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src/HOL/Probability/Regularity.thy

author | hoelzl |

Mon, 14 Jan 2013 17:29:04 +0100 | |

changeset 50881 | ae630bab13da |

parent 50530 | 6266e44b3396 |

child 51000 | c9adb50f74ad |

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

renamed countable_basis_space to second_countable_topology

(* 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 ereal_SUPI) 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 ereal_INFI) 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 ereal_INFI) (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 ereal_SUPI) (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!: ereal_INFI) 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 ereal_SUPI[symmetric]) auto } { case 2 show ?case by (intro ereal_INFI[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