added helper -- cf. SET616^5
(* Title: HOL/Probability/Borel_Space.thy
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
header {*Borel spaces*}
theory Borel_Space
imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
begin
section "Generic Borel spaces"
definition borel :: "'a::topological_space measure" where
"borel = sigma UNIV {S. open S}"
abbreviation "borel_measurable M \<equiv> measurable M borel"
lemma in_borel_measurable:
"f \<in> borel_measurable M \<longleftrightarrow>
(\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
by (auto simp add: measurable_def borel_def)
lemma in_borel_measurable_borel:
"f \<in> borel_measurable M \<longleftrightarrow>
(\<forall>S \<in> sets borel.
f -` S \<inter> space M \<in> sets M)"
by (auto simp add: measurable_def borel_def)
lemma space_borel[simp]: "space borel = UNIV"
unfolding borel_def by auto
lemma borel_open[simp]:
assumes "open A" shows "A \<in> sets borel"
proof -
have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
thus ?thesis unfolding borel_def by auto
qed
lemma borel_closed[simp]:
assumes "closed A" shows "A \<in> sets borel"
proof -
have "space borel - (- A) \<in> sets borel"
using assms unfolding closed_def by (blast intro: borel_open)
thus ?thesis by simp
qed
lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
unfolding Compl_eq_Diff_UNIV by (intro Diff) auto
lemma borel_measurable_vimage:
fixes f :: "'a \<Rightarrow> 'x::t2_space"
assumes borel: "f \<in> borel_measurable M"
shows "f -` {x} \<inter> space M \<in> sets M"
proof (cases "x \<in> f ` space M")
case True then obtain y where "x = f y" by auto
from closed_singleton[of "f y"]
have "{f y} \<in> sets borel" by (rule borel_closed)
with assms show ?thesis
unfolding in_borel_measurable_borel `x = f y` by auto
next
case False hence "f -` {x} \<inter> space M = {}" by auto
thus ?thesis by auto
qed
lemma borel_measurableI:
fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
shows "f \<in> borel_measurable M"
unfolding borel_def
proof (rule measurable_measure_of, simp_all)
fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
using assms[of S] by simp
qed
lemma borel_singleton[simp, intro]:
fixes x :: "'a::t1_space"
shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
proof (rule insert_in_sets)
show "{x} \<in> sets borel"
using closed_singleton[of x] by (rule borel_closed)
qed simp
lemma borel_measurable_const[simp, intro]:
"(\<lambda>x. c) \<in> borel_measurable M"
by auto
lemma borel_measurable_indicator[simp, intro!]:
assumes A: "A \<in> sets M"
shows "indicator A \<in> borel_measurable M"
unfolding indicator_def [abs_def] using A
by (auto intro!: measurable_If_set)
lemma borel_measurable_indicator_iff:
"(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
(is "?I \<in> borel_measurable M \<longleftrightarrow> _")
proof
assume "?I \<in> borel_measurable M"
then have "?I -` {1} \<inter> space M \<in> sets M"
unfolding measurable_def by auto
also have "?I -` {1} \<inter> space M = A \<inter> space M"
unfolding indicator_def [abs_def] by auto
finally show "A \<inter> space M \<in> sets M" .
next
assume "A \<inter> space M \<in> sets M"
moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
(indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
by (intro measurable_cong) (auto simp: indicator_def)
ultimately show "?I \<in> borel_measurable M" by auto
qed
lemma borel_measurable_subalgebra:
assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
shows "f \<in> borel_measurable M"
using assms unfolding measurable_def by auto
section "Borel spaces on euclidean spaces"
lemma lessThan_borel[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{..< a} \<in> sets borel"
by (blast intro: borel_open)
lemma greaterThan_borel[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{a <..} \<in> sets borel"
by (blast intro: borel_open)
lemma greaterThanLessThan_borel[simp, intro]:
fixes a b :: "'a\<Colon>ordered_euclidean_space"
shows "{a<..<b} \<in> sets borel"
by (blast intro: borel_open)
lemma atMost_borel[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{..a} \<in> sets borel"
by (blast intro: borel_closed)
lemma atLeast_borel[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{a..} \<in> sets borel"
by (blast intro: borel_closed)
lemma atLeastAtMost_borel[simp, intro]:
fixes a b :: "'a\<Colon>ordered_euclidean_space"
shows "{a..b} \<in> sets borel"
by (blast intro: borel_closed)
lemma greaterThanAtMost_borel[simp, intro]:
fixes a b :: "'a\<Colon>ordered_euclidean_space"
shows "{a<..b} \<in> sets borel"
unfolding greaterThanAtMost_def by blast
lemma atLeastLessThan_borel[simp, intro]:
fixes a b :: "'a\<Colon>ordered_euclidean_space"
shows "{a..<b} \<in> sets borel"
unfolding atLeastLessThan_def by blast
lemma hafspace_less_borel[simp, intro]:
fixes a :: real
shows "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel"
by (auto intro!: borel_open open_halfspace_component_gt)
lemma hafspace_greater_borel[simp, intro]:
fixes a :: real
shows "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel"
by (auto intro!: borel_open open_halfspace_component_lt)
lemma hafspace_less_eq_borel[simp, intro]:
fixes a :: real
shows "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel"
by (auto intro!: borel_closed closed_halfspace_component_ge)
lemma hafspace_greater_eq_borel[simp, intro]:
fixes a :: real
shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel"
by (auto intro!: borel_closed closed_halfspace_component_le)
lemma borel_measurable_less[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w < g w} \<in> sets M"
proof -
have "{w \<in> space M. f w < g w} =
(\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
using Rats_dense_in_real by (auto simp add: Rats_def)
then show ?thesis using f g
by simp (blast intro: measurable_sets)
qed
lemma borel_measurable_le[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
proof -
have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
by auto
thus ?thesis using f g
by simp blast
qed
lemma borel_measurable_eq[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w = g w} \<in> sets M"
proof -
have "{w \<in> space M. f w = g w} =
{w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
by auto
thus ?thesis using f g by auto
qed
lemma borel_measurable_neq[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
proof -
have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
by auto
thus ?thesis using f g by auto
qed
subsection "Borel space equals sigma algebras over intervals"
lemma rational_boxes:
fixes x :: "'a\<Colon>ordered_euclidean_space"
assumes "0 < e"
shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"
proof -
def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i")
proof
fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e
show "?th i" by auto
qed
from choice[OF this] guess a .. note a = this
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i")
proof
fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e
show "?th i" by auto
qed
from choice[OF this] guess b .. note b = this
{ fix y :: 'a assume *: "Chi a < y" "y < Chi b"
have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)"
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
fix i assume i: "i \<in> {..<DIM('a)}"
have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto
moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto
moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto
ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto
then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
by (rule power_strict_mono) auto
then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
by (simp add: power_divide)
qed auto
also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
finally have "dist x y < e" . }
with a b show ?thesis
apply (rule_tac exI[of _ "Chi a"])
apply (rule_tac exI[of _ "Chi b"])
using eucl_less[where 'a='a] by auto
qed
lemma ex_rat_list:
fixes x :: "'a\<Colon>ordered_euclidean_space"
assumes "\<And> i. x $$ i \<in> \<rat>"
shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)"
proof -
have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast
from choice[OF this] guess r ..
then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
qed
lemma open_UNION:
fixes M :: "'a\<Colon>ordered_euclidean_space set"
assumes "open M"
shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
(\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
(is "M = UNION ?idx ?box")
proof safe
fix x assume "x \<in> M"
obtain e where e: "e > 0" "ball x e \<subseteq> M"
using openE[OF assms `x \<in> M`] by auto
then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"
using rational_boxes[OF e(1)] by blast
then obtain p q where pq: "length p = DIM ('a)"
"length q = DIM ('a)"
"\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i"
using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
hence p: "Chi (of_rat \<circ> op ! p) = a"
using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
unfolding o_def by auto
from pq have q: "Chi (of_rat \<circ> op ! q) = b"
using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
unfolding o_def by auto
have "x \<in> ?box (p, q)"
using p q ab by auto
thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
qed auto
lemma borel_sigma_sets_subset:
"A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
using sigma_sets_subset[of A borel] by simp
lemma borel_eq_sigmaI1:
fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
assumes borel_eq: "borel = sigma UNIV X"
assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))"
assumes F: "\<And>i. F i \<in> sets borel"
shows "borel = sigma UNIV (range F)"
unfolding borel_def
proof (intro sigma_eqI antisym)
have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
unfolding borel_def by simp
also have "\<dots> = sigma_sets UNIV X"
unfolding borel_eq by simp
also have "\<dots> \<subseteq> sigma_sets UNIV (range F)"
using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" .
show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}"
unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
qed auto
lemma borel_eq_sigmaI2:
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
assumes F: "\<And>i j. F i j \<in> sets borel"
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto
lemma borel_eq_sigmaI3:
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
assumes borel_eq: "borel = sigma UNIV X"
assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
assumes F: "\<And>i j. F i j \<in> sets borel"
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
lemma borel_eq_sigmaI4:
fixes F :: "'i \<Rightarrow> 'a::topological_space set"
and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"
assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))"
assumes F: "\<And>i. F i \<in> sets borel"
shows "borel = sigma UNIV (range F)"
using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto
lemma borel_eq_sigmaI5:
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
assumes borel_eq: "borel = sigma UNIV (range G)"
assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
assumes F: "\<And>i j. F i j \<in> sets borel"
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
lemma halfspace_gt_in_halfspace:
"{x\<Colon>'a. a < x $$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))"
(is "?set \<in> ?SIGMA")
proof -
interpret sigma_algebra UNIV ?SIGMA
by (intro sigma_algebra_sigma_sets) simp_all
have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"
proof (safe, simp_all add: not_less)
fix x assume "a < x $$ i"
with reals_Archimedean[of "x $$ i - a"]
obtain n where "a + 1 / real (Suc n) < x $$ i"
by (auto simp: inverse_eq_divide field_simps)
then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i"
by (blast intro: less_imp_le)
next
fix x n
have "a < a + 1 / real (Suc n)" by auto
also assume "\<dots> \<le> x"
finally show "a < x" .
qed
show "?set \<in> ?SIGMA" unfolding *
by (auto intro!: Diff)
qed
lemma borel_eq_halfspace_less:
"borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x $$ i < a}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI3[OF borel_def])
fix S :: "'a set" assume "S \<in> {S. open S}"
then have "open S" by simp
from open_UNION[OF this]
obtain I where *: "S =
(\<Union>(a, b)\<in>I.
(\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>
(\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"
unfolding greaterThanLessThan_def
unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
by blast
show "S \<in> ?SIGMA"
unfolding *
by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace)
qed auto
lemma borel_eq_halfspace_le:
"borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i \<le> a}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
fix a i
have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"
proof (safe, simp_all)
fix x::'a assume *: "x$$i < a"
with reals_Archimedean[of "a - x$$i"]
obtain n where "x $$ i < a - 1 / (real (Suc n))"
by (auto simp: field_simps inverse_eq_divide)
then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"
by (blast intro: less_imp_le)
next
fix x::'a and n
assume "x$$i \<le> a - 1 / real (Suc n)"
also have "\<dots> < a" by auto
finally show "x$$i < a" .
qed
show "{x. x$$i < a} \<in> ?SIGMA" unfolding *
by (safe intro!: countable_UN) auto
qed auto
lemma borel_eq_halfspace_ge:
"borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto
show "{x. x$$i < a} \<in> ?SIGMA" unfolding *
by (safe intro!: compl_sets) auto
qed auto
lemma borel_eq_halfspace_greater:
"borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x $$ i}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *
by (safe intro!: compl_sets) auto
qed auto
lemma borel_eq_atMost:
"borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"
proof cases
assume "i < DIM('a)"
then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
fix x
from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat ..
then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k"
by (subst (asm) Max_le_iff) auto
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"
by (auto intro!: exI[of _ k])
qed
show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *
by (safe intro!: countable_UN) auto
qed (auto intro: sigma_sets_top sigma_sets.Empty)
qed auto
lemma borel_eq_greaterThan:
"borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"
proof cases
assume "i < DIM('a)"
have "{x::'a. x$$i \<le> a} = UNIV - {x::'a. a < x$$i}" by auto
also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)`
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
fix x
from reals_Archimedean2[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"]
guess k::nat .. note k = this
{ fix i assume "i < DIM('a)"
then have "-x$$i < real k"
using k by (subst (asm) Max_less_iff) auto
then have "- real k < x$$i" by simp }
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"
by (auto intro!: exI[of _ k])
qed
finally show "{x. x$$i \<le> a} \<in> ?SIGMA"
apply (simp only:)
apply (safe intro!: countable_UN Diff)
apply (auto intro: sigma_sets_top)
done
qed (auto intro: sigma_sets_top sigma_sets.Empty)
qed auto
lemma borel_eq_lessThan:
"borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
fix a i show "{x. a \<le> x$$i} \<in> ?SIGMA"
proof cases
fix a i assume "i < DIM('a)"
have "{x::'a. a \<le> x$$i} = UNIV - {x::'a. x$$i < a}" by auto
also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
fix x
from reals_Archimedean2[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"]
guess k::nat .. note k = this
{ fix i assume "i < DIM('a)"
then have "x$$i < real k"
using k by (subst (asm) Max_less_iff) auto
then have "x$$i < real k" by simp }
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k"
by (auto intro!: exI[of _ k])
qed
finally show "{x. a \<le> x$$i} \<in> ?SIGMA"
apply (simp only:)
apply (safe intro!: countable_UN Diff)
apply (auto intro: sigma_sets_top)
done
qed (auto intro: sigma_sets_top sigma_sets.Empty)
qed auto
lemma borel_eq_atLeastAtMost:
"borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
fix a::'a
have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
proof (safe, simp_all add: eucl_le[where 'a='a])
fix x
from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]
guess k::nat .. note k = this
{ fix i assume "i < DIM('a)"
with k have "- x$$i \<le> real k"
by (subst (asm) Max_le_iff) (auto simp: field_simps)
then have "- real k \<le> x$$i" by simp }
then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"
by (auto intro!: exI[of _ k])
qed
show "{..a} \<in> ?SIGMA" unfolding *
by (safe intro!: countable_UN)
(auto intro!: sigma_sets_top)
qed auto
lemma borel_eq_greaterThanLessThan:
"borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI1[OF borel_def])
fix M :: "'a set" assume "M \<in> {S. open S}"
then have "open M" by simp
show "M \<in> ?SIGMA"
apply (subst open_UNION[OF `open M`])
apply (safe intro!: countable_UN)
apply auto
done
qed auto
lemma borel_eq_atLeastLessThan:
"borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
fix x :: real
have "{..<x} = (\<Union>i::nat. {-real i ..< x})"
by (auto simp: move_uminus real_arch_simple)
then show "{..< x} \<in> ?SIGMA"
by (auto intro: sigma_sets.intros)
qed auto
lemma borel_measurable_halfspacesI:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
assumes F: "borel = sigma UNIV (range F)"
and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
proof safe
fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
then show "S a i \<in> sets M" unfolding assms
by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1))
next
assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
{ fix a i have "S a i \<in> sets M"
proof cases
assume "i < DIM('c)"
with a show ?thesis unfolding assms(2) by simp
next
assume "\<not> i < DIM('c)"
from S[OF this] show ?thesis .
qed }
then show "f \<in> borel_measurable M"
by (auto intro!: measurable_measure_of simp: S_eq F)
qed
lemma borel_measurable_iff_halfspace_le:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
lemma borel_measurable_iff_halfspace_less:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
lemma borel_measurable_iff_halfspace_ge:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
lemma borel_measurable_iff_halfspace_greater:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
lemma borel_measurable_iff_le:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
using borel_measurable_iff_halfspace_le[where 'c=real] by simp
lemma borel_measurable_iff_less:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
using borel_measurable_iff_halfspace_less[where 'c=real] by simp
lemma borel_measurable_iff_ge:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
lemma borel_measurable_iff_greater:
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
lemma borel_measurable_euclidean_component:
"(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel"
proof (rule borel_measurableI)
fix S::"real set" assume "open S"
from open_vimage_euclidean_component[OF this]
show "(\<lambda>x. x $$ i) -` S \<inter> space borel \<in> sets borel"
by (auto intro: borel_open)
qed
lemma borel_measurable_euclidean_space:
fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"
proof safe
fix i assume "f \<in> borel_measurable M"
then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"
using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def]
by (auto intro: borel_measurable_euclidean_component)
next
assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M"
then show "f \<in> borel_measurable M"
unfolding borel_measurable_iff_halfspace_le by auto
qed
subsection "Borel measurable operators"
lemma affine_borel_measurable_vector:
fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
proof (rule borel_measurableI)
fix S :: "'x set" assume "open S"
show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
proof cases
assume "b \<noteq> 0"
with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
by (auto intro!: open_affinity simp: scaleR_add_right)
hence "?S \<in> sets borel" by auto
moreover
from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
ultimately show ?thesis using assms unfolding in_borel_measurable_borel
by auto
qed simp
qed
lemma affine_borel_measurable:
fixes g :: "'a \<Rightarrow> real"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
lemma borel_measurable_add[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
proof -
have 1: "\<And>a. {w\<in>space M. a \<le> f w + g w} = {w \<in> space M. a + g w * -1 \<le> f w}"
by auto
have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
by (rule affine_borel_measurable [OF g])
then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
by auto
then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
by (simp add: 1)
then show ?thesis
by (simp add: borel_measurable_iff_ge)
qed
lemma borel_measurable_setsum[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
proof cases
assume "finite S"
thus ?thesis using assms by induct auto
qed simp
lemma borel_measurable_square:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
proof -
{
fix a
have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
proof (cases rule: linorder_cases [of a 0])
case less
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
by auto (metis less order_le_less_trans power2_less_0)
also have "... \<in> sets M"
by (rule empty_sets)
finally show ?thesis .
next
case equal
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
{w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
by auto
also have "... \<in> sets M"
apply (insert f)
apply (rule Int)
apply (simp add: borel_measurable_iff_le)
apply (simp add: borel_measurable_iff_ge)
done
finally show ?thesis .
next
case greater
have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a \<le> f x & f x \<le> sqrt a)"
by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
real_sqrt_le_iff real_sqrt_power)
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
{w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
using greater by auto
also have "... \<in> sets M"
apply (insert f)
apply (rule Int)
apply (simp add: borel_measurable_iff_ge)
apply (simp add: borel_measurable_iff_le)
done
finally show ?thesis .
qed
}
thus ?thesis by (auto simp add: borel_measurable_iff_le)
qed
lemma times_eq_sum_squares:
fixes x::real
shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
lemma borel_measurable_uminus[simp, intro]:
fixes g :: "'a \<Rightarrow> real"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. - g x) \<in> borel_measurable M"
proof -
have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
by simp
also have "... \<in> borel_measurable M"
by (fast intro: affine_borel_measurable g)
finally show ?thesis .
qed
lemma borel_measurable_times[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
proof -
have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
using assms by (fast intro: affine_borel_measurable borel_measurable_square)
have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
(\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
by (simp add: minus_divide_right)
also have "... \<in> borel_measurable M"
using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
show ?thesis
apply (simp add: times_eq_sum_squares diff_minus)
using 1 2 by simp
qed
lemma borel_measurable_setprod[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
proof cases
assume "finite S"
thus ?thesis using assms by induct auto
qed simp
lemma borel_measurable_diff[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
unfolding diff_minus using assms by fast
lemma borel_measurable_inverse[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
proof safe
fix a :: real
have *: "{w \<in> space M. a \<le> 1 / f w} =
({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
by (auto intro!: Int Un)
qed
lemma borel_measurable_divide[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
and "g \<in> borel_measurable M"
shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
unfolding field_divide_inverse
by (rule borel_measurable_inverse borel_measurable_times assms)+
lemma borel_measurable_max[intro, simp]:
fixes f g :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
unfolding borel_measurable_iff_le
proof safe
fix a
have "{x \<in> space M. max (g x) (f x) \<le> a} =
{x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
using assms unfolding borel_measurable_iff_le
by (auto intro!: Int)
qed
lemma borel_measurable_min[intro, simp]:
fixes f g :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
unfolding borel_measurable_iff_ge
proof safe
fix a
have "{x \<in> space M. a \<le> min (g x) (f x)} =
{x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
using assms unfolding borel_measurable_iff_ge
by (auto intro!: Int)
qed
lemma borel_measurable_abs[simp, intro]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
proof -
have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
show ?thesis unfolding * using assms by auto
qed
lemma borel_measurable_nth[simp, intro]:
"(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"
using borel_measurable_euclidean_component
unfolding nth_conv_component by auto
lemma borel_measurable_continuous_on1:
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
assumes "continuous_on UNIV f"
shows "f \<in> borel_measurable borel"
apply(rule borel_measurableI)
using continuous_open_preimage[OF assms] unfolding vimage_def by auto
lemma borel_measurable_continuous_on:
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
assumes cont: "continuous_on A f" "open A"
shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
proof (rule borel_measurableI)
fix S :: "'b set" assume "open S"
then have "open {x\<in>A. f x \<in> S}"
by (intro continuous_open_preimage[OF cont]) auto
then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto
have "?f -` S \<inter> space borel =
{x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"
by (auto split: split_if_asm)
also have "\<dots> \<in> sets borel"
using * `open A` by (auto simp del: space_borel intro!: Un)
finally show "?f -` S \<inter> space borel \<in> sets borel" .
qed
lemma convex_measurable:
fixes a b :: real
assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"
assumes q: "convex_on { a <..< b} q"
shows "q \<circ> X \<in> borel_measurable M"
proof -
have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<in> borel_measurable borel"
proof (rule borel_measurable_continuous_on)
show "open {a<..<b}" by auto
from this q show "continuous_on {a<..<b} q"
by (rule convex_on_continuous)
qed
then have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<circ> X \<in> borel_measurable M" (is ?qX)
using X by (intro measurable_comp) auto
moreover have "?qX \<longleftrightarrow> q \<circ> X \<in> borel_measurable M"
using X by (intro measurable_cong) auto
ultimately show ?thesis by simp
qed
lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"
proof -
{ fix x :: real assume x: "x \<le> 0"
{ fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
from this[of x] x this[of 0] have "log b 0 = log b x"
by (auto simp: ln_def log_def) }
note log_imp = this
have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"
proof (rule borel_measurable_continuous_on)
show "continuous_on {0<..} (log b)"
by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
simp: continuous_isCont[symmetric])
show "open ({0<..}::real set)" by auto
qed
also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"
by (simp add: fun_eq_iff not_less log_imp)
finally show ?thesis .
qed
lemma borel_measurable_log[simp,intro]:
assumes f: "f \<in> borel_measurable M" and "1 < b"
shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
by (simp add: comp_def)
lemma borel_measurable_real_floor:
"(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
unfolding borel_measurable_iff_ge
proof (intro allI)
fix a :: real
{ fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
unfolding real_eq_of_int by simp }
then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
qed
lemma borel_measurable_real_natfloor[intro, simp]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
proof -
have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
by (auto simp: max_def natfloor_def)
with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
show ?thesis by (simp add: comp_def)
qed
subsection "Borel space on the extended reals"
lemma borel_measurable_ereal_borel:
"ereal \<in> borel_measurable borel"
proof (rule borel_measurableI)
fix X :: "ereal set" assume "open X"
then have "open (ereal -` X \<inter> space borel)"
by (simp add: open_ereal_vimage)
then show "ereal -` X \<inter> space borel \<in> sets borel" by auto
qed
lemma borel_measurable_ereal[simp, intro]:
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
using measurable_comp[OF f borel_measurable_ereal_borel] unfolding comp_def .
lemma borel_measurable_real_of_ereal_borel:
"(real :: ereal \<Rightarrow> real) \<in> borel_measurable borel"
proof (rule borel_measurableI)
fix B :: "real set" assume "open B"
have *: "ereal -` real -` (B - {0}) = B - {0}" by auto
have open_real: "open (real -` (B - {0}) :: ereal set)"
unfolding open_ereal_def * using `open B` by auto
show "(real -` B \<inter> space borel :: ereal set) \<in> sets borel"
proof cases
assume "0 \<in> B"
then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0::ereal}"
by (auto simp add: real_of_ereal_eq_0)
then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
using open_real by auto
next
assume "0 \<notin> B"
then have *: "(real -` B :: ereal set) = real -` (B - {0})"
by (auto simp add: real_of_ereal_eq_0)
then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
using open_real by auto
qed
qed
lemma borel_measurable_real_of_ereal[simp, intro]:
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: ereal)) \<in> borel_measurable M"
using measurable_comp[OF f borel_measurable_real_of_ereal_borel] unfolding comp_def .
lemma borel_measurable_ereal_iff:
shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
proof
assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
from borel_measurable_real_of_ereal[OF this]
show "f \<in> borel_measurable M" by auto
qed auto
lemma borel_measurable_ereal_iff_real:
fixes f :: "'a \<Rightarrow> ereal"
shows "f \<in> borel_measurable M \<longleftrightarrow>
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
proof safe
assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
finally show "f \<in> borel_measurable M" .
qed (auto intro: measurable_sets borel_measurable_real_of_ereal)
lemma less_eq_ge_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
proof
assume "f -` {a <..} \<inter> space M \<in> sets M"
moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
next
assume "f -` {..a} \<inter> space M \<in> sets M"
moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
qed
lemma greater_eq_le_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
proof
assume "f -` {a ..} \<inter> space M \<in> sets M"
moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
next
assume "f -` {..< a} \<inter> space M \<in> sets M"
moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
qed
lemma borel_measurable_uminus_borel_ereal:
"(uminus :: ereal \<Rightarrow> ereal) \<in> borel_measurable borel"
proof (rule borel_measurableI)
fix X :: "ereal set" assume "open X"
have "uminus -` X = uminus ` X" by (force simp: image_iff)
then have "open (uminus -` X)" using `open X` ereal_open_uminus by auto
then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
qed
lemma borel_measurable_uminus_ereal[intro]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
using measurable_comp[OF assms borel_measurable_uminus_borel_ereal] by (simp add: comp_def)
lemma borel_measurable_uminus_eq_ereal[simp]:
"(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
proof
assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
qed auto
lemma borel_measurable_eq_atMost_ereal:
fixes f :: "'a \<Rightarrow> ereal"
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
proof (intro iffI allI)
assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
show "f \<in> borel_measurable M"
unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le
proof (intro conjI allI)
fix a :: real
{ fix x :: ereal assume *: "\<forall>i::nat. real i < x"
have "x = \<infinity>"
proof (rule ereal_top)
fix B from reals_Archimedean2[of B] guess n ..
then have "ereal B < real n" by auto
with * show "B \<le> x" by (metis less_trans less_imp_le)
qed }
then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
by (auto simp: not_le)
then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
moreover
have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto
then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"
using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)
moreover have "{w \<in> space M. real (f w) \<le> a} =
(if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M
else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
qed
qed (simp add: measurable_sets)
lemma borel_measurable_eq_atLeast_ereal:
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
proof
assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
by (auto simp: ereal_uminus_le_reorder)
ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
unfolding borel_measurable_eq_atMost_ereal by auto
then show "f \<in> borel_measurable M" by simp
qed (simp add: measurable_sets)
lemma borel_measurable_ereal_iff_less:
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable ..
lemma borel_measurable_ereal_iff_ge:
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable ..
lemma borel_measurable_ereal_eq_const:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x = c} \<in> sets M"
proof -
have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
lemma borel_measurable_ereal_neq_const:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
proof -
have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
lemma borel_measurable_ereal_le[intro,simp]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
proof -
have "{x \<in> space M. f x \<le> g x} =
{x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union>
f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")
proof (intro set_eqI)
fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: ereal2_cases[of "f x" "g x"]) auto
qed
with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
qed
lemma borel_measurable_ereal_less[intro,simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{x \<in> space M. f x < g x} \<in> sets M"
proof -
have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto
then show ?thesis using g f by auto
qed
lemma borel_measurable_ereal_eq[intro,simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w = g w} \<in> sets M"
proof -
have "{x \<in> space M. f x = g x} = {x \<in> space M. g x \<le> f x} \<inter> {x \<in> space M. f x \<le> g x}" by auto
then show ?thesis using g f by auto
qed
lemma borel_measurable_ereal_neq[intro,simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
proof -
have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto
thus ?thesis using f g by auto
qed
lemma split_sets:
"{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
"{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
by auto
lemma borel_measurable_ereal_add[intro, simp]:
fixes f :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
proof -
{ fix x assume "x \<in> space M" then have "f x + g x =
(if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>
else ereal (real (f x) + real (g x)))"
by (cases rule: ereal2_cases[of "f x" "g x"]) auto }
with assms show ?thesis
by (auto cong: measurable_cong simp: split_sets
intro!: Un measurable_If measurable_sets)
qed
lemma borel_measurable_ereal_setsum[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
proof cases
assume "finite S"
thus ?thesis using assms
by induct auto
qed (simp add: borel_measurable_const)
lemma borel_measurable_ereal_abs[intro, simp]:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
proof -
{ fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }
then show ?thesis using assms by (auto intro!: measurable_If)
qed
lemma borel_measurable_ereal_times[intro, simp]:
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
proof -
{ fix f g :: "'a \<Rightarrow> ereal"
assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"
{ fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0
else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
else ereal (real (f x) * real (g x)))"
apply (cases rule: ereal2_cases[of "f x" "g x"])
using pos[of x] by auto }
with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"
by (auto cong: measurable_cong simp: split_sets
intro!: Un measurable_If measurable_sets) }
note pos_times = this
have *: "(\<lambda>x. f x * g x) =
(\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))"
by (auto simp: fun_eq_iff)
show ?thesis using assms unfolding *
by (intro measurable_If pos_times borel_measurable_uminus_ereal)
(auto simp: split_sets intro!: Int)
qed
lemma borel_measurable_ereal_setprod[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
proof cases
assume "finite S"
thus ?thesis using assms by induct auto
qed simp
lemma borel_measurable_ereal_min[simp, intro]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
using assms unfolding min_def by (auto intro!: measurable_If)
lemma borel_measurable_ereal_max[simp, intro]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M"
and "g \<in> borel_measurable M"
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
using assms unfolding max_def by (auto intro!: measurable_If)
lemma borel_measurable_SUP[simp, intro]:
fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
unfolding borel_measurable_ereal_iff_ge
proof
fix a
have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
by (auto simp: less_SUP_iff)
then show "?sup -` {a<..} \<inter> space M \<in> sets M"
using assms by auto
qed
lemma borel_measurable_INF[simp, intro]:
fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
unfolding borel_measurable_ereal_iff_less
proof
fix a
have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
by (auto simp: INF_less_iff)
then show "?inf -` {..<a} \<inter> space M \<in> sets M"
using assms by auto
qed
lemma borel_measurable_liminf[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. f i \<in> borel_measurable M"
shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
unfolding liminf_SUPR_INFI using assms by auto
lemma borel_measurable_limsup[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. f i \<in> borel_measurable M"
shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
unfolding limsup_INFI_SUPR using assms by auto
lemma borel_measurable_ereal_diff[simp, intro]:
fixes f g :: "'a \<Rightarrow> ereal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
unfolding minus_ereal_def using assms by auto
lemma borel_measurable_ereal_inverse[simp, intro]:
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
proof -
{ fix x have "inverse (f x) = (if f x = 0 then \<infinity> else ereal (inverse (real (f x))))"
by (cases "f x") auto }
with f show ?thesis
by (auto intro!: measurable_If)
qed
lemma borel_measurable_ereal_divide[simp, intro]:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. f x / g x :: ereal) \<in> borel_measurable M"
unfolding divide_ereal_def by auto
lemma borel_measurable_psuminf[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
apply (subst measurable_cong)
apply (subst suminf_ereal_eq_SUPR)
apply (rule pos)
using assms by auto
section "LIMSEQ is borel measurable"
lemma borel_measurable_LIMSEQ:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
and u: "\<And>i. u i \<in> borel_measurable M"
shows "u' \<in> borel_measurable M"
proof -
have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
using u' by (simp add: lim_imp_Liminf)
moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
by auto
ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
qed
end