src/HOL/Probability/Finite_Product_Measure.thy
author hoelzl
Wed Dec 07 15:10:29 2011 +0100 (2011-12-07)
changeset 45777 c36637603821
parent 44928 7ef6505bde7f
child 46731 5302e932d1e5
permissions -rw-r--r--
remove unnecessary sublocale instantiations in HOL-Probability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
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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 Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
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  unfolding Pi_def 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_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
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  by safe (auto simp add: extensional_def fun_eq_iff)
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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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definition
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  "merge I x J y = (\<lambda>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 x J y i = x i"
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  "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
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  "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
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  "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
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  "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J 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 x J y = merge J y I x"
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  by (auto simp: merge_def intro!: ext)
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lemma Pi_cancel_merge_range[simp]:
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  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
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  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
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  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
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  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I 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 x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
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  "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
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  "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
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  "J \<inter> I = {} \<Longrightarrow> merge I x J 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 x J 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 x J y) I = restrict x I"
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  "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
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  "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
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  "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
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  by (auto simp: restrict_def intro!: ext)
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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 x {i} 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 E J 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 "(\<lambda>(x,y). merge I x J y) -` 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 intro!: ext)
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lemma merge_restrict[simp]:
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  "merge I (restrict x I) J y = merge I x J y"
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  "merge I x J (restrict y J) = merge I x J y"
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  unfolding merge_def by (auto intro!: ext)
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lemma merge_x_x_eq_restrict[simp]:
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  "merge I x J x = restrict x (I \<union> J)"
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  unfolding merge_def by (auto intro!: ext)
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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 = {}))"
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  then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
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    using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
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  with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
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next
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  assume "(\<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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  then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
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    using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
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qed
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section "Finite product spaces"
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section "Products"
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locale product_sigma_algebra =
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  fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme"
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  assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
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locale finite_product_sigma_algebra = product_sigma_algebra M
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  for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
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  fixes I :: "'i set"
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  assumes finite_index[simp, intro]: "finite I"
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definition
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  "product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)),
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    sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)),
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    measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>"
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definition product_algebra_def:
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  "Pi\<^isub>M I M = sigma (product_algebra_generator I M)
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    \<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and>
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      (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>"
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syntax
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  "_PiM"  :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
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              ('i => 'a, 'b) measure_space_scheme"  ("(3PIM _:_./ _)" 10)
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   258
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   259
syntax (xsymbols)
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   260
  "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
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   261
             ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)
hoelzl@40859
   262
hoelzl@40859
   263
syntax (HTML output)
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   264
  "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
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   265
             ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)
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   266
hoelzl@40859
   267
translations
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  "PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)"
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   269
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   270
abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M"
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   271
abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M"
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   272
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   273
sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
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   274
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   275
lemma sigma_into_space:
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   276
  assumes "sets M \<subseteq> Pow (space M)"
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   277
  shows "sets (sigma M) \<subseteq> Pow (space M)"
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   278
  using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto
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   279
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   280
lemma (in product_sigma_algebra) product_algebra_generator_into_space:
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   281
  "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
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   282
  using M.sets_into_space unfolding product_algebra_generator_def
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   283
  by auto blast
hoelzl@40859
   284
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   285
lemma (in product_sigma_algebra) product_algebra_into_space:
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   286
  "sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))"
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   287
  using product_algebra_generator_into_space
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   288
  by (auto intro!: sigma_into_space simp add: product_algebra_def)
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   289
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   290
lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)"
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   291
  using product_algebra_generator_into_space unfolding product_algebra_def
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   292
  by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all
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   293
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   294
sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
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   295
  using sigma_algebra_product_algebra .
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   296
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   297
lemma product_algebraE:
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   298
  assumes "A \<in> sets (product_algebra_generator I M)"
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   299
  obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
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   300
  using assms unfolding product_algebra_generator_def by auto
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   301
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   302
lemma product_algebra_generatorI[intro]:
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   303
  assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
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   304
  shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)"
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   305
  using assms unfolding product_algebra_generator_def by auto
hoelzl@41689
   306
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   307
lemma space_product_algebra_generator[simp]:
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   308
  "space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))"
hoelzl@41689
   309
  unfolding product_algebra_generator_def by simp
hoelzl@41095
   310
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   311
lemma space_product_algebra[simp]:
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   312
  "space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@41689
   313
  unfolding product_algebra_def product_algebra_generator_def by simp
hoelzl@40859
   314
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   315
lemma sets_product_algebra:
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   316
  "sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))"
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   317
  unfolding product_algebra_def sigma_def by simp
hoelzl@41689
   318
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   319
lemma product_algebra_generator_sets_into_space:
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   320
  assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
hoelzl@41689
   321
  shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
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   322
  using assms by (auto simp: product_algebra_generator_def) blast
hoelzl@40859
   323
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   324
lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
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   325
  "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
hoelzl@41689
   326
  by (auto simp: sets_product_algebra)
hoelzl@41026
   327
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   328
lemma Int_stable_product_algebra_generator:
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   329
  "(\<And>i. i \<in> I \<Longrightarrow> Int_stable (M i)) \<Longrightarrow> Int_stable (product_algebra_generator I M)"
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   330
  by (auto simp add: product_algebra_generator_def Int_stable_def PiE_Int Pi_iff)
hoelzl@42988
   331
hoelzl@40859
   332
section "Generating set generates also product algebra"
hoelzl@40859
   333
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   334
lemma sigma_product_algebra_sigma_eq:
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   335
  assumes "finite I"
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   336
  assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
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   337
  assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"
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   338
  assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
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   339
  and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
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   340
  shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"
hoelzl@41689
   341
    (is "sets ?S = sets ?E")
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   342
proof cases
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   343
  assume "I = {}" then show ?thesis
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   344
    by (simp add: product_algebra_def product_algebra_generator_def)
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   345
next
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   346
  assume "I \<noteq> {}"
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   347
  interpret E: sigma_algebra "sigma (E i)" for i
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   348
    using E by (rule sigma_algebra_sigma)
hoelzl@40859
   349
  have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)"
hoelzl@40859
   350
    using E by auto
hoelzl@40859
   351
  interpret G: sigma_algebra ?E
hoelzl@41689
   352
    unfolding product_algebra_def product_algebra_generator_def using E
hoelzl@41689
   353
    by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)
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   354
  { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
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   355
    then have "(\<lambda>x. x i) -` A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E"
hoelzl@41981
   356
      using mono union unfolding incseq_Suc_iff space_product_algebra
hoelzl@41689
   357
      by (auto dest: Pi_mem)
hoelzl@40859
   358
    also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
hoelzl@41689
   359
      unfolding space_product_algebra
hoelzl@40859
   360
      apply simp
hoelzl@40859
   361
      apply (subst Pi_UN[OF `finite I`])
hoelzl@41981
   362
      using mono[THEN incseqD] apply simp
hoelzl@40859
   363
      apply (simp add: PiE_Int)
hoelzl@40859
   364
      apply (intro PiE_cong)
hoelzl@40859
   365
      using A sets_into by (auto intro!: into_space)
hoelzl@41689
   366
    also have "\<dots> \<in> sets ?E"
hoelzl@40859
   367
      using sets_into `A \<in> sets (E i)`
hoelzl@41689
   368
      unfolding sets_product_algebra sets_sigma
hoelzl@40859
   369
      by (intro sigma_sets.Union)
hoelzl@40859
   370
         (auto simp: image_subset_iff intro!: sigma_sets.Basic)
hoelzl@40859
   371
    finally have "(\<lambda>x. x i) -` A \<inter> space ?E \<in> sets ?E" . }
hoelzl@40859
   372
  then have proj:
hoelzl@40859
   373
    "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
hoelzl@40859
   374
    using E by (subst G.measurable_iff_sigma)
hoelzl@41689
   375
               (auto simp: sets_product_algebra sets_sigma)
hoelzl@40859
   376
  { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
hoelzl@40859
   377
    with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) -` (A i) \<inter> space ?E \<in> sets ?E"
hoelzl@40859
   378
      unfolding measurable_def by simp
hoelzl@40859
   379
    have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) -` (A i) \<inter> space ?E)"
hoelzl@40859
   380
      using A E.sets_into_space `I \<noteq> {}` unfolding product_algebra_def by auto blast
hoelzl@40859
   381
    then have "Pi\<^isub>E I A \<in> sets ?E"
hoelzl@40859
   382
      using G.finite_INT[OF `finite I` `I \<noteq> {}` basic, of "\<lambda>i. i"] by simp }
hoelzl@41689
   383
  then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E"
hoelzl@41689
   384
    by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def)
hoelzl@40859
   385
  then have subset: "sets ?S \<subseteq> sets ?E"
hoelzl@41689
   386
    by (simp add: sets_sigma sets_product_algebra)
hoelzl@41689
   387
  show "sets ?S = sets ?E"
hoelzl@40859
   388
  proof (intro set_eqI iffI)
hoelzl@40859
   389
    fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
hoelzl@41689
   390
      unfolding sets_sigma sets_product_algebra
hoelzl@40859
   391
    proof induct
hoelzl@40859
   392
      case (Basic A) then show ?case
hoelzl@41689
   393
        by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic)
hoelzl@41689
   394
    qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def)
hoelzl@40859
   395
  next
hoelzl@40859
   396
    fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
hoelzl@40859
   397
  qed
hoelzl@41689
   398
qed
hoelzl@41689
   399
hoelzl@41689
   400
lemma product_algebraI[intro]:
hoelzl@41689
   401
    "E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M I M)"
hoelzl@41689
   402
  using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI)
hoelzl@41689
   403
hoelzl@41689
   404
lemma (in product_sigma_algebra) measurable_component_update:
hoelzl@41689
   405
  assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
hoelzl@41689
   406
  shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
hoelzl@41689
   407
  unfolding product_algebra_def apply simp
hoelzl@41689
   408
proof (intro measurable_sigma)
hoelzl@41689
   409
  let ?G = "product_algebra_generator (insert i I) M"
hoelzl@41689
   410
  show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space .
hoelzl@41689
   411
  show "?f \<in> space (M i) \<rightarrow> space ?G"
hoelzl@41689
   412
    using M.sets_into_space assms by auto
hoelzl@41689
   413
  fix A assume "A \<in> sets ?G"
hoelzl@41689
   414
  from product_algebraE[OF this] guess E . note E = this
hoelzl@41689
   415
  then have "?f -` A \<inter> space (M i) = E i \<or> ?f -` A \<inter> space (M i) = {}"
hoelzl@41689
   416
    using M.sets_into_space assms by auto
hoelzl@41689
   417
  then show "?f -` A \<inter> space (M i) \<in> sets (M i)"
hoelzl@41689
   418
    using E by (auto intro!: product_algebraI)
hoelzl@40859
   419
qed
hoelzl@40859
   420
hoelzl@41689
   421
lemma (in product_sigma_algebra) measurable_add_dim:
hoelzl@41689
   422
  assumes "i \<notin> I"
hoelzl@41689
   423
  shows "(\<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@41689
   424
proof -
hoelzl@41689
   425
  let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (insert i I) M"
hoelzl@41689
   426
  interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M i"
hoelzl@41689
   427
    unfolding pair_sigma_algebra_def
hoelzl@41689
   428
    by (intro sigma_algebra_product_algebra sigma_algebras conjI)
hoelzl@41689
   429
  have "?f \<in> measurable Ii.P (sigma ?G)"
hoelzl@41689
   430
  proof (rule Ii.measurable_sigma)
hoelzl@41689
   431
    show "sets ?G \<subseteq> Pow (space ?G)"
hoelzl@41689
   432
      using product_algebra_generator_into_space .
hoelzl@41689
   433
    show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G"
hoelzl@41689
   434
      by (auto simp: space_pair_measure)
hoelzl@41689
   435
  next
hoelzl@41689
   436
    fix A assume "A \<in> sets ?G"
hoelzl@41689
   437
    then obtain F where "A = Pi\<^isub>E (insert i I) F"
hoelzl@41689
   438
      and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M i)"
hoelzl@41689
   439
      by (auto elim!: product_algebraE)
hoelzl@41689
   440
    then have "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)"
hoelzl@41689
   441
      using sets_into_space `i \<notin> I`
hoelzl@41689
   442
      by (auto simp add: space_pair_measure) blast+
hoelzl@41689
   443
    then show "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)"
hoelzl@41689
   444
      using F by (auto intro!: pair_measureI)
hoelzl@41689
   445
  qed
hoelzl@41689
   446
  then show ?thesis
hoelzl@41689
   447
    by (simp add: product_algebra_def)
hoelzl@41689
   448
qed
hoelzl@41095
   449
hoelzl@41095
   450
lemma (in product_sigma_algebra) measurable_merge:
hoelzl@41095
   451
  assumes [simp]: "I \<inter> J = {}"
hoelzl@41689
   452
  shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
hoelzl@40859
   453
proof -
hoelzl@41689
   454
  let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M J M"
hoelzl@41689
   455
  interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J"
hoelzl@41689
   456
    by (intro sigma_algebra_pair_measure product_algebra_into_space)
hoelzl@41689
   457
  let ?f = "\<lambda>(x, y). merge I x J y"
hoelzl@41689
   458
  let ?G = "product_algebra_generator (I \<union> J) M"
hoelzl@41689
   459
  have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)"
hoelzl@41689
   460
  proof (rule P.measurable_sigma)
hoelzl@41689
   461
    fix A assume "A \<in> sets ?G"
hoelzl@41689
   462
    from product_algebraE[OF this]
hoelzl@41689
   463
    obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .
hoelzl@41689
   464
    then have *: "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
hoelzl@41689
   465
      using sets_into_space `I \<inter> J = {}`
hoelzl@41981
   466
      by (auto simp add: space_pair_measure) fast+
hoelzl@41689
   467
    then show "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"
hoelzl@41689
   468
      using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI
hoelzl@41689
   469
        simp: product_algebra_def)
hoelzl@41689
   470
  qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure)
hoelzl@41689
   471
  then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41689
   472
    unfolding product_algebra_def[of "I \<union> J"] by simp
hoelzl@40859
   473
qed
hoelzl@40859
   474
hoelzl@41095
   475
lemma (in product_sigma_algebra) measurable_component_singleton:
hoelzl@41689
   476
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
hoelzl@41689
   477
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   478
  fix A assume "A \<in> sets (M i)"
hoelzl@41689
   479
  then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
nipkow@44890
   480
    using M.sets_into_space `i \<in> I` by (fastforce dest: Pi_mem split: split_if_asm)
hoelzl@41689
   481
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
hoelzl@41689
   482
    using `A \<in> sets (M i)` by (auto intro!: product_algebraI)
hoelzl@41689
   483
qed (insert `i \<in> I`, auto)
hoelzl@41661
   484
hoelzl@42988
   485
lemma (in sigma_algebra) measurable_restrict:
hoelzl@42988
   486
  assumes I: "finite I"
hoelzl@42988
   487
  assumes "\<And>i. i \<in> I \<Longrightarrow> sets (N i) \<subseteq> Pow (space (N i))"
hoelzl@42988
   488
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (N i)"
hoelzl@42988
   489
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
hoelzl@42988
   490
  unfolding product_algebra_def
hoelzl@42988
   491
proof (simp, rule measurable_sigma)
hoelzl@42988
   492
  show "sets (product_algebra_generator I N) \<subseteq> Pow (space (product_algebra_generator I N))"
hoelzl@42988
   493
    by (rule product_algebra_generator_sets_into_space) fact
hoelzl@42988
   494
  show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> space M \<rightarrow> space (product_algebra_generator I N)"
hoelzl@42988
   495
    using X by (auto simp: measurable_def)
hoelzl@42988
   496
  fix E assume "E \<in> sets (product_algebra_generator I N)"
hoelzl@42988
   497
  then obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (N i)"
hoelzl@42988
   498
    by (auto simp: product_algebra_generator_def)
hoelzl@42988
   499
  then have "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M = (\<Inter>i\<in>I. X i -` F i \<inter> space M) \<inter> space M"
hoelzl@42988
   500
    by (auto simp: Pi_iff)
hoelzl@42988
   501
  also have "\<dots> \<in> sets M"
hoelzl@42988
   502
  proof cases
hoelzl@42988
   503
    assume "I = {}" then show ?thesis by simp
hoelzl@42988
   504
  next
hoelzl@42988
   505
    assume "I \<noteq> {}" with X F I show ?thesis
hoelzl@42988
   506
      by (intro finite_INT measurable_sets Int) auto
hoelzl@42988
   507
  qed
hoelzl@42988
   508
  finally show "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M \<in> sets M" .
hoelzl@42988
   509
qed
hoelzl@42988
   510
hoelzl@45777
   511
locale product_sigma_finite = product_sigma_algebra M
hoelzl@45777
   512
  for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" +
hoelzl@41689
   513
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   514
hoelzl@41689
   515
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   516
  by (rule sigma_finite_measures)
hoelzl@40859
   517
hoelzl@45777
   518
locale finite_product_sigma_finite = finite_product_sigma_algebra M I + product_sigma_finite M
hoelzl@45777
   519
  for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set"
hoelzl@40859
   520
hoelzl@41689
   521
lemma (in finite_product_sigma_finite) product_algebra_generator_measure:
hoelzl@41689
   522
  assumes "Pi\<^isub>E I F \<in> sets G"
hoelzl@41689
   523
  shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"
hoelzl@41689
   524
proof cases
hoelzl@41689
   525
  assume ne: "\<forall>i\<in>I. F i \<noteq> {}"
hoelzl@41689
   526
  have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') i = F i"
hoelzl@41689
   527
    by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
hoelzl@41689
   528
       (insert ne, auto simp: Pi_eq_iff)
hoelzl@41689
   529
  then show ?thesis
hoelzl@41689
   530
    unfolding product_algebra_generator_def by simp
hoelzl@41689
   531
next
hoelzl@41689
   532
  assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"
hoelzl@41689
   533
  then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"
hoelzl@43920
   534
    by (auto simp: setprod_ereal_0 intro!: bexI)
hoelzl@41689
   535
  moreover
hoelzl@41689
   536
  have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"
hoelzl@41689
   537
    by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
hoelzl@41689
   538
       (insert empty, auto simp: Pi_eq_empty_iff[symmetric])
hoelzl@41689
   539
  then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"
hoelzl@43920
   540
    by (auto simp: setprod_ereal_0 intro!: bexI)
hoelzl@41689
   541
  ultimately show ?thesis
hoelzl@41689
   542
    unfolding product_algebra_generator_def by simp
hoelzl@41689
   543
qed
hoelzl@41689
   544
hoelzl@40859
   545
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   546
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   547
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
hoelzl@41981
   548
    (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
hoelzl@41981
   549
    (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"
hoelzl@40859
   550
proof -
hoelzl@41981
   551
  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. \<mu> i (F k) \<noteq> \<infinity>)"
hoelzl@40859
   552
    using M.sigma_finite_up by simp
hoelzl@40859
   553
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@41981
   554
  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. \<mu> i (F i k) \<noteq> \<infinity>"
hoelzl@40859
   555
    by auto
hoelzl@40859
   556
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
hoelzl@40859
   557
  note space_product_algebra[simp]
hoelzl@40859
   558
  show ?thesis
hoelzl@41981
   559
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   560
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   561
  next
hoelzl@41981
   562
    fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   563
  next
hoelzl@40859
   564
    fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
hoelzl@41831
   565
      using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space
hoelzl@41831
   566
      by (force simp: image_subset_iff)
hoelzl@40859
   567
  next
hoelzl@40859
   568
    fix f assume "f \<in> space G"
hoelzl@41981
   569
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@41981
   570
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
hoelzl@40859
   571
  next
hoelzl@40859
   572
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   573
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   574
  qed
hoelzl@40859
   575
qed
hoelzl@40859
   576
hoelzl@41831
   577
lemma sets_pair_cancel_measure[simp]:
hoelzl@41831
   578
  "sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41831
   579
  "sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41831
   580
  unfolding pair_measure_def pair_measure_generator_def sets_sigma
hoelzl@41831
   581
  by simp_all
hoelzl@41831
   582
hoelzl@41831
   583
lemma measurable_pair_cancel_measure[simp]:
hoelzl@41831
   584
  "measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
hoelzl@41831
   585
  "measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
hoelzl@41831
   586
  "measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41831
   587
  "measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)"
hoelzl@41831
   588
  unfolding measurable_def by (auto simp add: space_pair_measure)
hoelzl@41831
   589
hoelzl@40859
   590
lemma (in product_sigma_finite) product_measure_exists:
hoelzl@40859
   591
  assumes "finite I"
hoelzl@41689
   592
  shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and>
hoelzl@41689
   593
    (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))"
hoelzl@40859
   594
using `finite I` proof induct
hoelzl@41689
   595
  case empty
hoelzl@41689
   596
  interpret finite_product_sigma_finite M "{}" by default simp
hoelzl@43920
   597
  let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> ereal"
hoelzl@41689
   598
  show ?case
hoelzl@41689
   599
  proof (intro exI conjI ballI)
hoelzl@41689
   600
    have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"
hoelzl@41689
   601
      by (rule sigma_algebra_cong) (simp_all add: product_algebra_def)
hoelzl@41689
   602
    then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
hoelzl@41689
   603
      by (rule finite_additivity_sufficient)
hoelzl@41689
   604
         (simp_all add: positive_def additive_def sets_sigma
hoelzl@41689
   605
                        product_algebra_generator_def image_constant)
hoelzl@41689
   606
    then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
hoelzl@41689
   607
      by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"]
hoelzl@41689
   608
               simp: sigma_finite_measure_def sigma_finite_measure_axioms_def
hoelzl@41689
   609
                     product_algebra_generator_def)
hoelzl@41689
   610
  qed auto
hoelzl@40859
   611
next
hoelzl@40859
   612
  case (insert i I)
hoelzl@41689
   613
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   614
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   615
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@40859
   616
  from insert obtain \<nu> where
hoelzl@41689
   617
    prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and
hoelzl@41689
   618
    "sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by auto
hoelzl@41689
   619
  then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp
hoelzl@41689
   620
  interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" ..
hoelzl@41661
   621
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@41689
   622
  let ?\<nu> = "\<lambda>A. P.\<mu> (?h -` A \<inter> space P.P)"
hoelzl@41689
   623
  have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)"
hoelzl@41689
   624
    by (rule I'.sigma_algebra_cong) simp_all
hoelzl@41689
   625
  interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>"
hoelzl@41689
   626
    using measurable_add_dim[OF `i \<notin> I`]
hoelzl@41831
   627
    by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def)
hoelzl@40859
   628
  show ?case
hoelzl@40859
   629
  proof (intro exI[of _ ?\<nu>] conjI ballI)
hoelzl@41689
   630
    let "?m A" = "measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h -` A \<inter> space P.P)"
hoelzl@40859
   631
    { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
hoelzl@41661
   632
      then have *: "?h -` Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"
hoelzl@41689
   633
        using `i \<notin> I` M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast
hoelzl@41689
   634
      show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))"
hoelzl@41661
   635
        unfolding * using A
hoelzl@40859
   636
        apply (subst P.pair_measure_times)
nipkow@44890
   637
        using A apply fastforce
nipkow@44890
   638
        using A apply fastforce
hoelzl@41661
   639
        using `i \<notin> I` `finite I` prod[of A] A by (auto simp: ac_simps) }
hoelzl@40859
   640
    note product = this
hoelzl@41689
   641
    have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>"
hoelzl@41689
   642
      by (simp add: product_algebra_def)
hoelzl@41689
   643
    show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)"
hoelzl@41689
   644
    proof (unfold *, default, simp)
hoelzl@40859
   645
      from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@40859
   646
      then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
hoelzl@41981
   647
        "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
hoelzl@41981
   648
        "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"
hoelzl@41981
   649
        "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"
hoelzl@40859
   650
        by blast+
hoelzl@40859
   651
      let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"
hoelzl@40859
   652
      show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
hoelzl@41981
   653
          (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"
hoelzl@40859
   654
      proof (intro exI[of _ ?F] conjI allI)
hoelzl@40859
   655
        show "range ?F \<subseteq> sets I'.P" using F(1) by auto
hoelzl@40859
   656
      next
hoelzl@41981
   657
        from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
hoelzl@40859
   658
      next
hoelzl@40859
   659
        fix j
hoelzl@41981
   660
        have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"
hoelzl@41981
   661
          using F(1) by auto
hoelzl@41981
   662
        with F `finite I` setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"
hoelzl@41981
   663
          by (subst product) auto
hoelzl@40859
   664
      qed
hoelzl@40859
   665
    qed
hoelzl@40859
   666
  qed
hoelzl@40859
   667
qed
hoelzl@40859
   668
hoelzl@41689
   669
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P
hoelzl@41689
   670
  unfolding product_algebra_def
hoelzl@41689
   671
  using product_measure_exists[OF finite_index]
hoelzl@41689
   672
  by (rule someI2_ex) auto
hoelzl@40859
   673
hoelzl@40859
   674
lemma (in finite_product_sigma_finite) measure_times:
hoelzl@40859
   675
  assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
hoelzl@41689
   676
  shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
hoelzl@41689
   677
  unfolding product_algebra_def
hoelzl@41689
   678
  using product_measure_exists[OF finite_index]
hoelzl@41689
   679
  proof (rule someI2_ex, elim conjE)
hoelzl@41689
   680
    fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
hoelzl@40859
   681
    have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
hoelzl@40859
   682
    then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
hoelzl@41689
   683
    also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
hoelzl@41689
   684
    finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
hoelzl@41689
   685
      by (simp add: sigma_def)
hoelzl@40859
   686
  qed
hoelzl@41096
   687
hoelzl@41096
   688
lemma (in product_sigma_finite) product_measure_empty[simp]:
hoelzl@41689
   689
  "measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
hoelzl@41096
   690
proof -
hoelzl@41689
   691
  interpret finite_product_sigma_finite M "{}" by default auto
hoelzl@41096
   692
  from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
hoelzl@41096
   693
qed
hoelzl@41096
   694
hoelzl@41689
   695
lemma (in finite_product_sigma_algebra) P_empty:
hoelzl@41689
   696
  assumes "I = {}"
hoelzl@41689
   697
  shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }"
hoelzl@41689
   698
  unfolding product_algebra_def product_algebra_generator_def `I = {}`
hoelzl@41689
   699
  by (simp_all add: sigma_def image_constant)
hoelzl@41689
   700
hoelzl@40859
   701
lemma (in product_sigma_finite) positive_integral_empty:
hoelzl@41981
   702
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
hoelzl@41981
   703
  shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
hoelzl@40859
   704
proof -
hoelzl@41689
   705
  interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
hoelzl@41689
   706
  have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
hoelzl@40859
   707
    using assms by (subst measure_times) auto
hoelzl@40859
   708
  then show ?thesis
hoelzl@40873
   709
    unfolding positive_integral_def simple_function_def simple_integral_def_raw
hoelzl@40859
   710
  proof (simp add: P_empty, intro antisym)
hoelzl@41981
   711
    show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
hoelzl@44928
   712
      by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
hoelzl@41981
   713
    show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
hoelzl@44928
   714
      by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
hoelzl@40859
   715
  qed
hoelzl@40859
   716
qed
hoelzl@40859
   717
hoelzl@41026
   718
lemma (in product_sigma_finite) measure_fold:
hoelzl@40859
   719
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@41689
   720
  assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41706
   721
  shows "measure (Pi\<^isub>M (I \<union> J) M) A =
hoelzl@41706
   722
    measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))"
hoelzl@40859
   723
proof -
hoelzl@41689
   724
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   725
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@40859
   726
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   727
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@41689
   728
  interpret P: pair_sigma_finite I.P J.P by default
hoelzl@41661
   729
  let ?g = "\<lambda>(x,y). merge I x J y"
hoelzl@41661
   730
  let "?X S" = "?g -` S \<inter> space P.P"
hoelzl@41661
   731
  from IJ.sigma_finite_pairs obtain F where
hoelzl@41661
   732
    F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
hoelzl@41981
   733
       "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
hoelzl@41981
   734
       "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"
hoelzl@41981
   735
       "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"
hoelzl@41661
   736
    by auto
hoelzl@41661
   737
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
hoelzl@41706
   738
  show "IJ.\<mu> A = P.\<mu> (?X A)"
hoelzl@41706
   739
  proof (rule measure_unique_Int_stable_vimage)
hoelzl@41706
   740
    show "measure_space IJ.P" "measure_space P.P" by default
hoelzl@41706
   741
    show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)"
hoelzl@41689
   742
      using A unfolding product_algebra_def by auto
hoelzl@41706
   743
  next
hoelzl@41689
   744
    show "Int_stable IJ.G"
hoelzl@42988
   745
      by (rule Int_stable_product_algebra_generator)
hoelzl@42988
   746
         (auto simp: Int_stable_def)
hoelzl@41689
   747
    show "range ?F \<subseteq> sets IJ.G" using F
hoelzl@41689
   748
      by (simp add: image_subset_iff product_algebra_def
hoelzl@41689
   749
                    product_algebra_generator_def)
hoelzl@41981
   750
    show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+
hoelzl@41981
   751
  next
hoelzl@41981
   752
    fix k
hoelzl@41981
   753
    have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"
hoelzl@41981
   754
      using F(1) by auto
hoelzl@41981
   755
    with F `finite I` setprod_PInf[of "I \<union> J", OF this]
hoelzl@41981
   756
    show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
hoelzl@41661
   757
  next
hoelzl@41661
   758
    fix A assume "A \<in> sets IJ.G"
hoelzl@41706
   759
    then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"
hoelzl@41661
   760
      and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"
hoelzl@41689
   761
      by (auto simp: product_algebra_generator_def)
hoelzl@41706
   762
    then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
hoelzl@41689
   763
      using sets_into_space by (auto simp: space_pair_measure) blast+
hoelzl@41689
   764
    then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
hoelzl@41661
   765
      using `finite J` `finite I` F
hoelzl@41661
   766
      by (simp add: P.pair_measure_times I.measure_times J.measure_times)
hoelzl@41661
   767
    also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
hoelzl@41661
   768
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
hoelzl@41689
   769
    also have "\<dots> = IJ.\<mu> A"
hoelzl@41661
   770
      using `finite J` `finite I` F unfolding A
hoelzl@41661
   771
      by (intro IJ.measure_times[symmetric]) auto
hoelzl@41706
   772
    finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)
hoelzl@41981
   773
  qed (rule measurable_merge[OF IJ])
hoelzl@41661
   774
qed
hoelzl@41026
   775
hoelzl@41831
   776
lemma (in product_sigma_finite) measure_preserving_merge:
hoelzl@41831
   777
  assumes IJ: "I \<inter> J = {}" and "finite I" "finite J"
hoelzl@41831
   778
  shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41831
   779
  by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms)
hoelzl@41831
   780
hoelzl@41026
   781
lemma (in product_sigma_finite) product_positive_integral_fold:
hoelzl@41831
   782
  assumes IJ[simp]: "I \<inter> J = {}" "finite I" "finite J"
hoelzl@41689
   783
  and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41689
   784
  shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
hoelzl@41689
   785
    (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
hoelzl@41026
   786
proof -
hoelzl@41689
   787
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   788
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41831
   789
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@41831
   790
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp
hoelzl@41026
   791
  have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
hoelzl@41831
   792
    using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def)
hoelzl@41661
   793
  show ?thesis
hoelzl@41026
   794
    unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
hoelzl@41661
   795
  proof (rule P.positive_integral_vimage)
hoelzl@41661
   796
    show "sigma_algebra IJ.P" by default
hoelzl@41831
   797
    show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P"
hoelzl@41831
   798
      using IJ by (rule measure_preserving_merge)
hoelzl@41689
   799
    show "f \<in> borel_measurable IJ.P" using f by simp
hoelzl@41661
   800
  qed
hoelzl@40859
   801
qed
hoelzl@40859
   802
hoelzl@41831
   803
lemma (in product_sigma_finite) measure_preserving_component_singelton:
hoelzl@41831
   804
  "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
hoelzl@41831
   805
proof (intro measure_preservingI measurable_component_singleton)
hoelzl@41831
   806
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@41831
   807
  fix A let ?P = "(\<lambda>x. x i) -` A \<inter> space I.P"
hoelzl@41831
   808
  assume A: "A \<in> sets (M i)"
hoelzl@41831
   809
  then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto
hoelzl@41831
   810
  show "I.\<mu> ?P = M.\<mu> i A" unfolding *
hoelzl@41831
   811
    using A I.measure_times[of "\<lambda>_. A"] by auto
hoelzl@41831
   812
qed auto
hoelzl@41831
   813
hoelzl@41026
   814
lemma (in product_sigma_finite) product_positive_integral_singleton:
hoelzl@40859
   815
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   816
  shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
hoelzl@40859
   817
proof -
hoelzl@41689
   818
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@41689
   819
  show ?thesis
hoelzl@41689
   820
  proof (rule I.positive_integral_vimage[symmetric])
hoelzl@41689
   821
    show "sigma_algebra (M i)" by (rule sigma_algebras)
hoelzl@41831
   822
    show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
hoelzl@41831
   823
      by (rule measure_preserving_component_singelton)
hoelzl@41689
   824
    show "f \<in> borel_measurable (M i)" by fact
hoelzl@41661
   825
  qed
hoelzl@40859
   826
qed
hoelzl@40859
   827
hoelzl@41096
   828
lemma (in product_sigma_finite) product_positive_integral_insert:
hoelzl@41096
   829
  assumes [simp]: "finite I" "i \<notin> I"
hoelzl@41689
   830
    and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@41689
   831
  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
   832
proof -
hoelzl@41689
   833
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41689
   834
  interpret i: finite_product_sigma_finite M "{i}" by default auto
hoelzl@41096
   835
  interpret P: pair_sigma_algebra I.P i.P ..
hoelzl@41689
   836
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
hoelzl@41689
   837
    using f by auto
hoelzl@41096
   838
  show ?thesis
hoelzl@41689
   839
    unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]
hoelzl@41096
   840
  proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)
hoelzl@41096
   841
    fix x assume x: "x \<in> space I.P"
hoelzl@41096
   842
    let "?f y" = "f (restrict (x(i := y)) (insert i I))"
hoelzl@41096
   843
    have f'_eq: "\<And>y. ?f y = f (x(i := y))"
hoelzl@41096
   844
      using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
hoelzl@41689
   845
    show "?f \<in> borel_measurable (M i)" unfolding f'_eq
hoelzl@41689
   846
      using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
hoelzl@41689
   847
      by (simp add: comp_def)
hoelzl@41689
   848
    show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"
hoelzl@41096
   849
      unfolding f'_eq by simp
hoelzl@41096
   850
  qed
hoelzl@41096
   851
qed
hoelzl@41096
   852
hoelzl@41096
   853
lemma (in product_sigma_finite) product_positive_integral_setprod:
hoelzl@43920
   854
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
hoelzl@41096
   855
  assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41981
   856
  and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
hoelzl@41689
   857
  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
   858
using assms proof induct
hoelzl@41096
   859
  case empty
hoelzl@41689
   860
  interpret finite_product_sigma_finite M "{}" by default auto
hoelzl@41096
   861
  then show ?case by simp
hoelzl@41096
   862
next
hoelzl@41096
   863
  case (insert i I)
hoelzl@41096
   864
  note `finite I`[intro, simp]
hoelzl@41689
   865
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41096
   866
  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
   867
    using insert by (auto intro!: setprod_cong)
hoelzl@41689
   868
  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
   869
    using sets_into_space insert
hoelzl@43920
   870
    by (intro sigma_algebra.borel_measurable_ereal_setprod sigma_algebra_product_algebra
hoelzl@41689
   871
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
hoelzl@41096
   872
       auto
hoelzl@41981
   873
  then show ?case
hoelzl@41981
   874
    apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
hoelzl@43920
   875
    apply (simp add: insert * pos borel setprod_ereal_pos M.positive_integral_multc)
hoelzl@41981
   876
    apply (subst I.positive_integral_cmult)
hoelzl@43920
   877
    apply (auto simp add: pos borel insert setprod_ereal_pos positive_integral_positive)
hoelzl@41981
   878
    done
hoelzl@41096
   879
qed
hoelzl@41096
   880
hoelzl@41026
   881
lemma (in product_sigma_finite) product_integral_singleton:
hoelzl@41026
   882
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   883
  shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
hoelzl@41026
   884
proof -
hoelzl@41689
   885
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@43920
   886
  have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
hoelzl@43920
   887
    "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
hoelzl@41026
   888
    using assms by auto
hoelzl@41026
   889
  show ?thesis
hoelzl@41689
   890
    unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
hoelzl@41026
   891
qed
hoelzl@41026
   892
hoelzl@41026
   893
lemma (in product_sigma_finite) product_integral_fold:
hoelzl@41026
   894
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@41689
   895
  and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
hoelzl@41689
   896
  shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I x J y) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
hoelzl@41026
   897
proof -
hoelzl@41689
   898
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   899
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41026
   900
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   901
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@41689
   902
  interpret P: pair_sigma_finite I.P J.P by default
hoelzl@41689
   903
  let ?M = "\<lambda>(x, y). merge I x J y"
hoelzl@41689
   904
  let ?f = "\<lambda>x. f (?M x)"
hoelzl@41026
   905
  show ?thesis
hoelzl@41689
   906
  proof (subst P.integrable_fst_measurable(2)[of ?f, simplified])
hoelzl@41689
   907
    have 1: "sigma_algebra IJ.P" by default
hoelzl@41831
   908
    have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] .
hoelzl@41831
   909
    have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact
hoelzl@41831
   910
    then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41831
   911
      by (simp add: integrable_def)
hoelzl@41689
   912
    show "integrable P.P ?f"
hoelzl@41831
   913
      by (rule P.integrable_vimage[where f=f, OF 1 2 3])
hoelzl@41689
   914
    show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f"
hoelzl@41831
   915
      by (rule P.integral_vimage[where f=f, OF 1 2 4])
hoelzl@41689
   916
  qed
hoelzl@41026
   917
qed
hoelzl@41026
   918
hoelzl@41096
   919
lemma (in product_sigma_finite) product_integral_insert:
hoelzl@41096
   920
  assumes [simp]: "finite I" "i \<notin> I"
hoelzl@41689
   921
    and f: "integrable (Pi\<^isub>M (insert i I) M) f"
hoelzl@41689
   922
  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
   923
proof -
hoelzl@41689
   924
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41689
   925
  interpret I': finite_product_sigma_finite M "insert i I" by default auto
hoelzl@41689
   926
  interpret i: finite_product_sigma_finite M "{i}" by default auto
hoelzl@41689
   927
  interpret P: pair_sigma_finite I.P i.P ..
hoelzl@41096
   928
  have IJ: "I \<inter> {i} = {}" by auto
hoelzl@41096
   929
  show ?thesis
hoelzl@41096
   930
    unfolding product_integral_fold[OF IJ, simplified, OF f]
hoelzl@41096
   931
  proof (rule I.integral_cong, subst product_integral_singleton)
hoelzl@41096
   932
    fix x assume x: "x \<in> space I.P"
hoelzl@41096
   933
    let "?f y" = "f (restrict (x(i := y)) (insert i I))"
hoelzl@41096
   934
    have f'_eq: "\<And>y. ?f y = f (x(i := y))"
hoelzl@41096
   935
      using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
hoelzl@41689
   936
    have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto
hoelzl@41096
   937
    show "?f \<in> borel_measurable (M i)"
hoelzl@41689
   938
      unfolding measurable_cong[OF f'_eq]
hoelzl@41689
   939
      using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
hoelzl@41689
   940
      by (simp add: comp_def)
hoelzl@41689
   941
    show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))"
hoelzl@41096
   942
      unfolding f'_eq by simp
hoelzl@41096
   943
  qed
hoelzl@41096
   944
qed
hoelzl@41096
   945
hoelzl@41096
   946
lemma (in product_sigma_finite) product_integrable_setprod:
hoelzl@41096
   947
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@41689
   948
  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
   949
  shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
hoelzl@41096
   950
proof -
hoelzl@41689
   951
  interpret finite_product_sigma_finite M I by default fact
hoelzl@41096
   952
  have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41689
   953
    using integrable unfolding integrable_def by auto
hoelzl@41096
   954
  then have borel: "?f \<in> borel_measurable P"
hoelzl@41689
   955
    using measurable_comp[OF measurable_component_singleton f]
hoelzl@41689
   956
    by (auto intro!: borel_measurable_setprod simp: comp_def)
hoelzl@41689
   957
  moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
hoelzl@41096
   958
  proof (unfold integrable_def, intro conjI)
hoelzl@41096
   959
    show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
hoelzl@41096
   960
      using borel by auto
hoelzl@43920
   961
    have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>P)"
hoelzl@43920
   962
      by (simp add: setprod_ereal abs_setprod)
hoelzl@43920
   963
    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
hoelzl@41096
   964
      using f by (subst product_positive_integral_setprod) auto
hoelzl@41981
   965
    also have "\<dots> < \<infinity>"
hoelzl@41096
   966
      using integrable[THEN M.integrable_abs]
hoelzl@41981
   967
      by (simp add: setprod_PInf integrable_def M.positive_integral_positive)
hoelzl@43920
   968
    finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto
hoelzl@43920
   969
    have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"
hoelzl@41981
   970
      by (intro positive_integral_cong_pos) auto
hoelzl@43920
   971
    then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp
hoelzl@41096
   972
  qed
hoelzl@41096
   973
  ultimately show ?thesis
hoelzl@41096
   974
    by (rule integrable_abs_iff[THEN iffD1])
hoelzl@41096
   975
qed
hoelzl@41096
   976
hoelzl@41096
   977
lemma (in product_sigma_finite) product_integral_setprod:
hoelzl@41096
   978
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@41689
   979
  assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
   980
  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@41096
   981
using assms proof (induct rule: finite_ne_induct)
hoelzl@41096
   982
  case (singleton i)
hoelzl@41096
   983
  then show ?case by (simp add: product_integral_singleton integrable_def)
hoelzl@41096
   984
next
hoelzl@41096
   985
  case (insert i I)
hoelzl@41096
   986
  then have iI: "finite (insert i I)" by auto
hoelzl@41096
   987
  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
hoelzl@41689
   988
    integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
hoelzl@41096
   989
    by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)
hoelzl@41689
   990
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41096
   991
  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
   992
    using `i \<notin> I` by (auto intro!: setprod_cong)
hoelzl@41096
   993
  show ?case
hoelzl@41096
   994
    unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
hoelzl@41096
   995
    by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)
hoelzl@41096
   996
qed
hoelzl@41096
   997
hoelzl@40859
   998
end