--- a/src/HOL/Analysis/Complete_Measure.thy Thu Sep 22 15:56:37 2016 +0100
+++ b/src/HOL/Analysis/Complete_Measure.thy Fri Sep 23 10:26:04 2016 +0200
@@ -6,6 +6,10 @@
imports Bochner_Integration
begin
+locale complete_measure =
+ fixes M :: "'a measure"
+ assumes complete: "\<And>A B. B \<subseteq> A \<Longrightarrow> A \<in> null_sets M \<Longrightarrow> B \<in> sets M"
+
definition
"split_completion M A p = (if A \<in> sets M then p = (A, {}) else
\<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)"
@@ -304,4 +308,522 @@
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)"
by (simp add: AE_iff_null null_sets_completion_iff)
+lemma sets_completion_AE: "(AE x in M. \<not> P x) \<Longrightarrow> Measurable.pred (completion M) P"
+ unfolding pred_def sets_completion eventually_ae_filter
+ by auto
+
+lemma null_sets_completion_iff2:
+ "A \<in> null_sets (completion M) \<longleftrightarrow> (\<exists>N'\<in>null_sets M. A \<subseteq> N')"
+proof safe
+ assume "A \<in> null_sets (completion M)"
+ then have A: "A \<in> sets (completion M)" and "main_part M A \<in> null_sets M"
+ by (auto simp: null_sets_def)
+ moreover obtain N where "N \<in> null_sets M" "null_part M A \<subseteq> N"
+ using null_part[OF A] by auto
+ ultimately show "\<exists>N'\<in>null_sets M. A \<subseteq> N'"
+ proof (intro bexI)
+ show "A \<subseteq> N \<union> main_part M A"
+ using \<open>null_part M A \<subseteq> N\<close> by (subst main_part_null_part_Un[OF A, symmetric]) auto
+ qed auto
+next
+ fix N assume "N \<in> null_sets M" "A \<subseteq> N"
+ then have "A \<in> sets (completion M)" and N: "N \<in> sets M" "A \<subseteq> N" "emeasure M N = 0"
+ by (auto intro: null_sets_completion)
+ moreover have "emeasure (completion M) A = 0"
+ using N by (intro emeasure_eq_0[of N _ A]) auto
+ ultimately show "A \<in> null_sets (completion M)"
+ by auto
+qed
+
+lemma null_sets_completion_subset:
+ "B \<subseteq> A \<Longrightarrow> A \<in> null_sets (completion M) \<Longrightarrow> B \<in> null_sets (completion M)"
+ unfolding null_sets_completion_iff2 by auto
+
+lemma null_sets_restrict_space:
+ "\<Omega> \<in> sets M \<Longrightarrow> A \<in> null_sets (restrict_space M \<Omega>) \<longleftrightarrow> A \<subseteq> \<Omega> \<and> A \<in> null_sets M"
+ by (auto simp: null_sets_def emeasure_restrict_space sets_restrict_space)
+lemma completion_ex_borel_measurable_real:
+ fixes g :: "'a \<Rightarrow> real"
+ assumes g: "g \<in> borel_measurable (completion M)"
+ shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
+proof -
+ 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"
+ using g by auto
+ from this[THEN completion_ex_borel_measurable]
+ obtain pf nf :: "'a \<Rightarrow> ennreal"
+ where [measurable]: "nf \<in> M \<rightarrow>\<^sub>M borel" "pf \<in> M \<rightarrow>\<^sub>M borel"
+ and ae: "AE x in M. pf x = ennreal (g x)" "AE x in M. nf x = ennreal (- g x)"
+ by (auto simp: eq_commute)
+ then have "AE x in M. pf x = ennreal (g x) \<and> nf x = ennreal (- g x)"
+ by auto
+ 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"
+ by (auto elim!: AE_E)
+ show ?thesis
+ proof
+ let ?F = "\<lambda>x. indicator (space M - N) x * (enn2real (pf x) - enn2real (nf x))"
+ show "?F \<in> M \<rightarrow>\<^sub>M borel"
+ using \<open>N \<in> null_sets M\<close> by auto
+ show "AE x in M. g x = ?F x"
+ using \<open>N \<in> null_sets M\<close>[THEN AE_not_in] ae AE_space
+ apply eventually_elim
+ subgoal for x
+ by (cases "0::real" "g x" rule: linorder_le_cases) (auto simp: ennreal_neg)
+ done
+ qed
+qed
+
+lemma simple_function_completion: "simple_function M f \<Longrightarrow> simple_function (completion M) f"
+ by (simp add: simple_function_def)
+
+lemma simple_integral_completion:
+ "simple_function M f \<Longrightarrow> simple_integral (completion M) f = simple_integral M f"
+ unfolding simple_integral_def by simp
+
+lemma nn_integral_completion: "nn_integral (completion M) f = nn_integral M f"
+ unfolding nn_integral_def
+proof (safe intro!: SUP_eq)
+ fix s assume s: "simple_function (completion M) s" and "s \<le> f"
+ then obtain s' where s': "simple_function M s'" "AE x in M. s x = s' x"
+ by (auto dest: completion_ex_simple_function)
+ then obtain N where N: "N \<in> null_sets M" "{x\<in>space M. s x \<noteq> s' x} \<subseteq> N"
+ by (auto elim!: AE_E)
+ then have ae_N: "AE x in M. (s x \<noteq> s' x \<longrightarrow> x \<in> N) \<and> x \<notin> N"
+ by (auto dest: AE_not_in)
+ define s'' where "s'' x = (if x \<in> N then 0 else s x)" for x
+ then have ae_s_eq_s'': "AE x in completion M. s x = s'' x"
+ using s' ae_N by (intro AE_completion) auto
+ have s'': "simple_function M s''"
+ proof (subst simple_function_cong)
+ show "t \<in> space M \<Longrightarrow> s'' t = (if t \<in> N then 0 else s' t)" for t
+ using N by (auto simp: s''_def dest: sets.sets_into_space)
+ show "simple_function M (\<lambda>t. if t \<in> N then 0 else s' t)"
+ unfolding s''_def[abs_def] using N by (auto intro!: simple_function_If s')
+ qed
+
+ 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"
+ proof (safe intro!: bexI[of _ s''])
+ have "integral\<^sup>S (completion M) s = integral\<^sup>S (completion M) s''"
+ by (intro simple_integral_cong_AE s simple_function_completion s'' ae_s_eq_s'')
+ then show "integral\<^sup>S (completion M) s \<le> integral\<^sup>S M s''"
+ using s'' by (simp add: simple_integral_completion)
+ from \<open>s \<le> f\<close> show "s'' \<le> f"
+ unfolding s''_def le_fun_def by auto
+ qed fact
+next
+ fix s assume "simple_function M s" "s \<le> f"
+ 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"
+ by (intro bexI[of _ s]) (auto simp: simple_integral_completion simple_function_completion)
+qed
+
+locale semifinite_measure =
+ fixes M :: "'a measure"
+ assumes semifinite:
+ "\<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>"
+
+locale locally_determined_measure = semifinite_measure +
+ assumes locally_determined:
+ "\<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"
+
+locale cld_measure = complete_measure M + locally_determined_measure M for M :: "'a measure"
+
+definition outer_measure_of :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ennreal"
+ where "outer_measure_of M A = (INF B : {B\<in>sets M. A \<subseteq> B}. emeasure M B)"
+
+lemma outer_measure_of_eq[simp]: "A \<in> sets M \<Longrightarrow> outer_measure_of M A = emeasure M A"
+ by (auto simp: outer_measure_of_def intro!: INF_eqI emeasure_mono)
+
+lemma outer_measure_of_mono: "A \<subseteq> B \<Longrightarrow> outer_measure_of M A \<le> outer_measure_of M B"
+ unfolding outer_measure_of_def by (intro INF_superset_mono) auto
+
+lemma outer_measure_of_attain:
+ assumes "A \<subseteq> space M"
+ shows "\<exists>E\<in>sets M. A \<subseteq> E \<and> outer_measure_of M A = emeasure M E"
+proof -
+ have "emeasure M ` {B \<in> sets M. A \<subseteq> B} \<noteq> {}"
+ using \<open>A \<subseteq> space M\<close> by auto
+ from ennreal_Inf_countable_INF[OF this]
+ obtain f
+ where f: "range f \<subseteq> emeasure M ` {B \<in> sets M. A \<subseteq> B}" "decseq f"
+ and "outer_measure_of M A = (INF i. f i)"
+ unfolding outer_measure_of_def by auto
+ 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"
+ proof (rule dependent_nat_choice)
+ show "\<exists>x. x \<in> sets M \<and> A \<subseteq> x \<and> emeasure M x \<le> f 0"
+ using f(1) by (fastforce simp: image_subset_iff image_iff intro: eq_refl[OF sym])
+ next
+ fix E n assume "E \<in> sets M \<and> A \<subseteq> E \<and> emeasure M E \<le> f n"
+ moreover obtain F where "F \<in> sets M" "A \<subseteq> F" "f (Suc n) = emeasure M F"
+ using f(1) by (auto simp: image_subset_iff image_iff)
+ ultimately show "\<exists>y. (y \<in> sets M \<and> A \<subseteq> y \<and> emeasure M y \<le> f (Suc n)) \<and> y \<subseteq> E"
+ by (auto intro!: exI[of _ "F \<inter> E"] emeasure_mono)
+ qed
+ then obtain E
+ where [simp]: "\<And>n. E n \<in> sets M"
+ and "\<And>n. A \<subseteq> E n"
+ and le_f: "\<And>n. emeasure M (E n) \<le> f n"
+ and "decseq E"
+ by (auto simp: decseq_Suc_iff)
+ show ?thesis
+ proof cases
+ assume fin: "\<exists>i. emeasure M (E i) < \<infinity>"
+ show ?thesis
+ proof (intro bexI[of _ "\<Inter>i. E i"] conjI)
+ show "A \<subseteq> (\<Inter>i. E i)" "(\<Inter>i. E i) \<in> sets M"
+ using \<open>\<And>n. A \<subseteq> E n\<close> by auto
+
+ have " (INF i. emeasure M (E i)) \<le> outer_measure_of M A"
+ unfolding \<open>outer_measure_of M A = (INF n. f n)\<close>
+ by (intro INF_superset_mono le_f) auto
+ moreover have "outer_measure_of M A \<le> (INF i. outer_measure_of M (E i))"
+ by (intro INF_greatest outer_measure_of_mono \<open>\<And>n. A \<subseteq> E n\<close>)
+ ultimately have "outer_measure_of M A = (INF i. emeasure M (E i))"
+ by auto
+ also have "\<dots> = emeasure M (\<Inter>i. E i)"
+ using fin by (intro INF_emeasure_decseq' \<open>decseq E\<close>) (auto simp: less_top)
+ finally show "outer_measure_of M A = emeasure M (\<Inter>i. E i)" .
+ qed
+ next
+ assume "\<nexists>i. emeasure M (E i) < \<infinity>"
+ then have "f n = \<infinity>" for n
+ using le_f by (auto simp: not_less top_unique)
+ moreover have "\<exists>E\<in>sets M. A \<subseteq> E \<and> f 0 = emeasure M E"
+ using f by auto
+ ultimately show ?thesis
+ unfolding \<open>outer_measure_of M A = (INF n. f n)\<close> by simp
+ qed
+qed
+
+lemma SUP_outer_measure_of_incseq:
+ assumes A: "\<And>n. A n \<subseteq> space M" and "incseq A"
+ shows "(SUP n. outer_measure_of M (A n)) = outer_measure_of M (\<Union>i. A i)"
+proof (rule antisym)
+ obtain E
+ 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)"
+ using outer_measure_of_attain[OF A] by metis
+
+ define F where "F n = (\<Inter>i\<in>{n ..}. E i)" for n
+ with E have F: "incseq F" "\<And>n. F n \<in> sets M"
+ by (auto simp: incseq_def)
+ have "A n \<subseteq> F n" for n
+ using incseqD[OF \<open>incseq A\<close>, of n] \<open>\<And>n. A n \<subseteq> E n\<close> by (auto simp: F_def)
+
+ have eq: "outer_measure_of M (A n) = outer_measure_of M (F n)" for n
+ proof (intro antisym)
+ have "outer_measure_of M (F n) \<le> outer_measure_of M (E n)"
+ by (intro outer_measure_of_mono) (auto simp add: F_def)
+ with E show "outer_measure_of M (F n) \<le> outer_measure_of M (A n)"
+ by auto
+ show "outer_measure_of M (A n) \<le> outer_measure_of M (F n)"
+ by (intro outer_measure_of_mono \<open>A n \<subseteq> F n\<close>)
+ qed
+
+ have "outer_measure_of M (\<Union>n. A n) \<le> outer_measure_of M (\<Union>n. F n)"
+ using \<open>\<And>n. A n \<subseteq> F n\<close> by (intro outer_measure_of_mono) auto
+ also have "\<dots> = (SUP n. emeasure M (F n))"
+ using F by (simp add: SUP_emeasure_incseq subset_eq)
+ finally show "outer_measure_of M (\<Union>n. A n) \<le> (SUP n. outer_measure_of M (A n))"
+ by (simp add: eq F)
+qed (auto intro: SUP_least outer_measure_of_mono)
+
+definition measurable_envelope :: "'a measure \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"
+ where "measurable_envelope M A E \<longleftrightarrow>
+ (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)))"
+
+lemma measurable_envelopeD:
+ assumes "measurable_envelope M A E"
+ shows "A \<subseteq> E"
+ and "E \<in> sets M"
+ and "\<And>F. F \<in> sets M \<Longrightarrow> emeasure M (F \<inter> E) = outer_measure_of M (F \<inter> A)"
+ and "A \<subseteq> space M"
+ using assms sets.sets_into_space[of E] by (auto simp: measurable_envelope_def)
+
+lemma measurable_envelopeD1:
+ assumes E: "measurable_envelope M A E" and F: "F \<in> sets M" "F \<subseteq> E - A"
+ shows "emeasure M F = 0"
+proof -
+ have "emeasure M F = emeasure M (F \<inter> E)"
+ using F by (intro arg_cong2[where f=emeasure]) auto
+ also have "\<dots> = outer_measure_of M (F \<inter> A)"
+ using measurable_envelopeD[OF E] \<open>F \<in> sets M\<close> by (auto simp: measurable_envelope_def)
+ also have "\<dots> = outer_measure_of M {}"
+ using \<open>F \<subseteq> E - A\<close> by (intro arg_cong2[where f=outer_measure_of]) auto
+ finally show "emeasure M F = 0"
+ by simp
+qed
+
+lemma measurable_envelope_eq1:
+ assumes "A \<subseteq> E" "E \<in> sets M"
+ shows "measurable_envelope M A E \<longleftrightarrow> (\<forall>F\<in>sets M. F \<subseteq> E - A \<longrightarrow> emeasure M F = 0)"
+proof safe
+ assume *: "\<forall>F\<in>sets M. F \<subseteq> E - A \<longrightarrow> emeasure M F = 0"
+ show "measurable_envelope M A E"
+ unfolding measurable_envelope_def
+ proof (rule ccontr, auto simp add: \<open>E \<in> sets M\<close> \<open>A \<subseteq> E\<close>)
+ fix F assume "F \<in> sets M" "emeasure M (F \<inter> E) \<noteq> outer_measure_of M (F \<inter> A)"
+ then have "outer_measure_of M (F \<inter> A) < emeasure M (F \<inter> E)"
+ 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)
+ then obtain G where G: "G \<in> sets M" "F \<inter> A \<subseteq> G" and less: "emeasure M G < emeasure M (E \<inter> F)"
+ unfolding outer_measure_of_def INF_less_iff by (auto simp: ac_simps)
+ have le: "emeasure M (G \<inter> E \<inter> F) \<le> emeasure M G"
+ using \<open>E \<in> sets M\<close> \<open>G \<in> sets M\<close> \<open>F \<in> sets M\<close> by (auto intro!: emeasure_mono)
+
+ from G have "E \<inter> F - G \<in> sets M" "E \<inter> F - G \<subseteq> E - A"
+ using \<open>F \<in> sets M\<close> \<open>E \<in> sets M\<close> by auto
+ with * have "0 = emeasure M (E \<inter> F - G)"
+ by auto
+ also have "E \<inter> F - G = E \<inter> F - (G \<inter> E \<inter> F)"
+ by auto
+ also have "emeasure M (E \<inter> F - (G \<inter> E \<inter> F)) = emeasure M (E \<inter> F) - emeasure M (G \<inter> E \<inter> F)"
+ using \<open>E \<in> sets M\<close> \<open>F \<in> sets M\<close> le less G by (intro emeasure_Diff) (auto simp: top_unique)
+ also have "\<dots> > 0"
+ using le less by (intro diff_gr0_ennreal) auto
+ finally show False by auto
+ qed
+qed (rule measurable_envelopeD1)
+
+lemma measurable_envelopeD2:
+ assumes E: "measurable_envelope M A E" shows "emeasure M E = outer_measure_of M A"
+proof -
+ from \<open>measurable_envelope M A E\<close> have "emeasure M (E \<inter> E) = outer_measure_of M (E \<inter> A)"
+ by (auto simp: measurable_envelope_def)
+ with measurable_envelopeD[OF E] show "emeasure M E = outer_measure_of M A"
+ by (auto simp: Int_absorb1)
+qed
+
+lemma measurable_envelope_eq2:
+ assumes "A \<subseteq> E" "E \<in> sets M" "emeasure M E < \<infinity>"
+ shows "measurable_envelope M A E \<longleftrightarrow> (emeasure M E = outer_measure_of M A)"
+proof safe
+ assume *: "emeasure M E = outer_measure_of M A"
+ show "measurable_envelope M A E"
+ unfolding measurable_envelope_eq1[OF \<open>A \<subseteq> E\<close> \<open>E \<in> sets M\<close>]
+ proof (intro conjI ballI impI assms)
+ fix F assume F: "F \<in> sets M" "F \<subseteq> E - A"
+ with \<open>E \<in> sets M\<close> have le: "emeasure M F \<le> emeasure M E"
+ by (intro emeasure_mono) auto
+ from F \<open>A \<subseteq> E\<close> have "outer_measure_of M A \<le> outer_measure_of M (E - F)"
+ by (intro outer_measure_of_mono) auto
+ then have "emeasure M E - 0 \<le> emeasure M (E - F)"
+ using * \<open>E \<in> sets M\<close> \<open>F \<in> sets M\<close> by simp
+ also have "\<dots> = emeasure M E - emeasure M F"
+ using \<open>E \<in> sets M\<close> \<open>emeasure M E < \<infinity>\<close> F le by (intro emeasure_Diff) (auto simp: top_unique)
+ finally show "emeasure M F = 0"
+ using ennreal_mono_minus_cancel[of "emeasure M E" 0 "emeasure M F"] le assms by auto
+ qed
+qed (auto intro: measurable_envelopeD2)
+
+lemma measurable_envelopeI_countable:
+ fixes A :: "nat \<Rightarrow> 'a set"
+ assumes E: "\<And>n. measurable_envelope M (A n) (E n)"
+ shows "measurable_envelope M (\<Union>n. A n) (\<Union>n. E n)"
+proof (subst measurable_envelope_eq1)
+ show "(\<Union>n. A n) \<subseteq> (\<Union>n. E n)" "(\<Union>n. E n) \<in> sets M"
+ using measurable_envelopeD(1,2)[OF E] by auto
+ show "\<forall>F\<in>sets M. F \<subseteq> (\<Union>n. E n) - (\<Union>n. A n) \<longrightarrow> emeasure M F = 0"
+ proof safe
+ fix F assume F: "F \<in> sets M" "F \<subseteq> (\<Union>n. E n) - (\<Union>n. A n)"
+ 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
+ using measurable_envelopeD(1,2)[OF E] by auto
+ then have "emeasure M (\<Union>n. F \<inter> E n) = 0"
+ by (intro emeasure_UN_eq_0 measurable_envelopeD1[OF E]) auto
+ then show "emeasure M F = 0"
+ using \<open>F \<subseteq> (\<Union>n. E n)\<close> by (auto simp: Int_absorb2)
+ qed
+qed
+
+lemma measurable_envelopeI_countable_cover:
+ fixes A and C :: "nat \<Rightarrow> 'a set"
+ assumes C: "A \<subseteq> (\<Union>n. C n)" "\<And>n. C n \<in> sets M" "\<And>n. emeasure M (C n) < \<infinity>"
+ shows "\<exists>E\<subseteq>(\<Union>n. C n). measurable_envelope M A E"
+proof -
+ have "A \<inter> C n \<subseteq> space M" for n
+ using \<open>C n \<in> sets M\<close> by (auto dest: sets.sets_into_space)
+ 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"
+ using outer_measure_of_attain[of "A \<inter> C n" M for n] by auto
+ then obtain E
+ where E: "\<And>n. E n \<in> sets M" "\<And>n. A \<inter> C n \<subseteq> E n"
+ and eq: "\<And>n. outer_measure_of M (A \<inter> C n) = emeasure M (E n)"
+ by metis
+
+ have "outer_measure_of M (A \<inter> C n) \<le> outer_measure_of M (E n \<inter> C n)" for n
+ using E by (intro outer_measure_of_mono) auto
+ moreover have "outer_measure_of M (E n \<inter> C n) \<le> outer_measure_of M (E n)" for n
+ by (intro outer_measure_of_mono) auto
+ ultimately have eq: "outer_measure_of M (A \<inter> C n) = emeasure M (E n \<inter> C n)" for n
+ using E C by (intro antisym) (auto simp: eq)
+
+ { fix n
+ have "outer_measure_of M (A \<inter> C n) \<le> outer_measure_of M (C n)"
+ by (intro outer_measure_of_mono) simp
+ also have "\<dots> < \<infinity>"
+ using assms by auto
+ finally have "emeasure M (E n \<inter> C n) < \<infinity>"
+ using eq by simp }
+ then have "measurable_envelope M (\<Union>n. A \<inter> C n) (\<Union>n. E n \<inter> C n)"
+ using E C by (intro measurable_envelopeI_countable measurable_envelope_eq2[THEN iffD2]) (auto simp: eq)
+ with \<open>A \<subseteq> (\<Union>n. C n)\<close> show ?thesis
+ by (intro exI[of _ "(\<Union>n. E n \<inter> C n)"]) (auto simp add: Int_absorb2)
+qed
+
+lemma (in complete_measure) complete_sets_sandwich:
+ assumes [measurable]: "A \<in> sets M" "C \<in> sets M" and subset: "A \<subseteq> B" "B \<subseteq> C"
+ and measure: "emeasure M A = emeasure M C" "emeasure M A < \<infinity>"
+ shows "B \<in> sets M"
+proof -
+ have "B - A \<in> sets M"
+ proof (rule complete)
+ show "B - A \<subseteq> C - A"
+ using subset by auto
+ show "C - A \<in> null_sets M"
+ using measure subset by(simp add: emeasure_Diff null_setsI)
+ qed
+ then have "A \<union> (B - A) \<in> sets M"
+ by measurable
+ also have "A \<union> (B - A) = B"
+ using \<open>A \<subseteq> B\<close> by auto
+ finally show ?thesis .
+qed
+
+lemma (in cld_measure) notin_sets_outer_measure_of_cover:
+ assumes E: "E \<subseteq> space M" "E \<notin> sets M"
+ shows "\<exists>B\<in>sets M. 0 < emeasure M B \<and> emeasure M B < \<infinity> \<and>
+ outer_measure_of M (B \<inter> E) = emeasure M B \<and> outer_measure_of M (B - E) = emeasure M B"
+proof -
+ from locally_determined[OF \<open>E \<subseteq> space M\<close>] \<open>E \<notin> sets M\<close>
+ obtain F
+ where [measurable]: "F \<in> sets M" and "emeasure M F < \<infinity>" "E \<inter> F \<notin> sets M"
+ by blast
+ then obtain H H'
+ where H: "measurable_envelope M (F \<inter> E) H" and H': "measurable_envelope M (F - E) H'"
+ using measurable_envelopeI_countable_cover[of "F \<inter> E" "\<lambda>_. F" M]
+ measurable_envelopeI_countable_cover[of "F - E" "\<lambda>_. F" M]
+ by auto
+ note measurable_envelopeD(2)[OF H', measurable] measurable_envelopeD(2)[OF H, measurable]
+
+ from measurable_envelopeD(1)[OF H'] measurable_envelopeD(1)[OF H]
+ have subset: "F - H' \<subseteq> F \<inter> E" "F \<inter> E \<subseteq> F \<inter> H"
+ by auto
+ moreover define G where "G = (F \<inter> H) - (F - H')"
+ ultimately have G: "G = F \<inter> H \<inter> H'"
+ by auto
+ have "emeasure M (F \<inter> H) \<noteq> 0"
+ proof
+ assume "emeasure M (F \<inter> H) = 0"
+ then have "F \<inter> H \<in> null_sets M"
+ by auto
+ with \<open>E \<inter> F \<notin> sets M\<close> show False
+ using complete[OF \<open>F \<inter> E \<subseteq> F \<inter> H\<close>] by (auto simp: Int_commute)
+ qed
+ moreover
+ have "emeasure M (F - H') \<noteq> emeasure M (F \<inter> H)"
+ proof
+ assume "emeasure M (F - H') = emeasure M (F \<inter> H)"
+ with \<open>E \<inter> F \<notin> sets M\<close> emeasure_mono[of "F \<inter> H" F M] \<open>emeasure M F < \<infinity>\<close>
+ have "F \<inter> E \<in> sets M"
+ by (intro complete_sets_sandwich[OF _ _ subset]) auto
+ with \<open>E \<inter> F \<notin> sets M\<close> show False
+ by (simp add: Int_commute)
+ qed
+ moreover have "emeasure M (F - H') \<le> emeasure M (F \<inter> H)"
+ using subset by (intro emeasure_mono) auto
+ ultimately have "emeasure M G \<noteq> 0"
+ unfolding G_def using subset
+ by (subst emeasure_Diff) (auto simp: top_unique diff_eq_0_iff_ennreal)
+ show ?thesis
+ proof (intro bexI conjI)
+ have "emeasure M G \<le> emeasure M F"
+ unfolding G by (auto intro!: emeasure_mono)
+ with \<open>emeasure M F < \<infinity>\<close> show "0 < emeasure M G" "emeasure M G < \<infinity>"
+ using \<open>emeasure M G \<noteq> 0\<close> by (auto simp: zero_less_iff_neq_zero)
+ show [measurable]: "G \<in> sets M"
+ unfolding G by auto
+
+ have "emeasure M G = outer_measure_of M (F \<inter> H' \<inter> (F \<inter> E))"
+ using measurable_envelopeD(3)[OF H, of "F \<inter> H'"] unfolding G by (simp add: ac_simps)
+ also have "\<dots> \<le> outer_measure_of M (G \<inter> E)"
+ using measurable_envelopeD(1)[OF H] by (intro outer_measure_of_mono) (auto simp: G)
+ finally show "outer_measure_of M (G \<inter> E) = emeasure M G"
+ using outer_measure_of_mono[of "G \<inter> E" G M] by auto
+
+ have "emeasure M G = outer_measure_of M (F \<inter> H \<inter> (F - E))"
+ using measurable_envelopeD(3)[OF H', of "F \<inter> H"] unfolding G by (simp add: ac_simps)
+ also have "\<dots> \<le> outer_measure_of M (G - E)"
+ using measurable_envelopeD(1)[OF H'] by (intro outer_measure_of_mono) (auto simp: G)
+ finally show "outer_measure_of M (G - E) = emeasure M G"
+ using outer_measure_of_mono[of "G - E" G M] by auto
+ qed
+qed
+
+text \<open>The following theorem is a specialization of D.H. Fremlin, Measure Theory vol 4I (413G). We
+ only show one direction and do not use a inner regular family $K$.\<close>
+
+lemma (in cld_measure) borel_measurable_cld:
+ fixes f :: "'a \<Rightarrow> real"
+ assumes "\<And>A a b. A \<in> sets M \<Longrightarrow> 0 < emeasure M A \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> a < b \<Longrightarrow>
+ 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"
+ shows "f \<in> M \<rightarrow>\<^sub>M borel"
+proof (rule ccontr)
+ let ?E = "\<lambda>a. {x\<in>space M. f x \<le> a}" and ?F = "\<lambda>a. {x\<in>space M. a \<le> f x}"
+
+ assume "f \<notin> M \<rightarrow>\<^sub>M borel"
+ then obtain a where "?E a \<notin> sets M"
+ unfolding borel_measurable_iff_le by blast
+ from notin_sets_outer_measure_of_cover[OF _ this]
+ obtain K
+ where K: "K \<in> sets M" "0 < emeasure M K" "emeasure M K < \<infinity>"
+ and eq1: "outer_measure_of M (K \<inter> ?E a) = emeasure M K"
+ and eq2: "outer_measure_of M (K - ?E a) = emeasure M K"
+ by auto
+ then have me_K: "measurable_envelope M (K \<inter> ?E a) K"
+ by (subst measurable_envelope_eq2) auto
+
+ define b where "b n = a + inverse (real (Suc n))" for n
+ have "(SUP n. outer_measure_of M (K \<inter> ?F (b n))) = outer_measure_of M (\<Union>n. K \<inter> ?F (b n))"
+ proof (intro SUP_outer_measure_of_incseq)
+ have "x \<le> y \<Longrightarrow> b y \<le> b x" for x y
+ by (auto simp: b_def field_simps)
+ then show "incseq (\<lambda>n. K \<inter> {x \<in> space M. b n \<le> f x})"
+ by (auto simp: incseq_def intro: order_trans)
+ qed auto
+ also have "(\<Union>n. K \<inter> ?F (b n)) = K - ?E a"
+ proof -
+ have "b \<longlonglongrightarrow> a"
+ unfolding b_def by (rule LIMSEQ_inverse_real_of_nat_add)
+ then have "\<forall>n. \<not> b n \<le> f x \<Longrightarrow> f x \<le> a" for x
+ by (rule LIMSEQ_le_const) (auto intro: less_imp_le simp: not_le)
+ moreover have "\<not> b n \<le> a" for n
+ by (auto simp: b_def)
+ ultimately show ?thesis
+ using \<open>K \<in> sets M\<close>[THEN sets.sets_into_space] by (auto simp: subset_eq intro: order_trans)
+ qed
+ finally have "0 < (SUP n. outer_measure_of M (K \<inter> ?F (b n)))"
+ using K by (simp add: eq2)
+ then obtain n where pos_b: "0 < outer_measure_of M (K \<inter> ?F (b n))" and "a < b n"
+ unfolding less_SUP_iff by (auto simp: b_def)
+ from measurable_envelopeI_countable_cover[of "K \<inter> ?F (b n)" "\<lambda>_. K" M] K
+ obtain K' where "K' \<subseteq> K" and me_K': "measurable_envelope M (K \<inter> ?F (b n)) K'"
+ by auto
+ then have K'_le_K: "emeasure M K' \<le> emeasure M K"
+ by (intro emeasure_mono K)
+ have "K' \<in> sets M"
+ using me_K' by (rule measurable_envelopeD)
+
+ 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'"
+ proof (rule assms)
+ show "0 < emeasure M K'" "emeasure M K' < \<infinity>"
+ using measurable_envelopeD2[OF me_K'] pos_b K K'_le_K by auto
+ qed fact+
+ also have "{x\<in>K'. f x \<le> a} = K' \<inter> (K \<inter> ?E a)"
+ using \<open>K' \<in> sets M\<close>[THEN sets.sets_into_space] \<open>K' \<subseteq> K\<close> by auto
+ also have "{x\<in>K'. b n \<le> f x} = K' \<inter> (K \<inter> ?F (b n))"
+ using \<open>K' \<in> sets M\<close>[THEN sets.sets_into_space] \<open>K' \<subseteq> K\<close> by auto
+ finally have "min (emeasure M K) (emeasure M K') < emeasure M K'"
+ unfolding
+ measurable_envelopeD(3)[OF me_K \<open>K' \<in> sets M\<close>, symmetric]
+ measurable_envelopeD(3)[OF me_K' \<open>K' \<in> sets M\<close>, symmetric]
+ using \<open>K' \<subseteq> K\<close> by (simp add: Int_absorb1 Int_absorb2)
+ with K'_le_K show False
+ by (auto simp: min_def split: if_split_asm)
+qed
+
end