author  hoelzl 
Fri, 16 Nov 2012 14:46:23 +0100  
changeset 50099  a58bb401af80 
parent 50095  94d7dfa9f404 
child 50123  69b35a75caf3 
permissions  rwrr 
42147  1 
(* Title: HOL/Probability/Infinite_Product_Measure.thy 
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Author: Johannes Hölzl, TU München 

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*) 

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header {*Infinite Product Measure*} 

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theory Infinite_Product_Measure 

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imports Probability_Measure Caratheodory Projective_Family 
42147  9 
begin 
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47694  11 
lemma (in product_prob_space) emeasure_PiM_emb_not_empty: 
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assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. X i \<in> sets (M i)" 

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shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)" 

42147  14 
proof cases 
47694  15 
assume "finite I" with X show ?thesis by simp 
42147  16 
next 
47694  17 
let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space (M i)" 
42147  18 
let ?G = generator 
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assume "\<not> finite I" 

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then have I_not_empty: "I \<noteq> {}" by auto 
47694  21 
interpret G!: algebra ?\<Omega> generator by (rule algebra_generator) fact 
42147  22 
note \<mu>G_mono = 
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G.additive_increasing[OF positive_\<mu>G[OF I_not_empty] additive_\<mu>G[OF I_not_empty], THEN increasingD] 
42147  24 

47694  25 
{ fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> ?G" 
42147  26 

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from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J" 

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by (metis rev_finite_subset subsetI) 

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moreover from Z guess K' X' by (rule generatorE) 

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moreover def K \<equiv> "insert k K'" 

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moreover def X \<equiv> "emb K K' X'" 

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ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X" 

47694  33 
"K  J \<noteq> {}" "K  J \<subseteq> I" "\<mu>G Z = emeasure (Pi\<^isub>M K M) X" 
42147  34 
by (auto simp: subset_insertI) 
49780  35 
let ?M = "\<lambda>y. (\<lambda>x. merge J (K  J) (y, x)) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M)" 
42147  36 
{ fix y assume y: "y \<in> space (Pi\<^isub>M J M)" 
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note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X] 

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moreover 

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have **: "?M y \<in> sets (Pi\<^isub>M (K  J) M)" 

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using J K y by (intro merge_sets) auto 

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ultimately 

49780  42 
have ***: "((\<lambda>x. merge J (I  J) (y, x)) ` Z \<inter> space (Pi\<^isub>M I M)) \<in> ?G" 
42147  43 
using J K by (intro generatorI) auto 
49780  44 
have "\<mu>G ((\<lambda>x. merge J (I  J) (y, x)) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = emeasure (Pi\<^isub>M (K  J) M) (?M y)" 
42147  45 
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto 
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note * ** *** this } 

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note merge_in_G = this 

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have "finite (K  J)" using K by auto 

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interpret J: finite_product_prob_space M J by default fact+ 

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interpret KmJ: finite_product_prob_space M "K  J" by default fact+ 

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47694  54 
have "\<mu>G Z = emeasure (Pi\<^isub>M (J \<union> (K  J)) M) (emb (J \<union> (K  J)) K X)" 
42147  55 
using K J by simp 
47694  56 
also have "\<dots> = (\<integral>\<^isup>+ x. emeasure (Pi\<^isub>M (K  J) M) (?M x) \<partial>Pi\<^isub>M J M)" 
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using K J by (subst emeasure_fold_integral) auto 

49780  58 
also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G ((\<lambda>x. merge J (I  J) (y, x)) ` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)" 
42147  59 
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)") 
47694  60 
proof (intro positive_integral_cong) 
42147  61 
fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 
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with K merge_in_G(2)[OF this] 

47694  63 
show "emeasure (Pi\<^isub>M (K  J) M) (?M x) = \<mu>G (?MZ x)" 
42147  64 
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto 
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qed 

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finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" . 

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

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then have "\<mu>G (?MZ x) \<le> 1" 

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unfolding merge_in_G(4)[OF x] `Z = emb I K X` 

71 
by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) } 

72 
note le_1 = this 

73 

49780  74 
let ?q = "\<lambda>y. \<mu>G ((\<lambda>x. merge J (I  J) (y,x)) ` Z \<inter> space (Pi\<^isub>M I M))" 
42147  75 
have "?q \<in> borel_measurable (Pi\<^isub>M J M)" 
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unfolding `Z = emb I K X` using J K merge_in_G(3) 

47694  77 
by (simp add: merge_in_G \<mu>G_eq emeasure_fold_measurable cong: measurable_cong) 
42147  78 
note this fold le_1 merge_in_G(3) } 
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note fold = this 

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47694  81 
have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>" 
42147  82 
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G]) 
47694  83 
fix A assume "A \<in> ?G" 
42147  84 
with generatorE guess J X . note JX = this 
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interpret JK: finite_product_prob_space M J by default fact+ 
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from JX show "\<mu>G A \<noteq> \<infinity>" by simp 
42147  87 
next 
47694  88 
fix A assume A: "range A \<subseteq> ?G" "decseq A" "(\<Inter>i. A i) = {}" 
42147  89 
then have "decseq (\<lambda>i. \<mu>G (A i))" 
90 
by (auto intro!: \<mu>G_mono simp: decseq_def) 

91 
moreover 

92 
have "(INF i. \<mu>G (A i)) = 0" 

93 
proof (rule ccontr) 

94 
assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0") 

95 
moreover have "0 \<le> ?a" 

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using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def) 
42147  97 
ultimately have "0 < ?a" by auto 
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have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = emeasure (limP J M (\<lambda>J. (Pi\<^isub>M J M))) X" 
42147  100 
using A by (intro allI generator_Ex) auto 
101 
then obtain J' X' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" "\<And>n. X' n \<in> sets (Pi\<^isub>M (J' n) M)" 

102 
and A': "\<And>n. A n = emb I (J' n) (X' n)" 

103 
unfolding choice_iff by blast 

104 
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)" 

105 
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)" 

106 
ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. X n \<in> sets (Pi\<^isub>M (J n) M)" 

107 
by auto 

47694  108 
with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> ?G" 
109 
unfolding J_def X_def by (subst prod_emb_trans) (insert A, auto) 

42147  110 

111 
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" 

112 
unfolding J_def by force 

113 

114 
interpret J: finite_product_prob_space M "J i" for i by default fact+ 

115 

116 
have a_le_1: "?a \<le> 1" 

117 
using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq 

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by (auto intro!: INF_lower2[of 0] J.measure_le_1) 
42147  119 

49780  120 
let ?M = "\<lambda>K Z y. (\<lambda>x. merge K (I  K) (y, x)) ` Z \<inter> space (Pi\<^isub>M I M)" 
42147  121 

47694  122 
{ fix Z k assume Z: "range Z \<subseteq> ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)" 
123 
then have Z_sets: "\<And>n. Z n \<in> ?G" by auto 

42147  124 
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I" 
125 
interpret J': finite_product_prob_space M J' by default fact+ 

126 

46731  127 
let ?q = "\<lambda>n y. \<mu>G (?M J' (Z n) y)" 
128 
let ?Q = "\<lambda>n. ?q n ` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)" 

42147  129 
{ fix n 
130 
have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)" 

131 
using Z J' by (intro fold(1)) auto 

132 
then have "?Q n \<in> sets (Pi\<^isub>M J' M)" 

133 
by (rule measurable_sets) auto } 

134 
note Q_sets = this 

135 

47694  136 
have "?a / 2^(k+1) \<le> (INF n. emeasure (Pi\<^isub>M J' M) (?Q n))" 
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proof (intro INF_greatest) 
42147  138 
fix n 
139 
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto 

140 
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)" 

47694  141 
unfolding fold(2)[OF J' `Z n \<in> ?G`] 
142 
proof (intro positive_integral_mono) 

42147  143 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 
144 
then have "?q n x \<le> 1 + 0" 

145 
using J' Z fold(3) Z_sets by auto 

146 
also have "\<dots> \<le> 1 + ?a / 2^(k+1)" 

147 
using `0 < ?a` by (intro add_mono) auto 

148 
finally have "?q n x \<le> 1 + ?a / 2^(k+1)" . 

149 
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)" 

150 
by (auto split: split_indicator simp del: power_Suc) 

151 
qed 

47694  152 
also have "\<dots> = emeasure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)" 
153 
using `0 \<le> ?a` Q_sets J'.emeasure_space_1 

154 
by (subst positive_integral_add) auto 

155 
finally show "?a / 2^(k+1) \<le> emeasure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1` 

156 
by (cases rule: ereal2_cases[of ?a "emeasure (Pi\<^isub>M J' M) (?Q n)"]) 

42147  157 
(auto simp: field_simps) 
158 
qed 

47694  159 
also have "\<dots> = emeasure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)" 
160 
proof (intro INF_emeasure_decseq) 

42147  161 
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto 
162 
show "decseq ?Q" 

163 
unfolding decseq_def 

164 
proof (safe intro!: vimageI[OF refl]) 

165 
fix m n :: nat assume "m \<le> n" 

166 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 

167 
assume "?a / 2^(k+1) \<le> ?q n x" 

168 
also have "?q n x \<le> ?q m x" 

169 
proof (rule \<mu>G_mono) 

170 
from fold(4)[OF J', OF Z_sets x] 

47694  171 
show "?M J' (Z n) x \<in> ?G" "?M J' (Z m) x \<in> ?G" by auto 
42147  172 
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x" 
173 
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto 

174 
qed 

175 
finally show "?a / 2^(k+1) \<le> ?q m x" . 

176 
qed 

47694  177 
qed simp 
42147  178 
finally have "(\<Inter>n. ?Q n) \<noteq> {}" 
179 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 

180 
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto } 

181 
note Ex_w = this 

182 

46731  183 
let ?q = "\<lambda>k n y. \<mu>G (?M (J k) (A n) y)" 
42147  184 

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have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_lower) 
42147  186 
from Ex_w[OF A(1,2) this J(13), of 0] guess w0 .. note w0 = this 
187 

46731  188 
let ?P = 
189 
"\<lambda>k wk w. w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and> 

190 
(\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)" 

42147  191 
def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))" 
192 

193 
{ fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and> 

194 
(\<forall>n. ?a / 2 ^ (k + 1) \<le> ?q k n (w k)) \<and> (k \<noteq> 0 \<longrightarrow> restrict (w k) (J (k  1)) = w (k  1))" 

195 
proof (induct k) 

196 
case 0 with w0 show ?case 

197 
unfolding w_def nat_rec_0 by auto 

198 
next 

199 
case (Suc k) 

200 
then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

201 
have "\<exists>w'. ?P k (w k) w'" 

202 
proof cases 

203 
assume [simp]: "J k = J (Suc k)" 

204 
show ?thesis 

205 
proof (intro exI[of _ "w k"] conjI allI) 

206 
fix n 

207 
have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)" 

208 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps) 

209 
also have "\<dots> \<le> ?q k n (w k)" using Suc by auto 

210 
finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp 

211 
next 

212 
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)" 

213 
using Suc by simp 

214 
then show "restrict (w k) (J k) = w k" 

47694  215 
by (simp add: extensional_restrict space_PiM) 
42147  216 
qed 
217 
next 

218 
assume "J k \<noteq> J (Suc k)" 

219 
with J_mono[of k "Suc k"] have "J (Suc k)  J k \<noteq> {}" (is "?D \<noteq> {}") by auto 

47694  220 
have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> ?G" 
42147  221 
"decseq (\<lambda>n. ?M (J k) (A n) (w k))" 
222 
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))" 

223 
using `decseq A` fold(4)[OF J(13) A_eq(2), of "w k" k] Suc 

224 
by (auto simp: decseq_def) 

225 
from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"] 

226 
obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)" 

227 
"\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto 

49780  228 
let ?w = "merge (J k) ?D (w k, w')" 
229 
have [simp]: "\<And>x. merge (J k) (I  J k) (w k, merge ?D (I  ?D) (w', x)) = 

230 
merge (J (Suc k)) (I  (J (Suc k))) (?w, x)" 

42147  231 
using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"] 
232 
by (auto intro!: ext split: split_merge) 

233 
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w" 

234 
using w'(1) J(3)[of "Suc k"] 

47694  235 
by (auto simp: space_PiM split: split_merge intro!: extensional_merge_sub) force+ 
42147  236 
show ?thesis 
237 
apply (rule exI[of _ ?w]) 

238 
using w' J_mono[of k "Suc k"] wk unfolding * 

47694  239 
apply (auto split: split_merge intro!: extensional_merge_sub ext simp: space_PiM) 
42147  240 
apply (force simp: extensional_def) 
241 
done 

242 
qed 

243 
then have "?P k (w k) (w (Suc k))" 

244 
unfolding w_def nat_rec_Suc unfolding w_def[symmetric] 

245 
by (rule someI_ex) 

246 
then show ?case by auto 

247 
qed 

248 
moreover 

249 
then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

250 
moreover 

251 
from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto 

252 
then have "?M (J k) (A k) (w k) \<noteq> {}" 

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parents:
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253 
using positive_\<mu>G[OF I_not_empty, unfolded positive_def] `0 < ?a` `?a \<le> 1` 
42147  254 
by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 
255 
then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto 

49780  256 
then have "merge (J k) (I  J k) (w k, x) \<in> A k" by auto 
42147  257 
then have "\<exists>x\<in>A k. restrict x (J k) = w k" 
258 
using `w k \<in> space (Pi\<^isub>M (J k) M)` 

47694  259 
by (intro rev_bexI) (auto intro!: ext simp: extensional_def space_PiM) 
42147  260 
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)" 
261 
"\<exists>x\<in>A k. restrict x (J k) = w k" 

262 
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k  1)) = w (k  1)" 

263 
by auto } 

264 
note w = this 

265 

266 
{ fix k l i assume "k \<le> l" "i \<in> J k" 

267 
{ fix l have "w k i = w (k + l) i" 

268 
proof (induct l) 

269 
case (Suc l) 

270 
from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto 

271 
with w(3)[of "k + Suc l"] 

272 
have "w (k + l) i = w (k + Suc l) i" 

273 
by (auto simp: restrict_def fun_eq_iff split: split_if_asm) 

274 
with Suc show ?case by simp 

275 
qed simp } 

276 
from this[of "l  k"] `k \<le> l` have "w l i = w k i" by simp } 

277 
note w_mono = this 

278 

279 
def w' \<equiv> "\<lambda>i. if i \<in> (\<Union>k. J k) then w (LEAST k. i \<in> J k) i else if i \<in> I then (SOME x. x \<in> space (M i)) else undefined" 

280 
{ fix i k assume k: "i \<in> J k" 

281 
have "w k i = w (LEAST k. i \<in> J k) i" 

282 
by (intro w_mono Least_le k LeastI[of _ k]) 

283 
then have "w' i = w k i" 

284 
unfolding w'_def using k by auto } 

285 
note w'_eq = this 

286 
have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined" 

287 
using J by (auto simp: w'_def) 

288 
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)" 

289 
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]]) 

290 
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)" 

47694  291 
using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq space_PiM)+ } 
42147  292 
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this 
293 

294 
have w': "w' \<in> space (Pi\<^isub>M I M)" 

47694  295 
using w(1) by (auto simp add: Pi_iff extensional_def space_PiM) 
42147  296 

297 
{ fix n 

298 
have "restrict w' (J n) = w n" using w(1) 

47694  299 
by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def space_PiM) 
42147  300 
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto 
47694  301 
then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: prod_emb_def space_PiM) } 
42147  302 
then have "w' \<in> (\<Inter>i. A i)" by auto 
303 
with `(\<Inter>i. A i) = {}` show False by auto 

304 
qed 

305 
ultimately show "(\<lambda>i. \<mu>G (A i)) > 0" 

43920  306 
using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp 
45777
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307 
qed fact+ 
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308 
then guess \<mu> .. note \<mu> = this 
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309 
show ?thesis 
47694  310 
proof (subst emeasure_extend_measure_Pair[OF PiM_def, of I M \<mu> J X]) 
311 
from assms show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))" 

312 
by (simp add: Pi_iff) 

313 
next 

314 
fix J X assume J: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))" 

315 
then show "emb I J (Pi\<^isub>E J X) \<in> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))" 

316 
by (auto simp: Pi_iff prod_emb_def dest: sets_into_space) 

317 
have "emb I J (Pi\<^isub>E J X) \<in> generator" 

50003  318 
using J `I \<noteq> {}` by (intro generatorI') (auto simp: Pi_iff) 
47694  319 
then have "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" 
320 
using \<mu> by simp 

321 
also have "\<dots> = (\<Prod> j\<in>J. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" 

322 
using J `I \<noteq> {}` by (subst \<mu>G_spec[OF _ _ _ refl]) (auto simp: emeasure_PiM Pi_iff) 

323 
also have "\<dots> = (\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. 

324 
if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" 

325 
using J `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1) 

326 
finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = \<dots>" . 

327 
next 

328 
let ?F = "\<lambda>j. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j))" 

329 
have "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = (\<Prod>j\<in>J. ?F j)" 

330 
using X `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1) 

331 
then show "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = 

332 
emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)" 

333 
using X by (auto simp add: emeasure_PiM) 

334 
next 

335 
show "positive (sets (Pi\<^isub>M I M)) \<mu>" "countably_additive (sets (Pi\<^isub>M I M)) \<mu>" 

49804  336 
using \<mu> unfolding sets_PiM_generator by (auto simp: measure_space_def) 
42147  337 
qed 
338 
qed 

339 

47694  340 
sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>M I M" 
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341 
proof 
47694  342 
show "emeasure (Pi\<^isub>M I M) (space (Pi\<^isub>M I M)) = 1" 
343 
proof cases 

344 
assume "I = {}" then show ?thesis by (simp add: space_PiM_empty) 

345 
next 

346 
assume "I \<noteq> {}" 

347 
then obtain i where "i \<in> I" by auto 

348 
moreover then have "emb I {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i)) = (space (Pi\<^isub>M I M))" 

349 
by (auto simp: prod_emb_def space_PiM) 

350 
ultimately show ?thesis 

351 
using emeasure_PiM_emb_not_empty[of "{i}" "\<lambda>i. space (M i)"] 

352 
by (simp add: emeasure_PiM emeasure_space_1) 

353 
qed 

42257
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354 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
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diff
changeset

355 

47694  356 
lemma (in product_prob_space) emeasure_PiM_emb: 
357 
assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" 

358 
shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. emeasure (M i) (X i))" 

359 
proof cases 

360 
assume "J = {}" 

361 
moreover have "emb I {} {\<lambda>x. undefined} = space (Pi\<^isub>M I M)" 

362 
by (auto simp: space_PiM prod_emb_def) 

363 
ultimately show ?thesis 

364 
by (simp add: space_PiM_empty P.emeasure_space_1) 

365 
next 

366 
assume "J \<noteq> {}" with X show ?thesis 

367 
by (subst emeasure_PiM_emb_not_empty) (auto simp: emeasure_PiM) 

42257
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diff
changeset

368 
qed 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
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diff
changeset

369 

50000
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370 
lemma (in product_prob_space) emeasure_PiM_Collect: 
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371 
assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" 
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changeset

372 
shows "emeasure (Pi\<^isub>M I M) {x\<in>space (Pi\<^isub>M I M). \<forall>i\<in>J. x i \<in> X i} = (\<Prod> i\<in>J. emeasure (M i) (X i))" 
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changeset

373 
proof  
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changeset

374 
have "{x\<in>space (Pi\<^isub>M I M). \<forall>i\<in>J. x i \<in> X i} = emb I J (Pi\<^isub>E J X)" 
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changeset

375 
unfolding prod_emb_def using assms by (auto simp: space_PiM Pi_iff) 
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376 
with emeasure_PiM_emb[OF assms] show ?thesis by simp 
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changeset

377 
qed 
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infinite product measure is invariant under adding prefixes
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parents:
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diff
changeset

378 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
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changeset

379 
lemma (in product_prob_space) emeasure_PiM_Collect_single: 
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changeset

380 
assumes X: "i \<in> I" "A \<in> sets (M i)" 
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diff
changeset

381 
shows "emeasure (Pi\<^isub>M I M) {x\<in>space (Pi\<^isub>M I M). x i \<in> A} = emeasure (M i) A" 
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changeset

382 
using emeasure_PiM_Collect[of "{i}" "\<lambda>i. A"] assms 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

383 
by simp 
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infinite product measure is invariant under adding prefixes
hoelzl
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diff
changeset

384 

47694  385 
lemma (in product_prob_space) measure_PiM_emb: 
386 
assumes "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" 

387 
shows "measure (PiM I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. measure (M i) (X i))" 

388 
using emeasure_PiM_emb[OF assms] 

389 
unfolding emeasure_eq_measure M.emeasure_eq_measure by (simp add: setprod_ereal) 

42865  390 

50000
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infinite product measure is invariant under adding prefixes
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changeset

391 
lemma sets_Collect_single': 
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infinite product measure is invariant under adding prefixes
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diff
changeset

392 
"i \<in> I \<Longrightarrow> {x\<in>space (M i). P x} \<in> sets (M i) \<Longrightarrow> {x\<in>space (PiM I M). P (x i)} \<in> sets (PiM I M)" 
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infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

393 
using sets_Collect_single[of i I "{x\<in>space (M i). P x}" M] 
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infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

394 
by (simp add: space_PiM Pi_iff cong: conj_cong) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

395 

47694  396 
lemma (in finite_product_prob_space) finite_measure_PiM_emb: 
397 
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> measure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))" 

398 
using measure_PiM_emb[of I A] finite_index prod_emb_PiE_same_index[OF sets_into_space, of I A M] 

399 
by auto 

42865  400 

50000
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infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

401 
lemma (in product_prob_space) PiM_component: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

402 
assumes "i \<in> I" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

403 
shows "distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

404 
proof (rule measure_eqI[symmetric]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

405 
fix A assume "A \<in> sets (M i)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

406 
moreover have "((\<lambda>\<omega>. \<omega> i) ` A \<inter> space (PiM I M)) = {x\<in>space (PiM I M). x i \<in> A}" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

407 
by auto 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

408 
ultimately show "emeasure (M i) A = emeasure (distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i)) A" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

409 
by (auto simp: `i\<in>I` emeasure_distr measurable_component_singleton emeasure_PiM_Collect_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

410 
qed simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

411 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

412 
lemma (in product_prob_space) PiM_eq: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

413 
assumes "I \<noteq> {}" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

414 
assumes "sets M' = sets (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

415 
assumes eq: "\<And>J F. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>j. j \<in> J \<Longrightarrow> F j \<in> sets (M j)) \<Longrightarrow> 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

416 
emeasure M' (prod_emb I M J (\<Pi>\<^isub>E j\<in>J. F j)) = (\<Prod>j\<in>J. emeasure (M j) (F j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

417 
shows "M' = (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

418 
proof (rule measure_eqI_generator_eq[symmetric, OF Int_stable_prod_algebra prod_algebra_sets_into_space]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

419 
show "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

420 
by (rule sets_PiM) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

421 
then show "sets M' = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

422 
unfolding `sets M' = sets (PiM I M)` by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

423 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

424 
def i \<equiv> "SOME i. i \<in> I" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

425 
with `I \<noteq> {}` have i: "i \<in> I" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

426 
by (auto intro: someI_ex) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

427 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

428 
def A \<equiv> "\<lambda>n::nat. prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. space (M i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

429 
then show "range A \<subseteq> prod_algebra I M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

430 
by (auto intro!: prod_algebraI i) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

431 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

432 
have A_eq: "\<And>i. A i = space (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

433 
by (auto simp: prod_emb_def space_PiM Pi_iff A_def i) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

434 
show "(\<Union>i. A i) = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

435 
unfolding A_eq by (auto simp: space_PiM) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

436 
show "\<And>i. emeasure (PiM I M) (A i) \<noteq> \<infinity>" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

437 
unfolding A_eq P.emeasure_space_1 by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

438 
next 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

439 
fix X assume X: "X \<in> prod_algebra I M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

440 
then obtain J E where X: "X = prod_emb I M J (PIE j:J. E j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

441 
and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

442 
by (force elim!: prod_algebraE) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

443 
from eq[OF J] have "emeasure M' X = (\<Prod>j\<in>J. emeasure (M j) (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

444 
by (simp add: X) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

445 
also have "\<dots> = emeasure (PiM I M) X" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

446 
unfolding X using J by (intro emeasure_PiM_emb[symmetric]) auto 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

447 
finally show "emeasure (PiM I M) X = emeasure M' X" .. 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

448 
qed 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

449 

42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

450 
subsection {* Sequence space *} 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

451 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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diff
changeset

452 
definition comb_seq :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a)" where 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

453 
"comb_seq i \<omega> \<omega>' j = (if j < i then \<omega> j else \<omega>' (j  i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

454 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

455 
lemma split_comb_seq: "P (comb_seq i \<omega> \<omega>' j) \<longleftrightarrow> (j < i \<longrightarrow> P (\<omega> j)) \<and> (\<forall>k. j = i + k \<longrightarrow> P (\<omega>' k))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

456 
by (auto simp: comb_seq_def not_less) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

457 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

458 
lemma split_comb_seq_asm: "P (comb_seq i \<omega> \<omega>' j) \<longleftrightarrow> \<not> ((j < i \<and> \<not> P (\<omega> j)) \<or> (\<exists>k. j = i + k \<and> \<not> P (\<omega>' k)))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

459 
by (auto simp: comb_seq_def) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

460 

50099  461 
lemma measurable_comb_seq: 
462 
"(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> measurable ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)) (\<Pi>\<^isub>M i\<in>UNIV. M)" 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

463 
proof (rule measurable_PiM_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

464 
show "(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> space ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)) \<rightarrow> (UNIV \<rightarrow>\<^isub>E space M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

465 
by (auto simp: space_pair_measure space_PiM Pi_iff split: split_comb_seq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

466 
fix j :: nat and A assume A: "A \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

467 
then have *: "{\<omega> \<in> space ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)). prod_case (comb_seq i) \<omega> j \<in> A} = 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

468 
(if j < i then {\<omega> \<in> space (\<Pi>\<^isub>M i\<in>UNIV. M). \<omega> j \<in> A} \<times> space (\<Pi>\<^isub>M i\<in>UNIV. M) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

469 
else space (\<Pi>\<^isub>M i\<in>UNIV. M) \<times> {\<omega> \<in> space (\<Pi>\<^isub>M i\<in>UNIV. M). \<omega> (j  i) \<in> A})" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

470 
by (auto simp: space_PiM space_pair_measure comb_seq_def dest: sets_into_space) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

471 
show "{\<omega> \<in> space ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)). prod_case (comb_seq i) \<omega> j \<in> A} \<in> sets ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

472 
unfolding * by (auto simp: A intro!: sets_Collect_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

473 
qed 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

474 

50099  475 
lemma measurable_comb_seq'[measurable (raw)]: 
50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

476 
assumes f: "f \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" and g: "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

477 
shows "(\<lambda>x. comb_seq i (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

478 
using measurable_compose[OF measurable_Pair[OF f g] measurable_comb_seq] by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

479 

50099  480 
lemma comb_seq_0: "comb_seq 0 \<omega> \<omega>' = \<omega>'" 
481 
by (auto simp add: comb_seq_def) 

482 

483 
lemma comb_seq_Suc: "comb_seq (Suc n) \<omega> \<omega>' = comb_seq n \<omega> (nat_case (\<omega> n) \<omega>')" 

484 
by (auto simp add: comb_seq_def not_less less_Suc_eq le_imp_diff_is_add intro!: ext split: nat.split) 

485 

486 
lemma comb_seq_Suc_0[simp]: "comb_seq (Suc 0) \<omega> = nat_case (\<omega> 0)" 

487 
by (intro ext) (simp add: comb_seq_Suc comb_seq_0) 

488 

489 
lemma comb_seq_less: "i < n \<Longrightarrow> comb_seq n \<omega> \<omega>' i = \<omega> i" 

490 
by (auto split: split_comb_seq) 

491 

492 
lemma comb_seq_add: "comb_seq n \<omega> \<omega>' (i + n) = \<omega>' i" 

493 
by (auto split: nat.split split_comb_seq) 

494 

495 
lemma nat_case_comb_seq: "nat_case s' (comb_seq n \<omega> \<omega>') (i + n) = nat_case (nat_case s' \<omega> n) \<omega>' i" 

496 
by (auto split: nat.split split_comb_seq) 

497 

498 
lemma nat_case_comb_seq': 

499 
"nat_case s (comb_seq i \<omega> \<omega>') = comb_seq (Suc i) (nat_case s \<omega>) \<omega>'" 

500 
by (auto split: split_comb_seq nat.split) 

501 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

502 
locale sequence_space = product_prob_space "\<lambda>i. M" "UNIV :: nat set" for M 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

503 
begin 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

504 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

505 
abbreviation "S \<equiv> \<Pi>\<^isub>M i\<in>UNIV::nat set. M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

506 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

507 
lemma infprod_in_sets[intro]: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

508 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

509 
shows "Pi UNIV E \<in> sets S" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

510 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

511 
have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))" 
47694  512 
using E E[THEN sets_into_space] 
513 
by (auto simp: prod_emb_def Pi_iff extensional_def) blast 

514 
with E show ?thesis by auto 

42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

515 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

516 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

517 
lemma measure_PiM_countable: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

518 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

519 
shows "(\<lambda>n. \<Prod>i\<le>n. measure M (E i)) > measure S (Pi UNIV E)" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

520 
proof  
46731  521 
let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^isub>E {.. n} E)" 
50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

522 
have "\<And>n. (\<Prod>i\<le>n. measure M (E i)) = measure S (?E n)" 
47694  523 
using E by (simp add: measure_PiM_emb) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

524 
moreover have "Pi UNIV E = (\<Inter>n. ?E n)" 
47694  525 
using E E[THEN sets_into_space] 
526 
by (auto simp: prod_emb_def extensional_def Pi_iff) blast 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

527 
moreover have "range ?E \<subseteq> sets S" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

528 
using E by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

529 
moreover have "decseq ?E" 
47694  530 
by (auto simp: prod_emb_def Pi_iff decseq_def) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

531 
ultimately show ?thesis 
47694  532 
by (simp add: finite_Lim_measure_decseq) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

533 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

534 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

535 
lemma nat_eq_diff_eq: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

536 
fixes a b c :: nat 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

537 
shows "c \<le> b \<Longrightarrow> a = b  c \<longleftrightarrow> a + c = b" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

538 
by auto 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

539 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

540 
lemma PiM_comb_seq: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

541 
"distr (S \<Otimes>\<^isub>M S) S (\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') = S" (is "?D = _") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

542 
proof (rule PiM_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

543 
let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

544 
let "distr _ _ ?f" = "?D" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

545 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

546 
fix J E assume J: "finite J" "J \<subseteq> ?I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

547 
let ?X = "prod_emb ?I ?M J (\<Pi>\<^isub>E j\<in>J. E j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

548 
have "\<And>j x. j \<in> J \<Longrightarrow> x \<in> E j \<Longrightarrow> x \<in> space M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

549 
using J(3)[THEN sets_into_space] by (auto simp: space_PiM Pi_iff subset_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

550 
with J have "?f ` ?X \<inter> space (S \<Otimes>\<^isub>M S) = 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

551 
(prod_emb ?I ?M (J \<inter> {..<i}) (PIE j:J \<inter> {..<i}. E j)) \<times> 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

552 
(prod_emb ?I ?M ((op + i) ` J) (PIE j:(op + i) ` J. E (i + j)))" (is "_ = ?E \<times> ?F") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

553 
by (auto simp: space_pair_measure space_PiM prod_emb_def all_conj_distrib Pi_iff 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

554 
split: split_comb_seq split_comb_seq_asm) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

555 
then have "emeasure ?D ?X = emeasure (S \<Otimes>\<^isub>M S) (?E \<times> ?F)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

556 
by (subst emeasure_distr[OF measurable_comb_seq]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

557 
(auto intro!: sets_PiM_I simp: split_beta' J) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

558 
also have "\<dots> = emeasure S ?E * emeasure S ?F" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

559 
using J by (intro P.emeasure_pair_measure_Times) (auto intro!: sets_PiM_I finite_vimageI simp: inj_on_def) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

560 
also have "emeasure S ?F = (\<Prod>j\<in>(op + i) ` J. emeasure M (E (i + j)))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

561 
using J by (intro emeasure_PiM_emb) (simp_all add: finite_vimageI inj_on_def) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

562 
also have "\<dots> = (\<Prod>j\<in>J  (J \<inter> {..<i}). emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

563 
by (rule strong_setprod_reindex_cong[where f="\<lambda>x. x  i"]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

564 
(auto simp: image_iff Bex_def not_less nat_eq_diff_eq ac_simps cong: conj_cong intro!: inj_onI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

565 
also have "emeasure S ?E = (\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

566 
using J by (intro emeasure_PiM_emb) simp_all 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

567 
also have "(\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j)) * (\<Prod>j\<in>J  (J \<inter> {..<i}). emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

568 
by (subst mult_commute) (auto simp: J setprod_subset_diff[symmetric]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

569 
finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" . 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

570 
qed simp_all 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

571 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

572 
lemma PiM_iter: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

573 
"distr (M \<Otimes>\<^isub>M S) S (\<lambda>(s, \<omega>). nat_case s \<omega>) = S" (is "?D = _") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

574 
proof (rule PiM_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

575 
let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

576 
let "distr _ _ ?f" = "?D" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

577 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

578 
fix J E assume J: "finite J" "J \<subseteq> ?I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

579 
let ?X = "prod_emb ?I ?M J (PIE j:J. E j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

580 
have "\<And>j x. j \<in> J \<Longrightarrow> x \<in> E j \<Longrightarrow> x \<in> space M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

581 
using J(3)[THEN sets_into_space] by (auto simp: space_PiM Pi_iff subset_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

582 
with J have "?f ` ?X \<inter> space (M \<Otimes>\<^isub>M S) = (if 0 \<in> J then E 0 else space M) \<times> 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

583 
(prod_emb ?I ?M (Suc ` J) (PIE j:Suc ` J. E (Suc j)))" (is "_ = ?E \<times> ?F") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

584 
by (auto simp: space_pair_measure space_PiM Pi_iff prod_emb_def all_conj_distrib 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

585 
split: nat.split nat.split_asm) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

586 
then have "emeasure ?D ?X = emeasure (M \<Otimes>\<^isub>M S) (?E \<times> ?F)" 
50099  587 
by (subst emeasure_distr) 
50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

588 
(auto intro!: sets_PiM_I simp: split_beta' J) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

589 
also have "\<dots> = emeasure M ?E * emeasure S ?F" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

590 
using J by (intro P.emeasure_pair_measure_Times) (auto intro!: sets_PiM_I finite_vimageI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

591 
also have "emeasure S ?F = (\<Prod>j\<in>Suc ` J. emeasure M (E (Suc j)))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

592 
using J by (intro emeasure_PiM_emb) (simp_all add: finite_vimageI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

593 
also have "\<dots> = (\<Prod>j\<in>J  {0}. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

594 
by (rule strong_setprod_reindex_cong[where f="\<lambda>x. x  1"]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
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595 
(auto simp: image_iff Bex_def not_less nat_eq_diff_eq ac_simps cong: conj_cong intro!: inj_onI) 
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596 
also have "emeasure M ?E * (\<Prod>j\<in>J  {0}. emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))" 
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infinite product measure is invariant under adding prefixes
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597 
by (auto simp: M.emeasure_space_1 setprod.remove J) 
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infinite product measure is invariant under adding prefixes
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598 
finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" . 
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infinite product measure is invariant under adding prefixes
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parents:
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599 
qed simp_all 
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infinite product measure is invariant under adding prefixes
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600 

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infinite product measure is invariant under adding prefixes
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601 
end 
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infinite product measure is invariant under adding prefixes
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602 

42147  603 
end 