src/HOL/Probability/Finite_Product_Measure.thy
author hoelzl
Mon May 19 12:04:45 2014 +0200 (2014-05-19)
changeset 56993 e5366291d6aa
parent 56154 f0a927235162
child 56994 8d5e5ec1cac3
permissions -rw-r--r--
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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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>\<^sub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" and ne: "(\<Pi>\<^sub>E i\<in>I. S i) \<noteq> {}"
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  and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
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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>\<^sub>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>\<^sub>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\<^sub>E I E \<times> Pi\<^sub>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\<^sub>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>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>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\<^sub>E I E) = Pi\<^sub>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\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
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  using assms sets.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>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>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\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>E i\<in>I. space (M i))"
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  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>\<^sub>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>\<^sub>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.sets_into_space) auto
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  show "A \<in> ?R" unfolding A using J sets.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>\<^sub>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>\<^sub>E i\<in>I. X i)"
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    by (simp add: prod_emb_PiE_same_index[OF sets.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\<^sub>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.sets_into_space] by simp
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lemma Int_stable_PiE: "Int_stable {Pi\<^sub>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\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
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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\<^sub>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\<^sub>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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    using sets.sets_into_space by auto
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  then have "A = (\<Pi>\<^sub>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)
wenzelm@53374
   238
  moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
immler@50244
   239
    using sets.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
wenzelm@53015
   249
  have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^sub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter> 
hoelzl@47694
   250
      (if i \<in> K then F i else space (M i)))"
immler@50244
   251
    unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4)
immler@50244
   252
      B(5)[THEN sets.sets_into_space]
hoelzl@47694
   253
    apply (subst (1 2 3) prod_emb_PiE)
hoelzl@47694
   254
    apply (simp_all add: subset_eq PiE_Int)
hoelzl@47694
   255
    apply blast
hoelzl@47694
   256
    apply (intro PiE_cong)
hoelzl@47694
   257
    apply auto
hoelzl@47694
   258
    done
hoelzl@47694
   259
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   260
    using A B by (auto intro!: prod_algebraI)
hoelzl@47694
   261
  finally show "A \<inter> B \<in> prod_algebra I M" .
hoelzl@47694
   262
qed
hoelzl@47694
   263
hoelzl@47694
   264
lemma prod_algebra_mono:
hoelzl@47694
   265
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
hoelzl@47694
   266
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
hoelzl@47694
   267
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
hoelzl@47694
   268
proof
hoelzl@47694
   269
  fix A assume "A \<in> prod_algebra I E"
hoelzl@47694
   270
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
wenzelm@53015
   271
    and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
hoelzl@47694
   272
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
hoelzl@47694
   273
    by (auto simp: prod_algebra_def)
hoelzl@47694
   274
  moreover
wenzelm@53015
   275
  from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
hoelzl@47694
   276
    by (rule PiE_cong)
wenzelm@53015
   277
  with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
hoelzl@47694
   278
    by (simp add: prod_emb_def)
hoelzl@47694
   279
  moreover
hoelzl@47694
   280
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
hoelzl@47694
   281
    by auto
hoelzl@47694
   282
  ultimately show "A \<in> prod_algebra I F"
hoelzl@47694
   283
    apply (simp add: prod_algebra_def image_iff)
hoelzl@47694
   284
    apply (intro exI[of _ J] exI[of _ G] conjI)
hoelzl@47694
   285
    apply auto
hoelzl@47694
   286
    done
hoelzl@41689
   287
qed
hoelzl@41689
   288
hoelzl@50104
   289
lemma prod_algebra_cong:
hoelzl@50104
   290
  assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
hoelzl@50104
   291
  shows "prod_algebra I M = prod_algebra J N"
hoelzl@50104
   292
proof -
hoelzl@50104
   293
  have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
hoelzl@50104
   294
    using sets_eq_imp_space_eq[OF sets] by auto
hoelzl@50104
   295
  with sets show ?thesis unfolding `I = J`
hoelzl@50104
   296
    by (intro antisym prod_algebra_mono) auto
hoelzl@50104
   297
qed
hoelzl@50104
   298
hoelzl@50104
   299
lemma space_in_prod_algebra:
wenzelm@53015
   300
  "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
hoelzl@50104
   301
proof cases
hoelzl@50104
   302
  assume "I = {}" then show ?thesis
hoelzl@50104
   303
    by (auto simp add: prod_algebra_def image_iff prod_emb_def)
hoelzl@50104
   304
next
hoelzl@50104
   305
  assume "I \<noteq> {}"
hoelzl@50104
   306
  then obtain i where "i \<in> I" by auto
wenzelm@53015
   307
  then have "(\<Pi>\<^sub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i))"
hoelzl@50123
   308
    by (auto simp: prod_emb_def)
hoelzl@50104
   309
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@50104
   310
    using `i \<in> I` by (intro prod_algebraI) auto
hoelzl@50104
   311
  finally show ?thesis .
hoelzl@50104
   312
qed
hoelzl@50104
   313
wenzelm@53015
   314
lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   315
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
hoelzl@47694
   316
wenzelm@53015
   317
lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
hoelzl@47694
   318
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
hoelzl@41689
   319
hoelzl@47694
   320
lemma sets_PiM_single: "sets (PiM I M) =
wenzelm@53015
   321
    sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@47694
   322
    (is "_ = sigma_sets ?\<Omega> ?R")
hoelzl@47694
   323
  unfolding sets_PiM
hoelzl@47694
   324
proof (rule sigma_sets_eqI)
hoelzl@47694
   325
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
hoelzl@47694
   326
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   327
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   328
  show "A \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   329
  proof cases
hoelzl@47694
   330
    assume "I = {}"
hoelzl@47694
   331
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
hoelzl@47694
   332
    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
hoelzl@47694
   333
  next
hoelzl@47694
   334
    assume "I \<noteq> {}"
wenzelm@53015
   335
    with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
hoelzl@50123
   336
      by (auto simp: prod_emb_def)
hoelzl@47694
   337
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   338
      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
hoelzl@47694
   339
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
hoelzl@47694
   340
  qed
hoelzl@47694
   341
next
hoelzl@47694
   342
  fix A assume "A \<in> ?R"
wenzelm@53015
   343
  then obtain i B where A: "A = {f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" 
hoelzl@47694
   344
    by auto
wenzelm@53015
   345
  then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
hoelzl@50123
   346
     by (auto simp: prod_emb_def)
hoelzl@47694
   347
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
hoelzl@47694
   348
    using A by (intro sigma_sets.Basic prod_algebraI) auto
hoelzl@47694
   349
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
hoelzl@47694
   350
qed
hoelzl@47694
   351
hoelzl@47694
   352
lemma sets_PiM_I:
hoelzl@47694
   353
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
hoelzl@47694
   354
  shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   355
proof cases
hoelzl@47694
   356
  assume "J = {}"
hoelzl@47694
   357
  then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
hoelzl@47694
   358
    by (auto simp: prod_emb_def)
hoelzl@47694
   359
  then show ?thesis
hoelzl@47694
   360
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
hoelzl@47694
   361
next
hoelzl@47694
   362
  assume "J \<noteq> {}" with assms show ?thesis
hoelzl@50003
   363
    by (force simp add: sets_PiM prod_algebra_def)
hoelzl@40859
   364
qed
hoelzl@40859
   365
hoelzl@47694
   366
lemma measurable_PiM:
wenzelm@53015
   367
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   368
  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>
wenzelm@53015
   369
    f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N" 
hoelzl@47694
   370
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   371
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   372
proof (rule measurable_sigma_sets)
hoelzl@47694
   373
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   374
  from prod_algebraE[OF this] guess J X .
hoelzl@47694
   375
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
hoelzl@47694
   376
qed
hoelzl@47694
   377
hoelzl@47694
   378
lemma measurable_PiM_Collect:
wenzelm@53015
   379
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   380
  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
   381
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 
hoelzl@47694
   382
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   383
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   384
proof (rule measurable_sigma_sets)
hoelzl@47694
   385
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   386
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@50123
   387
  then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
hoelzl@50123
   388
    using space by (auto simp: prod_emb_def del: PiE_I)
hoelzl@47694
   389
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
hoelzl@47694
   390
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@41689
   391
qed
hoelzl@41095
   392
hoelzl@47694
   393
lemma measurable_PiM_single:
wenzelm@53015
   394
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   395
  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
   396
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   397
  using sets_PiM_single
hoelzl@47694
   398
proof (rule measurable_sigma_sets)
wenzelm@53015
   399
  fix A assume "A \<in> {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
wenzelm@53015
   400
  then obtain B i where "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
hoelzl@47694
   401
    by auto
hoelzl@47694
   402
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
hoelzl@47694
   403
  also have "\<dots> \<in> sets N" using B by (rule sets)
hoelzl@47694
   404
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@47694
   405
qed (auto simp: space)
hoelzl@40859
   406
hoelzl@50099
   407
lemma measurable_PiM_single':
hoelzl@50099
   408
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
wenzelm@53015
   409
    and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
wenzelm@53015
   410
  shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
hoelzl@50099
   411
proof (rule measurable_PiM_single)
hoelzl@50099
   412
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@50099
   413
  then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
hoelzl@50099
   414
    by auto
hoelzl@50099
   415
  then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
hoelzl@50099
   416
    using A f by (auto intro!: measurable_sets)
hoelzl@50099
   417
qed fact
hoelzl@50099
   418
hoelzl@50003
   419
lemma sets_PiM_I_finite[measurable]:
hoelzl@47694
   420
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
hoelzl@47694
   421
  shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
immler@50244
   422
  using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto
hoelzl@47694
   423
hoelzl@50021
   424
lemma measurable_component_singleton:
wenzelm@53015
   425
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
hoelzl@41689
   426
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   427
  fix A assume "A \<in> sets (M i)"
wenzelm@53015
   428
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. A)"
immler@50244
   429
    using sets.sets_into_space `i \<in> I`
hoelzl@47694
   430
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
wenzelm@53015
   431
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
hoelzl@47694
   432
    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
hoelzl@47694
   433
qed (insert `i \<in> I`, auto simp: space_PiM)
hoelzl@47694
   434
hoelzl@50021
   435
lemma measurable_component_singleton'[measurable_app]:
wenzelm@53015
   436
  assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
hoelzl@50021
   437
  assumes i: "i \<in> I"
hoelzl@50021
   438
  shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
hoelzl@50021
   439
  using measurable_compose[OF f measurable_component_singleton, OF i] .
hoelzl@50021
   440
hoelzl@50099
   441
lemma measurable_PiM_component_rev[measurable (raw)]:
hoelzl@50099
   442
  "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
hoelzl@50099
   443
  by simp
hoelzl@50099
   444
blanchet@55415
   445
lemma measurable_case_nat[measurable (raw)]:
hoelzl@50021
   446
  assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
hoelzl@50021
   447
    "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
blanchet@55415
   448
  shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
hoelzl@50021
   449
  by (cases i) simp_all
hoelzl@50021
   450
blanchet@55415
   451
lemma measurable_case_nat'[measurable (raw)]:
wenzelm@53015
   452
  assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
blanchet@55415
   453
  shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
hoelzl@50099
   454
  using fg[THEN measurable_space]
hoelzl@50123
   455
  by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
hoelzl@50099
   456
hoelzl@50003
   457
lemma measurable_add_dim[measurable]:
wenzelm@53015
   458
  "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
hoelzl@47694
   459
    (is "?f \<in> measurable ?P ?I")
hoelzl@47694
   460
proof (rule measurable_PiM_single)
hoelzl@47694
   461
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
hoelzl@47694
   462
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
wenzelm@53015
   463
    (if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
immler@50244
   464
    using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
hoelzl@47694
   465
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   466
    using A j
hoelzl@47694
   467
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
blanchet@55414
   468
  finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
hoelzl@50123
   469
qed (auto simp: space_pair_measure space_PiM PiE_def)
hoelzl@41661
   470
hoelzl@50003
   471
lemma measurable_component_update:
wenzelm@53015
   472
  "x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)"
hoelzl@50003
   473
  by simp
hoelzl@50003
   474
hoelzl@50003
   475
lemma measurable_merge[measurable]:
wenzelm@53015
   476
  "merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)"
hoelzl@47694
   477
    (is "?f \<in> measurable ?P ?U")
hoelzl@47694
   478
proof (rule measurable_PiM_single)
hoelzl@47694
   479
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
hoelzl@49780
   480
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
hoelzl@47694
   481
    (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
   482
    by (auto simp: merge_def)
hoelzl@47694
   483
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   484
    using A
hoelzl@47694
   485
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@49780
   486
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
hoelzl@50123
   487
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
hoelzl@42988
   488
hoelzl@50003
   489
lemma measurable_restrict[measurable (raw)]:
hoelzl@47694
   490
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
wenzelm@53015
   491
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
hoelzl@47694
   492
proof (rule measurable_PiM_single)
hoelzl@47694
   493
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@47694
   494
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
hoelzl@47694
   495
    by auto
hoelzl@47694
   496
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
hoelzl@47694
   497
    using A X by (auto intro!: measurable_sets)
hoelzl@50123
   498
qed (insert X, auto simp add: PiE_def dest: measurable_space)
hoelzl@47694
   499
wenzelm@53015
   500
lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
immler@50038
   501
  by (intro measurable_restrict measurable_component_singleton) auto
immler@50038
   502
immler@50038
   503
lemma measurable_prod_emb[intro, simp]:
wenzelm@53015
   504
  "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
immler@50038
   505
  unfolding prod_emb_def space_PiM[symmetric]
immler@50038
   506
  by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
immler@50038
   507
hoelzl@50003
   508
lemma sets_in_Pi_aux:
hoelzl@50003
   509
  "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
   510
  {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
hoelzl@50003
   511
  by (simp add: subset_eq Pi_iff)
hoelzl@50003
   512
hoelzl@50003
   513
lemma sets_in_Pi[measurable (raw)]:
hoelzl@50003
   514
  "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
hoelzl@50003
   515
  (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50387
   516
  Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
hoelzl@50003
   517
  unfolding pred_def
hoelzl@50003
   518
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
hoelzl@50003
   519
hoelzl@50003
   520
lemma sets_in_extensional_aux:
hoelzl@50003
   521
  "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
hoelzl@50003
   522
proof -
hoelzl@50003
   523
  have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
hoelzl@50003
   524
    by (auto simp add: extensional_def space_PiM)
hoelzl@50003
   525
  then show ?thesis by simp
hoelzl@50003
   526
qed
hoelzl@50003
   527
hoelzl@50003
   528
lemma sets_in_extensional[measurable (raw)]:
hoelzl@50387
   529
  "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
hoelzl@50003
   530
  unfolding pred_def
hoelzl@50003
   531
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
hoelzl@50003
   532
hoelzl@47694
   533
locale product_sigma_finite =
hoelzl@47694
   534
  fixes M :: "'i \<Rightarrow> 'a measure"
hoelzl@41689
   535
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   536
hoelzl@41689
   537
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   538
  by (rule sigma_finite_measures)
hoelzl@40859
   539
hoelzl@47694
   540
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@47694
   541
  fixes I :: "'i set"
hoelzl@47694
   542
  assumes finite_index: "finite I"
hoelzl@41689
   543
hoelzl@40859
   544
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   545
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   546
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
wenzelm@53015
   547
    (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and>
wenzelm@53015
   548
    (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
hoelzl@40859
   549
proof -
hoelzl@47694
   550
  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
   551
    using M.sigma_finite_incseq by metis
hoelzl@40859
   552
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@47694
   553
  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
   554
    by auto
wenzelm@53015
   555
  let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
hoelzl@47694
   556
  note space_PiM[simp]
hoelzl@40859
   557
  show ?thesis
hoelzl@41981
   558
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   559
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   560
  next
hoelzl@47694
   561
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   562
  next
hoelzl@50123
   563
    fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
immler@50244
   564
      by (auto simp: PiE_def dest!: sets.sets_into_space)
hoelzl@40859
   565
  next
hoelzl@47694
   566
    fix f assume "f \<in> space (PiM I M)"
hoelzl@41981
   567
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@50123
   568
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
hoelzl@40859
   569
  next
hoelzl@40859
   570
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   571
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   572
  qed
hoelzl@40859
   573
qed
hoelzl@40859
   574
hoelzl@49780
   575
lemma
wenzelm@53015
   576
  shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
wenzelm@53015
   577
    and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
hoelzl@49780
   578
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
hoelzl@49780
   579
hoelzl@49780
   580
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
hoelzl@49780
   581
proof -
hoelzl@49780
   582
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
wenzelm@53015
   583
  have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
hoelzl@49780
   584
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49780
   585
    show "positive (PiM {} M) ?\<mu>"
hoelzl@49780
   586
      by (auto simp: positive_def)
hoelzl@49780
   587
    show "countably_additive (PiM {} M) ?\<mu>"
immler@50244
   588
      by (rule sets.countably_additiveI_finite)
hoelzl@49780
   589
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
hoelzl@49780
   590
  qed (auto simp: prod_emb_def)
wenzelm@53015
   591
  also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
hoelzl@49780
   592
    by (auto simp: prod_emb_def)
hoelzl@49780
   593
  finally show ?thesis
hoelzl@49780
   594
    by simp
hoelzl@49780
   595
qed
hoelzl@49780
   596
hoelzl@49780
   597
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
hoelzl@49780
   598
  by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
hoelzl@49780
   599
hoelzl@49776
   600
lemma (in product_sigma_finite) emeasure_PiM:
wenzelm@53015
   601
  "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
hoelzl@49776
   602
proof (induct I arbitrary: A rule: finite_induct)
hoelzl@40859
   603
  case (insert i I)
hoelzl@41689
   604
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   605
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   606
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@41661
   607
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@47694
   608
wenzelm@53015
   609
  let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
hoelzl@47694
   610
  let ?\<mu> = "emeasure ?P"
hoelzl@47694
   611
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
hoelzl@47694
   612
  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
   613
wenzelm@53015
   614
  have "emeasure (Pi\<^sub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^sub>E (insert i I) A)) =
hoelzl@49776
   615
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
hoelzl@49776
   616
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49776
   617
    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
   618
    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
wenzelm@53015
   619
    let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
wenzelm@53015
   620
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
hoelzl@49776
   621
    have "?\<mu> ?p =
wenzelm@53015
   622
      emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i))"
hoelzl@49776
   623
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
wenzelm@53015
   624
    also have "?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
immler@50244
   625
      using J E[rule_format, THEN sets.sets_into_space]
hoelzl@50123
   626
      by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
wenzelm@53015
   627
    also have "emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
wenzelm@53015
   628
      emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
hoelzl@49776
   629
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
wenzelm@53015
   630
    also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
immler@50244
   631
      using J E[rule_format, THEN sets.sets_into_space]
hoelzl@50123
   632
      by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
wenzelm@53015
   633
    also have "emeasure (Pi\<^sub>M I M) (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
hoelzl@49776
   634
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
hoelzl@49776
   635
      using E by (subst insert) (auto intro!: setprod_cong)
hoelzl@49776
   636
    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
   637
       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
   638
      using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
hoelzl@49776
   639
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
hoelzl@49776
   640
      using insert(1,2) J E by (intro setprod_mono_one_right) auto
hoelzl@49776
   641
    finally show "?\<mu> ?p = \<dots>" .
hoelzl@47694
   642
wenzelm@53015
   643
    show "prod_emb (insert i I) M J (Pi\<^sub>E J E) \<in> Pow (\<Pi>\<^sub>E i\<in>insert i I. space (M i))"
immler@50244
   644
      using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
hoelzl@49776
   645
  next
wenzelm@53015
   646
    show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
hoelzl@49776
   647
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
hoelzl@49776
   648
  next
hoelzl@49776
   649
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
hoelzl@49776
   650
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
hoelzl@49776
   651
      using insert by auto
hoelzl@49776
   652
  qed (auto intro!: setprod_cong)
hoelzl@49776
   653
  with insert show ?case
immler@50244
   654
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
hoelzl@50003
   655
qed simp
hoelzl@47694
   656
hoelzl@49776
   657
lemma (in product_sigma_finite) sigma_finite: 
hoelzl@49776
   658
  assumes "finite I"
hoelzl@49776
   659
  shows "sigma_finite_measure (PiM I M)"
hoelzl@49776
   660
proof -
hoelzl@49776
   661
  interpret finite_product_sigma_finite M I by default fact
hoelzl@49776
   662
hoelzl@49776
   663
  from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@49776
   664
  then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
wenzelm@53015
   665
    "incseq (\<lambda>k. \<Pi>\<^sub>E j \<in> I. F j k)"
wenzelm@53015
   666
    "(\<Union>k. \<Pi>\<^sub>E j \<in> I. F j k) = space (Pi\<^sub>M I M)"
hoelzl@49776
   667
    "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
hoelzl@47694
   668
    by blast+
wenzelm@53015
   669
  let ?F = "\<lambda>k. \<Pi>\<^sub>E j \<in> I. F j k"
hoelzl@47694
   670
hoelzl@49776
   671
  show ?thesis
hoelzl@47694
   672
  proof (unfold_locales, intro exI[of _ ?F] conjI allI)
wenzelm@53015
   673
    show "range ?F \<subseteq> sets (Pi\<^sub>M I M)" using F(1) `finite I` by auto
hoelzl@47694
   674
  next
wenzelm@53015
   675
    from F(3) show "(\<Union>i. ?F i) = space (Pi\<^sub>M I M)" by simp
hoelzl@47694
   676
  next
hoelzl@47694
   677
    fix j
hoelzl@49776
   678
    from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
wenzelm@53015
   679
    show "emeasure (\<Pi>\<^sub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
hoelzl@49776
   680
      by (subst emeasure_PiM) auto
hoelzl@40859
   681
  qed
hoelzl@40859
   682
qed
hoelzl@40859
   683
wenzelm@53015
   684
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
hoelzl@47694
   685
  using sigma_finite[OF finite_index] .
hoelzl@40859
   686
hoelzl@40859
   687
lemma (in finite_product_sigma_finite) measure_times:
wenzelm@53015
   688
  "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
hoelzl@47694
   689
  using emeasure_PiM[OF finite_index] by auto
hoelzl@41096
   690
hoelzl@40859
   691
lemma (in product_sigma_finite) positive_integral_empty:
hoelzl@41981
   692
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
wenzelm@53015
   693
  shows "integral\<^sup>P (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
hoelzl@40859
   694
proof -
hoelzl@41689
   695
  interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
wenzelm@53015
   696
  have "\<And>A. emeasure (Pi\<^sub>M {} M) (Pi\<^sub>E {} A) = 1"
hoelzl@40859
   697
    using assms by (subst measure_times) auto
hoelzl@40859
   698
  then show ?thesis
hoelzl@47694
   699
    unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
hoelzl@47694
   700
  proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
hoelzl@41981
   701
    show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
hoelzl@44928
   702
      by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
hoelzl@41981
   703
    show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
hoelzl@44928
   704
      by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
hoelzl@40859
   705
  qed
hoelzl@40859
   706
qed
hoelzl@40859
   707
hoelzl@47694
   708
lemma (in product_sigma_finite) distr_merge:
hoelzl@40859
   709
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
wenzelm@53015
   710
  shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
hoelzl@47694
   711
   (is "?D = ?P")
hoelzl@40859
   712
proof -
hoelzl@41689
   713
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   714
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@40859
   715
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   716
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
wenzelm@53015
   717
  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
hoelzl@49780
   718
  let ?g = "merge I J"
hoelzl@47694
   719
hoelzl@41661
   720
  from IJ.sigma_finite_pairs obtain F where
hoelzl@41661
   721
    F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
wenzelm@53015
   722
       "incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k)"
wenzelm@53015
   723
       "(\<Union>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k) = space ?P"
hoelzl@47694
   724
       "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
hoelzl@41661
   725
    by auto
wenzelm@53015
   726
  let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k"
hoelzl@47694
   727
  
hoelzl@47694
   728
  show ?thesis
hoelzl@47694
   729
  proof (rule measure_eqI_generator_eq[symmetric])
hoelzl@47694
   730
    show "Int_stable (prod_algebra (I \<union> J) M)"
hoelzl@47694
   731
      by (rule Int_stable_prod_algebra)
wenzelm@53015
   732
    show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^sub>E i \<in> I \<union> J. space (M i))"
hoelzl@47694
   733
      by (rule prod_algebra_sets_into_space)
wenzelm@53015
   734
    show "sets ?P = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   735
      by (rule sets_PiM)
wenzelm@53015
   736
    then show "sets ?D = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   737
      by simp
hoelzl@47694
   738
hoelzl@47694
   739
    show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
hoelzl@47694
   740
      using fin by (auto simp: prod_algebra_eq_finite)
wenzelm@53015
   741
    show "(\<Union>i. \<Pi>\<^sub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i))"
hoelzl@47694
   742
      using F(3) by (simp add: space_PiM)
hoelzl@41981
   743
  next
hoelzl@41981
   744
    fix k
hoelzl@47694
   745
    from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
hoelzl@47694
   746
    show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
hoelzl@41661
   747
  next
hoelzl@47694
   748
    fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
wenzelm@53015
   749
    with fin obtain F where A_eq: "A = (Pi\<^sub>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
   750
      by (auto simp add: prod_algebra_eq_finite)
wenzelm@53015
   751
    let ?B = "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M"
hoelzl@47694
   752
    let ?X = "?g -` A \<inter> space ?B"
wenzelm@53015
   753
    have "Pi\<^sub>E I F \<subseteq> space (Pi\<^sub>M I M)" "Pi\<^sub>E J F \<subseteq> space (Pi\<^sub>M J M)"
immler@50244
   754
      using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+
wenzelm@53015
   755
    then have X: "?X = (Pi\<^sub>E I F \<times> Pi\<^sub>E J F)"
hoelzl@47694
   756
      unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
hoelzl@47694
   757
    have "emeasure ?D A = emeasure ?B ?X"
hoelzl@47694
   758
      using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
hoelzl@47694
   759
    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
   760
      using `finite J` `finite I` F unfolding X
hoelzl@50123
   761
      by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
hoelzl@47694
   762
    also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
hoelzl@41661
   763
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
wenzelm@53015
   764
    also have "\<dots> = emeasure ?P (Pi\<^sub>E (I \<union> J) F)"
hoelzl@41661
   765
      using `finite J` `finite I` F unfolding A
hoelzl@41661
   766
      by (intro IJ.measure_times[symmetric]) auto
hoelzl@47694
   767
    finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
hoelzl@47694
   768
  qed
hoelzl@41661
   769
qed
hoelzl@41026
   770
hoelzl@41026
   771
lemma (in product_sigma_finite) product_positive_integral_fold:
hoelzl@47694
   772
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
wenzelm@53015
   773
  and f: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
wenzelm@53015
   774
  shows "integral\<^sup>P (Pi\<^sub>M (I \<union> J) M) f =
wenzelm@53015
   775
    (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
hoelzl@41026
   776
proof -
hoelzl@41689
   777
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   778
  interpret J: finite_product_sigma_finite M J by default fact
wenzelm@53015
   779
  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
wenzelm@53015
   780
  have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
hoelzl@49776
   781
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
hoelzl@41661
   782
  show ?thesis
hoelzl@47694
   783
    apply (subst distr_merge[OF IJ, symmetric])
hoelzl@49776
   784
    apply (subst positive_integral_distr[OF measurable_merge f])
hoelzl@56993
   785
    apply (subst J.positive_integral_fst[symmetric, OF P_borel])
hoelzl@47694
   786
    apply simp
hoelzl@47694
   787
    done
hoelzl@40859
   788
qed
hoelzl@40859
   789
hoelzl@47694
   790
lemma (in product_sigma_finite) distr_singleton:
wenzelm@53015
   791
  "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
hoelzl@47694
   792
proof (intro measure_eqI[symmetric])
hoelzl@41831
   793
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   794
  fix A assume A: "A \<in> sets (M i)"
wenzelm@53374
   795
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
immler@50244
   796
    using sets.sets_into_space by (auto simp: space_PiM)
wenzelm@53374
   797
  then show "emeasure (M i) A = emeasure ?D A"
hoelzl@47694
   798
    using A I.measure_times[of "\<lambda>_. A"]
hoelzl@47694
   799
    by (simp add: emeasure_distr measurable_component_singleton)
hoelzl@47694
   800
qed simp
hoelzl@41831
   801
hoelzl@41026
   802
lemma (in product_sigma_finite) product_positive_integral_singleton:
hoelzl@40859
   803
  assumes f: "f \<in> borel_measurable (M i)"
wenzelm@53015
   804
  shows "integral\<^sup>P (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>P (M i) f"
hoelzl@40859
   805
proof -
hoelzl@41689
   806
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   807
  from f show ?thesis
hoelzl@47694
   808
    apply (subst distr_singleton[symmetric])
hoelzl@47694
   809
    apply (subst positive_integral_distr[OF measurable_component_singleton])
hoelzl@47694
   810
    apply simp_all
hoelzl@47694
   811
    done
hoelzl@40859
   812
qed
hoelzl@40859
   813
hoelzl@41096
   814
lemma (in product_sigma_finite) product_positive_integral_insert:
hoelzl@49780
   815
  assumes I[simp]: "finite I" "i \<notin> I"
wenzelm@53015
   816
    and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
wenzelm@53015
   817
  shows "integral\<^sup>P (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
hoelzl@41096
   818
proof -
hoelzl@41689
   819
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41689
   820
  interpret i: finite_product_sigma_finite M "{i}" by default auto
hoelzl@41689
   821
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
hoelzl@41689
   822
    using f by auto
hoelzl@41096
   823
  show ?thesis
hoelzl@49780
   824
    unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
hoelzl@49780
   825
  proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
wenzelm@53015
   826
    fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
hoelzl@49780
   827
    let ?f = "\<lambda>y. f (x(i := y))"
hoelzl@49780
   828
    show "?f \<in> borel_measurable (M i)"
hoelzl@47694
   829
      using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
hoelzl@47694
   830
      unfolding comp_def .
wenzelm@53015
   831
    show "(\<integral>\<^sup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^sub>M {i} M) = (\<integral>\<^sup>+ y. f (x(i := y i)) \<partial>Pi\<^sub>M {i} M)"
hoelzl@49780
   832
      using x
hoelzl@49780
   833
      by (auto intro!: positive_integral_cong arg_cong[where f=f]
hoelzl@50123
   834
               simp add: space_PiM extensional_def PiE_def)
hoelzl@41096
   835
  qed
hoelzl@41096
   836
qed
hoelzl@41096
   837
hoelzl@41096
   838
lemma (in product_sigma_finite) product_positive_integral_setprod:
hoelzl@43920
   839
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
hoelzl@41096
   840
  assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41981
   841
  and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
wenzelm@53015
   842
  shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>P (M i) (f i))"
hoelzl@41096
   843
using assms proof induct
hoelzl@41096
   844
  case (insert i I)
hoelzl@41096
   845
  note `finite I`[intro, simp]
hoelzl@41689
   846
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41096
   847
  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
   848
    using insert by (auto intro!: setprod_cong)
wenzelm@53015
   849
  have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^sub>M J M)"
immler@50244
   850
    using sets.sets_into_space insert
hoelzl@47694
   851
    by (intro borel_measurable_ereal_setprod
hoelzl@41689
   852
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
hoelzl@41096
   853
       auto
hoelzl@41981
   854
  then show ?case
hoelzl@41981
   855
    apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
hoelzl@47694
   856
    apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
hoelzl@47694
   857
    apply (subst positive_integral_cmult)
hoelzl@47694
   858
    apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
hoelzl@41981
   859
    done
hoelzl@47694
   860
qed (simp add: space_PiM)
hoelzl@41096
   861
hoelzl@50104
   862
lemma (in product_sigma_finite) distr_component:
wenzelm@53015
   863
  "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
hoelzl@50104
   864
proof (intro measure_eqI[symmetric])
hoelzl@50104
   865
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@50104
   866
hoelzl@50104
   867
  have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
hoelzl@50104
   868
    by (auto simp: extensional_def restrict_def)
hoelzl@50104
   869
hoelzl@50104
   870
  fix A assume A: "A \<in> sets ?P"
wenzelm@53015
   871
  then have "emeasure ?P A = (\<integral>\<^sup>+x. indicator A x \<partial>?P)" 
hoelzl@50104
   872
    by simp
wenzelm@53015
   873
  also have "\<dots> = (\<integral>\<^sup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)" 
hoelzl@50123
   874
    by (intro positive_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
hoelzl@50104
   875
  also have "\<dots> = emeasure ?D A"
hoelzl@50104
   876
    using A by (simp add: product_positive_integral_singleton emeasure_distr)
wenzelm@53015
   877
  finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" .
hoelzl@50104
   878
qed simp
hoelzl@41026
   879
hoelzl@49776
   880
lemma (in product_sigma_finite)
wenzelm@53015
   881
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
hoelzl@49776
   882
  shows emeasure_fold_integral:
wenzelm@53015
   883
    "emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I)
hoelzl@49776
   884
    and emeasure_fold_measurable:
wenzelm@53015
   885
    "(\<lambda>x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M))) \<in> borel_measurable (Pi\<^sub>M I M)" (is ?B)
hoelzl@49776
   886
proof -
hoelzl@49776
   887
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@49776
   888
  interpret J: finite_product_sigma_finite M J by default fact
wenzelm@53015
   889
  interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
wenzelm@53015
   890
  have merge: "merge I J -` A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
hoelzl@49776
   891
    by (intro measurable_sets[OF _ A] measurable_merge assms)
hoelzl@49776
   892
hoelzl@49776
   893
  show ?I
hoelzl@49776
   894
    apply (subst distr_merge[symmetric, OF IJ])
hoelzl@49776
   895
    apply (subst emeasure_distr[OF measurable_merge A])
hoelzl@49776
   896
    apply (subst J.emeasure_pair_measure_alt[OF merge])
hoelzl@49776
   897
    apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49776
   898
    done
hoelzl@49776
   899
hoelzl@49776
   900
  show ?B
hoelzl@49776
   901
    using IJ.measurable_emeasure_Pair1[OF merge]
haftmann@56154
   902
    by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
hoelzl@49776
   903
qed
hoelzl@49776
   904
hoelzl@49776
   905
lemma sets_Collect_single:
wenzelm@53015
   906
  "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
hoelzl@50003
   907
  by simp
hoelzl@49776
   908
hoelzl@49776
   909
lemma sigma_prod_algebra_sigma_eq_infinite:
hoelzl@49776
   910
  fixes E :: "'i \<Rightarrow> 'a set set"
hoelzl@49779
   911
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@49776
   912
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@49776
   913
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@49776
   914
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
wenzelm@53015
   915
  defines "P == {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}"
hoelzl@49776
   916
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@49776
   917
proof
wenzelm@53015
   918
  let ?P = "sigma (space (Pi\<^sub>M I M)) P"
wenzelm@53015
   919
  have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))"
hoelzl@50123
   920
    using E_closed by (auto simp: space_PiM P_def subset_eq)
wenzelm@53015
   921
  then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@49776
   922
    by (simp add: space_PiM)
hoelzl@49776
   923
  have "sets (PiM I M) =
wenzelm@53015
   924
      sigma_sets (space ?P) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@49776
   925
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@49776
   926
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
immler@50244
   927
  proof (safe intro!: sets.sigma_sets_subset)
hoelzl@49776
   928
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@49776
   929
    then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@49776
   930
      apply (subst measurable_iff_measure_of)
hoelzl@49776
   931
      apply (simp_all add: P_closed)
hoelzl@49776
   932
      using E_closed
hoelzl@49776
   933
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
   934
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
   935
      apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
hoelzl@49776
   936
      apply (rule_tac x=Aa in exI)
hoelzl@49776
   937
      apply (auto simp: space_PiM)
hoelzl@49776
   938
      done
hoelzl@49776
   939
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@49776
   940
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@49776
   941
      by (simp add: E_generates)
wenzelm@53015
   942
    also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A}"
hoelzl@49776
   943
      using P_closed by (auto simp: space_PiM)
hoelzl@49776
   944
    finally show "\<dots> \<in> sets ?P" .
hoelzl@49776
   945
  qed
hoelzl@49776
   946
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@49776
   947
    by (simp add: P_closed)
hoelzl@49776
   948
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@49776
   949
    unfolding P_def space_PiM[symmetric]
immler@50244
   950
    by (intro sets.sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
hoelzl@49776
   951
qed
hoelzl@49776
   952
hoelzl@47694
   953
lemma sigma_prod_algebra_sigma_eq:
hoelzl@49779
   954
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
hoelzl@47694
   955
  assumes "finite I"
hoelzl@49779
   956
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@47694
   957
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@47694
   958
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@47694
   959
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
wenzelm@53015
   960
  defines "P == { Pi\<^sub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
hoelzl@47694
   961
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@47694
   962
proof
wenzelm@53015
   963
  let ?P = "sigma (space (Pi\<^sub>M I M)) P"
hoelzl@49779
   964
  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
hoelzl@49779
   965
  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
   966
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
wenzelm@53015
   967
  have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))"
hoelzl@50123
   968
    using E_closed by (auto simp: space_PiM P_def subset_eq)
wenzelm@53015
   969
  then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   970
    by (simp add: space_PiM)
hoelzl@47694
   971
  have "sets (PiM I M) =
wenzelm@53015
   972
      sigma_sets (space ?P) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@47694
   973
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@47694
   974
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
immler@50244
   975
  proof (safe intro!: sets.sigma_sets_subset)
hoelzl@47694
   976
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@47694
   977
    have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@47694
   978
    proof (subst measurable_iff_measure_of)
hoelzl@47694
   979
      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
hoelzl@50123
   980
      from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto
hoelzl@47694
   981
      show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
   982
      proof
hoelzl@47694
   983
        fix A assume A: "A \<in> E i"
wenzelm@53015
   984
        then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^sub>E j\<in>I. if i = j then A else space (M j))"
hoelzl@50123
   985
          using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm)
wenzelm@53015
   986
        also have "\<dots> = (\<Pi>\<^sub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
hoelzl@47694
   987
          by (intro PiE_cong) (simp add: S_union)
wenzelm@53015
   988
        also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^sub>E j\<in>I. if i = j then A else S j (xs ! T j))"
hoelzl@49779
   989
          using T
hoelzl@50123
   990
          apply (auto simp: PiE_iff bchoice_iff)
hoelzl@49779
   991
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
hoelzl@49779
   992
          apply (auto simp: bij_betw_def)
hoelzl@49779
   993
          done
hoelzl@47694
   994
        also have "\<dots> \<in> sets ?P"
immler@50244
   995
        proof (safe intro!: sets.countable_UN)
wenzelm@53015
   996
          fix xs show "(\<Pi>\<^sub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
hoelzl@47694
   997
            using A S_in_E
hoelzl@47694
   998
            by (simp add: P_closed)
hoelzl@49779
   999
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
hoelzl@47694
  1000
        qed
hoelzl@47694
  1001
        finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1002
          using P_closed by simp
hoelzl@47694
  1003
      qed
hoelzl@47694
  1004
    qed
hoelzl@47694
  1005
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@47694
  1006
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1007
      by (simp add: E_generates)
wenzelm@53015
  1008
    also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A}"
hoelzl@47694
  1009
      using P_closed by (auto simp: space_PiM)
hoelzl@47694
  1010
    finally show "\<dots> \<in> sets ?P" .
hoelzl@47694
  1011
  qed
hoelzl@47694
  1012
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1013
    by (simp add: P_closed)
hoelzl@47694
  1014
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@47694
  1015
    using `finite I`
immler@50244
  1016
    by (auto intro!: sets.sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
hoelzl@47694
  1017
qed
hoelzl@47694
  1018
hoelzl@50104
  1019
lemma pair_measure_eq_distr_PiM:
hoelzl@50104
  1020
  fixes M1 :: "'a measure" and M2 :: "'a measure"
hoelzl@50104
  1021
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
blanchet@55414
  1022
  shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))"
hoelzl@50104
  1023
    (is "?P = ?D")
hoelzl@50104
  1024
proof (rule pair_measure_eqI[OF assms])
blanchet@55414
  1025
  interpret B: product_sigma_finite "case_bool M1 M2"
hoelzl@50104
  1026
    unfolding product_sigma_finite_def using assms by (auto split: bool.split)
blanchet@55414
  1027
  let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"
hoelzl@50104
  1028
hoelzl@50104
  1029
  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
  1030
    by auto
hoelzl@50104
  1031
  fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
blanchet@55414
  1032
  have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
hoelzl@50104
  1033
    by (simp add: UNIV_bool ac_simps)
blanchet@55414
  1034
  also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
hoelzl@50104
  1035
    using A B by (subst B.emeasure_PiM) (auto split: bool.split)
blanchet@55414
  1036
  also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
immler@50244
  1037
    using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
hoelzl@50123
  1038
    by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
hoelzl@50104
  1039
  finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
hoelzl@50104
  1040
    using A B
blanchet@55414
  1041
      measurable_component_singleton[of True UNIV "case_bool M1 M2"]
blanchet@55414
  1042
      measurable_component_singleton[of False UNIV "case_bool M1 M2"]
hoelzl@50104
  1043
    by (subst emeasure_distr) (auto simp: measurable_pair_iff)
hoelzl@50104
  1044
qed simp
hoelzl@50104
  1045
hoelzl@47694
  1046
end