src/HOL/Probability/Finite_Product_Measure.thy
author hoelzl
Fri Nov 16 14:46:23 2012 +0100 (2012-11-16)
changeset 50099 a58bb401af80
parent 50087 635d73673b5e
child 50104 de19856feb54
permissions -rw-r--r--
measurability for nat_case and comb_seq
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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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abbreviation
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  "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
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syntax
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  "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
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syntax (xsymbols)
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  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
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syntax (HTML output)
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  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
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translations
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  "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
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abbreviation
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  funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
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    (infixr "->\<^isub>E" 60) where
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  "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
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notation (xsymbols)
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  funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
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lemma extensional_insert[intro, simp]:
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  assumes "a \<in> extensional (insert i I)"
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  shows "a(i := b) \<in> extensional (insert i I)"
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  using assms unfolding extensional_def by auto
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lemma extensional_Int[simp]:
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  "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
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  unfolding extensional_def by auto
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lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
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  by (auto simp: extensional_def)
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lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
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  unfolding restrict_def extensional_def by auto
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lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
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  unfolding restrict_def by (simp add: fun_eq_iff)
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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_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
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  by (auto simp: restrict_def Pi_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 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 injective_vimage_restrict:
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  assumes J: "J \<subseteq> I"
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  and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}"
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  and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
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  shows "A = B"
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proof  (intro set_eqI)
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  fix x
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  from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
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  have "J \<inter> (I - J) = {}" by auto
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  show "x \<in> A \<longleftrightarrow> x \<in> B"
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  proof cases
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    assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)"
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    have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
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      using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub split: split_merge)
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    then show "x \<in> A \<longleftrightarrow> x \<in> B"
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      using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub eq split: split_merge)
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  next
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    assume "x \<notin> (\<Pi>\<^isub>E i\<in>J. S i)" with sets show "x \<in> A \<longleftrightarrow> x \<in> B" by auto
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  qed
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qed
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lemma extensional_insert_undefined[intro, simp]:
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  assumes "a \<in> extensional (insert i I)"
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  shows "a(i := undefined) \<in> extensional I"
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  using assms unfolding extensional_def by auto
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lemma extensional_insert_cancel[intro, simp]:
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  assumes "a \<in> extensional I"
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  shows "a \<in> extensional (insert i I)"
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  using assms unfolding extensional_def by auto
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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 Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
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  by auto
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lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)"
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  by auto
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lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
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  by (auto simp: Pi_def)
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lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
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  by (auto simp: Pi_def)
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lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
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  by (auto simp: Pi_def)
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lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
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  by (auto simp: Pi_def)
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lemma restrict_vimage:
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  assumes "I \<inter> J = {}"
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  shows "(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I J (E, F))"
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  using assms by (auto simp: restrict_Pi_cancel)
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lemma merge_vimage:
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  assumes "I \<inter> J = {}"
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  shows "merge I J -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
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  using assms by (auto simp: restrict_Pi_cancel)
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lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
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  by (auto simp: restrict_def)
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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 Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
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  apply auto
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  apply (drule_tac x=x in Pi_mem)
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  apply (simp_all split: split_if_asm)
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  apply (drule_tac x=i in Pi_mem)
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  apply (auto dest!: Pi_mem)
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  done
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lemma Pi_UN:
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  fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
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  assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
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  shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
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proof (intro set_eqI iffI)
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  fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
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  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
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  from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
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  obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
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    using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
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  have "f \<in> Pi I (A k)"
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  proof (intro Pi_I)
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    fix i assume "i \<in> I"
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    from mono[OF this, of "n i" k] k[OF this] n[OF this]
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    show "f i \<in> A k i" by auto
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  qed
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  then show "f \<in> (\<Union>n. Pi I (A n))" by auto
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qed auto
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lemma PiE_cong:
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  assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
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  shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
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  using assms by (auto intro!: Pi_cong)
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lemma restrict_upd[simp]:
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  "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
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  by (auto simp: fun_eq_iff)
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lemma Pi_eq_subset:
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  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
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  assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"
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  shows "F i \<subseteq> F' i"
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proof
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  fix x assume "x \<in> F i"
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  with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto
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  from choice[OF this] guess f .. note f = this
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  then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
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  then have "f \<in> Pi\<^isub>E I F'" using assms by simp
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  then show "x \<in> F' i" using f `i \<in> I` by auto
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qed
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lemma Pi_eq_iff_not_empty:
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  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
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  shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
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proof (intro iffI ballI)
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  fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"
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  show "F i = F' i"
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    using Pi_eq_subset[of I F F', OF ne eq i]
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    using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
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    by auto
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qed auto
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lemma Pi_eq_empty_iff:
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  "Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
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proof
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  assume "Pi\<^isub>E I F = {}"
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  show "\<exists>i\<in>I. F i = {}"
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  proof (rule ccontr)
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    assume "\<not> ?thesis"
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    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
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    from choice[OF this] guess f ..
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    then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
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    with `Pi\<^isub>E I F = {}` show False by auto
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  qed
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qed auto
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lemma Pi_eq_iff:
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  "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
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proof (intro iffI disjCI)
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  assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"
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  assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
hoelzl@41689
   256
  then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
hoelzl@41689
   257
    using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
hoelzl@41689
   258
  with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
hoelzl@41689
   259
next
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   260
  assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
hoelzl@41689
   261
  then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
hoelzl@41689
   262
    using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
hoelzl@41689
   263
qed
hoelzl@41689
   264
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   265
section "Finite product spaces"
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   266
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   267
section "Products"
hoelzl@40859
   268
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   269
definition prod_emb where
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   270
  "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
hoelzl@47694
   271
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   272
lemma prod_emb_iff: 
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   273
  "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))"
hoelzl@47694
   274
  unfolding prod_emb_def by auto
hoelzl@40859
   275
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   276
lemma
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   277
  shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
hoelzl@47694
   278
    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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   279
    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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   280
    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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   281
    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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   282
    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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   283
  by (auto simp: prod_emb_def)
hoelzl@40859
   284
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   285
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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   286
    prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))"
hoelzl@47694
   287
  by (force simp: prod_emb_def Pi_iff split_if_mem2)
hoelzl@47694
   288
hoelzl@47694
   289
lemma prod_emb_PiE_same_index[simp]: "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E"
hoelzl@47694
   290
  by (auto simp: prod_emb_def Pi_iff)
hoelzl@41689
   291
immler@50038
   292
lemma prod_emb_trans[simp]:
immler@50038
   293
  "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"
immler@50038
   294
  by (auto simp add: Int_absorb1 prod_emb_def)
immler@50038
   295
immler@50038
   296
lemma prod_emb_Pi:
immler@50038
   297
  assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
immler@50038
   298
  shows "prod_emb K M J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))"
immler@50038
   299
  using assms space_closed
immler@50038
   300
  by (auto simp: prod_emb_def Pi_iff split: split_if_asm) blast+
immler@50038
   301
immler@50038
   302
lemma prod_emb_id:
immler@50038
   303
  "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
immler@50038
   304
  by (auto simp: prod_emb_def Pi_iff subset_eq extensional_restrict)
immler@50038
   305
immler@50041
   306
lemma prod_emb_mono:
immler@50041
   307
  "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
immler@50041
   308
  by (auto simp: prod_emb_def)
immler@50041
   309
hoelzl@47694
   310
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
hoelzl@47694
   311
  "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
hoelzl@47694
   312
    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
hoelzl@47694
   313
    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
hoelzl@47694
   314
    (\<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)))"
hoelzl@47694
   315
hoelzl@47694
   316
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
hoelzl@47694
   317
  "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
hoelzl@47694
   318
    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
hoelzl@47694
   319
hoelzl@47694
   320
abbreviation
hoelzl@47694
   321
  "Pi\<^isub>M I M \<equiv> PiM I M"
hoelzl@41689
   322
hoelzl@40859
   323
syntax
hoelzl@47694
   324
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
hoelzl@40859
   325
hoelzl@40859
   326
syntax (xsymbols)
hoelzl@47694
   327
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
hoelzl@40859
   328
hoelzl@40859
   329
syntax (HTML output)
hoelzl@47694
   330
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
hoelzl@40859
   331
hoelzl@40859
   332
translations
hoelzl@47694
   333
  "PIM x:I. M" == "CONST PiM I (%x. M)"
hoelzl@41689
   334
hoelzl@47694
   335
lemma prod_algebra_sets_into_space:
hoelzl@47694
   336
  "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   337
  using assms by (auto simp: prod_emb_def prod_algebra_def)
hoelzl@40859
   338
hoelzl@47694
   339
lemma prod_algebra_eq_finite:
hoelzl@47694
   340
  assumes I: "finite I"
hoelzl@47694
   341
  shows "prod_algebra I M = {(\<Pi>\<^isub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
hoelzl@47694
   342
proof (intro iffI set_eqI)
hoelzl@47694
   343
  fix A assume "A \<in> ?L"
hoelzl@47694
   344
  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)"
hoelzl@47694
   345
    and A: "A = prod_emb I M J (PIE j:J. E j)"
hoelzl@47694
   346
    by (auto simp: prod_algebra_def)
hoelzl@47694
   347
  let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
hoelzl@47694
   348
  have A: "A = ?A"
hoelzl@47694
   349
    unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
hoelzl@47694
   350
  show "A \<in> ?R" unfolding A using J top
hoelzl@47694
   351
    by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
hoelzl@47694
   352
next
hoelzl@47694
   353
  fix A assume "A \<in> ?R"
hoelzl@47694
   354
  then obtain X where "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
hoelzl@47694
   355
  then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
hoelzl@47694
   356
    using sets_into_space by (force simp: prod_emb_def Pi_iff)
hoelzl@47694
   357
  from X I show "A \<in> ?L" unfolding A
hoelzl@47694
   358
    by (auto simp: prod_algebra_def)
hoelzl@47694
   359
qed
hoelzl@41095
   360
hoelzl@47694
   361
lemma prod_algebraI:
hoelzl@47694
   362
  "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
hoelzl@47694
   363
    \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
hoelzl@47694
   364
  by (auto simp: prod_algebra_def Pi_iff)
hoelzl@41689
   365
immler@50038
   366
lemma prod_algebraI_finite:
immler@50038
   367
  "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M"
immler@50038
   368
  using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp
immler@50038
   369
immler@50038
   370
lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
immler@50038
   371
proof (safe intro!: Int_stableI)
immler@50038
   372
  fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
immler@50038
   373
  then show "\<exists>G. Pi\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
immler@50038
   374
    by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"])
immler@50038
   375
qed
immler@50038
   376
hoelzl@47694
   377
lemma prod_algebraE:
hoelzl@47694
   378
  assumes A: "A \<in> prod_algebra I M"
hoelzl@47694
   379
  obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
hoelzl@47694
   380
    "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" 
hoelzl@47694
   381
  using A by (auto simp: prod_algebra_def)
hoelzl@42988
   382
hoelzl@47694
   383
lemma prod_algebraE_all:
hoelzl@47694
   384
  assumes A: "A \<in> prod_algebra I M"
hoelzl@47694
   385
  obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
hoelzl@47694
   386
proof -
hoelzl@47694
   387
  from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
hoelzl@47694
   388
    and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
hoelzl@47694
   389
    by (auto simp: prod_algebra_def)
hoelzl@47694
   390
  from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
hoelzl@47694
   391
    using sets_into_space by auto
hoelzl@47694
   392
  then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
hoelzl@47694
   393
    using A J by (auto simp: prod_emb_PiE)
hoelzl@47694
   394
  moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
hoelzl@47694
   395
    using top E by auto
hoelzl@47694
   396
  ultimately show ?thesis using that by auto
hoelzl@47694
   397
qed
hoelzl@40859
   398
hoelzl@47694
   399
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
hoelzl@47694
   400
proof (unfold Int_stable_def, safe)
hoelzl@47694
   401
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   402
  from prod_algebraE[OF this] guess J E . note A = this
hoelzl@47694
   403
  fix B assume "B \<in> prod_algebra I M"
hoelzl@47694
   404
  from prod_algebraE[OF this] guess K F . note B = this
hoelzl@47694
   405
  have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^isub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter> 
hoelzl@47694
   406
      (if i \<in> K then F i else space (M i)))"
hoelzl@47694
   407
    unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space]
hoelzl@47694
   408
    apply (subst (1 2 3) prod_emb_PiE)
hoelzl@47694
   409
    apply (simp_all add: subset_eq PiE_Int)
hoelzl@47694
   410
    apply blast
hoelzl@47694
   411
    apply (intro PiE_cong)
hoelzl@47694
   412
    apply auto
hoelzl@47694
   413
    done
hoelzl@47694
   414
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   415
    using A B by (auto intro!: prod_algebraI)
hoelzl@47694
   416
  finally show "A \<inter> B \<in> prod_algebra I M" .
hoelzl@47694
   417
qed
hoelzl@47694
   418
hoelzl@47694
   419
lemma prod_algebra_mono:
hoelzl@47694
   420
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
hoelzl@47694
   421
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
hoelzl@47694
   422
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
hoelzl@47694
   423
proof
hoelzl@47694
   424
  fix A assume "A \<in> prod_algebra I E"
hoelzl@47694
   425
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
hoelzl@47694
   426
    and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   427
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
hoelzl@47694
   428
    by (auto simp: prod_algebra_def)
hoelzl@47694
   429
  moreover
hoelzl@47694
   430
  from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
hoelzl@47694
   431
    by (rule PiE_cong)
hoelzl@47694
   432
  with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   433
    by (simp add: prod_emb_def)
hoelzl@47694
   434
  moreover
hoelzl@47694
   435
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
hoelzl@47694
   436
    by auto
hoelzl@47694
   437
  ultimately show "A \<in> prod_algebra I F"
hoelzl@47694
   438
    apply (simp add: prod_algebra_def image_iff)
hoelzl@47694
   439
    apply (intro exI[of _ J] exI[of _ G] conjI)
hoelzl@47694
   440
    apply auto
hoelzl@47694
   441
    done
hoelzl@41689
   442
qed
hoelzl@41689
   443
hoelzl@47694
   444
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   445
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
hoelzl@47694
   446
hoelzl@47694
   447
lemma sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)"
hoelzl@47694
   448
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
hoelzl@41689
   449
hoelzl@47694
   450
lemma sets_PiM_single: "sets (PiM I M) =
hoelzl@47694
   451
    sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@47694
   452
    (is "_ = sigma_sets ?\<Omega> ?R")
hoelzl@47694
   453
  unfolding sets_PiM
hoelzl@47694
   454
proof (rule sigma_sets_eqI)
hoelzl@47694
   455
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
hoelzl@47694
   456
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   457
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   458
  show "A \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   459
  proof cases
hoelzl@47694
   460
    assume "I = {}"
hoelzl@47694
   461
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
hoelzl@47694
   462
    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
hoelzl@47694
   463
  next
hoelzl@47694
   464
    assume "I \<noteq> {}"
hoelzl@47694
   465
    with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})"
hoelzl@47694
   466
      using sets_into_space[OF X(5)]
hoelzl@47694
   467
      by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast
hoelzl@47694
   468
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   469
      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
hoelzl@47694
   470
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
hoelzl@47694
   471
  qed
hoelzl@47694
   472
next
hoelzl@47694
   473
  fix A assume "A \<in> ?R"
hoelzl@47694
   474
  then obtain i B where A: "A = {f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" 
hoelzl@47694
   475
    by auto
hoelzl@47694
   476
  then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
hoelzl@47694
   477
    using sets_into_space[OF A(3)]
hoelzl@47694
   478
    apply (subst prod_emb_PiE)
hoelzl@47694
   479
    apply (auto simp: Pi_iff split: split_if_asm)
hoelzl@47694
   480
    apply blast
hoelzl@47694
   481
    done
hoelzl@47694
   482
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
hoelzl@47694
   483
    using A by (intro sigma_sets.Basic prod_algebraI) auto
hoelzl@47694
   484
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
hoelzl@47694
   485
qed
hoelzl@47694
   486
hoelzl@47694
   487
lemma sets_PiM_I:
hoelzl@47694
   488
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
hoelzl@47694
   489
  shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   490
proof cases
hoelzl@47694
   491
  assume "J = {}"
hoelzl@47694
   492
  then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
hoelzl@47694
   493
    by (auto simp: prod_emb_def)
hoelzl@47694
   494
  then show ?thesis
hoelzl@47694
   495
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
hoelzl@47694
   496
next
hoelzl@47694
   497
  assume "J \<noteq> {}" with assms show ?thesis
hoelzl@50003
   498
    by (force simp add: sets_PiM prod_algebra_def)
hoelzl@40859
   499
qed
hoelzl@40859
   500
hoelzl@47694
   501
lemma measurable_PiM:
hoelzl@47694
   502
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   503
  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
   504
    f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" 
hoelzl@47694
   505
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   506
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   507
proof (rule measurable_sigma_sets)
hoelzl@47694
   508
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   509
  from prod_algebraE[OF this] guess J X .
hoelzl@47694
   510
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
hoelzl@47694
   511
qed
hoelzl@47694
   512
hoelzl@47694
   513
lemma measurable_PiM_Collect:
hoelzl@47694
   514
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   515
  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
   516
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 
hoelzl@47694
   517
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   518
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   519
proof (rule measurable_sigma_sets)
hoelzl@47694
   520
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   521
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   522
  have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
hoelzl@47694
   523
    using sets_into_space[OF X(5)] X(2-) space unfolding X(1)
hoelzl@47694
   524
    by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm)
hoelzl@47694
   525
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
hoelzl@47694
   526
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@41689
   527
qed
hoelzl@41095
   528
hoelzl@47694
   529
lemma measurable_PiM_single:
hoelzl@47694
   530
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   531
  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
   532
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   533
  using sets_PiM_single
hoelzl@47694
   534
proof (rule measurable_sigma_sets)
hoelzl@47694
   535
  fix A assume "A \<in> {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@47694
   536
  then obtain B i where "A = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
hoelzl@47694
   537
    by auto
hoelzl@47694
   538
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
hoelzl@47694
   539
  also have "\<dots> \<in> sets N" using B by (rule sets)
hoelzl@47694
   540
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@47694
   541
qed (auto simp: space)
hoelzl@40859
   542
hoelzl@50099
   543
lemma measurable_PiM_single':
hoelzl@50099
   544
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
hoelzl@50099
   545
    and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@50099
   546
  shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@50099
   547
proof (rule measurable_PiM_single)
hoelzl@50099
   548
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@50099
   549
  then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
hoelzl@50099
   550
    by auto
hoelzl@50099
   551
  then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
hoelzl@50099
   552
    using A f by (auto intro!: measurable_sets)
hoelzl@50099
   553
qed fact
hoelzl@50099
   554
hoelzl@50003
   555
lemma sets_PiM_I_finite[measurable]:
hoelzl@47694
   556
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
hoelzl@47694
   557
  shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   558
  using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
hoelzl@47694
   559
hoelzl@50021
   560
lemma measurable_component_singleton:
hoelzl@41689
   561
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
hoelzl@41689
   562
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   563
  fix A assume "A \<in> sets (M i)"
hoelzl@47694
   564
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)"
hoelzl@47694
   565
    using sets_into_space `i \<in> I`
hoelzl@47694
   566
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
hoelzl@41689
   567
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
hoelzl@47694
   568
    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
hoelzl@47694
   569
qed (insert `i \<in> I`, auto simp: space_PiM)
hoelzl@47694
   570
hoelzl@50021
   571
lemma measurable_component_singleton'[measurable_app]:
hoelzl@50021
   572
  assumes f: "f \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@50021
   573
  assumes i: "i \<in> I"
hoelzl@50021
   574
  shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
hoelzl@50021
   575
  using measurable_compose[OF f measurable_component_singleton, OF i] .
hoelzl@50021
   576
hoelzl@50099
   577
lemma measurable_PiM_component_rev[measurable (raw)]:
hoelzl@50099
   578
  "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
hoelzl@50099
   579
  by simp
hoelzl@50099
   580
hoelzl@50021
   581
lemma measurable_nat_case[measurable (raw)]:
hoelzl@50021
   582
  assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
hoelzl@50021
   583
    "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
hoelzl@50021
   584
  shows "(\<lambda>x. nat_case (f x) (g x) i) \<in> measurable M N"
hoelzl@50021
   585
  by (cases i) simp_all
hoelzl@50021
   586
hoelzl@50099
   587
lemma measurable_nat_case'[measurable (raw)]:
hoelzl@50099
   588
  assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
hoelzl@50099
   589
  shows "(\<lambda>x. nat_case (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
hoelzl@50099
   590
  using fg[THEN measurable_space]
hoelzl@50099
   591
  by (auto intro!: measurable_PiM_single' simp add: space_PiM Pi_iff split: nat.split)
hoelzl@50099
   592
hoelzl@50003
   593
lemma measurable_add_dim[measurable]:
hoelzl@49776
   594
  "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
hoelzl@47694
   595
    (is "?f \<in> measurable ?P ?I")
hoelzl@47694
   596
proof (rule measurable_PiM_single)
hoelzl@47694
   597
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
hoelzl@47694
   598
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
hoelzl@47694
   599
    (if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
hoelzl@47694
   600
    using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
hoelzl@47694
   601
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   602
    using A j
hoelzl@47694
   603
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@47694
   604
  finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
hoelzl@47694
   605
qed (auto simp: space_pair_measure space_PiM)
hoelzl@41661
   606
hoelzl@50003
   607
lemma measurable_component_update:
hoelzl@50003
   608
  "x \<in> space (Pi\<^isub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)"
hoelzl@50003
   609
  by simp
hoelzl@50003
   610
hoelzl@50003
   611
lemma measurable_merge[measurable]:
hoelzl@49780
   612
  "merge I J \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
hoelzl@47694
   613
    (is "?f \<in> measurable ?P ?U")
hoelzl@47694
   614
proof (rule measurable_PiM_single)
hoelzl@47694
   615
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
hoelzl@49780
   616
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
hoelzl@47694
   617
    (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
   618
    by (auto simp: merge_def)
hoelzl@47694
   619
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   620
    using A
hoelzl@47694
   621
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@49780
   622
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
hoelzl@49776
   623
qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def)
hoelzl@42988
   624
hoelzl@50003
   625
lemma measurable_restrict[measurable (raw)]:
hoelzl@47694
   626
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
hoelzl@47694
   627
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@47694
   628
proof (rule measurable_PiM_single)
hoelzl@47694
   629
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@47694
   630
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
hoelzl@47694
   631
    by auto
hoelzl@47694
   632
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
hoelzl@47694
   633
    using A X by (auto intro!: measurable_sets)
hoelzl@47694
   634
qed (insert X, auto dest: measurable_space)
hoelzl@47694
   635
immler@50038
   636
lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)"
immler@50038
   637
  by (intro measurable_restrict measurable_component_singleton) auto
immler@50038
   638
immler@50038
   639
lemma measurable_prod_emb[intro, simp]:
immler@50038
   640
  "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^isub>M L M)"
immler@50038
   641
  unfolding prod_emb_def space_PiM[symmetric]
immler@50038
   642
  by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
immler@50038
   643
hoelzl@50003
   644
lemma sets_in_Pi_aux:
hoelzl@50003
   645
  "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
   646
  {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
hoelzl@50003
   647
  by (simp add: subset_eq Pi_iff)
hoelzl@50003
   648
hoelzl@50003
   649
lemma sets_in_Pi[measurable (raw)]:
hoelzl@50003
   650
  "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
hoelzl@50003
   651
  (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50003
   652
  Sigma_Algebra.pred N (\<lambda>x. f x \<in> Pi I F)"
hoelzl@50003
   653
  unfolding pred_def
hoelzl@50003
   654
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
hoelzl@50003
   655
hoelzl@50003
   656
lemma sets_in_extensional_aux:
hoelzl@50003
   657
  "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
hoelzl@50003
   658
proof -
hoelzl@50003
   659
  have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
hoelzl@50003
   660
    by (auto simp add: extensional_def space_PiM)
hoelzl@50003
   661
  then show ?thesis by simp
hoelzl@50003
   662
qed
hoelzl@50003
   663
hoelzl@50003
   664
lemma sets_in_extensional[measurable (raw)]:
hoelzl@50003
   665
  "f \<in> measurable N (PiM I M) \<Longrightarrow> Sigma_Algebra.pred N (\<lambda>x. f x \<in> extensional I)"
hoelzl@50003
   666
  unfolding pred_def
hoelzl@50003
   667
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
hoelzl@50003
   668
hoelzl@47694
   669
locale product_sigma_finite =
hoelzl@47694
   670
  fixes M :: "'i \<Rightarrow> 'a measure"
hoelzl@41689
   671
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   672
hoelzl@41689
   673
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   674
  by (rule sigma_finite_measures)
hoelzl@40859
   675
hoelzl@47694
   676
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@47694
   677
  fixes I :: "'i set"
hoelzl@47694
   678
  assumes finite_index: "finite I"
hoelzl@41689
   679
hoelzl@40859
   680
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   681
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   682
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
hoelzl@47694
   683
    (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
hoelzl@47694
   684
    (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
hoelzl@40859
   685
proof -
hoelzl@47694
   686
  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
   687
    using M.sigma_finite_incseq by metis
hoelzl@40859
   688
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@47694
   689
  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
   690
    by auto
hoelzl@40859
   691
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
hoelzl@47694
   692
  note space_PiM[simp]
hoelzl@40859
   693
  show ?thesis
hoelzl@41981
   694
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   695
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   696
  next
hoelzl@47694
   697
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   698
  next
hoelzl@47694
   699
    fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)"
hoelzl@47694
   700
      using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space
hoelzl@47694
   701
      by auto blast
hoelzl@40859
   702
  next
hoelzl@47694
   703
    fix f assume "f \<in> space (PiM I M)"
hoelzl@41981
   704
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@41981
   705
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
hoelzl@40859
   706
  next
hoelzl@40859
   707
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   708
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   709
  qed
hoelzl@40859
   710
qed
hoelzl@40859
   711
hoelzl@49780
   712
lemma
hoelzl@49780
   713
  shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
hoelzl@49780
   714
    and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
hoelzl@49780
   715
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
hoelzl@49780
   716
hoelzl@49780
   717
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
hoelzl@49780
   718
proof -
hoelzl@49780
   719
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
hoelzl@49780
   720
  have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
hoelzl@49780
   721
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49780
   722
    show "positive (PiM {} M) ?\<mu>"
hoelzl@49780
   723
      by (auto simp: positive_def)
hoelzl@49780
   724
    show "countably_additive (PiM {} M) ?\<mu>"
hoelzl@49780
   725
      by (rule countably_additiveI_finite)
hoelzl@49780
   726
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
hoelzl@49780
   727
  qed (auto simp: prod_emb_def)
hoelzl@49780
   728
  also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
hoelzl@49780
   729
    by (auto simp: prod_emb_def)
hoelzl@49780
   730
  finally show ?thesis
hoelzl@49780
   731
    by simp
hoelzl@49780
   732
qed
hoelzl@49780
   733
hoelzl@49780
   734
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
hoelzl@49780
   735
  by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
hoelzl@49780
   736
hoelzl@49776
   737
lemma (in product_sigma_finite) emeasure_PiM:
hoelzl@49776
   738
  "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
hoelzl@49776
   739
proof (induct I arbitrary: A rule: finite_induct)
hoelzl@40859
   740
  case (insert i I)
hoelzl@41689
   741
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   742
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   743
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@41661
   744
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@47694
   745
hoelzl@47694
   746
  let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
hoelzl@47694
   747
  let ?\<mu> = "emeasure ?P"
hoelzl@47694
   748
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
hoelzl@47694
   749
  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
   750
hoelzl@49776
   751
  have "emeasure (Pi\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>E (insert i I) A)) =
hoelzl@49776
   752
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
hoelzl@49776
   753
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49776
   754
    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
   755
    then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto
hoelzl@49776
   756
    let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
hoelzl@49776
   757
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
hoelzl@49776
   758
    have "?\<mu> ?p =
hoelzl@49776
   759
      emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))"
hoelzl@49776
   760
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
hoelzl@49776
   761
    also have "?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
hoelzl@49776
   762
      using J E[rule_format, THEN sets_into_space]
hoelzl@49776
   763
      by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm)
hoelzl@49776
   764
    also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
hoelzl@49776
   765
      emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
hoelzl@49776
   766
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
hoelzl@49776
   767
    also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
hoelzl@49776
   768
      using J E[rule_format, THEN sets_into_space]
hoelzl@49776
   769
      by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+
hoelzl@49776
   770
    also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
hoelzl@49776
   771
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
hoelzl@49776
   772
      using E by (subst insert) (auto intro!: setprod_cong)
hoelzl@49776
   773
    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
   774
       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
   775
      using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
hoelzl@49776
   776
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
hoelzl@49776
   777
      using insert(1,2) J E by (intro setprod_mono_one_right) auto
hoelzl@49776
   778
    finally show "?\<mu> ?p = \<dots>" .
hoelzl@47694
   779
hoelzl@49776
   780
    show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>E i\<in>insert i I. space (M i))"
hoelzl@49776
   781
      using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff)
hoelzl@49776
   782
  next
hoelzl@49776
   783
    show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
hoelzl@49776
   784
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
hoelzl@49776
   785
  next
hoelzl@49776
   786
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
hoelzl@49776
   787
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
hoelzl@49776
   788
      using insert by auto
hoelzl@49776
   789
  qed (auto intro!: setprod_cong)
hoelzl@49776
   790
  with insert show ?case
hoelzl@49776
   791
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space)
hoelzl@50003
   792
qed simp
hoelzl@47694
   793
hoelzl@49776
   794
lemma (in product_sigma_finite) sigma_finite: 
hoelzl@49776
   795
  assumes "finite I"
hoelzl@49776
   796
  shows "sigma_finite_measure (PiM I M)"
hoelzl@49776
   797
proof -
hoelzl@49776
   798
  interpret finite_product_sigma_finite M I by default fact
hoelzl@49776
   799
hoelzl@49776
   800
  from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@49776
   801
  then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
hoelzl@49776
   802
    "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
hoelzl@49776
   803
    "(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
hoelzl@49776
   804
    "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
hoelzl@47694
   805
    by blast+
hoelzl@49776
   806
  let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"
hoelzl@47694
   807
hoelzl@49776
   808
  show ?thesis
hoelzl@47694
   809
  proof (unfold_locales, intro exI[of _ ?F] conjI allI)
hoelzl@49776
   810
    show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
hoelzl@47694
   811
  next
hoelzl@49776
   812
    from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
hoelzl@47694
   813
  next
hoelzl@47694
   814
    fix j
hoelzl@49776
   815
    from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
hoelzl@49776
   816
    show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
hoelzl@49776
   817
      by (subst emeasure_PiM) auto
hoelzl@40859
   818
  qed
hoelzl@40859
   819
qed
hoelzl@40859
   820
hoelzl@47694
   821
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
hoelzl@47694
   822
  using sigma_finite[OF finite_index] .
hoelzl@40859
   823
hoelzl@40859
   824
lemma (in finite_product_sigma_finite) measure_times:
hoelzl@47694
   825
  "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^isub>M I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
hoelzl@47694
   826
  using emeasure_PiM[OF finite_index] by auto
hoelzl@41096
   827
hoelzl@40859
   828
lemma (in product_sigma_finite) positive_integral_empty:
hoelzl@41981
   829
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
hoelzl@41981
   830
  shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
hoelzl@40859
   831
proof -
hoelzl@41689
   832
  interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
hoelzl@47694
   833
  have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
hoelzl@40859
   834
    using assms by (subst measure_times) auto
hoelzl@40859
   835
  then show ?thesis
hoelzl@47694
   836
    unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
hoelzl@47694
   837
  proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
hoelzl@41981
   838
    show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
hoelzl@44928
   839
      by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
hoelzl@41981
   840
    show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
hoelzl@44928
   841
      by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
hoelzl@40859
   842
  qed
hoelzl@40859
   843
qed
hoelzl@40859
   844
hoelzl@47694
   845
lemma (in product_sigma_finite) distr_merge:
hoelzl@40859
   846
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@49780
   847
  shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (merge I J) = Pi\<^isub>M (I \<union> J) M"
hoelzl@47694
   848
   (is "?D = ?P")
hoelzl@40859
   849
proof -
hoelzl@41689
   850
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   851
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@40859
   852
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   853
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   854
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   855
  let ?g = "merge I J"
hoelzl@47694
   856
hoelzl@41661
   857
  from IJ.sigma_finite_pairs obtain F where
hoelzl@41661
   858
    F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
hoelzl@41981
   859
       "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
hoelzl@47694
   860
       "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
hoelzl@47694
   861
       "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
hoelzl@41661
   862
    by auto
hoelzl@41661
   863
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
hoelzl@47694
   864
  
hoelzl@47694
   865
  show ?thesis
hoelzl@47694
   866
  proof (rule measure_eqI_generator_eq[symmetric])
hoelzl@47694
   867
    show "Int_stable (prod_algebra (I \<union> J) M)"
hoelzl@47694
   868
      by (rule Int_stable_prod_algebra)
hoelzl@47694
   869
    show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
hoelzl@47694
   870
      by (rule prod_algebra_sets_into_space)
hoelzl@47694
   871
    show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   872
      by (rule sets_PiM)
hoelzl@47694
   873
    then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   874
      by simp
hoelzl@47694
   875
hoelzl@47694
   876
    show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
hoelzl@47694
   877
      using fin by (auto simp: prod_algebra_eq_finite)
hoelzl@47694
   878
    show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))"
hoelzl@47694
   879
      using F(3) by (simp add: space_PiM)
hoelzl@41981
   880
  next
hoelzl@41981
   881
    fix k
hoelzl@47694
   882
    from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
hoelzl@47694
   883
    show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
hoelzl@41661
   884
  next
hoelzl@47694
   885
    fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
hoelzl@50003
   886
    with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
hoelzl@47694
   887
      by (auto simp add: prod_algebra_eq_finite)
hoelzl@47694
   888
    let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
hoelzl@47694
   889
    let ?X = "?g -` A \<inter> space ?B"
hoelzl@47694
   890
    have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)"
hoelzl@50003
   891
      using F[rule_format, THEN sets_into_space] by (force simp: space_PiM)+
hoelzl@47694
   892
    then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
hoelzl@47694
   893
      unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
hoelzl@47694
   894
    have "emeasure ?D A = emeasure ?B ?X"
hoelzl@47694
   895
      using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
hoelzl@47694
   896
    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
   897
      using `finite J` `finite I` F unfolding X
hoelzl@49776
   898
      by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff)
hoelzl@47694
   899
    also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
hoelzl@41661
   900
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
hoelzl@47694
   901
    also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
hoelzl@41661
   902
      using `finite J` `finite I` F unfolding A
hoelzl@41661
   903
      by (intro IJ.measure_times[symmetric]) auto
hoelzl@47694
   904
    finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
hoelzl@47694
   905
  qed
hoelzl@41661
   906
qed
hoelzl@41026
   907
hoelzl@41026
   908
lemma (in product_sigma_finite) product_positive_integral_fold:
hoelzl@47694
   909
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
hoelzl@41689
   910
  and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41689
   911
  shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
hoelzl@49780
   912
    (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
hoelzl@41026
   913
proof -
hoelzl@41689
   914
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   915
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41831
   916
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   917
  have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
hoelzl@49776
   918
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
hoelzl@41661
   919
  show ?thesis
hoelzl@47694
   920
    apply (subst distr_merge[OF IJ, symmetric])
hoelzl@49776
   921
    apply (subst positive_integral_distr[OF measurable_merge f])
hoelzl@49999
   922
    apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
hoelzl@47694
   923
    apply simp
hoelzl@47694
   924
    done
hoelzl@40859
   925
qed
hoelzl@40859
   926
hoelzl@47694
   927
lemma (in product_sigma_finite) distr_singleton:
hoelzl@47694
   928
  "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
hoelzl@47694
   929
proof (intro measure_eqI[symmetric])
hoelzl@41831
   930
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   931
  fix A assume A: "A \<in> sets (M i)"
hoelzl@47694
   932
  moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
hoelzl@47694
   933
    using sets_into_space by (auto simp: space_PiM)
hoelzl@47694
   934
  ultimately show "emeasure (M i) A = emeasure ?D A"
hoelzl@47694
   935
    using A I.measure_times[of "\<lambda>_. A"]
hoelzl@47694
   936
    by (simp add: emeasure_distr measurable_component_singleton)
hoelzl@47694
   937
qed simp
hoelzl@41831
   938
hoelzl@41026
   939
lemma (in product_sigma_finite) product_positive_integral_singleton:
hoelzl@40859
   940
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   941
  shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
hoelzl@40859
   942
proof -
hoelzl@41689
   943
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   944
  from f show ?thesis
hoelzl@47694
   945
    apply (subst distr_singleton[symmetric])
hoelzl@47694
   946
    apply (subst positive_integral_distr[OF measurable_component_singleton])
hoelzl@47694
   947
    apply simp_all
hoelzl@47694
   948
    done
hoelzl@40859
   949
qed
hoelzl@40859
   950
hoelzl@41096
   951
lemma (in product_sigma_finite) product_positive_integral_insert:
hoelzl@49780
   952
  assumes I[simp]: "finite I" "i \<notin> I"
hoelzl@41689
   953
    and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@41689
   954
  shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"
hoelzl@41096
   955
proof -
hoelzl@41689
   956
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41689
   957
  interpret i: finite_product_sigma_finite M "{i}" by default auto
hoelzl@41689
   958
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
hoelzl@41689
   959
    using f by auto
hoelzl@41096
   960
  show ?thesis
hoelzl@49780
   961
    unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
hoelzl@49780
   962
  proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
hoelzl@47694
   963
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@49780
   964
    let ?f = "\<lambda>y. f (x(i := y))"
hoelzl@49780
   965
    show "?f \<in> borel_measurable (M i)"
hoelzl@47694
   966
      using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
hoelzl@47694
   967
      unfolding comp_def .
hoelzl@49780
   968
    show "(\<integral>\<^isup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^isub>M {i} M) = (\<integral>\<^isup>+ y. f (x(i := y i)) \<partial>Pi\<^isub>M {i} M)"
hoelzl@49780
   969
      using x
hoelzl@49780
   970
      by (auto intro!: positive_integral_cong arg_cong[where f=f]
hoelzl@49780
   971
               simp add: space_PiM extensional_def)
hoelzl@41096
   972
  qed
hoelzl@41096
   973
qed
hoelzl@41096
   974
hoelzl@41096
   975
lemma (in product_sigma_finite) product_positive_integral_setprod:
hoelzl@43920
   976
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
hoelzl@41096
   977
  assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41981
   978
  and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
hoelzl@41689
   979
  shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
hoelzl@41096
   980
using assms proof induct
hoelzl@41096
   981
  case (insert i I)
hoelzl@41096
   982
  note `finite I`[intro, simp]
hoelzl@41689
   983
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41096
   984
  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
   985
    using insert by (auto intro!: setprod_cong)
hoelzl@41689
   986
  have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
hoelzl@41096
   987
    using sets_into_space insert
hoelzl@47694
   988
    by (intro borel_measurable_ereal_setprod
hoelzl@41689
   989
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
hoelzl@41096
   990
       auto
hoelzl@41981
   991
  then show ?case
hoelzl@41981
   992
    apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
hoelzl@47694
   993
    apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
hoelzl@47694
   994
    apply (subst positive_integral_cmult)
hoelzl@47694
   995
    apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
hoelzl@41981
   996
    done
hoelzl@47694
   997
qed (simp add: space_PiM)
hoelzl@41096
   998
hoelzl@41026
   999
lemma (in product_sigma_finite) product_integral_singleton:
hoelzl@41026
  1000
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
  1001
  shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
hoelzl@41026
  1002
proof -
hoelzl@41689
  1003
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@43920
  1004
  have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
hoelzl@43920
  1005
    "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
hoelzl@41026
  1006
    using assms by auto
hoelzl@41026
  1007
  show ?thesis
hoelzl@41689
  1008
    unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
hoelzl@41026
  1009
qed
hoelzl@41026
  1010
hoelzl@41026
  1011
lemma (in product_sigma_finite) product_integral_fold:
hoelzl@41026
  1012
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@41689
  1013
  and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
hoelzl@49780
  1014
  shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
hoelzl@41026
  1015
proof -
hoelzl@41689
  1016
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
  1017
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41026
  1018
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
  1019
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
  1020
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
  1021
  let ?M = "merge I J"
hoelzl@41689
  1022
  let ?f = "\<lambda>x. f (?M x)"
hoelzl@47694
  1023
  from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@47694
  1024
    by auto
hoelzl@49780
  1025
  have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
hoelzl@49776
  1026
    using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
hoelzl@47694
  1027
  have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
hoelzl@49776
  1028
    by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
hoelzl@41026
  1029
  show ?thesis
hoelzl@47694
  1030
    apply (subst distr_merge[symmetric, OF IJ fin])
hoelzl@49776
  1031
    apply (subst integral_distr[OF measurable_merge f_borel])
hoelzl@47694
  1032
    apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
hoelzl@47694
  1033
    apply simp
hoelzl@47694
  1034
    done
hoelzl@41026
  1035
qed
hoelzl@41026
  1036
hoelzl@49776
  1037
lemma (in product_sigma_finite)
hoelzl@49776
  1038
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
hoelzl@49776
  1039
  shows emeasure_fold_integral:
hoelzl@49780
  1040
    "emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
hoelzl@49776
  1041
    and emeasure_fold_measurable:
hoelzl@49780
  1042
    "(\<lambda>x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
hoelzl@49776
  1043
proof -
hoelzl@49776
  1044
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@49776
  1045
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@49776
  1046
  interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
hoelzl@49780
  1047
  have merge: "merge I J -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
hoelzl@49776
  1048
    by (intro measurable_sets[OF _ A] measurable_merge assms)
hoelzl@49776
  1049
hoelzl@49776
  1050
  show ?I
hoelzl@49776
  1051
    apply (subst distr_merge[symmetric, OF IJ])
hoelzl@49776
  1052
    apply (subst emeasure_distr[OF measurable_merge A])
hoelzl@49776
  1053
    apply (subst J.emeasure_pair_measure_alt[OF merge])
hoelzl@49776
  1054
    apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49776
  1055
    done
hoelzl@49776
  1056
hoelzl@49776
  1057
  show ?B
hoelzl@49776
  1058
    using IJ.measurable_emeasure_Pair1[OF merge]
hoelzl@49776
  1059
    by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
hoelzl@49776
  1060
qed
hoelzl@49776
  1061
hoelzl@41096
  1062
lemma (in product_sigma_finite) product_integral_insert:
hoelzl@47694
  1063
  assumes I: "finite I" "i \<notin> I"
hoelzl@41689
  1064
    and f: "integrable (Pi\<^isub>M (insert i I) M) f"
hoelzl@41689
  1065
  shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
hoelzl@41096
  1066
proof -
hoelzl@47694
  1067
  have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
hoelzl@47694
  1068
    by simp
hoelzl@49780
  1069
  also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)"
hoelzl@47694
  1070
    using f I by (intro product_integral_fold) auto
hoelzl@47694
  1071
  also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
hoelzl@47694
  1072
  proof (rule integral_cong, subst product_integral_singleton[symmetric])
hoelzl@47694
  1073
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@47694
  1074
    have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@47694
  1075
      using f by auto
hoelzl@47694
  1076
    show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
hoelzl@47694
  1077
      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
hoelzl@47694
  1078
      unfolding comp_def .
hoelzl@49780
  1079
    from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)"
hoelzl@47694
  1080
      by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def)
hoelzl@41096
  1081
  qed
hoelzl@47694
  1082
  finally show ?thesis .
hoelzl@41096
  1083
qed
hoelzl@41096
  1084
hoelzl@41096
  1085
lemma (in product_sigma_finite) product_integrable_setprod:
hoelzl@41096
  1086
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@41689
  1087
  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
  1088
  shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
hoelzl@41096
  1089
proof -
hoelzl@41689
  1090
  interpret finite_product_sigma_finite M I by default fact
hoelzl@41096
  1091
  have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41689
  1092
    using integrable unfolding integrable_def by auto
hoelzl@47694
  1093
  have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@47694
  1094
    using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
hoelzl@41689
  1095
  moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
hoelzl@41096
  1096
  proof (unfold integrable_def, intro conjI)
hoelzl@47694
  1097
    show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@41096
  1098
      using borel by auto
hoelzl@47694
  1099
    have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)"
hoelzl@43920
  1100
      by (simp add: setprod_ereal abs_setprod)
hoelzl@43920
  1101
    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
hoelzl@41096
  1102
      using f by (subst product_positive_integral_setprod) auto
hoelzl@41981
  1103
    also have "\<dots> < \<infinity>"
hoelzl@47694
  1104
      using integrable[THEN integrable_abs]
hoelzl@47694
  1105
      by (simp add: setprod_PInf integrable_def positive_integral_positive)
hoelzl@47694
  1106
    finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
hoelzl@47694
  1107
    have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
hoelzl@41981
  1108
      by (intro positive_integral_cong_pos) auto
hoelzl@47694
  1109
    then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
hoelzl@41096
  1110
  qed
hoelzl@41096
  1111
  ultimately show ?thesis
hoelzl@41096
  1112
    by (rule integrable_abs_iff[THEN iffD1])
hoelzl@41096
  1113
qed
hoelzl@41096
  1114
hoelzl@41096
  1115
lemma (in product_sigma_finite) product_integral_setprod:
hoelzl@41096
  1116
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@49780
  1117
  assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
  1118
  shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"
hoelzl@49780
  1119
using assms proof induct
hoelzl@49780
  1120
  case empty
hoelzl@49780
  1121
  interpret finite_measure "Pi\<^isub>M {} M"
hoelzl@49780
  1122
    by rule (simp add: space_PiM)
hoelzl@49780
  1123
  show ?case by (simp add: space_PiM measure_def)
hoelzl@41096
  1124
next
hoelzl@41096
  1125
  case (insert i I)
hoelzl@41096
  1126
  then have iI: "finite (insert i I)" by auto
hoelzl@41096
  1127
  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
hoelzl@41689
  1128
    integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
hoelzl@49780
  1129
    by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
hoelzl@41689
  1130
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41096
  1131
  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
  1132
    using `i \<notin> I` by (auto intro!: setprod_cong)
hoelzl@41096
  1133
  show ?case
hoelzl@49780
  1134
    unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
hoelzl@47694
  1135
    by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
hoelzl@41096
  1136
qed
hoelzl@41096
  1137
hoelzl@49776
  1138
lemma sets_Collect_single:
hoelzl@49776
  1139
  "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^isub>M I M). x i \<in> A } \<in> sets (Pi\<^isub>M I M)"
hoelzl@50003
  1140
  by simp
hoelzl@49776
  1141
hoelzl@49776
  1142
lemma sigma_prod_algebra_sigma_eq_infinite:
hoelzl@49776
  1143
  fixes E :: "'i \<Rightarrow> 'a set set"
hoelzl@49779
  1144
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@49776
  1145
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@49776
  1146
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@49776
  1147
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@49776
  1148
  defines "P == {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}"
hoelzl@49776
  1149
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1150
proof
hoelzl@49776
  1151
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49776
  1152
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@49776
  1153
    using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
hoelzl@49776
  1154
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@49776
  1155
    by (simp add: space_PiM)
hoelzl@49776
  1156
  have "sets (PiM I M) =
hoelzl@49776
  1157
      sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@49776
  1158
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@49776
  1159
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@49776
  1160
  proof (safe intro!: sigma_sets_subset)
hoelzl@49776
  1161
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@49776
  1162
    then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@49776
  1163
      apply (subst measurable_iff_measure_of)
hoelzl@49776
  1164
      apply (simp_all add: P_closed)
hoelzl@49776
  1165
      using E_closed
hoelzl@49776
  1166
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1167
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1168
      apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
hoelzl@49776
  1169
      apply (rule_tac x=Aa in exI)
hoelzl@49776
  1170
      apply (auto simp: space_PiM)
hoelzl@49776
  1171
      done
hoelzl@49776
  1172
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@49776
  1173
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@49776
  1174
      by (simp add: E_generates)
hoelzl@49776
  1175
    also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
hoelzl@49776
  1176
      using P_closed by (auto simp: space_PiM)
hoelzl@49776
  1177
    finally show "\<dots> \<in> sets ?P" .
hoelzl@49776
  1178
  qed
hoelzl@49776
  1179
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1180
    by (simp add: P_closed)
hoelzl@49776
  1181
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@49776
  1182
    unfolding P_def space_PiM[symmetric]
hoelzl@49776
  1183
    by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
hoelzl@49776
  1184
qed
hoelzl@49776
  1185
hoelzl@47694
  1186
lemma sigma_prod_algebra_sigma_eq:
hoelzl@49779
  1187
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
hoelzl@47694
  1188
  assumes "finite I"
hoelzl@49779
  1189
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@47694
  1190
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@47694
  1191
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@47694
  1192
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@47694
  1193
  defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
hoelzl@47694
  1194
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1195
proof
hoelzl@47694
  1196
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49779
  1197
  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
hoelzl@49779
  1198
  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
  1199
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
hoelzl@47694
  1200
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@47694
  1201
    using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
hoelzl@47694
  1202
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
  1203
    by (simp add: space_PiM)
hoelzl@47694
  1204
  have "sets (PiM I M) =
hoelzl@47694
  1205
      sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
hoelzl@47694
  1206
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@47694
  1207
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@47694
  1208
  proof (safe intro!: sigma_sets_subset)
hoelzl@47694
  1209
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@47694
  1210
    have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@47694
  1211
    proof (subst measurable_iff_measure_of)
hoelzl@47694
  1212
      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
hoelzl@47694
  1213
      from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)"
hoelzl@47694
  1214
        by (auto simp: Pi_iff)
hoelzl@47694
  1215
      show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1216
      proof
hoelzl@47694
  1217
        fix A assume A: "A \<in> E i"
hoelzl@47694
  1218
        then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
hoelzl@47694
  1219
          using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
hoelzl@47694
  1220
        also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
hoelzl@47694
  1221
          by (intro PiE_cong) (simp add: S_union)
hoelzl@49779
  1222
        also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j))"
hoelzl@49779
  1223
          using T
hoelzl@49779
  1224
          apply (auto simp: Pi_iff bchoice_iff)
hoelzl@49779
  1225
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
hoelzl@49779
  1226
          apply (auto simp: bij_betw_def)
hoelzl@49779
  1227
          done
hoelzl@47694
  1228
        also have "\<dots> \<in> sets ?P"
hoelzl@47694
  1229
        proof (safe intro!: countable_UN)
hoelzl@49779
  1230
          fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
hoelzl@47694
  1231
            using A S_in_E
hoelzl@47694
  1232
            by (simp add: P_closed)
hoelzl@49779
  1233
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
hoelzl@47694
  1234
        qed
hoelzl@47694
  1235
        finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1236
          using P_closed by simp
hoelzl@47694
  1237
      qed
hoelzl@47694
  1238
    qed
hoelzl@47694
  1239
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@47694
  1240
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1241
      by (simp add: E_generates)
hoelzl@47694
  1242
    also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
hoelzl@47694
  1243
      using P_closed by (auto simp: space_PiM)
hoelzl@47694
  1244
    finally show "\<dots> \<in> sets ?P" .
hoelzl@47694
  1245
  qed
hoelzl@47694
  1246
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1247
    by (simp add: P_closed)
hoelzl@47694
  1248
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@47694
  1249
    using `finite I`
hoelzl@50003
  1250
    by (auto intro!: sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
hoelzl@47694
  1251
qed
hoelzl@47694
  1252
hoelzl@47694
  1253
end