(* 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 Bochner_Integration 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>Every right continuous and nondecreasing function gives rise to a measure\<close>
definition 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 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 (auto simp: subset_eq Ball_def)
apply (metis Diff_iff less_le_trans leD linear singletonD)
apply (metis Diff_iff less_le_trans leD linear singletonD)
apply (metis order_trans less_le_not_le linear)
done
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 "(\<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) psubset.prems(1) j
apply (subst sum.union_disjoint)
apply simp_all
apply (subst sum.union_disjoint)
apply auto
apply (metis less_le_not_le)
done
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"]
thus "\<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
apply -
apply (subst (asm) continuous_at_right_real_increasing)
apply (rule mono_F, assumption)
apply (drule_tac x = "epsilon / 2 ^ (i + 2)" in spec)
apply (erule impE)
using egt0 by (auto simp add: field_simps)
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"
apply (insert right_cont_F [of a])
apply (subst (asm) continuous_at_right_real_increasing)
using mono_F apply force
apply (drule_tac x = "epsilon / 2" in spec)
using egt0 unfolding mult.commute [of 2] by force
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})"
apply (intro UN_mono)
apply (auto simp: S'_def)
apply (cut_tac i=i in deltai_gt0)
apply simp
done
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
from finS have "\<exists>n. \<forall>i \<in> S. i \<le> n"
by (subst finite_nat_set_iff_bounded_le [symmetric])
then obtain n where Sbound [rule_format]: "\<forall>i \<in> S. i \<le> n" ..
have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i + delta i) - F (l i))"
apply (rule claim2 [rule_format])
using finS Sprop apply auto
apply (frule Sprop2)
apply (subgoal_tac "delta i > 0")
apply arith
by (rule deltai_gt0)
also have "... \<le> (\<Sum>i \<in> S. F(r i) - F(l i) + epsilon / 2^(i+2))"
apply (rule sum_mono)
apply simp
apply (rule order_trans)
apply (rule less_imp_le)
apply (rule deltai_prop)
by auto
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)"
apply (rule add_left_mono)
apply (rule mult_left_mono)
apply (rule sum_mono2)
using egt0 apply auto
by (frule Sbound, auto)
also have "... \<le> ?t + (epsilon / 2)"
apply (rule add_left_mono)
apply (subst geometric_sum)
apply auto
apply (rule mult_left_mono)
using egt0 apply auto
done
finally have aux2: "F b - F a' \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2"
by simp
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"
apply (intro add_right_mono)
apply (rule aux2)
done
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: ennreal_plus[symmetric] sum_nonneg del: 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"
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 "measure (interval_measure F) {a <.. b} = F b - F a"
unfolding measure_def
apply (subst emeasure_interval_measure_Ioc)
apply fact+
apply (simp add: assms)
done
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 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)
apply (rule sigma_sets_eqI)
apply auto
apply (case_tac "a \<le> ba")
apply (auto intro: sigma_sets.Empty)
done
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 "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) \<longlonglongrightarrow> emeasure ?F (\<Inter>n. {X n <..b})"
apply (intro Lim_emeasure_decseq)
apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
apply force
apply (subst (asm ) *)
apply (auto intro: less_le_trans less_imp_le)
done
also have "(\<Inter>n. {X n <..b}) = {a..b}"
using \<open>\<And>n. X n < a\<close>
apply auto
apply (rule LIMSEQ_le_const2[OF \<open>X \<longlonglongrightarrow> a\<close>])
apply (auto intro: less_imp_le)
apply (auto intro: less_le_trans)
done
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 continuous_on_const)
qed (rule trivial_limit_at_left_real)
lemma 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 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 lebesgue :: "'a::euclidean_space measure"
where "lebesgue \<equiv> completion lborel"
abbreviation lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"
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 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_strong space_restrict_space)
text\<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)"
proof -
have "sets (restrict_space borel S) \<subseteq> sets (lebesgue_on S)"
by (simp add: mono_restrict_space subsetI)
then show ?thesis
by (simp add: borel_measurable_continuous_on_restrict [OF f] borel_measurable_subalgebra
space_restrict_space)
qed
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"
proof -
have "((\<lambda>f. f 1) -` {l..u} \<inter> space (Pi\<^sub>M {1} (\<lambda>b. interval_measure (\<lambda>x. x)))) = {1::real} \<rightarrow>\<^sub>E {l..u}"
by (auto simp: space_PiM)
then show ?thesis
by (simp add: lborel_def emeasure_distr emeasure_PiM emeasure_interval_measure_Icc continuous_on_id)
qed
lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l \<le> u then u - l else 0)"
by simp
lemma 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)"
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_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 simp add: prod_constant)
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 add: DIM_positive)
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 prod_constant
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 (auto simp add: )
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 lborel_neq_count_space[simp]: "lborel \<noteq> count_space (A::('a::ordered_euclidean_space) set)"
proof
assume asm: "lborel = count_space A"
have "space lborel = UNIV" by simp
hence [simp]: "A = UNIV" by (subst (asm) asm) (simp only: space_count_space)
have "emeasure lborel {undefined::'a} = 1"
by (subst asm, subst emeasure_count_space_finite) auto
moreover have "emeasure lborel {undefined} \<noteq> 1" by simp
ultimately show False by contradiction
qed
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 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
{ fix i show "emeasure lborel (?A i) \<noteq> \<infinity>" by auto }
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
apply (subst box_eq_empty(1)[THEN iffD2])
apply (auto intro: less_imp_le simp: not_le)
done }
qed
lemma 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_simps divide_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 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 (rule completion.measurable_completion2) (auto simp: 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
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 prod_constant)
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 divideR_right:
fixes x y :: "'a::real_normed_vector"
shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp
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: "distr lborel borel ((+) c) = (lborel :: real measure)"
by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])
interpretation lborel: sigma_finite_measure lborel
by (rule sigma_finite_lborel)
interpretation lborel_pair: pair_sigma_finite lborel lborel ..
lemma 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 -
from bounded_subset_cbox[OF \<open>bounded A\<close>] obtain a b where "A \<subseteq> cbox a b"
by auto
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 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 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 lmeasurable_compact: "compact S \<Longrightarrow> S \<in> lmeasurable"
using emeasure_compact_finite[of S] by (intro fmeasurableI) (auto simp: borel_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: "ball a r \<in> lmeasurable"
by (simp add: lmeasurable_open)
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"
proof -
have "emeasure lborel (cbox a b - box a b) = 0"
by (subst emeasure_Diff) (auto simp: emeasure_lborel_cbox_eq emeasure_lborel_box_eq box_subset_cbox)
then have "cbox a b - box a b \<in> null_sets lborel"
by (auto simp: null_sets_def)
then show ?thesis
by (auto dest!: AE_not_in)
qed
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)
subsection\<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)
then have "\<Omega> - a \<subseteq> \<Omega>" "a \<subseteq> \<Omega>"
by (auto simp: sigma_sets_into_sp [OF \<open>M \<subseteq> Pow \<Omega>\<close>])
then show ?case
by (auto simp: inj_on_image_set_diff [OF \<open>inj_on f \<Omega>\<close>] assms intro: Compl sigma_sets.Compl)
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> ((\<lambda>x. a + x) ` S) \<in> lmeasurable"
unfolding fmeasurable_def
apply (auto intro: lebesgue_sets_translation)
using emeasure_lebesgue_affine [of 1 a S]
by (auto simp: add.commute [of _ a])
lemma measure_translation:
"measure lebesgue ((\<lambda>x. a + x) ` S) = measure lebesgue S"
using measure_lebesgue_affine [of 1 a S]
by (auto simp: add.commute [of _ a])
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 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 vimage_eq_image: "( *\<^sub>R) (?f n) -` S = ( *\<^sub>R) (1 / ?f n) ` S" for n
apply safe
subgoal for x by (rule image_eqI[of _ _ "?f n *\<^sub>R x"]) auto
subgoal by auto
done
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)
lemma outer_regular_lborel:
assumes B: "B \<in> fmeasurable lborel" "0 < (e::real)"
shows "\<exists>U. open U \<and> B \<subseteq> U \<and> emeasure lborel U \<le> emeasure lborel B + e"
proof -
let ?\<mu> = "emeasure lborel"
let ?B = "\<lambda>n::nat. ball 0 n :: 'a set"
have B[measurable]: "B \<in> sets borel"
using B by auto
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 add: 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 add: emeasure_A)
interpret A: finite_measure ?A
by rule fact
have "emeasure ?A B + ?e n > (INF U:{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" "?\<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(1,2,3) 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
then show ?thesis
proof (intro exI conjI)
{ fix x assume "x \<in> B"
moreover
have "\<exists>n. norm x < real n"
by (simp add: reals_Archimedean2)
then guess n ..
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(1,2) by auto
have "?\<mu> (\<Union>n. U n) = ?\<mu> (B \<union> (\<Union>n. U n - B))"
using * U(2) by (intro arg_cong[where ?f="?\<mu>"]) auto
also have "\<dots> = ?\<mu> B + ?\<mu> (\<Union>n. U n - B)"
using U(1) by (intro plus_emeasure[symmetric]) auto
also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ?\<mu> (U n - B))"
using U(1) by (intro add_mono emeasure_subadditive_countably) auto
also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ennreal (?e n))"
using U(3) by (intro add_mono suminf_le) (auto intro: less_imp_le)
also have "(\<Sum>n. ennreal (?e n)) = 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) \<le> emeasure lborel B + ennreal e"
by simp
qed
qed
lemma lmeasurable_outer_open:
assumes S: "S \<in> lmeasurable" and "0 < e"
obtains T where "open T" "S \<subseteq> T" "T \<in> lmeasurable" "measure lebesgue T \<le> measure lebesgue S + e"
proof -
obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel" "emeasure lborel S' = emeasure lebesgue S"
using completion_upper[of S lborel] S by auto
then have f_S': "S' \<in> fmeasurable lborel"
using S by (auto simp: fmeasurable_def)
from outer_regular_lborel[OF this \<open>0<e\<close>] guess U .. note U = this
show thesis
proof (rule that)
show "open U" "S \<subseteq> U" "U \<in> lmeasurable"
using f_S' U S' by (auto simp: fmeasurable_def less_top[symmetric] top_unique)
then have "U \<in> fmeasurable lborel"
by (auto simp: fmeasurable_def)
with S U \<open>0<e\<close> show "measure lebesgue U \<le> measure lebesgue S + e"
unfolding S'(3) by (simp add: emeasure_eq_measure2 ennreal_plus[symmetric] del: ennreal_plus)
qed
qed
lemma lebesgue_openin:
"\<lbrakk>openin (subtopology euclidean 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 (subtopology euclidean 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)
end