Quelle  Equivalence_Measurable_On_Borel.thy   Sprache: Isabelle

(*  Title:      HOL/Analysis/Equivalence_Measurable_On_Borel
    Author: LC Paulson (some material ported from HOL Light)
*)

section‹Equivalence Between Classical Borel Measurability and HOL Light's\

theory Equivalence_Measurable_On_Borel
  imports Equivalence_Lebesgue_Henstock_Integration Improper_Integral Continuous_Extension
begin

subsection‹Austin's Lemma\

lemma Austin_Lemma:
  fixes D :: "'a::euclidean_space set set"
  assumes "finite \" and D: "\D. D \ \ \ \k a b. D = cbox a b \ (\i \ Basis. b\i - a\i = k)"
  obtains C where "\ \ \" "pairwise disjnt \"
                  "measure lebesgue (\\) \ measure lebesgue (\\) / 3 ^ (DIM('a))"
  using assms
proof (induction "card \" arbitrary: D thesis rule: less_induct)
  case less
  show ?case
  proof (cases "\ = {}")
    case True
    then show thesis
      using less by auto
  next
    case False
    then have "Max (Sigma_Algebra.measure lebesgue ` \) \ Sigma_Algebra.measure lebesgue ` \"
      using Max_in finite_imageI ‹finite D› by blast
    then obtain D where "D \ \" and "measure lebesgue D = Max (measure lebesgue ` \)"
      by auto
    then have D: "\C. C \ \ \ measure lebesgue C \ measure lebesgue D"
      by (simp add: ‹finite D›)
    let ?E = "{C. C \ \ - {D} \ disjnt C D}"
    obtain D' where \'sub: "\' \ ?\" and D'dis: "pairwise disjnt \'"
      and D'm: "measure lebesgue (\\') ≥ measure lebesgue (∪?E) / 3 ^ (DIM('a))"
    proof (rule less.hyps)
      have *: "?\ \ \"
        using ‹D ∈ D› by auto
      then show "card ?\ < card \" "finite ?\"
        by (auto simp: ‹finite D› psubset_card_mono)
      show "\k a b. D = cbox a b \ (\i\Basis. b \ i - a \ i = k)" if "D \ ?\" for D
        using less.prems(3) that by auto
    qed
    then have [simp]: "\\' - D = \\'"
      by (auto simp: disjnt_iff)
    show ?thesis
    proof (rule less.prems)
      show "insert D \' \ \"
        using D'sub \D \ \\ by blast
      show "disjoint (insert D \')"
        using D'dis \'sub by (fastforce simp add: pairwise_def disjnt_sym)
      obtain a3 b3 where  m3: "content (cbox a3 b3) = 3 ^ DIM('a) * measure lebesgue D"
        and sub3: "\C. \C \ \; \ disjnt C D\ \ C \ cbox a3 b3"
      proof -
        obtain k a b where ab: "D = cbox a b" and k: "\i. i \ Basis \ b\i - a\i = k"
          using less.prems ‹D ∈ D› by meson
        then have eqk: "\i. i \ Basis \ a \ i \ b \ i \ k \ 0"
          by force
        show thesis
        proof
          let ?a = "(a + b) /\<^sub>R 2 - (3/2) *\<^sub>R (b - a)"
          let ?b = "(a + b) /\<^sub>R 2 + (3/2) *\<^sub>R (b - a)"
          have eq: "(\i\Basis. b \ i * 3 - a \ i * 3) = (\i\Basis. b \ i - a \ i) * 3 ^ DIM('a)"
            by (simp add: comm_monoid_mult_class.prod.distrib flip: left_diff_distrib inner_diff_left)
          show "content (cbox ?a ?b) = 3 ^ DIM('a) * measure lebesgue D"
            by (simp add: content_cbox_if box_eq_empty algebra_simps eq ab k)
          show "C \ cbox ?a ?b" if "C \ \" and CD: "\ disjnt C D" for C
          proof -
            obtain k' a' b' where ab': "C = cbox a' b'" and k': "\i. i \ Basis \ b'∙i - a'\i = k'"
              using less.prems ‹C ∈ D› by meson
            then have eqk': "\i. i \ Basis \ a' ∙ i ≤ b' \ i \ k' ≥ 0"
              by force
            show ?thesis
            proof (clarsimp simp add: disjoint_interval disjnt_def ab ab' not_less subset_box algebra_simps)
              show "a \ i * 2 \ a' \ i + b \ i \ a \ i + b' \ i \ b \ i * 2"
                if * [rule_format]: "\j\Basis. a' \ j \ b' \ j" and "i \ Basis" for i
              proof -
                have "a' \ i \ b' \ i \ a \ i \ b \ i \ a \ i \ b' \ i \ a' \ i \ b \ i"
                  using ‹i ∈ Basis› CD by (simp_all add: disjoint_interval disjnt_def ab ab' not_less)
                then show ?thesis
                  using D [OF ‹C ∈ D›] ‹i ∈ Basis›
                  apply (simp add: ab ab' k k' eqk eqk' content_cbox_cases)
                  using k k' by fastforce
              qed
            qed
          qed
        qed
      qed
      have Dlm: "\D. D \ \ \ D \ lmeasurable"
        using less.prems(3) by blast
      have "measure lebesgue (\\) \ measure lebesgue (cbox a3 b3 \ (\\ - cbox a3 b3))"
      proof (rule measure_mono_fmeasurable)
        show "\\ \ sets lebesgue"
          using Dlm ‹finite D› by blast
        show "cbox a3 b3 \ (\\ - cbox a3 b3) \ lmeasurable"
          by (simp add: Dlm fmeasurable.Un fmeasurable.finite_Union less.prems(2) subset_eq)
      qed auto
      also have "\ = content (cbox a3 b3) + measure lebesgue (\\ - cbox a3 b3)"
        by (simp add: Dlm fmeasurable.finite_Union less.prems(2) measure_Un2 subsetI)
      also have "\ \ (measure lebesgue D + measure lebesgue (\\')) * 3 ^ DIM('a)"
      proof -
        have "(\\ - cbox a3 b3) \ \?\"
          using sub3 by fastforce
        then have "measure lebesgue (\\ - cbox a3 b3) \ measure lebesgue (\?\)"
        proof (rule measure_mono_fmeasurable)
          show "\ \ - cbox a3 b3 \ sets lebesgue"
            by (simp add: Dlm fmeasurableD less.prems(2) sets.Diff sets.finite_Union subsetI)
          show "\ {C \ \ - {D}. disjnt C D} \ lmeasurable"
            using Dlm less.prems(2) by auto
        qed
        then have "measure lebesgue (\\ - cbox a3 b3) / 3 ^ DIM('a) \ measure lebesgue (\ \')"
          using D'm by (simp add: field_split_simps)
        then show ?thesis
          by (simp add: m3 field_simps)
      qed
      also have "\ \ measure lebesgue (\(insert D \')) * 3 ^ DIM('a)"
      proof (simp add: Dlm ‹D ∈ D›)
        show "measure lebesgue D + measure lebesgue (\\') \ measure lebesgue (D \ \ \')"
        proof (subst measure_Un2)
          show "\ \' \ lmeasurable"
            by (meson Dlm ‹insert D D' \ \\ fmeasurable.finite_Union less.prems(2) finite_subset subset_eq subset_insertI)
          show "measure lebesgue D + measure lebesgue (\ \') \ measure lebesgue D + measure lebesgue (\ \' - D)"
            using ‹insert D D' \ \\ infinite_super less.prems(2) by force
        qed (simp add: Dlm ‹D ∈ D›)
      qed
      finally show "measure lebesgue (\\) / 3 ^ DIM('a) \ measure lebesgue (\(insert D \'))"
        by (simp add: field_split_simps)
    qed
  qed
qed


subsection‹A differentiability-like property of the indefinite integral.        ›

proposition integrable_ccontinuous_explicit:
  fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
  assumes "\a b::'a. f integrable_on cbox a b"
  obtains N where
       "negligible N"
       "\x e. \x \ N; 0 < e\ \
               ∃d>0. ∀h. 0 < h ∧ h < d ⟶
                         norm(integral (cbox x (x + h *🚫R One)) f /🚫R h ^ DIM('a) - f x) < e"
proof -
  define BOX where "BOX \ \h. \x::'a. cbox x (x + h *\<^sub>R One)"
  define BOX2 where "BOX2 \ \h. \x::'a. cbox (x - h *\<^sub>R One) (x + h *\<^sub>R One)"
  define i where "i \ \h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)"
  define Ψ where "\ \ \x r. \d>0. \h. 0 < h \ h < d \ r \ norm(i h x - f x)"
  let ?N = "{x. \e>0. \ x e}"
  have "\N. negligible N \ (\x e. x \ N \ 0 < e \ \ \ x e)"
  proof (rule exI ; intro conjI allI impI)
    let ?M =  "\n. {x. \ x (inverse(real n + 1))}"
    have "negligible ({x. \ x \} \ cbox a b)"
      if "\ > 0" for a b μ
    proof (cases "negligible(cbox a b)")
      case True
      then show ?thesis
        by (simp add: negligible_Int)
    next
      case False
      then have "box a b \ {}"
        by (simp add: negligible_interval)
      then have ab: "\i. i \ Basis \ a\i < b\i"
        by (simp add: box_ne_empty)
      show ?thesis
        unfolding negligible_outer_le
      proof (intro allI impI)
        fix e::real
        let ?ee = "(e * \) / 2 / 6 ^ (DIM('a))"
        assume "e > 0"
        then have gt0: "?ee > 0"
          using ‹μ > 0› by auto
        have f': "f integrable_on cbox (a - One) (b + One)"
          using assms by blast
        obtain γ where "gauge \"
          and γ: "\p. \p tagged_partial_division_of (cbox (a - One) (b + One)); \ fine p\
                    ==> (∑(x, k)∈p. norm (content k *🚫R f x - integral k f)) < ?ee"
          using Henstock_lemma [OF f' gt0] that by auto
        let ?E = "{x. x \ cbox a b \ \ x \}"
        have "\h>0. BOX h x \ \ x \
                    BOX h x ⊆ cbox (a - One) (b + One) ∧ μ ≤ norm (i h x - f x)"
          if "x \ cbox a b" "\ x \" for x
        proof -
          obtain d where "d > 0" and d: "ball x d \ \ x"
            using gaugeD [OF ‹gauge γ›, of x] openE by blast
          then obtain h where "0 < h" "h < 1" and hless: "h < d / real DIM('a)"
                          and mule: "\ \ norm (i h x - f x)"
            using ‹Ψ x μ› [unfolded Ψ_def, rule_format, of "min 1 (d / DIM('a))"]
            by auto
          show ?thesis
          proof (intro exI conjI)
            show "0 < h" "\ \ norm (i h x - f x)" by fact+
            have "BOX h x \ ball x d"
            proof (clarsimp simp: BOX_def mem_box dist_norm algebra_simps)
              fix y
              assume "\i\Basis. x \ i \ y \ i \ y \ i \ h + x \ i"
              then have lt: "\(x - y) \ i\ < d / real DIM('a)" if "i \ Basis" for i
                using hless that by (force simp: inner_diff_left)
              have "norm (x - y) \ (\i\Basis. \(x - y) \ i\)"
                using norm_le_l1 by blast
              also have "\ < d"
                using sum_bounded_above_strict [of Basis "\i. \(x - y) \ i\" "d / DIM('a)", OF lt]
                by auto
              finally show "norm (x - y) < d" .
            qed
            with d show "BOX h x \ \ x"
              by blast
            show "BOX h x \ cbox (a - One) (b + One)"
              using that ‹h < 1›
              by (force simp: BOX_def mem_box algebra_simps intro: subset_box_imp)
          qed
        qed
        then obtain 🚫 where h0: "\x. x \ ?E \ \ x > 0"
          and BOX_γ: "\x. x \ ?E \ BOX (\ x) x \ \ x"
          and "\x. x \ ?E \ BOX (\ x) x \ cbox (a - One) (b + One) \ \ \ norm (i (\ x) x - f x)"
          by simp metis
        then have BOX_cbox: "\x. x \ ?E \ BOX (\ x) x \ cbox (a - One) (b + One)"
             and μ_le: "\x. x \ ?E \ \ \ norm (i (\ x) x - f x)"
          by blast+
        define γ' where "\' ≡ λx. if x ∈ cbox a b ∧ Ψ x μ then ball x (🚫 x) else γ x"
        have "gauge \'"
          using ‹gauge γ› by (auto simp: h0 gauge_def γ'_def)
        obtain D where "countable \"
          and D: "\\ \ cbox a b"
          "\K. K \ \ \ interior K \ {} \ (\c d. K = cbox c d)"
          and Dcovered: "\K. K \ \ \ \x. x \ cbox a b \ \ x \ \ x \ K \ K \ \' x"
          and subUD: "?E \ \\"
          by (rule covering_lemma [of ?E a b γ']) (simp_all add: Bex_def \box a b \ {}\ \gauge \'›)
        then have "\ \ sets lebesgue"
          by fastforce
        show "\T. {x. \ x \} \ cbox a b \ T \ T \ lmeasurable \ measure lebesgue T \ e"
        proof (intro exI conjI)
          show "{x. \ x \} \ cbox a b \ \\"
            apply auto
            using subUD by auto
          have mUE: "measure lebesgue (\ \) \ measure lebesgue (cbox a b)"
            if "\ \ \" "finite \" for E
          proof (rule measure_mono_fmeasurable)
            show "\ \ \ cbox a b"
              using D(1) that(1) by blast
            show "\ \ \ sets lebesgue"
              by (metis D(2) fmeasurable.finite_Union fmeasurableD lmeasurable_cbox subset_eq that)
          qed auto
          then show "\\ \ lmeasurable"
            by (metis D(2) ‹countable D› fmeasurable_Union_bound lmeasurable_cbox)
          then have leab: "measure lebesgue (\\) \ measure lebesgue (cbox a b)"
            by (meson D(1) fmeasurableD lmeasurable_cbox measure_mono_fmeasurable)
          obtain F where "\ \ \" "finite \"
            and F: "measure lebesgue (\\) \ 2 * measure lebesgue (\\)"
          proof (cases "measure lebesgue (\\) = 0")
            case True
            then show ?thesis
              by (force intro: that [where F = "{}"])
          next
            case False
            obtain F where "\ \ \" "finite \"
              and F: "measure lebesgue (\\)/2 < measure lebesgue (\\)"
            proof (rule measure_countable_Union_approachable [of D "measure lebesgue (\\) / 2" "content (cbox a b)"])
              show "countable \"
                by fact
              show "0 < measure lebesgue (\ \) / 2"
                using False by (simp add: zero_less_measure_iff)
              show Dlm: "D \ lmeasurable" if "D \ \" for D
                using D(2) that by blast
              show "measure lebesgue (\ \) \ content (cbox a b)"
                if "\ \ \" "finite \" for F
              proof -
                have "measure lebesgue (\ \) \ measure lebesgue (\\)"
                proof (rule measure_mono_fmeasurable)
                  show "\ \ \ \ \"
                    by (simp add: Sup_subset_mono ‹F ⊆ D›)
                  show "\ \ \ sets lebesgue"
                    by (meson Dlm fmeasurableD sets.finite_Union subset_eq that)
                  show "\ \ \ lmeasurable"
                    by fact
                qed
                also have "\ \ measure lebesgue (cbox a b)"
                proof (rule measure_mono_fmeasurable)
                  show "\ \ \ sets lebesgue"
                    by (simp add: ‹∪ D ∈ lmeasurable› fmeasurableD)
                qed (auto simp:D(1))
                finally show ?thesis
                  by simp
              qed
            qed auto
            then show ?thesis
              using that by auto
          qed
          obtain tag where tag_in_E: "\D. D \ \ \ tag D \ ?E"
            and tag_in_self: "\D. D \ \ \ tag D \ D"
            and tag_sub: "\D. D \ \ \ D \ \' (tag D)"
            using Dcovered by simp metis
          then have sub_ball_tag: "\D. D \ \ \ D \ ball (tag D) (\ (tag D))"
            by (simp add: γ'_def)
          define Φ where "\ \ \D. BOX (\(tag D)) (tag D)"
          define Φ2 where "\2 \ \D. BOX2 (\(tag D)) (tag D)"
          obtain C where "\ \ \2 ` \" "pairwise disjnt \"
            "measure lebesgue (\\) \ measure lebesgue (\(\2`\)) / 3 ^ (DIM('a))"
          proof (rule Austin_Lemma)
            show "finite (\2`\)"
              using ‹finite F› by blast
            have "\k a b. \2 D = cbox a b \ (\i\Basis. b \ i - a \ i = k)" if "D \ \" for D
              apply (rule_tac x="2 * \(tag D)" in exI)
              apply (rule_tac x="tag D - \(tag D) *\<^sub>R One" in exI)
              apply (rule_tac x="tag D + \(tag D) *\<^sub>R One" in exI)
              using that
              apply (auto simp: Φ2_def BOX2_def algebra_simps)
              done
            then show "\D. D \ \2 ` \ \ \k a b. D = cbox a b \ (\i\Basis. b \ i - a \ i = k)"
              by blast
          qed auto
          then obtain G where "\ \ \" and disj: "pairwise disjnt (\2 ` \)"
            and "measure lebesgue (\(\2 ` \)) \ measure lebesgue (\(\2`\)) / 3 ^ (DIM('a))"
            unfolding Φ2_def subset_image_iff
            by (meson empty_subsetI equals0D pairwise_imageI)
          moreover
          have "measure lebesgue (\(\2 ` \)) * 3 ^ DIM('a) \ e/2"
          proof -
            have "finite \"
              using ‹finite F› ‹G ⊆ F› infinite_super by blast
            have BOX2_m: "\x. x \ tag ` \ \ BOX2 (\ x) x \ lmeasurable"
              by (auto simp: BOX2_def)
            have BOX_m: "\x. x \ tag ` \ \ BOX (\ x) x \ lmeasurable"
              by (auto simp: BOX_def)
            have BOX_sub: "BOX (\ x) x \ BOX2 (\ x) x" for x
              by (auto simp: BOX_def BOX2_def subset_box algebra_simps)
            have DISJ2: "BOX2 (\ (tag X)) (tag X) \ BOX2 (\ (tag Y)) (tag Y) = {}"
              if "X \ \" "Y \ \" "tag X \ tag Y" for X Y
            proof -
              obtain i where i: "i \ Basis" "tag X \ i \ tag Y \ i"
                using ‹tag X ≠ tag Y› by (auto simp: euclidean_eq_iff [of "tag X"])
              have XY: "X \ \" "Y \ \"
                using ‹F ⊆ D› ‹G ⊆ F› that by auto
              then have "0 \ \ (tag X)" "0 \ \ (tag Y)"
                by (meson h0 le_cases not_le tag_in_E)+
              with XY i have "BOX2 (\ (tag X)) (tag X) \ BOX2 (\ (tag Y)) (tag Y)"
                unfolding eq_iff
                by (fastforce simp add: BOX2_def subset_box algebra_simps)
              then show ?thesis
                using disj that by (auto simp: pairwise_def disjnt_def Φ2_def)
            qed
            then have BOX2_disj: "pairwise (\x y. negligible (BOX2 (\ x) x \ BOX2 (\ y) y)) (tag ` \)"
              by (simp add: pairwise_imageI)
            then have BOX_disj: "pairwise (\x y. negligible (BOX (\ x) x \ BOX (\ y) y)) (tag ` \)"
            proof (rule pairwise_mono)
              show "negligible (BOX (\ x) x \ BOX (\ y) y)"
                if "negligible (BOX2 (\ x) x \ BOX2 (\ y) y)" for x y
                by (metis (no_types, opaque_lifting) that Int_mono negligible_subset BOX_sub)
            qed auto

            have eq: "\box. (\D. box (\ (tag D)) (tag D)) ` \ = (\t. box (\ t) t) ` tag ` \"
              by (simp add: image_comp)
            have "measure lebesgue (BOX2 (\ t) t) * 3 ^ DIM('a)
                = measure lebesgue (BOX (🚫 t) t) * (2*3) ^ DIM('a)"
              if "t \ tag ` \" for t
            proof -
              have "content (cbox (t - \ t *\<^sub>R One) (t + \ t *\<^sub>R One))
                  = content (cbox t (t + 🚫 t *🚫R One)) * 2 ^ DIM('a)"
                using that by (simp add: algebra_simps content_cbox_if box_eq_empty)
              then show ?thesis
                by (simp add: BOX2_def BOX_def flip: power_mult_distrib)
            qed
            then have "measure lebesgue (\(\2 ` \)) * 3 ^ DIM('a) = measure lebesgue (\(\ ` \)) * 6 ^ DIM('a)"
              unfolding Φ_def Φ2_def eq
              by (simp add: measure_negligible_finite_Union_image
                  ‹finite G› BOX2_m BOX_m BOX2_disj BOX_disj sum_distrib_right
                  del: UN_simps)
            also have "\ \ e/2"
            proof -
              have "\ * measure lebesgue (\D\\. \ D) \ \ * (\D \ \`\. measure lebesgue D)"
                using ‹μ > 0› ‹finite G› by (force simp: BOX_m Φ_def fmeasurableD intro: measure_Union_le)
              also have "\ = (\D \ \`\. measure lebesgue D * \)"
                by (metis mult.commute sum_distrib_right)
              also have "\ \ (\(x, K) \ (\D. (tag D, \ D)) ` \. norm (content K *\<^sub>R f x - integral K f))"
              proof (rule sum_le_included; clarify?)
                fix D
                assume "D \ \"
                then have "\ (tag D) > 0"
                  using ‹F ⊆ D› ‹G ⊆ F› h0 tag_in_E by auto
                then have m_Φ: "measure lebesgue (\ D) > 0"
                  by (simp add: Φ_def BOX_def algebra_simps)
                have "\ \ norm (i (\(tag D)) (tag D) - f(tag D))"
                  using μ_le ‹D ∈ G› ‹F ⊆ D› ‹G ⊆ F› tag_in_E by auto
                also have "\ = norm ((content (\ D) *\<^sub>R f(tag D) - integral (\ D) f) /\<^sub>R measure lebesgue (\ D))"
                  using m_Φ
                  unfolding i_def Φ_def BOX_def
                  by (simp add: algebra_simps content_cbox_plus norm_minus_commute)
                finally have "measure lebesgue (\ D) * \ \ norm (content (\ D) *\<^sub>R f(tag D) - integral (\ D) f)"
                  using m_Φ by simp (simp add: field_simps)
                then show "\y\(\D. (tag D, \ D)) ` \.
                        snd y = Φ D ∧ measure lebesgue (Φ D) * μ ≤ (case y of (x, k) ==> norm (content k *🚫R f x - integral k f))"
                  using ‹D ∈ G› by auto
              qed (use ‹finite G› in auto)
              also have "\ < ?ee"
              proof (rule γ)
                show "(\D. (tag D, \ D)) ` \ tagged_partial_division_of cbox (a - One) (b + One)"
                  unfolding tagged_partial_division_of_def
                proof (intro conjI allI impI ; clarify ?)
                  show "tag D \ \ D"
                    if "D \ \" for D
                    using that ‹F ⊆ D› ‹G ⊆ F› h0 tag_in_E
                    by (auto simp: Φ_def BOX_def mem_box algebra_simps eucl_less_le_not_le in_mono)
                  show "y \ cbox (a - One) (b + One)" if "D \ \" "y \ \ D" for D y
                    using that BOX_cbox Φ_def ‹F ⊆ D› ‹G ⊆ F› tag_in_E by blast
                  show "tag D = tag E \ \ D = \ E"
                    if "D \ \" "E \ \" and ne: "interior (\ D) \ interior (\ E) \ {}" for D E
                  proof -
                    have "BOX2 (\ (tag D)) (tag D) \ BOX2 (\ (tag E)) (tag E) = {} \ tag E = tag D"
                      using DISJ2 ‹D ∈ G› ‹E ∈ G› by force
                    then have "BOX (\ (tag D)) (tag D) \ BOX (\ (tag E)) (tag E) = {} \ tag E = tag D"
                      using BOX_sub by blast
                    then show "tag D = tag E \ \ D = \ E"
                      by (metis Φ_def interior_Int interior_empty ne)
                  qed
                qed (use ‹finite G› Φ_def BOX_def in auto)
                show "\ fine (\D. (tag D, \ D)) ` \"
                  unfolding fine_def Φ_def using BOX_γ ‹F ⊆ D› ‹G ⊆ F› tag_in_E by blast
              qed
              finally show ?thesis
                using ‹μ > 0› by (auto simp: field_split_simps)
          qed
            finally show ?thesis .
          qed
          moreover
          have "measure lebesgue (\\) \ measure lebesgue (\(\2`\))"
          proof (rule measure_mono_fmeasurable)
            have "D \ ball (tag D) (\(tag D))" if "D \ \" for D
              using ‹F ⊆ D› sub_ball_tag that by blast
            moreover have "ball (tag D) (\(tag D)) \ BOX2 (\ (tag D)) (tag D)" if "D \ \" for D
            proof (clarsimp simp: Φ2_def BOX2_def mem_box algebra_simps dist_norm)
              fix x and i::'a
              assume "norm (tag D - x) < \ (tag D)" and "i \ Basis"
              then have "\tag D \ i - x \ i\ \ \ (tag D)"
                by (metis eucl_less_le_not_le inner_commute inner_diff_right norm_bound_Basis_le)
              then show "tag D \ i \ x \ i + \ (tag D) \ x \ i \ \ (tag D) + tag D \ i"
                by (simp add: abs_diff_le_iff)
            qed
            ultimately show "\\ \ \(\2`\)"
              by (force simp: Φ2_def)
            show "\\ \ sets lebesgue"
              using ‹finite F› ‹D ⊆ sets lebesgue› ‹F ⊆ D› by blast
            show "\(\2`\) \ lmeasurable"
              unfolding Φ2_def BOX2_def using ‹finite F› by blast
          qed
          ultimately
          have "measure lebesgue (\\) \ e/2"
            by (auto simp: field_split_simps)
          then show "measure lebesgue (\\) \ e"
            using F by linarith
        qed
      qed
    qed
    then have "\j. negligible {x. \ x (inverse(real j + 1))}"
      using negligible_on_intervals
      by (metis (full_types) inverse_positive_iff_positive le_add_same_cancel1 linorder_not_le nat_le_real_less not_add_less1 of_nat_0)
    then have "negligible ?M"
      by auto
    moreover have "?N \ ?M"
    proof (clarsimp simp: dist_norm)
      fix y e
      assume "0 < e"
        and ye [rule_format]: "\ y e"
      then obtain k where k: "0 < k" "inverse (real k + 1) < e"
        by (metis One_nat_def add.commute less_add_same_cancel2 less_imp_inverse_less less_trans neq0_conv of_nat_1 of_nat_Suc reals_Archimedean zero_less_one)
      with ye show "\n. \ y (inverse (real n + 1))"
        apply (rule_tac x=k in exI)
        unfolding Ψ_def
        by (force intro: less_le_trans)
    qed
    ultimately show "negligible ?N"
      by (blast intro: negligible_subset)
    show "\ \ x e" if "x \ ?N \ 0 < e" for x e
      using that by blast
  qed
  with that show ?thesis
    unfolding i_def BOX_def Ψ_def by (fastforce simp add: not_le)
qed


subsection‹HOL Light measurability›

definition measurable_on :: "('a::euclidean_space \ 'b::real_normed_vector) \ 'a set \ bool"
  (infixr ‹measurable'_on\ 46)
  where "f measurable_on S \
        ∃N g. negligible N ∧
              (∀n. continuous_on UNIV (g n)) ∧
              (∀x. x ∉ N ⟶ (λn. g n x) <---- (if x ∈ S then f x else 0))"

lemma measurable_on_UNIV:
  "(\x. if x \ S then f x else 0) measurable_on UNIV \ f measurable_on S"
  by (auto simp: measurable_on_def)

lemma measurable_on_spike_set:
  assumes f: "f measurable_on S" and neg: "negligible ((S - T) \ (T - S))"
  shows "f measurable_on T"
proof -
  obtain N and F
    where N: "negligible N"
      and conF: "\n. continuous_on UNIV (F n)"
      and tendsF: "\x. x \ N \ (\n. F n x) \ (if x \ S then f x else 0)"
    using f by (auto simp: measurable_on_def)
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "continuous_on UNIV (\x. F n x)" for n
      by (intro conF continuous_intros)
    show "negligible (N \ (S - T) \ (T - S))"
      by (metis (full_types) N neg negligible_Un_eq)
    show "(\n. F n x) \ (if x \ T then f x else 0)"
      if "x \ (N \ (S - T) \ (T - S))" for x
      using that tendsF [of x] by auto
  qed
qed

text‹ Various common equivalent forms of function measurability.                ›

lemma measurable_on_0 [simp]: "(\x. 0) measurable_on S"
  unfolding measurable_on_def
proof (intro exI conjI allI impI)
  show "(\n. 0) \ (if x \ S then 0::'b else 0)" for x
    by force
qed auto

lemma measurable_on_scaleR_const:
  assumes f: "f measurable_on S"
  shows "(\x. c *\<^sub>R f x) measurable_on S"
proof -
  obtain NF and F
    where NF: "negligible NF"
      and conF: "\n. continuous_on UNIV (F n)"
      and tendsF: "\x. x \ NF \ (\n. F n x) \ (if x \ S then f x else 0)"
    using f by (auto simp: measurable_on_def)
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "continuous_on UNIV (\x. c *\<^sub>R F n x)" for n
      by (intro conF continuous_intros)
    show "(\n. c *\<^sub>R F n x) \ (if x \ S then c *\<^sub>R f x else 0)"
      if "x \ NF" for x
      using tendsto_scaleR [OF tendsto_const tendsF, of x] that by auto
  qed (auto simp: NF)
qed


lemma measurable_on_cmul:
  fixes c :: real
  assumes "f measurable_on S"
  shows "(\x. c * f x) measurable_on S"
  using measurable_on_scaleR_const [OF assms] by simp

lemma measurable_on_cdivide:
  fixes c :: real
  assumes "f measurable_on S"
  shows "(\x. f x / c) measurable_on S"
proof (cases "c=0")
  case False
  then show ?thesis
    using measurable_on_cmul [of f S "1/c"]
    by (simp add: assms)
qed auto


lemma measurable_on_minus:
   "f measurable_on S \ (\x. -(f x)) measurable_on S"
  using measurable_on_scaleR_const [of f S "-1"] by auto


lemma continuous_imp_measurable_on:
   "continuous_on UNIV f \ f measurable_on UNIV"
  unfolding measurable_on_def
  apply (rule_tac x="{}" in exI)
  apply (rule_tac x="\n. f" in exI, auto)
  done

proposition integrable_subintervals_imp_measurable:
  fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
  assumes "\a b. f integrable_on cbox a b"
  shows "f measurable_on UNIV"
proof -
  define BOX where "BOX \ \h. \x::'a. cbox x (x + h *\<^sub>R One)"
  define i where "i \ \h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)"
  obtain N where "negligible N"
    and k: "\x e. \x \ N; 0 < e\
            ==> ∃d>0. ∀h. 0 < h ∧ h < d ⟶
                  norm (integral (cbox x (x + h *🚫R One)) f /🚫R h ^ DIM('a) - f x) < e"
    using integrable_ccontinuous_explicit assms by blast
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "continuous_on UNIV ((\n x. i (inverse(Suc n)) x) n)" for n
    proof (clarsimp simp: continuous_on_iff)
      show "\d>0. \x'. dist x' x < d \ dist (i (inverse (1 + real n)) x') (i (inverse (1 + real n)) x) < e"
        if "0 < e"
        for x e
      proof -
        let ?e = "e / (1 + real n) ^ DIM('a)"
        have "?e > 0"
          using ‹e > 0› by auto
        moreover have "x \ cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
          by (simp add: mem_box inner_diff_left inner_left_distrib)
        moreover have "x + One /\<^sub>R real (Suc n) \ cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
          by (auto simp: mem_box inner_diff_left inner_left_distrib field_simps)
        ultimately obtain δ where "\ > 0"
          and δ: "\c' d'. \c' \ cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One);
                           d' \ cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One);
                           norm(c' - x) \ \; norm(d' - (x + One /🚫R Suc n)) ≤ δ]
                          ==> norm(integral(cbox c' d') f - integral(cbox x (x + One /🚫R Suc n)) f) < ?e"
          by (blast intro: indefinite_integral_continuous [of f _ _ x] assms)
        show ?thesis
        proof (intro exI impI conjI allI)
          show "min \ 1 > 0"
            using ‹δ > 0› by auto
          show "dist (i (inverse (1 + real n)) y) (i (inverse (1 + real n)) x) < e"
            if "dist y x < min \ 1" for y
          proof -
            have no: "norm (y - x) < 1"
              using that by (auto simp: dist_norm)
            have le1: "inverse (1 + real n) \ 1"
              by (auto simp: field_split_simps)
            have "norm (integral (cbox y (y + One /\<^sub>R real (Suc n))) f
                - integral (cbox x (x + One /🚫R real (Suc n))) f)
                < e / (1 + real n) ^ DIM('a)"
            proof (rule δ)
              show "y \ cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
                using no by (auto simp: mem_box algebra_simps dest: Basis_le_norm [of _ "y-x"])
              show "y + One /\<^sub>R real (Suc n) \ cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
              proof (simp add: dist_norm mem_box algebra_simps, intro ballI conjI)
                fix i::'a
                assume "i \ Basis"
                then have 1: "\y \ i - x \ i\ < 1"
                  by (metis inner_commute inner_diff_right no norm_bound_Basis_lt)
                moreover have "\ < (2 + inverse (1 + real n))" "1 \ 2 - inverse (1 + real n)"
                  by (auto simp: field_simps)
                ultimately show "x \ i \ y \ i + (2 + inverse (1 + real n))"
                                "y \ i + inverse (1 + real n) \ x \ i + 2"
                  by linarith+
              qed
              show "norm (y - x) \ \" "norm (y + One /\<^sub>R real (Suc n) - (x + One /\<^sub>R real (Suc n))) \ \"
                using that by (auto simp: dist_norm)
            qed
            then show ?thesis
              using that by (simp add: dist_norm i_def BOX_def flip: scaleR_diff_right) (simp add: field_simps)
          qed
        qed
      qed
    qed
    show "negligible N"
      by (simp add: ‹negligible N›)
    show "(\n. i (inverse (Suc n)) x) \ (if x \ UNIV then f x else 0)"
      if "x \ N" for x
      unfolding lim_sequentially
    proof clarsimp
      show "\no. \n\no. dist (i (inverse (1 + real n)) x) (f x) < e"
        if "0 < e" for e
      proof -
        obtain d where "d > 0"
          and d: "\h. \0 < h; h < d\ \
              norm (integral (cbox x (x + h *🚫R One)) f /🚫R h ^ DIM('a) - f x) < e"
          using k [of x e] ‹x ∉ N› ‹0 < e› by blast
        then obtain M where M: "M \ 0" "0 < inverse (real M)" "inverse (real M) < d"
          using real_arch_invD by auto
        show ?thesis
        proof (intro exI allI impI)
          show "dist (i (inverse (1 + real n)) x) (f x) < e"
            if "M \ n" for n
          proof -
            have *: "0 < inverse (1 + real n)" "inverse (1 + real n) \ inverse M"
              using that ‹M ≠ 0› by auto
            show ?thesis
              using that M
              apply (simp add: i_def BOX_def dist_norm)
              apply (blast intro: le_less_trans * d)
              done
          qed
        qed
      qed
    qed
  qed
qed


subsection‹Composing continuous and measurable functions; a few variants›

lemma measurable_on_compose_continuous:
   assumes f: "f measurable_on UNIV" and g: "continuous_on UNIV g"
   shows "(g \ f) measurable_on UNIV"
proof -
  obtain N and F
    where "negligible N"
      and conF: "\n. continuous_on UNIV (F n)"
      and tendsF: "\x. x \ N \ (\n. F n x) \ f x"
    using f by (auto simp: measurable_on_def)
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "negligible N"
      by fact
    show "continuous_on UNIV (g \ (F n))" for n
      using conF continuous_on_compose continuous_on_subset g by blast
    show "(\n. (g \ F n) x) \ (if x \ UNIV then (g \ f) x else 0)"
      if "x \ N" for x :: 'a
      using that g tendsF by (auto simp: continuous_on_def intro: tendsto_compose)
  qed
qed

lemma measurable_on_compose_continuous_0:
   assumes f: "f measurable_on S" and g: "continuous_on UNIV g" and "g 0 = 0"
   shows "(g \ f) measurable_on S"
proof -
  have f': "(\x. if x \ S then f x else 0) measurable_on UNIV"
    using f measurable_on_UNIV by blast
  show ?thesis
    using measurable_on_compose_continuous [OF f' g]
    by (simp add: measurable_on_UNIV o_def if_distrib ‹g 0 = 0› cong: if_cong)
qed


lemma measurable_on_compose_continuous_box:
  assumes fm: "f measurable_on UNIV" and fab: "\x. f x \ box a b"
    and contg: "continuous_on (box a b) g"
  shows "(g \ f) measurable_on UNIV"
proof -
  have "\\. (\n. continuous_on UNIV (\ n)) \ (\x. x \ N \ (\n. \ n x) \ g (f x))"
    if "negligible N"
      and conth [rule_format]: "\n. continuous_on UNIV (\x. h n x)"
      and tends [rule_format]: "\x. x \ N \ (\n. h n x) \ f x"
    for N and h :: "nat \ 'a \ 'b"
  proof -
    define θ where "\ \ \n x. (\i\Basis. (max (a\i + (b\i - a\i) / real (n+2))
                                            (min ((h n x)∙i)
                                                 (b∙i - (b∙i - a∙i) / real (n+2)))) *🚫R i)"
    have aibi: "\i. i \ Basis \ a \ i < b \ i"
      using box_ne_empty(2) fab by auto
    then have *: "\i n. i \ Basis \ a \ i + real n * (a \ i) < b \ i + real n * (b \ i)"
      by (meson add_mono_thms_linordered_field(3) less_eq_real_def mult_left_mono of_nat_0_le_iff)
    show ?thesis
    proof (intro exI conjI allI impI)
      show "continuous_on UNIV (g \ (\ n))" for n :: nat
        unfolding θ_def
        apply (intro continuous_on_compose2 [OF contg] continuous_intros conth)
        apply (auto simp: aibi * mem_box less_max_iff_disj min_less_iff_disj field_split_simps)
        done
      show "(\n. (g \ \ n) x) \ g (f x)"
        if "x \ N" for x
        unfolding o_def
      proof (rule isCont_tendsto_compose [where g=g])
        show "isCont g (f x)"
          using contg fab continuous_on_eq_continuous_at by blast
        have "(\n. \ n x) \ (\i\Basis. max (a \ i) (min (f x \ i) (b \ i)) *\<^sub>R i)"
          unfolding θ_def
        proof (intro tendsto_intros ‹x ∉ N› tends)
          fix i::'b
          assume "i \ Basis"
          have a: "(\n. a \ i + (b \ i - a \ i) / real n) \ a\i + 0"
            by (intro tendsto_add lim_const_over_n tendsto_const)
          show "(\n. a \ i + (b \ i - a \ i) / real (n + 2)) \ a \ i"
            using LIMSEQ_ignore_initial_segment [where k=2, OF a] by simp
          have b: "(\n. b\i - (b \ i - a \ i) / (real n)) \ b\i - 0"
            by (intro tendsto_diff lim_const_over_n tendsto_const)
          show "(\n. b \ i - (b \ i - a \ i) / real (n + 2)) \ b \ i"
            using LIMSEQ_ignore_initial_segment [where k=2, OF b] by simp
        qed
        also have "(\i\Basis. max (a \ i) (min (f x \ i) (b \ i)) *\<^sub>R i) = (\i\Basis. (f x \ i) *\<^sub>R i)"
          using fab by (auto simp add: mem_box intro: sum.cong)
        also have "\ = f x"
          using euclidean_representation by blast
        finally show "(\n. \ n x) \ f x" .
      qed
    qed
  qed
  then show ?thesis
    using fm by (auto simp: measurable_on_def)
qed

lemma measurable_on_Pair:
  assumes f: "f measurable_on S" and g: "g measurable_on S"
  shows "(\x. (f x, g x)) measurable_on S"
proof -
  obtain NF and F
    where NF: "negligible NF"
      and conF: "\n. continuous_on UNIV (F n)"
      and tendsF: "\x. x \ NF \ (\n. F n x) \ (if x \ S then f x else 0)"
    using f by (auto simp: measurable_on_def)
  obtain NG and G
    where NG: "negligible NG"
      and conG: "\n. continuous_on UNIV (G n)"
      and tendsG: "\x. x \ NG \ (\n. G n x) \ (if x \ S then g x else 0)"
    using g by (auto simp: measurable_on_def)
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "negligible (NF \ NG)"
      by (simp add: NF NG)
    show "continuous_on UNIV (\x. (F n x, G n x))" for n
      using conF conG continuous_on_Pair by blast
    show "(\n. (F n x, G n x)) \ (if x \ S then (f x, g x) else 0)"
      if "x \ NF \ NG" for x
      using tendsto_Pair [OF tendsF tendsG, of x x] that unfolding zero_prod_def
      by (simp add: split: if_split_asm)
  qed
qed

lemma measurable_on_combine:
  assumes f: "f measurable_on S" and g: "g measurable_on S"
    and h: "continuous_on UNIV (\x. h (fst x) (snd x))" and "h 0 0 = 0"
  shows "(\x. h (f x) (g x)) measurable_on S"
proof -
  have *: "(\x. h (f x) (g x)) = (\x. h (fst x) (snd x)) \ (\x. (f x, g x))"
    by auto
  show ?thesis
    unfolding * by (auto simp: measurable_on_compose_continuous_0 measurable_on_Pair assms)
qed

lemma measurable_on_add:
  assumes f: "f measurable_on S" and g: "g measurable_on S"
  shows "(\x. f x + g x) measurable_on S"
  by (intro continuous_intros measurable_on_combine [OF assms]) auto

lemma measurable_on_diff:
  assumes f: "f measurable_on S" and g: "g measurable_on S"
  shows "(\x. f x - g x) measurable_on S"
  by (intro continuous_intros measurable_on_combine [OF assms]) auto

lemma measurable_on_scaleR:
  assumes f: "f measurable_on S" and g: "g measurable_on S"
  shows "(\x. f x *\<^sub>R g x) measurable_on S"
  by (intro continuous_intros measurable_on_combine [OF assms]) auto

lemma measurable_on_sum:
  assumes "finite I" "\i. i \ I \ f i measurable_on S"
  shows "(\x. sum (\i. f i x) I) measurable_on S"
  using assms by (induction I) (auto simp: measurable_on_add)

lemma measurable_on_spike:
  assumes f: "f measurable_on T" and "negligible S" and gf: "\x. x \ T - S \ g x = f x"
  shows "g measurable_on T"
proof -
  obtain NF and F
    where NF: "negligible NF"
      and conF: "\n. continuous_on UNIV (F n)"
      and tendsF: "\x. x \ NF \ (\n. F n x) \ (if x \ T then f x else 0)"
    using f by (auto simp: measurable_on_def)
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "negligible (NF \ S)"
      by (simp add: NF ‹negligible S›)
    show "\x. x \ NF \ S \ (\n. F n x) \ (if x \ T then g x else 0)"
      by (metis (full_types) Diff_iff Un_iff gf tendsF)
  qed (auto simp: conF)
qed

proposition indicator_measurable_on:
  assumes "S \ sets lebesgue"
  shows "indicat_real S measurable_on UNIV"
proof -
  { fix n::nat
    let ?ε = "(1::real) / (2 * 2^n)"
    have ε: "?\ > 0"
      by auto
    obtain T where "closed T" "T \ S" "S-T \ lmeasurable" and ST: "emeasure lebesgue (S - T) < ?\"
      by (meson ε assms sets_lebesgue_inner_closed)
    obtain U where "open U" "S \ U" "(U - S) \ lmeasurable" and US: "emeasure lebesgue (U - S) < ?\"
      by (meson ε assms sets_lebesgue_outer_open)
    have eq: "-T \ U = (S-T) \ (U - S)"
      using ‹T ⊆ S› ‹S ⊆ U› by auto
    have "emeasure lebesgue ((S-T) \ (U - S)) \ emeasure lebesgue (S - T) + emeasure lebesgue (U - S)"
      using ‹S - T ∈ lmeasurable› ‹U - S ∈ lmeasurable› emeasure_subadditive by blast
    also have "\ < ?\ + ?\"
      using ST US add_mono_ennreal by metis
    finally have le: "emeasure lebesgue (-T \ U) < ennreal (1 / 2^n)"
      by (simp add: eq)
    have 1: "continuous_on (T \ -U) (indicat_real S)"
      unfolding indicator_def of_bool_def
    proof (rule continuous_on_cases [OF ‹closed T›])
      show "closed (- U)"
        using ‹open U› by blast
      show "continuous_on T (\x. 1::real)" "continuous_on (- U) (\x. 0::real)"
        by (auto simp: continuous_on)
      show "\x. x \ T \ x \ S \ x \ - U \ x \ S \ (1::real) = 0"
        using ‹T ⊆ S› ‹S ⊆ U› by auto
    qed
    have 2: "closedin (top_of_set UNIV) (T \ -U)"
      using ‹closed T› ‹open U› by auto
    obtain g where "continuous_on UNIV g" "\x. x \ T \ -U \ g x = indicat_real S x" "\x. norm(g x) \ 1"
      by (rule Tietze [OF 1 2, of 1]) auto
    with le have "\g E. continuous_on UNIV g \ (\x \ -E. g x = indicat_real S x) \
                        (∀x. norm(g x) ≤ 1) ∧ E ∈ sets lebesgue ∧ emeasure lebesgue E < ennreal (1 / 2^n)"
      apply (rule_tac x=g in exI)
      apply (rule_tac x="-T \ U" in exI)
      using ‹S - T ∈ lmeasurable› ‹U - S ∈ lmeasurable› eq by auto
  }
  then obtain g E where cont: "\n. continuous_on UNIV (g n)"
    and geq: "\n x. x \ - E n \ g n x = indicat_real S x"
    and ng1: "\n x. norm(g n x) \ 1"
    and Eset: "\n. E n \ sets lebesgue"
    and Em: "\n. emeasure lebesgue (E n) < ennreal (1 / 2^n)"
    by metis
  have null: "limsup E \ null_sets lebesgue"
  proof (rule borel_cantelli_limsup1 [OF Eset])
    show "emeasure lebesgue (E n) < \" for n
      by (metis Em infinity_ennreal_def order.asym top.not_eq_extremum)
    show "summable (\n. measure lebesgue (E n))"
    proof (rule summable_comparison_test' [OF summable_geometric, of "1/2" 0])
      show "norm (measure lebesgue (E n)) \ (1/2) ^ n"  for n
        using Em [of n] by (simp add: measure_def enn2real_leI power_one_over)
    qed auto
  qed
  have tends: "(\n. g n x) \ indicat_real S x" if "x \ limsup E" for x
  proof -
    have "\\<^sub>F n in sequentially. x \ - E n"
      using that by (simp add: mem_limsup_iff not_frequently)
    then show ?thesis
      unfolding tendsto_iff dist_real_def
      by (simp add: eventually_mono geq)
  qed
  show ?thesis
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "negligible (limsup E)"
      using negligible_iff_null_sets null by blast
    show "continuous_on UNIV (g n)" for n
      using cont by blast
  qed (use tends in auto)
qed

lemma measurable_on_restrict:
  assumes f: "f measurable_on UNIV" and S: "S \ sets lebesgue"
  shows "(\x. if x \ S then f x else 0) measurable_on UNIV"
proof -
  have "indicat_real S measurable_on UNIV"
    by (simp add: S indicator_measurable_on)
  then show ?thesis
    using measurable_on_scaleR [OF _ f, of "indicat_real S"]
    by (simp add: indicator_scaleR_eq_if)
qed

lemma measurable_on_const_UNIV: "(\x. k) measurable_on UNIV"
  by (simp add: continuous_imp_measurable_on)

lemma measurable_on_const [simp]: "S \ sets lebesgue \ (\x. k) measurable_on S"
  using measurable_on_UNIV measurable_on_const_UNIV measurable_on_restrict by blast

lemma simple_function_indicator_representation_real:
  fixes f ::"'a \ real"
  assumes f: "simple_function M f" and x: "x \ space M" and nn: "\x. f x \ 0"
  shows "f x = (\y \ f ` space M. y * indicator (f -` {y} \ space M) x)"
proof -
  have f': "simple_function M (ennreal \ f)"
    by (simp add: f)
  have *: "f x =
     enn2real
      (∑y∈ ennreal ` f ` space M.
         y * indicator ((ennreal ∘ f) -` {y} ∩ space M) x)"
    using arg_cong [OF simple_function_indicator_representation [OF f' x], of enn2real, simplified nn o_def] nn
    unfolding o_def image_comp
    by (metis enn2real_ennreal)
  have "enn2real (\y\ennreal ` f ` space M. if ennreal (f x) = y \ x \ space M then y else 0)
      = sum (enn2real ∘ (λy. if ennreal (f x) = y ∧ x ∈ space M then y else 0))
            (ennreal ` f ` space M)"
    by (rule enn2real_sum) auto
  also have "\ = sum (enn2real \ (\y. if ennreal (f x) = y \ x \ space M then y else 0) \ ennreal)
                   (f ` space M)"
    by (rule sum.reindex) (use nn in ‹auto simp: inj_on_def intro: sum.cong›)
  also have "\ = (\y\f ` space M. if f x = y \ x \ space M then y else 0)"
    using nn
    by (auto simp: inj_on_def intro: sum.cong)
  finally show ?thesis
    by (subst *) (simp add: enn2real_sum indicator_def of_bool_def if_distrib cong: if_cong)
qed

lemma🍋‹tag important› simple_function_induct_real
    [consumes 1, case_names cong set mult add, induct set: simple_function]:
  fixes u :: "'a \ real"
  assumes u: "simple_function M u"
  assumes cong: "\f g. simple_function M f \ simple_function M g \ (AE x in M. f x = g x) \ P f \ P g"
  assumes set: "\A. A \ sets M \ P (indicator A)"
  assumes mult: "\u c. P u \ P (\x. c * u x)"
  assumes add: "\u v. P u \ P v \ P (\x. u x + v x)"
  and nn: "\x. u x \ 0"
  shows "P u"
proof (rule cong)
  from AE_space show "AE x in M. (\y\u ` space M. y * indicator (u -` {y} \ space M) x) = u x"
  proof eventually_elim
    fix x assume x: "x \ space M"
    from simple_function_indicator_representation_real[OF u x] nn
    show "(\y\u ` space M. y * indicator (u -` {y} \ space M) x) = u x"
      by metis
  qed
next
  from u have "finite (u ` space M)"
    unfolding simple_function_def by auto
  then show "P (\x. \y\u ` space M. y * indicator (u -` {y} \ space M) x)"
  proof induct
    case empty
    then show ?case
      using set[of "{}"] by (simp add: indicator_def[abs_def])
  next
    case (insert a F)
    have eq: "\ {y. u x = y \ (y = a \ y \ F) \ x \ space M}
            = (if u x = a ∧ x ∈ space M then a else 0) + ∑ {y. u x = y ∧ y ∈ F ∧ x ∈ space M}" for x
    proof (cases "x \ space M")
      case True
      have *: "{y. u x = y \ (y = a \ y \ F)} = {y. u x = a \ y = a} \ {y. u x = y \ y \ F}"
        by auto
      show ?thesis
        using insert by (simp add: * True)
    qed auto
    have a: "P (\x. a * indicator (u -` {a} \ space M) x)"
    proof (intro mult set)
      show "u -` {a} \ space M \ sets M"
        using u by auto
    qed
    show ?case
      using nn insert a
      by (simp add: eq indicator_times_eq_if [where f = "\x. a"] add)
  qed
next
  show "simple_function M (\x. (\y\u ` space M. y * indicator (u -` {y} \ space M) x))"
    apply (subst simple_function_cong)
    apply (rule simple_function_indicator_representation_real[symmetric])
    apply (auto intro: u nn)
    done
qed fact

proposition simple_function_measurable_on_UNIV:
  fixes f :: "'a::euclidean_space \ real"
  assumes f: "simple_function lebesgue f" and nn: "\x. f x \ 0"
  shows "f measurable_on UNIV"
  using f
proof (induction f)
  case (cong f g)
  then obtain N where "negligible N" "{x. g x \ f x} \ N"
    by (auto simp: eventually_ae_filter_negligible eq_commute)
  then show ?case
    by (blast intro: measurable_on_spike cong)
next
  case (set S)
  then show ?case
    by (simp add: indicator_measurable_on)
next
  case (mult u c)
  then show ?case
    by (simp add: measurable_on_cmul)
  case (add u v)
  then show ?case
    by (simp add: measurable_on_add)
qed (auto simp: nn)

lemma simple_function_lebesgue_if:
  fixes f :: "'a::euclidean_space \ real"
  assumes f: "simple_function lebesgue f" and S: "S \ sets lebesgue"
  shows "simple_function lebesgue (\x. if x \ S then f x else 0)"
proof -
  have ffin: "finite (range f)" and fsets: "\x. f -` {f x} \ sets lebesgue"
    using f by (auto simp: simple_function_def)
  have "finite (f ` S)"
    by (meson finite_subset subset_image_iff ffin top_greatest)
  moreover have "finite ((\x. 0::real) ` T)" for T :: "'a set"
    by (auto simp: image_def)
  moreover have if_sets: "(\x. if x \ S then f x else 0) -` {f a} \ sets lebesgue" for a
  proof -
    have *: "(\x. if x \ S then f x else 0) -` {f a}
           = (if f a = 0 then -S ∪ f -` {f a} else (f -` {f a}) ∩ S)"
      by (auto simp: split: if_split_asm)
    show ?thesis
      unfolding * by (metis Compl_in_sets_lebesgue S sets.Int sets.Un fsets)
  qed
  moreover have "(\x. if x \ S then f x else 0) -` {0} \ sets lebesgue"
  proof (cases "0 \ range f")
    case True
    then show ?thesis
      by (metis (no_types, lifting) if_sets rangeE)
  next
    case False
    then have "(\x. if x \ S then f x else 0) -` {0} = -S"
      by auto
    then show ?thesis
      by (simp add: Compl_in_sets_lebesgue S)
  qed
  ultimately show ?thesis
    by (auto simp: simple_function_def)
qed

corollary simple_function_measurable_on:
  fixes f :: "'a::euclidean_space \ real"
  assumes f: "simple_function lebesgue f" and nn: "\x. f x \ 0" and S: "S \ sets lebesgue"
  shows "f measurable_on S"
  by (simp add: measurable_on_UNIV [symmetric, of f] S f simple_function_lebesgue_if nn simple_function_measurable_on_UNIV)

lemma
  fixes f :: "'a::euclidean_space \ 'b::ordered_euclidean_space"
  assumes f: "f measurable_on S" and g: "g measurable_on S"
  shows measurable_on_sup: "(\x. sup (f x) (g x)) measurable_on S"
  and   measurable_on_inf: "(\x. inf (f x) (g x)) measurable_on S"
proof -
  obtain NF and F
    where NF: "negligible NF"
      and conF: "\n. continuous_on UNIV (F n)"
      and tendsF: "\x. x \ NF \ (\n. F n x) \ (if x \ S then f x else 0)"
    using f by (auto simp: measurable_on_def)
  obtain NG and G
    where NG: "negligible NG"
      and conG: "\n. continuous_on UNIV (G n)"
      and tendsG: "\x. x \ NG \ (\n. G n x) \ (if x \ S then g x else 0)"
    using g by (auto simp: measurable_on_def)
  show "(\x. sup (f x) (g x)) measurable_on S"
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "continuous_on UNIV (\x. sup (F n x) (G n x))" for n
      unfolding sup_max eucl_sup  by (intro conF conG continuous_intros)
    show "(\n. sup (F n x) (G n x)) \ (if x \ S then sup (f x) (g x) else 0)"
      if "x \ NF \ NG" for x
      using tendsto_sup [OF tendsF tendsG, of x x] that by auto
  qed (simp add: NF NG)
  show "(\x. inf (f x) (g x)) measurable_on S"
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "continuous_on UNIV (\x. inf (F n x) (G n x))" for n
      unfolding inf_min eucl_inf  by (intro conF conG continuous_intros)
    show "(\n. inf (F n x) (G n x)) \ (if x \ S then inf (f x) (g x) else 0)"
      if "x \ NF \ NG" for x
      using tendsto_inf [OF tendsF tendsG, of x x] that by auto
  qed (simp add: NF NG)
qed

proposition measurable_on_componentwise_UNIV:
  "f measurable_on UNIV \ (\i\Basis. (\x. (f x \ i) *\<^sub>R i) measurable_on UNIV)"
  (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  show ?rhs
  proof
    fix i::'b
    assume "i \ Basis"
    have cont: "continuous_on UNIV (\x. (x \ i) *\<^sub>R i)"
      by (intro continuous_intros)
    show "(\x. (f x \ i) *\<^sub>R i) measurable_on UNIV"
      using measurable_on_compose_continuous [OF L cont]
      by (simp add: o_def)
  qed
next
  assume ?rhs
  then have "\N g. negligible N \
              (∀n. continuous_on UNIV (g n)) ∧
              (∀x. x ∉ N ⟶ (λn. g n x) <---- (f x ∙ i) *🚫R i)"
    if "i \ Basis" for i
    by (simp add: measurable_on_def that)
  then obtain N g where N: "\i. i \ Basis \ negligible (N i)"
        and cont: "\i n. i \ Basis \ continuous_on UNIV (g i n)"
        and tends: "\i x. \i \ Basis; x \ N i\ \ (\n. g i n x) \ (f x \ i) *\<^sub>R i"
    by metis
  show ?lhs
    unfolding measurable_on_def
  proof (intro exI conjI allI impI)
    show "negligible (\i \ Basis. N i)"
      using N eucl.finite_Basis by blast
    show "continuous_on UNIV (\x. (\i\Basis. g i n x))" for n
      by (intro continuous_intros cont)
  next
    fix x
    assume "x \ (\i \ Basis. N i)"
    then have "\i. i \ Basis \ x \ N i"
      by auto
    then have "(\n. (\i\Basis. g i n x)) \ (\i\Basis. (f x \ i) *\<^sub>R i)"
      by (intro tends tendsto_intros)
    then show "(\n. (\i\Basis. g i n x)) \ (if x \ UNIV then f x else 0)"
      by (simp add: euclidean_representation)
  qed
qed

corollary measurable_on_componentwise:
  "f measurable_on S \ (\i\Basis. (\x. (f x \ i) *\<^sub>R i) measurable_on S)"
  apply (subst measurable_on_UNIV [symmetric])
  apply (subst measurable_on_componentwise_UNIV)
  apply (simp add: measurable_on_UNIV if_distrib [of "\x. inner x _"] if_distrib [of "\x. scaleR x _"] cong: if_cong)
  done


(*FIXME: avoid duplication of proofs WRT borel_measurable_implies_simple_function_sequence*)
lemma🍋‹tag important› borel_measurable_implies_simple_function_sequence_real:
  fixes u :: "'a \ real"
  assumes u[measurable]: "u \ borel_measurable M" and nn: "\x. u x \ 0"
  shows "\f. incseq f \ (\i. simple_function M (f i)) \ (\x. bdd_above (range (\i. f i x))) \
             (∀i x. 0 ≤ f i x) ∧ u = (SUP i. f i)"
proof -
  define f where [abs_def]:
    "f i x = real_of_int (floor ((min i (u x)) * 2^i)) / 2^i" for i x

  have [simp]: "0 \ f i x" for i x
    by (auto simp: f_def intro!: divide_nonneg_nonneg mult_nonneg_nonneg nn)

  have *: "2^n * real_of_int x = real_of_int (2^n * x)" for n x
    by simp

  have "real_of_int \real i * 2 ^ i\ = real_of_int \i * 2 ^ i\" for i
    by (intro arg_cong[where f=real_of_int]) simp
  then have [simp]: "real_of_int \real i * 2 ^ i\ = i * 2 ^ i" for i
    unfolding floor_of_nat by simp

  have bdd: "bdd_above (range (\i. f i x))" for x
    by (rule bdd_aboveI [where M = "u x"]) (auto simp: f_def field_simps min_def)

  have "incseq f"
  proof (intro monoI le_funI)
    fix m n :: nat and x assume "m \ n"
    moreover
    { fix d :: nat
      have "\2^d::real\ * \2^m * (min (of_nat m) (u x))\ \ \2^d * (2^m * (min (of_nat m) (u x)))\"
        by (rule le_mult_floor) (auto simp: nn)
      also have "\ \ \2^d * (2^m * (min (of_nat d + of_nat m) (u x)))\"
        by (intro floor_mono mult_mono min.mono)
           (auto simp: nn min_less_iff_disj of_nat_less_top)
      finally have "f m x \ f(m + d) x"
        unfolding f_def
        by (auto simp: field_simps power_add * simp del: of_int_mult) }
    ultimately show "f m x \ f n x"
      by (auto simp: le_iff_add)
  qed
  then have inc_f: "incseq (\i. f i x)" for x
    by (auto simp: incseq_def le_fun_def)
  moreover
  have "simple_function M (f i)" for i
  proof (rule simple_function_borel_measurable)
    have "\(min (of_nat i) (u x)) * 2 ^ i\ \ \int i * 2 ^ i\" for x
      by (auto split: split_min intro!: floor_mono)
    then have "f i ` space M \ (\n. real_of_int n / 2^i) ` {0 .. of_nat i * 2^i}"
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