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Quellcode-Bibliothek
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Datei:
Regularity.thy
Sprache: Isabelle
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(* Title: HOL/Analysis/Regularity.thy
Author: Fabian Immler, TU München
*)
section \<open>Regularity of Measures\<close>
theory Regularity (* FIX suggestion to rename e.g. RegularityMeasures and/ or move as
this theory consists of 1 result only *)
imports Measure_Space Borel_Space
begin
theorem
fixes M::"'a::{second_countable_topology, complete_space} measure"
assumes sb: "sets M = sets borel"
assumes "emeasure M (space M) \ \"
assumes "B \ sets borel"
shows inner_regular: "emeasure M B =
(SUP K \<in> {K. K \<subseteq> B \<and> compact K}. emeasure M K)" (is "?inner B")
and outer_regular: "emeasure M B =
(INF U \<in> {U. B \<subseteq> U \<and> open U}. emeasure M U)" (is "?outer B")
proof -
have Us: "UNIV = space M" by (metis assms(1) sets_eq_imp_space_eq space_borel)
hence sU: "space M = UNIV" by simp
interpret finite_measure M by rule fact
have approx_inner: "\A. A \ sets M \
(\<And>e. e > 0 \<Longrightarrow> \<exists>K. K \<subseteq> A \<and> compact K \<and> emeasure M A \<le> emeasure M K + ennreal e) \<Longrightarrow> ?inner A"
by (rule ennreal_approx_SUP)
(force intro!: emeasure_mono simp: compact_imp_closed emeasure_eq_measure)+
have approx_outer: "\A. A \ sets M \
(\<And>e. e > 0 \<Longrightarrow> \<exists>B. A \<subseteq> B \<and> open B \<and> emeasure M B \<le> emeasure M A + ennreal e) \<Longrightarrow> ?outer A"
by (rule ennreal_approx_INF)
(force intro!: emeasure_mono simp: emeasure_eq_measure sb)+
from countable_dense_setE guess X::"'a set" . note X = this
{
fix r::real assume "r > 0" hence "\y. open (ball y r)" "\y. ball y r \ {}" by auto
with X(2)[OF this]
have x: "space M = (\x\X. cball x r)"
by (auto simp add: sU) (metis dist_commute order_less_imp_le)
let ?U = "\k. (\n\{0..k}. cball (from_nat_into X n) r)"
have "(\k. emeasure M (\n\{0..k}. cball (from_nat_into X n) r)) \ M ?U"
by (rule Lim_emeasure_incseq) (auto intro!: borel_closed bexI simp: incseq_def Us sb)
also have "?U = space M"
proof safe
fix x from X(2)[OF open_ball[of x r]] \<open>r > 0\<close> obtain d where d: "d\<in>X" "d \<in> ball x r" by auto
show "x \ ?U"
using X(1) d
by simp (auto intro!: exI [where x = "to_nat_on X d"] simp: dist_commute Bex_def)
qed (simp add: sU)
finally have "(\k. M (\n\{0..k}. cball (from_nat_into X n) r)) \ M (space M)" .
} note M_space = this
{
fix e ::real and n :: nat assume "e > 0" "n > 0"
hence "1/n > 0" "e * 2 powr - n > 0" by (auto)
from M_space[OF \<open>1/n>0\<close>]
have "(\k. measure M (\i\{0..k}. cball (from_nat_into X i) (1/real n))) \ measure M (space M)"
unfolding emeasure_eq_measure by (auto)
from metric_LIMSEQ_D[OF this \<open>0 < e * 2 powr -n\<close>]
obtain k where "dist (measure M (\i\{0..k}. cball (from_nat_into X i) (1/real n))) (measure M (space M)) <
e * 2 powr -n"
by auto
hence "measure M (\i\{0..k}. cball (from_nat_into X i) (1/real n)) \
measure M (space M) - e * 2 powr -real n"
by (auto simp: dist_real_def)
hence "\k. measure M (\i\{0..k}. cball (from_nat_into X i) (1/real n)) \
measure M (space M) - e * 2 powr - real n" ..
} note k=this
hence "\e\{0<..}. \(n::nat)\{0<..}. \k.
measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n)) \<ge> measure M (space M) - e * 2 powr - real n"
by blast
then obtain k where k: "\e\{0<..}. \n\{0<..}. measure M (space M) - e * 2 powr - real (n::nat)
\<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"
by metis
hence k: "\e n. e > 0 \ n > 0 \ measure M (space M) - e * 2 powr - n
\<le> measure M (\<Union>i\<in>{0..k e n}. cball (from_nat_into X i) (1 / n))"
unfolding Ball_def by blast
have approx_space:
"\K \ {K. K \ space M \ compact K}. emeasure M (space M) \ emeasure M K + ennreal e"
(is "?thesis e") if "0 < e" for e :: real
proof -
define B where [abs_def]:
"B n = (\i\{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n))" for n
have "\n. closed (B n)" by (auto simp: B_def)
hence [simp]: "\n. B n \ sets M" by (simp add: sb)
from k[OF \<open>e > 0\<close> zero_less_Suc]
have "\n. measure M (space M) - measure M (B n) \ e * 2 powr - real (Suc n)"
by (simp add: algebra_simps B_def finite_measure_compl)
hence B_compl_le: "\n::nat. measure M (space M - B n) \ e * 2 powr - real (Suc n)"
by (simp add: finite_measure_compl)
define K where "K = (\n. B n)"
from \<open>closed (B _)\<close> have "closed K" by (auto simp: K_def)
hence [simp]: "K \ sets M" by (simp add: sb)
have "measure M (space M) - measure M K = measure M (space M - K)"
by (simp add: finite_measure_compl)
also have "\ = emeasure M (\n. space M - B n)" by (auto simp: K_def emeasure_eq_measure)
also have "\ \ (\n. emeasure M (space M - B n))"
by (rule emeasure_subadditive_countably) (auto simp: summable_def)
also have "\ \ (\n. ennreal (e*2 powr - real (Suc n)))"
using B_compl_le by (intro suminf_le) (simp_all add: emeasure_eq_measure ennreal_leI)
also have "\ \ (\n. ennreal (e * (1 / 2) ^ Suc n))"
by (simp add: powr_minus powr_realpow field_simps del: of_nat_Suc)
also have "\ = ennreal e * (\n. ennreal ((1 / 2) ^ Suc n))"
unfolding ennreal_power[symmetric]
using \<open>0 < e\<close>
by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def
ennreal_power[symmetric])
also have "\ = e"
by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
finally have "measure M (space M) \ measure M K + e"
using \<open>0 < e\<close> by simp
hence "emeasure M (space M) \ emeasure M K + e"
using \<open>0 < e\<close> by (simp add: emeasure_eq_measure flip: ennreal_plus)
moreover have "compact K"
unfolding compact_eq_totally_bounded
proof safe
show "complete K" using \<open>closed K\<close> by (simp add: complete_eq_closed)
fix e'::real assume "0 < e'"
from nat_approx_posE[OF this] guess n . note n = this
let ?k = "from_nat_into X ` {0..k e (Suc n)}"
have "finite ?k" by simp
moreover have "K \ (\x\?k. ball x e')" unfolding K_def B_def using n by force
ultimately show "\k. finite k \ K \ (\x\k. ball x e')" by blast
qed
ultimately
show ?thesis by (auto simp: sU)
qed
{ fix A::"'a set" assume "closed A" hence "A \ sets borel" by (simp add: compact_imp_closed)
hence [simp]: "A \ sets M" by (simp add: sb)
have "?inner A"
proof (rule approx_inner)
fix e::real assume "e > 0"
from approx_space[OF this] obtain K where
K: "K \ space M" "compact K" "emeasure M (space M) \ emeasure M K + e"
by (auto simp: emeasure_eq_measure)
hence [simp]: "K \ sets M" by (simp add: sb compact_imp_closed)
have "measure M A - measure M (A \ K) = measure M (A - A \ K)"
by (subst finite_measure_Diff) auto
also have "A - A \ K = A \ K - K" by auto
also have "measure M \ = measure M (A \ K) - measure M K"
by (subst finite_measure_Diff) auto
also have "\ \ measure M (space M) - measure M K"
by (simp add: emeasure_eq_measure sU sb finite_measure_mono)
also have "\ \ e"
using K \<open>0 < e\<close> by (simp add: emeasure_eq_measure flip: ennreal_plus)
finally have "emeasure M A \ emeasure M (A \ K) + ennreal e"
using \<open>0<e\<close> by (simp add: emeasure_eq_measure algebra_simps flip: ennreal_plus)
moreover have "A \ K \ A" "compact (A \ K)" using \closed A\ \compact K\ by auto
ultimately show "\K \ A. compact K \ emeasure M A \ emeasure M K + ennreal e"
by blast
qed simp
have "?outer A"
proof cases
assume "A \ {}"
let ?G = "\d. {x. infdist x A < d}"
{
fix d
have "?G d = (\x. infdist x A) -` {..
also have "open \"
by (intro continuous_open_vimage) (auto intro!: continuous_infdist continuous_ident)
finally have "open (?G d)" .
} note open_G = this
from in_closed_iff_infdist_zero[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>]
have "A = {x. infdist x A = 0}" by auto
also have "\ = (\i. ?G (1/real (Suc i)))"
proof (auto simp del: of_nat_Suc, rule ccontr)
fix x
assume "infdist x A \ 0"
hence pos: "infdist x A > 0" using infdist_nonneg[of x A] by simp
from nat_approx_posE[OF this] guess n .
moreover
assume "\i. infdist x A < 1 / real (Suc i)"
hence "infdist x A < 1 / real (Suc n)" by auto
ultimately show False by simp
qed
also have "M \ = (INF n. emeasure M (?G (1 / real (Suc n))))"
proof (rule INF_emeasure_decseq[symmetric], safe)
fix i::nat
from open_G[of "1 / real (Suc i)"]
show "?G (1 / real (Suc i)) \ sets M" by (simp add: sb borel_open)
next
show "decseq (\i. {x. infdist x A < 1 / real (Suc i)})"
by (auto intro: less_trans intro!: divide_strict_left_mono
simp: decseq_def le_eq_less_or_eq)
qed simp
finally
have "emeasure M A = (INF n. emeasure M {x. infdist x A < 1 / real (Suc n)})" .
moreover
have "\ \ (INF U\{U. A \ U \ open U}. emeasure M U)"
proof (intro INF_mono)
fix m
have "?G (1 / real (Suc m)) \ {U. A \ U \ open U}" using open_G by auto
moreover have "M (?G (1 / real (Suc m))) \ M (?G (1 / real (Suc m)))" by simp
ultimately show "\U\{U. A \ U \ open U}.
emeasure M U \<le> emeasure M {x. infdist x A < 1 / real (Suc m)}"
by blast
qed
moreover
have "emeasure M A \ (INF U\{U. A \ U \ open U}. emeasure M U)"
by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)
ultimately show ?thesis by simp
qed (auto intro!: INF_eqI)
note \<open>?inner A\<close> \<open>?outer A\<close> }
note closed_in_D = this
from \<open>B \<in> sets borel\<close>
have "Int_stable (Collect closed)" "Collect closed \ Pow UNIV" "B \ sigma_sets UNIV (Collect closed)"
by (auto simp: Int_stable_def borel_eq_closed)
then show "?inner B" "?outer B"
proof (induct B rule: sigma_sets_induct_disjoint)
case empty
{ case 1 show ?case by (intro SUP_eqI[symmetric]) auto }
{ case 2 show ?case by (intro INF_eqI[symmetric]) (auto elim!: meta_allE[of _ "{}"]) }
next
case (basic B)
{ case 1 from basic closed_in_D show ?case by auto }
{ case 2 from basic closed_in_D show ?case by auto }
next
case (compl B)
note inner = compl(2) and outer = compl(3)
from compl have [simp]: "B \ sets M" by (auto simp: sb borel_eq_closed)
case 2
have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
also have "\ = (INF K\{K. K \ B \ compact K}. M (space M) - M K)"
by (subst ennreal_SUP_const_minus) (auto simp: less_top[symmetric] inner)
also have "\ = (INF U\{U. U \ B \ compact U}. M (space M - U))"
by (auto simp add: emeasure_compl sb compact_imp_closed)
also have "\ \ (INF U\{U. U \ B \ closed U}. M (space M - U))"
by (rule INF_superset_mono) (auto simp add: compact_imp_closed)
also have "(INF U\{U. U \ B \ closed U}. M (space M - U)) =
(INF U\<in>{U. space M - B \<subseteq> U \<and> open U}. emeasure M U)"
apply (rule arg_cong [of _ _ Inf])
using sU
apply (auto simp add: image_iff)
apply (rule exI [of _ "UNIV - y" for y])
apply safe
apply (auto simp add: double_diff)
done
finally have
"(INF U\{U. space M - B \ U \ open U}. emeasure M U) \ emeasure M (space M - B)" .
moreover have
"(INF U\{U. space M - B \ U \ open U}. emeasure M U) \ emeasure M (space M - B)"
by (auto simp: sb sU intro!: INF_greatest emeasure_mono)
ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
case 1
have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
also have "\ = (SUP U\ {U. B \ U \ open U}. M (space M) - M U)"
unfolding outer by (subst ennreal_INF_const_minus) auto
also have "\ = (SUP U\{U. B \ U \ open U}. M (space M - U))"
by (auto simp add: emeasure_compl sb compact_imp_closed)
also have "\ = (SUP K\{K. K \ space M - B \ closed K}. emeasure M K)"
unfolding SUP_image [of _ "\u. space M - u" _, symmetric, unfolded comp_def]
apply (rule arg_cong [of _ _ Sup])
using sU apply (auto intro!: imageI)
done
also have "\ = (SUP K\{K. K \ space M - B \ compact K}. emeasure M K)"
proof (safe intro!: antisym SUP_least)
fix K assume "closed K" "K \ space M - B"
from closed_in_D[OF \<open>closed K\<close>]
have K_inner: "emeasure M K = (SUP K\{Ka. Ka \ K \ compact Ka}. emeasure M K)" by simp
show "emeasure M K \ (SUP K\{K. K \ space M - B \ compact K}. emeasure M K)"
unfolding K_inner using \<open>K \<subseteq> space M - B\<close>
by (auto intro!: SUP_upper SUP_least)
qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
next
case (union D)
then have "range D \ sets M" by (auto simp: sb borel_eq_closed)
with union have M[symmetric]: "(\i. M (D i)) = M (\i. D i)" by (intro suminf_emeasure)
also have "(\n. \i (\i. M (D i))"
by (intro summable_LIMSEQ) auto
finally have measure_LIMSEQ: "(\n. \i measure M (\i. D i)"
by (simp add: emeasure_eq_measure sum_nonneg)
have "(\i. D i) \ sets M" using \range D \ sets M\ by auto
case 1
show ?case
proof (rule approx_inner)
fix e::real assume "e > 0"
with measure_LIMSEQ
have "\no. \n\no. \(\ix. D x)\ < e/2"
by (auto simp: lim_sequentially dist_real_def simp del: less_divide_eq_numeral1)
hence "\n0. \(\ix. D x)\ < e/2" by auto
then obtain n0 where n0: "\(\ii. D i)\ < e/2"
unfolding choice_iff by blast
have "ennreal (\ii
by (auto simp add: emeasure_eq_measure)
also have "\ \ (\i. M (D i))" by (rule sum_le_suminf) auto
also have "\ = M (\i. D i)" by (simp add: M)
also have "\ = measure M (\i. D i)" by (simp add: emeasure_eq_measure)
finally have n0: "measure M (\i. D i) - (\i
using n0 by (auto simp: sum_nonneg)
have "\i. \K. K \ D i \ compact K \ emeasure M (D i) \ emeasure M K + e/(2*Suc n0)"
proof
fix i
from \<open>0 < e\<close> have "0 < e/(2*Suc n0)" by simp
have "emeasure M (D i) = (SUP K\{K. K \ (D i) \ compact K}. emeasure M K)"
using union by blast
from SUP_approx_ennreal[OF \<open>0 < e/(2*Suc n0)\<close> _ this]
show "\K. K \ D i \ compact K \ emeasure M (D i) \ emeasure M K + e/(2*Suc n0)"
by (auto simp: emeasure_eq_measure intro: less_imp_le compact_empty)
qed
then obtain K where K: "\i. K i \ D i" "\i. compact (K i)"
"\i. emeasure M (D i) \ emeasure M (K i) + e/(2*Suc n0)"
unfolding choice_iff by blast
let ?K = "\i\{..
have "disjoint_family_on K {.. using K \<open>disjoint_family D\<close>
unfolding disjoint_family_on_def by blast
hence mK: "measure M ?K = (\i
by (intro finite_measure_finite_Union) (auto simp: sb compact_imp_closed)
have "measure M (\i. D i) < (\i
also have "(\i (\i
using K \<open>0 < e\<close>
by (auto intro: sum_mono simp: emeasure_eq_measure simp flip: ennreal_plus)
also have "\ = (\ii
by (simp add: sum.distrib)
also have "\ \ (\i0 < e\
by (auto simp: field_simps intro!: mult_left_mono)
finally
have "measure M (\i. D i) < (\i
by auto
hence "M (\i. D i) < M ?K + e"
using \<open>0<e\<close> by (auto simp: mK emeasure_eq_measure sum_nonneg ennreal_less_iff simp flip: ennreal_plus)
moreover
have "?K \ (\i. D i)" using K by auto
moreover
have "compact ?K" using K by auto
ultimately
have "?K\(\i. D i) \ compact ?K \ emeasure M (\i. D i) \ emeasure M ?K + ennreal e" by simp
thus "\K\\i. D i. compact K \ emeasure M (\i. D i) \ emeasure M K + ennreal e" ..
qed fact
case 2
show ?case
proof (rule approx_outer[OF \<open>(\<Union>i. D i) \<in> sets M\<close>])
fix e::real assume "e > 0"
have "\i::nat. \U. D i \ U \ open U \ e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
proof
fix i::nat
from \<open>0 < e\<close> have "0 < e/(2 powr Suc i)" by simp
have "emeasure M (D i) = (INF U\{U. (D i) \ U \ open U}. emeasure M U)"
using union by blast
from INF_approx_ennreal[OF \<open>0 < e/(2 powr Suc i)\<close> this]
show "\U. D i \ U \ open U \ e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
using \<open>0<e\<close>
by (auto simp: emeasure_eq_measure sum_nonneg ennreal_less_iff ennreal_minus
finite_measure_mono sb
simp flip: ennreal_plus)
qed
then obtain U where U: "\i. D i \ U i" "\i. open (U i)"
"\i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)"
unfolding choice_iff by blast
let ?U = "\i. U i"
have "ennreal (measure M ?U - measure M (\i. D i)) = M ?U - M (\i. D i)"
using U(1,2)
by (subst ennreal_minus[symmetric])
(auto intro!: finite_measure_mono simp: sb emeasure_eq_measure)
also have "\ = M (?U - (\i. D i))" using U \(\i. D i) \ sets M\
by (subst emeasure_Diff) (auto simp: sb)
also have "\ \ M (\i. U i - D i)" using U \range D \ sets M\
by (intro emeasure_mono) (auto simp: sb intro!: sets.countable_nat_UN sets.Diff)
also have "\ \ (\i. M (U i - D i))" using U \range D \ sets M\
by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)
also have "\ \ (\i. ennreal e/(2 powr Suc i))" using U \range D \ sets M\
using \<open>0<e\<close>
by (intro suminf_le, subst emeasure_Diff)
(auto simp: emeasure_Diff emeasure_eq_measure sb ennreal_minus
finite_measure_mono divide_ennreal ennreal_less_iff
intro: less_imp_le)
also have "\ \ (\n. ennreal (e * (1 / 2) ^ Suc n))"
using \<open>0<e\<close>
by (simp add: powr_minus powr_realpow field_simps divide_ennreal del: of_nat_Suc)
also have "\ = ennreal e * (\n. ennreal ((1 / 2) ^ Suc n))"
unfolding ennreal_power[symmetric]
using \<open>0 < e\<close>
by (simp add: ac_simps ennreal_mult' divide_ennreal[symmetric] divide_ennreal_def
ennreal_power[symmetric])
also have "\ = ennreal e"
by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
finally have "emeasure M ?U \ emeasure M (\i. D i) + ennreal e"
using \<open>0<e\<close> by (simp add: emeasure_eq_measure flip: ennreal_plus)
moreover
have "(\i. D i) \ ?U" using U by auto
moreover
have "open ?U" using U by auto
ultimately
have "(\i. D i) \ ?U \ open ?U \ emeasure M ?U \ emeasure M (\i. D i) + ennreal e" by simp
thus "\B. (\i. D i) \ B \ open B \ emeasure M B \ emeasure M (\i. D i) + ennreal e" ..
qed
qed
qed
end
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