src/HOL/Analysis/Lebesgue_Measure.thy

More analysis / measure theory material

(*  Title:      HOL/Analysis/Lebesgue_Measure.thy
    Author:     Johannes Hölzl, TU München
    Author:     Robert Himmelmann, TU München
    Author:     Jeremy Avigad
    Author:     Luke Serafin
*)

section \<open>Lebesgue Measure\<close>

theory Lebesgue_Measure
imports
  Finite_Product_Measure
  Caratheodory
  Complete_Measure
  Summation_Tests
  Regularity
begin

lemma measure_eqI_lessThan:
  fixes M N :: "real measure"
  assumes sets: "sets M = sets borel" "sets N = sets borel"
  assumes fin: "\<And>x. emeasure M {x <..} < \<infinity>"
  assumes "\<And>x. emeasure M {x <..} = emeasure N {x <..}"
  shows "M = N"
proof (rule measure_eqI_generator_eq_countable)
  let ?LT = "\<lambda>a::real. {a <..}" let ?E = "range ?LT"
  show "Int_stable ?E"
    by (auto simp: Int_stable_def lessThan_Int_lessThan)

  show "?E \<subseteq> Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
    unfolding sets borel_Ioi by auto

  show "?LT`Rats \<subseteq> ?E" "(\<Union>i\<in>Rats. ?LT i) = UNIV" "\<And>a. a \<in> ?LT`Rats \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
    using fin by (auto intro: Rats_no_bot_less simp: less_top)
qed (auto intro: assms countable_rat)

subsection \<open>Measures defined by monotonous functions\<close>

text \<open>
  Every right-continuous and nondecreasing function gives rise to a measure on the reals:
\<close>

definition\<^marker>\<open>tag important\<close> interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
  "interval_measure F =
     extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a<..b}) (\<lambda>(a, b). ennreal (F b - F a))"

lemma\<^marker>\<open>tag important\<close> emeasure_interval_measure_Ioc:
  assumes "a \<le> b"
  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
  assumes right_cont_F : "\<And>a. continuous (at_right a) F"
  shows "emeasure (interval_measure F) {a<..b} = F b - F a"
proof (rule extend_measure_caratheodory_pair[OF interval_measure_def \<open>a \<le> b\<close>])
  show "semiring_of_sets UNIV {{a<..b} |a b :: real. a \<le> b}"
  proof (unfold_locales, safe)
    fix a b c d :: real assume *: "a \<le> b" "c \<le> d"
    then show "\<exists>C\<subseteq>{{a<..b} |a b. a \<le> b}. finite C \<and> disjoint C \<and> {a<..b} - {c<..d} = \<Union>C"
    proof cases
      let ?C = "{{a<..b}}"
      assume "b < c \<or> d \<le> a \<or> d \<le> c"
      with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
        by (auto simp add: disjoint_def)
      thus ?thesis ..
    next
      let ?C = "{{a<..c}, {d<..b}}"
      assume "\<not> (b < c \<or> d \<le> a \<or> d \<le> c)"
      with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
        by (auto simp add: disjoint_def Ioc_inj) (metis linear)+
      thus ?thesis ..
    qed
  qed (auto simp: Ioc_inj, metis linear)
next
  fix l r :: "nat \<Rightarrow> real" and a b :: real
  assume l_r[simp]: "\<And>n. l n \<le> r n" and "a \<le> b" and disj: "disjoint_family (\<lambda>n. {l n<..r n})"
  assume lr_eq_ab: "(\<Union>i. {l i<..r i}) = {a<..b}"

  have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> F a \<le> F b"
    by (auto intro!: l_r mono_F)

  { fix S :: "nat set" assume "finite S"
    moreover note \<open>a \<le> b\<close>
    moreover have "\<And>i. i \<in> S \<Longrightarrow> {l i <.. r i} \<subseteq> {a <.. b}"
      unfolding lr_eq_ab[symmetric] by auto
    ultimately have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
    proof (induction S arbitrary: a rule: finite_psubset_induct)
      case (psubset S)
      show ?case
      proof cases
        assume "\<exists>i\<in>S. l i < r i"
        with \<open>finite S\<close> have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
          by (intro Min_in) auto
        then obtain m where m: "m \<in> S" "l m < r m" "l m = Min (l ` {i\<in>S. l i < r i})"
          by fastforce

        have "(\<Sum>i\<in>S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\<Sum>i\<in>S - {m}. F (r i) - F (l i))"
          using m psubset by (intro sum.remove) auto
        also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
        proof (intro psubset.IH)
          show "S - {m} \<subset> S"
            using \<open>m\<in>S\<close> by auto
          show "r m \<le> b"
            using psubset.prems(2)[OF \<open>m\<in>S\<close>] \<open>l m < r m\<close> by auto
        next
          fix i assume "i \<in> S - {m}"
          then have i: "i \<in> S" "i \<noteq> m" by auto
          { assume i': "l i < r i" "l i < r m"
            with \<open>finite S\<close> i m have "l m \<le> l i"
              by auto
            with i' have "{l i <.. r i} \<inter> {l m <.. r m} \<noteq> {}"
              by auto
            then have False
              using disjoint_family_onD[OF disj, of i m] i by auto }
          then have "l i \<noteq> r i \<Longrightarrow> r m \<le> l i"
            unfolding not_less[symmetric] using l_r[of i] by auto
          then show "{l i <.. r i} \<subseteq> {r m <.. b}"
            using psubset.prems(2)[OF \<open>i\<in>S\<close>] by auto
        qed
        also have "F (r m) - F (l m) \<le> F (r m) - F a"
          using psubset.prems(2)[OF \<open>m \<in> S\<close>] \<open>l m < r m\<close>
          by (auto simp add: Ioc_subset_iff intro!: mono_F)
        finally show ?case
          by (auto intro: add_mono)
      qed (auto simp add: \<open>a \<le> b\<close> less_le)
    qed }
  note claim1 = this

  (* second key induction: a lower bound on the measures of any finite collection of Ai's
     that cover an interval {u..v} *)

  { fix S u v and l r :: "nat \<Rightarrow> real"
    assume "finite S" "\<And>i. i\<in>S \<Longrightarrow> l i < r i" "{u..v} \<subseteq> (\<Union>i\<in>S. {l i<..< r i})"
    then have "F v - F u \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
    proof (induction arbitrary: v u rule: finite_psubset_induct)
      case (psubset S)
      show ?case
      proof cases
        assume "S = {}" then show ?case
          using psubset by (simp add: mono_F)
      next
        assume "S \<noteq> {}"
        then obtain j where "j \<in> S"
          by auto

        let ?R = "r j < u \<or> l j > v \<or> (\<exists>i\<in>S-{j}. l i \<le> l j \<and> r j \<le> r i)"
        show ?case
        proof cases
          assume "?R"
          with \<open>j \<in> S\<close> psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
            apply (auto simp: subset_eq Ball_def)
            apply (metis Diff_iff less_le_trans leD linear singletonD)
            apply (metis Diff_iff less_le_trans leD linear singletonD)
            apply (metis order_trans less_le_not_le linear)
            done
          with \<open>j \<in> S\<close> have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
            by (intro psubset) auto
          also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
            using psubset.prems
            by (intro sum_mono2 psubset) (auto intro: less_imp_le)
          finally show ?thesis .
        next
          assume "\<not> ?R"
          then have j: "u \<le> r j" "l j \<le> v" "\<And>i. i \<in> S - {j} \<Longrightarrow> r i < r j \<or> l i > l j"
            by (auto simp: not_less)
          let ?S1 = "{i \<in> S. l i < l j}"
          let ?S2 = "{i \<in> S. r i > r j}"

          have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
            by (intro sum_mono2) (auto intro: less_imp_le)
          also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
            (\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
            using psubset(1) psubset.prems(1) j
            apply (subst sum.union_disjoint)
            apply simp_all
            apply (subst sum.union_disjoint)
            apply auto
            apply (metis less_le_not_le)
            done
          also (xtrans) have "(\<Sum>i\<in>?S1. F (r i) - F (l i)) \<ge> F (l j) - F u"
            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
            apply (intro psubset.IH psubset)
            apply (auto simp: subset_eq Ball_def)
            apply (metis less_le_trans not_le)
            done
          also (xtrans) have "(\<Sum>i\<in>?S2. F (r i) - F (l i)) \<ge> F v - F (r j)"
            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
            apply (intro psubset.IH psubset)
            apply (auto simp: subset_eq Ball_def)
            apply (metis le_less_trans not_le)
            done
          finally (xtrans) show ?case
            by (auto simp: add_mono)
        qed
      qed
    qed }
  note claim2 = this

  (* now prove the inequality going the other way *)
  have "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i)))"
  proof (rule ennreal_le_epsilon)
    fix epsilon :: real assume egt0: "epsilon > 0"
    have "\<forall>i. \<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
    proof
      fix i
      note right_cont_F [of "r i"]
      thus "\<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
        apply -
        apply (subst (asm) continuous_at_right_real_increasing)
        apply (rule mono_F, assumption)
        apply (drule_tac x = "epsilon / 2 ^ (i + 2)" in spec)
        apply (erule impE)
        using egt0 by (auto simp add: field_simps)
    qed
    then obtain delta where
        deltai_gt0: "\<And>i. delta i > 0" and
        deltai_prop: "\<And>i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
      by metis
    have "\<exists>a' > a. F a' - F a < epsilon / 2"
      apply (insert right_cont_F [of a])
      apply (subst (asm) continuous_at_right_real_increasing)
      using mono_F apply force
      apply (drule_tac x = "epsilon / 2" in spec)
      using egt0 unfolding mult.commute [of 2] by force
    then obtain a' where a'lea [arith]: "a' > a" and
      a_prop: "F a' - F a < epsilon / 2"
      by auto
    define S' where "S' = {i. l i < r i}"
    obtain S :: "nat set" where
      "S \<subseteq> S'" and finS: "finite S" and
      Sprop: "{a'..b} \<subseteq> (\<Union>i \<in> S. {l i<..<r i + delta i})"
    proof (rule compactE_image)
      show "compact {a'..b}"
        by (rule compact_Icc)
      show "\<And>i. i \<in> S' \<Longrightarrow> open ({l i<..<r i + delta i})" by auto
      have "{a'..b} \<subseteq> {a <.. b}"
        by auto
      also have "{a <.. b} = (\<Union>i\<in>S'. {l i<..r i})"
        unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
      also have "\<dots> \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})"
        apply (intro UN_mono)
        apply (auto simp: S'_def)
        apply (cut_tac i=i in deltai_gt0)
        apply simp
        done
      finally show "{a'..b} \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})" .
    qed
    with S'_def have Sprop2: "\<And>i. i \<in> S \<Longrightarrow> l i < r i" by auto
    from finS have "\<exists>n. \<forall>i \<in> S. i \<le> n"
      by (subst finite_nat_set_iff_bounded_le [symmetric])
    then obtain n where Sbound [rule_format]: "\<forall>i \<in> S. i \<le> n" ..
    have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i + delta i) - F (l i))"
      apply (rule claim2 [rule_format])
      using finS Sprop apply auto
      apply (frule Sprop2)
      apply (subgoal_tac "delta i > 0")
      apply arith
      by (rule deltai_gt0)
    also have "... \<le> (\<Sum>i \<in> S. F(r i) - F(l i) + epsilon / 2^(i+2))"
      apply (rule sum_mono)
      apply simp
      apply (rule order_trans)
      apply (rule less_imp_le)
      apply (rule deltai_prop)
      by auto
    also have "... = (\<Sum>i \<in> S. F(r i) - F(l i)) +
        (epsilon / 4) * (\<Sum>i \<in> S. (1 / 2)^i)" (is "_ = ?t + _")
      by (subst sum.distrib) (simp add: field_simps sum_distrib_left)
    also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
      apply (rule add_left_mono)
      apply (rule mult_left_mono)
      apply (rule sum_mono2)
      using egt0 apply auto
      by (frule Sbound, auto)
    also have "... \<le> ?t + (epsilon / 2)"
      apply (rule add_left_mono)
      apply (subst geometric_sum)
      apply auto
      apply (rule mult_left_mono)
      using egt0 apply auto
      done
    finally have aux2: "F b - F a' \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2"
      by simp

    have "F b - F a = (F b - F a') + (F a' - F a)"
      by auto
    also have "... \<le> (F b - F a') + epsilon / 2"
      using a_prop by (intro add_left_mono) simp
    also have "... \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
      apply (intro add_right_mono)
      apply (rule aux2)
      done
    also have "... = (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon"
      by auto
    also have "... \<le> (\<Sum>i\<le>n. F (r i) - F (l i)) + epsilon"
      using finS Sbound Sprop by (auto intro!: add_right_mono sum_mono2)
    finally have "ennreal (F b - F a) \<le> (\<Sum>i\<le>n. ennreal (F (r i) - F (l i))) + epsilon"
      using egt0 by (simp add: sum_nonneg flip: ennreal_plus)
    then show "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i))) + (epsilon :: real)"
      by (rule order_trans) (auto intro!: add_mono sum_le_suminf simp del: sum_ennreal)
  qed
  moreover have "(\<Sum>i. ennreal (F (r i) - F (l i))) \<le> ennreal (F b - F a)"
    using \<open>a \<le> b\<close> by (auto intro!: suminf_le_const ennreal_le_iff[THEN iffD2] claim1)
  ultimately show "(\<Sum>n. ennreal (F (r n) - F (l n))) = ennreal (F b - F a)"
    by (rule antisym[rotated])
qed (auto simp: Ioc_inj mono_F)

lemma measure_interval_measure_Ioc:
  assumes "a \<le> b" and "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y" and "\<And>a. continuous (at_right a) F"
  shows "measure (interval_measure F) {a <.. b} = F b - F a"
  unfolding measure_def
  by (simp add: assms emeasure_interval_measure_Ioc)

lemma emeasure_interval_measure_Ioc_eq:
  "(\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y) \<Longrightarrow> (\<And>a. continuous (at_right a) F) \<Longrightarrow>
    emeasure (interval_measure F) {a <.. b} = (if a \<le> b then F b - F a else 0)"
  using emeasure_interval_measure_Ioc[of a b F] by auto

lemma\<^marker>\<open>tag important\<close> sets_interval_measure [simp, measurable_cong]:
    "sets (interval_measure F) = sets borel"
  apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc)
  apply (rule sigma_sets_eqI)
  apply auto
  apply (case_tac "a \<le> ba")
  apply (auto intro: sigma_sets.Empty)
  done

lemma space_interval_measure [simp]: "space (interval_measure F) = UNIV"
  by (simp add: interval_measure_def space_extend_measure)

lemma emeasure_interval_measure_Icc:
  assumes "a \<le> b"
  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
  assumes cont_F : "continuous_on UNIV F"
  shows "emeasure (interval_measure F) {a .. b} = F b - F a"
proof (rule tendsto_unique)
  { fix a b :: real assume "a \<le> b" then have "emeasure (interval_measure F) {a <.. b} = F b - F a"
      using cont_F
      by (subst emeasure_interval_measure_Ioc)
         (auto intro: mono_F continuous_within_subset simp: continuous_on_eq_continuous_within) }
  note * = this

  let ?F = "interval_measure F"
  show "((\<lambda>a. F b - F a) \<longlongrightarrow> emeasure ?F {a..b}) (at_left a)"
  proof (rule tendsto_at_left_sequentially)
    show "a - 1 < a" by simp
    fix X assume "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
    with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) \<longlonglongrightarrow> emeasure ?F (\<Inter>n. {X n <..b})"
      apply (intro Lim_emeasure_decseq)
      apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
      apply force
      apply (subst (asm ) *)
      apply (auto intro: less_le_trans less_imp_le)
      done
    also have "(\<Inter>n. {X n <..b}) = {a..b}"
      using \<open>\<And>n. X n < a\<close>
      apply auto
      apply (rule LIMSEQ_le_const2[OF \<open>X \<longlonglongrightarrow> a\<close>])
      apply (auto intro: less_imp_le)
      apply (auto intro: less_le_trans)
      done
    also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
      using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
    finally show "(\<lambda>n. F b - F (X n)) \<longlonglongrightarrow> emeasure ?F {a..b}" .
  qed
  show "((\<lambda>a. ennreal (F b - F a)) \<longlongrightarrow> F b - F a) (at_left a)"
    by (rule continuous_on_tendsto_compose[where g="\<lambda>x. x" and s=UNIV])
       (auto simp: continuous_on_ennreal continuous_on_diff cont_F continuous_on_const)
qed (rule trivial_limit_at_left_real)

lemma\<^marker>\<open>tag important\<close> sigma_finite_interval_measure:
  assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
  assumes right_cont_F : "\<And>a. continuous (at_right a) F"
  shows "sigma_finite_measure (interval_measure F)"
  apply unfold_locales
  apply (intro exI[of _ "(\<lambda>(a, b). {a <.. b}) ` (\<rat> \<times> \<rat>)"])
  apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
  done

subsection \<open>Lebesgue-Borel measure\<close>

definition\<^marker>\<open>tag important\<close> lborel :: "('a :: euclidean_space) measure" where
  "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. interval_measure (\<lambda>x. x)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"

abbreviation\<^marker>\<open>tag important\<close> lebesgue :: "'a::euclidean_space measure"
  where "lebesgue \<equiv> completion lborel"

abbreviation\<^marker>\<open>tag important\<close> lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
  where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"

lemma lebesgue_on_mono:
  assumes major: "AE x in lebesgue_on S. P x" and minor: "\<And>x.\<lbrakk>P x; x \<in> S\<rbrakk> \<Longrightarrow> Q x"
  shows "AE x in lebesgue_on S. Q x"
proof -
  have "AE a in lebesgue_on S. P a \<longrightarrow> Q a"
    using minor space_restrict_space by fastforce
  then show ?thesis
    using major by auto
qed

lemma
  shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
    and space_lborel[simp]: "space lborel = space borel"
    and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
    and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"
  by (simp_all add: lborel_def)

lemma space_lebesgue_on [simp]: "space (lebesgue_on S) = S"
  by (simp add: space_restrict_space)

lemma sets_lebesgue_on_refl [iff]: "S \<in> sets (lebesgue_on S)"
    by (metis inf_top.right_neutral sets.top space_borel space_completion space_lborel space_restrict_space)

lemma Compl_in_sets_lebesgue: "-A \<in> sets lebesgue \<longleftrightarrow> A  \<in> sets lebesgue"
  by (metis Compl_eq_Diff_UNIV double_compl space_borel space_completion space_lborel Sigma_Algebra.sets.compl_sets)

lemma measurable_lebesgue_cong:
  assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
  shows "f \<in> measurable (lebesgue_on S) M \<longleftrightarrow> g \<in> measurable (lebesgue_on S) M"
  by (metis (mono_tags, lifting) IntD1 assms measurable_cong_simp space_restrict_space)

lemma integrable_lebesgue_on_UNIV_eq:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
  shows "integrable (lebesgue_on UNIV) f = integrable lebesgue f"
  by (auto simp: integrable_restrict_space)

text\<^marker>\<open>tag unimportant\<close> \<open>Measurability of continuous functions\<close>

lemma continuous_imp_measurable_on_sets_lebesgue:
  assumes f: "continuous_on S f" and S: "S \<in> sets lebesgue"
  shows "f \<in> borel_measurable (lebesgue_on S)"
proof -
  have "sets (restrict_space borel S) \<subseteq> sets (lebesgue_on S)"
    by (simp add: mono_restrict_space subsetI)
  then show ?thesis
    by (simp add: borel_measurable_continuous_on_restrict [OF f] borel_measurable_subalgebra 
                  space_restrict_space)
qed

lemma id_borel_measurable_lebesgue [iff]: "id \<in> borel_measurable lebesgue"
  by (simp add: measurable_completion)

lemma id_borel_measurable_lebesgue_on [iff]: "id \<in> borel_measurable (lebesgue_on S)"
  by (simp add: measurable_completion measurable_restrict_space1)

context
begin

interpretation sigma_finite_measure "interval_measure (\<lambda>x. x)"
  by (rule sigma_finite_interval_measure) auto
interpretation finite_product_sigma_finite "\<lambda>_. interval_measure (\<lambda>x. x)" Basis
  proof qed simp

lemma lborel_eq_real: "lborel = interval_measure (\<lambda>x. x)"
  unfolding lborel_def Basis_real_def
  using distr_id[of "interval_measure (\<lambda>x. x)"]
  by (subst distr_component[symmetric])
     (simp_all add: distr_distr comp_def del: distr_id cong: distr_cong)

lemma lborel_eq: "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. lborel) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
  by (subst lborel_def) (simp add: lborel_eq_real)

lemma nn_integral_lborel_prod:
  assumes [measurable]: "\<And>b. b \<in> Basis \<Longrightarrow> f b \<in> borel_measurable borel"
  assumes nn[simp]: "\<And>b x. b \<in> Basis \<Longrightarrow> 0 \<le> f b x"
  shows "(\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. f b (x \<bullet> b)) \<partial>lborel) = (\<Prod>b\<in>Basis. (\<integral>\<^sup>+x. f b x \<partial>lborel))"
  by (simp add: lborel_def nn_integral_distr product_nn_integral_prod
                product_nn_integral_singleton)

lemma emeasure_lborel_Icc[simp]:
  fixes l u :: real
  assumes [simp]: "l \<le> u"
  shows "emeasure lborel {l .. u} = u - l"
proof -
  have "((\<lambda>f. f 1) -` {l..u} \<inter> space (Pi\<^sub>M {1} (\<lambda>b. interval_measure (\<lambda>x. x)))) = {1::real} \<rightarrow>\<^sub>E {l..u}"
    by (auto simp: space_PiM)
  then show ?thesis
    by (simp add: lborel_def emeasure_distr emeasure_PiM emeasure_interval_measure_Icc continuous_on_id)
qed

lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l \<le> u then u - l else 0)"
  by simp

lemma\<^marker>\<open>tag important\<close> emeasure_lborel_cbox[simp]:
  assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
  shows "emeasure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
proof -
  have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (cbox l u)"
    by (auto simp: fun_eq_iff cbox_def split: split_indicator)
  then have "emeasure lborel (cbox l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
    by simp
  also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
    by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
  finally show ?thesis .
qed

lemma AE_lborel_singleton: "AE x in lborel::'a::euclidean_space measure. x \<noteq> c"
  using SOME_Basis AE_discrete_difference [of "{c}" lborel] emeasure_lborel_cbox [of c c]
  by (auto simp add: power_0_left)

lemma emeasure_lborel_Ioo[simp]:
  assumes [simp]: "l \<le> u"
  shows "emeasure lborel {l <..< u} = ennreal (u - l)"
proof -
  have "emeasure lborel {l <..< u} = emeasure lborel {l .. u}"
    using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
  then show ?thesis
    by simp
qed

lemma emeasure_lborel_Ioc[simp]:
  assumes [simp]: "l \<le> u"
  shows "emeasure lborel {l <.. u} = ennreal (u - l)"
proof -
  have "emeasure lborel {l <.. u} = emeasure lborel {l .. u}"
    using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
  then show ?thesis
    by simp
qed

lemma emeasure_lborel_Ico[simp]:
  assumes [simp]: "l \<le> u"
  shows "emeasure lborel {l ..< u} = ennreal (u - l)"
proof -
  have "emeasure lborel {l ..< u} = emeasure lborel {l .. u}"
    using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
  then show ?thesis
    by simp
qed

lemma emeasure_lborel_box[simp]:
  assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
  shows "emeasure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
proof -
  have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (box l u)"
    by (auto simp: fun_eq_iff box_def split: split_indicator)
  then have "emeasure lborel (box l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
    by simp
  also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
    by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
  finally show ?thesis .
qed

lemma emeasure_lborel_cbox_eq:
  "emeasure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
  using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)

lemma emeasure_lborel_box_eq:
  "emeasure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
  using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force

lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
  using emeasure_lborel_cbox[of x x] nonempty_Basis
  by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: prod_constant)

lemma fmeasurable_cbox [iff]: "cbox a b \<in> fmeasurable lborel"
  and fmeasurable_box [iff]: "box a b \<in> fmeasurable lborel"
  by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)

lemma
  fixes l u :: real
  assumes [simp]: "l \<le> u"
  shows measure_lborel_Icc[simp]: "measure lborel {l .. u} = u - l"
    and measure_lborel_Ico[simp]: "measure lborel {l ..< u} = u - l"
    and measure_lborel_Ioc[simp]: "measure lborel {l <.. u} = u - l"
    and measure_lborel_Ioo[simp]: "measure lborel {l <..< u} = u - l"
  by (simp_all add: measure_def)

lemma
  assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
  shows measure_lborel_box[simp]: "measure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
    and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
  by (simp_all add: measure_def inner_diff_left prod_nonneg)

lemma measure_lborel_cbox_eq:
  "measure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
  using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)

lemma measure_lborel_box_eq:
  "measure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
  using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force

lemma measure_lborel_singleton[simp]: "measure lborel {x} = 0"
  by (simp add: measure_def)

lemma sigma_finite_lborel: "sigma_finite_measure lborel"
proof
  show "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets lborel \<and> \<Union>A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
    by (intro exI[of _ "range (\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One))"])
       (auto simp: emeasure_lborel_cbox_eq UN_box_eq_UNIV)
qed

end

lemma emeasure_lborel_UNIV [simp]: "emeasure lborel (UNIV::'a::euclidean_space set) = \<infinity>"
proof -
  { fix n::nat
    let ?Ba = "Basis :: 'a set"
    have "real n \<le> (2::real) ^ card ?Ba * real n"
      by (simp add: mult_le_cancel_right1)
    also
    have "... \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
      apply (rule mult_left_mono)
      apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
      apply (simp add: DIM_positive)
      done
    finally have "real n \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
  } note [intro!] = this
  show ?thesis
    unfolding UN_box_eq_UNIV[symmetric]
    apply (subst SUP_emeasure_incseq[symmetric])
    apply (auto simp: incseq_def subset_box inner_add_left prod_constant
      simp del: Sup_eq_top_iff SUP_eq_top_iff
      intro!: ennreal_SUP_eq_top)
    done
qed

lemma emeasure_lborel_countable:
  fixes A :: "'a::euclidean_space set"
  assumes "countable A"
  shows "emeasure lborel A = 0"
proof -
  have "A \<subseteq> (\<Union>i. {from_nat_into A i})" using from_nat_into_surj assms by force
  then have "emeasure lborel A \<le> emeasure lborel (\<Union>i. {from_nat_into A i})"
    by (intro emeasure_mono) auto
  also have "emeasure lborel (\<Union>i. {from_nat_into A i}) = 0"
    by (rule emeasure_UN_eq_0) auto
  finally show ?thesis
    by (auto simp add: )
qed

lemma countable_imp_null_set_lborel: "countable A \<Longrightarrow> A \<in> null_sets lborel"
  by (simp add: null_sets_def emeasure_lborel_countable sets.countable)

lemma finite_imp_null_set_lborel: "finite A \<Longrightarrow> A \<in> null_sets lborel"
  by (intro countable_imp_null_set_lborel countable_finite)

lemma insert_null_sets_iff [simp]: "insert a N \<in> null_sets lebesgue \<longleftrightarrow> N \<in> null_sets lebesgue"     
    (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs
    by (meson completion.complete2 subset_insertI)
next
  assume ?rhs then show ?lhs
  by (simp add: null_sets.insert_in_sets null_setsI)
qed

lemma insert_null_sets_lebesgue_on_iff [simp]:
  assumes "a \<in> S" "S \<in> sets lebesgue"
  shows "insert a N \<in> null_sets (lebesgue_on S) \<longleftrightarrow> N \<in> null_sets (lebesgue_on S)"     
  by (simp add: assms null_sets_restrict_space)

lemma lborel_neq_count_space[simp]: "lborel \<noteq> count_space (A::('a::ordered_euclidean_space) set)"
proof
  assume asm: "lborel = count_space A"
  have "space lborel = UNIV" by simp
  hence [simp]: "A = UNIV" by (subst (asm) asm) (simp only: space_count_space)
  have "emeasure lborel {undefined::'a} = 1"
      by (subst asm, subst emeasure_count_space_finite) auto
  moreover have "emeasure lborel {undefined} \<noteq> 1" by simp
  ultimately show False by contradiction
qed

lemma mem_closed_if_AE_lebesgue_open:
  assumes "open S" "closed C"
  assumes "AE x \<in> S in lebesgue. x \<in> C"
  assumes "x \<in> S"
  shows "x \<in> C"
proof (rule ccontr)
  assume xC: "x \<notin> C"
  with openE[of "S - C"] assms
  obtain e where e: "0 < e" "ball x e \<subseteq> S - C"
    by blast
  then obtain a b where box: "x \<in> box a b" "box a b \<subseteq> S - C"
    by (metis rational_boxes order_trans)
  then have "0 < emeasure lebesgue (box a b)"
    by (auto simp: emeasure_lborel_box_eq mem_box algebra_simps intro!: prod_pos)
  also have "\<dots> \<le> emeasure lebesgue (S - C)"
    using assms box
    by (auto intro!: emeasure_mono)
  also have "\<dots> = 0"
    using assms
    by (auto simp: eventually_ae_filter completion.complete2 set_diff_eq null_setsD1)
  finally show False by simp
qed

lemma mem_closed_if_AE_lebesgue: "closed C \<Longrightarrow> (AE x in lebesgue. x \<in> C) \<Longrightarrow> x \<in> C"
  using mem_closed_if_AE_lebesgue_open[OF open_UNIV] by simp


subsection \<open>Affine transformation on the Lebesgue-Borel\<close>

lemma\<^marker>\<open>tag important\<close> lborel_eqI:
  fixes M :: "'a::euclidean_space measure"
  assumes emeasure_eq: "\<And>l u. (\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b) \<Longrightarrow> emeasure M (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
  assumes sets_eq: "sets M = sets borel"
  shows "lborel = M"
proof (rule measure_eqI_generator_eq)
  let ?E = "range (\<lambda>(a, b). box a b::'a set)"
  show "Int_stable ?E"
    by (auto simp: Int_stable_def box_Int_box)

  show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
    by (simp_all add: borel_eq_box sets_eq)

  let ?A = "\<lambda>n::nat. box (- (real n *\<^sub>R One)) (real n *\<^sub>R One) :: 'a set"
  show "range ?A \<subseteq> ?E" "(\<Union>i. ?A i) = UNIV"
    unfolding UN_box_eq_UNIV by auto

  { fix i show "emeasure lborel (?A i) \<noteq> \<infinity>" by auto }
  { fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
      apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
      apply (subst box_eq_empty(1)[THEN iffD2])
      apply (auto intro: less_imp_le simp: not_le)
      done }
qed

lemma\<^marker>\<open>tag important\<close> lborel_affine_euclidean:
  fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
  defines "T x \<equiv> t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
  assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
  shows "lborel = density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
proof (rule lborel_eqI)
  let ?B = "Basis :: 'a set"
  fix l u assume le: "\<And>b. b \<in> ?B \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
  have [measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
    by (simp add: T_def[abs_def])
  have eq: "T -` box l u = box
    (\<Sum>j\<in>Basis. (((if 0 < c j then l - t else u - t) \<bullet> j) / c j) *\<^sub>R j)
    (\<Sum>j\<in>Basis. (((if 0 < c j then u - t else l - t) \<bullet> j) / c j) *\<^sub>R j)"
    using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
  with le c show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
    by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
                   field_simps divide_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
             intro!: prod.cong)
qed simp

lemma lborel_affine:
  fixes t :: "'a::euclidean_space"
  shows "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))"
  using lborel_affine_euclidean[where c="\<lambda>_::'a. c" and t=t]
  unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation prod_constant by simp

lemma lborel_real_affine:
  "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
  using lborel_affine[of c t] by simp

lemma AE_borel_affine:
  fixes P :: "real \<Rightarrow> bool"
  shows "c \<noteq> 0 \<Longrightarrow> Measurable.pred borel P \<Longrightarrow> AE x in lborel. P x \<Longrightarrow> AE x in lborel. P (t + c * x)"
  by (subst lborel_real_affine[where t="- t / c" and c="1 / c"])
     (simp_all add: AE_density AE_distr_iff field_simps)

lemma nn_integral_real_affine:
  fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0"
  shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = \<bar>c\<bar> * (\<integral>\<^sup>+x. f (t + c * x) \<partial>lborel)"
  by (subst lborel_real_affine[OF c, of t])
     (simp add: nn_integral_density nn_integral_distr nn_integral_cmult)

lemma lborel_integrable_real_affine:
  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
  assumes f: "integrable lborel f"
  shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))"
  using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
  by (subst (asm) nn_integral_real_affine[where c=c and t=t]) (auto simp: ennreal_mult_less_top)

lemma lborel_integrable_real_affine_iff:
  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
  shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x)) \<longleftrightarrow> integrable lborel f"
  using
    lborel_integrable_real_affine[of f c t]
    lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "-t/c"]
  by (auto simp add: field_simps)

lemma\<^marker>\<open>tag important\<close> lborel_integral_real_affine:
  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real
  assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lborel) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f (t + c * x) \<partial>lborel)"
proof cases
  assume f[measurable]: "integrable lborel f" then show ?thesis
    using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
    by (subst lborel_real_affine[OF c, of t])
       (simp add: integral_density integral_distr)
next
  assume "\<not> integrable lborel f" with c show ?thesis
    by (simp add: lborel_integrable_real_affine_iff not_integrable_integral_eq)
qed

lemma
  fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
  assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
  defines "T == (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))"
  shows lebesgue_affine_euclidean: "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
    and lebesgue_affine_measurable: "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
proof -
  have T_borel[measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
    by (auto simp: T_def[abs_def])
  { fix A :: "'a set" assume A: "A \<in> sets borel"
    then have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))) A = 0"
      unfolding T_def using c by (subst lborel_affine_euclidean[symmetric]) auto
    also have "\<dots> \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0"
      using A c by (simp add: distr_completion emeasure_density nn_integral_cmult prod_nonneg cong: distr_cong)
    finally have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0" . }
  then have eq: "null_sets lborel = null_sets (distr lebesgue lborel T)"
    by (auto simp: null_sets_def)

  show "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
    by (rule completion.measurable_completion2) (auto simp: eq measurable_completion)

  have "lebesgue = completion (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>)))"
    using c by (subst lborel_affine_euclidean[of c t]) (simp_all add: T_def[abs_def])
  also have "\<dots> = density (completion (distr lebesgue lborel T)) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
    using c by (auto intro!: always_eventually prod_pos completion_density_eq simp: distr_completion cong: distr_cong)
  also have "\<dots> = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
    by (subst completion.completion_distr_eq) (auto simp: eq measurable_completion)
  finally show "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" .
qed

lemma lebesgue_measurable_scaling[measurable]: "(*\<^sub>R) x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
proof cases
  assume "x = 0"
  then have "(*\<^sub>R) x = (\<lambda>x. 0::'a)"
    by (auto simp: fun_eq_iff)
  then show ?thesis by auto
next
  assume "x \<noteq> 0" then show ?thesis
    using lebesgue_affine_measurable[of "\<lambda>_. x" 0]
    unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation
    by (auto simp add: ac_simps)
qed

lemma
  fixes m :: real and \<delta> :: "'a::euclidean_space"
  defines "T r d x \<equiv> r *\<^sub>R x + d"
  shows emeasure_lebesgue_affine: "emeasure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * emeasure lebesgue S" (is ?e)
    and measure_lebesgue_affine: "measure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * measure lebesgue S" (is ?m)
proof -
  show ?e
  proof cases
    assume "m = 0" then show ?thesis
      by (simp add: image_constant_conv T_def[abs_def])
  next
    let ?T = "T m \<delta>" and ?T' = "T (1 / m) (- ((1/m) *\<^sub>R \<delta>))"
    assume "m \<noteq> 0"
    then have s_comp_s: "?T' \<circ> ?T = id" "?T \<circ> ?T' = id"
      by (auto simp: T_def[abs_def] fun_eq_iff scaleR_add_right scaleR_diff_right)
    then have "inv ?T' = ?T" "bij ?T'"
      by (auto intro: inv_unique_comp o_bij)
    then have eq: "T m \<delta> ` S = T (1 / m) ((-1/m) *\<^sub>R \<delta>) -` S \<inter> space lebesgue"
      using bij_vimage_eq_inv_image[OF \<open>bij ?T'\<close>, of S] by auto

    have trans_eq_T: "(\<lambda>x. \<delta> + (\<Sum>j\<in>Basis. (m * (x \<bullet> j)) *\<^sub>R j)) = T m \<delta>" for m \<delta>
      unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
      by (auto simp add: euclidean_representation ac_simps)

    have T[measurable]: "T r d \<in> lebesgue \<rightarrow>\<^sub>M lebesgue" for r d
      using lebesgue_affine_measurable[of "\<lambda>_. r" d]
      by (cases "r = 0") (auto simp: trans_eq_T T_def[abs_def])

    show ?thesis
    proof cases
      assume "S \<in> sets lebesgue" with \<open>m \<noteq> 0\<close> show ?thesis
        unfolding eq
        apply (subst lebesgue_affine_euclidean[of "\<lambda>_. m" \<delta>])
        apply (simp_all add: emeasure_density trans_eq_T nn_integral_cmult emeasure_distr
                        del: space_completion emeasure_completion)
        apply (simp add: vimage_comp s_comp_s prod_constant)
        done
    next
      assume "S \<notin> sets lebesgue"
      moreover have "?T ` S \<notin> sets lebesgue"
      proof
        assume "?T ` S \<in> sets lebesgue"
        then have "?T -` (?T ` S) \<inter> space lebesgue \<in> sets lebesgue"
          by (rule measurable_sets[OF T])
        also have "?T -` (?T ` S) \<inter> space lebesgue = S"
          by (simp add: vimage_comp s_comp_s eq)
        finally show False using \<open>S \<notin> sets lebesgue\<close> by auto
      qed
      ultimately show ?thesis
        by (simp add: emeasure_notin_sets)
    qed
  qed
  show ?m
    unfolding measure_def \<open>?e\<close> by (simp add: enn2real_mult prod_nonneg)
qed

lemma lebesgue_real_scale:
  assumes "c \<noteq> 0"
  shows   "lebesgue = density (distr lebesgue lebesgue (\<lambda>x. c * x)) (\<lambda>x. ennreal \<bar>c\<bar>)"
  using assms by (subst lebesgue_affine_euclidean[of "\<lambda>_. c" 0]) simp_all

lemma divideR_right:
  fixes x y :: "'a::real_normed_vector"
  shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
  using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp

lemma lborel_has_bochner_integral_real_affine_iff:
  fixes x :: "'a :: {banach, second_countable_topology}"
  shows "c \<noteq> 0 \<Longrightarrow>
    has_bochner_integral lborel f x \<longleftrightarrow>
    has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
  unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
  by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)

lemma lborel_distr_uminus: "distr lborel borel uminus = (lborel :: real measure)"
  by (subst lborel_real_affine[of "-1" 0])
     (auto simp: density_1 one_ennreal_def[symmetric])

lemma lborel_distr_mult:
  assumes "(c::real) \<noteq> 0"
  shows "distr lborel borel ((*) c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
proof-
  have "distr lborel borel ((*) c) = distr lborel lborel ((*) c)" by (simp cong: distr_cong)
  also from assms have "... = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
    by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
  finally show ?thesis .
qed

lemma lborel_distr_mult':
  assumes "(c::real) \<noteq> 0"
  shows "lborel = density (distr lborel borel ((*) c)) (\<lambda>_. \<bar>c\<bar>)"
proof-
  have "lborel = density lborel (\<lambda>_. 1)" by (rule density_1[symmetric])
  also from assms have "(\<lambda>_. 1 :: ennreal) = (\<lambda>_. inverse \<bar>c\<bar> * \<bar>c\<bar>)" by (intro ext) simp
  also have "density lborel ... = density (density lborel (\<lambda>_. inverse \<bar>c\<bar>)) (\<lambda>_. \<bar>c\<bar>)"
    by (subst density_density_eq) (auto simp: ennreal_mult)
  also from assms have "density lborel (\<lambda>_. inverse \<bar>c\<bar>) = distr lborel borel ((*) c)"
    by (rule lborel_distr_mult[symmetric])
  finally show ?thesis .
qed

lemma lborel_distr_plus: "distr lborel borel ((+) c) = (lborel :: real measure)"
  by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])

interpretation lborel: sigma_finite_measure lborel
  by (rule sigma_finite_lborel)

interpretation lborel_pair: pair_sigma_finite lborel lborel ..

lemma\<^marker>\<open>tag important\<close> lborel_prod:
  "lborel \<Otimes>\<^sub>M lborel = (lborel :: ('a::euclidean_space \<times> 'b::euclidean_space) measure)"
proof (rule lborel_eqI[symmetric], clarify)
  fix la ua :: 'a and lb ub :: 'b
  assume lu: "\<And>a b. (a, b) \<in> Basis \<Longrightarrow> (la, lb) \<bullet> (a, b) \<le> (ua, ub) \<bullet> (a, b)"
  have [simp]:
    "\<And>b. b \<in> Basis \<Longrightarrow> la \<bullet> b \<le> ua \<bullet> b"
    "\<And>b. b \<in> Basis \<Longrightarrow> lb \<bullet> b \<le> ub \<bullet> b"
    "inj_on (\<lambda>u. (u, 0)) Basis" "inj_on (\<lambda>u. (0, u)) Basis"
    "(\<lambda>u. (u, 0)) ` Basis \<inter> (\<lambda>u. (0, u)) ` Basis = {}"
    "box (la, lb) (ua, ub) = box la ua \<times> box lb ub"
    using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
  show "emeasure (lborel \<Otimes>\<^sub>M lborel) (box (la, lb) (ua, ub)) =
      ennreal (prod ((\<bullet>) ((ua, ub) - (la, lb))) Basis)"
    by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def prod.union_disjoint
                  prod.reindex ennreal_mult inner_diff_left prod_nonneg)
qed (simp add: borel_prod[symmetric])

(* FIXME: conversion in measurable prover *)
lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel" 
  by simp

lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel"
  by simp

lemma emeasure_bounded_finite:
  assumes "bounded A" shows "emeasure lborel A < \<infinity>"
proof -
  obtain a b where "A \<subseteq> cbox a b"
    by (meson bounded_subset_cbox_symmetric \<open>bounded A\<close>)
  then have "emeasure lborel A \<le> emeasure lborel (cbox a b)"
    by (intro emeasure_mono) auto
  then show ?thesis
    by (auto simp: emeasure_lborel_cbox_eq prod_nonneg less_top[symmetric] top_unique split: if_split_asm)
qed

lemma emeasure_compact_finite: "compact A \<Longrightarrow> emeasure lborel A < \<infinity>"
  using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded)

lemma borel_integrable_compact:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
  assumes "compact S" "continuous_on S f"
  shows "integrable lborel (\<lambda>x. indicator S x *\<^sub>R f x)"
proof cases
  assume "S \<noteq> {}"
  have "continuous_on S (\<lambda>x. norm (f x))"
    using assms by (intro continuous_intros)
  from continuous_attains_sup[OF \<open>compact S\<close> \<open>S \<noteq> {}\<close> this]
  obtain M where M: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> M"
    by auto
  show ?thesis
  proof (rule integrable_bound)
    show "integrable lborel (\<lambda>x. indicator S x * M)"
      using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left)
    show "(\<lambda>x. indicator S x *\<^sub>R f x) \<in> borel_measurable lborel"
      using assms by (auto intro!: borel_measurable_continuous_on_indicator borel_compact)
    show "AE x in lborel. norm (indicator S x *\<^sub>R f x) \<le> norm (indicator S x * M)"
      by (auto split: split_indicator simp: abs_real_def dest!: M)
  qed
qed simp

lemma borel_integrable_atLeastAtMost:
  fixes f :: "real \<Rightarrow> real"
  assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
  shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
proof -
  have "integrable lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x)"
  proof (rule borel_integrable_compact)
    from f show "continuous_on {a..b} f"
      by (auto intro: continuous_at_imp_continuous_on)
  qed simp
  then show ?thesis
    by (auto simp: mult.commute)
qed

subsection \<open>Lebesgue measurable sets\<close>

abbreviation\<^marker>\<open>tag important\<close> lmeasurable :: "'a::euclidean_space set set"
where
  "lmeasurable \<equiv> fmeasurable lebesgue"

lemma not_measurable_UNIV [simp]: "UNIV \<notin> lmeasurable"
  by (simp add: fmeasurable_def)

lemma\<^marker>\<open>tag important\<close> lmeasurable_iff_integrable:
  "S \<in> lmeasurable \<longleftrightarrow> integrable lebesgue (indicator S :: 'a::euclidean_space \<Rightarrow> real)"
  by (auto simp: fmeasurable_def integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator)

lemma lmeasurable_cbox [iff]: "cbox a b \<in> lmeasurable"
  and lmeasurable_box [iff]: "box a b \<in> lmeasurable"
  by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)

lemma fmeasurable_compact: "compact S \<Longrightarrow> S \<in> fmeasurable lborel"
  using emeasure_compact_finite[of S] by (intro fmeasurableI) (auto simp: borel_compact)

lemma lmeasurable_compact: "compact S \<Longrightarrow> S \<in> lmeasurable"
  using fmeasurable_compact by (force simp: fmeasurable_def)

lemma measure_frontier:
   "bounded S \<Longrightarrow> measure lebesgue (frontier S) = measure lebesgue (closure S) - measure lebesgue (interior S)"
  using closure_subset interior_subset
  by (auto simp: frontier_def fmeasurable_compact intro!: measurable_measure_Diff)

lemma lmeasurable_closure:
   "bounded S \<Longrightarrow> closure S \<in> lmeasurable"
  by (simp add: lmeasurable_compact)

lemma lmeasurable_frontier:
   "bounded S \<Longrightarrow> frontier S \<in> lmeasurable"
  by (simp add: compact_frontier_bounded lmeasurable_compact)

lemma lmeasurable_open: "bounded S \<Longrightarrow> open S \<Longrightarrow> S \<in> lmeasurable"
  using emeasure_bounded_finite[of S] by (intro fmeasurableI) (auto simp: borel_open)

lemma lmeasurable_ball [simp]: "ball a r \<in> lmeasurable"
  by (simp add: lmeasurable_open)

lemma lmeasurable_cball [simp]: "cball a r \<in> lmeasurable"
  by (simp add: lmeasurable_compact)

lemma lmeasurable_interior: "bounded S \<Longrightarrow> interior S \<in> lmeasurable"
  by (simp add: bounded_interior lmeasurable_open)

lemma null_sets_cbox_Diff_box: "cbox a b - box a b \<in> null_sets lborel"
proof -
  have "emeasure lborel (cbox a b - box a b) = 0"
    by (subst emeasure_Diff) (auto simp: emeasure_lborel_cbox_eq emeasure_lborel_box_eq box_subset_cbox)
  then have "cbox a b - box a b \<in> null_sets lborel"
    by (auto simp: null_sets_def)
  then show ?thesis
    by (auto dest!: AE_not_in)
qed

lemma bounded_set_imp_lmeasurable:
  assumes "bounded S" "S \<in> sets lebesgue" shows "S \<in> lmeasurable"
  by (metis assms bounded_Un emeasure_bounded_finite emeasure_completion fmeasurableI main_part_null_part_Un)


subsection\<^marker>\<open>tag unimportant\<close>\<open>Translation preserves Lebesgue measure\<close>

lemma sigma_sets_image:
  assumes S: "S \<in> sigma_sets \<Omega> M" and "M \<subseteq> Pow \<Omega>" "f ` \<Omega> = \<Omega>" "inj_on f \<Omega>"
    and M: "\<And>y. y \<in> M \<Longrightarrow> f ` y \<in> M"
  shows "(f ` S) \<in> sigma_sets \<Omega> M"
  using S
proof (induct S rule: sigma_sets.induct)
  case (Basic a) then show ?case
    by (simp add: M)
next
  case Empty then show ?case
    by (simp add: sigma_sets.Empty)
next
  case (Compl a)
  then have "\<Omega> - a \<subseteq> \<Omega>" "a \<subseteq> \<Omega>"
    by (auto simp: sigma_sets_into_sp [OF \<open>M \<subseteq> Pow \<Omega>\<close>])
  then show ?case
    by (auto simp: inj_on_image_set_diff [OF \<open>inj_on f \<Omega>\<close>] assms intro: Compl sigma_sets.Compl)
next
  case (Union a) then show ?case
    by (metis image_UN sigma_sets.simps)
qed

lemma null_sets_translation:
  assumes "N \<in> null_sets lborel" shows "{x. x - a \<in> N} \<in> null_sets lborel"
proof -
  have [simp]: "(\<lambda>x. x + a) ` N = {x. x - a \<in> N}"
    by force
  show ?thesis
    using assms emeasure_lebesgue_affine [of 1 a N] by (auto simp: null_sets_def)
qed

lemma lebesgue_sets_translation:
  fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
  assumes S: "S \<in> sets lebesgue"
  shows "((\<lambda>x. a + x) ` S) \<in> sets lebesgue"
proof -
  have im_eq: "(+) a ` A = {x. x - a \<in> A}" for A
    by force
  have "((\<lambda>x. a + x) ` S) = ((\<lambda>x. -a + x) -` S) \<inter> (space lebesgue)"
    using image_iff by fastforce
  also have "\<dots> \<in> sets lebesgue"
  proof (rule measurable_sets [OF measurableI assms])
    fix A :: "'b set"
    assume A: "A \<in> sets lebesgue"
    have vim_eq: "(\<lambda>x. x - a) -` A = (+) a ` A" for A
      by force
    have "\<exists>s n N'. (+) a ` (S \<union> N) = s \<union> n \<and> s \<in> sets borel \<and> N' \<in> null_sets lborel \<and> n \<subseteq> N'"
      if "S \<in> sets borel" and "N' \<in> null_sets lborel" and "N \<subseteq> N'" for S N N'
    proof (intro exI conjI)
      show "(+) a ` (S \<union> N) = (\<lambda>x. a + x) ` S \<union> (\<lambda>x. a + x) ` N"
        by auto
      show "(\<lambda>x. a + x) ` N' \<in> null_sets lborel"
        using that by (auto simp: null_sets_translation im_eq)
    qed (use that im_eq in auto)
    with A have "(\<lambda>x. x - a) -` A \<in> sets lebesgue"
      by (force simp: vim_eq completion_def intro!: sigma_sets_image)
    then show "(+) (- a) -` A \<inter> space lebesgue \<in> sets lebesgue"
      by (auto simp: vimage_def im_eq)
  qed auto
  finally show ?thesis .
qed

lemma measurable_translation:
   "S \<in> lmeasurable \<Longrightarrow> ((+) a ` S) \<in> lmeasurable"
  using emeasure_lebesgue_affine [of 1 a S]
  apply (auto intro: lebesgue_sets_translation simp add: fmeasurable_def cong: image_cong_simp)
  apply (simp add: ac_simps)
  done

lemma measurable_translation_subtract:
   "S \<in> lmeasurable \<Longrightarrow> ((\<lambda>x. x - a) ` S) \<in> lmeasurable"
  using measurable_translation [of S "- a"] by (simp cong: image_cong_simp)

lemma measure_translation:
  "measure lebesgue ((+) a ` S) = measure lebesgue S"
  using measure_lebesgue_affine [of 1 a S] by (simp add: ac_simps cong: image_cong_simp)

lemma measure_translation_subtract:
  "measure lebesgue ((\<lambda>x. x - a) ` S) = measure lebesgue S"
  using measure_translation [of "- a"] by (simp cong: image_cong_simp)


subsection \<open>A nice lemma for negligibility proofs\<close>

lemma summable_iff_suminf_neq_top: "(\<And>n. f n \<ge> 0) \<Longrightarrow> \<not> summable f \<Longrightarrow> (\<Sum>i. ennreal (f i)) = top"
  by (metis summable_suminf_not_top)

proposition\<^marker>\<open>tag important\<close> starlike_negligible_bounded_gmeasurable:
  fixes S :: "'a :: euclidean_space set"
  assumes S: "S \<in> sets lebesgue" and "bounded S"
      and eq1: "\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1"
    shows "S \<in> null_sets lebesgue"
proof -
  obtain M where "0 < M" "S \<subseteq> ball 0 M"
    using \<open>bounded S\<close> by (auto dest: bounded_subset_ballD)

  let ?f = "\<lambda>n. root DIM('a) (Suc n)"

  have vimage_eq_image: "(*\<^sub>R) (?f n) -` S = (*\<^sub>R) (1 / ?f n) ` S" for n
    apply safe
    subgoal for x by (rule image_eqI[of _ _ "?f n *\<^sub>R x"]) auto
    subgoal by auto
    done

  have eq: "(1 / ?f n) ^ DIM('a) = 1 / Suc n" for n
    by (simp add: field_simps)

  { fix n x assume x: "root DIM('a) (1 + real n) *\<^sub>R x \<in> S"
    have "1 * norm x \<le> root DIM('a) (1 + real n) * norm x"
      by (rule mult_mono) auto
    also have "\<dots> < M"
      using x \<open>S \<subseteq> ball 0 M\<close> by auto
    finally have "norm x < M" by simp }
  note less_M = this

  have "(\<Sum>n. ennreal (1 / Suc n)) = top"
    using not_summable_harmonic[where 'a=real] summable_Suc_iff[where f="\<lambda>n. 1 / (real n)"]
    by (intro summable_iff_suminf_neq_top) (auto simp add: inverse_eq_divide)
  then have "top * emeasure lebesgue S = (\<Sum>n. (1 / ?f n)^DIM('a) * emeasure lebesgue S)"
    unfolding ennreal_suminf_multc eq by simp
  also have "\<dots> = (\<Sum>n. emeasure lebesgue ((*\<^sub>R) (?f n) -` S))"
    unfolding vimage_eq_image using emeasure_lebesgue_affine[of "1 / ?f n" 0 S for n] by simp
  also have "\<dots> = emeasure lebesgue (\<Union>n. (*\<^sub>R) (?f n) -` S)"
  proof (intro suminf_emeasure)
    show "disjoint_family (\<lambda>n. (*\<^sub>R) (?f n) -` S)"
      unfolding disjoint_family_on_def
    proof safe
      fix m n :: nat and x assume "m \<noteq> n" "?f m *\<^sub>R x \<in> S" "?f n *\<^sub>R x \<in> S"
      with eq1[of "?f m / ?f n" "?f n *\<^sub>R x"] show "x \<in> {}"
        by auto
    qed
    have "(*\<^sub>R) (?f i) -` S \<in> sets lebesgue" for i
      using measurable_sets[OF lebesgue_measurable_scaling[of "?f i"] S] by auto
    then show "range (\<lambda>i. (*\<^sub>R) (?f i) -` S) \<subseteq> sets lebesgue"
      by auto
  qed
  also have "\<dots> \<le> emeasure lebesgue (ball 0 M :: 'a set)"
    using less_M by (intro emeasure_mono) auto
  also have "\<dots> < top"
    using lmeasurable_ball by (auto simp: fmeasurable_def)
  finally have "emeasure lebesgue S = 0"
    by (simp add: ennreal_top_mult split: if_split_asm)
  then show "S \<in> null_sets lebesgue"
    unfolding null_sets_def using \<open>S \<in> sets lebesgue\<close> by auto
qed

corollary starlike_negligible_compact:
  "compact S \<Longrightarrow> (\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1) \<Longrightarrow> S \<in> null_sets lebesgue"
  using starlike_negligible_bounded_gmeasurable[of S] by (auto simp: compact_eq_bounded_closed)

proposition outer_regular_lborel_le:
  assumes B[measurable]: "B \<in> sets borel" and "0 < (e::real)"
  obtains U where "open U" "B \<subseteq> U" and "emeasure lborel (U - B) \<le> e"
proof -
  let ?\<mu> = "emeasure lborel"
  let ?B = "\<lambda>n::nat. ball 0 n :: 'a set"
  let ?e = "\<lambda>n. e*((1/2)^Suc n)"
  have "\<forall>n. \<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
  proof
    fix n :: nat
    let ?A = "density lborel (indicator (?B n))"
    have emeasure_A: "X \<in> sets borel \<Longrightarrow> emeasure ?A X = ?\<mu> (?B n \<inter> X)" for X
      by (auto simp: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])

    have finite_A: "emeasure ?A (space ?A) \<noteq> \<infinity>"
      using emeasure_bounded_finite[of "?B n"] by (auto simp: emeasure_A)
    interpret A: finite_measure ?A
      by rule fact
    have "emeasure ?A B + ?e n > (INF U\<in>{U. B \<subseteq> U \<and> open U}. emeasure ?A U)"
      using \<open>0<e\<close> by (auto simp: outer_regular[OF _ finite_A B, symmetric])
    then obtain U where U: "B \<subseteq> U" "open U" and muU: "?\<mu> (?B n \<inter> B) + ?e n > ?\<mu> (?B n \<inter> U)"
      unfolding INF_less_iff by (auto simp: emeasure_A)
    moreover
    { have "?\<mu> ((?B n \<inter> U) - B) = ?\<mu> ((?B n \<inter> U) - (?B n \<inter> B))"
        using U by (intro arg_cong[where f="?\<mu>"]) auto
      also have "\<dots> = ?\<mu> (?B n \<inter> U) - ?\<mu> (?B n \<inter> B)"
        using U A.emeasure_finite[of B]
        by (intro emeasure_Diff) (auto simp del: A.emeasure_finite simp: emeasure_A)
      also have "\<dots> < ?e n"
        using U muU A.emeasure_finite[of B]
        by (subst minus_less_iff_ennreal)
          (auto simp del: A.emeasure_finite simp: emeasure_A less_top ac_simps intro!: emeasure_mono)
      finally have "?\<mu> ((?B n \<inter> U) - B) < ?e n" . }
    ultimately show "\<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
      by (intro exI[of _ "?B n \<inter> U"]) auto
  qed
  then obtain U
    where U: "\<And>n. open (U n)" "\<And>n. ?B n \<inter> B \<subseteq> U n" "\<And>n. ?\<mu> (U n - B) < ?e n"
    by metis
  show ?thesis
  proof
    { fix x assume "x \<in> B"
      moreover
      obtain n where "norm x < real n"
        using reals_Archimedean2 by blast
      ultimately have "x \<in> (\<Union>n. U n)"
        using U(2)[of n] by auto }
    note * = this
    then show "open (\<Union>n. U n)" "B \<subseteq> (\<Union>n. U n)"
      using U by auto
    have "?\<mu> (\<Union>n. U n - B) \<le> (\<Sum>n. ?\<mu> (U n - B))"
      using U(1) by (intro emeasure_subadditive_countably) auto
    also have "\<dots> \<le> (\<Sum>n. ennreal (?e n))"
      using U(3) by (intro suminf_le) (auto intro: less_imp_le)
    also have "\<dots> = ennreal (e * 1)"
      using \<open>0<e\<close> by (intro suminf_ennreal_eq sums_mult power_half_series) auto
    finally show "emeasure lborel ((\<Union>n. U n) - B) \<le> ennreal e"
      by simp
  qed
qed

lemma\<^marker>\<open>tag important\<close> outer_regular_lborel:
  assumes B: "B \<in> sets borel" and "0 < (e::real)"
  obtains U where "open U" "B \<subseteq> U" "emeasure lborel (U - B) < e"
proof -
  obtain U where U: "open U" "B \<subseteq> U" and "emeasure lborel (U-B) \<le> e/2"
    using outer_regular_lborel_le [OF B, of "e/2"] \<open>e > 0\<close>
    by force
  moreover have "ennreal (e/2) < ennreal e"
    using \<open>e > 0\<close> by (simp add: ennreal_lessI)
  ultimately have "emeasure lborel (U-B) < e"
    by auto
  with U show ?thesis
    using that by auto
qed

lemma completion_upper:
  assumes A: "A \<in> sets (completion M)"
  obtains A' where "A \<subseteq> A'" "A' \<in> sets M" "A' - A \<in> null_sets (completion M)"
                   "emeasure (completion M) A = emeasure M A'"
proof -
  from AE_notin_null_part[OF A] obtain N where N: "N \<in> null_sets M" "null_part M A \<subseteq> N"
    unfolding eventually_ae_filter using null_part_null_sets[OF A, THEN null_setsD2, THEN sets.sets_into_space] by auto
  let ?A' = "main_part M A \<union> N"
  show ?thesis
  proof
    show "A \<subseteq> ?A'"
      using \<open>null_part M A \<subseteq> N\<close> by (subst main_part_null_part_Un[symmetric, OF A]) auto
    have "main_part M A \<subseteq> A"
      using assms main_part_null_part_Un by auto
    then have "?A' - A \<subseteq> N"
      by blast
    with N show "?A' - A \<in> null_sets (completion M)"
      by (blast intro: null_sets_completionI completion.complete_measure_axioms complete_measure.complete2)
    show "emeasure (completion M) A = emeasure M (main_part M A \<union> N)"
      using A \<open>N \<in> null_sets M\<close> by (simp add: emeasure_Un_null_set)
  qed (use A N in auto)
qed

lemma sets_lebesgue_outer_open:
  fixes e::real
  assumes S: "S \<in> sets lebesgue" and "e > 0"
  obtains T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable" "emeasure lebesgue (T - S) < ennreal e"
proof -
  obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel"
              and null: "S' - S \<in> null_sets lebesgue"
              and em: "emeasure lebesgue S = emeasure lborel S'"
    using completion_upper[of S lborel] S by auto
  then have f_S': "S' \<in> sets borel"
    using S by (auto simp: fmeasurable_def)
  with outer_regular_lborel[OF _ \<open>0<e\<close>]
  obtain U where U: "open U" "S' \<subseteq> U" "emeasure lborel (U - S') < e"
    by blast
  show thesis
  proof
    show "open U" "S \<subseteq> U"
      using f_S' U S' by auto
  have "(U - S) = (U - S') \<union> (S' - S)"
    using S' U by auto
  then have eq: "emeasure lebesgue (U - S) = emeasure lborel (U - S')"
    using null  by (simp add: U(1) emeasure_Un_null_set f_S' sets.Diff)
  have "(U - S) \<in> sets lebesgue"
    by (simp add: S U(1) sets.Diff)
  then show "(U - S) \<in> lmeasurable"
    unfolding fmeasurable_def using U(3) eq less_le_trans by fastforce
  with eq U show "emeasure lebesgue (U - S) < e"
    by (simp add: eq)
  qed
qed

lemma sets_lebesgue_inner_closed:
  fixes e::real
  assumes "S \<in> sets lebesgue" "e > 0"
  obtains T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" "emeasure lebesgue (S - T) < ennreal e"
proof -
  have "-S \<in> sets lebesgue"
    using assms by (simp add: Compl_in_sets_lebesgue)
  then obtain T where "open T" "-S \<subseteq> T"
          and T: "(T - -S) \<in> lmeasurable" "emeasure lebesgue (T - -S) < e"
    using sets_lebesgue_outer_open assms  by blast
  show thesis
  proof
    show "closed (-T)"
      using \<open>open T\<close> by blast
    show "-T \<subseteq> S"
      using \<open>- S \<subseteq> T\<close> by auto
    show "S - ( -T) \<in> lmeasurable" "emeasure lebesgue (S - (- T)) < e"
      using T by (auto simp: Int_commute)
  qed
qed

lemma lebesgue_openin:
   "\<lbrakk>openin (top_of_set S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
  by (metis borel_open openin_open sets.Int sets_completionI_sets sets_lborel)

lemma lebesgue_closedin:
   "\<lbrakk>closedin (top_of_set S) T; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
  by (metis borel_closed closedin_closed sets.Int sets_completionI_sets sets_lborel)


subsection\<open>\<open>F_sigma\<close> and \<open>G_delta\<close> sets.\<close> 

\<comment> \<open>\<^url>\<open>https://en.wikipedia.org/wiki/F-sigma_set\<close>\<close>
inductive\<^marker>\<open>tag important\<close> fsigma :: "'a::topological_space set \<Rightarrow> bool" where
  "(\<And>n::nat. closed (F n)) \<Longrightarrow> fsigma (\<Union>(F ` UNIV))"

inductive\<^marker>\<open>tag important\<close> gdelta :: "'a::topological_space set \<Rightarrow> bool" where
  "(\<And>n::nat. open (F n)) \<Longrightarrow> gdelta (\<Inter>(F ` UNIV))"

lemma fsigma_Union_compact:
  fixes S :: "'a::{real_normed_vector,heine_borel} set"
  shows "fsigma S \<longleftrightarrow> (\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = \<Union>(F ` UNIV))"
proof safe
  assume "fsigma S"
  then obtain F :: "nat \<Rightarrow> 'a set" where F: "range F \<subseteq> Collect closed" "S = \<Union>(F ` UNIV)"
    by (meson fsigma.cases image_subsetI mem_Collect_eq)
  then have "\<exists>D::nat \<Rightarrow> 'a set. range D \<subseteq> Collect compact \<and> \<Union>(D ` UNIV) = F i" for i
    using closed_Union_compact_subsets [of "F i"]
    by (metis image_subsetI mem_Collect_eq range_subsetD)
  then obtain D :: "nat \<Rightarrow> nat \<Rightarrow> 'a set"
    where D: "\<And>i. range (D i) \<subseteq> Collect compact \<and> \<Union>((D i) ` UNIV) = F i"
    by metis
  let ?DD = "\<lambda>n. (\<lambda>(i,j). D i j) (prod_decode n)"
  show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = \<Union>(F ` UNIV)"
  proof (intro exI conjI)
    show "range ?DD \<subseteq> Collect compact"
      using D by clarsimp (metis mem_Collect_eq rangeI split_conv subsetCE surj_pair)
    show "S = \<Union> (range ?DD)"
    proof
      show "S \<subseteq> \<Union> (range ?DD)"
        using D F
        by clarsimp (metis UN_iff old.prod.case prod_decode_inverse prod_encode_eq)
      show "\<Union> (range ?DD) \<subseteq> S"
        using D F  by fastforce
    qed
  qed
next
  fix F :: "nat \<Rightarrow> 'a set"
  assume "range F \<subseteq> Collect compact" and "S = \<Union>(F ` UNIV)"
  then show "fsigma (\<Union>(F ` UNIV))"
    by (simp add: compact_imp_closed fsigma.intros image_subset_iff)
qed

lemma gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
proof (induction rule: gdelta.induct)
  case (1 F)
  have "- \<Inter>(F ` UNIV) = (\<Union>i. -(F i))"
    by auto
  then show ?case
    by (simp add: fsigma.intros closed_Compl 1)
qed

lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
proof (induction rule: fsigma.induct)
  case (1 F)
  have "- \<Union>(F ` UNIV) = (\<Inter>i. -(F i))"
    by auto
  then show ?case
    by (simp add: 1 gdelta.intros open_closed)
qed

lemma gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
  using fsigma_imp_gdelta gdelta_imp_fsigma by force

lemma lebesgue_set_almost_fsigma:
  assumes "S \<in> sets lebesgue"
  obtains C T where "fsigma C" "T \<in> null_sets lebesgue" "C \<union> T = S" "disjnt C T"
proof -
  { fix n::nat
    obtain T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" "emeasure lebesgue (S - T) < ennreal (1 / Suc n)"
      using sets_lebesgue_inner_closed [OF assms]
      by (metis of_nat_0_less_iff zero_less_Suc zero_less_divide_1_iff)
    then have "\<exists>T. closed T \<and> T \<subseteq> S \<and> S - T \<in> lmeasurable \<and> measure lebesgue (S-T) < 1 / Suc n"
      by (metis emeasure_eq_measure2 ennreal_leI not_le)
  }
  then obtain F where F: "\<And>n::nat. closed (F n) \<and> F n \<subseteq> S \<and> S - F n \<in> lmeasurable \<and> measure lebesgue (S - F n) < 1 / Suc n"
    by metis
  let ?C = "\<Union>(F ` UNIV)"
  show thesis
  proof
    show "fsigma ?C"
      using F by (simp add: fsigma.intros)
    show "(S - ?C) \<in> null_sets lebesgue"
    proof (clarsimp simp add: completion.null_sets_outer_le)
      fix e :: "real"
      assume "0 < e"
      then obtain n where n: "1 / Suc n < e"
        using nat_approx_posE by metis
      show "\<exists>T \<in> lmeasurable. S - (\<Union>x. F x) \<subseteq> T \<and> measure lebesgue T \<le> e"
      proof (intro bexI conjI)
        show "measure lebesgue (S - F n) \<le> e"
          by (meson F n less_trans not_le order.asym)
      qed (use F in auto)
    qed
    show "?C \<union> (S - ?C) = S"
      using F by blast
    show "disjnt ?C (S - ?C)"
      by (auto simp: disjnt_def)
  qed
qed

lemma lebesgue_set_almost_gdelta:
  assumes "S \<in> sets lebesgue"
  obtains C T where "gdelta C" "T \<in> null_sets lebesgue" "S \<union> T = C" "disjnt S T"
proof -
  have "-S \<in> sets lebesgue"
    using assms Compl_in_sets_lebesgue by blast
  then obtain C T where C: "fsigma C" "T \<in> null_sets lebesgue" "C \<union> T = -S" "disjnt C T"
    using lebesgue_set_almost_fsigma by metis
  show thesis
  proof
    show "gdelta (-C)"
      by (simp add: \<open>fsigma C\<close> fsigma_imp_gdelta)
    show "S \<union> T = -C" "disjnt S T"
      using C by (auto simp: disjnt_def)
  qed (use C in auto)
qed

end