header {* Lebsegue measure *}
(* Author: Robert Himmelmann, TU Muenchen *)
theory Lebesgue_Measure
imports Gauge_Measure Measure Lebesgue_Integration
begin
subsection {* Various *}
lemma seq_offset_iff:"f ----> l \<longleftrightarrow> (\<lambda>i. f (i + k)) ----> l"
using seq_offset_rev seq_offset[of f l k] by auto
lemma has_integral_disjoint_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "(f has_integral i) s" "(f has_integral j) t" "s \<inter> t = {}"
shows "(f has_integral (i + j)) (s \<union> t)"
apply(rule has_integral_union[OF assms(1-2)]) unfolding assms by auto
lemma lim_eq: assumes "\<forall>n>N. f n = g n" shows "(f ----> l) \<longleftrightarrow> (g ----> l)" using assms
proof(induct N arbitrary: f g) case 0
hence *:"\<And>n. f (Suc n) = g (Suc n)" by auto
show ?case apply(subst LIMSEQ_Suc_iff[THEN sym]) apply(subst(2) LIMSEQ_Suc_iff[THEN sym])
unfolding * ..
next case (Suc n)
show ?case apply(subst LIMSEQ_Suc_iff[THEN sym]) apply(subst(2) LIMSEQ_Suc_iff[THEN sym])
apply(rule Suc(1)) using Suc(2) by auto
qed
subsection {* Standard Cubes *}
definition cube where
"cube (n::nat) \<equiv> {\<chi>\<chi> i. - real n .. (\<chi>\<chi> i. real n)::_::ordered_euclidean_space}"
lemma cube_subset[intro]:"n\<le>N \<Longrightarrow> cube n \<subseteq> (cube N::'a::ordered_euclidean_space set)"
apply(auto simp: eucl_le[where 'a='a] cube_def) apply(erule_tac[!] x=i in allE)+ by auto
lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n"
unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta'
proof- fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)"
thus "- real n \<le> x $$ i" "real n \<ge> x $$ i"
using component_le_norm[of x i] by(auto simp: dist_norm)
qed
lemma mem_big_cube: obtains n where "x \<in> cube n"
proof- from real_arch_lt[of "norm x"] guess n ..
thus ?thesis apply-apply(rule that[where n=n])
apply(rule ball_subset_cube[unfolded subset_eq,rule_format])
by (auto simp add:dist_norm)
qed
lemma Union_inter_cube:"\<Union>{s \<inter> cube n |n. n \<in> UNIV} = s"
proof safe case goal1
from mem_big_cube[of x] guess n . note n=this
show ?case unfolding Union_iff apply(rule_tac x="s \<inter> cube n" in bexI)
using n goal1 by auto
qed
lemma gmeasurable_cube[intro]:"gmeasurable (cube n)"
unfolding cube_def by auto
lemma gmeasure_le_inter_cube[intro]: fixes s::"'a::ordered_euclidean_space set"
assumes "gmeasurable (s \<inter> cube n)" shows "gmeasure (s \<inter> cube n) \<le> gmeasure (cube n::'a set)"
apply(rule has_gmeasure_subset[of "s\<inter>cube n" _ "cube n"])
unfolding has_gmeasure_measure[THEN sym] using assms by auto
subsection {* Measurability *}
definition lmeasurable :: "('a::ordered_euclidean_space) set => bool" where
"lmeasurable s \<equiv> (\<forall>n::nat. gmeasurable (s \<inter> cube n))"
lemma lmeasurableD[dest]:assumes "lmeasurable s"
shows "\<And>n. gmeasurable (s \<inter> cube n)"
using assms unfolding lmeasurable_def by auto
lemma measurable_imp_lmeasurable: assumes"gmeasurable s"
shows "lmeasurable s" unfolding lmeasurable_def cube_def
using assms gmeasurable_interval by auto
lemma lmeasurable_empty[intro]: "lmeasurable {}"
using gmeasurable_empty apply- apply(drule_tac measurable_imp_lmeasurable) .
lemma lmeasurable_union[intro]: assumes "lmeasurable s" "lmeasurable t"
shows "lmeasurable (s \<union> t)"
using assms unfolding lmeasurable_def Int_Un_distrib2
by(auto intro:gmeasurable_union)
lemma lmeasurable_countable_unions_strong:
fixes s::"nat => 'a::ordered_euclidean_space set"
assumes "\<And>n::nat. lmeasurable(s n)"
shows "lmeasurable(\<Union>{ s(n) |n. n \<in> UNIV })"
unfolding lmeasurable_def
proof fix n::nat
have *:"\<Union>{s n |n. n \<in> UNIV} \<inter> cube n = \<Union>{s k \<inter> cube n |k. k \<in> UNIV}" by auto
show "gmeasurable (\<Union>{s n |n. n \<in> UNIV} \<inter> cube n)" unfolding *
apply(rule gmeasurable_countable_unions_strong)
apply(rule assms[unfolded lmeasurable_def,rule_format])
proof- fix k::nat
show "gmeasure (\<Union>{s ka \<inter> cube n |ka. ka \<le> k}) \<le> gmeasure (cube n::'a set)"
apply(rule measure_subset) apply(rule gmeasurable_finite_unions)
using assms(1)[unfolded lmeasurable_def] by auto
qed
qed
lemma lmeasurable_inter[intro]: fixes s::"'a :: ordered_euclidean_space set"
assumes "lmeasurable s" "lmeasurable t" shows "lmeasurable (s \<inter> t)"
unfolding lmeasurable_def
proof fix n::nat
have *:"s \<inter> t \<inter> cube n = (s \<inter> cube n) \<inter> (t \<inter> cube n)" by auto
show "gmeasurable (s \<inter> t \<inter> cube n)"
using assms unfolding lmeasurable_def *
using gmeasurable_inter[of "s \<inter> cube n" "t \<inter> cube n"] by auto
qed
lemma lmeasurable_complement[intro]: assumes "lmeasurable s"
shows "lmeasurable (UNIV - s)"
unfolding lmeasurable_def
proof fix n::nat
have *:"(UNIV - s) \<inter> cube n = cube n - (s \<inter> cube n)" by auto
show "gmeasurable ((UNIV - s) \<inter> cube n)" unfolding *
apply(rule gmeasurable_diff) using assms unfolding lmeasurable_def by auto
qed
lemma lmeasurable_finite_unions:
assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> lmeasurable s"
shows "lmeasurable (\<Union> f)" unfolding lmeasurable_def
proof fix n::nat have *:"(\<Union>f \<inter> cube n) = \<Union>{x \<inter> cube n | x . x\<in>f}" by auto
show "gmeasurable (\<Union>f \<inter> cube n)" unfolding *
apply(rule gmeasurable_finite_unions) unfolding simple_image
using assms unfolding lmeasurable_def by auto
qed
lemma negligible_imp_lmeasurable[dest]: fixes s::"'a::ordered_euclidean_space set"
assumes "negligible s" shows "lmeasurable s"
unfolding lmeasurable_def
proof case goal1
have *:"\<And>x. (if x \<in> cube n then indicator s x else 0) = (if x \<in> s \<inter> cube n then 1 else 0)"
unfolding indicator_def_raw by auto
note assms[unfolded negligible_def,rule_format,of "(\<chi>\<chi> i. - real n)::'a" "\<chi>\<chi> i. real n"]
thus ?case apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
apply(subst(asm) has_integral_restrict_univ[THEN sym]) unfolding cube_def[symmetric]
apply(subst has_integral_restrict_univ[THEN sym]) unfolding * .
qed
section {* The Lebesgue Measure *}
definition has_lmeasure (infixr "has'_lmeasure" 80) where
"s has_lmeasure m \<equiv> lmeasurable s \<and> ((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
lemma has_lmeasureD: assumes "s has_lmeasure m"
shows "lmeasurable s" "gmeasurable (s \<inter> cube n)"
"((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
using assms unfolding has_lmeasure_def lmeasurable_def by auto
lemma has_lmeasureI: assumes "lmeasurable s" "((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
shows "s has_lmeasure m" using assms unfolding has_lmeasure_def by auto
definition lmeasure where
"lmeasure s \<equiv> SOME m. s has_lmeasure m"
lemma has_lmeasure_has_gmeasure: assumes "s has_lmeasure (Real m)" "m\<ge>0"
shows "s has_gmeasure m"
proof- note s = has_lmeasureD[OF assms(1)]
have *:"(\<lambda>n. (gmeasure (s \<inter> cube n))) ----> m"
using s(3) apply(subst (asm) lim_Real) using s(2) assms(2) by auto
have "(\<lambda>x. if x \<in> s then 1 else (0::real)) integrable_on UNIV \<and>
(\<lambda>k. integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)))
----> integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)"
proof(rule monotone_convergence_increasing)
have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m" apply(rule ccontr) unfolding not_all not_le
proof(erule exE) fix k assume k:"m < gmeasure (s \<inter> cube k)"
hence "gmeasure (s \<inter> cube k) - m > 0" by auto
from *[unfolded Lim_sequentially,rule_format,OF this] guess N ..
note this[unfolded dist_real_def,rule_format,of "N + k"]
moreover have "gmeasure (s \<inter> cube (N + k)) \<ge> gmeasure (s \<inter> cube k)" apply-
apply(rule measure_subset) prefer 3 using s(2)
using cube_subset[of k "N + k"] by auto
ultimately show False by auto
qed
thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}"
unfolding integral_measure_univ[OF s(2)] bounded_def apply-
apply(rule_tac x=0 in exI,rule_tac x=m in exI) unfolding dist_real_def
by (auto simp: measure_pos_le)
show "\<forall>k. (\<lambda>x. if x \<in> s \<inter> cube k then (1::real) else 0) integrable_on UNIV"
unfolding integrable_restrict_univ
using s(2) unfolding gmeasurable_def has_gmeasure_def by auto
have *:"\<And>n. n \<le> Suc n" by auto
show "\<forall>k. \<forall>x\<in>UNIV. (if x \<in> s \<inter> cube k then 1 else 0) \<le> (if x \<in> s \<inter> cube (Suc k) then 1 else (0::real))"
using cube_subset[OF *] by fastsimp
show "\<forall>x\<in>UNIV. (\<lambda>k. if x \<in> s \<inter> cube k then 1 else 0) ----> (if x \<in> s then 1 else (0::real))"
unfolding Lim_sequentially
proof safe case goal1 from real_arch_lt[of "norm x"] guess N .. note N = this
show ?case apply(rule_tac x=N in exI)
proof safe case goal1
have "x \<in> cube n" using cube_subset[OF goal1] N
using ball_subset_cube[of N] by(auto simp: dist_norm)
thus ?case using `e>0` by auto
qed
qed
qed note ** = conjunctD2[OF this]
hence *:"m = integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)" apply-
apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s(2)] using * .
show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto
qed
lemma has_lmeasure_unique: "s has_lmeasure m1 \<Longrightarrow> s has_lmeasure m2 \<Longrightarrow> m1 = m2"
unfolding has_lmeasure_def apply(rule Lim_unique) using trivial_limit_sequentially by auto
lemma lmeasure_unique[intro]: assumes "A has_lmeasure m" shows "lmeasure A = m"
using assms unfolding lmeasure_def lmeasurable_def apply-
apply(rule some_equality) defer apply(rule has_lmeasure_unique) by auto
lemma glmeasurable_finite: assumes "lmeasurable s" "lmeasure s \<noteq> \<omega>"
shows "gmeasurable s"
proof- have "\<exists>B. \<forall>n. gmeasure (s \<inter> cube n) \<le> B"
proof(rule ccontr) case goal1
note as = this[unfolded not_ex not_all not_le]
have "s has_lmeasure \<omega>" apply- apply(rule has_lmeasureI[OF assms(1)])
unfolding Lim_omega
proof fix B::real
from as[rule_format,of B] guess N .. note N = this
have "\<And>n. N \<le> n \<Longrightarrow> B \<le> gmeasure (s \<inter> cube n)"
apply(rule order_trans[where y="gmeasure (s \<inter> cube N)"]) defer
apply(rule measure_subset) prefer 3
using cube_subset N assms(1)[unfolded lmeasurable_def] by auto
thus "\<exists>N. \<forall>n\<ge>N. Real B \<le> Real (gmeasure (s \<inter> cube n))" apply-
apply(subst Real_max') apply(rule_tac x=N in exI,safe)
unfolding pinfreal_less_eq apply(subst if_P) by auto
qed note lmeasure_unique[OF this]
thus False using assms(2) by auto
qed then guess B .. note B=this
show ?thesis apply(rule gmeasurable_nested_unions[of "\<lambda>n. s \<inter> cube n",
unfolded Union_inter_cube,THEN conjunct1, where B1=B])
proof- fix n::nat
show " gmeasurable (s \<inter> cube n)" using assms by auto
show "gmeasure (s \<inter> cube n) \<le> B" using B by auto
show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)"
by (rule Int_mono) (simp_all add: cube_subset)
qed
qed
lemma lmeasure_empty[intro]:"lmeasure {} = 0"
apply(rule lmeasure_unique)
unfolding has_lmeasure_def by auto
lemma lmeasurableI[dest]:"s has_lmeasure m \<Longrightarrow> lmeasurable s"
unfolding has_lmeasure_def by auto
lemma has_gmeasure_has_lmeasure: assumes "s has_gmeasure m"
shows "s has_lmeasure (Real m)"
proof- have gmea:"gmeasurable s" using assms by auto
have m:"m \<ge> 0" using assms by auto
have *:"m = gmeasure (\<Union>{s \<inter> cube n |n. n \<in> UNIV})" unfolding Union_inter_cube
using assms by(rule measure_unique[THEN sym])
show ?thesis unfolding has_lmeasure_def
apply(rule,rule measurable_imp_lmeasurable[OF gmea])
apply(subst lim_Real) apply(rule,rule,rule m) unfolding *
apply(rule gmeasurable_nested_unions[THEN conjunct2, where B1="gmeasure s"])
proof- fix n::nat show *:"gmeasurable (s \<inter> cube n)"
using gmeasurable_inter[OF gmea gmeasurable_cube] .
show "gmeasure (s \<inter> cube n) \<le> gmeasure s" apply(rule measure_subset)
apply(rule * gmea)+ by auto
show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)" using cube_subset[of n "Suc n"] by auto
qed
qed
lemma gmeasure_lmeasure: assumes "gmeasurable s" shows "lmeasure s = Real (gmeasure s)"
proof- note has_gmeasure_measureI[OF assms]
note has_gmeasure_has_lmeasure[OF this]
thus ?thesis by(rule lmeasure_unique)
qed
lemma has_lmeasure_lmeasure: "lmeasurable s \<longleftrightarrow> s has_lmeasure (lmeasure s)" (is "?l = ?r")
proof assume ?l let ?f = "\<lambda>n. Real (gmeasure (s \<inter> cube n))"
have "\<forall>n m. n\<ge>m \<longrightarrow> ?f n \<ge> ?f m" unfolding pinfreal_less_eq apply safe
apply(subst if_P) defer apply(rule measure_subset) prefer 3
apply(drule cube_subset) using `?l` by auto
from lim_pinfreal_increasing[OF this] guess l . note l=this
hence "s has_lmeasure l" using `?l` apply-apply(rule has_lmeasureI) by auto
thus ?r using lmeasure_unique by auto
next assume ?r thus ?l unfolding has_lmeasure_def by auto
qed
lemma lmeasure_subset[dest]: assumes "lmeasurable s" "lmeasurable t" "s \<subseteq> t"
shows "lmeasure s \<le> lmeasure t"
proof(cases "lmeasure t = \<omega>")
case False have som:"lmeasure s \<noteq> \<omega>"
proof(rule ccontr,unfold not_not) assume as:"lmeasure s = \<omega>"
have "t has_lmeasure \<omega>" using assms(2) apply(rule has_lmeasureI)
unfolding Lim_omega
proof case goal1
note assms(1)[unfolded has_lmeasure_lmeasure]
note has_lmeasureD(3)[OF this,unfolded as Lim_omega,rule_format,of B]
then guess N .. note N = this
show ?case apply(rule_tac x=N in exI) apply safe
apply(rule order_trans) apply(rule N[rule_format],assumption)
unfolding pinfreal_less_eq apply(subst if_P)defer
apply(rule measure_subset) using assms by auto
qed
thus False using lmeasure_unique False by auto
qed
note assms(1)[unfolded has_lmeasure_lmeasure] note has_lmeasureD(3)[OF this]
hence "(\<lambda>n. Real (gmeasure (s \<inter> cube n))) ----> Real (real (lmeasure s))"
unfolding Real_real'[OF som] .
hence l1:"(\<lambda>n. gmeasure (s \<inter> cube n)) ----> real (lmeasure s)"
apply-apply(subst(asm) lim_Real) by auto
note assms(2)[unfolded has_lmeasure_lmeasure] note has_lmeasureD(3)[OF this]
hence "(\<lambda>n. Real (gmeasure (t \<inter> cube n))) ----> Real (real (lmeasure t))"
unfolding Real_real'[OF False] .
hence l2:"(\<lambda>n. gmeasure (t \<inter> cube n)) ----> real (lmeasure t)"
apply-apply(subst(asm) lim_Real) by auto
have "real (lmeasure s) \<le> real (lmeasure t)" apply(rule LIMSEQ_le[OF l1 l2])
apply(rule_tac x=0 in exI,safe) apply(rule measure_subset) using assms by auto
hence "Real (real (lmeasure s)) \<le> Real (real (lmeasure t))"
unfolding pinfreal_less_eq by auto
thus ?thesis unfolding Real_real'[OF som] Real_real'[OF False] .
qed auto
lemma has_lmeasure_negligible_unions_image:
assumes "finite s" "\<And>x. x \<in> s ==> lmeasurable(f x)"
"\<And>x y. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x \<noteq> y \<Longrightarrow> negligible((f x) \<inter> (f y))"
shows "(\<Union> (f ` s)) has_lmeasure (setsum (\<lambda>x. lmeasure(f x)) s)"
unfolding has_lmeasure_def
proof show lmeaf:"lmeasurable (\<Union>f ` s)" apply(rule lmeasurable_finite_unions)
using assms(1-2) by auto
show "(\<lambda>n. Real (gmeasure (\<Union>f ` s \<inter> cube n))) ----> (\<Sum>x\<in>s. lmeasure (f x))" (is ?l)
proof(cases "\<exists>x\<in>s. lmeasure (f x) = \<omega>")
case False hence *:"(\<Sum>x\<in>s. lmeasure (f x)) \<noteq> \<omega>" apply-
apply(rule setsum_neq_omega) using assms(1) by auto
have gmea:"\<And>x. x\<in>s \<Longrightarrow> gmeasurable (f x)" apply(rule glmeasurable_finite) using False assms(2) by auto
have "(\<Sum>x\<in>s. lmeasure (f x)) = (\<Sum>x\<in>s. Real (gmeasure (f x)))" apply(rule setsum_cong2)
apply(rule gmeasure_lmeasure) using False assms(2) gmea by auto
also have "... = Real (\<Sum>x\<in>s. (gmeasure (f x)))" apply(rule setsum_Real) by auto
finally have sum:"(\<Sum>x\<in>s. lmeasure (f x)) = Real (\<Sum>x\<in>s. gmeasure (f x))" .
have sum_0:"(\<Sum>x\<in>s. gmeasure (f x)) \<ge> 0" apply(rule setsum_nonneg) by auto
have int_un:"\<Union>f ` s has_gmeasure (\<Sum>x\<in>s. gmeasure (f x))"
apply(rule has_gmeasure_negligible_unions_image) using assms gmea by auto
have unun:"\<Union>{\<Union>f ` s \<inter> cube n |n. n \<in> UNIV} = \<Union>f ` s" unfolding simple_image
proof safe fix x y assume as:"x \<in> f y" "y \<in> s"
from mem_big_cube[of x] guess n . note n=this
thus "x \<in> \<Union>range (\<lambda>n. \<Union>f ` s \<inter> cube n)" unfolding Union_iff
apply-apply(rule_tac x="\<Union>f ` s \<inter> cube n" in bexI) using as by auto
qed
show ?l apply(subst Real_real'[OF *,THEN sym])apply(subst lim_Real)
apply rule apply rule unfolding sum real_Real if_P[OF sum_0] apply(rule sum_0)
unfolding measure_unique[OF int_un,THEN sym] apply(subst(2) unun[THEN sym])
apply(rule has_gmeasure_nested_unions[THEN conjunct2])
proof- fix n::nat
show *:"gmeasurable (\<Union>f ` s \<inter> cube n)" using lmeaf unfolding lmeasurable_def by auto
thus "gmeasure (\<Union>f ` s \<inter> cube n) \<le> gmeasure (\<Union>f ` s)"
apply(rule measure_subset) using int_un by auto
show "\<Union>f ` s \<inter> cube n \<subseteq> \<Union>f ` s \<inter> cube (Suc n)"
using cube_subset[of n "Suc n"] by auto
qed
next case True then guess X .. note X=this
hence sum:"(\<Sum>x\<in>s. lmeasure (f x)) = \<omega>" using setsum_\<omega>[THEN iffD2, of s] assms by fastsimp
show ?l unfolding sum Lim_omega
proof fix B::real
have Xm:"(f X) has_lmeasure \<omega>" using X by (metis assms(2) has_lmeasure_lmeasure)
note this[unfolded has_lmeasure_def,THEN conjunct2, unfolded Lim_omega]
from this[rule_format,of B] guess N .. note N=this[rule_format]
show "\<exists>N. \<forall>n\<ge>N. Real B \<le> Real (gmeasure (\<Union>f ` s \<inter> cube n))"
apply(rule_tac x=N in exI)
proof safe case goal1 show ?case apply(rule order_trans[OF N[OF goal1]])
unfolding pinfreal_less_eq apply(subst if_P) defer
apply(rule measure_subset) using has_lmeasureD(2)[OF Xm]
using lmeaf unfolding lmeasurable_def using X(1) by auto
qed qed qed qed
lemma has_lmeasure_negligible_unions:
assumes"finite f" "\<And>s. s \<in> f ==> s has_lmeasure (m s)"
"\<And>s t. s \<in> f \<Longrightarrow> t \<in> f \<Longrightarrow> s \<noteq> t ==> negligible (s\<inter>t)"
shows "(\<Union> f) has_lmeasure (setsum m f)"
proof- have *:"setsum m f = setsum lmeasure f" apply(rule setsum_cong2)
apply(subst lmeasure_unique[OF assms(2)]) by auto
show ?thesis unfolding *
apply(rule has_lmeasure_negligible_unions_image[where s=f and f=id,unfolded image_id id_apply])
using assms by auto
qed
lemma has_lmeasure_disjoint_unions:
assumes"finite f" "\<And>s. s \<in> f ==> s has_lmeasure (m s)"
"\<And>s t. s \<in> f \<Longrightarrow> t \<in> f \<Longrightarrow> s \<noteq> t ==> s \<inter> t = {}"
shows "(\<Union> f) has_lmeasure (setsum m f)"
proof- have *:"setsum m f = setsum lmeasure f" apply(rule setsum_cong2)
apply(subst lmeasure_unique[OF assms(2)]) by auto
show ?thesis unfolding *
apply(rule has_lmeasure_negligible_unions_image[where s=f and f=id,unfolded image_id id_apply])
using assms by auto
qed
lemma has_lmeasure_nested_unions:
assumes "\<And>n. lmeasurable(s n)" "\<And>n. s(n) \<subseteq> s(Suc n)"
shows "lmeasurable(\<Union> { s n | n. n \<in> UNIV }) \<and>
(\<lambda>n. lmeasure(s n)) ----> lmeasure(\<Union> { s(n) | n. n \<in> UNIV })" (is "?mea \<and> ?lim")
proof- have cube:"\<And>m. \<Union> { s(n) | n. n \<in> UNIV } \<inter> cube m = \<Union> { s(n) \<inter> cube m | n. n \<in> UNIV }" by blast
have 3:"\<And>n. \<forall>m\<ge>n. s n \<subseteq> s m" apply(rule transitive_stepwise_le) using assms(2) by auto
have mea:"?mea" unfolding lmeasurable_def cube apply rule
apply(rule_tac B1="gmeasure (cube n)" in has_gmeasure_nested_unions[THEN conjunct1])
prefer 3 apply rule using assms(1) unfolding lmeasurable_def
by(auto intro!:assms(2)[unfolded subset_eq,rule_format])
show ?thesis apply(rule,rule mea)
proof(cases "lmeasure(\<Union> { s(n) | n. n \<in> UNIV }) = \<omega>")
case True show ?lim unfolding True Lim_omega
proof(rule ccontr) case goal1 note this[unfolded not_all not_ex]
hence "\<exists>B. \<forall>n. \<exists>m\<ge>n. Real B > lmeasure (s m)" by(auto simp add:not_le)
from this guess B .. note B=this[rule_format]
have "\<forall>n. gmeasurable (s n) \<and> gmeasure (s n) \<le> max B 0"
proof safe fix n::nat from B[of n] guess m .. note m=this
hence *:"lmeasure (s n) < Real B" apply-apply(rule le_less_trans)
apply(rule lmeasure_subset[OF assms(1,1)]) apply(rule 3[rule_format]) by auto
thus **:"gmeasurable (s n)" apply-apply(rule glmeasurable_finite[OF assms(1)]) by auto
thus "gmeasure (s n) \<le> max B 0" using * unfolding gmeasure_lmeasure[OF **] Real_max'[of B]
unfolding pinfreal_less apply- apply(subst(asm) if_P) by auto
qed
hence "\<And>n. gmeasurable (s n)" "\<And>n. gmeasure (s n) \<le> max B 0" by auto
note g = conjunctD2[OF has_gmeasure_nested_unions[of s, OF this assms(2)]]
show False using True unfolding gmeasure_lmeasure[OF g(1)] by auto
qed
next let ?B = "lmeasure (\<Union>{s n |n. n \<in> UNIV})"
case False note gmea_lim = glmeasurable_finite[OF mea this]
have ls:"\<And>n. lmeasure (s n) \<le> lmeasure (\<Union>{s n |n. n \<in> UNIV})"
apply(rule lmeasure_subset) using assms(1) mea by auto
have "\<And>n. lmeasure (s n) \<noteq> \<omega>"
proof(rule ccontr,safe) case goal1
show False using False ls[of n] unfolding goal1 by auto
qed
note gmea = glmeasurable_finite[OF assms(1) this]
have "\<And>n. gmeasure (s n) \<le> real ?B" unfolding gmeasure_lmeasure[OF gmea_lim]
unfolding real_Real apply(subst if_P,rule) apply(rule measure_subset)
using gmea gmea_lim by auto
note has_gmeasure_nested_unions[of s, OF gmea this assms(2)]
thus ?lim unfolding gmeasure_lmeasure[OF gmea] gmeasure_lmeasure[OF gmea_lim]
apply-apply(subst lim_Real) by auto
qed
qed
lemma has_lmeasure_countable_negligible_unions:
assumes "\<And>n. lmeasurable(s n)" "\<And>m n. m \<noteq> n \<Longrightarrow> negligible(s m \<inter> s n)"
shows "(\<lambda>m. setsum (\<lambda>n. lmeasure(s n)) {..m}) ----> (lmeasure(\<Union> { s(n) |n. n \<in> UNIV }))"
proof- have *:"\<And>n. (\<Union> (s ` {0..n})) has_lmeasure (setsum (\<lambda>k. lmeasure(s k)) {0..n})"
apply(rule has_lmeasure_negligible_unions_image) using assms by auto
have **:"(\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV}) = (\<Union>{s n |n. n \<in> UNIV})" unfolding simple_image by fastsimp
have "lmeasurable (\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV}) \<and>
(\<lambda>n. lmeasure (\<Union>(s ` {0..n}))) ----> lmeasure (\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV})"
apply(rule has_lmeasure_nested_unions) apply(rule has_lmeasureD(1)[OF *])
apply(rule Union_mono,rule image_mono) by auto
note lem = conjunctD2[OF this,unfolded **]
show ?thesis using lem(2) unfolding lmeasure_unique[OF *] unfolding atLeast0AtMost .
qed
lemma lmeasure_eq_0: assumes "negligible s" shows "lmeasure s = 0"
proof- note mea=negligible_imp_lmeasurable[OF assms]
have *:"\<And>n. (gmeasure (s \<inter> cube n)) = 0"
unfolding gmeasurable_measure_eq_0[OF mea[unfolded lmeasurable_def,rule_format]]
using assms by auto
show ?thesis
apply(rule lmeasure_unique) unfolding has_lmeasure_def
apply(rule,rule mea) unfolding * by auto
qed
lemma negligible_img_gmeasurable: fixes s::"'a::ordered_euclidean_space set"
assumes "negligible s" shows "gmeasurable s"
apply(rule glmeasurable_finite)
using lmeasure_eq_0[OF assms] negligible_imp_lmeasurable[OF assms] by auto
section {* Instantiation of _::euclidean_space as measure_space *}
definition lebesgue_space :: "'a::ordered_euclidean_space algebra" where
"lebesgue_space = \<lparr> space = UNIV, sets = lmeasurable \<rparr>"
lemma lebesgue_measurable[simp]:"A \<in> sets lebesgue_space \<longleftrightarrow> lmeasurable A"
unfolding lebesgue_space_def by(auto simp: mem_def)
lemma mem_gmeasurable[simp]: "A \<in> gmeasurable \<longleftrightarrow> gmeasurable A"
unfolding mem_def ..
interpretation lebesgue: measure_space lebesgue_space lmeasure
apply(intro_locales) unfolding measure_space_axioms_def countably_additive_def
unfolding sigma_algebra_axioms_def algebra_def
unfolding lebesgue_measurable
proof safe
fix A::"nat => _" assume as:"range A \<subseteq> sets lebesgue_space" "disjoint_family A"
"lmeasurable (UNION UNIV A)"
have *:"UNION UNIV A = \<Union>range A" by auto
show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (UNION UNIV A)"
unfolding psuminf_def apply(rule SUP_Lim_pinfreal)
proof- fix n m::nat assume mn:"m\<le>n"
have *:"\<And>m. (\<Sum>n<m. lmeasure (A n)) = lmeasure (\<Union>A ` {..<m})"
apply(subst lmeasure_unique[OF has_lmeasure_negligible_unions[where m=lmeasure]])
apply(rule finite_imageI) apply rule apply(subst has_lmeasure_lmeasure[THEN sym])
proof- fix m::nat
show "(\<Sum>n<m. lmeasure (A n)) = setsum lmeasure (A ` {..<m})"
apply(subst setsum_reindex_nonzero) unfolding o_def apply rule
apply(rule lmeasure_eq_0) using as(2) unfolding disjoint_family_on_def
apply(erule_tac x=x in ballE,safe,erule_tac x=y in ballE) by auto
next fix m s assume "s \<in> A ` {..<m}"
hence "s \<in> range A" by auto thus "lmeasurable s" using as(1) by fastsimp
next fix m s t assume st:"s \<in> A ` {..<m}" "t \<in> A ` {..<m}" "s \<noteq> t"
from st(1-2) guess sa ta unfolding image_iff apply-by(erule bexE)+ note a=this
from st(3) have "sa \<noteq> ta" unfolding a by auto
thus "negligible (s \<inter> t)"
using as(2) unfolding disjoint_family_on_def a
apply(erule_tac x=sa in ballE,erule_tac x=ta in ballE) by auto
qed
have "\<And>m. lmeasurable (\<Union>A ` {..<m})" apply(rule lmeasurable_finite_unions)
apply(rule finite_imageI,rule) using as(1) by fastsimp
from this this show "(\<Sum>n<m. lmeasure (A n)) \<le> (\<Sum>n<n. lmeasure (A n))" unfolding *
apply(rule lmeasure_subset) apply(rule Union_mono) apply(rule image_mono) using mn by auto
next have *:"UNION UNIV A = \<Union>{A n |n. n \<in> UNIV}" by auto
show "(\<lambda>n. \<Sum>n<n. lmeasure (A n)) ----> lmeasure (UNION UNIV A)"
apply(rule LIMSEQ_imp_Suc) unfolding lessThan_Suc_atMost *
apply(rule has_lmeasure_countable_negligible_unions)
using as unfolding disjoint_family_on_def subset_eq by auto
qed
next show "lmeasure {} = 0" by auto
next fix A::"nat => _" assume as:"range A \<subseteq> sets lebesgue_space"
have *:"UNION UNIV A = (\<Union>{A n |n. n \<in> UNIV})" unfolding simple_image by auto
show "lmeasurable (UNION UNIV A)" unfolding * using as unfolding subset_eq
using lmeasurable_countable_unions_strong[of A] by auto
qed(auto simp: lebesgue_space_def mem_def)
lemma lmeasurbale_closed_interval[intro]:
"lmeasurable {a..b::'a::ordered_euclidean_space}"
unfolding lmeasurable_def cube_def inter_interval by auto
lemma space_lebesgue_space[simp]:"space lebesgue_space = UNIV"
unfolding lebesgue_space_def by auto
abbreviation "gintegral \<equiv> Integration.integral"
lemma lebesgue_simple_function_indicator:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
assumes f:"lebesgue.simple_function f"
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
lemma lmeasure_gmeasure:
"gmeasurable s \<Longrightarrow> gmeasure s = real (lmeasure s)"
apply(subst gmeasure_lmeasure) by auto
lemma lmeasure_finite: assumes "gmeasurable s" shows "lmeasure s \<noteq> \<omega>"
using gmeasure_lmeasure[OF assms] by auto
lemma negligible_lmeasure: assumes "lmeasurable s"
shows "lmeasure s = 0 \<longleftrightarrow> negligible s" (is "?l = ?r")
proof assume ?l
hence *:"gmeasurable s" using glmeasurable_finite[of s] assms by auto
show ?r unfolding gmeasurable_measure_eq_0[THEN sym,OF *]
unfolding lmeasure_gmeasure[OF *] using `?l` by auto
next assume ?r
note g=negligible_img_gmeasurable[OF this] and measure_eq_0[OF this]
hence "real (lmeasure s) = 0" using lmeasure_gmeasure[of s] by auto
thus ?l using lmeasure_finite[OF g] apply- apply(rule real_0_imp_eq_0) by auto
qed
end