src/HOL/Probability/Finite_Product_Measure.thy
author haftmann
Sat Jul 05 11:01:53 2014 +0200 (2014-07-05)
changeset 57514 bdc2c6b40bf2
parent 57512 cc97b347b301
child 58606 9c66f7c541fb
permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
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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 PiE_choice: "(\<exists>f\<in>PiE I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
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  by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1])
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     (force intro: exI[of _ "restrict f I" for f])
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lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
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  by auto
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subsubsection {* More about Function restricted by @{const extensional}  *}
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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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subsection {* Finite product spaces *}
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subsubsection {* 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)"
hoelzl@47694
   236
    and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
hoelzl@47694
   237
    by (auto simp: prod_algebra_def)
hoelzl@47694
   238
  from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
immler@50244
   239
    using sets.sets_into_space by auto
wenzelm@53015
   240
  then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))"
hoelzl@47694
   241
    using A J by (auto simp: prod_emb_PiE)
wenzelm@53374
   242
  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
   243
    using sets.top E by auto
hoelzl@47694
   244
  ultimately show ?thesis using that by auto
hoelzl@47694
   245
qed
hoelzl@40859
   246
hoelzl@47694
   247
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
hoelzl@47694
   248
proof (unfold Int_stable_def, safe)
hoelzl@47694
   249
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   250
  from prod_algebraE[OF this] guess J E . note A = this
hoelzl@47694
   251
  fix B assume "B \<in> prod_algebra I M"
hoelzl@47694
   252
  from prod_algebraE[OF this] guess K F . note B = this
wenzelm@53015
   253
  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
   254
      (if i \<in> K then F i else space (M i)))"
immler@50244
   255
    unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4)
immler@50244
   256
      B(5)[THEN sets.sets_into_space]
hoelzl@47694
   257
    apply (subst (1 2 3) prod_emb_PiE)
hoelzl@47694
   258
    apply (simp_all add: subset_eq PiE_Int)
hoelzl@47694
   259
    apply blast
hoelzl@47694
   260
    apply (intro PiE_cong)
hoelzl@47694
   261
    apply auto
hoelzl@47694
   262
    done
hoelzl@47694
   263
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   264
    using A B by (auto intro!: prod_algebraI)
hoelzl@47694
   265
  finally show "A \<inter> B \<in> prod_algebra I M" .
hoelzl@47694
   266
qed
hoelzl@47694
   267
hoelzl@47694
   268
lemma prod_algebra_mono:
hoelzl@47694
   269
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
hoelzl@47694
   270
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
hoelzl@47694
   271
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
hoelzl@47694
   272
proof
hoelzl@47694
   273
  fix A assume "A \<in> prod_algebra I E"
hoelzl@47694
   274
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
wenzelm@53015
   275
    and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
hoelzl@47694
   276
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
hoelzl@47694
   277
    by (auto simp: prod_algebra_def)
hoelzl@47694
   278
  moreover
wenzelm@53015
   279
  from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
hoelzl@47694
   280
    by (rule PiE_cong)
wenzelm@53015
   281
  with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
hoelzl@47694
   282
    by (simp add: prod_emb_def)
hoelzl@47694
   283
  moreover
hoelzl@47694
   284
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
hoelzl@47694
   285
    by auto
hoelzl@47694
   286
  ultimately show "A \<in> prod_algebra I F"
hoelzl@47694
   287
    apply (simp add: prod_algebra_def image_iff)
hoelzl@47694
   288
    apply (intro exI[of _ J] exI[of _ G] conjI)
hoelzl@47694
   289
    apply auto
hoelzl@47694
   290
    done
hoelzl@41689
   291
qed
hoelzl@41689
   292
hoelzl@50104
   293
lemma prod_algebra_cong:
hoelzl@50104
   294
  assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
hoelzl@50104
   295
  shows "prod_algebra I M = prod_algebra J N"
hoelzl@50104
   296
proof -
hoelzl@50104
   297
  have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
hoelzl@50104
   298
    using sets_eq_imp_space_eq[OF sets] by auto
hoelzl@50104
   299
  with sets show ?thesis unfolding `I = J`
hoelzl@50104
   300
    by (intro antisym prod_algebra_mono) auto
hoelzl@50104
   301
qed
hoelzl@50104
   302
hoelzl@50104
   303
lemma space_in_prod_algebra:
wenzelm@53015
   304
  "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
hoelzl@50104
   305
proof cases
hoelzl@50104
   306
  assume "I = {}" then show ?thesis
hoelzl@50104
   307
    by (auto simp add: prod_algebra_def image_iff prod_emb_def)
hoelzl@50104
   308
next
hoelzl@50104
   309
  assume "I \<noteq> {}"
hoelzl@50104
   310
  then obtain i where "i \<in> I" by auto
wenzelm@53015
   311
  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
   312
    by (auto simp: prod_emb_def)
hoelzl@50104
   313
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@50104
   314
    using `i \<in> I` by (intro prod_algebraI) auto
hoelzl@50104
   315
  finally show ?thesis .
hoelzl@50104
   316
qed
hoelzl@50104
   317
wenzelm@53015
   318
lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   319
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
hoelzl@47694
   320
wenzelm@53015
   321
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
   322
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
hoelzl@41689
   323
hoelzl@47694
   324
lemma sets_PiM_single: "sets (PiM I M) =
wenzelm@53015
   325
    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
   326
    (is "_ = sigma_sets ?\<Omega> ?R")
hoelzl@47694
   327
  unfolding sets_PiM
hoelzl@47694
   328
proof (rule sigma_sets_eqI)
hoelzl@47694
   329
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
hoelzl@47694
   330
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   331
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   332
  show "A \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   333
  proof cases
hoelzl@47694
   334
    assume "I = {}"
hoelzl@47694
   335
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
hoelzl@47694
   336
    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
hoelzl@47694
   337
  next
hoelzl@47694
   338
    assume "I \<noteq> {}"
wenzelm@53015
   339
    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
   340
      by (auto simp: prod_emb_def)
hoelzl@47694
   341
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   342
      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
hoelzl@47694
   343
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
hoelzl@47694
   344
  qed
hoelzl@47694
   345
next
hoelzl@47694
   346
  fix A assume "A \<in> ?R"
wenzelm@53015
   347
  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
   348
    by auto
wenzelm@53015
   349
  then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
hoelzl@50123
   350
     by (auto simp: prod_emb_def)
hoelzl@47694
   351
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
hoelzl@47694
   352
    using A by (intro sigma_sets.Basic prod_algebraI) auto
hoelzl@47694
   353
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
hoelzl@47694
   354
qed
hoelzl@47694
   355
hoelzl@47694
   356
lemma sets_PiM_I:
hoelzl@47694
   357
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
hoelzl@47694
   358
  shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   359
proof cases
hoelzl@47694
   360
  assume "J = {}"
hoelzl@47694
   361
  then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
hoelzl@47694
   362
    by (auto simp: prod_emb_def)
hoelzl@47694
   363
  then show ?thesis
hoelzl@47694
   364
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
hoelzl@47694
   365
next
hoelzl@47694
   366
  assume "J \<noteq> {}" with assms show ?thesis
hoelzl@50003
   367
    by (force simp add: sets_PiM prod_algebra_def)
hoelzl@40859
   368
qed
hoelzl@40859
   369
hoelzl@47694
   370
lemma measurable_PiM:
wenzelm@53015
   371
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   372
  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
   373
    f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N" 
hoelzl@47694
   374
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   375
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   376
proof (rule measurable_sigma_sets)
hoelzl@47694
   377
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   378
  from prod_algebraE[OF this] guess J X .
hoelzl@47694
   379
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
hoelzl@47694
   380
qed
hoelzl@47694
   381
hoelzl@47694
   382
lemma measurable_PiM_Collect:
wenzelm@53015
   383
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   384
  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
   385
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 
hoelzl@47694
   386
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   387
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   388
proof (rule measurable_sigma_sets)
hoelzl@47694
   389
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   390
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@50123
   391
  then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
hoelzl@50123
   392
    using space by (auto simp: prod_emb_def del: PiE_I)
hoelzl@47694
   393
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
hoelzl@47694
   394
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@41689
   395
qed
hoelzl@41095
   396
hoelzl@47694
   397
lemma measurable_PiM_single:
wenzelm@53015
   398
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   399
  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
   400
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   401
  using sets_PiM_single
hoelzl@47694
   402
proof (rule measurable_sigma_sets)
wenzelm@53015
   403
  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
   404
  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
   405
    by auto
hoelzl@47694
   406
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
hoelzl@47694
   407
  also have "\<dots> \<in> sets N" using B by (rule sets)
hoelzl@47694
   408
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@47694
   409
qed (auto simp: space)
hoelzl@40859
   410
hoelzl@50099
   411
lemma measurable_PiM_single':
hoelzl@50099
   412
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
wenzelm@53015
   413
    and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
wenzelm@53015
   414
  shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
hoelzl@50099
   415
proof (rule measurable_PiM_single)
hoelzl@50099
   416
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@50099
   417
  then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
hoelzl@50099
   418
    by auto
hoelzl@50099
   419
  then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
hoelzl@50099
   420
    using A f by (auto intro!: measurable_sets)
hoelzl@50099
   421
qed fact
hoelzl@50099
   422
hoelzl@50003
   423
lemma sets_PiM_I_finite[measurable]:
hoelzl@47694
   424
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
hoelzl@47694
   425
  shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
immler@50244
   426
  using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto
hoelzl@47694
   427
hoelzl@50021
   428
lemma measurable_component_singleton:
wenzelm@53015
   429
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
hoelzl@41689
   430
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   431
  fix A assume "A \<in> sets (M i)"
wenzelm@53015
   432
  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
   433
    using sets.sets_into_space `i \<in> I`
hoelzl@47694
   434
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
wenzelm@53015
   435
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
hoelzl@47694
   436
    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
hoelzl@47694
   437
qed (insert `i \<in> I`, auto simp: space_PiM)
hoelzl@47694
   438
hoelzl@50021
   439
lemma measurable_component_singleton'[measurable_app]:
wenzelm@53015
   440
  assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
hoelzl@50021
   441
  assumes i: "i \<in> I"
hoelzl@50021
   442
  shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
hoelzl@50021
   443
  using measurable_compose[OF f measurable_component_singleton, OF i] .
hoelzl@50021
   444
hoelzl@50099
   445
lemma measurable_PiM_component_rev[measurable (raw)]:
hoelzl@50099
   446
  "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
hoelzl@50099
   447
  by simp
hoelzl@50099
   448
blanchet@55415
   449
lemma measurable_case_nat[measurable (raw)]:
hoelzl@50021
   450
  assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
hoelzl@50021
   451
    "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
blanchet@55415
   452
  shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
hoelzl@50021
   453
  by (cases i) simp_all
hoelzl@50021
   454
blanchet@55415
   455
lemma measurable_case_nat'[measurable (raw)]:
wenzelm@53015
   456
  assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
blanchet@55415
   457
  shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
hoelzl@50099
   458
  using fg[THEN measurable_space]
hoelzl@50123
   459
  by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
hoelzl@50099
   460
hoelzl@50003
   461
lemma measurable_add_dim[measurable]:
wenzelm@53015
   462
  "(\<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
   463
    (is "?f \<in> measurable ?P ?I")
hoelzl@47694
   464
proof (rule measurable_PiM_single)
hoelzl@47694
   465
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
hoelzl@47694
   466
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
wenzelm@53015
   467
    (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
   468
    using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
hoelzl@47694
   469
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   470
    using A j
hoelzl@47694
   471
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
blanchet@55414
   472
  finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
hoelzl@50123
   473
qed (auto simp: space_pair_measure space_PiM PiE_def)
hoelzl@41661
   474
hoelzl@50003
   475
lemma measurable_component_update:
wenzelm@53015
   476
  "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
   477
  by simp
hoelzl@50003
   478
hoelzl@50003
   479
lemma measurable_merge[measurable]:
wenzelm@53015
   480
  "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
   481
    (is "?f \<in> measurable ?P ?U")
hoelzl@47694
   482
proof (rule measurable_PiM_single)
hoelzl@47694
   483
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
hoelzl@49780
   484
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
hoelzl@47694
   485
    (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
   486
    by (auto simp: merge_def)
hoelzl@47694
   487
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   488
    using A
hoelzl@47694
   489
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@49780
   490
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
hoelzl@50123
   491
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
hoelzl@42988
   492
hoelzl@50003
   493
lemma measurable_restrict[measurable (raw)]:
hoelzl@47694
   494
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
wenzelm@53015
   495
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
hoelzl@47694
   496
proof (rule measurable_PiM_single)
hoelzl@47694
   497
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@47694
   498
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
hoelzl@47694
   499
    by auto
hoelzl@47694
   500
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
hoelzl@47694
   501
    using A X by (auto intro!: measurable_sets)
hoelzl@50123
   502
qed (insert X, auto simp add: PiE_def dest: measurable_space)
hoelzl@47694
   503
hoelzl@57025
   504
lemma measurable_abs_UNIV: 
hoelzl@57025
   505
  "(\<And>n. (\<lambda>\<omega>. f n \<omega>) \<in> measurable M (N n)) \<Longrightarrow> (\<lambda>\<omega> n. f n \<omega>) \<in> measurable M (PiM UNIV N)"
hoelzl@57025
   506
  by (intro measurable_PiM_single) (auto dest: measurable_space)
hoelzl@57025
   507
wenzelm@53015
   508
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
   509
  by (intro measurable_restrict measurable_component_singleton) auto
immler@50038
   510
immler@50038
   511
lemma measurable_prod_emb[intro, simp]:
wenzelm@53015
   512
  "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
   513
  unfolding prod_emb_def space_PiM[symmetric]
immler@50038
   514
  by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
immler@50038
   515
hoelzl@50003
   516
lemma sets_in_Pi_aux:
hoelzl@50003
   517
  "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
   518
  {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
hoelzl@50003
   519
  by (simp add: subset_eq Pi_iff)
hoelzl@50003
   520
hoelzl@50003
   521
lemma sets_in_Pi[measurable (raw)]:
hoelzl@50003
   522
  "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
hoelzl@50003
   523
  (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50387
   524
  Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
hoelzl@50003
   525
  unfolding pred_def
hoelzl@50003
   526
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
hoelzl@50003
   527
hoelzl@50003
   528
lemma sets_in_extensional_aux:
hoelzl@50003
   529
  "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
hoelzl@50003
   530
proof -
hoelzl@50003
   531
  have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
hoelzl@50003
   532
    by (auto simp add: extensional_def space_PiM)
hoelzl@50003
   533
  then show ?thesis by simp
hoelzl@50003
   534
qed
hoelzl@50003
   535
hoelzl@50003
   536
lemma sets_in_extensional[measurable (raw)]:
hoelzl@50387
   537
  "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
hoelzl@50003
   538
  unfolding pred_def
hoelzl@50003
   539
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
hoelzl@50003
   540
hoelzl@47694
   541
locale product_sigma_finite =
hoelzl@47694
   542
  fixes M :: "'i \<Rightarrow> 'a measure"
hoelzl@41689
   543
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   544
hoelzl@41689
   545
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   546
  by (rule sigma_finite_measures)
hoelzl@40859
   547
hoelzl@47694
   548
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@47694
   549
  fixes I :: "'i set"
hoelzl@47694
   550
  assumes finite_index: "finite I"
hoelzl@41689
   551
hoelzl@40859
   552
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   553
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   554
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
wenzelm@53015
   555
    (\<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
   556
    (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
hoelzl@40859
   557
proof -
hoelzl@47694
   558
  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
   559
    using M.sigma_finite_incseq by metis
hoelzl@40859
   560
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@47694
   561
  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
   562
    by auto
wenzelm@53015
   563
  let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
hoelzl@47694
   564
  note space_PiM[simp]
hoelzl@40859
   565
  show ?thesis
hoelzl@41981
   566
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   567
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   568
  next
hoelzl@47694
   569
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   570
  next
hoelzl@50123
   571
    fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
immler@50244
   572
      by (auto simp: PiE_def dest!: sets.sets_into_space)
hoelzl@40859
   573
  next
hoelzl@47694
   574
    fix f assume "f \<in> space (PiM I M)"
hoelzl@41981
   575
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@50123
   576
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
hoelzl@40859
   577
  next
hoelzl@40859
   578
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   579
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   580
  qed
hoelzl@40859
   581
qed
hoelzl@40859
   582
hoelzl@49780
   583
lemma
wenzelm@53015
   584
  shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
wenzelm@53015
   585
    and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
hoelzl@49780
   586
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
hoelzl@49780
   587
hoelzl@49780
   588
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
hoelzl@49780
   589
proof -
hoelzl@49780
   590
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
wenzelm@53015
   591
  have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
hoelzl@49780
   592
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49780
   593
    show "positive (PiM {} M) ?\<mu>"
hoelzl@49780
   594
      by (auto simp: positive_def)
hoelzl@49780
   595
    show "countably_additive (PiM {} M) ?\<mu>"
immler@50244
   596
      by (rule sets.countably_additiveI_finite)
hoelzl@49780
   597
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
hoelzl@49780
   598
  qed (auto simp: prod_emb_def)
wenzelm@53015
   599
  also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
hoelzl@49780
   600
    by (auto simp: prod_emb_def)
hoelzl@49780
   601
  finally show ?thesis
hoelzl@49780
   602
    by simp
hoelzl@49780
   603
qed
hoelzl@49780
   604
hoelzl@49780
   605
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
hoelzl@49780
   606
  by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
hoelzl@49780
   607
hoelzl@49776
   608
lemma (in product_sigma_finite) emeasure_PiM:
wenzelm@53015
   609
  "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
   610
proof (induct I arbitrary: A rule: finite_induct)
hoelzl@40859
   611
  case (insert i I)
hoelzl@41689
   612
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   613
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   614
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@41661
   615
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@47694
   616
wenzelm@53015
   617
  let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
hoelzl@47694
   618
  let ?\<mu> = "emeasure ?P"
hoelzl@47694
   619
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
hoelzl@47694
   620
  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
   621
wenzelm@53015
   622
  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
   623
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
hoelzl@49776
   624
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49776
   625
    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
   626
    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
   627
    let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
wenzelm@53015
   628
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
hoelzl@49776
   629
    have "?\<mu> ?p =
wenzelm@53015
   630
      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
   631
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
wenzelm@53015
   632
    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
   633
      using J E[rule_format, THEN sets.sets_into_space]
hoelzl@50123
   634
      by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
wenzelm@53015
   635
    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
   636
      emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
hoelzl@49776
   637
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
wenzelm@53015
   638
    also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
immler@50244
   639
      using J E[rule_format, THEN sets.sets_into_space]
hoelzl@50123
   640
      by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
wenzelm@53015
   641
    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
   642
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
haftmann@57418
   643
      using E by (subst insert) (auto intro!: setprod.cong)
hoelzl@49776
   644
    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
   645
       emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
haftmann@57512
   646
      using insert by (auto simp: mult.commute intro!: arg_cong2[where f="op *"] setprod.cong)
hoelzl@49776
   647
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
haftmann@57418
   648
      using insert(1,2) J E by (intro setprod.mono_neutral_right) auto
hoelzl@49776
   649
    finally show "?\<mu> ?p = \<dots>" .
hoelzl@47694
   650
wenzelm@53015
   651
    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
   652
      using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
hoelzl@49776
   653
  next
wenzelm@53015
   654
    show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
hoelzl@49776
   655
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
hoelzl@49776
   656
  next
hoelzl@49776
   657
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
hoelzl@49776
   658
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
hoelzl@49776
   659
      using insert by auto
haftmann@57418
   660
  qed (auto intro!: setprod.cong)
hoelzl@49776
   661
  with insert show ?case
immler@50244
   662
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
hoelzl@50003
   663
qed simp
hoelzl@47694
   664
hoelzl@49776
   665
lemma (in product_sigma_finite) sigma_finite: 
hoelzl@49776
   666
  assumes "finite I"
hoelzl@49776
   667
  shows "sigma_finite_measure (PiM I M)"
hoelzl@57447
   668
proof
hoelzl@49776
   669
  interpret finite_product_sigma_finite M I by default fact
hoelzl@49776
   670
hoelzl@57447
   671
  obtain F where F: "\<And>j. countable (F j)" "\<And>j f. f \<in> F j \<Longrightarrow> f \<in> sets (M j)"
hoelzl@57447
   672
    "\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and
hoelzl@57447
   673
    in_space: "\<And>j. space (M j) = (\<Union>F j)"
hoelzl@57447
   674
    using sigma_finite_countable by (metis subset_eq)
hoelzl@57447
   675
  moreover have "(\<Union>(PiE I ` PiE I F)) = space (Pi\<^sub>M I M)"
hoelzl@57447
   676
    using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD2])
hoelzl@57447
   677
  ultimately show "\<exists>A. countable A \<and> A \<subseteq> sets (Pi\<^sub>M I M) \<and> \<Union>A = space (Pi\<^sub>M I M) \<and> (\<forall>a\<in>A. emeasure (Pi\<^sub>M I M) a \<noteq> \<infinity>)"
hoelzl@57447
   678
    by (intro exI[of _ "PiE I ` PiE I F"])
hoelzl@57447
   679
       (auto intro!: countable_PiE sets_PiM_I_finite
hoelzl@57447
   680
             simp: PiE_iff emeasure_PiM finite_index setprod_PInf emeasure_nonneg)
hoelzl@40859
   681
qed
hoelzl@40859
   682
wenzelm@53015
   683
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>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:
wenzelm@53015
   687
  "(\<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
   688
  using emeasure_PiM[OF finite_index] by auto
hoelzl@41096
   689
hoelzl@56996
   690
lemma (in product_sigma_finite) nn_integral_empty:
hoelzl@41981
   691
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
hoelzl@56996
   692
  shows "integral\<^sup>N (Pi\<^sub>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)
wenzelm@53015
   695
  have "\<And>A. emeasure (Pi\<^sub>M {} M) (Pi\<^sub>E {} A) = 1"
hoelzl@40859
   696
    using assms by (subst measure_times) auto
hoelzl@40859
   697
  then show ?thesis
hoelzl@56996
   698
    unfolding nn_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"
wenzelm@53015
   709
  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
   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
wenzelm@53015
   716
  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>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)"
wenzelm@53015
   721
       "incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k)"
wenzelm@53015
   722
       "(\<Union>k. \<Pi>\<^sub>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
wenzelm@53015
   725
  let ?F = "\<lambda>k. \<Pi>\<^sub>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)
wenzelm@53015
   731
    show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^sub>E i \<in> I \<union> J. space (M i))"
hoelzl@47694
   732
      by (rule prod_algebra_sets_into_space)
wenzelm@53015
   733
    show "sets ?P = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   734
      by (rule sets_PiM)
wenzelm@53015
   735
    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
   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)
wenzelm@53015
   740
    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
   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"
wenzelm@53015
   748
    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
   749
      by (auto simp add: prod_algebra_eq_finite)
wenzelm@53015
   750
    let ?B = "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M"
hoelzl@47694
   751
    let ?X = "?g -` A \<inter> space ?B"
wenzelm@53015
   752
    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
   753
      using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+
wenzelm@53015
   754
    then have X: "?X = (Pi\<^sub>E I F \<times> Pi\<^sub>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))"
haftmann@57418
   762
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod.union_inter_neutral)
wenzelm@53015
   763
    also have "\<dots> = emeasure ?P (Pi\<^sub>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@56996
   770
lemma (in product_sigma_finite) product_nn_integral_fold:
hoelzl@47694
   771
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
wenzelm@53015
   772
  and f: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
hoelzl@56996
   773
  shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
wenzelm@53015
   774
    (\<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
   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
wenzelm@53015
   778
  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
wenzelm@53015
   779
  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
   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@56996
   783
    apply (subst nn_integral_distr[OF measurable_merge f])
hoelzl@56996
   784
    apply (subst J.nn_integral_fst[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:
wenzelm@53015
   790
  "distr (Pi\<^sub>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)"
wenzelm@53374
   794
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
immler@50244
   795
    using sets.sets_into_space by (auto simp: space_PiM)
wenzelm@53374
   796
  then 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@56996
   801
lemma (in product_sigma_finite) product_nn_integral_singleton:
hoelzl@40859
   802
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@56996
   803
  shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (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@56996
   808
    apply (subst nn_integral_distr[OF measurable_component_singleton])
hoelzl@47694
   809
    apply simp_all
hoelzl@47694
   810
    done
hoelzl@40859
   811
qed
hoelzl@40859
   812
hoelzl@56996
   813
lemma (in product_sigma_finite) product_nn_integral_insert:
hoelzl@49780
   814
  assumes I[simp]: "finite I" "i \<notin> I"
wenzelm@53015
   815
    and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
hoelzl@56996
   816
  shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
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@56996
   823
    unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
hoelzl@56996
   824
  proof (rule nn_integral_cong, subst product_nn_integral_singleton[symmetric])
wenzelm@53015
   825
    fix x assume x: "x \<in> space (Pi\<^sub>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 .
wenzelm@53015
   830
    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
   831
      using x
hoelzl@56996
   832
      by (auto intro!: nn_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@56996
   837
lemma (in product_sigma_finite) product_nn_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@56996
   841
  shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
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))"
haftmann@57418
   847
    using insert by (auto intro!: setprod.cong)
wenzelm@53015
   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\<^sub>M J M)"
immler@50244
   849
    using sets.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@56996
   854
    apply (simp add: product_nn_integral_insert[OF insert(1,2) prod])
hoelzl@56996
   855
    apply (simp add: insert(2-) * pos borel setprod_ereal_pos nn_integral_multc)
hoelzl@56996
   856
    apply (subst nn_integral_cmult)
hoelzl@56996
   857
    apply (auto simp add: pos borel insert(2-) setprod_ereal_pos nn_integral_nonneg)
hoelzl@41981
   858
    done
hoelzl@47694
   859
qed (simp add: space_PiM)
hoelzl@41096
   860
hoelzl@50104
   861
lemma (in product_sigma_finite) distr_component:
wenzelm@53015
   862
  "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
   863
proof (intro measure_eqI[symmetric])
hoelzl@50104
   864
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@50104
   865
hoelzl@50104
   866
  have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
hoelzl@50104
   867
    by (auto simp: extensional_def restrict_def)
hoelzl@50104
   868
hoelzl@50104
   869
  fix A assume A: "A \<in> sets ?P"
wenzelm@53015
   870
  then have "emeasure ?P A = (\<integral>\<^sup>+x. indicator A x \<partial>?P)" 
hoelzl@50104
   871
    by simp
wenzelm@53015
   872
  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@56996
   873
    by (intro nn_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
hoelzl@50104
   874
  also have "\<dots> = emeasure ?D A"
hoelzl@56996
   875
    using A by (simp add: product_nn_integral_singleton emeasure_distr)
wenzelm@53015
   876
  finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" .
hoelzl@50104
   877
qed simp
hoelzl@41026
   878
hoelzl@49776
   879
lemma (in product_sigma_finite)
wenzelm@53015
   880
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
hoelzl@49776
   881
  shows emeasure_fold_integral:
wenzelm@53015
   882
    "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
   883
    and emeasure_fold_measurable:
wenzelm@53015
   884
    "(\<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
   885
proof -
hoelzl@49776
   886
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@49776
   887
  interpret J: finite_product_sigma_finite M J by default fact
wenzelm@53015
   888
  interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
wenzelm@53015
   889
  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
   890
    by (intro measurable_sets[OF _ A] measurable_merge assms)
hoelzl@49776
   891
hoelzl@49776
   892
  show ?I
hoelzl@49776
   893
    apply (subst distr_merge[symmetric, OF IJ])
hoelzl@49776
   894
    apply (subst emeasure_distr[OF measurable_merge A])
hoelzl@49776
   895
    apply (subst J.emeasure_pair_measure_alt[OF merge])
hoelzl@56996
   896
    apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49776
   897
    done
hoelzl@49776
   898
hoelzl@49776
   899
  show ?B
hoelzl@49776
   900
    using IJ.measurable_emeasure_Pair1[OF merge]
haftmann@56154
   901
    by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
hoelzl@49776
   902
qed
hoelzl@49776
   903
hoelzl@49776
   904
lemma sets_Collect_single:
wenzelm@53015
   905
  "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
   906
  by simp
hoelzl@49776
   907
hoelzl@49776
   908
lemma sigma_prod_algebra_sigma_eq_infinite:
hoelzl@49776
   909
  fixes E :: "'i \<Rightarrow> 'a set set"
hoelzl@49779
   910
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@49776
   911
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@49776
   912
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@49776
   913
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
wenzelm@53015
   914
  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
   915
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@49776
   916
proof
wenzelm@53015
   917
  let ?P = "sigma (space (Pi\<^sub>M I M)) P"
wenzelm@53015
   918
  have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))"
hoelzl@50123
   919
    using E_closed by (auto simp: space_PiM P_def subset_eq)
wenzelm@53015
   920
  then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@49776
   921
    by (simp add: space_PiM)
hoelzl@49776
   922
  have "sets (PiM I M) =
wenzelm@53015
   923
      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
   924
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@49776
   925
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
immler@50244
   926
  proof (safe intro!: sets.sigma_sets_subset)
hoelzl@49776
   927
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@49776
   928
    then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@49776
   929
      apply (subst measurable_iff_measure_of)
hoelzl@49776
   930
      apply (simp_all add: P_closed)
hoelzl@49776
   931
      using E_closed
hoelzl@49776
   932
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
   933
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
   934
      apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
hoelzl@49776
   935
      apply (rule_tac x=Aa in exI)
hoelzl@49776
   936
      apply (auto simp: space_PiM)
hoelzl@49776
   937
      done
hoelzl@49776
   938
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@49776
   939
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@49776
   940
      by (simp add: E_generates)
wenzelm@53015
   941
    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
   942
      using P_closed by (auto simp: space_PiM)
hoelzl@49776
   943
    finally show "\<dots> \<in> sets ?P" .
hoelzl@49776
   944
  qed
hoelzl@49776
   945
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@49776
   946
    by (simp add: P_closed)
hoelzl@49776
   947
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@49776
   948
    unfolding P_def space_PiM[symmetric]
immler@50244
   949
    by (intro sets.sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
hoelzl@49776
   950
qed
hoelzl@49776
   951
hoelzl@47694
   952
lemma sigma_prod_algebra_sigma_eq:
hoelzl@49779
   953
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
hoelzl@47694
   954
  assumes "finite I"
hoelzl@49779
   955
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@47694
   956
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@47694
   957
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@47694
   958
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
wenzelm@53015
   959
  defines "P == { Pi\<^sub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
hoelzl@47694
   960
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@47694
   961
proof
wenzelm@53015
   962
  let ?P = "sigma (space (Pi\<^sub>M I M)) P"
hoelzl@49779
   963
  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
hoelzl@49779
   964
  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
   965
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
wenzelm@53015
   966
  have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))"
hoelzl@50123
   967
    using E_closed by (auto simp: space_PiM P_def subset_eq)
wenzelm@53015
   968
  then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))"
hoelzl@47694
   969
    by (simp add: space_PiM)
hoelzl@47694
   970
  have "sets (PiM I M) =
wenzelm@53015
   971
      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
   972
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@47694
   973
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
immler@50244
   974
  proof (safe intro!: sets.sigma_sets_subset)
hoelzl@47694
   975
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@47694
   976
    have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@47694
   977
    proof (subst measurable_iff_measure_of)
hoelzl@47694
   978
      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
hoelzl@50123
   979
      from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto
hoelzl@47694
   980
      show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
   981
      proof
hoelzl@47694
   982
        fix A assume A: "A \<in> E i"
wenzelm@53015
   983
        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
   984
          using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm)
wenzelm@53015
   985
        also have "\<dots> = (\<Pi>\<^sub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
hoelzl@47694
   986
          by (intro PiE_cong) (simp add: S_union)
wenzelm@53015
   987
        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
   988
          using T
hoelzl@50123
   989
          apply (auto simp: PiE_iff bchoice_iff)
hoelzl@49779
   990
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
hoelzl@49779
   991
          apply (auto simp: bij_betw_def)
hoelzl@49779
   992
          done
hoelzl@47694
   993
        also have "\<dots> \<in> sets ?P"
immler@50244
   994
        proof (safe intro!: sets.countable_UN)
wenzelm@53015
   995
          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
   996
            using A S_in_E
hoelzl@47694
   997
            by (simp add: P_closed)
hoelzl@49779
   998
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
hoelzl@47694
   999
        qed
hoelzl@47694
  1000
        finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1001
          using P_closed by simp
hoelzl@47694
  1002
      qed
hoelzl@47694
  1003
    qed
hoelzl@47694
  1004
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@47694
  1005
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1006
      by (simp add: E_generates)
wenzelm@53015
  1007
    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
  1008
      using P_closed by (auto simp: space_PiM)
hoelzl@47694
  1009
    finally show "\<dots> \<in> sets ?P" .
hoelzl@47694
  1010
  qed
hoelzl@47694
  1011
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1012
    by (simp add: P_closed)
hoelzl@47694
  1013
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@47694
  1014
    using `finite I`
immler@50244
  1015
    by (auto intro!: sets.sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
hoelzl@47694
  1016
qed
hoelzl@47694
  1017
hoelzl@50104
  1018
lemma pair_measure_eq_distr_PiM:
hoelzl@50104
  1019
  fixes M1 :: "'a measure" and M2 :: "'a measure"
hoelzl@50104
  1020
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
blanchet@55414
  1021
  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
  1022
    (is "?P = ?D")
hoelzl@50104
  1023
proof (rule pair_measure_eqI[OF assms])
blanchet@55414
  1024
  interpret B: product_sigma_finite "case_bool M1 M2"
hoelzl@50104
  1025
    unfolding product_sigma_finite_def using assms by (auto split: bool.split)
blanchet@55414
  1026
  let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"
hoelzl@50104
  1027
hoelzl@50104
  1028
  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
  1029
    by auto
hoelzl@50104
  1030
  fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
blanchet@55414
  1031
  have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
hoelzl@50104
  1032
    by (simp add: UNIV_bool ac_simps)
blanchet@55414
  1033
  also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
hoelzl@50104
  1034
    using A B by (subst B.emeasure_PiM) (auto split: bool.split)
blanchet@55414
  1035
  also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
immler@50244
  1036
    using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
hoelzl@50123
  1037
    by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
hoelzl@50104
  1038
  finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
hoelzl@50104
  1039
    using A B
blanchet@55414
  1040
      measurable_component_singleton[of True UNIV "case_bool M1 M2"]
blanchet@55414
  1041
      measurable_component_singleton[of False UNIV "case_bool M1 M2"]
hoelzl@50104
  1042
    by (subst emeasure_distr) (auto simp: measurable_pair_iff)
hoelzl@50104
  1043
qed simp
hoelzl@50104
  1044
hoelzl@47694
  1045
end