author  hoelzl 
Tue, 19 Jul 2011 14:36:12 +0200  
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parent 42988  d8f3fc934ff6 
child 44890  22f665a2e91c 
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 
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begin 
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lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)" 
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unfolding Pi_def by auto 
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abbreviation 
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"Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A" 

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syntax 
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"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PIE _:_./ _)" 10) 
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syntax (xsymbols) 
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"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) 
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syntax (HTML output) 
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"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) 
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translations 
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"PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)" 
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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 

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notation (xsymbols) 

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

36 

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lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}" 

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by safe (auto simp add: extensional_def fun_eq_iff) 

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lemma extensional_insert[intro, simp]: 

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assumes "a \<in> extensional (insert i I)" 

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shows "a(i := b) \<in> extensional (insert i I)" 

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using assms unfolding extensional_def by auto 

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lemma extensional_Int[simp]: 

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

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unfolding extensional_def by auto 

38656  48 

35833  49 
definition 
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"merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)" 
51 

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lemma merge_apply[simp]: 

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"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i" 

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"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i" 

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"J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i" 

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"J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i" 

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"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined" 

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unfolding merge_def by auto 

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lemma merge_commute: 

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"I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x" 

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by (auto simp: merge_def intro!: ext) 

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lemma Pi_cancel_merge_range[simp]: 

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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A" 

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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A" 

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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A" 

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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A" 

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by (auto simp: Pi_def) 

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lemma Pi_cancel_merge[simp]: 

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"I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" 

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"J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" 

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"I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" 

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"J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" 

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by (auto simp: Pi_def) 

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lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)" 

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by (auto simp: extensional_def) 

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lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A" 

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by (auto simp: restrict_def Pi_def) 

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lemma restrict_merge[simp]: 

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"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I" 

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"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J" 

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"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I" 

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"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J" 

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by (auto simp: restrict_def intro!: ext) 

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lemma extensional_insert_undefined[intro, simp]: 

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assumes "a \<in> extensional (insert i I)" 

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shows "a(i := undefined) \<in> extensional I" 

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using assms unfolding extensional_def by auto 

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lemma extensional_insert_cancel[intro, simp]: 

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assumes "a \<in> extensional I" 

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shows "a \<in> extensional (insert i I)" 

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using assms unfolding extensional_def by auto 

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lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)" 
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unfolding merge_def by (auto simp: fun_eq_iff) 

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

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by auto 

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lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)" 
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by auto 

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lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B" 

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by (auto simp: Pi_def) 

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lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i" 

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by (auto simp: Pi_def) 

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lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X" 
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by (auto simp: Pi_def) 

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lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" 

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by (auto simp: Pi_def) 

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lemma restrict_vimage: 
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assumes "I \<inter> J = {}" 

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shows "(\<lambda>x. (restrict x I, restrict x J)) ` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I E J F)" 

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using assms by (auto simp: restrict_Pi_cancel) 

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lemma merge_vimage: 

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assumes "I \<inter> J = {}" 

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shows "(\<lambda>(x,y). merge I x J y) ` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E" 

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using assms by (auto simp: restrict_Pi_cancel) 

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lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I" 

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by (auto simp: restrict_def intro!: ext) 

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lemma merge_restrict[simp]: 

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"merge I (restrict x I) J y = merge I x J y" 

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"merge I x J (restrict y J) = merge I x J y" 

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unfolding merge_def by (auto intro!: ext) 

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lemma merge_x_x_eq_restrict[simp]: 

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"merge I x J x = restrict x (I \<union> J)" 

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unfolding merge_def by (auto intro!: ext) 

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lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I  {i}) B \<and> f i \<in> A" 

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apply auto 

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apply (drule_tac x=x in Pi_mem) 

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apply (simp_all split: split_if_asm) 

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apply (drule_tac x=i in Pi_mem) 

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apply (auto dest!: Pi_mem) 

150 
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)" 

156 
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)" 

163 
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] 

166 
show "f i \<in> A k i" by auto 

167 
qed 

168 
then show "f \<in> (\<Union>n. Pi I (A n))" by auto 

169 
qed auto 

170 

171 
lemma PiE_cong: 

172 
assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i" 

173 
shows "Pi\<^isub>E I A = Pi\<^isub>E I B" 

174 
using assms by (auto intro!: Pi_cong) 

175 

176 
lemma restrict_upd[simp]: 

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

178 
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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223 
then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})" 
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224 
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto 
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225 
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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226 
next 
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227 
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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228 
then show "Pi\<^isub>E I F = Pi\<^isub>E I F'" 
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229 
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto 
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230 
qed 
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231 

40859  232 
section "Finite product spaces" 
233 

234 
section "Products" 

235 

236 
locale product_sigma_algebra = 

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237 
fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" 
40859  238 
assumes sigma_algebras: "\<And>i. sigma_algebra (M i)" 
239 

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240 
locale finite_product_sigma_algebra = product_sigma_algebra M 
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241 
for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" + 
40859  242 
fixes I :: "'i set" 
243 
assumes finite_index: "finite I" 

244 

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245 
definition 
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246 
"product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)), 
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247 
sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)), 
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248 
measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>" 
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249 

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250 
definition product_algebra_def: 
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251 
"Pi\<^isub>M I M = sigma (product_algebra_generator I M) 
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252 
\<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and> 
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253 
(\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>" 
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254 

40859  255 
syntax 
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256 
"_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] => 
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257 
('i => 'a, 'b) measure_space_scheme" ("(3PIM _:_./ _)" 10) 
40859  258 

259 
syntax (xsymbols) 

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260 
"_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] => 
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261 
('i => 'a, 'b) measure_space_scheme" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) 
40859  262 

263 
syntax (HTML output) 

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264 
"_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] => 
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265 
('i => 'a, 'b) measure_space_scheme" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) 
40859  266 

267 
translations 

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268 
"PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)" 
40859  269 

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270 
abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M" 
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271 
abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M" 
40859  272 

273 
sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras) 

274 

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275 
lemma sigma_into_space: 
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276 
assumes "sets M \<subseteq> Pow (space M)" 
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277 
shows "sets (sigma M) \<subseteq> Pow (space M)" 
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278 
using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto 
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279 

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280 
lemma (in product_sigma_algebra) product_algebra_generator_into_space: 
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281 
"sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))" 
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282 
using M.sets_into_space unfolding product_algebra_generator_def 
40859  283 
by auto blast 
284 

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285 
lemma (in product_sigma_algebra) product_algebra_into_space: 
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286 
"sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))" 
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287 
using product_algebra_generator_into_space 
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288 
by (auto intro!: sigma_into_space simp add: product_algebra_def) 
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289 

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290 
lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)" 
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291 
using product_algebra_generator_into_space unfolding product_algebra_def 
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292 
by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all 
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293 

40859  294 
sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P 
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295 
using sigma_algebra_product_algebra . 
40859  296 

41095  297 
lemma product_algebraE: 
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298 
assumes "A \<in> sets (product_algebra_generator I M)" 
41095  299 
obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" 
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300 
using assms unfolding product_algebra_generator_def by auto 
41095  301 

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302 
lemma product_algebra_generatorI[intro]: 
41095  303 
assumes "E \<in> (\<Pi> i\<in>I. sets (M i))" 
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304 
shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)" 
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305 
using assms unfolding product_algebra_generator_def by auto 
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306 

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307 
lemma space_product_algebra_generator[simp]: 
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308 
"space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))" 
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309 
unfolding product_algebra_generator_def by simp 
41095  310 

40859  311 
lemma space_product_algebra[simp]: 
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312 
"space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
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313 
unfolding product_algebra_def product_algebra_generator_def by simp 
40859  314 

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315 
lemma sets_product_algebra: 
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316 
"sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))" 
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317 
unfolding product_algebra_def sigma_def by simp 
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318 

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319 
lemma product_algebra_generator_sets_into_space: 
41095  320 
assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))" 
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321 
shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))" 
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322 
using assms by (auto simp: product_algebra_generator_def) blast 
40859  323 

324 
lemma (in finite_product_sigma_algebra) in_P[simp, intro]: 

325 
"\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P" 

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326 
by (auto simp: sets_product_algebra) 
41026
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folding on arbitrary Lebesgue integrable functions
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327 

42988  328 
lemma Int_stable_product_algebra_generator: 
329 
"(\<And>i. i \<in> I \<Longrightarrow> Int_stable (M i)) \<Longrightarrow> Int_stable (product_algebra_generator I M)" 

330 
by (auto simp add: product_algebra_generator_def Int_stable_def PiE_Int Pi_iff) 

331 

40859  332 
section "Generating set generates also product algebra" 
333 

334 
lemma sigma_product_algebra_sigma_eq: 

335 
assumes "finite I" 

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336 
assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)" 
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337 
assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)" 
40859  338 
assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)" 
339 
and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))" 

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340 
shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)" 
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341 
(is "sets ?S = sets ?E") 
40859  342 
proof cases 
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343 
assume "I = {}" then show ?thesis 
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344 
by (simp add: product_algebra_def product_algebra_generator_def) 
40859  345 
next 
346 
assume "I \<noteq> {}" 

347 
interpret E: sigma_algebra "sigma (E i)" for i 

348 
using E by (rule sigma_algebra_sigma) 

349 
have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)" 

350 
using E by auto 

351 
interpret G: sigma_algebra ?E 

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352 
unfolding product_algebra_def product_algebra_generator_def using E 
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353 
by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem) 
40859  354 
{ fix A i assume "i \<in> I" and A: "A \<in> sets (E i)" 
355 
then have "(\<lambda>x. x i) ` A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E" 

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356 
using mono union unfolding incseq_Suc_iff space_product_algebra 
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357 
by (auto dest: Pi_mem) 
40859  358 
also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))" 
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359 
unfolding space_product_algebra 
40859  360 
apply simp 
361 
apply (subst Pi_UN[OF `finite I`]) 

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362 
using mono[THEN incseqD] apply simp 
40859  363 
apply (simp add: PiE_Int) 
364 
apply (intro PiE_cong) 

365 
using A sets_into by (auto intro!: into_space) 

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366 
also have "\<dots> \<in> sets ?E" 
40859  367 
using sets_into `A \<in> sets (E i)` 
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368 
unfolding sets_product_algebra sets_sigma 
40859  369 
by (intro sigma_sets.Union) 
370 
(auto simp: image_subset_iff intro!: sigma_sets.Basic) 

371 
finally have "(\<lambda>x. x i) ` A \<inter> space ?E \<in> sets ?E" . } 

372 
then have proj: 

373 
"\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))" 

374 
using E by (subst G.measurable_iff_sigma) 

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375 
(auto simp: sets_product_algebra sets_sigma) 
40859  376 
{ fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))" 
377 
with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) ` (A i) \<inter> space ?E \<in> sets ?E" 

378 
unfolding measurable_def by simp 

379 
have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) ` (A i) \<inter> space ?E)" 

380 
using A E.sets_into_space `I \<noteq> {}` unfolding product_algebra_def by auto blast 

381 
then have "Pi\<^isub>E I A \<in> sets ?E" 

382 
using G.finite_INT[OF `finite I` `I \<noteq> {}` basic, of "\<lambda>i. i"] by 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

383 
then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E" 
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

384 
by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def) 
40859  385 
then have subset: "sets ?S \<subseteq> sets ?E" 
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

386 
by (simp add: sets_sigma sets_product_algebra) 
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

387 
show "sets ?S = sets ?E" 
40859  388 
proof (intro set_eqI iffI) 
389 
fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S" 

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

390 
unfolding sets_sigma sets_product_algebra 
40859  391 
proof induct 
392 
case (Basic A) then show ?case 

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

393 
by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic) 
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

394 
qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def) 
40859  395 
next 
396 
fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto 

397 
qed 

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

398 
qed 
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

399 

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

400 
lemma product_algebraI[intro]: 
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

401 
"E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M 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

402 
using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI) 
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

403 

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

404 
lemma (in product_sigma_algebra) measurable_component_update: 
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

405 
assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> 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

406 
shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _") 
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

407 
unfolding product_algebra_def apply simp 
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

408 
proof (intro measurable_sigma) 
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

409 
let ?G = "product_algebra_generator (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

410 
show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space . 
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

411 
show "?f \<in> space (M i) \<rightarrow> space ?G" 
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

412 
using M.sets_into_space assms by 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

413 
fix A assume "A \<in> sets ?G" 
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

414 
from product_algebraE[OF this] guess E . note E = this 
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

415 
then have "?f ` A \<inter> space (M i) = E i \<or> ?f ` A \<inter> space (M 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

416 
using M.sets_into_space assms by 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

417 
then show "?f ` A \<inter> space (M i) \<in> sets (M 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

418 
using E by (auto intro!: product_algebraI) 
40859  419 
qed 
420 

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

421 
lemma (in product_sigma_algebra) measurable_add_dim: 
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

422 
assumes "i \<notin> 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

423 
shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)" 
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

424 
proof  
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

425 
let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (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

426 
interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M 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

427 
unfolding pair_sigma_algebra_def 
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

428 
by (intro sigma_algebra_product_algebra sigma_algebras conjI) 
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

429 
have "?f \<in> measurable Ii.P (sigma ?G)" 
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

430 
proof (rule Ii.measurable_sigma) 
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

431 
show "sets ?G \<subseteq> Pow (space ?G)" 
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

432 
using product_algebra_generator_into_space . 
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

433 
show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G" 
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

434 
by (auto simp: space_pair_measure) 
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

435 
next 
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

436 
fix A assume "A \<in> sets ?G" 
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

437 
then obtain F where "A = Pi\<^isub>E (insert i I) 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

438 
and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M 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

439 
by (auto elim!: product_algebraE) 
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

440 
then have "?f ` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)" 
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

441 
using sets_into_space `i \<notin> 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

442 
by (auto simp add: space_pair_measure) blast+ 
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

443 
then show "?f ` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)" 
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

444 
using F by (auto intro!: pair_measureI) 
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

445 
qed 
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

446 
then show ?thesis 
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

447 
by (simp add: product_algebra_def) 
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

448 
qed 
41095  449 

450 
lemma (in product_sigma_algebra) measurable_merge: 

451 
assumes [simp]: "I \<inter> 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

452 
shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)" 
40859  453 
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

454 
let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M 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

455 
interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J" 
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

456 
by (intro sigma_algebra_pair_measure product_algebra_into_space) 
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

457 
let ?f = "\<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

458 
let ?G = "product_algebra_generator (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

459 
have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)" 
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

460 
proof (rule P.measurable_sigma) 
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

461 
fix A assume "A \<in> sets ?G" 
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

462 
from product_algebraE[OF this] 
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

463 
obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M 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

464 
then have *: "?f ` A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E" 
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

465 
using sets_into_space `I \<inter> J = {}` 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

466 
by (auto simp add: space_pair_measure) fast+ 
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

467 
then show "?f ` A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)" 
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

468 
using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI 
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

469 
simp: product_algebra_def) 
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

470 
qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure) 
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

471 
then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (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

472 
unfolding product_algebra_def[of "I \<union> J"] by simp 
40859  473 
qed 
474 

41095  475 
lemma (in product_sigma_algebra) measurable_component_singleton: 
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

476 
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M 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

477 
proof (unfold measurable_def, intro CollectI conjI ballI) 
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

478 
fix A assume "A \<in> sets (M 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

479 
then have "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))" 
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

480 
using M.sets_into_space `i \<in> I` by (fastsimp dest: Pi_mem split: split_if_asm) 
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

481 
then show "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M 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

482 
using `A \<in> sets (M i)` by (auto intro!: product_algebraI) 
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

483 
qed (insert `i \<in> I`, auto) 
41661  484 

42988  485 
lemma (in sigma_algebra) measurable_restrict: 
486 
assumes I: "finite I" 

487 
assumes "\<And>i. i \<in> I \<Longrightarrow> sets (N i) \<subseteq> Pow (space (N i))" 

488 
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (N i)" 

489 
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)" 

490 
unfolding product_algebra_def 

491 
proof (simp, rule measurable_sigma) 

492 
show "sets (product_algebra_generator I N) \<subseteq> Pow (space (product_algebra_generator I N))" 

493 
by (rule product_algebra_generator_sets_into_space) fact 

494 
show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> space M \<rightarrow> space (product_algebra_generator I N)" 

495 
using X by (auto simp: measurable_def) 

496 
fix E assume "E \<in> sets (product_algebra_generator I N)" 

497 
then obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (N i)" 

498 
by (auto simp: product_algebra_generator_def) 

499 
then have "(\<lambda>x. \<lambda>i\<in>I. X i x) ` E \<inter> space M = (\<Inter>i\<in>I. X i ` F i \<inter> space M) \<inter> space M" 

500 
by (auto simp: Pi_iff) 

501 
also have "\<dots> \<in> sets M" 

502 
proof cases 

503 
assume "I = {}" then show ?thesis by simp 

504 
next 

505 
assume "I \<noteq> {}" with X F I show ?thesis 

506 
by (intro finite_INT measurable_sets Int) auto 

507 
qed 

508 
finally show "(\<lambda>x. \<lambda>i\<in>I. X i x) ` E \<inter> space M \<in> sets M" . 

509 
qed 

510 

40859  511 
locale product_sigma_finite = 
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

512 
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" 
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

513 
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" 
40859  514 

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

515 
locale finite_product_sigma_finite = product_sigma_finite 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

516 
for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" + 
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

517 
fixes I :: "'i set" assumes finite_index'[intro]: "finite I" 
40859  518 

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

519 
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i 
40859  520 
by (rule sigma_finite_measures) 
521 

522 
sublocale product_sigma_finite \<subseteq> product_sigma_algebra 

523 
by default 

524 

525 
sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra 

526 
by default (fact finite_index') 

527 

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

528 
lemma (in finite_product_sigma_finite) product_algebra_generator_measure: 
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

529 
assumes "Pi\<^isub>E I F \<in> sets G" 
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

530 
shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> 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

531 
proof cases 
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

532 
assume ne: "\<forall>i\<in>I. F i \<noteq> {}" 
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

533 
have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') 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

534 
by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I 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

535 
(insert ne, auto simp: Pi_eq_iff) 
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

536 
then show ?thesis 
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

537 
unfolding product_algebra_generator_def by simp 
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

538 
next 
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

539 
assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})" 
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

540 
then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0" 
43920  541 
by (auto simp: setprod_ereal_0 intro!: bexI) 
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

542 
moreover 
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

543 
have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}" 
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

544 
by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I 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

545 
(insert empty, auto simp: Pi_eq_empty_iff[symmetric]) 
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

546 
then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0" 
43920  547 
by (auto simp: setprod_ereal_0 intro!: bexI) 
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

548 
ultimately show ?thesis 
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 
unfolding product_algebra_generator_def by simp 
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

550 
qed 
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

551 

40859  552 
lemma (in finite_product_sigma_finite) sigma_finite_pairs: 
553 
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. 

554 
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> 

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

555 
(\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

556 
(\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G" 
40859  557 
proof  
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

558 
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)" 
40859  559 
using M.sigma_finite_up by simp 
560 
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. 

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

561 
then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>" 
40859  562 
by auto 
563 
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k" 

564 
note space_product_algebra[simp] 

565 
show ?thesis 

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

566 
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) 
40859  567 
fix i show "range (F i) \<subseteq> sets (M i)" by fact 
568 
next 

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

569 
fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact 
40859  570 
next 
571 
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G" 

41831  572 
using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space 
573 
by (force simp: image_subset_iff) 

40859  574 
next 
575 
fix f assume "f \<in> space G" 

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

576 
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

577 
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def) 
40859  578 
next 
579 
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

580 
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto 
40859  581 
qed 
582 
qed 

583 

41831  584 
lemma sets_pair_cancel_measure[simp]: 
585 
"sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)" 

586 
"sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)" 

587 
unfolding pair_measure_def pair_measure_generator_def sets_sigma 

588 
by simp_all 

589 

590 
lemma measurable_pair_cancel_measure[simp]: 

591 
"measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M" 

592 
"measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M" 

593 
"measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)" 

594 
"measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)" 

595 
unfolding measurable_def by (auto simp add: space_pair_measure) 

596 

40859  597 
lemma (in product_sigma_finite) product_measure_exists: 
598 
assumes "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

599 
shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and> 
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

600 
(\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))" 
40859  601 
using `finite I` proof 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

602 
case empty 
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

603 
interpret finite_product_sigma_finite M "{}" by default simp 
43920  604 
let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> ereal" 
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

605 
show ?case 
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

606 
proof (intro exI conjI ballI) 
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

607 
have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)" 
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

608 
by (rule sigma_algebra_cong) (simp_all add: product_algebra_def) 
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

609 
then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)" 
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

610 
by (rule finite_additivity_sufficient) 
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

611 
(simp_all add: positive_def additive_def sets_sigma 
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 
product_algebra_generator_def image_constant) 
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

613 
then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)" 
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 
by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"] 
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

615 
simp: sigma_finite_measure_def sigma_finite_measure_axioms_def 
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

616 
product_algebra_generator_def) 
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

617 
qed auto 
40859  618 
next 
619 
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

620 
interpret finite_product_sigma_finite M I by default fact 
40859  621 
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

622 
interpret I': finite_product_sigma_finite M "insert i I" by default fact 
40859  623 
from insert obtain \<nu> where 
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

624 
prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and 
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

625 
"sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by 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

626 
then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp 
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

627 
interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" .. 
41661  628 
let ?h = "(\<lambda>(f, y). f(i := y))" 
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

629 
let ?\<nu> = "\<lambda>A. P.\<mu> (?h ` A \<inter> space P.P)" 
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

630 
have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)" 
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

631 
by (rule I'.sigma_algebra_cong) simp_all 
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

632 
interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>" 
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

633 
using measurable_add_dim[OF `i \<notin> I`] 
41831  634 
by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def) 
40859  635 
show ?case 
636 
proof (intro exI[of _ ?\<nu>] conjI ballI) 

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

637 
let "?m A" = "measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h ` A \<inter> space P.P)" 
40859  638 
{ fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))" 
41661  639 
then have *: "?h ` Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A 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

640 
using `i \<notin> I` M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast 
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

641 
show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))" 
41661  642 
unfolding * using A 
40859  643 
apply (subst P.pair_measure_times) 
41661  644 
using A apply fastsimp 
645 
using A apply fastsimp 

646 
using `i \<notin> I` `finite I` prod[of A] A by (auto simp: ac_simps) } 

40859  647 
note product = this 
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

648 
have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>" 
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

649 
by (simp add: product_algebra_def) 
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

650 
show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)" 
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

651 
proof (unfold *, default, simp) 
40859  652 
from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. 
653 
then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)" 

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

654 
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

655 
"(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

656 
"\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>" 
40859  657 
by blast+ 
658 
let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k" 

659 
show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and> 

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

660 
(\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)" 
40859  661 
proof (intro exI[of _ ?F] conjI allI) 
662 
show "range ?F \<subseteq> sets I'.P" using F(1) by auto 

663 
next 

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

664 
from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp 
40859  665 
next 
666 
fix j 

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

667 
have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

668 
using F(1) by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

669 
with F `finite I` setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

670 
by (subst product) auto 
40859  671 
qed 
672 
qed 

673 
qed 

674 
qed 

675 

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

676 
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P 
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

677 
unfolding product_algebra_def 
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

678 
using product_measure_exists[OF finite_index] 
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

679 
by (rule someI2_ex) auto 
40859  680 

681 
lemma (in finite_product_sigma_finite) measure_times: 

682 
assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (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

683 
shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A 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

684 
unfolding product_algebra_def 
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

685 
using product_measure_exists[OF finite_index] 
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

686 
proof (rule someI2_ex, elim conjE) 
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

687 
fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" 
40859  688 
have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem) 
689 
then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by 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

690 
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by 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

691 
finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A 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

692 
by (simp add: sigma_def) 
40859  693 
qed 
41096  694 

695 
lemma (in product_sigma_finite) product_measure_empty[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

696 
"measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1" 
41096  697 
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

698 
interpret finite_product_sigma_finite M "{}" by default auto 
41096  699 
from measure_times[of "\<lambda>x. {}"] show ?thesis by simp 
700 
qed 

701 

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 
lemma (in finite_product_sigma_algebra) P_empty: 
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

703 
assumes "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

704 
shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }" 
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

705 
unfolding product_algebra_def product_algebra_generator_def `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

706 
by (simp_all add: sigma_def image_constant) 
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

707 

40859  708 
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

709 
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

710 
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)" 
40859  711 
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

712 
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) 
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

713 
have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" 
40859  714 
using assms by (subst measure_times) auto 
715 
then show ?thesis 

40873  716 
unfolding positive_integral_def simple_function_def simple_integral_def_raw 
40859  717 
proof (simp add: P_empty, intro antisym) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

718 
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

719 
by (intro le_SUPI) (auto simp: le_fun_def split: split_max) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

720 
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

721 
by (intro SUP_leI) (auto simp: le_fun_def simp: max_def split: split_if_asm) 
40859  722 
qed 
723 
qed 

724 

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

725 
lemma (in product_sigma_finite) measure_fold: 
40859  726 
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

727 
assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" 
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

728 
shows "measure (Pi\<^isub>M (I \<union> J) M) A = 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

729 
measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) ` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))" 
40859  730 
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

731 
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

732 
interpret J: finite_product_sigma_finite M J by default fact 
40859  733 
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

734 
interpret IJ: finite_product_sigma_finite M "I \<union> J" 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

735 
interpret P: pair_sigma_finite I.P J.P by default 
41661  736 
let ?g = "\<lambda>(x,y). merge I x J y" 
737 
let "?X S" = "?g ` S \<inter> space P.P" 

738 
from IJ.sigma_finite_pairs obtain F where 

739 
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

740 
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

741 
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

742 
"\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>" 
41661  743 
by auto 
744 
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k" 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

745 
show "IJ.\<mu> A = P.\<mu> (?X A)" 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

746 
proof (rule measure_unique_Int_stable_vimage) 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

747 
show "measure_space IJ.P" "measure_space P.P" by default 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

748 
show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)" 
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

749 
using A unfolding product_algebra_def by auto 
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

750 
next 
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

751 
show "Int_stable IJ.G" 
42988  752 
by (rule Int_stable_product_algebra_generator) 
753 
(auto simp: Int_stable_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

754 
show "range ?F \<subseteq> sets IJ.G" using 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

755 
by (simp add: image_subset_iff product_algebra_def 
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

756 
product_algebra_generator_def) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

757 
show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+ 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

758 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

759 
fix k 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

760 
have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

761 
using F(1) by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

762 
with F `finite I` setprod_PInf[of "I \<union> J", OF this] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

763 
show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto 
41661  764 
next 
765 
fix A assume "A \<in> sets IJ.G" 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

766 
then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F" 
41661  767 
and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (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

768 
by (auto simp: product_algebra_generator_def) 
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

769 
then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)" 
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

770 
using sets_into_space by (auto simp: space_pair_measure) blast+ 
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

771 
then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))" 
41661  772 
using `finite J` `finite I` F 
773 
by (simp add: P.pair_measure_times I.measure_times J.measure_times) 

774 
also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))" 

775 
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one) 

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

776 
also have "\<dots> = IJ.\<mu> A" 
41661  777 
using `finite J` `finite I` F unfolding A 
778 
by (intro IJ.measure_times[symmetric]) auto 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset

779 
finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

780 
qed (rule measurable_merge[OF IJ]) 
41661  781 
qed 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

782 

41831  783 
lemma (in product_sigma_finite) measure_preserving_merge: 
784 
assumes IJ: "I \<inter> J = {}" and "finite I" "finite J" 

785 
shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)" 

786 
by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms) 

787 

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

788 
lemma (in product_sigma_finite) product_positive_integral_fold: 
41831  789 
assumes IJ[simp]: "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

790 
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

791 
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

792 
(\<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

793 
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

794 
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

795 
interpret J: finite_product_sigma_finite M J by default fact 
41831  796 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
797 
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp 

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

798 
have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P" 
41831  799 
using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def) 
41661  800 
show ?thesis 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

801 
unfolding P.positive_integral_fst_measurable[OF P_borel, simplified] 
41661  802 
proof (rule P.positive_integral_vimage) 
803 
show "sigma_algebra IJ.P" by default 

41831  804 
show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P" 
805 
using IJ by (rule measure_preserving_merge) 

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

806 
show "f \<in> borel_measurable IJ.P" using f by simp 
41661  807 
qed 
40859  808 
qed 
809 

41831  810 
lemma (in product_sigma_finite) measure_preserving_component_singelton: 
811 
"(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)" 

812 
proof (intro measure_preservingI measurable_component_singleton) 

813 
interpret I: finite_product_sigma_finite M "{i}" by default simp 

814 
fix A let ?P = "(\<lambda>x. x i) ` A \<inter> space I.P" 

815 
assume A: "A \<in> sets (M i)" 

816 
then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto 

817 
show "I.\<mu> ?P = M.\<mu> i A" unfolding * 

818 
using A I.measure_times[of "\<lambda>_. A"] by auto 

819 
qed auto 

820 

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

821 
lemma (in product_sigma_finite) product_positive_integral_singleton: 
40859  822 
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

823 
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f" 
40859  824 
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

825 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
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

826 
show ?thesis 
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 
proof (rule I.positive_integral_vimage[symmetric]) 
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

828 
show "sigma_algebra (M i)" by (rule sigma_algebras) 
41831  829 
show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)" 
830 
by (rule measure_preserving_component_singelton) 

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

831 
show "f \<in> borel_measurable (M i)" by fact 
41661  832 
qed 
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 
41096  842 
interpret P: pair_sigma_algebra I.P i.P .. 
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

843 
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

844 
using f by auto 
41096  845 
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

846 
unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f] 
41096  847 
proof (rule I.positive_integral_cong, subst product_positive_integral_singleton) 
848 
fix x assume x: "x \<in> space I.P" 

849 
let "?f y" = "f (restrict (x(i := y)) (insert i I))" 

850 
have f'_eq: "\<And>y. ?f y = f (x(i := y))" 

851 
using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_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

852 
show "?f \<in> borel_measurable (M i)" unfolding f'_eq 
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

853 
using measurable_comp[OF measurable_component_update f] x `i \<notin> 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

854 
by (simp add: comp_def) 
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

855 
show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i" 
41096  856 
unfolding f'_eq by simp 
857 
qed 

858 
qed 

859 

860 
lemma (in product_sigma_finite) product_positive_integral_setprod: 

43920  861 
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" 
41096  862 
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

863 
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

864 
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  865 
using assms proof induct 
866 
case empty 

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 finite_product_sigma_finite M "{}" by default auto 
41096  868 
then show ?case by simp 
869 
next 

870 
case (insert i I) 

871 
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

872 
interpret I: finite_product_sigma_finite M I by default auto 
41096  873 
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))" 
874 
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

875 
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  876 
using sets_into_space insert 
43920  877 
by (intro sigma_algebra.borel_measurable_ereal_setprod sigma_algebra_product_algebra 
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

878 
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) 
41096  879 
auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

880 
then show ?case 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

881 
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod]) 
43920  882 
apply (simp add: insert * pos borel setprod_ereal_pos M.positive_integral_multc) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

883 
apply (subst I.positive_integral_cmult) 
43920  884 
apply (auto simp add: pos borel insert 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

885 
done 
41096  886 
qed 
887 

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

888 
lemma (in product_sigma_finite) product_integral_singleton: 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

889 
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

890 
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

891 
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

892 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
43920  893 
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)" 
894 
"(\<lambda>x. ereal ( f x)) \<in> borel_measurable (M i)" 

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

895 
using assms by auto 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

896 
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

897 
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] .. 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

898 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

899 

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

900 
lemma (in product_sigma_finite) product_integral_fold: 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

901 
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

902 
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

903 
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

904 
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

905 
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

906 
interpret J: finite_product_sigma_finite M J by default fact 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

907 
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

908 
interpret IJ: finite_product_sigma_finite M "I \<union> J" 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

909 
interpret P: pair_sigma_finite I.P J.P by default 
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

910 
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

911 
let ?f = "\<lambda>x. f (?M x)" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

912 
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

913 
proof (subst P.integrable_fst_measurable(2)[of ?f, simplified]) 
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

914 
have 1: "sigma_algebra IJ.P" by default 
41831  915 
have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] . 
916 
have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact 

917 
then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" 

918 
by (simp add: integrable_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

919 
show "integrable P.P ?f" 
41831  920 
by (rule P.integrable_vimage[where f=f, OF 1 2 3]) 
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

921 
show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f" 
41831  922 
by (rule P.integral_vimage[where f=f, OF 1 2 4]) 
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

923 
qed 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

924 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

925 

41096  926 
lemma (in product_sigma_finite) product_integral_insert: 
927 
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

928 
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

929 
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  930 
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

931 
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

932 
interpret I': finite_product_sigma_finite M "insert i 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

933 
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

934 
interpret P: pair_sigma_finite I.P i.P .. 
41096  935 
have IJ: "I \<inter> {i} = {}" by auto 
936 
show ?thesis 

937 
unfolding product_integral_fold[OF IJ, simplified, OF f] 

938 
proof (rule I.integral_cong, subst product_integral_singleton) 

939 
fix x assume x: "x \<in> space I.P" 

940 
let "?f y" = "f (restrict (x(i := y)) (insert i I))" 

941 
have f'_eq: "\<And>y. ?f y = f (x(i := y))" 

942 
using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_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

943 
have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto 
41096  944 
show "?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

945 
unfolding measurable_cong[OF f'_eq] 
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

946 
using measurable_comp[OF measurable_component_update f] x `i \<notin> 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

947 
by (simp add: comp_def) 
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 
show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))" 
41096  949 
unfolding f'_eq by simp 
950 
qed 

951 
qed 

952 

953 
lemma (in product_sigma_finite) product_integrable_setprod: 

954 
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

955 
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

956 
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f") 
41096  957 
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

958 
interpret finite_product_sigma_finite M I by default fact 
41096  959 
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

960 
using integrable unfolding integrable_def by auto 
41096  961 
then have borel: "?f \<in> borel_measurable P" 
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

962 
using measurable_comp[OF measurable_component_singleton 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

963 
by (auto intro!: borel_measurable_setprod simp: comp_def) 
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

964 
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)" 
41096  965 
proof (unfold integrable_def, intro conjI) 
966 
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P" 

967 
using borel by auto 

43920  968 
have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>P)" 
969 
by (simp add: setprod_ereal abs_setprod) 

970 
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))" 

41096  971 
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

972 
also have "\<dots> < \<infinity>" 
41096  973 
using integrable[THEN M.integrable_abs] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

974 
by (simp add: setprod_PInf integrable_def M.positive_integral_positive) 
43920  975 
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto 
976 
have "(\<integral>\<^isup>+x. ereal ( abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)" 

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

977 
by (intro positive_integral_cong_pos) auto 
43920  978 
then show "(\<integral>\<^isup>+x. ereal ( abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp 
41096  979 
qed 
980 
ultimately show ?thesis 

981 
by (rule integrable_abs_iff[THEN iffD1]) 

982 
qed 

983 

984 
lemma (in product_sigma_finite) product_integral_setprod: 

985 
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

986 
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

987 
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  988 
using assms proof (induct rule: finite_ne_induct) 
989 
case (singleton i) 

990 
then show ?case by (simp add: product_integral_singleton integrable_def) 

991 
next 

992 
case (insert i I) 

993 
then have iI: "finite (insert i I)" by auto 

994 
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

995 
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))" 
41096  996 
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

997 
interpret I: finite_product_sigma_finite M I by default fact 
41096  998 
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))" 
999 
using `i \<notin> I` by (auto intro!: setprod_cong) 

1000 
show ?case 

1001 
unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]] 

1002 
by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI) 

1003 
qed 

1004 

40859  1005 
end 