tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/Probability/Regularity.thy
Author: Fabian Immler, TU München
*)
header {* Regularity of Measures *}
theory Regularity
imports Measure_Space Borel_Space
begin
lemma ereal_approx_SUP:
fixes x::ereal
assumes A_notempty: "A \<noteq> {}"
assumes f_bound: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x"
assumes f_fin: "\<And>i. i \<in> A \<Longrightarrow> f i \<noteq> \<infinity>"
assumes f_nonneg: "\<And>i. 0 \<le> f i"
assumes approx: "\<And>e. (e::real) > 0 \<Longrightarrow> \<exists>i \<in> A. x \<le> f i + e"
shows "x = (SUP i : A. f i)"
proof (subst eq_commute, rule SUP_eqI)
show "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x" using f_bound by simp
next
fix y :: ereal assume f_le_y: "(\<And>i::'a. i \<in> A \<Longrightarrow> f i \<le> y)"
with A_notempty f_nonneg have "y \<ge> 0" by auto (metis order_trans)
show "x \<le> y"
proof (rule ccontr)
assume "\<not> x \<le> y" hence "x > y" by simp
hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using `y \<ge> 0` by auto
have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using `x > y` f_fin approx[where e = 1] by auto
def e \<equiv> "real ((x - y) / 2)"
have e: "x > y + e" "e > 0" using `x > y` y_fin x_fin by (auto simp: e_def field_simps)
note e(1)
also from approx[OF `e > 0`] obtain i where i: "i \<in> A" "x \<le> f i + e" by blast
note i(2)
finally have "y < f i" using y_fin f_fin by (metis add_right_mono linorder_not_le)
moreover have "f i \<le> y" by (rule f_le_y) fact
ultimately show False by simp
qed
qed
lemma ereal_approx_INF:
fixes x::ereal
assumes A_notempty: "A \<noteq> {}"
assumes f_bound: "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i"
assumes f_fin: "\<And>i. i \<in> A \<Longrightarrow> f i \<noteq> \<infinity>"
assumes f_nonneg: "\<And>i. 0 \<le> f i"
assumes approx: "\<And>e. (e::real) > 0 \<Longrightarrow> \<exists>i \<in> A. f i \<le> x + e"
shows "x = (INF i : A. f i)"
proof (subst eq_commute, rule INF_eqI)
show "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i" using f_bound by simp
next
fix y :: ereal assume f_le_y: "(\<And>i::'a. i \<in> A \<Longrightarrow> y \<le> f i)"
with A_notempty f_fin have "y \<noteq> \<infinity>" by force
show "y \<le> x"
proof (rule ccontr)
assume "\<not> y \<le> x" hence "y > x" by simp hence "y \<noteq> - \<infinity>" by auto
hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using `y \<noteq> \<infinity>` by auto
have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using `y > x` f_fin f_nonneg approx[where e = 1] A_notempty
apply auto by (metis ereal_infty_less_eq(2) f_le_y)
def e \<equiv> "real ((y - x) / 2)"
have e: "y > x + e" "e > 0" using `y > x` y_fin x_fin by (auto simp: e_def field_simps)
from approx[OF `e > 0`] obtain i where i: "i \<in> A" "x + e \<ge> f i" by blast
note i(2)
also note e(1)
finally have "y > f i" .
moreover have "y \<le> f i" by (rule f_le_y) fact
ultimately show False by simp
qed
qed
lemma INF_approx_ereal:
fixes x::ereal and e::real
assumes "e > 0"
assumes INF: "x = (INF i : A. f i)"
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
shows "\<exists>i \<in> A. f i < x + e"
proof (rule ccontr, clarsimp)
assume "\<forall>i\<in>A. \<not> f i < x + e"
moreover
from INF have "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x" by (auto intro: INF_greatest)
ultimately
have "(INF i : A. f i) = x + e" using `e > 0`
by (intro INF_eqI)
(force, metis add.comm_neutral add_left_mono ereal_less(1)
linorder_not_le not_less_iff_gr_or_eq)
thus False using assms by auto
qed
lemma SUP_approx_ereal:
fixes x::ereal and e::real
assumes "e > 0"
assumes SUP: "x = (SUP i : A. f i)"
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
shows "\<exists>i \<in> A. x \<le> f i + e"
proof (rule ccontr, clarsimp)
assume "\<forall>i\<in>A. \<not> x \<le> f i + e"
moreover
from SUP have "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> y \<ge> x" by (auto intro: SUP_least)
ultimately
have "(SUP i : A. f i) = x - e" using `e > 0` `\<bar>x\<bar> \<noteq> \<infinity>`
by (intro SUP_eqI)
(metis PInfty_neq_ereal(2) abs_ereal.simps(1) ereal_minus_le linorder_linear,
metis ereal_between(1) ereal_less(2) less_eq_ereal_def order_trans)
thus False using assms by auto
qed
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 + ereal e) \<Longrightarrow> ?inner A"
by (rule ereal_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 + ereal e) \<Longrightarrow> ?outer A"
by (rule ereal_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)) ----> M ?U"
by (rule Lim_emeasure_incseq)
(auto intro!: borel_closed bexI simp: closed_cball incseq_def Us sb)
also have "?U = space M"
proof safe
fix x from X(2)[OF open_ball[of x r]] `r > 0` obtain d where d: "d\<in>X" "d \<in> ball x r" by auto
show "x \<in> ?U"
using X(1) d by (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)) ----> 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 intro: mult_pos_pos)
from M_space[OF `1/n>0`]
have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) ----> measure M (space M)"
unfolding emeasure_eq_measure by simp
from metric_LIMSEQ_D[OF this `0 < e * 2 powr -n`]
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))"
apply atomize_elim unfolding bchoice_iff .
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:
"\<And>e. e > 0 \<Longrightarrow>
\<exists>K \<in> {K. K \<subseteq> space M \<and> compact K}. emeasure M (space M) \<le> emeasure M K + ereal e"
(is "\<And>e. _ \<Longrightarrow> ?thesis e")
proof -
fix e :: real assume "e > 0"
def B \<equiv> "\<lambda>n. \<Union>i\<in>{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n)"
have "\<And>n. closed (B n)" by (auto simp: B_def closed_cball)
hence [simp]: "\<And>n. B n \<in> sets M" by (simp add: sb)
from k[OF `e > 0` 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)
def K \<equiv> "\<Inter>n. B n"
from `closed (B _)` 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. ereal (e*2 powr - real (Suc n)))"
using B_compl_le by (intro suminf_le_pos) (simp_all add: measure_nonneg emeasure_eq_measure)
also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
by (simp add: powr_minus inverse_eq_divide powr_realpow field_simps power_divide)
also have "\<dots> = (\<Sum>n. ereal e * ((1 / 2) ^ Suc n))"
unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
by simp
also have "\<dots> = ereal e * (\<Sum>n. ((1 / 2) ^ Suc n))"
by (rule suminf_cmult_ereal) (auto simp: `0 < e` less_imp_le)
also have "\<dots> = e" unfolding suminf_half_series_ereal by simp
finally have "measure M (space M) \<le> measure M K + e" by simp
hence "emeasure M (space M) \<le> emeasure M K + e" by (simp add: emeasure_eq_measure)
moreover have "compact K"
unfolding compact_eq_totally_bounded
proof safe
show "complete K" using `closed K` 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>((\<lambda>x. ball x e') ` ?k)" unfolding K_def B_def using n by force
ultimately show "\<exists>k. finite k \<and> K \<subseteq> \<Union>((\<lambda>x. ball x e') ` k)" by blast
qed
ultimately
show "?thesis e " 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 "M A - M (A \<inter> K) = measure M A - measure M (A \<inter> K)"
by (simp add: emeasure_eq_measure)
also have "\<dots> = 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 by (simp add: emeasure_eq_measure)
finally have "emeasure M A \<le> emeasure M (A \<inter> K) + ereal e"
by (simp add: emeasure_eq_measure algebra_simps)
moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using `closed A` `compact K` by auto
ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ereal 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_at_id)
finally have "open (?G d)" .
} note open_G = this
from in_closed_iff_infdist_zero[OF `closed A` `A \<noteq> {}`]
have "A = {x. infdist x A = 0}" by auto
also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"
proof (auto, 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 mult_pos_pos
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 `?inner A` `?outer A` }
note closed_in_D = this
from `B \<in> sets borel`
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)"
unfolding inner by (subst INFI_ereal_cminus) force+
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)"
by (subst INF_image[of "\<lambda>u. space M - u", symmetric])
(rule INF_cong, auto simp add: sU intro!: INF_cong)
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 SUPR_ereal_cminus) 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)"
by (subst SUP_image[of "\<lambda>u. space M - u", symmetric])
(rule SUP_cong, auto simp: 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 `closed K`]
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 `K \<subseteq> space M - B`
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\<in>{0..<n}. M (D i)) ----> (\<Sum>i. M (D i))"
by (intro summable_sumr_LIMSEQ_suminf summable_ereal_pos emeasure_nonneg)
finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i = 0..<n. measure M (D i)) ----> measure M (\<Union>i. D i)"
by (simp add: emeasure_eq_measure)
have "(\<Union>i. D i) \<in> sets M" using `range D \<subseteq> sets M` 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 = 0..<n. measure M (D i)) -measure M (\<Union>x. D x)\<bar> < e/2"
by (auto simp: LIMSEQ_def dist_real_def simp del: less_divide_eq_numeral1)
hence "\<exists>n0. \<bar>(\<Sum>i = 0..<n0. measure M (D i)) - measure M (\<Union>x. D x)\<bar> < e/2" by auto
then obtain n0 where n0: "\<bar>(\<Sum>i = 0..<n0. measure M (D i)) - measure M (\<Union>i. D i)\<bar> < e/2"
unfolding choice_iff by blast
have "ereal (\<Sum>i = 0..<n0. measure M (D i)) = (\<Sum>i = 0..<n0. M (D i))"
by (auto simp add: emeasure_eq_measure)
also have "\<dots> = (\<Sum>i<n0. M (D i))" by (rule setsum_cong) auto
also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule suminf_upper) (auto simp: emeasure_nonneg)
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 = 0..<n0. measure M (D i)) < e/2"
using n0 by auto
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 `0 < e` have "0 < e/(2*Suc n0)" by (auto intro: divide_pos_pos)
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_ereal[OF `0 < e/(2*Suc n0)` 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)
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>{0..<n0}. K i"
have "disjoint_family_on K {0..<n0}" using K `disjoint_family D`
unfolding disjoint_family_on_def by blast
hence mK: "measure M ?K = (\<Sum>i = 0..<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 = 0..<n0. measure M (D i)) + e/2" using n0 by simp
also have "(\<Sum>i = 0..<n0. measure M (D i)) \<le> (\<Sum>i = 0..<n0. measure M (K i) + e/(2*Suc n0))"
using K by (auto intro: setsum_mono simp: emeasure_eq_measure)
also have "\<dots> = (\<Sum>i = 0..<n0. measure M (K i)) + (\<Sum>i = 0..<n0. e/(2*Suc n0))"
by (simp add: setsum.distrib)
also have "\<dots> \<le> (\<Sum>i = 0..<n0. measure M (K i)) + e / 2" using `0 < e`
by (auto simp: real_of_nat_def[symmetric] field_simps intro!: mult_left_mono)
finally
have "measure M (\<Union>i. D i) < (\<Sum>i = 0..<n0. measure M (K i)) + e / 2 + e / 2"
by auto
hence "M (\<Union>i. D i) < M ?K + e" by (auto simp: mK emeasure_eq_measure)
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 + ereal e" by simp
thus "\<exists>K\<subseteq>\<Union>i. D i. compact K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M K + ereal e" ..
qed fact
case 2
show ?case
proof (rule approx_outer[OF `(\<Union>i. D i) \<in> sets M`])
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 `0 < e` have "0 < e/(2 powr Suc i)" by (auto intro: divide_pos_pos)
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_ereal[OF `0 < e/(2 powr Suc i)` this]
show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
by (auto simp: emeasure_eq_measure)
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 "M ?U - M (\<Union>i. D i) = M (?U - (\<Union>i. D i))" using U `(\<Union>i. D i) \<in> sets M`
by (subst emeasure_Diff) (auto simp: sb)
also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U `range D \<subseteq> sets M`
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 `range D \<subseteq> sets M`
by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)
also have "\<dots> \<le> (\<Sum>i. ereal e/(2 powr Suc i))" using U `range D \<subseteq> sets M`
by (intro suminf_le_pos, subst emeasure_Diff)
(auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg intro: less_imp_le)
also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
by (simp add: powr_minus inverse_eq_divide powr_realpow field_simps power_divide)
also have "\<dots> = (\<Sum>n. ereal e * ((1 / 2) ^ Suc n))"
unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
by simp
also have "\<dots> = ereal e * (\<Sum>n. ((1 / 2) ^ Suc n))"
by (rule suminf_cmult_ereal) (auto simp: `0 < e` less_imp_le)
also have "\<dots> = e" unfolding suminf_half_series_ereal by simp
finally
have "emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ereal e" by (simp add: emeasure_eq_measure)
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) + ereal 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) + ereal e" ..
qed
qed
qed
end