| author | wenzelm |
| Fri, 05 Aug 2022 13:23:52 +0200 | |
| changeset 75760 | f8be63d2ec6f |
| parent 75607 | 3c544d64c218 |
| child 78258 | 71366be2c647 |
| permissions | -rw-r--r-- |
(* Title: HOL/Analysis/Equivalence_Measurable_On_Borel Author: LC Paulson (some material ported from HOL Light) *) section\<open>Equivalence Between Classical Borel Measurability and HOL Light's\<close> theory Equivalence_Measurable_On_Borel imports Equivalence_Lebesgue_Henstock_Integration Improper_Integral Continuous_Extension begin (*Borrowed from Ergodic_Theory/SG_Library_Complement*) abbreviation sym_diff :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" where "sym_diff A B \<equiv> ((A - B) \<union> (B-A))" subsection\<open>Austin's Lemma\<close> lemma Austin_Lemma: fixes \<D> :: "'a::euclidean_space set set" assumes "finite \<D>" and \<D>: "\<And>D. D \<in> \<D> \<Longrightarrow> \<exists>k a b. D = cbox a b \<and> (\<forall>i \<in> Basis. b\<bullet>i - a\<bullet>i = k)" obtains \<C> where "\<C> \<subseteq> \<D>" "pairwise disjnt \<C>" "measure lebesgue (\<Union>\<C>) \<ge> measure lebesgue (\<Union>\<D>) / 3 ^ (DIM('a))" using assms proof (induction "card \<D>" arbitrary: \<D> thesis rule: less_induct) case less show ?case proof (cases "\<D> = {}") case True then show thesis using less by auto next case False then have "Max (Sigma_Algebra.measure lebesgue ` \<D>) \<in> Sigma_Algebra.measure lebesgue ` \<D>" using Max_in finite_imageI \<open>finite \<D>\<close> by blast then obtain D where "D \<in> \<D>" and "measure lebesgue D = Max (measure lebesgue ` \<D>)" by auto then have D: "\<And>C. C \<in> \<D> \<Longrightarrow> measure lebesgue C \<le> measure lebesgue D" by (simp add: \<open>finite \<D>\<close>) let ?\<E> = "{C. C \<in> \<D> - {D} \<and> disjnt C D}" obtain \<D>' where \<D>'sub: "\<D>' \<subseteq> ?\<E>" and \<D>'dis: "pairwise disjnt \<D>'" and \<D>'m: "measure lebesgue (\<Union>\<D>') \<ge> measure lebesgue (\<Union>?\<E>) / 3 ^ (DIM('a))" proof (rule less.hyps) have *: "?\<E> \<subset> \<D>" using \<open>D \<in> \<D>\<close> by auto then show "card ?\<E> < card \<D>" "finite ?\<E>" by (auto simp: \<open>finite \<D>\<close> psubset_card_mono) show "\<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)" if "D \<in> ?\<E>" for D using less.prems(3) that by auto qed then have [simp]: "\<Union>\<D>' - D = \<Union>\<D>'" by (auto simp: disjnt_iff) show ?thesis proof (rule less.prems) show "insert D \<D>' \<subseteq> \<D>" using \<D>'sub \<open>D \<in> \<D>\<close> by blast show "disjoint (insert D \<D>')" using \<D>'dis \<D>'sub by (fastforce simp add: pairwise_def disjnt_sym) obtain a3 b3 where m3: "content (cbox a3 b3) = 3 ^ DIM('a) * measure lebesgue D" and sub3: "\<And>C. \<lbrakk>C \<in> \<D>; \<not> disjnt C D\<rbrakk> \<Longrightarrow> C \<subseteq> cbox a3 b3" proof - obtain k a b where ab: "D = cbox a b" and k: "\<And>i. i \<in> Basis \<Longrightarrow> b\<bullet>i - a\<bullet>i = k" using less.prems \<open>D \<in> \<D>\<close> by meson then have eqk: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i \<longleftrightarrow> k \<ge> 0" by force show thesis proof let ?a = "(a + b) /\<^sub>R 2 - (3/2) *\<^sub>R (b - a)" let ?b = "(a + b) /\<^sub>R 2 + (3/2) *\<^sub>R (b - a)" have eq: "(\<Prod>i\<in>Basis. b \<bullet> i * 3 - a \<bullet> i * 3) = (\<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i) * 3 ^ DIM('a)" by (simp add: comm_monoid_mult_class.prod.distrib flip: left_diff_distrib inner_diff_left) show "content (cbox ?a ?b) = 3 ^ DIM('a) * measure lebesgue D" by (simp add: content_cbox_if box_eq_empty algebra_simps eq ab k) show "C \<subseteq> cbox ?a ?b" if "C \<in> \<D>" and CD: "\<not> disjnt C D" for C proof - obtain k' a' b' where ab': "C = cbox a' b'" and k': "\<And>i. i \<in> Basis \<Longrightarrow> b'\<bullet>i - a'\<bullet>i = k'" using less.prems \<open>C \<in> \<D>\<close> by meson then have eqk': "\<And>i. i \<in> Basis \<Longrightarrow> a' \<bullet> i \<le> b' \<bullet> i \<longleftrightarrow> k' \<ge> 0" by force show ?thesis proof (clarsimp simp add: disjoint_interval disjnt_def ab ab' not_less subset_box algebra_simps) show "a \<bullet> i * 2 \<le> a' \<bullet> i + b \<bullet> i \<and> a \<bullet> i + b' \<bullet> i \<le> b \<bullet> i * 2" if * [rule_format]: "\<forall>j\<in>Basis. a' \<bullet> j \<le> b' \<bullet> j" and "i \<in> Basis" for i proof - have "a' \<bullet> i \<le> b' \<bullet> i \<and> a \<bullet> i \<le> b \<bullet> i \<and> a \<bullet> i \<le> b' \<bullet> i \<and> a' \<bullet> i \<le> b \<bullet> i" using \<open>i \<in> Basis\<close> CD by (simp_all add: disjoint_interval disjnt_def ab ab' not_less) then show ?thesis using D [OF \<open>C \<in> \<D>\<close>] \<open>i \<in> Basis\<close> apply (simp add: ab ab' k k' eqk eqk' content_cbox_cases) using k k' by fastforce qed qed qed qed qed have \<D>lm: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<in> lmeasurable" using less.prems(3) by blast have "measure lebesgue (\<Union>\<D>) \<le> measure lebesgue (cbox a3 b3 \<union> (\<Union>\<D> - cbox a3 b3))" proof (rule measure_mono_fmeasurable) show "\<Union>\<D> \<in> sets lebesgue" using \<D>lm \<open>finite \<D>\<close> by blast show "cbox a3 b3 \<union> (\<Union>\<D> - cbox a3 b3) \<in> lmeasurable" by (simp add: \<D>lm fmeasurable.Un fmeasurable.finite_Union less.prems(2) subset_eq) qed auto also have "\<dots> = content (cbox a3 b3) + measure lebesgue (\<Union>\<D> - cbox a3 b3)" by (simp add: \<D>lm fmeasurable.finite_Union less.prems(2) measure_Un2 subsetI) also have "\<dots> \<le> (measure lebesgue D + measure lebesgue (\<Union>\<D>')) * 3 ^ DIM('a)" proof - have "(\<Union>\<D> - cbox a3 b3) \<subseteq> \<Union>?\<E>" using sub3 by fastforce then have "measure lebesgue (\<Union>\<D> - cbox a3 b3) \<le> measure lebesgue (\<Union>?\<E>)" proof (rule measure_mono_fmeasurable) show "\<Union> \<D> - cbox a3 b3 \<in> sets lebesgue" by (simp add: \<D>lm fmeasurableD less.prems(2) sets.Diff sets.finite_Union subsetI) show "\<Union> {C \<in> \<D> - {D}. disjnt C D} \<in> lmeasurable" using \<D>lm less.prems(2) by auto qed then have "measure lebesgue (\<Union>\<D> - cbox a3 b3) / 3 ^ DIM('a) \<le> measure lebesgue (\<Union> \<D>')" using \<D>'m by (simp add: field_split_simps) then show ?thesis by (simp add: m3 field_simps) qed also have "\<dots> \<le> measure lebesgue (\<Union>(insert D \<D>')) * 3 ^ DIM('a)" proof (simp add: \<D>lm \<open>D \<in> \<D>\<close>) show "measure lebesgue D + measure lebesgue (\<Union>\<D>') \<le> measure lebesgue (D \<union> \<Union> \<D>')" proof (subst measure_Un2) show "\<Union> \<D>' \<in> lmeasurable" by (meson \<D>lm \<open>insert D \<D>' \<subseteq> \<D>\<close> fmeasurable.finite_Union less.prems(2) finite_subset subset_eq subset_insertI) show "measure lebesgue D + measure lebesgue (\<Union> \<D>') \<le> measure lebesgue D + measure lebesgue (\<Union> \<D>' - D)" using \<open>insert D \<D>' \<subseteq> \<D>\<close> infinite_super less.prems(2) by force qed (simp add: \<D>lm \<open>D \<in> \<D>\<close>) qed finally show "measure lebesgue (\<Union>\<D>) / 3 ^ DIM('a) \<le> measure lebesgue (\<Union>(insert D \<D>'))" by (simp add: field_split_simps) qed qed qed subsection\<open>A differentiability-like property of the indefinite integral. \<close> proposition integrable_ccontinuous_explicit: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "\<And>a b::'a. f integrable_on cbox a b" obtains N where "negligible N" "\<And>x e. \<lbrakk>x \<notin> N; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow> norm(integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e" proof - define BOX where "BOX \<equiv> \<lambda>h. \<lambda>x::'a. cbox x (x + h *\<^sub>R One)" define BOX2 where "BOX2 \<equiv> \<lambda>h. \<lambda>x::'a. cbox (x - h *\<^sub>R One) (x + h *\<^sub>R One)" define i where "i \<equiv> \<lambda>h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)" define \<Psi> where "\<Psi> \<equiv> \<lambda>x r. \<forall>d>0. \<exists>h. 0 < h \<and> h < d \<and> r \<le> norm(i h x - f x)" let ?N = "{x. \<exists>e>0. \<Psi> x e}" have "\<exists>N. negligible N \<and> (\<forall>x e. x \<notin> N \<and> 0 < e \<longrightarrow> \<not> \<Psi> x e)" proof (rule exI ; intro conjI allI impI) let ?M = "\<Union>n. {x. \<Psi> x (inverse(real n + 1))}" have "negligible ({x. \<Psi> x \<mu>} \<inter> cbox a b)" if "\<mu> > 0" for a b \<mu> proof (cases "negligible(cbox a b)") case True then show ?thesis by (simp add: negligible_Int) next case False then have "box a b \<noteq> {}" by (simp add: negligible_interval) then have ab: "\<And>i. i \<in> Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i" by (simp add: box_ne_empty) show ?thesis unfolding negligible_outer_le proof (intro allI impI) fix e::real let ?ee = "(e * \<mu>) / 2 / 6 ^ (DIM('a))" assume "e > 0" then have gt0: "?ee > 0" using \<open>\<mu> > 0\<close> by auto have f': "f integrable_on cbox (a - One) (b + One)" using assms by blast obtain \<gamma> where "gauge \<gamma>" and \<gamma>: "\<And>p. \<lbrakk>p tagged_partial_division_of (cbox (a - One) (b + One)); \<gamma> fine p\<rbrakk> \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < ?ee" using Henstock_lemma [OF f' gt0] that by auto let ?E = "{x. x \<in> cbox a b \<and> \<Psi> x \<mu>}" have "\<exists>h>0. BOX h x \<subseteq> \<gamma> x \<and> BOX h x \<subseteq> cbox (a - One) (b + One) \<and> \<mu> \<le> norm (i h x - f x)" if "x \<in> cbox a b" "\<Psi> x \<mu>" for x proof - obtain d where "d > 0" and d: "ball x d \<subseteq> \<gamma> x" using gaugeD [OF \<open>gauge \<gamma>\<close>, of x] openE by blast then obtain h where "0 < h" "h < 1" and hless: "h < d / real DIM('a)" and mule: "\<mu> \<le> norm (i h x - f x)" using \<open>\<Psi> x \<mu>\<close> [unfolded \<Psi>_def, rule_format, of "min 1 (d / DIM('a))"] by auto show ?thesis proof (intro exI conjI) show "0 < h" "\<mu> \<le> norm (i h x - f x)" by fact+ have "BOX h x \<subseteq> ball x d" proof (clarsimp simp: BOX_def mem_box dist_norm algebra_simps) fix y assume "\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> h + x \<bullet> i" then have lt: "\<bar>(x - y) \<bullet> i\<bar> < d / real DIM('a)" if "i \<in> Basis" for i using hless that by (force simp: inner_diff_left) have "norm (x - y) \<le> (\<Sum>i\<in>Basis. \<bar>(x - y) \<bullet> i\<bar>)" using norm_le_l1 by blast also have "\<dots> < d" using sum_bounded_above_strict [of Basis "\<lambda>i. \<bar>(x - y) \<bullet> i\<bar>" "d / DIM('a)", OF lt] by auto finally show "norm (x - y) < d" . qed with d show "BOX h x \<subseteq> \<gamma> x" by blast show "BOX h x \<subseteq> cbox (a - One) (b + One)" using that \<open>h < 1\<close> by (force simp: BOX_def mem_box algebra_simps intro: subset_box_imp) qed qed then obtain \<eta> where h0: "\<And>x. x \<in> ?E \<Longrightarrow> \<eta> x > 0" and BOX_\<gamma>: "\<And>x. x \<in> ?E \<Longrightarrow> BOX (\<eta> x) x \<subseteq> \<gamma> x" and "\<And>x. x \<in> ?E \<Longrightarrow> BOX (\<eta> x) x \<subseteq> cbox (a - One) (b + One) \<and> \<mu> \<le> norm (i (\<eta> x) x - f x)" by simp metis then have BOX_cbox: "\<And>x. x \<in> ?E \<Longrightarrow> BOX (\<eta> x) x \<subseteq> cbox (a - One) (b + One)" and \<mu>_le: "\<And>x. x \<in> ?E \<Longrightarrow> \<mu> \<le> norm (i (\<eta> x) x - f x)" by blast+ define \<gamma>' where "\<gamma>' \<equiv> \<lambda>x. if x \<in> cbox a b \<and> \<Psi> x \<mu> then ball x (\<eta> x) else \<gamma> x" have "gauge \<gamma>'" using \<open>gauge \<gamma>\<close> by (auto simp: h0 gauge_def \<gamma>'_def) obtain \<D> where "countable \<D>" and \<D>: "\<Union>\<D> \<subseteq> cbox a b" "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" and Dcovered: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x. x \<in> cbox a b \<and> \<Psi> x \<mu> \<and> x \<in> K \<and> K \<subseteq> \<gamma>' x" and subUD: "?E \<subseteq> \<Union>\<D>" by (rule covering_lemma [of ?E a b \<gamma>']) (simp_all add: Bex_def \<open>box a b \<noteq> {}\<close> \<open>gauge \<gamma>'\<close>) then have "\<D> \<subseteq> sets lebesgue" by fastforce show "\<exists>T. {x. \<Psi> x \<mu>} \<inter> cbox a b \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e" proof (intro exI conjI) show "{x. \<Psi> x \<mu>} \<inter> cbox a b \<subseteq> \<Union>\<D>" apply auto using subUD by auto have mUE: "measure lebesgue (\<Union> \<E>) \<le> measure lebesgue (cbox a b)" if "\<E> \<subseteq> \<D>" "finite \<E>" for \<E> proof (rule measure_mono_fmeasurable) show "\<Union> \<E> \<subseteq> cbox a b" using \<D>(1) that(1) by blast show "\<Union> \<E> \<in> sets lebesgue" by (metis \<D>(2) fmeasurable.finite_Union fmeasurableD lmeasurable_cbox subset_eq that) qed auto then show "\<Union>\<D> \<in> lmeasurable" by (metis \<D>(2) \<open>countable \<D>\<close> fmeasurable_Union_bound lmeasurable_cbox) then have leab: "measure lebesgue (\<Union>\<D>) \<le> measure lebesgue (cbox a b)" by (meson \<D>(1) fmeasurableD lmeasurable_cbox measure_mono_fmeasurable) obtain \<F> where "\<F> \<subseteq> \<D>" "finite \<F>" and \<F>: "measure lebesgue (\<Union>\<D>) \<le> 2 * measure lebesgue (\<Union>\<F>)" proof (cases "measure lebesgue (\<Union>\<D>) = 0") case True then show ?thesis by (force intro: that [where \<F> = "{}"]) next case False obtain \<F> where "\<F> \<subseteq> \<D>" "finite \<F>" and \<F>: "measure lebesgue (\<Union>\<D>)/2 < measure lebesgue (\<Union>\<F>)" proof (rule measure_countable_Union_approachable [of \<D> "measure lebesgue (\<Union>\<D>) / 2" "content (cbox a b)"]) show "countable \<D>" by fact show "0 < measure lebesgue (\<Union> \<D>) / 2" using False by (simp add: zero_less_measure_iff) show Dlm: "D \<in> lmeasurable" if "D \<in> \<D>" for D using \<D>(2) that by blast show "measure lebesgue (\<Union> \<F>) \<le> content (cbox a b)" if "\<F> \<subseteq> \<D>" "finite \<F>" for \<F> proof - have "measure lebesgue (\<Union> \<F>) \<le> measure lebesgue (\<Union>\<D>)" proof (rule measure_mono_fmeasurable) show "\<Union> \<F> \<subseteq> \<Union> \<D>" by (simp add: Sup_subset_mono \<open>\<F> \<subseteq> \<D>\<close>) show "\<Union> \<F> \<in> sets lebesgue" by (meson Dlm fmeasurableD sets.finite_Union subset_eq that) show "\<Union> \<D> \<in> lmeasurable" by fact qed also have "\<dots> \<le> measure lebesgue (cbox a b)" proof (rule measure_mono_fmeasurable) show "\<Union> \<D> \<in> sets lebesgue" by (simp add: \<open>\<Union> \<D> \<in> lmeasurable\<close> fmeasurableD) qed (auto simp:\<D>(1)) finally show ?thesis by simp qed qed auto then show ?thesis using that by auto qed obtain tag where tag_in_E: "\<And>D. D \<in> \<D> \<Longrightarrow> tag D \<in> ?E" and tag_in_self: "\<And>D. D \<in> \<D> \<Longrightarrow> tag D \<in> D" and tag_sub: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<subseteq> \<gamma>' (tag D)" using Dcovered by simp metis then have sub_ball_tag: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<subseteq> ball (tag D) (\<eta> (tag D))" by (simp add: \<gamma>'_def) define \<Phi> where "\<Phi> \<equiv> \<lambda>D. BOX (\<eta>(tag D)) (tag D)" define \<Phi>2 where "\<Phi>2 \<equiv> \<lambda>D. BOX2 (\<eta>(tag D)) (tag D)" obtain \<C> where "\<C> \<subseteq> \<Phi>2 ` \<F>" "pairwise disjnt \<C>" "measure lebesgue (\<Union>\<C>) \<ge> measure lebesgue (\<Union>(\<Phi>2`\<F>)) / 3 ^ (DIM('a))" proof (rule Austin_Lemma) show "finite (\<Phi>2`\<F>)" using \<open>finite \<F>\<close> by blast have "\<exists>k a b. \<Phi>2 D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)" if "D \<in> \<F>" for D apply (rule_tac x="2 * \<eta>(tag D)" in exI) apply (rule_tac x="tag D - \<eta>(tag D) *\<^sub>R One" in exI) apply (rule_tac x="tag D + \<eta>(tag D) *\<^sub>R One" in exI) using that apply (auto simp: \<Phi>2_def BOX2_def algebra_simps) done then show "\<And>D. D \<in> \<Phi>2 ` \<F> \<Longrightarrow> \<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)" by blast qed auto then obtain \<G> where "\<G> \<subseteq> \<F>" and disj: "pairwise disjnt (\<Phi>2 ` \<G>)" and "measure lebesgue (\<Union>(\<Phi>2 ` \<G>)) \<ge> measure lebesgue (\<Union>(\<Phi>2`\<F>)) / 3 ^ (DIM('a))" unfolding \<Phi>2_def subset_image_iff by (meson empty_subsetI equals0D pairwise_imageI) moreover have "measure lebesgue (\<Union>(\<Phi>2 ` \<G>)) * 3 ^ DIM('a) \<le> e/2" proof - have "finite \<G>" using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> infinite_super by blast have BOX2_m: "\<And>x. x \<in> tag ` \<G> \<Longrightarrow> BOX2 (\<eta> x) x \<in> lmeasurable" by (auto simp: BOX2_def) have BOX_m: "\<And>x. x \<in> tag ` \<G> \<Longrightarrow> BOX (\<eta> x) x \<in> lmeasurable" by (auto simp: BOX_def) have BOX_sub: "BOX (\<eta> x) x \<subseteq> BOX2 (\<eta> x) x" for x by (auto simp: BOX_def BOX2_def subset_box algebra_simps) have DISJ2: "BOX2 (\<eta> (tag X)) (tag X) \<inter> BOX2 (\<eta> (tag Y)) (tag Y) = {}" if "X \<in> \<G>" "Y \<in> \<G>" "tag X \<noteq> tag Y" for X Y proof - obtain i where i: "i \<in> Basis" "tag X \<bullet> i \<noteq> tag Y \<bullet> i" using \<open>tag X \<noteq> tag Y\<close> by (auto simp: euclidean_eq_iff [of "tag X"]) have XY: "X \<in> \<D>" "Y \<in> \<D>" using \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> that by auto then have "0 \<le> \<eta> (tag X)" "0 \<le> \<eta> (tag Y)" by (meson h0 le_cases not_le tag_in_E)+ with XY i have "BOX2 (\<eta> (tag X)) (tag X) \<noteq> BOX2 (\<eta> (tag Y)) (tag Y)" unfolding eq_iff by (fastforce simp add: BOX2_def subset_box algebra_simps) then show ?thesis using disj that by (auto simp: pairwise_def disjnt_def \<Phi>2_def) qed then have BOX2_disj: "pairwise (\<lambda>x y. negligible (BOX2 (\<eta> x) x \<inter> BOX2 (\<eta> y) y)) (tag ` \<G>)" by (simp add: pairwise_imageI) then have BOX_disj: "pairwise (\<lambda>x y. negligible (BOX (\<eta> x) x \<inter> BOX (\<eta> y) y)) (tag ` \<G>)" proof (rule pairwise_mono) show "negligible (BOX (\<eta> x) x \<inter> BOX (\<eta> y) y)" if "negligible (BOX2 (\<eta> x) x \<inter> BOX2 (\<eta> y) y)" for x y by (metis (no_types, opaque_lifting) that Int_mono negligible_subset BOX_sub) qed auto have eq: "\<And>box. (\<lambda>D. box (\<eta> (tag D)) (tag D)) ` \<G> = (\<lambda>t. box (\<eta> t) t) ` tag ` \<G>" by (simp add: image_comp) have "measure lebesgue (BOX2 (\<eta> t) t) * 3 ^ DIM('a) = measure lebesgue (BOX (\<eta> t) t) * (2*3) ^ DIM('a)" if "t \<in> tag ` \<G>" for t proof - have "content (cbox (t - \<eta> t *\<^sub>R One) (t + \<eta> t *\<^sub>R One)) = content (cbox t (t + \<eta> t *\<^sub>R One)) * 2 ^ DIM('a)" using that by (simp add: algebra_simps content_cbox_if box_eq_empty) then show ?thesis by (simp add: BOX2_def BOX_def flip: power_mult_distrib) qed then have "measure lebesgue (\<Union>(\<Phi>2 ` \<G>)) * 3 ^ DIM('a) = measure lebesgue (\<Union>(\<Phi> ` \<G>)) * 6 ^ DIM('a)" unfolding \<Phi>_def \<Phi>2_def eq by (simp add: measure_negligible_finite_Union_image \<open>finite \<G>\<close> BOX2_m BOX_m BOX2_disj BOX_disj sum_distrib_right del: UN_simps) also have "\<dots> \<le> e/2" proof - have "\<mu> * measure lebesgue (\<Union>D\<in>\<G>. \<Phi> D) \<le> \<mu> * (\<Sum>D \<in> \<Phi>`\<G>. measure lebesgue D)" using \<open>\<mu> > 0\<close> \<open>finite \<G>\<close> by (force simp: BOX_m \<Phi>_def fmeasurableD intro: measure_Union_le) also have "\<dots> = (\<Sum>D \<in> \<Phi>`\<G>. measure lebesgue D * \<mu>)" by (metis mult.commute sum_distrib_right) also have "\<dots> \<le> (\<Sum>(x, K) \<in> (\<lambda>D. (tag D, \<Phi> D)) ` \<G>. norm (content K *\<^sub>R f x - integral K f))" proof (rule sum_le_included; clarify?) fix D assume "D \<in> \<G>" then have "\<eta> (tag D) > 0" using \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> h0 tag_in_E by auto then have m_\<Phi>: "measure lebesgue (\<Phi> D) > 0" by (simp add: \<Phi>_def BOX_def algebra_simps) have "\<mu> \<le> norm (i (\<eta>(tag D)) (tag D) - f(tag D))" using \<mu>_le \<open>D \<in> \<G>\<close> \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by auto also have "\<dots> = norm ((content (\<Phi> D) *\<^sub>R f(tag D) - integral (\<Phi> D) f) /\<^sub>R measure lebesgue (\<Phi> D))" using m_\<Phi> unfolding i_def \<Phi>_def BOX_def by (simp add: algebra_simps content_cbox_plus norm_minus_commute) finally have "measure lebesgue (\<Phi> D) * \<mu> \<le> norm (content (\<Phi> D) *\<^sub>R f(tag D) - integral (\<Phi> D) f)" using m_\<Phi> by simp (simp add: field_simps) then show "\<exists>y\<in>(\<lambda>D. (tag D, \<Phi> D)) ` \<G>. snd y = \<Phi> D \<and> measure lebesgue (\<Phi> D) * \<mu> \<le> (case y of (x, k) \<Rightarrow> norm (content k *\<^sub>R f x - integral k f))" using \<open>D \<in> \<G>\<close> by auto qed (use \<open>finite \<G>\<close> in auto) also have "\<dots> < ?ee" proof (rule \<gamma>) show "(\<lambda>D. (tag D, \<Phi> D)) ` \<G> tagged_partial_division_of cbox (a - One) (b + One)" unfolding tagged_partial_division_of_def proof (intro conjI allI impI ; clarify ?) show "tag D \<in> \<Phi> D" if "D \<in> \<G>" for D using that \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> h0 tag_in_E by (auto simp: \<Phi>_def BOX_def mem_box algebra_simps eucl_less_le_not_le in_mono) show "y \<in> cbox (a - One) (b + One)" if "D \<in> \<G>" "y \<in> \<Phi> D" for D y using that BOX_cbox \<Phi>_def \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by blast show "tag D = tag E \<and> \<Phi> D = \<Phi> E" if "D \<in> \<G>" "E \<in> \<G>" and ne: "interior (\<Phi> D) \<inter> interior (\<Phi> E) \<noteq> {}" for D E proof - have "BOX2 (\<eta> (tag D)) (tag D) \<inter> BOX2 (\<eta> (tag E)) (tag E) = {} \<or> tag E = tag D" using DISJ2 \<open>D \<in> \<G>\<close> \<open>E \<in> \<G>\<close> by force then have "BOX (\<eta> (tag D)) (tag D) \<inter> BOX (\<eta> (tag E)) (tag E) = {} \<or> tag E = tag D" using BOX_sub by blast then show "tag D = tag E \<and> \<Phi> D = \<Phi> E" by (metis \<Phi>_def interior_Int interior_empty ne) qed qed (use \<open>finite \<G>\<close> \<Phi>_def BOX_def in auto) show "\<gamma> fine (\<lambda>D. (tag D, \<Phi> D)) ` \<G>" unfolding fine_def \<Phi>_def using BOX_\<gamma> \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by blast qed finally show ?thesis using \<open>\<mu> > 0\<close> by (auto simp: field_split_simps) qed finally show ?thesis . qed moreover have "measure lebesgue (\<Union>\<F>) \<le> measure lebesgue (\<Union>(\<Phi>2`\<F>))" proof (rule measure_mono_fmeasurable) have "D \<subseteq> ball (tag D) (\<eta>(tag D))" if "D \<in> \<F>" for D using \<open>\<F> \<subseteq> \<D>\<close> sub_ball_tag that by blast moreover have "ball (tag D) (\<eta>(tag D)) \<subseteq> BOX2 (\<eta> (tag D)) (tag D)" if "D \<in> \<F>" for D proof (clarsimp simp: \<Phi>2_def BOX2_def mem_box algebra_simps dist_norm) fix x and i::'a assume "norm (tag D - x) < \<eta> (tag D)" and "i \<in> Basis" then have "\<bar>tag D \<bullet> i - x \<bullet> i\<bar> \<le> \<eta> (tag D)" by (metis eucl_less_le_not_le inner_commute inner_diff_right norm_bound_Basis_le) then show "tag D \<bullet> i \<le> x \<bullet> i + \<eta> (tag D) \<and> x \<bullet> i \<le> \<eta> (tag D) + tag D \<bullet> i" by (simp add: abs_diff_le_iff) qed ultimately show "\<Union>\<F> \<subseteq> \<Union>(\<Phi>2`\<F>)" by (force simp: \<Phi>2_def) show "\<Union>\<F> \<in> sets lebesgue" using \<open>finite \<F>\<close> \<open>\<D> \<subseteq> sets lebesgue\<close> \<open>\<F> \<subseteq> \<D>\<close> by blast show "\<Union>(\<Phi>2`\<F>) \<in> lmeasurable" unfolding \<Phi>2_def BOX2_def using \<open>finite \<F>\<close> by blast qed ultimately have "measure lebesgue (\<Union>\<F>) \<le> e/2" by (auto simp: field_split_simps) then show "measure lebesgue (\<Union>\<D>) \<le> e" using \<F> by linarith qed qed qed then have "\<And>j. negligible {x. \<Psi> x (inverse(real j + 1))}" using negligible_on_intervals by (metis (full_types) inverse_positive_iff_positive le_add_same_cancel1 linorder_not_le nat_le_real_less not_add_less1 of_nat_0) then have "negligible ?M" by auto moreover have "?N \<subseteq> ?M" proof (clarsimp simp: dist_norm) fix y e assume "0 < e" and ye [rule_format]: "\<Psi> y e" then obtain k where k: "0 < k" "inverse (real k + 1) < e" by (metis One_nat_def add.commute less_add_same_cancel2 less_imp_inverse_less less_trans neq0_conv of_nat_1 of_nat_Suc reals_Archimedean zero_less_one) with ye show "\<exists>n. \<Psi> y (inverse (real n + 1))" apply (rule_tac x=k in exI) unfolding \<Psi>_def by (force intro: less_le_trans) qed ultimately show "negligible ?N" by (blast intro: negligible_subset) show "\<not> \<Psi> x e" if "x \<notin> ?N \<and> 0 < e" for x e using that by blast qed with that show ?thesis unfolding i_def BOX_def \<Psi>_def by (fastforce simp add: not_le) qed subsection\<open>HOL Light measurability\<close> definition measurable_on :: "('a::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "measurable'_on" 46) where "f measurable_on S \<equiv> \<exists>N g. negligible N \<and> (\<forall>n. continuous_on UNIV (g n)) \<and> (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. g n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0))" lemma measurable_on_UNIV: "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV \<longleftrightarrow> f measurable_on S" by (auto simp: measurable_on_def) lemma measurable_on_spike_set: assumes f: "f measurable_on S" and neg: "negligible ((S - T) \<union> (T - S))" shows "f measurable_on T" proof - obtain N and F where N: "negligible N" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)" using f by (auto simp: measurable_on_def) show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "continuous_on UNIV (\<lambda>x. F n x)" for n by (intro conF continuous_intros) show "negligible (N \<union> (S - T) \<union> (T - S))" by (metis (full_types) N neg negligible_Un_eq) show "(\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then f x else 0)" if "x \<notin> (N \<union> (S - T) \<union> (T - S))" for x using that tendsF [of x] by auto qed qed text\<open> Various common equivalent forms of function measurability. \<close> lemma measurable_on_0 [simp]: "(\<lambda>x. 0) measurable_on S" unfolding measurable_on_def proof (intro exI conjI allI impI) show "(\<lambda>n. 0) \<longlonglongrightarrow> (if x \<in> S then 0::'b else 0)" for x by force qed auto lemma measurable_on_scaleR_const: assumes f: "f measurable_on S" shows "(\<lambda>x. c *\<^sub>R f x) measurable_on S" proof - obtain NF and F where NF: "negligible NF" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)" using f by (auto simp: measurable_on_def) show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "continuous_on UNIV (\<lambda>x. c *\<^sub>R F n x)" for n by (intro conF continuous_intros) show "(\<lambda>n. c *\<^sub>R F n x) \<longlonglongrightarrow> (if x \<in> S then c *\<^sub>R f x else 0)" if "x \<notin> NF" for x using tendsto_scaleR [OF tendsto_const tendsF, of x] that by auto qed (auto simp: NF) qed lemma measurable_on_cmul: fixes c :: real assumes "f measurable_on S" shows "(\<lambda>x. c * f x) measurable_on S" using measurable_on_scaleR_const [OF assms] by simp lemma measurable_on_cdivide: fixes c :: real assumes "f measurable_on S" shows "(\<lambda>x. f x / c) measurable_on S" proof (cases "c=0") case False then show ?thesis using measurable_on_cmul [of f S "1/c"] by (simp add: assms) qed auto lemma measurable_on_minus: "f measurable_on S \<Longrightarrow> (\<lambda>x. -(f x)) measurable_on S" using measurable_on_scaleR_const [of f S "-1"] by auto lemma continuous_imp_measurable_on: "continuous_on UNIV f \<Longrightarrow> f measurable_on UNIV" unfolding measurable_on_def apply (rule_tac x="{}" in exI) apply (rule_tac x="\<lambda>n. f" in exI, auto) done proposition integrable_subintervals_imp_measurable: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "\<And>a b. f integrable_on cbox a b" shows "f measurable_on UNIV" proof - define BOX where "BOX \<equiv> \<lambda>h. \<lambda>x::'a. cbox x (x + h *\<^sub>R One)" define i where "i \<equiv> \<lambda>h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)" obtain N where "negligible N" and k: "\<And>x e. \<lbrakk>x \<notin> N; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow> norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e" using integrable_ccontinuous_explicit assms by blast show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "continuous_on UNIV ((\<lambda>n x. i (inverse(Suc n)) x) n)" for n proof (clarsimp simp: continuous_on_iff) show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow> dist (i (inverse (1 + real n)) x') (i (inverse (1 + real n)) x) < e" if "0 < e" for x e proof - let ?e = "e / (1 + real n) ^ DIM('a)" have "?e > 0" using \<open>e > 0\<close> by auto moreover have "x \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)" by (simp add: mem_box inner_diff_left inner_left_distrib) moreover have "x + One /\<^sub>R real (Suc n) \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)" by (auto simp: mem_box inner_diff_left inner_left_distrib field_simps) ultimately obtain \<delta> where "\<delta> > 0" and \<delta>: "\<And>c' d'. \<lbrakk>c' \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One); d' \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One); norm(c' - x) \<le> \<delta>; norm(d' - (x + One /\<^sub>R Suc n)) \<le> \<delta>\<rbrakk> \<Longrightarrow> norm(integral(cbox c' d') f - integral(cbox x (x + One /\<^sub>R Suc n)) f) < ?e" by (blast intro: indefinite_integral_continuous [of f _ _ x] assms) show ?thesis proof (intro exI impI conjI allI) show "min \<delta> 1 > 0" using \<open>\<delta> > 0\<close> by auto show "dist (i (inverse (1 + real n)) y) (i (inverse (1 + real n)) x) < e" if "dist y x < min \<delta> 1" for y proof - have no: "norm (y - x) < 1" using that by (auto simp: dist_norm) have le1: "inverse (1 + real n) \<le> 1" by (auto simp: field_split_simps) have "norm (integral (cbox y (y + One /\<^sub>R real (Suc n))) f - integral (cbox x (x + One /\<^sub>R real (Suc n))) f) < e / (1 + real n) ^ DIM('a)" proof (rule \<delta>) show "y \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)" using no by (auto simp: mem_box algebra_simps dest: Basis_le_norm [of _ "y-x"]) show "y + One /\<^sub>R real (Suc n) \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)" proof (simp add: dist_norm mem_box algebra_simps, intro ballI conjI) fix i::'a assume "i \<in> Basis" then have 1: "\<bar>y \<bullet> i - x \<bullet> i\<bar> < 1" by (metis inner_commute inner_diff_right no norm_bound_Basis_lt) moreover have "\<dots> < (2 + inverse (1 + real n))" "1 \<le> 2 - inverse (1 + real n)" by (auto simp: field_simps) ultimately show "x \<bullet> i \<le> y \<bullet> i + (2 + inverse (1 + real n))" "y \<bullet> i + inverse (1 + real n) \<le> x \<bullet> i + 2" by linarith+ qed show "norm (y - x) \<le> \<delta>" "norm (y + One /\<^sub>R real (Suc n) - (x + One /\<^sub>R real (Suc n))) \<le> \<delta>" using that by (auto simp: dist_norm) qed then show ?thesis using that by (simp add: dist_norm i_def BOX_def flip: scaleR_diff_right) (simp add: field_simps) qed qed qed qed show "negligible N" by (simp add: \<open>negligible N\<close>) show "(\<lambda>n. i (inverse (Suc n)) x) \<longlonglongrightarrow> (if x \<in> UNIV then f x else 0)" if "x \<notin> N" for x unfolding lim_sequentially proof clarsimp show "\<exists>no. \<forall>n\<ge>no. dist (i (inverse (1 + real n)) x) (f x) < e" if "0 < e" for e proof - obtain d where "d > 0" and d: "\<And>h. \<lbrakk>0 < h; h < d\<rbrakk> \<Longrightarrow> norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e" using k [of x e] \<open>x \<notin> N\<close> \<open>0 < e\<close> by blast then obtain M where M: "M \<noteq> 0" "0 < inverse (real M)" "inverse (real M) < d" using real_arch_invD by auto show ?thesis proof (intro exI allI impI) show "dist (i (inverse (1 + real n)) x) (f x) < e" if "M \<le> n" for n proof - have *: "0 < inverse (1 + real n)" "inverse (1 + real n) \<le> inverse M" using that \<open>M \<noteq> 0\<close> by auto show ?thesis using that M apply (simp add: i_def BOX_def dist_norm) apply (blast intro: le_less_trans * d) done qed qed qed qed qed qed subsection\<open>Composing continuous and measurable functions; a few variants\<close> lemma measurable_on_compose_continuous: assumes f: "f measurable_on UNIV" and g: "continuous_on UNIV g" shows "(g \<circ> f) measurable_on UNIV" proof - obtain N and F where "negligible N" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> f x" using f by (auto simp: measurable_on_def) show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "negligible N" by fact show "continuous_on UNIV (g \<circ> (F n))" for n using conF continuous_on_compose continuous_on_subset g by blast show "(\<lambda>n. (g \<circ> F n) x) \<longlonglongrightarrow> (if x \<in> UNIV then (g \<circ> f) x else 0)" if "x \<notin> N" for x :: 'a using that g tendsF by (auto simp: continuous_on_def intro: tendsto_compose) qed qed lemma measurable_on_compose_continuous_0: assumes f: "f measurable_on S" and g: "continuous_on UNIV g" and "g 0 = 0" shows "(g \<circ> f) measurable_on S" proof - have f': "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV" using f measurable_on_UNIV by blast show ?thesis using measurable_on_compose_continuous [OF f' g] by (simp add: measurable_on_UNIV o_def if_distrib \<open>g 0 = 0\<close> cong: if_cong) qed lemma measurable_on_compose_continuous_box: assumes fm: "f measurable_on UNIV" and fab: "\<And>x. f x \<in> box a b" and contg: "continuous_on (box a b) g" shows "(g \<circ> f) measurable_on UNIV" proof - have "\<exists>\<gamma>. (\<forall>n. continuous_on UNIV (\<gamma> n)) \<and> (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. \<gamma> n x) \<longlonglongrightarrow> g (f x))" if "negligible N" and conth [rule_format]: "\<forall>n. continuous_on UNIV (\<lambda>x. h n x)" and tends [rule_format]: "\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. h n x) \<longlonglongrightarrow> f x" for N and h :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" proof - define \<theta> where "\<theta> \<equiv> \<lambda>n x. (\<Sum>i\<in>Basis. (max (a\<bullet>i + (b\<bullet>i - a\<bullet>i) / real (n+2)) (min ((h n x)\<bullet>i) (b\<bullet>i - (b\<bullet>i - a\<bullet>i) / real (n+2)))) *\<^sub>R i)" have aibi: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i < b \<bullet> i" using box_ne_empty(2) fab by auto then have *: "\<And>i n. i \<in> Basis \<Longrightarrow> a \<bullet> i + real n * (a \<bullet> i) < b \<bullet> i + real n * (b \<bullet> i)" by (meson add_mono_thms_linordered_field(3) less_eq_real_def mult_left_mono of_nat_0_le_iff) show ?thesis proof (intro exI conjI allI impI) show "continuous_on UNIV (g \<circ> (\<theta> n))" for n :: nat unfolding \<theta>_def apply (intro continuous_on_compose2 [OF contg] continuous_intros conth) apply (auto simp: aibi * mem_box less_max_iff_disj min_less_iff_disj field_split_simps) done show "(\<lambda>n. (g \<circ> \<theta> n) x) \<longlonglongrightarrow> g (f x)" if "x \<notin> N" for x unfolding o_def proof (rule isCont_tendsto_compose [where g=g]) show "isCont g (f x)" using contg fab continuous_on_eq_continuous_at by blast have "(\<lambda>n. \<theta> n x) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. max (a \<bullet> i) (min (f x \<bullet> i) (b \<bullet> i)) *\<^sub>R i)" unfolding \<theta>_def proof (intro tendsto_intros \<open>x \<notin> N\<close> tends) fix i::'b assume "i \<in> Basis" have a: "(\<lambda>n. a \<bullet> i + (b \<bullet> i - a \<bullet> i) / real n) \<longlonglongrightarrow> a\<bullet>i + 0" by (intro tendsto_add lim_const_over_n tendsto_const) show "(\<lambda>n. a \<bullet> i + (b \<bullet> i - a \<bullet> i) / real (n + 2)) \<longlonglongrightarrow> a \<bullet> i" using LIMSEQ_ignore_initial_segment [where k=2, OF a] by simp have b: "(\<lambda>n. b\<bullet>i - (b \<bullet> i - a \<bullet> i) / (real n)) \<longlonglongrightarrow> b\<bullet>i - 0" by (intro tendsto_diff lim_const_over_n tendsto_const) show "(\<lambda>n. b \<bullet> i - (b \<bullet> i - a \<bullet> i) / real (n + 2)) \<longlonglongrightarrow> b \<bullet> i" using LIMSEQ_ignore_initial_segment [where k=2, OF b] by simp qed also have "(\<Sum>i\<in>Basis. max (a \<bullet> i) (min (f x \<bullet> i) (b \<bullet> i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)" using fab by (auto simp add: mem_box intro: sum.cong) also have "\<dots> = f x" using euclidean_representation by blast finally show "(\<lambda>n. \<theta> n x) \<longlonglongrightarrow> f x" . qed qed qed then show ?thesis using fm by (auto simp: measurable_on_def) qed lemma measurable_on_Pair: assumes f: "f measurable_on S" and g: "g measurable_on S" shows "(\<lambda>x. (f x, g x)) measurable_on S" proof - obtain NF and F where NF: "negligible NF" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)" using f by (auto simp: measurable_on_def) obtain NG and G where NG: "negligible NG" and conG: "\<And>n. continuous_on UNIV (G n)" and tendsG: "\<And>x. x \<notin> NG \<Longrightarrow> (\<lambda>n. G n x) \<longlonglongrightarrow> (if x \<in> S then g x else 0)" using g by (auto simp: measurable_on_def) show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "negligible (NF \<union> NG)" by (simp add: NF NG) show "continuous_on UNIV (\<lambda>x. (F n x, G n x))" for n using conF conG continuous_on_Pair by blast show "(\<lambda>n. (F n x, G n x)) \<longlonglongrightarrow> (if x \<in> S then (f x, g x) else 0)" if "x \<notin> NF \<union> NG" for x using tendsto_Pair [OF tendsF tendsG, of x x] that unfolding zero_prod_def by (simp add: split: if_split_asm) qed qed lemma measurable_on_combine: assumes f: "f measurable_on S" and g: "g measurable_on S" and h: "continuous_on UNIV (\<lambda>x. h (fst x) (snd x))" and "h 0 0 = 0" shows "(\<lambda>x. h (f x) (g x)) measurable_on S" proof - have *: "(\<lambda>x. h (f x) (g x)) = (\<lambda>x. h (fst x) (snd x)) \<circ> (\<lambda>x. (f x, g x))" by auto show ?thesis unfolding * by (auto simp: measurable_on_compose_continuous_0 measurable_on_Pair assms) qed lemma measurable_on_add: assumes f: "f measurable_on S" and g: "g measurable_on S" shows "(\<lambda>x. f x + g x) measurable_on S" by (intro continuous_intros measurable_on_combine [OF assms]) auto lemma measurable_on_diff: assumes f: "f measurable_on S" and g: "g measurable_on S" shows "(\<lambda>x. f x - g x) measurable_on S" by (intro continuous_intros measurable_on_combine [OF assms]) auto lemma measurable_on_scaleR: assumes f: "f measurable_on S" and g: "g measurable_on S" shows "(\<lambda>x. f x *\<^sub>R g x) measurable_on S" by (intro continuous_intros measurable_on_combine [OF assms]) auto lemma measurable_on_sum: assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i measurable_on S" shows "(\<lambda>x. sum (\<lambda>i. f i x) I) measurable_on S" using assms by (induction I) (auto simp: measurable_on_add) lemma measurable_on_spike: assumes f: "f measurable_on T" and "negligible S" and gf: "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g measurable_on T" proof - obtain NF and F where NF: "negligible NF" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then f x else 0)" using f by (auto simp: measurable_on_def) show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "negligible (NF \<union> S)" by (simp add: NF \<open>negligible S\<close>) show "\<And>x. x \<notin> NF \<union> S \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then g x else 0)" by (metis (full_types) Diff_iff Un_iff gf tendsF) qed (auto simp: conF) qed proposition indicator_measurable_on: assumes "S \<in> sets lebesgue" shows "indicat_real S measurable_on UNIV" proof - { fix n::nat let ?\<epsilon> = "(1::real) / (2 * 2^n)" have \<epsilon>: "?\<epsilon> > 0" by auto obtain T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" and ST: "emeasure lebesgue (S - T) < ?\<epsilon>" by (meson \<epsilon> assms sets_lebesgue_inner_closed) obtain U where "open U" "S \<subseteq> U" "(U - S) \<in> lmeasurable" and US: "emeasure lebesgue (U - S) < ?\<epsilon>" by (meson \<epsilon> assms sets_lebesgue_outer_open) have eq: "-T \<inter> U = (S-T) \<union> (U - S)" using \<open>T \<subseteq> S\<close> \<open>S \<subseteq> U\<close> by auto have "emeasure lebesgue ((S-T) \<union> (U - S)) \<le> emeasure lebesgue (S - T) + emeasure lebesgue (U - S)" using \<open>S - T \<in> lmeasurable\<close> \<open>U - S \<in> lmeasurable\<close> emeasure_subadditive by blast also have "\<dots> < ?\<epsilon> + ?\<epsilon>" using ST US add_mono_ennreal by metis finally have le: "emeasure lebesgue (-T \<inter> U) < ennreal (1 / 2^n)" by (simp add: eq) have 1: "continuous_on (T \<union> -U) (indicat_real S)" unfolding indicator_def of_bool_def proof (rule continuous_on_cases [OF \<open>closed T\<close>]) show "closed (- U)" using \<open>open U\<close> by blast show "continuous_on T (\<lambda>x. 1::real)" "continuous_on (- U) (\<lambda>x. 0::real)" by (auto simp: continuous_on) show "\<forall>x. x \<in> T \<and> x \<notin> S \<or> x \<in> - U \<and> x \<in> S \<longrightarrow> (1::real) = 0" using \<open>T \<subseteq> S\<close> \<open>S \<subseteq> U\<close> by auto qed have 2: "closedin (top_of_set UNIV) (T \<union> -U)" using \<open>closed T\<close> \<open>open U\<close> by auto obtain g where "continuous_on UNIV g" "\<And>x. x \<in> T \<union> -U \<Longrightarrow> g x = indicat_real S x" "\<And>x. norm(g x) \<le> 1" by (rule Tietze [OF 1 2, of 1]) auto with le have "\<exists>g E. continuous_on UNIV g \<and> (\<forall>x \<in> -E. g x = indicat_real S x) \<and> (\<forall>x. norm(g x) \<le> 1) \<and> E \<in> sets lebesgue \<and> emeasure lebesgue E < ennreal (1 / 2^n)" apply (rule_tac x=g in exI) apply (rule_tac x="-T \<inter> U" in exI) using \<open>S - T \<in> lmeasurable\<close> \<open>U - S \<in> lmeasurable\<close> eq by auto } then obtain g E where cont: "\<And>n. continuous_on UNIV (g n)" and geq: "\<And>n x. x \<in> - E n \<Longrightarrow> g n x = indicat_real S x" and ng1: "\<And>n x. norm(g n x) \<le> 1" and Eset: "\<And>n. E n \<in> sets lebesgue" and Em: "\<And>n. emeasure lebesgue (E n) < ennreal (1 / 2^n)" by metis have null: "limsup E \<in> null_sets lebesgue" proof (rule borel_cantelli_limsup1 [OF Eset]) show "emeasure lebesgue (E n) < \<infinity>" for n by (metis Em infinity_ennreal_def order.asym top.not_eq_extremum) show "summable (\<lambda>n. measure lebesgue (E n))" proof (rule summable_comparison_test' [OF summable_geometric, of "1/2" 0]) show "norm (measure lebesgue (E n)) \<le> (1/2) ^ n" for n using Em [of n] by (simp add: measure_def enn2real_leI power_one_over) qed auto qed have tends: "(\<lambda>n. g n x) \<longlonglongrightarrow> indicat_real S x" if "x \<notin> limsup E" for x proof - have "\<forall>\<^sub>F n in sequentially. x \<in> - E n" using that by (simp add: mem_limsup_iff not_frequently) then show ?thesis unfolding tendsto_iff dist_real_def by (simp add: eventually_mono geq) qed show ?thesis unfolding measurable_on_def proof (intro exI conjI allI impI) show "negligible (limsup E)" using negligible_iff_null_sets null by blast show "continuous_on UNIV (g n)" for n using cont by blast qed (use tends in auto) qed lemma measurable_on_restrict: assumes f: "f measurable_on UNIV" and S: "S \<in> sets lebesgue" shows "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV" proof - have "indicat_real S measurable_on UNIV" by (simp add: S indicator_measurable_on) then show ?thesis using measurable_on_scaleR [OF _ f, of "indicat_real S"] by (simp add: indicator_scaleR_eq_if) qed lemma measurable_on_const_UNIV: "(\<lambda>x. k) measurable_on UNIV" by (simp add: continuous_imp_measurable_on) lemma measurable_on_const [simp]: "S \<in> sets lebesgue \<Longrightarrow> (\<lambda>x. k) measurable_on S" using measurable_on_UNIV measurable_on_const_UNIV measurable_on_restrict by blast lemma simple_function_indicator_representation_real: fixes f ::"'a \<Rightarrow> real" assumes f: "simple_function M f" and x: "x \<in> space M" and nn: "\<And>x. f x \<ge> 0" shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)" proof - have f': "simple_function M (ennreal \<circ> f)" by (simp add: f) have *: "f x = enn2real (\<Sum>y\<in> ennreal ` f ` space M. y * indicator ((ennreal \<circ> f) -` {y} \<inter> space M) x)" using arg_cong [OF simple_function_indicator_representation [OF f' x], of enn2real, simplified nn o_def] nn unfolding o_def image_comp by (metis enn2real_ennreal) have "enn2real (\<Sum>y\<in>ennreal ` f ` space M. if ennreal (f x) = y \<and> x \<in> space M then y else 0) = sum (enn2real \<circ> (\<lambda>y. if ennreal (f x) = y \<and> x \<in> space M then y else 0)) (ennreal ` f ` space M)" by (rule enn2real_sum) auto also have "\<dots> = sum (enn2real \<circ> (\<lambda>y. if ennreal (f x) = y \<and> x \<in> space M then y else 0) \<circ> ennreal) (f ` space M)" by (rule sum.reindex) (use nn in \<open>auto simp: inj_on_def intro: sum.cong\<close>) also have "\<dots> = (\<Sum>y\<in>f ` space M. if f x = y \<and> x \<in> space M then y else 0)" using nn by (auto simp: inj_on_def intro: sum.cong) finally show ?thesis by (subst *) (simp add: enn2real_sum indicator_def of_bool_def if_distrib cong: if_cong) qed lemma\<^marker>\<open>tag important\<close> simple_function_induct_real [consumes 1, case_names cong set mult add, induct set: simple_function]: fixes u :: "'a \<Rightarrow> real" assumes u: "simple_function M u" assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g" assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)" assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)" assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. u x + v x)" and nn: "\<And>x. u x \<ge> 0" shows "P u" proof (rule cong) from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" proof eventually_elim fix x assume x: "x \<in> space M" from simple_function_indicator_representation_real[OF u x] nn show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" by metis qed next from u have "finite (u ` space M)" unfolding simple_function_def by auto then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)" proof induct case empty then show ?case using set[of "{}"] by (simp add: indicator_def[abs_def]) next case (insert a F) have eq: "\<Sum> {y. u x = y \<and> (y = a \<or> y \<in> F) \<and> x \<in> space M} = (if u x = a \<and> x \<in> space M then a else 0) + \<Sum> {y. u x = y \<and> y \<in> F \<and> x \<in> space M}" for x proof (cases "x \<in> space M") case True have *: "{y. u x = y \<and> (y = a \<or> y \<in> F)} = {y. u x = a \<and> y = a} \<union> {y. u x = y \<and> y \<in> F}" by auto show ?thesis using insert by (simp add: * True) qed auto have a: "P (\<lambda>x. a * indicator (u -` {a} \<inter> space M) x)" proof (intro mult set) show "u -` {a} \<inter> space M \<in> sets M" using u by auto qed show ?case using nn insert a by (simp add: eq indicator_times_eq_if [where f = "\<lambda>x. a"] add) qed next show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))" apply (subst simple_function_cong) apply (rule simple_function_indicator_representation_real[symmetric]) apply (auto intro: u nn) done qed fact proposition simple_function_measurable_on_UNIV: fixes f :: "'a::euclidean_space \<Rightarrow> real" assumes f: "simple_function lebesgue f" and nn: "\<And>x. f x \<ge> 0" shows "f measurable_on UNIV" using f proof (induction f) case (cong f g) then obtain N where "negligible N" "{x. g x \<noteq> f x} \<subseteq> N" by (auto simp: eventually_ae_filter_negligible eq_commute) then show ?case by (blast intro: measurable_on_spike cong) next case (set S) then show ?case by (simp add: indicator_measurable_on) next case (mult u c) then show ?case by (simp add: measurable_on_cmul) case (add u v) then show ?case by (simp add: measurable_on_add) qed (auto simp: nn) lemma simple_function_lebesgue_if: fixes f :: "'a::euclidean_space \<Rightarrow> real" assumes f: "simple_function lebesgue f" and S: "S \<in> sets lebesgue" shows "simple_function lebesgue (\<lambda>x. if x \<in> S then f x else 0)" proof - have ffin: "finite (range f)" and fsets: "\<forall>x. f -` {f x} \<in> sets lebesgue" using f by (auto simp: simple_function_def) have "finite (f ` S)" by (meson finite_subset subset_image_iff ffin top_greatest) moreover have "finite ((\<lambda>x. 0::real) ` T)" for T :: "'a set" by (auto simp: image_def) moreover have if_sets: "(\<lambda>x. if x \<in> S then f x else 0) -` {f a} \<in> sets lebesgue" for a proof - have *: "(\<lambda>x. if x \<in> S then f x else 0) -` {f a} = (if f a = 0 then -S \<union> f -` {f a} else (f -` {f a}) \<inter> S)" by (auto simp: split: if_split_asm) show ?thesis unfolding * by (metis Compl_in_sets_lebesgue S sets.Int sets.Un fsets) qed moreover have "(\<lambda>x. if x \<in> S then f x else 0) -` {0} \<in> sets lebesgue" proof (cases "0 \<in> range f") case True then show ?thesis by (metis (no_types, lifting) if_sets rangeE) next case False then have "(\<lambda>x. if x \<in> S then f x else 0) -` {0} = -S" by auto then show ?thesis by (simp add: Compl_in_sets_lebesgue S) qed ultimately show ?thesis by (auto simp: simple_function_def) qed corollary simple_function_measurable_on: fixes f :: "'a::euclidean_space \<Rightarrow> real" assumes f: "simple_function lebesgue f" and nn: "\<And>x. f x \<ge> 0" and S: "S \<in> sets lebesgue" shows "f measurable_on S" by (simp add: measurable_on_UNIV [symmetric, of f] S f simple_function_lebesgue_if nn simple_function_measurable_on_UNIV) lemma fixes f :: "'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space" assumes f: "f measurable_on S" and g: "g measurable_on S" shows measurable_on_sup: "(\<lambda>x. sup (f x) (g x)) measurable_on S" and measurable_on_inf: "(\<lambda>x. inf (f x) (g x)) measurable_on S" proof - obtain NF and F where NF: "negligible NF" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)" using f by (auto simp: measurable_on_def) obtain NG and G where NG: "negligible NG" and conG: "\<And>n. continuous_on UNIV (G n)" and tendsG: "\<And>x. x \<notin> NG \<Longrightarrow> (\<lambda>n. G n x) \<longlonglongrightarrow> (if x \<in> S then g x else 0)" using g by (auto simp: measurable_on_def) show "(\<lambda>x. sup (f x) (g x)) measurable_on S" unfolding measurable_on_def proof (intro exI conjI allI impI) show "continuous_on UNIV (\<lambda>x. sup (F n x) (G n x))" for n unfolding sup_max eucl_sup by (intro conF conG continuous_intros) show "(\<lambda>n. sup (F n x) (G n x)) \<longlonglongrightarrow> (if x \<in> S then sup (f x) (g x) else 0)" if "x \<notin> NF \<union> NG" for x using tendsto_sup [OF tendsF tendsG, of x x] that by auto qed (simp add: NF NG) show "(\<lambda>x. inf (f x) (g x)) measurable_on S" unfolding measurable_on_def proof (intro exI conjI allI impI) show "continuous_on UNIV (\<lambda>x. inf (F n x) (G n x))" for n unfolding inf_min eucl_inf by (intro conF conG continuous_intros) show "(\<lambda>n. inf (F n x) (G n x)) \<longlonglongrightarrow> (if x \<in> S then inf (f x) (g x) else 0)" if "x \<notin> NF \<union> NG" for x using tendsto_inf [OF tendsF tendsG, of x x] that by auto qed (simp add: NF NG) qed proposition measurable_on_componentwise_UNIV: "f measurable_on UNIV \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on UNIV)" (is "?lhs = ?rhs") proof assume L: ?lhs show ?rhs proof fix i::'b assume "i \<in> Basis" have cont: "continuous_on UNIV (\<lambda>x. (x \<bullet> i) *\<^sub>R i)" by (intro continuous_intros) show "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on UNIV" using measurable_on_compose_continuous [OF L cont] by (simp add: o_def) qed next assume ?rhs then have "\<exists>N g. negligible N \<and> (\<forall>n. continuous_on UNIV (g n)) \<and> (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. g n x) \<longlonglongrightarrow> (f x \<bullet> i) *\<^sub>R i)" if "i \<in> Basis" for i by (simp add: measurable_on_def that) then obtain N g where N: "\<And>i. i \<in> Basis \<Longrightarrow> negligible (N i)" and cont: "\<And>i n. i \<in> Basis \<Longrightarrow> continuous_on UNIV (g i n)" and tends: "\<And>i x. \<lbrakk>i \<in> Basis; x \<notin> N i\<rbrakk> \<Longrightarrow> (\<lambda>n. g i n x) \<longlonglongrightarrow> (f x \<bullet> i) *\<^sub>R i" by metis show ?lhs unfolding measurable_on_def proof (intro exI conjI allI impI) show "negligible (\<Union>i \<in> Basis. N i)" using N eucl.finite_Basis by blast show "continuous_on UNIV (\<lambda>x. (\<Sum>i\<in>Basis. g i n x))" for n by (intro continuous_intros cont) next fix x assume "x \<notin> (\<Union>i \<in> Basis. N i)" then have "\<And>i. i \<in> Basis \<Longrightarrow> x \<notin> N i" by auto then have "(\<lambda>n. (\<Sum>i\<in>Basis. g i n x)) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)" by (intro tends tendsto_intros) then show "(\<lambda>n. (\<Sum>i\<in>Basis. g i n x)) \<longlonglongrightarrow> (if x \<in> UNIV then f x else 0)" by (simp add: euclidean_representation) qed qed corollary measurable_on_componentwise: "f measurable_on S \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on S)" apply (subst measurable_on_UNIV [symmetric]) apply (subst measurable_on_componentwise_UNIV) apply (simp add: measurable_on_UNIV if_distrib [of "\<lambda>x. inner x _"] if_distrib [of "\<lambda>x. scaleR x _"] cong: if_cong) done (*FIXME: avoid duplication of proofs WRT borel_measurable_implies_simple_function_sequence*) lemma\<^marker>\<open>tag important\<close> borel_measurable_implies_simple_function_sequence_real: fixes u :: "'a \<Rightarrow> real" assumes u[measurable]: "u \<in> borel_measurable M" and nn: "\<And>x. u x \<ge> 0" shows "\<exists>f. incseq f \<and> (\<forall>i. simple_function M (f i)) \<and> (\<forall>x. bdd_above (range (\<lambda>i. f i x))) \<and> (\<forall>i x. 0 \<le> f i x) \<and> u = (SUP i. f i)" proof - define f where [abs_def]: "f i x = real_of_int (floor ((min i (u x)) * 2^i)) / 2^i" for i x have [simp]: "0 \<le> f i x" for i x by (auto simp: f_def intro!: divide_nonneg_nonneg mult_nonneg_nonneg nn) have *: "2^n * real_of_int x = real_of_int (2^n * x)" for n x by simp have "real_of_int \<lfloor>real i * 2 ^ i\<rfloor> = real_of_int \<lfloor>i * 2 ^ i\<rfloor>" for i by (intro arg_cong[where f=real_of_int]) simp then have [simp]: "real_of_int \<lfloor>real i * 2 ^ i\<rfloor> = i * 2 ^ i" for i unfolding floor_of_nat by simp have bdd: "bdd_above (range (\<lambda>i. f i x))" for x by (rule bdd_aboveI [where M = "u x"]) (auto simp: f_def field_simps min_def) have "incseq f" proof (intro monoI le_funI) fix m n :: nat and x assume "m \<le> n" moreover { fix d :: nat have "\<lfloor>2^d::real\<rfloor> * \<lfloor>2^m * (min (of_nat m) (u x))\<rfloor> \<le> \<lfloor>2^d * (2^m * (min (of_nat m) (u x)))\<rfloor>" by (rule le_mult_floor) (auto simp: nn) also have "\<dots> \<le> \<lfloor>2^d * (2^m * (min (of_nat d + of_nat m) (u x)))\<rfloor>" by (intro floor_mono mult_mono min.mono) (auto simp: nn min_less_iff_disj of_nat_less_top) finally have "f m x \<le> f(m + d) x" unfolding f_def by (auto simp: field_simps power_add * simp del: of_int_mult) } ultimately show "f m x \<le> f n x" by (auto simp: le_iff_add) qed then have inc_f: "incseq (\<lambda>i. f i x)" for x by (auto simp: incseq_def le_fun_def) moreover have "simple_function M (f i)" for i proof (rule simple_function_borel_measurable) have "\<lfloor>(min (of_nat i) (u x)) * 2 ^ i\<rfloor> \<le> \<lfloor>int i * 2 ^ i\<rfloor>" for x by (auto split: split_min intro!: floor_mono) then have "f i ` space M \<subseteq> (\<lambda>n. real_of_int n / 2^i) ` {0 .. of_nat i * 2^i}" unfolding floor_of_int by (auto simp: f_def nn intro!: imageI) then show "finite (f i ` space M)" by (rule finite_subset) auto show "f i \<in> borel_measurable M" unfolding f_def enn2real_def by measurable qed moreover { fix x have "(SUP i. (f i x)) = u x" proof - obtain n where "u x \<le> of_nat n" using real_arch_simple by auto then have min_eq_r: "\<forall>\<^sub>F i in sequentially. min (real i) (u x) = u x" by (auto simp: eventually_sequentially intro!: exI[of _ n] split: split_min) have "(\<lambda>i. real_of_int \<lfloor>min (real i) (u x) * 2^i\<rfloor> / 2^i) \<longlonglongrightarrow> u x" proof (rule tendsto_sandwich) show "(\<lambda>n. u x - (1/2)^n) \<longlonglongrightarrow> u x" by (auto intro!: tendsto_eq_intros LIMSEQ_power_zero) show "\<forall>\<^sub>F n in sequentially. real_of_int \<lfloor>min (real n) (u x) * 2 ^ n\<rfloor> / 2 ^ n \<le> u x" using min_eq_r by eventually_elim (auto simp: field_simps) have *: "u x * (2 ^ n * 2 ^ n) \<le> 2^n + 2^n * real_of_int \<lfloor>u x * 2 ^ n\<rfloor>" for n using real_of_int_floor_ge_diff_one[of "u x * 2^n", THEN mult_left_mono, of "2^n"] by (auto simp: field_simps) show "\<forall>\<^sub>F n in sequentially. u x - (1/2)^n \<le> real_of_int \<lfloor>min (real n) (u x) * 2 ^ n\<rfloor> / 2 ^ n" using min_eq_r by eventually_elim (insert *, auto simp: field_simps) qed auto then have "(\<lambda>i. (f i x)) \<longlonglongrightarrow> u x" by (simp add: f_def) from LIMSEQ_unique LIMSEQ_incseq_SUP [OF bdd inc_f] this show ?thesis by blast qed } ultimately show ?thesis by (intro exI [of _ "\<lambda>i x. f i x"]) (auto simp: \<open>incseq f\<close> bdd image_comp) qed lemma homeomorphic_open_interval_UNIV: fixes a b:: real assumes "a < b" shows "{a<..<b} homeomorphic (UNIV::real set)" proof - have "{a<..<b} = ball ((b+a) / 2) ((b-a) / 2)" using assms by (auto simp: dist_real_def abs_if field_split_simps split: if_split_asm) then show ?thesis by (simp add: homeomorphic_ball_UNIV assms) qed proposition homeomorphic_box_UNIV: fixes a b:: "'a::euclidean_space" assumes "box a b \<noteq> {}" shows "box a b homeomorphic (UNIV::'a set)" proof - have "{a \<bullet> i <..<b \<bullet> i} homeomorphic (UNIV::real set)" if "i \<in> Basis" for i using assms box_ne_empty that by (blast intro: homeomorphic_open_interval_UNIV) then have "\<exists>f g. (\<forall>x. a \<bullet> i < x \<and> x < b \<bullet> i \<longrightarrow> g (f x) = x) \<and> (\<forall>y. a \<bullet> i < g y \<and> g y < b \<bullet> i \<and> f(g y) = y) \<and> continuous_on {a \<bullet> i<..<b \<bullet> i} f \<and> continuous_on (UNIV::real set) g" if "i \<in> Basis" for i using that by (auto simp: homeomorphic_minimal mem_box Ball_def) then obtain f g where gf: "\<And>i x. \<lbrakk>i \<in> Basis; a \<bullet> i < x; x < b \<bullet> i\<rbrakk> \<Longrightarrow> g i (f i x) = x" and fg: "\<And>i y. i \<in> Basis \<Longrightarrow> a \<bullet> i < g i y \<and> g i y < b \<bullet> i \<and> f i (g i y) = y" and contf: "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on {a \<bullet> i<..<b \<bullet> i} (f i)" and contg: "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on (UNIV::real set) (g i)" by metis define F where "F \<equiv> \<lambda>x. \<Sum>i\<in>Basis. (f i (x \<bullet> i)) *\<^sub>R i" define G where "G \<equiv> \<lambda>x. \<Sum>i\<in>Basis. (g i (x \<bullet> i)) *\<^sub>R i" show ?thesis unfolding homeomorphic_minimal proof (intro exI conjI ballI) show "G y \<in> box a b" for y using fg by (simp add: G_def mem_box) show "G (F x) = x" if "x \<in> box a b" for x using that by (simp add: F_def G_def gf mem_box euclidean_representation) show "F (G y) = y" for y by (simp add: F_def G_def fg mem_box euclidean_representation) show "continuous_on (box a b) F" unfolding F_def proof (intro continuous_intros continuous_on_compose2 [OF contf continuous_on_inner]) show "(\<lambda>x. x \<bullet> i) ` box a b \<subseteq> {a \<bullet> i<..<b \<bullet> i}" if "i \<in> Basis" for i using that by (auto simp: mem_box) qed show "continuous_on UNIV G" unfolding G_def by (intro continuous_intros continuous_on_compose2 [OF contg continuous_on_inner]) auto qed auto qed lemma diff_null_sets_lebesgue: "\<lbrakk>N \<in> null_sets (lebesgue_on S); X-N \<in> sets (lebesgue_on S); N \<subseteq> X\<rbrakk> \<Longrightarrow> X \<in> sets (lebesgue_on S)" by (metis Int_Diff_Un inf.commute inf.orderE null_setsD2 sets.Un) lemma borel_measurable_diff_null: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes N: "N \<in> null_sets (lebesgue_on S)" and S: "S \<in> sets lebesgue" shows "f \<in> borel_measurable (lebesgue_on (S-N)) \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)" unfolding in_borel_measurable space_lebesgue_on sets_restrict_UNIV proof (intro ball_cong iffI) show "f -` T \<inter> S \<in> sets (lebesgue_on S)" if "f -` T \<inter> (S-N) \<in> sets (lebesgue_on (S-N))" for T proof - have "N \<inter> S = N" by (metis N S inf.orderE null_sets_restrict_space) moreover have "N \<inter> S \<in> sets lebesgue" by (metis N S inf.orderE null_setsD2 null_sets_restrict_space) moreover have "f -` T \<inter> S \<inter> (f -` T \<inter> N) \<in> sets lebesgue" by (metis N S completion.complete inf.absorb2 inf_le2 inf_mono null_sets_restrict_space) ultimately show ?thesis by (metis Diff_Int_distrib Int_Diff_Un S inf_le2 sets.Diff sets.Un sets_restrict_space_iff space_lebesgue_on space_restrict_space that) qed show "f -` T \<inter> (S-N) \<in> sets (lebesgue_on (S-N))" if "f -` T \<inter> S \<in> sets (lebesgue_on S)" for T proof - have "(S - N) \<inter> f -` T = (S - N) \<inter> (f -` T \<inter> S)" by blast then have "(S - N) \<inter> f -` T \<in> sets.restricted_space lebesgue (S - N)" by (metis S image_iff sets.Int_space_eq2 sets_restrict_space_iff that) then show ?thesis by (simp add: inf.commute sets_restrict_space) qed qed auto lemma lebesgue_measurable_diff_null: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "N \<in> null_sets lebesgue" shows "f \<in> borel_measurable (lebesgue_on (-N)) \<longleftrightarrow> f \<in> borel_measurable lebesgue" by (simp add: Compl_eq_Diff_UNIV assms borel_measurable_diff_null lebesgue_on_UNIV_eq) proposition measurable_on_imp_borel_measurable_lebesgue_UNIV: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "f measurable_on UNIV" shows "f \<in> borel_measurable lebesgue" proof - obtain N and F where NF: "negligible N" and conF: "\<And>n. continuous_on UNIV (F n)" and tendsF: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> f x" using assms by (auto simp: measurable_on_def) obtain N where "N \<in> null_sets lebesgue" "f \<in> borel_measurable (lebesgue_on (-N))" proof show "f \<in> borel_measurable (lebesgue_on (- N))" proof (rule borel_measurable_LIMSEQ_metric) show "F i \<in> borel_measurable (lebesgue_on (- N))" for i by (meson Compl_in_sets_lebesgue NF conF continuous_imp_measurable_on_sets_lebesgue continuous_on_subset negligible_imp_sets subset_UNIV) show "(\<lambda>i. F i x) \<longlonglongrightarrow> f x" if "x \<in> space (lebesgue_on (- N))" for x using that by (simp add: tendsF) qed show "N \<in> null_sets lebesgue" using NF negligible_iff_null_sets by blast qed then show ?thesis using lebesgue_measurable_diff_null by blast qed corollary measurable_on_imp_borel_measurable_lebesgue: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "f measurable_on S" and S: "S \<in> sets lebesgue" shows "f \<in> borel_measurable (lebesgue_on S)" proof - have "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV" using assms(1) measurable_on_UNIV by blast then show ?thesis by (simp add: borel_measurable_if_D measurable_on_imp_borel_measurable_lebesgue_UNIV) qed proposition measurable_on_limit: fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes f: "\<And>n. f n measurable_on S" and N: "negligible N" and lim: "\<And>x. x \<in> S - N \<Longrightarrow> (\<lambda>n. f n x) \<longlonglongrightarrow> g x" shows "g measurable_on S" proof - have "box (0::'b) One homeomorphic (UNIV::'b set)" by (simp add: homeomorphic_box_UNIV) then obtain h h':: "'b\<Rightarrow>'b" where hh': "\<And>x. x \<in> box 0 One \<Longrightarrow> h (h' x) = x" and h'im: "h' ` box 0 One = UNIV" and conth: "continuous_on UNIV h" and conth': "continuous_on (box 0 One) h'" and h'h: "\<And>y. h' (h y) = y" and rangeh: "range h = box 0 One" by (auto simp: homeomorphic_def homeomorphism_def) have "norm y \<le> DIM('b)" if y: "y \<in> box 0 One" for y::'b proof - have y01: "0 < y \<bullet> i" "y \<bullet> i < 1" if "i \<in> Basis" for i using that y by (auto simp: mem_box) have "norm y \<le> (\<Sum>i\<in>Basis. \<bar>y \<bullet> i\<bar>)" using norm_le_l1 by blast also have "\<dots> \<le> (\<Sum>i::'b\<in>Basis. 1)" proof (rule sum_mono) show "\<bar>y \<bullet> i\<bar> \<le> 1" if "i \<in> Basis" for i using y01 that by fastforce qed also have "\<dots> \<le> DIM('b)" by auto finally show ?thesis . qed then have norm_le: "norm(h y) \<le> DIM('b)" for y by (metis UNIV_I image_eqI rangeh) have "(h' \<circ> (h \<circ> (\<lambda>x. if x \<in> S then g x else 0))) measurable_on UNIV" proof (rule measurable_on_compose_continuous_box) let ?\<chi> = "h \<circ> (\<lambda>x. if x \<in> S then g x else 0)" let ?f = "\<lambda>n. h \<circ> (\<lambda>x. if x \<in> S then f n x else 0)" show "?\<chi> measurable_on UNIV" proof (rule integrable_subintervals_imp_measurable) show "?\<chi> integrable_on cbox a b" for a b proof (rule integrable_spike_set) show "?\<chi> integrable_on (cbox a b - N)" proof (rule dominated_convergence_integrable) show const: "(\<lambda>x. DIM('b)) integrable_on cbox a b - N" by (simp add: N has_integral_iff integrable_const integrable_negligible integrable_setdiff negligible_diff) show "norm ((h \<circ> (\<lambda>x. if x \<in> S then g x else 0)) x) \<le> DIM('b)" if "x \<in> cbox a b - N" for x using that norm_le by (simp add: o_def) show "(\<lambda>k. ?f k x) \<longlonglongrightarrow> ?\<chi> x" if "x \<in> cbox a b - N" for x using that lim [of x] conth by (auto simp: continuous_on_def intro: tendsto_compose) show "(?f n) absolutely_integrable_on cbox a b - N" for n proof (rule measurable_bounded_by_integrable_imp_absolutely_integrable) show "?f n \<in> borel_measurable (lebesgue_on (cbox a b - N))" proof (rule measurable_on_imp_borel_measurable_lebesgue [OF measurable_on_spike_set]) show "?f n measurable_on cbox a b" unfolding measurable_on_UNIV [symmetric, of _ "cbox a b"] proof (rule measurable_on_restrict) have f': "(\<lambda>x. if x \<in> S then f n x else 0) measurable_on UNIV" by (simp add: f measurable_on_UNIV) show "?f n measurable_on UNIV" using measurable_on_compose_continuous [OF f' conth] by auto qed auto show "negligible (sym_diff (cbox a b) (cbox a b - N))" by (auto intro: negligible_subset [OF N]) show "cbox a b - N \<in> sets lebesgue" by (simp add: N negligible_imp_sets sets.Diff) qed show "cbox a b - N \<in> sets lebesgue" by (simp add: N negligible_imp_sets sets.Diff) show "norm (?f n x) \<le> DIM('b)" if "x \<in> cbox a b - N" for x using that local.norm_le by simp qed (auto simp: const) qed show "negligible {x \<in> cbox a b - N - cbox a b. ?\<chi> x \<noteq> 0}" by (auto simp: empty_imp_negligible) have "{x \<in> cbox a b - (cbox a b - N). ?\<chi> x \<noteq> 0} \<subseteq> N" by auto then show "negligible {x \<in> cbox a b - (cbox a b - N). ?\<chi> x \<noteq> 0}" using N negligible_subset by blast qed qed show "?\<chi> x \<in> box 0 One" for x using rangeh by auto show "continuous_on (box 0 One) h'" by (rule conth') qed then show ?thesis by (simp add: o_def h'h measurable_on_UNIV) qed lemma measurable_on_if_simple_function_limit: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" shows "\<lbrakk>\<And>n. g n measurable_on UNIV; \<And>n. finite (range (g n)); \<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x\<rbrakk> \<Longrightarrow> f measurable_on UNIV" by (force intro: measurable_on_limit [where N="{}"]) lemma lebesgue_measurable_imp_measurable_on_nnreal_UNIV: fixes u :: "'a::euclidean_space \<Rightarrow> real" assumes u: "u \<in> borel_measurable lebesgue" and nn: "\<And>x. u x \<ge> 0" shows "u measurable_on UNIV" proof - obtain f where "incseq f" and f: "\<forall>i. simple_function lebesgue (f i)" and bdd: "\<And>x. bdd_above (range (\<lambda>i. f i x))" and nnf: "\<And>i x. 0 \<le> f i x" and *: "u = (SUP i. f i)" using borel_measurable_implies_simple_function_sequence_real nn u by metis show ?thesis unfolding * proof (rule measurable_on_if_simple_function_limit [of concl: "Sup (range f)"]) show "(f i) measurable_on UNIV" for i by (simp add: f nnf simple_function_measurable_on_UNIV) show "finite (range (f i))" for i by (metis f simple_function_def space_borel space_completion space_lborel) show "(\<lambda>i. f i x) \<longlonglongrightarrow> Sup (range f) x" for x proof - have "incseq (\<lambda>i. f i x)" using \<open>incseq f\<close> apply (auto simp: incseq_def) by (simp add: le_funD) then show ?thesis by (metis SUP_apply bdd LIMSEQ_incseq_SUP) qed qed qed lemma lebesgue_measurable_imp_measurable_on_nnreal: fixes u :: "'a::euclidean_space \<Rightarrow> real" assumes "u \<in> borel_measurable lebesgue" "\<And>x. u x \<ge> 0""S \<in> sets lebesgue" shows "u measurable_on S" unfolding measurable_on_UNIV [symmetric, of u] using assms by (auto intro: lebesgue_measurable_imp_measurable_on_nnreal_UNIV) lemma lebesgue_measurable_imp_measurable_on_real: fixes u :: "'a::euclidean_space \<Rightarrow> real" assumes u: "u \<in> borel_measurable lebesgue" and S: "S \<in> sets lebesgue" shows "u measurable_on S" proof - let ?f = "\<lambda>x. \<bar>u x\<bar> + u x" let ?g = "\<lambda>x. \<bar>u x\<bar> - u x" have "?f measurable_on S" "?g measurable_on S" using S u by (auto intro: lebesgue_measurable_imp_measurable_on_nnreal) then have "(\<lambda>x. (?f x - ?g x) / 2) measurable_on S" using measurable_on_cdivide measurable_on_diff by blast then show ?thesis by auto qed proposition lebesgue_measurable_imp_measurable_on: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes f: "f \<in> borel_measurable lebesgue" and S: "S \<in> sets lebesgue" shows "f measurable_on S" unfolding measurable_on_componentwise [of f] proof fix i::'b assume "i \<in> Basis" have "(\<lambda>x. (f x \<bullet> i)) \<in> borel_measurable lebesgue" using \<open>i \<in> Basis\<close> borel_measurable_euclidean_space f by blast then have "(\<lambda>x. (f x \<bullet> i)) measurable_on S" using S lebesgue_measurable_imp_measurable_on_real by blast then show "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on S" by (intro measurable_on_scaleR measurable_on_const S) qed proposition measurable_on_iff_borel_measurable: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "S \<in> sets lebesgue" shows "f measurable_on S \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)" (is "?lhs = ?rhs") proof show "f \<in> borel_measurable (lebesgue_on S)" if "f measurable_on S" using that by (simp add: assms measurable_on_imp_borel_measurable_lebesgue) next assume "f \<in> borel_measurable (lebesgue_on S)" then have "(\<lambda>a. if a \<in> S then f a else 0) measurable_on UNIV" by (simp add: assms borel_measurable_if lebesgue_measurable_imp_measurable_on) then show "f measurable_on S" using measurable_on_UNIV by blast qed subsection \<open>Monotonic functions are Lebesgue integrable\<close> (*Can these be generalised from type real?*) lemma integrable_mono_on_nonneg: fixes f :: "real \<Rightarrow> real" assumes mon: "mono_on {a..b} f" and 0: "\<And>x. 0 \<le> f x" shows "integrable (lebesgue_on {a..b}) f" proof - have "space lborel = space lebesgue" "sets borel \<subseteq> sets lebesgue" by force+ then have fborel: "f \<in> borel_measurable (lebesgue_on {a..b})" by (metis mon borel_measurable_mono_on_fnc borel_measurable_subalgebra mono_restrict_space space_lborel space_restrict_space) then obtain g where g: "incseq g" and simple: "\<And>i. simple_function (lebesgue_on {a..b}) (g i)" and bdd: " (\<forall>x. bdd_above (range (\<lambda>i. g i x)))" and nonneg: "\<forall>i x. 0 \<le> g i x" and fsup: "f = (SUP i. g i)" by (metis borel_measurable_implies_simple_function_sequence_real 0) have "f ` {a..b} \<subseteq> {f a..f b}" using assms by (auto simp: mono_on_def) have g_le_f: "g i x \<le> f x" for i x proof - have "bdd_above ((\<lambda>h. h x) ` range g)" using bdd cSUP_lessD linorder_not_less by fastforce then show ?thesis by (metis SUP_apply UNIV_I bdd cSUP_upper fsup) qed then have gfb: "g i x \<le> f b" if "x \<in> {a..b}" for i x by (smt (verit, best) mon atLeastAtMost_iff mono_on_def that) have g_le: "g i x \<le> g j x" if "i\<le>j" for i j x using g by (simp add: incseq_def le_funD that) show "integrable (lebesgue_on {a..b}) ( f)" proof (rule integrable_dominated_convergence) show "f \<in> borel_measurable (lebesgue_on {a..b})" using fborel by blast have "\<And>x. (\<lambda>i. g i x) \<longlonglongrightarrow> (SUP h \<in> range g. h x)" proof (rule order_tendstoI) show "\<forall>\<^sub>F i in sequentially. y < g i x" if "y < (SUP h\<in>range g. h x)" for x y proof - from that obtain h where h: "h \<in> range g" "y < h x" using g_le_f by (subst (asm)less_cSUP_iff) fastforce+ then show ?thesis by (smt (verit, ccfv_SIG) eventually_sequentially g_le imageE) qed show "\<forall>\<^sub>F i in sequentially. g i x < y" if "(SUP h\<in>range g. h x) < y" for x y by (smt (verit, best) that Sup_apply g_le_f always_eventually fsup image_cong) qed then show "AE x in lebesgue_on {a..b}. (\<lambda>i. g i x) \<longlonglongrightarrow> f x" by (simp add: fsup) fix i show "g i \<in> borel_measurable (lebesgue_on {a..b})" using borel_measurable_simple_function simple by blast show "AE x in lebesgue_on {a..b}. norm (g i x) \<le> f b" by (simp add: gfb nonneg Measure_Space.AE_I' [of "{}"]) qed auto qed lemma integrable_mono_on: fixes f :: "real \<Rightarrow> real" assumes "mono_on {a..b} f" shows "integrable (lebesgue_on {a..b}) f" proof - define f' where "f' \<equiv> \<lambda>x. if x \<in> {a..b} then f x - f a else 0" have "mono_on {a..b} f'" by (smt (verit, best) assms f'_def mono_on_def) moreover have 0: "\<And>x. 0 \<le> f' x" by (smt (verit, best) assms atLeastAtMost_iff f'_def mono_on_def) ultimately have "integrable (lebesgue_on {a..b}) f'" using integrable_mono_on_nonneg by presburger then have "integrable (lebesgue_on {a..b}) (\<lambda>x. f' x + f a)" by force moreover have "space lborel = space lebesgue" "sets borel \<subseteq> sets lebesgue" by force+ then have fborel: "f \<in> borel_measurable (lebesgue_on {a..b})" by (metis assms borel_measurable_mono_on_fnc borel_measurable_subalgebra mono_restrict_space space_lborel space_restrict_space) ultimately show ?thesis by (rule integrable_cong_AE_imp) (auto simp add: f'_def) qed lemma integrable_on_mono_on: fixes f :: "real \<Rightarrow> real" assumes "mono_on {a..b} f" shows "f integrable_on {a..b}" by (simp add: assms integrable_mono_on integrable_on_lebesgue_on) subsection \<open>Measurability on generalisations of the binary product\<close> lemma measurable_on_bilinear: fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space" assumes h: "bilinear h" and f: "f measurable_on S" and g: "g measurable_on S" shows "(\<lambda>x. h (f x) (g x)) measurable_on S" proof (rule measurable_on_combine [where h = h]) show "continuous_on UNIV (\<lambda>x. h (fst x) (snd x))" by (simp add: bilinear_continuous_on_compose [OF continuous_on_fst continuous_on_snd h]) show "h 0 0 = 0" by (simp add: bilinear_lzero h) qed (auto intro: assms) lemma borel_measurable_bilinear: fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space" assumes "bilinear h" "f \<in> borel_measurable (lebesgue_on S)" "g \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue" shows "(\<lambda>x. h (f x) (g x)) \<in> borel_measurable (lebesgue_on S)" using assms measurable_on_bilinear [of h f S g] by (simp flip: measurable_on_iff_borel_measurable) lemma absolutely_integrable_bounded_measurable_product: fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space" assumes "bilinear h" and f: "f \<in> borel_measurable (lebesgue_on S)" "S \<in> sets lebesgue" and bou: "bounded (f ` S)" and g: "g absolutely_integrable_on S" shows "(\<lambda>x. h (f x) (g x)) absolutely_integrable_on S" proof - obtain B where "B > 0" and B: "\<And>x y. norm (h x y) \<le> B * norm x * norm y" using bilinear_bounded_pos \<open>bilinear h\<close> by blast obtain C where "C > 0" and C: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> C" using bounded_pos by (metis bou imageI) show ?thesis proof (rule measurable_bounded_by_integrable_imp_absolutely_integrable [OF _ \<open>S \<in> sets lebesgue\<close>]) show "norm (h (f x) (g x)) \<le> B * C * norm(g x)" if "x \<in> S" for x by (meson less_le mult_left_mono mult_right_mono norm_ge_zero order_trans that \<open>B > 0\<close> B C) show "(\<lambda>x. h (f x) (g x)) \<in> borel_measurable (lebesgue_on S)" using \<open>bilinear h\<close> f g by (blast intro: borel_measurable_bilinear dest: absolutely_integrable_measurable) show "(\<lambda>x. B * C * norm(g x)) integrable_on S" using \<open>0 < B\<close> \<open>0 < C\<close> absolutely_integrable_on_def g by auto qed qed lemma absolutely_integrable_bounded_measurable_product_real: fixes f :: "real \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)" "S \<in> sets lebesgue" and "bounded (f ` S)" and "g absolutely_integrable_on S" shows "(\<lambda>x. f x * g x) absolutely_integrable_on S" using absolutely_integrable_bounded_measurable_product bilinear_times assms by blast lemma borel_measurable_AE: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "f \<in> borel_measurable lebesgue" and ae: "AE x in lebesgue. f x = g x" shows "g \<in> borel_measurable lebesgue" proof - obtain N where N: "N \<in> null_sets lebesgue" "\<And>x. x \<notin> N \<Longrightarrow> f x = g x" using ae unfolding completion.AE_iff_null_sets by auto have "f measurable_on UNIV" by (simp add: assms lebesgue_measurable_imp_measurable_on) then have "g measurable_on UNIV" by (metis Diff_iff N measurable_on_spike negligible_iff_null_sets) then show ?thesis using measurable_on_imp_borel_measurable_lebesgue_UNIV by blast qed lemma has_bochner_integral_combine: fixes f :: "real \<Rightarrow> 'a::euclidean_space" assumes "a \<le> c" "c \<le> b" and ac: "has_bochner_integral (lebesgue_on {a..c}) f i" and cb: "has_bochner_integral (lebesgue_on {c..b}) f j" shows "has_bochner_integral (lebesgue_on {a..b}) f(i + j)" proof - have i: "has_bochner_integral lebesgue (\<lambda>x. indicator {a..c} x *\<^sub>R f x) i" and j: "has_bochner_integral lebesgue (\<lambda>x. indicator {c..b} x *\<^sub>R f x) j" using assms by (auto simp: has_bochner_integral_restrict_space) have AE: "AE x in lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" proof (rule AE_I') have eq: "indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" if "x \<noteq> c" for x using assms that by (auto simp: indicator_def) then show "{x \<in> space lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x \<noteq> indicat_real {a..b} x *\<^sub>R f x} \<subseteq> {c}" by auto qed auto have "has_bochner_integral lebesgue (\<lambda>x. indicator {a..b} x *\<^sub>R f x) (i + j)" proof (rule has_bochner_integralI_AE [OF has_bochner_integral_add [OF i j] _ AE]) have eq: "indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" if "x \<noteq> c" for x using assms that by (auto simp: indicator_def) show "(\<lambda>x. indicat_real {a..b} x *\<^sub>R f x) \<in> borel_measurable lebesgue" proof (rule borel_measurable_AE [OF borel_measurable_add AE]) show "(\<lambda>x. indicator {a..c} x *\<^sub>R f x) \<in> borel_measurable lebesgue" "(\<lambda>x. indicator {c..b} x *\<^sub>R f x) \<in> borel_measurable lebesgue" using i j by auto qed qed then show ?thesis by (simp add: has_bochner_integral_restrict_space) qed lemma integrable_combine: fixes f :: "real \<Rightarrow> 'a::euclidean_space" assumes "integrable (lebesgue_on {a..c}) f" "integrable (lebesgue_on {c..b}) f" and "a \<le> c" "c \<le> b" shows "integrable (lebesgue_on {a..b}) f" using assms has_bochner_integral_combine has_bochner_integral_iff by blast lemma integral_combine: fixes f :: "real \<Rightarrow> 'a::euclidean_space" assumes f: "integrable (lebesgue_on {a..b}) f" and "a \<le> c" "c \<le> b" shows "integral\<^sup>L (lebesgue_on {a..b}) f = integral\<^sup>L (lebesgue_on {a..c}) f + integral\<^sup>L (lebesgue_on {c..b}) f" proof - have i: "has_bochner_integral (lebesgue_on {a..c}) f(integral\<^sup>L (lebesgue_on {a..c}) f)" using integrable_subinterval \<open>c \<le> b\<close> f has_bochner_integral_iff by fastforce have j: "has_bochner_integral (lebesgue_on {c..b}) f(integral\<^sup>L (lebesgue_on {c..b}) f)" using integrable_subinterval \<open>a \<le> c\<close> f has_bochner_integral_iff by fastforce show ?thesis by (meson \<open>a \<le> c\<close> \<open>c \<le> b\<close> has_bochner_integral_combine has_bochner_integral_iff i j) qed lemma has_bochner_integral_null [intro]: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "N \<in> null_sets lebesgue" shows "has_bochner_integral (lebesgue_on N) f 0" unfolding has_bochner_integral_iff \<comment>\<open>strange that the proof's so long\<close> proof show "integrable (lebesgue_on N) f" proof (subst integrable_restrict_space) show "N \<inter> space lebesgue \<in> sets lebesgue" using assms by force show "integrable lebesgue (\<lambda>x. indicat_real N x *\<^sub>R f x)" proof (rule integrable_cong_AE_imp) show "integrable lebesgue (\<lambda>x. 0)" by simp show *: "AE x in lebesgue. 0 = indicat_real N x *\<^sub>R f x" using assms by (simp add: indicator_def completion.null_sets_iff_AE eventually_mono) show "(\<lambda>x. indicat_real N x *\<^sub>R f x) \<in> borel_measurable lebesgue" by (auto intro: borel_measurable_AE [OF _ *]) qed qed show "integral\<^sup>L (lebesgue_on N) f = 0" proof (rule integral_eq_zero_AE) show "AE x in lebesgue_on N. f x = 0" by (rule AE_I' [where N=N]) (auto simp: assms null_setsD2 null_sets_restrict_space) qed qed lemma has_bochner_integral_null_eq[simp]: fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" assumes "N \<in> null_sets lebesgue" shows "has_bochner_integral (lebesgue_on N) f i \<longleftrightarrow> i = 0" using assms has_bochner_integral_eq by blast end