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

subsectiontag 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 (insert assms, simp)
  ultimately show ?thesis using assms
    by (subst suminf_eq_SUP) (auto simp: indicator_def)
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 "(iP. 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 theoryHOL-Analysis.Sigma_Algebra, as it
  is also used to represent sigma algebras (with an arbitrary emeasure).


subsectiontag 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< Suc (Suc n). f (binaryset A B i))  f A + f B"
    by metis
  hence "(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)"
    by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)

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)

subsectiontag unimportant Properties of a premeasure termμ

text 
  The definitions for constpositive and constcountably_additive should be here, by they are
  necessary to define typ'a measure in theoryHOL-Analysis.Sigma_Algebra.


definition subadditive where
  "subadditive M f  (xM. yM. 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  (xM. yM. x  y = {}  μ (x  y) = μ x + μ y)"

definition increasing where
  "increasing M μ  (xM. yM. 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  xM  yM  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 "(in. f (disjointed A i)) = f (A n)"
proof (induct n)
  case (Suc n)
  then have "(iSuc 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  "(iS. f (A i)) = f (iS. 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  (iS. A i) = {}"
    by (auto simp add: disjoint_family_on_def neq_iff)
  moreover
  have "A s  M" using insert by blast
  moreover have "(iS. A i)  M"
    using insert finite S by auto
  ultimately have "f (A s  (iS. A i)) = f (A s) + f(iS. 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))" using addf
    by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
  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 (iS. A i)  (iS. 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" "(iF. A i)  M" "(iF. A i) - A x  M" using A by force+
  have subs: "(iF. A i) - A x  (iF. A i)" by auto
  have "(i(insert x F). A i) = A x  ((iF. A i) - A x)" by auto
  hence "f (i(insert x F). A i) = f (A x  ((iF. A i) - A x))"
    by simp
  also have " = f (A x) + f ((iF. A i) - A x)"
    using f(2) by (rule additiveD) (insert in_M, auto)
  also have "  f (A x) + f (iF. A i)"
    using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)
  also have "  f (A x) + (iF. 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))" using insert by simp
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{i. F i  {}}. x. x  F i" by auto
  from bchoice[OF this] obtain f where f: "i. F i  {}  f i  F i" by auto

  have inj_f: "inj_on f {i. F i  {}}"
  proof (rule inj_onI, simp)
    fix i j a b assume *: "f i = f j" "F i  {}" "F j  {}"
    then have "f i  F i" "f j  F j" using f by force+
    with disj * show "i = j" by (auto simp: disjoint_family_on_def)
  qed
  have "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 (rule finite_subset) fact
  qed fact
  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))"
    using f(1)[unfolded positive_def] dA
    by (auto intro!: summable_LIMSEQ)
  from LIMSEQ_Suc[OF this]
  have "(λn. (in. f (disjointed A i)))  (i. f (disjointed A i))"
    unfolding lessThan_Suc_atMost .
  moreover have "n. (in. 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: "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 "(λn. f (i<n. A i))  f (i. A i)"
  proof (unfold *[symmetric], intro cont[rule_format])
    show "range (λi. i<i. A i)  M" "(i. i<i. A i)  M"
      using A * by auto
  qed (force intro!: incseq_SucI)
  moreover have "n. f (i<n. A i) = (i<n. f (A i))"
    using A
    by (intro additive_sum[OF f, of _ A, symmetric])
       (auto intro: disjoint_family_on_mono[where B=UNIV])
  ultimately
  have "(λi. f (A i)) sums f (i. A i)"
    unfolding sums_def by simp
  from sums_unique[OF this]
  show "(i. f (A i)) = f (i. A i)" by simp
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: "(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[THEN spec, of A] show "(λi. f (A i))  0"
    using positive M f[unfolded positive_def] by auto
next
  assume cont: "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)
  { fix i
    have "f (A i - (i. A i))  f (A i)" using A by (auto intro!: f_mono)
    then have "f (A i - (i. A i))  "
      using A by (auto simp: top_unique) }
  note f_fin = this
  have "(λi. f (A i - (i. A i)))  0"
  proof (intro cont[rule_format, OF _ decseq _ f_fin])
    show "range (λi. A i - (i. A i))  M" "(i. A i - (i. A i)) = {}"
      using A by auto
  qed
  from INF_Lim[OF decseq_f this]
  have "(INF n. f (A n - (i. A i))) = 0" .
  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
    by (subst INF_ennreal_add_const) (auto simp: decseq_f)
  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" "AM. 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 f(3)[rule_format, of "A i"] A by (auto simp: ennreal_add_diff_cancel subset_eq) }
  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: "AM. 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

subsectiontag unimportant Properties of constemeasure

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 
    (iI. emeasure M (F i)) = emeasure M (iI. 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 finite: "emeasure M B  "
  and [measurable]: "A  sets M" "B  sets M" and "B  A"
  shows "emeasure M (A - B) = emeasure M A - emeasure M B"
proof -
  have "(A - B)  B = A" using B  A by auto
  then have "emeasure M A = emeasure M ((A - B)  B)" by simp
  also have " = emeasure M (A - B) + emeasure M B"
    by (subst plus_emeasure[symmetric]) auto
  finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
    using finite by simp
qed

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 (rule ennreal_minus_cancel[rotated 3])
       (insert finite A, auto intro: INF_lower emeasure_mono)
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 "jUNIV. emeasure M (A (j + i))  emeasure M (A i')" for i'
      by (intro bexI[of _ i'] emeasure_mono decseqD[OF decseq A] A) auto
  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 (iI. F i) = (INF iI. emeasure M (F i))"
proof cases
  assume "finite I"
  have "(iI. F i) = F (Max I)"
    using I finite I by (intro antisym INF_lower INF_greatest F) auto
  moreover have "(INF iI. 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 iI. 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)) = (iI. 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 {xspace M. lfp F x} = (SUP i. emeasure M {xspace M. (F ^^ i) (λx. False) x})"
proof -
  have "emeasure M {xspace M. lfp F x} = emeasure M (i. {xspace 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 "{xspace M. (F ^^ i) (λx. False) x}  sets M"
    by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
  moreover have "incseq (λi. {xspace 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) {xspace N. F P x} = f (λs. emeasure (M s) {xspace N. P x}) s"
  shows "emeasure (M s) {xspace N. lfp F x} = lfp f s"
proof (subst lfp_transfer_bounded[where α="λF s. emeasure (M s) {xspace N. F x}" and g=f and f=F and P="Measurable.pred N", 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[abs_def]
    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[abs_def] intro!: meas cont)

lemma emeasure_subadditive_finite:
  "finite I  A ` I  sets M  emeasure M (iI. A i)  (iI. 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 = (xS. 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  (iS. B i)"
  assumes disj: "disjoint_family_on B S"
  shows "emeasure M A = (iS. emeasure M (A  (B i)))"
proof -
  have "(iS. emeasure M (A  (B i))) = emeasure M (iS. 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 "(iS. 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 = (aX. emeasure M {a})"
    using finite A by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
  also have " = (aX. 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   = "emeasure M" and  = "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 " F  "
    then have [intro]: "F  sigma_sets Ω E" by auto
    have " F  " using  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 " (F  A)   F" by (auto intro!: emeasure_mono simp: M N)
      then have " (F  A)  " using  F   by (auto simp: top_unique)
      have " (F  A)   F" by (auto intro!: emeasure_mono simp: M N)
      then have " (F  A)  " using  F   by (auto simp: top_unique)
      then have " (F  (Ω - A)) =  F -  (F  A)" unfolding **
        using F  A  sigma_sets Ω E by (auto intro!: emeasure_Diff simp: M N)
      also have " =  F -  (F  A)" using eq F  E compl by simp
      also have " =  (F  (Ω - A))" unfolding **
        using F  A  sigma_sets Ω E  (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 " (x. F  A x) =  (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

definitiontag important null_sets :: "'a measure  'a set set" where
  "null_sets M = {Nsets 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 "(iI. N i) = (i. N (from_nat_into I i))"
proof -
  have "(iI. 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 "(iI. 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 "(iI. N i)  sets M"
      using assms by (intro sets.countable_UN') auto
    have "emeasure M (iI. 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 (iI. 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)

definitiontag important ae_filter :: "'a measure  'a filter" where
  "ae_filter M = (INF Nnull_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 xA in M. P"  "CONST set_almost_everywhere A M (λx. P)"

lemma eventually_ae_filter: "eventually P (ae_filter M)  (Nnull_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  {xspace M. ¬ P x}  N  (AE x in M. P x)"
  unfolding eventually_ae_filter by auto

lemma AE_iff_null:
  assumes "{xspace M. ¬ P x}  sets M" (is "?P  sets M")
  shows "(AE x in M. P x)  {xspace 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_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  {xspace 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" "{xspace M. P x}  sets M"
  shows "emeasure M {xspace M. ¬ P x} = 0" (is "emeasure M ?P = 0")
proof -
  have "{xspace M. ¬ P x} = space M - {xspace M. P x}" by auto
  with AE_iff_null[of M P] assms show ?thesis by auto
qed

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"
proof -
  from AE_P obtain A where P: "{xspace M. ¬ P x}  A"
    and A: "A  sets M" "emeasure M A = 0"
    by (auto elim!: AE_E)

  from AE_imp obtain B where imp: "{xspace M. P x  ¬ Q x}  B"
    and B: "B  sets M" "emeasure M B = 0"
    by (auto elim!: AE_E)

  show ?thesis
  proof (intro AE_I)
    have "emeasure M (A  B)  0"
      using emeasure_subadditive[of A M B] A B by auto
    then show "A  B  sets M" "emeasure M (A  B) = 0"
      using A B by auto
    show "{xspace M. ¬ Q x}  A  B"
      using P imp by auto
  qed
qed

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: "{xspace M. P x}  sets M" (is "?P  sets M")
  shows "emeasure M {xspace 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:
  "xspace 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. {xspace M. ¬ P i x}  N i" by auto
  have "{xspace M. ¬ (i. P i x)}  (i. {xspace M. ¬ P i x})" by auto
  also have "  (i. N i)" using N by auto
  finally have "{xspace 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. yX. P x y)  (yX. AE x in M. P x y)"
proof
  assume "yX. 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  {xspace M. ¬ P x y}  N y"
    by auto
  have "{xspace M. ¬ (yX. P x y)}  (yX. {xspace M. ¬ P x y})"
    by auto
  also have "  (yX. N y)"
    using N by auto
  finally have "{xspace M. ¬ (yX. P x y)}  (yX. N y)" .
  moreover from N have "(yX. N y)  null_sets M"
    by (intro null_sets_UN') auto
  ultimately show "AE x in M. yX. 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 "(xX. {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. iS. P i x)  (iS. 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. sS. 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: "{xspace 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 {xspace M. P x} = emeasure M {xspace M. Q x}"
   by (intro emeasure_eq_AE) auto

lemma emeasure_eq_0_AE: "AE x in M. ¬ P x  emeasure M {xspace M. P x} = 0"
  using AE_iff_measurable[OF _ refl, of M "λx. ¬ P x"]
  by (cases "{xspace 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

subsection