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Quellcode-Bibliothek
© Kompilation durch diese Firma
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Datei:
QueueT.vdmpp
Sprache: Isabelle
(* Title: HOL/Analysis/Lebesgue_Measure.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
Author: Jeremy Avigad
Author: Luke Serafin
*)
section \<open>Lebesgue Measure\<close>
theory Lebesgue_Measure
imports
Finite_Product_Measure
Caratheodory
Complete_Measure
Summation_Tests
Regularity
begin
lemma measure_eqI_lessThan:
fixes M N :: "real measure"
assumes sets: "sets M = sets borel" "sets N = sets borel"
assumes fin: "\x. emeasure M {x <..} < \"
assumes "\x. emeasure M {x <..} = emeasure N {x <..}"
shows "M = N"
proof (rule measure_eqI_generator_eq_countable)
let ?LT = "\a::real. {a <..}" let ?E = "range ?LT"
show "Int_stable ?E"
by (auto simp: Int_stable_def lessThan_Int_lessThan)
show "?E \ Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
unfolding sets borel_Ioi by auto
show "?LT`Rats \ ?E" "(\i\Rats. ?LT i) = UNIV" "\a. a \ ?LT`Rats \ emeasure M a \ \"
using fin by (auto intro: Rats_no_bot_less simp: less_top)
qed (auto intro: assms countable_rat)
subsection \<open>Measures defined by monotonous functions\<close>
text \<open>
Every right-continuous and nondecreasing function gives rise to a measure on the reals:
\<close>
definition\<^marker>\<open>tag important\<close> interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
"interval_measure F =
extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a<..b}) (\<lambda>(a, b). ennreal (F b - F a))"
lemma\<^marker>\<open>tag important\<close> emeasure_interval_measure_Ioc:
assumes "a \ b"
assumes mono_F: "\x y. x \ y \ F x \ F y"
assumes right_cont_F : "\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 \ b}"
proof (unfold_locales, safe)
fix a b c d :: real assume *: "a \ b" "c \ d"
then show "\C\{{a<..b} |a b. a \ b}. finite C \ disjoint C \ {a<..b} - {c<..d} = \C"
proof cases
let ?C = "{{a<..b}}"
assume "b < c \ d \ a \ d \ c"
with * have "?C \ {{a<..b} |a b. a \ b} \ finite ?C \ disjoint ?C \ {a<..b} - {c<..d} = \?C"
by (auto simp add: disjoint_def)
thus ?thesis ..
next
let ?C = "{{a<..c}, {d<..b}}"
assume "\ (b < c \ d \ a \ d \ c)"
with * have "?C \ {{a<..b} |a b. a \ b} \ finite ?C \ disjoint ?C \ {a<..b} - {c<..d} = \?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 \ real" and a b :: real
assume l_r[simp]: "\n. l n \ r n" and "a \ b" and disj: "disjoint_family (\n. {l n<..r n})"
assume lr_eq_ab: "(\i. {l i<..r i}) = {a<..b}"
have [intro, simp]: "\a b. a \ b \ F a \ 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 "\i. i \ S \ {l i <.. r i} \ {a <.. b}"
unfolding lr_eq_ab[symmetric] by auto
ultimately have "(\i\S. F (r i) - F (l i)) \ F b - F a"
proof (induction S arbitrary: a rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "\i\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 \ S" "l m < r m" "l m = Min (l ` {i\S. l i < r i})"
by fastforce
have "(\i\S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\i\S - {m}. F (r i) - F (l i))"
using m psubset by (intro sum.remove) auto
also have "(\i\S - {m}. F (r i) - F (l i)) \ F b - F (r m)"
proof (intro psubset.IH)
show "S - {m} \ S"
using \<open>m\<in>S\<close> by auto
show "r m \ b"
using psubset.prems(2)[OF \<open>m\<in>S\<close>] \<open>l m < r m\<close> by auto
next
fix i assume "i \ S - {m}"
then have i: "i \ S" "i \ 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} \ {l m <.. r m} \ {}"
by auto
then have False
using disjoint_family_onD[OF disj, of i m] i by auto }
then have "l i \ r i \ r m \ l i"
unfolding not_less[symmetric] using l_r[of i] by auto
then show "{l i <.. r i} \ {r m <.. b}"
using psubset.prems(2)[OF \<open>i\<in>S\<close>] by auto
qed
also have "F (r m) - F (l m) \ 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 \ real"
assume "finite S" "\i. i\S \ l i < r i" "{u..v} \ (\i\S. {l i<..< r i})"
then have "F v - F u \ (\i\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 \ {}"
then obtain j where "j \ S"
by auto
let ?R = "r j < u \ l j > v \ (\i\S-{j}. l i \ l j \ r j \ 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 "\ \ (\i\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 "\ ?R"
then have j: "u \ r j" "l j \ v" "\i. i \ S - {j} \ r i < r j \ l i > l j"
by (auto simp: not_less)
let ?S1 = "{i \ S. l i < l j}"
let ?S2 = "{i \ S. r i > r j}"
have "(\i\S. F (r i) - F (l i)) \ (\i\?S1 \ ?S2 \ {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 "(\i\?S1 \ ?S2 \ {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 "(\i\?S1. F (r i) - F (l i)) \ 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 "(\i\?S2. F (r i) - F (l i)) \ 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) \ (\i. ennreal (F (r i) - F (l i)))"
proof (rule ennreal_le_epsilon)
fix epsilon :: real assume egt0: "epsilon > 0"
have "\i. \d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
proof
fix i
note right_cont_F [of "r i"]
thus "\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: "\i. delta i > 0" and
deltai_prop: "\i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
by metis
have "\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 \ S'" and finS: "finite S" and
Sprop: "{a'..b} \ (\i \ S. {l i<..
proof (rule compactE_image)
show "compact {a'..b}"
by (rule compact_Icc)
show "\i. i \ S' \ open ({l i<..
have "{a'..b} \ {a <.. b}"
by auto
also have "{a <.. b} = (\i\S'. {l i<..r i})"
unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
also have "\ \ (\i \ S'. {l 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} \ (\i \ S'. {l i<..
qed
with S'_def have Sprop2: "\i. i \ S \ l i < r i" by auto
from finS have "\n. \i \ S. i \ n"
by (subst finite_nat_set_iff_bounded_le [symmetric])
then obtain n where Sbound [rule_format]: "\i \ S. i \ n" ..
have "F b - F a' \ (\i\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 "... \ (\i \ 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 "... = (\i \ 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 "... \ ?t + (epsilon / 4) * (\ 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 "... \ ?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' \ (\i\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 "... \ (F b - F a') + epsilon / 2"
using a_prop by (intro add_left_mono) simp
also have "... \ (\i\S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
apply (intro add_right_mono)
apply (rule aux2)
done
also have "... = (\i\S. F (r i) - F (l i)) + epsilon"
by auto
also have "... \ (\i\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) \ (\i\n. ennreal (F (r i) - F (l i))) + epsilon"
using egt0 by (simp add: sum_nonneg flip: ennreal_plus)
then show "ennreal (F b - F a) \ (\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 "(\i. ennreal (F (r i) - F (l i))) \ 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 "(\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 \ b" and "\x y. x \ y \ F x \ F y" and "\a. continuous (at_right a) F"
shows "measure (interval_measure F) {a <.. b} = F b - F a"
unfolding measure_def
by (simp add: assms emeasure_interval_measure_Ioc)
lemma emeasure_interval_measure_Ioc_eq:
"(\x y. x \ y \ F x \ F y) \ (\a. continuous (at_right a) F) \
emeasure (interval_measure F) {a <.. b} = (if a \<le> b then F b - F a else 0)"
using emeasure_interval_measure_Ioc[of a b F] by auto
lemma\<^marker>\<open>tag important\<close> sets_interval_measure [simp, measurable_cong]:
"sets (interval_measure F) = sets borel"
apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc)
apply (rule sigma_sets_eqI)
apply auto
apply (case_tac "a \ 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 \ b"
assumes mono_F: "\x y. x \ y \ F x \ 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 \ 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 "((\a. F b - F a) \ emeasure ?F {a..b}) (at_left a)"
proof (rule tendsto_at_left_sequentially)
show "a - 1 < a" by simp
fix X assume "\n. X n < a" "incseq X" "X \ 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 "(\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 "(\n. emeasure ?F {X n<..b}) = (\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 "(\n. F b - F (X n)) \ emeasure ?F {a..b}" .
qed
show "((\a. ennreal (F b - F a)) \ F b - F a) (at_left a)"
by (rule continuous_on_tendsto_compose[where g="\x. x" and s=UNIV])
(auto simp: continuous_on_ennreal continuous_on_diff cont_F)
qed (rule trivial_limit_at_left_real)
lemma\<^marker>\<open>tag important\<close> sigma_finite_interval_measure:
assumes mono_F: "\x y. x \ y \ F x \ F y"
assumes right_cont_F : "\a. continuous (at_right a) F"
shows "sigma_finite_measure (interval_measure F)"
apply unfold_locales
apply (intro exI[of _ "(\(a, b). {a <.. b}) ` (\ \ \)"])
apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
done
subsection \<open>Lebesgue-Borel measure\<close>
definition\<^marker>\<open>tag important\<close> lborel :: "('a :: euclidean_space) measure" where
"lborel = distr (\\<^sub>M b\Basis. interval_measure (\x. x)) borel (\f. \b\Basis. f b *\<^sub>R b)"
abbreviation\<^marker>\<open>tag important\<close> lebesgue :: "'a::euclidean_space measure"
where "lebesgue \ completion lborel"
abbreviation\<^marker>\<open>tag important\<close> lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
where "lebesgue_on \ \ restrict_space (completion lborel) \"
lemma lebesgue_on_mono:
assumes major: "AE x in lebesgue_on S. P x" and minor: "\x.\P x; x \ S\ \ Q x"
shows "AE x in lebesgue_on S. Q x"
proof -
have "AE a in lebesgue_on S. P a \ Q a"
using minor space_restrict_space by fastforce
then show ?thesis
using major by auto
qed
lemma integral_eq_zero_null_sets:
assumes "S \ null_sets lebesgue"
shows "integral\<^sup>L (lebesgue_on S) f = 0"
proof (rule integral_eq_zero_AE)
show "AE x in lebesgue_on S. f x = 0"
by (metis (no_types, lifting) assms AE_not_in lebesgue_on_mono null_setsD2 null_sets_restrict_space order_refl)
qed
lemma
shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
and space_lborel[simp]: "space lborel = space borel"
and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"
by (simp_all add: lborel_def)
lemma space_lebesgue_on [simp]: "space (lebesgue_on S) = S"
by (simp add: space_restrict_space)
lemma sets_lebesgue_on_refl [iff]: "S \ 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 \ sets lebesgue \ A \ 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 "\x. x \ S \ f x = g x"
shows "f \ measurable (lebesgue_on S) M \ g \ measurable (lebesgue_on S) M"
by (metis (mono_tags, lifting) IntD1 assms measurable_cong_simp space_restrict_space)
lemma lebesgue_on_UNIV_eq: "lebesgue_on UNIV = lebesgue"
proof -
have "measure_of UNIV (sets lebesgue) (emeasure lebesgue) = lebesgue"
by (metis measure_of_of_measure space_borel space_completion space_lborel)
then show ?thesis
by (auto simp: restrict_space_def)
qed
lemma integral_restrict_Int:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "S \ sets lebesgue" "T \ sets lebesgue"
shows "integral\<^sup>L (lebesgue_on T) (\x. if x \ S then f x else 0) = integral\<^sup>L (lebesgue_on (S \ T)) f"
proof -
have "(\x. indicat_real T x *\<^sub>R (if x \ S then f x else 0)) = (\x. indicat_real (S \ T) x *\<^sub>R f x)"
by (force simp: indicator_def)
then show ?thesis
by (simp add: assms sets.Int Bochner_Integration.integral_restrict_space)
qed
lemma integral_restrict:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "S \ T" "S \ sets lebesgue" "T \ sets lebesgue"
shows "integral\<^sup>L (lebesgue_on T) (\x. if x \ S then f x else 0) = integral\<^sup>L (lebesgue_on S) f"
using integral_restrict_Int [of S T f] assms
by (simp add: Int_absorb2)
lemma integral_restrict_UNIV:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "S \ sets lebesgue"
shows "integral\<^sup>L lebesgue (\x. if x \ S then f x else 0) = integral\<^sup>L (lebesgue_on S) f"
using integral_restrict_Int [of S UNIV f] assms
by (simp add: lebesgue_on_UNIV_eq)
lemma integrable_lebesgue_on_empty [iff]:
fixes f :: "'a::euclidean_space \ 'b::{second_countable_topology,banach}"
shows "integrable (lebesgue_on {}) f"
by (simp add: integrable_restrict_space)
lemma integral_lebesgue_on_empty [simp]:
fixes f :: "'a::euclidean_space \ 'b::{second_countable_topology,banach}"
shows "integral\<^sup>L (lebesgue_on {}) f = 0"
by (simp add: Bochner_Integration.integral_empty)
lemma has_bochner_integral_restrict_space:
fixes f :: "'a \ 'b::{banach, second_countable_topology}"
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
shows "has_bochner_integral (restrict_space M \) f i
\<longleftrightarrow> has_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) i"
by (simp add: integrable_restrict_space [OF assms] integral_restrict_space [OF assms] has_bochner_integral_iff)
lemma integrable_restrict_UNIV:
fixes f :: "'a::euclidean_space \ 'b::{banach, second_countable_topology}"
assumes S: "S \ sets lebesgue"
shows "integrable lebesgue (\x. if x \ S then f x else 0) \ integrable (lebesgue_on S) f"
using has_bochner_integral_restrict_space [of S lebesgue f] assms
by (simp add: integrable.simps indicator_scaleR_eq_if)
lemma integral_mono_lebesgue_on_AE:
fixes f::"_ \ real"
assumes f: "integrable (lebesgue_on T) f"
and gf: "AE x in (lebesgue_on S). g x \ f x"
and f0: "AE x in (lebesgue_on T). 0 \ f x"
and "S \ T" and S: "S \ sets lebesgue" and T: "T \ sets lebesgue"
shows "(\x. g x \(lebesgue_on S)) \ (\x. f x \(lebesgue_on T))"
proof -
have "(\x. g x \(lebesgue_on S)) = (\x. (if x \ S then g x else 0) \lebesgue)"
by (simp add: Lebesgue_Measure.integral_restrict_UNIV S)
also have "\ \ (\x. (if x \ T then f x else 0) \lebesgue)"
proof (rule Bochner_Integration.integral_mono_AE')
show "integrable lebesgue (\x. if x \ T then f x else 0)"
by (simp add: integrable_restrict_UNIV T f)
show "AE x in lebesgue. (if x \ S then g x else 0) \ (if x \ T then f x else 0)"
using assms by (auto simp: AE_restrict_space_iff)
show "AE x in lebesgue. 0 \ (if x \ T then f x else 0)"
using f0 by (simp add: AE_restrict_space_iff T)
qed
also have "\ = (\x. f x \(lebesgue_on T))"
using Lebesgue_Measure.integral_restrict_UNIV T by blast
finally show ?thesis .
qed
subsection \<open>Borel measurability\<close>
lemma borel_measurable_if_I:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes f: "f \ borel_measurable (lebesgue_on S)" and S: "S \ sets lebesgue"
shows "(\x. if x \ S then f x else 0) \ borel_measurable lebesgue"
proof -
have eq: "{x. x \ S} \ {x. f x \ Y} = {x. x \ S} \ {x. f x \ Y} \ S" for Y
by blast
show ?thesis
using f S
apply (simp add: vimage_def in_borel_measurable_borel Ball_def)
apply (elim all_forward imp_forward asm_rl)
apply (simp only: Collect_conj_eq Collect_disj_eq imp_conv_disj eq)
apply (auto simp: Compl_eq [symmetric] Compl_in_sets_lebesgue sets_restrict_space_iff)
done
qed
lemma borel_measurable_if_D:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "(\x. if x \ S then f x else 0) \ borel_measurable lebesgue"
shows "f \ borel_measurable (lebesgue_on S)"
using assms
apply (simp add: in_borel_measurable_borel Ball_def)
apply (elim all_forward imp_forward asm_rl)
apply (force simp: space_restrict_space sets_restrict_space image_iff intro: rev_bexI)
done
lemma borel_measurable_if:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "S \ sets lebesgue"
shows "(\x. if x \ S then f x else 0) \ borel_measurable lebesgue \ f \ borel_measurable (lebesgue_on S)"
using assms borel_measurable_if_D borel_measurable_if_I by blast
lemma borel_measurable_if_lebesgue_on:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "S \ sets lebesgue" "T \ sets lebesgue" "S \ T"
shows "(\x. if x \ S then f x else 0) \ borel_measurable (lebesgue_on T) \ f \ borel_measurable (lebesgue_on S)"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using measurable_restrict_mono [OF _ \<open>S \<subseteq> T\<close>]
by (subst measurable_lebesgue_cong [where g = "(\x. if x \ S then f x else 0)"]) auto
next
assume ?rhs then show ?lhs
by (simp add: \<open>S \<in> sets lebesgue\<close> borel_measurable_if_I measurable_restrict_space1)
qed
lemma borel_measurable_vimage_halfspace_component_lt:
"f \ borel_measurable (lebesgue_on S) \
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_less])
apply (fastforce simp add: space_restrict_space)
done
lemma borel_measurable_vimage_halfspace_component_ge:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable (lebesgue_on S) \
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<ge> a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_ge])
apply (fastforce simp add: space_restrict_space)
done
lemma borel_measurable_vimage_halfspace_component_gt:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable (lebesgue_on S) \
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i > a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_greater])
apply (fastforce simp add: space_restrict_space)
done
lemma borel_measurable_vimage_halfspace_component_le:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable (lebesgue_on S) \
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<le> a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_le])
apply (fastforce simp add: space_restrict_space)
done
lemma
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows borel_measurable_vimage_open_interval:
"f \ borel_measurable (lebesgue_on S) \
(\<forall>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S))" (is ?thesis1)
and borel_measurable_vimage_open:
"f \ borel_measurable (lebesgue_on S) \
(\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))" (is ?thesis2)
proof -
have "{x \ S. f x \ box a b} \ sets (lebesgue_on S)" if "f \ borel_measurable (lebesgue_on S)" for a b
proof -
have "S = S \ space lebesgue"
by simp
then have "S \ (f -` box a b) \ sets (lebesgue_on S)"
by (metis (no_types) box_borel in_borel_measurable_borel inf_sup_aci(1) space_restrict_space that)
then show ?thesis
by (simp add: Collect_conj_eq vimage_def)
qed
moreover
have "{x \ S. f x \ T} \ sets (lebesgue_on S)"
if T: "\a b. {x \ S. f x \ box a b} \ sets (lebesgue_on S)" "open T" for T
proof -
obtain \<D> where "countable \<D>" and \<D>: "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = T"
using open_countable_Union_open_box that \<open>open T\<close> by metis
then have eq: "{x \ S. f x \ T} = (\U \ \. {x \ S. f x \ U})"
by blast
have "{x \ S. f x \ U} \ sets (lebesgue_on S)" if "U \ \" for U
using that T \<D> by blast
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \countable \\])
qed
moreover
have eq: "{x \ S. f x \ i < a} = {x \ S. f x \ {y. y \ i < a}}" for i a
by auto
have "f \ borel_measurable (lebesgue_on S)"
if "\T. open T \ {x \ S. f x \ T} \ sets (lebesgue_on S)"
by (metis (no_types) eq borel_measurable_vimage_halfspace_component_lt open_halfspace_component_lt that)
ultimately show "?thesis1" "?thesis2"
by blast+
qed
lemma borel_measurable_vimage_closed:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable (lebesgue_on S) \
(\<forall>T. closed T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof -
have eq: "{x \ S. f x \ T} = S - {x \ S. f x \ (- T)}" for T
by auto
show ?thesis
apply (simp add: borel_measurable_vimage_open, safe)
apply (simp_all (no_asm) add: eq)
apply (intro sets.Diff sets_lebesgue_on_refl, force simp: closed_open)
apply (intro sets.Diff sets_lebesgue_on_refl, force simp: open_closed)
done
qed
lemma borel_measurable_vimage_closed_interval:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable (lebesgue_on S) \
(\<forall>a b. {x \<in> S. f x \<in> cbox a b} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using borel_measurable_vimage_closed by blast
next
assume RHS: ?rhs
have "{x \ S. f x \ T} \ sets (lebesgue_on S)" if "open T" for T
proof -
obtain \<D> where "countable \<D>" and \<D>: "\<D> \<subseteq> Pow T" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = T"
using open_countable_Union_open_cbox that \<open>open T\<close> by metis
then have eq: "{x \ S. f x \ T} = (\U \ \. {x \ S. f x \ U})"
by blast
have "{x \ S. f x \ U} \ sets (lebesgue_on S)" if "U \ \" for U
using that \<D> by (metis RHS)
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \countable \\])
qed
then show ?lhs
by (simp add: borel_measurable_vimage_open)
qed
lemma borel_measurable_vimage_borel:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable (lebesgue_on S) \
(\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume f: ?lhs
then show ?rhs
using measurable_sets [OF f]
by (simp add: Collect_conj_eq inf_sup_aci(1) space_restrict_space vimage_def)
qed (simp add: borel_measurable_vimage_open_interval)
lemma lebesgue_measurable_vimage_borel:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "f \ borel_measurable lebesgue" "T \ sets borel"
shows "{x. f x \ T} \ sets lebesgue"
using assms borel_measurable_vimage_borel [of f UNIV] by auto
lemma borel_measurable_lebesgue_preimage_borel:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
shows "f \ borel_measurable lebesgue \
(\<forall>T. T \<in> sets borel \<longrightarrow> {x. f x \<in> T} \<in> sets lebesgue)"
apply (intro iffI allI impI lebesgue_measurable_vimage_borel)
apply (auto simp: in_borel_measurable_borel vimage_def)
done
subsection \<^marker>\<open>tag unimportant\<close> \<open>Measurability of continuous functions\<close>
lemma continuous_imp_measurable_on_sets_lebesgue:
assumes f: "continuous_on S f" and S: "S \ sets lebesgue"
shows "f \ borel_measurable (lebesgue_on S)"
proof -
have "sets (restrict_space borel S) \ 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
lemma id_borel_measurable_lebesgue [iff]: "id \ borel_measurable lebesgue"
by (simp add: measurable_completion)
lemma id_borel_measurable_lebesgue_on [iff]: "id \ borel_measurable (lebesgue_on S)"
by (simp add: measurable_completion measurable_restrict_space1)
context
begin
interpretation sigma_finite_measure "interval_measure (\x. x)"
by (rule sigma_finite_interval_measure) auto
interpretation finite_product_sigma_finite "\_. interval_measure (\x. x)" Basis
proof qed simp
lemma lborel_eq_real: "lborel = interval_measure (\x. x)"
unfolding lborel_def Basis_real_def
using distr_id[of "interval_measure (\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 (\\<^sub>M b\Basis. lborel) borel (\f. \b\Basis. f b *\<^sub>R b)"
by (subst lborel_def) (simp add: lborel_eq_real)
lemma nn_integral_lborel_prod:
assumes [measurable]: "\b. b \ Basis \ f b \ borel_measurable borel"
assumes nn[simp]: "\b x. b \ Basis \ 0 \ f b x"
shows "(\\<^sup>+x. (\b\Basis. f b (x \ b)) \lborel) = (\b\Basis. (\\<^sup>+x. f b x \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 \ u"
shows "emeasure lborel {l .. u} = u - l"
proof -
have "((\f. f 1) -` {l..u} \ space (Pi\<^sub>M {1} (\b. interval_measure (\x. x)))) = {1::real} \\<^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)
qed
lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l \ u then u - l else 0)"
by simp
lemma\<^marker>\<open>tag important\<close> emeasure_lborel_cbox[simp]:
assumes [simp]: "\b. b \ Basis \ l \ b \ u \ b"
shows "emeasure lborel (cbox l u) = (\b\Basis. (u - l) \ b)"
proof -
have "(\x. \b\Basis. indicator {l\b .. u\b} (x \ b) :: ennreal) = indicator (cbox l u)"
by (auto simp: fun_eq_iff cbox_def split: split_indicator)
then have "emeasure lborel (cbox l u) = (\\<^sup>+x. (\b\Basis. indicator {l\b .. u\b} (x \ b)) \lborel)"
by simp
also have "\ = (\b\Basis. (u - l) \ 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 \ 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 \ 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 \ 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 \ 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]: "\b. b \ Basis \ l \ b \ u \ b"
shows "emeasure lborel (box l u) = (\b\Basis. (u - l) \ b)"
proof -
have "(\x. \b\Basis. indicator {l\b <..< u\b} (x \ b) :: ennreal) = indicator (box l u)"
by (auto simp: fun_eq_iff box_def split: split_indicator)
then have "emeasure lborel (box l u) = (\\<^sup>+x. (\b\Basis. indicator {l\b <..< u\b} (x \ b)) \lborel)"
by simp
also have "\ = (\b\Basis. (u - l) \ 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 \b\Basis. l \ b \ u \ b then \b\Basis. (u - l) \ 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 \b\Basis. l \ b \ u \ b then \b\Basis. (u - l) \ b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
using emeasure_lborel_cbox[of x x] nonempty_Basis
by (auto simp del: emeasure_lborel_cbox nonempty_Basis)
lemma emeasure_lborel_cbox_finite: "emeasure lborel (cbox a b) < \"
by (auto simp: emeasure_lborel_cbox_eq)
lemma emeasure_lborel_box_finite: "emeasure lborel (box a b) < \"
by (auto simp: emeasure_lborel_box_eq)
lemma emeasure_lborel_ball_finite: "emeasure lborel (ball c r) < \"
proof -
have "bounded (ball c r)" by simp
from bounded_subset_cbox_symmetric[OF this] obtain a where a: "ball c r \ cbox (-a) a"
by auto
hence "emeasure lborel (ball c r) \ emeasure lborel (cbox (-a) a)"
by (intro emeasure_mono) auto
also have "\ < \" by (simp add: emeasure_lborel_cbox_eq)
finally show ?thesis .
qed
lemma emeasure_lborel_cball_finite: "emeasure lborel (cball c r) < \"
proof -
have "bounded (cball c r)" by simp
from bounded_subset_cbox_symmetric[OF this] obtain a where a: "cball c r \ cbox (-a) a"
by auto
hence "emeasure lborel (cball c r) \ emeasure lborel (cbox (-a) a)"
by (intro emeasure_mono) auto
also have "\ < \" by (simp add: emeasure_lborel_cbox_eq)
finally show ?thesis .
qed
lemma fmeasurable_cbox [iff]: "cbox a b \ fmeasurable lborel"
and fmeasurable_box [iff]: "box a b \ fmeasurable lborel"
by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)
lemma
fixes l u :: real
assumes [simp]: "l \ 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]: "\b. b \ Basis \ l \ b \ u \ b"
shows measure_lborel_box[simp]: "measure lborel (box l u) = (\b\Basis. (u - l) \ b)"
and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\b\Basis. (u - l) \ b)"
by (simp_all add: measure_def inner_diff_left prod_nonneg)
lemma measure_lborel_cbox_eq:
"measure lborel (cbox l u) = (if \b\Basis. l \ b \ u \ b then \b\Basis. (u - l) \ 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 \b\Basis. l \ b \ u \ b then \b\Basis. (u - l) \ 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 "\A::'a set set. countable A \ A \ sets lborel \ \A = space lborel \ (\a\A. emeasure lborel a \ \)"
by (intro exI[of _ "range (\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) = \"
proof -
{ fix n::nat
let ?Ba = "Basis :: 'a set"
have "real n \ (2::real) ^ card ?Ba * real n"
by (simp add: mult_le_cancel_right1)
also
have "... \ (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
apply (rule mult_left_mono)
apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
apply (simp)
done
finally have "real n \ (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
} note [intro!] = this
show ?thesis
unfolding UN_box_eq_UNIV[symmetric]
apply (subst SUP_emeasure_incseq[symmetric])
apply (auto simp: incseq_def subset_box inner_add_left
simp del: Sup_eq_top_iff SUP_eq_top_iff
intro!: ennreal_SUP_eq_top)
done
qed
lemma emeasure_lborel_countable:
fixes A :: "'a::euclidean_space set"
assumes "countable A"
shows "emeasure lborel A = 0"
proof -
have "A \ (\i. {from_nat_into A i})" using from_nat_into_surj assms by force
then have "emeasure lborel A \ emeasure lborel (\i. {from_nat_into A i})"
by (intro emeasure_mono) auto
also have "emeasure lborel (\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 \ A \ null_sets lborel"
by (simp add: null_sets_def emeasure_lborel_countable sets.countable)
lemma finite_imp_null_set_lborel: "finite A \ A \ null_sets lborel"
by (intro countable_imp_null_set_lborel countable_finite)
lemma insert_null_sets_iff [simp]: "insert a N \ null_sets lebesgue \ N \ null_sets lebesgue"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (meson completion.complete2 subset_insertI)
next
assume ?rhs then show ?lhs
by (simp add: null_sets.insert_in_sets null_setsI)
qed
lemma insert_null_sets_lebesgue_on_iff [simp]:
assumes "a \ S" "S \ sets lebesgue"
shows "insert a N \ null_sets (lebesgue_on S) \ N \ null_sets (lebesgue_on S)"
by (simp add: assms null_sets_restrict_space)
lemma lborel_neq_count_space[simp]: "lborel \ 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} \ 1" by simp
ultimately show False by contradiction
qed
lemma mem_closed_if_AE_lebesgue_open:
assumes "open S" "closed C"
assumes "AE x \ S in lebesgue. x \ C"
assumes "x \ S"
shows "x \ C"
proof (rule ccontr)
assume xC: "x \ C"
with openE[of "S - C"] assms
obtain e where e: "0 < e" "ball x e \ S - C"
by blast
then obtain a b where box: "x \ box a b" "box a b \ 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 "\ \ emeasure lebesgue (S - C)"
using assms box
by (auto intro!: emeasure_mono)
also have "\ = 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 \ (AE x in lebesgue. x \ C) \ x \ C"
using mem_closed_if_AE_lebesgue_open[OF open_UNIV] by simp
subsection \<open>Affine transformation on the Lebesgue-Borel\<close>
lemma\<^marker>\<open>tag important\<close> lborel_eqI:
fixes M :: "'a::euclidean_space measure"
assumes emeasure_eq: "\l u. (\b. b \ Basis \ l \ b \ u \ b) \ emeasure M (box l u) = (\b\Basis. (u - l) \ b)"
assumes sets_eq: "sets M = sets borel"
shows "lborel = M"
proof (rule measure_eqI_generator_eq)
let ?E = "range (\(a, b). box a b::'a set)"
show "Int_stable ?E"
by (auto simp: Int_stable_def box_Int_box)
show "?E \ Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
by (simp_all add: borel_eq_box sets_eq)
let ?A = "\n::nat. box (- (real n *\<^sub>R One)) (real n *\<^sub>R One) :: 'a set"
show "range ?A \ ?E" "(\i. ?A i) = UNIV"
unfolding UN_box_eq_UNIV by auto
{ fix i show "emeasure lborel (?A i) \ \" by auto }
{ fix X assume "X \ ?E" then show "emeasure lborel X = emeasure M X"
apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
apply (subst box_eq_empty(1)[THEN iffD2])
apply (auto intro: less_imp_le simp: not_le)
done }
qed
lemma\<^marker>\<open>tag important\<close> lborel_affine_euclidean:
fixes c :: "'a::euclidean_space \ real" and t
defines "T x \ t + (\j\Basis. (c j * (x \ j)) *\<^sub>R j)"
assumes c: "\j. j \ Basis \ c j \ 0"
shows "lborel = density (distr lborel borel T) (\_. (\j\Basis. \c j\))" (is "_ = ?D")
proof (rule lborel_eqI)
let ?B = "Basis :: 'a set"
fix l u assume le: "\b. b \ ?B \ l \ b \ u \ b"
have [measurable]: "T \ borel \\<^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) = (\b\?B. (u - l) \ b)"
by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
field_split_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
intro!: prod.cong)
qed simp
lemma lborel_affine:
fixes t :: "'a::euclidean_space"
shows "c \ 0 \ lborel = density (distr lborel borel (\x. t + c *\<^sub>R x)) (\_. \c\^DIM('a))"
using lborel_affine_euclidean[where c="\_::'a. c" and t=t]
unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation prod_constant by simp
lemma lborel_real_affine:
"c \ 0 \ lborel = density (distr lborel borel (\x. t + c * x)) (\_. ennreal (abs c))"
using lborel_affine[of c t] by simp
lemma AE_borel_affine:
fixes P :: "real \ bool"
shows "c \ 0 \ Measurable.pred borel P \ AE x in lborel. P x \ 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 \ borel_measurable borel" and c: "c \ 0"
shows "(\\<^sup>+x. f x \lborel) = \c\ * (\\<^sup>+x. f (t + c * x) \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 \ 'a :: {banach, second_countable_topology}"
assumes f: "integrable lborel f"
shows "c \ 0 \ integrable lborel (\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 \ 'a :: {banach, second_countable_topology}"
shows "c \ 0 \ integrable lborel (\x. f (t + c * x)) \ integrable lborel f"
using
lborel_integrable_real_affine[of f c t]
lborel_integrable_real_affine[of "\x. f (t + c * x)" "1/c" "-t/c"]
by (auto simp add: field_simps)
lemma\<^marker>\<open>tag important\<close> lborel_integral_real_affine:
fixes f :: "real \ 'a :: {banach, second_countable_topology}" and c :: real
assumes c: "c \ 0" shows "(\x. f x \ lborel) = \c\ *\<^sub>R (\x. f (t + c * x) \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 "\ 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 \ real" and t
assumes c: "\j. j \ Basis \ c j \ 0"
defines "T == (\x. t + (\j\Basis. (c j * (x \ j)) *\<^sub>R j))"
shows lebesgue_affine_euclidean: "lebesgue = density (distr lebesgue lebesgue T) (\_. (\j\Basis. \c j\))" (is "_ = ?D")
and lebesgue_affine_measurable: "T \ lebesgue \\<^sub>M lebesgue"
proof -
have T_borel[measurable]: "T \ borel \\<^sub>M borel"
by (auto simp: T_def[abs_def])
{ fix A :: "'a set" assume A: "A \ sets borel"
then have "emeasure lborel A = 0 \ emeasure (density (distr lborel borel T) (\_. (\j\Basis. \c j\))) A = 0"
unfolding T_def using c by (subst lborel_affine_euclidean[symmetric]) auto
also have "\ \ 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 \ 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 \ lebesgue \\<^sub>M lebesgue"
by (rule completion.measurable_completion2) (auto simp: eq measurable_completion)
have "lebesgue = completion (density (distr lborel borel T) (\_. (\j\Basis. \c j\)))"
using c by (subst lborel_affine_euclidean[of c t]) (simp_all add: T_def[abs_def])
also have "\ = density (completion (distr lebesgue lborel T)) (\_. (\j\Basis. \c j\))"
using c by (auto intro!: always_eventually prod_pos completion_density_eq simp: distr_completion cong: distr_cong)
also have "\ = density (distr lebesgue lebesgue T) (\_. (\j\Basis. \c j\))"
by (subst completion.completion_distr_eq) (auto simp: eq measurable_completion)
finally show "lebesgue = density (distr lebesgue lebesgue T) (\_. (\j\Basis. \c j\))" .
qed
corollary lebesgue_real_affine:
"c \ 0 \ lebesgue = density (distr lebesgue lebesgue (\x. t + c * x)) (\_. ennreal (abs c))"
using lebesgue_affine_euclidean [where c= "\x::real. c"] by simp
lemma nn_integral_real_affine_lebesgue:
fixes c :: real assumes f[measurable]: "f \ borel_measurable lebesgue" and c: "c \ 0"
shows "(\\<^sup>+x. f x \lebesgue) = ennreal\c\ * (\\<^sup>+x. f(t + c * x) \lebesgue)"
proof -
have "(\\<^sup>+x. f x \lebesgue) = (\\<^sup>+x. f x \density (distr lebesgue lebesgue (\x. t + c * x)) (\x. ennreal \c\))"
using lebesgue_real_affine c by auto
also have "\ = \\<^sup>+ x. ennreal \c\ * f x \distr lebesgue lebesgue (\x. t + c * x)"
by (subst nn_integral_density) auto
also have "\ = ennreal \c\ * integral\<^sup>N (distr lebesgue lebesgue (\x. t + c * x)) f"
using f measurable_distr_eq1 nn_integral_cmult by blast
also have "\ = \c\ * (\\<^sup>+x. f(t + c * x) \lebesgue)"
using lebesgue_affine_measurable[where c= "\x::real. c"]
by (subst nn_integral_distr) (force+)
finally show ?thesis .
qed
lemma lebesgue_measurable_scaling[measurable]: "(*\<^sub>R) x \ lebesgue \\<^sub>M lebesgue"
proof cases
assume "x = 0"
then have "(*\<^sub>R) x = (\x. 0::'a)"
by (auto simp: fun_eq_iff)
then show ?thesis by auto
next
assume "x \ 0" then show ?thesis
using lebesgue_affine_measurable[of "\_. 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 \ r *\<^sub>R x + d"
shows emeasure_lebesgue_affine: "emeasure lebesgue (T m \ ` S) = \m\ ^ DIM('a) * emeasure lebesgue S" (is ?e)
and measure_lebesgue_affine: "measure lebesgue (T m \ ` S) = \m\ ^ 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 \" and ?T' = "T (1 / m) (- ((1/m) *\<^sub>R \))"
assume "m \ 0"
then have s_comp_s: "?T' \ ?T = id" "?T \ ?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 \ ` S = T (1 / m) ((-1/m) *\<^sub>R \) -` S \ space lebesgue"
using bij_vimage_eq_inv_image[OF \<open>bij ?T'\<close>, of S] by auto
have trans_eq_T: "(\x. \ + (\j\Basis. (m * (x \ j)) *\<^sub>R j)) = T m \" for m \
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 \ lebesgue \\<^sub>M lebesgue" for r d
using lebesgue_affine_measurable[of "\_. r" d]
by (cases "r = 0") (auto simp: trans_eq_T T_def[abs_def])
show ?thesis
proof cases
assume "S \ sets lebesgue" with \m \ 0\ show ?thesis
unfolding eq
apply (subst lebesgue_affine_euclidean[of "\_. m" \])
apply (simp_all add: emeasure_density trans_eq_T nn_integral_cmult emeasure_distr
del: space_completion emeasure_completion)
apply (simp add: vimage_comp s_comp_s)
done
next
assume "S \ sets lebesgue"
moreover have "?T ` S \ sets lebesgue"
proof
assume "?T ` S \ sets lebesgue"
then have "?T -` (?T ` S) \ space lebesgue \ sets lebesgue"
by (rule measurable_sets[OF T])
also have "?T -` (?T ` S) \ 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 \ 0"
shows "lebesgue = density (distr lebesgue lebesgue (\x. c * x)) (\x. ennreal \c\)"
using assms by (subst lebesgue_affine_euclidean[of "\_. c" 0]) simp_all
lemma divideR_right:
fixes x y :: "'a::real_normed_vector"
shows "r \ 0 \ y = x /\<^sub>R r \ 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 \ 0 \
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) \ 0"
shows "distr lborel borel ((*) c) = density lborel (\_. inverse \c\)"
proof-
have "distr lborel borel ((*) c) = distr lborel lborel ((*) c)" by (simp cong: distr_cong)
also from assms have "... = density lborel (\_. inverse \c\)"
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) \ 0"
shows "lborel = density (distr lborel borel ((*) c)) (\_. \c\)"
proof-
have "lborel = density lborel (\_. 1)" by (rule density_1[symmetric])
also from assms have "(\_. 1 :: ennreal) = (\_. inverse \c\ * \c\)" by (intro ext) simp
also have "density lborel ... = density (density lborel (\_. inverse \c\)) (\_. \c\)"
by (subst density_density_eq) (auto simp: ennreal_mult)
also from assms have "density lborel (\_. inverse \c\) = distr lborel borel ((*) c)"
by (rule lborel_distr_mult[symmetric])
finally show ?thesis .
qed
lemma lborel_distr_plus:
fixes c :: "'a::euclidean_space"
shows "distr lborel borel ((+) c) = lborel"
by (subst lborel_affine[of 1 c], auto simp: density_1)
interpretation lborel: sigma_finite_measure lborel
by (rule sigma_finite_lborel)
interpretation lborel_pair: pair_sigma_finite lborel lborel ..
lemma\<^marker>\<open>tag important\<close> lborel_prod:
"lborel \\<^sub>M lborel = (lborel :: ('a::euclidean_space \ 'b::euclidean_space) measure)"
proof (rule lborel_eqI[symmetric], clarify)
fix la ua :: 'a and lb ub :: 'b
assume lu: "\a b. (a, b) \ Basis \ (la, lb) \ (a, b) \ (ua, ub) \ (a, b)"
have [simp]:
"\b. b \ Basis \ la \ b \ ua \ b"
"\b. b \ Basis \ lb \ b \ ub \ b"
"inj_on (\u. (u, 0)) Basis" "inj_on (\u. (0, u)) Basis"
"(\u. (u, 0)) ` Basis \ (\u. (0, u)) ` Basis = {}"
"box (la, lb) (ua, ub) = box la ua \ 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 \\<^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\space borel. P x} \ sets borel \ {x\space lborel. P x} \ sets lborel"
by simp
lemma lborelD[measurable (raw)]: "A \ sets borel \ A \ sets lborel"
by simp
lemma emeasure_bounded_finite:
assumes "bounded A" shows "emeasure lborel A < \"
proof -
obtain a b where "A \ cbox a b"
by (meson bounded_subset_cbox_symmetric \<open>bounded A\<close>)
then have "emeasure lborel A \ 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 \ emeasure lborel A < \"
using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded)
lemma borel_integrable_compact:
fixes f :: "'a::euclidean_space \ 'b::{banach, second_countable_topology}"
assumes "compact S" "continuous_on S f"
shows "integrable lborel (\x. indicator S x *\<^sub>R f x)"
proof cases
assume "S \ {}"
have "continuous_on S (\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: "\x. x \ S \ norm (f x) \ M"
by auto
show ?thesis
proof (rule integrable_bound)
show "integrable lborel (\x. indicator S x * M)"
using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left)
show "(\x. indicator S x *\<^sub>R f x) \ 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) \ 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 \ real"
assumes f: "\x. a \ x \ x \ b \ isCont f x"
shows "integrable lborel (\x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
proof -
have "integrable lborel (\x. indicator {a .. b} x *\<^sub>R f x)"
proof (rule borel_integrable_compact)
from f show "continuous_on {a..b} f"
by (auto intro: continuous_at_imp_continuous_on)
qed simp
then show ?thesis
by (auto simp: mult.commute)
qed
subsection \<open>Lebesgue measurable sets\<close>
abbreviation\<^marker>\<open>tag important\<close> lmeasurable :: "'a::euclidean_space set set"
where
"lmeasurable \ fmeasurable lebesgue"
lemma not_measurable_UNIV [simp]: "UNIV \ lmeasurable"
by (simp add: fmeasurable_def)
lemma\<^marker>\<open>tag important\<close> lmeasurable_iff_integrable:
"S \ lmeasurable \ integrable lebesgue (indicator S :: 'a::euclidean_space \ real)"
by (auto simp: fmeasurable_def integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator)
lemma lmeasurable_cbox [iff]: "cbox a b \ lmeasurable"
and lmeasurable_box [iff]: "box a b \ lmeasurable"
by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)
lemma
fixes a::real
shows lmeasurable_interval [iff]: "{a..b} \ lmeasurable" "{a<.. lmeasurable"
apply (metis box_real(2) lmeasurable_cbox)
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