src/HOL/Probability/Borel_Space.thy
 author hoelzl Mon, 14 Mar 2011 14:37:33 +0100 changeset 41969 1cf3e4107a2a parent 41830 719b0a517c33 child 41981 cdf7693bbe08 permissions -rw-r--r--
moved t2_spaces to HOL image
```
(* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)

theory Borel_Space
imports Sigma_Algebra Positive_Extended_Real Multivariate_Analysis
begin

lemma LIMSEQ_max:
"u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)

section "Generic Borel spaces"

definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
abbreviation "borel_measurable M \<equiv> measurable M borel"

interpretation borel: sigma_algebra borel
by (auto simp: borel_def intro!: sigma_algebra_sigma)

lemma in_borel_measurable:
"f \<in> borel_measurable M \<longleftrightarrow>
(\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = open\<rparr>).
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> open" unfolding mem_def using assms .
thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
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 borel.Diff) auto

lemma (in sigma_algebra) 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 (in sigma_algebra) 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_sigma, simp_all)
fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"
using assms[of S] by (simp add: mem_def)
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 borel.insert_in_sets)
show "{x} \<in> sets borel"
using closed_singleton[of x] by (rule borel_closed)
qed simp

lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
"(\<lambda>x. c) \<in> borel_measurable M"
by (auto intro!: measurable_const)

lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
assumes A: "A \<in> sets M"
shows "indicator A \<in> borel_measurable M"
unfolding indicator_def_raw using A
by (auto intro!: measurable_If_set borel_measurable_const)

lemma (in sigma_algebra) 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_raw 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_translate:
assumes "A \<in> sets borel" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel"
shows "f -` A \<in> sets borel"
proof -
have "A \<in> sigma_sets UNIV open" using assms
thus ?thesis
proof (induct rule: sigma_sets.induct)
case (Basic a) thus ?case using trans[of a] by (simp add: mem_def)
next
case (Compl a)
moreover have "UNIV \<in> sets borel"
using borel.top by simp
ultimately show ?case
by (auto simp: vimage_Diff borel.Diff)
qed

lemma (in sigma_algebra) borel_measurable_restricted:
fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" assumes "A \<in> sets M"
shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
(is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
proof -
interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
by (auto intro!: measurable_cong)
show ?thesis unfolding *
unfolding in_borel_measurable_borel
proof (simp, safe)
fix S :: "'x set" assume "S \<in> sets borel"
"\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
then have f: "?f -` S \<inter> A \<in> sets M"
using `A \<in> sets M` sets_into_space by fastsimp
show "?f -` S \<inter> space M \<in> sets M"
proof cases
assume "0 \<in> S"
then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
using `A \<in> sets M` sets_into_space by auto
then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
next
assume "0 \<notin> S"
then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
using `A \<in> sets M` sets_into_space
by (auto simp: indicator_def split: split_if_asm)
then show ?thesis using f by auto
qed
next
fix S :: "'x set" assume "S \<in> sets borel"
"\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
then have f: "?f -` S \<inter> space M \<in> sets M" by auto
then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
using `A \<in> sets M` sets_into_space
apply (rule bexI[OF _ f])
by auto
qed
qed

lemma (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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)"
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 halfspace_span_open:
"sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a}))
\<subseteq> sets borel"
by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
open_halfspace_component_lt)

lemma halfspace_lt_in_halfspace:
"{x\<Colon>'a. x \$\$ i < a} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
by (auto intro!: sigma_sets.Basic simp: sets_sigma)

lemma halfspace_gt_in_halfspace:
"{x\<Colon>'a. a < x \$\$ i} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
(is "?set \<in> sets ?SIGMA")
proof -
interpret sigma_algebra "?SIGMA"
by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x \$\$ i < a + 1 / real (Suc n)})"
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> sets ?SIGMA" unfolding *
by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
qed

lemma open_span_halfspace:
"sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x \$\$ i < a})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
then interpret sigma_algebra ?SIGMA .
{ fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .
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
have "S \<in> sets ?SIGMA"
unfolding *
by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
then show ?thesis unfolding borel_def
by (intro sets_sigma_subset) auto
qed

lemma halfspace_span_halfspace_le:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x \$\$ i \<le> a})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
{ 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
have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
by (safe intro!: countable_UN)
(auto simp: sets_sigma intro!: sigma_sets.Basic) }
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma halfspace_span_halfspace_ge:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i < a})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x \$\$ i})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
{ fix a i have *: "{x::'a. x\$\$i < a} = space ?SIGMA - {x::'a. a \<le> x\$\$i}" by auto
have "{x. x\$\$i < a} \<in> sets ?SIGMA" unfolding *
by (safe intro!: Diff)
(auto simp: sets_sigma intro!: sigma_sets.Basic) }
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma halfspace_le_span_halfspace_gt:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x \$\$ i})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
{ fix a i have *: "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {x::'a. a < x\$\$i}" by auto
have "{x. x\$\$i \<le> a} \<in> sets ?SIGMA" unfolding *
by (safe intro!: Diff)
(auto simp: sets_sigma intro!: sigma_sets.Basic) }
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma halfspace_le_span_atMost:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
proof cases
fix a i 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> sets ?SIGMA" unfolding *
by (safe intro!: countable_UN)
(auto simp: sets_sigma intro!: sigma_sets.Basic)
next
fix a i assume "\<not> i < DIM('a)"
then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
using top by auto
qed
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma halfspace_le_span_greaterThan:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x \$\$ i \<le> a})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
have "\<And>a i. {x. x\$\$i \<le> a} \<in> sets ?SIGMA"
proof cases
fix a i assume "i < DIM('a)"
have "{x::'a. x\$\$i \<le> a} = space ?SIGMA - {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 real_arch_lt[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> sets ?SIGMA"
apply (simp only:)
apply (safe intro!: countable_UN Diff)
by (auto simp: sets_sigma intro!: sigma_sets.Basic)
next
fix a i assume "\<not> i < DIM('a)"
then show "{x. x\$\$i \<le> a} \<in> sets ?SIGMA"
using top by auto
qed
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma halfspace_le_span_lessThan:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x \$\$ i})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
have "\<And>a i. {x. a \<le> x\$\$i} \<in> sets ?SIGMA"
proof cases
fix a i assume "i < DIM('a)"
have "{x::'a. a \<le> x\$\$i} = space ?SIGMA - {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 real_arch_lt[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> sets ?SIGMA"
apply (simp only:)
apply (safe intro!: countable_UN Diff)
by (auto simp: sets_sigma intro!: sigma_sets.Basic)
next
fix a i assume "\<not> i < DIM('a)"
then show "{x. a \<le> x\$\$i} \<in> sets ?SIGMA"
using top by auto
qed
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma atMost_span_atLeastAtMost:
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
(is "_ \<subseteq> sets ?SIGMA")
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
{ 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
have "{..a} \<in> sets ?SIGMA" unfolding *
by (safe intro!: countable_UN)
(auto simp: sets_sigma intro!: sigma_sets.Basic) }
then show ?thesis by (intro sets_sigma_subset) auto
qed

lemma borel_eq_atMost:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_atLeastAtMost:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using atMost_span_atLeastAtMost halfspace_le_span_atMost
halfspace_span_halfspace_le open_span_halfspace
by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_greaterThan:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_greaterThan
halfspace_span_halfspace_le open_span_halfspace
by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_lessThan:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_lessThan
halfspace_span_halfspace_ge open_span_halfspace
by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_greaterThanLessThan:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
show "sets borel \<subseteq> sets ?SIGMA"
proof -
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
then interpret sigma_algebra ?SIGMA .
{ fix M :: "'a set" assume "M \<in> open"
then have "open M" by (simp add: mem_def)
have "M \<in> sets ?SIGMA"
apply (subst open_UNION[OF `open M`])
apply (safe intro!: countable_UN)
by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
then show ?thesis
unfolding borel_def by (intro sets_sigma_subset) auto
qed
qed auto

lemma borel_eq_halfspace_le:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i \<le> a})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using open_span_halfspace halfspace_span_halfspace_le by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_halfspace_less:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x\$\$i < a})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using open_span_halfspace .
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_halfspace_gt:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x\$\$i})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma borel_eq_halfspace_ge:
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x\$\$i})\<rparr>)"
(is "_ = ?SIGMA")
proof (intro algebra.equality antisym)
show "sets borel \<subseteq> sets ?SIGMA"
using halfspace_span_halfspace_ge open_span_halfspace by auto
show "sets ?SIGMA \<subseteq> sets borel"
by (rule borel.sets_sigma_subset) auto
qed auto

lemma (in sigma_algebra) borel_measurable_halfspacesI:
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
and "\<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) sigma_def)
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 assms(3)[OF this] show ?thesis .
qed }
then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
by (auto intro!: measurable_sigma simp: assms(2))
then show "f \<in> borel_measurable M" unfolding measurable_def
unfolding assms(1) by simp
qed

lemma (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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_gt]) auto

lemma (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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"
unfolding borel_def[where 'a=real]
proof (rule borel.measurable_sigma, simp_all)
fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
from open_vimage_euclidean_component[OF this]
show "(\<lambda>x. x \$\$ i) -` S \<in> sets borel"
by (auto intro: borel_open)
qed

lemma (in sigma_algebra) 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 (in sigma_algebra) 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 "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")
by (auto intro!: open_affinity simp: scaleR.add_right mem_def)
hence "?S \<in> sets borel"
unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
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 (in sigma_algebra) 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)

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"
then show ?thesis
qed

lemma (in sigma_algebra) 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 (in sigma_algebra) 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)
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)
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 (in sigma_algebra) 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 (in sigma_algebra) 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))"
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
using 1 2 by simp
qed

lemma (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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 (in sigma_algebra) 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

section "Borel space over the real line with infinity"

lemma borel_Real_measurable:
"A \<in> sets borel \<Longrightarrow> Real -` A \<in> sets borel"
proof (rule borel_measurable_translate)
fix B :: "pextreal set" assume "open B"
then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and
x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"
unfolding open_pextreal_def by blast
have "Real -` B = Real -` (B - {\<omega>})" by auto
also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp
also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"
apply (auto simp add: Real_eq_Real image_iff)
apply (rule_tac x="max 0 x" in bexI)
by (auto simp: max_def)
finally show "Real -` B \<in> sets borel"
using `open T` by auto
qed simp

lemma borel_real_measurable:
"A \<in> sets borel \<Longrightarrow> (real -` A :: pextreal set) \<in> sets borel"
proof (rule borel_measurable_translate)
fix B :: "real set" assume "open B"
{ fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto }
note Ex_less_real = this
have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))"
by (force simp: Ex_less_real)

have "open (real -` (B \<inter> {0 <..}) :: pextreal set)"
unfolding open_pextreal_def using `open B`
by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real)
then show "(real -` B :: pextreal set) \<in> sets borel" unfolding * by auto
qed simp

lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
unfolding in_borel_measurable_borel
proof safe
fix S :: "pextreal set" assume "S \<in> sets borel"
from borel_Real_measurable[OF this]
have "(Real \<circ> f) -` S \<inter> space M \<in> sets M"
using assms
unfolding vimage_compose in_borel_measurable_borel
by auto
thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
qed

lemma (in sigma_algebra) borel_measurable_real[intro, simp]:
fixes f :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
unfolding in_borel_measurable_borel
proof safe
fix S :: "real set" assume "S \<in> sets borel"
from borel_real_measurable[OF this]
have "(real \<circ> f) -` S \<inter> space M \<in> sets M"
using assms
unfolding vimage_compose in_borel_measurable_borel
by auto
thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
qed

lemma (in sigma_algebra) borel_measurable_Real_eq:
assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
proof
have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}"
by auto
assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M"
by (rule borel_measurable_real)
moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x"
using assms by auto
ultimately show "f \<in> borel_measurable M"
by (simp cong: measurable_cong)
qed auto

lemma (in sigma_algebra) borel_measurable_pextreal_eq_real:
"f \<in> borel_measurable M \<longleftrightarrow>
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)"
proof safe
assume "f \<in> borel_measurable M"
then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
by (auto intro: borel_measurable_vimage borel_measurable_real)
next
assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto
with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp
have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))"
show "f \<in> borel_measurable M"
apply (subst f)
apply (rule measurable_If)
using * ** by auto
qed

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.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" and f: "f -` {c} \<inter> A \<in> sets borel"
shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
proof (rule borel.borel_measurableI)
fix S :: "'b set" assume "open S"
then have "open {x\<in>A. f x \<in> S - {c}}"
by (intro continuous_open_preimage[OF cont]) auto
then have *: "{x\<in>A. f x \<in> S - {c}} \<in> sets borel" by auto
show "?f -` S \<inter> space borel \<in> sets borel"
proof cases
assume "c \<in> S"
then have "?f -` S = {x\<in>A. f x \<in> S - {c}} \<union> (f -` {c} \<inter> A) \<union> -A"
by auto
with * show "?f -` S \<inter> space borel \<in> sets borel"
using `open A` f by (auto intro!: borel.Un)
next
assume "c \<notin> S"
then have "?f -` S = {x\<in>A. f x \<in> S - {c}}" by (auto split: split_if_asm)
with * show "?f -` S \<inter> space borel \<in> sets borel" by auto
qed
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
show "log b -` {log b 0} \<inter> {0<..} \<in> sets borel"
proof cases
assume "log b -` {log b 0} \<inter> {0<..} = {}"
then show ?thesis by simp
next
assume "log b -` {log b 0} \<inter> {0<..} \<noteq> {}"
then obtain x where "0 < x" "log b x = log b 0" by auto
with log_inj[OF `1 < b`] have "log b -` {log b 0} \<inter> {0<..} = {x}"
by (auto simp: inj_on_def)
then show ?thesis by simp
qed
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 (in sigma_algebra) 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`]]

lemma (in sigma_algebra) less_eq_ge_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M"
proof
assume "{x\<in>space M. f x \<le> a} \<in> sets M"
moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto
ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto
next
assume "{x\<in>space M. a < f x} \<in> sets M"
moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto
ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto
qed

lemma (in sigma_algebra) greater_eq_le_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M"
proof
assume "{x\<in>space M. a \<le> f x} \<in> sets M"
moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto
ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto
next
assume "{x\<in>space M. f x < a} \<in> sets M"
moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto
ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto
qed

lemma (in sigma_algebra) less_eq_le_pextreal_measurable:
fixes f :: "'a \<Rightarrow> pextreal"
shows "(\<forall>a. {x\<in>space M. a < f x} \<in> sets M) \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
proof
assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M"
show "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
proof
fix a show "{x \<in> space M. a < f x} \<in> sets M"
proof (cases a)
case (preal r)
have "{x\<in>space M. a < f x} = (\<Union>i. {x\<in>space M. a + inverse (of_nat (Suc i)) \<le> f x})"
proof safe
fix x assume "a < f x" and [simp]: "x \<in> space M"
with ex_pextreal_inverse_of_nat_Suc_less[of "f x - a"]
obtain n where "a + inverse (of_nat (Suc n)) < f x"
by (cases "f x", auto simp: pextreal_minus_order)
then have "a + inverse (of_nat (Suc n)) \<le> f x" by simp
then show "x \<in> (\<Union>i. {x \<in> space M. a + inverse (of_nat (Suc i)) \<le> f x})"
by auto
next
fix i x assume [simp]: "x \<in> space M"
have "a < a + inverse (of_nat (Suc i))" using preal by auto
also assume "a + inverse (of_nat (Suc i)) \<le> f x"
finally show "a < f x" .
qed
with a show ?thesis by auto
qed simp
qed
next
assume a': "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
then have a: "\<forall>a. {x \<in> space M. f x \<le> a} \<in> sets M" unfolding less_eq_ge_measurable .
show "\<forall>a. {x \<in> space M. a \<le> f x} \<in> sets M" unfolding greater_eq_le_measurable[symmetric]
proof
fix a show "{x \<in> space M. f x < a} \<in> sets M"
proof (cases a)
case (preal r)
show ?thesis
proof cases
assume "a = 0" then show ?thesis by simp
next
assume "a \<noteq> 0"
have "{x\<in>space M. f x < a} = (\<Union>i. {x\<in>space M. f x \<le> a - inverse (of_nat (Suc i))})"
proof safe
fix x assume "f x < a" and [simp]: "x \<in> space M"
with ex_pextreal_inverse_of_nat_Suc_less[of "a - f x"]
obtain n where "inverse (of_nat (Suc n)) < a - f x"
using preal by (cases "f x") auto
then have "f x \<le> a - inverse (of_nat (Suc n)) "
using preal by (cases "f x") (auto split: split_if_asm)
then show "x \<in> (\<Union>i. {x \<in> space M. f x \<le> a - inverse (of_nat (Suc i))})"
by auto
next
fix i x assume [simp]: "x \<in> space M"
assume "f x \<le> a - inverse (of_nat (Suc i))"
also have "\<dots> < a" using `a \<noteq> 0` preal by auto
finally show "f x < a" .
qed
with a show ?thesis by auto
qed
next
case infinite
have "f -` {\<omega>} \<inter> space M = (\<Inter>n. {x\<in>space M. of_nat n < f x})"
proof (safe, simp_all, safe)
fix x assume *: "\<forall>n::nat. Real (real n) < f x"
show "f x = \<omega>"    proof (rule ccontr)
assume "f x \<noteq> \<omega>"
with real_arch_lt[of "real (f x)"] obtain n where "f x < of_nat n"
by (auto simp: pextreal_noteq_omega_Ex)
with *[THEN spec, of n] show False by auto
qed
qed
with a' have \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" by auto
moreover have "{x \<in> space M. f x < a} = space M - f -` {\<omega>} \<inter> space M"
using infinite by auto
ultimately show ?thesis by auto
qed
qed
qed

lemma (in sigma_algebra) borel_measurable_pextreal_iff_greater:
"(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)"
proof safe
fix a assume f: "f \<in> borel_measurable M"
have "{x\<in>space M. a < f x} = f -` {a <..} \<inter> space M" by auto
with f show "{x\<in>space M. a < f x} \<in> sets M"
by (auto intro!: measurable_sets)
next
assume *: "\<forall>a. {x\<in>space M. a < f x} \<in> sets M"
hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M"
unfolding less_eq_le_pextreal_measurable
unfolding greater_eq_le_measurable .
show "f \<in> borel_measurable M" unfolding borel_measurable_pextreal_eq_real borel_measurable_iff_greater
proof safe
have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto
then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto
fix a
have "{w \<in> space M. a < real (f w)} =
(if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)"
proof (split split_if, safe del: notI)
fix x assume "0 \<le> a"
{ assume "a < real (f x)" then show "Real a < f x" "x \<notin> f -` {\<omega>} \<inter> space M"
using `0 \<le> a` by (cases "f x", auto) }
{ assume "Real a < f x" "x \<notin> f -` {\<omega>}" then show "a < real (f x)"
using `0 \<le> a` by (cases "f x", auto) }
next
fix x assume "\<not> 0 \<le> a" then show "a < real (f x)" by (cases "f x") auto
qed
then show "{w \<in> space M. a < real (f w)} \<in> sets M"
using \<omega> * by (auto intro!: Diff)
qed
qed

lemma (in sigma_algebra) borel_measurable_pextreal_iff_less:
"(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)"
using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable greater_eq_le_measurable .

lemma (in sigma_algebra) borel_measurable_pextreal_iff_le:
"(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)"
using borel_measurable_pextreal_iff_greater unfolding less_eq_ge_measurable .

lemma (in sigma_algebra) borel_measurable_pextreal_iff_ge:
"(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable .

lemma (in sigma_algebra) borel_measurable_pextreal_eq_const:
fixes f :: "'a \<Rightarrow> pextreal" 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 (in sigma_algebra) borel_measurable_pextreal_neq_const:
fixes f :: "'a \<Rightarrow> pextreal"
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 (in sigma_algebra) borel_measurable_pextreal_less[intro,simp]:
fixes f g :: "'a \<Rightarrow> pextreal"
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 "(\<lambda>x. real (f x)) \<in> borel_measurable M"
"(\<lambda>x. real (g x)) \<in> borel_measurable M"
using assms by (auto intro!: borel_measurable_real)
from borel_measurable_less[OF this]
have "{x \<in> space M. real (f x) < real (g x)} \<in> sets M" .
moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pextreal_neq_const)
moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_eq_const)
moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_neq_const)
moreover have "{x \<in> space M. f x < g x} = ({x \<in> space M. g x = \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>}) \<union>
({x \<in> space M. g x \<noteq> \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>} \<inter> {x \<in> space M. real (f x) < real (g x)})"
by (auto simp: real_of_pextreal_strict_mono_iff)
ultimately show ?thesis by auto
qed

lemma (in sigma_algebra) borel_measurable_pextreal_le[intro,simp]:
fixes f :: "'a \<Rightarrow> pextreal"
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} = space M - {x \<in> space M. g x < f x}" by auto
then show ?thesis using g f by auto
qed

lemma (in sigma_algebra) borel_measurable_pextreal_eq[intro,simp]:
fixes f :: "'a \<Rightarrow> pextreal"
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 (in sigma_algebra) borel_measurable_pextreal_neq[intro,simp]:
fixes f :: "'a \<Rightarrow> pextreal"
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

fixes f :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
proof -
have *: "(\<lambda>x. f x + g x) =
(\<lambda>x. if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))"
by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
show ?thesis using assms unfolding *
by (auto intro!: measurable_If)
qed

lemma (in sigma_algebra) borel_measurable_pextreal_setsum[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
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

lemma (in sigma_algebra) borel_measurable_pextreal_times[intro, simp]:
fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
proof -
have *: "(\<lambda>x. f x * g x) =
(\<lambda>x. if f x = 0 then 0 else if g x = 0 then 0 else if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else
Real (real (f x) * real (g x)))"
by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
show ?thesis using assms unfolding *
by (auto intro!: measurable_If)
qed

lemma (in sigma_algebra) borel_measurable_pextreal_setprod[simp, intro]:
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
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 (in sigma_algebra) borel_measurable_pextreal_min[simp, intro]:
fixes f g :: "'a \<Rightarrow> pextreal"
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 (in sigma_algebra) borel_measurable_pextreal_max[simp, intro]:
fixes f g :: "'a \<Rightarrow> pextreal"
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 (in sigma_algebra) borel_measurable_SUP[simp, intro]:
fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pextreal"
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_pextreal_iff_greater
proof safe
fix a
have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
by (auto simp: less_SUP_iff SUPR_apply)
then show "{x\<in>space M. a < ?sup x} \<in> sets M"
using assms by auto
qed

lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pextreal"
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_pextreal_iff_less
proof safe
fix a
have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
by (auto simp: INF_less_iff INFI_apply)
then show "{x\<in>space M. ?inf x < a} \<in> sets M"
using assms by auto
qed

lemma (in sigma_algebra) borel_measurable_pextreal_diff[simp, intro]:
fixes f g :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
unfolding borel_measurable_pextreal_iff_greater
proof safe
fix a
have "{x \<in> space M. a < f x - g x} = {x \<in> space M. g x + a < f x}"
then show "{x \<in> space M. a < f x - g x} \<in> sets M"
using assms by auto
qed

lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
assumes "\<And>i. f i \<in> borel_measurable M"
shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
using assms unfolding psuminf_def by auto

section "LIMSEQ is borel measurable"

lemma (in sigma_algebra) 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 -
let "?pu x i" = "max (u i x) 0"
let "?nu x i" = "max (- u i x) 0"
{ fix x assume x: "x \<in> space M"
have "(?pu x) ----> max (u' x) 0"
"(?nu x) ----> max (- u' x) 0"
using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
"(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
note eq = this
have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
by auto
have "(\<lambda>x. SUP n. INF m. Real (u (n + m) x)) \<in> borel_measurable M"
"(\<lambda>x. SUP n. INF m. Real (- u (n + m) x)) \<in> borel_measurable M"
using u by auto
with eq[THEN measurable_cong, of M "\<lambda>x. x" borel]
have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
"(\<lambda>x. Real (- u' x)) \<in> borel_measurable M" by auto
note this[THEN borel_measurable_real]
from borel_measurable_diff[OF this]
show ?thesis unfolding * .
qed

end
```