# Theory Borel_Space

```(*  Title:      HOL/Analysis/Borel_Space.thy
Author:     Johannes Hölzl, TU München
Author:     Armin Heller, TU München
*)

section ‹Borel Space›

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<..<M::ereal})"
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} = ((λ(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' = "(λy. y - x) ` interior A"
from deriv show "((λh. (f (x + h) - f x) / h) ⤏ D) (at 0)"
from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset)

show "eventually (λh. (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 ‹x ∈ interior A› show "0 ∈ ?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 ‹mono_on f A› by (simp add: mono_on_def)
have "∀a ∈ {a∈A. ¬ continuous (at a within A) f}. ∃q :: nat × rat.
(fst q = 0 ∧ of_rat (snd q) < f a ∧ (∀x ∈ A. x < a ⟶ f x < of_rat (snd q))) ∨
(fst q = 1 ∧ of_rat (snd q) > f a ∧ (∀x ∈ A. x > a ⟶ 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 ∧ real_of_rat q2 < f a ∧ (∀x∈A. x < a ⟶ f x < real_of_rat q2) ∨
q1 = 1 ∧ f a < real_of_rat q2 ∧ (∀x∈A. a < x ⟶ 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 (subgoal_tac "f x ≥ f xa")
by (auto intro: mono)
qed
thus ?thesis by auto
qed
qed
then obtain g :: "real ⇒ nat × rat" where "∀a ∈ {a∈A. ¬ continuous (at a within A) f}.
(fst (g a) = 0 ∧ of_rat (snd (g a)) < f a ∧ (∀x ∈ A. x < a ⟶ f x < of_rat (snd (g a)))) |
(fst (g a) = 1 ∧ of_rat (snd (g a)) > f a ∧ (∀x ∈ A. x > a ⟶ f x > of_rat (snd (g a))))"
by (rule bchoice [THEN exE]) blast
hence g: "⋀a x. a ∈ A ⟹ ¬ continuous (at a within A) f ⟹ x ∈ A ⟹
(fst (g a) = 0 ∧ of_rat (snd (g a)) < f a ∧ (x < a ⟶ f x < of_rat (snd (g a)))) |
(fst (g a) = 1 ∧ of_rat (snd (g a)) > f a ∧ (x > a ⟶ f x > of_rat (snd (g a))))"
by auto
have "inj_on g {a∈A. ¬ continuous (at a within A) f}"
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 A›])
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 ‹Generic Borel spaces›

definition✐‹tag important› (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 ⟷
(∀S ∈ sigma_sets UNIV {S. open S}. f -` S ∩ space M ∈ sets M)"
by (auto simp add: measurable_def borel_def)

lemma in_borel_measurable_borel:
"f ∈ borel_measurable M ⟷
(∀S ∈ sets borel.
f -` S ∩ space M ∈ 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:
"(λx. 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 "(λx. 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 ∩ space M) :: 'a ⇒ 'x) ∈ 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 Ω[measurable, simp]: "Ω ∩ space M ∈ sets M"
shows "f ∈ borel_measurable (restrict_space M Ω) ⟷
(λx. f x * indicator Ω x) ∈ borel_measurable M"
by (subst measurable_restrict_space_iff)
(auto simp: indicator_def of_bool_def if_distrib[where f="λx. a * x" for a] cong del: if_weak_cong)

lemma borel_measurable_restrict_space_iff_ennreal:
fixes f :: "'a ⇒ ennreal"
assumes Ω[measurable, simp]: "Ω ∩ space M ∈ sets M"
shows "f ∈ borel_measurable (restrict_space M Ω) ⟷
(λx. f x * indicator Ω x) ∈ borel_measurable M"
by (subst measurable_restrict_space_iff)
(auto simp: indicator_def of_bool_def if_distrib[where f="λx. a * x" for a] cong del: if_weak_cong)

lemma borel_measurable_restrict_space_iff:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes Ω[measurable, simp]: "Ω ∩ space M ∈ sets M"
shows "f ∈ borel_measurable (restrict_space M Ω) ⟷
(λx. indicator Ω x *⇩R f x) ∈ borel_measurable M"
by (subst measurable_restrict_space_iff)
(auto simp: indicator_def of_bool_def if_distrib[where f="λx. x *⇩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 ((λ(i, j). G i j)`B)"
assumes X: "⋀i j. (i, j) ∈ B ⟹ G i j ∈ sets (sigma UNIV ((λ(i, j). F i j) ` A))"
assumes F: "⋀i j. (i, j) ∈ A ⟹ F i j ∈ sets borel"
shows "borel = sigma UNIV ((λ(i, j). F i j) ` A)"
using assms
by (intro borel_eq_sigmaI1[where X="(λ(i, j). G i j) ` B" and F="(λ(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 ((λ(i, j). F i j) ` A))"
assumes F: "⋀i j. (i, j) ∈ A ⟹ F i j ∈ sets borel"
shows "borel = sigma UNIV ((λ(i, j). F i j) ` A)"
using assms by (intro borel_eq_sigmaI1[where X=X and F="(λ(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 ((λ(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="(λ(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 (λ(i, j). F i j)))"
assumes F: "⋀i j. F i j ∈ sets borel"
shows "borel = sigma UNIV (range (λ(i, j). F i j))"
using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(λ(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 _ ‹countable B›]) 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 ‹⋃U = ⋃K› 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 x›)
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: ‹closed x›)
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 ‹countable B› 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 ⟹
(λx. if x ∈ A then f x else g x) ∈ 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 "(λx. 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 "(λx. 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 ⟹ (λx. indicator A x *⇩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 "(λx. (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 = "(λ(b, c). b × c) ` (?B × ?C)"
show "countable ?P" "topological_basis ?P"
by (auto intro!: countable_basis topological_basis_prod is_basis)

show "(λx. (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 "(λx. (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 "(λx. (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 (λx. H (fst x) (snd x))"
shows "(λx. H (f x) (g x)) ∈ borel_measurable M"
proof -
have eq: "(λx. H (f x) (g x)) = (λx. (λx. 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 ‹Borel spaces on order topologies›

lemma [measurable]:
fixes a b :: "'a::linorder_topology"
shows lessThan_borel: "{..< a} ∈ sets borel"
and greaterThan_borel: "{a <..} ∈ sets borel"
and greaterThanLessThan_borel: "{a<..<b} ∈ 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..<b} ∈ 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)
obtain D :: "'a set" where D: "countable D" "⋀X. open X ⟹ X ≠ {} ⟹ ∃d∈D. d ∈ X"
by (rule countable_dense_setE) blast

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)
obtain D :: "'a set" where D: "countable D" "⋀X. open X ⟹ X ≠ {} ⟹ ∃d∈D. d ∈ X"
by (rule countable_dense_setE) blast

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 "∀x<y. ∃d. x < d ∧ d < y"
with D(2)[of "{x <..< y}" for x] have "∀x<y. ∃d∈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 "¬ (∀x<y. ∃d. 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} = (λx. (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 "(λx. 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 "(λx. 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 ((λi. F i x) ` I)"
shows "(λx. SUP i∈I. F i x) ∈ borel_measurable M"
proof cases
assume "I = {}" then show ?thesis
unfolding ‹I = {}› 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 ‹I ≠ {}› 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 ((λi. F i x) ` I)"
shows "(λx. INF i∈I. F i x) ∈ borel_measurable M"
proof cases
assume "I = {}" then show ?thesis
unfolding ‹I = {}› 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 ‹I ≠ {}› 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 "(λx. SUP i. (F ^^ i) bot x) ∈ borel_measurable M"
by measurable
also have "(λx. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)"
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 "(λx. INF i. (F ^^ i) top x) ∈ borel_measurable M"
by measurable
also have "(λx. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)"
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 ⟹ (λx. 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 ⟹ (λx. 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) ⟹ (λx. Min ((λi. 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) ⟹ (λx. Max ((λi. 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 ⟹ (λx. 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 ⟹ (λx. 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: "(λx. liminf (λi. f i x)) ∈ borel_measurable M"
and borel_measurable_limsup: "(λx. limsup (λi. 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 (λx. convergent (λi. 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 (λi. 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 "(λx. lim (λi. f i x)) ∈ borel_measurable M"
proof -
have "⋀x. lim (λi. f i x) = (if convergent (λi. f i x) then limsup (λi. 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 ⟹ (λi. 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 (λn. u n x)"
using u' by (simp add: lim_imp_Liminf[symmetric])
with u show ?thesis by (simp cong: measurable_cong)
qed

subsection ‹Borel spaces on topological monoids›

fixes f g :: "'a ⇒ 'b::{second_countable_topology, topological_monoid_add}"
assumes f: "f ∈ borel_measurable M"
assumes g: "g ∈ borel_measurable M"
shows "(λx. 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 "(λx. ∑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 "(λx. suminf (λi. f i x)) ∈ borel_measurable M"
unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp

subsection ‹Borel spaces on Euclidean spaces›

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 "(λx. f x ∙ g x) ∈ borel_measurable M"
using assms
by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

notation
eucl_less (infix "<e" 50)

lemma box_oc: "{x. a <e x ∧ x ≤ b} = {x. a <e x} ∩ {..b}"
and box_co: "{x. a ≤ x ∧ x <e b} = {a..} ∩ {x. x <e b}"
by auto

lemma eucl_ivals[measurable]:
fixes a b :: "'a::ordered_euclidean_space"
shows "{x. x <e a} ∈ sets borel"
and "{x. a <e x} ∈ sets borel"
and "{..a} ∈ sets borel"
and "{a..} ∈ sets borel"
and "{a..b} ∈ sets borel"
and  "{x. a <e x ∧ x ≤ b} ∈ sets borel"
and "{x. a ≤ x ∧  x <e b} ∈ 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 (λ (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 M›])
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 ((λ (a, i). {x::'a::euclidean_space. x ∙ i < a}) ` (UNIV × 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 ((λ(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 ((λ (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 ((λ (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 ((λ (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 (λa. {..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)*⇩R n)})"
proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm)
fix x :: 'a
obtain k where "Max ((∙) x ` Basis) ≤ real k"
using real_arch_simple by blast
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 (λa::'a::ordered_euclidean_space. {x. a <e x}))"
(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} =
(⋃k::nat. {x. (∑n∈Basis. (if n = i then a else -real k) *⇩R n) <e x})" using i
proof (safe, simp_all add: eucl_less_def split: if_split_asm)
fix x :: 'a
obtain k where k: "Max ((∙) (- x) ` Basis) < real k"
using reals_Archimedean2 by blast
{ 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 (λa::'a::ordered_euclidean_space. {x. x <e a}))"
(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 <e (∑n∈Basis. (if n = i then a else real k) *⇩R n)})" using ‹i∈ Basis›
proof (safe, simp_all add: eucl_less_def split: if_split_asm)
fix x :: 'a
obtain k where k: "Max ((∙) x ` Basis) < real k"
using reals_Archimedean2 by blast
{ 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 (λ(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 *⇩R One .. a})"
proof (safe, simp_all add: eucl_le[where 'a='a])
fix x :: 'a
obtain k where k: "Max ((∙) (- x) ` Basis) ≤ real k"
using real_arch_simple by blast
{ 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 (λ(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 (λ(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 (λ(i, j). {i<..j})))"
by (intro sets.countable_nat_UN) auto
finally show "{..i} ∈ sets (sigma UNIV (range (λ(i, j). {i<..j})))" .
qed simp

lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"

lemma borel_eq_atLeastLessThan:
"borel = sigma UNIV (range (λ(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 "{..<x} = (⋃i::nat. {-real i ..< x})"
by (auto simp: move_uminus real_arch_simple)
then show "{y. y <e x} ∈ ?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. (λx. f x ∙ i) ∈ borel_measurable M)"
proof safe
assume f: "∀i∈Basis. (λx. 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 "(λx. - 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 "(λx. 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 "(λx. 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 "(λx. ∏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 "(λx. 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 "(λx. f x *⇩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 "(λx. - 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 "(λx. a + b *⇩R f x) ∈ borel_measurable M"
proof (rule borel_measurableI)
fix S :: "'x set" assume "open S"
show "(λx. a + b *⇩R f x) -` S ∩ space M ∈ sets M"
proof cases
assume "b ≠ 0"
with ‹open S› have "open ((λx. (- a + x) /⇩R b) ` S)" (is "open ?S")
using open_affinity [of S "inverse b" "- a /⇩R b"]
by (auto simp: algebra_simps)
hence "?S ∈ sets borel" by auto
moreover
from ‹b ≠ 0› have "(λx. a + b *⇩R f x) -` S = f -` ?S"
apply auto by (rule_tac x="a + b *⇩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 ⟹ (λx. b *⇩R f x ::'a::real_normed_vector) ∈ borel_measurable M"
using affine_borel_measurable_vector[of f M 0 b] by simp

"f ∈ borel_measurable M ⟹ (λx. 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 "(λx. 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 ⟹
(λx. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) ∈ borel_measurable M"