src/HOL/Probability/Regularity.thy
author hoelzl
Thu Apr 14 15:48:11 2016 +0200 (2016-04-14)
changeset 62975 1d066f6ab25d
parent 62533 bc25f3916a99
child 63040 eb4ddd18d635
permissions -rw-r--r--
Probability: move emeasure and nn_integral from ereal to ennreal
     1 (*  Title:      HOL/Probability/Regularity.thy
     2     Author:     Fabian Immler, TU M√ľnchen
     3 *)
     4 
     5 section \<open>Regularity of Measures\<close>
     6 
     7 theory Regularity
     8 imports Measure_Space Borel_Space
     9 begin
    10 
    11 lemma
    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 : {K. K \<subseteq> B \<and> compact K}. emeasure M K)" (is "?inner B")
    18   and outer_regular: "emeasure M B =
    19     (INF U : {U. B \<subseteq> U \<and> open U}. emeasure M U)" (is "?outer B")
    20 proof -
    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     def B \<equiv> "\<lambda>n. \<Union>i\<in>{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n)"
    80     have "\<And>n. closed (B n)" by (auto simp: B_def)
    81     hence [simp]: "\<And>n. B n \<in> sets M" by (simp add: sb)
    82     from k[OF \<open>e > 0\<close> zero_less_Suc]
    83     have "\<And>n. measure M (space M) - measure M (B n) \<le> e * 2 powr - real (Suc n)"
    84       by (simp add: algebra_simps B_def finite_measure_compl)
    85     hence B_compl_le: "\<And>n::nat. measure M (space M - B n) \<le> e * 2 powr - real (Suc n)"
    86       by (simp add: finite_measure_compl)
    87     def K \<equiv> "\<Inter>n. B n"
    88     from \<open>closed (B _)\<close> have "closed K" by (auto simp: K_def)
    89     hence [simp]: "K \<in> sets M" by (simp add: sb)
    90     have "measure M (space M) - measure M K = measure M (space M - K)"
    91       by (simp add: finite_measure_compl)
    92     also have "\<dots> = emeasure M (\<Union>n. space M - B n)" by (auto simp: K_def emeasure_eq_measure)
    93     also have "\<dots> \<le> (\<Sum>n. emeasure M (space M - B n))"
    94       by (rule emeasure_subadditive_countably) (auto simp: summable_def)
    95     also have "\<dots> \<le> (\<Sum>n. ennreal (e*2 powr - real (Suc n)))"
    96       using B_compl_le by (intro suminf_le) (simp_all add: measure_nonneg emeasure_eq_measure ennreal_leI)
    97     also have "\<dots> \<le> (\<Sum>n. ennreal (e * (1 / 2) ^ Suc n))"
    98       by (simp add: powr_minus powr_realpow field_simps del: of_nat_Suc)
    99     also have "\<dots> = ennreal e * (\<Sum>n. ennreal ((1 / 2) ^ Suc n))"
   100       unfolding ennreal_power[symmetric]
   101       using \<open>0 < e\<close>
   102       by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def
   103                     ennreal_power[symmetric])
   104     also have "\<dots> = e"
   105       by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
   106     finally have "measure M (space M) \<le> measure M K + e"
   107       using \<open>0 < e\<close> by simp
   108     hence "emeasure M (space M) \<le> emeasure M K + e"
   109       using \<open>0 < e\<close> by (simp add: emeasure_eq_measure measure_nonneg ennreal_plus[symmetric] del: ennreal_plus)
   110     moreover have "compact K"
   111       unfolding compact_eq_totally_bounded
   112     proof safe
   113       show "complete K" using \<open>closed K\<close> by (simp add: complete_eq_closed)
   114       fix e'::real assume "0 < e'"
   115       from nat_approx_posE[OF this] guess n . note n = this
   116       let ?k = "from_nat_into X ` {0..k e (Suc n)}"
   117       have "finite ?k" by simp
   118       moreover have "K \<subseteq> (\<Union>x\<in>?k. ball x e')" unfolding K_def B_def using n by force
   119       ultimately show "\<exists>k. finite k \<and> K \<subseteq> (\<Union>x\<in>k. ball x e')" by blast
   120     qed
   121     ultimately
   122     show ?thesis by (auto simp: sU)
   123   qed
   124   { fix A::"'a set" assume "closed A" hence "A \<in> sets borel" by (simp add: compact_imp_closed)
   125     hence [simp]: "A \<in> sets M" by (simp add: sb)
   126     have "?inner A"
   127     proof (rule approx_inner)
   128       fix e::real assume "e > 0"
   129       from approx_space[OF this] obtain K where
   130         K: "K \<subseteq> space M" "compact K" "emeasure M (space M) \<le> emeasure M K + e"
   131         by (auto simp: emeasure_eq_measure)
   132       hence [simp]: "K \<in> sets M" by (simp add: sb compact_imp_closed)
   133       have "measure M A - measure M (A \<inter> K) = measure M (A - A \<inter> K)"
   134         by (subst finite_measure_Diff) auto
   135       also have "A - A \<inter> K = A \<union> K - K" by auto
   136       also have "measure M \<dots> = measure M (A \<union> K) - measure M K"
   137         by (subst finite_measure_Diff) auto
   138       also have "\<dots> \<le> measure M (space M) - measure M K"
   139         by (simp add: emeasure_eq_measure sU sb finite_measure_mono)
   140       also have "\<dots> \<le> e"
   141         using K \<open>0 < e\<close> by (simp add: emeasure_eq_measure ennreal_plus[symmetric] measure_nonneg del: ennreal_plus)
   142       finally have "emeasure M A \<le> emeasure M (A \<inter> K) + ennreal e"
   143         using \<open>0<e\<close> by (simp add: emeasure_eq_measure algebra_simps ennreal_plus[symmetric] measure_nonneg del: ennreal_plus)
   144       moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using \<open>closed A\<close> \<open>compact K\<close> by auto
   145       ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ennreal e"
   146         by blast
   147     qed simp
   148     have "?outer A"
   149     proof cases
   150       assume "A \<noteq> {}"
   151       let ?G = "\<lambda>d. {x. infdist x A < d}"
   152       {
   153         fix d
   154         have "?G d = (\<lambda>x. infdist x A) -` {..<d}" by auto
   155         also have "open \<dots>"
   156           by (intro continuous_open_vimage) (auto intro!: continuous_infdist continuous_ident)
   157         finally have "open (?G d)" .
   158       } note open_G = this
   159       from in_closed_iff_infdist_zero[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>]
   160       have "A = {x. infdist x A = 0}" by auto
   161       also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"
   162       proof (auto simp del: of_nat_Suc, rule ccontr)
   163         fix x
   164         assume "infdist x A \<noteq> 0"
   165         hence pos: "infdist x A > 0" using infdist_nonneg[of x A] by simp
   166         from nat_approx_posE[OF this] guess n .
   167         moreover
   168         assume "\<forall>i. infdist x A < 1 / real (Suc i)"
   169         hence "infdist x A < 1 / real (Suc n)" by auto
   170         ultimately show False by simp
   171       qed
   172       also have "M \<dots> = (INF n. emeasure M (?G (1 / real (Suc n))))"
   173       proof (rule INF_emeasure_decseq[symmetric], safe)
   174         fix i::nat
   175         from open_G[of "1 / real (Suc i)"]
   176         show "?G (1 / real (Suc i)) \<in> sets M" by (simp add: sb borel_open)
   177       next
   178         show "decseq (\<lambda>i. {x. infdist x A < 1 / real (Suc i)})"
   179           by (auto intro: less_trans intro!: divide_strict_left_mono
   180             simp: decseq_def le_eq_less_or_eq)
   181       qed simp
   182       finally
   183       have "emeasure M A = (INF n. emeasure M {x. infdist x A < 1 / real (Suc n)})" .
   184       moreover
   185       have "\<dots> \<ge> (INF U:{U. A \<subseteq> U \<and> open U}. emeasure M U)"
   186       proof (intro INF_mono)
   187         fix m
   188         have "?G (1 / real (Suc m)) \<in> {U. A \<subseteq> U \<and> open U}" using open_G by auto
   189         moreover have "M (?G (1 / real (Suc m))) \<le> M (?G (1 / real (Suc m)))" by simp
   190         ultimately show "\<exists>U\<in>{U. A \<subseteq> U \<and> open U}.
   191           emeasure M U \<le> emeasure M {x. infdist x A < 1 / real (Suc m)}"
   192           by blast
   193       qed
   194       moreover
   195       have "emeasure M A \<le> (INF U:{U. A \<subseteq> U \<and> open U}. emeasure M U)"
   196         by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)
   197       ultimately show ?thesis by simp
   198     qed (auto intro!: INF_eqI)
   199     note \<open>?inner A\<close> \<open>?outer A\<close> }
   200   note closed_in_D = this
   201   from \<open>B \<in> sets borel\<close>
   202   have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"
   203     by (auto simp: Int_stable_def borel_eq_closed)
   204   then show "?inner B" "?outer B"
   205   proof (induct B rule: sigma_sets_induct_disjoint)
   206     case empty
   207     { case 1 show ?case by (intro SUP_eqI[symmetric]) auto }
   208     { case 2 show ?case by (intro INF_eqI[symmetric]) (auto elim!: meta_allE[of _ "{}"]) }
   209   next
   210     case (basic B)
   211     { case 1 from basic closed_in_D show ?case by auto }
   212     { case 2 from basic closed_in_D show ?case by auto }
   213   next
   214     case (compl B)
   215     note inner = compl(2) and outer = compl(3)
   216     from compl have [simp]: "B \<in> sets M" by (auto simp: sb borel_eq_closed)
   217     case 2
   218     have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
   219     also have "\<dots> = (INF K:{K. K \<subseteq> B \<and> compact K}. M (space M) -  M K)"
   220       by (subst ennreal_SUP_const_minus) (auto simp: less_top[symmetric] inner)
   221     also have "\<dots> = (INF U:{U. U \<subseteq> B \<and> compact U}. M (space M - U))"
   222       by (rule INF_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
   223     also have "\<dots> \<ge> (INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U))"
   224       by (rule INF_superset_mono) (auto simp add: compact_imp_closed)
   225     also have "(INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U)) =
   226         (INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"
   227       unfolding INF_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]
   228         by (rule INF_cong) (auto simp add: sU Compl_eq_Diff_UNIV [symmetric, simp])
   229     finally have
   230       "(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<le> emeasure M (space M - B)" .
   231     moreover have
   232       "(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<ge> emeasure M (space M - B)"
   233       by (auto simp: sb sU intro!: INF_greatest emeasure_mono)
   234     ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
   235 
   236     case 1
   237     have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
   238     also have "\<dots> = (SUP U: {U. B \<subseteq> U \<and> open U}. M (space M) -  M U)"
   239       unfolding outer by (subst ennreal_INF_const_minus) auto
   240     also have "\<dots> = (SUP U:{U. B \<subseteq> U \<and> open U}. M (space M - U))"
   241       by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
   242     also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"
   243       unfolding SUP_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]
   244         by (rule SUP_cong) (auto simp add: sU)
   245     also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
   246     proof (safe intro!: antisym SUP_least)
   247       fix K assume "closed K" "K \<subseteq> space M - B"
   248       from closed_in_D[OF \<open>closed K\<close>]
   249       have K_inner: "emeasure M K = (SUP K:{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp
   250       show "emeasure M K \<le> (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
   251         unfolding K_inner using \<open>K \<subseteq> space M - B\<close>
   252         by (auto intro!: SUP_upper SUP_least)
   253     qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
   254     finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
   255   next
   256     case (union D)
   257     then have "range D \<subseteq> sets M" by (auto simp: sb borel_eq_closed)
   258     with union have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (intro suminf_emeasure)
   259     also have "(\<lambda>n. \<Sum>i<n. M (D i)) \<longlonglongrightarrow> (\<Sum>i. M (D i))"
   260       by (intro summable_LIMSEQ) auto
   261     finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) \<longlonglongrightarrow> measure M (\<Union>i. D i)"
   262       by (simp add: emeasure_eq_measure measure_nonneg setsum_nonneg)
   263     have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto
   264 
   265     case 1
   266     show ?case
   267     proof (rule approx_inner)
   268       fix e::real assume "e > 0"
   269       with measure_LIMSEQ
   270       have "\<exists>no. \<forall>n\<ge>no. \<bar>(\<Sum>i<n. measure M (D i)) -measure M (\<Union>x. D x)\<bar> < e/2"
   271         by (auto simp: lim_sequentially dist_real_def simp del: less_divide_eq_numeral1)
   272       hence "\<exists>n0. \<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>x. D x)\<bar> < e/2" by auto
   273       then obtain n0 where n0: "\<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>i. D i)\<bar> < e/2"
   274         unfolding choice_iff by blast
   275       have "ennreal (\<Sum>i<n0. measure M (D i)) = (\<Sum>i<n0. M (D i))"
   276         by (auto simp add: emeasure_eq_measure setsum_nonneg measure_nonneg)
   277       also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule setsum_le_suminf) auto
   278       also have "\<dots> = M (\<Union>i. D i)" by (simp add: M)
   279       also have "\<dots> = measure M (\<Union>i. D i)" by (simp add: emeasure_eq_measure)
   280       finally have n0: "measure M (\<Union>i. D i) - (\<Sum>i<n0. measure M (D i)) < e/2"
   281         using n0 by (auto simp: measure_nonneg setsum_nonneg)
   282       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)"
   283       proof
   284         fix i
   285         from \<open>0 < e\<close> have "0 < e/(2*Suc n0)" by simp
   286         have "emeasure M (D i) = (SUP K:{K. K \<subseteq> (D i) \<and> compact K}. emeasure M K)"
   287           using union by blast
   288         from SUP_approx_ennreal[OF \<open>0 < e/(2*Suc n0)\<close> _ this]
   289         show "\<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
   290           by (auto simp: emeasure_eq_measure intro: less_imp_le compact_empty)
   291       qed
   292       then obtain K where K: "\<And>i. K i \<subseteq> D i" "\<And>i. compact (K i)"
   293         "\<And>i. emeasure M (D i) \<le> emeasure M (K i) + e/(2*Suc n0)"
   294         unfolding choice_iff by blast
   295       let ?K = "\<Union>i\<in>{..<n0}. K i"
   296       have "disjoint_family_on K {..<n0}" using K \<open>disjoint_family D\<close>
   297         unfolding disjoint_family_on_def by blast
   298       hence mK: "measure M ?K = (\<Sum>i<n0. measure M (K i))" using K
   299         by (intro finite_measure_finite_Union) (auto simp: sb compact_imp_closed)
   300       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (D i)) + e/2" using n0 by simp
   301       also have "(\<Sum>i<n0. measure M (D i)) \<le> (\<Sum>i<n0. measure M (K i) + e/(2*Suc n0))"
   302         using K \<open>0 < e\<close>
   303         by (auto intro: setsum_mono simp: emeasure_eq_measure measure_nonneg ennreal_plus[symmetric] simp del: ennreal_plus)
   304       also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"
   305         by (simp add: setsum.distrib)
   306       also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) +  e / 2" using \<open>0 < e\<close>
   307         by (auto simp: field_simps intro!: mult_left_mono)
   308       finally
   309       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"
   310         by auto
   311       hence "M (\<Union>i. D i) < M ?K + e"
   312         using \<open>0<e\<close> by (auto simp: mK emeasure_eq_measure measure_nonneg setsum_nonneg ennreal_less_iff ennreal_plus[symmetric] simp del: ennreal_plus)
   313       moreover
   314       have "?K \<subseteq> (\<Union>i. D i)" using K by auto
   315       moreover
   316       have "compact ?K" using K by auto
   317       ultimately
   318       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
   319       thus "\<exists>K\<subseteq>\<Union>i. D i. compact K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M K + ennreal e" ..
   320     qed fact
   321     case 2
   322     show ?case
   323     proof (rule approx_outer[OF \<open>(\<Union>i. D i) \<in> sets M\<close>])
   324       fix e::real assume "e > 0"
   325       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)"
   326       proof
   327         fix i::nat
   328         from \<open>0 < e\<close> have "0 < e/(2 powr Suc i)" by simp
   329         have "emeasure M (D i) = (INF U:{U. (D i) \<subseteq> U \<and> open U}. emeasure M U)"
   330           using union by blast
   331         from INF_approx_ennreal[OF \<open>0 < e/(2 powr Suc i)\<close> this]
   332         show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
   333           using \<open>0<e\<close>
   334           by (auto simp: emeasure_eq_measure measure_nonneg setsum_nonneg ennreal_less_iff ennreal_plus[symmetric] ennreal_minus
   335                          finite_measure_mono sb
   336                    simp del: ennreal_plus)
   337       qed
   338       then obtain U where U: "\<And>i. D i \<subseteq> U i" "\<And>i. open (U i)"
   339         "\<And>i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)"
   340         unfolding choice_iff by blast
   341       let ?U = "\<Union>i. U i"
   342       have "ennreal (measure M ?U - measure M (\<Union>i. D i)) = M ?U - M (\<Union>i. D i)"
   343         using U(1,2)
   344         by (subst ennreal_minus[symmetric])
   345            (auto intro!: finite_measure_mono simp: sb measure_nonneg emeasure_eq_measure)
   346       also have "\<dots> = M (?U - (\<Union>i. D i))" using U  \<open>(\<Union>i. D i) \<in> sets M\<close>
   347         by (subst emeasure_Diff) (auto simp: sb)
   348       also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U  \<open>range D \<subseteq> sets M\<close>
   349         by (intro emeasure_mono) (auto simp: sb intro!: sets.countable_nat_UN sets.Diff)
   350       also have "\<dots> \<le> (\<Sum>i. M (U i - D i))" using U  \<open>range D \<subseteq> sets M\<close>
   351         by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)
   352       also have "\<dots> \<le> (\<Sum>i. ennreal e/(2 powr Suc i))" using U \<open>range D \<subseteq> sets M\<close>
   353         using \<open>0<e\<close>
   354         by (intro suminf_le, subst emeasure_Diff)
   355            (auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg ennreal_minus
   356                        finite_measure_mono divide_ennreal ennreal_less_iff
   357                  intro: less_imp_le)
   358       also have "\<dots> \<le> (\<Sum>n. ennreal (e * (1 / 2) ^ Suc n))"
   359         using \<open>0<e\<close>
   360         by (simp add: powr_minus powr_realpow field_simps divide_ennreal del: of_nat_Suc)
   361       also have "\<dots> = ennreal e * (\<Sum>n. ennreal ((1 / 2) ^  Suc n))"
   362         unfolding ennreal_power[symmetric]
   363         using \<open>0 < e\<close>
   364         by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def
   365                       ennreal_power[symmetric])
   366       also have "\<dots> = ennreal e"
   367         by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
   368       finally have "emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ennreal e"
   369         using \<open>0<e\<close> by (simp add: emeasure_eq_measure ennreal_plus[symmetric] measure_nonneg del: ennreal_plus)
   370       moreover
   371       have "(\<Union>i. D i) \<subseteq> ?U" using U by auto
   372       moreover
   373       have "open ?U" using U by auto
   374       ultimately
   375       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
   376       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" ..
   377     qed
   378   qed
   379 qed
   380 
   381 end
   382