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src/HOL/Analysis/Regularity.thy

author | nipkow |

Sat Dec 29 15:43:53 2018 +0100 (6 months ago) | |

changeset 69529 | 4ab9657b3257 |

parent 69260 | 0a9688695a1b |

child 69661 | a03a63b81f44 |

permissions | -rw-r--r-- |

capitalize proper names in lemma names

1 (* Title: HOL/Analysis/Regularity.thy

2 Author: Fabian Immler, TU München

3 *)

5 section \<open>Regularity of Measures\<close>

7 theory Regularity (* FIX suggestion to rename to e.g. RegularityMeasures *)

8 imports Measure_Space Borel_Space

9 begin

11 lemma%important (*FIX needs name *)

12 fixes M::"'a::{second_countable_topology, complete_space} measure"

13 assumes sb: "sets M = sets borel"

14 assumes "emeasure M (space M) \<noteq> \<infinity>"

15 assumes "B \<in> sets borel"

16 shows inner_regular: "emeasure M B =

17 (SUP K \<in> {K. K \<subseteq> B \<and> compact K}. emeasure M K)" (is "?inner B")

18 and outer_regular: "emeasure M B =

19 (INF U \<in> {U. B \<subseteq> U \<and> open U}. emeasure M U)" (is "?outer B")

20 proof%unimportant -

21 have Us: "UNIV = space M" by (metis assms(1) sets_eq_imp_space_eq space_borel)

22 hence sU: "space M = UNIV" by simp

23 interpret finite_measure M by rule fact

24 have approx_inner: "\<And>A. A \<in> sets M \<Longrightarrow>

25 (\<And>e. e > 0 \<Longrightarrow> \<exists>K. K \<subseteq> A \<and> compact K \<and> emeasure M A \<le> emeasure M K + ennreal e) \<Longrightarrow> ?inner A"

26 by (rule ennreal_approx_SUP)

27 (force intro!: emeasure_mono simp: compact_imp_closed emeasure_eq_measure)+

28 have approx_outer: "\<And>A. A \<in> sets M \<Longrightarrow>

29 (\<And>e. e > 0 \<Longrightarrow> \<exists>B. A \<subseteq> B \<and> open B \<and> emeasure M B \<le> emeasure M A + ennreal e) \<Longrightarrow> ?outer A"

30 by (rule ennreal_approx_INF)

31 (force intro!: emeasure_mono simp: emeasure_eq_measure sb)+

32 from countable_dense_setE guess X::"'a set" . note X = this

33 {

34 fix r::real assume "r > 0" hence "\<And>y. open (ball y r)" "\<And>y. ball y r \<noteq> {}" by auto

35 with X(2)[OF this]

36 have x: "space M = (\<Union>x\<in>X. cball x r)"

37 by (auto simp add: sU) (metis dist_commute order_less_imp_le)

38 let ?U = "\<Union>k. (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)"

39 have "(\<lambda>k. emeasure M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M ?U"

40 by (rule Lim_emeasure_incseq) (auto intro!: borel_closed bexI simp: incseq_def Us sb)

41 also have "?U = space M"

42 proof safe

43 fix x from X(2)[OF open_ball[of x r]] \<open>r > 0\<close> obtain d where d: "d\<in>X" "d \<in> ball x r" by auto

44 show "x \<in> ?U"

45 using X(1) d

46 by simp (auto intro!: exI [where x = "to_nat_on X d"] simp: dist_commute Bex_def)

47 qed (simp add: sU)

48 finally have "(\<lambda>k. M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M (space M)" .

49 } note M_space = this

50 {

51 fix e ::real and n :: nat assume "e > 0" "n > 0"

52 hence "1/n > 0" "e * 2 powr - n > 0" by (auto)

53 from M_space[OF \<open>1/n>0\<close>]

54 have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) \<longlonglongrightarrow> measure M (space M)"

55 unfolding emeasure_eq_measure by (auto simp: measure_nonneg)

56 from metric_LIMSEQ_D[OF this \<open>0 < e * 2 powr -n\<close>]

57 obtain k where "dist (measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) (measure M (space M)) <

58 e * 2 powr -n"

59 by auto

60 hence "measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n)) \<ge>

61 measure M (space M) - e * 2 powr -real n"

62 by (auto simp: dist_real_def)

63 hence "\<exists>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n)) \<ge>

64 measure M (space M) - e * 2 powr - real n" ..

65 } note k=this

66 hence "\<forall>e\<in>{0<..}. \<forall>(n::nat)\<in>{0<..}. \<exists>k.

67 measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n)) \<ge> measure M (space M) - e * 2 powr - real n"

68 by blast

69 then obtain k where k: "\<forall>e\<in>{0<..}. \<forall>n\<in>{0<..}. measure M (space M) - e * 2 powr - real (n::nat)

70 \<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"

71 by metis

72 hence k: "\<And>e n. e > 0 \<Longrightarrow> n > 0 \<Longrightarrow> measure M (space M) - e * 2 powr - n

73 \<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"

74 unfolding Ball_def by blast

75 have approx_space:

76 "\<exists>K \<in> {K. K \<subseteq> space M \<and> compact K}. emeasure M (space M) \<le> emeasure M K + ennreal e"

77 (is "?thesis e") if "0 < e" for e :: real

78 proof -

79 define B where [abs_def]:

80 "B n = (\<Union>i\<in>{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n))" for n

81 have "\<And>n. closed (B n)" by (auto simp: B_def)

82 hence [simp]: "\<And>n. B n \<in> sets M" by (simp add: sb)

83 from k[OF \<open>e > 0\<close> zero_less_Suc]

84 have "\<And>n. measure M (space M) - measure M (B n) \<le> e * 2 powr - real (Suc n)"

85 by (simp add: algebra_simps B_def finite_measure_compl)

86 hence B_compl_le: "\<And>n::nat. measure M (space M - B n) \<le> e * 2 powr - real (Suc n)"

87 by (simp add: finite_measure_compl)

88 define K where "K = (\<Inter>n. B n)"

89 from \<open>closed (B _)\<close> have "closed K" by (auto simp: K_def)

90 hence [simp]: "K \<in> sets M" by (simp add: sb)

91 have "measure M (space M) - measure M K = measure M (space M - K)"

92 by (simp add: finite_measure_compl)

93 also have "\<dots> = emeasure M (\<Union>n. space M - B n)" by (auto simp: K_def emeasure_eq_measure)

94 also have "\<dots> \<le> (\<Sum>n. emeasure M (space M - B n))"

95 by (rule emeasure_subadditive_countably) (auto simp: summable_def)

96 also have "\<dots> \<le> (\<Sum>n. ennreal (e*2 powr - real (Suc n)))"

97 using B_compl_le by (intro suminf_le) (simp_all add: measure_nonneg emeasure_eq_measure ennreal_leI)

98 also have "\<dots> \<le> (\<Sum>n. ennreal (e * (1 / 2) ^ Suc n))"

99 by (simp add: powr_minus powr_realpow field_simps del: of_nat_Suc)

100 also have "\<dots> = ennreal e * (\<Sum>n. ennreal ((1 / 2) ^ Suc n))"

101 unfolding ennreal_power[symmetric]

102 using \<open>0 < e\<close>

103 by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def

104 ennreal_power[symmetric])

105 also have "\<dots> = e"

106 by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto

107 finally have "measure M (space M) \<le> measure M K + e"

108 using \<open>0 < e\<close> by simp

109 hence "emeasure M (space M) \<le> emeasure M K + e"

110 using \<open>0 < e\<close> by (simp add: emeasure_eq_measure flip: ennreal_plus)

111 moreover have "compact K"

112 unfolding compact_eq_totally_bounded

113 proof safe

114 show "complete K" using \<open>closed K\<close> by (simp add: complete_eq_closed)

115 fix e'::real assume "0 < e'"

116 from nat_approx_posE[OF this] guess n . note n = this

117 let ?k = "from_nat_into X ` {0..k e (Suc n)}"

118 have "finite ?k" by simp

119 moreover have "K \<subseteq> (\<Union>x\<in>?k. ball x e')" unfolding K_def B_def using n by force

120 ultimately show "\<exists>k. finite k \<and> K \<subseteq> (\<Union>x\<in>k. ball x e')" by blast

121 qed

122 ultimately

123 show ?thesis by (auto simp: sU)

124 qed

125 { fix A::"'a set" assume "closed A" hence "A \<in> sets borel" by (simp add: compact_imp_closed)

126 hence [simp]: "A \<in> sets M" by (simp add: sb)

127 have "?inner A"

128 proof (rule approx_inner)

129 fix e::real assume "e > 0"

130 from approx_space[OF this] obtain K where

131 K: "K \<subseteq> space M" "compact K" "emeasure M (space M) \<le> emeasure M K + e"

132 by (auto simp: emeasure_eq_measure)

133 hence [simp]: "K \<in> sets M" by (simp add: sb compact_imp_closed)

134 have "measure M A - measure M (A \<inter> K) = measure M (A - A \<inter> K)"

135 by (subst finite_measure_Diff) auto

136 also have "A - A \<inter> K = A \<union> K - K" by auto

137 also have "measure M \<dots> = measure M (A \<union> K) - measure M K"

138 by (subst finite_measure_Diff) auto

139 also have "\<dots> \<le> measure M (space M) - measure M K"

140 by (simp add: emeasure_eq_measure sU sb finite_measure_mono)

141 also have "\<dots> \<le> e"

142 using K \<open>0 < e\<close> by (simp add: emeasure_eq_measure flip: ennreal_plus)

143 finally have "emeasure M A \<le> emeasure M (A \<inter> K) + ennreal e"

144 using \<open>0<e\<close> by (simp add: emeasure_eq_measure algebra_simps flip: ennreal_plus)

145 moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using \<open>closed A\<close> \<open>compact K\<close> by auto

146 ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ennreal e"

147 by blast

148 qed simp

149 have "?outer A"

150 proof cases

151 assume "A \<noteq> {}"

152 let ?G = "\<lambda>d. {x. infdist x A < d}"

153 {

154 fix d

155 have "?G d = (\<lambda>x. infdist x A) -` {..<d}" by auto

156 also have "open \<dots>"

157 by (intro continuous_open_vimage) (auto intro!: continuous_infdist continuous_ident)

158 finally have "open (?G d)" .

159 } note open_G = this

160 from in_closed_iff_infdist_zero[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>]

161 have "A = {x. infdist x A = 0}" by auto

162 also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"

163 proof (auto simp del: of_nat_Suc, rule ccontr)

164 fix x

165 assume "infdist x A \<noteq> 0"

166 hence pos: "infdist x A > 0" using infdist_nonneg[of x A] by simp

167 from nat_approx_posE[OF this] guess n .

168 moreover

169 assume "\<forall>i. infdist x A < 1 / real (Suc i)"

170 hence "infdist x A < 1 / real (Suc n)" by auto

171 ultimately show False by simp

172 qed

173 also have "M \<dots> = (INF n. emeasure M (?G (1 / real (Suc n))))"

174 proof (rule INF_emeasure_decseq[symmetric], safe)

175 fix i::nat

176 from open_G[of "1 / real (Suc i)"]

177 show "?G (1 / real (Suc i)) \<in> sets M" by (simp add: sb borel_open)

178 next

179 show "decseq (\<lambda>i. {x. infdist x A < 1 / real (Suc i)})"

180 by (auto intro: less_trans intro!: divide_strict_left_mono

181 simp: decseq_def le_eq_less_or_eq)

182 qed simp

183 finally

184 have "emeasure M A = (INF n. emeasure M {x. infdist x A < 1 / real (Suc n)})" .

185 moreover

186 have "\<dots> \<ge> (INF U\<in>{U. A \<subseteq> U \<and> open U}. emeasure M U)"

187 proof (intro INF_mono)

188 fix m

189 have "?G (1 / real (Suc m)) \<in> {U. A \<subseteq> U \<and> open U}" using open_G by auto

190 moreover have "M (?G (1 / real (Suc m))) \<le> M (?G (1 / real (Suc m)))" by simp

191 ultimately show "\<exists>U\<in>{U. A \<subseteq> U \<and> open U}.

192 emeasure M U \<le> emeasure M {x. infdist x A < 1 / real (Suc m)}"

193 by blast

194 qed

195 moreover

196 have "emeasure M A \<le> (INF U\<in>{U. A \<subseteq> U \<and> open U}. emeasure M U)"

197 by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)

198 ultimately show ?thesis by simp

199 qed (auto intro!: INF_eqI)

200 note \<open>?inner A\<close> \<open>?outer A\<close> }

201 note closed_in_D = this

202 from \<open>B \<in> sets borel\<close>

203 have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"

204 by (auto simp: Int_stable_def borel_eq_closed)

205 then show "?inner B" "?outer B"

206 proof (induct B rule: sigma_sets_induct_disjoint)

207 case empty

208 { case 1 show ?case by (intro SUP_eqI[symmetric]) auto }

209 { case 2 show ?case by (intro INF_eqI[symmetric]) (auto elim!: meta_allE[of _ "{}"]) }

210 next

211 case (basic B)

212 { case 1 from basic closed_in_D show ?case by auto }

213 { case 2 from basic closed_in_D show ?case by auto }

214 next

215 case (compl B)

216 note inner = compl(2) and outer = compl(3)

217 from compl have [simp]: "B \<in> sets M" by (auto simp: sb borel_eq_closed)

218 case 2

219 have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)

220 also have "\<dots> = (INF K\<in>{K. K \<subseteq> B \<and> compact K}. M (space M) - M K)"

221 by (subst ennreal_SUP_const_minus) (auto simp: less_top[symmetric] inner)

222 also have "\<dots> = (INF U\<in>{U. U \<subseteq> B \<and> compact U}. M (space M - U))"

223 by (rule INF_cong) (auto simp add: emeasure_compl sb compact_imp_closed)

224 also have "\<dots> \<ge> (INF U\<in>{U. U \<subseteq> B \<and> closed U}. M (space M - U))"

225 by (rule INF_superset_mono) (auto simp add: compact_imp_closed)

226 also have "(INF U\<in>{U. U \<subseteq> B \<and> closed U}. M (space M - U)) =

227 (INF U\<in>{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"

228 unfolding INF_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]

229 by (rule INF_cong) (auto simp add: sU Compl_eq_Diff_UNIV [symmetric, simp])

230 finally have

231 "(INF U\<in>{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<le> emeasure M (space M - B)" .

232 moreover have

233 "(INF U\<in>{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<ge> emeasure M (space M - B)"

234 by (auto simp: sb sU intro!: INF_greatest emeasure_mono)

235 ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])

237 case 1

238 have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)

239 also have "\<dots> = (SUP U\<in> {U. B \<subseteq> U \<and> open U}. M (space M) - M U)"

240 unfolding outer by (subst ennreal_INF_const_minus) auto

241 also have "\<dots> = (SUP U\<in>{U. B \<subseteq> U \<and> open U}. M (space M - U))"

242 by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)

243 also have "\<dots> = (SUP K\<in>{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"

244 unfolding SUP_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]

245 by (rule SUP_cong) (auto simp add: sU)

246 also have "\<dots> = (SUP K\<in>{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"

247 proof (safe intro!: antisym SUP_least)

248 fix K assume "closed K" "K \<subseteq> space M - B"

249 from closed_in_D[OF \<open>closed K\<close>]

250 have K_inner: "emeasure M K = (SUP K\<in>{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp

251 show "emeasure M K \<le> (SUP K\<in>{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"

252 unfolding K_inner using \<open>K \<subseteq> space M - B\<close>

253 by (auto intro!: SUP_upper SUP_least)

254 qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)

255 finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])

256 next

257 case (union D)

258 then have "range D \<subseteq> sets M" by (auto simp: sb borel_eq_closed)

259 with union have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (intro suminf_emeasure)

260 also have "(\<lambda>n. \<Sum>i<n. M (D i)) \<longlonglongrightarrow> (\<Sum>i. M (D i))"

261 by (intro summable_LIMSEQ) auto

262 finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) \<longlonglongrightarrow> measure M (\<Union>i. D i)"

263 by (simp add: emeasure_eq_measure measure_nonneg sum_nonneg)

264 have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto

266 case 1

267 show ?case

268 proof (rule approx_inner)

269 fix e::real assume "e > 0"

270 with measure_LIMSEQ

271 have "\<exists>no. \<forall>n\<ge>no. \<bar>(\<Sum>i<n. measure M (D i)) -measure M (\<Union>x. D x)\<bar> < e/2"

272 by (auto simp: lim_sequentially dist_real_def simp del: less_divide_eq_numeral1)

273 hence "\<exists>n0. \<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>x. D x)\<bar> < e/2" by auto

274 then obtain n0 where n0: "\<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>i. D i)\<bar> < e/2"

275 unfolding choice_iff by blast

276 have "ennreal (\<Sum>i<n0. measure M (D i)) = (\<Sum>i<n0. M (D i))"

277 by (auto simp add: emeasure_eq_measure sum_nonneg measure_nonneg)

278 also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule sum_le_suminf) auto

279 also have "\<dots> = M (\<Union>i. D i)" by (simp add: M)

280 also have "\<dots> = measure M (\<Union>i. D i)" by (simp add: emeasure_eq_measure)

281 finally have n0: "measure M (\<Union>i. D i) - (\<Sum>i<n0. measure M (D i)) < e/2"

282 using n0 by (auto simp: measure_nonneg sum_nonneg)

283 have "\<forall>i. \<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"

284 proof

285 fix i

286 from \<open>0 < e\<close> have "0 < e/(2*Suc n0)" by simp

287 have "emeasure M (D i) = (SUP K\<in>{K. K \<subseteq> (D i) \<and> compact K}. emeasure M K)"

288 using union by blast

289 from SUP_approx_ennreal[OF \<open>0 < e/(2*Suc n0)\<close> _ this]

290 show "\<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"

291 by (auto simp: emeasure_eq_measure intro: less_imp_le compact_empty)

292 qed

293 then obtain K where K: "\<And>i. K i \<subseteq> D i" "\<And>i. compact (K i)"

294 "\<And>i. emeasure M (D i) \<le> emeasure M (K i) + e/(2*Suc n0)"

295 unfolding choice_iff by blast

296 let ?K = "\<Union>i\<in>{..<n0}. K i"

297 have "disjoint_family_on K {..<n0}" using K \<open>disjoint_family D\<close>

298 unfolding disjoint_family_on_def by blast

299 hence mK: "measure M ?K = (\<Sum>i<n0. measure M (K i))" using K

300 by (intro finite_measure_finite_Union) (auto simp: sb compact_imp_closed)

301 have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (D i)) + e/2" using n0 by simp

302 also have "(\<Sum>i<n0. measure M (D i)) \<le> (\<Sum>i<n0. measure M (K i) + e/(2*Suc n0))"

303 using K \<open>0 < e\<close>

304 by (auto intro: sum_mono simp: emeasure_eq_measure simp flip: ennreal_plus)

305 also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"

306 by (simp add: sum.distrib)

307 also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) + e / 2" using \<open>0 < e\<close>

308 by (auto simp: field_simps intro!: mult_left_mono)

309 finally

310 have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"

311 by auto

312 hence "M (\<Union>i. D i) < M ?K + e"

313 using \<open>0<e\<close> by (auto simp: mK emeasure_eq_measure sum_nonneg ennreal_less_iff simp flip: ennreal_plus)

314 moreover

315 have "?K \<subseteq> (\<Union>i. D i)" using K by auto

316 moreover

317 have "compact ?K" using K by auto

318 ultimately

319 have "?K\<subseteq>(\<Union>i. D i) \<and> compact ?K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M ?K + ennreal e" by simp

320 thus "\<exists>K\<subseteq>\<Union>i. D i. compact K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M K + ennreal e" ..

321 qed fact

322 case 2

323 show ?case

324 proof (rule approx_outer[OF \<open>(\<Union>i. D i) \<in> sets M\<close>])

325 fix e::real assume "e > 0"

326 have "\<forall>i::nat. \<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"

327 proof

328 fix i::nat

329 from \<open>0 < e\<close> have "0 < e/(2 powr Suc i)" by simp

330 have "emeasure M (D i) = (INF U\<in>{U. (D i) \<subseteq> U \<and> open U}. emeasure M U)"

331 using union by blast

332 from INF_approx_ennreal[OF \<open>0 < e/(2 powr Suc i)\<close> this]

333 show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"

334 using \<open>0<e\<close>

335 by (auto simp: emeasure_eq_measure sum_nonneg ennreal_less_iff ennreal_minus

336 finite_measure_mono sb

337 simp flip: ennreal_plus)

338 qed

339 then obtain U where U: "\<And>i. D i \<subseteq> U i" "\<And>i. open (U i)"

340 "\<And>i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)"

341 unfolding choice_iff by blast

342 let ?U = "\<Union>i. U i"

343 have "ennreal (measure M ?U - measure M (\<Union>i. D i)) = M ?U - M (\<Union>i. D i)"

344 using U(1,2)

345 by (subst ennreal_minus[symmetric])

346 (auto intro!: finite_measure_mono simp: sb measure_nonneg emeasure_eq_measure)

347 also have "\<dots> = M (?U - (\<Union>i. D i))" using U \<open>(\<Union>i. D i) \<in> sets M\<close>

348 by (subst emeasure_Diff) (auto simp: sb)

349 also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U \<open>range D \<subseteq> sets M\<close>

350 by (intro emeasure_mono) (auto simp: sb intro!: sets.countable_nat_UN sets.Diff)

351 also have "\<dots> \<le> (\<Sum>i. M (U i - D i))" using U \<open>range D \<subseteq> sets M\<close>

352 by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)

353 also have "\<dots> \<le> (\<Sum>i. ennreal e/(2 powr Suc i))" using U \<open>range D \<subseteq> sets M\<close>

354 using \<open>0<e\<close>

355 by (intro suminf_le, subst emeasure_Diff)

356 (auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg ennreal_minus

357 finite_measure_mono divide_ennreal ennreal_less_iff

358 intro: less_imp_le)

359 also have "\<dots> \<le> (\<Sum>n. ennreal (e * (1 / 2) ^ Suc n))"

360 using \<open>0<e\<close>

361 by (simp add: powr_minus powr_realpow field_simps divide_ennreal del: of_nat_Suc)

362 also have "\<dots> = ennreal e * (\<Sum>n. ennreal ((1 / 2) ^ Suc n))"

363 unfolding ennreal_power[symmetric]

364 using \<open>0 < e\<close>

365 by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def

366 ennreal_power[symmetric])

367 also have "\<dots> = ennreal e"

368 by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto

369 finally have "emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ennreal e"

370 using \<open>0<e\<close> by (simp add: emeasure_eq_measure flip: ennreal_plus)

371 moreover

372 have "(\<Union>i. D i) \<subseteq> ?U" using U by auto

373 moreover

374 have "open ?U" using U by auto

375 ultimately

376 have "(\<Union>i. D i) \<subseteq> ?U \<and> open ?U \<and> emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ennreal e" by simp

377 thus "\<exists>B. (\<Union>i. D i) \<subseteq> B \<and> open B \<and> emeasure M B \<le> emeasure M (\<Union>i. D i) + ennreal e" ..

378 qed

379 qed

380 qed

382 end