Theory Measure_Space

(*  Title:      HOL/Analysis/Measure_Space.thy
    Author:     Lawrence C Paulson
    Author:     Johannes Hölzl, TU München
    Author:     Armin Heller, TU München
*)

section ‹Measure Spaces›

theory Measure_Space
imports
  Measurable "HOL-Library.Extended_Nonnegative_Real"
begin

subsection✐‹tag unimportant› "Relate extended reals and the indicator function"

lemma suminf_cmult_indicator:
  fixes f :: "nat ⇒ ennreal"
  assumes "disjoint_family A" "x ∈ A i"
  shows "(∑n. f n * indicator (A n) x) = f i"
proof -
  have **: "⋀n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)"
    using ‹x ∈ A i› assms unfolding disjoint_family_on_def indicator_def by auto
  then have "⋀n. (∑j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ennreal)"
    by (auto simp: sum.If_cases)
  moreover have "(SUP n. if i < n then f i else 0) = (f i :: ennreal)"
  proof (rule SUP_eqI)
    fix y :: ennreal assume "⋀n. n ∈ UNIV ⟹ (if i < n then f i else 0) ≤ y"
    from this[of "Suc i"] show "f i ≤ y" by auto
  qed (use assms in simp)
  ultimately show ?thesis using assms
    by (simp add: suminf_eq_SUP)
qed

lemma suminf_indicator:
  assumes "disjoint_family A"
  shows "(∑n. indicator (A n) x :: ennreal) = indicator (⋃i. A i) x"
proof cases
  assume *: "x ∈ (⋃i. A i)"
  then obtain i where "x ∈ A i" by auto
  from suminf_cmult_indicator[OF assms(1), OF ‹x ∈ A i›, of "λk. 1"]
  show ?thesis using * by simp
qed simp

lemma sum_indicator_disjoint_family:
  fixes f :: "'d ⇒ 'e::semiring_1"
  assumes d: "disjoint_family_on A P" and "x ∈ A j" and "finite P" and "j ∈ P"
  shows "(∑i∈P. f i * indicator (A i) x) = f j"
proof -
  have "P ∩ {i. x ∈ A i} = {j}"
    using d ‹x ∈ A j› ‹j ∈ P› unfolding disjoint_family_on_def
    by auto
  with ‹finite P› show ?thesis
    by (simp add: indicator_def)
qed

text ‹
  The type for emeasure spaces is already defined in \<^theory>‹HOL-Analysis.Sigma_Algebra›, as it
  is also used to represent sigma algebras (with an arbitrary emeasure).
›

subsection✐‹tag unimportant› "Extend binary sets"

lemma LIMSEQ_binaryset:
  assumes f: "f {} = 0"
  shows  "(λn. ∑i<n. f (binaryset A B i)) ⇢ f A + f B"
proof -
  have "(λn. ∑i < Suc (Suc n). f (binaryset A B i)) = (λn. f A + f B)"
    proof
      fix n
      show "(∑i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
        by (induct n)  (auto simp add: binaryset_def f)
    qed
  moreover
  have "… ⇢ f A + f B" by (rule tendsto_const)
  ultimately have "(λn. ∑i< n+2. f (binaryset A B i)) ⇢ f A + f B"
    by simp
  thus ?thesis by (rule LIMSEQ_offset [where k=2])
qed

lemma binaryset_sums:
  assumes f: "f {} = 0"
  shows  "(λn. f (binaryset A B n)) sums (f A + f B)"
  using LIMSEQ_binaryset f sums_def by blast

lemma suminf_binaryset_eq:
  fixes f :: "'a set ⇒ 'b::{comm_monoid_add, t2_space}"
  shows "f {} = 0 ⟹ (∑n. f (binaryset A B n)) = f A + f B"
  by (metis binaryset_sums sums_unique)

subsection✐‹tag unimportant› ‹Properties of a premeasure \<^term>‹μ››

text ‹
  The definitions for \<^const>‹positive› and \<^const>‹countably_additive› should be here, by they are
  necessary to define \<^typ>‹'a measure› in \<^theory>‹HOL-Analysis.Sigma_Algebra›.
›

definition subadditive where
  "subadditive M f ⟷ (∀x∈M. ∀y∈M. x ∩ y = {} ⟶ f (x ∪ y) ≤ f x + f y)"

lemma subadditiveD: "subadditive M f ⟹ x ∩ y = {} ⟹ x ∈ M ⟹ y ∈ M ⟹ f (x ∪ y) ≤ f x + f y"
  by (auto simp add: subadditive_def)

definition countably_subadditive where
  "countably_subadditive M f ⟷
    (∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i. A i) ∈ M ⟶ (f (⋃i. A i) ≤ (∑i. f (A i))))"

lemma (in ring_of_sets) countably_subadditive_subadditive:
  fixes f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" and cs: "countably_subadditive M f"
  shows  "subadditive M f"
proof (auto simp add: subadditive_def)
  fix x y
  assume x: "x ∈ M" and y: "y ∈ M" and "x ∩ y = {}"
  hence "disjoint_family (binaryset x y)"
    by (auto simp add: disjoint_family_on_def binaryset_def)
  hence "range (binaryset x y) ⊆ M ⟶
         (⋃i. binaryset x y i) ∈ M ⟶
         f (⋃i. binaryset x y i) ≤ (∑ n. f (binaryset x y n))"
    using cs by (auto simp add: countably_subadditive_def)
  hence "{x,y,{}} ⊆ M ⟶ x ∪ y ∈ M ⟶
         f (x ∪ y) ≤ (∑ n. f (binaryset x y n))"
    by (simp add: range_binaryset_eq UN_binaryset_eq)
  thus "f (x ∪ y) ≤  f x + f y" using f x y
    by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
qed

definition additive where
  "additive M μ ⟷ (∀x∈M. ∀y∈M. x ∩ y = {} ⟶ μ (x ∪ y) = μ x + μ y)"

definition increasing where
  "increasing M μ ⟷ (∀x∈M. ∀y∈M. x ⊆ y ⟶ μ x ≤ μ y)"

lemma positiveD1: "positive M f ⟹ f {} = 0" by (auto simp: positive_def)

lemma positiveD_empty:
  "positive M f ⟹ f {} = 0"
  by (auto simp add: positive_def)

lemma additiveD:
  "additive M f ⟹ x ∩ y = {} ⟹ x ∈ M ⟹ y ∈ M ⟹ f (x ∪ y) = f x + f y"
  by (auto simp add: additive_def)

lemma increasingD:
  "increasing M f ⟹ x ⊆ y ⟹ x∈M ⟹ y∈M ⟹ f x ≤ f y"
  by (auto simp add: increasing_def)

lemma countably_additiveI[case_names countably]:
  "(⋀A. ⟦range A ⊆ M; disjoint_family A; (⋃i. A i) ∈ M⟧ ⟹ (∑i. f(A i)) = f(⋃i. A i))
  ⟹ countably_additive M f"
  by (simp add: countably_additive_def)

lemma (in ring_of_sets) disjointed_additive:
  assumes f: "positive M f" "additive M f" and A: "range A ⊆ M" "incseq A"
  shows "(∑i≤n. f (disjointed A i)) = f (A n)"
proof (induct n)
  case (Suc n)
  then have "(∑i≤Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
    by simp
  also have "… = f (A n ∪ disjointed A (Suc n))"
    using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_mono)
  also have "A n ∪ disjointed A (Suc n) = A (Suc n)"
    using ‹incseq A› by (auto dest: incseq_SucD simp: disjointed_mono)
  finally show ?case .
qed simp

lemma (in ring_of_sets) additive_sum:
  fixes A:: "'i ⇒ 'a set"
  assumes f: "positive M f" and ad: "additive M f" and "finite S"
      and A: "A`S ⊆ M"
      and disj: "disjoint_family_on A S"
  shows  "(∑i∈S. f (A i)) = f (⋃i∈S. A i)"
  using ‹finite S› disj A
proof induct
  case empty show ?case using f by (simp add: positive_def)
next
  case (insert s S)
  then have "A s ∩ (⋃i∈S. A i) = {}"
    by (auto simp add: disjoint_family_on_def neq_iff)
  moreover
  have "A s ∈ M" using insert by blast
  moreover have "(⋃i∈S. A i) ∈ M"
    using insert ‹finite S› by auto
  ultimately have "f (A s ∪ (⋃i∈S. A i)) = f (A s) + f(⋃i∈S. A i)"
    using ad UNION_in_sets A by (auto simp add: additive_def)
  with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
    by (auto simp add: additive_def subset_insertI)
qed

lemma (in ring_of_sets) additive_increasing:
  fixes f :: "'a set ⇒ ennreal"
  assumes posf: "positive M f" and addf: "additive M f"
  shows "increasing M f"
proof (auto simp add: increasing_def)
  fix x y
  assume xy: "x ∈ M" "y ∈ M" "x ⊆ y"
  then have "y - x ∈ M" by auto
  then have "f x + 0 ≤ f x + f (y-x)" by (intro add_left_mono zero_le)
  also have "… = f (x ∪ (y-x))"
    by (metis addf Diff_disjoint ‹y - x ∈ M› additiveD xy(1))
  also have "… = f y"
    by (metis Un_Diff_cancel Un_absorb1 xy(3))
  finally show "f x ≤ f y" by simp
qed

lemma (in ring_of_sets) subadditive:
  fixes f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" "additive M f" and A: "A`S ⊆ M" and S: "finite S"
  shows "f (⋃i∈S. A i) ≤ (∑i∈S. f (A i))"
using S A
proof (induct S)
  case empty thus ?case using f by (auto simp: positive_def)
next
  case (insert x F)
  hence in_M: "A x ∈ M" "(⋃i∈F. A i) ∈ M" "(⋃i∈F. A i) - A x ∈ M" using A by force+
  have subs: "(⋃i∈F. A i) - A x ⊆ (⋃i∈F. A i)" by auto
  have "(⋃i∈(insert x F). A i) = A x ∪ ((⋃i∈F. A i) - A x)" by auto
  hence "f (⋃i∈(insert x F). A i) = f (A x ∪ ((⋃i∈F. A i) - A x))"
    by simp
  also have "… = f (A x) + f ((⋃i∈F. A i) - A x)"
    using f(2) by (rule additiveD) (insert in_M, auto)
  also have "… ≤ f (A x) + f (⋃i∈F. A i)"
    using additive_increasing[OF f] in_M subs 
    by (simp add: increasingD)
  also have "… ≤ f (A x) + (∑i∈F. f (A i))" 
    using insert by (auto intro: add_left_mono)
  finally show "f (⋃i∈(insert x F). A i) ≤ (∑i∈(insert x F). f (A i))"
    by (simp add: insert)
qed

lemma (in ring_of_sets) countably_additive_additive:
  fixes f :: "'a set ⇒ ennreal"
  assumes posf: "positive M f" and ca: "countably_additive M f"
  shows "additive M f"
proof (auto simp add: additive_def)
  fix x y
  assume x: "x ∈ M" and y: "y ∈ M" and "x ∩ y = {}"
  hence "disjoint_family (binaryset x y)"
    by (auto simp add: disjoint_family_on_def binaryset_def)
  hence "range (binaryset x y) ⊆ M ⟶
         (⋃i. binaryset x y i) ∈ M ⟶
         f (⋃i. binaryset x y i) = (∑ n. f (binaryset x y n))"
    using ca by (simp add: countably_additive_def)
  hence "{x,y,{}} ⊆ M ⟶ x ∪ y ∈ M ⟶ f (x ∪ y) = (∑n. f (binaryset x y n))"
    by (simp add: range_binaryset_eq UN_binaryset_eq)
  thus "f (x ∪ y) = f x + f y" using posf x y
    by (auto simp add: Un suminf_binaryset_eq positive_def)
qed

lemma (in algebra) increasing_additive_bound:
  fixes A:: "nat ⇒ 'a set" and  f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" and ad: "additive M f"
      and inc: "increasing M f"
      and A: "range A ⊆ M"
      and disj: "disjoint_family A"
  shows  "(∑i. f (A i)) ≤ f Ω"
proof (safe intro!: suminf_le_const)
  fix N
  note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
  have "(∑i<N. f (A i)) = f (⋃i∈{..<N}. A i)"
    using A by (intro additive_sum [OF f ad]) (auto simp: disj_N)
  also have "… ≤ f Ω" using space_closed A
    by (intro increasingD[OF inc] finite_UN) auto
  finally show "(∑i<N. f (A i)) ≤ f Ω" by simp
qed (insert f A, auto simp: positive_def)

lemma (in ring_of_sets) countably_additiveI_finite:
  fixes μ :: "'a set ⇒ ennreal"
  assumes "finite Ω" "positive M μ" "additive M μ"
  shows "countably_additive M μ"
proof (rule countably_additiveI)
  fix F :: "nat ⇒ 'a set" assume F: "range F ⊆ M" "(⋃i. F i) ∈ M" and disj: "disjoint_family F"

  have "∀i. F i ≠ {} ⟶ (∃x. x ∈ F i)" by auto
  then obtain f where f: "⋀i. F i ≠ {} ⟹ f i ∈ F i" by metis

  have finU: "finite (⋃i. F i)"
    by (metis F(2) assms(1) infinite_super sets_into_space)

  have F_subset: "{i. μ (F i) ≠ 0} ⊆ {i. F i ≠ {}}"
    by (auto simp: positiveD_empty[OF ‹positive M μ›])
  moreover have fin_not_empty: "finite {i. F i ≠ {}}"
  proof (rule finite_imageD)
    from f have "f`{i. F i ≠ {}} ⊆ (⋃i. F i)" by auto
    then show "finite (f`{i. F i ≠ {}})"
      by (simp add: finU finite_subset)
    show inj_f: "inj_on f {i. F i ≠ {}}" 
      using f disj
      by (simp add: inj_on_def disjoint_family_on_def disjoint_iff) metis
  qed 
  ultimately have fin_not_0: "finite {i. μ (F i) ≠ 0}"
    by (rule finite_subset)

  have disj_not_empty: "disjoint_family_on F {i. F i ≠ {}}"
    using disj by (auto simp: disjoint_family_on_def)

  from fin_not_0 have "(∑i. μ (F i)) = (∑i | μ (F i) ≠ 0. μ (F i))"
    by (rule suminf_finite) auto
  also have "… = (∑i | F i ≠ {}. μ (F i))"
    using fin_not_empty F_subset by (rule sum.mono_neutral_left) auto
  also have "… = μ (⋃i∈{i. F i ≠ {}}. F i)"
    using ‹positive M μ› ‹additive M μ› fin_not_empty disj_not_empty F by (intro additive_sum) auto
  also have "… = μ (⋃i. F i)"
    by (rule arg_cong[where f=μ]) auto
  finally show "(∑i. μ (F i)) = μ (⋃i. F i)" .
qed

lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
  fixes f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" "additive M f"
  shows "countably_additive M f ⟷
    (∀A. range A ⊆ M ⟶ incseq A ⟶ (⋃i. A i) ∈ M ⟶ (λi. f (A i)) ⇢ f (⋃i. A i))"
  unfolding countably_additive_def
proof safe
  assume count_sum: "∀A. range A ⊆ M ⟶ disjoint_family A ⟶ ⋃(A ` UNIV) ∈ M ⟶ (∑i. f (A i)) = f (⋃(A ` UNIV))"
  fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ M" "incseq A" "(⋃i. A i) ∈ M"
  then have dA: "range (disjointed A) ⊆ M" by (auto simp: range_disjointed_sets)
  with count_sum[THEN spec, of "disjointed A"] A(3)
  have f_UN: "(∑i. f (disjointed A i)) = f (⋃i. A i)"
    by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
  moreover have "(λn. (∑i<n. f (disjointed A i))) ⇢ (∑i. f (disjointed A i))"
    by (simp add: summable_LIMSEQ)
  from LIMSEQ_Suc[OF this]
  have "(λn. (∑i≤n. f (disjointed A i))) ⇢ (∑i. f (disjointed A i))"
    unfolding lessThan_Suc_atMost .
  moreover have "⋀n. (∑i≤n. f (disjointed A i)) = f (A n)"
    using disjointed_additive[OF f A(1,2)] .
  ultimately show "(λi. f (A i)) ⇢ f (⋃i. A i)" by simp
next
  assume cont[rule_format]: "∀A. range A ⊆ M ⟶ incseq A ⟶ (⋃i. A i) ∈ M ⟶ (λi. f (A i)) ⇢ f (⋃i. A i)"
  fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ M" "disjoint_family A" "(⋃i. A i) ∈ M"
  have *: "(⋃n. (⋃i<n. A i)) = (⋃i. A i)" by auto
  have "range (λi. ⋃i<i. A i) ⊆ M" "(⋃i. ⋃i<i. A i) ∈ M"
    using A * by auto
  then have "(λn. f (⋃i<n. A i)) ⇢ f (⋃i. A i)"
    unfolding *[symmetric] by (force intro!: cont incseq_SucI)+
  moreover have "⋀n. f (⋃i<n. A i) = (∑i<n. f (A i))"
    using A
    by (intro additive_sum[OF f, symmetric]) (auto intro: disjoint_family_on_mono)
  ultimately
  have "(λi. f (A i)) sums f (⋃i. A i)"
    unfolding sums_def by simp
  then show "(∑i. f (A i)) = f (⋃i. A i)"
    by (metis sums_unique)
qed

lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
  fixes f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" "additive M f"
  shows "(∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) ∈ M ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ f (⋂i. A i))
     ⟷ (∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) = {} ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ 0)"
proof safe
  assume cont[rule_format]: "(∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) ∈ M ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ f (⋂i. A i))"
  fix A :: "nat ⇒ 'a set" 
  assume A: "range A ⊆ M" "decseq A" "(⋂i. A i) = {}" "∀i. f (A i) ≠ ∞"
  with cont[of A] show "(λi. f (A i)) ⇢ 0"
    using ‹positive M f›[unfolded positive_def] by auto
next
  assume cont[rule_format]: "∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) = {} ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ 0"
  fix A :: "nat ⇒ 'a set" 
  assume A: "range A ⊆ M" "decseq A" "(⋂i. A i) ∈ M" "∀i. f (A i) ≠ ∞"

  have f_mono: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ⊆ b ⟹ f a ≤ f b"
    using additive_increasing[OF f] unfolding increasing_def by simp

  have decseq_fA: "decseq (λi. f (A i))"
    using A by (auto simp: decseq_def intro!: f_mono)
  have decseq: "decseq (λi. A i - (⋂i. A i))"
    using A by (auto simp: decseq_def)
  then have decseq_f: "decseq (λi. f (A i - (⋂i. A i)))"
    using A unfolding decseq_def by (auto intro!: f_mono Diff)
  have "f (⋂x. A x) ≤ f (A 0)"
    using A by (auto intro!: f_mono)
  then have f_Int_fin: "f (⋂x. A x) ≠ ∞"
    using A by (auto simp: top_unique)
  have f_fin: "f (A i - (⋂i. A i)) ≠ ∞" for i
    using A by (metis Diff Diff_subset f_mono infinity_ennreal_def range_subsetD top_unique)
  have "(λi. f (A i - (⋂i. A i))) ⇢ 0"
  proof (intro cont[ OF _ decseq _ f_fin])
    show "range (λi. A i - (⋂i. A i)) ⊆ M" "(⋂i. A i - (⋂i. A i)) = {}"
      using A by auto
  qed
  with INF_Lim decseq_f have "(INF n. f (A n - (⋂i. A i))) = 0" by metis
  moreover have "(INF n. f (⋂i. A i)) = f (⋂i. A i)"
    by auto
  ultimately have "(INF n. f (A n - (⋂i. A i)) + f (⋂i. A i)) = 0 + f (⋂i. A i)"
    using A(4) f_fin f_Int_fin
    using INF_ennreal_add_const by presburger
  moreover {
    fix n
    have "f (A n - (⋂i. A i)) + f (⋂i. A i) = f ((A n - (⋂i. A i)) ∪ (⋂i. A i))"
      using A by (subst f(2)[THEN additiveD]) auto
    also have "(A n - (⋂i. A i)) ∪ (⋂i. A i) = A n"
      by auto
    finally have "f (A n - (⋂i. A i)) + f (⋂i. A i) = f (A n)" . }
  ultimately have "(INF n. f (A n)) = f (⋂i. A i)"
    by simp
  with LIMSEQ_INF[OF decseq_fA]
  show "(λi. f (A i)) ⇢ f (⋂i. A i)" by simp
qed

lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
  fixes f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" "additive M f" "∀A∈M. f A ≠ ∞"
  assumes cont: "∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) = {} ⟶ (λi. f (A i)) ⇢ 0"
  assumes A: "range A ⊆ M" "incseq A" "(⋃i. A i) ∈ M"
  shows "(λi. f (A i)) ⇢ f (⋃i. A i)"
proof -
  from A have "(λi. f ((⋃i. A i) - A i)) ⇢ 0"
    by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)
  moreover
  { fix i
    have "f ((⋃i. A i) - A i ∪ A i) = f ((⋃i. A i) - A i) + f (A i)"
      using A by (intro f(2)[THEN additiveD]) auto
    also have "((⋃i. A i) - A i) ∪ A i = (⋃i. A i)"
      by auto
    finally have "f ((⋃i. A i) - A i) = f (⋃i. A i) - f (A i)"
      using assms f by fastforce
  }
  moreover have "∀⇩F i in sequentially. f (A i) ≤ f (⋃i. A i)"
    using increasingD[OF additive_increasing[OF f(1, 2)], of "A _" "⋃i. A i"] A
    by (auto intro!: always_eventually simp: subset_eq)
  ultimately show "(λi. f (A i)) ⇢ f (⋃i. A i)"
    by (auto intro: ennreal_tendsto_const_minus)
qed

lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
  fixes f :: "'a set ⇒ ennreal"
  assumes f: "positive M f" "additive M f" and fin: "∀A∈M. f A ≠ ∞"
  assumes cont: "⋀A. range A ⊆ M ⟹ decseq A ⟹ (⋂i. A i) = {} ⟹ (λi. f (A i)) ⇢ 0"
  shows "countably_additive M f"
  using countably_additive_iff_continuous_from_below[OF f]
  using empty_continuous_imp_continuous_from_below[OF f fin] cont
  by blast

subsection✐‹tag unimportant› ‹Properties of \<^const>‹emeasure››

lemma emeasure_positive: "positive (sets M) (emeasure M)"
  by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
  using emeasure_positive[of M] by (simp add: positive_def)

lemma emeasure_single_in_space: "emeasure M {x} ≠ 0 ⟹ x ∈ space M"
  using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space zero_less_iff_neq_zero[THEN iffD2])

lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
  by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

lemma suminf_emeasure:
  "range A ⊆ sets M ⟹ disjoint_family A ⟹ (∑i. emeasure M (A i)) = emeasure M (⋃i. A i)"
  using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]
  by (simp add: countably_additive_def)

lemma sums_emeasure:
  "disjoint_family F ⟹ (⋀i. F i ∈ sets M) ⟹ (λi. emeasure M (F i)) sums emeasure M (⋃i. F i)"
  unfolding sums_iff by (intro conjI suminf_emeasure) auto

lemma emeasure_additive: "additive (sets M) (emeasure M)"
  by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive)

lemma plus_emeasure:
  "a ∈ sets M ⟹ b ∈ sets M ⟹ a ∩ b = {} ⟹ emeasure M a + emeasure M b = emeasure M (a ∪ b)"
  using additiveD[OF emeasure_additive] ..

lemma emeasure_Un:
  "A ∈ sets M ⟹ B ∈ sets M ⟹ emeasure M (A ∪ B) = emeasure M A + emeasure M (B - A)"
  using plus_emeasure[of A M "B - A"] by auto

lemma emeasure_Un_Int:
  assumes "A ∈ sets M" "B ∈ sets M"
  shows "emeasure M A + emeasure M B = emeasure M (A ∪ B) + emeasure M (A ∩ B)"
proof -
  have "A = (A-B) ∪ (A ∩ B)" by auto
  then have "emeasure M A = emeasure M (A-B) + emeasure M (A ∩ B)"
    by (metis Diff_Diff_Int Diff_disjoint assms plus_emeasure sets.Diff)
  moreover have "A ∪ B = (A-B) ∪ B" by auto
  then have "emeasure M (A ∪ B) = emeasure M (A-B) + emeasure M B"
    by (metis Diff_disjoint Int_commute assms plus_emeasure sets.Diff)
  ultimately show ?thesis by (metis add.assoc add.commute)
qed

lemma sum_emeasure:
  "F`I ⊆ sets M ⟹ disjoint_family_on F I ⟹ finite I ⟹
    (∑i∈I. emeasure M (F i)) = emeasure M (⋃i∈I. F i)"
  by (metis sets.additive_sum emeasure_positive emeasure_additive)

lemma emeasure_mono:
  "a ⊆ b ⟹ b ∈ sets M ⟹ emeasure M a ≤ emeasure M b"
  by (metis zero_le sets.additive_increasing emeasure_additive emeasure_notin_sets emeasure_positive increasingD)

lemma emeasure_space:
  "emeasure M A ≤ emeasure M (space M)"
  by (metis emeasure_mono emeasure_notin_sets sets.sets_into_space sets.top zero_le)

lemma emeasure_Diff:
  assumes "emeasure M B ≠ ∞"
  and "A ∈ sets M" "B ∈ sets M" and "B ⊆ A"
  shows "emeasure M (A - B) = emeasure M A - emeasure M B"
  by (smt (verit, best) add_diff_self_ennreal assms emeasure_Un emeasure_mono 
      ennreal_add_left_cancel le_iff_sup)

lemma emeasure_compl:
  "s ∈ sets M ⟹ emeasure M s ≠ ∞ ⟹ emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
  by (rule emeasure_Diff) (auto dest: sets.sets_into_space)

lemma Lim_emeasure_incseq:
  "range A ⊆ sets M ⟹ incseq A ⟹ (λi. (emeasure M (A i))) ⇢ emeasure M (⋃i. A i)"
  using emeasure_countably_additive
  by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive
    emeasure_additive)

lemma incseq_emeasure:
  assumes "range B ⊆ sets M" "incseq B"
  shows "incseq (λi. emeasure M (B i))"
  using assms by (auto simp: incseq_def intro!: emeasure_mono)

lemma SUP_emeasure_incseq:
  assumes A: "range A ⊆ sets M" "incseq A"
  shows "(SUP n. emeasure M (A n)) = emeasure M (⋃i. A i)"
  using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]
  by (simp add: LIMSEQ_unique)

lemma decseq_emeasure:
  assumes "range B ⊆ sets M" "decseq B"
  shows "decseq (λi. emeasure M (B i))"
  using assms by (auto simp: decseq_def intro!: emeasure_mono)

lemma INF_emeasure_decseq:
  assumes A: "range A ⊆ sets M" and "decseq A"
  and finite: "⋀i. emeasure M (A i) ≠ ∞"
  shows "(INF n. emeasure M (A n)) = emeasure M (⋂i. A i)"
proof -
  have le_MI: "emeasure M (⋂i. A i) ≤ emeasure M (A 0)"
    using A by (auto intro!: emeasure_mono)
  hence *: "emeasure M (⋂i. A i) ≠ ∞" using finite[of 0] by (auto simp: top_unique)

  have "emeasure M (A 0) - (INF n. emeasure M (A n)) = (SUP n. emeasure M (A 0) - emeasure M (A n))"
    by (simp add: ennreal_INF_const_minus)
  also have "… = (SUP n. emeasure M (A 0 - A n))"
    using A finite ‹decseq A›[unfolded decseq_def] by (subst emeasure_Diff) auto
  also have "… = emeasure M (⋃i. A 0 - A i)"
  proof (rule SUP_emeasure_incseq)
    show "range (λn. A 0 - A n) ⊆ sets M"
      using A by auto
    show "incseq (λn. A 0 - A n)"
      using ‹decseq A› by (auto simp add: incseq_def decseq_def)
  qed
  also have "… = emeasure M (A 0) - emeasure M (⋂i. A i)"
    using A finite * by (simp, subst emeasure_Diff) auto
  finally show ?thesis
    by (smt (verit, best) Inf_lower diff_diff_ennreal le_MI finite range_eqI)
qed

lemma INF_emeasure_decseq':
  assumes A: "⋀i. A i ∈ sets M" and "decseq A"
  and finite: "∃i. emeasure M (A i) ≠ ∞"
  shows "(INF n. emeasure M (A n)) = emeasure M (⋂i. A i)"
proof -
  from finite obtain i where i: "emeasure M (A i) < ∞"
    by (auto simp: less_top)
  have fin: "i ≤ j ⟹ emeasure M (A j) < ∞" for j
    by (rule le_less_trans[OF emeasure_mono i]) (auto intro!: decseqD[OF ‹decseq A›] A)

  have "(INF n. emeasure M (A n)) = (INF n. emeasure M (A (n + i)))"
  proof (rule INF_eq)
    show "∃j∈UNIV. emeasure M (A (j + i)) ≤ emeasure M (A i')" for i'
      by (meson A ‹decseq A› decseq_def emeasure_mono iso_tuple_UNIV_I nat_le_iff_add)
  qed auto
  also have "… = emeasure M (INF n. (A (n + i)))"
    using A ‹decseq A› fin by (intro INF_emeasure_decseq) (auto simp: decseq_def less_top)
  also have "(INF n. (A (n + i))) = (INF n. A n)"
    by (meson INF_eq UNIV_I assms(2) decseqD le_add1)
  finally show ?thesis .
qed

lemma emeasure_INT_decseq_subset:
  fixes F :: "nat ⇒ 'a set"
  assumes I: "I ≠ {}" and F: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ i ≤ j ⟹ F j ⊆ F i"
  assumes F_sets[measurable]: "⋀i. i ∈ I ⟹ F i ∈ sets M"
    and fin: "⋀i. i ∈ I ⟹ emeasure M (F i) ≠ ∞"
  shows "emeasure M (⋂i∈I. F i) = (INF i∈I. emeasure M (F i))"
proof cases
  assume "finite I"
  have "(⋂i∈I. F i) = F (Max I)"
    using I ‹finite I› by (intro antisym INF_lower INF_greatest F) auto
  moreover have "(INF i∈I. emeasure M (F i)) = emeasure M (F (Max I))"
    using I ‹finite I› by (intro antisym INF_lower INF_greatest F emeasure_mono) auto
  ultimately show ?thesis
    by simp
next
  assume "infinite I"
  define L where "L n = (LEAST i. i ∈ I ∧ i ≥ n)" for n
  have L: "L n ∈ I ∧ n ≤ L n" for n
    unfolding L_def
  proof (rule LeastI_ex)
    show "∃x. x ∈ I ∧ n ≤ x"
      using ‹infinite I› finite_subset[of I "{..< n}"]
      by (rule_tac ccontr) (auto simp: not_le)
  qed
  have L_eq[simp]: "i ∈ I ⟹ L i = i" for i
    unfolding L_def by (intro Least_equality) auto
  have L_mono: "i ≤ j ⟹ L i ≤ L j" for i j
    using L[of j] unfolding L_def by (intro Least_le) (auto simp: L_def)

  have "emeasure M (⋂i. F (L i)) = (INF i. emeasure M (F (L i)))"
  proof (intro INF_emeasure_decseq[symmetric])
    show "decseq (λi. F (L i))"
      using L by (intro antimonoI F L_mono) auto
  qed (insert L fin, auto)
  also have "… = (INF i∈I. emeasure M (F i))"
  proof (intro antisym INF_greatest)
    show "i ∈ I ⟹ (INF i. emeasure M (F (L i))) ≤ emeasure M (F i)" for i
      by (intro INF_lower2[of i]) auto
  qed (insert L, auto intro: INF_lower)
  also have "(⋂i. F (L i)) = (⋂i∈I. F i)"
  proof (intro antisym INF_greatest)
    show "i ∈ I ⟹ (⋂i. F (L i)) ⊆ F i" for i
      by (intro INF_lower2[of i]) auto
  qed (insert L, auto)
  finally show ?thesis .
qed

lemma Lim_emeasure_decseq:
  assumes A: "range A ⊆ sets M" "decseq A" and fin: "⋀i. emeasure M (A i) ≠ ∞"
  shows "(λi. emeasure M (A i)) ⇢ emeasure M (⋂i. A i)"
  using LIMSEQ_INF[OF decseq_emeasure, OF A]
  using INF_emeasure_decseq[OF A fin] by simp

lemma emeasure_lfp'[consumes 1, case_names cont measurable]:
  assumes "P M"
  assumes cont: "sup_continuous F"
  assumes *: "⋀M A. P M ⟹ (⋀N. P N ⟹ Measurable.pred N A) ⟹ Measurable.pred M (F A)"
  shows "emeasure M {x∈space M. lfp F x} = (SUP i. emeasure M {x∈space M. (F ^^ i) (λx. False) x})"
proof -
  have "emeasure M {x∈space M. lfp F x} = emeasure M (⋃i. {x∈space M. (F ^^ i) (λx. False) x})"
    using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
  moreover { fix i from ‹P M› have "{x∈space M. (F ^^ i) (λx. False) x} ∈ sets M"
    by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
  moreover have "incseq (λi. {x∈space M. (F ^^ i) (λx. False) x})"
  proof (rule incseq_SucI)
    fix i
    have "(F ^^ i) (λx. False) ≤ (F ^^ (Suc i)) (λx. False)"
    proof (induct i)
      case 0 show ?case by (simp add: le_fun_def)
    next
      case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto
    qed
    then show "{x ∈ space M. (F ^^ i) (λx. False) x} ⊆ {x ∈ space M. (F ^^ Suc i) (λx. False) x}"
      by auto
  qed
  ultimately show ?thesis
    by (subst SUP_emeasure_incseq) auto
qed

lemma emeasure_lfp:
  assumes [simp]: "⋀s. sets (M s) = sets N"
  assumes cont: "sup_continuous F" "sup_continuous f"
  assumes meas: "⋀P. Measurable.pred N P ⟹ Measurable.pred N (F P)"
  assumes iter: "⋀P s. Measurable.pred N P ⟹ P ≤ lfp F ⟹ emeasure (M s) {x∈space N. F P x} = f (λs. emeasure (M s) {x∈space N. P x}) s"
  shows "emeasure (M s) {x∈space N. lfp F x} = lfp f s"
proof (subst lfp_transfer_bounded[where α="λF s. emeasure (M s) {x∈space N. F x}" and f=F , symmetric])
  fix C assume "incseq C" "⋀i. Measurable.pred N (C i)"
  then show "(λs. emeasure (M s) {x ∈ space N. (SUP i. C i) x}) = (SUP i. (λs. emeasure (M s) {x ∈ space N. C i x}))"
    unfolding SUP_apply
    by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
qed (auto simp add: iter le_fun_def SUP_apply intro!: meas cont)

lemma emeasure_subadditive_finite:
  "finite I ⟹ A ` I ⊆ sets M ⟹ emeasure M (⋃i∈I. A i) ≤ (∑i∈I. emeasure M (A i))"
  by (rule sets.subadditive[OF emeasure_positive emeasure_additive]) auto

lemma emeasure_subadditive:
  "A ∈ sets M ⟹ B ∈ sets M ⟹ emeasure M (A ∪ B) ≤ emeasure M A + emeasure M B"
  using emeasure_subadditive_finite[of "{True, False}" "λTrue ⇒ A | False ⇒ B" M] by simp

lemma emeasure_subadditive_countably:
  assumes "range f ⊆ sets M"
  shows "emeasure M (⋃i. f i) ≤ (∑i. emeasure M (f i))"
proof -
  have "emeasure M (⋃i. f i) = emeasure M (⋃i. disjointed f i)"
    unfolding UN_disjointed_eq ..
  also have "… = (∑i. emeasure M (disjointed f i))"
    using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
    by (simp add:  disjoint_family_disjointed comp_def)
  also have "… ≤ (∑i. emeasure M (f i))"
    using sets.range_disjointed_sets[OF assms] assms
    by (auto intro!: suminf_le emeasure_mono disjointed_subset)
  finally show ?thesis .
qed

lemma emeasure_insert:
  assumes sets: "{x} ∈ sets M" "A ∈ sets M" and "x ∉ A"
  shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
proof -
  have "{x} ∩ A = {}" using ‹x ∉ A› by auto
  from plus_emeasure[OF sets this] show ?thesis by simp
qed

lemma emeasure_insert_ne:
  "A ≠ {} ⟹ {x} ∈ sets M ⟹ A ∈ sets M ⟹ x ∉ A ⟹ emeasure M (insert x A) = emeasure M {x} + emeasure M A"
  by (rule emeasure_insert)

lemma emeasure_eq_sum_singleton:
  assumes "finite S" "⋀x. x ∈ S ⟹ {x} ∈ sets M"
  shows "emeasure M S = (∑x∈S. emeasure M {x})"
  using sum_emeasure[of "λx. {x}" S M] assms
  by (auto simp: disjoint_family_on_def subset_eq)

lemma sum_emeasure_cover:
  assumes "finite S" and "A ∈ sets M" and br_in_M: "B ` S ⊆ sets M"
  assumes A: "A ⊆ (⋃i∈S. B i)"
  assumes disj: "disjoint_family_on B S"
  shows "emeasure M A = (∑i∈S. emeasure M (A ∩ (B i)))"
proof -
  have "(∑i∈S. emeasure M (A ∩ (B i))) = emeasure M (⋃i∈S. A ∩ (B i))"
  proof (rule sum_emeasure)
    show "disjoint_family_on (λi. A ∩ B i) S"
      using ‹disjoint_family_on B S›
      unfolding disjoint_family_on_def by auto
  qed (insert assms, auto)
  also have "(⋃i∈S. A ∩ (B i)) = A"
    using A by auto
  finally show ?thesis by simp
qed

lemma emeasure_eq_0:
  "N ∈ sets M ⟹ emeasure M N = 0 ⟹ K ⊆ N ⟹ emeasure M K = 0"
  by (metis emeasure_mono order_eq_iff zero_le)

lemma emeasure_UN_eq_0:
  assumes "⋀i::nat. emeasure M (N i) = 0" and "range N ⊆ sets M"
  shows "emeasure M (⋃i. N i) = 0"
proof -
  have "emeasure M (⋃i. N i) ≤ 0"
    using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
  then show ?thesis
    by (auto intro: antisym zero_le)
qed

lemma measure_eqI_finite:
  assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
  assumes eq: "⋀a. a ∈ A ⟹ emeasure M {a} = emeasure N {a}"
  shows "M = N"
proof (rule measure_eqI)
  fix X assume "X ∈ sets M"
  then have X: "X ⊆ A" by auto
  then have "emeasure M X = (∑a∈X. emeasure M {a})"
    using ‹finite A› by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
  also have "… = (∑a∈X. emeasure N {a})"
    using X eq by (auto intro!: sum.cong)
  also have "… = emeasure N X"
    using X ‹finite A› by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
  finally show "emeasure M X = emeasure N X" .
qed simp

lemma measure_eqI_generator_eq:
  fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat ⇒ 'a set"
  assumes "Int_stable E" "E ⊆ Pow Ω"
  and eq: "⋀X. X ∈ E ⟹ emeasure M X = emeasure N X"
  and M: "sets M = sigma_sets Ω E"
  and N: "sets N = sigma_sets Ω E"
  and A: "range A ⊆ E" "(⋃i. A i) = Ω" "⋀i. emeasure M (A i) ≠ ∞"
  shows "M = N"
proof -
  let ?μ  = "emeasure M" and ?ν = "emeasure N"
  interpret S: sigma_algebra Ω "sigma_sets Ω E" by (rule sigma_algebra_sigma_sets) fact
  have "space M = Ω"
    using sets.top[of M] sets.space_closed[of M] S.top S.space_closed ‹sets M = sigma_sets Ω E›
    by blast

  { fix F D assume "F ∈ E" and "?μ F ≠ ∞"
    then have [intro]: "F ∈ sigma_sets Ω E" by auto
    have "?ν F ≠ ∞" using ‹?μ F ≠ ∞› ‹F ∈ E› eq by simp
    assume "D ∈ sets M"
    with ‹Int_stable E› ‹E ⊆ Pow Ω› have "emeasure M (F ∩ D) = emeasure N (F ∩ D)"
      unfolding M
    proof (induct rule: sigma_sets_induct_disjoint)
      case (basic A)
      then have "F ∩ A ∈ E" using ‹Int_stable E› ‹F ∈ E› by (auto simp: Int_stable_def)
      then show ?case using eq by auto
    next
      case empty then show ?case by simp
    next
      case (compl A)
      then have **: "F ∩ (Ω - A) = F - (F ∩ A)"
        and [intro]: "F ∩ A ∈ sigma_sets Ω E"
        using ‹F ∈ E› S.sets_into_space by (auto simp: M)
      have "?ν (F ∩ A) ≤ ?ν F" by (auto intro!: emeasure_mono simp: M N)
      then have "?ν (F ∩ A) ≠ ∞" using ‹?ν F ≠ ∞› by (auto simp: top_unique)
      have "?μ (F ∩ A) ≤ ?μ F" by (auto intro!: emeasure_mono simp: M N)
      then have "?μ (F ∩ A) ≠ ∞" using ‹?μ F ≠ ∞› by (auto simp: top_unique)
      then have "?μ (F ∩ (Ω - A)) = ?μ F - ?μ (F ∩ A)" unfolding **
        using ‹F ∩ A ∈ sigma_sets Ω E› by (auto intro!: emeasure_Diff simp: M N)
      also have "… = ?ν F - ?ν (F ∩ A)" using eq ‹F ∈ E› compl by simp
      also have "… = ?ν (F ∩ (Ω - A))" unfolding **
        using ‹F ∩ A ∈ sigma_sets Ω E› ‹?ν (F ∩ A) ≠ ∞›
        by (auto intro!: emeasure_Diff[symmetric] simp: M N)
      finally show ?case
        using ‹space M = Ω› by auto
    next
      case (union A)
      then have "?μ (⋃x. F ∩ A x) = ?ν (⋃x. F ∩ A x)"
        by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)
      with A show ?case
        by auto
    qed }
  note * = this
  show "M = N"
  proof (rule measure_eqI)
    show "sets M = sets N"
      using M N by simp
    have [simp, intro]: "⋀i. A i ∈ sets M"
      using A(1) by (auto simp: subset_eq M)
    fix F assume "F ∈ sets M"
    let ?D = "disjointed (λi. F ∩ A i)"
    from ‹space M = Ω› have F_eq: "F = (⋃i. ?D i)"
      using ‹F ∈ sets M›[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
    have [simp, intro]: "⋀i. ?D i ∈ sets M"
      using sets.range_disjointed_sets[of "λi. F ∩ A i" M] ‹F ∈ sets M›
      by (auto simp: subset_eq)
    have "disjoint_family ?D"
      by (auto simp: disjoint_family_disjointed)
    moreover
    have "(∑i. emeasure M (?D i)) = (∑i. emeasure N (?D i))"
    proof (intro arg_cong[where f=suminf] ext)
      fix i
      have "A i ∩ ?D i = ?D i"
        by (auto simp: disjointed_def)
      then show "emeasure M (?D i) = emeasure N (?D i)"
        using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
    qed
    ultimately show "emeasure M F = emeasure N F"
      by (simp add: image_subset_iff ‹sets M = sets N›[symmetric] F_eq[symmetric] suminf_emeasure)
  qed
qed

lemma space_empty: "space M = {} ⟹ M = count_space {}"
  by (rule measure_eqI) (simp_all add: space_empty_iff)

lemma measure_eqI_generator_eq_countable:
  fixes M N :: "'a measure" and E :: "'a set set" and A :: "'a set set"
  assumes E: "Int_stable E" "E ⊆ Pow Ω" "⋀X. X ∈ E ⟹ emeasure M X = emeasure N X"
    and sets: "sets M = sigma_sets Ω E" "sets N = sigma_sets Ω E"
  and A: "A ⊆ E" "(⋃A) = Ω" "countable A" "⋀a. a ∈ A ⟹ emeasure M a ≠ ∞"
  shows "M = N"
proof cases
  assume "Ω = {}"
  have *: "sigma_sets Ω E = sets (sigma Ω E)"
    using E(2) by simp
  have "space M = Ω" "space N = Ω"
    using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of)
  then show "M = N"
    unfolding ‹Ω = {}› by (auto dest: space_empty)
next
  assume "Ω ≠ {}" with ‹⋃A = Ω› have "A ≠ {}" by auto
  from this ‹countable A› have rng: "range (from_nat_into A) = A"
    by (rule range_from_nat_into)
  show "M = N"
  proof (rule measure_eqI_generator_eq[OF E sets])
    show "range (from_nat_into A) ⊆ E"
      unfolding rng using ‹A ⊆ E› .
    show "(⋃i. from_nat_into A i) = Ω"
      unfolding rng using ‹⋃A = Ω› .
    show "emeasure M (from_nat_into A i) ≠ ∞" for i
      using rng by (intro A) auto
  qed
qed

lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
proof (intro measure_eqI emeasure_measure_of_sigma)
  show "sigma_algebra (space M) (sets M)" ..
  show "positive (sets M) (emeasure M)"
    by (simp add: positive_def)
  show "countably_additive (sets M) (emeasure M)"
    by (simp add: emeasure_countably_additive)
qed simp_all

subsection ‹‹μ›-null sets›

definition✐‹tag important› null_sets :: "'a measure ⇒ 'a set set" where
  "null_sets M = {N∈sets M. emeasure M N = 0}"

lemma null_setsD1[dest]: "A ∈ null_sets M ⟹ emeasure M A = 0"
  by (simp add: null_sets_def)

lemma null_setsD2[dest]: "A ∈ null_sets M ⟹ A ∈ sets M"
  unfolding null_sets_def by simp

lemma null_setsI[intro]: "emeasure M A = 0 ⟹ A ∈ sets M ⟹ A ∈ null_sets M"
  unfolding null_sets_def by simp

interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
proof (rule ring_of_setsI)
  show "null_sets M ⊆ Pow (space M)"
    using sets.sets_into_space by auto
  show "{} ∈ null_sets M"
    by auto
  fix A B assume null_sets: "A ∈ null_sets M" "B ∈ null_sets M"
  then have sets: "A ∈ sets M" "B ∈ sets M"
    by auto
  then have *: "emeasure M (A ∪ B) ≤ emeasure M A + emeasure M B"
    "emeasure M (A - B) ≤ emeasure M A"
    by (auto intro!: emeasure_subadditive emeasure_mono)
  then have "emeasure M B = 0" "emeasure M A = 0"
    using null_sets by auto
  with sets * show "A - B ∈ null_sets M" "A ∪ B ∈ null_sets M"
    by (auto intro!: antisym zero_le)
qed

lemma UN_from_nat_into:
  assumes I: "countable I" "I ≠ {}"
  shows "(⋃i∈I. N i) = (⋃i. N (from_nat_into I i))"
proof -
  have "(⋃i∈I. N i) = ⋃(N ` range (from_nat_into I))"
    using I by simp
  also have "… = (⋃i. (N ∘ from_nat_into I) i)"
    by simp
  finally show ?thesis by simp
qed

lemma null_sets_UN':
  assumes "countable I"
  assumes "⋀i. i ∈ I ⟹ N i ∈ null_sets M"
  shows "(⋃i∈I. N i) ∈ null_sets M"
proof cases
  assume "I = {}" then show ?thesis by simp
next
  assume "I ≠ {}"
  show ?thesis
  proof (intro conjI CollectI null_setsI)
    show "(⋃i∈I. N i) ∈ sets M"
      using assms by (intro sets.countable_UN') auto
    have "emeasure M (⋃i∈I. N i) ≤ (∑n. emeasure M (N (from_nat_into I n)))"
      unfolding UN_from_nat_into[OF ‹countable I› ‹I ≠ {}›]
      using assms ‹I ≠ {}› by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
    also have "(λn. emeasure M (N (from_nat_into I n))) = (λ_. 0)"
      using assms ‹I ≠ {}› by (auto intro: from_nat_into)
    finally show "emeasure M (⋃i∈I. N i) = 0"
      by (intro antisym zero_le) simp
  qed
qed

lemma null_sets_UN[intro]:
  "(⋀i::'i::countable. N i ∈ null_sets M) ⟹ (⋃i. N i) ∈ null_sets M"
  by (rule null_sets_UN') auto

lemma null_set_Int1:
  assumes "B ∈ null_sets M" "A ∈ sets M" shows "A ∩ B ∈ null_sets M"
proof (intro CollectI conjI null_setsI)
  show "emeasure M (A ∩ B) = 0" using assms
    by (intro emeasure_eq_0[of B _ "A ∩ B"]) auto
qed (insert assms, auto)

lemma null_set_Int2:
  assumes "B ∈ null_sets M" "A ∈ sets M" shows "B ∩ A ∈ null_sets M"
  using assms by (subst Int_commute) (rule null_set_Int1)

lemma emeasure_Diff_null_set:
  assumes "B ∈ null_sets M" "A ∈ sets M"
  shows "emeasure M (A - B) = emeasure M A"
proof -
  have *: "A - B = (A - (A ∩ B))" by auto
  have "A ∩ B ∈ null_sets M" using assms by (rule null_set_Int1)
  then show ?thesis
    unfolding * using assms
    by (subst emeasure_Diff) auto
qed

lemma null_set_Diff:
  assumes "B ∈ null_sets M" "A ∈ sets M" shows "B - A ∈ null_sets M"
proof (intro CollectI conjI null_setsI)
  show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
qed (insert assms, auto)

lemma emeasure_Un_null_set:
  assumes "A ∈ sets M" "B ∈ null_sets M"
  shows "emeasure M (A ∪ B) = emeasure M A"
proof -
  have *: "A ∪ B = A ∪ (B - A)" by auto
  have "B - A ∈ null_sets M" using assms(2,1) by (rule null_set_Diff)
  then show ?thesis
    unfolding * using assms
    by (subst plus_emeasure[symmetric]) auto
qed

lemma emeasure_Un':
  assumes "A ∈ sets M" "B ∈ sets M" "A ∩ B ∈ null_sets M"
  shows   "emeasure M (A ∪ B) = emeasure M A + emeasure M B"
proof -
  have "A ∪ B = A ∪ (B - A ∩ B)" by blast
  also have "emeasure M … = emeasure M A + emeasure M (B - A ∩ B)"
    using assms by (subst plus_emeasure) auto
  also have "emeasure M (B - A ∩ B) = emeasure M B"
    using assms by (intro emeasure_Diff_null_set) auto
  finally show ?thesis .
qed

subsection ‹The almost everywhere filter (i.e.\ quantifier)›

definition✐‹tag important› ae_filter :: "'a measure ⇒ 'a filter" where
  "ae_filter M = (INF N∈null_sets M. principal (space M - N))"

abbreviation almost_everywhere :: "'a measure ⇒ ('a ⇒ bool) ⇒ bool" where
  "almost_everywhere M P ≡ eventually P (ae_filter M)"

syntax
  "_almost_everywhere" :: "pttrn ⇒ 'a ⇒ bool ⇒ bool" ("AE _ in _. _" [0,0,10] 10)

translations
  "AE x in M. P" ⇌ "CONST almost_everywhere M (λx. P)"

abbreviation
  "set_almost_everywhere A M P ≡ AE x in M. x ∈ A ⟶ P x"

syntax
  "_set_almost_everywhere" :: "pttrn ⇒ 'a set ⇒ 'a ⇒ bool ⇒ bool"
  ("AE _∈_ in _./ _" [0,0,0,10] 10)

translations
  "AE x∈A in M. P" ⇌ "CONST set_almost_everywhere A M (λx. P)"

lemma eventually_ae_filter: "eventually P (ae_filter M) ⟷ (∃N∈null_sets M. {x ∈ space M. ¬ P x} ⊆ N)"
  unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)

lemma AE_I':
  "N ∈ null_sets M ⟹ {x∈space M. ¬ P x} ⊆ N ⟹ (AE x in M. P x)"
  unfolding eventually_ae_filter by auto

lemma AE_iff_null:
  assumes "{x∈space M. ¬ P x} ∈ sets M" (is "?P ∈ sets M")
  shows "(AE x in M. P x) ⟷ {x∈space M. ¬ P x} ∈ null_sets M"
proof
  assume "AE x in M. P x" then obtain N where N: "N ∈ sets M" "?P ⊆ N" "emeasure M N = 0"
    unfolding eventually_ae_filter by auto
  have "emeasure M ?P ≤ emeasure M N"
    using assms N(1,2) by (auto intro: emeasure_mono)
  then have "emeasure M ?P = 0"
    unfolding ‹emeasure M N = 0› by auto
  then show "?P ∈ null_sets M" using assms by auto
next
  assume "?P ∈ null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
qed

lemma AE_iff_null_sets:
  "N ∈ sets M ⟹ N ∈ null_sets M ⟷ (AE x in M. x ∉ N)"
  using Int_absorb1[OF sets.sets_into_space, of N M]
  by (subst AE_iff_null) (auto simp: Int_def[symmetric])

lemma ae_filter_eq_bot_iff: "ae_filter M = bot ⟷ emeasure M (space M) = 0"
proof -
  have "ae_filter M = bot ⟷ (AE x in M. False)"
    using trivial_limit_def by blast
  also have "… ⟷ space M ∈ null_sets M"
    by (simp add: AE_iff_null_sets eventually_ae_filter)
  also have "… ⟷ emeasure M (space M) = 0"
    by auto
  finally show ?thesis .
qed

lemma AE_not_in:
  "N ∈ null_sets M ⟹ AE x in M. x ∉ N"
  by (metis AE_iff_null_sets null_setsD2)

lemma AE_iff_measurable:
  "N ∈ sets M ⟹ {x∈space M. ¬ P x} = N ⟹ (AE x in M. P x) ⟷ emeasure M N = 0"
  using AE_iff_null[of _ P] by auto

lemma AE_E[consumes 1]:
  assumes "AE x in M. P x"
  obtains N where "{x ∈ space M. ¬ P x} ⊆ N" "emeasure M N = 0" "N ∈ sets M"
  using assms unfolding eventually_ae_filter by auto

lemma AE_E2:
  assumes "AE x in M. P x"
  shows "emeasure M {x∈space M. ¬ P x} = 0"
  by (metis (mono_tags, lifting) AE_iff_null assms emeasure_notin_sets null_setsD1)

lemma AE_E3:
  assumes "AE x in M. P x"
  obtains N where "⋀x. x ∈ space M - N ⟹ P x" "N ∈ null_sets M"
using assms unfolding eventually_ae_filter by auto

lemma AE_I:
  assumes "{x ∈ space M. ¬ P x} ⊆ N" "emeasure M N = 0" "N ∈ sets M"
  shows "AE x in M. P x"
  using assms unfolding eventually_ae_filter by auto

lemma AE_mp[elim!]:
  assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x ⟶ Q x"
  shows "AE x in M. Q x"
  using assms by (fact eventually_rev_mp)

text ‹The next lemma is convenient to combine with a lemma whose conclusion is of the
form ‹AE x in M. P x = Q x›: for such a lemma, there is no ‹[symmetric]› variant,
but using ‹AE_symmetric[OF…]› will replace it.›

(* depricated replace by laws about eventually *)
lemma
  shows AE_iffI: "AE x in M. P x ⟹ AE x in M. P x ⟷ Q x ⟹ AE x in M. Q x"
    and AE_disjI1: "AE x in M. P x ⟹ AE x in M. P x ∨ Q x"
    and AE_disjI2: "AE x in M. Q x ⟹ AE x in M. P x ∨ Q x"
    and AE_conjI: "AE x in M. P x ⟹ AE x in M. Q x ⟹ AE x in M. P x ∧ Q x"
    and AE_conj_iff[simp]: "(AE x in M. P x ∧ Q x) ⟷ (AE x in M. P x) ∧ (AE x in M. Q x)"
  by auto

lemma AE_symmetric:
  assumes "AE x in M. P x = Q x"
  shows "AE x in M. Q x = P x"
  using assms by auto

lemma AE_impI:
  "(P ⟹ AE x in M. Q x) ⟹ AE x in M. P ⟶ Q x"
  by fastforce

lemma AE_measure:
  assumes AE: "AE x in M. P x" and sets: "{x∈space M. P x} ∈ sets M" (is "?P ∈ sets M")
  shows "emeasure M {x∈space M. P x} = emeasure M (space M)"
proof -
  from AE_E[OF AE] obtain N
    where N: "{x ∈ space M. ¬ P x} ⊆ N" "emeasure M N = 0" "N ∈ sets M"
    by auto
  with sets have "emeasure M (space M) ≤ emeasure M (?P ∪ N)"
    by (intro emeasure_mono) auto
  also have "… ≤ emeasure M ?P + emeasure M N"
    using sets N by (intro emeasure_subadditive) auto
  also have "… = emeasure M ?P" using N by simp
  finally show "emeasure M ?P = emeasure M (space M)"
    using emeasure_space[of M "?P"] by auto
qed

lemma AE_space: "AE x in M. x ∈ space M"
  by (rule AE_I[where N="{}"]) auto

lemma AE_I2[simp, intro]:
  "(⋀x. x ∈ space M ⟹ P x) ⟹ AE x in M. P x"
  using AE_space by force

lemma AE_Ball_mp:
  "∀x∈space M. P x ⟹ AE x in M. P x ⟶ Q x ⟹ AE x in M. Q x"
  by auto

lemma AE_cong[cong]:
  "(⋀x. x ∈ space M ⟹ P x ⟷ Q x) ⟹ (AE x in M. P x) ⟷ (AE x in M. Q x)"
  by auto

lemma AE_cong_simp: "M = N ⟹ (⋀x. x ∈ space N =simp=> P x = Q x) ⟹ (AE x in M. P x) ⟷ (AE x in N. Q x)"
  by (auto simp: simp_implies_def)

lemma AE_all_countable:
  "(AE x in M. ∀i. P i x) ⟷ (∀i::'i::countable. AE x in M. P i x)"
proof
  assume "∀i. AE x in M. P i x"
  from this[unfolded eventually_ae_filter Bex_def, THEN choice]
  obtain N where N: "⋀i. N i ∈ null_sets M" "⋀i. {x∈space M. ¬ P i x} ⊆ N i" by auto
  have "{x∈space M. ¬ (∀i. P i x)} ⊆ (⋃i. {x∈space M. ¬ P i x})" by auto
  also have "… ⊆ (⋃i. N i)" using N by auto
  finally have "{x∈space M. ¬ (∀i. P i x)} ⊆ (⋃i. N i)" .
  moreover from N have "(⋃i. N i) ∈ null_sets M"
    by (intro null_sets_UN) auto
  ultimately show "AE x in M. ∀i. P i x"
    unfolding eventually_ae_filter by auto
qed auto

lemma AE_ball_countable:
  assumes [intro]: "countable X"
  shows "(AE x in M. ∀y∈X. P x y) ⟷ (∀y∈X. AE x in M. P x y)"
proof
  assume "∀y∈X. AE x in M. P x y"
  from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]
  obtain N where N: "⋀y. y ∈ X ⟹ N y ∈ null_sets M" "⋀y. y ∈ X ⟹ {x∈space M. ¬ P x y} ⊆ N y"
    by auto
  have "{x∈space M. ¬ (∀y∈X. P x y)} ⊆ (⋃y∈X. {x∈space M. ¬ P x y})"
    by auto
  also have "… ⊆ (⋃y∈X. N y)"
    using N by auto
  finally have "{x∈space M. ¬ (∀y∈X. P x y)} ⊆ (⋃y∈X. N y)" .
  moreover from N have "(⋃y∈X. N y) ∈ null_sets M"
    by (intro null_sets_UN') auto
  ultimately show "AE x in M. ∀y∈X. P x y"
    unfolding eventually_ae_filter by auto
qed auto

lemma AE_ball_countable':
  "(⋀N. N ∈ I ⟹ AE x in M. P N x) ⟹ countable I ⟹ AE x in M. ∀N ∈ I. P N x"
  unfolding AE_ball_countable by simp

lemma AE_pairwise: "countable F ⟹ pairwise (λA B. AE x in M. R x A B) F ⟷ (AE x in M. pairwise (R x) F)"
  unfolding pairwise_alt by (simp add: AE_ball_countable)

lemma AE_discrete_difference:
  assumes X: "countable X"
  assumes null: "⋀x. x ∈ X ⟹ emeasure M {x} = 0"
  assumes sets: "⋀x. x ∈ X ⟹ {x} ∈ sets M"
  shows "AE x in M. x ∉ X"
proof -
  have "(⋃x∈X. {x}) ∈ null_sets M"
    using assms by (intro null_sets_UN') auto
  from AE_not_in[OF this] show "AE x in M. x ∉ X"
    by auto
qed

lemma AE_finite_all:
  assumes f: "finite S" shows "(AE x in M. ∀i∈S. P i x) ⟷ (∀i∈S. AE x in M. P i x)"
  using f by induct auto

lemma AE_finite_allI:
  assumes "finite S"
  shows "(⋀s. s ∈ S ⟹ AE x in M. Q s x) ⟹ AE x in M. ∀s∈S. Q s x"
  using AE_finite_all[OF ‹finite S›] by auto

lemma emeasure_mono_AE:
  assumes imp: "AE x in M. x ∈ A ⟶ x ∈ B"
    and B: "B ∈ sets M"
  shows "emeasure M A ≤ emeasure M B"
proof cases
  assume A: "A ∈ sets M"
  from imp obtain N where N: "{x∈space M. ¬ (x ∈ A ⟶ x ∈ B)} ⊆ N" "N ∈ null_sets M"
    by (auto simp: eventually_ae_filter)
  have "emeasure M A = emeasure M (A - N)"
    using N A by (subst emeasure_Diff_null_set) auto
  also have "emeasure M (A - N) ≤ emeasure M (B - N)"
    using N A B sets.sets_into_space by (auto intro!: emeasure_mono)
  also have "emeasure M (B - N) = emeasure M B"
    using N B by (subst emeasure_Diff_null_set) auto
  finally show ?thesis .
qed (simp add: emeasure_notin_sets)

lemma emeasure_eq_AE:
  assumes iff: "AE x in M. x ∈ A ⟷ x ∈ B"
  assumes A: "A ∈ sets M" and B: "B ∈ sets M"
  shows "emeasure M A = emeasure M B"
  using assms by (safe intro!: antisym emeasure_mono_AE) auto

lemma emeasure_Collect_eq_AE:
  "AE x in M. P x ⟷ Q x ⟹ Measurable.pred M Q ⟹ Measurable.pred M P ⟹
   emeasure M {x∈space M. P x} = emeasure M {x∈space M. Q x}"
   by (intro emeasure_eq_AE) auto

lemma emeasure_eq_0_AE: "AE x in M. ¬ P x ⟹ emeasure M {x∈space M. P x} = 0"
  using AE_iff_measurable[OF _ refl, of M "λx. ¬ P x"]
  by (cases "{x∈space M. P x} ∈ sets M") (simp_all add: emeasure_notin_sets)

lemma emeasure_0_AE:
  assumes "emeasure M (space M) = 0"
  shows "AE x in M. P x"
using eventually_ae_filter assms by blast

lemma emeasure_add_AE:
  assumes [measurable]: "A ∈ sets M" "B ∈ sets M" "C ∈ sets M"
  assumes 1: "AE x in M. x ∈ C ⟷ x ∈ A ∨ x ∈ B"
  assumes 2: "AE x in M. ¬ (x ∈ A ∧ x ∈ B)"
  shows "emeasure M C = emeasure M A + emeasure M B"
proof -
  have "emeasure M C = emeasure M (A ∪ B)"
    by (rule emeasure_eq_AE) (insert 1, auto)
  also have "… = emeasure M A + emeasure M (B - A)"
    by (subst plus_emeasure) auto
  also have "emeasure M (B - A) = emeasure M B"
    by (rule emeasure_eq_AE) (insert 2, auto)
  finally show ?thesis .
qed