src/HOL/Probability/Finite_Product_Measure.thy
author hoelzl
Mon Nov 19 12:29:02 2012 +0100 (2012-11-19)
changeset 50123 69b35a75caf3
parent 50104 de19856feb54
child 50244 de72bbe42190
permissions -rw-r--r--
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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(*  Title:      HOL/Probability/Finite_Product_Measure.thy
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    Author:     Johannes Hölzl, TU München
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*)
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header {*Finite product measures*}
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theory Finite_Product_Measure
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imports Binary_Product_Measure
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begin
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lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
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  by auto
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subsubsection {* Merge two extensional functions *}
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definition
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  "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
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lemma merge_apply[simp]:
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  "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
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  "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
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  "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
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  "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
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  "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined"
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  unfolding merge_def by auto
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lemma merge_commute:
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  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)"
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  by (force simp: merge_def)
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lemma Pi_cancel_merge_range[simp]:
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  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
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  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
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  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
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  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
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  by (auto simp: Pi_def)
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lemma Pi_cancel_merge[simp]:
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  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
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  "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
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  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
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  "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
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  by (auto simp: Pi_def)
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lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)"
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  by (auto simp: extensional_def)
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lemma restrict_merge[simp]:
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  "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
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  "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
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  "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
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  "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
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  by (auto simp: restrict_def)
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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)"
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  unfolding merge_def by auto
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lemma PiE_cancel_merge[simp]:
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  "I \<inter> J = {} \<Longrightarrow>
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    merge I J (x, y) \<in> PiE (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B"
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  by (auto simp: PiE_def restrict_Pi_cancel)
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lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)"
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  unfolding merge_def by (auto simp: fun_eq_iff)
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lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K"
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  unfolding merge_def extensional_def by auto
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lemma merge_restrict[simp]:
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  "merge I J (restrict x I, y) = merge I J (x, y)"
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  "merge I J (x, restrict y J) = merge I J (x, y)"
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  unfolding merge_def by auto
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lemma merge_x_x_eq_restrict[simp]:
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  "merge I J (x, x) = restrict x (I \<union> J)"
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  unfolding merge_def by auto
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lemma injective_vimage_restrict:
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  assumes J: "J \<subseteq> I"
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  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> {}"
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  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)"
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  shows "A = B"
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proof  (intro set_eqI)
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  fix x
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  from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
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  have "J \<inter> (I - J) = {}" by auto
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  show "x \<in> A \<longleftrightarrow> x \<in> B"
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  proof cases
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    assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)"
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    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)"
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      using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S]
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      by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
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    then show "x \<in> A \<longleftrightarrow> x \<in> B"
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      using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S]
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      by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq)
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  qed (insert sets, auto)
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qed
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lemma restrict_vimage:
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  "I \<inter> J = {} \<Longrightarrow>
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    (\<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))"
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  by (auto simp: restrict_Pi_cancel PiE_def)
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lemma merge_vimage:
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  "I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
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  by (auto simp: restrict_Pi_cancel PiE_def)
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section "Finite product spaces"
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section "Products"
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definition prod_emb where
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  "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
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lemma prod_emb_iff: 
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  "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))"
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  unfolding prod_emb_def PiE_def by auto
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lemma
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  shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
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    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"
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    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"
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    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))"
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    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))"
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    and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
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  by (auto simp: prod_emb_def)
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lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
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    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))"
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  by (force simp: prod_emb_def PiE_iff split_if_mem2)
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lemma prod_emb_PiE_same_index[simp]:
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    "(\<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"
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  by (auto simp: prod_emb_def PiE_iff)
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lemma prod_emb_trans[simp]:
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  "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"
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  by (auto simp add: Int_absorb1 prod_emb_def PiE_def)
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lemma prod_emb_Pi:
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  assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
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  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))"
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  using assms space_closed
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  by (auto simp: prod_emb_def PiE_iff split: split_if_asm) blast+
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lemma prod_emb_id:
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  "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
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  by (auto simp: prod_emb_def subset_eq extensional_restrict)
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lemma prod_emb_mono:
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  "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
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  by (auto simp: prod_emb_def)
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definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
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  "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
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    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
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    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
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    (\<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)))"
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definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
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  "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
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    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
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abbreviation
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  "Pi\<^isub>M I M \<equiv> PiM I M"
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syntax
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  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
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syntax (xsymbols)
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  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
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syntax (HTML output)
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  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
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translations
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  "PIM x:I. M" == "CONST PiM I (%x. M)"
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lemma prod_algebra_sets_into_space:
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  "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
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  using assms by (auto simp: prod_emb_def prod_algebra_def)
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lemma prod_algebra_eq_finite:
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  assumes I: "finite I"
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  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")
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proof (intro iffI set_eqI)
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  fix A assume "A \<in> ?L"
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  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)"
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    and A: "A = prod_emb I M J (PIE j:J. E j)"
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    by (auto simp: prod_algebra_def)
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  let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
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  have A: "A = ?A"
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    unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
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  show "A \<in> ?R" unfolding A using J top
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    by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
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next
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  fix A assume "A \<in> ?R"
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  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
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  then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
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    by (simp add: prod_emb_PiE_same_index[OF sets_into_space] Pi_iff)
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  from X I show "A \<in> ?L" unfolding A
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    by (auto simp: prod_algebra_def)
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qed
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lemma prod_algebraI:
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  "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
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    \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
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  by (auto simp: prod_algebra_def)
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lemma prod_algebraI_finite:
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  "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M"
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  using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp
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lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
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proof (safe intro!: Int_stableI)
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  fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
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  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))"
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    by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
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qed
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lemma prod_algebraE:
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  assumes A: "A \<in> prod_algebra I M"
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  obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
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    "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" 
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  using A by (auto simp: prod_algebra_def)
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lemma prod_algebraE_all:
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  assumes A: "A \<in> prod_algebra I M"
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  obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
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proof -
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  from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
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    and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
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    by (auto simp: prod_algebra_def)
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  from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
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   235
    using sets_into_space by auto
hoelzl@47694
   236
  then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
hoelzl@47694
   237
    using A J by (auto simp: prod_emb_PiE)
hoelzl@47694
   238
  moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
hoelzl@47694
   239
    using top E by auto
hoelzl@47694
   240
  ultimately show ?thesis using that by auto
hoelzl@47694
   241
qed
hoelzl@40859
   242
hoelzl@47694
   243
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
hoelzl@47694
   244
proof (unfold Int_stable_def, safe)
hoelzl@47694
   245
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   246
  from prod_algebraE[OF this] guess J E . note A = this
hoelzl@47694
   247
  fix B assume "B \<in> prod_algebra I M"
hoelzl@47694
   248
  from prod_algebraE[OF this] guess K F . note B = this
hoelzl@47694
   249
  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> 
hoelzl@47694
   250
      (if i \<in> K then F i else space (M i)))"
hoelzl@47694
   251
    unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space]
hoelzl@47694
   252
    apply (subst (1 2 3) prod_emb_PiE)
hoelzl@47694
   253
    apply (simp_all add: subset_eq PiE_Int)
hoelzl@47694
   254
    apply blast
hoelzl@47694
   255
    apply (intro PiE_cong)
hoelzl@47694
   256
    apply auto
hoelzl@47694
   257
    done
hoelzl@47694
   258
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   259
    using A B by (auto intro!: prod_algebraI)
hoelzl@47694
   260
  finally show "A \<inter> B \<in> prod_algebra I M" .
hoelzl@47694
   261
qed
hoelzl@47694
   262
hoelzl@47694
   263
lemma prod_algebra_mono:
hoelzl@47694
   264
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
hoelzl@47694
   265
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
hoelzl@47694
   266
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
hoelzl@47694
   267
proof
hoelzl@47694
   268
  fix A assume "A \<in> prod_algebra I E"
hoelzl@47694
   269
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
hoelzl@47694
   270
    and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   271
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
hoelzl@47694
   272
    by (auto simp: prod_algebra_def)
hoelzl@47694
   273
  moreover
hoelzl@47694
   274
  from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
hoelzl@47694
   275
    by (rule PiE_cong)
hoelzl@47694
   276
  with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   277
    by (simp add: prod_emb_def)
hoelzl@47694
   278
  moreover
hoelzl@47694
   279
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
hoelzl@47694
   280
    by auto
hoelzl@47694
   281
  ultimately show "A \<in> prod_algebra I F"
hoelzl@47694
   282
    apply (simp add: prod_algebra_def image_iff)
hoelzl@47694
   283
    apply (intro exI[of _ J] exI[of _ G] conjI)
hoelzl@47694
   284
    apply auto
hoelzl@47694
   285
    done
hoelzl@41689
   286
qed
hoelzl@41689
   287
hoelzl@50104
   288
lemma prod_algebra_cong:
hoelzl@50104
   289
  assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
hoelzl@50104
   290
  shows "prod_algebra I M = prod_algebra J N"
hoelzl@50104
   291
proof -
hoelzl@50104
   292
  have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
hoelzl@50104
   293
    using sets_eq_imp_space_eq[OF sets] by auto
hoelzl@50104
   294
  with sets show ?thesis unfolding `I = J`
hoelzl@50104
   295
    by (intro antisym prod_algebra_mono) auto
hoelzl@50104
   296
qed
hoelzl@50104
   297
hoelzl@50104
   298
lemma space_in_prod_algebra:
hoelzl@50104
   299
  "(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
hoelzl@50104
   300
proof cases
hoelzl@50104
   301
  assume "I = {}" then show ?thesis
hoelzl@50104
   302
    by (auto simp add: prod_algebra_def image_iff prod_emb_def)
hoelzl@50104
   303
next
hoelzl@50104
   304
  assume "I \<noteq> {}"
hoelzl@50104
   305
  then obtain i where "i \<in> I" by auto
hoelzl@50104
   306
  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))"
hoelzl@50123
   307
    by (auto simp: prod_emb_def)
hoelzl@50104
   308
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@50104
   309
    using `i \<in> I` by (intro prod_algebraI) auto
hoelzl@50104
   310
  finally show ?thesis .
hoelzl@50104
   311
qed
hoelzl@50104
   312
hoelzl@47694
   313
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   314
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
hoelzl@47694
   315
hoelzl@47694
   316
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)"
hoelzl@47694
   317
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
hoelzl@41689
   318
hoelzl@47694
   319
lemma sets_PiM_single: "sets (PiM I M) =
hoelzl@47694
   320
    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)}"
hoelzl@47694
   321
    (is "_ = sigma_sets ?\<Omega> ?R")
hoelzl@47694
   322
  unfolding sets_PiM
hoelzl@47694
   323
proof (rule sigma_sets_eqI)
hoelzl@47694
   324
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
hoelzl@47694
   325
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   326
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   327
  show "A \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   328
  proof cases
hoelzl@47694
   329
    assume "I = {}"
hoelzl@47694
   330
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
hoelzl@47694
   331
    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
hoelzl@47694
   332
  next
hoelzl@47694
   333
    assume "I \<noteq> {}"
hoelzl@47694
   334
    with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})"
hoelzl@50123
   335
      by (auto simp: prod_emb_def)
hoelzl@47694
   336
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   337
      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
hoelzl@47694
   338
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
hoelzl@47694
   339
  qed
hoelzl@47694
   340
next
hoelzl@47694
   341
  fix A assume "A \<in> ?R"
hoelzl@47694
   342
  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)" 
hoelzl@47694
   343
    by auto
hoelzl@47694
   344
  then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
hoelzl@50123
   345
     by (auto simp: prod_emb_def)
hoelzl@47694
   346
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
hoelzl@47694
   347
    using A by (intro sigma_sets.Basic prod_algebraI) auto
hoelzl@47694
   348
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
hoelzl@47694
   349
qed
hoelzl@47694
   350
hoelzl@47694
   351
lemma sets_PiM_I:
hoelzl@47694
   352
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
hoelzl@47694
   353
  shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   354
proof cases
hoelzl@47694
   355
  assume "J = {}"
hoelzl@47694
   356
  then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
hoelzl@47694
   357
    by (auto simp: prod_emb_def)
hoelzl@47694
   358
  then show ?thesis
hoelzl@47694
   359
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
hoelzl@47694
   360
next
hoelzl@47694
   361
  assume "J \<noteq> {}" with assms show ?thesis
hoelzl@50003
   362
    by (force simp add: sets_PiM prod_algebra_def)
hoelzl@40859
   363
qed
hoelzl@40859
   364
hoelzl@47694
   365
lemma measurable_PiM:
hoelzl@47694
   366
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   367
  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>
hoelzl@47694
   368
    f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" 
hoelzl@47694
   369
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   370
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   371
proof (rule measurable_sigma_sets)
hoelzl@47694
   372
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   373
  from prod_algebraE[OF this] guess J X .
hoelzl@47694
   374
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
hoelzl@47694
   375
qed
hoelzl@47694
   376
hoelzl@47694
   377
lemma measurable_PiM_Collect:
hoelzl@47694
   378
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   379
  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>
hoelzl@47694
   380
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 
hoelzl@47694
   381
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   382
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   383
proof (rule measurable_sigma_sets)
hoelzl@47694
   384
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   385
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@50123
   386
  then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
hoelzl@50123
   387
    using space by (auto simp: prod_emb_def del: PiE_I)
hoelzl@47694
   388
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
hoelzl@47694
   389
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@41689
   390
qed
hoelzl@41095
   391
hoelzl@47694
   392
lemma measurable_PiM_single:
hoelzl@47694
   393
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   394
  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" 
hoelzl@47694
   395
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   396
  using sets_PiM_single
hoelzl@47694
   397
proof (rule measurable_sigma_sets)
hoelzl@47694
   398
  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)}"
hoelzl@47694
   399
  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)"
hoelzl@47694
   400
    by auto
hoelzl@47694
   401
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
hoelzl@47694
   402
  also have "\<dots> \<in> sets N" using B by (rule sets)
hoelzl@47694
   403
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@47694
   404
qed (auto simp: space)
hoelzl@40859
   405
hoelzl@50099
   406
lemma measurable_PiM_single':
hoelzl@50099
   407
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
hoelzl@50099
   408
    and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@50099
   409
  shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@50099
   410
proof (rule measurable_PiM_single)
hoelzl@50099
   411
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@50099
   412
  then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
hoelzl@50099
   413
    by auto
hoelzl@50099
   414
  then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
hoelzl@50099
   415
    using A f by (auto intro!: measurable_sets)
hoelzl@50099
   416
qed fact
hoelzl@50099
   417
hoelzl@50003
   418
lemma sets_PiM_I_finite[measurable]:
hoelzl@47694
   419
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
hoelzl@47694
   420
  shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   421
  using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
hoelzl@47694
   422
hoelzl@50021
   423
lemma measurable_component_singleton:
hoelzl@41689
   424
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
hoelzl@41689
   425
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   426
  fix A assume "A \<in> sets (M i)"
hoelzl@47694
   427
  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)"
hoelzl@47694
   428
    using sets_into_space `i \<in> I`
hoelzl@47694
   429
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
hoelzl@41689
   430
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
hoelzl@47694
   431
    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
hoelzl@47694
   432
qed (insert `i \<in> I`, auto simp: space_PiM)
hoelzl@47694
   433
hoelzl@50021
   434
lemma measurable_component_singleton'[measurable_app]:
hoelzl@50021
   435
  assumes f: "f \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@50021
   436
  assumes i: "i \<in> I"
hoelzl@50021
   437
  shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
hoelzl@50021
   438
  using measurable_compose[OF f measurable_component_singleton, OF i] .
hoelzl@50021
   439
hoelzl@50099
   440
lemma measurable_PiM_component_rev[measurable (raw)]:
hoelzl@50099
   441
  "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
hoelzl@50099
   442
  by simp
hoelzl@50099
   443
hoelzl@50021
   444
lemma measurable_nat_case[measurable (raw)]:
hoelzl@50021
   445
  assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
hoelzl@50021
   446
    "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
hoelzl@50021
   447
  shows "(\<lambda>x. nat_case (f x) (g x) i) \<in> measurable M N"
hoelzl@50021
   448
  by (cases i) simp_all
hoelzl@50021
   449
hoelzl@50099
   450
lemma measurable_nat_case'[measurable (raw)]:
hoelzl@50099
   451
  assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
hoelzl@50099
   452
  shows "(\<lambda>x. nat_case (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
hoelzl@50099
   453
  using fg[THEN measurable_space]
hoelzl@50123
   454
  by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
hoelzl@50099
   455
hoelzl@50003
   456
lemma measurable_add_dim[measurable]:
hoelzl@49776
   457
  "(\<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)"
hoelzl@47694
   458
    (is "?f \<in> measurable ?P ?I")
hoelzl@47694
   459
proof (rule measurable_PiM_single)
hoelzl@47694
   460
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
hoelzl@47694
   461
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
hoelzl@47694
   462
    (if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
hoelzl@47694
   463
    using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
hoelzl@47694
   464
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   465
    using A j
hoelzl@47694
   466
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@47694
   467
  finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
hoelzl@50123
   468
qed (auto simp: space_pair_measure space_PiM PiE_def)
hoelzl@41661
   469
hoelzl@50003
   470
lemma measurable_component_update:
hoelzl@50003
   471
  "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)"
hoelzl@50003
   472
  by simp
hoelzl@50003
   473
hoelzl@50003
   474
lemma measurable_merge[measurable]:
hoelzl@49780
   475
  "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)"
hoelzl@47694
   476
    (is "?f \<in> measurable ?P ?U")
hoelzl@47694
   477
proof (rule measurable_PiM_single)
hoelzl@47694
   478
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
hoelzl@49780
   479
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
hoelzl@47694
   480
    (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)"
hoelzl@49776
   481
    by (auto simp: merge_def)
hoelzl@47694
   482
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   483
    using A
hoelzl@47694
   484
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@49780
   485
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
hoelzl@50123
   486
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
hoelzl@42988
   487
hoelzl@50003
   488
lemma measurable_restrict[measurable (raw)]:
hoelzl@47694
   489
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
hoelzl@47694
   490
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@47694
   491
proof (rule measurable_PiM_single)
hoelzl@47694
   492
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@47694
   493
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
hoelzl@47694
   494
    by auto
hoelzl@47694
   495
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
hoelzl@47694
   496
    using A X by (auto intro!: measurable_sets)
hoelzl@50123
   497
qed (insert X, auto simp add: PiE_def dest: measurable_space)
hoelzl@47694
   498
immler@50038
   499
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)"
immler@50038
   500
  by (intro measurable_restrict measurable_component_singleton) auto
immler@50038
   501
immler@50038
   502
lemma measurable_prod_emb[intro, simp]:
immler@50038
   503
  "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)"
immler@50038
   504
  unfolding prod_emb_def space_PiM[symmetric]
immler@50038
   505
  by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
immler@50038
   506
hoelzl@50003
   507
lemma sets_in_Pi_aux:
hoelzl@50003
   508
  "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50003
   509
  {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
hoelzl@50003
   510
  by (simp add: subset_eq Pi_iff)
hoelzl@50003
   511
hoelzl@50003
   512
lemma sets_in_Pi[measurable (raw)]:
hoelzl@50003
   513
  "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
hoelzl@50003
   514
  (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50003
   515
  Sigma_Algebra.pred N (\<lambda>x. f x \<in> Pi I F)"
hoelzl@50003
   516
  unfolding pred_def
hoelzl@50003
   517
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
hoelzl@50003
   518
hoelzl@50003
   519
lemma sets_in_extensional_aux:
hoelzl@50003
   520
  "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
hoelzl@50003
   521
proof -
hoelzl@50003
   522
  have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
hoelzl@50003
   523
    by (auto simp add: extensional_def space_PiM)
hoelzl@50003
   524
  then show ?thesis by simp
hoelzl@50003
   525
qed
hoelzl@50003
   526
hoelzl@50003
   527
lemma sets_in_extensional[measurable (raw)]:
hoelzl@50003
   528
  "f \<in> measurable N (PiM I M) \<Longrightarrow> Sigma_Algebra.pred N (\<lambda>x. f x \<in> extensional I)"
hoelzl@50003
   529
  unfolding pred_def
hoelzl@50003
   530
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
hoelzl@50003
   531
hoelzl@47694
   532
locale product_sigma_finite =
hoelzl@47694
   533
  fixes M :: "'i \<Rightarrow> 'a measure"
hoelzl@41689
   534
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   535
hoelzl@41689
   536
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   537
  by (rule sigma_finite_measures)
hoelzl@40859
   538
hoelzl@47694
   539
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@47694
   540
  fixes I :: "'i set"
hoelzl@47694
   541
  assumes finite_index: "finite I"
hoelzl@41689
   542
hoelzl@40859
   543
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   544
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   545
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
hoelzl@47694
   546
    (\<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>
hoelzl@47694
   547
    (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
hoelzl@40859
   548
proof -
hoelzl@47694
   549
  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>)"
hoelzl@47694
   550
    using M.sigma_finite_incseq by metis
hoelzl@40859
   551
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@47694
   552
  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>"
hoelzl@40859
   553
    by auto
hoelzl@40859
   554
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
hoelzl@47694
   555
  note space_PiM[simp]
hoelzl@40859
   556
  show ?thesis
hoelzl@41981
   557
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   558
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   559
  next
hoelzl@47694
   560
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   561
  next
hoelzl@50123
   562
    fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
hoelzl@50123
   563
      by (auto simp: PiE_def dest!: sets_into_space)
hoelzl@40859
   564
  next
hoelzl@47694
   565
    fix f assume "f \<in> space (PiM I M)"
hoelzl@41981
   566
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@50123
   567
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
hoelzl@40859
   568
  next
hoelzl@40859
   569
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   570
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   571
  qed
hoelzl@40859
   572
qed
hoelzl@40859
   573
hoelzl@49780
   574
lemma
hoelzl@49780
   575
  shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
hoelzl@49780
   576
    and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
hoelzl@49780
   577
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
hoelzl@49780
   578
hoelzl@49780
   579
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
hoelzl@49780
   580
proof -
hoelzl@49780
   581
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
hoelzl@49780
   582
  have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
hoelzl@49780
   583
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49780
   584
    show "positive (PiM {} M) ?\<mu>"
hoelzl@49780
   585
      by (auto simp: positive_def)
hoelzl@49780
   586
    show "countably_additive (PiM {} M) ?\<mu>"
hoelzl@49780
   587
      by (rule countably_additiveI_finite)
hoelzl@49780
   588
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
hoelzl@49780
   589
  qed (auto simp: prod_emb_def)
hoelzl@49780
   590
  also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
hoelzl@49780
   591
    by (auto simp: prod_emb_def)
hoelzl@49780
   592
  finally show ?thesis
hoelzl@49780
   593
    by simp
hoelzl@49780
   594
qed
hoelzl@49780
   595
hoelzl@49780
   596
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
hoelzl@49780
   597
  by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
hoelzl@49780
   598
hoelzl@49776
   599
lemma (in product_sigma_finite) emeasure_PiM:
hoelzl@49776
   600
  "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))"
hoelzl@49776
   601
proof (induct I arbitrary: A rule: finite_induct)
hoelzl@40859
   602
  case (insert i I)
hoelzl@41689
   603
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   604
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   605
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@41661
   606
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@47694
   607
hoelzl@47694
   608
  let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
hoelzl@47694
   609
  let ?\<mu> = "emeasure ?P"
hoelzl@47694
   610
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
hoelzl@47694
   611
  let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
hoelzl@47694
   612
hoelzl@49776
   613
  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)) =
hoelzl@49776
   614
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
hoelzl@49776
   615
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49776
   616
    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))"
hoelzl@49776
   617
    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
hoelzl@49776
   618
    let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
hoelzl@49776
   619
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
hoelzl@49776
   620
    have "?\<mu> ?p =
hoelzl@49776
   621
      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))"
hoelzl@49776
   622
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
hoelzl@49776
   623
    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))"
hoelzl@49776
   624
      using J E[rule_format, THEN sets_into_space]
hoelzl@50123
   625
      by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
hoelzl@49776
   626
    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))) =
hoelzl@49776
   627
      emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
hoelzl@49776
   628
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
hoelzl@49776
   629
    also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
hoelzl@49776
   630
      using J E[rule_format, THEN sets_into_space]
hoelzl@50123
   631
      by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
hoelzl@49776
   632
    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)) =
hoelzl@49776
   633
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
hoelzl@49776
   634
      using E by (subst insert) (auto intro!: setprod_cong)
hoelzl@49776
   635
    also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
hoelzl@49776
   636
       emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
hoelzl@49776
   637
      using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
hoelzl@49776
   638
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
hoelzl@49776
   639
      using insert(1,2) J E by (intro setprod_mono_one_right) auto
hoelzl@49776
   640
    finally show "?\<mu> ?p = \<dots>" .
hoelzl@47694
   641
hoelzl@49776
   642
    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))"
hoelzl@50123
   643
      using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff PiE_def)
hoelzl@49776
   644
  next
hoelzl@49776
   645
    show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
hoelzl@49776
   646
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
hoelzl@49776
   647
  next
hoelzl@49776
   648
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
hoelzl@49776
   649
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
hoelzl@49776
   650
      using insert by auto
hoelzl@49776
   651
  qed (auto intro!: setprod_cong)
hoelzl@49776
   652
  with insert show ?case
hoelzl@49776
   653
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space)
hoelzl@50003
   654
qed simp
hoelzl@47694
   655
hoelzl@49776
   656
lemma (in product_sigma_finite) sigma_finite: 
hoelzl@49776
   657
  assumes "finite I"
hoelzl@49776
   658
  shows "sigma_finite_measure (PiM I M)"
hoelzl@49776
   659
proof -
hoelzl@49776
   660
  interpret finite_product_sigma_finite M I by default fact
hoelzl@49776
   661
hoelzl@49776
   662
  from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@49776
   663
  then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
hoelzl@49776
   664
    "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
hoelzl@49776
   665
    "(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
hoelzl@49776
   666
    "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
hoelzl@47694
   667
    by blast+
hoelzl@49776
   668
  let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"
hoelzl@47694
   669
hoelzl@49776
   670
  show ?thesis
hoelzl@47694
   671
  proof (unfold_locales, intro exI[of _ ?F] conjI allI)
hoelzl@49776
   672
    show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
hoelzl@47694
   673
  next
hoelzl@49776
   674
    from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
hoelzl@47694
   675
  next
hoelzl@47694
   676
    fix j
hoelzl@49776
   677
    from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
hoelzl@49776
   678
    show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
hoelzl@49776
   679
      by (subst emeasure_PiM) auto
hoelzl@40859
   680
  qed
hoelzl@40859
   681
qed
hoelzl@40859
   682
hoelzl@47694
   683
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
hoelzl@47694
   684
  using sigma_finite[OF finite_index] .
hoelzl@40859
   685
hoelzl@40859
   686
lemma (in finite_product_sigma_finite) measure_times:
hoelzl@47694
   687
  "(\<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))"
hoelzl@47694
   688
  using emeasure_PiM[OF finite_index] by auto
hoelzl@41096
   689
hoelzl@40859
   690
lemma (in product_sigma_finite) positive_integral_empty:
hoelzl@41981
   691
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
hoelzl@41981
   692
  shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
hoelzl@40859
   693
proof -
hoelzl@41689
   694
  interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
hoelzl@47694
   695
  have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
hoelzl@40859
   696
    using assms by (subst measure_times) auto
hoelzl@40859
   697
  then show ?thesis
hoelzl@47694
   698
    unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
hoelzl@47694
   699
  proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
hoelzl@41981
   700
    show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
hoelzl@44928
   701
      by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
hoelzl@41981
   702
    show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
hoelzl@44928
   703
      by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
hoelzl@40859
   704
  qed
hoelzl@40859
   705
qed
hoelzl@40859
   706
hoelzl@47694
   707
lemma (in product_sigma_finite) distr_merge:
hoelzl@40859
   708
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@49780
   709
  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"
hoelzl@47694
   710
   (is "?D = ?P")
hoelzl@40859
   711
proof -
hoelzl@41689
   712
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   713
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@40859
   714
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   715
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   716
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   717
  let ?g = "merge I J"
hoelzl@47694
   718
hoelzl@41661
   719
  from IJ.sigma_finite_pairs obtain F where
hoelzl@41661
   720
    F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
hoelzl@41981
   721
       "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
hoelzl@47694
   722
       "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
hoelzl@47694
   723
       "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
hoelzl@41661
   724
    by auto
hoelzl@41661
   725
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
hoelzl@47694
   726
  
hoelzl@47694
   727
  show ?thesis
hoelzl@47694
   728
  proof (rule measure_eqI_generator_eq[symmetric])
hoelzl@47694
   729
    show "Int_stable (prod_algebra (I \<union> J) M)"
hoelzl@47694
   730
      by (rule Int_stable_prod_algebra)
hoelzl@47694
   731
    show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
hoelzl@47694
   732
      by (rule prod_algebra_sets_into_space)
hoelzl@47694
   733
    show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   734
      by (rule sets_PiM)
hoelzl@47694
   735
    then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   736
      by simp
hoelzl@47694
   737
hoelzl@47694
   738
    show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
hoelzl@47694
   739
      using fin by (auto simp: prod_algebra_eq_finite)
hoelzl@47694
   740
    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))"
hoelzl@47694
   741
      using F(3) by (simp add: space_PiM)
hoelzl@41981
   742
  next
hoelzl@41981
   743
    fix k
hoelzl@47694
   744
    from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
hoelzl@47694
   745
    show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
hoelzl@41661
   746
  next
hoelzl@47694
   747
    fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
hoelzl@50003
   748
    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)"
hoelzl@47694
   749
      by (auto simp add: prod_algebra_eq_finite)
hoelzl@47694
   750
    let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
hoelzl@47694
   751
    let ?X = "?g -` A \<inter> space ?B"
hoelzl@47694
   752
    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)"
hoelzl@50003
   753
      using F[rule_format, THEN sets_into_space] by (force simp: space_PiM)+
hoelzl@47694
   754
    then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
hoelzl@47694
   755
      unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
hoelzl@47694
   756
    have "emeasure ?D A = emeasure ?B ?X"
hoelzl@47694
   757
      using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
hoelzl@47694
   758
    also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
hoelzl@50003
   759
      using `finite J` `finite I` F unfolding X
hoelzl@50123
   760
      by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
hoelzl@47694
   761
    also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
hoelzl@41661
   762
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
hoelzl@47694
   763
    also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
hoelzl@41661
   764
      using `finite J` `finite I` F unfolding A
hoelzl@41661
   765
      by (intro IJ.measure_times[symmetric]) auto
hoelzl@47694
   766
    finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
hoelzl@47694
   767
  qed
hoelzl@41661
   768
qed
hoelzl@41026
   769
hoelzl@41026
   770
lemma (in product_sigma_finite) product_positive_integral_fold:
hoelzl@47694
   771
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
hoelzl@41689
   772
  and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41689
   773
  shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
hoelzl@49780
   774
    (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
hoelzl@41026
   775
proof -
hoelzl@41689
   776
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   777
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41831
   778
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   779
  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)"
hoelzl@49776
   780
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
hoelzl@41661
   781
  show ?thesis
hoelzl@47694
   782
    apply (subst distr_merge[OF IJ, symmetric])
hoelzl@49776
   783
    apply (subst positive_integral_distr[OF measurable_merge f])
hoelzl@49999
   784
    apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
hoelzl@47694
   785
    apply simp
hoelzl@47694
   786
    done
hoelzl@40859
   787
qed
hoelzl@40859
   788
hoelzl@47694
   789
lemma (in product_sigma_finite) distr_singleton:
hoelzl@47694
   790
  "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
hoelzl@47694
   791
proof (intro measure_eqI[symmetric])
hoelzl@41831
   792
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   793
  fix A assume A: "A \<in> sets (M i)"
hoelzl@47694
   794
  moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
hoelzl@47694
   795
    using sets_into_space by (auto simp: space_PiM)
hoelzl@47694
   796
  ultimately show "emeasure (M i) A = emeasure ?D A"
hoelzl@47694
   797
    using A I.measure_times[of "\<lambda>_. A"]
hoelzl@47694
   798
    by (simp add: emeasure_distr measurable_component_singleton)
hoelzl@47694
   799
qed simp
hoelzl@41831
   800
hoelzl@41026
   801
lemma (in product_sigma_finite) product_positive_integral_singleton:
hoelzl@40859
   802
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   803
  shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
hoelzl@40859
   804
proof -
hoelzl@41689
   805
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   806
  from f show ?thesis
hoelzl@47694
   807
    apply (subst distr_singleton[symmetric])
hoelzl@47694
   808
    apply (subst positive_integral_distr[OF measurable_component_singleton])
hoelzl@47694
   809
    apply simp_all
hoelzl@47694
   810
    done
hoelzl@40859
   811
qed
hoelzl@40859
   812
hoelzl@41096
   813
lemma (in product_sigma_finite) product_positive_integral_insert:
hoelzl@49780
   814
  assumes I[simp]: "finite I" "i \<notin> I"
hoelzl@41689
   815
    and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@41689
   816
  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))"
hoelzl@41096
   817
proof -
hoelzl@41689
   818
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41689
   819
  interpret i: finite_product_sigma_finite M "{i}" by default auto
hoelzl@41689
   820
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
hoelzl@41689
   821
    using f by auto
hoelzl@41096
   822
  show ?thesis
hoelzl@49780
   823
    unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
hoelzl@49780
   824
  proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
hoelzl@47694
   825
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@49780
   826
    let ?f = "\<lambda>y. f (x(i := y))"
hoelzl@49780
   827
    show "?f \<in> borel_measurable (M i)"
hoelzl@47694
   828
      using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
hoelzl@47694
   829
      unfolding comp_def .
hoelzl@49780
   830
    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)"
hoelzl@49780
   831
      using x
hoelzl@49780
   832
      by (auto intro!: positive_integral_cong arg_cong[where f=f]
hoelzl@50123
   833
               simp add: space_PiM extensional_def PiE_def)
hoelzl@41096
   834
  qed
hoelzl@41096
   835
qed
hoelzl@41096
   836
hoelzl@41096
   837
lemma (in product_sigma_finite) product_positive_integral_setprod:
hoelzl@43920
   838
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
hoelzl@41096
   839
  assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41981
   840
  and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
hoelzl@41689
   841
  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))"
hoelzl@41096
   842
using assms proof induct
hoelzl@41096
   843
  case (insert i I)
hoelzl@41096
   844
  note `finite I`[intro, simp]
hoelzl@41689
   845
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41096
   846
  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))"
hoelzl@41096
   847
    using insert by (auto intro!: setprod_cong)
hoelzl@41689
   848
  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)"
hoelzl@41096
   849
    using sets_into_space insert
hoelzl@47694
   850
    by (intro borel_measurable_ereal_setprod
hoelzl@41689
   851
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
hoelzl@41096
   852
       auto
hoelzl@41981
   853
  then show ?case
hoelzl@41981
   854
    apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
hoelzl@47694
   855
    apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
hoelzl@47694
   856
    apply (subst positive_integral_cmult)
hoelzl@47694
   857
    apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
hoelzl@41981
   858
    done
hoelzl@47694
   859
qed (simp add: space_PiM)
hoelzl@41096
   860
hoelzl@41026
   861
lemma (in product_sigma_finite) product_integral_singleton:
hoelzl@41026
   862
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   863
  shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
hoelzl@41026
   864
proof -
hoelzl@41689
   865
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@43920
   866
  have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
hoelzl@43920
   867
    "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
hoelzl@41026
   868
    using assms by auto
hoelzl@41026
   869
  show ?thesis
hoelzl@41689
   870
    unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
hoelzl@41026
   871
qed
hoelzl@50104
   872
lemma (in product_sigma_finite) distr_component:
hoelzl@50104
   873
  "distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P")
hoelzl@50104
   874
proof (intro measure_eqI[symmetric])
hoelzl@50104
   875
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@50104
   876
hoelzl@50104
   877
  have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
hoelzl@50104
   878
    by (auto simp: extensional_def restrict_def)
hoelzl@50104
   879
hoelzl@50104
   880
  fix A assume A: "A \<in> sets ?P"
hoelzl@50104
   881
  then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)" 
hoelzl@50104
   882
    by simp
hoelzl@50104
   883
  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)" 
hoelzl@50123
   884
    by (intro positive_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
hoelzl@50104
   885
  also have "\<dots> = emeasure ?D A"
hoelzl@50104
   886
    using A by (simp add: product_positive_integral_singleton emeasure_distr)
hoelzl@50104
   887
  finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" .
hoelzl@50104
   888
qed simp
hoelzl@41026
   889
hoelzl@41026
   890
lemma (in product_sigma_finite) product_integral_fold:
hoelzl@41026
   891
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@41689
   892
  and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
hoelzl@49780
   893
  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)"
hoelzl@41026
   894
proof -
hoelzl@41689
   895
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   896
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41026
   897
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   898
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   899
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   900
  let ?M = "merge I J"
hoelzl@41689
   901
  let ?f = "\<lambda>x. f (?M x)"
hoelzl@47694
   902
  from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@47694
   903
    by auto
hoelzl@49780
   904
  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)"
hoelzl@49776
   905
    using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
hoelzl@47694
   906
  have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
hoelzl@49776
   907
    by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
hoelzl@41026
   908
  show ?thesis
hoelzl@47694
   909
    apply (subst distr_merge[symmetric, OF IJ fin])
hoelzl@49776
   910
    apply (subst integral_distr[OF measurable_merge f_borel])
hoelzl@47694
   911
    apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
hoelzl@47694
   912
    apply simp
hoelzl@47694
   913
    done
hoelzl@41026
   914
qed
hoelzl@41026
   915
hoelzl@49776
   916
lemma (in product_sigma_finite)
hoelzl@49776
   917
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
hoelzl@49776
   918
  shows emeasure_fold_integral:
hoelzl@49780
   919
    "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)
hoelzl@49776
   920
    and emeasure_fold_measurable:
hoelzl@49780
   921
    "(\<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)
hoelzl@49776
   922
proof -
hoelzl@49776
   923
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@49776
   924
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@49776
   925
  interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
hoelzl@49780
   926
  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)"
hoelzl@49776
   927
    by (intro measurable_sets[OF _ A] measurable_merge assms)
hoelzl@49776
   928
hoelzl@49776
   929
  show ?I
hoelzl@49776
   930
    apply (subst distr_merge[symmetric, OF IJ])
hoelzl@49776
   931
    apply (subst emeasure_distr[OF measurable_merge A])
hoelzl@49776
   932
    apply (subst J.emeasure_pair_measure_alt[OF merge])
hoelzl@49776
   933
    apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49776
   934
    done
hoelzl@49776
   935
hoelzl@49776
   936
  show ?B
hoelzl@49776
   937
    using IJ.measurable_emeasure_Pair1[OF merge]
hoelzl@49776
   938
    by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
hoelzl@49776
   939
qed
hoelzl@49776
   940
hoelzl@41096
   941
lemma (in product_sigma_finite) product_integral_insert:
hoelzl@47694
   942
  assumes I: "finite I" "i \<notin> I"
hoelzl@41689
   943
    and f: "integrable (Pi\<^isub>M (insert i I) M) f"
hoelzl@41689
   944
  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)"
hoelzl@41096
   945
proof -
hoelzl@47694
   946
  have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
hoelzl@47694
   947
    by simp
hoelzl@49780
   948
  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)"
hoelzl@47694
   949
    using f I by (intro product_integral_fold) auto
hoelzl@47694
   950
  also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
hoelzl@47694
   951
  proof (rule integral_cong, subst product_integral_singleton[symmetric])
hoelzl@47694
   952
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@47694
   953
    have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@47694
   954
      using f by auto
hoelzl@47694
   955
    show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
hoelzl@47694
   956
      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
hoelzl@47694
   957
      unfolding comp_def .
hoelzl@49780
   958
    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)"
hoelzl@50123
   959
      by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
hoelzl@41096
   960
  qed
hoelzl@47694
   961
  finally show ?thesis .
hoelzl@41096
   962
qed
hoelzl@41096
   963
hoelzl@41096
   964
lemma (in product_sigma_finite) product_integrable_setprod:
hoelzl@41096
   965
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@41689
   966
  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
   967
  shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
hoelzl@41096
   968
proof -
hoelzl@41689
   969
  interpret finite_product_sigma_finite M I by default fact
hoelzl@41096
   970
  have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41689
   971
    using integrable unfolding integrable_def by auto
hoelzl@47694
   972
  have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@47694
   973
    using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
hoelzl@41689
   974
  moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
hoelzl@41096
   975
  proof (unfold integrable_def, intro conjI)
hoelzl@47694
   976
    show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@41096
   977
      using borel by auto
hoelzl@47694
   978
    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)"
hoelzl@43920
   979
      by (simp add: setprod_ereal abs_setprod)
hoelzl@43920
   980
    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
hoelzl@41096
   981
      using f by (subst product_positive_integral_setprod) auto
hoelzl@41981
   982
    also have "\<dots> < \<infinity>"
hoelzl@47694
   983
      using integrable[THEN integrable_abs]
hoelzl@47694
   984
      by (simp add: setprod_PInf integrable_def positive_integral_positive)
hoelzl@47694
   985
    finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
hoelzl@47694
   986
    have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
hoelzl@41981
   987
      by (intro positive_integral_cong_pos) auto
hoelzl@47694
   988
    then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
hoelzl@41096
   989
  qed
hoelzl@41096
   990
  ultimately show ?thesis
hoelzl@41096
   991
    by (rule integrable_abs_iff[THEN iffD1])
hoelzl@41096
   992
qed
hoelzl@41096
   993
hoelzl@41096
   994
lemma (in product_sigma_finite) product_integral_setprod:
hoelzl@41096
   995
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@49780
   996
  assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
   997
  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))"
hoelzl@49780
   998
using assms proof induct
hoelzl@49780
   999
  case empty
hoelzl@49780
  1000
  interpret finite_measure "Pi\<^isub>M {} M"
hoelzl@49780
  1001
    by rule (simp add: space_PiM)
hoelzl@49780
  1002
  show ?case by (simp add: space_PiM measure_def)
hoelzl@41096
  1003
next
hoelzl@41096
  1004
  case (insert i I)
hoelzl@41096
  1005
  then have iI: "finite (insert i I)" by auto
hoelzl@41096
  1006
  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
hoelzl@41689
  1007
    integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
hoelzl@49780
  1008
    by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
hoelzl@41689
  1009
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41096
  1010
  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))"
hoelzl@41096
  1011
    using `i \<notin> I` by (auto intro!: setprod_cong)
hoelzl@41096
  1012
  show ?case
hoelzl@49780
  1013
    unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
hoelzl@47694
  1014
    by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
hoelzl@41096
  1015
qed
hoelzl@41096
  1016
hoelzl@49776
  1017
lemma sets_Collect_single:
hoelzl@49776
  1018
  "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)"
hoelzl@50003
  1019
  by simp
hoelzl@49776
  1020
hoelzl@49776
  1021
lemma sigma_prod_algebra_sigma_eq_infinite:
hoelzl@49776
  1022
  fixes E :: "'i \<Rightarrow> 'a set set"
hoelzl@49779
  1023
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@49776
  1024
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@49776
  1025
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@49776
  1026
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@49776
  1027
  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}"
hoelzl@49776
  1028
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1029
proof
hoelzl@49776
  1030
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49776
  1031
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@50123
  1032
    using E_closed by (auto simp: space_PiM P_def subset_eq)
hoelzl@49776
  1033
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@49776
  1034
    by (simp add: space_PiM)
hoelzl@49776
  1035
  have "sets (PiM I M) =
hoelzl@49776
  1036
      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)}"
hoelzl@49776
  1037
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@49776
  1038
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@49776
  1039
  proof (safe intro!: sigma_sets_subset)
hoelzl@49776
  1040
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@49776
  1041
    then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@49776
  1042
      apply (subst measurable_iff_measure_of)
hoelzl@49776
  1043
      apply (simp_all add: P_closed)
hoelzl@49776
  1044
      using E_closed
hoelzl@49776
  1045
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1046
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1047
      apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
hoelzl@49776
  1048
      apply (rule_tac x=Aa in exI)
hoelzl@49776
  1049
      apply (auto simp: space_PiM)
hoelzl@49776
  1050
      done
hoelzl@49776
  1051
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@49776
  1052
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@49776
  1053
      by (simp add: E_generates)
hoelzl@49776
  1054
    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}"
hoelzl@49776
  1055
      using P_closed by (auto simp: space_PiM)
hoelzl@49776
  1056
    finally show "\<dots> \<in> sets ?P" .
hoelzl@49776
  1057
  qed
hoelzl@49776
  1058
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1059
    by (simp add: P_closed)
hoelzl@49776
  1060
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@49776
  1061
    unfolding P_def space_PiM[symmetric]
hoelzl@49776
  1062
    by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
hoelzl@49776
  1063
qed
hoelzl@49776
  1064
hoelzl@47694
  1065
lemma sigma_prod_algebra_sigma_eq:
hoelzl@49779
  1066
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
hoelzl@47694
  1067
  assumes "finite I"
hoelzl@49779
  1068
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@47694
  1069
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@47694
  1070
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@47694
  1071
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@47694
  1072
  defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
hoelzl@47694
  1073
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1074
proof
hoelzl@47694
  1075
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49779
  1076
  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
hoelzl@49779
  1077
  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"
hoelzl@49779
  1078
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
hoelzl@47694
  1079
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@50123
  1080
    using E_closed by (auto simp: space_PiM P_def subset_eq)
hoelzl@47694
  1081
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
  1082
    by (simp add: space_PiM)
hoelzl@47694
  1083
  have "sets (PiM I M) =
hoelzl@47694
  1084
      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)}"
hoelzl@47694
  1085
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@47694
  1086
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@47694
  1087
  proof (safe intro!: sigma_sets_subset)
hoelzl@47694
  1088
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@47694
  1089
    have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@47694
  1090
    proof (subst measurable_iff_measure_of)
hoelzl@47694
  1091
      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
hoelzl@50123
  1092
      from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto
hoelzl@47694
  1093
      show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1094
      proof
hoelzl@47694
  1095
        fix A assume A: "A \<in> E i"
hoelzl@47694
  1096
        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))"
hoelzl@50123
  1097
          using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm)
hoelzl@47694
  1098
        also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
hoelzl@47694
  1099
          by (intro PiE_cong) (simp add: S_union)
hoelzl@49779
  1100
        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))"
hoelzl@49779
  1101
          using T
hoelzl@50123
  1102
          apply (auto simp: PiE_iff bchoice_iff)
hoelzl@49779
  1103
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
hoelzl@49779
  1104
          apply (auto simp: bij_betw_def)
hoelzl@49779
  1105
          done
hoelzl@47694
  1106
        also have "\<dots> \<in> sets ?P"
hoelzl@47694
  1107
        proof (safe intro!: countable_UN)
hoelzl@49779
  1108
          fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
hoelzl@47694
  1109
            using A S_in_E
hoelzl@47694
  1110
            by (simp add: P_closed)
hoelzl@49779
  1111
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
hoelzl@47694
  1112
        qed
hoelzl@47694
  1113
        finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1114
          using P_closed by simp
hoelzl@47694
  1115
      qed
hoelzl@47694
  1116
    qed
hoelzl@47694
  1117
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@47694
  1118
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1119
      by (simp add: E_generates)
hoelzl@47694
  1120
    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}"
hoelzl@47694
  1121
      using P_closed by (auto simp: space_PiM)
hoelzl@47694
  1122
    finally show "\<dots> \<in> sets ?P" .
hoelzl@47694
  1123
  qed
hoelzl@47694
  1124
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1125
    by (simp add: P_closed)
hoelzl@47694
  1126
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@47694
  1127
    using `finite I`
hoelzl@50003
  1128
    by (auto intro!: sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
hoelzl@47694
  1129
qed
hoelzl@47694
  1130
hoelzl@50104
  1131
lemma pair_measure_eq_distr_PiM:
hoelzl@50104
  1132
  fixes M1 :: "'a measure" and M2 :: "'a measure"
hoelzl@50104
  1133
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
hoelzl@50104
  1134
  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))"
hoelzl@50104
  1135
    (is "?P = ?D")
hoelzl@50104
  1136
proof (rule pair_measure_eqI[OF assms])
hoelzl@50104
  1137
  interpret B: product_sigma_finite "bool_case M1 M2"
hoelzl@50104
  1138
    unfolding product_sigma_finite_def using assms by (auto split: bool.split)
hoelzl@50104
  1139
  let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)"
hoelzl@50104
  1140
hoelzl@50104
  1141
  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)"
hoelzl@50104
  1142
    by auto
hoelzl@50104
  1143
  fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
hoelzl@50104
  1144
  have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))"
hoelzl@50104
  1145
    by (simp add: UNIV_bool ac_simps)
hoelzl@50104
  1146
  also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))"
hoelzl@50104
  1147
    using A B by (subst B.emeasure_PiM) (auto split: bool.split)
hoelzl@50104
  1148
  also have "Pi\<^isub>E UNIV (bool_case A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
hoelzl@50104
  1149
    using A[THEN sets_into_space] B[THEN sets_into_space]
hoelzl@50123
  1150
    by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
hoelzl@50104
  1151
  finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
hoelzl@50104
  1152
    using A B
hoelzl@50104
  1153
      measurable_component_singleton[of True UNIV "bool_case M1 M2"]
hoelzl@50104
  1154
      measurable_component_singleton[of False UNIV "bool_case M1 M2"]
hoelzl@50104
  1155
    by (subst emeasure_distr) (auto simp: measurable_pair_iff)
hoelzl@50104
  1156
qed simp
hoelzl@50104
  1157
hoelzl@47694
  1158
end