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\<^isub>E J E)"
let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
have "?\<mu> ?p =
emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))"
by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
also have "?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>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: split_if_asm)
also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
emeasure (Pi\<^isub>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>\<^isub>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: split_if_asm) blast+
also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>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!: setprod_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="op *"] setprod_cong)
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
using insert(1,2) J E by (intro setprod_mono_one_right) auto
finally show "?\<mu> ?p = \<dots>" .

show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>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\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>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!: setprod_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) sigma_finite:
assumes "finite I"
shows "sigma_finite_measure (PiM I M)"
proof -
interpret finite_product_sigma_finite M I by default fact

from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
"(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
"\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
by blast+
let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"

show ?thesis
proof (unfold_locales, intro exI[of _ ?F] conjI allI)
show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
next
from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
next
fix j
from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
by (subst emeasure_PiM) auto
qed
qed

sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>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\<^isub>M I M) (Pi\<^isub>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) positive_integral_empty:
assumes pos: "0 \<le> f (\<lambda>k. undefined)"
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
proof -
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
using assms by (subst measure_times) auto
then show ?thesis
unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
qed
qed

lemma (in product_sigma_finite) distr_merge:
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (merge I J) = Pi\<^isub>M (I \<union> J) M"
(is "?D = ?P")
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
let ?g = "merge I J"

from IJ.sigma_finite_pairs obtain F where
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
"\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"

show ?thesis
proof (rule measure_eqI_generator_eq[symmetric])
show "Int_stable (prod_algebra (I \<union> J) M)"
by (rule Int_stable_prod_algebra)
show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
by (rule prod_algebra_sets_into_space)
show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
by (rule sets_PiM)
then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
by simp

show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
using fin by (auto simp: prod_algebra_eq_finite)
show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))"
using F(3) by (simp add: space_PiM)
next
fix k
from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
next
fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
let ?X = "?g -` A \<inter> space ?B"
have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)"
using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+
then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
have "emeasure ?D A = emeasure ?B ?X"
using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
using `finite J` `finite I` F unfolding X
by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
using `finite J` `finite I` F unfolding A
by (intro IJ.measure_times[symmetric]) auto
finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
qed
qed

lemma (in product_sigma_finite) product_positive_integral_fold:
assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>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 positive_integral_distr[OF measurable_merge f])
apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
apply simp
done
qed

lemma (in product_sigma_finite) distr_singleton:
"distr (Pi\<^isub>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 default simp
fix A assume A: "A \<in> sets (M i)"
moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
using sets.sets_into_space by (auto simp: space_PiM)
ultimately show "emeasure (M i) A = emeasure ?D A"
using A I.measure_times[of "\<lambda>_. A"]
qed simp

lemma (in product_sigma_finite) product_positive_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
proof -
interpret I: finite_product_sigma_finite M "{i}" by default simp
from f show ?thesis
apply (subst distr_singleton[symmetric])
apply (subst positive_integral_distr[OF measurable_component_singleton])
apply simp_all
done
qed

lemma (in product_sigma_finite) product_positive_integral_insert:
assumes I[simp]: "finite I" "i \<notin> I"
and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"
proof -
interpret I: finite_product_sigma_finite M I by default auto
interpret i: finite_product_sigma_finite M "{i}" by default auto
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
using f by auto
show ?thesis
unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
fix x assume x: "x \<in> space (Pi\<^isub>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 `i \<notin> I`]
unfolding comp_def .
show "(\<integral>\<^isup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^isub>M {i} M) = (\<integral>\<^isup>+ y. f (x(i := y i)) \<partial>Pi\<^isub>M {i} M)"
using x
by (auto intro!: positive_integral_cong arg_cong[where f=f]
qed
qed

lemma (in product_sigma_finite) product_positive_integral_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
using assms proof induct
case (insert i I)
note `finite I`[intro, simp]
interpret I: finite_product_sigma_finite M I by default 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!: setprod_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\<^isub>M J M)"
using sets.sets_into_space insert
by (intro borel_measurable_ereal_setprod
measurable_comp[OF measurable_component_singleton, unfolded comp_def])
auto
then show ?case
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
apply (subst positive_integral_cmult)
apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
done

lemma (in product_sigma_finite) product_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
proof -
interpret I: finite_product_sigma_finite M "{i}" by default simp
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
"(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
using assms by auto
show ?thesis
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
qed
lemma (in product_sigma_finite) distr_component:
"distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P")
proof (intro measure_eqI[symmetric])
interpret I: finite_product_sigma_finite M "{i}" by default simp

have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
by (auto simp: extensional_def restrict_def)

fix A assume A: "A \<in> sets ?P"
then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)"
by simp
also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)"
by (intro positive_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
also have "\<dots> = emeasure ?D A"
using A by (simp add: product_positive_integral_singleton emeasure_distr)
finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" .
qed simp

lemma (in product_sigma_finite) product_integral_fold:
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
let ?M = "merge I J"
let ?f = "\<lambda>x. f (?M x)"
from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
by auto
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
show ?thesis
apply (subst distr_merge[symmetric, OF IJ fin])
apply (subst integral_distr[OF measurable_merge f_borel])
apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
apply simp
done
qed

lemma (in product_sigma_finite)
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
shows emeasure_fold_integral:
"emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
and emeasure_fold_measurable:
"(\<lambda>x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
have merge: "merge I J -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>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!: positive_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_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
qed

lemma (in product_sigma_finite) product_integral_insert:
assumes I: "finite I" "i \<notin> I"
and f: "integrable (Pi\<^isub>M (insert i I) M) f"
shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
proof -
have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
by simp
also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)"
using f I by (intro product_integral_fold) auto
also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
proof (rule integral_cong, subst product_integral_singleton[symmetric])
fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
using f by auto
show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
unfolding comp_def .
from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)"
by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
qed
finally show ?thesis .
qed

lemma (in product_sigma_finite) product_integrable_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
proof -
interpret finite_product_sigma_finite M I by default fact
have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
using integrable unfolding integrable_def by auto
have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
proof (unfold integrable_def, intro conjI)
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
using borel by auto
have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)"
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
using f by (subst product_positive_integral_setprod) auto
also have "\<dots> < \<infinity>"
using integrable[THEN integrable_abs]
by (simp add: setprod_PInf integrable_def positive_integral_positive)
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
by (intro positive_integral_cong_pos) auto
then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
qed
ultimately show ?thesis
by (rule integrable_abs_iff[THEN iffD1])
qed

lemma (in product_sigma_finite) product_integral_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"
using assms proof induct
case empty
interpret finite_measure "Pi\<^isub>M {} M"
show ?case by (simp add: space_PiM measure_def)
next
case (insert i I)
then have iI: "finite (insert i I)" by auto
then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
interpret I: finite_product_sigma_finite M I by default fact
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 `i \<notin> I` by (auto intro!: setprod_cong)
show ?case
unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
qed

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

lemma sigma_prod_algebra_sigma_eq_infinite:
fixes E :: "'i \<Rightarrow> 'a set set"
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
defines "P == {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}"
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
proof
let ?P = "sigma (space (Pi\<^isub>M I M)) P"
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
using E_closed by (auto simp: space_PiM P_def subset_eq)
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
have "sets (PiM I M) =
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
using sets_PiM_single[of I M] by (simp add: space_P)
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
proof (safe intro!: sets.sigma_sets_subset)
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
apply (subst measurable_iff_measure_of)
using E_closed
apply (force simp: subset_eq space_PiM)
apply (force simp: subset_eq space_PiM)
apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
apply (rule_tac x=Aa in exI)
apply (auto simp: space_PiM)
done
from measurable_sets[OF this, of A] A `i \<in> I` E_closed
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
using P_closed by (auto simp: space_PiM)
finally show "\<dots> \<in> sets ?P" .
qed
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
unfolding P_def space_PiM[symmetric]
by (intro sets.sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
qed

lemma sigma_prod_algebra_sigma_eq:
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
assumes "finite I"
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
proof
let ?P = "sigma (space (Pi\<^isub>M I M)) P"
from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i"
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
using E_closed by (auto simp: space_PiM P_def subset_eq)
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
have "sets (PiM I M) =
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
using sets_PiM_single[of I M] by (simp add: space_P)
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
proof (safe intro!: sets.sigma_sets_subset)
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
proof (subst measurable_iff_measure_of)
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto
show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
proof
fix A assume A: "A \<in> E i"
then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm)
also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
by (intro PiE_cong) (simp add: S_union)
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j))"
using T
apply (auto simp: PiE_iff bchoice_iff)
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
apply (auto simp: bij_betw_def)
done
also have "\<dots> \<in> sets ?P"
proof (safe intro!: sets.countable_UN)
fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
using A S_in_E
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
qed
finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
using P_closed by simp
qed
qed
from measurable_sets[OF this, of A] A `i \<in> I` E_closed
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
using P_closed by (auto simp: space_PiM)
finally show "\<dots> \<in> sets ?P" .
qed
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
using `finite I`
by (auto intro!: sets.sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
qed

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>\<^isub>M M2) = distr (Pi\<^isub>M UNIV (bool_case M1 M2)) (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. (x True, x False))"
(is "?P = ?D")
proof (rule pair_measure_eqI[OF assms])
interpret B: product_sigma_finite "bool_case M1 M2"
unfolding product_sigma_finite_def using assms by (auto split: bool.split)
let ?B = "Pi\<^isub>M UNIV (bool_case 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 (bool_case M1 M2 i) (bool_case A B i))"
also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))"
using A B by (subst B.emeasure_PiM) (auto split: bool.split)
also have "Pi\<^isub>E UNIV (bool_case 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 "bool_case M1 M2"]
measurable_component_singleton[of False UNIV "bool_case M1 M2"]
by (subst emeasure_distr) (auto simp: measurable_pair_iff)
qed simp

end
```