author  hoelzl 
Wed, 10 Oct 2012 12:12:18 +0200  
changeset 49776  199d1d5bb17e 
parent 47761  dfe747e72fa8 
child 49779  1484b4b82855 
permissions  rwrr 
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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 
35833  9 
begin 
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47694  11 
lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)" 
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by auto 

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40859  14 
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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40859  29 
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)" 

33 

34 
notation (xsymbols) 

35 
funcset_extensional (infixr "\<rightarrow>\<^isub>E" 60) 

36 

37 
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 

41 

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lemma extensional_Int[simp]: 

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"extensional I \<inter> extensional I' = extensional (I \<inter> I')" 

44 
unfolding extensional_def by auto 

38656  45 

35833  46 
definition 
40859  47 
"merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)" 
48 

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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) 

60 

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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) 

74 

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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) 

80 

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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) 
40859  87 

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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 

92 

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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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101 
lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)" 

102 
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 

106 

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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) 

115 

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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) 
41095  131 

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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 
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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 
41095  140 

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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) 

147 
done 

148 

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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)" 

160 
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 

164 
qed 

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then show "f \<in> (\<Union>n. Pi I (A n))" by auto 

166 
qed auto 

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168 
lemma PiE_cong: 

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assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i" 

170 
shows "Pi\<^isub>E I A = Pi\<^isub>E I B" 

171 
using assms by (auto intro!: Pi_cong) 

172 

173 
lemma restrict_upd[simp]: 

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"i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)" 

175 
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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224 
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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40859  229 
section "Finite product spaces" 
230 

231 
section "Products" 

232 

47694  233 
definition prod_emb where 
234 
"prod_emb I M K X = (\<lambda>x. restrict x K) ` X \<inter> (PIE i:I. space (M i))" 

235 

236 
lemma prod_emb_iff: 

237 
"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))" 

238 
unfolding prod_emb_def by auto 

40859  239 

47694  240 
lemma 
241 
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}" 

242 
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" 

243 
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" 

244 
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))" 

245 
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))" 

246 
and prod_emb_Diff[simp]: "prod_emb M L K (A  B) = prod_emb M L K A  prod_emb M L K B" 

247 
by (auto simp: prod_emb_def) 

40859  248 

47694  249 
lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> 
250 
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))" 

251 
by (force simp: prod_emb_def Pi_iff split_if_mem2) 

252 

253 
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" 

254 
by (auto simp: prod_emb_def Pi_iff) 

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47694  256 
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where 
257 
"PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) 

258 
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} 

259 
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) 

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)))" 

261 

262 
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where 

263 
"prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) ` 

264 
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

265 

266 
abbreviation 

267 
"Pi\<^isub>M I M \<equiv> PiM I M" 

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40859  269 
syntax 
47694  270 
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIM _:_./ _)" 10) 
40859  271 

272 
syntax (xsymbols) 

47694  273 
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) 
40859  274 

275 
syntax (HTML output) 

47694  276 
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) 
40859  277 

278 
translations 

47694  279 
"PIM x:I. M" == "CONST PiM I (%x. M)" 
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47694  281 
lemma prod_algebra_sets_into_space: 
282 
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))" 

283 
using assms by (auto simp: prod_emb_def prod_algebra_def) 

40859  284 

47694  285 
lemma prod_algebra_eq_finite: 
286 
assumes I: "finite I" 

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") 

288 
proof (intro iffI set_eqI) 

289 
fix A assume "A \<in> ?L" 

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)" 

291 
and A: "A = prod_emb I M J (PIE j:J. E j)" 

292 
by (auto simp: prod_algebra_def) 

293 
let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)" 

294 
have A: "A = ?A" 

295 
unfolding A using J by (intro prod_emb_PiE sets_into_space) auto 

296 
show "A \<in> ?R" unfolding A using J top 

297 
by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp 

298 
next 

299 
fix A assume "A \<in> ?R" 

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 

301 
then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)" 

302 
using sets_into_space by (force simp: prod_emb_def Pi_iff) 

303 
from X I show "A \<in> ?L" unfolding A 

304 
by (auto simp: prod_algebra_def) 

305 
qed 

41095  306 

47694  307 
lemma prod_algebraI: 
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)) 

309 
\<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M" 

310 
by (auto simp: prod_algebra_def Pi_iff) 

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47694  312 
lemma prod_algebraE: 
313 
assumes A: "A \<in> prod_algebra I M" 

314 
obtains J E where "A = prod_emb I M J (PIE j:J. E j)" 

315 
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" 

316 
using A by (auto simp: prod_algebra_def) 

42988  317 

47694  318 
lemma prod_algebraE_all: 
319 
assumes A: "A \<in> prod_algebra I M" 

320 
obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" 

321 
proof  

322 
from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)" 

323 
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))" 

324 
by (auto simp: prod_algebra_def) 

325 
from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" 

326 
using sets_into_space by auto 

327 
then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))" 

328 
using A J by (auto simp: prod_emb_PiE) 

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))" 

330 
using top E by auto 

331 
ultimately show ?thesis using that by auto 

332 
qed 

40859  333 

47694  334 
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)" 
335 
proof (unfold Int_stable_def, safe) 

336 
fix A assume "A \<in> prod_algebra I M" 

337 
from prod_algebraE[OF this] guess J E . note A = this 

338 
fix B assume "B \<in> prod_algebra I M" 

339 
from prod_algebraE[OF this] guess K F . note B = this 

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> 

341 
(if i \<in> K then F i else space (M i)))" 

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] 

343 
apply (subst (1 2 3) prod_emb_PiE) 

344 
apply (simp_all add: subset_eq PiE_Int) 

345 
apply blast 

346 
apply (intro PiE_cong) 

347 
apply auto 

348 
done 

349 
also have "\<dots> \<in> prod_algebra I M" 

350 
using A B by (auto intro!: prod_algebraI) 

351 
finally show "A \<inter> B \<in> prod_algebra I M" . 

352 
qed 

353 

354 
lemma prod_algebra_mono: 

355 
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)" 

356 
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" 

357 
shows "prod_algebra I E \<subseteq> prod_algebra I F" 

358 
proof 

359 
fix A assume "A \<in> prod_algebra I E" 

360 
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" 

361 
and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)" 

362 
and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)" 

363 
by (auto simp: prod_algebra_def) 

364 
moreover 

365 
from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))" 

366 
by (rule PiE_cong) 

367 
with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)" 

368 
by (simp add: prod_emb_def) 

369 
moreover 

370 
from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)" 

371 
by auto 

372 
ultimately show "A \<in> prod_algebra I F" 

373 
apply (simp add: prod_algebra_def image_iff) 

374 
apply (intro exI[of _ J] exI[of _ G] conjI) 

375 
apply auto 

376 
done 

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377 
qed 
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378 

47694  379 
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
380 
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp 

381 

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)" 

383 
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp 

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384 

47694  385 
lemma sets_PiM_single: "sets (PiM I M) = 
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)}" 

387 
(is "_ = sigma_sets ?\<Omega> ?R") 

388 
unfolding sets_PiM 

389 
proof (rule sigma_sets_eqI) 

390 
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto 

391 
fix A assume "A \<in> prod_algebra I M" 

392 
from prod_algebraE[OF this] guess J X . note X = this 

393 
show "A \<in> sigma_sets ?\<Omega> ?R" 

394 
proof cases 

395 
assume "I = {}" 

396 
with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def) 

397 
with `I = {}` show ?thesis by (auto intro!: sigma_sets_top) 

398 
next 

399 
assume "I \<noteq> {}" 

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})" 

401 
using sets_into_space[OF X(5)] 

402 
by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast 

403 
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" 

404 
using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto 

405 
finally show "A \<in> sigma_sets ?\<Omega> ?R" . 

406 
qed 

407 
next 

408 
fix A assume "A \<in> ?R" 

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)" 

410 
by auto 

411 
then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)" 

412 
using sets_into_space[OF A(3)] 

413 
apply (subst prod_emb_PiE) 

414 
apply (auto simp: Pi_iff split: split_if_asm) 

415 
apply blast 

416 
done 

417 
also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)" 

418 
using A by (intro sigma_sets.Basic prod_algebraI) auto 

419 
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" . 

420 
qed 

421 

422 
lemma sets_PiM_I: 

423 
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" 

424 
shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)" 

425 
proof cases 

426 
assume "J = {}" 

427 
then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))" 

428 
by (auto simp: prod_emb_def) 

429 
then show ?thesis 

430 
by (auto simp add: sets_PiM intro!: sigma_sets_top) 

431 
next 

432 
assume "J \<noteq> {}" with assms show ?thesis 

433 
by (auto simp add: sets_PiM prod_algebra_def intro!: sigma_sets.Basic) 

40859  434 
qed 
435 

47694  436 
lemma measurable_PiM: 
437 
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

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> 

439 
f ` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" 

440 
shows "f \<in> measurable N (PiM I M)" 

441 
using sets_PiM prod_algebra_sets_into_space space 

442 
proof (rule measurable_sigma_sets) 

443 
fix A assume "A \<in> prod_algebra I M" 

444 
from prod_algebraE[OF this] guess J X . 

445 
with sets[of J X] show "f ` A \<inter> space N \<in> sets N" by auto 

446 
qed 

447 

448 
lemma measurable_PiM_Collect: 

449 
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

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> 

451 
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 

452 
shows "f \<in> measurable N (PiM I M)" 

453 
using sets_PiM prod_algebra_sets_into_space space 

454 
proof (rule measurable_sigma_sets) 

455 
fix A assume "A \<in> prod_algebra I M" 

456 
from prod_algebraE[OF this] guess J X . note X = this 

457 
have "f ` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}" 

458 
using sets_into_space[OF X(5)] X(2) space unfolding X(1) 

459 
by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm) 

460 
also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets) 

461 
finally show "f ` A \<inter> space N \<in> sets N" . 

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462 
qed 
41095  463 

47694  464 
lemma measurable_PiM_single: 
465 
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

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" 

467 
shows "f \<in> measurable N (PiM I M)" 

468 
using sets_PiM_single 

469 
proof (rule measurable_sigma_sets) 

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)}" 

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)" 

472 
by auto 

473 
with space have "f ` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto 

474 
also have "\<dots> \<in> sets N" using B by (rule sets) 

475 
finally show "f ` A \<inter> space N \<in> sets N" . 

476 
qed (auto simp: space) 

40859  477 

47694  478 
lemma sets_PiM_I_finite[simp, intro]: 
479 
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))" 

480 
shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)" 

481 
using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto 

482 

483 
lemma measurable_component_update: 

484 
assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I" 

485 
shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _") 

486 
proof (intro measurable_PiM_single) 

487 
fix j A assume "j \<in> insert i I" "A \<in> sets (M j)" 

488 
moreover have "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} = 

489 
(if i = j then space (M i) \<inter> A else if x j \<in> A then space (M i) else {})" 

490 
by auto 

491 
ultimately show "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} \<in> sets (M i)" 

492 
by auto 

493 
qed (insert sets_into_space assms, auto simp: space_PiM) 

494 

495 
lemma measurable_component_singleton: 

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496 
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)" 
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497 
proof (unfold measurable_def, intro CollectI conjI ballI) 
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498 
fix A assume "A \<in> sets (M i)" 
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)" 
500 
using sets_into_space `i \<in> I` 

501 
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm) 

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502 
then show "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)" 
47694  503 
using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I) 
504 
qed (insert `i \<in> I`, auto simp: space_PiM) 

505 

506 
lemma measurable_add_dim: 

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)" 
47694  508 
(is "?f \<in> measurable ?P ?I") 
509 
proof (rule measurable_PiM_single) 

510 
fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)" 

511 
have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} = 

512 
(if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) ` A \<inter> space ?P)" 

513 
using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM) 

514 
also have "\<dots> \<in> sets ?P" 

515 
using A j 

516 
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) 

517 
finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" . 

518 
qed (auto simp: space_pair_measure space_PiM) 

41661  519 

47694  520 
lemma measurable_merge: 
49776  521 
"(\<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)" 
47694  522 
(is "?f \<in> measurable ?P ?U") 
523 
proof (rule measurable_PiM_single) 

524 
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J" 

49776  525 
then have "{\<omega> \<in> space ?P. prod_case (\<lambda>x y. merge I x J y) \<omega> i \<in> A} = 
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)" 
49776  527 
by (auto simp: merge_def) 
47694  528 
also have "\<dots> \<in> sets ?P" 
529 
using A 

530 
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) 

49776  531 
finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>x y. merge I x J y) \<omega> i \<in> A} \<in> sets ?P" . 
532 
qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def) 

42988  533 

47694  534 
lemma measurable_restrict: 
535 
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" 

536 
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)" 

537 
proof (rule measurable_PiM_single) 

538 
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" 

539 
then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i ` A \<inter> space N" 

540 
by auto 

541 
then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N" 

542 
using A X by (auto intro!: measurable_sets) 

543 
qed (insert X, auto dest: measurable_space) 

544 

545 
locale product_sigma_finite = 

546 
fixes M :: "'i \<Rightarrow> 'a measure" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

547 
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" 
40859  548 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

549 
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i 
40859  550 
by (rule sigma_finite_measures) 
551 

47694  552 
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" + 
553 
fixes I :: "'i set" 

554 
assumes finite_index: "finite I" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

555 

40859  556 
lemma (in finite_product_sigma_finite) sigma_finite_pairs: 
557 
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. 

558 
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> 

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> 
560 
(\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)" 

40859  561 
proof  
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>)" 
563 
using M.sigma_finite_incseq by metis 

40859  564 
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. 
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>" 
40859  566 
by auto 
567 
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k" 

47694  568 
note space_PiM[simp] 
40859  569 
show ?thesis 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

570 
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) 
40859  571 
fix i show "range (F i) \<subseteq> sets (M i)" by fact 
572 
next 

47694  573 
fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact 
40859  574 
next 
47694  575 
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)" 
576 
using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space 

577 
by auto blast 

40859  578 
next 
47694  579 
fix f assume "f \<in> space (PiM I M)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

580 
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

581 
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def) 
40859  582 
next 
583 
fix i show "?F i \<subseteq> ?F (Suc i)" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

584 
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto 
40859  585 
qed 
586 
qed 

587 

49776  588 
lemma (in product_sigma_finite) emeasure_PiM: 
589 
"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))" 

590 
proof (induct I arbitrary: A rule: finite_induct) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

591 
case empty 
47694  592 
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)" 
593 
have "prod_algebra {} M = {{\<lambda>_. undefined}}" 

594 
by (auto simp: prod_algebra_def prod_emb_def intro!: image_eqI) 

595 
then have sets_empty: "sets (PiM {} M) = {{}, {\<lambda>_. undefined}}" 

596 
by (simp add: sets_PiM) 

597 
have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1" 

598 
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) 

599 
have "finite (space (PiM {} M))" 

600 
by (simp add: space_PiM) 

601 
moreover show "positive (PiM {} M) ?\<mu>" 

602 
by (auto simp: positive_def) 

603 
ultimately show "countably_additive (PiM {} M) ?\<mu>" 

604 
by (rule countably_additiveI_finite) (auto simp: additive_def space_PiM sets_empty) 

605 
qed (auto simp: prod_emb_def) 

606 
also have *: "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}" 

607 
by (auto simp: prod_emb_def) 

49776  608 
finally show ?case 
609 
using * by simp 

40859  610 
next 
611 
case (insert i I) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

612 
interpret finite_product_sigma_finite M I by default fact 
40859  613 
have "finite (insert i I)" using `finite I` by auto 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

614 
interpret I': finite_product_sigma_finite M "insert i I" by default fact 
41661  615 
let ?h = "(\<lambda>(f, y). f(i := y))" 
47694  616 

617 
let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h" 

618 
let ?\<mu> = "emeasure ?P" 

619 
let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}" 

620 
let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))" 

621 

49776  622 
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)) = 
623 
(\<Prod>i\<in>insert i I. emeasure (M i) (A i))" 

624 
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) 

625 
fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))" 

626 
then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto 

627 
let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)" 

628 
let ?p' = "prod_emb I M (J  {i}) (\<Pi>\<^isub>E j\<in>J{i}. E j)" 

629 
have "?\<mu> ?p = 

630 
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))" 

631 
by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+ 

632 
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))" 

633 
using J E[rule_format, THEN sets_into_space] 

634 
by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm) 

635 
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))) = 

636 
emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))" 

637 
using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto 

638 
also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J{i} then E j else space (M j))" 

639 
using J E[rule_format, THEN sets_into_space] 

640 
by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+ 

641 
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)) = 

642 
(\<Prod> j\<in>I. if j \<in> J{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))" 

643 
using E by (subst insert) (auto intro!: setprod_cong) 

644 
also have "(\<Prod>j\<in>I. if j \<in> J  {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) * 

645 
emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)" 

646 
using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong) 

647 
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)" 

648 
using insert(1,2) J E by (intro setprod_mono_one_right) auto 

649 
finally show "?\<mu> ?p = \<dots>" . 

47694  650 

49776  651 
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))" 
652 
using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff) 

653 
next 

654 
show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" 

655 
using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all 

656 
next 

657 
show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and> 

658 
insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))" 

659 
using insert by auto 

660 
qed (auto intro!: setprod_cong) 

661 
with insert show ?case 

662 
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space) 

663 
qed 

47694  664 

49776  665 
lemma (in product_sigma_finite) sigma_finite: 
666 
assumes "finite I" 

667 
shows "sigma_finite_measure (PiM I M)" 

668 
proof  

669 
interpret finite_product_sigma_finite M I by default fact 

670 

671 
from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. 

672 
then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)" 

673 
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)" 

674 
"(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)" 

675 
"\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>" 

47694  676 
by blast+ 
49776  677 
let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k" 
47694  678 

49776  679 
show ?thesis 
47694  680 
proof (unfold_locales, intro exI[of _ ?F] conjI allI) 
49776  681 
show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto 
47694  682 
next 
49776  683 
from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp 
47694  684 
next 
685 
fix j 

49776  686 
from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M] 
687 
show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>" 

688 
by (subst emeasure_PiM) auto 

40859  689 
qed 
690 
qed 

691 

47694  692 
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M" 
693 
using sigma_finite[OF finite_index] . 

40859  694 

695 
lemma (in finite_product_sigma_finite) measure_times: 

47694  696 
"(\<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))" 
697 
using emeasure_PiM[OF finite_index] by auto 

41096  698 

699 
lemma (in product_sigma_finite) product_measure_empty[simp]: 

47694  700 
"emeasure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1" 
41096  701 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

702 
interpret finite_product_sigma_finite M "{}" by default auto 
41096  703 
from measure_times[of "\<lambda>x. {}"] show ?thesis by simp 
704 
qed 

705 

47694  706 
lemma 
707 
shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}" 

708 
and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }" 

709 
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

710 

40859  711 
lemma (in product_sigma_finite) positive_integral_empty: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

712 
assumes pos: "0 \<le> f (\<lambda>k. undefined)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

713 
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)" 
40859  714 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

715 
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) 
47694  716 
have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" 
40859  717 
using assms by (subst measure_times) auto 
718 
then show ?thesis 

47694  719 
unfolding positive_integral_def simple_function_def simple_integral_def[abs_def] 
720 
proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

721 
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset

722 
by (intro SUP_upper) (auto simp: le_fun_def split: split_max) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

723 
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset

724 
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) 
40859  725 
qed 
726 
qed 

727 

47694  728 
lemma (in product_sigma_finite) distr_merge: 
40859  729 
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" 
47694  730 
shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (\<lambda>(x,y). merge I x J y) = Pi\<^isub>M (I \<union> J) M" 
731 
(is "?D = ?P") 

40859  732 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

733 
interpret I: finite_product_sigma_finite M I by default fact 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

734 
interpret J: finite_product_sigma_finite M J by default fact 
40859  735 
have "finite (I \<union> J)" using fin by auto 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

736 
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact 
47694  737 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
41661  738 
let ?g = "\<lambda>(x,y). merge I x J y" 
47694  739 

41661  740 
from IJ.sigma_finite_pairs obtain F where 
741 
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

742 
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)" 
47694  743 
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P" 
744 
"\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>" 

41661  745 
by auto 
746 
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k" 

47694  747 

748 
show ?thesis 

749 
proof (rule measure_eqI_generator_eq[symmetric]) 

750 
show "Int_stable (prod_algebra (I \<union> J) M)" 

751 
by (rule Int_stable_prod_algebra) 

752 
show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))" 

753 
by (rule prod_algebra_sets_into_space) 

754 
show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" 

755 
by (rule sets_PiM) 

756 
then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" 

757 
by simp 

758 

759 
show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F 

760 
using fin by (auto simp: prod_algebra_eq_finite) 

761 
show "incseq ?F" by fact 

762 
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))" 

763 
using F(3) by (simp add: space_PiM) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

764 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

765 
fix k 
47694  766 
from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M] 
767 
show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto 

41661  768 
next 
47694  769 
fix A assume A: "A \<in> prod_algebra (I \<union> J) M" 
770 
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)" 

771 
by (auto simp add: prod_algebra_eq_finite) 

772 
let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M" 

773 
let ?X = "?g ` A \<inter> space ?B" 

774 
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)" 

775 
using F[rule_format, THEN sets_into_space] by (auto simp: space_PiM) 

776 
then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)" 

777 
unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM) 

778 
have "emeasure ?D A = emeasure ?B ?X" 

779 
using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM) 

780 
also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))" 

781 
using `finite J` `finite I` F X 

49776  782 
by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff) 
47694  783 
also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))" 
41661  784 
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one) 
47694  785 
also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)" 
41661  786 
using `finite J` `finite I` F unfolding A 
787 
by (intro IJ.measure_times[symmetric]) auto 

47694  788 
finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp 
789 
qed 

41661  790 
qed 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

791 

bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

792 
lemma (in product_sigma_finite) product_positive_integral_fold: 
47694  793 
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

794 
and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

795 
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f = 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

796 
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

797 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

798 
interpret I: finite_product_sigma_finite M I by default fact 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

799 
interpret J: finite_product_sigma_finite M J by default fact 
41831  800 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
47694  801 
have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" 
49776  802 
using measurable_comp[OF measurable_merge f] by (simp add: comp_def) 
41661  803 
show ?thesis 
47694  804 
apply (subst distr_merge[OF IJ, symmetric]) 
49776  805 
apply (subst positive_integral_distr[OF measurable_merge f]) 
47694  806 
apply (subst P.positive_integral_fst_measurable(2)[symmetric, OF P_borel]) 
807 
apply simp 

808 
done 

40859  809 
qed 
810 

47694  811 
lemma (in product_sigma_finite) distr_singleton: 
812 
"distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _") 

813 
proof (intro measure_eqI[symmetric]) 

41831  814 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
47694  815 
fix A assume A: "A \<in> sets (M i)" 
816 
moreover then have "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)" 

817 
using sets_into_space by (auto simp: space_PiM) 

818 
ultimately show "emeasure (M i) A = emeasure ?D A" 

819 
using A I.measure_times[of "\<lambda>_. A"] 

820 
by (simp add: emeasure_distr measurable_component_singleton) 

821 
qed simp 

41831  822 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

823 
lemma (in product_sigma_finite) product_positive_integral_singleton: 
40859  824 
assumes f: "f \<in> borel_measurable (M i)" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

825 
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f" 
40859  826 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

827 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
47694  828 
from f show ?thesis 
829 
apply (subst distr_singleton[symmetric]) 

830 
apply (subst positive_integral_distr[OF measurable_component_singleton]) 

831 
apply simp_all 

832 
done 

40859  833 
qed 
834 

41096  835 
lemma (in product_sigma_finite) product_positive_integral_insert: 
836 
assumes [simp]: "finite I" "i \<notin> I" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

837 
and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

838 
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))" 
41096  839 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

840 
interpret I: finite_product_sigma_finite M I by default auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

841 
interpret i: finite_product_sigma_finite M "{i}" by default auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

842 
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

843 
using f by auto 
41096  844 
show ?thesis 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

845 
unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f] 
47694  846 
proof (rule positive_integral_cong, subst product_positive_integral_singleton) 
847 
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" 

46731  848 
let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))" 
41096  849 
have f'_eq: "\<And>y. ?f y = f (x(i := y))" 
47694  850 
using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def space_PiM) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

851 
show "?f \<in> borel_measurable (M i)" unfolding f'_eq 
47694  852 
using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`] 
853 
unfolding comp_def . 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

854 
show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i" 
41096  855 
unfolding f'_eq by simp 
856 
qed 

857 
qed 

858 

859 
lemma (in product_sigma_finite) product_positive_integral_setprod: 

43920  860 
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" 
41096  861 
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

862 
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

863 
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))" 
41096  864 
using assms proof induct 
865 
case (insert i I) 

866 
note `finite I`[intro, simp] 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

867 
interpret I: finite_product_sigma_finite M I by default auto 
41096  868 
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))" 
869 
using insert by (auto intro!: setprod_cong) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

870 
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)" 
41096  871 
using sets_into_space insert 
47694  872 
by (intro borel_measurable_ereal_setprod 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

873 
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) 
41096  874 
auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

875 
then show ?case 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

876 
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod]) 
47694  877 
apply (simp add: insert(2) * pos borel setprod_ereal_pos positive_integral_multc) 
878 
apply (subst positive_integral_cmult) 

879 
apply (auto simp add: pos borel insert(2) setprod_ereal_pos positive_integral_positive) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

880 
done 
47694  881 
qed (simp add: space_PiM) 
41096  882 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

883 
lemma (in product_sigma_finite) product_integral_singleton: 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

884 
assumes f: "f \<in> borel_measurable (M i)" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

885 
shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

886 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

887 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
43920  888 
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)" 
889 
"(\<lambda>x. ereal ( f x)) \<in> borel_measurable (M i)" 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

890 
using assms by auto 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

891 
show ?thesis 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

892 
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] .. 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

893 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

894 

bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

895 
lemma (in product_sigma_finite) product_integral_fold: 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

896 
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

897 
and f: "integrable (Pi\<^isub>M (I \<union> J) M) f" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

898 
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)" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

899 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

900 
interpret I: finite_product_sigma_finite M I by default fact 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

901 
interpret J: finite_product_sigma_finite M J by default fact 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

902 
have "finite (I \<union> J)" using fin by auto 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

903 
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact 
47694  904 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

905 
let ?M = "\<lambda>(x, y). merge I x J y" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

906 
let ?f = "\<lambda>x. f (?M x)" 
47694  907 
from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" 
908 
by auto 

909 
have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" 

49776  910 
using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def) 
47694  911 
have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f" 
49776  912 
by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f) 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

913 
show ?thesis 
47694  914 
apply (subst distr_merge[symmetric, OF IJ fin]) 
49776  915 
apply (subst integral_distr[OF measurable_merge f_borel]) 
47694  916 
apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int]) 
917 
apply simp 

918 
done 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

919 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

920 

49776  921 
lemma (in product_sigma_finite) 
922 
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" 

923 
shows emeasure_fold_integral: 

924 
"emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) (merge I x J ` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I) 

925 
and emeasure_fold_measurable: 

926 
"(\<lambda>x. emeasure (Pi\<^isub>M J M) (merge I x J ` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B) 

927 
proof  

928 
interpret I: finite_product_sigma_finite M I by default fact 

929 
interpret J: finite_product_sigma_finite M J by default fact 

930 
interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" .. 

931 
have merge: "(\<lambda>(x, y). merge I x J y) ` 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)" 

932 
by (intro measurable_sets[OF _ A] measurable_merge assms) 

933 

934 
show ?I 

935 
apply (subst distr_merge[symmetric, OF IJ]) 

936 
apply (subst emeasure_distr[OF measurable_merge A]) 

937 
apply (subst J.emeasure_pair_measure_alt[OF merge]) 

938 
apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure) 

939 
done 

940 

941 
show ?B 

942 
using IJ.measurable_emeasure_Pair1[OF merge] 

943 
by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong) 

944 
qed 

945 

41096  946 
lemma (in product_sigma_finite) product_integral_insert: 
47694  947 
assumes I: "finite I" "i \<notin> I" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

948 
and f: "integrable (Pi\<^isub>M (insert i I) M) f" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

949 
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)" 
41096  950 
proof  
47694  951 
have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f" 
952 
by simp 

953 
also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I x {i} y) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)" 

954 
using f I by (intro product_integral_fold) auto 

955 
also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)" 

956 
proof (rule integral_cong, subst product_integral_singleton[symmetric]) 

957 
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" 

958 
have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" 

959 
using f by auto 

960 
show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)" 

961 
using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`] 

962 
unfolding comp_def . 

963 
from x I show "(\<integral> y. f (merge I x {i} y) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)" 

964 
by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def) 

41096  965 
qed 
47694  966 
finally show ?thesis . 
41096  967 
qed 
968 

969 
lemma (in product_sigma_finite) product_integrable_setprod: 

970 
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

971 
assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

972 
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f") 
41096  973 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

974 
interpret finite_product_sigma_finite M I by default fact 
41096  975 
have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

976 
using integrable unfolding integrable_def by auto 
47694  977 
have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)" 
978 
using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def) 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

979 
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)" 
41096  980 
proof (unfold integrable_def, intro conjI) 
47694  981 
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)" 
41096  982 
using borel by auto 
47694  983 
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)" 
43920  984 
by (simp add: setprod_ereal abs_setprod) 
985 
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))" 

41096  986 
using f by (subst product_positive_integral_setprod) auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

987 
also have "\<dots> < \<infinity>" 
47694  988 
using integrable[THEN integrable_abs] 
989 
by (simp add: setprod_PInf integrable_def positive_integral_positive) 

990 
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto 

991 
have "(\<integral>\<^isup>+x. ereal ( abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

992 
by (intro positive_integral_cong_pos) auto 
47694  993 
then show "(\<integral>\<^isup>+x. ereal ( abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp 
41096  994 
qed 
995 
ultimately show ?thesis 

996 
by (rule integrable_abs_iff[THEN iffD1]) 

997 
qed 

998 

999 
lemma (in product_sigma_finite) product_integral_setprod: 

1000 
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

1001 
assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

1002 
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))" 
41096  1003 
using assms proof (induct rule: finite_ne_induct) 
1004 
case (singleton i) 

1005 
then show ?case by (simp add: product_integral_singleton integrable_def) 

1006 
next 

1007 
case (insert i I) 

1008 
then have iI: "finite (insert i I)" by auto 

1009 
then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

1010 
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))" 
41096  1011 
by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

1012 
interpret I: finite_product_sigma_finite M I by default fact 
41096  1013 
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))" 
1014 
using `i \<notin> I` by (auto intro!: setprod_cong) 

1015 
show ?case 

1016 
unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]] 

47694  1017 
by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI) 
41096  1018 
qed 
1019 

49776  1020 
lemma sets_Collect_single: 
1021 
"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)" 

1022 
unfolding sets_PiM_single 

1023 
by (auto intro!: sigma_sets.Basic exI[of _ i] exI[of _ A]) (auto simp: space_PiM) 

1024 

1025 
lemma sigma_prod_algebra_sigma_eq_infinite: 

1026 
fixes E :: "'i \<Rightarrow> 'a set set" 

1027 
assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)" 

1028 
and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 

1029 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 

1030 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

1031 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

1032 
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}" 

1033 
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" 

1034 
proof 

1035 
let ?P = "sigma (space (Pi\<^isub>M I M)) P" 

1036 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" 

1037 
using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq) 

1038 
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" 

1039 
by (simp add: space_PiM) 

1040 
have "sets (PiM I M) = 

1041 
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)}" 

1042 
using sets_PiM_single[of I M] by (simp add: space_P) 

1043 
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" 

1044 
proof (safe intro!: sigma_sets_subset) 

1045 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 

1046 
then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

1047 
apply (subst measurable_iff_measure_of) 

1048 
apply (simp_all add: P_closed) 

1049 
using E_closed 

1050 
apply (force simp: subset_eq space_PiM) 

1051 
apply (force simp: subset_eq space_PiM) 

1052 
apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i]) 

1053 
apply (rule_tac x=Aa in exI) 

1054 
apply (auto simp: space_PiM) 

1055 
done 

1056 
from measurable_sets[OF this, of A] A `i \<in> I` E_closed 

1057 
have "(\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 

1058 
by (simp add: E_generates) 

1059 
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}" 

1060 
using P_closed by (auto simp: space_PiM) 

1061 
finally show "\<dots> \<in> sets ?P" . 

1062 
qed 

1063 
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" 

1064 
by (simp add: P_closed) 

1065 
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" 

1066 
unfolding P_def space_PiM[symmetric] 

1067 
by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single) 

1068 
qed 

1069 

47694  1070 
lemma sigma_prod_algebra_sigma_eq: 
1071 
fixes E :: "'i \<Rightarrow> 'a set set" 

1072 
assumes "finite I" 

1073 
assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)" 

1074 
and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 

1075 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 

1076 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

1077 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

1078 
defines "P == { Pi\<^isub>E I F  F. \<forall>i\<in>I. F i \<in> E i }" 

1079 
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" 

1080 
proof 

1081 
let ?P = "sigma (space (Pi\<^isub>M I M)) P" 

1082 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" 

1083 
using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq) 

1084 
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" 

1085 
by (simp add: space_PiM) 

1086 
have "sets (PiM I M) = 

1087 
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)}" 

1088 
using sets_PiM_single[of I M] by (simp add: space_P) 

1089 
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" 

1090 
proof (safe intro!: sigma_sets_subset) 

1091 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 

1092 
have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

1093 
proof (subst measurable_iff_measure_of) 

1094 
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact 

1095 
from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" 

1096 
by (auto simp: Pi_iff) 

1097 
show "\<forall>A\<in>E i. (\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 

1098 
proof 

1099 
fix A assume A: "A \<in> E i" 

1100 
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))" 

1101 
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm) 

1102 
also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)" 

1103 
by (intro PiE_cong) (simp add: S_union) 

1104 
also have "\<dots> = (\<Union>n. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j n)" 

1105 
using S_mono 

1106 
by (subst Pi_UN[symmetric, OF `finite I`]) (auto simp: incseq_def) 

1107 
also have "\<dots> \<in> sets ?P" 

1108 
proof (safe intro!: countable_UN) 

1109 
fix n show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j n) \<in> sets ?P" 

1110 
using A S_in_E 

1111 
by (simp add: P_closed) 

1112 
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j n"]) 

1113 
qed 

1114 
finally show "(\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 

1115 
using P_closed by simp 

1116 
qed 

1117 
qed 

1118 
from measurable_sets[OF this, of A] A `i \<in> I` E_closed 

1119 
have "(\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 

1120 
by (simp add: E_generates) 

1121 
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}" 

1122 
using P_closed by (auto simp: space_PiM) 

1123 
finally show "\<dots> \<in> sets ?P" . 

1124 
qed 

1125 
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" 

1126 
by (simp add: P_closed) 

1127 
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" 

1128 
using `finite I` 

1129 
by (auto intro!: sigma_sets_subset simp: E_generates P_def) 

1130 
qed 

1131 

1132 
end 