(* Title: HOL/Analysis/Lebesgue_Measure.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
Author: Jeremy Avigad
Author: Luke Serafin
*)
section \<open>Lebesgue Measure\<close>
theory Lebesgue_Measure
imports
Finite_Product_Measure
Caratheodory
Complete_Measure
Summation_Tests
Regularity
begin
lemma measure_eqI_lessThan:
fixes M N :: "real measure"
assumes sets: "sets M = sets borel" "sets N = sets borel"
assumes fin: "\<And>x. emeasure M {x <..} < \<infinity>"
assumes "\<And>x. emeasure M {x <..} = emeasure N {x <..}"
shows "M = N"
proof (rule measure_eqI_generator_eq_countable)
let ?LT = "\<lambda>a::real. {a <..}" let ?E = "range ?LT"
show "Int_stable ?E"
by (auto simp: Int_stable_def lessThan_Int_lessThan)
show "?E \<subseteq> Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
unfolding sets borel_Ioi by auto
show "?LT`Rats \<subseteq> ?E" "(\<Union>i\<in>Rats. ?LT i) = UNIV" "\<And>a. a \<in> ?LT`Rats \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
using fin by (auto intro: Rats_no_bot_less simp: less_top)
qed (auto intro: assms countable_rat)
subsection \<open>Measures defined by monotonous functions\<close>
text \<open>
Every right-continuous and nondecreasing function gives rise to a measure on the reals:
\<close>
definition\<^marker>\<open>tag important\<close> interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
"interval_measure F =
extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a<..b}) (\<lambda>(a, b). ennreal (F b - F a))"
lemma\<^marker>\<open>tag important\<close> emeasure_interval_measure_Ioc:
assumes "a \<le> b"
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes right_cont_F : "\<And>a. continuous (at_right a) F"
shows "emeasure (interval_measure F) {a<..b} = F b - F a"
proof (rule extend_measure_caratheodory_pair[OF interval_measure_def \<open>a \<le> b\<close>])
show "semiring_of_sets UNIV {{a<..b} |a b :: real. a \<le> b}"
proof (unfold_locales, safe)
fix a b c d :: real assume *: "a \<le> b" "c \<le> d"
then show "\<exists>C\<subseteq>{{a<..b} |a b. a \<le> b}. finite C \<and> disjoint C \<and> {a<..b} - {c<..d} = \<Union>C"
proof cases
let ?C = "{{a<..b}}"
assume "b < c \<or> d \<le> a \<or> d \<le> c"
with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
by (auto simp add: disjoint_def)
thus ?thesis ..
next
let ?C = "{{a<..c}, {d<..b}}"
assume "\<not> (b < c \<or> d \<le> a \<or> d \<le> c)"
with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
by (auto simp add: disjoint_def Ioc_inj) (metis linear)+
thus ?thesis ..
qed
qed (auto simp: Ioc_inj, metis linear)
next
fix l r :: "nat \<Rightarrow> real" and a b :: real
assume l_r[simp]: "\<And>n. l n \<le> r n" and "a \<le> b" and disj: "disjoint_family (\<lambda>n. {l n<..r n})"
assume lr_eq_ab: "(\<Union>i. {l i<..r i}) = {a<..b}"
have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> F a \<le> F b"
by (auto intro!: l_r mono_F)
{ fix S :: "nat set" assume "finite S"
moreover note \<open>a \<le> b\<close>
moreover have "\<And>i. i \<in> S \<Longrightarrow> {l i <.. r i} \<subseteq> {a <.. b}"
unfolding lr_eq_ab[symmetric] by auto
ultimately have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
proof (induction S arbitrary: a rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "\<exists>i\<in>S. l i < r i"
with \<open>finite S\<close> have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
by (intro Min_in) auto
then obtain m where m: "m \<in> S" "l m < r m" "l m = Min (l ` {i\<in>S. l i < r i})"
by fastforce
have "(\<Sum>i\<in>S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\<Sum>i\<in>S - {m}. F (r i) - F (l i))"
using m psubset by (intro sum.remove) auto
also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
proof (intro psubset.IH)
show "S - {m} \<subset> S"
using \<open>m\<in>S\<close> by auto
show "r m \<le> b"
using psubset.prems(2)[OF \<open>m\<in>S\<close>] \<open>l m < r m\<close> by auto
next
fix i assume "i \<in> S - {m}"
then have i: "i \<in> S" "i \<noteq> m" by auto
{ assume i': "l i < r i" "l i < r m"
with \<open>finite S\<close> i m have "l m \<le> l i"
by auto
with i' have "{l i <.. r i} \<inter> {l m <.. r m} \<noteq> {}"
by auto
then have False
using disjoint_family_onD[OF disj, of i m] i by auto }
then have "l i \<noteq> r i \<Longrightarrow> r m \<le> l i"
unfolding not_less[symmetric] using l_r[of i] by auto
then show "{l i <.. r i} \<subseteq> {r m <.. b}"
using psubset.prems(2)[OF \<open>i\<in>S\<close>] by auto
qed
also have "F (r m) - F (l m) \<le> F (r m) - F a"
using psubset.prems(2)[OF \<open>m \<in> S\<close>] \<open>l m < r m\<close>
by (auto simp add: Ioc_subset_iff intro!: mono_F)
finally show ?case
by (auto intro: add_mono)
qed (auto simp add: \<open>a \<le> b\<close> less_le)
qed }
note claim1 = this
(* second key induction: a lower bound on the measures of any finite collection of Ai's
that cover an interval {u..v} *)
{ fix S u v and l r :: "nat \<Rightarrow> real"
assume "finite S" "\<And>i. i\<in>S \<Longrightarrow> l i < r i" "{u..v} \<subseteq> (\<Union>i\<in>S. {l i<..< r i})"
then have "F v - F u \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
proof (induction arbitrary: v u rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "S = {}" then show ?case
using psubset by (simp add: mono_F)
next
assume "S \<noteq> {}"
then obtain j where "j \<in> S"
by auto
let ?R = "r j < u \<or> l j > v \<or> (\<exists>i\<in>S-{j}. l i \<le> l j \<and> r j \<le> r i)"
show ?case
proof cases
assume "?R"
with \<open>j \<in> S\<close> psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
apply (simp add: subset_eq Ball_def Bex_def)
by (metis order_le_less_trans order_less_le_trans order_less_not_sym)
with \<open>j \<in> S\<close> have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
by (intro psubset) auto
also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
using psubset.prems
by (intro sum_mono2 psubset) (auto intro: less_imp_le)
finally show ?thesis .
next
assume "\<not> ?R"
then have j: "u \<le> r j" "l j \<le> v" "\<And>i. i \<in> S - {j} \<Longrightarrow> r i < r j \<or> l i > l j"
by (auto simp: not_less)
let ?S1 = "{i \<in> S. l i < l j}"
let ?S2 = "{i \<in> S. r i > r j}"
have *: "?S1 \<inter> ?S2 = {}"
using j by (fastforce simp add: disjoint_iff)
have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
by (intro sum_mono2) (auto intro: less_imp_le)
also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
(\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
using psubset(1) by (simp add: * sum.union_disjoint disjoint_iff_not_equal)
also (xtrans) have "(\<Sum>i\<in>?S1. F (r i) - F (l i)) \<ge> F (l j) - F u"
using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
apply (intro psubset.IH psubset)
apply (auto simp: subset_eq Ball_def)
apply (metis less_le_trans not_le)
done
also (xtrans) have "(\<Sum>i\<in>?S2. F (r i) - F (l i)) \<ge> F v - F (r j)"
using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
apply (intro psubset.IH psubset)
apply (auto simp: subset_eq Ball_def)
apply (metis le_less_trans not_le)
done
finally (xtrans) show ?case
by (auto simp: add_mono)
qed
qed
qed }
note claim2 = this
(* now prove the inequality going the other way *)
have "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i)))"
proof (rule ennreal_le_epsilon)
fix epsilon :: real assume egt0: "epsilon > 0"
have "\<forall>i. \<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
proof
fix i
note right_cont_F [of "r i"]
then have "\<exists>d>0. F (r i + d) - F (r i) < epsilon / 2 ^ (i + 2)"
by (simp add: continuous_at_right_real_increasing egt0)
thus "\<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
by force
qed
then obtain delta where
deltai_gt0: "\<And>i. delta i > 0" and
deltai_prop: "\<And>i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
by metis
have "\<exists>a' > a. F a' - F a < epsilon / 2"
using right_cont_F [of a]
by (metis continuous_at_right_real_increasing egt0 half_gt_zero less_add_same_cancel1 mono_F)
then obtain a' where a'lea [arith]: "a' > a" and
a_prop: "F a' - F a < epsilon / 2"
by auto
define S' where "S' = {i. l i < r i}"
obtain S :: "nat set" where "S \<subseteq> S'" and finS: "finite S"
and Sprop: "{a'..b} \<subseteq> (\<Union>i \<in> S. {l i<..<r i + delta i})"
proof (rule compactE_image)
show "compact {a'..b}"
by (rule compact_Icc)
show "\<And>i. i \<in> S' \<Longrightarrow> open ({l i<..<r i + delta i})" by auto
have "{a'..b} \<subseteq> {a <.. b}"
by auto
also have "{a <.. b} = (\<Union>i\<in>S'. {l i<..r i})"
unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
also have "\<dots> \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})"
by (intro UN_mono; simp add: add.commute add_strict_increasing deltai_gt0 subset_iff)
finally show "{a'..b} \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})" .
qed
with S'_def have Sprop2: "\<And>i. i \<in> S \<Longrightarrow> l i < r i" by auto
obtain n where Sbound: "\<And>i. i \<in> S \<Longrightarrow> i \<le> n"
using Max_ge finS by blast
have "F b - F a = (F b - F a') + (F a' - F a)"
by auto
also have "... \<le> (F b - F a') + epsilon / 2"
using a_prop by (intro add_left_mono) simp
also have "... \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
proof -
have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i + delta i) - F (l i))"
using claim2 l_r Sprop by (simp add: deltai_gt0 finS add.commute add_strict_increasing)
also have "... \<le> (\<Sum>i \<in> S. F(r i) - F(l i) + epsilon / 2^(i+2))"
by (smt (verit) sum_mono deltai_prop)
also have "... = (\<Sum>i \<in> S. F(r i) - F(l i)) +
(epsilon / 4) * (\<Sum>i \<in> S. (1 / 2)^i)" (is "_ = ?t + _")
by (subst sum.distrib) (simp add: field_simps sum_distrib_left)
also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
using egt0 Sbound by (intro add_left_mono mult_left_mono sum_mono2) force+
also have "... \<le> ?t + (epsilon / 2)"
using egt0 by (simp add: geometric_sum add_left_mono mult_left_mono)
finally have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2"
by simp
then show ?thesis
by linarith
qed
also have "... = (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon"
by auto
also have "... \<le> (\<Sum>i\<le>n. F (r i) - F (l i)) + epsilon"
using finS Sbound Sprop by (auto intro!: add_right_mono sum_mono2)
finally have "ennreal (F b - F a) \<le> (\<Sum>i\<le>n. ennreal (F (r i) - F (l i))) + epsilon"
using egt0 by (simp add: sum_nonneg flip: ennreal_plus)
then show "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i))) + (epsilon :: real)"
by (rule order_trans) (auto intro!: add_mono sum_le_suminf simp del: sum_ennreal)
qed
moreover have "(\<Sum>i. ennreal (F (r i) - F (l i))) \<le> ennreal (F b - F a)"
using \<open>a \<le> b\<close> by (auto intro!: suminf_le_const ennreal_le_iff[THEN iffD2] claim1)
ultimately show "(\<Sum>n. ennreal (F (r n) - F (l n))) = ennreal (F b - F a)"
by (rule antisym[rotated])
qed (auto simp: Ioc_inj mono_F)
lemma measure_interval_measure_Ioc:
assumes "a \<le> b" and "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y" and "\<And>a. continuous (at_right a) F"
shows "measure (interval_measure F) {a <.. b} = F b - F a"
unfolding measure_def
by (simp add: assms emeasure_interval_measure_Ioc)
lemma emeasure_interval_measure_Ioc_eq:
"(\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y) \<Longrightarrow> (\<And>a. continuous (at_right a) F) \<Longrightarrow>
emeasure (interval_measure F) {a <.. b} = (if a \<le> b then F b - F a else 0)"
using emeasure_interval_measure_Ioc[of a b F] by auto
lemma\<^marker>\<open>tag important\<close> sets_interval_measure [simp, measurable_cong]:
"sets (interval_measure F) = sets borel"
apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc image_def split: prod.split)
by (metis greaterThanAtMost_empty nle_le)
lemma space_interval_measure [simp]: "space (interval_measure F) = UNIV"
by (simp add: interval_measure_def space_extend_measure)
lemma emeasure_interval_measure_Icc:
assumes "a \<le> b"
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes cont_F : "continuous_on UNIV F"
shows "emeasure (interval_measure F) {a .. b} = F b - F a"
proof (rule tendsto_unique)
{ fix a b :: real assume "a \<le> b" then have "emeasure (interval_measure F) {a <.. b} = F b - F a"
using cont_F
by (subst emeasure_interval_measure_Ioc)
(auto intro: mono_F continuous_within_subset simp: continuous_on_eq_continuous_within) }
note * = this
let ?F = "interval_measure F"
show "((\<lambda>a. F b - F a) \<longlongrightarrow> emeasure ?F {a..b}) (at_left a)"
proof (rule tendsto_at_left_sequentially)
show "a - 1 < a" by simp
fix X assume X: "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
then have "emeasure (interval_measure F) {X n<..b} \<noteq> \<infinity>" for n
by (smt (verit) "*" \<open>a \<le> b\<close> ennreal_neq_top infinity_ennreal_def)
with X have "(\<lambda>n. emeasure ?F {X n<..b}) \<longlonglongrightarrow> emeasure ?F (\<Inter>n. {X n <..b})"
by (intro Lim_emeasure_decseq; force simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
also have "(\<Inter>n. {X n <..b}) = {a..b}"
apply auto
apply (rule LIMSEQ_le_const2[OF \<open>X \<longlonglongrightarrow> a\<close>])
using less_eq_real_def apply presburger
using X(1) order_less_le_trans by blast
also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
finally show "(\<lambda>n. F b - F (X n)) \<longlonglongrightarrow> emeasure ?F {a..b}" .
qed
show "((\<lambda>a. ennreal (F b - F a)) \<longlongrightarrow> F b - F a) (at_left a)"
by (rule continuous_on_tendsto_compose[where g="\<lambda>x. x" and s=UNIV])
(auto simp: continuous_on_ennreal continuous_on_diff cont_F)
qed (rule trivial_limit_at_left_real)
lemma\<^marker>\<open>tag important\<close> sigma_finite_interval_measure:
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes right_cont_F : "\<And>a. continuous (at_right a) F"
shows "sigma_finite_measure (interval_measure F)"
apply unfold_locales
apply (intro exI[of _ "(\<lambda>(a, b). {a <.. b}) ` (\<rat> \<times> \<rat>)"])
apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
done
subsection \<open>Lebesgue-Borel measure\<close>
definition\<^marker>\<open>tag important\<close> lborel :: "('a :: euclidean_space) measure" where
"lborel = distr (\<Pi>\<^sub>M b\<in>Basis. interval_measure (\<lambda>x. x)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
abbreviation\<^marker>\<open>tag important\<close> lebesgue :: "'a::euclidean_space measure"
where "lebesgue \<equiv> completion lborel"
abbreviation\<^marker>\<open>tag important\<close> lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"
lemma lebesgue_on_mono:
assumes major: "AE x in lebesgue_on S. P x" and minor: "\<And>x.\<lbrakk>P x; x \<in> S\<rbrakk> \<Longrightarrow> Q x"
shows "AE x in lebesgue_on S. Q x"
proof -
have "AE a in lebesgue_on S. P a \<longrightarrow> Q a"
using minor space_restrict_space by fastforce
then show ?thesis
using major by auto
qed
lemma integral_eq_zero_null_sets:
assumes "S \<in> null_sets lebesgue"
shows "integral\<^sup>L (lebesgue_on S) f = 0"
proof (rule integral_eq_zero_AE)
show "AE x in lebesgue_on S. f x = 0"
by (metis (no_types, lifting) assms AE_not_in lebesgue_on_mono null_setsD2 null_sets_restrict_space order_refl)
qed
lemma
shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
and space_lborel[simp]: "space lborel = space borel"
and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"
by (simp_all add: lborel_def)
lemma space_lebesgue_on [simp]: "space (lebesgue_on S) = S"
by (simp add: space_restrict_space)
lemma sets_lebesgue_on_refl [iff]: "S \<in> sets (lebesgue_on S)"
by (metis inf_top.right_neutral sets.top space_borel space_completion space_lborel space_restrict_space)
lemma Compl_in_sets_lebesgue: "-A \<in> sets lebesgue \<longleftrightarrow> A \<in> sets lebesgue"
by (metis Compl_eq_Diff_UNIV double_compl space_borel space_completion space_lborel Sigma_Algebra.sets.compl_sets)
lemma measurable_lebesgue_cong:
assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
shows "f \<in> measurable (lebesgue_on S) M \<longleftrightarrow> g \<in> measurable (lebesgue_on S) M"
by (metis (mono_tags, lifting) IntD1 assms measurable_cong_simp space_restrict_space)
lemma lebesgue_on_UNIV_eq: "lebesgue_on UNIV = lebesgue"
by (simp add: emeasure_restrict_space measure_eqI)
lemma integral_restrict_Int:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<in> sets lebesgue" "T \<in> sets lebesgue"
shows "integral\<^sup>L (lebesgue_on T) (\<lambda>x. if x \<in> S then f x else 0) = integral\<^sup>L (lebesgue_on (S \<inter> T)) f"
proof -
have "(\<lambda>x. indicat_real T x *\<^sub>R (if x \<in> S then f x else 0)) = (\<lambda>x. indicat_real (S \<inter> T) x *\<^sub>R f x)"
by (force simp: indicator_def)
then show ?thesis
by (simp add: assms sets.Int Bochner_Integration.integral_restrict_space)
qed
lemma integral_restrict:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<subseteq> T" "S \<in> sets lebesgue" "T \<in> sets lebesgue"
shows "integral\<^sup>L (lebesgue_on T) (\<lambda>x. if x \<in> S then f x else 0) = integral\<^sup>L (lebesgue_on S) f"
using integral_restrict_Int [of S T f] assms
by (simp add: Int_absorb2)
lemma integral_restrict_UNIV:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<in> sets lebesgue"
shows "integral\<^sup>L lebesgue (\<lambda>x. if x \<in> S then f x else 0) = integral\<^sup>L (lebesgue_on S) f"
using integral_restrict_Int [of S UNIV f] assms
by (simp add: lebesgue_on_UNIV_eq)
lemma integrable_lebesgue_on_empty [iff]:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{second_countable_topology,banach}"
shows "integrable (lebesgue_on {}) f"
by (simp add: integrable_restrict_space)
lemma integral_lebesgue_on_empty [simp]:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{second_countable_topology,banach}"
shows "integral\<^sup>L (lebesgue_on {}) f = 0"
by (simp add: Bochner_Integration.integral_empty)
lemma has_bochner_integral_restrict_space:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
shows "has_bochner_integral (restrict_space M \<Omega>) f i
\<longleftrightarrow> has_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) i"
by (simp add: integrable_restrict_space [OF assms] integral_restrict_space [OF assms] has_bochner_integral_iff)
lemma integrable_restrict_UNIV:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes S: "S \<in> sets lebesgue"
shows "integrable lebesgue (\<lambda>x. if x \<in> S then f x else 0) \<longleftrightarrow> integrable (lebesgue_on S) f"
using has_bochner_integral_restrict_space [of S lebesgue f] assms
by (simp add: integrable.simps indicator_scaleR_eq_if)
lemma integral_mono_lebesgue_on_AE:
fixes f::"_ \<Rightarrow> real"
assumes f: "integrable (lebesgue_on T) f"
and gf: "AE x in (lebesgue_on S). g x \<le> f x"
and f0: "AE x in (lebesgue_on T). 0 \<le> f x"
and "S \<subseteq> T" and S: "S \<in> sets lebesgue" and T: "T \<in> sets lebesgue"
shows "(\<integral>x. g x \<partial>(lebesgue_on S)) \<le> (\<integral>x. f x \<partial>(lebesgue_on T))"
proof -
have "(\<integral>x. g x \<partial>(lebesgue_on S)) = (\<integral>x. (if x \<in> S then g x else 0) \<partial>lebesgue)"
by (simp add: Lebesgue_Measure.integral_restrict_UNIV S)
also have "\<dots> \<le> (\<integral>x. (if x \<in> T then f x else 0) \<partial>lebesgue)"
proof (rule Bochner_Integration.integral_mono_AE')
show "integrable lebesgue (\<lambda>x. if x \<in> T then f x else 0)"
by (simp add: integrable_restrict_UNIV T f)
show "AE x in lebesgue. (if x \<in> S then g x else 0) \<le> (if x \<in> T then f x else 0)"
using assms by (auto simp: AE_restrict_space_iff)
show "AE x in lebesgue. 0 \<le> (if x \<in> T then f x else 0)"
using f0 by (simp add: AE_restrict_space_iff T)
qed
also have "\<dots> = (\<integral>x. f x \<partial>(lebesgue_on T))"
using Lebesgue_Measure.integral_restrict_UNIV T by blast
finally show ?thesis .
qed
subsection \<open>Borel measurability\<close>
lemma borel_measurable_if_I:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue"
shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
proof -
have eq: "{x. x \<notin> S} \<union> {x. f x \<in> Y} = {x. x \<notin> S} \<union> {x. f x \<in> Y} \<inter> S" for Y
by blast
show ?thesis
using f S
apply (simp add: vimage_def in_borel_measurable_borel Ball_def)
apply (elim all_forward imp_forward asm_rl)
apply (simp only: Collect_conj_eq Collect_disj_eq imp_conv_disj eq)
apply (auto simp: Compl_eq [symmetric] Compl_in_sets_lebesgue sets_restrict_space_iff)
done
qed
lemma borel_measurable_if_D:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
shows "f \<in> borel_measurable (lebesgue_on S)"
using assms by (smt (verit) measurable_lebesgue_cong measurable_restrict_space1)
lemma borel_measurable_if:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<in> sets lebesgue"
shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)"
using assms borel_measurable_if_D borel_measurable_if_I by blast
lemma borel_measurable_if_lebesgue_on:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<in> sets lebesgue" "T \<in> sets lebesgue" "S \<subseteq> T"
shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable (lebesgue_on T) \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using measurable_restrict_mono [OF _ \<open>S \<subseteq> T\<close>]
by (subst measurable_lebesgue_cong [where g = "(\<lambda>x. if x \<in> S then f x else 0)"]) auto
next
assume ?rhs then show ?lhs
by (simp add: \<open>S \<in> sets lebesgue\<close> borel_measurable_if_I measurable_restrict_space1)
qed
lemma borel_measurable_vimage_halfspace_component_lt:
"f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_less])
lemma borel_measurable_vimage_halfspace_component_ge:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<ge> a} \<in> sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_ge])
lemma borel_measurable_vimage_halfspace_component_gt:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i > a} \<in> sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_greater])
lemma borel_measurable_vimage_halfspace_component_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<le> a} \<in> sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_le])
lemma
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows borel_measurable_vimage_open_interval:
"f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S))" (is ?thesis1)
and borel_measurable_vimage_open:
"f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))" (is ?thesis2)
proof -
have "{x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S)" if "f \<in> borel_measurable (lebesgue_on S)" for a b
proof -
have "S = S \<inter> space lebesgue"
by simp
then have "S \<inter> (f -` box a b) \<in> sets (lebesgue_on S)"
by (metis (no_types) box_borel in_borel_measurable_borel inf_sup_aci(1) space_restrict_space that)
then show ?thesis
by (simp add: Collect_conj_eq vimage_def)
qed
moreover
have "{x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)"
if T: "\<And>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S)" "open T" for T
proof -
obtain \<D> where "countable \<D>" and \<D>: "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = T"
using open_countable_Union_open_box that \<open>open T\<close> by metis
then have eq: "{x \<in> S. f x \<in> T} = (\<Union>U \<in> \<D>. {x \<in> S. f x \<in> U})"
by blast
have "{x \<in> S. f x \<in> U} \<in> sets (lebesgue_on S)" if "U \<in> \<D>" for U
using that T \<D> by blast
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \<open>countable \<D>\<close>])
qed
moreover
have eq: "{x \<in> S. f x \<bullet> i < a} = {x \<in> S. f x \<in> {y. y \<bullet> i < a}}" for i a
by auto
have "f \<in> borel_measurable (lebesgue_on S)"
if "\<And>T. open T \<Longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)"
by (metis (no_types) eq borel_measurable_vimage_halfspace_component_lt open_halfspace_component_lt that)
ultimately show "?thesis1" "?thesis2"
by blast+
qed
lemma borel_measurable_vimage_closed:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. closed T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
proof -
have eq: "{x \<in> S. f x \<in> T} = S - (S \<inter> f -` (- T))" for T
by auto
show ?thesis
unfolding borel_measurable_vimage_open eq
by (metis Diff_Diff_Int closed_Compl diff_eq open_Compl sets.Diff sets_lebesgue_on_refl vimage_Compl)
qed
lemma borel_measurable_vimage_closed_interval:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a b. {x \<in> S. f x \<in> cbox a b} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using borel_measurable_vimage_closed by blast
next
assume RHS: ?rhs
have "{x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)" if "open T" for T
proof -
obtain \<D> where "countable \<D>" and \<D>: "\<D> \<subseteq> Pow T" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = T"
using open_countable_Union_open_cbox that \<open>open T\<close> by metis
then have eq: "{x \<in> S. f x \<in> T} = (\<Union>U \<in> \<D>. {x \<in> S. f x \<in> U})"
by blast
have "{x \<in> S. f x \<in> U} \<in> sets (lebesgue_on S)" if "U \<in> \<D>" for U
using that \<D> by (metis RHS)
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \<open>countable \<D>\<close>])
qed
then show ?lhs
by (simp add: borel_measurable_vimage_open)
qed
lemma borel_measurable_vimage_borel:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume f: ?lhs
then show ?rhs
using measurable_sets [OF f]
by (simp add: Collect_conj_eq inf_sup_aci(1) space_restrict_space vimage_def)
qed (simp add: borel_measurable_vimage_open_interval)
lemma lebesgue_measurable_vimage_borel:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "f \<in> borel_measurable lebesgue" "T \<in> sets borel"
shows "{x. f x \<in> T} \<in> sets lebesgue"
using assms borel_measurable_vimage_borel [of f UNIV] by auto
lemma borel_measurable_lebesgue_preimage_borel:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable lebesgue \<longleftrightarrow>
(\<forall>T. T \<in> sets borel \<longrightarrow> {x. f x \<in> T} \<in> sets lebesgue)"
by (smt (verit, best) Collect_cong UNIV_I borel_measurable_vimage_borel lebesgue_on_UNIV_eq)
subsection \<^marker>\<open>tag unimportant\<close> \<open>Measurability of continuous functions\<close>
lemma continuous_imp_measurable_on_sets_lebesgue:
assumes f: "continuous_on S f" and S: "S \<in> sets lebesgue"
shows "f \<in> borel_measurable (lebesgue_on S)"
by (metis borel_measurable_continuous_on_restrict borel_measurable_subalgebra f
lebesgue_on_UNIV_eq mono_restrict_space sets_completionI_sets sets_lborel space_borel
space_lebesgue_on space_restrict_space subsetI)
lemma id_borel_measurable_lebesgue [iff]: "id \<in> borel_measurable lebesgue"
by (simp add: measurable_completion)
lemma id_borel_measurable_lebesgue_on [iff]: "id \<in> borel_measurable (lebesgue_on S)"
by (simp add: measurable_completion measurable_restrict_space1)
context
begin
interpretation sigma_finite_measure "interval_measure (\<lambda>x. x)"
by (rule sigma_finite_interval_measure) auto
interpretation finite_product_sigma_finite "\<lambda>_. interval_measure (\<lambda>x. x)" Basis
proof qed simp
lemma lborel_eq_real: "lborel = interval_measure (\<lambda>x. x)"
unfolding lborel_def Basis_real_def
using distr_id[of "interval_measure (\<lambda>x. x)"]
by (subst distr_component[symmetric])
(simp_all add: distr_distr comp_def del: distr_id cong: distr_cong)
lemma lborel_eq: "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. lborel) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
by (subst lborel_def) (simp add: lborel_eq_real)
lemma nn_integral_lborel_prod:
assumes [measurable]: "\<And>b. b \<in> Basis \<Longrightarrow> f b \<in> borel_measurable borel"
assumes nn[simp]: "\<And>b x. b \<in> Basis \<Longrightarrow> 0 \<le> f b x"
shows "(\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. f b (x \<bullet> b)) \<partial>lborel) = (\<Prod>b\<in>Basis. (\<integral>\<^sup>+x. f b x \<partial>lborel))"
by (simp add: lborel_def nn_integral_distr product_nn_integral_prod
product_nn_integral_singleton)
lemma emeasure_lborel_Icc[simp]:
fixes l u :: real
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l .. u} = u - l"
by (simp add: emeasure_interval_measure_Icc lborel_eq_real)
lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l \<le> u then u - l else 0)"
by simp
lemma\<^marker>\<open>tag important\<close> emeasure_lborel_cbox[simp]:
assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
shows "emeasure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
proof -
have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (cbox l u)"
by (auto simp: fun_eq_iff cbox_def split: split_indicator)
then have "emeasure lborel (cbox l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
by simp
also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
finally show ?thesis .
qed
lemma AE_lborel_singleton: "AE x in lborel::'a::euclidean_space measure. x \<noteq> c"
using SOME_Basis AE_discrete_difference [of "{c}" lborel] emeasure_lborel_cbox [of c c]
by (auto simp add: power_0_left)
lemma emeasure_lborel_Ioo[simp]:
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l <..< u} = ennreal (u - l)"
proof -
have "emeasure lborel {l <..< u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed
lemma emeasure_lborel_Ioc[simp]:
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l <.. u} = ennreal (u - l)"
by (simp add: emeasure_interval_measure_Ioc lborel_eq_real)
lemma emeasure_lborel_Ico[simp]:
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l ..< u} = ennreal (u - l)"
proof -
have "emeasure lborel {l ..< u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed
lemma emeasure_lborel_box[simp]:
assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
shows "emeasure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
proof -
have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (box l u)"
by (auto simp: fun_eq_iff box_def split: split_indicator)
then have "emeasure lborel (box l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
by simp
also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
finally show ?thesis .
qed
lemma emeasure_lborel_cbox_eq:
"emeasure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
lemma emeasure_lborel_box_eq:
"emeasure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
using emeasure_lborel_cbox[of x x] nonempty_Basis
by (auto simp del: emeasure_lborel_cbox nonempty_Basis)
lemma emeasure_lborel_cbox_finite: "emeasure lborel (cbox a b) < \<infinity>"
by (auto simp: emeasure_lborel_cbox_eq)
lemma emeasure_lborel_box_finite: "emeasure lborel (box a b) < \<infinity>"
by (auto simp: emeasure_lborel_box_eq)
lemma emeasure_lborel_ball_finite: "emeasure lborel (ball c r) < \<infinity>"
by (metis bounded_ball bounded_subset_cbox_symmetric cbox_borel emeasure_lborel_cbox_finite
emeasure_mono order_le_less_trans sets_lborel)
lemma emeasure_lborel_cball_finite: "emeasure lborel (cball c r) < \<infinity>"
by (metis bounded_cball bounded_subset_cbox_symmetric cbox_borel emeasure_lborel_cbox_finite
emeasure_mono order_le_less_trans sets_lborel)
lemma fmeasurable_cbox [iff]: "cbox a b \<in> fmeasurable lborel"
and fmeasurable_box [iff]: "box a b \<in> fmeasurable lborel"
by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)
lemma
fixes l u :: real
assumes [simp]: "l \<le> u"
shows measure_lborel_Icc[simp]: "measure lborel {l .. u} = u - l"
and measure_lborel_Ico[simp]: "measure lborel {l ..< u} = u - l"
and measure_lborel_Ioc[simp]: "measure lborel {l <.. u} = u - l"
and measure_lborel_Ioo[simp]: "measure lborel {l <..< u} = u - l"
by (simp_all add: measure_def)
lemma
assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
shows measure_lborel_box[simp]: "measure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (simp_all add: measure_def inner_diff_left prod_nonneg)
lemma measure_lborel_cbox_eq:
"measure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
lemma measure_lborel_box_eq:
"measure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
lemma measure_lborel_singleton[simp]: "measure lborel {x} = 0"
by (simp add: measure_def)
lemma sigma_finite_lborel: "sigma_finite_measure lborel"
proof
show "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets lborel \<and> \<Union>A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
by (intro exI[of _ "range (\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One))"])
(auto simp: emeasure_lborel_cbox_eq UN_box_eq_UNIV)
qed
end
lemma emeasure_lborel_UNIV [simp]: "emeasure lborel (UNIV::'a::euclidean_space set) = \<infinity>"
proof -
{ fix n::nat
let ?Ba = "Basis :: 'a set"
have "real n \<le> (2::real) ^ card ?Ba * real n"
by (simp add: mult_le_cancel_right1)
also
have "... \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
apply (rule mult_left_mono)
apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
apply (simp)
done
finally have "real n \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
} note [intro!] = this
show ?thesis
unfolding UN_box_eq_UNIV[symmetric]
apply (subst SUP_emeasure_incseq[symmetric])
apply (auto simp: incseq_def subset_box inner_add_left
simp del: Sup_eq_top_iff SUP_eq_top_iff
intro!: ennreal_SUP_eq_top)
done
qed
lemma emeasure_lborel_countable:
fixes A :: "'a::euclidean_space set"
assumes "countable A"
shows "emeasure lborel A = 0"
proof -
have "A \<subseteq> (\<Union>i. {from_nat_into A i})" using from_nat_into_surj assms by force
then have "emeasure lborel A \<le> emeasure lborel (\<Union>i. {from_nat_into A i})"
by (intro emeasure_mono) auto
also have "emeasure lborel (\<Union>i. {from_nat_into A i}) = 0"
by (rule emeasure_UN_eq_0) auto
finally show ?thesis
by simp
qed
lemma countable_imp_null_set_lborel: "countable A \<Longrightarrow> A \<in> null_sets lborel"
by (simp add: null_sets_def emeasure_lborel_countable sets.countable)
lemma finite_imp_null_set_lborel: "finite A \<Longrightarrow> A \<in> null_sets lborel"
by (intro countable_imp_null_set_lborel countable_finite)
lemma insert_null_sets_iff [simp]: "insert a N \<in> null_sets lebesgue \<longleftrightarrow> N \<in> null_sets lebesgue"
by (meson completion.complete2 finite.simps finite_imp_null_set_lborel null_sets.insert_in_sets
null_sets_completionI subset_insertI)
lemma insert_null_sets_lebesgue_on_iff [simp]:
assumes "a \<in> S" "S \<in> sets lebesgue"
shows "insert a N \<in> null_sets (lebesgue_on S) \<longleftrightarrow> N \<in> null_sets (lebesgue_on S)"
by (simp add: assms null_sets_restrict_space)
lemma lborel_neq_count_space[simp]:
fixes A :: "('a::ordered_euclidean_space) set"
shows "lborel \<noteq> count_space A"
by (metis finite.simps finite_imp_null_set_lborel insert_not_empty null_sets_count_space singleton_iff)
lemma mem_closed_if_AE_lebesgue_open:
assumes "open S" "closed C"
assumes "AE x \<in> S in lebesgue. x \<in> C"
assumes "x \<in> S"
shows "x \<in> C"
proof (rule ccontr)
assume xC: "x \<notin> C"
with openE[of "S - C"] assms
obtain e where e: "0 < e" "ball x e \<subseteq> S - C"
by blast
then obtain a b where box: "x \<in> box a b" "box a b \<subseteq> S - C"
by (metis rational_boxes order_trans)
then have "0 < emeasure lebesgue (box a b)"
by (auto simp: emeasure_lborel_box_eq mem_box algebra_simps intro!: prod_pos)
also have "\<dots> \<le> emeasure lebesgue (S - C)"
using assms box
by (auto intro!: emeasure_mono)
also have "\<dots> = 0"
using assms
by (auto simp: eventually_ae_filter completion.complete2 set_diff_eq null_setsD1)
finally show False by simp
qed
lemma mem_closed_if_AE_lebesgue: "closed C \<Longrightarrow> (AE x in lebesgue. x \<in> C) \<Longrightarrow> x \<in> C"
using mem_closed_if_AE_lebesgue_open[OF open_UNIV] by simp
subsection \<open>Affine transformation on the Lebesgue-Borel\<close>
lemma\<^marker>\<open>tag important\<close> lborel_eqI:
fixes M :: "'a::euclidean_space measure"
assumes emeasure_eq: "\<And>l u. (\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b) \<Longrightarrow> emeasure M (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
assumes sets_eq: "sets M = sets borel"
shows "lborel = M"
proof (rule measure_eqI_generator_eq)
let ?E = "range (\<lambda>(a, b). box a b::'a set)"
show "Int_stable ?E"
by (auto simp: Int_stable_def box_Int_box)
show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
by (simp_all add: borel_eq_box sets_eq)
let ?A = "\<lambda>n::nat. box (- (real n *\<^sub>R One)) (real n *\<^sub>R One) :: 'a set"
show "range ?A \<subseteq> ?E" "(\<Union>i. ?A i) = UNIV"
unfolding UN_box_eq_UNIV by auto
show "emeasure lborel (?A i) \<noteq> \<infinity>" for i by auto
show "emeasure lborel X = emeasure M X" if "X \<in> ?E" for X
using that box_eq_empty(1) by (fastforce simp: emeasure_eq emeasure_lborel_box_eq)
qed
lemma\<^marker>\<open>tag important\<close> lborel_affine_euclidean:
fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
defines "T x \<equiv> t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
shows "lborel = density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
proof (rule lborel_eqI)
let ?B = "Basis :: 'a set"
fix l u assume le: "\<And>b. b \<in> ?B \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
have [measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
by (simp add: T_def[abs_def])
have eq: "T -` box l u = box
(\<Sum>j\<in>Basis. (((if 0 < c j then l - t else u - t) \<bullet> j) / c j) *\<^sub>R j)
(\<Sum>j\<in>Basis. (((if 0 < c j then u - t else l - t) \<bullet> j) / c j) *\<^sub>R j)"
using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
with le c show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
field_split_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
intro!: prod.cong)
qed simp
lemma lborel_affine:
fixes t :: "'a::euclidean_space"
shows "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))"
using lborel_affine_euclidean[where c="\<lambda>_::'a. c" and t=t]
unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation prod_constant by simp
lemma lborel_real_affine:
"c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
using lborel_affine[of c t] by simp
lemma AE_borel_affine:
fixes P :: "real \<Rightarrow> bool"
shows "c \<noteq> 0 \<Longrightarrow> Measurable.pred borel P \<Longrightarrow> AE x in lborel. P x \<Longrightarrow> AE x in lborel. P (t + c * x)"
by (subst lborel_real_affine[where t="- t / c" and c="1 / c"])
(simp_all add: AE_density AE_distr_iff field_simps)
lemma nn_integral_real_affine:
fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0"
shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = \<bar>c\<bar> * (\<integral>\<^sup>+x. f (t + c * x) \<partial>lborel)"
by (subst lborel_real_affine[OF c, of t])
(simp add: nn_integral_density nn_integral_distr nn_integral_cmult)
lemma lborel_integrable_real_affine:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes f: "integrable lborel f"
shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))"
using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
by (subst (asm) nn_integral_real_affine[where c=c and t=t]) (auto simp: ennreal_mult_less_top)
lemma lborel_integrable_real_affine_iff:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x)) \<longleftrightarrow> integrable lborel f"
using
lborel_integrable_real_affine[of f c t]
lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "-t/c"]
by (auto simp add: field_simps)
lemma\<^marker>\<open>tag important\<close> lborel_integral_real_affine:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real
assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lborel) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f (t + c * x) \<partial>lborel)"
proof cases
assume f[measurable]: "integrable lborel f" then show ?thesis
using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
by (subst lborel_real_affine[OF c, of t])
(simp add: integral_density integral_distr)
next
assume "\<not> integrable lborel f" with c show ?thesis
by (simp add: lborel_integrable_real_affine_iff not_integrable_integral_eq)
qed
lemma
fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
defines "T == (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))"
shows lebesgue_affine_euclidean: "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
and lebesgue_affine_measurable: "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
proof -
have T_borel[measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
by (auto simp: T_def[abs_def])
{ fix A :: "'a set" assume A: "A \<in> sets borel"
then have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))) A = 0"
unfolding T_def using c by (subst lborel_affine_euclidean[symmetric]) auto
also have "\<dots> \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0"
using A c by (simp add: distr_completion emeasure_density nn_integral_cmult prod_nonneg cong: distr_cong)
finally have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0" . }
then have eq: "null_sets lborel = null_sets (distr lebesgue lborel T)"
by (auto simp: null_sets_def)
show "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
by (simp add: completion.measurable_completion2 eq measurable_completion)
have "lebesgue = completion (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>)))"
using c by (subst lborel_affine_euclidean[of c t]) (simp_all add: T_def[abs_def])
also have "\<dots> = density (completion (distr lebesgue lborel T)) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
using c by (auto intro!: always_eventually prod_pos completion_density_eq simp: distr_completion cong: distr_cong)
also have "\<dots> = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
by (subst completion.completion_distr_eq) (auto simp: eq measurable_completion)
finally show "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" .
qed
corollary lebesgue_real_affine:
"c \<noteq> 0 \<Longrightarrow> lebesgue = density (distr lebesgue lebesgue (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
using lebesgue_affine_euclidean [where c= "\<lambda>x::real. c"] by simp
lemma nn_integral_real_affine_lebesgue:
fixes c :: real assumes f[measurable]: "f \<in> borel_measurable lebesgue" and c: "c \<noteq> 0"
shows "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = ennreal\<bar>c\<bar> * (\<integral>\<^sup>+x. f(t + c * x) \<partial>lebesgue)"
proof -
have "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = (\<integral>\<^sup>+x. f x \<partial>density (distr lebesgue lebesgue (\<lambda>x. t + c * x)) (\<lambda>x. ennreal \<bar>c\<bar>))"
using lebesgue_real_affine c by auto
also have "\<dots> = \<integral>\<^sup>+ x. ennreal \<bar>c\<bar> * f x \<partial>distr lebesgue lebesgue (\<lambda>x. t + c * x)"
by (subst nn_integral_density) auto
also have "\<dots> = ennreal \<bar>c\<bar> * integral\<^sup>N (distr lebesgue lebesgue (\<lambda>x. t + c * x)) f"
using f measurable_distr_eq1 nn_integral_cmult by blast
also have "\<dots> = \<bar>c\<bar> * (\<integral>\<^sup>+x. f(t + c * x) \<partial>lebesgue)"
using lebesgue_affine_measurable[where c= "\<lambda>x::real. c"]
by (subst nn_integral_distr) (force+)
finally show ?thesis .
qed
lemma lebesgue_measurable_scaling[measurable]: "(*\<^sub>R) x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
proof cases
assume "x = 0"
then have "(*\<^sub>R) x = (\<lambda>x. 0::'a)"
by (auto simp: fun_eq_iff)
then show ?thesis by auto
next
assume "x \<noteq> 0" then show ?thesis
using lebesgue_affine_measurable[of "\<lambda>_. x" 0]
unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation
by (auto simp add: ac_simps)
qed
lemma
fixes m :: real and \<delta> :: "'a::euclidean_space"
defines "T r d x \<equiv> r *\<^sub>R x + d"
shows emeasure_lebesgue_affine: "emeasure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * emeasure lebesgue S" (is ?e)
and measure_lebesgue_affine: "measure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * measure lebesgue S" (is ?m)
proof -
show ?e
proof cases
assume "m = 0" then show ?thesis
by (simp add: image_constant_conv T_def[abs_def])
next
let ?T = "T m \<delta>" and ?T' = "T (1 / m) (- ((1/m) *\<^sub>R \<delta>))"
assume "m \<noteq> 0"
then have s_comp_s: "?T' \<circ> ?T = id" "?T \<circ> ?T' = id"
by (auto simp: T_def[abs_def] fun_eq_iff scaleR_add_right scaleR_diff_right)
then have "inv ?T' = ?T" "bij ?T'"
by (auto intro: inv_unique_comp o_bij)
then have eq: "T m \<delta> ` S = T (1 / m) ((-1/m) *\<^sub>R \<delta>) -` S \<inter> space lebesgue"
using bij_vimage_eq_inv_image[OF \<open>bij ?T'\<close>, of S] by auto
have trans_eq_T: "(\<lambda>x. \<delta> + (\<Sum>j\<in>Basis. (m * (x \<bullet> j)) *\<^sub>R j)) = T m \<delta>" for m \<delta>
unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
by (auto simp add: euclidean_representation ac_simps)
have T[measurable]: "T r d \<in> lebesgue \<rightarrow>\<^sub>M lebesgue" for r d
using lebesgue_affine_measurable[of "\<lambda>_. r" d]
by (cases "r = 0") (auto simp: trans_eq_T T_def[abs_def])
show ?thesis
proof cases
assume "S \<in> sets lebesgue" with \<open>m \<noteq> 0\<close> show ?thesis
unfolding eq
apply (subst lebesgue_affine_euclidean[of "\<lambda>_. m" \<delta>])
apply (simp_all add: emeasure_density trans_eq_T nn_integral_cmult emeasure_distr
del: space_completion emeasure_completion)
apply (simp add: vimage_comp s_comp_s)
done
next
assume "S \<notin> sets lebesgue"
moreover have "?T ` S \<notin> sets lebesgue"
proof
assume "?T ` S \<in> sets lebesgue"
then have "?T -` (?T ` S) \<inter> space lebesgue \<in> sets lebesgue"
by (rule measurable_sets[OF T])
also have "?T -` (?T ` S) \<inter> space lebesgue = S"
by (simp add: vimage_comp s_comp_s eq)
finally show False using \<open>S \<notin> sets lebesgue\<close> by auto
qed
ultimately show ?thesis
by (simp add: emeasure_notin_sets)
qed
qed
show ?m
unfolding measure_def \<open>?e\<close> by (simp add: enn2real_mult prod_nonneg)
qed
lemma lebesgue_real_scale:
assumes "c \<noteq> 0"
shows "lebesgue = density (distr lebesgue lebesgue (\<lambda>x. c * x)) (\<lambda>x. ennreal \<bar>c\<bar>)"
using assms by (subst lebesgue_affine_euclidean[of "\<lambda>_. c" 0]) simp_all
lemma lborel_has_bochner_integral_real_affine_iff:
fixes x :: "'a :: {banach, second_countable_topology}"
shows "c \<noteq> 0 \<Longrightarrow>
has_bochner_integral lborel f x \<longleftrightarrow>
has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)
lemma lborel_distr_uminus: "distr lborel borel uminus = (lborel :: real measure)"
by (subst lborel_real_affine[of "-1" 0])
(auto simp: density_1 one_ennreal_def[symmetric])
lemma lborel_distr_mult:
assumes "(c::real) \<noteq> 0"
shows "distr lborel borel ((*) c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
proof-
have "distr lborel borel ((*) c) = distr lborel lborel ((*) c)" by (simp cong: distr_cong)
also from assms have "... = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
finally show ?thesis .
qed
lemma lborel_distr_mult':
assumes "(c::real) \<noteq> 0"
shows "lborel = density (distr lborel borel ((*) c)) (\<lambda>_. \<bar>c\<bar>)"
proof-
have "lborel = density lborel (\<lambda>_. 1)" by (rule density_1[symmetric])
also from assms have "(\<lambda>_. 1 :: ennreal) = (\<lambda>_. inverse \<bar>c\<bar> * \<bar>c\<bar>)" by (intro ext) simp
also have "density lborel ... = density (density lborel (\<lambda>_. inverse \<bar>c\<bar>)) (\<lambda>_. \<bar>c\<bar>)"
by (subst density_density_eq) (auto simp: ennreal_mult)
also from assms have "density lborel (\<lambda>_. inverse \<bar>c\<bar>) = distr lborel borel ((*) c)"
by (rule lborel_distr_mult[symmetric])
finally show ?thesis .
qed
lemma lborel_distr_plus:
fixes c :: "'a::euclidean_space"
shows "distr lborel borel ((+) c) = lborel"
by (subst lborel_affine[of 1 c], auto simp: density_1)
interpretation lborel: sigma_finite_measure lborel
by (rule sigma_finite_lborel)
interpretation lborel_pair: pair_sigma_finite lborel lborel ..
lemma\<^marker>\<open>tag important\<close> lborel_prod:
"lborel \<Otimes>\<^sub>M lborel = (lborel :: ('a::euclidean_space \<times> 'b::euclidean_space) measure)"
proof (rule lborel_eqI[symmetric], clarify)
fix la ua :: 'a and lb ub :: 'b
assume lu: "\<And>a b. (a, b) \<in> Basis \<Longrightarrow> (la, lb) \<bullet> (a, b) \<le> (ua, ub) \<bullet> (a, b)"
have [simp]:
"\<And>b. b \<in> Basis \<Longrightarrow> la \<bullet> b \<le> ua \<bullet> b"
"\<And>b. b \<in> Basis \<Longrightarrow> lb \<bullet> b \<le> ub \<bullet> b"
"inj_on (\<lambda>u. (u, 0)) Basis" "inj_on (\<lambda>u. (0, u)) Basis"
"(\<lambda>u. (u, 0)) ` Basis \<inter> (\<lambda>u. (0, u)) ` Basis = {}"
"box (la, lb) (ua, ub) = box la ua \<times> box lb ub"
using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
show "emeasure (lborel \<Otimes>\<^sub>M lborel) (box (la, lb) (ua, ub)) =
ennreal (prod ((\<bullet>) ((ua, ub) - (la, lb))) Basis)"
by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def prod.union_disjoint
prod.reindex ennreal_mult inner_diff_left prod_nonneg)
qed (simp add: borel_prod[symmetric])
(* FIXME: conversion in measurable prover *)
lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel"
by simp
lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel"
by simp
lemma emeasure_bounded_finite:
assumes "bounded A" shows "emeasure lborel A < \<infinity>"
proof -
obtain a b where "A \<subseteq> cbox a b"
by (meson bounded_subset_cbox_symmetric \<open>bounded A\<close>)
then have "emeasure lborel A \<le> emeasure lborel (cbox a b)"
by (intro emeasure_mono) auto
then show ?thesis
by (auto simp: emeasure_lborel_cbox_eq prod_nonneg less_top[symmetric] top_unique split: if_split_asm)
qed
lemma emeasure_compact_finite: "compact A \<Longrightarrow> emeasure lborel A < \<infinity>"
using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded)
lemma borel_integrable_compact:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes "compact S" "continuous_on S f"
shows "integrable lborel (\<lambda>x. indicator S x *\<^sub>R f x)"
proof cases
assume "S \<noteq> {}"
have "continuous_on S (\<lambda>x. norm (f x))"
using assms by (intro continuous_intros)
from continuous_attains_sup[OF \<open>compact S\<close> \<open>S \<noteq> {}\<close> this]
obtain M where M: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> M"
by auto
show ?thesis
proof (rule integrable_bound)
show "integrable lborel (\<lambda>x. indicator S x * M)"
using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left)
show "(\<lambda>x. indicator S x *\<^sub>R f x) \<in> borel_measurable lborel"
using assms by (auto intro!: borel_measurable_continuous_on_indicator borel_compact)
show "AE x in lborel. norm (indicator S x *\<^sub>R f x) \<le> norm (indicator S x * M)"
by (auto split: split_indicator simp: abs_real_def dest!: M)
qed
qed simp
lemma borel_integrable_atLeastAtMost:
fixes f :: "real \<Rightarrow> real"
assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
proof -
have "integrable lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x)"
proof (rule borel_integrable_compact)
from f show "continuous_on {a..b} f"
by (auto intro: continuous_at_imp_continuous_on)
qed simp
then show ?thesis
by (auto simp: mult.commute)
qed
subsection \<open>Lebesgue measurable sets\<close>
abbreviation\<^marker>\<open>tag important\<close> lmeasurable :: "'a::euclidean_space set set"
where
"lmeasurable \<equiv> fmeasurable lebesgue"
lemma not_measurable_UNIV [simp]: "UNIV \<notin> lmeasurable"
by (simp add: fmeasurable_def)
lemma\<^marker>\<open>tag important\<close> lmeasurable_iff_integrable:
"S \<in> lmeasurable \<longleftrightarrow> integrable lebesgue (indicator S :: 'a::euclidean_space \<Rightarrow> real)"
by (auto simp: fmeasurable_def integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator)
lemma lmeasurable_cbox [iff]: "cbox a b \<in> lmeasurable"
and lmeasurable_box [iff]: "box a b \<in> lmeasurable"
by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)
lemma
fixes a::real
shows lmeasurable_interval [iff]: "{a..b} \<in> lmeasurable" "{a<..<b} \<in> lmeasurable"
by (metis box_real lmeasurable_box lmeasurable_cbox)+
lemma fmeasurable_compact: "compact S \<Longrightarrow> S \<in> fmeasurable lborel"
using emeasure_compact_finite[of S] by (intro fmeasurableI) (auto simp: borel_compact)
lemma lmeasurable_compact: "compact S \<Longrightarrow> S \<in> lmeasurable"
using fmeasurable_compact by (force simp: fmeasurable_def)
lemma measure_frontier:
"bounded S \<Longrightarrow> measure lebesgue (frontier S) = measure lebesgue (closure S) - measure lebesgue (interior S)"
using closure_subset interior_subset
by (auto simp: frontier_def fmeasurable_compact intro!: measurable_measure_Diff)
lemma lmeasurable_closure:
"bounded S \<Longrightarrow> closure S \<in> lmeasurable"
by (simp add: lmeasurable_compact)
lemma lmeasurable_frontier:
"bounded S \<Longrightarrow> frontier S \<in> lmeasurable"
by (simp add: compact_frontier_bounded lmeasurable_compact)
lemma lmeasurable_open: "bounded S \<Longrightarrow> open S \<Longrightarrow> S \<in> lmeasurable"
using emeasure_bounded_finite[of S] by (intro fmeasurableI) (auto simp: borel_open)
lemma lmeasurable_ball [simp]: "ball a r \<in> lmeasurable"
by (simp add: lmeasurable_open)
lemma lmeasurable_cball [simp]: "cball a r \<in> lmeasurable"
by (simp add: lmeasurable_compact)
lemma lmeasurable_interior: "bounded S \<Longrightarrow> interior S \<in> lmeasurable"
by (simp add: bounded_interior lmeasurable_open)
lemma null_sets_cbox_Diff_box: "cbox a b - box a b \<in> null_sets lborel"
by (simp add: emeasure_Diff emeasure_lborel_box_eq emeasure_lborel_cbox_eq null_setsI subset_box)
lemma bounded_set_imp_lmeasurable:
assumes "bounded S" "S \<in> sets lebesgue" shows "S \<in> lmeasurable"
by (metis assms bounded_Un emeasure_bounded_finite emeasure_completion fmeasurableI main_part_null_part_Un)
lemma finite_measure_lebesgue_on: "S \<in> lmeasurable \<Longrightarrow> finite_measure (lebesgue_on S)"
by (auto simp: finite_measureI fmeasurable_def emeasure_restrict_space)
lemma integrable_const_ivl [iff]:
fixes a::"'a::ordered_euclidean_space"
shows "integrable (lebesgue_on {a..b}) (\<lambda>x. c)"
by (metis cbox_interval finite_measure.integrable_const finite_measure_lebesgue_on lmeasurable_cbox)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Translation preserves Lebesgue measure\<close>
lemma sigma_sets_image:
assumes S: "S \<in> sigma_sets \<Omega> M" and "M \<subseteq> Pow \<Omega>" "f ` \<Omega> = \<Omega>" "inj_on f \<Omega>"
and M: "\<And>y. y \<in> M \<Longrightarrow> f ` y \<in> M"
shows "(f ` S) \<in> sigma_sets \<Omega> M"
using S
proof (induct S rule: sigma_sets.induct)
case (Basic a) then show ?case
by (simp add: M)
next
case Empty then show ?case
by (simp add: sigma_sets.Empty)
next
case (Compl a)
with assms show ?case
by (metis inj_on_image_set_diff sigma_sets.Compl sigma_sets_into_sp)
next
case (Union a) then show ?case
by (metis image_UN sigma_sets.simps)
qed
lemma null_sets_translation:
assumes "N \<in> null_sets lborel" shows "{x. x - a \<in> N} \<in> null_sets lborel"
proof -
have [simp]: "(\<lambda>x. x + a) ` N = {x. x - a \<in> N}"
by force
show ?thesis
using assms emeasure_lebesgue_affine [of 1 a N] by (auto simp: null_sets_def)
qed
lemma lebesgue_sets_translation:
fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
assumes S: "S \<in> sets lebesgue"
shows "((\<lambda>x. a + x) ` S) \<in> sets lebesgue"
proof -
have im_eq: "(+) a ` A = {x. x - a \<in> A}" for A
by force
have "((\<lambda>x. a + x) ` S) = ((\<lambda>x. -a + x) -` S) \<inter> (space lebesgue)"
using image_iff by fastforce
also have "\<dots> \<in> sets lebesgue"
proof (rule measurable_sets [OF measurableI assms])
fix A :: "'b set"
assume A: "A \<in> sets lebesgue"
have vim_eq: "(\<lambda>x. x - a) -` A = (+) a ` A" for A
by force
have "\<exists>s n N'. (+) a ` (S \<union> N) = s \<union> n \<and> s \<in> sets borel \<and> N' \<in> null_sets lborel \<and> n \<subseteq> N'"
if "S \<in> sets borel" and "N' \<in> null_sets lborel" and "N \<subseteq> N'" for S N N'
proof (intro exI conjI)
show "(+) a ` (S \<union> N) = (\<lambda>x. a + x) ` S \<union> (\<lambda>x. a + x) ` N"
by auto
show "(\<lambda>x. a + x) ` N' \<in> null_sets lborel"
using that by (auto simp: null_sets_translation im_eq)
qed (use that im_eq in auto)
with A have "(\<lambda>x. x - a) -` A \<in> sets lebesgue"
by (force simp: vim_eq completion_def intro!: sigma_sets_image)
then show "(+) (- a) -` A \<inter> space lebesgue \<in> sets lebesgue"
by (auto simp: vimage_def im_eq)
qed auto
finally show ?thesis .
qed
lemma measurable_translation:
"S \<in> lmeasurable \<Longrightarrow> ((+) a ` S) \<in> lmeasurable"
using emeasure_lebesgue_affine [of 1 a S]
by (smt (verit, best) add.commute ennreal_1 fmeasurable_def image_cong lambda_one
lebesgue_sets_translation mem_Collect_eq power_one scaleR_one)
lemma measurable_translation_subtract:
"S \<in> lmeasurable \<Longrightarrow> ((\<lambda>x. x - a) ` S) \<in> lmeasurable"
using measurable_translation [of S "- a"] by (simp cong: image_cong_simp)
lemma measure_translation:
"measure lebesgue ((+) a ` S) = measure lebesgue S"
using measure_lebesgue_affine [of 1 a S] by (simp add: ac_simps cong: image_cong_simp)
lemma measure_translation_subtract:
"measure lebesgue ((\<lambda>x. x - a) ` S) = measure lebesgue S"
using measure_translation [of "- a"] by (simp cong: image_cong_simp)
subsection \<open>A nice lemma for negligibility proofs\<close>
lemma summable_iff_suminf_neq_top: "(\<And>n. f n \<ge> 0) \<Longrightarrow> \<not> summable f \<Longrightarrow> (\<Sum>i. ennreal (f i)) = top"
by (metis summable_suminf_not_top)
proposition\<^marker>\<open>tag important\<close> starlike_negligible_bounded_gmeasurable:
fixes S :: "'a :: euclidean_space set"
assumes S: "S \<in> sets lebesgue" and "bounded S"
and eq1: "\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1"
shows "S \<in> null_sets lebesgue"
proof -
obtain M where "0 < M" "S \<subseteq> ball 0 M"
using \<open>bounded S\<close> by (auto dest: bounded_subset_ballD)
let ?f = "\<lambda>n. root DIM('a) (Suc n)"
have "?f n *\<^sub>R x \<in> S \<Longrightarrow> x \<in> (*\<^sub>R) (1 / ?f n) ` S" for x n
by (rule image_eqI[of _ _ "?f n *\<^sub>R x"]) auto
then have vimage_eq_image: "(*\<^sub>R) (?f n) -` S = (*\<^sub>R) (1 / ?f n) ` S" for n
by auto
have eq: "(1 / ?f n) ^ DIM('a) = 1 / Suc n" for n
by (simp add: field_simps)
{ fix n x assume x: "root DIM('a) (1 + real n) *\<^sub>R x \<in> S"
have "1 * norm x \<le> root DIM('a) (1 + real n) * norm x"
by (rule mult_mono) auto
also have "\<dots> < M"
using x \<open>S \<subseteq> ball 0 M\<close> by auto
finally have "norm x < M" by simp }
note less_M = this
have "(\<Sum>n. ennreal (1 / Suc n)) = top"
using not_summable_harmonic[where 'a=real] summable_Suc_iff[where f="\<lambda>n. 1 / (real n)"]
by (intro summable_iff_suminf_neq_top) (auto simp add: inverse_eq_divide)
then have "top * emeasure lebesgue S = (\<Sum>n. (1 / ?f n)^DIM('a) * emeasure lebesgue S)"
unfolding ennreal_suminf_multc eq by simp
also have "\<dots> = (\<Sum>n. emeasure lebesgue ((*\<^sub>R) (?f n) -` S))"
unfolding vimage_eq_image using emeasure_lebesgue_affine[of "1 / ?f n" 0 S for n] by simp
also have "\<dots> = emeasure lebesgue (\<Union>n. (*\<^sub>R) (?f n) -` S)"
proof (intro suminf_emeasure)
show "disjoint_family (\<lambda>n. (*\<^sub>R) (?f n) -` S)"
unfolding disjoint_family_on_def
proof safe
fix m n :: nat and x assume "m \<noteq> n" "?f m *\<^sub>R x \<in> S" "?f n *\<^sub>R x \<in> S"
with eq1[of "?f m / ?f n" "?f n *\<^sub>R x"] show "x \<in> {}"
by auto
qed
have "(*\<^sub>R) (?f i) -` S \<in> sets lebesgue" for i
using measurable_sets[OF lebesgue_measurable_scaling[of "?f i"] S] by auto
then show "range (\<lambda>i. (*\<^sub>R) (?f i) -` S) \<subseteq> sets lebesgue"
by auto
qed
also have "\<dots> \<le> emeasure lebesgue (ball 0 M :: 'a set)"
using less_M by (intro emeasure_mono) auto
also have "\<dots> < top"
using lmeasurable_ball by (auto simp: fmeasurable_def)
finally have "emeasure lebesgue S = 0"
by (simp add: ennreal_top_mult split: if_split_asm)
then show "S \<in> null_sets lebesgue"
unfolding null_sets_def using \<open>S \<in> sets lebesgue\<close> by auto
qed
corollary starlike_negligible_compact:
"compact S \<Longrightarrow> (\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1) \<Longrightarrow> S \<in> null_sets lebesgue"
using starlike_negligible_bounded_gmeasurable[of S] by (auto simp: compact_eq_bounded_closed)
proposition outer_regular_lborel_le:
assumes B[measurable]: "B \<in> sets borel" and "0 < (e::real)"
obtains U where "open U" "B \<subseteq> U" and "emeasure lborel (U - B) \<le> e"
proof -
let ?\<mu> = "emeasure lborel"
let ?B = "\<lambda>n::nat. ball 0 n :: 'a set"
let ?e = "\<lambda>n. e*((1/2)^Suc n)"
have "\<forall>n. \<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
proof
fix n :: nat
let ?A = "density lborel (indicator (?B n))"
have emeasure_A: "X \<in> sets borel \<Longrightarrow> emeasure ?A X = ?\<mu> (?B n \<inter> X)" for X
by (auto simp: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])
have finite_A: "emeasure ?A (space ?A) \<noteq> \<infinity>"
using emeasure_bounded_finite[of "?B n"] by (auto simp: emeasure_A)
interpret A: finite_measure ?A
by rule fact
have "emeasure ?A B + ?e n > (INF U\<in>{U. B \<subseteq> U \<and> open U}. emeasure ?A U)"
using \<open>0<e\<close> by (auto simp: outer_regular[OF _ finite_A B, symmetric])
then obtain U where U: "B \<subseteq> U" "open U" and muU: "?\<mu> (?B n \<inter> B) + ?e n > ?\<mu> (?B n \<inter> U)"
unfolding INF_less_iff by (auto simp: emeasure_A)
moreover
{ have "?\<mu> ((?B n \<inter> U) - B) = ?\<mu> ((?B n \<inter> U) - (?B n \<inter> B))"
using U by (intro arg_cong[where f="?\<mu>"]) auto
also have "\<dots> = ?\<mu> (?B n \<inter> U) - ?\<mu> (?B n \<inter> B)"
using U A.emeasure_finite[of B]
by (intro emeasure_Diff) (auto simp del: A.emeasure_finite simp: emeasure_A)
also have "\<dots> < ?e n"
using U muU A.emeasure_finite[of B]
by (subst minus_less_iff_ennreal)
(auto simp del: A.emeasure_finite simp: emeasure_A less_top ac_simps intro!: emeasure_mono)
finally have "?\<mu> ((?B n \<inter> U) - B) < ?e n" . }
ultimately show "\<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
by (intro exI[of _ "?B n \<inter> U"]) auto
qed
then obtain U
where U: "\<And>n. open (U n)" "\<And>n. ?B n \<inter> B \<subseteq> U n" "\<And>n. ?\<mu> (U n - B) < ?e n"
by metis
show ?thesis
proof
{ fix x assume "x \<in> B"
moreover
obtain n where "norm x < real n"
using reals_Archimedean2 by blast
ultimately have "x \<in> (\<Union>n. U n)"
using U(2)[of n] by auto }
note * = this
then show "open (\<Union>n. U n)" "B \<subseteq> (\<Union>n. U n)"
using U by auto
have "?\<mu> (\<Union>n. U n - B) \<le> (\<Sum>n. ?\<mu> (U n - B))"
using U(1) by (intro emeasure_subadditive_countably) auto
also have "\<dots> \<le> (\<Sum>n. ennreal (?e n))"
using U(3) by (intro suminf_le) (auto intro: less_imp_le)
also have "\<dots> = ennreal (e * 1)"
using \<open>0<e\<close> by (intro suminf_ennreal_eq sums_mult power_half_series) auto
finally show "emeasure lborel ((\<Union>n. U n) - B) \<le> ennreal e"
by simp
qed
qed
lemma\<^marker>\<open>tag important\<close> outer_regular_lborel:
assumes B: "B \<in> sets borel" and "0 < (e::real)"
obtains U where "open U" "B \<subseteq> U" "emeasure lborel (U - B) < e"
proof -
obtain U where U: "open U" "B \<subseteq> U" and "emeasure lborel (U-B) \<le> e/2"
using outer_regular_lborel_le [OF B, of "e/2"] \<open>e > 0\<close>
by force
moreover have "ennreal (e/2) < ennreal e"
using \<open>e > 0\<close> by (simp add: ennreal_lessI)
ultimately have "emeasure lborel (U-B) < e"
by auto
with U show ?thesis
using that by auto
qed
lemma completion_upper:
assumes A: "A \<in> sets (completion M)"
obtains A' where "A \<subseteq> A'" "A' \<in> sets M" "A' - A \<in> null_sets (completion M)"
"emeasure (completion M) A = emeasure M A'"
proof -
from AE_notin_null_part[OF A] obtain N where N: "N \<in> null_sets M" "null_part M A \<subseteq> N"
by (meson assms null_part)
let ?A' = "main_part M A \<union> N"
show ?thesis
proof
show "A \<subseteq> ?A'"
using \<open>null_part M A \<subseteq> N\<close> assms main_part_null_part_Un by blast
have "main_part M A \<subseteq> A"
using assms main_part_null_part_Un by auto
then have "?A' - A \<subseteq> N"
by blast
with N show "?A' - A \<in> null_sets (completion M)"
by (blast intro: null_sets_completionI completion.complete_measure_axioms complete_measure.complete2)
show "emeasure (completion M) A = emeasure M (main_part M A \<union> N)"
using A \<open>N \<in> null_sets M\<close> by (simp add: emeasure_Un_null_set)
qed (use A N in auto)
qed
lemma sets_lebesgue_outer_open:
fixes e::real
assumes S: "S \<in> sets lebesgue" and "e > 0"
obtains T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable" "emeasure lebesgue (T - S) < ennreal e"
proof -
obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel"
and null: "S' - S \<in> null_sets lebesgue"
and em: "emeasure lebesgue S = emeasure lborel S'"
using completion_upper[of S lborel] S by auto
then have f_S': "S' \<in> sets borel"
using S by (auto simp: fmeasurable_def)
with outer_regular_lborel[OF _ \<open>0<e\<close>]
obtain U where U: "open U" "S' \<subseteq> U" "emeasure lborel (U - S') < e"
by blast
show thesis
proof
show "open U" "S \<subseteq> U"
using f_S' U S' by auto
have "(U - S) = (U - S') \<union> (S' - S)"
using S' U by auto
then have eq: "emeasure lebesgue (U - S) = emeasure lborel (U - S')"
using null by (simp add: U(1) emeasure_Un_null_set f_S' sets.Diff)
have "(U - S) \<in> sets lebesgue"
by (simp add: S U(1) sets.Diff)
then show "(U - S) \<in> lmeasurable"
unfolding fmeasurable_def using U(3) eq less_le_trans by fastforce
with eq U show "emeasure lebesgue (U - S) < e"
by (simp add: eq)
qed
qed
lemma sets_lebesgue_inner_closed:
fixes e::real
assumes "S \<in> sets lebesgue" "e > 0"
obtains T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" "emeasure lebesgue (S - T) < ennreal e"
proof -
have "-S \<in> sets lebesgue"
using assms by (simp add: Compl_in_sets_lebesgue)
then obtain T where "open T" "-S \<subseteq> T"
and T: "(T - -S) \<in> lmeasurable" "emeasure lebesgue (T - -S) < e"
using sets_lebesgue_outer_open assms by blast
show thesis
proof
show "closed (-T)"
using \<open>open T\<close> by blast
show "-T \<subseteq> S"
using \<open>- S \<subseteq> T\<close> by auto
show "S - ( -T) \<in> lmeasurable" "emeasure lebesgue (S - (- T)) < e"
using T by (auto simp: Int_commute)
qed
qed
lemma lebesgue_openin:
"\<lbrakk>openin (top_of_set S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
by (metis borel_open openin_open sets.Int sets_completionI_sets sets_lborel)
lemma lebesgue_closedin:
"\<lbrakk>closedin (top_of_set S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
by (metis borel_closed closedin_closed sets.Int sets_completionI_sets sets_lborel)
subsection\<open>\<open>F_sigma\<close> and \<open>G_delta\<close> sets.\<close>
\<comment> \<open>\<^url>\<open>https://en.wikipedia.org/wiki/F-sigma_set\<close>\<close>
inductive\<^marker>\<open>tag important\<close> fsigma :: "'a::topological_space set \<Rightarrow> bool" where
"(\<And>n::nat. closed (F n)) \<Longrightarrow> fsigma (\<Union>(F ` UNIV))"
inductive\<^marker>\<open>tag important\<close> gdelta :: "'a::topological_space set \<Rightarrow> bool" where
"(\<And>n::nat. open (F n)) \<Longrightarrow> gdelta (\<Inter>(F ` UNIV))"
lemma fsigma_UNIV [iff]: "fsigma (UNIV :: 'a::real_inner set)"
proof -
have "(UNIV ::'a set) = (\<Union>i. cball 0 (of_nat i))"
by (auto simp: real_arch_simple)
then show ?thesis
by (metis closed_cball fsigma.intros)
qed
lemma fsigma_Union_compact:
fixes S :: "'a::{real_normed_vector,heine_borel} set"
shows "fsigma S \<longleftrightarrow> (\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = \<Union>(F ` UNIV))"
proof safe
assume "fsigma S"
then obtain F :: "nat \<Rightarrow> 'a set" where F: "range F \<subseteq> Collect closed" "S = \<Union>(F ` UNIV)"
by (meson fsigma.cases image_subsetI mem_Collect_eq)
then have "\<exists>D::nat \<Rightarrow> 'a set. range D \<subseteq> Collect compact \<and> \<Union>(D ` UNIV) = F i" for i
using closed_Union_compact_subsets [of "F i"]
by (metis image_subsetI mem_Collect_eq range_subsetD)
then obtain D :: "nat \<Rightarrow> nat \<Rightarrow> 'a set"
where D: "\<And>i. range (D i) \<subseteq> Collect compact \<and> \<Union>((D i) ` UNIV) = F i"
by metis
let ?DD = "\<lambda>n. (\<lambda>(i,j). D i j) (prod_decode n)"
show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = \<Union>(F ` UNIV)"
proof (intro exI conjI)
show "range ?DD \<subseteq> Collect compact"
using D by clarsimp (metis mem_Collect_eq rangeI split_conv subsetCE surj_pair)
show "S = \<Union> (range ?DD)"
proof
show "S \<subseteq> \<Union> (range ?DD)"
using D F
by clarsimp (metis UN_iff old.prod.case prod_decode_inverse prod_encode_eq)
show "\<Union> (range ?DD) \<subseteq> S"
using D F by fastforce
qed
qed
next
fix F :: "nat \<Rightarrow> 'a set"
assume "range F \<subseteq> Collect compact" and "S = \<Union>(F ` UNIV)"
then show "fsigma (\<Union>(F ` UNIV))"
by (simp add: compact_imp_closed fsigma.intros image_subset_iff)
qed
lemma gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
proof (induction rule: gdelta.induct)
case (1 F)
have "- \<Inter>(F ` UNIV) = (\<Union>i. -(F i))"
by auto
then show ?case
by (simp add: fsigma.intros closed_Compl 1)
qed
lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
proof (induction rule: fsigma.induct)
case (1 F)
have "- \<Union>(F ` UNIV) = (\<Inter>i. -(F i))"
by auto
then show ?case
by (simp add: 1 gdelta.intros open_closed)
qed
lemma gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
using fsigma_imp_gdelta gdelta_imp_fsigma by force
lemma lebesgue_set_almost_fsigma:
assumes "S \<in> sets lebesgue"
obtains C T where "fsigma C" "T \<in> null_sets lebesgue" "C \<union> T = S" "disjnt C T"
proof -
{ fix n::nat
obtain T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" "emeasure lebesgue (S - T) < ennreal (1 / Suc n)"
using sets_lebesgue_inner_closed [OF assms]
by (metis of_nat_0_less_iff zero_less_Suc zero_less_divide_1_iff)
then have "\<exists>T. closed T \<and> T \<subseteq> S \<and> S - T \<in> lmeasurable \<and> measure lebesgue (S-T) < 1 / Suc n"
by (metis emeasure_eq_measure2 ennreal_leI not_le)
}
then obtain F where F: "\<And>n::nat. closed (F n) \<and> F n \<subseteq> S \<and> S - F n \<in> lmeasurable \<and> measure lebesgue (S - F n) < 1 / Suc n"
by metis
let ?C = "\<Union>(F ` UNIV)"
show thesis
proof
show "fsigma ?C"
using F by (simp add: fsigma.intros)
show "(S - ?C) \<in> null_sets lebesgue"
proof (clarsimp simp add: completion.null_sets_outer_le)
fix e :: "real"
assume "0 < e"
then obtain n where n: "1 / Suc n < e"
using nat_approx_posE by metis
show "\<exists>T \<in> lmeasurable. S - (\<Union>x. F x) \<subseteq> T \<and> measure lebesgue T \<le> e"
proof (intro bexI conjI)
show "measure lebesgue (S - F n) \<le> e"
by (meson F n less_trans not_le order.asym)
qed (use F in auto)
qed
show "?C \<union> (S - ?C) = S"
using F by blast
show "disjnt ?C (S - ?C)"
by (auto simp: disjnt_def)
qed
qed
lemma lebesgue_set_almost_gdelta:
assumes "S \<in> sets lebesgue"
obtains C T where "gdelta C" "T \<in> null_sets lebesgue" "S \<union> T = C" "disjnt S T"
proof -
have "-S \<in> sets lebesgue"
using assms Compl_in_sets_lebesgue by blast
then obtain C T where C: "fsigma C" "T \<in> null_sets lebesgue" "C \<union> T = -S" "disjnt C T"
using lebesgue_set_almost_fsigma by metis
show thesis
proof
show "gdelta (-C)"
by (simp add: \<open>fsigma C\<close> fsigma_imp_gdelta)
show "S \<union> T = -C" "disjnt S T"
using C by (auto simp: disjnt_def)
qed (use C in auto)
qed
end