author  wenzelm 
Thu, 18 Apr 2013 17:07:01 +0200  
changeset 51717  9e7d1c139569 
parent 50387  3d8863c41fe8 
child 53015  a1119cf551e8 
permissions  rwrr 
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(* Title: HOL/Probability/Finite_Product_Measure.thy 
42067  2 
Author: Johannes Hölzl, TU München 
3 
*) 

4 

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header {*Finite product measures*} 
42067  6 

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theory Finite_Product_Measure 
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imports Binary_Product_Measure 
35833  9 
begin 
10 

47694  11 
lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)" 
12 
by auto 

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subsubsection {* Merge two extensional functions *} 
50038  15 

35833  16 
definition 
49780  17 
"merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)" 
40859  18 

19 
lemma merge_apply[simp]: 

49780  20 
"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" 
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"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" 

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

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

24 
"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined" 

40859  25 
unfolding merge_def by auto 
26 

27 
lemma merge_commute: 

49780  28 
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)" 
50003  29 
by (force simp: merge_def) 
40859  30 

31 
lemma Pi_cancel_merge_range[simp]: 

49780  32 
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" 
33 
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" 

34 
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" 

35 
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" 

40859  36 
by (auto simp: Pi_def) 
37 

38 
lemma Pi_cancel_merge[simp]: 

49780  39 
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" 
40 
"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" 

41 
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" 

42 
"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" 

40859  43 
by (auto simp: Pi_def) 
44 

49780  45 
lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)" 
40859  46 
by (auto simp: extensional_def) 
47 

48 
lemma restrict_merge[simp]: 

49780  49 
"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" 
50 
"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" 

51 
"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" 

52 
"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" 

47694  53 
by (auto simp: restrict_def) 
40859  54 

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lemma split_merge: "P (merge I J (x,y) i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J  I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)" 
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unfolding merge_def by auto 
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lemma PiE_cancel_merge[simp]: 
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"I \<inter> J = {} \<Longrightarrow> 
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merge I J (x, y) \<in> PiE (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B" 
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by (auto simp: PiE_def restrict_Pi_cancel) 
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lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)" 
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unfolding merge_def by (auto simp: fun_eq_iff) 
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lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K" 
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unfolding merge_def extensional_def by auto 
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lemma merge_restrict[simp]: 
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"merge I J (restrict x I, y) = merge I J (x, y)" 
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"merge I J (x, restrict y J) = merge I J (x, y)" 
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unfolding merge_def by auto 
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lemma merge_x_x_eq_restrict[simp]: 
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"merge I J (x, x) = restrict x (I \<union> J)" 
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unfolding merge_def by auto 
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lemma injective_vimage_restrict: 
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assumes J: "J \<subseteq> I" 
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and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}" 
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and eq: "(\<lambda>x. restrict x J) ` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) ` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" 
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shows "A = B" 
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proof (intro set_eqI) 
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fix x 
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from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto 
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have "J \<inter> (I  J) = {}" by auto 
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show "x \<in> A \<longleftrightarrow> x \<in> B" 
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proof cases 
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assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)" 
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have "x \<in> A \<longleftrightarrow> merge J (I  J) (x,y) \<in> (\<lambda>x. restrict x J) ` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" 
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using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I  J" x y S] 
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by (auto simp del: PiE_cancel_merge simp add: Un_absorb1) 
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then show "x \<in> A \<longleftrightarrow> x \<in> B" 
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using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I  J" x y S] 
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by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq) 
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qed (insert sets, auto) 
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qed 
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41095  99 
lemma restrict_vimage: 
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"I \<inter> J = {} \<Longrightarrow> 
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(\<lambda>x. (restrict x I, restrict x J)) ` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I J (E, F))" 
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by (auto simp: restrict_Pi_cancel PiE_def) 
41095  103 

104 
lemma merge_vimage: 

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"I \<inter> J = {} \<Longrightarrow> merge I J ` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E" 
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by (auto simp: restrict_Pi_cancel PiE_def) 
50104  107 

40859  108 
section "Finite product spaces" 
109 

110 
section "Products" 

111 

47694  112 
definition prod_emb where 
113 
"prod_emb I M K X = (\<lambda>x. restrict x K) ` X \<inter> (PIE i:I. space (M i))" 

114 

115 
lemma prod_emb_iff: 

116 
"f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))" 

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unfolding prod_emb_def PiE_def by auto 
40859  118 

47694  119 
lemma 
120 
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}" 

121 
and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B" 

122 
and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B" 

123 
and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))" 

124 
and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))" 

125 
and prod_emb_Diff[simp]: "prod_emb M L K (A  B) = prod_emb M L K A  prod_emb M L K B" 

126 
by (auto simp: prod_emb_def) 

40859  127 

47694  128 
lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> 
129 
prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))" 

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by (force simp: prod_emb_def PiE_iff split_if_mem2) 
47694  131 

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lemma prod_emb_PiE_same_index[simp]: 
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"(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E" 
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by (auto simp: prod_emb_def PiE_iff) 
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50038  136 
lemma prod_emb_trans[simp]: 
137 
"J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X" 

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by (auto simp add: Int_absorb1 prod_emb_def PiE_def) 
50038  139 

140 
lemma prod_emb_Pi: 

141 
assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K" 

142 
shows "prod_emb K M J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))" 

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using assms sets.space_closed 
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by (auto simp: prod_emb_def PiE_iff split: split_if_asm) blast+ 
50038  145 

146 
lemma prod_emb_id: 

147 
"B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B" 

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by (auto simp: prod_emb_def subset_eq extensional_restrict) 
50038  149 

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lemma prod_emb_mono: 
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"F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G" 
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by (auto simp: prod_emb_def) 
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47694  154 
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where 
155 
"PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) 

156 
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} 

157 
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) 

158 
(\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" 

159 

160 
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where 

161 
"prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) ` 

162 
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

163 

164 
abbreviation 

165 
"Pi\<^isub>M I M \<equiv> PiM I M" 

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40859  167 
syntax 
47694  168 
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIM _:_./ _)" 10) 
40859  169 

170 
syntax (xsymbols) 

47694  171 
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) 
40859  172 

173 
syntax (HTML output) 

47694  174 
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) 
40859  175 

176 
translations 

47694  177 
"PIM x:I. M" == "CONST PiM I (%x. M)" 
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178 

47694  179 
lemma prod_algebra_sets_into_space: 
180 
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))" 

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by (auto simp: prod_emb_def prod_algebra_def) 
40859  182 

47694  183 
lemma prod_algebra_eq_finite: 
184 
assumes I: "finite I" 

185 
shows "prod_algebra I M = {(\<Pi>\<^isub>E i\<in>I. X i) X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R") 

186 
proof (intro iffI set_eqI) 

187 
fix A assume "A \<in> ?L" 

188 
then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" 

189 
and A: "A = prod_emb I M J (PIE j:J. E j)" 

190 
by (auto simp: prod_algebra_def) 

191 
let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)" 

192 
have A: "A = ?A" 

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unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto 
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show "A \<in> ?R" unfolding A using J sets.top 
47694  195 
by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp 
196 
next 

197 
fix A assume "A \<in> ?R" 

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then obtain X where A: "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto 
47694  199 
then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)" 
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by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff) 
47694  201 
from X I show "A \<in> ?L" unfolding A 
202 
by (auto simp: prod_algebra_def) 

203 
qed 

41095  204 

47694  205 
lemma prod_algebraI: 
206 
"finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)) 

207 
\<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M" 

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by (auto simp: prod_algebra_def) 
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50038  210 
lemma prod_algebraI_finite: 
211 
"finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M" 

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using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp 
50038  213 

214 
lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E  E. \<forall>i\<in>I. E i \<in> sets (M i)}" 

215 
proof (safe intro!: Int_stableI) 

216 
fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" 

217 
then show "\<exists>G. Pi\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))" 

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by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int) 
50038  219 
qed 
220 

47694  221 
lemma prod_algebraE: 
222 
assumes A: "A \<in> prod_algebra I M" 

223 
obtains J E where "A = prod_emb I M J (PIE j:J. E j)" 

224 
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" 

225 
using A by (auto simp: prod_algebra_def) 

42988  226 

47694  227 
lemma prod_algebraE_all: 
228 
assumes A: "A \<in> prod_algebra I M" 

229 
obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" 

230 
proof  

231 
from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)" 

232 
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))" 

233 
by (auto simp: prod_algebra_def) 

234 
from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" 

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using sets.sets_into_space by auto 
47694  236 
then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))" 
237 
using A J by (auto simp: prod_emb_PiE) 

238 
moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))" 

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239 
using sets.top E by auto 
47694  240 
ultimately show ?thesis using that by auto 
241 
qed 

40859  242 

47694  243 
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)" 
244 
proof (unfold Int_stable_def, safe) 

245 
fix A assume "A \<in> prod_algebra I M" 

246 
from prod_algebraE[OF this] guess J E . note A = this 

247 
fix B assume "B \<in> prod_algebra I M" 

248 
from prod_algebraE[OF this] guess K F . note B = this 

249 
have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^isub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter> 

250 
(if i \<in> K then F i else space (M i)))" 

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unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4) 
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B(5)[THEN sets.sets_into_space] 
47694  253 
apply (subst (1 2 3) prod_emb_PiE) 
254 
apply (simp_all add: subset_eq PiE_Int) 

255 
apply blast 

256 
apply (intro PiE_cong) 

257 
apply auto 

258 
done 

259 
also have "\<dots> \<in> prod_algebra I M" 

260 
using A B by (auto intro!: prod_algebraI) 

261 
finally show "A \<inter> B \<in> prod_algebra I M" . 

262 
qed 

263 

264 
lemma prod_algebra_mono: 

265 
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)" 

266 
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" 

267 
shows "prod_algebra I E \<subseteq> prod_algebra I F" 

268 
proof 

269 
fix A assume "A \<in> prod_algebra I E" 

270 
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" 

271 
and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)" 

272 
and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)" 

273 
by (auto simp: prod_algebra_def) 

274 
moreover 

275 
from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))" 

276 
by (rule PiE_cong) 

277 
with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)" 

278 
by (simp add: prod_emb_def) 

279 
moreover 

280 
from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)" 

281 
by auto 

282 
ultimately show "A \<in> prod_algebra I F" 

283 
apply (simp add: prod_algebra_def image_iff) 

284 
apply (intro exI[of _ J] exI[of _ G] conjI) 

285 
apply auto 

286 
done 

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qed 
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50104  289 
lemma prod_algebra_cong: 
290 
assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))" 

291 
shows "prod_algebra I M = prod_algebra J N" 

292 
proof  

293 
have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)" 

294 
using sets_eq_imp_space_eq[OF sets] by auto 

295 
with sets show ?thesis unfolding `I = J` 

296 
by (intro antisym prod_algebra_mono) auto 

297 
qed 

298 

299 
lemma space_in_prod_algebra: 

300 
"(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M" 

301 
proof cases 

302 
assume "I = {}" then show ?thesis 

303 
by (auto simp add: prod_algebra_def image_iff prod_emb_def) 

304 
next 

305 
assume "I \<noteq> {}" 

306 
then obtain i where "i \<in> I" by auto 

307 
then have "(\<Pi>\<^isub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i))" 

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by (auto simp: prod_emb_def) 
50104  309 
also have "\<dots> \<in> prod_algebra I M" 
310 
using `i \<in> I` by (intro prod_algebraI) auto 

311 
finally show ?thesis . 

312 
qed 

313 

47694  314 
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
315 
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp 

316 

317 
lemma sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" 

318 
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp 

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47694  320 
lemma sets_PiM_single: "sets (PiM I M) = 
321 
sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}  i A. i \<in> I \<and> A \<in> sets (M i)}" 

322 
(is "_ = sigma_sets ?\<Omega> ?R") 

323 
unfolding sets_PiM 

324 
proof (rule sigma_sets_eqI) 

325 
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto 

326 
fix A assume "A \<in> prod_algebra I M" 

327 
from prod_algebraE[OF this] guess J X . note X = this 

328 
show "A \<in> sigma_sets ?\<Omega> ?R" 

329 
proof cases 

330 
assume "I = {}" 

331 
with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def) 

332 
with `I = {}` show ?thesis by (auto intro!: sigma_sets_top) 

333 
next 

334 
assume "I \<noteq> {}" 

335 
with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})" 

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by (auto simp: prod_emb_def) 
47694  337 
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" 
338 
using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto 

339 
finally show "A \<in> sigma_sets ?\<Omega> ?R" . 

340 
qed 

341 
next 

342 
fix A assume "A \<in> ?R" 

343 
then obtain i B where A: "A = {f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" 

344 
by auto 

345 
then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)" 

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by (auto simp: prod_emb_def) 
47694  347 
also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)" 
348 
using A by (intro sigma_sets.Basic prod_algebraI) auto 

349 
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" . 

350 
qed 

351 

352 
lemma sets_PiM_I: 

353 
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" 

354 
shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)" 

355 
proof cases 

356 
assume "J = {}" 

357 
then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))" 

358 
by (auto simp: prod_emb_def) 

359 
then show ?thesis 

360 
by (auto simp add: sets_PiM intro!: sigma_sets_top) 

361 
next 

362 
assume "J \<noteq> {}" with assms show ?thesis 

50003  363 
by (force simp add: sets_PiM prod_algebra_def) 
40859  364 
qed 
365 

47694  366 
lemma measurable_PiM: 
367 
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

368 
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow> 

369 
f ` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" 

370 
shows "f \<in> measurable N (PiM I M)" 

371 
using sets_PiM prod_algebra_sets_into_space space 

372 
proof (rule measurable_sigma_sets) 

373 
fix A assume "A \<in> prod_algebra I M" 

374 
from prod_algebraE[OF this] guess J X . 

375 
with sets[of J X] show "f ` A \<inter> space N \<in> sets N" by auto 

376 
qed 

377 

378 
lemma measurable_PiM_Collect: 

379 
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

380 
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow> 

381 
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 

382 
shows "f \<in> measurable N (PiM I M)" 

383 
using sets_PiM prod_algebra_sets_into_space space 

384 
proof (rule measurable_sigma_sets) 

385 
fix A assume "A \<in> prod_algebra I M" 

386 
from prod_algebraE[OF this] guess J X . note X = this 

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then have "f ` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}" 
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388 
using space by (auto simp: prod_emb_def del: PiE_I) 
47694  389 
also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets) 
390 
finally show "f ` A \<inter> space N \<in> sets N" . 

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qed 
41095  392 

47694  393 
lemma measurable_PiM_single: 
394 
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

395 
assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N" 

396 
shows "f \<in> measurable N (PiM I M)" 

397 
using sets_PiM_single 

398 
proof (rule measurable_sigma_sets) 

399 
fix A assume "A \<in> {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} i A. i \<in> I \<and> A \<in> sets (M i)}" 

400 
then obtain B i where "A = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)" 

401 
by auto 

402 
with space have "f ` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto 

403 
also have "\<dots> \<in> sets N" using B by (rule sets) 

404 
finally show "f ` A \<inter> space N \<in> sets N" . 

405 
qed (auto simp: space) 

40859  406 

50099  407 
lemma measurable_PiM_single': 
408 
assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)" 

409 
and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" 

410 
shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^isub>M I M)" 

411 
proof (rule measurable_PiM_single) 

412 
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" 

413 
then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i ` A \<inter> space N" 

414 
by auto 

415 
then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N" 

416 
using A f by (auto intro!: measurable_sets) 

417 
qed fact 

418 

50003  419 
lemma sets_PiM_I_finite[measurable]: 
47694  420 
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))" 
421 
shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)" 

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using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto 
47694  423 

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lemma measurable_component_singleton: 
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assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)" 
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proof (unfold measurable_def, intro CollectI conjI ballI) 
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427 
fix A assume "A \<in> sets (M i)" 
47694  428 
then have "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)" 
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429 
using sets.sets_into_space `i \<in> I` 
47694  430 
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm) 
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431 
then show "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)" 
47694  432 
using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I) 
433 
qed (insert `i \<in> I`, auto simp: space_PiM) 

434 

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435 
lemma measurable_component_singleton'[measurable_app]: 
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436 
assumes f: "f \<in> measurable N (Pi\<^isub>M I M)" 
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assumes i: "i \<in> I" 
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shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)" 
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439 
using measurable_compose[OF f measurable_component_singleton, OF i] . 
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440 

50099  441 
lemma measurable_PiM_component_rev[measurable (raw)]: 
442 
"i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N" 

443 
by simp 

444 

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445 
lemma measurable_nat_case[measurable (raw)]: 
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446 
assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N" 
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447 
"\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N" 
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448 
shows "(\<lambda>x. nat_case (f x) (g x) i) \<in> measurable M N" 
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changeset

449 
by (cases i) simp_all 
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450 

50099  451 
lemma measurable_nat_case'[measurable (raw)]: 
452 
assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 

453 
shows "(\<lambda>x. nat_case (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 

454 
using fg[THEN measurable_space] 

50123
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455 
by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split) 
50099  456 

50003  457 
lemma measurable_add_dim[measurable]: 
49776  458 
"(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)" 
47694  459 
(is "?f \<in> measurable ?P ?I") 
460 
proof (rule measurable_PiM_single) 

461 
fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)" 

462 
have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} = 

463 
(if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) ` A \<inter> space ?P)" 

50244
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464 
using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM) 
47694  465 
also have "\<dots> \<in> sets ?P" 
466 
using A j 

467 
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) 

468 
finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" . 

50123
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469 
qed (auto simp: space_pair_measure space_PiM PiE_def) 
41661  470 

50003  471 
lemma measurable_component_update: 
472 
"x \<in> space (Pi\<^isub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" 

473 
by simp 

474 

475 
lemma measurable_merge[measurable]: 

49780  476 
"merge I J \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)" 
47694  477 
(is "?f \<in> measurable ?P ?U") 
478 
proof (rule measurable_PiM_single) 

479 
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J" 

49780  480 
then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} = 
47694  481 
(if i \<in> I then ((\<lambda>x. x i) \<circ> fst) ` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) ` A \<inter> space ?P)" 
49776  482 
by (auto simp: merge_def) 
47694  483 
also have "\<dots> \<in> sets ?P" 
484 
using A 

485 
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) 

49780  486 
finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" . 
50123
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487 
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def) 
42988  488 

50003  489 
lemma measurable_restrict[measurable (raw)]: 
47694  490 
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" 
491 
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)" 

492 
proof (rule measurable_PiM_single) 

493 
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" 

494 
then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i ` A \<inter> space N" 

495 
by auto 

496 
then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N" 

497 
using A X by (auto intro!: measurable_sets) 

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498 
qed (insert X, auto simp add: PiE_def dest: measurable_space) 
47694  499 

50038  500 
lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)" 
501 
by (intro measurable_restrict measurable_component_singleton) auto 

502 

503 
lemma measurable_prod_emb[intro, simp]: 

504 
"J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^isub>M L M)" 

505 
unfolding prod_emb_def space_PiM[symmetric] 

506 
by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton) 

507 

50003  508 
lemma sets_in_Pi_aux: 
509 
"finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> 

510 
{x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)" 

511 
by (simp add: subset_eq Pi_iff) 

512 

513 
lemma sets_in_Pi[measurable (raw)]: 

514 
"finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow> 

515 
(\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> 

50387  516 
Measurable.pred N (\<lambda>x. f x \<in> Pi I F)" 
50003  517 
unfolding pred_def 
518 
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto 

519 

520 
lemma sets_in_extensional_aux: 

521 
"{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)" 

522 
proof  

523 
have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)" 

524 
by (auto simp add: extensional_def space_PiM) 

525 
then show ?thesis by simp 

526 
qed 

527 

528 
lemma sets_in_extensional[measurable (raw)]: 

50387  529 
"f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)" 
50003  530 
unfolding pred_def 
531 
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto 

532 

47694  533 
locale product_sigma_finite = 
534 
fixes M :: "'i \<Rightarrow> 'a measure" 

41689
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changeset

535 
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" 
40859  536 

41689
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changeset

537 
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i 
40859  538 
by (rule sigma_finite_measures) 
539 

47694  540 
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" + 
541 
fixes I :: "'i set" 

542 
assumes finite_index: "finite I" 

41689
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hoelzl
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changeset

543 

40859  544 
lemma (in finite_product_sigma_finite) sigma_finite_pairs: 
545 
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. 

546 
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> 

47694  547 
(\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and> 
548 
(\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)" 

40859  549 
proof  
47694  550 
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)" 
551 
using M.sigma_finite_incseq by metis 

40859  552 
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. 
47694  553 
then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. emeasure (M i) (F i k) \<noteq> \<infinity>" 
40859  554 
by auto 
555 
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k" 

47694  556 
note space_PiM[simp] 
40859  557 
show ?thesis 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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558 
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) 
40859  559 
fix i show "range (F i) \<subseteq> sets (M i)" by fact 
560 
next 

47694  561 
fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact 
40859  562 
next 
50123
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hoelzl
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diff
changeset

563 
fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)" 
50244
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qualified interpretation of sigma_algebra, to avoid name clashes
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parents:
50123
diff
changeset

564 
by (auto simp: PiE_def dest!: sets.sets_into_space) 
40859  565 
next 
47694  566 
fix f assume "f \<in> space (PiM I M)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

567 
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F 
50123
69b35a75caf3
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hoelzl
parents:
50104
diff
changeset

568 
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def) 
40859  569 
next 
570 
fix i show "?F i \<subseteq> ?F (Suc i)" 

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

571 
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto 
40859  572 
qed 
573 
qed 

574 

49780  575 
lemma 
576 
shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}" 

577 
and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }" 

578 
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq) 

579 

580 
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1" 

581 
proof  

582 
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)" 

583 
have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1" 

584 
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) 

585 
show "positive (PiM {} M) ?\<mu>" 

586 
by (auto simp: positive_def) 

587 
show "countably_additive (PiM {} M) ?\<mu>" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

588 
by (rule sets.countably_additiveI_finite) 
49780  589 
(auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: ) 
590 
qed (auto simp: prod_emb_def) 

591 
also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}" 

592 
by (auto simp: prod_emb_def) 

593 
finally show ?thesis 

594 
by simp 

595 
qed 

596 

597 
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}" 

598 
by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def) 

599 

49776  600 
lemma (in product_sigma_finite) emeasure_PiM: 
601 
"finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" 

602 
proof (induct I arbitrary: A rule: finite_induct) 

40859  603 
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

604 
interpret finite_product_sigma_finite M I by default fact 
40859  605 
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

606 
interpret I': finite_product_sigma_finite M "insert i I" by default fact 
41661  607 
let ?h = "(\<lambda>(f, y). f(i := y))" 
47694  608 

609 
let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h" 

610 
let ?\<mu> = "emeasure ?P" 

611 
let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}" 

612 
let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))" 

613 

49776  614 
have "emeasure (Pi\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>E (insert i I) A)) = 
615 
(\<Prod>i\<in>insert i I. emeasure (M i) (A i))" 

616 
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) 

617 
fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))" 

618 
then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto 

619 
let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)" 

620 
let ?p' = "prod_emb I M (J  {i}) (\<Pi>\<^isub>E j\<in>J{i}. E j)" 

621 
have "?\<mu> ?p = 

622 
emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h ` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))" 

623 
by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+ 

624 
also have "?h ` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

625 
using J E[rule_format, THEN sets.sets_into_space] 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

626 
by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm) 
49776  627 
also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) = 
628 
emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))" 

629 
using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto 

630 
also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J{i} then E j else space (M j))" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

631 
using J E[rule_format, THEN sets.sets_into_space] 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

632 
by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+ 
49776  633 
also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>E j\<in>I. if j \<in> J{i} then E j else space (M j)) = 
634 
(\<Prod> j\<in>I. if j \<in> J{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))" 

635 
using E by (subst insert) (auto intro!: setprod_cong) 

636 
also have "(\<Prod>j\<in>I. if j \<in> J  {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) * 

637 
emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)" 

638 
using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong) 

639 
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)" 

640 
using insert(1,2) J E by (intro setprod_mono_one_right) auto 

641 
finally show "?\<mu> ?p = \<dots>" . 

47694  642 

49776  643 
show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

644 
using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def) 
49776  645 
next 
646 
show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" 

647 
using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all 

648 
next 

649 
show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and> 

650 
insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))" 

651 
using insert by auto 

652 
qed (auto intro!: setprod_cong) 

653 
with insert show ?case 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

654 
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space) 
50003  655 
qed simp 
47694  656 

49776  657 
lemma (in product_sigma_finite) sigma_finite: 
658 
assumes "finite I" 

659 
shows "sigma_finite_measure (PiM I M)" 

660 
proof  

661 
interpret finite_product_sigma_finite M I by default fact 

662 

663 
from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. 

664 
then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)" 

665 
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)" 

666 
"(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)" 

667 
"\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>" 

47694  668 
by blast+ 
49776  669 
let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k" 
47694  670 

49776  671 
show ?thesis 
47694  672 
proof (unfold_locales, intro exI[of _ ?F] conjI allI) 
49776  673 
show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto 
47694  674 
next 
49776  675 
from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp 
47694  676 
next 
677 
fix j 

49776  678 
from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M] 
679 
show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>" 

680 
by (subst emeasure_PiM) auto 

40859  681 
qed 
682 
qed 

683 

47694  684 
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M" 
685 
using sigma_finite[OF finite_index] . 

40859  686 

687 
lemma (in finite_product_sigma_finite) measure_times: 

47694  688 
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^isub>M I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" 
689 
using emeasure_PiM[OF finite_index] by auto 

41096  690 

40859  691 
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

692 
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

693 
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)" 
40859  694 
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

695 
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) 
47694  696 
have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" 
40859  697 
using assms by (subst measure_times) auto 
698 
then show ?thesis 

47694  699 
unfolding positive_integral_def simple_function_def simple_integral_def[abs_def] 
700 
proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym) 

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

701 
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset

702 
by (intro SUP_upper) (auto simp: le_fun_def split: split_max) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

703 
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset

704 
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) 
40859  705 
qed 
706 
qed 

707 

47694  708 
lemma (in product_sigma_finite) distr_merge: 
40859  709 
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" 
49780  710 
shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (merge I J) = Pi\<^isub>M (I \<union> J) M" 
47694  711 
(is "?D = ?P") 
40859  712 
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

713 
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

714 
interpret J: finite_product_sigma_finite M J by default fact 
40859  715 
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

716 
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact 
47694  717 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
49780  718 
let ?g = "merge I J" 
47694  719 

41661  720 
from IJ.sigma_finite_pairs obtain F where 
721 
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

722 
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)" 
47694  723 
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P" 
724 
"\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>" 

41661  725 
by auto 
726 
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k" 

47694  727 

728 
show ?thesis 

729 
proof (rule measure_eqI_generator_eq[symmetric]) 

730 
show "Int_stable (prod_algebra (I \<union> J) M)" 

731 
by (rule Int_stable_prod_algebra) 

732 
show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))" 

733 
by (rule prod_algebra_sets_into_space) 

734 
show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" 

735 
by (rule sets_PiM) 

736 
then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" 

737 
by simp 

738 

739 
show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F 

740 
using fin by (auto simp: prod_algebra_eq_finite) 

741 
show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))" 

742 
using F(3) by (simp add: space_PiM) 

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

743 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

744 
fix k 
47694  745 
from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M] 
746 
show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto 

41661  747 
next 
47694  748 
fix A assume A: "A \<in> prod_algebra (I \<union> J) M" 
50003  749 
with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" 
47694  750 
by (auto simp add: prod_algebra_eq_finite) 
751 
let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M" 

752 
let ?X = "?g ` A \<inter> space ?B" 

753 
have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

754 
using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+ 
47694  755 
then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)" 
756 
unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM) 

757 
have "emeasure ?D A = emeasure ?B ?X" 

758 
using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM) 

759 
also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))" 

50003  760 
using `finite J` `finite I` F unfolding X 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

761 
by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times) 
47694  762 
also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))" 
41661  763 
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one) 
47694  764 
also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)" 
41661  765 
using `finite J` `finite I` F unfolding A 
766 
by (intro IJ.measure_times[symmetric]) auto 

47694  767 
finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp 
768 
qed 

41661  769 
qed 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

770 

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

771 
lemma (in product_sigma_finite) product_positive_integral_fold: 
47694  772 
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

773 
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

774 
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f = 
49780  775 
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, 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

776 
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

777 
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

778 
interpret J: finite_product_sigma_finite M J by default fact 
41831  779 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
49780  780 
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" 
49776  781 
using measurable_comp[OF measurable_merge f] by (simp add: comp_def) 
41661  782 
show ?thesis 
47694  783 
apply (subst distr_merge[OF IJ, symmetric]) 
49776  784 
apply (subst positive_integral_distr[OF measurable_merge f]) 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49784
diff
changeset

785 
apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel]) 
47694  786 
apply simp 
787 
done 

40859  788 
qed 
789 

47694  790 
lemma (in product_sigma_finite) distr_singleton: 
791 
"distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _") 

792 
proof (intro measure_eqI[symmetric]) 

41831  793 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
47694  794 
fix A assume A: "A \<in> sets (M i)" 
795 
moreover then have "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

796 
using sets.sets_into_space by (auto simp: space_PiM) 
47694  797 
ultimately show "emeasure (M i) A = emeasure ?D A" 
798 
using A I.measure_times[of "\<lambda>_. A"] 

799 
by (simp add: emeasure_distr measurable_component_singleton) 

800 
qed simp 

41831  801 

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

802 
lemma (in product_sigma_finite) product_positive_integral_singleton: 
40859  803 
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

804 
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f" 
40859  805 
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

806 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
47694  807 
from f show ?thesis 
808 
apply (subst distr_singleton[symmetric]) 

809 
apply (subst positive_integral_distr[OF measurable_component_singleton]) 

810 
apply simp_all 

811 
done 

40859  812 
qed 
813 

41096  814 
lemma (in product_sigma_finite) product_positive_integral_insert: 
49780  815 
assumes I[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

816 
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

817 
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  818 
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

819 
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

820 
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

821 
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

822 
using f by auto 
41096  823 
show ?thesis 
49780  824 
unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f] 
825 
proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric]) 

47694  826 
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" 
49780  827 
let ?f = "\<lambda>y. f (x(i := y))" 
828 
show "?f \<in> borel_measurable (M i)" 

47694  829 
using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`] 
830 
unfolding comp_def . 

49780  831 
show "(\<integral>\<^isup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^isub>M {i} M) = (\<integral>\<^isup>+ y. f (x(i := y i)) \<partial>Pi\<^isub>M {i} M)" 
832 
using x 

833 
by (auto intro!: positive_integral_cong arg_cong[where f=f] 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

834 
simp add: space_PiM extensional_def PiE_def) 
41096  835 
qed 
836 
qed 

837 

838 
lemma (in product_sigma_finite) product_positive_integral_setprod: 

43920  839 
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" 
41096  840 
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

841 
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

842 
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  843 
using assms proof induct 
844 
case (insert i I) 

845 
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

846 
interpret I: finite_product_sigma_finite M I by default auto 
41096  847 
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))" 
848 
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

849 
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)" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

850 
using sets.sets_into_space insert 
47694  851 
by (intro borel_measurable_ereal_setprod 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

852 
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) 
41096  853 
auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

854 
then show ?case 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

855 
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod]) 
47694  856 
apply (simp add: insert(2) * pos borel setprod_ereal_pos positive_integral_multc) 
857 
apply (subst positive_integral_cmult) 

858 
apply (auto simp add: pos borel insert(2) setprod_ereal_pos positive_integral_positive) 

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

859 
done 
47694  860 
qed (simp add: space_PiM) 
41096  861 

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

862 
lemma (in product_sigma_finite) product_integral_singleton: 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

863 
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

864 
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

865 
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

866 
interpret I: finite_product_sigma_finite M "{i}" by default simp 
43920  867 
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)" 
868 
"(\<lambda>x. ereal ( f x)) \<in> borel_measurable (M i)" 

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

869 
using assms by auto 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

870 
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

871 
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] .. 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

872 
qed 
50104  873 
lemma (in product_sigma_finite) distr_component: 
874 
"distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P") 

875 
proof (intro measure_eqI[symmetric]) 

876 
interpret I: finite_product_sigma_finite M "{i}" by default simp 

877 

878 
have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x" 

879 
by (auto simp: extensional_def restrict_def) 

880 

881 
fix A assume A: "A \<in> sets ?P" 

882 
then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)" 

883 
by simp 

884 
also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) ` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)" 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

885 
by (intro positive_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq) 
50104  886 
also have "\<dots> = emeasure ?D A" 
887 
using A by (simp add: product_positive_integral_singleton emeasure_distr) 

888 
finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" . 

889 
qed simp 

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

890 

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

891 
lemma (in product_sigma_finite) product_integral_fold: 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

892 
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

893 
and f: "integrable (Pi\<^isub>M (I \<union> J) M) f" 
49780  894 
shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

895 
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

896 
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

897 
interpret J: finite_product_sigma_finite M J by default fact 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

898 
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

899 
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact 
47694  900 
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default 
49780  901 
let ?M = "merge I 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 
let ?f = "\<lambda>x. f (?M x)" 
47694  903 
from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" 
904 
by auto 

49780  905 
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" 
49776  906 
using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def) 
47694  907 
have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f" 
49776  908 
by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f) 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

909 
show ?thesis 
47694  910 
apply (subst distr_merge[symmetric, OF IJ fin]) 
49776  911 
apply (subst integral_distr[OF measurable_merge f_borel]) 
47694  912 
apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int]) 
913 
apply simp 

914 
done 

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

915 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

916 

49776  917 
lemma (in product_sigma_finite) 
918 
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" 

919 
shows emeasure_fold_integral: 

49780  920 
"emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) ` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I) 
49776  921 
and emeasure_fold_measurable: 
49780  922 
"(\<lambda>x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) ` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B) 
49776  923 
proof  
924 
interpret I: finite_product_sigma_finite M I by default fact 

925 
interpret J: finite_product_sigma_finite M J by default fact 

926 
interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" .. 

49780  927 
have merge: "merge I J ` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" 
49776  928 
by (intro measurable_sets[OF _ A] measurable_merge assms) 
929 

930 
show ?I 

931 
apply (subst distr_merge[symmetric, OF IJ]) 

932 
apply (subst emeasure_distr[OF measurable_merge A]) 

933 
apply (subst J.emeasure_pair_measure_alt[OF merge]) 

934 
apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure) 

935 
done 

936 

937 
show ?B 

938 
using IJ.measurable_emeasure_Pair1[OF merge] 

939 
by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong) 

940 
qed 

941 

41096  942 
lemma (in product_sigma_finite) product_integral_insert: 
47694  943 
assumes I: "finite I" "i \<notin> I" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

944 
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

945 
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  946 
proof  
47694  947 
have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f" 
948 
by simp 

49780  949 
also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)" 
47694  950 
using f I by (intro product_integral_fold) auto 
951 
also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)" 

952 
proof (rule integral_cong, subst product_integral_singleton[symmetric]) 

953 
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" 

954 
have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" 

955 
using f by auto 

956 
show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)" 

957 
using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`] 

958 
unfolding comp_def . 

49780  959 
from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)" 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

960 
by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def) 
41096  961 
qed 
47694  962 
finally show ?thesis . 
41096  963 
qed 
964 

965 
lemma (in product_sigma_finite) product_integrable_setprod: 

966 
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

967 
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

968 
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f") 
41096  969 
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

970 
interpret finite_product_sigma_finite M I by default fact 
41096  971 
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

972 
using integrable unfolding integrable_def by auto 
47694  973 
have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)" 
974 
using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def) 

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

975 
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)" 
41096  976 
proof (unfold integrable_def, intro conjI) 
47694  977 
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)" 
41096  978 
using borel by auto 
47694  979 
have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)" 
43920  980 
by (simp add: setprod_ereal abs_setprod) 
981 
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))" 

41096  982 
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

983 
also have "\<dots> < \<infinity>" 
47694  984 
using integrable[THEN integrable_abs] 
985 
by (simp add: setprod_PInf integrable_def positive_integral_positive) 

986 
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto 

987 
have "(\<integral>\<^isup>+x. ereal ( abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))" 

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

988 
by (intro positive_integral_cong_pos) auto 
47694  989 
then show "(\<integral>\<^isup>+x. ereal ( abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp 
41096  990 
qed 
991 
ultimately show ?thesis 

992 
by (rule integrable_abs_iff[THEN iffD1]) 

993 
qed 

994 

995 
lemma (in product_sigma_finite) product_integral_setprod: 

996 
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" 

49780  997 
assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f 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

998 
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))" 
49780  999 
using assms proof induct 
1000 
case empty 

1001 
interpret finite_measure "Pi\<^isub>M {} M" 

1002 
by rule (simp add: space_PiM) 

1003 
show ?case by (simp add: space_PiM measure_def) 

41096  1004 
next 
1005 
case (insert i I) 

1006 
then have iI: "finite (insert i I)" by auto 

1007 
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

1008 
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))" 
49780  1009 
by (intro product_integrable_setprod insert(4)) (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

1010 
interpret I: finite_product_sigma_finite M I by default fact 
41096  1011 
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))" 
1012 
using `i \<notin> I` by (auto intro!: setprod_cong) 

1013 
show ?case 

49780  1014 
unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]] 
47694  1015 
by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI) 
41096  1016 
qed 
1017 

49776  1018 
lemma sets_Collect_single: 
1019 
"i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^isub>M I M). x i \<in> A } \<in> sets (Pi\<^isub>M I M)" 

50003  1020 
by simp 
49776  1021 

1022 
lemma sigma_prod_algebra_sigma_eq_infinite: 

1023 
fixes E :: "'i \<Rightarrow> 'a set set" 

49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1024 
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 
49776  1025 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 
1026 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

1027 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

1028 
defines "P == {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}  i A. i \<in> I \<and> A \<in> E i}" 

1029 
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" 

1030 
proof 

1031 
let ?P = "sigma (space (Pi\<^isub>M I M)) P" 

1032 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

1033 
using E_closed by (auto simp: space_PiM P_def subset_eq) 
49776  1034 
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
1035 
by (simp add: space_PiM) 

1036 
have "sets (PiM I M) = 

1037 
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} i A. i \<in> I \<and> A \<in> sets (M i)}" 

1038 
using sets_PiM_single[of I M] by (simp add: space_P) 

1039 
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

1040 
proof (safe intro!: sets.sigma_sets_subset) 
49776  1041 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 
1042 
then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

1043 
apply (subst measurable_iff_measure_of) 

1044 
apply (simp_all add: P_closed) 

1045 
using E_closed 

1046 
apply (force simp: subset_eq space_PiM) 

1047 
apply (force simp: subset_eq space_PiM) 

1048 
apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i]) 

1049 
apply (rule_tac x=Aa in exI) 

1050 
apply (auto simp: space_PiM) 

1051 
done 

1052 
from measurable_sets[OF this, of A] A `i \<in> I` E_closed 

1053 
have "(\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 

1054 
by (simp add: E_generates) 

1055 
also have "(\<lambda>x. x i) ` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}" 

1056 
using P_closed by (auto simp: space_PiM) 

1057 
finally show "\<dots> \<in> sets ?P" . 

1058 
qed 

1059 
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" 

1060 
by (simp add: P_closed) 

1061 
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" 

1062 
unfolding P_def space_PiM[symmetric] 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

1063 
by (intro sets.sigma_sets_subset) (auto simp: E_generates sets_Collect_single) 
49776  1064 
qed 
1065 

47694  1066 
lemma sigma_prod_algebra_sigma_eq: 
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1067 
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" 
47694  1068 
assumes "finite I" 
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1069 
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 
47694  1070 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 
1071 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

1072 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

1073 
defines "P == { Pi\<^isub>E I F  F. \<forall>i\<in>I. F i \<in> E i }" 

1074 
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" 

1075 
proof 

1076 
let ?P = "sigma (space (Pi\<^isub>M I M)) P" 

49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1077 
from `finite I`[THEN ex_bij_betw_finite_nat] guess T .. 
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1078 
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i" 
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1079 
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f) 
47694  1080 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

1081 
using E_closed by (auto simp: space_PiM P_def subset_eq) 
47694  1082 
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
1083 
by (simp add: space_PiM) 

1084 
have "sets (PiM I M) = 

1085 
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} i A. i \<in> I \<and> A \<in> sets (M i)}" 

1086 
using sets_PiM_single[of I M] by (simp add: space_P) 

1087 
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

1088 
proof (safe intro!: sets.sigma_sets_subset) 
47694  1089 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 
1090 
have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

1091 
proof (subst measurable_iff_measure_of) 

1092 
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

1093 
from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto 
47694  1094 
show "\<forall>A\<in>E i. (\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 
1095 
proof 

1096 
fix A assume A: "A \<in> E i" 

1097 
then have "(\<lambda>x. x i) ` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))" 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

1098 
using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm) 
47694  1099 
also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)" 
1100 
by (intro PiE_cong) (simp add: S_union) 

49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1101 
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j))" 
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1102 
using T 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset

1103 
apply (auto simp: PiE_iff bchoice_iff) 
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1104 
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI) 
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1105 
apply (auto simp: bij_betw_def) 
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1106 
done 
47694  1107 
also have "\<dots> \<in> sets ?P" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

1108 
proof (safe intro!: sets.countable_UN) 
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1109 
fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P" 
47694  1110 
using A S_in_E 
1111 
by (simp add: P_closed) 

49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset

1112 
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"]) 
47694  1113 
qed 
1114 
finally show "(\<lambda>x. x i) ` A \<inter> space ?P \<in> sets ?P" 

1115 
using P_closed by simp 

1116 
qed 

1117 
qed 

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

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

1120 
by (simp add: E_generates) 

1121 
also have "(\<lambda>x. x i) ` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}" 

1122 
using P_closed by (auto simp: space_PiM) 

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

1124 
qed 

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

1126 
by (simp add: P_closed) 

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

1128 
using `finite I` 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset

1129 
by (auto intro!: sets.sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def) 
47694  1130 
qed 
1131 

50104  1132 
lemma pair_measure_eq_distr_PiM: 
1133 
fixes M1 :: "'a measure" and M2 :: "'a measure" 

1134 
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" 

1135 
shows "(M1 \<Otimes>\<^isub>M M2) = distr (Pi\<^isub>M UNIV (bool_case M1 M2)) (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. (x True, x False))" 

1136 
(is "?P = ?D") 

1137 
proof (rule pair_measure_eqI[OF assms]) 

1138 
interpret B: product_sigma_finite "bool_case M1 M2" 

1139 
unfolding product_sigma_finite_def using assms by (auto split: bool.split) 

1140 
let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)" 

1141 

1142 
have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)" 

1143 
by auto 

1144 
fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2" 

1145 
have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))" 

1146 
by (simp add: UNIV_bool ac_simps) 

1147 
also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))" 

1148 
using A B by (subst B.emeasure_PiM) (auto split: bool.split) 

de19856feb54
move theorems to be more generally useable
hoe 