author  hoelzl 
Tue, 05 Apr 2011 19:55:04 +0200  
changeset 42257  08d717c82828 
parent 42166  efd229daeb2c 
child 42865  36353d913424 
permissions  rwrr 
42147  1 
(* Title: HOL/Probability/Infinite_Product_Measure.thy 
2 
Author: Johannes Hölzl, TU München 

3 
*) 

4 

5 
header {*Infinite Product Measure*} 

6 

7 
theory Infinite_Product_Measure 

42148  8 
imports Probability_Measure 
42147  9 
begin 
10 

11 
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B" 

12 
unfolding restrict_def extensional_def by auto 

13 

14 
lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)" 

15 
unfolding restrict_def by (simp add: fun_eq_iff) 

16 

17 
lemma split_merge: "P (merge I x J 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)" 

18 
unfolding merge_def by auto 

19 

20 
lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I x J y \<in> extensional K" 

21 
unfolding merge_def extensional_def by auto 

22 

23 
lemma injective_vimage_restrict: 

24 
assumes J: "J \<subseteq> I" 

25 
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> {}" 

26 
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)" 

27 
shows "A = B" 

28 
proof (intro set_eqI) 

29 
fix x 

30 
from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto 

31 
have "J \<inter> (I  J) = {}" by auto 

32 
show "x \<in> A \<longleftrightarrow> x \<in> B" 

33 
proof cases 

34 
assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)" 

35 
have "x \<in> A \<longleftrightarrow> merge J x (I  J) y \<in> (\<lambda>x. restrict x J) ` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" 

36 
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub split: split_merge) 

37 
then show "x \<in> A \<longleftrightarrow> x \<in> B" 

38 
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub eq split: split_merge) 

39 
next 

40 
assume "x \<notin> (\<Pi>\<^isub>E i\<in>J. S i)" with sets show "x \<in> A \<longleftrightarrow> x \<in> B" by auto 

41 
qed 

42 
qed 

43 

44 
locale product_prob_space = 

45 
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set" 

46 
assumes prob_spaces: "\<And>i. prob_space (M i)" 

47 
and I_not_empty: "I \<noteq> {}" 

48 

49 
locale finite_product_prob_space = product_prob_space M I 

50 
for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set" + 

51 
assumes finite_index'[intro]: "finite I" 

52 

53 
sublocale product_prob_space \<subseteq> M: prob_space "M i" for i 

54 
by (rule prob_spaces) 

55 

56 
sublocale product_prob_space \<subseteq> product_sigma_finite 

57 
by default 

58 

59 
sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite 

60 
by default (fact finite_index') 

61 

62 
sublocale finite_product_prob_space \<subseteq> prob_space "Pi\<^isub>M I M" 

63 
proof 

64 
show "measure P (space P) = 1" 

65 
by (simp add: measure_times measure_space_1 setprod_1) 

66 
qed 

67 

68 
lemma (in product_prob_space) measure_preserving_restrict: 

69 
assumes "J \<noteq> {}" "J \<subseteq> K" "finite K" 

70 
shows "(\<lambda>f. restrict f J) \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" (is "?R \<in> _") 

71 
proof  

72 
interpret K: finite_product_prob_space M K 

73 
by default (insert assms, auto) 

74 
have J: "J \<noteq> {}" "finite J" using assms by (auto simp add: finite_subset) 

75 
interpret J: finite_product_prob_space M J 

76 
by default (insert J, auto) 

77 
from J.sigma_finite_pairs guess F .. note F = this 

78 
then have [simp,intro]: "\<And>k i. k \<in> J \<Longrightarrow> F k i \<in> sets (M k)" 

79 
by auto 

80 
let "?F i" = "\<Pi>\<^isub>E k\<in>J. F k i" 

81 
let ?J = "product_algebra_generator J M \<lparr> measure := measure (Pi\<^isub>M J M) \<rparr>" 

82 
have "?R \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (sigma ?J)" 

83 
proof (rule K.measure_preserving_Int_stable) 

84 
show "Int_stable ?J" 

85 
by (auto simp: Int_stable_def product_algebra_generator_def PiE_Int) 

86 
show "range ?F \<subseteq> sets ?J" "incseq ?F" "(\<Union>i. ?F i) = space ?J" 

87 
using F by auto 

88 
show "\<And>i. measure ?J (?F i) \<noteq> \<infinity>" 

89 
using F by (simp add: J.measure_times setprod_PInf) 

90 
have "measure_space (Pi\<^isub>M J M)" by default 

91 
then show "measure_space (sigma ?J)" 

92 
by (simp add: product_algebra_def sigma_def) 

93 
show "?R \<in> measure_preserving (Pi\<^isub>M K M) ?J" 

94 
proof (simp add: measure_preserving_def measurable_def product_algebra_generator_def del: vimage_Int, 

95 
safe intro!: restrict_extensional) 

96 
fix x k assume "k \<in> J" "x \<in> (\<Pi> i\<in>K. space (M i))" 

97 
then show "x k \<in> space (M k)" using `J \<subseteq> K` by auto 

98 
next 

99 
fix E assume "E \<in> (\<Pi> i\<in>J. sets (M i))" 

100 
then have E: "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" by auto 

101 
then have *: "?R ` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i)) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))" 

102 
(is "?X = Pi\<^isub>E K ?M") 

103 
using `J \<subseteq> K` sets_into_space by (auto simp: Pi_iff split: split_if_asm) blast+ 

104 
with E show "?X \<in> sets (Pi\<^isub>M K M)" 

105 
by (auto intro!: product_algebra_generatorI) 

106 
have "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = (\<Prod>i\<in>J. measure (M i) (?M i))" 

107 
using E by (simp add: J.measure_times) 

108 
also have "\<dots> = measure (Pi\<^isub>M K M) ?X" 

109 
unfolding * using E `finite K` `J \<subseteq> K` 

110 
by (auto simp: K.measure_times M.measure_space_1 

111 
cong del: setprod_cong 

112 
intro!: setprod_mono_one_left) 

113 
finally show "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = measure (Pi\<^isub>M K M) ?X" . 

114 
qed 

115 
qed 

116 
then show ?thesis 

117 
by (simp add: product_algebra_def sigma_def) 

118 
qed 

119 

120 
lemma (in product_prob_space) measurable_restrict: 

121 
assumes *: "J \<noteq> {}" "J \<subseteq> K" "finite K" 

122 
shows "(\<lambda>f. restrict f J) \<in> measurable (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" 

123 
using measure_preserving_restrict[OF *] 

124 
by (rule measure_preservingD2) 

125 

126 
definition (in product_prob_space) 

127 
"emb J K X = (\<lambda>x. restrict x K) ` X \<inter> space (Pi\<^isub>M J M)" 

128 

129 
lemma (in product_prob_space) emb_trans[simp]: 

130 
"J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> emb L K (emb K J X) = emb L J X" 

131 
by (auto simp add: Int_absorb1 emb_def) 

132 

133 
lemma (in product_prob_space) emb_empty[simp]: 

134 
"emb K J {} = {}" 

135 
by (simp add: emb_def) 

136 

137 
lemma (in product_prob_space) emb_Pi: 

138 
assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K" 

139 
shows "emb K J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))" 

140 
using assms space_closed 

141 
by (auto simp: emb_def Pi_iff split: split_if_asm) blast+ 

142 

143 
lemma (in product_prob_space) emb_injective: 

144 
assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)" 

145 
assumes "emb L J X = emb L J Y" 

146 
shows "X = Y" 

147 
proof  

148 
interpret J: finite_product_sigma_finite M J by default fact 

149 
show "X = Y" 

150 
proof (rule injective_vimage_restrict) 

151 
show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" 

152 
using J.sets_into_space sets by auto 

153 
have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)" 

154 
using M.not_empty by auto 

155 
from bchoice[OF this] 

156 
show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto 

157 
show "(\<lambda>x. restrict x J) ` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) ` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))" 

158 
using `emb L J X = emb L J Y` by (simp add: emb_def) 

159 
qed fact 

160 
qed 

161 

162 
lemma (in product_prob_space) emb_id: 

163 
"B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> emb L L B = B" 

164 
by (auto simp: emb_def Pi_iff subset_eq extensional_restrict) 

165 

166 
lemma (in product_prob_space) emb_simps: 

167 
shows "emb L K (A \<union> B) = emb L K A \<union> emb L K B" 

168 
and "emb L K (A \<inter> B) = emb L K A \<inter> emb L K B" 

169 
and "emb L K (A  B) = emb L K A  emb L K B" 

170 
by (auto simp: emb_def) 

171 

172 
lemma (in product_prob_space) measurable_emb[intro,simp]: 

173 
assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)" 

174 
shows "emb L J X \<in> sets (Pi\<^isub>M L M)" 

175 
using measurable_restrict[THEN measurable_sets, OF *] by (simp add: emb_def) 

176 

177 
lemma (in product_prob_space) measure_emb[intro,simp]: 

178 
assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)" 

179 
shows "measure (Pi\<^isub>M L M) (emb L J X) = measure (Pi\<^isub>M J M) X" 

180 
using measure_preserving_restrict[THEN measure_preservingD, OF *] 

181 
by (simp add: emb_def) 

182 

183 
definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) measure_space" where 

184 
"generator = \<lparr> 

185 
space = (\<Pi>\<^isub>E i\<in>I. space (M i)), 

186 
sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)), 

187 
measure = undefined 

188 
\<rparr>" 

189 

190 
lemma (in product_prob_space) generatorI: 

191 
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> sets generator" 

192 
unfolding generator_def by auto 

193 

194 
lemma (in product_prob_space) generatorI': 

195 
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> sets generator" 

196 
unfolding generator_def by auto 

197 

198 
lemma (in product_sigma_finite) 

199 
assumes "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" 

200 
shows measure_fold_integral: 

201 
"measure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. measure (Pi\<^isub>M J M) (merge I x J ` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I) 

202 
and measure_fold_measurable: 

203 
"(\<lambda>x. measure (Pi\<^isub>M J M) (merge I x J ` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B) 

204 
proof  

205 
interpret I: finite_product_sigma_finite M I by default fact 

206 
interpret J: finite_product_sigma_finite M J by default fact 

207 
interpret IJ: pair_sigma_finite I.P J.P .. 

208 
show ?I 

209 
unfolding measure_fold[OF assms] 

210 
apply (subst IJ.pair_measure_alt) 

211 
apply (intro measurable_sets[OF _ A] measurable_merge assms) 

212 
apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure 

213 
intro!: I.positive_integral_cong) 

214 
done 

215 

216 
have "(\<lambda>(x, y). merge I x J y) ` A \<inter> space (I.P \<Otimes>\<^isub>M J.P) \<in> sets (I.P \<Otimes>\<^isub>M J.P)" 

217 
by (intro measurable_sets[OF _ A] measurable_merge assms) 

218 
from IJ.measure_cut_measurable_fst[OF this] 

219 
show ?B 

220 
apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure) 

221 
apply (subst (asm) measurable_cong) 

222 
apply auto 

223 
done 

224 
qed 

225 

226 
lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1" 

227 
unfolding measure_space_1[symmetric] 

228 
using sets_into_space 

229 
by (intro measure_mono) auto 

230 

231 
definition (in product_prob_space) 

232 
"\<mu>G A = 

233 
(THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = measure (Pi\<^isub>M J M) X))" 

234 

235 
lemma (in product_prob_space) \<mu>G_spec: 

236 
assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" 

237 
shows "\<mu>G A = measure (Pi\<^isub>M J M) X" 

238 
unfolding \<mu>G_def 

239 
proof (intro the_equality allI impI ballI) 

240 
fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)" 

241 
have "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)" 

242 
using K J by simp 

243 
also have "emb (K \<union> J) K Y = emb (K \<union> J) J X" 

244 
using K J by (simp add: emb_injective[of "K \<union> J" I]) 

245 
also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X" 

246 
using K J by simp 

247 
finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" .. 

248 
qed (insert J, force) 

249 

250 
lemma (in product_prob_space) \<mu>G_eq: 

251 
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = measure (Pi\<^isub>M J M) X" 

252 
by (intro \<mu>G_spec) auto 

253 

254 
lemma (in product_prob_space) generator_Ex: 

255 
assumes *: "A \<in> sets generator" 

256 
shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = measure (Pi\<^isub>M J M) X" 

257 
proof  

258 
from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" 

259 
unfolding generator_def by auto 

260 
with \<mu>G_spec[OF this] show ?thesis by auto 

261 
qed 

262 

263 
lemma (in product_prob_space) generatorE: 

264 
assumes A: "A \<in> sets generator" 

265 
obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = measure (Pi\<^isub>M J M) X" 

266 
proof  

267 
from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" 

268 
"\<mu>G A = measure (Pi\<^isub>M J M) X" by auto 

269 
then show thesis by (intro that) auto 

270 
qed 

271 

272 
lemma (in product_prob_space) merge_sets: 

273 
assumes "finite J" "finite K" "J \<inter> K = {}" and A: "A \<in> sets (Pi\<^isub>M (J \<union> K) M)" and x: "x \<in> space (Pi\<^isub>M J M)" 

274 
shows "merge J x K ` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)" 

275 
proof  

276 
interpret J: finite_product_sigma_algebra M J by default fact 

277 
interpret K: finite_product_sigma_algebra M K by default fact 

278 
interpret JK: pair_sigma_algebra J.P K.P .. 

279 

280 
from JK.measurable_cut_fst[OF 

281 
measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x 

282 
show ?thesis 

283 
by (simp add: space_pair_measure comp_def vimage_compose[symmetric]) 

284 
qed 

285 

286 
lemma (in product_prob_space) merge_emb: 

287 
assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)" 

288 
shows "(merge J y (I  J) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = 

289 
emb I (K  J) (merge J y (K  J) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M))" 

290 
proof  

291 
have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x" 

292 
by (auto simp: restrict_def merge_def) 

293 
have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x" 

294 
by (auto simp: restrict_def merge_def) 

295 
have [simp]: "(I  J) \<inter> K = K  J" using `K \<subseteq> I` `J \<subseteq> I` by auto 

296 
have [simp]: "(K  J) \<inter> (K \<union> J) = K  J" by auto 

297 
have [simp]: "(K  J) \<inter> K = K  J" by auto 

298 
from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis 

299 
by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto 

300 
qed 

301 

302 
definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where 

303 
"infprod_algebra = sigma generator \<lparr> measure := 

304 
(SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and> 

305 
measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>" 

306 

307 
syntax 

308 
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3PIP _:_./ _)" 10) 

309 

310 
syntax (xsymbols) 

311 
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10) 

312 

313 
syntax (HTML output) 

314 
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10) 

315 

316 
abbreviation 

317 
"Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I" 

318 

319 
translations 

320 
"PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)" 

321 

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

322 
sublocale product_prob_space \<subseteq> G!: algebra generator 
42147  323 
proof 
324 
let ?G = generator 

325 
show "sets ?G \<subseteq> Pow (space ?G)" 

326 
by (auto simp: generator_def emb_def) 

327 
from I_not_empty 

328 
obtain i where "i \<in> I" by auto 

329 
then show "{} \<in> sets ?G" 

330 
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"] 

331 
simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def) 

332 
from `i \<in> I` show "space ?G \<in> sets ?G" 

333 
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"] 

334 
simp: generator_def emb_def) 

335 
fix A assume "A \<in> sets ?G" 

336 
then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA" 

337 
by (auto simp: generator_def) 

338 
fix B assume "B \<in> sets ?G" 

339 
then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB" 

340 
by (auto simp: generator_def) 

341 
let ?RA = "emb (JA \<union> JB) JA XA" 

342 
let ?RB = "emb (JA \<union> JB) JB XB" 

343 
interpret JAB: finite_product_sigma_algebra M "JA \<union> JB" 

344 
by default (insert XA XB, auto) 

345 
have *: "A  B = emb I (JA \<union> JB) (?RA  ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)" 

346 
using XA A XB B by (auto simp: emb_simps) 

347 
then show "A  B \<in> sets ?G" "A \<union> B \<in> sets ?G" 

348 
using XA XB by (auto intro!: generatorI') 

349 
qed 

350 

351 
lemma (in product_prob_space) positive_\<mu>G: "positive generator \<mu>G" 

352 
proof (intro positive_def[THEN iffD2] conjI ballI) 

353 
from generatorE[OF G.empty_sets] guess J X . note this[simp] 

354 
interpret J: finite_product_sigma_finite M J by default fact 

355 
have "X = {}" 

356 
by (rule emb_injective[of J I]) simp_all 

357 
then show "\<mu>G {} = 0" by simp 

358 
next 

359 
fix A assume "A \<in> sets generator" 

360 
from generatorE[OF this] guess J X . note this[simp] 

361 
interpret J: finite_product_sigma_finite M J by default fact 

362 
show "0 \<le> \<mu>G A" by simp 

363 
qed 

364 

365 
lemma (in product_prob_space) additive_\<mu>G: "additive generator \<mu>G" 

366 
proof (intro additive_def[THEN iffD2] ballI impI) 

367 
fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this 

368 
fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this 

369 
assume "A \<inter> B = {}" 

370 
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)" 

371 
using J K by auto 

372 
interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact 

373 
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}" 

374 
apply (rule emb_injective[of "J \<union> K" I]) 

375 
apply (insert `A \<inter> B = {}` JK J K) 

376 
apply (simp_all add: JK.Int emb_simps) 

377 
done 

378 
have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)" 

379 
using J K by simp_all 

380 
then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))" 

381 
by (simp add: emb_simps) 

382 
also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" 

383 
using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un) 

384 
also have "\<dots> = \<mu>G A + \<mu>G B" 

385 
using J K JK_disj by (simp add: JK.measure_additive[symmetric]) 

386 
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" . 

387 
qed 

388 

389 
lemma (in product_prob_space) finite_index_eq_finite_product: 

390 
assumes "finite I" 

391 
shows "sets (sigma generator) = sets (Pi\<^isub>M I M)" 

392 
proof safe 

393 
interpret I: finite_product_sigma_algebra M I by default fact 

394 
have [simp]: "space generator = space (Pi\<^isub>M I M)" 

395 
by (simp add: generator_def product_algebra_def) 

396 
{ fix A assume "A \<in> sets (sigma generator)" 

397 
then show "A \<in> sets I.P" unfolding sets_sigma 

398 
proof induct 

399 
case (Basic A) 

400 
from generatorE[OF this] guess J X . note J = this 

401 
with `finite I` have "emb I J X \<in> sets I.P" by auto 

402 
with `emb I J X = A` show "A \<in> sets I.P" by simp 

403 
qed auto } 

404 
{ fix A assume "A \<in> sets I.P" 

405 
moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto 

406 
ultimately show "A \<in> sets (sigma generator)" 

407 
using `finite I` I_not_empty unfolding sets_sigma 

408 
by (intro sigma_sets.Basic generatorI[of I A]) auto } 

409 
qed 

410 

411 
lemma (in product_prob_space) extend_\<mu>G: 

412 
"\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and> 

413 
measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>" 

414 
proof cases 

415 
assume "finite I" 

416 
interpret I: finite_product_sigma_finite M I by default fact 

417 
show ?thesis 

418 
proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI) 

419 
fix A assume "A \<in> sets generator" 

420 
from generatorE[OF this] guess J X . note J = this 

421 
from J(14) `finite I` show "measure I.P A = \<mu>G A" 

422 
unfolding J(6) 

423 
by (subst J(5)[symmetric]) (simp add: measure_emb) 

424 
next 

425 
have [simp]: "space generator = space (Pi\<^isub>M I M)" 

426 
by (simp add: generator_def product_algebra_def) 

427 
have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr> 

428 
= I.P" (is "?P = _") 

429 
by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`]) 

430 
then show "measure_space ?P" by simp default 

431 
qed 

432 
next 

433 
let ?G = generator 

434 
assume "\<not> finite I" 

435 
note \<mu>G_mono = 

436 
G.additive_increasing[OF positive_\<mu>G additive_\<mu>G, THEN increasingD] 

437 

438 
{ fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G" 

439 

440 
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J" 

441 
by (metis rev_finite_subset subsetI) 

442 
moreover from Z guess K' X' by (rule generatorE) 

443 
moreover def K \<equiv> "insert k K'" 

444 
moreover def X \<equiv> "emb K K' X'" 

445 
ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X" 

446 
"K  J \<noteq> {}" "K  J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X" 

447 
by (auto simp: subset_insertI) 

448 

449 
let "?M y" = "merge J y (K  J) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M)" 

450 
{ fix y assume y: "y \<in> space (Pi\<^isub>M J M)" 

451 
note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X] 

452 
moreover 

453 
have **: "?M y \<in> sets (Pi\<^isub>M (K  J) M)" 

454 
using J K y by (intro merge_sets) auto 

455 
ultimately 

456 
have ***: "(merge J y (I  J) ` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G" 

457 
using J K by (intro generatorI) auto 

458 
have "\<mu>G (merge J y (I  J) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = measure (Pi\<^isub>M (K  J) M) (?M y)" 

459 
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto 

460 
note * ** *** this } 

461 
note merge_in_G = this 

462 

463 
have "finite (K  J)" using K by auto 

464 

465 
interpret J: finite_product_prob_space M J by default fact+ 

466 
interpret KmJ: finite_product_prob_space M "K  J" by default fact+ 

467 

468 
have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K  J)) M) (emb (J \<union> (K  J)) K X)" 

469 
using K J by simp 

470 
also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K  J) M) (?M x) \<partial>Pi\<^isub>M J M)" 

471 
using K J by (subst measure_fold_integral) auto 

472 
also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G (merge J y (I  J) ` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)" 

473 
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)") 

474 
proof (intro J.positive_integral_cong) 

475 
fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 

476 
with K merge_in_G(2)[OF this] 

477 
show "measure (Pi\<^isub>M (K  J) M) (?M x) = \<mu>G (?MZ x)" 

478 
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto 

479 
qed 

480 
finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" . 

481 

482 
{ fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 

483 
then have "\<mu>G (?MZ x) \<le> 1" 

484 
unfolding merge_in_G(4)[OF x] `Z = emb I K X` 

485 
by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) } 

486 
note le_1 = this 

487 

488 
let "?q y" = "\<mu>G (merge J y (I  J) ` Z \<inter> space (Pi\<^isub>M I M))" 

489 
have "?q \<in> borel_measurable (Pi\<^isub>M J M)" 

490 
unfolding `Z = emb I K X` using J K merge_in_G(3) 

491 
by (simp add: merge_in_G \<mu>G_eq measure_fold_measurable 

492 
del: space_product_algebra cong: measurable_cong) 

493 
note this fold le_1 merge_in_G(3) } 

494 
note fold = this 

495 

496 
show ?thesis 

497 
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G]) 

498 
fix A assume "A \<in> sets ?G" 

499 
with generatorE guess J X . note JX = this 

500 
interpret JK: finite_product_prob_space M J by default fact+ 

501 
with JX show "\<mu>G A \<noteq> \<infinity>" by simp 

502 
next 

503 
fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}" 

504 
then have "decseq (\<lambda>i. \<mu>G (A i))" 

505 
by (auto intro!: \<mu>G_mono simp: decseq_def) 

506 
moreover 

507 
have "(INF i. \<mu>G (A i)) = 0" 

508 
proof (rule ccontr) 

509 
assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0") 

510 
moreover have "0 \<le> ?a" 

511 
using A positive_\<mu>G by (auto intro!: le_INFI simp: positive_def) 

512 
ultimately have "0 < ?a" by auto 

513 

514 
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) = measure (Pi\<^isub>M J M) X" 

515 
using A by (intro allI generator_Ex) auto 

516 
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)" 

517 
and A': "\<And>n. A n = emb I (J' n) (X' n)" 

518 
unfolding choice_iff by blast 

519 
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)" 

520 
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)" 

521 
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)" 

522 
by auto 

523 
with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G" 

524 
unfolding J_def X_def by (subst emb_trans) (insert A, auto) 

525 

526 
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" 

527 
unfolding J_def by force 

528 

529 
interpret J: finite_product_prob_space M "J i" for i by default fact+ 

530 

531 
have a_le_1: "?a \<le> 1" 

532 
using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq 

533 
by (auto intro!: INF_leI2[of 0] J.measure_le_1) 

534 

535 
let "?M K Z y" = "merge K y (I  K) ` Z \<inter> space (Pi\<^isub>M I M)" 

536 

537 
{ fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)" 

538 
then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto 

539 
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I" 

540 
interpret J': finite_product_prob_space M J' by default fact+ 

541 

542 
let "?q n y" = "\<mu>G (?M J' (Z n) y)" 

543 
let "?Q n" = "?q n ` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)" 

544 
{ fix n 

545 
have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)" 

546 
using Z J' by (intro fold(1)) auto 

547 
then have "?Q n \<in> sets (Pi\<^isub>M J' M)" 

548 
by (rule measurable_sets) auto } 

549 
note Q_sets = this 

550 

551 
have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))" 

552 
proof (intro le_INFI) 

553 
fix n 

554 
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto 

555 
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)" 

556 
unfolding fold(2)[OF J' `Z n \<in> sets ?G`] 

557 
proof (intro J'.positive_integral_mono) 

558 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 

559 
then have "?q n x \<le> 1 + 0" 

560 
using J' Z fold(3) Z_sets by auto 

561 
also have "\<dots> \<le> 1 + ?a / 2^(k+1)" 

562 
using `0 < ?a` by (intro add_mono) auto 

563 
finally have "?q n x \<le> 1 + ?a / 2^(k+1)" . 

564 
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)" 

565 
by (auto split: split_indicator simp del: power_Suc) 

566 
qed 

567 
also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)" 

568 
using `0 \<le> ?a` Q_sets J'.measure_space_1 

569 
by (subst J'.positive_integral_add) auto 

570 
finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1` 

571 
by (cases rule: extreal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"]) 

572 
(auto simp: field_simps) 

573 
qed 

574 
also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)" 

575 
proof (intro J'.continuity_from_above) 

576 
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto 

577 
show "decseq ?Q" 

578 
unfolding decseq_def 

579 
proof (safe intro!: vimageI[OF refl]) 

580 
fix m n :: nat assume "m \<le> n" 

581 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 

582 
assume "?a / 2^(k+1) \<le> ?q n x" 

583 
also have "?q n x \<le> ?q m x" 

584 
proof (rule \<mu>G_mono) 

585 
from fold(4)[OF J', OF Z_sets x] 

586 
show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto 

587 
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x" 

588 
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto 

589 
qed 

590 
finally show "?a / 2^(k+1) \<le> ?q m x" . 

591 
qed 

592 
qed (intro J'.finite_measure Q_sets) 

593 
finally have "(\<Inter>n. ?Q n) \<noteq> {}" 

594 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 

595 
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto } 

596 
note Ex_w = this 

597 

598 
let "?q k n y" = "\<mu>G (?M (J k) (A n) y)" 

599 

600 
have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_leI) 

601 
from Ex_w[OF A(1,2) this J(13), of 0] guess w0 .. note w0 = this 

602 

603 
let "?P k wk w" = 

604 
"w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and> (\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)" 

605 
def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))" 

606 

607 
{ fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and> 

608 
(\<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))" 

609 
proof (induct k) 

610 
case 0 with w0 show ?case 

611 
unfolding w_def nat_rec_0 by auto 

612 
next 

613 
case (Suc k) 

614 
then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

615 
have "\<exists>w'. ?P k (w k) w'" 

616 
proof cases 

617 
assume [simp]: "J k = J (Suc k)" 

618 
show ?thesis 

619 
proof (intro exI[of _ "w k"] conjI allI) 

620 
fix n 

621 
have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)" 

622 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps) 

623 
also have "\<dots> \<le> ?q k n (w k)" using Suc by auto 

624 
finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp 

625 
next 

626 
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)" 

627 
using Suc by simp 

628 
then show "restrict (w k) (J k) = w k" 

629 
by (simp add: extensional_restrict) 

630 
qed 

631 
next 

632 
assume "J k \<noteq> J (Suc k)" 

633 
with J_mono[of k "Suc k"] have "J (Suc k)  J k \<noteq> {}" (is "?D \<noteq> {}") by auto 

634 
have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G" 

635 
"decseq (\<lambda>n. ?M (J k) (A n) (w k))" 

636 
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))" 

637 
using `decseq A` fold(4)[OF J(13) A_eq(2), of "w k" k] Suc 

638 
by (auto simp: decseq_def) 

639 
from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"] 

640 
obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)" 

641 
"\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto 

642 
let ?w = "merge (J k) (w k) ?D w'" 

643 
have [simp]: "\<And>x. merge (J k) (w k) (I  J k) (merge ?D w' (I  ?D) x) = 

644 
merge (J (Suc k)) ?w (I  (J (Suc k))) x" 

645 
using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"] 

646 
by (auto intro!: ext split: split_merge) 

647 
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w" 

648 
using w'(1) J(3)[of "Suc k"] 

649 
by (auto split: split_merge intro!: extensional_merge_sub) force+ 

650 
show ?thesis 

651 
apply (rule exI[of _ ?w]) 

652 
using w' J_mono[of k "Suc k"] wk unfolding * 

653 
apply (auto split: split_merge intro!: extensional_merge_sub ext) 

654 
apply (force simp: extensional_def) 

655 
done 

656 
qed 

657 
then have "?P k (w k) (w (Suc k))" 

658 
unfolding w_def nat_rec_Suc unfolding w_def[symmetric] 

659 
by (rule someI_ex) 

660 
then show ?case by auto 

661 
qed 

662 
moreover 

663 
then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

664 
moreover 

665 
from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto 

666 
then have "?M (J k) (A k) (w k) \<noteq> {}" 

667 
using positive_\<mu>G[unfolded positive_def] `0 < ?a` `?a \<le> 1` 

668 
by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 

669 
then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto 

670 
then have "merge (J k) (w k) (I  J k) x \<in> A k" by auto 

671 
then have "\<exists>x\<in>A k. restrict x (J k) = w k" 

672 
using `w k \<in> space (Pi\<^isub>M (J k) M)` 

673 
by (intro rev_bexI) (auto intro!: ext simp: extensional_def) 

674 
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)" 

675 
"\<exists>x\<in>A k. restrict x (J k) = w k" 

676 
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k  1)) = w (k  1)" 

677 
by auto } 

678 
note w = this 

679 

680 
{ fix k l i assume "k \<le> l" "i \<in> J k" 

681 
{ fix l have "w k i = w (k + l) i" 

682 
proof (induct l) 

683 
case (Suc l) 

684 
from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto 

685 
with w(3)[of "k + Suc l"] 

686 
have "w (k + l) i = w (k + Suc l) i" 

687 
by (auto simp: restrict_def fun_eq_iff split: split_if_asm) 

688 
with Suc show ?case by simp 

689 
qed simp } 

690 
from this[of "l  k"] `k \<le> l` have "w l i = w k i" by simp } 

691 
note w_mono = this 

692 

693 
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" 

694 
{ fix i k assume k: "i \<in> J k" 

695 
have "w k i = w (LEAST k. i \<in> J k) i" 

696 
by (intro w_mono Least_le k LeastI[of _ k]) 

697 
then have "w' i = w k i" 

698 
unfolding w'_def using k by auto } 

699 
note w'_eq = this 

700 
have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined" 

701 
using J by (auto simp: w'_def) 

702 
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)" 

703 
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]]) 

704 
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)" 

705 
using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ } 

706 
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this 

707 

708 
have w': "w' \<in> space (Pi\<^isub>M I M)" 

709 
using w(1) by (auto simp add: Pi_iff extensional_def) 

710 

711 
{ fix n 

712 
have "restrict w' (J n) = w n" using w(1) 

713 
by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def) 

714 
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto 

715 
then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) } 

716 
then have "w' \<in> (\<Inter>i. A i)" by auto 

717 
with `(\<Inter>i. A i) = {}` show False by auto 

718 
qed 

719 
ultimately show "(\<lambda>i. \<mu>G (A i)) > 0" 

720 
using LIMSEQ_extreal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp 

721 
qed 

722 
qed 

723 

724 
lemma (in product_prob_space) infprod_spec: 

725 
shows "(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> measure_space (Pi\<^isub>P I M)" 

726 
proof  

727 
let ?P = "\<lambda>\<mu>. (\<forall>A\<in>sets generator. \<mu> A = \<mu>G A) \<and> 

728 
measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>" 

729 
have **: "measure infprod_algebra = (SOME \<mu>. ?P \<mu>)" 

730 
unfolding infprod_algebra_def by simp 

731 
have *: "Pi\<^isub>P I M = \<lparr>space = space generator, sets = sets (sigma generator), measure = measure (Pi\<^isub>P I M)\<rparr>" 

732 
unfolding infprod_algebra_def by auto 

733 
show ?thesis 

734 
apply (subst (2) *) 

735 
apply (unfold **) 

736 
apply (rule someI_ex[where P="?P"]) 

737 
apply (rule extend_\<mu>G) 

738 
done 

739 
qed 

740 

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

741 
sublocale product_prob_space \<subseteq> P: measure_space "Pi\<^isub>P I M" 
42147  742 
using infprod_spec by auto 
743 

744 
lemma (in product_prob_space) measure_infprod_emb: 

745 
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" 

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

746 
shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X" 
42147  747 
proof  
748 
have "emb I J X \<in> sets generator" 

749 
using assms by (rule generatorI') 

750 
with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto 

751 
qed 

752 

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

753 
sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M" 
42166
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

754 
proof 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

755 
obtain i where "i \<in> I" using I_not_empty by auto 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

756 
interpret i: finite_product_sigma_finite M "{i}" by default auto 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

757 
let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

758 
have "?X \<in> sets (Pi\<^isub>M {i} M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

759 
by auto 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

760 
from measure_infprod_emb[OF _ _ _ this] `i \<in> I` 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

761 
have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))" 
42166
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

762 
by (simp add: i.measure_times) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

763 
also have "emb I {i} ?X = space (Pi\<^isub>P I M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

764 
using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

765 
finally show "\<mu> (space (Pi\<^isub>P I M)) = 1" 
42166
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

766 
using M.measure_space_1 by simp 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

767 
qed 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

768 

efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

769 
lemma (in product_prob_space) measurable_component: 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

770 
assumes "i \<in> I" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

771 
shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

772 
proof (unfold measurable_def, safe) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

773 
fix x assume "x \<in> space (Pi\<^isub>P I M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

774 
then show "x i \<in> space (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

775 
using `i \<in> I` by (auto simp: infprod_algebra_def generator_def) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

776 
next 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

777 
fix A assume "A \<in> sets (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

778 
with `i \<in> I` have 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

779 
"(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

780 
"(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

781 
by (auto simp: infprod_algebra_def generator_def emb_def) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

782 
from generatorI[OF _ _ _ this] `i \<in> I` 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

783 
show "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

784 
unfolding infprod_algebra_def by auto 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

785 
qed 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

786 

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

787 
lemma (in product_prob_space) emb_in_infprod_algebra[intro]: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

788 
fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

789 
shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

790 
proof cases 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

791 
assume "J = {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

792 
with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

793 
by (auto simp: emb_def infprod_algebra_def generator_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

794 
product_algebra_def product_algebra_generator_def image_constant sigma_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

795 
then show ?thesis by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

796 
next 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

797 
assume "J \<noteq> {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

798 
show ?thesis unfolding infprod_algebra_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

799 
by simp (intro in_sigma generatorI' `J \<noteq> {}` J X) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

800 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

801 

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

802 
lemma (in product_prob_space) finite_measure_infprod_emb: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

803 
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

804 
shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

805 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

806 
interpret J: finite_product_prob_space M J by default fact+ 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

807 
from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

808 
with assms show "\<mu>' (emb I J X) = J.\<mu>' X" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

809 
unfolding \<mu>'_def J.\<mu>'_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

810 
unfolding measure_infprod_emb[OF assms] 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

811 
by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

812 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

813 

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

814 
lemma (in finite_product_prob_space) finite_measure_times: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

815 
assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

816 
shows "\<mu>' (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu>' i (A i))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

817 
using assms 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

818 
unfolding \<mu>'_def M.\<mu>'_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

819 
by (subst measure_times[OF assms]) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

820 
(auto simp: finite_measure_eq M.finite_measure_eq setprod_extreal) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

821 

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

822 
lemma (in product_prob_space) finite_measure_infprod_emb_Pi: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

823 
assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

824 
shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

825 
proof cases 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

826 
assume "J = {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

827 
then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

828 
by (auto simp: infprod_algebra_def generator_def sigma_def emb_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

829 
then show ?thesis using `J = {}` prob_space by simp 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

830 
next 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

831 
assume "J \<noteq> {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

832 
interpret J: finite_product_prob_space M J by default fact+ 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

833 
have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

834 
using J `J \<noteq> {}` by (subst J.finite_measure_times) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

835 
also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

836 
using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

837 
finally show ?thesis by simp 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

838 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

839 

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

840 
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

841 
proof 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

842 
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

843 
by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

844 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

845 

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

846 
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

847 
proof 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

848 
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

849 
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

850 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

851 

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

852 
lemma sigma_sets_subseteq: "A \<subseteq> sigma_sets X A" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

853 
by (auto intro: sigma_sets.Basic) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

854 

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

855 
lemma (in product_prob_space) infprod_algebra_alt: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

856 
"Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

857 
sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i))), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

858 
measure = measure (Pi\<^isub>P I M) \<rparr>" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

859 
(is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>") 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

860 
proof (rule measure_space.equality) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

861 
let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

862 
have "sigma_sets ?O ?M = sigma_sets ?O ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

863 
proof (intro equalityI sigma_sets_mono UN_least) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

864 
fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

865 
have "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> emb I J ` sets (Pi\<^isub>M J M)" by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

866 
also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

867 
also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_subseteq) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

868 
finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" . 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

869 
have "emb I J ` sets (Pi\<^isub>M J M) = emb I J ` sigma_sets (space (Pi\<^isub>M J M)) (Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

870 
by (simp add: sets_sigma product_algebra_generator_def product_algebra_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

871 
also have "\<dots> = sigma_sets (space (Pi\<^isub>M I M)) (emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

872 
using J M.sets_into_space 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

873 
by (auto simp: emb_def_raw intro!: sigma_sets_vimage[symmetric]) blast 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

874 
also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

875 
using J by (intro sigma_sets_mono') auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

876 
finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

877 
by (simp add: infprod_algebra_def generator_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

878 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

879 
then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

880 
by (simp_all add: infprod_algebra_def generator_def sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

881 
qed simp_all 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

882 

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

883 
lemma (in product_prob_space) infprod_algebra_alt2: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

884 
"Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

885 
sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

886 
measure = measure (Pi\<^isub>P I M) \<rparr>" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

887 
(is "_ = ?S") 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

888 
proof (rule measure_space.equality) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

889 
let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

890 
let ?G = "(\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

891 
have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

892 
by (subst infprod_algebra_alt) (simp add: sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

893 
also have "\<dots> = sigma_sets ?O ?A" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

894 
proof (intro equalityI sigma_sets_mono subsetI) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

895 
interpret A: sigma_algebra ?S 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

896 
by (rule sigma_algebra_sigma) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

897 
fix A assume "A \<in> ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

898 
then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

899 
and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

900 
by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

901 
then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) ` (B j) \<inter> space (Pi\<^isub>P I M))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

902 
by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

903 
{ fix j assume "j\<in>J" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

904 
with `J \<subseteq> I` have "j \<in> I" by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

905 
with `j \<in> J` B have "(\<lambda>x. x j) ` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

906 
by (auto simp: sets_sigma intro: sigma_sets.Basic) } 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

907 
with `finite J` `J \<noteq> {}` have "A \<in> sets ?S" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

908 
unfolding A by (intro A.finite_INT) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

909 
then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

910 
next 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

911 
fix A assume "A \<in> ?A" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

912 
then obtain i B where i: "i \<in> I" "B \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

913 
and "A = (\<lambda>x. x i) ` B \<inter> space (Pi\<^isub>P I M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

914 
by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

915 
then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

916 
by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

917 
with i show "A \<in> sigma_sets ?O ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

918 
by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

919 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

920 
finally show "sets (Pi\<^isub>P I M) = sets ?S" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

921 
by (simp add: sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

922 
qed simp_all 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

923 

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

924 
lemma (in product_prob_space) measurable_into_infprod_algebra: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

925 
assumes "sigma_algebra N" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

926 
assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

927 
assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

928 
shows "f \<in> measurable N (Pi\<^isub>P I M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

929 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

930 
interpret N: sigma_algebra N by fact 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

931 
have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

932 
using f by (auto simp: measurable_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

933 
{ fix i A assume i: "i \<in> I" "A \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

934 
then have "f ` (\<lambda>x. x i) ` A \<inter> f ` space infprod_algebra \<inter> space N = (\<lambda>x. f x i) ` A \<inter> space N" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

935 
using f_in ext by (auto simp: infprod_algebra_def generator_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

936 
also have "\<dots> \<in> sets N" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

937 
by (rule measurable_sets f i)+ 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

938 
finally have "f ` (\<lambda>x. x i) ` A \<inter> f ` space infprod_algebra \<inter> space N \<in> sets N" . } 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

939 
with f_in ext show ?thesis 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

940 
by (subst infprod_algebra_alt2) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

941 
(auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

942 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

943 

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

944 
subsection {* Sequence space *} 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

945 

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

946 
locale sequence_space = product_prob_space M "UNIV :: nat set" for M 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

947 

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

948 
lemma (in sequence_space) infprod_in_sets[intro]: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

949 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

950 
shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

951 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

952 
have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

953 
using E E[THEN M.sets_into_space] 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

954 
by (auto simp: emb_def Pi_iff extensional_def) blast 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

955 
with E show ?thesis 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

956 
by (auto intro: emb_in_infprod_algebra) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

957 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

958 

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

959 
lemma (in sequence_space) measure_infprod: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

960 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

961 
shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) > \<mu>' (Pi UNIV E)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

962 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

963 
let "?E n" = "emb UNIV {..n} (Pi\<^isub>E {.. n} E)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

964 
{ fix n :: nat 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

965 
interpret n: finite_product_prob_space M "{..n}" by default auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

966 
have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

967 
using E by (subst n.finite_measure_times) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

968 
also have "\<dots> = \<mu>' (?E n)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

969 
using E by (intro finite_measure_infprod_emb[symmetric]) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

970 
finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . } 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

971 
moreover have "Pi UNIV E = (\<Inter>n. ?E n)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

972 
using E E[THEN M.sets_into_space] 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

973 
by (auto simp: emb_def extensional_def Pi_iff) blast 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

974 
moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

975 
using E by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

976 
moreover have "decseq ?E" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

977 
by (auto simp: emb_def Pi_iff decseq_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

978 
ultimately show ?thesis 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

979 
by (simp add: finite_continuity_from_above) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

980 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

981 

42147  982 
end 