--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Regularity.thy Thu Nov 15 10:49:58 2012 +0100
@@ -0,0 +1,516 @@
+(* Title: HOL/Probability/Projective_Family.thy
+ Author: Fabian Immler, TU München
+*)
+
+theory Regularity
+imports Measure_Space Borel_Space
+begin
+
+instantiation nat::topological_space
+begin
+
+definition open_nat::"nat set \<Rightarrow> bool"
+ where "open_nat s = True"
+
+instance proof qed (auto simp: open_nat_def)
+end
+
+instantiation nat::metric_space
+begin
+
+definition dist_nat::"nat \<Rightarrow> nat \<Rightarrow> real"
+ where "dist_nat n m = (if n = m then 0 else 1)"
+
+instance proof qed (auto simp: open_nat_def dist_nat_def intro: exI[where x=1])
+end
+
+instance nat::complete_space
+proof
+ fix X::"nat\<Rightarrow>nat" assume "Cauchy X"
+ hence "\<exists>n. \<forall>m\<ge>n. X m = X n"
+ by (force simp: dist_nat_def Cauchy_def split: split_if_asm dest:spec[where x=1])
+ then guess n ..
+ thus "convergent X"
+ apply (intro convergentI[where L="X n"] tendstoI)
+ unfolding eventually_sequentially dist_nat_def
+ apply (intro exI[where x=n])
+ apply (intro allI)
+ apply (drule_tac x=na in spec)
+ apply simp
+ done
+qed
+
+instance nat::enumerable_basis
+proof
+ have "topological_basis (range (\<lambda>n::nat. {n}))"
+ by (intro topological_basisI) (auto simp: open_nat_def)
+ thus "\<exists>f::nat\<Rightarrow>nat set. topological_basis (range f)" by blast
+qed
+
+subsection {* Regularity of Measures *}
+
+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::{enumerable_basis, 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::"nat \<Rightarrow> 'a" . 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[OF this]
+ have x: "space M = (\<Union>n. cball (x n) r)"
+ by (auto simp add: sU) (metis dist_commute order_less_imp_le)
+ have "(\<lambda>k. emeasure M (\<Union>n\<in>{0..k}. cball (x n) r)) ----> M (\<Union>k. (\<Union>n\<in>{0..k}. cball (x n) r))"
+ by (rule Lim_emeasure_incseq)
+ (auto intro!: borel_closed bexI simp: closed_cball incseq_def Us sb)
+ also have "(\<Union>k. (\<Union>n\<in>{0..k}. cball (x n) r)) = space M"
+ unfolding x by force
+ finally have "(\<lambda>k. M (\<Union>n\<in>{0..k}. cball (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 (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 (x i) (1/real n))) (measure M (space M)) <
+ e * 2 powr -n"
+ by auto
+ hence "measure M (\<Union>i\<in>{0..k}. cball (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 (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 (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 (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 (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 (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 = "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
+ have closed_in_D: "\<And>A. closed A \<Longrightarrow> ?inner A \<and> ?outer A"
+ proof
+ 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)
+ show "?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
+ show "?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)
+ qed
+ let ?D = "{B \<in> sets M. ?inner B \<and> ?outer B}"
+ interpret dynkin: dynkin_system "space M" ?D
+ proof (rule dynkin_systemI)
+ have "{U::'a set. space M \<subseteq> U \<and> open U} = {space M}" by (auto simp add: sU)
+ hence "?outer (space M)" by (simp add: min_def INF_def)
+ moreover
+ have "?inner (space M)"
+ proof (rule ereal_approx_SUP)
+ fix e::real assume "0 < e"
+ thus "\<exists>K\<in>{K. K \<subseteq> space M \<and> compact K}. emeasure M (space M) \<le> emeasure M K + ereal e"
+ by (rule approx_space)
+ qed (auto intro: emeasure_mono simp: sU sb intro!: exI[where x="{}"])
+ ultimately show "space M \<in> ?D" by (simp add: sU sb)
+ next
+ fix B assume "B \<in> ?D" thus "B \<subseteq> space M" by (simp add: sU)
+ from `B \<in> ?D` have [simp]: "B \<in> sets M" and "?inner B" "?outer B" by auto
+ hence inner: "emeasure M B = (SUP K:{K. K \<subseteq> B \<and> compact K}. emeasure M K)"
+ and outer: "emeasure M B = (INF U:{U. B \<subseteq> U \<and> open U}. emeasure M U)" by auto
+ 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 have "?outer (space M - B)" by simp
+ moreover
+ {
+ 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 have "?inner (space M - B)" .
+ } hence "?inner (space M - B)" .
+ ultimately show "space M - B \<in> ?D" by auto
+ next
+ fix D :: "nat \<Rightarrow> _"
+ assume "range D \<subseteq> ?D" hence "range D \<subseteq> sets M" by auto
+ moreover assume "disjoint_family D"
+ ultimately have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (rule 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
+ moreover
+ hence "?inner (\<Union>i. D i)"
+ 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 `range D \<subseteq> ?D` 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
+ moreover have "?outer (\<Union>i. D i)"
+ 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 `range D \<subseteq> ?D` 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!: countable_nat_UN 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!: 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
+ ultimately show "(\<Union>i. D i) \<in> ?D" by safe
+ qed
+ have "sets borel = sigma_sets (space M) (Collect closed)" by (simp add: borel_eq_closed sU)
+ also have "\<dots> = dynkin (space M) (Collect closed)"
+ proof (rule sigma_eq_dynkin)
+ show "Collect closed \<subseteq> Pow (space M)" using Sigma_Algebra.sets_into_space by (auto simp: sU)
+ show "Int_stable (Collect closed)" by (auto simp: Int_stable_def)
+ qed
+ also have "\<dots> \<subseteq> ?D" using closed_in_D
+ by (intro dynkin.dynkin_subset) (auto simp add: compact_imp_closed sb)
+ finally have "sets borel \<subseteq> ?D" .
+ moreover have "?D \<subseteq> sets borel" by (auto simp: sb)
+ ultimately have "sets borel = ?D" by simp
+ with assms show "?inner B" and "?outer B" by auto
+qed
+
+end
+