src/HOL/Probability/Regularity.thy
 author paulson Mon Mar 07 14:34:45 2016 +0000 (2016-03-07) changeset 62533 bc25f3916a99 parent 62343 24106dc44def child 62975 1d066f6ab25d permissions -rw-r--r--
new material to Blochj's theorem, as well as supporting lemmas
```     1 (*  Title:      HOL/Probability/Regularity.thy
```
```     2     Author:     Fabian Immler, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Regularity of Measures\<close>
```
```     6
```
```     7 theory Regularity
```
```     8 imports Measure_Space Borel_Space
```
```     9 begin
```
```    10
```
```    11 lemma ereal_approx_SUP:
```
```    12   fixes x::ereal
```
```    13   assumes A_notempty: "A \<noteq> {}"
```
```    14   assumes f_bound: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x"
```
```    15   assumes f_fin: "\<And>i. i \<in> A \<Longrightarrow> f i \<noteq> \<infinity>"
```
```    16   assumes f_nonneg: "\<And>i. 0 \<le> f i"
```
```    17   assumes approx: "\<And>e. (e::real) > 0 \<Longrightarrow> \<exists>i \<in> A. x \<le> f i + e"
```
```    18   shows "x = (SUP i : A. f i)"
```
```    19 proof (subst eq_commute, rule SUP_eqI)
```
```    20   show "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x" using f_bound by simp
```
```    21 next
```
```    22   fix y :: ereal assume f_le_y: "(\<And>i::'a. i \<in> A \<Longrightarrow> f i \<le> y)"
```
```    23   with A_notempty f_nonneg have "y \<ge> 0" by auto (metis order_trans)
```
```    24   show "x \<le> y"
```
```    25   proof (rule ccontr)
```
```    26     assume "\<not> x \<le> y" hence "x > y" by simp
```
```    27     hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using \<open>y \<ge> 0\<close> by auto
```
```    28     have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using \<open>x > y\<close> f_fin approx[where e = 1] by auto
```
```    29     def e \<equiv> "real_of_ereal ((x - y) / 2)"
```
```    30     have e: "x > y + e" "e > 0" using \<open>x > y\<close> y_fin x_fin by (auto simp: e_def field_simps)
```
```    31     note e(1)
```
```    32     also from approx[OF \<open>e > 0\<close>] obtain i where i: "i \<in> A" "x \<le> f i + e" by blast
```
```    33     note i(2)
```
```    34     finally have "y < f i" using y_fin f_fin by (metis add_right_mono linorder_not_le)
```
```    35     moreover have "f i \<le> y" by (rule f_le_y) fact
```
```    36     ultimately show False by simp
```
```    37   qed
```
```    38 qed
```
```    39
```
```    40 lemma ereal_approx_INF:
```
```    41   fixes x::ereal
```
```    42   assumes A_notempty: "A \<noteq> {}"
```
```    43   assumes f_bound: "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i"
```
```    44   assumes f_fin: "\<And>i. i \<in> A \<Longrightarrow> f i \<noteq> \<infinity>"
```
```    45   assumes f_nonneg: "\<And>i. 0 \<le> f i"
```
```    46   assumes approx: "\<And>e. (e::real) > 0 \<Longrightarrow> \<exists>i \<in> A. f i \<le> x + e"
```
```    47   shows "x = (INF i : A. f i)"
```
```    48 proof (subst eq_commute, rule INF_eqI)
```
```    49   show "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i" using f_bound by simp
```
```    50 next
```
```    51   fix y :: ereal assume f_le_y: "(\<And>i::'a. i \<in> A \<Longrightarrow> y \<le> f i)"
```
```    52   with A_notempty f_fin have "y \<noteq> \<infinity>" by force
```
```    53   show "y \<le> x"
```
```    54   proof (rule ccontr)
```
```    55     assume "\<not> y \<le> x" hence "y > x" by simp hence "y \<noteq> - \<infinity>" by auto
```
```    56     hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using \<open>y \<noteq> \<infinity>\<close> by auto
```
```    57     have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using \<open>y > x\<close> f_fin f_nonneg approx[where e = 1] A_notempty
```
```    58       by auto
```
```    59     def e \<equiv> "real_of_ereal ((y - x) / 2)"
```
```    60     have e: "y > x + e" "e > 0" using \<open>y > x\<close> y_fin x_fin by (auto simp: e_def field_simps)
```
```    61     from approx[OF \<open>e > 0\<close>] obtain i where i: "i \<in> A" "x + e \<ge> f i" by blast
```
```    62     note i(2)
```
```    63     also note e(1)
```
```    64     finally have "y > f i" .
```
```    65     moreover have "y \<le> f i" by (rule f_le_y) fact
```
```    66     ultimately show False by simp
```
```    67   qed
```
```    68 qed
```
```    69
```
```    70 lemma INF_approx_ereal:
```
```    71   fixes x::ereal and e::real
```
```    72   assumes "e > 0"
```
```    73   assumes INF: "x = (INF i : A. f i)"
```
```    74   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
```
```    75   shows "\<exists>i \<in> A. f i < x + e"
```
```    76 proof (rule ccontr, clarsimp)
```
```    77   assume "\<forall>i\<in>A. \<not> f i < x + e"
```
```    78   moreover
```
```    79   from INF have "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x" by (auto intro: INF_greatest)
```
```    80   ultimately
```
```    81   have "(INF i : A. f i) = x + e" using \<open>e > 0\<close>
```
```    82     by (intro INF_eqI)
```
```    83       (force, metis add.comm_neutral add_left_mono ereal_less(1)
```
```    84         linorder_not_le not_less_iff_gr_or_eq)
```
```    85   thus False using assms by auto
```
```    86 qed
```
```    87
```
```    88 lemma SUP_approx_ereal:
```
```    89   fixes x::ereal and e::real
```
```    90   assumes "e > 0"
```
```    91   assumes SUP: "x = (SUP i : A. f i)"
```
```    92   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
```
```    93   shows "\<exists>i \<in> A. x \<le> f i + e"
```
```    94 proof (rule ccontr, clarsimp)
```
```    95   assume "\<forall>i\<in>A. \<not> x \<le> f i + e"
```
```    96   moreover
```
```    97   from SUP have "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> y \<ge> x" by (auto intro: SUP_least)
```
```    98   ultimately
```
```    99   have "(SUP i : A. f i) = x - e" using \<open>e > 0\<close> \<open>\<bar>x\<bar> \<noteq> \<infinity>\<close>
```
```   100     by (intro SUP_eqI)
```
```   101        (metis PInfty_neq_ereal(2) abs_ereal.simps(1) ereal_minus_le linorder_linear,
```
```   102         metis ereal_between(1) ereal_less(2) less_eq_ereal_def order_trans)
```
```   103   thus False using assms by auto
```
```   104 qed
```
```   105
```
```   106 lemma
```
```   107   fixes M::"'a::{second_countable_topology, complete_space} measure"
```
```   108   assumes sb: "sets M = sets borel"
```
```   109   assumes "emeasure M (space M) \<noteq> \<infinity>"
```
```   110   assumes "B \<in> sets borel"
```
```   111   shows inner_regular: "emeasure M B =
```
```   112     (SUP K : {K. K \<subseteq> B \<and> compact K}. emeasure M K)" (is "?inner B")
```
```   113   and outer_regular: "emeasure M B =
```
```   114     (INF U : {U. B \<subseteq> U \<and> open U}. emeasure M U)" (is "?outer B")
```
```   115 proof -
```
```   116   have Us: "UNIV = space M" by (metis assms(1) sets_eq_imp_space_eq space_borel)
```
```   117   hence sU: "space M = UNIV" by simp
```
```   118   interpret finite_measure M by rule fact
```
```   119   have approx_inner: "\<And>A. A \<in> sets M \<Longrightarrow>
```
```   120     (\<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"
```
```   121     by (rule ereal_approx_SUP)
```
```   122       (force intro!: emeasure_mono simp: compact_imp_closed emeasure_eq_measure)+
```
```   123   have approx_outer: "\<And>A. A \<in> sets M \<Longrightarrow>
```
```   124     (\<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"
```
```   125     by (rule ereal_approx_INF)
```
```   126        (force intro!: emeasure_mono simp: emeasure_eq_measure sb)+
```
```   127   from countable_dense_setE guess X::"'a set"  . note X = this
```
```   128   {
```
```   129     fix r::real assume "r > 0" hence "\<And>y. open (ball y r)" "\<And>y. ball y r \<noteq> {}" by auto
```
```   130     with X(2)[OF this]
```
```   131     have x: "space M = (\<Union>x\<in>X. cball x r)"
```
```   132       by (auto simp add: sU) (metis dist_commute order_less_imp_le)
```
```   133     let ?U = "\<Union>k. (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)"
```
```   134     have "(\<lambda>k. emeasure M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M ?U"
```
```   135       by (rule Lim_emeasure_incseq)
```
```   136         (auto intro!: borel_closed bexI simp: closed_cball incseq_def Us sb)
```
```   137     also have "?U = space M"
```
```   138     proof safe
```
```   139       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
```
```   140       show "x \<in> ?U"
```
```   141         using X(1) d
```
```   142         by simp (auto intro!: exI [where x = "to_nat_on X d"] simp: dist_commute Bex_def)
```
```   143     qed (simp add: sU)
```
```   144     finally have "(\<lambda>k. M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M (space M)" .
```
```   145   } note M_space = this
```
```   146   {
```
```   147     fix e ::real and n :: nat assume "e > 0" "n > 0"
```
```   148     hence "1/n > 0" "e * 2 powr - n > 0" by (auto)
```
```   149     from M_space[OF \<open>1/n>0\<close>]
```
```   150     have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) \<longlonglongrightarrow> measure M (space M)"
```
```   151       unfolding emeasure_eq_measure by simp
```
```   152     from metric_LIMSEQ_D[OF this \<open>0 < e * 2 powr -n\<close>]
```
```   153     obtain k where "dist (measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) (measure M (space M)) <
```
```   154       e * 2 powr -n"
```
```   155       by auto
```
```   156     hence "measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n)) \<ge>
```
```   157       measure M (space M) - e * 2 powr -real n"
```
```   158       by (auto simp: dist_real_def)
```
```   159     hence "\<exists>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n)) \<ge>
```
```   160       measure M (space M) - e * 2 powr - real n" ..
```
```   161   } note k=this
```
```   162   hence "\<forall>e\<in>{0<..}. \<forall>(n::nat)\<in>{0<..}. \<exists>k.
```
```   163     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"
```
```   164     by blast
```
```   165   then obtain k where k: "\<forall>e\<in>{0<..}. \<forall>n\<in>{0<..}. measure M (space M) - e * 2 powr - real (n::nat)
```
```   166     \<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"
```
```   167     by metis
```
```   168   hence k: "\<And>e n. e > 0 \<Longrightarrow> n > 0 \<Longrightarrow> measure M (space M) - e * 2 powr - n
```
```   169     \<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"
```
```   170     unfolding Ball_def by blast
```
```   171   have approx_space:
```
```   172     "\<And>e. e > 0 \<Longrightarrow>
```
```   173       \<exists>K \<in> {K. K \<subseteq> space M \<and> compact K}. emeasure M (space M) \<le> emeasure M K + ereal e"
```
```   174       (is "\<And>e. _ \<Longrightarrow> ?thesis e")
```
```   175   proof -
```
```   176     fix e :: real assume "e > 0"
```
```   177     def B \<equiv> "\<lambda>n. \<Union>i\<in>{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n)"
```
```   178     have "\<And>n. closed (B n)" by (auto simp: B_def closed_cball)
```
```   179     hence [simp]: "\<And>n. B n \<in> sets M" by (simp add: sb)
```
```   180     from k[OF \<open>e > 0\<close> zero_less_Suc]
```
```   181     have "\<And>n. measure M (space M) - measure M (B n) \<le> e * 2 powr - real (Suc n)"
```
```   182       by (simp add: algebra_simps B_def finite_measure_compl)
```
```   183     hence B_compl_le: "\<And>n::nat. measure M (space M - B n) \<le> e * 2 powr - real (Suc n)"
```
```   184       by (simp add: finite_measure_compl)
```
```   185     def K \<equiv> "\<Inter>n. B n"
```
```   186     from \<open>closed (B _)\<close> have "closed K" by (auto simp: K_def)
```
```   187     hence [simp]: "K \<in> sets M" by (simp add: sb)
```
```   188     have "measure M (space M) - measure M K = measure M (space M - K)"
```
```   189       by (simp add: finite_measure_compl)
```
```   190     also have "\<dots> = emeasure M (\<Union>n. space M - B n)" by (auto simp: K_def emeasure_eq_measure)
```
```   191     also have "\<dots> \<le> (\<Sum>n. emeasure M (space M - B n))"
```
```   192       by (rule emeasure_subadditive_countably) (auto simp: summable_def)
```
```   193     also have "\<dots> \<le> (\<Sum>n. ereal (e*2 powr - real (Suc n)))"
```
```   194       using B_compl_le by (intro suminf_le_pos) (simp_all add: measure_nonneg emeasure_eq_measure)
```
```   195     also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
```
```   196       by (simp add: Transcendental.powr_minus powr_realpow field_simps del: of_nat_Suc)
```
```   197     also have "\<dots> = (\<Sum>n. ereal e * ((1 / 2) ^ Suc n))"
```
```   198       unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
```
```   199       by simp
```
```   200     also have "\<dots> = ereal e * (\<Sum>n. ((1 / 2) ^ Suc n))"
```
```   201       by (rule suminf_cmult_ereal) (auto simp: \<open>0 < e\<close> less_imp_le)
```
```   202     also have "\<dots> = e" unfolding suminf_half_series_ereal by simp
```
```   203     finally have "measure M (space M) \<le> measure M K + e" by simp
```
```   204     hence "emeasure M (space M) \<le> emeasure M K + e" by (simp add: emeasure_eq_measure)
```
```   205     moreover have "compact K"
```
```   206       unfolding compact_eq_totally_bounded
```
```   207     proof safe
```
```   208       show "complete K" using \<open>closed K\<close> by (simp add: complete_eq_closed)
```
```   209       fix e'::real assume "0 < e'"
```
```   210       from nat_approx_posE[OF this] guess n . note n = this
```
```   211       let ?k = "from_nat_into X ` {0..k e (Suc n)}"
```
```   212       have "finite ?k" by simp
```
```   213       moreover have "K \<subseteq> (\<Union>x\<in>?k. ball x e')" unfolding K_def B_def using n by force
```
```   214       ultimately show "\<exists>k. finite k \<and> K \<subseteq> (\<Union>x\<in>k. ball x e')" by blast
```
```   215     qed
```
```   216     ultimately
```
```   217     show "?thesis e " by (auto simp: sU)
```
```   218   qed
```
```   219   { fix A::"'a set" assume "closed A" hence "A \<in> sets borel" by (simp add: compact_imp_closed)
```
```   220     hence [simp]: "A \<in> sets M" by (simp add: sb)
```
```   221     have "?inner A"
```
```   222     proof (rule approx_inner)
```
```   223       fix e::real assume "e > 0"
```
```   224       from approx_space[OF this] obtain K where
```
```   225         K: "K \<subseteq> space M" "compact K" "emeasure M (space M) \<le> emeasure M K + e"
```
```   226         by (auto simp: emeasure_eq_measure)
```
```   227       hence [simp]: "K \<in> sets M" by (simp add: sb compact_imp_closed)
```
```   228       have "M A - M (A \<inter> K) = measure M A - measure M (A \<inter> K)"
```
```   229         by (simp add: emeasure_eq_measure)
```
```   230       also have "\<dots> = measure M (A - A \<inter> K)"
```
```   231         by (subst finite_measure_Diff) auto
```
```   232       also have "A - A \<inter> K = A \<union> K - K" by auto
```
```   233       also have "measure M \<dots> = measure M (A \<union> K) - measure M K"
```
```   234         by (subst finite_measure_Diff) auto
```
```   235       also have "\<dots> \<le> measure M (space M) - measure M K"
```
```   236         by (simp add: emeasure_eq_measure sU sb finite_measure_mono)
```
```   237       also have "\<dots> \<le> e" using K by (simp add: emeasure_eq_measure)
```
```   238       finally have "emeasure M A \<le> emeasure M (A \<inter> K) + ereal e"
```
```   239         by (simp add: emeasure_eq_measure algebra_simps)
```
```   240       moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using \<open>closed A\<close> \<open>compact K\<close> by auto
```
```   241       ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ereal e"
```
```   242         by blast
```
```   243     qed simp
```
```   244     have "?outer A"
```
```   245     proof cases
```
```   246       assume "A \<noteq> {}"
```
```   247       let ?G = "\<lambda>d. {x. infdist x A < d}"
```
```   248       {
```
```   249         fix d
```
```   250         have "?G d = (\<lambda>x. infdist x A) -` {..<d}" by auto
```
```   251         also have "open \<dots>"
```
```   252           by (intro continuous_open_vimage) (auto intro!: continuous_infdist continuous_ident)
```
```   253         finally have "open (?G d)" .
```
```   254       } note open_G = this
```
```   255       from in_closed_iff_infdist_zero[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>]
```
```   256       have "A = {x. infdist x A = 0}" by auto
```
```   257       also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"
```
```   258       proof (auto simp del: of_nat_Suc, rule ccontr)
```
```   259         fix x
```
```   260         assume "infdist x A \<noteq> 0"
```
```   261         hence pos: "infdist x A > 0" using infdist_nonneg[of x A] by simp
```
```   262         from nat_approx_posE[OF this] guess n .
```
```   263         moreover
```
```   264         assume "\<forall>i. infdist x A < 1 / real (Suc i)"
```
```   265         hence "infdist x A < 1 / real (Suc n)" by auto
```
```   266         ultimately show False by simp
```
```   267       qed
```
```   268       also have "M \<dots> = (INF n. emeasure M (?G (1 / real (Suc n))))"
```
```   269       proof (rule INF_emeasure_decseq[symmetric], safe)
```
```   270         fix i::nat
```
```   271         from open_G[of "1 / real (Suc i)"]
```
```   272         show "?G (1 / real (Suc i)) \<in> sets M" by (simp add: sb borel_open)
```
```   273       next
```
```   274         show "decseq (\<lambda>i. {x. infdist x A < 1 / real (Suc i)})"
```
```   275           by (auto intro: less_trans intro!: divide_strict_left_mono
```
```   276             simp: decseq_def le_eq_less_or_eq)
```
```   277       qed simp
```
```   278       finally
```
```   279       have "emeasure M A = (INF n. emeasure M {x. infdist x A < 1 / real (Suc n)})" .
```
```   280       moreover
```
```   281       have "\<dots> \<ge> (INF U:{U. A \<subseteq> U \<and> open U}. emeasure M U)"
```
```   282       proof (intro INF_mono)
```
```   283         fix m
```
```   284         have "?G (1 / real (Suc m)) \<in> {U. A \<subseteq> U \<and> open U}" using open_G by auto
```
```   285         moreover have "M (?G (1 / real (Suc m))) \<le> M (?G (1 / real (Suc m)))" by simp
```
```   286         ultimately show "\<exists>U\<in>{U. A \<subseteq> U \<and> open U}.
```
```   287           emeasure M U \<le> emeasure M {x. infdist x A < 1 / real (Suc m)}"
```
```   288           by blast
```
```   289       qed
```
```   290       moreover
```
```   291       have "emeasure M A \<le> (INF U:{U. A \<subseteq> U \<and> open U}. emeasure M U)"
```
```   292         by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)
```
```   293       ultimately show ?thesis by simp
```
```   294     qed (auto intro!: INF_eqI)
```
```   295     note \<open>?inner A\<close> \<open>?outer A\<close> }
```
```   296   note closed_in_D = this
```
```   297   from \<open>B \<in> sets borel\<close>
```
```   298   have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"
```
```   299     by (auto simp: Int_stable_def borel_eq_closed)
```
```   300   then show "?inner B" "?outer B"
```
```   301   proof (induct B rule: sigma_sets_induct_disjoint)
```
```   302     case empty
```
```   303     { case 1 show ?case by (intro SUP_eqI[symmetric]) auto }
```
```   304     { case 2 show ?case by (intro INF_eqI[symmetric]) (auto elim!: meta_allE[of _ "{}"]) }
```
```   305   next
```
```   306     case (basic B)
```
```   307     { case 1 from basic closed_in_D show ?case by auto }
```
```   308     { case 2 from basic closed_in_D show ?case by auto }
```
```   309   next
```
```   310     case (compl B)
```
```   311     note inner = compl(2) and outer = compl(3)
```
```   312     from compl have [simp]: "B \<in> sets M" by (auto simp: sb borel_eq_closed)
```
```   313     case 2
```
```   314     have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
```
```   315     also have "\<dots> = (INF K:{K. K \<subseteq> B \<and> compact K}. M (space M) -  M K)"
```
```   316       unfolding inner by (subst INF_ereal_minus_right) force+
```
```   317     also have "\<dots> = (INF U:{U. U \<subseteq> B \<and> compact U}. M (space M - U))"
```
```   318       by (rule INF_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
```
```   319     also have "\<dots> \<ge> (INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U))"
```
```   320       by (rule INF_superset_mono) (auto simp add: compact_imp_closed)
```
```   321     also have "(INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U)) =
```
```   322         (INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"
```
```   323       unfolding INF_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]
```
```   324         by (rule INF_cong) (auto simp add: sU open_Compl Compl_eq_Diff_UNIV [symmetric, simp])
```
```   325     finally have
```
```   326       "(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<le> emeasure M (space M - B)" .
```
```   327     moreover have
```
```   328       "(INF U:{U. space M - B \<subseteq> U \<and> open U}. emeasure M U) \<ge> emeasure M (space M - B)"
```
```   329       by (auto simp: sb sU intro!: INF_greatest emeasure_mono)
```
```   330     ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
```
```   331
```
```   332     case 1
```
```   333     have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
```
```   334     also have "\<dots> = (SUP U: {U. B \<subseteq> U \<and> open U}. M (space M) -  M U)"
```
```   335       unfolding outer by (subst SUP_ereal_minus_right) auto
```
```   336     also have "\<dots> = (SUP U:{U. B \<subseteq> U \<and> open U}. M (space M - U))"
```
```   337       by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
```
```   338     also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"
```
```   339       unfolding SUP_image [of _ "\<lambda>u. space M - u" _, symmetric, unfolded comp_def]
```
```   340         by (rule SUP_cong) (auto simp add: sU)
```
```   341     also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
```
```   342     proof (safe intro!: antisym SUP_least)
```
```   343       fix K assume "closed K" "K \<subseteq> space M - B"
```
```   344       from closed_in_D[OF \<open>closed K\<close>]
```
```   345       have K_inner: "emeasure M K = (SUP K:{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp
```
```   346       show "emeasure M K \<le> (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
```
```   347         unfolding K_inner using \<open>K \<subseteq> space M - B\<close>
```
```   348         by (auto intro!: SUP_upper SUP_least)
```
```   349     qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
```
```   350     finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
```
```   351   next
```
```   352     case (union D)
```
```   353     then have "range D \<subseteq> sets M" by (auto simp: sb borel_eq_closed)
```
```   354     with union have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (intro suminf_emeasure)
```
```   355     also have "(\<lambda>n. \<Sum>i<n. M (D i)) \<longlonglongrightarrow> (\<Sum>i. M (D i))"
```
```   356       by (intro summable_LIMSEQ summable_ereal_pos emeasure_nonneg)
```
```   357     finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) \<longlonglongrightarrow> measure M (\<Union>i. D i)"
```
```   358       by (simp add: emeasure_eq_measure)
```
```   359     have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto
```
```   360
```
```   361     case 1
```
```   362     show ?case
```
```   363     proof (rule approx_inner)
```
```   364       fix e::real assume "e > 0"
```
```   365       with measure_LIMSEQ
```
```   366       have "\<exists>no. \<forall>n\<ge>no. \<bar>(\<Sum>i<n. measure M (D i)) -measure M (\<Union>x. D x)\<bar> < e/2"
```
```   367         by (auto simp: lim_sequentially dist_real_def simp del: less_divide_eq_numeral1)
```
```   368       hence "\<exists>n0. \<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>x. D x)\<bar> < e/2" by auto
```
```   369       then obtain n0 where n0: "\<bar>(\<Sum>i<n0. measure M (D i)) - measure M (\<Union>i. D i)\<bar> < e/2"
```
```   370         unfolding choice_iff by blast
```
```   371       have "ereal (\<Sum>i<n0. measure M (D i)) = (\<Sum>i<n0. M (D i))"
```
```   372         by (auto simp add: emeasure_eq_measure)
```
```   373       also have "\<dots> \<le> (\<Sum>i. M (D i))" by (rule suminf_upper) (auto simp: emeasure_nonneg)
```
```   374       also have "\<dots> = M (\<Union>i. D i)" by (simp add: M)
```
```   375       also have "\<dots> = measure M (\<Union>i. D i)" by (simp add: emeasure_eq_measure)
```
```   376       finally have n0: "measure M (\<Union>i. D i) - (\<Sum>i<n0. measure M (D i)) < e/2"
```
```   377         using n0 by auto
```
```   378       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)"
```
```   379       proof
```
```   380         fix i
```
```   381         from \<open>0 < e\<close> have "0 < e/(2*Suc n0)" by simp
```
```   382         have "emeasure M (D i) = (SUP K:{K. K \<subseteq> (D i) \<and> compact K}. emeasure M K)"
```
```   383           using union by blast
```
```   384         from SUP_approx_ereal[OF \<open>0 < e/(2*Suc n0)\<close> this]
```
```   385         show "\<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
```
```   386           by (auto simp: emeasure_eq_measure)
```
```   387       qed
```
```   388       then obtain K where K: "\<And>i. K i \<subseteq> D i" "\<And>i. compact (K i)"
```
```   389         "\<And>i. emeasure M (D i) \<le> emeasure M (K i) + e/(2*Suc n0)"
```
```   390         unfolding choice_iff by blast
```
```   391       let ?K = "\<Union>i\<in>{..<n0}. K i"
```
```   392       have "disjoint_family_on K {..<n0}" using K \<open>disjoint_family D\<close>
```
```   393         unfolding disjoint_family_on_def by blast
```
```   394       hence mK: "measure M ?K = (\<Sum>i<n0. measure M (K i))" using K
```
```   395         by (intro finite_measure_finite_Union) (auto simp: sb compact_imp_closed)
```
```   396       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (D i)) + e/2" using n0 by simp
```
```   397       also have "(\<Sum>i<n0. measure M (D i)) \<le> (\<Sum>i<n0. measure M (K i) + e/(2*Suc n0))"
```
```   398         using K by (auto intro: setsum_mono simp: emeasure_eq_measure)
```
```   399       also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"
```
```   400         by (simp add: setsum.distrib)
```
```   401       also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) +  e / 2" using \<open>0 < e\<close>
```
```   402         by (auto simp: field_simps intro!: mult_left_mono)
```
```   403       finally
```
```   404       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"
```
```   405         by auto
```
```   406       hence "M (\<Union>i. D i) < M ?K + e" by (auto simp: mK emeasure_eq_measure)
```
```   407       moreover
```
```   408       have "?K \<subseteq> (\<Union>i. D i)" using K by auto
```
```   409       moreover
```
```   410       have "compact ?K" using K by auto
```
```   411       ultimately
```
```   412       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
```
```   413       thus "\<exists>K\<subseteq>\<Union>i. D i. compact K \<and> emeasure M (\<Union>i. D i) \<le> emeasure M K + ereal e" ..
```
```   414     qed fact
```
```   415     case 2
```
```   416     show ?case
```
```   417     proof (rule approx_outer[OF \<open>(\<Union>i. D i) \<in> sets M\<close>])
```
```   418       fix e::real assume "e > 0"
```
```   419       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)"
```
```   420       proof
```
```   421         fix i::nat
```
```   422         from \<open>0 < e\<close> have "0 < e/(2 powr Suc i)" by simp
```
```   423         have "emeasure M (D i) = (INF U:{U. (D i) \<subseteq> U \<and> open U}. emeasure M U)"
```
```   424           using union by blast
```
```   425         from INF_approx_ereal[OF \<open>0 < e/(2 powr Suc i)\<close> this]
```
```   426         show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
```
```   427           by (auto simp: emeasure_eq_measure)
```
```   428       qed
```
```   429       then obtain U where U: "\<And>i. D i \<subseteq> U i" "\<And>i. open (U i)"
```
```   430         "\<And>i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)"
```
```   431         unfolding choice_iff by blast
```
```   432       let ?U = "\<Union>i. U i"
```
```   433       have "M ?U - M (\<Union>i. D i) = M (?U - (\<Union>i. D i))" using U  \<open>(\<Union>i. D i) \<in> sets M\<close>
```
```   434         by (subst emeasure_Diff) (auto simp: sb)
```
```   435       also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U  \<open>range D \<subseteq> sets M\<close>
```
```   436         by (intro emeasure_mono) (auto simp: sb intro!: sets.countable_nat_UN sets.Diff)
```
```   437       also have "\<dots> \<le> (\<Sum>i. M (U i - D i))" using U  \<open>range D \<subseteq> sets M\<close>
```
```   438         by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)
```
```   439       also have "\<dots> \<le> (\<Sum>i. ereal e/(2 powr Suc i))" using U \<open>range D \<subseteq> sets M\<close>
```
```   440         by (intro suminf_le_pos, subst emeasure_Diff)
```
```   441            (auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg intro: less_imp_le)
```
```   442       also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
```
```   443         by (simp add: powr_minus inverse_eq_divide powr_realpow field_simps power_divide del: of_nat_Suc)
```
```   444       also have "\<dots> = (\<Sum>n. ereal e * ((1 / 2) ^  Suc n))"
```
```   445         unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
```
```   446         by simp
```
```   447       also have "\<dots> = ereal e * (\<Sum>n. ((1 / 2) ^ Suc n))"
```
```   448         by (rule suminf_cmult_ereal) (auto simp: \<open>0 < e\<close> less_imp_le)
```
```   449       also have "\<dots> = e" unfolding suminf_half_series_ereal by simp
```
```   450       finally
```
```   451       have "emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ereal e" by (simp add: emeasure_eq_measure)
```
```   452       moreover
```
```   453       have "(\<Union>i. D i) \<subseteq> ?U" using U by auto
```
```   454       moreover
```
```   455       have "open ?U" using U by auto
```
```   456       ultimately
```
```   457       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
```
```   458       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" ..
```
```   459     qed
```
```   460   qed
```
```   461 qed
```
```   462
```
```   463 end
```
```   464
```