Quelle  Complete_Measure.thy

(*  Title:      HOL/Analysis/Complete_Measure.thy
  Author: Robert Himmelmann, Johannes Hoelzl, TU Muenchen
*)
section ‹Complete Measures›
theory Complete_Measure
  imports Bochner_Integration
begin

locale🍋‹tag important› complete_measure =
  fixes M :: "'a measure"
  assumes complete: "∧A B. B ⊆ A ==> A ∈ null_sets M ==> B ∈ sets M"

definition🍋‹tag important›
  "split_completion M A p = (if A ∈ sets M then p = (A, {}) else
   ∃N'. A = fst p ∪ snd p ∧ fst p ∩ snd p = {} ∧ fst p ∈ sets M ∧ snd p ⊆ N' ∧ N' ∈ null_sets M)"

definition🍋‹tag important›
  "main_part M A = fst (Eps (split_completion M A))"

definition🍋‹tag important›
  "null_part M A = snd (Eps (split_completion M A))"

definition🍋‹tag important› completion :: "'a measure ==> 'a measure" where
  "completion M = measure_of (space M) { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }
    (emeasure M ∘ main_part M)"

lemma completion_into_space:
  "{ S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' } ⊆ Pow (space M)"
  using sets.sets_into_space by auto

lemma space_completion[simp]: "space (completion M) = space M"
  unfolding completion_def using space_measure_of[OF completion_into_space] by simp

lemma completionI:
  assumes "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
  shows "A ∈ { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
  using assms by auto

lemma completionE:
  assumes "A ∈ { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
  obtains S N N' where "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
  using assms by auto

lemma sigma_algebra_completion:
  "sigma_algebra (space M) { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
    (is "sigma_algebra _ ?A")
  unfolding sigma_algebra_iff2
proof (intro conjI ballI allI impI)
  show "?A ⊆ Pow (space M)"
    using sets.sets_into_space by auto
next
  show "{} ∈ ?A" by auto
next
  let ?C = "space M"
  fix A assume "A ∈ ?A"
  then obtain S N N'
    where "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
    by (rule completionE)
  then show "space M - A ∈ ?A"
    by (intro completionI[of _ "(?C - S) ∩ (?C - N')" "(?C - S) ∩ N' ∩ (?C - N)"]) auto
next
  fix A :: "nat ==> 'a set" assume A: "range A ⊆ ?A"
  then have "∀n. ∃S N N'. A n = S ∪ N ∧ S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N'"
    by (auto simp: image_subset_iff)
  then obtain S N N' where "∀x. A x = S x ∪ N x ∧ S x ∈ sets M ∧ N' x ∈ null_sets M ∧ N x ⊆ N' x"
    by metis
  then show "∪(A ` UNIV) ∈ ?A"
    using null_sets_UN[of N']
    by (intro completionI[of _ "∪(S ` UNIV)" "∪(N ` UNIV)" "∪(N' ` UNIV)"]) auto
qed

lemma🍋‹tag important› sets_completion:
  "sets (completion M) = { S ∪ N |S N N'. S ∈ sets M ∧ N' ∈ null_sets M ∧ N ⊆ N' }"
  using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] 
  by (simp add: completion_def)

lemma sets_completionE:
  assumes "A ∈ sets (completion M)"
  obtains S N N' where "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
  using assms unfolding sets_completion by auto

lemma sets_completionI:
  assumes "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
  shows "A ∈ sets (completion M)"
  using assms unfolding sets_completion by auto

lemma sets_completionI_sets[intro, simp]:
  "A ∈ sets M ==> A ∈ sets (completion M)"
  unfolding sets_completion by force

lemma🍋‹tag important› measurable_completion: "f ∈ M →🪙M N ==> f ∈ completion M →??M N"
  by (auto simp: measurable_def)

lemma null_sets_completion:
  assumes "N' ∈ null_sets M" "N ⊆ N'" shows "N ∈ sets (completion M)"
  using assms by (intro sets_completionI[of N "{}" N N']) auto

lemma🍋‹tag important› split_completion:
  assumes "A ∈ sets (completion M)"
  shows "split_completion M A (main_part M A, null_part M A)"
proof cases
  assume "A ∈ sets M" then show ?thesis
    by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
next
  assume nA: "A ∉ sets M"
  show ?thesis
    unfolding main_part_def null_part_def if_not_P[OF nA]
  proof (rule someI2_ex)
    from assms obtain S N N' where A: "A = S ∪ N" "N ⊆ N'" "N' ∈ null_sets M" "S ∈ sets M"
      by (blast intro: sets_completionE)
    let ?P = "(S, N - S)"
    show "∃p. split_completion M A p"
      unfolding split_completion_def if_not_P[OF nA] using A
    proof (intro exI conjI)
      show "A = fst ?P ∪ snd ?P" using A by auto
      show "snd ?P ⊆ N'" using A by auto
   qed auto
  qed auto
qed

lemma sets_restrict_space_subset:
  assumes S: "S ∈ sets (completion M)"
  shows "sets (restrict_space (completion M) S) ⊆ sets (completion M)"
  by (metis assms sets.Int_space_eq2 sets_restrict_space_iff subsetI)

lemma
  assumes "S ∈ sets (completion M)"
  shows main_part_sets[intro, simp]: "main_part M S ∈ sets M"
    and main_part_null_part_Un[simp]: "main_part M S ∪ null_part M S = S"
    and main_part_null_part_Int[simp]: "main_part M S ∩ null_part M S = {}"
  using split_completion[OF assms]
  by (auto simp: split_completion_def split: if_split_asm)

lemma main_part[simp]: "S ∈ sets M ==> main_part M S = S"
  using split_completion[of S M]
  by (auto simp: split_completion_def split: if_split_asm)

lemma null_part:
  assumes "S ∈ sets (completion M)" shows "∃N. N∈null_sets M ∧ null_part M S ⊆ N"
  using split_completion[OF assms] by (auto simp: split_completion_def split: if_split_asm)

lemma null_part_sets[intro, simp]:
  assumes "S ∈ sets M" shows "null_part M S ∈ sets M" "emeasure M (null_part M S) = 0"
proof -
  have S: "S ∈ sets (completion M)" using assms by auto
  have *: "S - main_part M S ∈ sets M" using assms by auto
  from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
  have "S - main_part M S = null_part M S" by auto
  with * show sets: "null_part M S ∈ sets M" by auto
  from null_part[OF S] obtain N where "N ∈ null_sets M ∧ null_part M S ⊆ N" ..
  with emeasure_eq_0[of N _ "null_part M S"] sets
  show "emeasure M (null_part M S) = 0" by auto
qed

lemma emeasure_main_part_UN:
  fixes S :: "nat ==> 'a set"
  assumes "range S ⊆ sets (completion M)"
  shows "emeasure M (main_part M (∪i. (S i))) = emeasure M (∪i. main_part M (S i))"
proof -
  have S: "∧i. S i ∈ sets (completion M)" using assms by auto
  then have UN: "(∪i. S i) ∈ sets (completion M)" by auto
  have "∀i. ∃N. N ∈ null_sets M ∧ null_part M (S i) ⊆ N"
    using null_part[OF S] by auto
  then obtain N where N: "∀x. N x ∈ null_sets M ∧ null_part M (S x) ⊆ N x" by metis
  then have UN_N: "(∪i. N i) ∈ null_sets M" by (intro null_sets_UN) auto
  from S have "(∪i. S i) ∈ sets (completion M)" by auto
  from null_part[OF this] obtain N' where N': "N' ∈ null_sets M ∧ null_part M (∪ (range S)) ⊆ N'" ..
  let ?N = "(∪i. N i) ∪ N'"
  have null_set: "?N ∈ null_sets M" using N' UN_N by (intro null_sets.Un) auto
  have "main_part M (∪i. S i) ∪ ?N = (main_part M (∪i. S i) ∪ null_part M (∪i. S i)) ∪ ?N"
    using N' by auto
  also have "… = (∪i. main_part M (S i) ∪ null_part M (S i)) ∪ ?N"
    unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
  also have "… = (∪i. main_part M (S i)) ∪ ?N"
    using N by auto
  finally have *: "main_part M (∪i. S i) ∪ ?N = (∪i. main_part M (S i)) ∪ ?N" .
  have "emeasure M (main_part M (∪i. S i)) = emeasure M (main_part M (∪i. S i) ∪ ?N)"
    using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
  also have "… = emeasure M ((∪i. main_part M (S i)) ∪ ?N)"
    unfolding * ..
  also have "… = emeasure M (∪i. main_part M (S i))"
    using null_set S by (intro emeasure_Un_null_set) auto
  finally show ?thesis .
qed

lemma🍋‹tag important› emeasure_completion[simp]:
  assumes S: "S ∈ sets (completion M)"
  shows "emeasure (completion M) S = emeasure M (main_part M S)"
proof (subst emeasure_measure_of[OF completion_def completion_into_space])
  let ?μ = "emeasure M ∘ main_part M"
  show "S ∈ sets (completion M)" "?μ S = emeasure M (main_part M S) " using S by simp_all
  show "positive (sets (completion M)) ?μ"
    by (simp add: positive_def)
  show "countably_additive (sets (completion M)) ?μ"
  proof (intro countably_additiveI)
    fix A :: "nat ==> 'a set" assume A: "range A ⊆ sets (completion M)" "disjoint_family A"
    have "disjoint_family (λi. main_part M (A i))"
    proof (intro disjoint_family_on_bisimulation[OF A(2)])
      fix n m assume "A n ∩ A m = {}"
      then have "(main_part M (A n) ∪ null_part M (A n)) ∩ (main_part M (A m) ∪ null_part M (A m)) = {}"
        using A by (subst (1 2) main_part_null_part_Un) auto
      then show "main_part M (A n) ∩ main_part M (A m) = {}" by auto
    qed
    then have "(∑n. emeasure M (main_part M (A n))) = emeasure M (∪i. main_part M (A i))"
      using A by (auto intro!: suminf_emeasure)
    then show "(∑n. ?μ (A n)) = ?μ (∪(A ` UNIV))"
      by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
  qed
qed

lemma measure_completion[simp]: "S ∈ sets M ==> measure (completion M) S = measure M S"
  unfolding measure_def by auto

lemma emeasure_completion_UN:
  "range S ⊆ sets (completion M) ==>
    emeasure (completion M) (∪i::nat. (S i)) = emeasure M (∪i. main_part M (S i))"
  by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)

lemma emeasure_completion_Un:
  assumes S: "S ∈ sets (completion M)" and T: "T ∈ sets (completion M)"
  shows "emeasure (completion M) (S ∪ T) = emeasure M (main_part M S ∪ main_part M T)"
proof (subst emeasure_completion)
  have UN: "(∪i. binary (main_part M S) (main_part M T) i) = (∪i. main_part M (binary S T i))"
    unfolding binary_def by (auto split: if_split_asm)
  show "emeasure M (main_part M (S ∪ T)) = emeasure M (main_part M S ∪ main_part M T)"
    using emeasure_main_part_UN[of "binary S T" M] assms
    by (simp add: range_binary_eq, simp add: Un_range_binary UN)
qed (auto intro: S T)

lemma sets_completionI_sub:
  assumes N: "N' ∈ null_sets M" "N ⊆ N'"
  shows "N ∈ sets (completion M)"
  using assms by (intro sets_completionI[of _ "{}" N N']) auto

lemma completion_ex_simple_function:
  assumes f: "simple_function (completion M) f"
  shows "∃f'. simple_function M f' ∧ (AE x in M. f x = f' x)"
proof -
  let ?F = "λx. f -` {x} ∩ space M"
  have F: "∧x. ?F x ∈ sets (completion M)" and fin: "finite (f`space M)"
    using simple_functionD[OF f] simple_functionD[OF f] by simp_all
  have "∀x. ∃N. N ∈ null_sets M ∧ null_part M (?F x) ⊆ N"
    using F null_part by auto
  from choice[OF this] obtain N where
    N: "∧x. null_part M (?F x) ⊆ N x" "∧x. N x ∈ null_sets M" by auto
  let ?N = "∪x∈f`space M. N x"
  let ?f' = "λx. if x ∈ ?N then undefined else f x"
  have sets: "?N ∈ null_sets M" using N fin by (intro null_sets.finite_UN) auto
  show ?thesis unfolding simple_function_def
  proof (safe intro!: exI[of _ ?f'])
    have "?f' ` space M ⊆ f`space M ∪ {undefined}" by auto
    from finite_subset[OF this] simple_functionD(1)[OF f]
    show "finite (?f' ` space M)" by auto
  next
    fix x assume "x ∈ space M"
    have "?f' -` {?f' x} ∩ space M =
      (if x ∈ ?N then ?F undefined ∪ ?N
       else if f x = undefined then ?F (f x) ∪ ?N
       else ?F (f x) - ?N)"
      using N(2) sets.sets_into_space by (auto split: if_split_asm simp: null_sets_def)
    moreover { fix y have "?F y ∪ ?N ∈ sets M"
      proof cases
        assume y: "y ∈ f`space M"
        have "?F y ∪ ?N = (main_part M (?F y) ∪ null_part M (?F y)) ∪ ?N"
          using main_part_null_part_Un[OF F] by auto
        also have "… = main_part M (?F y) ∪ ?N"
          using y N by auto
        finally show ?thesis
          using F sets by auto
      next
        assume "y ∉ f`space M" then have "?F y = {}" by auto
        then show ?thesis using sets by auto
      qed }
    moreover {
      have "?F (f x) - ?N = main_part M (?F (f x)) ∪ null_part M (?F (f x)) - ?N"
        using main_part_null_part_Un[OF F] by auto
      also have "… = main_part M (?F (f x)) - ?N"
        using N ‹x ∈ space M› by auto
      finally have "?F (f x) - ?N ∈ sets M"
        using F sets by auto }
    ultimately show "?f' -` {?f' x} ∩ space M ∈ sets M" by auto
  next
    show "AE x in M. f x = ?f' x"
      by (rule AE_I', rule sets) auto
  qed
qed

lemma🍋‹tag important› completion_ex_borel_measurable:
  fixes g :: "'a ==> ennreal"
  assumes g: "g ∈ borel_measurable (completion M)"
  shows "∃g'∈borel_measurable M. (AE x in M. g x = g' x)"
proof -
  obtain f :: "nat ==> 'a ==> ennreal"
    where *: "∧i. simple_function (completion M) (f i)"
      and **: "∧x. (SUP i. f i x) = g x"
    using g[THEN borel_measurable_implies_simple_function_sequence'] by metis
  from *[THEN completion_ex_simple_function]
  have "∀i. ∃f'. simple_function M f' ∧ (AE x in M. f i x = f' x)" ..
  then obtain f'
    where sf: "∧i. simple_function M (f' i)"
      and AE: "∀i. AE x in M. f i x = f' i x"
    by metis
  show ?thesis
  proof (intro bexI)
    from AE[unfolded AE_all_countable[symmetric]]
    show "AE x in M. g x = (SUP i. f' i x)" (is "AE x in M. g x = ?f x")
    proof (elim AE_mp, safe intro!: AE_I2)
      fix x assume eq: "∀i. f i x = f' i x"
      have "g x = (SUP i. f i x)" by (auto simp: ** split: split_max)
      with eq show "g x = ?f x" by auto
    qed
    show "?f ∈ borel_measurable M"
      using sf[THEN borel_measurable_simple_function] by auto
  qed
qed

lemma null_sets_completionI: "N ∈ null_sets M ==> N ∈ null_sets (completion M)"
  by (auto simp: null_sets_def)

lemma AE_completion: "(AE x in M. P x) ==> (AE x in completion M. P x)"
  unfolding eventually_ae_filter by (auto intro: null_sets_completionI)

lemma null_sets_completion_iff: "N ∈ sets M ==> N ∈ null_sets (completion M) ⟷ N ∈ null_sets M"
  by (auto simp: null_sets_def)

lemma sets_completion_AE: "(AE x in M. ¬ P x) ==> Measurable.pred (completion M) P"
  unfolding pred_def sets_completion eventually_ae_filter
  by auto

lemma null_sets_completion_iff2:
  "A ∈ null_sets (completion M) ⟷ (∃N'∈null_sets M. A ⊆ N')"
proof safe
  assume "A ∈ null_sets (completion M)"
  then have A: "A ∈ sets (completion M)" and "main_part M A ∈ null_sets M"
    by (auto simp: null_sets_def)
  moreover obtain N where "N ∈ null_sets M" "null_part M A ⊆ N"
    using null_part[OF A] by auto
  ultimately show "∃N'∈null_sets M. A ⊆ N'"
  proof (intro bexI)
    show "A ⊆ N ∪ main_part M A"
      using ‹null_part M A ⊆ N› by (subst main_part_null_part_Un[OF A, symmetric]) auto
  qed auto
next
  fix N assume "N ∈ null_sets M" "A ⊆ N"
  then have "A ∈ sets (completion M)" and N: "N ∈ sets M" "A ⊆ N" "emeasure M N = 0"
    by (auto intro: null_sets_completion)
  moreover have "emeasure (completion M) A = 0"
    using N by (intro emeasure_eq_0[of N _ A]) auto
  ultimately show "A ∈ null_sets (completion M)"
    by auto
qed

lemma null_sets_completion_subset:
  "B ⊆ A ==> A ∈ null_sets (completion M) ==> B ∈ null_sets (completion M)"
  unfolding null_sets_completion_iff2 by auto

interpretation completion: complete_measure "completion M" for M
proof
  show "B ⊆ A ==> A ∈ null_sets (completion M) ==> B ∈ sets (completion M)" for B A
    using null_sets_completion_subset[of B A M] by (simp add: null_sets_def)
qed

lemma null_sets_restrict_space:
  "Ω ∈ sets M ==> A ∈ null_sets (restrict_space M Ω) ⟷ A ⊆ Ω ∧ A ∈ null_sets M"
  by (auto simp: null_sets_def emeasure_restrict_space sets_restrict_space)

lemma completion_ex_borel_measurable_real:
  fixes g :: "'a ==> real"
  assumes g: "g ∈ borel_measurable (completion M)"
  shows "∃g'∈borel_measurable M. (AE x in M. g x = g' x)"
proof -
  have "(λx. ennreal (g x)) ∈ completion M →🪙M borel" "(λx. ennreal (- g x)) ∈ completion M →🪙M borel"
    using g by auto
  from this[THEN completion_ex_borel_measurable]
  obtain pf nf :: "'a ==> ennreal"
    where [measurable]: "nf ∈ M →🪙M borel" "pf ∈ M →🪙M borel"
      and ae: "AE x in M. pf x = ennreal (g x)" "AE x in M. nf x = ennreal (- g x)"
    by (auto simp: eq_commute)
  then have "AE x in M. pf x = ennreal (g x) ∧ nf x = ennreal (- g x)"
    by auto
  then obtain N where "N ∈ null_sets M" "{x∈space M. pf x ≠ ennreal (g x) ∧ nf x ≠ ennreal (- g x)} ⊆ N"
    by (auto elim!: AE_E)
  show ?thesis
  proof
    let ?F = "λx. indicator (space M - N) x * (enn2real (pf x) - enn2real (nf x))"
    show "?F ∈ M →🪙M borel"
      using ‹N ∈ null_sets M› by auto
    show "AE x in M. g x = ?F x"
      using ‹N ∈ null_sets M›[THEN AE_not_in] ae AE_space
      apply eventually_elim
      subgoal for x
        by (cases "0::real" "g x" rule: linorder_le_cases) (auto simp: ennreal_neg)
      done
  qed
qed

lemma simple_function_completion: "simple_function M f ==> simple_function (completion M) f"
  by (simp add: simple_function_def)

lemma simple_integral_completion:
  "simple_function M f ==> simple_integral (completion M) f = simple_integral M f"
  unfolding simple_integral_def by simp

lemma nn_integral_completion: "nn_integral (completion M) f = nn_integral M f"
  unfolding nn_integral_def
proof (safe intro!: SUP_eq)
  fix s assume s: "simple_function (completion M) s" and "s ≤ f"
  then obtain s' where s': "simple_function M s'" "AE x in M. s x = s' x"
    by (auto dest: completion_ex_simple_function)
  then obtain N where N: "N ∈ null_sets M" "{x∈space M. s x ≠ s' x} ⊆ N"
    by (auto elim!: AE_E)
  then have ae_N: "AE x in M. (s x ≠ s' x ⟶ x ∈ N) ∧ x ∉ N"
    by (auto dest: AE_not_in)
  define s'' where "s'' x = (if x ∈ N then 0 else s x)" for x
  then have ae_s_eq_s'': "AE x in completion M. s x = s'' x"
    using s' ae_N by (intro AE_completion) auto
  have s'': "simple_function M s''"
  proof (subst simple_function_cong)
    show "t ∈ space M ==> s'' t = (if t ∈ N then 0 else s' t)" for t
      using N by (auto simp: s''_def dest: sets.sets_into_space)
    show "simple_function M (λt. if t ∈ N then 0 else s' t)"
      unfolding s''_def[abs_def] using N by (auto intro!: simple_function_If s')
  qed

  show "∃j∈{g. simple_function M g ∧ g ≤ f}. integral🪙S (completion M) s ≤ integral🪙S M j"
  proof (safe intro!: bexI[of _ s''])
    have "integral🪙S (completion M) s = integral🪙S (completion M) s''"
      by (intro simple_integral_cong_AE s simple_function_completion s'' ae_s_eq_s'')
    then show "integral🪙S (completion M) s ≤ integral🪙S M s''"
      using s'' by (simp add: simple_integral_completion)
    from ‹s ≤ f› show "s'' ≤ f"
      unfolding s''_def le_fun_def by auto
  qed fact
next
  fix s assume "simple_function M s" "s ≤ f"
  then show "∃j∈{g. simple_function (completion M) g ∧ g ≤ f}. integral🪙S M s ≤ integral🪙S (completion M) j"
    by (intro bexI[of _ s]) (auto simp: simple_integral_completion simple_function_completion)
qed

lemma integrable_completion:
  fixes f :: "'a ==> 'b::{banach, second_countable_topology}"
  shows "f ∈ M →🪙M borel ==> integrable (completion M) f ⟷ integrable M f"
  by (rule integrable_subalgebra[symmetric]) auto

lemma integral_completion:
  fixes f :: "'a ==> 'b::{banach, second_countable_topology}"
  assumes f: "f ∈ M →🪙M borel" shows "integral🪙L (completion M) f = integral🪙L M f"
  by (rule integral_subalgebra[symmetric]) (auto intro: f)

locale🍋‹tag important› semifinite_measure =
  fixes M :: "'a measure"
  assumes semifinite:
    "∧A. A ∈ sets M ==> emeasure M A = ∞ ==> ∃B∈sets M. B ⊆ A ∧ emeasure M B < ∞"

locale🍋‹tag important› locally_determined_measure = semifinite_measure +
  assumes locally_determined:
    "∧A. A ⊆ space M ==> (∧B. B ∈ sets M ==> emeasure M B < ∞ ==> A ∩ B ∈ sets M) ==> A ∈ sets M"

locale🍋‹tag important› cld_measure =
  complete_measure M + locally_determined_measure M for M :: "'a measure"

definition🍋‹tag important› outer_measure_of :: "'a measure ==> 'a set ==> ennreal"
  where "outer_measure_of M A = (INF B ∈ {B∈sets M. A ⊆ B}. emeasure M B)"

lemma outer_measure_of_eq[simp]: "A ∈ sets M ==> outer_measure_of M A = emeasure M A"
  by (auto simp: outer_measure_of_def intro!: INF_eqI emeasure_mono)

lemma outer_measure_of_mono: "A ⊆ B ==> outer_measure_of M A ≤ outer_measure_of M B"
  unfolding outer_measure_of_def by (intro INF_superset_mono) auto

lemma outer_measure_of_attain:
  assumes "A ⊆ space M"
  shows "∃E∈sets M. A ⊆ E ∧ outer_measure_of M A = emeasure M E"
proof -
  have "emeasure M ` {B ∈ sets M. A ⊆ B} ≠ {}"
    using ‹A ⊆ space M› by auto
  from ennreal_Inf_countable_INF[OF this]
  obtain f
    where f: "range f ⊆ emeasure M ` {B ∈ sets M. A ⊆ B}" "decseq f"
      and "outer_measure_of M A = (INF i. f i)"
    unfolding outer_measure_of_def by auto
  have "∃E. ∀n. (E n ∈ sets M ∧ A ⊆ E n ∧ emeasure M (E n) ≤ f n) ∧ E (Suc n) ⊆ E n"
  proof (rule dependent_nat_choice)
    show "∃x. x ∈ sets M ∧ A ⊆ x ∧ emeasure M x ≤ f 0"
      using f(1) by (fastforce simp: image_subset_iff image_iff intro: eq_refl[OF sym])
  next
    fix E n assume "E ∈ sets M ∧ A ⊆ E ∧ emeasure M E ≤ f n"
    moreover obtain F where "F ∈ sets M" "A ⊆ F" "f (Suc n) = emeasure M F"
      using f(1) by (auto simp: image_subset_iff image_iff)
    ultimately show "∃y. (y ∈ sets M ∧ A ⊆ y ∧ emeasure M y ≤ f (Suc n)) ∧ y ⊆ E"
      by (auto intro!: exI[of _ "F ∩ E"] emeasure_mono)
  qed
  then obtain E
    where [simp]: "∧n. E n ∈ sets M"
      and "∧n. A ⊆ E n"
      and le_f: "∧n. emeasure M (E n) ≤ f n"
      and "decseq E"
    by (auto simp: decseq_Suc_iff)
  show ?thesis
  proof cases
    assume fin: "∃i. emeasure M (E i) < ∞"
    show ?thesis
    proof (intro bexI[of _ "∩i. E i"] conjI)
      show "A ⊆ (∩i. E i)" "(∩i. E i) ∈ sets M"
        using ‹∧n. A ⊆ E n› by auto

      have " (INF i. emeasure M (E i)) ≤ outer_measure_of M A"
        unfolding ‹outer_measure_of M A = (INF n. f n)›
        by (intro INF_superset_mono le_f) auto
      moreover have "outer_measure_of M A ≤ (INF i. outer_measure_of M (E i))"
        by (intro INF_greatest outer_measure_of_mono ‹∧n. A ⊆ E n›)
      ultimately have "outer_measure_of M A = (INF i. emeasure M (E i))"
        by auto
      also have "… = emeasure M (∩i. E i)"
        using fin by (intro INF_emeasure_decseq' ‹decseq E›) (auto simp: less_top)
      finally show "outer_measure_of M A = emeasure M (∩i. E i)" .
    qed
  next
    assume "∄i. emeasure M (E i) < ∞"
    then have "f n = ∞" for n
      using le_f by (auto simp: not_less top_unique)
    moreover have "∃E∈sets M. A ⊆ E ∧ f 0 = emeasure M E"
      using f by auto
    ultimately show ?thesis
      unfolding ‹outer_measure_of M A = (INF n. f n)› by simp
  qed
qed

lemma SUP_outer_measure_of_incseq:
  assumes A: "∧n. A n ⊆ space M" and "incseq A"
  shows "(SUP n. outer_measure_of M (A n)) = outer_measure_of M (∪i. A i)"
proof (rule antisym)
  obtain E
    where E: "∧n. E n ∈ sets M" "∧n. A n ⊆ E n" "∧n. outer_measure_of M (A n) = emeasure M (E n)"
    using outer_measure_of_attain[OF A] by metis

  define F where "F n = (∩i∈{n ..}. E i)" for n
  with E have F: "incseq F" "∧n. F n ∈ sets M"
    by (auto simp: incseq_def)
  have "A n ⊆ F n" for n
    using incseqD[OF ‹incseq A›, of n] ‹∧n. A n ⊆ E n› by (auto simp: F_def)

  have eq: "outer_measure_of M (A n) = outer_measure_of M (F n)" for n
  proof (intro antisym)
    have "outer_measure_of M (F n) ≤ outer_measure_of M (E n)"
      by (intro outer_measure_of_mono) (auto simp add: F_def)
    with E show "outer_measure_of M (F n) ≤ outer_measure_of M (A n)"
      by auto
    show "outer_measure_of M (A n) ≤ outer_measure_of M (F n)"
      by (intro outer_measure_of_mono ‹A n ⊆ F n›)
  qed

  have "outer_measure_of M (∪n. A n) ≤ outer_measure_of M (∪n. F n)"
    using ‹∧n. A n ⊆ F n› by (intro outer_measure_of_mono) auto
  also have "… = (SUP n. emeasure M (F n))"
    using F by (simp add: SUP_emeasure_incseq subset_eq)
  finally show "outer_measure_of M (∪n. A n) ≤ (SUP n. outer_measure_of M (A n))"
    by (simp add: eq F)
qed (auto intro: SUP_least outer_measure_of_mono)

definition🍋‹tag important› measurable_envelope :: "'a measure ==> 'a set ==> 'a set ==> bool"
  where "measurable_envelope M A E ⟷
    (A ⊆ E ∧ E ∈ sets M ∧ (∀F∈sets M. emeasure M (F ∩ E) = outer_measure_of M (F ∩ A)))"

lemma measurable_envelopeD:
  assumes "measurable_envelope M A E"
  shows "A ⊆ E"
    and "E ∈ sets M"
    and "∧F. F ∈ sets M ==> emeasure M (F ∩ E) = outer_measure_of M (F ∩ A)"
    and "A ⊆ space M"
  using assms sets.sets_into_space[of E] by (auto simp: measurable_envelope_def)

lemma measurable_envelopeD1:
  assumes E: "measurable_envelope M A E" and F: "F ∈ sets M" "F ⊆ E - A"
  shows "emeasure M F = 0"
proof -
  have "emeasure M F = emeasure M (F ∩ E)"
    using F by (intro arg_cong2[where f=emeasure]) auto
  also have "… = outer_measure_of M (F ∩ A)"
    using measurable_envelopeD[OF E] ‹F ∈ sets M› by (auto simp: measurable_envelope_def)
  also have "… = outer_measure_of M {}"
    using ‹F ⊆ E - A› by (intro arg_cong2[where f=outer_measure_of]) auto
  finally show "emeasure M F = 0"
    by simp
qed

lemma measurable_envelope_eq1:
  assumes "A ⊆ E" "E ∈ sets M"
  shows "measurable_envelope M A E ⟷ (∀F∈sets M. F ⊆ E - A ⟶ emeasure M F = 0)"
proof safe
  assume *: "∀F∈sets M. F ⊆ E - A ⟶ emeasure M F = 0"
  show "measurable_envelope M A E"
    unfolding measurable_envelope_def
  proof (rule ccontr, auto simp add: ‹E ∈ sets M› ‹A ⊆ E›)
    fix F assume "F ∈ sets M" "emeasure M (F ∩ E) ≠ outer_measure_of M (F ∩ A)"
    then have "outer_measure_of M (F ∩ A) < emeasure M (F ∩ E)"
      using outer_measure_of_mono[of "F ∩ A" "F ∩ E" M] ‹A ⊆ E› ‹E ∈ sets M› by (auto simp: less_le)
    then obtain G where G: "G ∈ sets M" "F ∩ A ⊆ G" and less: "emeasure M G < emeasure M (E ∩ F)"
      unfolding outer_measure_of_def INF_less_iff by (auto simp: ac_simps)
    have le: "emeasure M (G ∩ E ∩ F) ≤ emeasure M G"
      using ‹E ∈ sets M› ‹G ∈ sets M› ‹F ∈ sets M› by (auto intro!: emeasure_mono)

    from G have "E ∩ F - G ∈ sets M" "E ∩ F - G ⊆ E - A"
      using ‹F ∈ sets M› ‹E ∈ sets M› by auto
    with * have "0 = emeasure M (E ∩ F - G)"
      by auto
    also have "E ∩ F - G = E ∩ F - (G ∩ E ∩ F)"
      by auto
    also have "emeasure M (E ∩ F - (G ∩ E ∩ F)) = emeasure M (E ∩ F) - emeasure M (G ∩ E ∩ F)"
      using ‹E ∈ sets M› ‹F ∈ sets M› le less G by (intro emeasure_Diff) (auto simp: top_unique)
    also have "… > 0"
      using le less by (intro diff_gr0_ennreal) auto
    finally show False by auto
  qed
qed (rule measurable_envelopeD1)

lemma measurable_envelopeD2:
  assumes E: "measurable_envelope M A E" shows "emeasure M E = outer_measure_of M A"
proof -
  from ‹measurable_envelope M A E› have "emeasure M (E ∩ E) = outer_measure_of M (E ∩ A)"
    by (auto simp: measurable_envelope_def)
  with measurable_envelopeD[OF E] show "emeasure M E = outer_measure_of M A"
    by (auto simp: Int_absorb1)
qed

lemma🍋‹tag important› measurable_envelope_eq2:
  assumes "A ⊆ E" "E ∈ sets M" "emeasure M E < ∞"
  shows "measurable_envelope M A E ⟷ (emeasure M E = outer_measure_of M A)"
proof safe
  assume *: "emeasure M E = outer_measure_of M A"
  show "measurable_envelope M A E"
    unfolding measurable_envelope_eq1[OF ‹A ⊆ E› ‹E ∈ sets M›]
  proof (intro conjI ballI impI assms)
    fix F assume F: "F ∈ sets M" "F ⊆ E - A"
    with ‹E ∈ sets M› have le: "emeasure M F ≤ emeasure M E"
      by (intro emeasure_mono) auto
    from F ‹A ⊆ E› have "outer_measure_of M A ≤ outer_measure_of M (E - F)"
      by (intro outer_measure_of_mono) auto
    then have "emeasure M E - 0 ≤ emeasure M (E - F)"
      using * ‹E ∈ sets M› ‹F ∈ sets M› by simp
    also have "… = emeasure M E - emeasure M F"
      using ‹E ∈ sets M› ‹emeasure M E 🚫∞› F le by (intro emeasure_Diff) (auto simp: top_unique)
    finally show "emeasure M F = 0"
      using ennreal_mono_minus_cancel[of "emeasure M E" 0 "emeasure M F"] le assms by auto
  qed
qed (auto intro: measurable_envelopeD2)

lemma measurable_envelopeI_countable:
  fixes A :: "nat ==> 'a set"
  assumes E: "∧n. measurable_envelope M (A n) (E n)"
  shows "measurable_envelope M (∪n. A n) (∪n. E n)"
proof (subst measurable_envelope_eq1)
  show "(∪n. A n) ⊆ (∪n. E n)" "(∪n. E n) ∈ sets M"
    using measurable_envelopeD(1,2)[OF E] by auto
  show "∀F∈sets M. F ⊆ (∪n. E n) - (∪n. A n) ⟶ emeasure M F = 0"
  proof safe
    fix F assume F: "F ∈ sets M" "F ⊆ (∪n. E n) - (∪n. A n)"
    then have "F ∩ E n ∈ sets M" "F ∩ E n ⊆ E n - A n" "F ⊆ (∪n. E n)" for n
      using measurable_envelopeD(1,2)[OF E] by auto
    then have "emeasure M (∪n. F ∩ E n) = 0"
      by (intro emeasure_UN_eq_0 measurable_envelopeD1[OF E]) auto
    then show "emeasure M F = 0"
      using ‹F ⊆ (∪n. E n)› by (auto simp: Int_absorb2)
  qed
qed

lemma measurable_envelopeI_countable_cover:
  fixes A and C :: "nat ==> 'a set"
  assumes C: "A ⊆ (∪n. C n)" "∧n. C n ∈ sets M" "∧n. emeasure M (C n) < ∞"
  shows "∃E⊆(∪n. C n). measurable_envelope M A E"
proof -
  have "A ∩ C n ⊆ space M" for n
    using ‹C n ∈ sets M› by (auto dest: sets.sets_into_space)
  then have "∀n. ∃E∈sets M. A ∩ C n ⊆ E ∧ outer_measure_of M (A ∩ C n) = emeasure M E"
    using outer_measure_of_attain[of "A ∩ C n" M for n] by auto
  then obtain E
    where E: "∧n. E n ∈ sets M" "∧n. A ∩ C n ⊆ E n"
      and eq: "∧n. outer_measure_of M (A ∩ C n) = emeasure M (E n)"
    by metis

  have "outer_measure_of M (A ∩ C n) ≤ outer_measure_of M (E n ∩ C n)" for n
    using E by (intro outer_measure_of_mono) auto
  moreover have "outer_measure_of M (E n ∩ C n) ≤ outer_measure_of M (E n)" for n
    by (intro outer_measure_of_mono) auto
  ultimately have eq: "outer_measure_of M (A ∩ C n) = emeasure M (E n ∩ C n)" for n
    using E C by (intro antisym) (auto simp: eq)

  { fix n
    have "outer_measure_of M (A ∩ C n) ≤ outer_measure_of M (C n)"
      by (intro outer_measure_of_mono) simp
    also have "… < ∞"
      using assms by auto
    finally have "emeasure M (E n ∩ C n) < ∞"
      using eq by simp }
  then have "measurable_envelope M (∪n. A ∩ C n) (∪n. E n ∩ C n)"
    using E C by (intro measurable_envelopeI_countable measurable_envelope_eq2[THEN iffD2]) (auto simp: eq)
  with ‹A ⊆ (∪n. C n)› show ?thesis
    by (intro exI[of _ "(∪n. E n ∩ C n)"]) (auto simp add: Int_absorb2)
qed

lemma (in complete_measure) complete_sets_sandwich:
  assumes [measurable]: "A ∈ sets M" "C ∈ sets M" and subset: "A ⊆ B" "B ⊆ C"
    and measure: "emeasure M A = emeasure M C" "emeasure M A < ∞"
  shows "B ∈ sets M"
proof -
  have "B - A ∈ sets M"
  proof (rule complete)
    show "B - A ⊆ C - A"
      using subset by auto
    show "C - A ∈ null_sets M"
      using measure subset by(simp add: emeasure_Diff null_setsI)
  qed
  then have "A ∪ (B - A) ∈ sets M"
    by measurable
  also have "A ∪ (B - A) = B"
    using ‹A ⊆ B› by auto
  finally show ?thesis .
qed

lemma (in complete_measure) complete_sets_sandwich_fmeasurable:
  assumes [measurable]: "A ∈ fmeasurable M" "C ∈ fmeasurable M" and subset: "A ⊆ B" "B ⊆ C"
    and measure: "measure M A = measure M C"
  shows "B ∈ fmeasurable M"
proof (rule fmeasurableI2)
  show "B ⊆ C" "C ∈ fmeasurable M" by fact+
  show "B ∈ sets M"
  proof (rule complete_sets_sandwich)
    show "A ∈ sets M" "C ∈ sets M" "A ⊆ B" "B ⊆ C"
      using assms by auto
    show "emeasure M A < ∞"
      using ‹A ∈ fmeasurable M› by (auto simp: fmeasurable_def)
    show "emeasure M A = emeasure M C"
      using assms by (simp add: emeasure_eq_measure2)
  qed
qed

lemma AE_completion_iff: "(AE x in completion M. P x) ⟷ (AE x in M. P x)"
proof
  assume "AE x in completion M. P x"
  then obtain N where "N ∈ null_sets (completion M)" and P: "{x∈space M. ¬ P x} ⊆ N"
    unfolding eventually_ae_filter by auto
  then obtain N' where N': "N' ∈ null_sets M" and "N ⊆ N'"
    unfolding null_sets_completion_iff2 by auto
  from P ‹N ⊆ N'› have "{x∈space M. ¬ P x} ⊆ N'"
    by auto
  with N' show "AE x in M. P x"
    unfolding eventually_ae_filter by auto
qed (rule AE_completion)

lemma null_part_null_sets: "S ∈ completion M ==> null_part M S ∈ null_sets (completion M)"
  by (auto dest!: null_part intro: null_sets_completionI null_sets_completion_subset)

lemma AE_notin_null_part: "S ∈ completion M ==> (AE x in M. x ∉ null_part M S)"
  by (auto dest!: null_part_null_sets AE_not_in simp: AE_completion_iff)

lemma completion_upper:
  assumes A: "A ∈ sets (completion M)"
  shows "∃A'∈sets M. A ⊆ A' ∧ emeasure (completion M) A = emeasure M A'"
proof -
  from AE_notin_null_part[OF A] obtain N where N: "N ∈ null_sets M" "null_part M A ⊆ N"
    unfolding eventually_ae_filter using null_part_null_sets[OF A, THEN null_setsD2, THEN sets.sets_into_space] by auto
  show ?thesis
  proof (intro bexI conjI)
    show "A ⊆ main_part M A ∪ N"
      using ‹null_part M A ⊆ N› by (subst main_part_null_part_Un[symmetric, OF A]) auto
    show "emeasure (completion M) A = emeasure M (main_part M A ∪ N)"
      using A ‹N ∈ null_sets M› by (simp add: emeasure_Un_null_set)
  qed (use A N in auto)
qed

lemma AE_in_main_part:
  assumes A: "A ∈ completion M" shows "AE x in M. x ∈ main_part M A ⟷ x ∈ A"
  using AE_notin_null_part[OF A]
  by (subst (2) main_part_null_part_Un[symmetric, OF A]) auto

lemma completion_density_eq:
  assumes ae: "AE x in M. 0 < f x" and [measurable]: "f ∈ M →🪙M borel"
  shows "completion (density M f) = density (completion M) f"
proof (intro measure_eqI)
  have "N' ∈ sets M ∧ (AE x∈N' in M. f x = 0) ⟷ N' ∈ null_sets M" for N'
  proof safe
    assume N': "N' ∈ sets M" and ae_N': "AE x∈N' in M. f x = 0"
    from ae_N' ae have "AE x in M. x ∉ N'"
      by eventually_elim auto
    then show "N' ∈ null_sets M"
      using N' by (simp add: AE_iff_null_sets)
  next
    assume N': "N' ∈ null_sets M" then show "N' ∈ sets M" "AE x∈N' in M. f x = 0"
      using ae AE_not_in[OF N'] by (auto simp: less_le)
  qed
  then show sets_eq: "sets (completion (density M f)) = sets (density (completion M) f)"
    by (simp add: sets_completion null_sets_density_iff)

  fix A assume A: ‹A ∈ completion (density M f)›
  moreover
  have "A ∈ completion M"
    using A unfolding sets_eq by simp
  moreover
  have "main_part (density M f) A ∈ M"
    using A main_part_sets[of A "density M f"] unfolding sets_density sets_eq by simp
  moreover have "AE x in M. x ∈ main_part (density M f) A ⟷ x ∈ A"
    using AE_in_main_part[OF ‹A ∈ completion (density M f)›] ae by (auto simp add: AE_density)
  ultimately show "emeasure (completion (density M f)) A = emeasure (density (completion M) f) A"
    by (auto simp add: emeasure_density measurable_completion nn_integral_completion intro!: nn_integral_cong_AE)
qed

lemma null_sets_subset: "B ∈ null_sets M ==> A ∈ sets M ==> A ⊆ B ==> A ∈ null_sets M"
  using emeasure_mono[of A B M] by (simp add: null_sets_def)

lemma (in complete_measure) complete2: "A ⊆ B ==> B ∈ null_sets M ==> A ∈ null_sets M"
  using complete[of A B] null_sets_subset[of B M A] by simp

lemma (in complete_measure) AE_iff_null_sets: "(AE x in M. P x) ⟷ {x∈space M. ¬ P x} ∈ null_sets M"
  unfolding eventually_ae_filter by (auto intro: complete2)

lemma (in complete_measure) null_sets_iff_AE: "A ∈ null_sets M ⟷ ((AE x in M. x ∉ A) ∧ A ⊆ space M)"
  unfolding AE_iff_null_sets by (auto cong: rev_conj_cong dest: sets.sets_into_space simp: subset_eq)

lemma (in complete_measure) in_sets_AE:
  assumes ae: "AE x in M. x ∈ A ⟷ x ∈ B" and A: "A ∈ sets M" and B: "B ⊆ space M"
  shows "B ∈ sets M"
proof -
  have "(AE x in M. x ∉ B - A ∧ x ∉ A - B)"
    using ae by eventually_elim auto
  then have "B - A ∈ null_sets M" "A - B ∈ null_sets M"
    using A B unfolding null_sets_iff_AE by (auto dest: sets.sets_into_space)
  then have "A ∪ (B - A) - (A - B) ∈ sets M"
    using A by blast
  also have "A ∪ (B - A) - (A - B) = B"
    by auto
  finally show "B ∈ sets M" .
qed

lemma (in complete_measure) vimage_null_part_null_sets:
  assumes f: "f ∈ M →🪙M N" and eq: "null_sets N ⊆ null_sets (distr M N f)"
    and A: "A ∈ completion N"
  shows "f -` null_part N A ∩ space M ∈ null_sets M"
proof -
  obtain N' where "N' ∈ null_sets N" "null_part N A ⊆ N'"
    using null_part[OF A] by auto
  then have N': "N' ∈ null_sets (distr M N f)"
    using eq by auto
  show ?thesis
  proof (rule complete2)
    show "f -` null_part N A ∩ space M ⊆ f -` N' ∩ space M"
      using ‹null_part N A ⊆ N'› by auto
    show "f -` N' ∩ space M ∈ null_sets M"
      using f N' by (auto simp: null_sets_def emeasure_distr)
  qed
qed

lemma (in complete_measure) vimage_null_part_sets:
  "f ∈ M →🪙M N ==> null_sets N ⊆ null_sets (distr M N f) ==> A ∈ completion N ==>
  f -` null_part N A ∩ space M ∈ sets M"
  using vimage_null_part_null_sets[of f N A] by auto

lemma (in complete_measure) measurable_completion2:
  assumes f: "f ∈ M →🪙M N" and null_sets: "null_sets N ⊆ null_sets (distr M N f)"
  shows "f ∈ M →🪙M completion N"
proof (rule measurableI)
  show "x ∈ space M ==> f x ∈ space (completion N)" for x
    using f[THEN measurable_space] by auto
  fix A assume A: "A ∈ sets (completion N)"
  have "f -` A ∩ space M = (f -` main_part N A ∩ space M) ∪ (f -` null_part N A ∩ space M)"
    using main_part_null_part_Un[OF A] by auto
  then show "f -` A ∩ space M ∈ sets M"
    using f A null_sets by (auto intro: vimage_null_part_sets measurable_sets)
qed

lemma (in complete_measure) completion_distr_eq:
  assumes X: "X ∈ M →🪙M N" and null_sets: "null_sets (distr M N X) = null_sets N"
  shows "completion (distr M N X) = distr M (completion N) X"
proof (rule measure_eqI)
  show eq: "sets (completion (distr M N X)) = sets (distr M (completion N) X)"
    by (simp add: sets_completion null_sets)

  fix A assume A: "A ∈ completion (distr M N X)"
  moreover have A': "A ∈ completion N"
    using A by (simp add: eq)
  moreover have "main_part (distr M N X) A ∈ sets N"
    using main_part_sets[OF A] by simp
  moreover have "X -` A ∩ space M = (X -` main_part (distr M N X) A ∩ space M) ∪ (X -` null_part (distr M N X) A ∩ space M)"
    using main_part_null_part_Un[OF A] by auto
  moreover have "X -` null_part (distr M N X) A ∩ space M ∈ null_sets M"
    using X A by (intro vimage_null_part_null_sets) (auto cong: distr_cong)
  ultimately show "emeasure (completion (distr M N X)) A = emeasure (distr M (completion N) X) A"
    using X by (auto simp: emeasure_distr measurable_completion null_sets measurable_completion2
                     intro!: emeasure_Un_null_set[symmetric])
qed

lemma distr_completion: "X ∈ M →🪙M N ==> distr (completion M) N X = distr M N X"
  by (intro measure_eqI) (auto simp: emeasure_distr measurable_completion)

proposition (in complete_measure) fmeasurable_inner_outer:
  "S ∈ fmeasurable M ⟷
    (∀e>0. ∃T∈fmeasurable M. ∃U∈fmeasurable M. T ⊆ S ∧ S ⊆ U ∧ ∣measure M T - measure M U∣ < e)"
  (is "_ ⟷ ?approx")
proof safe
  let ?μ = "measure M" let ?D = "λT U . ∣?μ T - ?μ U∣"
  assume ?approx
  have "∃A. ∀n. (fst (A n) ∈ fmeasurable M ∧ snd (A n) ∈ fmeasurable M ∧ fst (A n) ⊆ S ∧ S ⊆ snd (A n) ∧
    ?D (fst (A n)) (snd (A n)) < 1/Suc n) ∧ (fst (A n) ⊆ fst (A (Suc n)) ∧ snd (A (Suc n)) ⊆ snd (A n))"
    (is "∃A. ∀n. ?P n (A n) ∧ ?Q (A n) (A (Suc n))")
  proof (intro dependent_nat_choice)
    show "∃A. ?P 0 A"
      using ‹?approx›[THEN spec, of 1] by auto
  next
    fix A n assume "?P n A"
    moreover
    from ‹?approx›[THEN spec, of "1/Suc (Suc n)"]
    obtain T U where *: "T ∈ fmeasurable M" "U ∈ fmeasurable M" "T ⊆ S" "S ⊆ U" "?D T U < 1 / Suc (Suc n)"
      by auto
    ultimately have "?μ T ≤ ?μ (T ∪ fst A)" "?μ (U ∩ snd A) ≤ ?μ U"
      "?μ T ≤ ?μ U" "?μ (T ∪ fst A) ≤ ?μ (U ∩ snd A)"
      by (auto intro!: measure_mono_fmeasurable)
    then have "?D (T ∪ fst A) (U ∩ snd A) ≤ ?D T U"
      by auto
    also have "?D T U < 1/Suc (Suc n)" by fact
    finally show "∃B. ?P (Suc n) B ∧ ?Q A B"
      using ‹?P n A› *
      by (intro exI[of _ "(T ∪ fst A, U ∩ snd A)"] conjI) auto
  qed
  then obtain A
    where lm: "∧n. fst (A n) ∈ fmeasurable M" "∧n. snd (A n) ∈ fmeasurable M"
      and set_bound: "∧n. fst (A n) ⊆ S" "∧n. S ⊆ snd (A n)"
      and mono: "∧n. fst (A n) ⊆ fst (A (Suc n))" "∧n. snd (A (Suc n)) ⊆ snd (A n)"
      and bound: "∧n. ?D (fst (A n)) (snd (A n)) < 1/Suc n"
    by metis

  have INT_sA: "(∩n. snd (A n)) ∈ fmeasurable M"
    using lm by (intro fmeasurable_INT[of _ 0]) auto
  have UN_fA: "(∪n. fst (A n)) ∈ fmeasurable M"
    using lm order_trans[OF set_bound(1) set_bound(2)[of 0]] by (intro fmeasurable_UN[of _ _ "snd (A 0)"]) auto

  have "(λn. ?μ (fst (A n)) - ?μ (snd (A n))) <---- 0"
    using bound
    by (subst tendsto_rabs_zero_iff[symmetric])
       (intro tendsto_sandwich[OF _ _ tendsto_const LIMSEQ_inverse_real_of_nat];
        auto intro!: always_eventually less_imp_le simp: divide_inverse)
  moreover
  have "(λn. ?μ (fst (A n)) - ?μ (snd (A n))) <---- ?μ (∪n. fst (A n)) - ?μ (∩n. snd (A n))"
  proof (intro tendsto_diff Lim_measure_incseq Lim_measure_decseq)
    show "range (λi. fst (A i)) ⊆ sets M" "range (λi. snd (A i)) ⊆ sets M"
      "incseq (λi. fst (A i))" "decseq (λn. snd (A n))"
      using mono lm by (auto simp: incseq_Suc_iff decseq_Suc_iff intro!: measure_mono_fmeasurable)
    show "emeasure M (∪x. fst (A x)) ≠ ∞" "emeasure M (snd (A n)) ≠ ∞" for n
      using lm(2)[of n] UN_fA by (auto simp: fmeasurable_def)
  qed
  ultimately have eq: "0 = ?μ (∪n. fst (A n)) - ?μ (∩n. snd (A n))"
    by (rule LIMSEQ_unique)

  show "S ∈ fmeasurable M"
    using UN_fA INT_sA
  proof (rule complete_sets_sandwich_fmeasurable)
    show "(∪n. fst (A n)) ⊆ S" "S ⊆ (∩n. snd (A n))"
      using set_bound by auto
    show "?μ (∪n. fst (A n)) = ?μ (∩n. snd (A n))"
      using eq by auto
  qed
qed (auto intro!: bexI[of _ S])

lemma (in complete_measure) fmeasurable_measure_inner_outer:
   "(S ∈ fmeasurable M ∧ m = measure M S) ⟷
      (∀e>0. ∃T∈fmeasurable M. T ⊆ S ∧ m - e < measure M T) ∧
      (∀e>0. ∃U∈fmeasurable M. S ⊆ U ∧ measure M U < m + e)"
    (is "?lhs = ?rhs")
proof
  assume RHS: ?rhs
  then have T: "∧e. 0 < e ⟶ (∃T∈fmeasurable M. T ⊆ S ∧ m - e < measure M T)"
        and U: "∧e. 0 < e ⟶ (∃U∈fmeasurable M. S ⊆ U ∧ measure M U < m + e)"
    by auto
  have "S ∈ fmeasurable M"
  proof (subst fmeasurable_inner_outer, safe)
    fix e::real assume "0 < e"
    with RHS obtain T U where T: "T ∈ fmeasurable M" "T ⊆ S" "m - e/2 < measure M T"
                          and U: "U ∈ fmeasurable M" "S ⊆ U" "measure M U < m + e/2"
      by (meson half_gt_zero)+
    moreover have "measure M U - measure M T < (m + e/2) - (m - e/2)"
      by (intro diff_strict_mono) fact+
    moreover have "measure M T ≤ measure M U"
      using T U by (intro measure_mono_fmeasurable) auto
    ultimately show "∃T∈fmeasurable M. ∃U∈fmeasurable M. T ⊆ S ∧ S ⊆ U ∧ ∣measure M T - measure M U∣ < e"
      apply (rule_tac bexI[OF _ ‹T ∈ fmeasurable M›])
      apply (rule_tac bexI[OF _ ‹U ∈ fmeasurable M›])
      by auto
  qed
  moreover have "m = measure M S"
    using ‹S ∈ fmeasurable M› U[of "measure M S - m"] T[of "m - measure M S"]
    by (cases m "measure M S" rule: linorder_cases)
       (auto simp: not_le[symmetric] measure_mono_fmeasurable)
  ultimately show ?lhs
    by simp
qed (auto intro!: bexI[of _ S])

lemma (in complete_measure) null_sets_outer:
  "S ∈ null_sets M ⟷ (∀e>0. ∃T∈fmeasurable M. S ⊆ T ∧ measure M T < e)"
proof -
  have "S ∈ null_sets M ⟷ (S ∈ fmeasurable M ∧ 0 = measure M S)"
    by (auto simp: null_sets_def emeasure_eq_measure2 intro: fmeasurableI) (simp add: measure_def)
  also have "… = (∀e>0. ∃T∈fmeasurable M. S ⊆ T ∧ measure M T < e)"
    unfolding fmeasurable_measure_inner_outer by auto
  finally show ?thesis .
qed

lemma (in complete_measure) fmeasurable_measure_inner_outer_le:
     "(S ∈ fmeasurable M ∧ m = measure M S) ⟷
        (∀e>0. ∃T∈fmeasurable M. T ⊆ S ∧ m - e ≤ measure M T) ∧
        (∀e>0. ∃U∈fmeasurable M. S ⊆ U ∧ measure M U ≤ m + e)" (is ?T1)
  and null_sets_outer_le:
     "S ∈ null_sets M ⟷ (∀e>0. ∃T∈fmeasurable M. S ⊆ T ∧ measure M T ≤ e)" (is ?T2)
proof -
  have "e > 0 ∧ m - e/2 ≤ t ==> m - e < t"
       "e > 0 ∧ t ≤ m + e/2 ==> t < m + e"
       "e > 0 ⟷ e/2 > 0"
       for e t
    by auto
  then show ?T1 ?T2
    unfolding fmeasurable_measure_inner_outer null_sets_outer
    by (meson dense le_less_trans less_imp_le)+
qed

lemma (in cld_measure) notin_sets_outer_measure_of_cover:
  assumes E: "E ⊆ space M" "E ∉ sets M"
  shows "∃B∈sets M. 0 < emeasure M B ∧ emeasure M B < ∞ ∧
    outer_measure_of M (B ∩ E) = emeasure M B ∧ outer_measure_of M (B - E) = emeasure M B"
proof -
  from locally_determined[OF ‹E ⊆ space M›] ‹E ∉ sets M›
  obtain F
    where [measurable]: "F ∈ sets M" and "emeasure M F < ∞" "E ∩ F ∉ sets M"
    by blast
  then obtain H H'
    where H: "measurable_envelope M (F ∩ E) H" and H': "measurable_envelope M (F - E) H'"
    using measurable_envelopeI_countable_cover[of "F ∩ E" "λ_. F" M]
       measurable_envelopeI_countable_cover[of "F - E" "λ_. F" M]
    by auto
  note measurable_envelopeD(2)[OF H', measurable] measurable_envelopeD(2)[OF H, measurable]

  from measurable_envelopeD(1)[OF H'] measurable_envelopeD(1)[OF H]
  have subset: "F - H' ⊆ F ∩ E" "F ∩ E ⊆ F ∩ H"
    by auto
  moreover define G where "G = (F ∩ H) - (F - H')"
  ultimately have G: "G = F ∩ H ∩ H'"
    by auto
  have "emeasure M (F ∩ H) ≠ 0"
  proof
    assume "emeasure M (F ∩ H) = 0"
    then have "F ∩ H ∈ null_sets M"
      by auto
    with ‹E ∩ F ∉ sets M› show False
      using complete[OF ‹F ∩ E ⊆ F ∩ H›] by (auto simp: Int_commute)
  qed
  moreover
  have "emeasure M (F - H') ≠ emeasure M (F ∩ H)"
  proof
    assume "emeasure M (F - H') = emeasure M (F ∩ H)"
    with ‹E ∩ F ∉ sets M› emeasure_mono[of "F ∩ H" F M] ‹emeasure M F 🚫∞›
    have "F ∩ E ∈ sets M"
      by (intro complete_sets_sandwich[OF _ _ subset]) auto
    with ‹E ∩ F ∉ sets M› show False
      by (simp add: Int_commute)
  qed
  moreover have "emeasure M (F - H') ≤ emeasure M (F ∩ H)"
    using subset by (intro emeasure_mono) auto
  ultimately have "emeasure M G ≠ 0"
    unfolding G_def using subset
    by (subst emeasure_Diff) (auto simp: top_unique diff_eq_0_iff_ennreal)
  show ?thesis
  proof (intro bexI conjI)
    have "emeasure M G ≤ emeasure M F"
      unfolding G by (auto intro!: emeasure_mono)
    with ‹emeasure M F 🚫∞› show "0 < emeasure M G" "emeasure M G < ∞"
      using ‹emeasure M G ≠ 0› by (auto simp: zero_less_iff_neq_zero)
    show [measurable]: "G ∈ sets M"
      unfolding G by auto

    have "emeasure M G = outer_measure_of M (F ∩ H' ∩ (F ∩ E))"
      using measurable_envelopeD(3)[OF H, of "F ∩ H'"] unfolding G by (simp add: ac_simps)
    also have "… ≤ outer_measure_of M (G ∩ E)"
      using measurable_envelopeD(1)[OF H] by (intro outer_measure_of_mono) (auto simp: G)
    finally show "outer_measure_of M (G ∩ E) = emeasure M G"
      using outer_measure_of_mono[of "G ∩ E" G M] by auto

    have "emeasure M G = outer_measure_of M (F ∩ H ∩ (F - E))"
      using measurable_envelopeD(3)[OF H', of "F ∩ H"] unfolding G by (simp add: ac_simps)
    also have "… ≤ outer_measure_of M (G - E)"
      using measurable_envelopeD(1)[OF H'] by (intro outer_measure_of_mono) (auto simp: G)
    finally show "outer_measure_of M (G - E) = emeasure M G"
      using outer_measure_of_mono[of "G - E" G M] by auto
  qed
qed

text ‹The following theorem is a specialization of D.H. Fremlin, Measure Theory vol 4I (413G). We
  only show one direction and do not use a inner regular family ‹K›.›

lemma (in cld_measure) borel_measurable_cld:
  fixes f :: "'a ==> real"
  assumes "∧A a b. A ∈ sets M ==> 0 < emeasure M A ==> emeasure M A < ∞ ==> a < b ==>
      min (outer_measure_of M {x∈A. f x ≤ a}) (outer_measure_of M {x∈A. b ≤ f x}) < emeasure M A"
  shows "f ∈ M →🪙M borel"
proof (rule ccontr)
  let ?E = "λa. {x∈space M. f x ≤ a}" and ?F = "λa. {x∈space M. a ≤ f x}"

  assume "f ∉ M →🪙M borel"
  then obtain a where "?E a ∉ sets M"
    unfolding borel_measurable_iff_le by blast
  from notin_sets_outer_measure_of_cover[OF _ this]
  obtain K
    where K: "K ∈ sets M" "0 < emeasure M K" "emeasure M K < ∞"
      and eq1: "outer_measure_of M (K ∩ ?E a) = emeasure M K"
      and eq2: "outer_measure_of M (K - ?E a) = emeasure M K"
    by auto
  then have me_K: "measurable_envelope M (K ∩ ?E a) K"
    by (subst measurable_envelope_eq2) auto

  define b where "b n = a + inverse (real (Suc n))" for n
  have "(SUP n. outer_measure_of M (K ∩ ?F (b n))) = outer_measure_of M (∪n. K ∩ ?F (b n))"
  proof (intro SUP_outer_measure_of_incseq)
    have "x ≤ y ==> b y ≤ b x" for x y
      by (auto simp: b_def field_simps)
    then show "incseq (λn. K ∩ {x ∈ space M. b n ≤ f x})"
      by (auto simp: incseq_def intro: order_trans)
  qed auto
  also have "(∪n. K ∩ ?F (b n)) = K - ?E a"
  proof -
    have "b <---- a"
      unfolding b_def by (rule LIMSEQ_inverse_real_of_nat_add)
    then have "∀n. ¬ b n ≤ f x ==> f x ≤ a" for x
      by (rule LIMSEQ_le_const) (auto intro: less_imp_le simp: not_le)
    moreover have "¬ b n ≤ a" for n
      by (auto simp: b_def)
    ultimately show ?thesis
      using ‹K ∈ sets M›[THEN sets.sets_into_space] by (auto simp: subset_eq intro: order_trans)
  qed
  finally have "0 < (SUP n. outer_measure_of M (K ∩ ?F (b n)))"
    using K by (simp add: eq2)
  then obtain n where pos_b: "0 < outer_measure_of M (K ∩ ?F (b n))" and "a < b n"
    unfolding less_SUP_iff by (auto simp: b_def)
  from measurable_envelopeI_countable_cover[of "K ∩ ?F (b n)" "λ_. K" M] K
  obtain K' where "K' ⊆ K" and me_K': "measurable_envelope M (K ∩ ?F (b n)) K'"
    by auto
  then have K'_le_K: "emeasure M K' ≤ emeasure M K"
    by (intro emeasure_mono K)
  have "K' ∈ sets M"
    using me_K' by (rule measurable_envelopeD)

  have "min (outer_measure_of M {x∈K'. f x ≤ a}) (outer_measure_of M {x∈K'. b n ≤ f x}) < emeasure M K'"
  proof (rule assms)
    show "0 < emeasure M K'" "emeasure M K' < ∞"
      using measurable_envelopeD2[OF me_K'] pos_b K K'_le_K by auto
  qed fact+
  also have "{x∈K'. f x ≤ a} = K' ∩ (K ∩ ?E a)"
    using ‹K' ∈ sets M›[THEN sets.sets_into_space] ‹K' ⊆ K› by auto
  also have "{x∈K'. b n ≤ f x} = K' ∩ (K ∩ ?F (b n))"
    using ‹K' ∈ sets M›[THEN sets.sets_into_space] ‹K' ⊆ K› by auto
  finally have "min (emeasure M K) (emeasure M K') < emeasure M K'"
    unfolding
      measurable_envelopeD(3)[OF me_K ‹K' ∈ sets M›, symmetric]
      measurable_envelopeD(3)[OF me_K' ‹K' ∈ sets M›, symmetric]
    using ‹K' ⊆ K› by (simp add: Int_absorb1 Int_absorb2)
  with K'_le_K show False
    by (auto simp: min_def split: if_split_asm)
qed

end