Theory Borel_Space

theory Borel_Space
imports Measurable Multivariate_Analysis
(*  Title:      HOL/Probability/Borel_Space.thy
    Author:     Johannes Hölzl, TU München
    Author:     Armin Heller, TU München
*)

section ‹Borel spaces›

theory Borel_Space
imports
  Measurable
  "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
begin

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 open_Collect_less:
  fixes f g :: "'i::topological_space ⇒ 'a :: {dense_linorder, linorder_topology}"
  assumes "continuous_on UNIV f"
  assumes "continuous_on UNIV g"
  shows "open {x. f x < g x}"
proof -
  have "open (⋃y. {x ∈ UNIV. f x ∈ {..< y}} ∩ {x ∈ UNIV. g x ∈ {y <..}})" (is "open ?X")
    by (intro open_UN ballI open_Int continuous_open_preimage assms) auto
  also have "?X = {x. f x < g x}"
    by (auto intro: dense)
  finally show ?thesis .
qed

lemma closed_Collect_le:
  fixes f g :: "'i::topological_space ⇒ 'a :: {dense_linorder, linorder_topology}"
  assumes f: "continuous_on UNIV f"
  assumes g: "continuous_on UNIV g"
  shows "closed {x. f x ≤ g x}"
  using open_Collect_less[OF g f] unfolding not_less[symmetric] Collect_neg_eq open_closed .

lemma topological_basis_trivial: "topological_basis {A. open A}"
  by (auto simp: topological_basis_def)

lemma 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

definition "mono_on f A ≡ ∀r s. r ∈ A ∧ s ∈ A ∧ r ≤ s ⟶ f r ≤ f s"

lemma mono_onI:
  "(⋀r s. r ∈ A ⟹ s ∈ A ⟹ r ≤ s ⟹ f r ≤ f s) ⟹ mono_on f A"
  unfolding mono_on_def by simp

lemma mono_onD:
  "⟦mono_on f A; r ∈ A; s ∈ A; r ≤ s⟧ ⟹ f r ≤ f s"
  unfolding mono_on_def by simp

lemma mono_imp_mono_on: "mono f ⟹ mono_on f A"
  unfolding mono_def mono_on_def by auto

lemma mono_on_subset: "mono_on f A ⟹ B ⊆ A ⟹ mono_on f B"
  unfolding mono_on_def by auto

definition "strict_mono_on f A ≡ ∀r s. r ∈ A ∧ s ∈ A ∧ r < s ⟶ f r < f s"

lemma strict_mono_onI:
  "(⋀r s. r ∈ A ⟹ s ∈ A ⟹ r < s ⟹ f r < f s) ⟹ strict_mono_on f A"
  unfolding strict_mono_on_def by simp

lemma strict_mono_onD:
  "⟦strict_mono_on f A; r ∈ A; s ∈ A; r < s⟧ ⟹ f r < f s"
  unfolding strict_mono_on_def by simp

lemma mono_on_greaterD:
  assumes "mono_on g A" "x ∈ A" "y ∈ A" "g x > (g (y::_::linorder) :: _ :: linorder)"
  shows "x > y"
proof (rule ccontr)
  assume "¬x > y"
  hence "x ≤ y" by (simp add: not_less)
  from assms(1-3) and this have "g x ≤ g y" by (rule mono_onD)
  with assms(4) show False by simp
qed

lemma strict_mono_inv:
  fixes f :: "('a::linorder) ⇒ ('b::linorder)"
  assumes "strict_mono f" and "surj f" and inv: "⋀x. g (f x) = x"
  shows "strict_mono g"
proof
  fix x y :: 'b assume "x < y"
  from ‹surj f› obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
  with ‹x < y› and ‹strict_mono f› have "x' < y'" by (simp add: strict_mono_less)
  with inv show "g x < g y" by simp
qed

lemma strict_mono_on_imp_inj_on:
  assumes "strict_mono_on (f :: (_ :: linorder) ⇒ (_ :: preorder)) A"
  shows "inj_on f A"
proof (rule inj_onI)
  fix x y assume "x ∈ A" "y ∈ A" "f x = f y"
  thus "x = y"
    by (cases x y rule: linorder_cases)
       (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
qed

lemma strict_mono_on_leD:
  assumes "strict_mono_on (f :: (_ :: linorder) ⇒ _ :: preorder) A" "x ∈ A" "y ∈ A" "x ≤ y"
  shows "f x ≤ f y"
proof (insert le_less_linear[of y x], elim disjE)
  assume "x < y"
  with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all
  thus ?thesis by (rule less_imp_le)
qed (insert assms, simp)

lemma strict_mono_on_eqD:
  fixes f :: "(_ :: linorder) ⇒ (_ :: preorder)"
  assumes "strict_mono_on f A" "f x = f y" "x ∈ A" "y ∈ A"
  shows "y = x"
  using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)

lemma 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_le_const)
  let ?A' = "(λy. y - x) ` interior A"
  from deriv show "((λh. (f (x + h) - f x) / h) ⤏ D) (at 0)"
      by (simp add: field_has_derivative_at has_field_derivative_def)
  from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset)

  show "eventually (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 (op + (-x) ` interior A)" by (rule open_translation)
    also have "(op + (-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

lemma strict_mono_on_imp_mono_on:
  "strict_mono_on (f :: (_ :: linorder) ⇒ _ :: preorder) A ⟹ mono_on f A"
  by (rule mono_onI, rule strict_mono_on_leD)

lemma 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 (auto simp add: dist_real_def)
          apply (subgoal_tac "f x ≥ f xa")
          by (auto intro: mono)
      qed
      thus ?thesis by auto
    qed
  qed
  hence "∃g :: real ⇒ nat × rat . ∀a ∈ {a∈A. ¬ continuous (at a within A) f}.
      (fst (g a) = 0 ∧ 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 guess g ..
  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}"
  proof (auto simp add: inj_on_def)
    fix w z
    assume 1: "w ∈ A" and 2: "¬ continuous (at w within A) f" and
           3: "z ∈ A" and 4: "¬ continuous (at z within A) f" and
           5: "g w = g z"
    from g [OF 1 2 3] g [OF 3 4 1] 5
    show "w = z" by auto
  qed
  thus ?thesis
    by (rule countableI')
qed

lemma mono_on_ctble_discont_open:
  fixes f :: "real ⇒ real"
  fixes A :: "real set"
  assumes "open A" "mono_on f A"
  shows "countable {a∈A. ¬isCont f a}"
proof -
  have "{a∈A. ¬isCont f a} = {a∈A. ¬(continuous (at a within A) f)}"
    by (auto simp add: continuous_within_open [OF _ `open 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)
  apply (intro ballI Deriv.differentiableI)
  apply (rule has_field_derivative_subset[OF assms])
  apply simp_all
  done

lemma closure_contains_Sup:
  fixes S :: "real set"
  assumes "S ≠ {}" "bdd_above S"
  shows "Sup S ∈ closure S"
proof-
  have "Inf (uminus ` S) ∈ closure (uminus ` S)"
      using assms by (intro closure_contains_Inf) auto
  also have "Inf (uminus ` S) = -Sup S" by (simp add: Inf_real_def)
  also have "closure (uminus ` S) = uminus ` closure S"
      by (rule sym, intro closure_injective_linear_image) (auto intro: linearI)
  finally show ?thesis by auto
qed

lemma closed_contains_Sup:
  fixes S :: "real set"
  shows "S ≠ {} ⟹ bdd_above S ⟹ closed S ⟹ Sup S ∈ S"
  by (subst closure_closed[symmetric], assumption, rule closure_contains_Sup)

lemma deriv_nonneg_imp_mono:
  assumes deriv: "⋀x. x ∈ {a..b} ⟹ (g has_real_derivative g' x) (at x)"
  assumes nonneg: "⋀x. x ∈ {a..b} ⟹ g' x ≥ 0"
  assumes ab: "a ≤ b"
  shows "g a ≤ g b"
proof (cases "a < b")
  assume "a < b"
  from deriv have "∀x. x ≥ a ∧ x ≤ b ⟶ (g has_real_derivative g' x) (at x)" by simp
  from MVT2[OF ‹a < b› this] and deriv
    obtain ξ where ξ_ab: "ξ > a" "ξ < b" and g_ab: "g b - g a = (b - a) * g' ξ" by blast
  from ξ_ab ab nonneg have "(b - a) * g' ξ ≥ 0" by simp
  with g_ab show ?thesis by simp
qed (insert ab, simp)

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

    def c'  "Inf ?A" and 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 (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 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 if_distrib[where f="λx. a * x" for a] cong del: if_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 if_distrib[where f="λx. a * x" for a] cong del: if_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 if_distrib[where f="λx. x *R a" for a] ac_simps cong del: if_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

lemma 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
    def 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

lemma 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 using assms by unfold_locales
  fix X::"'a set" assume "open X"
  from open_countable_basisE[OF this] guess B' . note B' = this
  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_on1: "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_on1[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)
  from countable_dense_setE guess D :: "'a set" . note D = this

  interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)"
    by (rule sigma_algebra_sigma_sets) simp

  fix A :: "'a set" assume "A ∈ range lessThan ∪ range greaterThan"
  then obtain y where "A = {y <..} ∨ A = {..< y}"
    by blast
  then show "A ∈ sigma_sets UNIV (range lessThan)"
  proof
    assume A: "A = {y <..}"
    show ?thesis
    proof cases
      assume "∀x>y. ∃d. y < d ∧ d < x"
      with D(2)[of "{y <..< x}" for x] have "∀x>y. ∃d∈D. y < d ∧ d < x"
        by (auto simp: set_eq_iff)
      then have "A = UNIV - (⋂d∈{d∈D. y < d}. {..< d})"
        by (auto simp: A) (metis less_asym)
      also have "… ∈ sigma_sets UNIV (range lessThan)"
        using D(1) by (intro L.Diff L.top L.countable_INT'') auto
      finally show ?thesis .
    next
      assume "¬ (∀x>y. ∃d. y < d ∧ d < x)"
      then obtain x where "y < x"  "⋀d. y < d ⟹ ¬ d < x"
        by auto
      then have "A = UNIV - {..< x}"
        unfolding A by (auto simp: not_less[symmetric])
      also have "… ∈ sigma_sets UNIV (range lessThan)"
        by auto
      finally show ?thesis .
    qed
  qed auto
qed auto

lemma borel_Ioi:
  "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
  unfolding second_countable_borel_measurable[OF open_generated_order]
proof (intro sigma_eqI sigma_sets_eqI)
  from countable_dense_setE guess D :: "'a set" . note D = this

  interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)"
    by (rule sigma_algebra_sigma_sets) simp

  fix A :: "'a set" assume "A ∈ range lessThan ∪ range greaterThan"
  then obtain y where "A = {y <..} ∨ A = {..< y}"
    by blast
  then show "A ∈ sigma_sets UNIV (range greaterThan)"
  proof
    assume A: "A = {..< y}"
    show ?thesis
    proof cases
      assume "∀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, dense_linorder, 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, dense_linorder, 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)"
    by auto
  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)"
    by auto
  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, dense_linorder, 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, dense_linorder, 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, dense_linorder, 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, dense_linorder, 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›

lemma borel_measurable_add[measurable (raw)]:
  fixes f g :: "'a ⇒ 'b::{second_countable_topology, topological_monoid_add}"
  assumes f: "f ∈ borel_measurable M"
  assumes g: "g ∈ borel_measurable M"
  shows "(λx. f x + g x) ∈ borel_measurable M"
  using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)

lemma borel_measurable_setsum[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, dense_linorder, 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
    from real_arch_simple[of "Max ((λi. x∙i)`Basis)"] guess k::nat ..
    then have "⋀i. i ∈ Basis ⟹ x∙i ≤ real k"
      by (subst (asm) Max_le_iff) auto
    then show "∃k::nat. ∀ia∈Basis. ia ≠ i ⟶ x ∙ ia ≤ real k"
      by (auto intro!: exI[of _ k])
  qed
  show "{x. x∙i ≤ a} ∈ ?SIGMA" unfolding *
    by (intro sets.countable_UN) auto
qed auto

lemma borel_eq_greaterThan:
  "borel = sigma UNIV (range (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
    from reals_Archimedean2[of "Max ((λi. -x∙i)`Basis)"]
    guess k::nat .. note k = this
    { fix i :: 'a assume "i ∈ Basis"
      then have "-x∙i < real k"
        using k by (subst (asm) Max_less_iff) auto
      then have "- real k < x∙i" by simp }
    then show "∃k::nat. ∀ia∈Basis. ia ≠ i ⟶ -real k < x ∙ ia"
      by (auto intro!: exI[of _ k])
  qed
  finally show "{x. x∙i ≤ a} ∈ ?SIGMA"
    apply (simp only:)
    apply (intro sets.countable_UN sets.Diff)
    apply (auto intro: sigma_sets_top)
    done
qed auto

lemma borel_eq_lessThan:
  "borel = sigma UNIV (range (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
    from reals_Archimedean2[of "Max ((λi. x∙i)`Basis)"]
    guess k::nat .. note k = this
    { fix i :: 'a assume "i ∈ Basis"
      then have "x∙i < real k"
        using k by (subst (asm) Max_less_iff) auto
      then have "x∙i < real k" by simp }
    then show "∃k::nat. ∀ia∈Basis. ia ≠ i ⟶ x ∙ ia < real k"
      by (auto intro!: exI[of _ k])
  qed
  finally show "{x. a ≤ x∙i} ∈ ?SIGMA"
    apply (simp only:)
    apply (intro sets.countable_UN sets.Diff)
    apply (auto intro: sigma_sets_top )
    done
qed auto

lemma borel_eq_atLeastAtMost:
  "borel = sigma UNIV (range (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
    from real_arch_simple[of "Max ((λi. - x∙i)`Basis)"]
    guess k::nat .. note k = this
    { fix i :: 'a assume "i ∈ Basis"
      with k have "- x∙i ≤ real k"
        by (subst (asm) Max_le_iff) (auto simp: field_simps)
      then have "- real k ≤ x∙i" by simp }
    then show "∃n::nat. ∀i∈Basis. - real n ≤ x ∙ i"
      by (auto intro!: exI[of _ k])
  qed
  show "{..a} ∈ ?SIGMA" unfolding *
    by (intro sets.countable_UN)
       (auto intro!: sigma_sets_top)
qed auto

lemma borel_set_induct[consumes 1, case_names empty interval compl union]:
  assumes "A ∈ sets borel"
  assumes empty: "P {}" and int: "⋀a b. a ≤ b ⟹ P {a..b}" and compl: "⋀A. A ∈ sets borel ⟹ P A ⟹ P (-A)" and
          un: "⋀f. disjoint_family f ⟹ (⋀i. f i ∈ sets borel) ⟹  (⋀i. P (f i)) ⟹ P (⋃i::nat. f i)"
  shows "P (A::real set)"
proof-
  let ?G = "range (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"
  by (simp add: eucl_less_def lessThan_def)

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_on1 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_setprod[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 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

lemma borel_measurable_const_add[measurable (raw)]:
  "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"
  by (simp add: divide_inverse)

lemma borel_measurable_abs[measurable (raw)]:
  "f ∈ borel_measurable M ⟹ (λx. ¦f x :: real¦) ∈ borel_measurable M"
  unfolding abs_real_def by simp

lemma borel_measurable_nth[measurable (raw)]:
  "(λx::real^'n. x $ i) ∈ borel_measurable borel"
  by (simp add: cart_eq_inner_axis)

lemma convex_measurable:
  fixes A :: "'a :: euclidean_space set"
  shows "X ∈ borel_measurable M ⟹ X ` space M ⊆ A ⟹ open A ⟹ convex_on A q ⟹
    (λx. q (X x)) ∈ borel_measurable M"
  by (rule measurable_compose[where f=X and N="restrict_space borel A"])
     (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)

lemma borel_measurable_ln[measurable (raw)]:
  assumes f: "f ∈ borel_measurable M"
  shows "(λx. ln (f x :: real)) ∈ borel_measurable M"
  apply (rule measurable_compose[OF f])
  apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"])
  apply (auto intro!: continuous_on_ln continuous_on_id)
  done

lemma borel_measurable_log[measurable (raw)]:
  "f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ (λx. log (g x) (f x)) ∈ borel_measurable M"
  unfolding log_def by auto

lemma borel_measurable_exp[measurable]:
  "(exp::'a::{real_normed_field,banach}⇒'a) ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)

lemma measurable_real_floor[measurable]:
  "(floor :: real ⇒ int) ∈ measurable borel (count_space UNIV)"
proof -
  have "⋀a x. ⌊x⌋ = a ⟷ (real_of_int a ≤ x ∧ x < real_of_int (a + 1))"
    by (auto intro: floor_eq2)
  then show ?thesis
    by (auto simp: vimage_def measurable_count_space_eq2_countable)
qed

lemma measurable_real_ceiling[measurable]:
  "(ceiling :: real ⇒ int) ∈ measurable borel (count_space UNIV)"
  unfolding ceiling_def[abs_def] by simp

lemma borel_measurable_real_floor: "(λx::real. real_of_int ⌊x⌋) ∈ borel_measurable borel"
  by simp

lemma borel_measurable_root [measurable]: "root n ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_sqrt [measurable]: "sqrt ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_power [measurable (raw)]:
  fixes f :: "_ ⇒ 'b::{power,real_normed_algebra}"
  assumes f: "f ∈ borel_measurable M"
  shows "(λx. (f x) ^ n) ∈ borel_measurable M"
  by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)

lemma borel_measurable_Re [measurable]: "Re ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_Im [measurable]: "Im ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_of_real [measurable]: "(of_real :: _ ⇒ (_::real_normed_algebra)) ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_sin [measurable]: "(sin :: _ ⇒ (_::{real_normed_field,banach})) ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_cos [measurable]: "(cos :: _ ⇒ (_::{real_normed_field,banach})) ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_arctan [measurable]: "arctan ∈ borel_measurable borel"
  by (intro borel_measurable_continuous_on1 continuous_intros)

lemma borel_measurable_complex_iff:
  "f ∈ borel_measurable M ⟷
    (λx. Re (f x)) ∈ borel_measurable M ∧ (λx. Im (f x)) ∈ borel_measurable M"
  apply auto
  apply (subst fun_complex_eq)
  apply (intro borel_measurable_add)
  apply auto
  done

subsection "Borel space on the extended reals"

lemma borel_measurable_ereal[measurable (raw)]:
  assumes f: "f ∈ borel_measurable M" shows "(λx. ereal (f x)) ∈ borel_measurable M"
  using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)

lemma borel_measurable_real_of_ereal[measurable (raw)]:
  fixes f :: "'a ⇒ ereal"
  assumes f: "f ∈ borel_measurable M"
  shows "(λx. real_of_ereal (f x)) ∈ borel_measurable M"
  apply (rule measurable_compose[OF f])
  apply (rule borel_measurable_continuous_countable_exceptions[of "{∞, -∞ }"])
  apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
  done

lemma borel_measurable_ereal_cases:
  fixes f :: "'a ⇒ ereal"
  assumes f: "f ∈ borel_measurable M"
  assumes H: "(λx. H (ereal (real_of_ereal (f x)))) ∈ borel_measurable M"
  shows "(λx. H (f x)) ∈ borel_measurable M"
proof -
  let ?F = "λx. if f x = ∞ then H ∞ else if f x = - ∞ then H (-∞) else H (ereal (real_of_ereal (f x)))"
  { fix x have "H (f x) = ?F x" by (cases "f x") auto }
  with f H show ?thesis by simp
qed

lemma
  fixes f :: "'a ⇒ ereal" assumes f[measurable]: "f ∈ borel_measurable M"
  shows borel_measurable_ereal_abs[measurable(raw)]: "(λx. ¦f x¦) ∈ borel_measurable M"
    and borel_measurable_ereal_inverse[measurable(raw)]: "(λx. inverse (f x) :: ereal) ∈ borel_measurable M"
    and borel_measurable_uminus_ereal[measurable(raw)]: "(λx. - f x :: ereal) ∈ borel_measurable M"
  by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)

lemma borel_measurable_uminus_eq_ereal[simp]:
  "(λx. - f x :: ereal) ∈ borel_measurable M ⟷ f ∈ borel_measurable M" (is "?l = ?r")
proof
  assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
qed auto

lemma set_Collect_ereal2:
  fixes f g :: "'a ⇒ ereal"
  assumes f: "f ∈ borel_measurable M"
  assumes g: "g ∈ borel_measurable M"
  assumes H: "{x ∈ space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} ∈ sets M"
    "{x ∈ space borel. H (-∞) (ereal x)} ∈ sets borel"
    "{x ∈ space borel. H (∞) (ereal x)} ∈ sets borel"
    "{x ∈ space borel. H (ereal x) (-∞)} ∈ sets borel"
    "{x ∈ space borel. H (ereal x) (∞)} ∈ sets borel"
  shows "{x ∈ space M. H (f x) (g x)} ∈ sets M"
proof -
  let ?G = "λy x. if g x = ∞ then H y ∞ else if g x = -∞ then H y (-∞) else H y (ereal (real_of_ereal (g x)))"
  let ?F = "λx. if f x = ∞ then ?G ∞ x else if f x = -∞ then ?G (-∞) x else ?G (ereal (real_of_ereal (f x))) x"
  { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
  note * = this
  from assms show ?thesis
    by (subst *) (simp del: space_borel split del: if_split)
qed

lemma borel_measurable_ereal_iff:
  shows "(λx. ereal (f x)) ∈ borel_measurable M ⟷ f ∈ borel_measurable M"
proof
  assume "(λx. ereal (f x)) ∈ borel_measurable M"
  from borel_measurable_real_of_ereal[OF this]
  show "f ∈ borel_measurable M" by auto
qed auto

lemma borel_measurable_erealD[measurable_dest]:
  "(λx. ereal (f x)) ∈ borel_measurable M ⟹ g ∈ measurable N M ⟹ (λx. f (g x)) ∈ borel_measurable N"
  unfolding borel_measurable_ereal_iff by simp

lemma borel_measurable_ereal_iff_real:
  fixes f :: "'a ⇒ ereal"
  shows "f ∈ borel_measurable M ⟷
    ((λx. real_of_ereal (f x)) ∈ borel_measurable M ∧ f -` {∞} ∩ space M ∈ sets M ∧ f -` {-∞} ∩ space M ∈ sets M)"
proof safe
  assume *: "(λx. real_of_ereal (f x)) ∈ borel_measurable M" "f -` {∞} ∩ space M ∈ sets M" "f -` {-∞} ∩ space M ∈ sets M"
  have "f -` {∞} ∩ space M = {x∈space M. f x = ∞}" "f -` {-∞} ∩ space M = {x∈space M. f x = -∞}" by auto
  with * have **: "{x∈space M. f x = ∞} ∈ sets M" "{x∈space M. f x = -∞} ∈ sets M" by simp_all
  let ?f = "λx. if f x = ∞ then ∞ else if f x = -∞ then -∞ else ereal (real_of_ereal (f x))"
  have "?f ∈ borel_measurable M" using * ** by (intro measurable_If) auto
  also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
  finally show "f ∈ borel_measurable M" .
qed simp_all

lemma borel_measurable_ereal_iff_Iio:
  "(f::'a ⇒ ereal) ∈ borel_measurable M ⟷ (∀a. f -` {..< a} ∩ space M ∈ sets M)"
  by (auto simp: borel_Iio measurable_iff_measure_of)

lemma borel_measurable_ereal_iff_Ioi:
  "(f::'a ⇒ ereal) ∈ borel_measurable M ⟷ (∀a. f -` {a <..} ∩ space M ∈ sets M)"
  by (auto simp: borel_Ioi measurable_iff_measure_of)

lemma vimage_sets_compl_iff:
  "f -` A ∩ space M ∈ sets M ⟷ f -` (- A) ∩ space M ∈ sets M"
proof -
  { fix A assume "f -` A ∩ space M ∈ sets M"
    moreover have "f -` (- A) ∩ space M = space M - f -` A ∩ space M" by auto
    ultimately have "f -` (- A) ∩ space M ∈ sets M" by auto }
  from this[of A] this[of "-A"] show ?thesis
    by (metis double_complement)
qed

lemma borel_measurable_iff_Iic_ereal:
  "(f::'a⇒ereal) ∈ borel_measurable M ⟷ (∀a. f -` {..a} ∩ space M ∈ sets M)"
  unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp

lemma borel_measurable_iff_Ici_ereal:
  "(f::'a ⇒ ereal) ∈ borel_measurable M ⟷ (∀a. f -` {a..} ∩ space M ∈ sets M)"
  unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp

lemma borel_measurable_ereal2:
  fixes f g :: "'a ⇒ ereal"
  assumes f: "f ∈ borel_measurable M"
  assumes g: "g ∈ borel_measurable M"
  assumes H: "(λx. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) ∈ borel_measurable M"
    "(λx. H (-∞) (ereal (real_of_ereal (g x)))) ∈ borel_measurable M"
    "(λx. H (∞) (ereal (real_of_ereal (g x)))) ∈ borel_measurable M"
    "(λx. H (ereal (real_of_ereal (f x))) (-∞)) ∈ borel_measurable M"
    "(λx. H (ereal (real_of_ereal (f x))) (∞)) ∈ borel_measurable M"
  shows "(λx. H (f x) (g x)) ∈ borel_measurable M"
proof -
  let ?G = "λy x. if g x = ∞ then H y ∞ else if g x = - ∞ then H y (-∞) else H y (ereal (real_of_ereal (g x)))"
  let ?F = "λx. if f x = ∞ then ?G ∞ x else if f x = - ∞ then ?G (-∞) x else ?G (ereal (real_of_ereal (f x))) x"
  { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
  note * = this
  from assms show ?thesis unfolding * by simp
qed

lemma [measurable(raw)]:
  fixes f :: "'a ⇒ ereal"
  assumes [measurable]: "f ∈ borel_measurable M" "g ∈ borel_measurable M"
  shows borel_measurable_ereal_add: "(λx. f x + g x) ∈ borel_measurable M"
    and borel_measurable_ereal_times: "(λx. f x * g x) ∈ borel_measurable M"
  by (simp_all add: borel_measurable_ereal2)

lemma [measurable(raw)]:
  fixes f g :: "'a ⇒ ereal"
  assumes "f ∈ borel_measurable M"
  assumes "g ∈ borel_measurable M"
  shows borel_measurable_ereal_diff: "(λx. f x - g x) ∈ borel_measurable M"
    and borel_measurable_ereal_divide: "(λx. f x / g x) ∈ borel_measurable M"
  using assms by (simp_all add: minus_ereal_def divide_ereal_def)

lemma borel_measurable_ereal_setsum[measurable (raw)]:
  fixes f :: "'c ⇒ 'a ⇒ ereal"
  assumes "⋀i. i ∈ S ⟹ f i ∈ borel_measurable M"
  shows "(λx. ∑i∈S. f i x) ∈ borel_measurable M"
  using assms by (induction S rule: infinite_finite_induct) auto

lemma borel_measurable_ereal_setprod[measurable (raw)]:
  fixes f :: "'c ⇒ 'a ⇒ ereal"
  assumes "⋀i. i ∈ S ⟹ f i ∈ borel_measurable M"
  shows "(λx. ∏i∈S. f i x) ∈ borel_measurable M"
  using assms by (induction S rule: infinite_finite_induct) auto

lemma borel_measurable_extreal_suminf[measurable (raw)]:
  fixes f :: "nat ⇒ 'a ⇒ ereal"
  assumes [measurable]: "⋀i. f i ∈ borel_measurable M"
  shows "(λx. (∑i. f i x)) ∈ borel_measurable M"
  unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp

subsection "Borel space on the extended non-negative reals"

text ‹ @{type ennreal} is a topological monoid, so no rules for plus are required, also all order
  statements are usually done on type classes. ›

lemma measurable_enn2ereal[measurable]: "enn2ereal ∈ borel →M borel"
  by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal)

lemma measurable_e2ennreal[measurable]: "e2ennreal ∈ borel →M borel"
  by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal)

lemma borel_measurable_enn2real[measurable (raw)]:
  "f ∈ M →M borel ⟹ (λx. enn2real (f x)) ∈ M →M borel"
  unfolding enn2real_def[abs_def] by measurable

definition [simp]: "is_borel f M ⟷ f ∈ borel_measurable M"

lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun op = pcr_ennreal) op = is_borel is_borel"
  unfolding is_borel_def[abs_def]
proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
  fix f and M :: "'a measure"
  show "f ∈ borel_measurable M" if f: "enn2ereal ∘ f ∈ borel_measurable M"
    using measurable_compose[OF f measurable_e2ennreal] by simp
qed simp

context
  includes ennreal.lifting
begin

lemma measurable_ennreal[measurable]: "ennreal ∈ borel →M borel"
  unfolding is_borel_def[symmetric]
  by transfer simp

lemma borel_measurable_ennreal_iff[simp]:
  assumes [simp]: "⋀x. x ∈ space M ⟹ 0 ≤ f x"
  shows "(λx. ennreal (f x)) ∈ M →M borel ⟷ f ∈ M →M borel"
proof safe
  assume "(λx. ennreal (f x)) ∈ M →M borel"
  then have "(λx. enn2real (ennreal (f x))) ∈ M →M borel"
    by measurable
  then show "f ∈ M →M borel"
    by (rule measurable_cong[THEN iffD1, rotated]) auto
qed measurable

lemma borel_measurable_times_ennreal[measurable (raw)]:
  fixes f g :: "'a ⇒ ennreal"
  shows "f ∈ M →M borel ⟹ g ∈ M →M borel ⟹ (λx. f x * g x) ∈ M →M borel"
  unfolding is_borel_def[symmetric] by transfer simp

lemma borel_measurable_inverse_ennreal[measurable (raw)]:
  fixes f :: "'a ⇒ ennreal"
  shows "f ∈ M →M borel ⟹ (λx. inverse (f x)) ∈ M →M borel"
  unfolding is_borel_def[symmetric] by transfer simp

lemma borel_measurable_divide_ennreal[measurable (raw)]:
  fixes f :: "'a ⇒ ennreal"
  shows "f ∈ M →M borel ⟹ g ∈ M →M borel ⟹ (λx. f x / g x) ∈ M →M borel"
  unfolding divide_ennreal_def by simp

lemma borel_measurable_minus_ennreal[measurable (raw)]:
  fixes f :: "'a ⇒ ennreal"
  shows "f ∈ M →M borel ⟹ g ∈ M →M borel ⟹ (λx. f x - g x) ∈ M →M borel"
  unfolding is_borel_def[symmetric] by transfer simp

lemma borel_measurable_setprod_ennreal[measurable (raw)]:
  fixes f :: "'c ⇒ 'a ⇒ ennreal"
  assumes "⋀i. i ∈ S ⟹ f i ∈ borel_measurable M"
  shows "(λx. ∏i∈S. f i x) ∈ borel_measurable M"
  using assms by (induction S rule: infinite_finite_induct) auto

end

hide_const (open) is_borel

subsection ‹LIMSEQ is borel measurable›

lemma borel_measurable_LIMSEQ_real:
  fixes u :: "nat ⇒ 'a ⇒ real"
  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 ⟹ liminf (λn. ereal (u n x)) = ereal (u' x)"
    using u' by (simp add: lim_imp_Liminf)
  moreover from u have "(λx. liminf (λn. ereal (u n x))) ∈ borel_measurable M"
    by auto
  ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
qed

lemma borel_measurable_LIMSEQ_metric:
  fixes f :: "nat ⇒ 'a ⇒ 'b :: metric_space"
  assumes [measurable]: "⋀i. f i ∈ borel_measurable M"
  assumes lim: "⋀x. x ∈ space M ⟹ (λi. f i x) ⇢ g x"
  shows "g ∈ borel_measurable M"
  unfolding borel_eq_closed
proof (safe intro!: measurable_measure_of)
  fix A :: "'b set" assume "closed A"

  have [measurable]: "(λx. infdist (g x) A) ∈ borel_measurable M"
  proof (rule borel_measurable_LIMSEQ_real)
    show "⋀x. x ∈ space M ⟹ (λi. infdist (f i x) A) ⇢ infdist (g x) A"
      by (intro tendsto_infdist lim)
    show "⋀i. (λx. infdist (f i x) A) ∈ borel_measurable M"
      by (intro borel_measurable_continuous_on[where f="λx. infdist x A"]
        continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto
  qed

  show "g -` A ∩ space M ∈ sets M"
  proof cases
    assume "A ≠ {}"
    then have "⋀x. infdist x A = 0 ⟷ x ∈ A"
      using ‹closed A› by (simp add: in_closed_iff_infdist_zero)
    then have "g -` A ∩ space M = {x∈space M. infdist (g x) A = 0}"
      by auto
    also have "… ∈ sets M"
      by measurable
    finally show ?thesis .
  qed simp
qed auto

lemma sets_Collect_Cauchy[measurable]:
  fixes f :: "nat ⇒ 'a => 'b::{metric_space, second_countable_topology}"
  assumes f[measurable]: "⋀i. f i ∈ borel_measurable M"
  shows "{x∈space M. Cauchy (λi. f i x)} ∈ sets M"
  unfolding metric_Cauchy_iff2 using f by auto

lemma borel_measurable_lim_metric[measurable (raw)]:
  fixes f :: "nat ⇒ 'a ⇒ 'b::{banach, second_countable_topology}"
  assumes f[measurable]: "⋀i. f i ∈ borel_measurable M"
  shows "(λx. lim (λi. f i x)) ∈ borel_measurable M"
proof -
  def u'  "λx. lim (λi. if Cauchy (λi. f i x) then f i x else 0)"
  then have *: "⋀x. lim (λi. f i x) = (if Cauchy (λi. f i x) then u' x else (THE x. False))"
    by (auto simp: lim_def convergent_eq_cauchy[symmetric])
  have "u' ∈ borel_measurable M"
  proof (rule borel_measurable_LIMSEQ_metric)
    fix x
    have "convergent (λi. if Cauchy (λi. f i x) then f i x else 0)"
      by (cases "Cauchy (λi. f i x)")
         (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
    then show "(λi. if Cauchy (λi. f i x) then f i x else 0) ⇢ u' x"
      unfolding u'_def
      by (rule convergent_LIMSEQ_iff[THEN iffD1])
  qed measurable
  then show ?thesis
    unfolding * by measurable
qed

lemma borel_measurable_suminf[measurable (raw)]:
  fixes f :: "nat ⇒ 'a ⇒ 'b::{banach, second_countable_topology}"
  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

(* Proof by Jeremy Avigad and Luke Serafin *)
lemma isCont_borel:
  fixes f :: "'b::metric_space ⇒ 'a::metric_space"
  shows "{x. isCont f x} ∈ sets borel"
proof -
  let ?I = "λj. inverse(real (Suc j))"

  { fix x
    have "isCont f x = (∀i. ∃j. ∀y z. dist x y < ?I j ∧ dist x z < ?I j ⟶ dist (f y) (f z) ≤ ?I i)"
      unfolding continuous_at_eps_delta
    proof safe
      fix i assume "∀e>0. ∃d>0. ∀y. dist y x < d ⟶ dist (f y) (f x) < e"
      moreover have "0 < ?I i / 2"
        by simp
      ultimately obtain d where d: "0 < d" "⋀y. dist x y < d ⟹ dist (f y) (f x) < ?I i / 2"
        by (metis dist_commute)
      then obtain j where j: "?I j < d"
        by (metis reals_Archimedean)

      show "∃j. ∀y z. dist x y < ?I j ∧ dist x z < ?I j ⟶ dist (f y) (f z) ≤ ?I i"
      proof (safe intro!: exI[where x=j])
        fix y z assume *: "dist x y < ?I j" "dist x z < ?I j"
        have "dist (f y) (f z) ≤ dist (f y) (f x) + dist (f z) (f x)"
          by (rule dist_triangle2)
        also have "… < ?I i / 2 + ?I i / 2"
          by (intro add_strict_mono d less_trans[OF _ j] *)
        also have "… ≤ ?I i"
          by (simp add: field_simps of_nat_Suc)
        finally show "dist (f y) (f z) ≤ ?I i"
          by simp
      qed
    next
      fix e::real assume "0 < e"
      then obtain n where n: "?I n < e"
        by (metis reals_Archimedean)
      assume "∀i. ∃j. ∀y z. dist x y < ?I j ∧ dist x z < ?I j ⟶ dist (f y) (f z) ≤ ?I i"
      from this[THEN spec, of "Suc n"]
      obtain j where j: "⋀y z. dist x y < ?I j ⟹ dist x z < ?I j ⟹ dist (f y) (f z) ≤ ?I (Suc n)"
        by auto

      show "∃d>0. ∀y. dist y x < d ⟶ dist (f y) (f x) < e"
      proof (safe intro!: exI[of _ "?I j"])
        fix y assume "dist y x < ?I j"
        then have "dist (f y) (f x) ≤ ?I (Suc n)"
          by (intro j) (auto simp: dist_commute)
        also have "?I (Suc n) < ?I n"
          by simp
        also note n
        finally show "dist (f y) (f x) < e" .
      qed simp
    qed }
  note * = this

  have **: "⋀e y. open {x. dist x y < e}"
    using open_ball by (simp_all add: ball_def dist_commute)

  have "{x∈space borel. isCont f x} ∈ sets borel"
    unfolding *
    apply (intro sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex)
    apply (simp add: Collect_all_eq)
    apply (intro borel_closed closed_INT ballI closed_Collect_imp open_Collect_conj **)
    apply auto
    done
  then show ?thesis
    by simp
qed

lemma isCont_borel_pred[measurable]:
  fixes f :: "'b::metric_space ⇒ 'a::metric_space"
  shows "Measurable.pred borel (isCont f)"
  unfolding pred_def by (simp add: isCont_borel)

lemma is_real_interval:
  assumes S: "is_interval S"
  shows "∃a b::real. S = {} ∨ S = UNIV ∨ S = {..<b} ∨ S = {..b} ∨ S = {a<..} ∨ S = {a..} ∨
    S = {a<..<b} ∨ S = {a<..b} ∨ S = {a..<b} ∨ S = {a..b}"
  using S unfolding is_interval_1 by (blast intro: interval_cases)

lemma real_interval_borel_measurable:
  assumes "is_interval (S::real set)"
  shows "S ∈ sets borel"
proof -
  from assms is_real_interval have "∃a b::real. S = {} ∨ S = UNIV ∨ S = {..<b} ∨ S = {..b} ∨
    S = {a<..} ∨ S = {a..} ∨ S = {a<..<b} ∨ S = {a<..b} ∨ S = {a..<b} ∨ S = {a..b}" by auto
  then guess a ..
  then guess b ..
  thus ?thesis
    by auto
qed

lemma borel_measurable_mono_on_fnc:
  fixes f :: "real ⇒ real" and A :: "real set"
  assumes "mono_on f A"
  shows "f ∈ borel_measurable (restrict_space borel A)"
  apply (rule measurable_restrict_countable[OF mono_on_ctble_discont[OF assms]])
  apply (auto intro!: image_eqI[where x="{x}" for x] simp: sets_restrict_space)
  apply (auto simp add: sets_restrict_restrict_space continuous_on_eq_continuous_within
              cong: measurable_cong_sets
              intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset)
  done

lemma borel_measurable_mono:
  fixes f :: "real ⇒ real"
  shows "mono f ⟹ f ∈ borel_measurable borel"
  using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)

no_notation
  eucl_less (infix "<e" 50)

end