(* 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: divide_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: divide_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, hide_lams) 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: divide_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: divide_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: divide_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 algebra_simps divide_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)"
apply (rule sum.cong)
using fab
apply auto
apply (intro order_antisym)
apply (auto simp: mem_box)
using less_imp_le apply blast
by (metis (full_types) linear max_less_iff_conj min.bounded_iff not_le)
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
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 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 divide_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>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