author hoelzl Fri Sep 23 10:26:04 2016 +0200 (2016-09-23) changeset 63940 0d82c4c94014 parent 63939 d4b89572ae71 child 63941 f353674c2528
prove HK-integrable implies Lebesgue measurable; prove HK-integral equals Lebesgue integral for nonneg functions
1.1 --- a/src/HOL/Analysis/Complete_Measure.thy	Thu Sep 22 15:56:37 2016 +0100
1.2 +++ b/src/HOL/Analysis/Complete_Measure.thy	Fri Sep 23 10:26:04 2016 +0200
1.3 @@ -6,6 +6,10 @@
1.4    imports Bochner_Integration
1.5  begin
1.7 +locale complete_measure =
1.8 +  fixes M :: "'a measure"
1.9 +  assumes complete: "\<And>A B. B \<subseteq> A \<Longrightarrow> A \<in> null_sets M \<Longrightarrow> B \<in> sets M"
1.10 +
1.11  definition
1.12    "split_completion M A p = (if A \<in> sets M then p = (A, {}) else
1.13     \<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets M)"
1.14 @@ -304,4 +308,522 @@
1.15  lemma AE_completion_iff: "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in completion M. P x)"
1.16    by (simp add: AE_iff_null null_sets_completion_iff)
1.18 +lemma sets_completion_AE: "(AE x in M. \<not> P x) \<Longrightarrow> Measurable.pred (completion M) P"
1.19 +  unfolding pred_def sets_completion eventually_ae_filter
1.20 +  by auto
1.21 +
1.22 +lemma null_sets_completion_iff2:
1.23 +  "A \<in> null_sets (completion M) \<longleftrightarrow> (\<exists>N'\<in>null_sets M. A \<subseteq> N')"
1.24 +proof safe
1.25 +  assume "A \<in> null_sets (completion M)"
1.26 +  then have A: "A \<in> sets (completion M)" and "main_part M A \<in> null_sets M"
1.27 +    by (auto simp: null_sets_def)
1.28 +  moreover obtain N where "N \<in> null_sets M" "null_part M A \<subseteq> N"
1.29 +    using null_part[OF A] by auto
1.30 +  ultimately show "\<exists>N'\<in>null_sets M. A \<subseteq> N'"
1.31 +  proof (intro bexI)
1.32 +    show "A \<subseteq> N \<union> main_part M A"
1.33 +      using \<open>null_part M A \<subseteq> N\<close> by (subst main_part_null_part_Un[OF A, symmetric]) auto
1.34 +  qed auto
1.35 +next
1.36 +  fix N assume "N \<in> null_sets M" "A \<subseteq> N"
1.37 +  then have "A \<in> sets (completion M)" and N: "N \<in> sets M" "A \<subseteq> N" "emeasure M N = 0"
1.38 +    by (auto intro: null_sets_completion)
1.39 +  moreover have "emeasure (completion M) A = 0"
1.40 +    using N by (intro emeasure_eq_0[of N _ A]) auto
1.41 +  ultimately show "A \<in> null_sets (completion M)"
1.42 +    by auto
1.43 +qed
1.44 +
1.45 +lemma null_sets_completion_subset:
1.46 +  "B \<subseteq> A \<Longrightarrow> A \<in> null_sets (completion M) \<Longrightarrow> B \<in> null_sets (completion M)"
1.47 +  unfolding null_sets_completion_iff2 by auto
1.48 +
1.49 +lemma null_sets_restrict_space:
1.50 +  "\<Omega> \<in> sets M \<Longrightarrow> A \<in> null_sets (restrict_space M \<Omega>) \<longleftrightarrow> A \<subseteq> \<Omega> \<and> A \<in> null_sets M"
1.51 +  by (auto simp: null_sets_def emeasure_restrict_space sets_restrict_space)
1.52 +lemma completion_ex_borel_measurable_real:
1.53 +  fixes g :: "'a \<Rightarrow> real"
1.54 +  assumes g: "g \<in> borel_measurable (completion M)"
1.55 +  shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
1.56 +proof -
1.57 +  have "(\<lambda>x. ennreal (g x)) \<in> completion M \<rightarrow>\<^sub>M borel" "(\<lambda>x. ennreal (- g x)) \<in> completion M \<rightarrow>\<^sub>M borel"
1.58 +    using g by auto
1.59 +  from this[THEN completion_ex_borel_measurable]
1.60 +  obtain pf nf :: "'a \<Rightarrow> ennreal"
1.61 +    where [measurable]: "nf \<in> M \<rightarrow>\<^sub>M borel" "pf \<in> M \<rightarrow>\<^sub>M borel"
1.62 +      and ae: "AE x in M. pf x = ennreal (g x)" "AE x in M. nf x = ennreal (- g x)"
1.63 +    by (auto simp: eq_commute)
1.64 +  then have "AE x in M. pf x = ennreal (g x) \<and> nf x = ennreal (- g x)"
1.65 +    by auto
1.66 +  then obtain N where "N \<in> null_sets M" "{x\<in>space M. pf x \<noteq> ennreal (g x) \<and> nf x \<noteq> ennreal (- g x)} \<subseteq> N"
1.67 +    by (auto elim!: AE_E)
1.68 +  show ?thesis
1.69 +  proof
1.70 +    let ?F = "\<lambda>x. indicator (space M - N) x * (enn2real (pf x) - enn2real (nf x))"
1.71 +    show "?F \<in> M \<rightarrow>\<^sub>M borel"
1.72 +      using \<open>N \<in> null_sets M\<close> by auto
1.73 +    show "AE x in M. g x = ?F x"
1.74 +      using \<open>N \<in> null_sets M\<close>[THEN AE_not_in] ae AE_space
1.75 +      apply eventually_elim
1.76 +      subgoal for x
1.77 +        by (cases "0::real" "g x" rule: linorder_le_cases) (auto simp: ennreal_neg)
1.78 +      done
1.79 +  qed
1.80 +qed
1.81 +
1.82 +lemma simple_function_completion: "simple_function M f \<Longrightarrow> simple_function (completion M) f"
1.83 +  by (simp add: simple_function_def)
1.84 +
1.85 +lemma simple_integral_completion:
1.86 +  "simple_function M f \<Longrightarrow> simple_integral (completion M) f = simple_integral M f"
1.87 +  unfolding simple_integral_def by simp
1.88 +
1.89 +lemma nn_integral_completion: "nn_integral (completion M) f = nn_integral M f"
1.90 +  unfolding nn_integral_def
1.91 +proof (safe intro!: SUP_eq)
1.92 +  fix s assume s: "simple_function (completion M) s" and "s \<le> f"
1.93 +  then obtain s' where s': "simple_function M s'" "AE x in M. s x = s' x"
1.94 +    by (auto dest: completion_ex_simple_function)
1.95 +  then obtain N where N: "N \<in> null_sets M" "{x\<in>space M. s x \<noteq> s' x} \<subseteq> N"
1.96 +    by (auto elim!: AE_E)
1.97 +  then have ae_N: "AE x in M. (s x \<noteq> s' x \<longrightarrow> x \<in> N) \<and> x \<notin> N"
1.98 +    by (auto dest: AE_not_in)
1.99 +  define s'' where "s'' x = (if x \<in> N then 0 else s x)" for x
1.100 +  then have ae_s_eq_s'': "AE x in completion M. s x = s'' x"
1.101 +    using s' ae_N by (intro AE_completion) auto
1.102 +  have s'': "simple_function M s''"
1.103 +  proof (subst simple_function_cong)
1.104 +    show "t \<in> space M \<Longrightarrow> s'' t = (if t \<in> N then 0 else s' t)" for t
1.105 +      using N by (auto simp: s''_def dest: sets.sets_into_space)
1.106 +    show "simple_function M (\<lambda>t. if t \<in> N then 0 else s' t)"
1.107 +      unfolding s''_def[abs_def] using N by (auto intro!: simple_function_If s')
1.108 +  qed
1.110 +  show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> f}. integral\<^sup>S (completion M) s \<le> integral\<^sup>S M j"
1.111 +  proof (safe intro!: bexI[of _ s''])
1.112 +    have "integral\<^sup>S (completion M) s = integral\<^sup>S (completion M) s''"
1.113 +      by (intro simple_integral_cong_AE s simple_function_completion s'' ae_s_eq_s'')
1.114 +    then show "integral\<^sup>S (completion M) s \<le> integral\<^sup>S M s''"
1.115 +      using s'' by (simp add: simple_integral_completion)
1.116 +    from \<open>s \<le> f\<close> show "s'' \<le> f"
1.117 +      unfolding s''_def le_fun_def by auto
1.118 +  qed fact
1.119 +next
1.120 +  fix s assume "simple_function M s" "s \<le> f"
1.121 +  then show "\<exists>j\<in>{g. simple_function (completion M) g \<and> g \<le> f}. integral\<^sup>S M s \<le> integral\<^sup>S (completion M) j"
1.122 +    by (intro bexI[of _ s]) (auto simp: simple_integral_completion simple_function_completion)
1.123 +qed
1.125 +locale semifinite_measure =
1.126 +  fixes M :: "'a measure"
1.127 +  assumes semifinite:
1.128 +    "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = \<infinity> \<Longrightarrow> \<exists>B\<in>sets M. B \<subseteq> A \<and> emeasure M B < \<infinity>"
1.130 +locale locally_determined_measure = semifinite_measure +
1.131 +  assumes locally_determined:
1.132 +    "\<And>A. A \<subseteq> space M \<Longrightarrow> (\<And>B. B \<in> sets M \<Longrightarrow> emeasure M B < \<infinity> \<Longrightarrow> A \<inter> B \<in> sets M) \<Longrightarrow> A \<in> sets M"
1.134 +locale cld_measure = complete_measure M + locally_determined_measure M for M :: "'a measure"
1.136 +definition outer_measure_of :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ennreal"
1.137 +  where "outer_measure_of M A = (INF B : {B\<in>sets M. A \<subseteq> B}. emeasure M B)"
1.139 +lemma outer_measure_of_eq[simp]: "A \<in> sets M \<Longrightarrow> outer_measure_of M A = emeasure M A"
1.140 +  by (auto simp: outer_measure_of_def intro!: INF_eqI emeasure_mono)
1.142 +lemma outer_measure_of_mono: "A \<subseteq> B \<Longrightarrow> outer_measure_of M A \<le> outer_measure_of M B"
1.143 +  unfolding outer_measure_of_def by (intro INF_superset_mono) auto
1.145 +lemma outer_measure_of_attain:
1.146 +  assumes "A \<subseteq> space M"
1.147 +  shows "\<exists>E\<in>sets M. A \<subseteq> E \<and> outer_measure_of M A = emeasure M E"
1.148 +proof -
1.149 +  have "emeasure M ` {B \<in> sets M. A \<subseteq> B} \<noteq> {}"
1.150 +    using \<open>A \<subseteq> space M\<close> by auto
1.151 +  from ennreal_Inf_countable_INF[OF this]
1.152 +  obtain f
1.153 +    where f: "range f \<subseteq> emeasure M ` {B \<in> sets M. A \<subseteq> B}" "decseq f"
1.154 +      and "outer_measure_of M A = (INF i. f i)"
1.155 +    unfolding outer_measure_of_def by auto
1.156 +  have "\<exists>E. \<forall>n. (E n \<in> sets M \<and> A \<subseteq> E n \<and> emeasure M (E n) \<le> f n) \<and> E (Suc n) \<subseteq> E n"
1.157 +  proof (rule dependent_nat_choice)
1.158 +    show "\<exists>x. x \<in> sets M \<and> A \<subseteq> x \<and> emeasure M x \<le> f 0"
1.159 +      using f(1) by (fastforce simp: image_subset_iff image_iff intro: eq_refl[OF sym])
1.160 +  next
1.161 +    fix E n assume "E \<in> sets M \<and> A \<subseteq> E \<and> emeasure M E \<le> f n"
1.162 +    moreover obtain F where "F \<in> sets M" "A \<subseteq> F" "f (Suc n) = emeasure M F"
1.163 +      using f(1) by (auto simp: image_subset_iff image_iff)
1.164 +    ultimately show "\<exists>y. (y \<in> sets M \<and> A \<subseteq> y \<and> emeasure M y \<le> f (Suc n)) \<and> y \<subseteq> E"
1.165 +      by (auto intro!: exI[of _ "F \<inter> E"] emeasure_mono)
1.166 +  qed
1.167 +  then obtain E
1.168 +    where [simp]: "\<And>n. E n \<in> sets M"
1.169 +      and "\<And>n. A \<subseteq> E n"
1.170 +      and le_f: "\<And>n. emeasure M (E n) \<le> f n"
1.171 +      and "decseq E"
1.172 +    by (auto simp: decseq_Suc_iff)
1.173 +  show ?thesis
1.174 +  proof cases
1.175 +    assume fin: "\<exists>i. emeasure M (E i) < \<infinity>"
1.176 +    show ?thesis
1.177 +    proof (intro bexI[of _ "\<Inter>i. E i"] conjI)
1.178 +      show "A \<subseteq> (\<Inter>i. E i)" "(\<Inter>i. E i) \<in> sets M"
1.179 +        using \<open>\<And>n. A \<subseteq> E n\<close> by auto
1.181 +      have " (INF i. emeasure M (E i)) \<le> outer_measure_of M A"
1.182 +        unfolding \<open>outer_measure_of M A = (INF n. f n)\<close>
1.183 +        by (intro INF_superset_mono le_f) auto
1.184 +      moreover have "outer_measure_of M A \<le> (INF i. outer_measure_of M (E i))"
1.185 +        by (intro INF_greatest outer_measure_of_mono \<open>\<And>n. A \<subseteq> E n\<close>)
1.186 +      ultimately have "outer_measure_of M A = (INF i. emeasure M (E i))"
1.187 +        by auto
1.188 +      also have "\<dots> = emeasure M (\<Inter>i. E i)"
1.189 +        using fin by (intro INF_emeasure_decseq' \<open>decseq E\<close>) (auto simp: less_top)
1.190 +      finally show "outer_measure_of M A = emeasure M (\<Inter>i. E i)" .
1.191 +    qed
1.192 +  next
1.193 +    assume "\<nexists>i. emeasure M (E i) < \<infinity>"
1.194 +    then have "f n = \<infinity>" for n
1.195 +      using le_f by (auto simp: not_less top_unique)
1.196 +    moreover have "\<exists>E\<in>sets M. A \<subseteq> E \<and> f 0 = emeasure M E"
1.197 +      using f by auto
1.198 +    ultimately show ?thesis
1.199 +      unfolding \<open>outer_measure_of M A = (INF n. f n)\<close> by simp
1.200 +  qed
1.201 +qed
1.203 +lemma SUP_outer_measure_of_incseq:
1.204 +  assumes A: "\<And>n. A n \<subseteq> space M" and "incseq A"
1.205 +  shows "(SUP n. outer_measure_of M (A n)) = outer_measure_of M (\<Union>i. A i)"
1.206 +proof (rule antisym)
1.207 +  obtain E
1.208 +    where E: "\<And>n. E n \<in> sets M" "\<And>n. A n \<subseteq> E n" "\<And>n. outer_measure_of M (A n) = emeasure M (E n)"
1.209 +    using outer_measure_of_attain[OF A] by metis
1.211 +  define F where "F n = (\<Inter>i\<in>{n ..}. E i)" for n
1.212 +  with E have F: "incseq F" "\<And>n. F n \<in> sets M"
1.213 +    by (auto simp: incseq_def)
1.214 +  have "A n \<subseteq> F n" for n
1.215 +    using incseqD[OF \<open>incseq A\<close>, of n] \<open>\<And>n. A n \<subseteq> E n\<close> by (auto simp: F_def)
1.217 +  have eq: "outer_measure_of M (A n) = outer_measure_of M (F n)" for n
1.218 +  proof (intro antisym)
1.219 +    have "outer_measure_of M (F n) \<le> outer_measure_of M (E n)"
1.220 +      by (intro outer_measure_of_mono) (auto simp add: F_def)
1.221 +    with E show "outer_measure_of M (F n) \<le> outer_measure_of M (A n)"
1.222 +      by auto
1.223 +    show "outer_measure_of M (A n) \<le> outer_measure_of M (F n)"
1.224 +      by (intro outer_measure_of_mono \<open>A n \<subseteq> F n\<close>)
1.225 +  qed
1.227 +  have "outer_measure_of M (\<Union>n. A n) \<le> outer_measure_of M (\<Union>n. F n)"
1.228 +    using \<open>\<And>n. A n \<subseteq> F n\<close> by (intro outer_measure_of_mono) auto
1.229 +  also have "\<dots> = (SUP n. emeasure M (F n))"
1.230 +    using F by (simp add: SUP_emeasure_incseq subset_eq)
1.231 +  finally show "outer_measure_of M (\<Union>n. A n) \<le> (SUP n. outer_measure_of M (A n))"
1.232 +    by (simp add: eq F)
1.233 +qed (auto intro: SUP_least outer_measure_of_mono)
1.235 +definition measurable_envelope :: "'a measure \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"
1.236 +  where "measurable_envelope M A E \<longleftrightarrow>
1.237 +    (A \<subseteq> E \<and> E \<in> sets M \<and> (\<forall>F\<in>sets M. emeasure M (F \<inter> E) = outer_measure_of M (F \<inter> A)))"
1.239 +lemma measurable_envelopeD:
1.240 +  assumes "measurable_envelope M A E"
1.241 +  shows "A \<subseteq> E"
1.242 +    and "E \<in> sets M"
1.243 +    and "\<And>F. F \<in> sets M \<Longrightarrow> emeasure M (F \<inter> E) = outer_measure_of M (F \<inter> A)"
1.244 +    and "A \<subseteq> space M"
1.245 +  using assms sets.sets_into_space[of E] by (auto simp: measurable_envelope_def)
1.247 +lemma measurable_envelopeD1:
1.248 +  assumes E: "measurable_envelope M A E" and F: "F \<in> sets M" "F \<subseteq> E - A"
1.249 +  shows "emeasure M F = 0"
1.250 +proof -
1.251 +  have "emeasure M F = emeasure M (F \<inter> E)"
1.252 +    using F by (intro arg_cong2[where f=emeasure]) auto
1.253 +  also have "\<dots> = outer_measure_of M (F \<inter> A)"
1.254 +    using measurable_envelopeD[OF E] \<open>F \<in> sets M\<close> by (auto simp: measurable_envelope_def)
1.255 +  also have "\<dots> = outer_measure_of M {}"
1.256 +    using \<open>F \<subseteq> E - A\<close> by (intro arg_cong2[where f=outer_measure_of]) auto
1.257 +  finally show "emeasure M F = 0"
1.258 +    by simp
1.259 +qed
1.261 +lemma measurable_envelope_eq1:
1.262 +  assumes "A \<subseteq> E" "E \<in> sets M"
1.263 +  shows "measurable_envelope M A E \<longleftrightarrow> (\<forall>F\<in>sets M. F \<subseteq> E - A \<longrightarrow> emeasure M F = 0)"
1.264 +proof safe
1.265 +  assume *: "\<forall>F\<in>sets M. F \<subseteq> E - A \<longrightarrow> emeasure M F = 0"
1.266 +  show "measurable_envelope M A E"
1.267 +    unfolding measurable_envelope_def
1.268 +  proof (rule ccontr, auto simp add: \<open>E \<in> sets M\<close> \<open>A \<subseteq> E\<close>)
1.269 +    fix F assume "F \<in> sets M" "emeasure M (F \<inter> E) \<noteq> outer_measure_of M (F \<inter> A)"
1.270 +    then have "outer_measure_of M (F \<inter> A) < emeasure M (F \<inter> E)"
1.271 +      using outer_measure_of_mono[of "F \<inter> A" "F \<inter> E" M] \<open>A \<subseteq> E\<close> \<open>E \<in> sets M\<close> by (auto simp: less_le)
1.272 +    then obtain G where G: "G \<in> sets M" "F \<inter> A \<subseteq> G" and less: "emeasure M G < emeasure M (E \<inter> F)"
1.273 +      unfolding outer_measure_of_def INF_less_iff by (auto simp: ac_simps)
1.274 +    have le: "emeasure M (G \<inter> E \<inter> F) \<le> emeasure M G"
1.275 +      using \<open>E \<in> sets M\<close> \<open>G \<in> sets M\<close> \<open>F \<in> sets M\<close> by (auto intro!: emeasure_mono)
1.277 +    from G have "E \<inter> F - G \<in> sets M" "E \<inter> F - G \<subseteq> E - A"
1.278 +      using \<open>F \<in> sets M\<close> \<open>E \<in> sets M\<close> by auto
1.279 +    with * have "0 = emeasure M (E \<inter> F - G)"
1.280 +      by auto
1.281 +    also have "E \<inter> F - G = E \<inter> F - (G \<inter> E \<inter> F)"
1.282 +      by auto
1.283 +    also have "emeasure M (E \<inter> F - (G \<inter> E \<inter> F)) = emeasure M (E \<inter> F) - emeasure M (G \<inter> E \<inter> F)"
1.284 +      using \<open>E \<in> sets M\<close> \<open>F \<in> sets M\<close> le less G by (intro emeasure_Diff) (auto simp: top_unique)
1.285 +    also have "\<dots> > 0"
1.286 +      using le less by (intro diff_gr0_ennreal) auto
1.287 +    finally show False by auto
1.288 +  qed
1.289 +qed (rule measurable_envelopeD1)
1.291 +lemma measurable_envelopeD2:
1.292 +  assumes E: "measurable_envelope M A E" shows "emeasure M E = outer_measure_of M A"
1.293 +proof -
1.294 +  from \<open>measurable_envelope M A E\<close> have "emeasure M (E \<inter> E) = outer_measure_of M (E \<inter> A)"
1.295 +    by (auto simp: measurable_envelope_def)
1.296 +  with measurable_envelopeD[OF E] show "emeasure M E = outer_measure_of M A"
1.297 +    by (auto simp: Int_absorb1)
1.298 +qed
1.300 +lemma measurable_envelope_eq2:
1.301 +  assumes "A \<subseteq> E" "E \<in> sets M" "emeasure M E < \<infinity>"
1.302 +  shows "measurable_envelope M A E \<longleftrightarrow> (emeasure M E = outer_measure_of M A)"
1.303 +proof safe
1.304 +  assume *: "emeasure M E = outer_measure_of M A"
1.305 +  show "measurable_envelope M A E"
1.306 +    unfolding measurable_envelope_eq1[OF \<open>A \<subseteq> E\<close> \<open>E \<in> sets M\<close>]
1.307 +  proof (intro conjI ballI impI assms)
1.308 +    fix F assume F: "F \<in> sets M" "F \<subseteq> E - A"
1.309 +    with \<open>E \<in> sets M\<close> have le: "emeasure M F \<le> emeasure M  E"
1.310 +      by (intro emeasure_mono) auto
1.311 +    from F \<open>A \<subseteq> E\<close> have "outer_measure_of M A \<le> outer_measure_of M (E - F)"
1.312 +      by (intro outer_measure_of_mono) auto
1.313 +    then have "emeasure M E - 0 \<le> emeasure M (E - F)"
1.314 +      using * \<open>E \<in> sets M\<close> \<open>F \<in> sets M\<close> by simp
1.315 +    also have "\<dots> = emeasure M E - emeasure M F"
1.316 +      using \<open>E \<in> sets M\<close> \<open>emeasure M E < \<infinity>\<close> F le by (intro emeasure_Diff) (auto simp: top_unique)
1.317 +    finally show "emeasure M F = 0"
1.318 +      using ennreal_mono_minus_cancel[of "emeasure M E" 0 "emeasure M F"] le assms by auto
1.319 +  qed
1.320 +qed (auto intro: measurable_envelopeD2)
1.322 +lemma measurable_envelopeI_countable:
1.323 +  fixes A :: "nat \<Rightarrow> 'a set"
1.324 +  assumes E: "\<And>n. measurable_envelope M (A n) (E n)"
1.325 +  shows "measurable_envelope M (\<Union>n. A n) (\<Union>n. E n)"
1.326 +proof (subst measurable_envelope_eq1)
1.327 +  show "(\<Union>n. A n) \<subseteq> (\<Union>n. E n)" "(\<Union>n. E n) \<in> sets M"
1.328 +    using measurable_envelopeD(1,2)[OF E] by auto
1.329 +  show "\<forall>F\<in>sets M. F \<subseteq> (\<Union>n. E n) - (\<Union>n. A n) \<longrightarrow> emeasure M F = 0"
1.330 +  proof safe
1.331 +    fix F assume F: "F \<in> sets M" "F \<subseteq> (\<Union>n. E n) - (\<Union>n. A n)"
1.332 +    then have "F \<inter> E n \<in> sets M" "F \<inter> E n \<subseteq> E n - A n" "F \<subseteq> (\<Union>n. E n)" for n
1.333 +      using measurable_envelopeD(1,2)[OF E] by auto
1.334 +    then have "emeasure M (\<Union>n. F \<inter> E n) = 0"
1.335 +      by (intro emeasure_UN_eq_0 measurable_envelopeD1[OF E]) auto
1.336 +    then show "emeasure M F = 0"
1.337 +      using \<open>F \<subseteq> (\<Union>n. E n)\<close> by (auto simp: Int_absorb2)
1.338 +  qed
1.339 +qed
1.341 +lemma measurable_envelopeI_countable_cover:
1.342 +  fixes A and C :: "nat \<Rightarrow> 'a set"
1.343 +  assumes C: "A \<subseteq> (\<Union>n. C n)" "\<And>n. C n \<in> sets M" "\<And>n. emeasure M (C n) < \<infinity>"
1.344 +  shows "\<exists>E\<subseteq>(\<Union>n. C n). measurable_envelope M A E"
1.345 +proof -
1.346 +  have "A \<inter> C n \<subseteq> space M" for n
1.347 +    using \<open>C n \<in> sets M\<close> by (auto dest: sets.sets_into_space)
1.348 +  then have "\<forall>n. \<exists>E\<in>sets M. A \<inter> C n \<subseteq> E \<and> outer_measure_of M (A \<inter> C n) = emeasure M E"
1.349 +    using outer_measure_of_attain[of "A \<inter> C n" M for n] by auto
1.350 +  then obtain E
1.351 +    where E: "\<And>n. E n \<in> sets M" "\<And>n. A \<inter> C n \<subseteq> E n"
1.352 +      and eq: "\<And>n. outer_measure_of M (A \<inter> C n) = emeasure M (E n)"
1.353 +    by metis
1.355 +  have "outer_measure_of M (A \<inter> C n) \<le> outer_measure_of M (E n \<inter> C n)" for n
1.356 +    using E by (intro outer_measure_of_mono) auto
1.357 +  moreover have "outer_measure_of M (E n \<inter> C n) \<le> outer_measure_of M (E n)" for n
1.358 +    by (intro outer_measure_of_mono) auto
1.359 +  ultimately have eq: "outer_measure_of M (A \<inter> C n) = emeasure M (E n \<inter> C n)" for n
1.360 +    using E C by (intro antisym) (auto simp: eq)
1.362 +  { fix n
1.363 +    have "outer_measure_of M (A \<inter> C n) \<le> outer_measure_of M (C n)"
1.364 +      by (intro outer_measure_of_mono) simp
1.365 +    also have "\<dots> < \<infinity>"
1.366 +      using assms by auto
1.367 +    finally have "emeasure M (E n \<inter> C n) < \<infinity>"
1.368 +      using eq by simp }
1.369 +  then have "measurable_envelope M (\<Union>n. A \<inter> C n) (\<Union>n. E n \<inter> C n)"
1.370 +    using E C by (intro measurable_envelopeI_countable measurable_envelope_eq2[THEN iffD2]) (auto simp: eq)
1.371 +  with \<open>A \<subseteq> (\<Union>n. C n)\<close> show ?thesis
1.372 +    by (intro exI[of _ "(\<Union>n. E n \<inter> C n)"]) (auto simp add: Int_absorb2)
1.373 +qed
1.375 +lemma (in complete_measure) complete_sets_sandwich:
1.376 +  assumes [measurable]: "A \<in> sets M" "C \<in> sets M" and subset: "A \<subseteq> B" "B \<subseteq> C"
1.377 +    and measure: "emeasure M A = emeasure M C" "emeasure M A < \<infinity>"
1.378 +  shows "B \<in> sets M"
1.379 +proof -
1.380 +  have "B - A \<in> sets M"
1.381 +  proof (rule complete)
1.382 +    show "B - A \<subseteq> C - A"
1.383 +      using subset by auto
1.384 +    show "C - A \<in> null_sets M"
1.385 +      using measure subset by(simp add: emeasure_Diff null_setsI)
1.386 +  qed
1.387 +  then have "A \<union> (B - A) \<in> sets M"
1.388 +    by measurable
1.389 +  also have "A \<union> (B - A) = B"
1.390 +    using \<open>A \<subseteq> B\<close> by auto
1.391 +  finally show ?thesis .
1.392 +qed
1.394 +lemma (in cld_measure) notin_sets_outer_measure_of_cover:
1.395 +  assumes E: "E \<subseteq> space M" "E \<notin> sets M"
1.396 +  shows "\<exists>B\<in>sets M. 0 < emeasure M B \<and> emeasure M B < \<infinity> \<and>
1.397 +    outer_measure_of M (B \<inter> E) = emeasure M B \<and> outer_measure_of M (B - E) = emeasure M B"
1.398 +proof -
1.399 +  from locally_determined[OF \<open>E \<subseteq> space M\<close>] \<open>E \<notin> sets M\<close>
1.400 +  obtain F
1.401 +    where [measurable]: "F \<in> sets M" and "emeasure M F < \<infinity>" "E \<inter> F \<notin> sets M"
1.402 +    by blast
1.403 +  then obtain H H'
1.404 +    where H: "measurable_envelope M (F \<inter> E) H" and H': "measurable_envelope M (F - E) H'"
1.405 +    using measurable_envelopeI_countable_cover[of "F \<inter> E" "\<lambda>_. F" M]
1.406 +       measurable_envelopeI_countable_cover[of "F - E" "\<lambda>_. F" M]
1.407 +    by auto
1.408 +  note measurable_envelopeD(2)[OF H', measurable] measurable_envelopeD(2)[OF H, measurable]
1.410 +  from measurable_envelopeD(1)[OF H'] measurable_envelopeD(1)[OF H]
1.411 +  have subset: "F - H' \<subseteq> F \<inter> E" "F \<inter> E \<subseteq> F \<inter> H"
1.412 +    by auto
1.413 +  moreover define G where "G = (F \<inter> H) - (F - H')"
1.414 +  ultimately have G: "G = F \<inter> H \<inter> H'"
1.415 +    by auto
1.416 +  have "emeasure M (F \<inter> H) \<noteq> 0"
1.417 +  proof
1.418 +    assume "emeasure M (F \<inter> H) = 0"
1.419 +    then have "F \<inter> H \<in> null_sets M"
1.420 +      by auto
1.421 +    with \<open>E \<inter> F \<notin> sets M\<close> show False
1.422 +      using complete[OF \<open>F \<inter> E \<subseteq> F \<inter> H\<close>] by (auto simp: Int_commute)
1.423 +  qed
1.424 +  moreover
1.425 +  have "emeasure M (F - H') \<noteq> emeasure M (F \<inter> H)"
1.426 +  proof
1.427 +    assume "emeasure M (F - H') = emeasure M (F \<inter> H)"
1.428 +    with \<open>E \<inter> F \<notin> sets M\<close> emeasure_mono[of "F \<inter> H" F M] \<open>emeasure M F < \<infinity>\<close>
1.429 +    have "F \<inter> E \<in> sets M"
1.430 +      by (intro complete_sets_sandwich[OF _ _ subset]) auto
1.431 +    with \<open>E \<inter> F \<notin> sets M\<close> show False
1.432 +      by (simp add: Int_commute)
1.433 +  qed
1.434 +  moreover have "emeasure M (F - H') \<le> emeasure M (F \<inter> H)"
1.435 +    using subset by (intro emeasure_mono) auto
1.436 +  ultimately have "emeasure M G \<noteq> 0"
1.437 +    unfolding G_def using subset
1.438 +    by (subst emeasure_Diff) (auto simp: top_unique diff_eq_0_iff_ennreal)
1.439 +  show ?thesis
1.440 +  proof (intro bexI conjI)
1.441 +    have "emeasure M G \<le> emeasure M F"
1.442 +      unfolding G by (auto intro!: emeasure_mono)
1.443 +    with \<open>emeasure M F < \<infinity>\<close> show "0 < emeasure M G" "emeasure M G < \<infinity>"
1.444 +      using \<open>emeasure M G \<noteq> 0\<close> by (auto simp: zero_less_iff_neq_zero)
1.445 +    show [measurable]: "G \<in> sets M"
1.446 +      unfolding G by auto
1.448 +    have "emeasure M G = outer_measure_of M (F \<inter> H' \<inter> (F \<inter> E))"
1.449 +      using measurable_envelopeD(3)[OF H, of "F \<inter> H'"] unfolding G by (simp add: ac_simps)
1.450 +    also have "\<dots> \<le> outer_measure_of M (G \<inter> E)"
1.451 +      using measurable_envelopeD(1)[OF H] by (intro outer_measure_of_mono) (auto simp: G)
1.452 +    finally show "outer_measure_of M (G \<inter> E) = emeasure M G"
1.453 +      using outer_measure_of_mono[of "G \<inter> E" G M] by auto
1.455 +    have "emeasure M G = outer_measure_of M (F \<inter> H \<inter> (F - E))"
1.456 +      using measurable_envelopeD(3)[OF H', of "F \<inter> H"] unfolding G by (simp add: ac_simps)
1.457 +    also have "\<dots> \<le> outer_measure_of M (G - E)"
1.458 +      using measurable_envelopeD(1)[OF H'] by (intro outer_measure_of_mono) (auto simp: G)
1.459 +    finally show "outer_measure_of M (G - E) = emeasure M G"
1.460 +      using outer_measure_of_mono[of "G - E" G M] by auto
1.461 +  qed
1.462 +qed
1.464 +text \<open>The following theorem is a specialization of D.H. Fremlin, Measure Theory vol 4I (413G). We
1.465 +  only show one direction and do not use a inner regular family \$K\$.\<close>
1.467 +lemma (in cld_measure) borel_measurable_cld:
1.468 +  fixes f :: "'a \<Rightarrow> real"
1.469 +  assumes "\<And>A a b. A \<in> sets M \<Longrightarrow> 0 < emeasure M A \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> a < b \<Longrightarrow>
1.470 +      min (outer_measure_of M {x\<in>A. f x \<le> a}) (outer_measure_of M {x\<in>A. b \<le> f x}) < emeasure M A"
1.471 +  shows "f \<in> M \<rightarrow>\<^sub>M borel"
1.472 +proof (rule ccontr)
1.473 +  let ?E = "\<lambda>a. {x\<in>space M. f x \<le> a}" and ?F = "\<lambda>a. {x\<in>space M. a \<le> f x}"
1.475 +  assume "f \<notin> M \<rightarrow>\<^sub>M borel"
1.476 +  then obtain a where "?E a \<notin> sets M"
1.477 +    unfolding borel_measurable_iff_le by blast
1.478 +  from notin_sets_outer_measure_of_cover[OF _ this]
1.479 +  obtain K
1.480 +    where K: "K \<in> sets M" "0 < emeasure M K" "emeasure M K < \<infinity>"
1.481 +      and eq1: "outer_measure_of M (K \<inter> ?E a) = emeasure M K"
1.482 +      and eq2: "outer_measure_of M (K - ?E a) = emeasure M K"
1.483 +    by auto
1.484 +  then have me_K: "measurable_envelope M (K \<inter> ?E a) K"
1.485 +    by (subst measurable_envelope_eq2) auto
1.487 +  define b where "b n = a + inverse (real (Suc n))" for n
1.488 +  have "(SUP n. outer_measure_of M (K \<inter> ?F (b n))) = outer_measure_of M (\<Union>n. K \<inter> ?F (b n))"
1.489 +  proof (intro SUP_outer_measure_of_incseq)
1.490 +    have "x \<le> y \<Longrightarrow> b y \<le> b x" for x y
1.491 +      by (auto simp: b_def field_simps)
1.492 +    then show "incseq (\<lambda>n. K \<inter> {x \<in> space M. b n \<le> f x})"
1.493 +      by (auto simp: incseq_def intro: order_trans)
1.494 +  qed auto
1.495 +  also have "(\<Union>n. K \<inter> ?F (b n)) = K - ?E a"
1.496 +  proof -
1.497 +    have "b \<longlonglongrightarrow> a"
1.498 +      unfolding b_def by (rule LIMSEQ_inverse_real_of_nat_add)
1.499 +    then have "\<forall>n. \<not> b n \<le> f x \<Longrightarrow> f x \<le> a" for x
1.500 +      by (rule LIMSEQ_le_const) (auto intro: less_imp_le simp: not_le)
1.501 +    moreover have "\<not> b n \<le> a" for n
1.502 +      by (auto simp: b_def)
1.503 +    ultimately show ?thesis
1.504 +      using \<open>K \<in> sets M\<close>[THEN sets.sets_into_space] by (auto simp: subset_eq intro: order_trans)
1.505 +  qed
1.506 +  finally have "0 < (SUP n. outer_measure_of M (K \<inter> ?F (b n)))"
1.507 +    using K by (simp add: eq2)
1.508 +  then obtain n where pos_b: "0 < outer_measure_of M (K \<inter> ?F (b n))" and "a < b n"
1.509 +    unfolding less_SUP_iff by (auto simp: b_def)
1.510 +  from measurable_envelopeI_countable_cover[of "K \<inter> ?F (b n)" "\<lambda>_. K" M] K
1.511 +  obtain K' where "K' \<subseteq> K" and me_K': "measurable_envelope M (K \<inter> ?F (b n)) K'"
1.512 +    by auto
1.513 +  then have K'_le_K: "emeasure M K' \<le> emeasure M K"
1.514 +    by (intro emeasure_mono K)
1.515 +  have "K' \<in> sets M"
1.516 +    using me_K' by (rule measurable_envelopeD)
1.518 +  have "min (outer_measure_of M {x\<in>K'. f x \<le> a}) (outer_measure_of M {x\<in>K'. b n \<le> f x}) < emeasure M K'"
1.519 +  proof (rule assms)
1.520 +    show "0 < emeasure M K'" "emeasure M K' < \<infinity>"
1.521 +      using measurable_envelopeD2[OF me_K'] pos_b K K'_le_K by auto
1.522 +  qed fact+
1.523 +  also have "{x\<in>K'. f x \<le> a} = K' \<inter> (K \<inter> ?E a)"
1.524 +    using \<open>K' \<in> sets M\<close>[THEN sets.sets_into_space] \<open>K' \<subseteq> K\<close> by auto
1.525 +  also have "{x\<in>K'. b n \<le> f x} = K' \<inter> (K \<inter> ?F (b n))"
1.526 +    using \<open>K' \<in> sets M\<close>[THEN sets.sets_into_space] \<open>K' \<subseteq> K\<close> by auto
1.527 +  finally have "min (emeasure M K) (emeasure M K') < emeasure M K'"
1.528 +    unfolding
1.529 +      measurable_envelopeD(3)[OF me_K \<open>K' \<in> sets M\<close>, symmetric]
1.530 +      measurable_envelopeD(3)[OF me_K' \<open>K' \<in> sets M\<close>, symmetric]
1.531 +    using \<open>K' \<subseteq> K\<close> by (simp add: Int_absorb1 Int_absorb2)
1.532 +  with K'_le_K show False
1.533 +    by (auto simp: min_def split: if_split_asm)
1.534 +qed
1.536  end
2.1 --- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Thu Sep 22 15:56:37 2016 +0100
2.2 +++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Fri Sep 23 10:26:04 2016 +0200
2.3 @@ -1,7 +1,340 @@
2.4  theory Equivalence_Lebesgue_Henstock_Integration
2.5 -  imports Lebesgue_Measure Henstock_Kurzweil_Integration
2.6 +  imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure
2.7  begin
2.9 +lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
2.10 +  by (auto intro: order_trans)
2.11 +
2.12 +lemma ball_trans:
2.13 +  assumes "y \<in> ball z q" "r + q \<le> s" shows "ball y r \<subseteq> ball z s"
2.14 +proof safe
2.15 +  fix x assume x: "x \<in> ball y r"
2.16 +  have "dist z x \<le> dist z y + dist y x"
2.17 +    by (rule dist_triangle)
2.18 +  also have "\<dots> < s"
2.19 +    using assms x by auto
2.20 +  finally show "x \<in> ball z s"
2.21 +    by simp
2.22 +qed
2.23 +
2.24 +abbreviation lebesgue :: "'a::euclidean_space measure"
2.25 +  where "lebesgue \<equiv> completion lborel"
2.26 +
2.27 +abbreviation lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
2.28 +  where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"
2.29 +
2.30 +lemma has_integral_implies_lebesgue_measurable_cbox:
2.31 +  fixes f :: "'a :: euclidean_space \<Rightarrow> real"
2.32 +  assumes f: "(f has_integral I) (cbox x y)"
2.33 +  shows "f \<in> lebesgue_on (cbox x y) \<rightarrow>\<^sub>M borel"
2.34 +proof (rule cld_measure.borel_measurable_cld)
2.35 +  let ?L = "lebesgue_on (cbox x y)"
2.36 +  let ?\<mu> = "emeasure ?L"
2.37 +  let ?\<mu>' = "outer_measure_of ?L"
2.38 +  interpret L: finite_measure ?L
2.39 +  proof
2.40 +    show "?\<mu> (space ?L) \<noteq> \<infinity>"
2.41 +      by (simp add: emeasure_restrict_space space_restrict_space emeasure_lborel_cbox_eq)
2.42 +  qed
2.43 +
2.44 +  show "cld_measure ?L"
2.45 +  proof
2.46 +    fix B A assume "B \<subseteq> A" "A \<in> null_sets ?L"
2.47 +    then show "B \<in> sets ?L"
2.48 +      using null_sets_completion_subset[OF \<open>B \<subseteq> A\<close>, of lborel]
2.49 +      by (auto simp add: null_sets_restrict_space sets_restrict_space_iff intro: )
2.50 +  next
2.51 +    fix A assume "A \<subseteq> space ?L" "\<And>B. B \<in> sets ?L \<Longrightarrow> ?\<mu> B < \<infinity> \<Longrightarrow> A \<inter> B \<in> sets ?L"
2.52 +    from this(1) this(2)[of "space ?L"] show "A \<in> sets ?L"
2.53 +      by (auto simp: Int_absorb2 less_top[symmetric])
2.54 +  qed auto
2.55 +  then interpret cld_measure ?L
2.56 +    .
2.57 +
2.58 +  have content_eq_L: "A \<in> sets borel \<Longrightarrow> A \<subseteq> cbox x y \<Longrightarrow> content A = measure ?L A" for A
2.59 +    by (subst measure_restrict_space) (auto simp: measure_def)
2.60 +
2.61 +  fix E and a b :: real assume "E \<in> sets ?L" "a < b" "0 < ?\<mu> E" "?\<mu> E < \<infinity>"
2.62 +  then obtain M :: real where "?\<mu> E = M" "0 < M"
2.63 +    by (cases "?\<mu> E") auto
2.64 +  define e where "e = M / (4 + 2 / (b - a))"
2.65 +  from \<open>a < b\<close> \<open>0<M\<close> have "0 < e"
2.66 +    by (auto intro!: divide_pos_pos simp: field_simps e_def)
2.67 +
2.68 +  have "e < M / (3 + 2 / (b - a))"
2.69 +    using \<open>a < b\<close> \<open>0 < M\<close>
2.70 +    unfolding e_def by (intro divide_strict_left_mono add_strict_right_mono mult_pos_pos) (auto simp: field_simps)
2.71 +  then have "2 * e < (b - a) * (M - e * 3)"
2.72 +    using \<open>0<M\<close> \<open>0 < e\<close> \<open>a < b\<close> by (simp add: field_simps)
2.73 +
2.74 +  have e_less_M: "e < M / 1"
2.75 +    unfolding e_def using \<open>a < b\<close> \<open>0<M\<close> by (intro divide_strict_left_mono) (auto simp: field_simps)
2.76 +
2.77 +  obtain d
2.78 +    where "gauge d"
2.79 +      and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow>
2.80 +        norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
2.81 +    using \<open>0<e\<close> f unfolding has_integral by auto
2.82 +
2.83 +  define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m
2.84 +  have "incseq (C X)" for X
2.85 +    unfolding C_def [abs_def]
2.86 +    by (intro monoI Collect_mono conj_mono imp_refl le_left_mono subset_ball divide_left_mono Int_mono) auto
2.87 +
2.88 +  { fix X assume "X \<subseteq> space ?L" and eq: "?\<mu>' X = ?\<mu> E"
2.89 +    have "(SUP m. outer_measure_of ?L (C X m)) = outer_measure_of ?L (\<Union>m. C X m)"
2.90 +      using \<open>X \<subseteq> space ?L\<close> by (intro SUP_outer_measure_of_incseq \<open>incseq (C X)\<close>) (auto simp: C_def)
2.91 +    also have "(\<Union>m. C X m) = X"
2.92 +    proof -
2.93 +      { fix x
2.94 +        obtain e where "0 < e" "ball x e \<subseteq> d x"
2.95 +          using gaugeD[OF \<open>gauge d\<close>, of x] unfolding open_contains_ball by auto
2.96 +        moreover
2.97 +        obtain n where "1 / (1 + real n) < e"
2.98 +          using reals_Archimedean[OF \<open>0<e\<close>] by (auto simp: inverse_eq_divide)
2.99 +        then have "ball x (1 / (1 + real n)) \<subseteq> ball x e"
2.100 +          by (intro subset_ball) auto
2.101 +        ultimately have "\<exists>n. ball x (1 / (1 + real n)) \<subseteq> d x"
2.102 +          by blast }
2.103 +      then show ?thesis
2.104 +        by (auto simp: C_def)
2.105 +    qed
2.106 +    finally have "(SUP m. outer_measure_of ?L (C X m)) = ?\<mu> E"
2.107 +      using eq by auto
2.108 +    also have "\<dots> > M - e"
2.109 +      using \<open>0 < M\<close> \<open>?\<mu> E = M\<close> \<open>0<e\<close> by (auto intro!: ennreal_lessI)
2.110 +    finally have "\<exists>m. M - e < outer_measure_of ?L (C X m)"
2.111 +      unfolding less_SUP_iff by auto }
2.112 +  note C = this
2.114 +  let ?E = "{x\<in>E. f x \<le> a}" and ?F = "{x\<in>E. b \<le> f x}"
2.116 +  have "\<not> (?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E)"
2.117 +  proof
2.118 +    assume eq: "?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E"
2.119 +    with C[of ?E] C[of ?F] \<open>E \<in> sets ?L\<close>[THEN sets.sets_into_space] obtain ma mb
2.120 +      where "M - e < outer_measure_of ?L (C ?E ma)" "M - e < outer_measure_of ?L (C ?F mb)"
2.121 +      by auto
2.122 +    moreover define m where "m = max ma mb"
2.123 +    ultimately have M_minus_e: "M - e < outer_measure_of ?L (C ?E m)" "M - e < outer_measure_of ?L (C ?F m)"
2.124 +      using
2.125 +        incseqD[OF \<open>incseq (C ?E)\<close>, of ma m, THEN outer_measure_of_mono]
2.126 +        incseqD[OF \<open>incseq (C ?F)\<close>, of mb m, THEN outer_measure_of_mono]
2.127 +      by (auto intro: less_le_trans)
2.128 +    define d' where "d' x = d x \<inter> ball x (1 / (3 * Suc m))" for x
2.129 +    have "gauge d'"
2.130 +      unfolding d'_def by (intro gauge_inter \<open>gauge d\<close> gauge_ball) auto
2.131 +    then obtain p where p: "p tagged_division_of cbox x y" "d' fine p"
2.132 +      by (rule fine_division_exists)
2.133 +    then have "d fine p"
2.134 +      unfolding d'_def[abs_def] fine_def by auto
2.136 +    define s where "s = {(x::'a, k). k \<inter> (C ?E m) \<noteq> {} \<and> k \<inter> (C ?F m) \<noteq> {}}"
2.137 +    define T where "T E k = (SOME x. x \<in> k \<inter> C E m)" for E k
2.138 +    let ?A = "(\<lambda>(x, k). (T ?E k, k)) ` (p \<inter> s) \<union> (p - s)"
2.139 +    let ?B = "(\<lambda>(x, k). (T ?F k, k)) ` (p \<inter> s) \<union> (p - s)"
2.141 +    { fix X assume X_eq: "X = ?E \<or> X = ?F"
2.142 +      let ?T = "(\<lambda>(x, k). (T X k, k))"
2.143 +      let ?p = "?T ` (p \<inter> s) \<union> (p - s)"
2.145 +      have in_s: "(x, k) \<in> s \<Longrightarrow> T X k \<in> k \<inter> C X m" for x k
2.146 +        using someI_ex[of "\<lambda>x. x \<in> k \<inter> C X m"] X_eq unfolding ex_in_conv by (auto simp: T_def s_def)
2.148 +      { fix x k assume "(x, k) \<in> p" "(x, k) \<in> s"
2.149 +        have k: "k \<subseteq> ball x (1 / (3 * Suc m))"
2.150 +          using \<open>d' fine p\<close>[THEN fineD, OF \<open>(x, k) \<in> p\<close>] by (auto simp: d'_def)
2.151 +        then have "x \<in> ball (T X k) (1 / (3 * Suc m))"
2.152 +          using in_s[OF \<open>(x, k) \<in> s\<close>] by (auto simp: C_def subset_eq dist_commute)
2.153 +        then have "ball x (1 / (3 * Suc m)) \<subseteq> ball (T X k) (1 / Suc m)"
2.154 +          by (rule ball_trans) (auto simp: divide_simps)
2.155 +        with k in_s[OF \<open>(x, k) \<in> s\<close>] have "k \<subseteq> d (T X k)"
2.156 +          by (auto simp: C_def) }
2.157 +      then have "d fine ?p"
2.158 +        using \<open>d fine p\<close> by (auto intro!: fineI)
2.159 +      moreover
2.160 +      have "?p tagged_division_of cbox x y"
2.161 +      proof (rule tagged_division_ofI)
2.162 +        show "finite ?p"
2.163 +          using p(1) by auto
2.164 +      next
2.165 +        fix z k assume *: "(z, k) \<in> ?p"
2.166 +        then consider "(z, k) \<in> p" "(z, k) \<notin> s"
2.167 +          | x' where "(x', k) \<in> p" "(x', k) \<in> s" "z = T X k"
2.168 +          by (auto simp: T_def)
2.169 +        then have "z \<in> k \<and> k \<subseteq> cbox x y \<and> (\<exists>a b. k = cbox a b)"
2.170 +          using p(1) by cases (auto dest: in_s)
2.171 +        then show "z \<in> k" "k \<subseteq> cbox x y" "\<exists>a b. k = cbox a b"
2.172 +          by auto
2.173 +      next
2.174 +        fix z k z' k' assume "(z, k) \<in> ?p" "(z', k') \<in> ?p" "(z, k) \<noteq> (z', k')"
2.175 +        with tagged_division_ofD(5)[OF p(1), of _ k _ k']
2.176 +        show "interior k \<inter> interior k' = {}"
2.177 +          by (auto simp: T_def dest: in_s)
2.178 +      next
2.179 +        have "{k. \<exists>x. (x, k) \<in> ?p} = {k. \<exists>x. (x, k) \<in> p}"
2.180 +          by (auto simp: T_def image_iff Bex_def)
2.181 +        then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y"
2.182 +          using p(1) by auto
2.183 +      qed
2.184 +      ultimately have I: "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - I) < e"
2.185 +        using integral_f by auto
2.187 +      have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
2.188 +        (\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)"
2.189 +        using p(1)[THEN tagged_division_ofD(1)]
2.190 +        by (safe intro!: setsum.union_inter_neutral) (auto simp: s_def T_def)
2.191 +      also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))"
2.192 +      proof (subst setsum.reindex_nontrivial, safe)
2.193 +        fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
2.194 +          and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
2.195 +        with tagged_division_ofD(5)[OF p(1), of x1 k x2 k] tagged_division_ofD(4)[OF p(1), of x1 k]
2.196 +        show "x1 = x2"
2.197 +          by (auto simp: content_eq_0_interior)
2.198 +      qed (use p in \<open>auto intro!: setsum.cong\<close>)
2.199 +      finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
2.200 +        (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" .
2.202 +      have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k
2.203 +        using in_s[of x k] by (auto simp: C_def)
2.205 +      note I eq in_T }
2.206 +    note parts = this
2.208 +    have p_in_L: "(x, k) \<in> p \<Longrightarrow> k \<in> sets ?L" for x k
2.209 +      using tagged_division_ofD(3, 4)[OF p(1), of x k] by (auto simp: sets_restrict_space)
2.211 +    have [simp]: "finite p"
2.212 +      using tagged_division_ofD(1)[OF p(1)] .
2.214 +    have "(M - 3*e) * (b - a) \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k) * (b - a)"
2.215 +    proof (intro mult_right_mono)
2.216 +      have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X
2.217 +        using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>)
2.218 +      have sets: "(E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<in> sets ?L" for X
2.219 +        using tagged_division_ofD(1)[OF p(1)] by (intro sets.Diff \<open>E \<in> sets ?L\<close> sets.finite_Union sets.Int) (auto intro: p_in_L)
2.220 +      { fix X assume "X \<subseteq> E" "M - e < ?\<mu>' (C X m)"
2.221 +        have "M - e \<le> ?\<mu>' (C X m)"
2.222 +          by (rule less_imp_le) fact
2.223 +        also have "\<dots> \<le> ?\<mu>' (E - (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}))"
2.224 +        proof (intro outer_measure_of_mono subsetI)
2.225 +          fix v assume "v \<in> C X m"
2.226 +          then have "v \<in> cbox x y" "v \<in> E"
2.227 +            using \<open>E \<subseteq> space ?L\<close> \<open>X \<subseteq> E\<close> by (auto simp: space_restrict_space C_def)
2.228 +          then obtain z k where "(z, k) \<in> p" "v \<in> k"
2.229 +            using tagged_division_ofD(6)[OF p(1), symmetric] by auto
2.230 +          then show "v \<in> E - E \<inter> (\<Union>{k\<in>snd`p. k \<inter> C X m = {}})"
2.231 +            using \<open>v \<in> C X m\<close> \<open>v \<in> E\<close> by auto
2.232 +        qed
2.233 +        also have "\<dots> = ?\<mu> E - ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})"
2.234 +          using \<open>E \<in> sets ?L\<close> fin[of X] sets[of X] by (auto intro!: emeasure_Diff)
2.235 +        finally have "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<le> e"
2.236 +          using \<open>0 < e\<close> e_less_M apply (cases "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})")
2.237 +          by (auto simp add: \<open>?\<mu> E = M\<close> ennreal_minus ennreal_le_iff2)
2.238 +        note this }
2.239 +      note upper_bound = this
2.241 +      have "?\<mu> (E \<inter> \<Union>(snd`(p - s))) =
2.242 +        ?\<mu> ((E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) \<union> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}}))"
2.243 +        by (intro arg_cong[where f="?\<mu>"]) (auto simp: s_def image_def Bex_def)
2.244 +      also have "\<dots> \<le> ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) + ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}})"
2.245 +        using sets[of ?E] sets[of ?F] M_minus_e by (intro emeasure_subadditive) auto
2.246 +      also have "\<dots> \<le> e + ennreal e"
2.247 +        using upper_bound[of ?E] upper_bound[of ?F] M_minus_e by (intro add_mono) auto
2.248 +      finally have "?\<mu> E - 2*e \<le> ?\<mu> (E - (E \<inter> \<Union>(snd`(p - s))))"
2.249 +        using \<open>0 < e\<close> \<open>E \<in> sets ?L\<close> tagged_division_ofD(1)[OF p(1)]
2.250 +        by (subst emeasure_Diff)
2.251 +           (auto simp: ennreal_plus[symmetric] top_unique simp del: ennreal_plus
2.252 +                 intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
2.253 +      also have "\<dots> \<le> ?\<mu> (\<Union>x\<in>p \<inter> s. snd x)"
2.254 +      proof (safe intro!: emeasure_mono subsetI)
2.255 +        fix v assume "v \<in> E" and not: "v \<notin> (\<Union>x\<in>p \<inter> s. snd x)"
2.256 +        then have "v \<in> cbox x y"
2.257 +          using \<open>E \<subseteq> space ?L\<close> by (auto simp: space_restrict_space)
2.258 +        then obtain z k where "(z, k) \<in> p" "v \<in> k"
2.259 +          using tagged_division_ofD(6)[OF p(1), symmetric] by auto
2.260 +        with not show "v \<in> UNION (p - s) snd"
2.261 +          by (auto intro!: bexI[of _ "(z, k)"] elim: ballE[of _ _ "(z, k)"])
2.262 +      qed (auto intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
2.263 +      also have "\<dots> = measure ?L (\<Union>x\<in>p \<inter> s. snd x)"
2.264 +        by (auto intro!: emeasure_eq_ennreal_measure)
2.265 +      finally have "M - 2 * e \<le> measure ?L (\<Union>x\<in>p \<inter> s. snd x)"
2.266 +        unfolding \<open>?\<mu> E = M\<close> using \<open>0 < e\<close> by (simp add: ennreal_minus)
2.267 +      also have "measure ?L (\<Union>x\<in>p \<inter> s. snd x) = content (\<Union>x\<in>p \<inter> s. snd x)"
2.268 +        using tagged_division_ofD(1,3,4) [OF p(1)]
2.269 +        by (intro content_eq_L[symmetric])
2.270 +           (fastforce intro!: sets.finite_UN UN_least del: subsetI)+
2.271 +      also have "content (\<Union>x\<in>p \<inter> s. snd x) \<le> (\<Sum>k\<in>p \<inter> s. content (snd k))"
2.272 +        using p(1) by (auto simp: emeasure_lborel_cbox_eq intro!: measure_subadditive_finite
2.273 +                            dest!: p(1)[THEN tagged_division_ofD(4)])
2.274 +      finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)"
2.275 +        using \<open>0 < e\<close> by (simp add: split_beta)
2.276 +    qed (use \<open>a < b\<close> in auto)
2.277 +    also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))"
2.278 +      by (simp add: setsum_distrib_right split_beta')
2.279 +    also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
2.280 +      using parts(3) by (auto intro!: setsum_mono mult_left_mono diff_mono)
2.281 +    also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))"
2.282 +      by (auto intro!: setsum.cong simp: field_simps setsum_subtractf[symmetric])
2.283 +    also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)"
2.284 +      by (subst (1 2) parts) auto
2.285 +    also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))"
2.286 +      by auto
2.287 +    also have "\<dots> \<le> e + e"
2.288 +      using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
2.289 +    finally show False
2.290 +      using \<open>2 * e < (b - a) * (M - e * 3)\<close> by (auto simp: field_simps)
2.291 +  qed
2.292 +  moreover have "?\<mu>' ?E \<le> ?\<mu> E" "?\<mu>' ?F \<le> ?\<mu> E"
2.293 +    unfolding outer_measure_of_eq[OF \<open>E \<in> sets ?L\<close>, symmetric] by (auto intro!: outer_measure_of_mono)
2.294 +  ultimately show "min (?\<mu>' ?E) (?\<mu>' ?F) < ?\<mu> E"
2.295 +    unfolding min_less_iff_disj by (auto simp: less_le)
2.296 +qed
2.298 +lemma has_integral_implies_lebesgue_measurable_real:
2.299 +  fixes f :: "'a :: euclidean_space \<Rightarrow> real"
2.300 +  assumes f: "(f has_integral I) \<Omega>"
2.301 +  shows "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
2.302 +proof -
2.303 +  define B :: "nat \<Rightarrow> 'a set" where "B n = cbox (- real n *\<^sub>R One) (real n *\<^sub>R One)" for n
2.304 +  show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
2.305 +  proof (rule measurable_piecewise_restrict)
2.306 +    have "(\<Union>n. box (- real n *\<^sub>R One) (real n *\<^sub>R One)) \<subseteq> UNION UNIV B"
2.307 +      unfolding B_def by (intro UN_mono box_subset_cbox order_refl)
2.308 +    then show "countable (range B)" "space lebesgue \<subseteq> UNION UNIV B"
2.309 +      by (auto simp: B_def UN_box_eq_UNIV)
2.310 +  next
2.311 +    fix \<Omega>' assume "\<Omega>' \<in> range B"
2.312 +    then obtain n where \<Omega>': "\<Omega>' = B n" by auto
2.313 +    then show "\<Omega>' \<inter> space lebesgue \<in> sets lebesgue"
2.314 +      by (auto simp: B_def)
2.316 +    have "f integrable_on \<Omega>"
2.317 +      using f by auto
2.318 +    then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on \<Omega>"
2.319 +      by (auto simp: integrable_on_def cong: has_integral_cong)
2.320 +    then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on (\<Omega> \<union> B n)"
2.321 +      by (rule integrable_on_superset[rotated 2]) auto
2.322 +    then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on B n"
2.323 +      unfolding B_def by (rule integrable_on_subcbox) auto
2.324 +    then show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue_on \<Omega>' \<rightarrow>\<^sub>M borel"
2.325 +      unfolding B_def \<Omega>' by (auto intro: has_integral_implies_lebesgue_measurable_cbox simp: integrable_on_def)
2.326 +  qed
2.327 +qed
2.329 +lemma has_integral_implies_lebesgue_measurable:
2.330 +  fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
2.331 +  assumes f: "(f has_integral I) \<Omega>"
2.332 +  shows "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
2.333 +proof (intro borel_measurable_euclidean_space[where 'c='b, THEN iffD2] ballI)
2.334 +  fix i :: "'b" assume "i \<in> Basis"
2.335 +  have "(\<lambda>x. (f x \<bullet> i) * indicator \<Omega> x) \<in> borel_measurable (completion lborel)"
2.336 +    using has_integral_linear[OF f bounded_linear_inner_left, of i]
2.337 +    by (intro has_integral_implies_lebesgue_measurable_real) (auto simp: comp_def)
2.338 +  then show "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x \<bullet> i) \<in> borel_measurable (completion lborel)"
2.339 +    by (simp add: ac_simps)
2.340 +qed
2.342  subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close>
2.344  lemma has_integral_measure_lborel:
2.345 @@ -347,6 +680,82 @@
2.346    qed
2.347  qed
2.349 +lemma has_integral_AE:
2.350 +  assumes ae: "AE x in lborel. x \<in> \<Omega> \<longrightarrow> f x = g x"
2.351 +  shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>"
2.352 +proof -
2.353 +  from ae obtain N
2.354 +    where N: "N \<in> sets borel" "emeasure lborel N = 0" "{x. \<not> (x \<in> \<Omega> \<longrightarrow> f x = g x)} \<subseteq> N"
2.355 +    by (auto elim!: AE_E)
2.356 +  then have not_N: "AE x in lborel. x \<notin> N"
2.357 +    by (simp add: AE_iff_measurable)
2.358 +  show ?thesis
2.359 +  proof (rule has_integral_spike_eq[symmetric])
2.360 +    show "\<forall>x\<in>\<Omega> - N. f x = g x" using N(3) by auto
2.361 +    show "negligible N"
2.362 +      unfolding negligible_def
2.363 +    proof (intro allI)
2.364 +      fix a b :: "'a"
2.365 +      let ?F = "\<lambda>x::'a. if x \<in> cbox a b then indicator N x else 0 :: real"
2.366 +      have "integrable lborel ?F = integrable lborel (\<lambda>x::'a. 0::real)"
2.367 +        using not_N N(1) by (intro integrable_cong_AE) auto
2.368 +      moreover have "(LINT x|lborel. ?F x) = (LINT x::'a|lborel. 0::real)"
2.369 +        using not_N N(1) by (intro integral_cong_AE) auto
2.370 +      ultimately have "(?F has_integral 0) UNIV"
2.371 +        using has_integral_integral_real[of ?F] by simp
2.372 +      then show "(indicator N has_integral (0::real)) (cbox a b)"
2.373 +        unfolding has_integral_restrict_univ .
2.374 +    qed
2.375 +  qed
2.376 +qed
2.378 +lemma nn_integral_has_integral_lebesgue:
2.379 +  fixes f :: "'a::euclidean_space \<Rightarrow> real"
2.380 +  assumes nonneg: "\<And>x. 0 \<le> f x" and I: "(f has_integral I) \<Omega>"
2.381 +  shows "integral\<^sup>N lborel (\<lambda>x. indicator \<Omega> x * f x) = I"
2.382 +proof -
2.383 +  from I have "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
2.384 +    by (rule has_integral_implies_lebesgue_measurable)
2.385 +  then obtain f' :: "'a \<Rightarrow> real"
2.386 +    where [measurable]: "f' \<in> borel \<rightarrow>\<^sub>M borel" and eq: "AE x in lborel. indicator \<Omega> x * f x = f' x"
2.387 +    by (auto dest: completion_ex_borel_measurable_real)
2.389 +  from I have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV"
2.390 +    using nonneg by (simp add: indicator_def if_distrib[of "\<lambda>x. x * f y" for y] cong: if_cong)
2.391 +  also have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV \<longleftrightarrow> ((\<lambda>x. abs (f' x)) has_integral I) UNIV"
2.392 +    using eq by (intro has_integral_AE) auto
2.393 +  finally have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = I"
2.394 +    by (rule nn_integral_has_integral_lborel[rotated 2]) auto
2.395 +  also have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = integral\<^sup>N lborel (\<lambda>x. abs (indicator \<Omega> x * f x))"
2.396 +    using eq by (intro nn_integral_cong_AE) auto
2.397 +  finally show ?thesis
2.398 +    using nonneg by auto
2.399 +qed
2.401 +lemma has_integral_iff_nn_integral_lebesgue:
2.402 +  assumes f: "\<And>x. 0 \<le> f x"
2.403 +  shows "(f has_integral r) UNIV \<longleftrightarrow> (f \<in> lebesgue \<rightarrow>\<^sub>M borel \<and> integral\<^sup>N lebesgue f = r \<and> 0 \<le> r)" (is "?I = ?N")
2.404 +proof
2.405 +  assume ?I
2.406 +  have "0 \<le> r"
2.407 +    using has_integral_nonneg[OF \<open>?I\<close>] f by auto
2.408 +  then show ?N
2.409 +    using nn_integral_has_integral_lebesgue[OF f \<open>?I\<close>]
2.410 +      has_integral_implies_lebesgue_measurable[OF \<open>?I\<close>]
2.411 +    by (auto simp: nn_integral_completion)
2.412 +next
2.413 +  assume ?N
2.414 +  then obtain f' where f': "f' \<in> borel \<rightarrow>\<^sub>M borel" "AE x in lborel. f x = f' x"
2.415 +    by (auto dest: completion_ex_borel_measurable_real)
2.416 +  moreover have "(\<integral>\<^sup>+ x. ennreal \<bar>f' x\<bar> \<partial>lborel) = (\<integral>\<^sup>+ x. ennreal \<bar>f x\<bar> \<partial>lborel)"
2.417 +    using f' by (intro nn_integral_cong_AE) auto
2.418 +  moreover have "((\<lambda>x. \<bar>f' x\<bar>) has_integral r) UNIV \<longleftrightarrow> ((\<lambda>x. \<bar>f x\<bar>) has_integral r) UNIV"
2.419 +    using f' by (intro has_integral_AE) auto
2.420 +  moreover note nn_integral_has_integral[of "\<lambda>x. \<bar>f' x\<bar>" r] \<open>?N\<close>
2.421 +  ultimately show ?I
2.422 +    using f by (auto simp: nn_integral_completion)
2.423 +qed
2.425  context
2.426    fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
2.427  begin
3.1 --- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Thu Sep 22 15:56:37 2016 +0100
3.2 +++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Fri Sep 23 10:26:04 2016 +0200
3.3 @@ -1393,10 +1393,7 @@
3.4  proof (rule tagged_division_ofI)
3.5    note assm = tagged_division_ofD[OF assms(2)[rule_format]]
3.6    show "finite (\<Union>(pfn ` iset))"
3.7 -    apply (rule finite_Union)
3.8 -    using assms
3.9 -    apply auto
3.10 -    done
3.11 +    using assms by auto
3.12    have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
3.13      by blast
3.14    also have "\<dots> = \<Union>iset"
3.15 @@ -1936,8 +1933,7 @@
3.16  definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
3.17    where "(f has_integral_compact_interval y) i \<longleftrightarrow>
3.18      (\<forall>e>0. \<exists>d. gauge d \<and>
3.19 -      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
3.20 -        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
3.21 +      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow> norm ((\<Sum>(x,k)\<in>p. content k *\<^sub>R f x) - y) < e))"
3.23  definition has_integral ::
3.24      "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
4.1 --- a/src/HOL/Analysis/Measure_Space.thy	Thu Sep 22 15:56:37 2016 +0100
4.2 +++ b/src/HOL/Analysis/Measure_Space.thy	Fri Sep 23 10:26:04 2016 +0200
4.3 @@ -551,6 +551,28 @@
4.4         (insert finite A, auto intro: INF_lower emeasure_mono)
4.5  qed
4.7 +lemma INF_emeasure_decseq':
4.8 +  assumes A: "\<And>i. A i \<in> sets M" and "decseq A"
4.9 +  and finite: "\<exists>i. emeasure M (A i) \<noteq> \<infinity>"
4.10 +  shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
4.11 +proof -
4.12 +  from finite obtain i where i: "emeasure M (A i) < \<infinity>"
4.13 +    by (auto simp: less_top)
4.14 +  have fin: "i \<le> j \<Longrightarrow> emeasure M (A j) < \<infinity>" for j
4.15 +    by (rule le_less_trans[OF emeasure_mono i]) (auto intro!: decseqD[OF \<open>decseq A\<close>] A)
4.16 +
4.17 +  have "(INF n. emeasure M (A n)) = (INF n. emeasure M (A (n + i)))"
4.18 +  proof (rule INF_eq)
4.19 +    show "\<exists>j\<in>UNIV. emeasure M (A (j + i)) \<le> emeasure M (A i')" for i'
4.20 +      by (intro bexI[of _ i'] emeasure_mono decseqD[OF \<open>decseq A\<close>] A) auto
4.21 +  qed auto
4.22 +  also have "\<dots> = emeasure M (INF n. (A (n + i)))"
4.23 +    using A \<open>decseq A\<close> fin by (intro INF_emeasure_decseq) (auto simp: decseq_def less_top)
4.24 +  also have "(INF n. (A (n + i))) = (INF n. A n)"
4.25 +    by (meson INF_eq UNIV_I assms(2) decseqD le_add1)
4.26 +  finally show ?thesis .
4.27 +qed
4.28 +
4.29  lemma emeasure_INT_decseq_subset:
4.30    fixes F :: "nat \<Rightarrow> 'a set"
4.31    assumes I: "I \<noteq> {}" and F: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<le> j \<Longrightarrow> F j \<subseteq> F i"
5.1 --- a/src/HOL/Library/Extended_Nonnegative_Real.thy	Thu Sep 22 15:56:37 2016 +0100
5.2 +++ b/src/HOL/Library/Extended_Nonnegative_Real.thy	Fri Sep 23 10:26:04 2016 +0200
5.3 @@ -1574,6 +1574,19 @@
5.4      done
5.5    done
5.7 +lemma ennreal_Inf_countable_INF:
5.8 +  "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ennreal. decseq f \<and> range f \<subseteq> A \<and> Inf A = (INF i. f i)"
5.9 +  including ennreal.lifting
5.10 +  unfolding decseq_def
5.11 +  apply transfer
5.12 +  subgoal for A
5.13 +    using Inf_countable_INF[of A]
5.14 +    apply (clarsimp simp add: decseq_def[symmetric])
5.15 +    subgoal for f
5.16 +      by (intro exI[of _ f]) auto
5.17 +    done
5.18 +  done
5.19 +
5.20  lemma ennreal_SUP_countable_SUP:
5.21    "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ennreal. range f \<subseteq> g`A \<and> SUPREMUM A g = SUPREMUM UNIV f"
5.22    using ennreal_Sup_countable_SUP [of "g`A"] by auto
6.1 --- a/src/HOL/Library/Extended_Real.thy	Thu Sep 22 15:56:37 2016 +0100
6.2 +++ b/src/HOL/Library/Extended_Real.thy	Fri Sep 23 10:26:04 2016 +0200
6.3 @@ -2228,6 +2228,16 @@
6.4      by auto
6.5  qed
6.7 +lemma Inf_countable_INF:
6.8 +  assumes "A \<noteq> {}" shows "\<exists>f::nat \<Rightarrow> ereal. decseq f \<and> range f \<subseteq> A \<and> Inf A = (INF i. f i)"
6.9 +proof -
6.10 +  obtain f where "incseq f" "range f \<subseteq> uminus`A" "Sup (uminus`A) = (SUP i. f i)"
6.11 +    using Sup_countable_SUP[of "uminus ` A"] \<open>A \<noteq> {}\<close> by auto
6.12 +  then show ?thesis
6.13 +    by (intro exI[of _ "\<lambda>x. - f x"])
6.14 +       (auto simp: ereal_Sup_uminus_image_eq ereal_INF_uminus_eq eq_commute[of "- _"])
6.15 +qed
6.16 +
6.17  lemma SUP_countable_SUP:
6.18    "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPREMUM A g = SUPREMUM UNIV f"
6.19    using Sup_countable_SUP [of "g`A"] by auto