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Quellcode-Bibliothek
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Prinzipien der Anforderungsanalyse
Sprache: Isabelle
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(* Title: HOL/Analysis/Borel_Space.thy
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
section \<open>Borel Space\<close>
theory Borel_Space
imports
Measurable Derivative Ordered_Euclidean_Space Extended_Real_Limits
begin
lemma is_interval_real_ereal_oo: "is_interval (real_of_ereal ` {N<..
by (auto simp: real_atLeastGreaterThan_eq)
lemma sets_Collect_eventually_sequentially[measurable]:
"(\i. {x\space M. P x i} \ sets M) \ {x\space M. eventually (P x) sequentially} \ sets M"
unfolding eventually_sequentially by simp
lemma topological_basis_trivial: "topological_basis {A. open A}"
by (auto simp: topological_basis_def)
proposition open_prod_generated: "open = generate_topology {A \ B | A B. open A \ open B}"
proof -
have "{A \ B :: ('a \ 'b) set | A B. open A \ open B} = ((\(a, b). a \ b) ` ({A. open A} \ {A. open A}))"
by auto
then show ?thesis
by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)
qed
proposition mono_on_imp_deriv_nonneg:
assumes mono: "mono_on f A" and deriv: "(f has_real_derivative D) (at x)"
assumes "x \ interior A"
shows "D \ 0"
proof (rule tendsto_lowerbound)
let ?A' = "(\y. y - x) ` interior A"
from deriv show "((\h. (f (x + h) - f x) / h) \ D) (at 0)"
by (simp add: field_has_derivative_at has_field_derivative_def)
from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset)
show "eventually (\h. (f (x + h) - f x) / h \ 0) (at 0)"
proof (subst eventually_at_topological, intro exI conjI ballI impI)
have "open (interior A)" by simp
hence "open ((+) (-x) ` interior A)" by (rule open_translation)
also have "((+) (-x) ` interior A) = ?A'" by auto
finally show "open ?A'" .
next
from \<open>x \<in> interior A\<close> show "0 \<in> ?A'" by auto
next
fix h assume "h \ ?A'"
hence "x + h \ interior A" by auto
with mono' and \x \ interior A\ show "(f (x + h) - f x) / h \ 0"
by (cases h rule: linorder_cases[of _ 0])
(simp_all add: divide_nonpos_neg divide_nonneg_pos mono_onD field_simps)
qed
qed simp
proposition mono_on_ctble_discont:
fixes f :: "real \ real"
fixes A :: "real set"
assumes "mono_on f A"
shows "countable {a\A. \ continuous (at a within A) f}"
proof -
have mono: "\x y. x \ A \ y \ A \ x \ y \ f x \ f y"
using \<open>mono_on f A\<close> by (simp add: mono_on_def)
have "\a \ {a\A. \ continuous (at a within A) f}. \q :: nat \ rat.
(fst q = 0 \<and> of_rat (snd q) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd q))) \<or>
(fst q = 1 \<and> of_rat (snd q) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd q)))"
proof (clarsimp simp del: One_nat_def)
fix a assume "a \ A" assume "\ continuous (at a within A) f"
thus "\q1 q2.
q1 = 0 \<and> real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2) \<or>
q1 = 1 \<and> f a < real_of_rat q2 \<and> (\<forall>x\<in>A. a < x \<longrightarrow> real_of_rat q2 < f x)"
proof (auto simp add: continuous_within order_tendsto_iff eventually_at)
fix l assume "l < f a"
then obtain q2 where q2: "l < of_rat q2" "of_rat q2 < f a"
using of_rat_dense by blast
assume * [rule_format]: "\d>0. \x\A. x \ a \ dist x a < d \ \ l < f x"
from q2 have "real_of_rat q2 < f a \ (\x\A. x < a \ f x < real_of_rat q2)"
proof auto
fix x assume "x \ A" "x < a"
with q2 *[of "a - x"] show "f x < real_of_rat q2"
apply (auto simp add: dist_real_def not_less)
apply (subgoal_tac "f x \ f xa")
by (auto intro: mono)
qed
thus ?thesis by auto
next
fix u assume "u > f a"
then obtain q2 where q2: "f a < of_rat q2" "of_rat q2 < u"
using of_rat_dense by blast
assume *[rule_format]: "\d>0. \x\A. x \ a \ dist x a < d \ \ u > f x"
from q2 have "real_of_rat q2 > f a \ (\x\A. x > a \ f x > real_of_rat q2)"
proof auto
fix x assume "x \ A" "x > a"
with q2 *[of "x - a"] show "f x > real_of_rat q2"
apply (auto simp add: dist_real_def)
apply (subgoal_tac "f x \ f xa")
by (auto intro: mono)
qed
thus ?thesis by auto
qed
qed
hence "\g :: real \ nat \ rat . \a \ {a\A. \ continuous (at a within A) f}.
(fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd (g a)))) |
(fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd (g a))))"
by (rule bchoice)
then guess g ..
hence g: "\a x. a \ A \ \ continuous (at a within A) f \ x \ A \
(fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (x < a \<longrightarrow> f x < of_rat (snd (g a)))) |
(fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (x > a \<longrightarrow> f x > of_rat (snd (g a))))"
by auto
have "inj_on g {a\A. \ continuous (at a within A) f}"
proof (auto simp add: inj_on_def)
fix w z
assume 1: "w \ A" and 2: "\ continuous (at w within A) f" and
3: "z \ A" and 4: "\ continuous (at z within A) f" and
5: "g w = g z"
from g [OF 1 2 3] g [OF 3 4 1] 5
show "w = z" by auto
qed
thus ?thesis
by (rule countableI')
qed
lemma mono_on_ctble_discont_open:
fixes f :: "real \ real"
fixes A :: "real set"
assumes "open A" "mono_on f A"
shows "countable {a\A. \isCont f a}"
proof -
have "{a\A. \isCont f a} = {a\A. \(continuous (at a within A) f)}"
by (auto simp add: continuous_within_open [OF _ \<open>open A\<close>])
thus ?thesis
apply (elim ssubst)
by (rule mono_on_ctble_discont, rule assms)
qed
lemma mono_ctble_discont:
fixes f :: "real \ real"
assumes "mono f"
shows "countable {a. \ isCont f a}"
using assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto
lemma has_real_derivative_imp_continuous_on:
assumes "\x. x \ A \ (f has_real_derivative f' x) (at x)"
shows "continuous_on A f"
apply (intro differentiable_imp_continuous_on, unfold differentiable_on_def)
using assms differentiable_at_withinI real_differentiable_def by blast
lemma continuous_interval_vimage_Int:
assumes "continuous_on {a::real..b} g" and mono: "\x y. a \ x \ x \ y \ y \ b \ g x \ g y"
assumes "a \ b" "(c::real) \ d" "{c..d} \ {g a..g b}"
obtains c' d' where "{a..b} \ g -` {c..d} = {c'..d'}" "c' \ d'" "g c' = c" "g d' = d"
proof-
let ?A = "{a..b} \ g -` {c..d}"
from IVT'[of g a c b, OF _ _ \a \ b\ assms(1)] assms(4,5)
obtain c'' where c'': "c'' \ ?A" "g c'' = c" by auto
from IVT'[of g a d b, OF _ _ \a \ b\ assms(1)] assms(4,5)
obtain d'' where d'': "d'' \ ?A" "g d'' = d" by auto
hence [simp]: "?A \ {}" by blast
define c' where "c' = Inf ?A"
define d' where "d' = Sup ?A"
have "?A \ {c'..d'}" unfolding c'_def d'_def
by (intro subsetI) (auto intro: cInf_lower cSup_upper)
moreover from assms have "closed ?A"
using continuous_on_closed_vimage[of "{a..b}" g] by (subst Int_commute) simp
hence c'd'_in_set: "c' \ ?A" "d' \ ?A" unfolding c'_def d'_def
by ((intro closed_contains_Inf closed_contains_Sup, simp_all)[])+
hence "{c'..d'} \ ?A" using assms
by (intro subsetI)
(auto intro!: order_trans[of c "g c'" "g x" for x] order_trans[of "g x" "g d'" d for x]
intro!: mono)
moreover have "c' \ d'" using c'd'_in_set(2) unfolding c'_def by (intro cInf_lower) auto
moreover have "g c' \ c" "g d' \ d"
apply (insert c'' d'' c'd'_in_set)
apply (subst c''(2)[symmetric])
apply (auto simp: c'_def intro!: mono cInf_lower c'') []
apply (subst d''(2)[symmetric])
apply (auto simp: d'_def intro!: mono cSup_upper d'') []
done
with c'd'_in_set have "g c' = c" "g d' = d" by auto
ultimately show ?thesis using that by blast
qed
subsection \<open>Generic Borel spaces\<close>
definition\<^marker>\<open>tag important\<close> (in topological_space) borel :: "'a measure" where
"borel = sigma UNIV {S. open S}"
abbreviation "borel_measurable M \ measurable M borel"
lemma in_borel_measurable:
"f \ borel_measurable M \
(\<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 \ borel_measurable M \
(\<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 space_in_borel[measurable]: "UNIV \ sets borel"
unfolding borel_def by auto
lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
unfolding borel_def by (rule sets_measure_of) simp
lemma measurable_sets_borel:
"\f \ measurable borel M; A \ sets M\ \ f -` A \ sets borel"
by (drule (1) measurable_sets) simp
lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \ {x. P x} \ sets borel"
unfolding borel_def pred_def by auto
lemma borel_open[measurable (raw generic)]:
assumes "open A" shows "A \ sets borel"
proof -
have "A \ {S. open S}" unfolding mem_Collect_eq using assms .
thus ?thesis unfolding borel_def by auto
qed
lemma borel_closed[measurable (raw generic)]:
assumes "closed A" shows "A \ sets borel"
proof -
have "space borel - (- A) \ sets borel"
using assms unfolding closed_def by (blast intro: borel_open)
thus ?thesis by simp
qed
lemma borel_singleton[measurable]:
"A \ sets borel \ insert x A \ sets (borel :: 'a::t1_space measure)"
unfolding insert_def by (rule sets.Un) auto
lemma sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV"
proof -
have "(\a\A. {a}) \ sets borel" for A :: "'a set"
by (intro sets.countable_UN') auto
then show ?thesis
by auto
qed
lemma borel_comp[measurable]: "A \ sets borel \ - A \ sets borel"
unfolding Compl_eq_Diff_UNIV by simp
lemma borel_measurable_vimage:
fixes f :: "'a \ 'x::t2_space"
assumes borel[measurable]: "f \ borel_measurable M"
shows "f -` {x} \ space M \ sets M"
by simp
lemma borel_measurableI:
fixes f :: "'a \ 'x::topological_space"
assumes "\S. open S \ f -` S \ space M \ sets M"
shows "f \ borel_measurable M"
unfolding borel_def
proof (rule measurable_measure_of, simp_all)
fix S :: "'x set" assume "open S" thus "f -` S \ space M \ sets M"
using assms[of S] by simp
qed
lemma borel_measurable_const:
"(\x. c) \ borel_measurable M"
by auto
lemma borel_measurable_indicator:
assumes A: "A \ sets M"
shows "indicator A \ borel_measurable M"
unfolding indicator_def [abs_def] using A
by (auto intro!: measurable_If_set)
lemma borel_measurable_count_space[measurable (raw)]:
"f \ borel_measurable (count_space S)"
unfolding measurable_def by auto
lemma borel_measurable_indicator'[measurable (raw)]:
assumes [measurable]: "{x\space M. f x \ A x} \ sets M"
shows "(\x. indicator (A x) (f x)) \ borel_measurable M"
unfolding indicator_def[abs_def]
by (auto intro!: measurable_If)
lemma borel_measurable_indicator_iff:
"(indicator A :: 'a \ 'x::{t1_space, zero_neq_one}) \ borel_measurable M \ A \ space M \ sets M"
(is "?I \ borel_measurable M \ _")
proof
assume "?I \ borel_measurable M"
then have "?I -` {1} \ space M \ sets M"
unfolding measurable_def by auto
also have "?I -` {1} \ space M = A \ space M"
unfolding indicator_def [abs_def] by auto
finally show "A \ space M \ sets M" .
next
assume "A \ space M \ sets M"
moreover have "?I \ borel_measurable M \
(indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
by (intro measurable_cong) (auto simp: indicator_def)
ultimately show "?I \ borel_measurable M" by auto
qed
lemma borel_measurable_subalgebra:
assumes "sets N \ sets M" "space N = space M" "f \ borel_measurable N"
shows "f \ borel_measurable M"
using assms unfolding measurable_def by auto
lemma borel_measurable_restrict_space_iff_ereal:
fixes f :: "'a \ ereal"
assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
shows "f \ borel_measurable (restrict_space M \) \
(\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
by (subst measurable_restrict_space_iff)
(auto simp: indicator_def if_distrib[where f="\x. a * x" for a] cong del: if_weak_cong)
lemma borel_measurable_restrict_space_iff_ennreal:
fixes f :: "'a \ ennreal"
assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
shows "f \ borel_measurable (restrict_space M \) \
(\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M"
by (subst measurable_restrict_space_iff)
(auto simp: indicator_def if_distrib[where f="\x. a * x" for a] cong del: if_weak_cong)
lemma borel_measurable_restrict_space_iff:
fixes f :: "'a \ 'b::real_normed_vector"
assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
shows "f \ borel_measurable (restrict_space M \) \
(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M"
by (subst measurable_restrict_space_iff)
(auto simp: indicator_def if_distrib[where f="\x. x *\<^sub>R a" for a] ac_simps
cong del: if_weak_cong)
lemma cbox_borel[measurable]: "cbox a b \ sets borel"
by (auto intro: borel_closed)
lemma box_borel[measurable]: "box a b \ sets borel"
by (auto intro: borel_open)
lemma borel_compact: "compact (A::'a::t2_space set) \ A \ sets borel"
by (auto intro: borel_closed dest!: compact_imp_closed)
lemma borel_sigma_sets_subset:
"A \ sets borel \ sigma_sets UNIV A \ sets borel"
using sets.sigma_sets_subset[of A borel] by simp
lemma borel_eq_sigmaI1:
fixes F :: "'i \ 'a::topological_space set" and X :: "'a::topological_space set set"
assumes borel_eq: "borel = sigma UNIV X"
assumes X: "\x. x \ X \ x \ sets (sigma UNIV (F ` A))"
assumes F: "\i. i \ A \ F i \ sets borel"
shows "borel = sigma UNIV (F ` A)"
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 "\ = sigma_sets UNIV X"
unfolding borel_eq by simp
also have "\ \ sigma_sets UNIV (F`A)"
using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto
finally show "sigma_sets UNIV {S. open S} \ sigma_sets UNIV (F`A)" .
show "sigma_sets UNIV (F`A) \ 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 \ 'j \ 'a::topological_space set"
and G :: "'l \ 'k \ 'a::topological_space set"
assumes borel_eq: "borel = sigma UNIV ((\(i, j). G i j)`B)"
assumes X: "\i j. (i, j) \ B \ G i j \ sets (sigma UNIV ((\(i, j). F i j) ` A))"
assumes F: "\i j. (i, j) \ A \ F i j \ sets borel"
shows "borel = sigma UNIV ((\(i, j). F i j) ` A)"
using assms
by (intro borel_eq_sigmaI1[where X="(\(i, j). G i j) ` B" and F="(\(i, j). F i j)"]) auto
lemma borel_eq_sigmaI3:
fixes F :: "'i \ 'j \ 'a::topological_space set" and X :: "'a::topological_space set set"
assumes borel_eq: "borel = sigma UNIV X"
assumes X: "\x. x \ X \ x \ sets (sigma UNIV ((\(i, j). F i j) ` A))"
assumes F: "\i j. (i, j) \ A \ F i j \ sets borel"
shows "borel = sigma UNIV ((\(i, j). F i j) ` A)"
using assms by (intro borel_eq_sigmaI1[where X=X and F="(\(i, j). F i j)"]) auto
lemma borel_eq_sigmaI4:
fixes F :: "'i \ 'a::topological_space set"
and G :: "'l \ 'k \ 'a::topological_space set"
assumes borel_eq: "borel = sigma UNIV ((\(i, j). G i j)`A)"
assumes X: "\i j. (i, j) \ A \ G i j \ sets (sigma UNIV (range F))"
assumes F: "\i. F i \ sets borel"
shows "borel = sigma UNIV (range F)"
using assms by (intro borel_eq_sigmaI1[where X="(\(i, j). G i j) ` A" and F=F]) auto
lemma borel_eq_sigmaI5:
fixes F :: "'i \ 'j \ 'a::topological_space set" and G :: "'l \ 'a::topological_space set"
assumes borel_eq: "borel = sigma UNIV (range G)"
assumes X: "\i. G i \ sets (sigma UNIV (range (\(i, j). F i j)))"
assumes F: "\i j. F i j \ sets borel"
shows "borel = sigma UNIV (range (\(i, j). F i j))"
using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\(i, j). F i j)"]) auto
theorem second_countable_borel_measurable:
fixes X :: "'a::second_countable_topology set set"
assumes eq: "open = generate_topology X"
shows "borel = sigma UNIV X"
unfolding borel_def
proof (intro sigma_eqI sigma_sets_eqI)
interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
by (rule sigma_algebra_sigma_sets) simp
fix S :: "'a set" assume "S \ Collect open"
then have "generate_topology X S"
by (auto simp: eq)
then show "S \ sigma_sets UNIV X"
proof induction
case (UN K)
then have K: "\k. k \ K \ open k"
unfolding eq by auto
from ex_countable_basis obtain B :: "'a set set" where
B: "\b. b \ B \ open b" "\X. open X \ \b\B. (\b) = X" and "countable B"
by (auto simp: topological_basis_def)
from B(2)[OF K] obtain m where m: "\k. k \ K \ m k \ B" "\k. k \ K \ \(m k) = k"
by metis
define U where "U = (\k\K. m k)"
with m have "countable U"
by (intro countable_subset[OF _ \<open>countable B\<close>]) auto
have "\U = (\A\U. A)" by simp
also have "\ = \K"
unfolding U_def UN_simps by (simp add: m)
finally have "\U = \K" .
have "\b\U. \k\K. b \ k"
using m by (auto simp: U_def)
then obtain u where u: "\b. b \ U \ u b \ K" and "\b. b \ U \ b \ u b"
by metis
then have "(\b\U. u b) \ \K" "\U \ (\b\U. u b)"
by auto
then have "\K = (\b\U. u b)"
unfolding \<open>\<Union>U = \<Union>K\<close> by auto
also have "\ \ sigma_sets UNIV X"
using u UN by (intro X.countable_UN' \countable U\) auto
finally show "\K \ sigma_sets UNIV X" .
qed auto
qed (auto simp: eq intro: generate_topology.Basis)
lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"
unfolding borel_def
proof (intro sigma_eqI sigma_sets_eqI, safe)
fix x :: "'a set" assume "open x"
hence "x = UNIV - (UNIV - x)" by auto
also have "\ \ sigma_sets UNIV (Collect closed)"
by (force intro: sigma_sets.Compl simp: \<open>open x\<close>)
finally show "x \ sigma_sets UNIV (Collect closed)" by simp
next
fix x :: "'a set" assume "closed x"
hence "x = UNIV - (UNIV - x)" by auto
also have "\ \ sigma_sets UNIV (Collect open)"
by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>)
finally show "x \ sigma_sets UNIV (Collect open)" by simp
qed simp_all
proposition borel_eq_countable_basis:
fixes B::"'a::topological_space set set"
assumes "countable B"
assumes "topological_basis B"
shows "borel = sigma UNIV B"
unfolding borel_def
proof (intro sigma_eqI sigma_sets_eqI, safe)
interpret countable_basis "open" B using assms by (rule countable_basis_openI)
fix X::"'a set" assume "open X"
from open_countable_basisE[OF this] obtain B' where B': "B' \ B" "X = \ B'" .
then show "X \ sigma_sets UNIV B"
by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset)
next
fix b assume "b \ B"
hence "open b" by (rule topological_basis_open[OF assms(2)])
thus "b \ sigma_sets UNIV (Collect open)" by auto
qed simp_all
lemma borel_measurable_continuous_on_restrict:
fixes f :: "'a::topological_space \ 'b::topological_space"
assumes f: "continuous_on A f"
shows "f \ borel_measurable (restrict_space borel A)"
proof (rule borel_measurableI)
fix S :: "'b set" assume "open S"
with f obtain T where "f -` S \ A = T \ A" "open T"
by (metis continuous_on_open_invariant)
then show "f -` S \ space (restrict_space borel A) \ sets (restrict_space borel A)"
by (force simp add: sets_restrict_space space_restrict_space)
qed
lemma borel_measurable_continuous_onI: "continuous_on UNIV f \ f \ borel_measurable borel"
by (drule borel_measurable_continuous_on_restrict) simp
lemma borel_measurable_continuous_on_if:
"A \ sets borel \ continuous_on A f \ continuous_on (- A) g \
(\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
intro!: borel_measurable_continuous_on_restrict)
lemma borel_measurable_continuous_countable_exceptions:
fixes f :: "'a::t1_space \ 'b::topological_space"
assumes X: "countable X"
assumes "continuous_on (- X) f"
shows "f \ borel_measurable borel"
proof (rule measurable_discrete_difference[OF _ X])
have "X \ sets borel"
by (rule sets.countable[OF _ X]) auto
then show "(\x. if x \ X then undefined else f x) \ borel_measurable borel"
by (intro borel_measurable_continuous_on_if assms continuous_intros)
qed auto
lemma borel_measurable_continuous_on:
assumes f: "continuous_on UNIV f" and g: "g \ borel_measurable M"
shows "(\x. f (g x)) \ borel_measurable M"
using measurable_comp[OF g borel_measurable_continuous_onI[OF f]] by (simp add: comp_def)
lemma borel_measurable_continuous_on_indicator:
fixes f g :: "'a::topological_space \ 'b::real_normed_vector"
shows "A \ sets borel \ continuous_on A f \ (\x. indicator A x *\<^sub>R f x) \ borel_measurable borel"
by (subst borel_measurable_restrict_space_iff[symmetric])
(auto intro: borel_measurable_continuous_on_restrict)
lemma borel_measurable_Pair[measurable (raw)]:
fixes f :: "'a \ 'b::second_countable_topology" and g :: "'a \ 'c::second_countable_topology"
assumes f[measurable]: "f \ borel_measurable M"
assumes g[measurable]: "g \ borel_measurable M"
shows "(\x. (f x, g x)) \ borel_measurable M"
proof (subst borel_eq_countable_basis)
let ?B = "SOME B::'b set set. countable B \ topological_basis B"
let ?C = "SOME B::'c set set. countable B \ topological_basis B"
let ?P = "(\(b, c). b \ c) ` (?B \ ?C)"
show "countable ?P" "topological_basis ?P"
by (auto intro!: countable_basis topological_basis_prod is_basis)
show "(\x. (f x, g x)) \ measurable M (sigma UNIV ?P)"
proof (rule measurable_measure_of)
fix S assume "S \ ?P"
then obtain b c where "b \ ?B" "c \ ?C" and S: "S = b \ c" by auto
then have borel: "open b" "open c"
by (auto intro: is_basis topological_basis_open)
have "(\x. (f x, g x)) -` S \ space M = (f -` b \ space M) \ (g -` c \ space M)"
unfolding S by auto
also have "\ \ sets M"
using borel by simp
finally show "(\x. (f x, g x)) -` S \ space M \ sets M" .
qed auto
qed
lemma borel_measurable_continuous_Pair:
fixes f :: "'a \ 'b::second_countable_topology" and g :: "'a \ 'c::second_countable_topology"
assumes [measurable]: "f \ borel_measurable M"
assumes [measurable]: "g \ borel_measurable M"
assumes H: "continuous_on UNIV (\x. H (fst x) (snd x))"
shows "(\x. H (f x) (g x)) \ borel_measurable M"
proof -
have eq: "(\x. H (f x) (g x)) = (\x. (\x. H (fst x) (snd x)) (f x, g x))" by auto
show ?thesis
unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
qed
subsection \<open>Borel spaces on order topologies\<close>
lemma [measurable]:
fixes a b :: "'a::linorder_topology"
shows lessThan_borel: "{..< a} \ sets borel"
and greaterThan_borel: "{a <..} \ sets borel"
and greaterThanLessThan_borel: "{a<.. sets borel"
and atMost_borel: "{..a} \ sets borel"
and atLeast_borel: "{a..} \ sets borel"
and atLeastAtMost_borel: "{a..b} \ sets borel"
and greaterThanAtMost_borel: "{a<..b} \ sets borel"
and atLeastLessThan_borel: "{a.. sets borel"
unfolding greaterThanAtMost_def atLeastLessThan_def
by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
closed_atMost closed_atLeast closed_atLeastAtMost)+
lemma borel_Iio:
"borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
unfolding second_countable_borel_measurable[OF open_generated_order]
proof (intro sigma_eqI sigma_sets_eqI)
from countable_dense_setE guess D :: "'a set" . note D = this
interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
by (rule sigma_algebra_sigma_sets) simp
fix A :: "'a set" assume "A \ range lessThan \ range greaterThan"
then obtain y where "A = {y <..} \ A = {..< y}"
by blast
then show "A \ sigma_sets UNIV (range lessThan)"
proof
assume A: "A = {y <..}"
show ?thesis
proof cases
assume "\x>y. \d. y < d \ d < x"
with D(2)[of "{y <..< x}" for x] have "\x>y. \d\D. y < d \ d < x"
by (auto simp: set_eq_iff)
then have "A = UNIV - (\d\{d\D. y < d}. {..< d})"
by (auto simp: A) (metis less_asym)
also have "\ \ sigma_sets UNIV (range lessThan)"
using D(1) by (intro L.Diff L.top L.countable_INT'') auto
finally show ?thesis .
next
assume "\ (\x>y. \d. y < d \ d < x)"
then obtain x where "y < x" "\d. y < d \ \ d < x"
by auto
then have "A = UNIV - {..< x}"
unfolding A by (auto simp: not_less[symmetric])
also have "\ \ sigma_sets UNIV (range lessThan)"
by auto
finally show ?thesis .
qed
qed auto
qed auto
lemma borel_Ioi:
"borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
unfolding second_countable_borel_measurable[OF open_generated_order]
proof (intro sigma_eqI sigma_sets_eqI)
from countable_dense_setE guess D :: "'a set" . note D = this
interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
by (rule sigma_algebra_sigma_sets) simp
fix A :: "'a set" assume "A \ range lessThan \ range greaterThan"
then obtain y where "A = {y <..} \ A = {..< y}"
by blast
then show "A \ sigma_sets UNIV (range greaterThan)"
proof
assume A: "A = {..< y}"
show ?thesis
proof cases
assume "\xd. x < d \ d < y"
with D(2)[of "{x <..< y}" for x] have "\xd\D. x < d \ d < y"
by (auto simp: set_eq_iff)
then have "A = UNIV - (\d\{d\D. d < y}. {d <..})"
by (auto simp: A) (metis less_asym)
also have "\ \ sigma_sets UNIV (range greaterThan)"
using D(1) by (intro L.Diff L.top L.countable_INT'') auto
finally show ?thesis .
next
assume "\ (\xd. x < d \ d < y)"
then obtain x where "x < y" "\d. y > d \ x \ d"
by (auto simp: not_less[symmetric])
then have "A = UNIV - {x <..}"
unfolding A Compl_eq_Diff_UNIV[symmetric] by auto
also have "\ \ sigma_sets UNIV (range greaterThan)"
by auto
finally show ?thesis .
qed
qed auto
qed auto
lemma borel_measurableI_less:
fixes f :: "'a \ 'b::{linorder_topology, second_countable_topology}"
shows "(\y. {x\space M. f x < y} \ sets M) \ f \ borel_measurable M"
unfolding borel_Iio
by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
lemma borel_measurableI_greater:
fixes f :: "'a \ 'b::{linorder_topology, second_countable_topology}"
shows "(\y. {x\space M. y < f x} \ sets M) \ f \ borel_measurable M"
unfolding borel_Ioi
by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
lemma borel_measurableI_le:
fixes f :: "'a \ 'b::{linorder_topology, second_countable_topology}"
shows "(\y. {x\space M. f x \ y} \ sets M) \ f \ borel_measurable M"
by (rule borel_measurableI_greater) (auto simp: not_le[symmetric])
lemma borel_measurableI_ge:
fixes f :: "'a \ 'b::{linorder_topology, second_countable_topology}"
shows "(\y. {x\space M. y \ f x} \ sets M) \ f \ borel_measurable M"
by (rule borel_measurableI_less) (auto simp: not_le[symmetric])
lemma borel_measurable_less[measurable]:
fixes f :: "'a \ 'b::{second_countable_topology, linorder_topology}"
assumes "f \ borel_measurable M"
assumes "g \ borel_measurable M"
shows "{w \ space M. f w < g w} \ sets M"
proof -
have "{w \ space M. f w < g w} = (\x. (f x, g x)) -` {x. fst x < snd x} \ space M"
by auto
also have "\ \ sets M"
by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less]
continuous_intros)
finally show ?thesis .
qed
lemma
fixes f :: "'a \ 'b::{second_countable_topology, linorder_topology}"
assumes f[measurable]: "f \ borel_measurable M"
assumes g[measurable]: "g \ borel_measurable M"
shows borel_measurable_le[measurable]: "{w \ space M. f w \ g w} \ sets M"
and borel_measurable_eq[measurable]: "{w \ space M. f w = g w} \ sets M"
and borel_measurable_neq: "{w \ space M. f w \ g w} \ sets M"
unfolding eq_iff not_less[symmetric]
by measurable
lemma borel_measurable_SUP[measurable (raw)]:
fixes F :: "_ \ _ \ _::{complete_linorder, linorder_topology, second_countable_topology}"
assumes [simp]: "countable I"
assumes [measurable]: "\i. i \ I \ F i \ borel_measurable M"
shows "(\x. SUP i\I. F i x) \ borel_measurable M"
by (rule borel_measurableI_greater) (simp add: less_SUP_iff)
lemma borel_measurable_INF[measurable (raw)]:
fixes F :: "_ \ _ \ _::{complete_linorder, linorder_topology, second_countable_topology}"
assumes [simp]: "countable I"
assumes [measurable]: "\i. i \ I \ F i \ borel_measurable M"
shows "(\x. INF i\I. F i x) \ borel_measurable M"
by (rule borel_measurableI_less) (simp add: INF_less_iff)
lemma borel_measurable_cSUP[measurable (raw)]:
fixes F :: "_ \ _ \ 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
assumes [simp]: "countable I"
assumes [measurable]: "\i. i \ I \ F i \ borel_measurable M"
assumes bdd: "\x. x \ space M \ bdd_above ((\i. F i x) ` I)"
shows "(\x. SUP i\I. F i x) \ borel_measurable M"
proof cases
assume "I = {}" then show ?thesis
unfolding \<open>I = {}\<close> image_empty by simp
next
assume "I \ {}"
show ?thesis
proof (rule borel_measurableI_le)
fix y
have "{x \ space M. \i\I. F i x \ y} \ sets M"
by measurable
also have "{x \ space M. \i\I. F i x \ y} = {x \ space M. (SUP i\I. F i x) \ y}"
by (simp add: cSUP_le_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
finally show "{x \ space M. (SUP i\I. F i x) \ y} \ sets M" .
qed
qed
lemma borel_measurable_cINF[measurable (raw)]:
fixes F :: "_ \ _ \ 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
assumes [simp]: "countable I"
assumes [measurable]: "\i. i \ I \ F i \ borel_measurable M"
assumes bdd: "\x. x \ space M \ bdd_below ((\i. F i x) ` I)"
shows "(\x. INF i\I. F i x) \ borel_measurable M"
proof cases
assume "I = {}" then show ?thesis
unfolding \<open>I = {}\<close> image_empty by simp
next
assume "I \ {}"
show ?thesis
proof (rule borel_measurableI_ge)
fix y
have "{x \ space M. \i\I. y \ F i x} \ sets M"
by measurable
also have "{x \ space M. \i\I. y \ F i x} = {x \ space M. y \ (INF i\I. F i x)}"
by (simp add: le_cINF_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong)
finally show "{x \ space M. y \ (INF i\I. F i x)} \ sets M" .
qed
qed
lemma borel_measurable_lfp[consumes 1, case_names continuity step]:
fixes F :: "('a \ 'b) \ ('a \ 'b::{complete_linorder, linorder_topology, second_countable_topology})"
assumes "sup_continuous F"
assumes *: "\f. f \ borel_measurable M \ F f \ borel_measurable M"
shows "lfp F \ borel_measurable M"
proof -
{ fix i have "((F ^^ i) bot) \ borel_measurable M"
by (induct i) (auto intro!: *) }
then have "(\x. SUP i. (F ^^ i) bot x) \ borel_measurable M"
by measurable
also have "(\x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
by (auto simp add: image_comp)
also have "(SUP i. (F ^^ i) bot) = lfp F"
by (rule sup_continuous_lfp[symmetric]) fact
finally show ?thesis .
qed
lemma borel_measurable_gfp[consumes 1, case_names continuity step]:
fixes F :: "('a \ 'b) \ ('a \ 'b::{complete_linorder, linorder_topology, second_countable_topology})"
assumes "inf_continuous F"
assumes *: "\f. f \ borel_measurable M \ F f \ borel_measurable M"
shows "gfp F \ borel_measurable M"
proof -
{ fix i have "((F ^^ i) top) \ borel_measurable M"
by (induct i) (auto intro!: * simp: bot_fun_def) }
then have "(\x. INF i. (F ^^ i) top x) \ borel_measurable M"
by measurable
also have "(\x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
by (auto simp add: image_comp)
also have "\ = gfp F"
by (rule inf_continuous_gfp[symmetric]) fact
finally show ?thesis .
qed
lemma borel_measurable_max[measurable (raw)]:
"f \ borel_measurable M \ g \ borel_measurable M \ (\x. max (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \ borel_measurable M"
by (rule borel_measurableI_less) simp
lemma borel_measurable_min[measurable (raw)]:
"f \ borel_measurable M \ g \ borel_measurable M \ (\x. min (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \ borel_measurable M"
by (rule borel_measurableI_greater) simp
lemma borel_measurable_Min[measurable (raw)]:
"finite I \ (\i. i \ I \ f i \ borel_measurable M) \ (\x. Min ((\i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \ borel_measurable M"
proof (induct I rule: finite_induct)
case (insert i I) then show ?case
by (cases "I = {}") auto
qed auto
lemma borel_measurable_Max[measurable (raw)]:
"finite I \ (\i. i \ I \ f i \ borel_measurable M) \ (\x. Max ((\i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \ borel_measurable M"
proof (induct I rule: finite_induct)
case (insert i I) then show ?case
by (cases "I = {}") auto
qed auto
lemma borel_measurable_sup[measurable (raw)]:
"f \ borel_measurable M \ g \ borel_measurable M \ (\x. sup (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \ borel_measurable M"
unfolding sup_max by measurable
lemma borel_measurable_inf[measurable (raw)]:
"f \ borel_measurable M \ g \ borel_measurable M \ (\x. inf (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \ borel_measurable M"
unfolding inf_min by measurable
lemma [measurable (raw)]:
fixes f :: "nat \ 'a \ 'b::{complete_linorder, second_countable_topology, linorder_topology}"
assumes "\i. f i \ borel_measurable M"
shows borel_measurable_liminf: "(\x. liminf (\i. f i x)) \ borel_measurable M"
and borel_measurable_limsup: "(\x. limsup (\i. f i x)) \ borel_measurable M"
unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
lemma measurable_convergent[measurable (raw)]:
fixes f :: "nat \ 'a \ 'b::{complete_linorder, second_countable_topology, linorder_topology}"
assumes [measurable]: "\i. f i \ borel_measurable M"
shows "Measurable.pred M (\x. convergent (\i. f i x))"
unfolding convergent_ereal by measurable
lemma sets_Collect_convergent[measurable]:
fixes f :: "nat \ 'a \ 'b::{complete_linorder, second_countable_topology, linorder_topology}"
assumes f[measurable]: "\i. f i \ borel_measurable M"
shows "{x\space M. convergent (\i. f i x)} \ sets M"
by measurable
lemma borel_measurable_lim[measurable (raw)]:
fixes f :: "nat \ 'a \ 'b::{complete_linorder, second_countable_topology, linorder_topology}"
assumes [measurable]: "\i. f i \ borel_measurable M"
shows "(\x. lim (\i. f i x)) \ borel_measurable M"
proof -
have "\x. lim (\i. f i x) = (if convergent (\i. f i x) then limsup (\i. f i x) else (THE i. False))"
by (simp add: lim_def convergent_def convergent_limsup_cl)
then show ?thesis
by simp
qed
lemma borel_measurable_LIMSEQ_order:
fixes u :: "nat \ 'a \ 'b::{complete_linorder, second_countable_topology, linorder_topology}"
assumes u': "\x. x \ space M \ (\i. u i x) \ u' x"
and u: "\i. u i \ borel_measurable M"
shows "u' \ borel_measurable M"
proof -
have "\x. x \ space M \ u' x = liminf (\n. u n x)"
using u' by (simp add: lim_imp_Liminf[symmetric])
with u show ?thesis by (simp cong: measurable_cong)
qed
subsection \<open>Borel spaces on topological monoids\<close>
lemma borel_measurable_add[measurable (raw)]:
fixes f g :: "'a \ 'b::{second_countable_topology, topological_monoid_add}"
assumes f: "f \ borel_measurable M"
assumes g: "g \ borel_measurable M"
shows "(\x. f x + g x) \ borel_measurable M"
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
lemma borel_measurable_sum[measurable (raw)]:
fixes f :: "'c \ 'a \ 'b::{second_countable_topology, topological_comm_monoid_add}"
assumes "\i. i \ S \ f i \ borel_measurable M"
shows "(\x. \i\S. f i x) \ borel_measurable M"
proof cases
assume "finite S"
thus ?thesis using assms by induct auto
qed simp
lemma borel_measurable_suminf_order[measurable (raw)]:
fixes f :: "nat \ 'a \ 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}"
assumes f[measurable]: "\i. f i \ borel_measurable M"
shows "(\x. suminf (\i. f i x)) \ borel_measurable M"
unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
subsection \<open>Borel spaces on Euclidean spaces\<close>
lemma borel_measurable_inner[measurable (raw)]:
fixes f g :: "'a \ 'b::{second_countable_topology, real_inner}"
assumes "f \ borel_measurable M"
assumes "g \ borel_measurable M"
shows "(\x. f x \ g x) \ borel_measurable M"
using assms
by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
notation
eucl_less (infix " 50)
lemma box_oc: "{x. a x \ b} = {x. a {..b}"
and box_co: "{x. a \ x \ x {x. x
by auto
lemma eucl_ivals[measurable]:
fixes a b :: "'a::ordered_euclidean_space"
shows "{x. x sets borel"
and "{x. a sets borel"
and "{..a} \ sets borel"
and "{a..} \ sets borel"
and "{a..b} \ sets borel"
and "{x. a x \ b} \ sets borel"
and "{x. a \ x \ x sets borel"
unfolding box_oc box_co
by (auto intro: borel_open borel_closed)
lemma
fixes i :: "'a::{second_countable_topology, real_inner}"
shows hafspace_less_borel: "{x. a < x \ i} \ sets borel"
and hafspace_greater_borel: "{x. x \ i < a} \ sets borel"
and hafspace_less_eq_borel: "{x. a \ x \ i} \ sets borel"
and hafspace_greater_eq_borel: "{x. x \ i \ a} \ sets borel"
by simp_all
lemma borel_eq_box:
"borel = sigma UNIV (range (\ (a, b). box a b :: 'a :: euclidean_space set))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI1[OF borel_def])
fix M :: "'a set" assume "M \ {S. open S}"
then have "open M" by simp
show "M \ ?SIGMA"
apply (subst open_UNION_box[OF \<open>open M\<close>])
apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
apply (auto intro: countable_rat)
done
qed (auto simp: box_def)
lemma halfspace_gt_in_halfspace:
assumes i: "i \ A"
shows "{x::'a. a < x \ i} \
sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
(is "?set \ ?SIGMA")
proof -
interpret sigma_algebra UNIV ?SIGMA
by (intro sigma_algebra_sigma_sets) simp_all
have *: "?set = (\n. UNIV - {x::'a. x \ i < a + 1 / real (Suc n)})"
proof (safe, simp_all add: not_less del: of_nat_Suc)
fix x :: 'a assume "a < x \ i"
with reals_Archimedean[of "x \ i - a"]
obtain n where "a + 1 / real (Suc n) < x \ i"
by (auto simp: field_simps)
then show "\n. a + 1 / real (Suc n) \ x \ i"
by (blast intro: less_imp_le)
next
fix x n
have "a < a + 1 / real (Suc n)" by auto
also assume "\ \ x"
finally show "a < x" .
qed
show "?set \ ?SIGMA" unfolding *
by (auto intro!: Diff sigma_sets_Inter i)
qed
lemma borel_eq_halfspace_less:
"borel = sigma UNIV ((\(a, i). {x::'a::euclidean_space. x \ i < a}) ` (UNIV \ Basis))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_box])
fix a b :: 'a
have "box a b = {x\space ?SIGMA. \i\Basis. a \ i < x \ i \ x \ i < b \ i}"
by (auto simp: box_def)
also have "\ \ sets ?SIGMA"
by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const)
(auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat)
finally show "box a b \ sets ?SIGMA" .
qed auto
lemma borel_eq_halfspace_le:
"borel = sigma UNIV ((\ (a, i). {x::'a::euclidean_space. x \ i \ a}) ` (UNIV \ Basis))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
fix a :: real and i :: 'a assume "(a, i) \ UNIV \ Basis"
then have i: "i \ Basis" by auto
have *: "{x::'a. x\i < a} = (\n. {x. x\i \ a - 1/real (Suc n)})"
proof (safe, simp_all del: of_nat_Suc)
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)
then show "\n. x \ i \ a - 1 / (real (Suc n))"
by (blast intro: less_imp_le)
next
fix x::'a and n
assume "x\i \ a - 1 / real (Suc n)"
also have "\ < a" by auto
finally show "x\i < a" .
qed
show "{x. x\i < a} \ ?SIGMA" unfolding *
by (intro sets.countable_UN) (auto intro: i)
qed auto
lemma borel_eq_halfspace_ge:
"borel = sigma UNIV ((\ (a, i). {x::'a::euclidean_space. a \ x \ i}) ` (UNIV \ Basis))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
fix a :: real and i :: 'a assume i: "(a, i) \ UNIV \ Basis"
have *: "{x::'a. x\i < a} = space ?SIGMA - {x::'a. a \ x\i}" by auto
show "{x. x\i < a} \ ?SIGMA" unfolding *
using i by (intro sets.compl_sets) auto
qed auto
lemma borel_eq_halfspace_greater:
"borel = sigma UNIV ((\ (a, i). {x::'a::euclidean_space. a < x \ i}) ` (UNIV \ Basis))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
fix a :: real and i :: 'a assume "(a, i) \ (UNIV \ Basis)"
then have i: "i \ Basis" by auto
have *: "{x::'a. x\i \ a} = space ?SIGMA - {x::'a. a < x\i}" by auto
show "{x. x\i \ a} \ ?SIGMA" unfolding *
by (intro sets.compl_sets) (auto intro: i)
qed auto
lemma borel_eq_atMost:
"borel = sigma UNIV (range (\a. {..a::'a::ordered_euclidean_space}))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
fix a :: real and i :: 'a assume "(a, i) \ UNIV \ Basis"
then have "i \ Basis" by auto
then have *: "{x::'a. x\i \ a} = (\k::nat. {.. (\n\Basis. (if n = i then a else real k)*\<^sub>R n)})"
proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm)
fix x :: 'a
from real_arch_simple[of "Max ((\i. x\i)`Basis)"] guess k::nat ..
then have "\i. i \ Basis \ x\i \ real k"
by (subst (asm) Max_le_iff) auto
then show "\k::nat. \ia\Basis. ia \ i \ x \ ia \ real k"
by (auto intro!: exI[of _ k])
qed
show "{x. x\i \ a} \ ?SIGMA" unfolding *
by (intro sets.countable_UN) auto
qed auto
lemma borel_eq_greaterThan:
"borel = sigma UNIV (range (\a::'a::ordered_euclidean_space. {x. a
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
fix a :: real and i :: 'a assume "(a, i) \ UNIV \ Basis"
then have i: "i \ Basis" by auto
have "{x::'a. x\i \ a} = UNIV - {x::'a. a < x\i}" by auto
also have *: "{x::'a. a < x\i} =
(\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i
proof (safe, simp_all add: eucl_less_def split: if_split_asm)
fix x :: 'a
from reals_Archimedean2[of "Max ((\i. -x\i)`Basis)"]
guess k::nat .. note k = this
{ fix i :: 'a assume "i \ Basis"
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 "\k::nat. \ia\Basis. ia \ i \ -real k < x \ ia"
by (auto intro!: exI[of _ k])
qed
finally show "{x. x\i \ a} \ ?SIGMA"
apply (simp only:)
apply (intro sets.countable_UN sets.Diff)
apply (auto intro: sigma_sets_top)
done
qed auto
lemma borel_eq_lessThan:
"borel = sigma UNIV (range (\a::'a::ordered_euclidean_space. {x. x
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
fix a :: real and i :: 'a assume "(a, i) \ UNIV \ Basis"
then have i: "i \ Basis" by auto
have "{x::'a. a \ x\i} = UNIV - {x::'a. x\i < a}" by auto
also have *: "{x::'a. x\i < a} = (\k::nat. {x. x n\Basis. (if n = i then a else real k) *\<^sub>R n)})" using \i\ Basis\
proof (safe, simp_all add: eucl_less_def split: if_split_asm)
fix x :: 'a
from reals_Archimedean2[of "Max ((\i. x\i)`Basis)"]
guess k::nat .. note k = this
{ fix i :: 'a assume "i \ Basis"
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 "\k::nat. \ia\Basis. ia \ i \ x \ ia < real k"
by (auto intro!: exI[of _ k])
qed
finally show "{x. a \ x\i} \ ?SIGMA"
apply (simp only:)
apply (intro sets.countable_UN sets.Diff)
apply (auto intro: sigma_sets_top )
done
qed auto
lemma borel_eq_atLeastAtMost:
"borel = sigma UNIV (range (\(a,b). {a..b} ::'a::ordered_euclidean_space set))"
(is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
fix a::'a
have *: "{..a} = (\n::nat. {- real n *\<^sub>R One .. a})"
proof (safe, simp_all add: eucl_le[where 'a='a])
fix x :: 'a
from real_arch_simple[of "Max ((\i. - x\i)`Basis)"]
guess k::nat .. note k = this
{ fix i :: 'a assume "i \ Basis"
with k have "- x\i \ real k"
by (subst (asm) Max_le_iff) (auto simp: field_simps)
then have "- real k \ x\i" by simp }
then show "\n::nat. \i\Basis. - real n \ x \ i"
by (auto intro!: exI[of _ k])
qed
show "{..a} \ ?SIGMA" unfolding *
by (intro sets.countable_UN)
(auto intro!: sigma_sets_top)
qed auto
lemma borel_set_induct[consumes 1, case_names empty interval compl union]:
assumes "A \ sets borel"
assumes empty: "P {}" and int: "\a b. a \ b \ P {a..b}" and compl: "\A. A \ sets borel \ P A \ P (-A)" and
un: "\f. disjoint_family f \ (\i. f i \ sets borel) \ (\i. P (f i)) \ P (\i::nat. f i)"
shows "P (A::real set)"
proof -
let ?G = "range (\(a,b). {a..b::real})"
have "Int_stable ?G" "?G \ Pow UNIV" "A \ sigma_sets UNIV ?G"
using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def)
thus ?thesis
proof (induction rule: sigma_sets_induct_disjoint)
case (union f)
from union.hyps(2) have "\i. f i \ sets borel" by (auto simp: borel_eq_atLeastAtMost)
with union show ?case by (auto intro: un)
next
case (basic A)
then obtain a b where "A = {a .. b}" by auto
then show ?case
by (cases "a \ b") (auto intro: int empty)
qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost)
qed
lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\(a, b). {a <.. b::real}))"
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
fix i :: real
have "{..i} = (\j::nat. {-j <.. i})"
by (auto simp: minus_less_iff reals_Archimedean2)
also have "\ \ sets (sigma UNIV (range (\(i, j). {i<..j})))"
by (intro sets.countable_nat_UN) auto
finally show "{..i} \ sets (sigma UNIV (range (\(i, j). {i<..j})))" .
qed simp
lemma eucl_lessThan: "{x::real. x
by (simp add: eucl_less_def lessThan_def)
lemma borel_eq_atLeastLessThan:
"borel = sigma UNIV (range (\(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
have move_uminus: "\x y::real. -x \ y \ -y \ x" by auto
fix x :: real
have "{..i::nat. {-real i ..< x})"
by (auto simp: move_uminus real_arch_simple)
then show "{y. y ?SIGMA"
by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
qed auto
lemma borel_measurable_halfspacesI:
fixes f :: "'a \ 'c::euclidean_space"
assumes F: "borel = sigma UNIV (F ` (UNIV \ Basis))"
and S_eq: "\a i. S a i = f -` F (a,i) \ space M"
shows "f \ borel_measurable M = (\i\Basis. \a::real. S a i \ sets M)"
proof safe
fix a :: real and i :: 'b assume i: "i \ Basis" and f: "f \ borel_measurable M"
then show "S a i \ sets M" unfolding assms
by (auto intro!: measurable_sets simp: assms(1))
next
assume a: "\i\Basis. \a. S a i \ sets M"
then show "f \ borel_measurable M"
by (auto intro!: measurable_measure_of simp: S_eq F)
qed
lemma borel_measurable_iff_halfspace_le:
fixes f :: "'a \ 'c::euclidean_space"
shows "f \ borel_measurable M = (\i\Basis. \a. {w \ space M. f w \ i \ a} \ sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
lemma borel_measurable_iff_halfspace_less:
fixes f :: "'a \ 'c::euclidean_space"
shows "f \ borel_measurable M \ (\i\Basis. \a. {w \ space M. f w \ i < a} \ sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
lemma borel_measurable_iff_halfspace_ge:
fixes f :: "'a \ 'c::euclidean_space"
shows "f \ borel_measurable M = (\i\Basis. \a. {w \ space M. a \ f w \ i} \ sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
lemma borel_measurable_iff_halfspace_greater:
fixes f :: "'a \ 'c::euclidean_space"
shows "f \ borel_measurable M \ (\i\Basis. \a. {w \ space M. a < f w \ i} \ sets M)"
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
lemma borel_measurable_iff_le:
"(f::'a \ real) \ borel_measurable M = (\a. {w \ space M. f w \ a} \ sets M)"
using borel_measurable_iff_halfspace_le[where 'c=real] by simp
lemma borel_measurable_iff_less:
"(f::'a \ real) \ borel_measurable M = (\a. {w \ space M. f w < a} \ sets M)"
using borel_measurable_iff_halfspace_less[where 'c=real] by simp
lemma borel_measurable_iff_ge:
"(f::'a \ real) \ borel_measurable M = (\a. {w \ space M. a \ f w} \ sets M)"
using borel_measurable_iff_halfspace_ge[where 'c=real]
by simp
lemma borel_measurable_iff_greater:
"(f::'a \ real) \ borel_measurable M = (\a. {w \ space M. a < f w} \ sets M)"
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
lemma borel_measurable_euclidean_space:
fixes f :: "'a \ 'c::euclidean_space"
shows "f \ borel_measurable M \ (\i\Basis. (\x. f x \ i) \ borel_measurable M)"
proof safe
assume f: "\i\Basis. (\x. f x \ i) \ borel_measurable M"
then show "f \ borel_measurable M"
by (subst borel_measurable_iff_halfspace_le) auto
qed auto
subsection "Borel measurable operators"
lemma borel_measurable_norm[measurable]: "norm \ borel_measurable borel"
by (intro borel_measurable_continuous_onI continuous_intros)
lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \ 'a) \ borel_measurable borel"
by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
(auto intro!: continuous_on_sgn continuous_on_id)
lemma borel_measurable_uminus[measurable (raw)]:
fixes g :: "'a \ 'b::{second_countable_topology, real_normed_vector}"
assumes g: "g \ borel_measurable M"
shows "(\x. - g x) \ borel_measurable M"
by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
lemma borel_measurable_diff[measurable (raw)]:
fixes f :: "'a \ 'b::{second_countable_topology, real_normed_vector}"
assumes f: "f \ borel_measurable M"
assumes g: "g \ borel_measurable M"
shows "(\x. f x - g x) \ borel_measurable M"
using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
lemma borel_measurable_times[measurable (raw)]:
fixes f :: "'a \ 'b::{second_countable_topology, real_normed_algebra}"
assumes f: "f \ borel_measurable M"
assumes g: "g \ borel_measurable M"
shows "(\x. f x * g x) \ borel_measurable M"
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
lemma borel_measurable_prod[measurable (raw)]:
fixes f :: "'c \ 'a \ 'b::{second_countable_topology, real_normed_field}"
assumes "\i. i \ S \ f i \ borel_measurable M"
shows "(\x. \i\S. f i x) \ borel_measurable M"
proof cases
assume "finite S"
thus ?thesis using assms by induct auto
qed simp
lemma borel_measurable_dist[measurable (raw)]:
fixes g f :: "'a \ 'b::{second_countable_topology, metric_space}"
assumes f: "f \ borel_measurable M"
assumes g: "g \ borel_measurable M"
shows "(\x. dist (f x) (g x)) \ borel_measurable M"
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
lemma borel_measurable_scaleR[measurable (raw)]:
fixes g :: "'a \ 'b::{second_countable_topology, real_normed_vector}"
assumes f: "f \ borel_measurable M"
assumes g: "g \ borel_measurable M"
shows "(\x. f x *\<^sub>R g x) \ borel_measurable M"
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
lemma borel_measurable_uminus_eq [simp]:
fixes f :: "'a \ 'b::{second_countable_topology, real_normed_vector}"
shows "(\x. - f x) \ borel_measurable M \ f \ borel_measurable M" (is "?l = ?r")
proof
assume ?l from borel_measurable_uminus[OF this] show ?r by simp
qed auto
lemma affine_borel_measurable_vector:
fixes f :: "'a \ 'x::real_normed_vector"
assumes "f \ borel_measurable M"
shows "(\x. a + b *\<^sub>R f x) \ borel_measurable M"
proof (rule borel_measurableI)
fix S :: "'x set" assume "open S"
show "(\x. a + b *\<^sub>R f x) -` S \ space M \ sets M"
proof cases
assume "b \ 0"
with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
by (auto simp: algebra_simps)
hence "?S \ sets borel" by auto
moreover
from \<open>b \<noteq> 0\<close> 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 borel_measurable_const_scaleR[measurable (raw)]:
"f \ borel_measurable M \ (\x. b *\<^sub>R f x ::'a::real_normed_vector) \ borel_measurable M"
using affine_borel_measurable_vector[of f M 0 b] by simp
lemma borel_measurable_const_add[measurable (raw)]:
"f \ borel_measurable M \ (\x. a + f x ::'a::real_normed_vector) \ borel_measurable M"
using affine_borel_measurable_vector[of f M a 1] by simp
lemma borel_measurable_inverse[measurable (raw)]:
fixes f :: "'a \ 'b::real_normed_div_algebra"
assumes f: "f \ borel_measurable M"
shows "(\x. inverse (f x)) \ borel_measurable M"
apply (rule measurable_compose[OF f])
apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
apply (auto intro!: continuous_on_inverse continuous_on_id)
done
lemma borel_measurable_divide[measurable (raw)]:
"f \ borel_measurable M \ g \ borel_measurable M \
(\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
by (simp add: divide_inverse)
lemma borel_measurable_abs[measurable (raw)]:
"f \ borel_measurable M \ (\x. \f x :: real\) \ borel_measurable M"
unfolding abs_real_def by simp
lemma borel_measurable_nth[measurable (raw)]:
"(\x::real^'n. x $ i) \ borel_measurable borel"
by (simp add: cart_eq_inner_axis)
lemma convex_measurable:
fixes A :: "'a :: euclidean_space set"
shows "X \ borel_measurable M \ X ` space M \ A \ open A \ convex_on A q \
(\<lambda>x. q (X x)) \<in> borel_measurable M"
by (rule measurable_compose[where f=X and N="restrict_space borel A"])
(auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
lemma borel_measurable_ln[measurable (raw)]:
assumes f: "f \ borel_measurable M"
shows "(\x. ln (f x :: real)) \ borel_measurable M"
apply (rule measurable_compose[OF f])
apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
apply (auto intro!: continuous_on_ln continuous_on_id)
done
lemma borel_measurable_log[measurable (raw)]:
"f \ borel_measurable M \ g \ borel_measurable M \ (\x. log (g x) (f x)) \ borel_measurable M"
unfolding log_def by auto
lemma borel_measurable_exp[measurable]:
"(exp::'a::{real_normed_field,banach}\'a) \ borel_measurable borel"
by (intro borel_measurable_continuous_onI continuous_at_imp_continuous_on ballI isCont_exp)
lemma measurable_real_floor[measurable]:
"(floor :: real \ int) \ measurable borel (count_space UNIV)"
proof -
have "\a x. \x\ = a \ (real_of_int a \ x \ x < real_of_int (a + 1))"
by (auto intro: floor_eq2)
then show ?thesis
by (auto simp: vimage_def measurable_count_space_eq2_countable)
qed
lemma measurable_real_ceiling[measurable]:
"(ceiling :: real \ int) \ measurable borel (count_space UNIV)"
unfolding ceiling_def[abs_def] by simp
lemma borel_measurable_real_floor: "(\x::real. real_of_int \x\) \ borel_measurable borel"
by simp
lemma borel_measurable_root [measurable]: "root n \ borel_measurable borel"
by (intro borel_measurable_continuous_onI continuous_intros)
lemma borel_measurable_sqrt [measurable]: "sqrt \ borel_measurable borel"
by (intro borel_measurable_continuous_onI continuous_intros)
lemma borel_measurable_power [measurable (raw)]:
fixes f :: "_ \ 'b::{power,real_normed_algebra}"
--> --------------------
--> maximum size reached
--> --------------------
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