src/HOL/Probability/Finite_Product_Measure.thy
author hoelzl
Wed Oct 10 12:12:21 2012 +0200 (2012-10-10)
changeset 49780 92a58f80b20c
parent 49779 1484b4b82855
child 49784 5e5b2da42a69
permissions -rw-r--r--
merge should operate on pairs
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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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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 (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 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 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 = {}))"
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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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definition prod_emb where
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  "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
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lemma prod_emb_iff: 
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  "f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))"
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  unfolding prod_emb_def by auto
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lemma
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  shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
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    and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
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    and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
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    and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))"
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    and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
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    and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
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  by (auto simp: prod_emb_def)
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lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
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    prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))"
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  by (force simp: prod_emb_def Pi_iff split_if_mem2)
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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"
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   254
  by (auto simp: prod_emb_def Pi_iff)
hoelzl@41689
   255
hoelzl@47694
   256
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
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   257
  "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
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   258
    {(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
   259
    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
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   260
    (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
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   261
hoelzl@47694
   262
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
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   263
  "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
hoelzl@47694
   264
    {(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
   265
hoelzl@47694
   266
abbreviation
hoelzl@47694
   267
  "Pi\<^isub>M I M \<equiv> PiM I M"
hoelzl@41689
   268
hoelzl@40859
   269
syntax
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   270
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
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   271
hoelzl@40859
   272
syntax (xsymbols)
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   273
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
hoelzl@40859
   274
hoelzl@40859
   275
syntax (HTML output)
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   276
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
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   277
hoelzl@40859
   278
translations
hoelzl@47694
   279
  "PIM x:I. M" == "CONST PiM I (%x. M)"
hoelzl@41689
   280
hoelzl@47694
   281
lemma prod_algebra_sets_into_space:
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   282
  "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   283
  using assms by (auto simp: prod_emb_def prod_algebra_def)
hoelzl@40859
   284
hoelzl@47694
   285
lemma prod_algebra_eq_finite:
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   286
  assumes I: "finite I"
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   287
  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
   288
proof (intro iffI set_eqI)
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   289
  fix A assume "A \<in> ?L"
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   290
  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
   291
    and A: "A = prod_emb I M J (PIE j:J. E j)"
hoelzl@47694
   292
    by (auto simp: prod_algebra_def)
hoelzl@47694
   293
  let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
hoelzl@47694
   294
  have A: "A = ?A"
hoelzl@47694
   295
    unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
hoelzl@47694
   296
  show "A \<in> ?R" unfolding A using J top
hoelzl@47694
   297
    by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
hoelzl@47694
   298
next
hoelzl@47694
   299
  fix A assume "A \<in> ?R"
hoelzl@47694
   300
  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
   301
  then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
hoelzl@47694
   302
    using sets_into_space by (force simp: prod_emb_def Pi_iff)
hoelzl@47694
   303
  from X I show "A \<in> ?L" unfolding A
hoelzl@47694
   304
    by (auto simp: prod_algebra_def)
hoelzl@47694
   305
qed
hoelzl@41095
   306
hoelzl@47694
   307
lemma prod_algebraI:
hoelzl@47694
   308
  "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
   309
    \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
hoelzl@47694
   310
  by (auto simp: prod_algebra_def Pi_iff)
hoelzl@41689
   311
hoelzl@47694
   312
lemma prod_algebraE:
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   313
  assumes A: "A \<in> prod_algebra I M"
hoelzl@47694
   314
  obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
hoelzl@47694
   315
    "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" 
hoelzl@47694
   316
  using A by (auto simp: prod_algebra_def)
hoelzl@42988
   317
hoelzl@47694
   318
lemma prod_algebraE_all:
hoelzl@47694
   319
  assumes A: "A \<in> prod_algebra I M"
hoelzl@47694
   320
  obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
hoelzl@47694
   321
proof -
hoelzl@47694
   322
  from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
hoelzl@47694
   323
    and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
hoelzl@47694
   324
    by (auto simp: prod_algebra_def)
hoelzl@47694
   325
  from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
hoelzl@47694
   326
    using sets_into_space by auto
hoelzl@47694
   327
  then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
hoelzl@47694
   328
    using A J by (auto simp: prod_emb_PiE)
hoelzl@47694
   329
  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
   330
    using top E by auto
hoelzl@47694
   331
  ultimately show ?thesis using that by auto
hoelzl@47694
   332
qed
hoelzl@40859
   333
hoelzl@47694
   334
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
hoelzl@47694
   335
proof (unfold Int_stable_def, safe)
hoelzl@47694
   336
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   337
  from prod_algebraE[OF this] guess J E . note A = this
hoelzl@47694
   338
  fix B assume "B \<in> prod_algebra I M"
hoelzl@47694
   339
  from prod_algebraE[OF this] guess K F . note B = this
hoelzl@47694
   340
  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
   341
      (if i \<in> K then F i else space (M i)))"
hoelzl@47694
   342
    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
   343
    apply (subst (1 2 3) prod_emb_PiE)
hoelzl@47694
   344
    apply (simp_all add: subset_eq PiE_Int)
hoelzl@47694
   345
    apply blast
hoelzl@47694
   346
    apply (intro PiE_cong)
hoelzl@47694
   347
    apply auto
hoelzl@47694
   348
    done
hoelzl@47694
   349
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   350
    using A B by (auto intro!: prod_algebraI)
hoelzl@47694
   351
  finally show "A \<inter> B \<in> prod_algebra I M" .
hoelzl@47694
   352
qed
hoelzl@47694
   353
hoelzl@47694
   354
lemma prod_algebra_mono:
hoelzl@47694
   355
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
hoelzl@47694
   356
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
hoelzl@47694
   357
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
hoelzl@47694
   358
proof
hoelzl@47694
   359
  fix A assume "A \<in> prod_algebra I E"
hoelzl@47694
   360
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
hoelzl@47694
   361
    and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   362
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
hoelzl@47694
   363
    by (auto simp: prod_algebra_def)
hoelzl@47694
   364
  moreover
hoelzl@47694
   365
  from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
hoelzl@47694
   366
    by (rule PiE_cong)
hoelzl@47694
   367
  with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   368
    by (simp add: prod_emb_def)
hoelzl@47694
   369
  moreover
hoelzl@47694
   370
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
hoelzl@47694
   371
    by auto
hoelzl@47694
   372
  ultimately show "A \<in> prod_algebra I F"
hoelzl@47694
   373
    apply (simp add: prod_algebra_def image_iff)
hoelzl@47694
   374
    apply (intro exI[of _ J] exI[of _ G] conjI)
hoelzl@47694
   375
    apply auto
hoelzl@47694
   376
    done
hoelzl@41689
   377
qed
hoelzl@41689
   378
hoelzl@47694
   379
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   380
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
hoelzl@47694
   381
hoelzl@47694
   382
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
   383
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
hoelzl@41689
   384
hoelzl@47694
   385
lemma sets_PiM_single: "sets (PiM I M) =
hoelzl@47694
   386
    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
   387
    (is "_ = sigma_sets ?\<Omega> ?R")
hoelzl@47694
   388
  unfolding sets_PiM
hoelzl@47694
   389
proof (rule sigma_sets_eqI)
hoelzl@47694
   390
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
hoelzl@47694
   391
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   392
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   393
  show "A \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   394
  proof cases
hoelzl@47694
   395
    assume "I = {}"
hoelzl@47694
   396
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
hoelzl@47694
   397
    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
hoelzl@47694
   398
  next
hoelzl@47694
   399
    assume "I \<noteq> {}"
hoelzl@47694
   400
    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
   401
      using sets_into_space[OF X(5)]
hoelzl@47694
   402
      by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast
hoelzl@47694
   403
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   404
      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
hoelzl@47694
   405
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
hoelzl@47694
   406
  qed
hoelzl@47694
   407
next
hoelzl@47694
   408
  fix A assume "A \<in> ?R"
hoelzl@47694
   409
  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
   410
    by auto
hoelzl@47694
   411
  then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
hoelzl@47694
   412
    using sets_into_space[OF A(3)]
hoelzl@47694
   413
    apply (subst prod_emb_PiE)
hoelzl@47694
   414
    apply (auto simp: Pi_iff split: split_if_asm)
hoelzl@47694
   415
    apply blast
hoelzl@47694
   416
    done
hoelzl@47694
   417
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
hoelzl@47694
   418
    using A by (intro sigma_sets.Basic prod_algebraI) auto
hoelzl@47694
   419
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
hoelzl@47694
   420
qed
hoelzl@47694
   421
hoelzl@47694
   422
lemma sets_PiM_I:
hoelzl@47694
   423
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
hoelzl@47694
   424
  shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   425
proof cases
hoelzl@47694
   426
  assume "J = {}"
hoelzl@47694
   427
  then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
hoelzl@47694
   428
    by (auto simp: prod_emb_def)
hoelzl@47694
   429
  then show ?thesis
hoelzl@47694
   430
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
hoelzl@47694
   431
next
hoelzl@47694
   432
  assume "J \<noteq> {}" with assms show ?thesis
hoelzl@47694
   433
    by (auto simp add: sets_PiM prod_algebra_def intro!: sigma_sets.Basic)
hoelzl@40859
   434
qed
hoelzl@40859
   435
hoelzl@47694
   436
lemma measurable_PiM:
hoelzl@47694
   437
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   438
  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
   439
    f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" 
hoelzl@47694
   440
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   441
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   442
proof (rule measurable_sigma_sets)
hoelzl@47694
   443
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   444
  from prod_algebraE[OF this] guess J X .
hoelzl@47694
   445
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
hoelzl@47694
   446
qed
hoelzl@47694
   447
hoelzl@47694
   448
lemma measurable_PiM_Collect:
hoelzl@47694
   449
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   450
  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
   451
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 
hoelzl@47694
   452
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   453
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   454
proof (rule measurable_sigma_sets)
hoelzl@47694
   455
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   456
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   457
  have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
hoelzl@47694
   458
    using sets_into_space[OF X(5)] X(2-) space unfolding X(1)
hoelzl@47694
   459
    by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm)
hoelzl@47694
   460
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
hoelzl@47694
   461
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@41689
   462
qed
hoelzl@41095
   463
hoelzl@47694
   464
lemma measurable_PiM_single:
hoelzl@47694
   465
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   466
  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
   467
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   468
  using sets_PiM_single
hoelzl@47694
   469
proof (rule measurable_sigma_sets)
hoelzl@47694
   470
  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
   471
  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
   472
    by auto
hoelzl@47694
   473
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
hoelzl@47694
   474
  also have "\<dots> \<in> sets N" using B by (rule sets)
hoelzl@47694
   475
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@47694
   476
qed (auto simp: space)
hoelzl@40859
   477
hoelzl@47694
   478
lemma sets_PiM_I_finite[simp, intro]:
hoelzl@47694
   479
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
hoelzl@47694
   480
  shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   481
  using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
hoelzl@47694
   482
hoelzl@47694
   483
lemma measurable_component_update:
hoelzl@47694
   484
  assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
hoelzl@47694
   485
  shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
hoelzl@47694
   486
proof (intro measurable_PiM_single)
hoelzl@47694
   487
  fix j A assume "j \<in> insert i I" "A \<in> sets (M j)"
hoelzl@47694
   488
  moreover have "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} =
hoelzl@47694
   489
    (if i = j then space (M i) \<inter> A else if x j \<in> A then space (M i) else {})"
hoelzl@47694
   490
    by auto
hoelzl@47694
   491
  ultimately show "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} \<in> sets (M i)"
hoelzl@47694
   492
    by auto
hoelzl@47694
   493
qed (insert sets_into_space assms, auto simp: space_PiM)
hoelzl@47694
   494
hoelzl@47694
   495
lemma measurable_component_singleton:
hoelzl@41689
   496
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
hoelzl@41689
   497
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   498
  fix A assume "A \<in> sets (M i)"
hoelzl@47694
   499
  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
   500
    using sets_into_space `i \<in> I`
hoelzl@47694
   501
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
hoelzl@41689
   502
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
hoelzl@47694
   503
    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
hoelzl@47694
   504
qed (insert `i \<in> I`, auto simp: space_PiM)
hoelzl@47694
   505
hoelzl@47694
   506
lemma measurable_add_dim:
hoelzl@49776
   507
  "(\<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
   508
    (is "?f \<in> measurable ?P ?I")
hoelzl@47694
   509
proof (rule measurable_PiM_single)
hoelzl@47694
   510
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
hoelzl@47694
   511
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
hoelzl@47694
   512
    (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
   513
    using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
hoelzl@47694
   514
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   515
    using A j
hoelzl@47694
   516
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@47694
   517
  finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
hoelzl@47694
   518
qed (auto simp: space_pair_measure space_PiM)
hoelzl@41661
   519
hoelzl@47694
   520
lemma measurable_merge:
hoelzl@49780
   521
  "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
   522
    (is "?f \<in> measurable ?P ?U")
hoelzl@47694
   523
proof (rule measurable_PiM_single)
hoelzl@47694
   524
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
hoelzl@49780
   525
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
hoelzl@47694
   526
    (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
   527
    by (auto simp: merge_def)
hoelzl@47694
   528
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   529
    using A
hoelzl@47694
   530
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@49780
   531
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
hoelzl@49776
   532
qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def)
hoelzl@42988
   533
hoelzl@47694
   534
lemma measurable_restrict:
hoelzl@47694
   535
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
hoelzl@47694
   536
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@47694
   537
proof (rule measurable_PiM_single)
hoelzl@47694
   538
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@47694
   539
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
hoelzl@47694
   540
    by auto
hoelzl@47694
   541
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
hoelzl@47694
   542
    using A X by (auto intro!: measurable_sets)
hoelzl@47694
   543
qed (insert X, auto dest: measurable_space)
hoelzl@47694
   544
hoelzl@47694
   545
locale product_sigma_finite =
hoelzl@47694
   546
  fixes M :: "'i \<Rightarrow> 'a measure"
hoelzl@41689
   547
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   548
hoelzl@41689
   549
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   550
  by (rule sigma_finite_measures)
hoelzl@40859
   551
hoelzl@47694
   552
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@47694
   553
  fixes I :: "'i set"
hoelzl@47694
   554
  assumes finite_index: "finite I"
hoelzl@41689
   555
hoelzl@40859
   556
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   557
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   558
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
hoelzl@47694
   559
    (\<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
   560
    (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
hoelzl@40859
   561
proof -
hoelzl@47694
   562
  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
   563
    using M.sigma_finite_incseq by metis
hoelzl@40859
   564
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@47694
   565
  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
   566
    by auto
hoelzl@40859
   567
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
hoelzl@47694
   568
  note space_PiM[simp]
hoelzl@40859
   569
  show ?thesis
hoelzl@41981
   570
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   571
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   572
  next
hoelzl@47694
   573
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   574
  next
hoelzl@47694
   575
    fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)"
hoelzl@47694
   576
      using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space
hoelzl@47694
   577
      by auto blast
hoelzl@40859
   578
  next
hoelzl@47694
   579
    fix f assume "f \<in> space (PiM I M)"
hoelzl@41981
   580
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@41981
   581
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
hoelzl@40859
   582
  next
hoelzl@40859
   583
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   584
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   585
  qed
hoelzl@40859
   586
qed
hoelzl@40859
   587
hoelzl@49780
   588
lemma
hoelzl@49780
   589
  shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
hoelzl@49780
   590
    and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
hoelzl@49780
   591
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
hoelzl@49780
   592
hoelzl@49780
   593
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
hoelzl@49780
   594
proof -
hoelzl@49780
   595
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
hoelzl@49780
   596
  have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
hoelzl@49780
   597
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49780
   598
    show "positive (PiM {} M) ?\<mu>"
hoelzl@49780
   599
      by (auto simp: positive_def)
hoelzl@49780
   600
    show "countably_additive (PiM {} M) ?\<mu>"
hoelzl@49780
   601
      by (rule countably_additiveI_finite)
hoelzl@49780
   602
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
hoelzl@49780
   603
  qed (auto simp: prod_emb_def)
hoelzl@49780
   604
  also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
hoelzl@49780
   605
    by (auto simp: prod_emb_def)
hoelzl@49780
   606
  finally show ?thesis
hoelzl@49780
   607
    by simp
hoelzl@49780
   608
qed
hoelzl@49780
   609
hoelzl@49780
   610
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
hoelzl@49780
   611
  by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
hoelzl@49780
   612
hoelzl@49776
   613
lemma (in product_sigma_finite) emeasure_PiM:
hoelzl@49776
   614
  "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
   615
proof (induct I arbitrary: A rule: finite_induct)
hoelzl@40859
   616
  case (insert i I)
hoelzl@41689
   617
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   618
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   619
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@41661
   620
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@47694
   621
hoelzl@47694
   622
  let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
hoelzl@47694
   623
  let ?\<mu> = "emeasure ?P"
hoelzl@47694
   624
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
hoelzl@47694
   625
  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
   626
hoelzl@49776
   627
  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
   628
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
hoelzl@49776
   629
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49776
   630
    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
   631
    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
   632
    let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
hoelzl@49776
   633
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
hoelzl@49776
   634
    have "?\<mu> ?p =
hoelzl@49776
   635
      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
   636
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
hoelzl@49776
   637
    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
   638
      using J E[rule_format, THEN sets_into_space]
hoelzl@49776
   639
      by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm)
hoelzl@49776
   640
    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
   641
      emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
hoelzl@49776
   642
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
hoelzl@49776
   643
    also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
hoelzl@49776
   644
      using J E[rule_format, THEN sets_into_space]
hoelzl@49776
   645
      by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+
hoelzl@49776
   646
    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
   647
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
hoelzl@49776
   648
      using E by (subst insert) (auto intro!: setprod_cong)
hoelzl@49776
   649
    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
   650
       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
   651
      using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
hoelzl@49776
   652
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
hoelzl@49776
   653
      using insert(1,2) J E by (intro setprod_mono_one_right) auto
hoelzl@49776
   654
    finally show "?\<mu> ?p = \<dots>" .
hoelzl@47694
   655
hoelzl@49776
   656
    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
   657
      using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff)
hoelzl@49776
   658
  next
hoelzl@49776
   659
    show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
hoelzl@49776
   660
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
hoelzl@49776
   661
  next
hoelzl@49776
   662
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
hoelzl@49776
   663
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
hoelzl@49776
   664
      using insert by auto
hoelzl@49776
   665
  qed (auto intro!: setprod_cong)
hoelzl@49776
   666
  with insert show ?case
hoelzl@49776
   667
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space)
hoelzl@49780
   668
qed (simp add: emeasure_PiM_empty)
hoelzl@47694
   669
hoelzl@49776
   670
lemma (in product_sigma_finite) sigma_finite: 
hoelzl@49776
   671
  assumes "finite I"
hoelzl@49776
   672
  shows "sigma_finite_measure (PiM I M)"
hoelzl@49776
   673
proof -
hoelzl@49776
   674
  interpret finite_product_sigma_finite M I by default fact
hoelzl@49776
   675
hoelzl@49776
   676
  from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@49776
   677
  then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
hoelzl@49776
   678
    "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
hoelzl@49776
   679
    "(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
hoelzl@49776
   680
    "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
hoelzl@47694
   681
    by blast+
hoelzl@49776
   682
  let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"
hoelzl@47694
   683
hoelzl@49776
   684
  show ?thesis
hoelzl@47694
   685
  proof (unfold_locales, intro exI[of _ ?F] conjI allI)
hoelzl@49776
   686
    show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
hoelzl@47694
   687
  next
hoelzl@49776
   688
    from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
hoelzl@47694
   689
  next
hoelzl@47694
   690
    fix j
hoelzl@49776
   691
    from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
hoelzl@49776
   692
    show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
hoelzl@49776
   693
      by (subst emeasure_PiM) auto
hoelzl@40859
   694
  qed
hoelzl@40859
   695
qed
hoelzl@40859
   696
hoelzl@47694
   697
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
hoelzl@47694
   698
  using sigma_finite[OF finite_index] .
hoelzl@40859
   699
hoelzl@40859
   700
lemma (in finite_product_sigma_finite) measure_times:
hoelzl@47694
   701
  "(\<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
   702
  using emeasure_PiM[OF finite_index] by auto
hoelzl@41096
   703
hoelzl@40859
   704
lemma (in product_sigma_finite) positive_integral_empty:
hoelzl@41981
   705
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
hoelzl@41981
   706
  shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
hoelzl@40859
   707
proof -
hoelzl@41689
   708
  interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
hoelzl@47694
   709
  have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
hoelzl@40859
   710
    using assms by (subst measure_times) auto
hoelzl@40859
   711
  then show ?thesis
hoelzl@47694
   712
    unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
hoelzl@47694
   713
  proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
hoelzl@41981
   714
    show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
hoelzl@44928
   715
      by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
hoelzl@41981
   716
    show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
hoelzl@44928
   717
      by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
hoelzl@40859
   718
  qed
hoelzl@40859
   719
qed
hoelzl@40859
   720
hoelzl@47694
   721
lemma (in product_sigma_finite) distr_merge:
hoelzl@40859
   722
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@49780
   723
  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
   724
   (is "?D = ?P")
hoelzl@40859
   725
proof -
hoelzl@41689
   726
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   727
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@40859
   728
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   729
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   730
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   731
  let ?g = "merge I J"
hoelzl@47694
   732
hoelzl@41661
   733
  from IJ.sigma_finite_pairs obtain F where
hoelzl@41661
   734
    F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
hoelzl@41981
   735
       "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
hoelzl@47694
   736
       "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
hoelzl@47694
   737
       "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
hoelzl@41661
   738
    by auto
hoelzl@41661
   739
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
hoelzl@47694
   740
  
hoelzl@47694
   741
  show ?thesis
hoelzl@47694
   742
  proof (rule measure_eqI_generator_eq[symmetric])
hoelzl@47694
   743
    show "Int_stable (prod_algebra (I \<union> J) M)"
hoelzl@47694
   744
      by (rule Int_stable_prod_algebra)
hoelzl@47694
   745
    show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
hoelzl@47694
   746
      by (rule prod_algebra_sets_into_space)
hoelzl@47694
   747
    show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   748
      by (rule sets_PiM)
hoelzl@47694
   749
    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
   750
      by simp
hoelzl@47694
   751
hoelzl@47694
   752
    show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
hoelzl@47694
   753
      using fin by (auto simp: prod_algebra_eq_finite)
hoelzl@47694
   754
    show "incseq ?F" by fact
hoelzl@47694
   755
    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
   756
      using F(3) by (simp add: space_PiM)
hoelzl@41981
   757
  next
hoelzl@41981
   758
    fix k
hoelzl@47694
   759
    from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
hoelzl@47694
   760
    show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
hoelzl@41661
   761
  next
hoelzl@47694
   762
    fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
hoelzl@47694
   763
    with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>I \<union> J. F i \<in> sets (M i)"
hoelzl@47694
   764
      by (auto simp add: prod_algebra_eq_finite)
hoelzl@47694
   765
    let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
hoelzl@47694
   766
    let ?X = "?g -` A \<inter> space ?B"
hoelzl@47694
   767
    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@47694
   768
      using F[rule_format, THEN sets_into_space] by (auto simp: space_PiM)
hoelzl@47694
   769
    then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
hoelzl@47694
   770
      unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
hoelzl@47694
   771
    have "emeasure ?D A = emeasure ?B ?X"
hoelzl@47694
   772
      using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
hoelzl@47694
   773
    also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
hoelzl@47694
   774
      using `finite J` `finite I` F X
hoelzl@49776
   775
      by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff)
hoelzl@47694
   776
    also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
hoelzl@41661
   777
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
hoelzl@47694
   778
    also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
hoelzl@41661
   779
      using `finite J` `finite I` F unfolding A
hoelzl@41661
   780
      by (intro IJ.measure_times[symmetric]) auto
hoelzl@47694
   781
    finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
hoelzl@47694
   782
  qed
hoelzl@41661
   783
qed
hoelzl@41026
   784
hoelzl@41026
   785
lemma (in product_sigma_finite) product_positive_integral_fold:
hoelzl@47694
   786
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
hoelzl@41689
   787
  and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41689
   788
  shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
hoelzl@49780
   789
    (\<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
   790
proof -
hoelzl@41689
   791
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   792
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41831
   793
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   794
  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
   795
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
hoelzl@41661
   796
  show ?thesis
hoelzl@47694
   797
    apply (subst distr_merge[OF IJ, symmetric])
hoelzl@49776
   798
    apply (subst positive_integral_distr[OF measurable_merge f])
hoelzl@47694
   799
    apply (subst P.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
hoelzl@47694
   800
    apply simp
hoelzl@47694
   801
    done
hoelzl@40859
   802
qed
hoelzl@40859
   803
hoelzl@47694
   804
lemma (in product_sigma_finite) distr_singleton:
hoelzl@47694
   805
  "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
hoelzl@47694
   806
proof (intro measure_eqI[symmetric])
hoelzl@41831
   807
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   808
  fix A assume A: "A \<in> sets (M i)"
hoelzl@47694
   809
  moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
hoelzl@47694
   810
    using sets_into_space by (auto simp: space_PiM)
hoelzl@47694
   811
  ultimately show "emeasure (M i) A = emeasure ?D A"
hoelzl@47694
   812
    using A I.measure_times[of "\<lambda>_. A"]
hoelzl@47694
   813
    by (simp add: emeasure_distr measurable_component_singleton)
hoelzl@47694
   814
qed simp
hoelzl@41831
   815
hoelzl@41026
   816
lemma (in product_sigma_finite) product_positive_integral_singleton:
hoelzl@40859
   817
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   818
  shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
hoelzl@40859
   819
proof -
hoelzl@41689
   820
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   821
  from f show ?thesis
hoelzl@47694
   822
    apply (subst distr_singleton[symmetric])
hoelzl@47694
   823
    apply (subst positive_integral_distr[OF measurable_component_singleton])
hoelzl@47694
   824
    apply simp_all
hoelzl@47694
   825
    done
hoelzl@40859
   826
qed
hoelzl@40859
   827
hoelzl@41096
   828
lemma (in product_sigma_finite) product_positive_integral_insert:
hoelzl@49780
   829
  assumes I[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@41689
   835
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
hoelzl@41689
   836
    using f by auto
hoelzl@41096
   837
  show ?thesis
hoelzl@49780
   838
    unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
hoelzl@49780
   839
  proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
hoelzl@47694
   840
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@49780
   841
    let ?f = "\<lambda>y. f (x(i := y))"
hoelzl@49780
   842
    show "?f \<in> borel_measurable (M i)"
hoelzl@47694
   843
      using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
hoelzl@47694
   844
      unfolding comp_def .
hoelzl@49780
   845
    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
   846
      using x
hoelzl@49780
   847
      by (auto intro!: positive_integral_cong arg_cong[where f=f]
hoelzl@49780
   848
               simp add: space_PiM extensional_def)
hoelzl@41096
   849
  qed
hoelzl@41096
   850
qed
hoelzl@41096
   851
hoelzl@41096
   852
lemma (in product_sigma_finite) product_positive_integral_setprod:
hoelzl@43920
   853
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
hoelzl@41096
   854
  assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41981
   855
  and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
hoelzl@41689
   856
  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
   857
using assms proof induct
hoelzl@41096
   858
  case (insert i I)
hoelzl@41096
   859
  note `finite I`[intro, simp]
hoelzl@41689
   860
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41096
   861
  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
   862
    using insert by (auto intro!: setprod_cong)
hoelzl@41689
   863
  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
   864
    using sets_into_space insert
hoelzl@47694
   865
    by (intro borel_measurable_ereal_setprod
hoelzl@41689
   866
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
hoelzl@41096
   867
       auto
hoelzl@41981
   868
  then show ?case
hoelzl@41981
   869
    apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
hoelzl@47694
   870
    apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
hoelzl@47694
   871
    apply (subst positive_integral_cmult)
hoelzl@47694
   872
    apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
hoelzl@41981
   873
    done
hoelzl@47694
   874
qed (simp add: space_PiM)
hoelzl@41096
   875
hoelzl@41026
   876
lemma (in product_sigma_finite) product_integral_singleton:
hoelzl@41026
   877
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   878
  shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
hoelzl@41026
   879
proof -
hoelzl@41689
   880
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@43920
   881
  have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
hoelzl@43920
   882
    "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
hoelzl@41026
   883
    using assms by auto
hoelzl@41026
   884
  show ?thesis
hoelzl@41689
   885
    unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
hoelzl@41026
   886
qed
hoelzl@41026
   887
hoelzl@41026
   888
lemma (in product_sigma_finite) product_integral_fold:
hoelzl@41026
   889
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@41689
   890
  and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
hoelzl@49780
   891
  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
   892
proof -
hoelzl@41689
   893
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   894
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41026
   895
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   896
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   897
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   898
  let ?M = "merge I J"
hoelzl@41689
   899
  let ?f = "\<lambda>x. f (?M x)"
hoelzl@47694
   900
  from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@47694
   901
    by auto
hoelzl@49780
   902
  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
   903
    using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
hoelzl@47694
   904
  have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
hoelzl@49776
   905
    by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
hoelzl@41026
   906
  show ?thesis
hoelzl@47694
   907
    apply (subst distr_merge[symmetric, OF IJ fin])
hoelzl@49776
   908
    apply (subst integral_distr[OF measurable_merge f_borel])
hoelzl@47694
   909
    apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
hoelzl@47694
   910
    apply simp
hoelzl@47694
   911
    done
hoelzl@41026
   912
qed
hoelzl@41026
   913
hoelzl@49776
   914
lemma (in product_sigma_finite)
hoelzl@49776
   915
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
hoelzl@49776
   916
  shows emeasure_fold_integral:
hoelzl@49780
   917
    "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
   918
    and emeasure_fold_measurable:
hoelzl@49780
   919
    "(\<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
   920
proof -
hoelzl@49776
   921
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@49776
   922
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@49776
   923
  interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
hoelzl@49780
   924
  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
   925
    by (intro measurable_sets[OF _ A] measurable_merge assms)
hoelzl@49776
   926
hoelzl@49776
   927
  show ?I
hoelzl@49776
   928
    apply (subst distr_merge[symmetric, OF IJ])
hoelzl@49776
   929
    apply (subst emeasure_distr[OF measurable_merge A])
hoelzl@49776
   930
    apply (subst J.emeasure_pair_measure_alt[OF merge])
hoelzl@49776
   931
    apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49776
   932
    done
hoelzl@49776
   933
hoelzl@49776
   934
  show ?B
hoelzl@49776
   935
    using IJ.measurable_emeasure_Pair1[OF merge]
hoelzl@49776
   936
    by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
hoelzl@49776
   937
qed
hoelzl@49776
   938
hoelzl@41096
   939
lemma (in product_sigma_finite) product_integral_insert:
hoelzl@47694
   940
  assumes I: "finite I" "i \<notin> I"
hoelzl@41689
   941
    and f: "integrable (Pi\<^isub>M (insert i I) M) f"
hoelzl@41689
   942
  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
   943
proof -
hoelzl@47694
   944
  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
   945
    by simp
hoelzl@49780
   946
  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
   947
    using f I by (intro product_integral_fold) auto
hoelzl@47694
   948
  also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
hoelzl@47694
   949
  proof (rule integral_cong, subst product_integral_singleton[symmetric])
hoelzl@47694
   950
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@47694
   951
    have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@47694
   952
      using f by auto
hoelzl@47694
   953
    show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
hoelzl@47694
   954
      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
hoelzl@47694
   955
      unfolding comp_def .
hoelzl@49780
   956
    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
   957
      by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def)
hoelzl@41096
   958
  qed
hoelzl@47694
   959
  finally show ?thesis .
hoelzl@41096
   960
qed
hoelzl@41096
   961
hoelzl@41096
   962
lemma (in product_sigma_finite) product_integrable_setprod:
hoelzl@41096
   963
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@41689
   964
  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
   965
  shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
hoelzl@41096
   966
proof -
hoelzl@41689
   967
  interpret finite_product_sigma_finite M I by default fact
hoelzl@41096
   968
  have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41689
   969
    using integrable unfolding integrable_def by auto
hoelzl@47694
   970
  have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@47694
   971
    using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
hoelzl@41689
   972
  moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
hoelzl@41096
   973
  proof (unfold integrable_def, intro conjI)
hoelzl@47694
   974
    show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@41096
   975
      using borel by auto
hoelzl@47694
   976
    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
   977
      by (simp add: setprod_ereal abs_setprod)
hoelzl@43920
   978
    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
hoelzl@41096
   979
      using f by (subst product_positive_integral_setprod) auto
hoelzl@41981
   980
    also have "\<dots> < \<infinity>"
hoelzl@47694
   981
      using integrable[THEN integrable_abs]
hoelzl@47694
   982
      by (simp add: setprod_PInf integrable_def positive_integral_positive)
hoelzl@47694
   983
    finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
hoelzl@47694
   984
    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
   985
      by (intro positive_integral_cong_pos) auto
hoelzl@47694
   986
    then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
hoelzl@41096
   987
  qed
hoelzl@41096
   988
  ultimately show ?thesis
hoelzl@41096
   989
    by (rule integrable_abs_iff[THEN iffD1])
hoelzl@41096
   990
qed
hoelzl@41096
   991
hoelzl@41096
   992
lemma (in product_sigma_finite) product_integral_setprod:
hoelzl@41096
   993
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@49780
   994
  assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
   995
  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
   996
using assms proof induct
hoelzl@49780
   997
  case empty
hoelzl@49780
   998
  interpret finite_measure "Pi\<^isub>M {} M"
hoelzl@49780
   999
    by rule (simp add: space_PiM)
hoelzl@49780
  1000
  show ?case by (simp add: space_PiM measure_def)
hoelzl@41096
  1001
next
hoelzl@41096
  1002
  case (insert i I)
hoelzl@41096
  1003
  then have iI: "finite (insert i I)" by auto
hoelzl@41096
  1004
  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
hoelzl@41689
  1005
    integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
hoelzl@49780
  1006
    by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
hoelzl@41689
  1007
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41096
  1008
  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
  1009
    using `i \<notin> I` by (auto intro!: setprod_cong)
hoelzl@41096
  1010
  show ?case
hoelzl@49780
  1011
    unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
hoelzl@47694
  1012
    by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
hoelzl@41096
  1013
qed
hoelzl@41096
  1014
hoelzl@49776
  1015
lemma sets_Collect_single:
hoelzl@49776
  1016
  "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@49776
  1017
  unfolding sets_PiM_single
hoelzl@49776
  1018
  by (auto intro!: sigma_sets.Basic exI[of _ i] exI[of _ A]) (auto simp: space_PiM)
hoelzl@49776
  1019
hoelzl@49776
  1020
lemma sigma_prod_algebra_sigma_eq_infinite:
hoelzl@49776
  1021
  fixes E :: "'i \<Rightarrow> 'a set set"
hoelzl@49779
  1022
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@49776
  1023
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@49776
  1024
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@49776
  1025
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@49776
  1026
  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
  1027
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1028
proof
hoelzl@49776
  1029
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49776
  1030
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@49776
  1031
    using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
hoelzl@49776
  1032
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@49776
  1033
    by (simp add: space_PiM)
hoelzl@49776
  1034
  have "sets (PiM I M) =
hoelzl@49776
  1035
      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
  1036
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@49776
  1037
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@49776
  1038
  proof (safe intro!: sigma_sets_subset)
hoelzl@49776
  1039
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@49776
  1040
    then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@49776
  1041
      apply (subst measurable_iff_measure_of)
hoelzl@49776
  1042
      apply (simp_all add: P_closed)
hoelzl@49776
  1043
      using E_closed
hoelzl@49776
  1044
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1045
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1046
      apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
hoelzl@49776
  1047
      apply (rule_tac x=Aa in exI)
hoelzl@49776
  1048
      apply (auto simp: space_PiM)
hoelzl@49776
  1049
      done
hoelzl@49776
  1050
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@49776
  1051
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@49776
  1052
      by (simp add: E_generates)
hoelzl@49776
  1053
    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
  1054
      using P_closed by (auto simp: space_PiM)
hoelzl@49776
  1055
    finally show "\<dots> \<in> sets ?P" .
hoelzl@49776
  1056
  qed
hoelzl@49776
  1057
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1058
    by (simp add: P_closed)
hoelzl@49776
  1059
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@49776
  1060
    unfolding P_def space_PiM[symmetric]
hoelzl@49776
  1061
    by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
hoelzl@49776
  1062
qed
hoelzl@49776
  1063
hoelzl@49779
  1064
lemma bchoice_iff: "(\<forall>a\<in>A. \<exists>b. P a b) \<longleftrightarrow> (\<exists>f. \<forall>a\<in>A. P a (f a))"
hoelzl@49779
  1065
  by metis
hoelzl@49779
  1066
hoelzl@47694
  1067
lemma sigma_prod_algebra_sigma_eq:
hoelzl@49779
  1068
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
hoelzl@47694
  1069
  assumes "finite I"
hoelzl@49779
  1070
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@47694
  1071
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@47694
  1072
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@47694
  1073
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@47694
  1074
  defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
hoelzl@47694
  1075
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1076
proof
hoelzl@47694
  1077
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49779
  1078
  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
hoelzl@49779
  1079
  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
  1080
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
hoelzl@47694
  1081
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@47694
  1082
    using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
hoelzl@47694
  1083
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
  1084
    by (simp add: space_PiM)
hoelzl@47694
  1085
  have "sets (PiM I M) =
hoelzl@47694
  1086
      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
  1087
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@47694
  1088
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@47694
  1089
  proof (safe intro!: sigma_sets_subset)
hoelzl@47694
  1090
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@47694
  1091
    have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@47694
  1092
    proof (subst measurable_iff_measure_of)
hoelzl@47694
  1093
      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
hoelzl@47694
  1094
      from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)"
hoelzl@47694
  1095
        by (auto simp: Pi_iff)
hoelzl@47694
  1096
      show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1097
      proof
hoelzl@47694
  1098
        fix A assume A: "A \<in> E i"
hoelzl@47694
  1099
        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
  1100
          using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
hoelzl@47694
  1101
        also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
hoelzl@47694
  1102
          by (intro PiE_cong) (simp add: S_union)
hoelzl@49779
  1103
        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
  1104
          using T
hoelzl@49779
  1105
          apply (auto simp: Pi_iff bchoice_iff)
hoelzl@49779
  1106
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
hoelzl@49779
  1107
          apply (auto simp: bij_betw_def)
hoelzl@49779
  1108
          done
hoelzl@47694
  1109
        also have "\<dots> \<in> sets ?P"
hoelzl@47694
  1110
        proof (safe intro!: countable_UN)
hoelzl@49779
  1111
          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
  1112
            using A S_in_E
hoelzl@47694
  1113
            by (simp add: P_closed)
hoelzl@49779
  1114
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
hoelzl@47694
  1115
        qed
hoelzl@47694
  1116
        finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1117
          using P_closed by simp
hoelzl@47694
  1118
      qed
hoelzl@47694
  1119
    qed
hoelzl@47694
  1120
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@47694
  1121
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1122
      by (simp add: E_generates)
hoelzl@47694
  1123
    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
  1124
      using P_closed by (auto simp: space_PiM)
hoelzl@47694
  1125
    finally show "\<dots> \<in> sets ?P" .
hoelzl@47694
  1126
  qed
hoelzl@47694
  1127
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1128
    by (simp add: P_closed)
hoelzl@47694
  1129
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@47694
  1130
    using `finite I`
hoelzl@47694
  1131
    by (auto intro!: sigma_sets_subset simp: E_generates P_def)
hoelzl@47694
  1132
qed
hoelzl@47694
  1133
hoelzl@47694
  1134
end