(* Title: HOL/Analysis/Regularity.thy
Author: Fabian Immler, TU München
*)
section \<open>Regularity of Measures\<close>
theory Regularity
imports Measure_Space Borel_Space
begin
lemma
fixes M::"'a::{second_countable_topology, complete_space} measure"
assumes sb: "sets M = sets borel"
assumes "emeasure M (space M) \<noteq> \<infinity>"
assumes "B \<in> sets borel"
shows inner_regular: "emeasure M B =
(SUP K : {K. K \<subseteq> B \<and> compact K}. emeasure M K)" (is "?inner B")
and outer_regular: "emeasure M B =
(INF U : {U. B \<subseteq> U \<and> open U}. emeasure M U)" (is "?outer B")
proof -
have Us: "UNIV = space M" by (metis assms(1) sets_eq_imp_space_eq space_borel)
hence sU: "space M = UNIV" by simp
interpret finite_measure M by rule fact
have approx_inner: "\<And>A. A \<in> sets M \<Longrightarrow>
(\<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"
by (rule ennreal_approx_SUP)
(force intro!: emeasure_mono simp: compact_imp_closed emeasure_eq_measure)+
have approx_outer: "\<And>A. A \<in> sets M \<Longrightarrow>
(\<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"
by (rule ennreal_approx_INF)
(force intro!: emeasure_mono simp: emeasure_eq_measure sb)+
from countable_dense_setE guess X::"'a set" . note X = this
{
fix r::real assume "r > 0" hence "\<And>y. open (ball y r)" "\<And>y. ball y r \<noteq> {}" by auto
with X(2)[OF this]
have x: "space M = (\<Union>x\<in>X. cball x r)"
by (auto simp add: sU) (metis dist_commute order_less_imp_le)
let ?U = "\<Union>k. (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)"
have "(\<lambda>k. emeasure M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M ?U"
by (rule Lim_emeasure_incseq) (auto intro!: borel_closed bexI simp: incseq_def Us sb)
also have "?U = space M"
proof safe
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
show "x \<in> ?U"
using X(1) d
by simp (auto intro!: exI [where x = "to_nat_on X d"] simp: dist_commute Bex_def)
qed (simp add: sU)
finally have "(\<lambda>k. M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M (space M)" .
} note M_space = this
{
fix e ::real and n :: nat assume "e > 0" "n > 0"
hence "1/n > 0" "e * 2 powr - n > 0" by (auto)
from M_space[OF \<open>1/n>0\<close>]
have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) \<longlonglongrightarrow> measure M (space M)"
unfolding emeasure_eq_measure by (auto simp: measure_nonneg)
from metric_LIMSEQ_D[OF this \<open>0 < e * 2 powr -n\<close>]
obtain k where "dist (measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) (measure M (space M)) <
e * 2 powr -n"
by auto
hence "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"
by (auto simp: dist_real_def)
hence "\<exists>k. 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" ..
} note k=this
hence "\<forall>e\<in>{0<..}. \<forall>(n::nat)\<in>{0<..}. \<exists>k.
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"
by blast
then obtain k where k: "\<forall>e\<in>{0<..}. \<forall>n\<in>{0<..}. measure M (space M) - e * 2 powr - real (n::nat)
\<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"
by metis
hence k: "\<And>e n. e > 0 \<Longrightarrow> n > 0 \<Longrightarrow> measure M (space M) - e * 2 powr - n
\<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"
unfolding Ball_def by blast
have approx_space:
"\<exists>K \<in> {K. K \<subseteq> space M \<and> compact K}. emeasure M (space M) \<le> emeasure M K + ennreal e"
(is "?thesis e") if "0 < e" for e :: real
proof -
define B where [abs_def]:
"B n = (\<Union>i\<in>{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n))" for n
have "\<And>n. closed (B n)" by (auto simp: B_def)
hence [simp]: "\<And>n. B n \<in> sets M" by (simp add: sb)
from k[OF \<open>e > 0\<close> zero_less_Suc]
have "\<And>n. measure M (space M) - measure M (B n) \<le> e * 2 powr - real (Suc n)"
by (simp add: algebra_simps B_def finite_measure_compl)
hence B_compl_le: "\<And>n::nat. measure M (space M - B n) \<le> e * 2 powr - real (Suc n)"
by (simp add: finite_measure_compl)
define K where "K = (\<Inter>n. B n)"
from \<open>closed (B _)\<close> have "closed K" by (auto simp: K_def)
hence [simp]: "K \<in> sets M" by (simp add: sb)
have "measure M (space M) - measure M K = measure M (space M - K)"
by (simp add: finite_measure_compl)
also have "\<dots> = emeasure M (\<Union>n. space M - B n)" by (auto simp: K_def emeasure_eq_measure)
also have "\<dots> \<le> (\<Sum>n. emeasure M (space M - B n))"
by (rule emeasure_subadditive_countably) (auto simp: summable_def)
also have "\<dots> \<le> (\<Sum>n. ennreal (e*2 powr - real (Suc n)))"
using B_compl_le by (intro suminf_le) (simp_all add: measure_nonneg emeasure_eq_measure ennreal_leI)
also have "\<dots> \<le> (\<Sum>n. ennreal (e * (1 / 2) ^ Suc n))"
by (simp add: powr_minus powr_realpow field_simps del: of_nat_Suc)
also have "\<dots> = ennreal e * (\<Sum>n. ennreal ((1 / 2) ^ Suc n))"
unfolding ennreal_power[symmetric]
using \<open>0 < e\<close>
by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def
ennreal_power[symmetric])
also have "\<dots> = e"
by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
finally have "measure M (space M) \<le> measure M K + e"
using \<open>0 < e\<close> by simp
hence "emeasure M (space M) \<le> emeasure M K + e"
using \<open>0 < e\<close> by (simp add: emeasure_eq_measure reorient: ennreal_plus)
moreover have "compact K"
unfolding compact_eq_totally_bounded
proof safe
show "complete K" using \<open>closed K\<close> by (simp add: complete_eq_closed)
fix e'::real assume "0 < e'"
from nat_approx_posE[OF this] guess n . note n = this
let ?k = "from_nat_into X ` {0..k e (Suc n)}"
have "finite ?k" by simp
moreover have "K \<subseteq> (\<Union>x\<in>?k. ball x e')" unfolding K_def B_def using n by force
ultimately show "\<exists>k. finite k \<and> K \<subseteq> (\<Union>x\<in>k. ball x e')" by blast
qed
ultimately
show ?thesis by (auto simp: sU)
qed
{ fix A::"'a set" assume "closed A" hence "A \<in> sets borel" by (simp add: compact_imp_closed)
hence [simp]: "A \<in> sets M" by (simp add: sb)
have "?inner A"
proof (rule approx_inner)
fix e::real assume "e > 0"
from approx_space[OF this] obtain K where
K: "K \<subseteq> space M" "compact K" "emeasure M (space M) \<le> emeasure M K + e"
by (auto simp: emeasure_eq_measure)
hence [simp]: "K \<in> sets M" by (simp add: sb compact_imp_closed)
have "measure M A - measure M (A \<inter> K) = measure M (A - A \<inter> K)"
by (subst finite_measure_Diff) auto
also have "A - A \<inter> K = A \<union> K - K" by auto
also have "measure M \<dots> = measure M (A \<union> K) - measure M K"
by (subst finite_measure_Diff) auto
also have "\<dots> \<le> measure M (space M) - measure M K"
by (simp add: emeasure_eq_measure sU sb finite_measure_mono)
also have "\<dots> \<le> e"
using K \<open>0 < e\<close> by (simp add: emeasure_eq_measure reorient: ennreal_plus)
finally have "emeasure M A \<le> emeasure M (A \<inter> K) + ennreal e"
using \<open>0<e\<close> by (simp add: emeasure_eq_measure algebra_simps reorient: ennreal_plus)
moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using \<open>closed A\<close> \<open>compact K\<close> by auto
ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ennreal e"
by blast
qed simp
have "?outer A"
proof cases
assume "A \<noteq> {}"
let ?G = "\<lambda>d. {x. infdist x A < d}"
{
fix d
have "?G d = (\<lambda>x. infdist x A) -` {..<d}" by auto
also have "open \<dots>"
by (intro continuous_open_vimage) (auto intro!: continuous_infdist continuous_ident)
finally have "open (?G d)" .
} note open_G = this
from in_closed_iff_infdist_zero[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>]
have "A = {x. infdist x A = 0}" by auto
also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"
proof (auto simp del: of_nat_Suc, rule ccontr)
fix x
assume "infdist x A \<noteq> 0"
hence pos: "infdist x A > 0" using infdist_nonneg[of x A] by simp
from nat_approx_posE[OF this] guess n .
moreover
assume "\<forall>i. infdist x A < 1 / real (Suc i)"
hence "infdist x A < 1 / real (Suc n)" by auto
ultimately show False by simp
qed
also have "M \<dots> = (INF n. emeasure M (?G (1 / real (Suc n))))"
proof (rule INF_emeasure_decseq[symmetric], safe)
fix i::nat
from open_G[of "1 / real (Suc i)"]
show "?G (1 / real (Suc i)) \<in> sets M" by (simp add: sb borel_open)
next
show "decseq (\<lambda>i. {x. infdist x A < 1 / real (Suc i)})"
by (auto intro: less_trans intro!: divide_strict_left_mono
simp: decseq_def le_eq_less_or_eq)
qed simp
finally
have "emeasure M A = (INF n. emeasure M {x. infdist x A < 1 / real (Suc n)})" .
moreover
have "\<dots> \<ge> (INF U:{U. A \<subseteq> U \<and> open U}. emeasure M U)"
proof (intro INF_mono)
fix m
have "?G (1 / real (Suc m)) \<in> {U. A \<subseteq> U \<and> open U}" using open_G by auto
moreover have "M (?G (1 / real (Suc m))) \<le> M (?G (1 / real (Suc m)))" by simp
ultimately show "\<exists>U\<in>{U. A \<subseteq> U \<and> open U}.
emeasure M U \<le> emeasure M {x. infdist x A < 1 / real (Suc m)}"
by blast
qed
moreover
have "emeasure M A \<le> (INF U:{U. A \<subseteq> U \<and> open U}. emeasure M U)"
by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)
ultimately show ?thesis by simp
qed (auto intro!: INF_eqI)
note \<open>?inner A\<close> \<open>?outer A\<close> }
note closed_in_D = this
from \<open>B \<in> sets borel\<close>
have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"
by (auto simp: Int_stable_def borel_eq_closed)
then show "?inner B" "?outer B"
proof (induct B rule: sigma_sets_induct_disjoint)
case empty
{ case 1 show ?case by (intro SUP_eqI[symmetric]) auto }
{ case 2 show ?case by (intro INF_eqI[symmetric]) (auto elim!: meta_allE[of _ "{}"]) }
next
case (basic B)
{ case 1 from basic closed_in_D show ?case by auto }
{ case 2 from basic closed_in_D show ?case by auto }
next
case (compl B)
note inner = compl(2) and outer = compl(3)
from compl have [simp]: "B \<in> sets M" by (auto simp: sb borel_eq_closed)
case 2
have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
also have "\<dots> = (INF K:{K. K \<subseteq> B \<and> compact K}. M (space M) - M K)"
by (subst ennreal_SUP_const_minus) (auto simp: less_top[symmetric] inner)
also have "\<dots> = (INF U:{U. U \<subseteq> B \<and> compact U}. M (space M - U))"
by (rule INF_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
also have "\<dots> \<ge> (INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U))"
by (rule INF_superset_mono) (auto simp add: compact_imp_closed)
also have "(INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U)) =
(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"
unfolding INF_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]
by (rule INF_cong) (auto simp add: sU Compl_eq_Diff_UNIV [symmetric, simp])
finally have
"(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<le> emeasure M (space M - B)" .
moreover have
"(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<ge> emeasure M (space M - B)"
by (auto simp: sb sU intro!: INF_greatest emeasure_mono)
ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
case 1
have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
also have "\<dots> = (SUP U: {U. B \<subseteq> U \<and> open U}. M (space M) - M U)"
unfolding outer by (subst ennreal_INF_const_minus) auto
also have "\<dots> = (SUP U:{U. B \<subseteq> U \<and> open U}. M (space M - U))"
by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"
unfolding SUP_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]
by (rule SUP_cong) (auto simp add: sU)
also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
proof (safe intro!: antisym SUP_least)
fix K assume "closed K" "K \<subseteq> space M - B"
from closed_in_D[OF \<open>closed K\<close>]
have K_inner: "emeasure M K = (SUP K:{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp
show "emeasure M K \<le> (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
unfolding K_inner using \<open>K \<subseteq> space M - B\<close>
by (auto intro!: SUP_upper SUP_least)
qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
next
case (union D)
then have "range D \<subseteq> sets M" by (auto simp: sb borel_eq_closed)
with union have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (intro suminf_emeasure)
also have "(\<lambda>n. \<Sum>i<n. M (D i)) \<longlonglongrightarrow> (\<Sum>i. M (D i))"
by (intro summable_LIMSEQ) auto
finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) \<longlonglongrightarrow> measure M (\<Union>i. D i)"
by (simp add: emeasure_eq_measure measure_nonneg sum_nonneg)
have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto
case 1
show ?case
proof (rule approx_inner)
fix e::real assume "e > 0"
with measure_LIMSEQ
have "\<exists>no. \<forall>n\<ge>no. \<bar>(\<Sum>i<n. measure M (D i)) -measure M (\<Union>x. D x)\<bar> < e/2"
by (auto simp: lim_sequentially dist_real_def simp del: less_divide_eq_numeral1)
hence "\<exists>n0. \<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>x. D x)\<bar> < e/2" by auto
then obtain n0 where n0: "\<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>i. D i)\<bar> < e/2"
unfolding choice_iff by blast
have "ennreal (\<Sum>i<n0. measure M (D i)) = (\<Sum>i<n0. M (D i))"
by (auto simp add: emeasure_eq_measure sum_nonneg measure_nonneg)
also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule sum_le_suminf) auto
also have "\<dots> = M (\<Union>i. D i)" by (simp add: M)
also have "\<dots> = measure M (\<Union>i. D i)" by (simp add: emeasure_eq_measure)
finally have n0: "measure M (\<Union>i. D i) - (\<Sum>i<n0. measure M (D i)) < e/2"
using n0 by (auto simp: measure_nonneg sum_nonneg)
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)"
proof
fix i
from \<open>0 < e\<close> have "0 < e/(2*Suc n0)" by simp
have "emeasure M (D i) = (SUP K:{K. K \<subseteq> (D i) \<and> compact K}. emeasure M K)"
using union by blast
from SUP_approx_ennreal[OF \<open>0 < e/(2*Suc n0)\<close> _ this]
show "\<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
by (auto simp: emeasure_eq_measure intro: less_imp_le compact_empty)
qed
then obtain K where K: "\<And>i. K i \<subseteq> D i" "\<And>i. compact (K i)"
"\<And>i. emeasure M (D i) \<le> emeasure M (K i) + e/(2*Suc n0)"
unfolding choice_iff by blast
let ?K = "\<Union>i\<in>{..<n0}. K i"
have "disjoint_family_on K {..<n0}" using K \<open>disjoint_family D\<close>
unfolding disjoint_family_on_def by blast
hence mK: "measure M ?K = (\<Sum>i<n0. measure M (K i))" using K
by (intro finite_measure_finite_Union) (auto simp: sb compact_imp_closed)
have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (D i)) + e/2" using n0 by simp
also have "(\<Sum>i<n0. measure M (D i)) \<le> (\<Sum>i<n0. measure M (K i) + e/(2*Suc n0))"
using K \<open>0 < e\<close>
by (auto intro: sum_mono simp: emeasure_eq_measure reorient: ennreal_plus)
also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"
by (simp add: sum.distrib)
also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) + e / 2" using \<open>0 < e\<close>
by (auto simp: field_simps intro!: mult_left_mono)
finally
have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"
by auto
hence "M (\<Union>i. D i) < M ?K + e"
using \<open>0<e\<close> by (auto simp: mK emeasure_eq_measure sum_nonneg ennreal_less_iff reorient: ennreal_plus)
moreover
have "?K \<subseteq> (\<Union>i. D i)" using K by auto
moreover
have "compact ?K" using K by auto
ultimately
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
thus "\<exists>K\<subseteq>\<Union>i. D i. compact K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M K + ennreal e" ..
qed fact
case 2
show ?case
proof (rule approx_outer[OF \<open>(\<Union>i. D i) \<in> sets M\<close>])
fix e::real assume "e > 0"
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)"
proof
fix i::nat
from \<open>0 < e\<close> have "0 < e/(2 powr Suc i)" by simp
have "emeasure M (D i) = (INF U:{U. (D i) \<subseteq> U \<and> open U}. emeasure M U)"
using union by blast
from INF_approx_ennreal[OF \<open>0 < e/(2 powr Suc i)\<close> this]
show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
using \<open>0<e\<close>
by (auto simp: emeasure_eq_measure sum_nonneg ennreal_less_iff ennreal_minus
finite_measure_mono sb
reorient: ennreal_plus)
qed
then obtain U where U: "\<And>i. D i \<subseteq> U i" "\<And>i. open (U i)"
"\<And>i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)"
unfolding choice_iff by blast
let ?U = "\<Union>i. U i"
have "ennreal (measure M ?U - measure M (\<Union>i. D i)) = M ?U - M (\<Union>i. D i)"
using U(1,2)
by (subst ennreal_minus[symmetric])
(auto intro!: finite_measure_mono simp: sb measure_nonneg emeasure_eq_measure)
also have "\<dots> = M (?U - (\<Union>i. D i))" using U \<open>(\<Union>i. D i) \<in> sets M\<close>
by (subst emeasure_Diff) (auto simp: sb)
also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U \<open>range D \<subseteq> sets M\<close>
by (intro emeasure_mono) (auto simp: sb intro!: sets.countable_nat_UN sets.Diff)
also have "\<dots> \<le> (\<Sum>i. M (U i - D i))" using U \<open>range D \<subseteq> sets M\<close>
by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)
also have "\<dots> \<le> (\<Sum>i. ennreal e/(2 powr Suc i))" using U \<open>range D \<subseteq> sets M\<close>
using \<open>0<e\<close>
by (intro suminf_le, subst emeasure_Diff)
(auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg ennreal_minus
finite_measure_mono divide_ennreal ennreal_less_iff
intro: less_imp_le)
also have "\<dots> \<le> (\<Sum>n. ennreal (e * (1 / 2) ^ Suc n))"
using \<open>0<e\<close>
by (simp add: powr_minus powr_realpow field_simps divide_ennreal del: of_nat_Suc)
also have "\<dots> = ennreal e * (\<Sum>n. ennreal ((1 / 2) ^ Suc n))"
unfolding ennreal_power[symmetric]
using \<open>0 < e\<close>
by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def
ennreal_power[symmetric])
also have "\<dots> = ennreal e"
by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
finally have "emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ennreal e"
using \<open>0<e\<close> by (simp add: emeasure_eq_measure reorient: ennreal_plus)
moreover
have "(\<Union>i. D i) \<subseteq> ?U" using U by auto
moreover
have "open ?U" using U by auto
ultimately
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
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" ..
qed
qed
qed
end