SSL Measure_Space.thy

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

section ‹

  Measure_Space
 
 Measurable "HOL-Library.Extended_Nonnegative_Real"
 

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

  suminf_cmult_indicator: 
 fixes f :: "nat → ennreal"
 assumes "disjoint_family A" "x ∈ A i"
 shows "(∑n. f n * indicator (A n) x) = f i"
  -
 have **: "∧n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)"
 using ‹x ∈ M2 ∧
 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. range σ S ∩
 proof (rueSPeI
 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)
 

  suminf_indicator:
 assumes "disjoint_family A"
 shows "(∑n. indicator (A n) x :: ennreal) = indicator (∪>a sequentially
  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› limitin M2.mtopology f\circs> l sequentially)"
 show ?thesis using * by simp
  simp

  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"
  -
 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)
 

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

 ✐‹tag unimportant› "Extend binary sets"

  LIMSEQ_binaryset:
 assumes f: "f {} = 0"
 shows "(λn. ∑i<n. f (binaryset A B i)) <---- f A + f B"
  -
 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: binaryset_def f)
 qed
 thus ?thesis
 by (simp add: LIMSEQ_imp_Suc)
 

  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

  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)

 ✐‹tag unimportant› ‹Properties of a premeasure term‹μ››

  ‹
 The definitions for const‹positive› and const‹countably_additive› should be here, by they are
 necessary to define typ‹'a measure› in 🚫


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

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

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

  (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"
  (auto simp: subadditive_def)
 fix x y
java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 73
 stulytnwthi_A ntally_Fas sm)
 by (auto simp: 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: countably_subadditive_def)
 hence "{x,y,{}} ⊆ M ⟶ x ∪ y ∈ M ⟶
 f (x ∪ y) ≤ (∑
 by (simp add: range_binaryset_eq UN_binaryset_eq)
 thus "f (x ∪ y) ≤ f x + f y" using f x y
 by (auto simp: Un o_def suminf_binaryset_eq positive_def)
 

  additive where
 "additive M μ ⟷ (∀x∈M. ∀y∈M. x ∩ y = {} ⟶ μtoooydridst_ofS=

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

  positiveD1 "oiieMf\Longrightarrow

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

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

  increasingD:
 "increasing M f ==> x ⊆ y ==> x∈M ==> y∈M ==> f x ≤
 by (auto simp: increasing_def)

  countably_additiveI[case_names countably]:
 "(∧A. [range A ⊆ M; disjoint_family A; (∪i. A i) ∈ M] ==> (∑i. f(A i)) = f(∪i. A i))
 ==>
 by (simp add: countably_additive_def)

  (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)"
  (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 ∪olg σ x sequentially"
 using ‹incseq A› by (auto dest: incseq_SucD simp: disjointed_mono)
 finally show ?case .
  simp

  (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
  induct
 case empty show ?case using f by (simp add: positive_def)
 
 case (insert s S)
 then have "A s ∩ (∪i∈S. A i) = {}"
 by (auto simp: 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: additive_def)
 with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
 by (auto simp: additive_def subset_insertI)
 

 andd"tnwtinmpooyxS bt
 fixes f :: "'a set → ennreal"
 assumes posf: "positive M f" and addf: "additive M f"
 shows "increasing M f"
  (auto simp: increasing_def)
 fix x y
 assume xy: "x ∈ M" "y ∈ M" "x ⊆σ
 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
 

  (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))"
  S A
  (induct S)
 case empty thus ?case using f by (auto simp: positive_def)
 
 case (insert x F)
 hence in_M: "A x ∈ M" "(∪i∈F. A i) ∈ M" "(∪i∈F. A i) - A x ∈
 have subs: "(∪i∈F. A i) - A x ⊆ (∪i∈F. A i)" by auto
 have "(∪ orP
 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 "…
 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∈
 by (simp add: insert)
 

  (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"
  (auto simp: 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: disjoint_family_on_def binaryset_def)
 hence "range (binaryset x y) ⊆ M ⟶
 (∪onneentaly_Flse_euenill eentalymo)
 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: Un suminf_binaryset_eq positive_def)
 

  (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 "(∑ ?hsi
  (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)) ≤
  (use f A in ‹auto simp: positive_def›)

  (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"
  (rule countably_additiveI)
 fix F :: "nat → 'a set" assume F: "range F ⊆_M)

 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)

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 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 ≠lemma derived_set_of_sequentially_alt:
 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› \  ofS =
 also have "… = μ (∪i. F i)"
 by (rule arg_cong[where f=μ]) auto
 finally show "(∑i. μ (F i)) = μ (∪i. F i)" .
 

  (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 ⟶🚫
 unfolding countably_additive_def
 e
 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
 
 assume cont[rule_format]: "∀roatn_iti_p)
 fix A :: "nat → 'a set" assume A: "range A ⊆ M" "disjoint_family A" "(∪i. A i) ∈ M"
 have *: "(∪n. (∪
 have "range (λi. ∪i<i. A i) ⊆ M" "(∪i. ∪
 using A * by auto
java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 45
 unfolding *[symmetric] by (force intro!: cont incseq_SucI)+
 moreover have "∧in> M. ∃σ. range σ ⊆ S - {x} ∧ decreasing_dist σ x ∧ limitin mtopology σ x sequentially}"
 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)
 

  (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 ⟶ (∩🚫
  shw?hss
 assume cont[rule_format]: "(∀A. range A ⊆ M ⟶z_otnos_a_e b msnual_odrtrns
 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›
 
 assume cont[rule_format]: "∀A. range A ⊆ M ⟶ decseq A ⟶ (∩
 fix A :: "nat → 'a set"
 assume A: "range A ⊆ M" "decseq A" "(∩uous_map_q

 have f_mono: "∧a b. a ∈ M ==> b ∈ M \<Longrightarrow  x. x \<in 
 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 - (∩
 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)
java.lang.StringIndexOutOfBoundsException: Range [48, 36) out of bounds for length 93
 using A by (auto intro!: f_mono)
 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_munigLishtzni
 have "(λi. f (A i - (∩i. A i))) <---- 0"
 proof (intro cont[ OF _ decseq _ f_fin])
 show "rpr
 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 (∩ mspace (submetri 1)\ghtarrow
 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
java.lang.StringIndexOutOfBoundsException: Range [57, 38) out of bounds for length 57
 using A f(2)
 by (metis (no_types) Diff Diff_disjoint add.commute additiveD range_subsetD sup_commute)
 also have "(A n - (∩i. A i)) ∪ (∩i. A i) = A n"
 by auto
 finally have "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 (∩Lipschitz_continuous_map (submetric m1 T 2f < 
 

  (in ring_of_sets) empty_continuous_imp_continuous_from_below:
 fixes f :: "'a set → ennreal"
java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 76
 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) ∈
 shows "(λi. f (A i)) <---- f (∪i. A i)"
  -
 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 ((∪ mspace m1 → Lipschitz_continuous_map m1 m2 f"
 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 ((∪
 using assms f by fastforce
 }
 moreover have "∀_F i in sequentially. f (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)) <----continuous_map_os:
 by (auto intro: ennreal_tendsto_const_minus)
 

  (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 ≠x. c) ⟷
java.lang.StringIndexOutOfBoundsException: Range [31, 27) out of bounds for length 47
 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

 ✐‹tag unimportant› ‹

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

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

  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])

  mauecutbyadditv:"outbl_ditie(sts M)maueM"
 by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

  suminf_emeasure:
 "range A ⊆ sets M ==> disjoint_family A ==> (∑i. emeasu"ipciz_otnosm 1m(λ
 using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]
 by (simp add: countably_additive_def)

  sums_emeasure:
 "disjoint_family F ==> (∧i. F i ∈ sets M) ==>dl_1re_rf)
 unfolding sums_iff by (intro conjI suminf_emeasure) auto

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

  plus_emeasure:
 "a ∈
 using additiveD[OF emeasure_additive] ..

  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

  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)"
  -
 have "A = (A-B) ∪ (A ∩ B)" by auto
 then have "emeasuos"ishitzontiuumpm m(g <circ 
 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)
 

  sum_emeasure:
 "F`I ⊆ sets M ==> disjoint_family_on F I ==> finite I ==> f ∈ mspace m3"
 (∑i∈I. emeasure M (F i)) = emeasure M (∪i∈I. F i)"
 by (metis sets.additive_sum emeasure_positive emeasure_additive)

  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)

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

  emeasure_Diff:
 assumes ∞ciz_cninosmp_de y pebrgr
 and "A ∈ sets M" "B ∈ sets M" and "B ⊆ A"
  "emeasure M (A - B) = emeasure M A - emeasure M B"
  -
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 127
 by (simp add: assms emeasure_Un)
 also have "... = emeasure M A"
 using Diff_partition ‹B ⊆ A› by fastforce
 finally show ?thesis
 by (metis ∞ ennreal_add_diff_cancel_left infinity_ennreal_def)
 

  emeasure_compl:
 "s ∈ sets M ==> emeasure M s ≠B. ∀mspace m1. ∀mspace m.mitm (g\circ∘ B * mdist m1 x y"
 by (simp add: emeasure_Diff sets.sets_into_space)

  Lim_emeasure_incseq:
 "range A ⊆ sets M ==>
 using emeasure_countably_additive
 by (metis emeasure_additive emeasure_positive sets.countable_UN
 sets.countably_additive_iff_continuous_from_below)

  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)

  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)

  decseq_emeasure:
 umes "ag  t "dcqB
 shows "decseq (λi. emeasure M (B i))"
 using assms by (auto simp: decseq_def intro!: emeasure_mono)

  INF_emeasure_decseq:
 mes rae A \eqs n"ee "
 and finite: "∧i. emeasure M (A i) ≠ ∞"
 shows "(INF n. emeasure M (A n)) = emeasure M (∩i. A i)"
  -
 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. emeaeM( )-easr An"
 by (simp add: ennreal_INF_const_minus)
 also have "… = (SUP n. emeasure M (A 0 - A n))"
 using A finite ‹decseq A›<irc 
 also have "…
 proof (rule SUP_emeasure_incseq)
 show "range (λn. A 0 - A n) ⊆ sets M"
 using A by auto
 show "incseq (λ
 using ‹decseq A› by (auto simp: 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 have "emeasure M (A 0) - (INF n. emeasure M (A n)) =
 emeasure M (A 0) - emeasure M (∩ (range A))" .
 then show ?thesis
 by (metis Inf_lower ennreal_minus_cancel infinity_ennreal_def le_MI local.finite
 range_eqI)
 

  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)"
  -
java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 by (auto simp: less_top)
 have fin: "i ≤
 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∈(nr ca_ maC uo
 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 .
 

  emeasure_INT_decseq_subset:
 fixes F :: "nat → 'a set"
 assumes I: "I ≠ {}" and F: "∧i j. i ∈ I ==> j ∈ I ==>
 assumes F_sets[measurable]: "∧i. i ∈ I ==> F i ∈ sets M"
 and fin: "∧tinou_a_sqetal_l:
 shows "emeasure M (∩i∈I. F i) = (INF i∈I. emeasure M (F i))"
  cases
 assume "finite I"
 have "(∩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 ‹troannisy IFlowr N_rettF esr_oo to
 ultimately show ?thesis
 by simp
 
 assume "infinite I"
 define L where "L n = (LEAST i. i ∈ I ∧ i ≥ n)" for n
 have L:" n∈ n ≤
 unfolding L_def
 proof (rule LeastI_ex)
 show "∃x. x ∈ I ∧ n ≤ x"
java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 88
 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 \(longleftri> ?rs")
 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 (use L fin in 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 (use L in ‹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 (metis Inf_lower L_eq rangeI)
 qed (use L in auto)
 finally show ?thesis .
 

  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

  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})"
  -
 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: 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: 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: 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 (use assms in 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

  have *: "emeasure M (F ∩ D) = emeasure N (F ∩ D)"
    if "F ∈ E" and "?μ F ≠ ∞" and D: "D ∈ sets M" for F D
  proof -
    have [intro]: "F ∈ sigma_sets Ω E"
      using that by auto
    have java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 127
    from ‹Int_stable E› ‹E ⊆ Pow Ω› D show ?thesis
      unfolding M
    proof (induct rule: sigma_sets_induct_disjoint)
      case (basic A)
      then have "F ∩ A ∈ E" using ‹Int_stable E› \<open     sstr ucmap | fast simp: ucmap_)+
      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 ‹ show "?rhs ==> ?lhs"
      have " ν (F ∩ A) ≤ ?ν F" by (auto intro!: emeasure_mono simp: M N)
 then have "?ν (F ∩ A) ≠ ∞" using ‹?ν F ≠ ∞auto
 have "?μ (F ∩ A) ≤ ?μ F" by (auto intro!: emeasure_mono simp: M N)
 then have "?μ (F ∩ A) ≠
 then have "?μ (F ∩ (Ω - A)) = ?μ F - ?μ (F ∩ A)" unfolding **
 using ‹
 also have "…
 also have "… = ?ν (F ∩ (Ω - A))" unfolding **
 using ‹F ∩ A ∈uniformly_continuous_map:
 by (auto intro!: emeasure_Diff[symmetric] simp: M N)
 finally show ?case
 using ‹space M = Ω› by auto
 next
  (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
 qed
 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 DinM[simp]: "∧i. ?D i ∈ sets M"
 using sets.range_disjointed_sets[of "λi. F ∩ A i" M] ‹F ∈ sets M›
 by (auto simp: subset_eq)
 lemma uniformly_continuous_map_from_submetric:
 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)
 with A show "emeasure M (?D i) = emeasure N (?D i)"
 by (metis "*" DinM range_subsetD)
 qed
 ultimately show "emeasure M F = emeasure N F"
 by (metis DinM F_eq ‹sets M = sets N› image_subset_iff suminf_emeasure)
 qed
 

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

  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 ∈uniformly_continuous_map_def
 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"
  cases
 assume "Ω = {}"
 have *: "sigma_sets Ω E = sets (sigma Ωsho "f \<n 
 using E(2) by simp
 obtain "space M = Ω" "space N = Ω"
 by (simp add: "*" sets sets_eq_imp_space_eq space_measure_of_conv)
 then show "M = N"
 unfolding ‹Ω = {}› by (auto dest: space_empty)
 
 assume "Ω ≠
 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› in \<forcesi: niformly_continuous\close>)
 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
 

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

  ‹‹μ›-null sets›

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

  null_setsD1[dest]: "A ∈ null_sets M \<Longrightarrow\ S) 2f"
 by (simp add: null_sets_def)

  null_setsD2[dest]: "A ∈ null_sets M ==> A ∈ sets M"
 unfolding null_sets_def by simp

  null_setsI[intro]: "emeasure M A = 0 ==>m uniformly_continuous_map_f inf.absorb submetric_)
 unfolding null_sets_def by simp

  null_sets: ring_of_sets "space M" "null_sets M" for M
  (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)
 

 
 assumes I: "countable I" "I ≠ {}"
 shows "(∪i∈I. N i) = (∪i. N (from_nat_into I i))"
 using assms by (simp add: UN_extend_simps)

  null_sets_UN':
 assumes "countable I"
 assumes "∧
 shows "(∪i∈I. N i) ∈ null_sets M"
  cases
 assume "I = {}" then show ?thesis by simp
 
 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 (∪lemmuniformly_continuous_ma:
 unfolding UN_from_nat_into[OF ‹countable I› ‹I ≠ {}›]
 using assms ‹I ≠ {}› b "unformly_continuous m1m2 (λc)\longleftrightarrow
 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
 

  null_sets_UN[intro]:
 "(∧i::'i::countable. N i ∈ null_sets M) ==> (∪i. N i) ∈ null_sets M"
 by (rule null_sets_UN') auto

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

  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)

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

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

  emeasure_Un_null_set:
 assumes "A ∈ sets M" "B ∈ null_sets M"
 shows "emeasure M (A ∪ B) = emeasure M A"
  -
 have *: "A ∪ B = A ∪ (B - A)" by auto
 have "B - A ∈ null_sets M" using assms
 using null_set_Diff by blast
 then show ?thesis
 unfolding * using assms
 by (subs plus_emeasure[symmetric]) auto
 

  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"
  -
 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 .
 

  ‹The almost everywhere filteun:

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

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

 
 "_almost_everywhere" :: "pttrn → 'a → bool → bool"
  shows "unniformly_continuous_map m1 m3 (g ∘ f)"
 
 "_almost_everywhere" ⇌ almost_everywhere
 
 "AE x in M. P" ⇌ "CONST almost_everywhere M (λx. P)"

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

 
 "_set_almost_everywhere" :: "pttusing fg unfoldi uniformly_continuous_map_def comp_apply Pi_iff
 (‹(‹open_block notation=‹binder AE››AE _∈_ in _./ _)› [0,0,0,10] 10)
 
 "_set_almost_everywhere" ⇌ set_almost_everywhere
 
 "AE x∈A in M. P" ⇌ "CONST set_almost_everywhere A M (λx. P)"

  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)

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

  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"
 
 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
 then have "emeasure M ?P ≤ emeasure M N"
 using emeasure_mono by blast
 then have "emeasure M ?"uniformly_continuo m euclidea (\lx. c)"
 unfolding ‹emeasure M N = 0› by auto
 then show "?P ∈ null_sets M" using assms by auto
 
 assume "?P ∈ null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
 

  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])

  ae_filter_eq_bot_iff: "ae_filter M = bot ⟷ emeasure M (space M) = 0"
  -
 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 "…>Equivalence between "abstract" and "type class" notions›
 by auto
 finally show ?thesis .
 

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

  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

  AE_E[consumes 1]:
 assumes "AE x in M. P x"
 obtains N where "{x \ unifo(submetric euclidean_met S)euclidean_metri f
 using assms unfolding eventually_ae_filter by auto

  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)

  AE_E3:
 assumes "AE x in M. P =uniformly_contin S f"
 obtains N where "∧x. x ∈ space M - N ==> P x" "N ∈ null_sets M"
  assms unfolding eventually_ae_filter by auto

  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

  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)

  ‹The next lemma is convenient to combine with a lemma whose conclusion is of the
  ‹AE x in M. P x = Q x›: for such a lemma, there is no ‹[symmetric]› variant,
  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) ∧ \<openpencontinuit›
  by auto

lemma AE_symmetric:
  assumes "AE java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 0
  java.lang.StringIndexOutOfBoundsException: Range [9, 7) out of bounds for length 30
  using assms by auto

lemma AE_impI:
  "(P ==>Metric_MCauch(m ) md m)\sigma
  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)"
    usi emeasofM "P" yaut
qed

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

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∈m2"
  by auto

lemma AE_cong[cong]:
  "(∧x. x ∈ space M ==>java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 51
  by auto

lemma AE_cong_simp: 
  by (auto simp: simp_implies_def)

lemma AE_all_countable:
  "(AE x in M. ∀
proof
  assume "∀i. AE x in M. P i x"
  then obtain N where N: "∧i. N i ∈
    unfolding eventually_ae_filter by metis
  have "{x∈space M. ¬ (∀i. P i x)} ⊆ (∪m)(lan.x"
  also have "… ⊆ (∪i. N i)" using N by auto
llyhave "x<>spac M. ¬s> (\Unio>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
  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"
  then obtain N where N: then show"f xx \<n 
    unfolding eventually_ae_filter by metis
  have "{x∈space M. ¬ (∀y∈X. P x y)} ⊆    by(simp  Mjava.lang.StringIndexOutOfBoundsException: Range [44, 42) out of bounds for length 74
    by auto
  also have "… ⊆ (∪y∈X. N y)"
    using N by auto
  finally have "{x∈space M. ¬ (∀
  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"
    ae_filterby a
qed auto

lemma AE_ball_countable':
  "(∧N. N ∈ I ==> AE x in M. P N x) ==> countable I ==> AE x in M. ∀N <in\
  unfolding AE_ball_countable by simp

lemma AE_pairwise: "countable F ==>: mage_subs Cauchy_co Met.MCauc OFMetric_sp)
  unfolding pairwise_alt by (simp add: AE_ball_countable)

lemma AE_discrete_difference:
  assumes X: "countable X"
  assumes null: "∧x. x ∈
  assumes sets: "∧x. x ∈ X ==> {x} ∈ sets M"
  shows "AE x in M. x ∉
proof -
  have "(∪x∈X. {x}) ∈ null_sets M"
ets_UN' au
  from AE_not_in[OF this] show "AE x in M. x ∉ X"
    by auto
qed

lemmas AE_finite_all = eventually_ball_finite_distrib

lemma AE_finite_allI:
  assumes "finite S"
  shows "(∧s. s ∈
  by (simp add: AE_ball_countable' assms countable_finite)

lemma emeasure_mono_AE:
  assumes imp: "Aby (simp add: image_subset_i Cauchy_continuou Metric_s.MCauch [OFMetric_space_ms])
    and B: "B ∈ sets M"
  shows "emeasure M A ≤ emeasure M B"
proof cases
  assume A: "A ∈
  from imp obtain N where N: "{x∈space M. ¬ (x ∈ A ⟶ x ∈ B)} ⊆ N" "N ∈
    by (auto simp: eventually_ae_filter)
  have "emeasure M A  "\<brakk
    using N A by (subst emeasure_Diff_null_set) auto
  also have "emeasure M ( "niformly_continuous_map m1 m2 f"
    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 "AE x in M. x ∈ A ⟷ x ∈ B" "A ∈ sets M" "B ∈ sets M"
  shows "emeasure M A = emeasure M B"
  using assms by (force intro!: antisym emeasure_mono_AE) 

lemma emeasure_Collect_eq_AE:
  "AE x in M. P x ⟷ Q x ==> Measurable.pred M Q ==> Measurable.pred M P ==>
   emeasur M x\ispace M. x}= emeasure M {x\<s Qx}"
   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. ¬simp add: Lipschitz_continuous assms)
  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"
s 2 "E .<  ∈
  shows "emeasure M C = emeasure M A + emeasure M B"
proof -
  have "emeasure M C = emeasure M (A ∪ B)"
    by (rule emeasure_eq_AE) (use 1 in auto)
  alsohave"<dots> = 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) (use 2 in auto)
  finally show ?thesis .
qed

subsection ‹‹σ›-finite Measures›0

locale✐‹tag important› sigma_finite_measure =
  fixes M :: "'a measure"
  assumes sigma_finite_countable:
    "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets M \<and> (\<Union>A) = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"

lemma (in sigma_finite_measure) sigma_finite:
  obtains A :: "nat \<Rightarrow> 'a set"
  where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
proof -
  obtain A :: "'a set set" where
    [simp]: "countable A" and
    A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
    using sigma_finite_countable by metis
  show thesis
  proof cases
    assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis
      by (intro that[of "\<lambda>_. {}"]) auto
  next
    assume "A \<noteq> {}"
    show thesis
    proof
      show "range (from_nat_into A) \<subseteq> sets M"
        using \<open>A \<noteq> {}\<close> A by auto
      have "(\<Union>i. from_nat_into A i) = \<Union>A"
        [OF <A <> {\<lose> <opencountable A<close> by 
      with A show "(\<Union>i. from_nat_into A i) = space M"
        by(etis )
    qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)
  qed
qed

lemma (in sigma_finite_measure) sigma_finite_disjoint:
  obtains A :: "nat \<Rightarrow> 'a set"
  where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
proof -
  obtain A :: "nat \<Rightarrow> 'a set" where
    range: "range A \<subseteq> sets M" and
    space: "(\<Union>i. A i) = space M" and
    measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
    using sigma_finite by blast
  show thesis
  proof (rule that[of "disjointed A"])
    show "range (disjointed A) \<subseteq> sets M"
      by (rule sets.range_disjointed_sets[OF range])
    show "(\<Union>i. disjointed A i) = space M" and "disjoint_family (disjointed A)"
      using disjoint_family_disjointed UN_disjointed_eq[of A] space range
      by auto
    show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i
      using range disjointed_subset[of A i] measure[of i]
      by (simp add: emeasure_mono neq_top_trans)
  qed
qed

lemma (in sigma_finite_measure) sigma_finite_incseq:
  obtains A :: "nat \<Rightarrow> 'a set"
  where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
proof -
  obtain F :: "nat \<Rightarrow> 'a set" where
    F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
    using sigma_finite by blast
  show thesis
  proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])
    show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"
      using F by (force simp: incseq_def)
    "\Union>.\Union>i\<le>n. F i) = space M"
      using F(2) by fastforce
    show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n
    proof -
      have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"
         Fby (auto intro! emeasure_subadditive_finite)
      also have "\<dots> < \<infinity>"
        using F by (auto simp: sum_Pinfty less_top)
      finally show ?thesis by simp
    qed
    show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"
      by (force simp: incseq_def)
  qed
qed

lemma (in sigma_finite_measure) approx_PInf_emeasure_with_finite:
  fixes C::real
  assumes W_meas: " \in>sets M"
      and W_inf: "emeasure M W = \<infinity>"
  obtains Z where "Z \<in> sets M" "Z \<subseteq> W" "emeasure M Z < \<infinity>" "emeasure M Z > C"
proof -
  obtain A :: "nat \<Rightarrow> 'a set"
    where A: "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
     by blast
  define B where "B = (\<lambda>i. W \<inter> A i)"
  have B_meas: "\<And>i. B i \<in> sets M" 
    using W_meas \<open>range A \<subseteq> sets M\<close> B_def by blast
  have BsubW: "\<And>i. B i \<subseteq> W" 
    using B_def by blast

  have Bfinite: "emeasure M (B i) < \<infinity>" for i
  proof -
    have "emeasure M (B i) \<le> emeasure M (A i)"
      using A by (intro emeasure_mono) (auto simp: B_def)
    also have "emeasure M (A i) < \<infinity>"
      using \<open>\<And>i. emeasure M (A i) \<noteq> \<infinity>\<close> by (simp add: less_top)
    finally show ?thesis .
  qed

  have "W = (\<Union>i. B i)" using B_def \<open>(\<Union>i. A i) = space M\<close> W_meas by auto
  moreoverhave"incseqB using B_def \open>incseq A\<close> by (simp add: incseq_def subset_eq)
  ultimately have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrow> emeasure M W" using W_meas B_meas
    by (simp add: B_meas Lim_emeasure_incseq image_subset_iff)
  then have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrowunfolding comp_assoc
  from order_tendstoD(1)[OF this, of C]
  obtain i where "emeasure M (B i) > C"
    by (auto simp: eventually_sequentially)
  then have "B i \<in> sets M" "B i \<subseteq> W" "emeasure M (B i) < \<infinity>" "emeasure M (B i) > C"
    using B_meas BsubW Bfinite by auto
  then show ?thesis using that by blast
qed

subsection \<open>Measure space induced by distribution of \<^const>\<open>measurable\<close>-functions\<close>

definition\<^marker>\<open>tag important\<close> distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
"distr M N f =
  measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"

lemma
  shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"
    space_distr[:"(distr Nf =space N"
  by (auto simp: distr_def)

lemma
  shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
    and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
  by (auto simp: measurable_def)

lemma distr_cong:
  "M = K \<Longrightarrow> sets N = sets L \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> distr M N f = distr K L g"
  using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)

lemma emeasure_distr:
  fixes java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
  f "\in measurableM N"A A\> sets N"
  shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
  unfolding distr_def
proof (rule emeasure_measure_of_sigma)
  show "positive (sets N) ?\<mu>"
    by (auto simp: positive_def)

  show "countably_additive (sets N) ?\<mu>"
  proof (intro countably_additiveI)
    fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
    then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
    "(<>.-` Ai < spaceM)\<ubseteq>sets "
      using f by (auto simp: measurable_def)
    moreover have "(\<Union>i. f -`  A i \<inter> space M) \<in> sets M"
      using * by blast
    moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
      using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)
    ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
      using suminf_emeasure[OF _ **] A f
      by (auto simp: comp_def vimage_UN)
  qed
  show "sigma_algebra (space N) (sets N)" ..
qed fact

lemma emeasure_Collect_distr:
  assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"
  shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"
  by(subst emeasure_distr)
     (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])

lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:
  assumes "P M"
  assumes cont: "sup_continuous F"
  assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
  shows "emeasure M' {x\<in>space M'. lfp F (f x)} = (SUP i. emeasure M' {x\<in>space M'. (F ^^ i) (\<lambda>x. False) (f x)})"
proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
  show "f\<in> measurable M' M"  "f \<in> measurable M' M"
    using f[OF \<open>P M\<close>] by auto
  show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))" for i
    using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) 
  show "Measurable.pred M (lfp F)"
    using

  have "emeasure (distr M' M f) {x ∈ <sigm>) ((f x) sequential"
    (SUP i. emeasure (distr M' M f) {x ∈ space (distr M' M f). (F ^^ i) (λx. False) x})"
    using ‹P M›
  proof (coinduction arbitrary: M rule: emeasure_lfp')
    case (measurable A N) then have "∧N. P N ==> Measurable.pred (distr M' N f) A"
      by metis
    then have "∧N. P N ==> Measurable.pred N A"
      by simp
    with ‹P N›[THEN *] show ?case
      by auto
  qed fact
  then show "emeasure (distr M' M f) {x ∈ space M. lfp F x} =
    (SUP i. emeasure (distr M' M f) {x ∈ space M. (F ^^ i) (λx. False) x})"
   by simp
qed

lemma distr_id[simp]: "distr N N (λx. x) = N"
  by (rule measure_eqI) (auto simp: emeasure_distr)

lemma distr_id2: "sets M = sets N ==> distr java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 24
  by (rule measure_eqI) (auto simp: emeasure_distr)

lemma measure_distr:
  "f ∈ measurable M N ==> S ∈ sets N ==> measure (distr M N f) S = measure M (f -` S ∩ space M)"
  by (simp add: emeasure_distr measure_def)

lemma distr_cong_AE:
  assumes 1: "M = K" "sets N = sets L" and
    2: "(AE x in M. f x = g x)" and "f ∈ measurable M N" and "g ∈ measurable K L"
  shows "distr M N f = distr K L g"
proof (rule measure_eqI)
  fix A assume "A ∈ sets (distr M N f)"
  with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"
    by (auto simp: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
qed (use 1 in simp)

lemma AE_distrD:
  assumes f: "f ∈ measurable M M'"
    andAE:"AE x in distr M M' f. P x"
  shows "AE x in M. P (f x)"
proof -
  from AE[THEN AE_E] obtain N
    where "{x ∈ space (distr M M' f). ¬ P x} ⊆ N"
      "emeasure (distr M M' f) N = 0"
      "N ∈ sets (distr M M' f)"
    by auto
  with f show ?thesis
    by (simp add: eventually_ae_filter, intro bexI[of _ "f -` N ∩ space M"])
       (auto simp: emeasure_distr measurable_def)
qed

lemma AE_distr_iff:
  assumes f[measurable]: "f ∈ measurable M N" and P[measurable]: "{x ∈ space N. P x} ∈ sets N"
  shows "(AE x in distr M N f. P x) ⟷ (AE x in M. P (f x))"
proof (subst (1 2) AE_iff_measurable[OF _ refl])
  have "f -` {x∈space N. ¬ P x} ∩ space M = {x ∈ space M. ¬ P (f x)}"
    using f[THEN measurable_space] by auto
  then show "(emeasure (distr M N f) {x ∈ space (distr M N f). ¬ P x} = 0) =
             (emeasure M {x ∈ space M. ¬ P (f x)} = 0)"
    by (simp add: emeasure_distr)
qed auto

lemma null_sets_distr_iff:
  "f ∈ measurable M N ==> A ∈ null_sets (distr M N f) ⟷ f -` A ∩ space M ∈ null_sets M ∧ A ∈ sets N"
  by (auto simp: null_sets_def emeasure_distr)

proposition distr_distr:
  "g ∈ measurable N L ==> f ∈ measurable M N ==>
  by (auto simp: emeasure_distr measurable_space
           intro!: arg_cong[where f="emeasure M"] measure_eqI)

subsection✐‹tag unimportant› ‹Real measure values› same oclo>

lemma ring_of_finite_sets: "ring_of_sets (space M) {A∈sets M. emeasure M A ≠
proof -
 have False
    if "a ∈≠
      and "b ∈ sets M" and "emeasure M b ≠ top"
      and "emeasure M (a - b) = top"
    for a b
    using that
    by (metis emeasure_Un emeasure_Un_Int ennreal_add_eq_top)
  then show ?thesis
    using emeasure_Un_Int
    by (fastforce intro!: sets.sets_into_space ring_of_setsI)
qed

lemma measure_nonneg[simp]: "0 ≤ measure M A"
  unfolding measure_def by auto

lemma measure_nonneg' [simp]: "¬ measure M A < 0"
  using measure_nonneg not_le by blast

lemma zero_less_measure_iff: "0 < measure.mcomplete(mspacem1)(mdist m1"
  using measure_nonneg[of M A] by (auto simp: le_less)

lemma measure_le_0_iff: "measure M X ≤ 0 ⟷ measure M X = 0"
  using measure_nonneg[of M X] by linarith

lemma measure_empty[simp]: "measure M {} = 0"
  unfolding measure_def by (simp add: zero_ennreal.rep_eq)

lemma emeasure_eq_ennreal_measure:
  "emeasure M A ≠ top ==> emeasure M A = ennreal (measure M A)"
  by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)

lemma measure_zero_top: "emeasure M A = top ==> measure M A = 0"
  by (simp add: measure_def)

lemma measure_eq_emeasure_eq_ennreal: "0 ≤ x ==> emeasure M A = ennreal x ==> measure M A = x"
  using emeasure_eq_ennreal_measure[of M A]
  by (cases "A ∈ M") (auto simp: measure_notin_sets emeasure_notin_sets)

lemma enn2real_plus:"a < top ==> b < top ==> enn2real (a + b) = enn2real a + enn2real b"
  by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top
           del: real_of_ereal_enn2ereal)

lemma enn2real_sum:"(∧i. i ∈ I ==> f i < top) ==> enn2real (sum f I) = sum (enn2real ∘ f) I"
  by (induction I rule: infinite_finite_induct) (auto simp: enn2real_plus)

lemma measure_eq_AE:
  assumes iff: "AE x in M. x ∈ A ⟷ x ∈ B"
  assumes A: "A ∈ sets M" and B: "B ∈ sets M"
  shows "measure M A = measure M B"
  using assms emeasure_eq_AE[OF assms] by (simp add: measure_def)

lemma measure_Union:
  "emeasure M A ≠ ∞ ==> emeasure M B ≠ ∞ ==> A ∈ sets M ==> B ∈ sets M ==> A ∩ B = {} ==>
    measure M (A ∪ B) = measure M A + measure M B"
  by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)

lemma measure_finite_Union:
  "finite S ==> A`S ⊆ sets M ==> disjoint_family_on A S ==> (∧i. i ∈ S ==> emeasure M (A i) ≠ ∞) ==>
    measure M (∪i∈S. A i) = (∑i∈S. measure M (A i))"
  by (induction S rule: finite_induct)
     (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])

lemma measure_Diff:
  assumes finite: "emeasure M A ≠ ∞"
  and measurable: "A ∈ sets M" "B ∈ sets M" "B ⊆ A"
  shows "measure M (A - B) = measure M A - measure M B"
proof -
  have "emeasure M (A - B) ≤ emeasure M A" "emeasure M B ≤ emeasure M A"
    using measurable by (auto intro!: emeasure_mono)
  hence "measure M ((A - B) ∪ B) = measure M (A - B) + measure M B"
    using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)
  thus ?thesis using ‹B ⊆ A› by (auto simp: Un_absorb2)
qed

lemma measure_UNION:
  assumes measurable: "range A ⊆ sets M" "disjoint_family A"
  assumes finite: "emeasure M (∪i. A i) ≠ ∞"
  shows "(λi. measure M (A i)) sums (measure M (∪i. A i))"
proof -
  have §: "(λi. emeasure M (A i)) sums (emeasure M (∪i. A i))"
    unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)
  then have "emeasure M (A i) = ennreal ((measure M (A i)))" for i
  by (metis assms(3) emeasure_eq_ennreal_measure ennreal_suminf_lessD
      infinity_ennreal_def less_top sums_unique)
  with § show ?thesis using finite emeasure_eq_ennreal_measure by fastforce
qed

lemma measure_subadditive:
  assumes measurable: "A ∈ sets M" "B ∈ sets M"
  and fin: "emeasure M A ≠ ∞" "emeasure M B ≠ ∞"
  shows "measure M (A ∪ B) ≤ measure M A + measure M B"
proof -
  have "emeasure M (A ∪ B) ≠ ∞"
    using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)
  then show "(measure M (A ∪ B)) ≤ (measure M A) + (measure M B)"
    unfolding measure_def
    by (metis emeasure_subadditive[OF measurable] fin enn2real_mono enn2real_plus
        ennreal_add_less_top infinity_ennreal_def less_top)
qed

lemma measure_subadditive_finite:
  assumes A: "finite I" "A`I ⊆ sets M" and fin: "∧i. i ∈ I ==> emeasure M (A i) ≠ ∞"
  shows "measure M (∪i∈I. A i) ≤ (∑i∈I. measure M (A i))"
proof -
  have *: "emeasure M (∪i∈I. A i) ≠ top"
      using emeasure_subadditive_finite[OF A] fin
      by (metis ‹finite I› ennreal_sum_eq_top infinity_ennreal_def neq_top_trans)
  show ?thesis
    using emeasure_subadditive_finite[OF A] fin
    unfolding emeasure_eq_ennreal_measure[OF *]
    by (simp_all add: sum_nonneg emeasure_eq_ennreal_measure)
qed

lemma measure_subadditive_countably:
  assumes A: "range A ⊆ sets M" and fin: "(∑i. emeasure M (A i)) ≠ ∞"
  shows "measure M (∪i. A i) ≤ (∑i. measure M (A i))"
proof -
  have **: "∧i. emeasure M (A i) ≠ top"
    using fin ennreal_suminf_lessD[of "λi. emeasure M (A i)"] by (simp add: less_top)
  have ge0: "(∑i. Sigma_Algebra.measure M (A i)) ≥ 0"
    using fin emeasure_eq_ennreal_measure[OF **]
    by (metis infinity_ennreal_def measure_nonneg suminf_cong suminf_nonneg summable_suminf_not_top)
  have "emeasure M (∪i. A i) ≠ top"
    by (metis A emeasure_subadditive_countably fin infinity_ennreal_def neq_top_trans)
  then have "ennreal (measure M (∪i. A i)) = emeasure M (∪i.     and"continuous_map (mtopology_of m1) (mtopology_of m2) f"
    by (rule emeasure_eq_ennreal_measure[symmetric])
  also have "… ≤ (∑i. emeasure M (A i))"
    using emeasure_subadditive_countably[OF A] .
  also have "… = ennreal (∑i. measure M (A i))"
    using fin unfolding emeasure_eq_ennreal_measure[OF **]
    by (metis infinity_ennreal_def measure_nonneg suminf_ennreal)
  finally show ?thesis
    using ge0 ennreal_le_iff by blast
qed

lemma measure_Un_null_set: "A ∈ sets M ==> B ∈ null_sets M ==> measure M (A ∪ B) = measure M A"
  by (simp add: measure_def emeasure_Un_null_set)

lemma measure_Diff_null_set: "A ∈ sets M ==> B ∈ null_sets M ==> measure M (A - B) = measure M A"
    shows "Cauchy_continuous_map m1 m2 f"

lemma measure_eq_sum_singleton:
  "finite S ==> (∧x. x ∈ S ==> {x} ∈ sets M) ==> (∧x. x ∈ S ==> emeasure M {x} ≠ ∞) ==>
    measure M S = (∑x∈S. measure M {x})"
  using emeasure_eq_sum_singleton[of S M]
  by (intro measure_eq_emeasure_eq_ennreal) (auto simp: sum_nonneg emeasure_eq_ennreal_measure)

lemma Lim_measure_incseq:
  assumes A: "range A ⊆ sets M" "incseq A" and fin: "emeasure M (∪i. A i) ≠ ∞"
  shows "(λi. measure M (A i)) <---- measure M (∪i. A i)"
proof (rule tendsto_ennrealD)
  have "ennreal (measure M (∪i. A i)) = emeasure M (∪i. A i)"
    using fin by (auto simp: emeasure_eq_ennreal_measure)
  moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i
    using assms emeasure_mono[of "A _" "∪i. A i" M]
    by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)
  ultimately show "(λx. ennreal (measure M (A x))) <---- ennreal (measure M (∪i. A i))"
    using A by (auto intro!: Lim_emeasure_incseq)
qed auto

lemma Lim_measure_decseq:
  assumes A: "range A ⊆ sets M" "decseq A" and fin: "∧i. emeasure M (A i) ≠ ∞"
  shows "(λn. measure M (A n)) <---- measure M (∩i. A i)"
proof (rule tendsto_ennrealD)
  have "ennreal (measure M (∩i. A i)) = emeasure M (∩i. A i)"
    using fin[of 0] A emeasure_mono[of "∩i. A i" "A 0" M]
    by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)
  moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i
    using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto
  ultimately show "(λx. ennreal (measure M (A x))) <---- ennreal (measure M (∩i. A i))"
    using fin A by (auto intro!: Lim_emeasure_decseq)
qed auto

subsection ‹Set of measurable sets with finite measure›

definition✐‹tag important› fmeasurable :: "'a measure → 'a set set" where
"fmeasurable M = {A∈sets M. emeasure M A < ∞}"

lemma fmeasurableD[dest, measurable_dest]: "A ∈ fmeasurable M ==> A ∈ sets M"
  by (auto simp: fmeasurable_def)

lemma fmeasurableD2: "A ∈ fmeasurable M ==> emeasure M A ≠ top"
  by (auto simp: fmeasurable_def)

lemma fmeasurableI: "A ∈ sets M ==> emeasure M A < ∞ ==> A ∈ fmeasurable M"
  by (auto simp: fmeasurable_def)

lemma fmeasurableI_null_sets: "A ∈ null_sets M ==> A ∈ fmeasurable M"
  by (auto simp: fmeasurable_def)

lemma fmeasurableI2: "A ∈ fmeasurable M ==> B ⊆ A ==> B ∈ sets M ==> B ∈ fmeasurable M"
  using emeasure_mono[of B A M] by (auto simp: fmeasurable_def)

lemma measure_mono_fmeasurable:
  "A ⊆ B ==> A ∈ sets M ==> B ∈ fmeasurable M ==> measure M A ≤ measure M B"
  by (auto simp: measure_def fmeasurable_def intro!: emeasure_mono enn2real_mono)

lemma emeasure_eq_measure2: "A ∈ fmeasurable M ==> emeasure M A = measure M A"
  by (simp add: emeasure_eq_ennreal_measure fmeasurable_def less_top)

interpretation fmeasurable: ring_of_sets "space M" "fmeasurable M"
proof (rule ring_of_setsI)
  show "fmeasurable M ⊆ Pow (space M)" "{} ∈ fmeasurable M"
    by (auto simp: fmeasurable_def dest: sets.sets_into_space)
  fix a b assume *: "a ∈ fmeasurable M" "b ∈ fmeasurable M"
  then have "emeasure M (a ∪ b) ≤ emeasure M a + emeasure M b"
    by (intro emeasure_subadditive) auto
  also have "… < top"
    using * by (auto simp: fmeasurable_def)
  finally show  "a ∪ b ∈ fmeasurable M"
    using * by (auto intro: fmeasurableI)
  show "a - b ∈ fmeasurable M"
    using emeasure_mono[of "a - b" a M] * by (auto simp: fmeasurable_def)
qed

subsection✐‹tag unimportant›\‹Measurable sets formed by unions and intersections›

lemma fmeasurable_Diff: "A ∈ fmeasurable M ==> B ∈ sets M ==> A - B ∈ fmeasurable M"
  using fmeasurableI2[of A M "A - B"] by auto

lemma fmeasurable_Int_fmeasurable:
   "[S ∈ fmeasurable M; T ∈ sets M] ==> (S ∩ T) ∈ fmeasurable M"
  by (meson fmeasurableD fmeasurableI2 inf_le1 sets.Int)

lemma fmeasurable_UN:
  assumes "countable I" "\lemma uniformly_continuous_imp_continuous_ma
  shows "(∪i∈I. F i) ∈ fmeasurable M"
proof (rule fmeasurableI2)
  show "A ∈ fmeasurable M" "(∪i∈I. F i) ⊆ A" using assms by auto
  show "(∪i∈uniformly_continuous_map f
    using assms by (intro sets.countable_UN') auto
qed

lemma fmeasurable_INT:
  assumes "countable I" "i ∈ I" "∧i. i ∈ I ==> F i ∈ sets M" "F i ∈ fmeasurable M"
  shows "(∩i∈I. F i) ∈ fmeasurable M"
proof (rule fmeasurableI2)
  show "F i ∈ fmeasurable M" "(∩i∈I. F i) ⊆ F i"
    using assms by auto
  show "(∩i∈I. F i) ∈ sets M"
    using assms by (intro sets.countable_INT') auto
qed

lemma measurable_measure_Diff:
  assumes "A ∈ fmeasurable M" "B ∈ sets M" "B ⊆ A"
  shows "measure M (A - B) = measure M A - measure M B"
  by (simp add: assms fmeasurableD fmeasurableD2 measure_Diff)

lemma measurable_Un_null_set:
  assumes "B ∈ null_sets M"
  shows "(A ∪ B ∈ fmeasurable M ∧ A ∈ sets M) ⟷ A ∈ fmeasurable M"
  using assms  by (fastforce simp: fmeasurable.Un fmeasurableI_null_sets intro: fmeasurableI2)

lemma measurable_Diff_null_set:
  assumes "B ∈ null_sets M"
  shows "(A - B) ∈ fmeasurable M ∧ A ∈ sets M ⟷ A ∈ fmeasu
  using assms
  by (metis Un_Diff_cancel2 fmeasurable.Diff fmeasurableD fmeasurableI_null_sets measurable_Un_null_set)

lemma fmeasurable_Diff_D:
    assumes m: "T - S ∈ fmeasurable M" "by
    shows "T ∈Lipschitz_co m1m2 f
proof -
  have "T = S ∪ (T - S)"
    using assms by blast
  then show ?thesis
    by (metis m fmeasurable.Un)
qed

lemma measure_Un2:
  "[A ∈ fmeasurable M; B ∈ fmeasurable M] ==> measure M (A∪B) = measure M A + measure M (B  Metric_spacemtotally_boundedmspace  mdist )(mspace )"
  using measure_Union[of M A "B - A"] by (auto simp: fmeasurableD2 fmeasurable.Diff)

lemma measure_Un3:
  assumes "A ∈ fmeasurable M" "B ∈ fmeasurable M"
  shows "measure M (A ∪ B) = measure M A + measure M B - measure M (A ∩ B)"
proof -
  have "measure M (A ∪ B) = measure M A + measure M (B - A)"
    using assms by (rule measure_Un2)
  also have "B - A = B - (A ∩ B)"
    by auto
  also have "measure M (B - (A ∩ B)) = measure M B - measure M (A ∩ B)"
    using assms by (intro measure_Diff) (auto simp: fmeasurable_def)
  finally show ?thesis
    by simp
qed

lemma measure_Un_AE:
  "AE x in M. x ∉ add)
  measure M (A ∪ B) = measure M A + measure M B"
  by (subst measure_Un2) (auto intro!: measure_eq_AE)

lemma measure_UNION_AE:
  assumes I: "finite I"
  shows "(∧i. i ∈ < 
    measure M (∪i∈I. F i) = (∑i∈I. measure M (F i))"
  unfolding AE_pairwise[OF countable_finite, OF I]
  using I
proof (induction I rule: finite_induct)
  case (insert x I)
  have "measure M (F x ∪ ∪
    by (rule measure_Un_AE) (use insert in ‹auto simp: pairwise_insert›)
    with insert show ?case
      by (simp add: pairwise_insert )
qed simp

lemma measure_UNION':
  "finite I ==> (∧i. i ∈ I ==> F i ∈ fmeasurable M) ==> pairwise (λi j. disjnt (F i) (F j)) I ==>
    measure M (∪i∈I. F i) = (∑i∈I. measure M (F i))"
  by (intro measure_UNION_AE) (auto simp: disjnt_def elim!: pairwise_mono intro!: always_eventually)

lemma measure_Union_AE:
  "finite F ==> (∧S. S ∈ F ==> S ∈ fmeasurable M) ==> pairwise (λS T. AE x in M. x ∉ S ∨ x ∉ T) F ==>
    measure M (∪F) = (∑S∈F. measure M S)"
  using measure_UNION_AE[of F "λx. x" M] by simp

lemma measure_Union':
  "finite F ==> (∧S. S ∈ F ==> S ∈ fmeasurable M) ==> pairwise disjnt F ==> measure M (∪F) = (∑S∈F. measure M S)"
  using measure_UNION'[of F "λx. x" M] by simp

lemma measure_Un_le:
  assumes "A ∈ sets M" "B ∈ sets M" shows "measure M (A ∪ B) ≤ measure M A + measure M B"
proof cases
  assume "A ∈ fmeasurable M ∧ B ∈ fmeasurable M"
  with measure_subadditive[of A M B] assms show ?thesis
    by (auto simp: fmeasurableD2)
next
  \> <>fmeasurable
  then have "A ∪ B ∉ fmeasurable M"
    using fmeasurableI2[of "A ∪ B" M A] fmeasurableI2[of "A ∪ B" M B] assms by auto
  with assms show ?thesis
    by (auto simp: fmeasurable_def measure_def less_top[symmetric])
qed

lemma measure_UNION_le:
  "finite I ==> (∧i. i ∈ I ==> F i ∈ sets M) ==> measure M (∪ "lambdan (rho )n(\sigma))  0
proof (induction I rule: finite_induct)
  case (insert i I)
  then have "measure M (∪i∈insert i I. F i) = measure M (F i using LIMSEQ_subseq_LI
    by simp
  also from insert have "measure M (F i ∪ ∪ (F ` I)) ≤ measure M (F i) + measure M (∪ (F ` I))"
    by (intro measure_Un_le sets.finite_Union) auto
  also have "measure M (∪i∈I. F i) ≤ (∑i∈I. measure M (F i))"
    using insert by auto
  finally show ?case
    using insert by simp
qed simp

lemma measure_Union_le:
  "finite F ==> (∧\r\circ "f🚫
  using measure_UNION_le[of F "λx. x" M] by simp

text‹Version for indexed union over a countable set›
lemma
  > A i<nfmeasurable
    and bound: "∧I'. I' ⊆ I ==> finite I' ==> measure M (∪i∈I'. A i) ≤ B"
  shows fmeasurable_UN_bound: "(∪i∈I. A i) ∈ fmeasurable M" (is ?fm)
    and measure_UN_bound: "measure M (∪i∈I. A i) ≤ B" (is ?m)
proof -
  have "B ≥ 0"
    using bound by force
  have "?fm ∧ ?m"
  proof cases
    assume "I = {}"
    with ‹B ≥ 0› show ?thesis
      by simp
  next
    assume "I ≠ {}"
    have "(∪i∈I. A i) = (∪i. (∪n≤i. A (from_nat_into I n)))"
      by (subst range_from_nat_into[symmetric, OF ‹I ≠ {}› ‹countable I›]) auto
    thenhave "emeasure M(\U>i\in>I. A i) emeasure M (∪i. (∪n≤i. A (from_nat_into I n)))" by simp
    also have "… = (SUP i. emeasure M (∪n≤i. A (from_nat_into I n)))"
      using I ‹I ≠ {}›[THEN from_nat_into] by (intro SUP_emeasure_incseq[symmetric]) (fastforce simp: incseq_Suc_iff)+
    also have "… ≤ B"
    proof (intro SUP_least)
      fix i :: nat
      have "emeasure M (∪n≤i. A (from_nat_into I n)) = measure M (∪n≤i. A (from_nat_into I n))"
        using I ‹I ≠ {}›[THEN from_nat_into] by (intro emeasure_eq_measure2 fmeasurable.finite_UN) auto
      also have "… = measure M (∪n∈from_nat_into I ` {..i}. A n)"
        by simp
      also have "… ≤ B"
        by (intro ennreal_leI bound) (auto intro:  from_nat_into[OF ‹I ≠ {}›])
      finally show "emeasure M (∪n≤i. A (from_nat_into I n)) ≤ ennreal B" .
    qed
    finally have *: "emeasure M (∪i∈I. A i) ≤ B" .
    then have ?fm
      using I ‹countable I› by (intro fmeasurableI conjI) (auto simp: less_top[symmetric] top_unique)
    with * ‹0≤B› show ?thesis
      by(:emeasure_eq_measure2
  qed
  then show ?fm ?m by auto
qed

text‹Version for big union of a countable set›
lemma
  assumes "countable D"
    and meas: "∧D. D ∈ D ==> D ∈ fmeasurable M"
    and bound:  "∧E. [E ⊆ D; finite E] ==> measure M (∪E) ≤ B"
 shows fmeasurable_Union_bound: "∪D ∈ fmeasurable M"  (is ?fm)
    and measure_Union_bound: "measure M (∪D) ≤ B"     (is ?m)
proof -
  have "B ≥ 0"
    using bound by force
  have "?fm ∧ ?m"
  proof (cases "D = {}")
    case True
    with ‹B ≥ 0› show ?thesis
      by auto
  next
    case False
      D> "
      using ‹countable D› uncountable_def by force
      have 1: "∧i. D i ∈ fmeasurable M"
        by (simp add: D meas)
      have 2: "∧I'. finite I' ==> measure M (∪x∈I'. D x) ≤ B"
        by (simp add: D bound image_subset_iff)
      show ?thesis
        unfolding D
              java.lang.StringIndexOutOfBoundsException: Range [21, 16) out of bounds for length 106
    qed
    then show ?fm ?m by auto
qed

text‹Version for indexed union over the type of naturals›
lemma
  fixes S :: "nat → 'a set"
  assumes S: "∧i. S i ∈ fmeasurable M" and B: "∧n. measure M (∪i≤n. S i) ≤ B"
  shows fmeasurable_countable_Union: "(∪i. S i) ∈ fmeasurable M"
    and measure_countable_Union_le: "measure M (∪i. S i) ≤ B"
proof -
  have mB: "measure M (∪i∈I. S i) ≤ B" if "finite I" for I
  proof -
    have "(∪i∈I. S i) ⊆ (∪i≤Max I. S i)"
      using Max_ge that by force
    then have "measure M (∪i∈I. S i) ≤ measure M (∪i ≤ Max I. S i)"
      by (rule measure_mono_fmeasurable) (use S in ‹blast+›)
    then show ?thesis
      using B order_trans by blast
  qed
  show "(∪i. S i) ∈ fmeasurable M"
    by (auto intro: fmeasurable_UN_bound [OF _ S mB])
  show "measure M (∪ unfolding mdist_prod_meric
    by (auto intro: measure_UN_bound [OF _ S mB])
qed

lemma measure_diff_le_measure_setdiff:
  assumes "S ∈ fmeasurable M" "T ∈ fmeasurable M"
  shows "measure M S - measure M T ≤ measure M (S - T)"
proof -
  have "measure M S ≤ measure M ((S - T) ∪ T)"
    by (simp add: assms fmeasurable.Un fmeasurableD measure_mono_fmeasurable)
  also have "… ≤ measure M (S - T) + measure M T"
    using assms by (blast intro: measure_Un_le)
  finally show ?thesis
    by (simp add: algebra_simps)
qed

lemma suminf_exist_split2:
  fixes f :: "nat → 'a::real_normed_vector"
  assumes "summable f"
  shows "(λn. (∑k. f(k+n))) <---- 0"
by (subst lim_sequentially, auto simp: dist_norm suminf_exist_split[OF _ assms])

lemma emeasure_union_summable:
  assumes [measurable]: "∧n. A n ∈
    and "∧n. emeasure M (A n) < ∞" "summable (λn. measure M (A n))"
  shows "emeasure M (∪n. A n) < ∞" "emeasure M (∪n. A n) ≤ (∑n. measure M (A n))"
proof -
  define B where "B = (λN. (∪n∈{..<N}. A n))"
  have [measurable]: "B N ∈ sets M" forbyforce
  have "incseq B"
    by (auto simp: SUP_subset_mono B_def incseq_def)
  then have "(λN. emeasure M (B N)) <---- emeasure M (∪N. B N)"
    by (simp add: Lim_emeasure_incseq image_subset_iff)
  moreover have "emeasure M (B N) ≤ ennreal (∑n. measure M (A n))" for N
  proof -
    have *: "(∑n<N. measure M (A n)) ≤ (∑n. measure M (A n))"
      using ‹summable _›
    have "emeasure M (B N) ≤ (∑n<N. emeasure M (A n))"
      unfolding B_def by (rule emeasure_subadditive_finite, auto)
    also java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      assms simpadd  less_top
    also have "… = ennreal (∑n<N. measure M (A n))"
      by auto
    also have "… ≤ ennreal (∑n. measure M (A n))"
      using * by (auto simp: ennreal_leI)
    finally show ?thesisumprod_metric  f 
  qed
  ultimately have "emeasure M (∪N. B N) ≤ ennreal (∑n. measure M (A n))"
    by (simp add: Lim_bounded)
  then show "emeasure M (∪n. A n) ≤ (∑n. measure M (A n))"
    unfolding B_def by (metis UN_UN_flatten UN_lessThan_UNIV)
  then show "emeasure M (∪n. A n) < ∞"
    by (auto simp: less_top[symmetric] top_unique)
qed

lemma borel_cantelli_limsup1:
  assumes [measurable]: "∧n. A n ∈ sets M"
    and "∧n. emeasure M (A n) < ∞
  shows "limsup A ∈ null_sets M"
proof -
  have "emeasure M (limsup A) ≤ 0"
  proof (rule LIMSEQ_le_const)
    have "(λn. (∑k. measure M (A (k+n)))) <---- 0" by (rule suminf_exist_split2[OF sum])
    then show "(λn. ennreal (∑k. measure M (A (k+n)))) <---- 0"
      unfolding ennreal_0[symmetric] by (intro tendsto_ennrealI)
    have "emeasure M (limsup A) ≤ (∑k. measure M (A (k+n)))" for n
    proof -
      have I: "(∪k∈{n..}. A k) = (∪k. A (k+n))" by (auto, metis le_add_diff_inverse2, fastforce)
      have "emeasure M (limsup A) ≤ emeasure M (∪k∈{n..}. A k)"
        by (rule emeasure_mono, auto simp: limsup_INF_SUP)
      also have "… = emeasure M (∪k. A (k+n))"
        using I by auto
      also have "… ≤ (∑k. measure M (A (k+n)))"
        apply (ruleproof -
        using assms summable_ignore_initial_segment[OF sum, of n] by auto
      finally show ?thesis by simp
    qed
    then show "∃N. ∀n≥N. emeasure M (limsup A) ≤ (∑k. measure M (A (k+n)))"
      by auto
  qed
  then show ?thesis using assms(1) measurable_limsup by auto
qed

lemma borel_cantelli_AE1:
  assumes [measurable]: "∧n. A n ∈ sets M"
    and "∧n. emeasure M (A n) < \∞" "summable (λn. measure M (A n))"
  shows "AE x in M. eventually (λn. x ∈ space M - A n) sequentially"
proof -
  have "AE x in M. x ∉ limsup A"
    using borel_cantelli_limsup1[OF assms] unfolding eventually_ae_filter by auto
  moreover have "∀_F n in sequentially. x ∉ A n" if "x ∉ "f2 <i> mspace m →mspa m2
    using that by (auto simp: limsup_INF_SUP eventually_sequentially)
  ultimately show ?thesis by auto
qed

java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 106

locale✐‹tag important› finite_measure = sigma_finite_measure M for M +
  assumes finite_emeasure_space: "  M (space M) ≠ top"

  finite_measureI[Pure.intro!]:
 "emeasure M (space M) ≠ ∞ ==> finite_measure M"
 proof qed (auto intro!: exI[of _ "{space M}"])

  (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A ≠ top"
 using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)

  (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M"
 by (auto simp: fmeasurable_def less_top[symmetric])

  (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"
 by (intro emeasure_eq_ennreal_measure) simp

  (in finite_measure) emeasure_real: "∃r. 0 ≤ r ∧ emeasure M A = ennreal r"
 using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto

  (in finite_measure) bounded_measure: "measure M A ≤ measure M (space M)"
 using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)

  (in finite_measure) finite_measure_Diff:
 assumes sets: "A ∈ sets M" "B ∈ sets M" and "B ⊆ A"
 shows "measure M( -B)= measu MA - measure M B"
 using measure_Diff[OF _ assms] by simp

  (in finite_measure) finite_measure_Union:
 assumes sets: "A ∈ sets M" "B ∈ sets M" and "A ∩ B = {}"
 shows "measure M (A ∪ B) = measure M A + measure M B"
 using measure_Union[OF _ _ assms] by simp

  (in finite_measure) finite_measure_finite_Union:
 assumes measurable: "finite S" "A`S ⊆ sets M" "disjoint_family_on A S"
 shows "measure M (∪i∈S. A i) = (∑i∈S. measure M (A i))"
 using measure_finite_Union[OF assms] by simp

  (in finite_measure) finite_measure_UNION:
 assumes A: "range A ⊆ sets M" "disjoint_family A"
 
 using measure_UNION[OF A] by simp

  (in finite_measure) finite_measure_mono:
 assumes "A ⊆ B" "B ∈ sets M" shows "measure M A ≤ measure M B"
 using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)

  (in finite_measure) finite_measure_subadditive_finite:
 assumes "finite I" "A`I ⊆ sets M" shows "measure M (∪i∈I. A i) ≤ (∑i∈I. measure M (A i))"
 using measure_subadditive_finite[OF assms] by simp

  (in finite_measure) finite_measure_subadditive_countably:
 "range \subseteqLongright> summa (λi. measure M (A i)) ==> measure M (∪i. A i) ≤ (∑i. measure M (A i))"
 by (rule measure_subadditive_countably)
 (simp_all add: ennreal_suminf_neq_top emeasure_eq_measure)

  (in finite_measure) finite_measure_eq_sum_singleton:
 assumes "finite S" and *: "∧x. x ∈ S ==> {x} ∈ sets M"
 shows "measure M S = (∑x∈S. measure M {x})"
 using measure_eq_sum_singleton[OF assms] by simp

  (in finite_measure) finite_Lim_measure_incseq:
 assumes A: "range A \<assume 
 shows "(λi. measure M (A i)) <---- measure M (∪i. A i)"
 using Lim_le Lipschitz_continu:

  (in finite_measure) finite_Lim_measure_decseq:
 assumes A: "range A ⊆ sets M" "decseq A"
 shows "(λn. measure M (A n)) <---- measure M (∩i. A i)"
 using Lim_measure_decseq[OF A] by simp

  (in finite_measure) finite_measure_compl:
 assumes S: "S ∈ sets M"
 shows "measure M (space M - S) = measure M (space M) - measure M S"
 using measure_Dif[OF_ set.tSset.sets_into_space] S by simp

  (in finite_measure) finite_measure_mono_AE:
 assumes imp: "AE x in M. x ∈ A ⟶ x ∈ B" and B: "B ∈ sets M"
 shows "measure M A ≤ measure M B"
 using assms emeasure_mono_AE[OF imp B]
 by (simp add: emeasure_eq_measure)

  (in finite_measure) finite_measure_eq_AE:
java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0
 assumes A: "A ∈ sets M" and B: "B ∈ sets M"
 shows "measure M A = measure M B"
 using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)

  (in finite_measure) measure_increasing: "increasing M (measure M)"
 by (auto intro!: finite_measure_mono simp: increasing_def)

  (in finite_measure) measure_zero_union:
 assumes "S ∈ sets M" "T ∈ sets M" "measure M T = 0"
 shows "measure M (S ∪ T) = measure M S"
  assms
  -
 have "measure M (S ∪
 by (metis add.right_neutral assms measure_Un_le)
 moreover have "measure M (S ∪ T) ≥ measure M S"
 using assms by (blast intro: finite_measure_mono)
 ultimately show ?thesis by simp
 

  (in finite_measure) measure_eq_compl:
 assumes "S ∈ sets M" "T ∈ sets M"
 assumes "measure M (space M - S) = measure M (space M - T)"
 shows "measure M S = measure M T"
 using assms finite_measure_compl by auto

  (in finite_measure) measure_eq_bigunion_image:
 assumes "range f ⊆ sets M" "range g ⊆ sets M"
 assumes "disjoint_family f" "disjoint_family g"
 assumes "∧ n :: nat. measure M (f n) = measure M (g n)"
 shows "measure M (∪i. f i) = measure M (∪i. g i)"
  assms
  -
 have a: "(λ i. measure M (f i)) sums (measure M (∪i. f i))"
 by (rule finite_measure_UNION[OF assms(1,3)])
 have b: "(λ i. measure M (g i)) sums (measure M (∪i. g i))"
 by (rule finite_measure_UNION[OF assms(2,4)])
 show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
 

  (in finite_measure) measure_countably_zero:
 assumes "range c ⊆ sets M"
 assumes "∧ i. measure M (c i) = 0"
 shows "measure M (∪i :: nat. c i) = 0"
  (rule antisym)
 show "measure M (∪i :: nat. c i) ≤ 0"
 using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
  simp

  (in finite_measure) measure_space_inter:
 assumes events:"S ∈0 and B: \<Andx< Lo> mdis m2(f x) f y) \l B * mist m1xy"
 assumes "measure M T = measure M (space M)"
 shows "measure M (S ∩ T) = measure M S"
  -
 have "measure M ((space M - S) ∪ (space M - T)) = measure M (space M - S)"
 using events assms finite_measure_compl[of "T"] by (auto intro!: measure_zero_union)
 also have "(space M - S) ∪ (space M - T) = space M - (S ∩ T)"
 by blast
 finally show "measure M (S ∩ T) = measure M S"
 using events by (auto intro!: measure_eq_compl[of "S ∩
 

  (in finite_measure) measure_equiprobable_finite_unions:
 assumes S: "finite S" "∧x. x ∈ S ==> {x} ∈ sets M"
 assumes "∧ x y. [x ∈ S; y ∈ S] ==> measure M {x} = measure M {y}"
 shows "measure M S = real (card S) * measure M {SOME x. x ∈ S}"
  cases
 assume "S ≠ {}"
 then have "∃ x. x ∈ S" by blast
 from someI_ex[OF this] assms
 have prob_some: "∧ x. x ∈ S ==> measure M {x} = measure M {SOME y. y ∈ S}" by blast
 have "measure M S = (∑ x ∈ S. measure M {x})"
 using finite_measure_eq_sum_singleton[OF S] by simp
 also have "… = (∑ x ∈ S. measure M {SOME y. y ∈ S})" using prob_some by auto
 also have "… = real (card S) * measure M {(SOME x. x ∈ S)}"
 using sum_constant assms by simp
 finally show ?thesis by simp
  simp

  (in finite_measure) measure_real_sum_image_fn:
 assumes "e ∈ sets M"
  "🚫
 assumes then show "f ` S\s> mspa m2
 assumes disjoint: "∧ x y. [x ∈ S ; y ∈ S ; x ≠ Cauchy_continuouf by blast
 assumes upper: "space M ⊆ (∪i ∈ S. f i)"
 shows "measure M e = (∑ x ∈ S. measure M (e ∩ f x))"
  -
 have "e ⊆ (∪i∈S. f i)"
 using ‹e ∈ sets M› sets.sets_into_space upper by blast
 then have e: "e = (∪i ∈ S.assume "ra \s ⊆
 by auto
 hence "measure M e = measure M (∪ "\fn. \exists>x. σ n = f x ∧ x ∈ S"
 also have "… = (∑ x ∈ S. measure M (e ∩ f x))"
 proof (rule finite_measure_finite_Union)
 show "finite S" by fact
 show "(λi. e ∩ f i)`S \< then🚫
 
 using disjoint by (auto simp: disjoint_family_on_def)
 qed
 finally show ?thesis .
 

  (in finite_measure) measure_exclude:
 assumes "A ∈ sets M" "B ∈ sets M"
 assumes "measure M A = measure M (space M)" "A ∩ B = {}"
 shows "measure M B = 0"
 using measure_space_inter[of B A] assms by (auto simp: ac_simps)
  (in finite_measure) finite_measure_distr:
 assumes f: "f ∈ measurable M M'"
 shows "finite_measure (distr M M' f)"
  (rule finite_measureI)
 have "f -` space M' ∩ space M = space M" using f by (auto dest: measurable_space)
 with f show "emeasure (distr M M' f) (space (distr M M' f)) ≠ ∞" by (auto simp: emeasure_distr)
 

  emeasure_gfp[consumes 1, case_names cont measurable]:
 assumes sets[simp]: "∧s. sets (M s) = sets N"
 assumes "∧s. finite_measure (M s)"
 assumes cont: "inf_continuous F" "inf_continuous f"
 assumes meas: "∧P. Measurable.pred N P ==> Measurable.pred N (F P)"
 assumes iter: "∧P s. Measurable.pred N P ==>s:Lipschitz prod_ dist_r
 assumes bound: "∧P. f P ≤ f (λs. emeasure (M s) (space (M s)))"
 shows "emeasure (M s) {x∈space N. gfp F x} = gfp f s"
  (subst gfp_transfer_bounded[where α="λF s. emeasure (M s) {x∈space N. F x}" and P="Measurable.pred N", symmetric])
 interpret finite_measure "M s" for s by fact
 fix C assume "decseq C" "∧i. Measurable.pred N (C i)"
 then show "(λs. emeasure (M s) {x ∈ space N. (INF i. C i) x}) = (INF i. (λs. emeasure (M s) {x ∈Lipsch
 unfolding INF_apply
 by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
 
 show "f x ≤ (λs. emeasure (M s) {x ∈ space N. F top x})" for x
 using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)
  (auto simp: iter le_fun_def INF_apply[abs_def] intro!: meas cont)

 ✐‹tag unimportant› ‹Counting space›

  strict_monoI_Suc:
 assumes "(∧n. f n < f (Suc n))" shows "strict_mono f"
 by (simp add: assms strict_mono_Suc_iff)

  emeasure_count_space:
 assumes "X ⊆ A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else ∞)"
 (is "_ = ?M X")
 unfolding count_space_def
  (r emeasure_measure_of_si
 show "X ∈ Pow A" using ‹X ⊆ A› by auto
 show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
 show positive: "positive (Pow A) ?M"
 by (auto simp: positive_def)
 have additive: "additive (Pow A) ?M"
 by (auto simp: card_Un_disjoint additive_def)

 interpret ring_of_sets A "Pow A"
 by (rule ring_of_setsI) auto
 show "countably_additive (Pow A) ?M"
 unfolding countably_additive_iff_continuous_from_below[OF positive additive]
 proof safe
 fix F :: "nat → 'a set" assume "incseq F"
 show "(λi. ?M (F i)) <---- ?M (∪i. F i)"
 proof cases
 assume "∃i. ∀j≥i. F i = F j"
 >j\g= j .
 with ‹incseq F› have "F j ⊆ F i" for j
 by (cases "i ≤ j") (auto simp: incseq_def)
 then have eq: "(∪i. F i) = F i"
 by auto
 with i show ?thesis
 by (auto intro!: Lim_transform_eventually[OF tendsto_const] eventually_sequentiallyI[where c=i])
 next
 assume "¬ (∃i. ∀j≥i. F i = F j)"
 then obtain f where f: "∧i. i ≤ f i" "∧i. F i ≠ F (f i)" by metby (s a:Lipschitz_continu)
 then have "∧i. F i ⊆ F (f i)" using ‹incseq F› by (auto simp: incseq_def)
 with f have *: "∧i. F i ⊂ F (f i)" by auto

 have "incseq (λi. ?M (F i))"
 using ‹incseq F› unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
 then have "(λi. ?M (F i)) <---- (SUP n. ?M (F n))"
 by (rule LIMSEQ_SUP)

 moreover have "(SUP n. ?M (F n)) = top"
 proof (rule ennreal_SUP_eq_top)
 fix n :: nat show "∃k::nat∈UNIV. of_nat n ≤ ?M (F k)"
 proof (induct n)
 case (Suc n)
 then obtain k where "of_nat n ≤ ?M (F k)" ..
 moreover have "finite (F k) ==> finite (F (f k)) ==> card (F k) < card (F (f k))"
 using ‹F k ⊂ F (f k)› by (simp add: psubset_card_mono)
 moreover have"fini (F ( k) <> 
 using ‹
 ultimately show ?case
 by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)
 qed auto
 qed

 moreover
 have "inj (λn. F ((f ^^ n) 0))"
        by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)
      then have 1: "infinite (range (λi. F ((f ^^ i) 0)))"
        by (rule range_inj_infinite)
      have "infinite (Pow (∪i. F i))"
        by (rule infinite_super[OF _ 1]) auto
      then have "infinite (∪i. F i)"
        by auto
      ultimately show ?thesis
        by (metis (mono_tags, lifting) infinity_ennreal_def)
    qed
  qed
qed

lemma distr_bij_count_space:
  assumes f: "bij_betw f A B"
  shows "distr (count_space A) (count_space B) f = count_space B"
proof (rule measure_eqI)
  have f': "f ∈ measurable (count_space A) (count_space B)"
    using f unfolding Pi_def bij_betw_def by auto
  fix X assume "X ∈ sets (distr (count_space A) (count_space B) f)"
  then have X: "X ∈ sets (count_space B)" by auto
  moreover from X have "f -` X ∩ A = the_inv_into A f ` X"
    using f by (auto simp: bij_betw_def subset_image_iff image_iff the_inv_into_f_f intro: the_inv_into_f_f[symmetric])
  moreover have "inj_on (the_inv_into A f) B"
    using X f by (auto simp: bij_betw_def inj_on_the_inv_into)
  with X have "inj_on (the_inv_into A f) X"
    by (auto intro: inj_on_subset)
  ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"
    using f unfolding emeasure_distr[OF f' X]
    by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)
qed simp

lemma emeasure_count_space_finite[simp]:
  "X ⊆ A ==> finite X ==> emeasure (count_space A) X = of_nat (card X)"
  using emeasure_count_space[of X A] by simp

lemma emeasure_count_space_infinite[simp]:
  "X ⊆ A ==> infinite X ==> emeasure (count_space A) X = ∞"
  using assumes"contiX (pr (

lemma measure_count_space: "measure (count_space A) X = (if X ⊆ A then of_nat (card X) else 0)"
  by (cases "finite X")
     (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat
                 measure_zero_top measure_eq_emeasure_eq_ennreal)

lemma emeasure_count_space_eq_0:
  "emeasure (count_space A) X = 0 ⟷ (X ⊆ A ⟶ X = {})"
proof cases
  assume X: "X ⊆ A"
  then show ?thesis
  proof (intro iffI impI)
    assume "emeasure (count_space A) X = 0"
    with X show "X = {}"
      by (subst (asm) emeasure_count_space) (auto split: if_split_asm)
  qed simp
qed (simp add: emeasure_notin_sets)

lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
  unfolding null_sets_def by (auto simp: emeasure_count_space_eq_0)

lemma AE_count_space: "(AE x in count_space A. P x) ⟷ (∀x∈A. P x)"
  unfolding eventually_ae_filter by (auto simp: null_sets_count_space)

lemma sigma_finite_measure_count_space_countable:
  assumes A: "countable A"
  shows "sigma_finite_measure (count_space A)"
  proof qed (use A in ‹auto intro!: exI[of _ "(λa. {a}) ` A"]›)

lemma sigma_finite_measure_count_space:
  fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"
  using countableI_type sigma_finite_measure_count_space_countable by blast

lemma finite_measure_count_space:
  assumes [simp]: "finite A"
  shows "finite_measure (count_space A)"
  by rule simp

lemma sigma_finite_measure_count_space_finite:
  assumes A: "finiteA" sho "sigma_finite_measure count_space A)
  by (simp add: assms finite_measure.axioms(1) finite_measure_count_space)

subsection✐‹tag unimportant› ‹Measure restricted to space›

lemma emeasure_restrict_space:
  assumes "Ω ∩ space M ∈ sets M" "A ⊆(is"java.lang.StringIndexOutOfBoundsException: Range [26, 23) out of bounds for length 32
  shows "emeasure (restrict_space M Ω) A = emeasure M A"
proof (cases "A ∈ sets M")
  case True
  show ?thesis
  
    show "(∩) Ω ` sets M ⊆ Pow (Ω ∩ space M)" "A ∈ sets (restrict_space M Ω)"
      using ‹A ⊆ Ω› ‹A ∈ sets M› sets.space_closed by (auto simp: sets_restrict_space)
    show "positive (sets (restrict_space M Ω)) (emeasure M)"
      by (auto simp: positive_def)
    show "countably_additive (sets (restrict_space M Ω)) (emeasure M)"
    proof (rule countably_additiveI)
      fix A :: "nat → _" assume "range A ⊆
      with assms have "∧i. A i ∈ sets M" "∧i. A i ⊆ space M" "disjoint_family A"
        by(fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff
                      dest: sets.sets_into_space)+
      then show "(∑i. emeasure M (A i)) = emeasure M (∪i. A i)"
        by (simp add: image_subset_iff suminf_emeasure)
    qed
  qed
next
  case False
  with assms show ?thesis
    by (metis emeasure_notin_sets sets_restrict_space_iff)
qed

lemma measure_restrict_space:
  assumes "Ω qed
  shows "measure (restrict_space M Ω) A = measure M A"
  using emeasure_restrict_space[OF assms] by (simp add: measure_def)

lemma AE_restrict_space_iff:
  assumes "Ω ∩ space M ∈ sets M"
  shows "(AE x in restrict_space M Ω. P x) ⟷ (AE x in M. x ∈ Ω ⟶ P x)"
proof -
  have ex_cong: "∧P Q f. (∧x. P x ==> Q x) ==> (∧x. Q x ==> P (f x)) ==> (∃x. P x) ⟷ (∃x. Q x)"
    by auto
  have> \inter M\inter> X0"
    if "X ∈ sets M" "emeasure M X = 0" for X
    by (meson emeasure_eq_0 inf_le2 that)
  with assms show ?thesis
    unfolding eventually_ae_filter
    by (auto simp: space_restrict_space null_sets_def sets_restrict_space_iff
                       emeasure_restrict_space cong: conj_cong
             intro!: ex_cong[where f="λX. (Ω ∩ space M) ∩ X"])
qed

lemma restrict_restrict_space:
  assumes "A ∩ space M ∈ sets M" "B ∩ space M ∈ sets M"
  shows "restrict_space (restrict_space M A) B = restrict_space M (A ∩ B)" (is "?l = ?r")
proof (rule measure_eqI[symmetric])
  show "sets ?r = sets ?l"
    unfolding sets_restrict_space image_comp by (intro image_cong) auto
next
  fix X assume "X ∈ sets (restrict_space M (A ∩ B))"
  then obtain Y where "Y ∈ sets M" "X = Y ∩ A ∩ B"
    by (auto simp: sets_restrict_space)
  with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"
    by (subst (1 2) emeasure_restrict_space)
       (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)
qed

lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A ∩ B)"
proof (rule measure_eqI)
  show "sets (restrict_space (count_space B) A) = sets (count_space (A ∩ B))"
    by (subst sets_restrict_space) auto
  moreover fix X assume "X ∈ sets (restrict_space (count_space B) A)"
  ultimately have "X ⊆ A ∩ B" by auto
  then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A ∩ B)) X"
    by (cases "finite X") (auto simp: emeasure_restrict_space)
qed

lemma sigma_finite_measure_restrict_space:
  assumes "sigma_finite_measure M"
  and A: "A ∈ sets M"
  shows "sigma_finite_measure (restrict_space M A)"
proof -
  interpret sigma_finite_measure M by fact
  from sigma_finite_countable obtain C
    where C: "countable C" "C ⊆ sets M" "(∪C) = space M" "∀a∈C. emeasure M a ≠ ∞"
    by blast
  let ?C = "(∩) A ` C"
  from C have "countable ?C" "?C ⊆ sets (restrict_space M A)" "(∪?C) = space (restrict_space M A)"
    by(auto simp: sets_restrict_space space_restrict_space)
  moreover {
    fix a
    assume "a ∈ ?C"
    then obtain a' where "a = A ∩ a'" "a' ∈ C" ..
    then have "emeasure (restrict_space M A) a ≤ emeasure M a'"
      using A C by(auto simp: emeasure_restrict_space intro: emeasure_mono)
    also have "… < \∞" using C(4) ‹a' ∈ C› top.not_eq_extremum by auto
    finally have "emeasure (restrict_space M A) a ≠ ∞" by simp }
  ultimately show ?thesis
    by (meson sigma_finite_measure_def)
qed

lemma finite_measure_restrict_space:
  assumes "finite_measure M"
  and A: "A ∈ sets M"
  shows "finite_measure (restrict_space M A)"
  by (simad emeasure_r finite_meemea fin)

lemma restrict_distr:
  assumes [measurable]: "f ∈ measurable M N"
  assumes [simp]: "Ω ∩ space N ∈ sets N" and restrict: "f ∈ space M → Ω"
  shows "restrict_space (distr M N f) Ω = distr M (restrict_space N Ω) f"
  (is "?l = ?r")
proof (rule measure_eqI)
  fix A assume "A ∈ sets (restrict_space (distr M N f) Ω)"
  with restrict show "emeasure ?l A = emeasure ?r A"
    by (simp add: emeasure_distr emeasure_restrict_space measur unfolding homeomorphic_maps_def
        sets_restrict_space_iff)
qed (simp add: sets_restrict_space)

lemma measure_eqI_restrict_generator:
  assumes E: "Int_stable E" "E ⊆ Pow Ω" "∧X. X ∈ E ==> emeasure M X = emeasure N X"
  assumes sets_eq: "sets M = sets N" and Ω: "Ω ∈ sets M"
  assumes "sets (restrict_space M Ω) = sigma_sets Ω E"
  assumes "sets (restrict_space N Ω) = sigma_sets Ω E"
  assumes ae: "AE x in M. x ∈ Ω" "AE x in N. x ∈ Ω"
  assumes A: "countable A" "A ≠ {}" "A ⊆ E" "∪A = Ω" "∧a. a ∈ A ==> emeasure M a ≠ ∞"
  shows "M = N"
proof (rule measure_eqI)
  fix X assume X: "X ∈ sets M"
  then have "emeasure M X = emeasure (restrict_space M Ω) (X ∩ Ω)"
    using ae Ω by (auto simp: emeasure_restrict_space intro!: emeasure_eq_AE)
  also have "restrict_space M Ω = restrict_space N Ω"
  proof (rule measure_eqI_generator_eq)
    fix X assume "X ∈qed
    then show "emeasure (restrict_space M Ω) X = emeasure (restrict_space N Ω) X"
       java.lang.StringIndexOutOfBoundsException: Range [22, 23) out of bounds for length 22
      by (metis Pow_iff emeasure_restrict_space inf.orderE sets.sets_into_space sets_eq subsetD)
  next
    show "range (from_nat_into A) ⊆ E" "(∪i. from_nat_into A i) = Ω"
      using A by (auto cong del: SUP_cong_simp)
  next
    fix i
    have "emeasure (restrict_space M Ω) (from_nat_into A i) = emeasure M (from_nat_into A i)"
      using A Ω by (subst emeasure_restrict_space)
                   (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)
    with A show "emeasure (restrict_space M Ω) (from_nat_into A i) ≠ ∞"
      by (auto intro: from_nat_into)
  qed fact+
  also have "emeasure (restrict_space N Ω) (X ∩ Ω) = emeasure N X"
    using X ae Ω by (auto simp: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
  java.lang.StringIndexOutOfBoundsException: Range [23, 9) out of bounds for length 46
qed fact

subsection✐‹tag unimportant› ‹Null measure›

definition null_measure : using  byforce
"null_measure M = sigma (space M) (sets M)"

lemma space_null_measure[simp]: "space (null_measure M) = space M"
  by (simp add: null_measure_def)

lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"
  by (simp add: null_measure_def)

lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"
  by (cases "X ∈ sets M", rule emeasure_measure_of)
     _notin_setsnull_measure_def
           dest: sets.sets_into_space)

lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"
  by (intro measure_eq_emeasure_eq_ennreal) auto

lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"
java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 31

subsection ‹Scaling a measure›

definition✐‹tag important› scale_measure :: "ennreal → 'a measure → 'a measure" where
"scale_measure r M = measure_of (space M) (sets M) (λA. r * emeasure M A)"

lemma space_scale_measure: "space (scale_measure r M) = space M"
  by (simp add: scale_measure_def)

lemma sets_scale_measure [simp, java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 45
  by (simp add: scale_measure_def)

lemma emeasure_scale_measure [simp]:
   yauto simp: capped_metric_def capped_dist_def)
  (is "_ = ?μ A")
proof(cases "A ∈ sets M")
  case True
  show ?thesis unfolding scale_measure_def
  proof(rule emeasure_measure_of_sigma)
    show "sigma_algebra (space M) (sets M)" ..
    show "positive (sets M) ?μ" by (simp add: positive_def)
    show "countably_additive (sets M) ?μ"
    proof (rule countably_additiveI)
      fix A :: "nat → _"  assume *: "range A ⊆ sets M" "disjoint_family A"
      have "(∑i. ?μ (A i)) = r * (∑i. emeasure M (A i))"
        by simp
      also have "… = ?μ (∪i. A i)" using * by(simp add: suminf_emeasure)
      finally show "(∑i. ?μ (A i)) = ?μ (∪i. A i)" .
    qed
  qed(fact True)
qed(simp add: emeasure_notin_sets)

lemma scale_measure_1 [simp]: "scale_measure 1 M = M"
  by(rule measure_eqI) simp_all

lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"
  by(rule measure_eqI) simp_all

lemma measure_scale_measure [simp]: "0 ≤ r ==> measure (scale_measure r M) A = r * measure M A"
  using emeasure_scale_measure[of r M A]
    emeasure_eq_ennreal_measure[of M A]
    measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]
  by (cases "emeasure (scale_measure r M) A = top")
     (auto simp del: emeasure_scale_measure
           simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])

lemma scale_scale_measure [simp]:
  "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"
  by (rule measure_eqI) (simp_all lemma capped_metric_mdist:

lemma scale_null_measure [simp]: java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 123
  by (rule measure_eqI) simp_all


subsection ‹Complete lattice structure on measures›:dt(ape_erc\<elta 

lemma (in finite_measure) finite_measure_Diff':
  "A ∈
  using finite_measure_Diff[of A "A ∩ B"] by (auto simp: Diff_Int)

lemma (in finite_measure) finite_measure_Union':
 A∈ B ∈M\Longrightarrow measure M A\union>B) ==measure M A + measure M (B - A)"
  using finite_measure_Union[of A "B - A"] by auto

lemma finite_unsigned_Hahn_decomposition:
  assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"
  s y (metis Metric_spa.mball_subse
proof -
  interpret M: finite_measure M by fact
  interpret N: finite_measure N by fact

  define d where "d X = measure M X - measure N X" for X

  have [intro]: "bdd_above (d`sets M)"
    using sets.sets_into_space[of _ M]
    by (intro bdd_aboveI[where M="measure M (space M)"])
       (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)

  define γ where "γ = (SUP X∈:mball_of_def)
  have le_γ[intro]: "X ∈ sets M ==> d X ≤ γ" for X
    by (auto simp: γ_def intro!: cSUP_upper)

  have "∃X∈sets M. γ - 1 / 2^n < d X" for n
     γintro less_cSUP_iffiffD1) 
  then have "∃f. ∀n. f n ∈ sets M ∧ d (f n) > γ - 1 / 2^n"
    by metis
  then obtain E where [measurable]: "E n ∈ sets M" and E: "d (E n) > γ - 1 / 2^n" .capped_dist by blast
    by auto

  define F where "F m n = (if m ≤ n then ∩i∈{m..n}. E i else space M)" for m n

  have [measurable]: "m ≤ n ==> F m n ∈ sets M" for m n
    by (auto simp: F_def)

  have 1: "γ - 2 / 2 ^ m + 1 / 2 ^ n ≤(cases "\delta > 0")
    using that
  proof (induct rule: dec_induct)
    case base with E[of m] show ?case
      by (simp add: F_def field_simps)
  next
    case (step i)
    have F_Suc: "F m (Suc i) = F m i ∩ E (Suc i)"
      using ‹m ≤ i› by (auto simp: F_def le_Suc_eq)

    have "γ + (γ - 2 / 2^m + 1 / 2 ^ Suc i) ≤ (γ - 1 / 2^Suc i) + (γ - 2 / 2^m + 1 / 2^i)"
      by (simp add: field_simps)
    also have "… ≤ d (E (Suc i)) + d (F m i)"
      using E[of "Suc i"] by (intro add_mono step) auto
    also have "… = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"
      using ‹m ≤ i› by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')
    also have "… = d (E (Suc i) ∪ F m i) + d (F m (Suc i))"
      using ‹m ≤show "openin (mtopology_of (capped_metric δ m)) S = openin (mtopology_of m) S"
    also have " … ≤ γ + d (F m (Suc i))"
 using ‹m ≤ i› by auto
 finally show ?case
 by auto
 qed

 define F' where "F' m = (∩i∈{m..}. E i)" for m
 have F'_eq: "F' m = (∩i. F m (i + m))" for m
 by (fastforce simp: le_iff_add[of m] F'_def F_def)

 have [measurable]: "F' m ∈ sets M" for m
 by (auto simp: F'_def)

 have γ_le: "γ - 0 ≤ d (∪m. F' m)"
  (rule LIMSEQ_le)
 show "(λn. γ - 2 / 2 ^ n) <----p (intro exI conjI)
 by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto
 have "ins F'
 by (auto simp: incseq_def F'_def)
 >m F ) <----m. F' m)"
 unfolding d_def
 by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto

 have "γ - 2 / 2 ^ m + 0 ≤ d (F' m)" for m
 proof (rule LIMSEQ_le)
 have *: "decseq (λn. F m (n + m))"
 by (auto simp: decseq_def F_def)
 show "(λm n)) <---- d (F' m)"
 unfolding d_def F'_eq
 by (rule LIMSEQ_offset[where k=m])
 (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)
 show "(λ qed
 by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto
 show "∃N. ∀n≥N. γ - 2 / 2 ^ m + 1 / 2 ^ n ≤ d (F m n)"
 using 1[of m] by (intro exI[of _ m]) auto
 qed
 then show "∃N. ∀n≥N. γ - 2 / 2 ^ n ≤ d (F' n)"
 by auto
 qed

 show ?thesis
 proof (safe intro!: bexI[of _ "∪m. F' m"])
 fix X assume [measurable]: "X ∈ sets M" and X: "X ⊆ (∪m. F' m)"
 have "d (∪m. F' m) - d X = d ((∪m. F' m) - X)"
 using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)
 also have "… ≤ γ"
 by auto
 finally have "0 ≤ d X"
 using γ_le by auto
 then show "emeasure N X ≤ emeasure M X"
 by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)
 next
 fix X assume [measurable]: "X ∈ sets M" and X: "X ∩ (∪m. F' m) = {}"
 then have "d (∪m. F' m) + d X = d (X ∪ (∪m. F' m))"
 by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)
 also have "… ≤ γ"
 by auto
 finally have "d X ≤ 0"
 using γ_le by auto
 then show "emeasure M X ≤ emeasure N X"
 by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)
 qed auto
 

  unsigned_Hahn_decomposition:
 assumes [simp]: "sets N = sets M" and [measurable]: "A ∈ sets M"
 and [ by (simp add: Msimp add: Metric_space.mtopology_of cappd_dist capped_metric_def)
 shows "∃Y∈sets M. Y ⊆ A ∧ (∀X∈sets M. X ⊆ Y ⟶ N X ≤ M X) ∧ (∀X∈sets M. X ⊆ A ⟶ X ∩ Y = {} ⟶ M X ≤ N X)"
  -
 have "∃Y∈.Cauchy M (capped_dist δ) σ ⟷ MCauchy σ"
 (∀X∈sets (restrict_space M A). X ⊆ Y ⟶ (restrict_space N A) X ≤ (restrict_space M A) X) ∧
 (∀X∈sets (restrict_space M A). X ∩ Y = {} ⟶ (restrict_space M A) X ≤ (restrict_space N A) X)"
 proof (rule finite_unsigned_Hahn_decomposition)
 show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"
 by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)
 qed (simp add: sets_restrict_space)
 with assms show ?thesis
 by (metis Int_subset_iff emeasure_restrict_space sets.Int_space_eq2 sets_restrict_space_iff space_restrict_space)
 

 ✐‹.MCauchy σ"
 Define a lexicographical order on 🚫
 of the lexicographical order are point-wise ordered.
 ›

  measure :: (type) order_bot
 

  less_eq_measure :: "'a measure → 'a measure → bool" where
 "space M ⊂ space N ==> less_eq_measure M N"
  "space M = space N ==>lemma bounded_equivale:
  "space M = space N ==> sets M = sets N ==> emeasure M ≤ emeasure N ==> less_eq_measure M N"

  le_measure_iff:
 "M ≤"mto m' =mto m "<x y. mdist m' x y < <
 if sets M = sets N then emeasure M ≤ emeasure N else sets M ⊆ sets N else space M ⊆ space N)"
 by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)

 ✐tag important› less_measure ' bool"e
  "less_measure M N ⟷ (M ≤ N ∧ ¬ N ≤ M)"

definition✐‹tag important› bot_measure :: "'a measure" where
  "bot_measure = sigma {} {}"

lemma
  shows space_bot[simp]: "space bot auto java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 40
    and sets_bot[simp]: "sets bot = {{}}"
    and emeasure_bot[simp]: "emeasure bot X = 0"
  by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)

instance
proof standard
  show "bot ≤ a" for a :java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 47
    by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma   defines "D \<equiv> \<lambda>x y. if x \<in> S \<and> y \<in> S then SUP i\<in>I. mdist (m i
qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)

end

proposition le_measure: "sets M = sets N ==> M ≤ N ⟷ (∀A∈sets M. emeasure M A ≤ emeasure N A)"
  by (metis emeasure_neq_0_sets le_fun_def le_measure_iff order_class.order_eq_iff sets_eq_imp_space_eq)

definition✐‹tag important› sup_measure' :: "'a measure → 'a measure → 'a measure" where
"sup_measure' A B =
  measure_of (space A) (sets A)
    (λX. SUP Y∈sets A. emeasure A (X ∩ Y) + emeasure B (X ∩ - Y))"

lemma assumes [simp]: "sets B = sets A"
  shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"
    and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"
  using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)

lemma emeasure_sup_measure':
  assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X ∈ sets A"
  shows "emeasure (sup_measure' A B) X = (SUP Y∈sets A. emeasure A (X ∩ Y) + emeasure B (X ∩ - Y))"
    (is "_ = ?S X")
proof -
  note sets_eq_imp_space_eq[OF sets_eq, simp]
  show ?thesis
    using sup_measure'_def
  proof (rule emeasure_measure_of)
    let ?d = "λ
    show "countably_additive (sets (sup_measure' A B)) (λX. SUP Y ∈ sets A. emeasure A (X ∩ Y) + emeasure B (X ∩ - Y))"
    proof (rule countably_additiveI, goal_cases)
      case (1 X)
      then have [measurable]: "∧i. X i ∈ sets A" and "disjoint_family X"
        o
      have disjoint: "disjoint_family (λi. X i ∩ Y)" "disjoint_family (λi. X i - Y)" for Y
        using "1"(2) disjoint_family_subset by fastforce+
      have "(∑i. ?S (X i)) = (SUP Y∈sets A. ∑i. ?d (X i) Y)"
      proof (rule ennreal_suminf_SUP_eq_directed)
        fix J :: "nat set" and a b assume "finite J" and [measurable]: "a ∈ sets A" "b ∈ sets A"
        have "∃c∈sets A. c ⊆ X i ∧ (∀a∈sets A. ?d (X i) a ≤ ?d (X i) c)" for i
        proof cases
          assume "emeasure A ( i) = top \or> emeasureB (X i) = top"
          then show ?thesis
            by force
        next
          assume finite: "\n (emeasure A (X i) = top ∨ emeasure B (X i) = top)"
          then have "∃Y∈sets A. Y ⊆ X i ∧ (∀C∈sets A. C ⊆ Y ⟶ B C ≤ A C) ∧ (∀C∈sets A. C ⊆ X i ⟶ C ∩ Y = {} ⟶ A C ≤ B C)"
            using unsigned_Hahn_decomposition[of B A "X i"] by simp
          then obtain Y where [measurable]: "Y ∈ sets A" and [simp]: "Y ⊆ X i"
            and B_le_A: "∧C. C ∈ sets A ==> C ⊆ Y ==> B C ≤ A C"
            and A_le_B: "∧C. C ∈ sets A ==> C ⊆ X i ==> C ∩ Y = {} ==> A C ≤ B C"
            by auto

          show ?thesis
          proof (intro bexI ballI conjI)
            fix a assume [measurable]: "a ∈ sets A"
            have *: "(X i ∩ a ∩ Y ∪ (X i ∩ a - Y)) = X i ∩ a" "(X i - a) ∩ Y ∪ (X i - a - Y) = X i ∩ - a"
              for a Y by auto
            then have "?d (X i) a =
              (A (X i ∩ a ∩ Y) + A (X i ∩ a ∩ - Y)) + (B (X i ∩ - a ∩ Y) + B (X i ∩ - a ∩ - Y))"
              by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])
            also have "… ≤ (A (X i ∩ a ∩ Y) + B (X i ∩ a ∩ - Y)) + (A (X i ∩ - a ∩ Y) + B (X i ∩ - a ∩ - Y))"
              by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])
            also have "… ≤ (A (X i ∩ Y ∩ a) + A (X i ∩ Y ∩ - a)) + (B (X i ∩ - Y ∩ a) + B (X i ∩ - Y ∩ - a))"
              by (simp add: ac_simps)
            also have "… ≤ A (X i ∩ Y) + B (X i ∩ - Y)"
              by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)
            finally show "?d (X i) a ≤ ?d (X i) Y" .
          qed auto
        qed
        then obtain C where [measurable]: "C i ∈ sets A" and "C i ⊆ X i"
          and C: "∧a. a ∈ sets A ==> ?d (X i) a ≤ ?d (X i) (C i)" for i
          by metis
        have *: "X i ∩ (∪i. C i) = X i ∩ C i" for i
          using ‹disjoint_family X› ‹∧i. C i ⊆ X i›
          by (simp add: disjoint_family_on_def disjoint_iff_not_equal set_eq_iff) (metis subsetD)
        then have **: "X i ∩ - (∪i. C i) = X i ∩ - C i" for i by blast
        moreover have "(∪i. C i) ∈ sets A"
          by fastforce
        ultimately show "∃c∈sets A. ∀i∈J. ?d (X i) a ≤ ?d (X i) c ∧ ?d (X i) b ≤ ?d (X i) c"
          by (metis "*" C ‹a ∈ sets A› ‹b ∈ sets A›)
      qed
      also have "… = ?S (∪i. X i)"
      proof -
        have "∧Y. Y ∈ sets A ==> (∑i. emeasure A (X i ∩ Y) + emeasure B (X i ∩ -Y)) 
                              = emeasure A (∪i. X i ∩ Y) + emeasure B (∪i. X i ∩ -Y)"
          using disjoint
          by (auto simp flip: suminf_add Diff_eq simp add: image_subset_iff suminf_emeasure)
        then show ?thesis by force
      qed
      finally show "(∑i. ?S (X i)) = ?S (∪i. X i)" .
    qed
  qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)
qed

lemma le_emeasure_sup_measure'1:
  assumes "sets B = sets A" "X ∈ sets A" shows "emeasure A X ≤ emeasure (sup_measure' A B) X"
  by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)

lemma le_emeasure_sup_measure'2:
  assumes "sets B = sets A" "X ∈ sets A" shows "emeasure B X ≤ emeasure (sup_measure' A B) X"
  by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)

lemma emeasure_sup_measure'_le2:
  assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X ∈ sets C"
  assumes A: "∧Y. Y ⊆ X ==> Y ∈ sets A ==> emeasure A Y ≤ emeasure C Y"
  assumes B: "∧Y. Y ⊆ X ==> Y ∈ sets A ==> emeasurejava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 51
  shows "emeasure (sup_measure' A B) X ≤ emeasure C X"
proof (subst emeasure_sup_measure')
  ow<>
    unfolding ‹sets A = sets C›
  proof (intro SUP_least)
    fix Y assume [measurable]: "Y ∈ sets C"
    have [simp]: "X ∩ Y ∪ (X - Y) = X"
      by auto
    have "emeasure A (X ∩ Y) + emeasure B (X ∩ - Y) ≤ emeasure C (X ∩ Y) + emeasure C (X ∩ - Y)"
      by (intro add_mono A B) (auto simp: Diff_eq[symmetric])
    also have "… = emeasure C X"
      by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])
    finally show "emeasure A (X ∩ Y) + emeasure B (X ∩ - Y) ≤ emeasure C X" .
  qed
qed simp_all

definition<   finally
"sup_lexord A B k s c =
  (if k A = k B then c else
   if ¬
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0

lemma sup_lexord:
  "(k A < k B ==> P B) <: "metrizable_space (product_topology X I"
    (¬ k B <usinget
  by (auto simp: sup_lexord_def)

lemmas le_sup_lexord = sup_lexord[where P="λa. c ≤ a" for c]

lemma sup_lexord1: "k A = k B ==> sup_lexord A B k s c = c"
   djava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 31

lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"
  by (auto simp: sup_lexord_def)

lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) A ⊆ sets x) = (A ⊆ sets x)"
  using sets.sigma_sets_subset[of A

java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 220
  apply (simp add: le_measure_iff le_fun_def emeasure_sigma)
  by (metis order_refl sets_measure_of sigma_sets_le_sets_iff)

instantiation measure :: (type) semilattice_sup
begin

definition✐‹tag important› sup_measure :: "'a measure → 'a measure → 'a measure" where
  "sup_measure A B =
    sup_lexord A B space (sigma (space A ∪ space B) {})
      (sup_lexord A B sets (sigma (space A) (sets A ∪ sets B)) (sup_measure' A B))"

instance
proof
  fix x y z :: "'a measure"
  show "x ≤
    unfolding sup_measure_def
  proof (intro le_sup_lexord)
    assume "space x = space y"
    then have *: "sets x ∪ sets y ⊆ Pow (space x)"
      using sets.space_closed by auto
    assume "¬ sets y ⊆ sets x" "¬
    then have "sets x ⊂ sets x ∪ sets y"
      by auto
    also have "… ≤ sigma (space x) (sets x ∪ sets y)"
      by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)
    finally show "x ≤ sigma (space x) (sets x ∪ sets y)"
      java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 79
  next
    assume "¬ space y ⊆ spac define cm wre c <equiv> 🚫
    then show "x ≤ sigma (space x ∪ space y) {}"
      by (intro less_eq_measure.intros) auto
  next
    assume "sets x = sets y" then show "x ≤
      by (simp add: le_measure le_emeasure_sup_measure'1)
  qed (auto intro: less_eq_measure.intros)
  show "y ≤ sup x y"
    unfolding sup_measure_def
  proof (intro le_sup_lexord)
    assume **: "space x = space y"
    then have *: "sets x ∪ sets y ⊆ Pow (space y)"
      using sets.space_closed by auto
    assume "¬ sets y ⊆ sets x" "¬ sets x ⊆ sets y"
    then have "sets y ⊂ sets x ∪ sets y"
      by auto
    also have "… ≤ sigma (space y) (sets x ∪ sets y)"
      by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)
    finally show "y ≤ sigma (space x) (sets x ∪ sets y)"
      byusing\open>0 < <
  next
    assume "¬ space y ⊆ space x" "¬ space x ⊆ space y"
    then show "y ≤ sigma (space x ∪ space y) {}"
      by (intro less_eq_measure.intros) auto
  next
    assume "sets x = sets y" then show "y ≤ sup_measure' x y"
      by (simp add: le_measure le_emeasure_sup_measure'2)
  qed (auto intro: less_eq_measure.intros)
  show " ha \existsm. m ∈ m ≤ i = nk m"
    unfolding sup_measure_def
  proof (intro sup_lexord[where P="λx. x ≤ y"])
    assume "x ≤ y" "z ≤ y" and [simp]: "space x = space z" "sets x = sets z"
    from ‹x ≤ y› show "sup_measure' x z ≤ y"
    proof cases
      case 1 then show ?thesis
        by (intro less_eq_measure.intros(1)) simp
    next
      case 2 then show ?thesis
        by (intro less_eq_measure.intros(2)) simp_all
    next
      case 3 with ‹z ≤ y› ‹x ≤ y› show ?thesis
        by (auto simp: le_measure intro!: emeasure_sup_measure'_le2)
    qed
  next
    assume **: "x ≤ y" "z ≤ y" "space x = space z" "¬ sets z ⊆ sets x" "¬ sets x ⊆ sets z"
    then have *: "sets x ∪ sets z ⊆ Pow (space x)"
      using sets.space_closed by auto
    show "sigma (space x) (sets x ∪ sets z) ≤ y"
      unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)
  next
    assume "x ≤ y" "z ≤ y" "¬ space z ⊆ space x" "¬ space x ⊆ space z"
    then have "space x ⊆ space y" "space z ⊆ space y"
      by (auto simp: easure_iff _asmjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    then show "sigma (space x ∪ space z) {} ≤ y"
      by (simp add: sigma_le_iff)
  qed
qed

end

lemma space_empty_eq_bot: "space a = {} ⟷ a = bot"
  using space_empty[of a] by (auto intro!: measure_eqI)

lemma sets_eq_iff_bounded: "A ≤ B ==> B ≤ C ==> sets A = sets C ==> sets B = sets A"
  by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)

lemma sets_sup: "sets A = sets M ==> sets B = sets M ==> sets (sup A B) = sets M"
  by (auto simp: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)

lemma le_measureD1: "A ≤ B ==> space A ≤ space B"
  by (auto simp: le_measure_iff split: if_split_asm)

lemma le_measureD2: "A ≤ B ==> space A = space B ==> sets A ≤ sets B"
  by (auto simp: le_measure_iff split: if_split_asm)

lemma le_measureD3: "A ≤ B ==> sets A = sets B ==> emeasure A X ≤ emeasure B X"
  by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)

lemma UN_space_closed: "∪(sets ` S) ⊆ Pow (∪(space ` S))"
  using sets.space_closed by auto

definition✐‹tag important›
  Sup_lexordRightarrow>':atticeRightarrow>(a<tarrow 
where
  "Sup_lexord k c s A =
  (let U = (SUP a\<in>A. k a)
   in if \<exists>a\<in>A. k a = U then c {a\<in>A. k a = U} else s A)"

lemma Sup_lexord:
  "(\<And>a S. a \<in> A \<Longrightarrow> k a = (SUP a\<in>A. k a) \<Longrightarrow> S = {a'\<in>A. k a' = k a} \<Longrightarrow> P (c S)) \<Longrightarrow> ((\<And>a. a \<in> A \<Longrightarrow> k a \<noteq> (SUP a\<in>A. k a)) \<Longrightarrow> P (s A)) \<Longrightarrow>
    P (Sup_lexord k c s A)"
  by (auto simp: Sup_lexord_def Let_def)

lemma Sup_lexord1:
  assumes A: "A \<noteq> {}" and eq: "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" and P: "P (c A)"
  shows "P (Sup_lexord k c s A)"
proof -
  have "{a \<in> A. k a = \<Union>(k ` A)} = A" for a :: 'a
    by (metis (mono_tags, lifting) Collect_cong Collect_mem_eq eq)
  then show ?thesis
    using A P by (auto simp: Sup_lexord_def Let_def)
qed

instantiation measure :: (type) complete_lattice
begin

interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"
  by standard (auto intro!: antisym)

lemma sup_measure_F_mono':
  "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
proof (induction J rule: finite_induct)
  case empty then show ?case
    by simp
next
  case (insert i J)
  then show ?case
    by (metis finite.insertI sup.orderE sup_ge1 sup_ge2 sup_measure.union_diff2 sup_measure.union_inter)
qed

lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"
  using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)

lemma sets_sup_measure_F:
  "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> sets i = sets M) \<Longrightarrow> sets (sup_measure.F id I) = sets M"
  by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)

definition\<^marker>\<open>tag important\<close> Sup_measure' :: "'a measure set \<Rightarrow> 'a measure" where
"Sup_measure' M =
  measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)
    (\<lambda>X. (SUP P\<in>{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"

lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"
  unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])

lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"
  unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])

lemma sets_Sup_measure':

  shows "sets (Sup_measure' M) = sets A"
  using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)

lemma space_Sup_measure':
  assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"
  shows "space (Sup_measure' M) = space A"
THEN ]<> < {\cjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  by (simp add: Sup_measure'_def )

lemma emeasure_Sup_measure':
  assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"
  shows "emeasure (Sup_measure' M) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"
    (is "_ = ?S X")
  using Sup_measure'_def
proof (rule emeasure_measure_of)
  note sets_eq[THEN sets_eq_imp_space_eq, simp]
M) = space A"
    using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)
  let ?\<mu> = "sup_measure.F id"
  show"ountably_additive (sets (Sup_measure' M)) ?S"
  proof (rule countably_additiveI, goal_cases)
    case (1 F)
    then have **: "range F \<subseteq> sets A"
      by (auto simp: *)
    show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"
    proof (subst ennreal_suminf_SUP_eq_directed)
      fix i j and N :: "nat set" assume ij: "i \<in> {P. finite P \<and> P \<subseteq> M}" "j \<in> {P. finite P \<and> P \<subseteq> M}"
      have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>
        (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"
        using ij by (intro impI sets_sup_measure_F conjI) auto
      then have "?\<mu> j (F n) \<le> ?\<mu> (i \<union> j) (F n) \<and> ?\<mu> i (F n) \<le> ?\<mu> (i \<union> j) (F n)" for n
        using ij
        by (cases "i = {}"; cases "j = {}")
           (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F
                 simp del: id_apply)
      with ij show "\<exists>k\<in>{P. finite P \<and> P \<subseteq> M}. \<forall>n\<in>N. ?\<mu> i (F n) \<le> ?\<mu> k (F n) \<and> ?\<mu> j (F n) \<le> ?\<mu> k (F n)"
         "<exists>N.\forall> '. \le> n <>N\le ' <>  m ) \sigma>n i)(<igma>n' i)<\epsilon>"
    next
      show "(SUP P \<in> {P. finite P \<and> P \<subseteq> M}. \<Sum>n. ?\<mu> P (F n)) = (SUP P \<in> {P. finite P \<and> P \<subseteq> M}. ?\<mu> P (\<Union>(F ` UNIV)))"
      proof (intro arg_cong [of _ _ Sup] image_cong refl)
        fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"
        show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (\<Union>(F ` UNIV))"
        proof (cases "i = {}")
          case False 
          with i ** sets_eq show ?thesis
            by (smt (verit, best) "1"(2) Measure_Space.sets_sup_measure_F mem_Collect_eq subset_eq suminf_cong suminf_emeasure)
        qed simp
      qed
    qed
  qed
  show "positive (sets (Sup_measure' M)) ?S"
    by (auto simp: positive_def bot_ennreal[symmetric])
  show "X \<in> sets (Sup_measure' M)"
    using assms * by auto
qed (rule UN_space_closed)

definition\<^marker>\<open>tag important\<close> Sup_measure :: "'a measure set \<Rightarrow> 'a measure" where
"Sup_measure =
  Sup_lexord space
    (Sup_lexord sets Sup_measure'
      (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u)))
    (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"

definition\<^marker>\<open>tag important\<close> Inf_measure :: "'a measure set \<Rightarrow> 'a measure" where
  "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"

definition\<^marker>\<open>tag important\<close> inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure" where
  "inf_measure a b = Inf {a, b}"

definition\<^marker>\<open>tag important\<close> top_measure :: "'a measure" where
  "top_measure = Inf {}"

instance
proof
  note UN_space_closed [simp]
  show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A
    unfolding Sup_measure_def
  proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])
    assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"
    from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"
      by (intro less_eq_measure.intros) auto
  next
    fix a S assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"
      and neq: " ed auto
    have sp_a: "space a = (\<Union>(space ` S))"
      using \<open>a\<in>A\<close> by (auto simp: S)
    show "x \<le> sigma (\<Union>(space ` S)) (\<Union>(sets ` S))"
    proof cases
      assume [simp]: "space x = space a"
      have "sets x \<subset> (\<Union>a\<in>S. sets a)"
        using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)
      also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"
        by (rule sigma_sets_superset_generator)
      finally show ?thesis
        by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)
    next
      assume "space x \<noteq> space a"
moreoverhave "space x <le>space a"
        unfolding a using \<open>x\<in>A\<close> by auto
      ultimately show ?thesis
        by (intro less_eq_measure.intros) (simp add: less_le sp_a)
    qed
  next
    fix a b S S' assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"
      and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"
    then have "S' \<noteq> {}" "space b = space a"
      by auto
    have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"
      by (auto simp: S')
    note sets_eq[THEN sets_eq_imp_space_eq,         by metiseq eventually_at_top_linorder  topspace_mtopology
    have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"
      using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)
    show "x \<le> Sup_measure' S'"
    proof cases
      assume "x \<in> S"
      with \<open>b \<in> S\<close> have "space x = space b"
        by (simp add: S)
      show ?thesis
      proof cases
        assume "x \<in> S'"
        show "x \<le> Sup_measure' S'"
        proof (intro le_measure[THEN iffD2] ballI)
          show "sets x = sets (Sup_measure' S')"
            using \<open>x\<in>S'\<close> * by (simp add: S')
          fix X assume "X \<in> sets x"
          show "emeasure x X \<le> emeasure (Sup_measure' S') X"
          proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])
            show "emeasure x X \<le> (SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"
              using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto
          qed (use \<open>x\<in>S'\<close> S' in auto)
        qed
      next
        assume "x \<notin> S'"
        then have "sets x \<noteq> sets b"
          using \<open>x\<in>S\<close> by (auto simp: S')
        moreover have "sets x \<le> sets b"
          using \<open>x\<in>S\<close> unfolding b by auto
        ultimately show ?thesis
          using * \<open>x \<in> S\<close> by (simp add: le_measure_iff sets_le_imp_space_le)
      qed
    next
      assume "x \<notin> S"
      with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis
        by (simp add: "*" S SUP_upper2 a le_measure_iff)
    qed
  qed
  show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A
    unfolding Sup_measure_def
  proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])
    assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"
    show "sigma (\<Union>(space ` A)) {} \<le> x"
      using x[THEN le_measureD1] by (subst sigma_le_iff) auto
  next
    fix a S assume "a \<in> A" "space a = (\<Union>a\<in>Ajava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111
      "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"
    have`<>x
      using S le_measureD1[OF x] by auto
    moreover
    have "\<Union>(space ` S) = space a"
       <>a<>A<   
    then have "space x = \<Union>(space ` S) \<Longrightarrow> \<Union>(sets ` S) \<subseteq> sets x"
      using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)
    ultimately show "sigma (\<Union>(space ` S)) (\<Union>(sets ` S)) \<le> x"
      by (subst sigma_le_iff) simp_all
  next
    fix a b S S' assume        product_topology I)=trivial_topology\<r>
      and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> countable i \in  \not>(<a topspace(X i)\s>{}}\and>
    then have "S' \<noteq> {}" "space b = space a"
      by auto
    have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"
      by (auto simp: S')
    note sets_eq[THEN sets_eq_imp_space_eq, simp]
    have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"
      using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)
    show "Sup_measure' S' \<le> x"
    proof cases
      assume "space x = space a"
      show ?thesis
      proof cases
        assume **: "sets x = sets b"
        show ?thesis
        proof (intro le_measure[THEN iffD2] ballI)
          show ***: "sets (Sup_measure' S') = sets x"
            simpadd:***)
          fix X assume "X \<in> sets (Sup_measure' S')"
          show "emeasure (Sup_measure' S') X \<le> emeasure x X"
            **
          proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])
            show "(SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"
            proof (safe intro!: SUP_least)
              fix P assume P: "finite P" "P \<subseteq> S'"
              show "emeasure (sup_measure.F id P) X \<le> emeasure x X"
              proof cases
                assume "P = {}" then show ?thesis
                  by auto
              next
                P\noteq }
                from P have "finite P" "P \<subseteq> A"
                  unfolding S' S by (simp_all add: subset_eq)
                then have "sup_measure.F id P \<le> x"
                  by (induction P) (auto simp: x)
                moreover have "sets (sup_measure.F id P) = sets x"
                  using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>
                  by (intro sets_sup_measure_F) (auto simp: S')
                ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"
                  by (rule le_measureD3)
              qed
            qed
            show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" java.lang.StringIndexOutOfBoundsException: Range [0, 83) out of bounds for length 0
              unfolding * by (simp add: S')
          qed fact
        qed
      next
        assume "sets x \<noteq> sets b"
        moreover have "sets b \<le> sets x"
          unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto
        ultimately show "Sup_measure' S' \<le> x"
          using \<open>space x = space a\<close> \<open>b \<in> S\<close>
          by (intro less_eq_measure.intros(2)) (simp_all add: * S)
      qed
    next
      assume "space x \<noteq> space a"
      then have "space a < space x"
        by (simp add: \<open>a \<in> A\<close> le_measureD1 psubsetI x)
      then show "Sup_measure' S' \<le> x"
        by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)
    qed
  qed
  show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"
    by (auto intro!: antisym least simp: top_measure_def)
  show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A
    unfolding Inf_measure_def by (intro least) auto
  show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A
    unfolding Inf_measure_def by (intro upper) auto
  show "inf x y \<le> x" "inf x y \<le> y" "x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> inf y z" for x y z :: "'a measure"
    by (auto simp: inf_measure_def intro!: lower greatest)
qed

end

lemma sets_SUP:
  assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"
  shows "I \<noteq> {} \<Longrightarrow> sets (SUP i\<in>I. M i) = sets N"
  unfolding Sup_measure_def
  using assms assms[THEN sets_eq_imp_space_eq]
    sets_Sup_measure'[where A=N and M="M`I"]
  by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto

lemma emeasure_SUP:
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"
  shows "emeasure (SUP i\<in>I. M i) X = (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (SUP i\<in>J. M i) X)"
proof -
  interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"
    by standard (auto intro!: antisym)
  have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i\<in>J. i)" for J :: "'b measure set"
    by (induction J rule: finite_induct) auto
  have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x\<in>J. M x) = sets N" for J
    by (intro sets_SUP sets) (auto )
  from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto
  have "Sup_measure' (M`I) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> M`I}. sup_measure.F id P X)"
    using sets by (intro emeasure_Sup_measure') auto
  also have "Sup_measure' (M`I) = (SUP i\<in>I. M i)"
    unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]
    by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto
  also have "(SUP P\<in>{P. finite P \<and> P \<subseteq> M`I}. sup_measure.F id P X) =
    (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i\<in>J. M i) X)"
  proof (intro SUP_eq)
    fix J assume "J \<in> {P. finite P \<and> P \<subseteq> M`I}"
    then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = M`J'" and "finite J"
      using finite_subset_image[of J M I] by auto
    show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i\<in>j. M i) X"
    proof cases
      assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis
        by (auto simp: J)
    next
      assume "J' \<noteq> {}" with J J' show ?thesis
        using eq by auto
    qed
  next
    fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"
    show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> M`I}. (SUP i\<in>J. M i) X \<le> sup_measure.F id J' X"
      using J by (intro bexI[of _ "M`J"]) (auto simp: eq simp del: id_apply)
  qed
  finally show ?thesis .
qed

lemma emeasure_SUP_chain:
  assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"
  assumes ch: "Complete_Partial_Order.chain (\<le>) (M ` A)" and "A \<noteq> {}"
  shows "emeasure (SUP i\<in>A. M i) X = (SUP i\<in>A. emeasure (M i) X)"
proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])
  show "(SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (Sup (M ` J)) X) = (SUP i\<in>A. emeasure (M i) X)"
  proof (rule SUP_eq)
    fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"
    then have J: "Complete_Partial_Order.chain (\<le>) (M ` J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"
      using ch[THEN chain_subset, of "M`J"] by auto
    with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j\<in>J. M j) = M j"
      by auto
    with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (Sup (M ` J)) X \<le> emeasure (M j) X"
      by auto
  next
    fix j assume "j\<in>A" then show "\<exists>i\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (M j) X \<le> emeasure (Sup (M ` i)) X"
      by (intro bexI[of _ "{j}"]) auto
  qed
qed

subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Supremum of a set of \<open>\<sigma>\<close>-algebras\<close>

lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)" (is "?L=?R")
proof
  show "?L \<subseteq> ?R"
  proof -
    define A where "A \<equiv> {a \<in> M. space a = \<Union> (space ` M)}"
    have "\<exists>x\<in>M. a \<in> space x"
      if "a \<in> space (Sup_measure' {a \<in> A. sets a = \<Union> (sets ` A)})"
      for a
      by (metis (no_types, lifting) A_def UN_E mem_Collect_eq space_Sup_measure'2 that)
    then show ?thesis
      by (auto simp: A_def space_measure_of_conv Sup_measure_def Sup_lexord_def Let_def split: if_splits)
  qed
qed (use Sup_upper le_measureD1 in fastforce)


lemma sets_Sup_eq:
  assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"
  shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"
  unfolding Sup_measure_def
proof (rule Sup_lexord1 [OF \<open>M \<noteq> {}\<close>])
  show "sets (Sup_lexord sets Sup_measure' (\<lambda>U. sigma (\<Union> (space ` U)) (\<Union> (sets ` U))) M)
      = sigma_sets X (\<Union> (sets ` M))"
  apply (rule Sup_lexord)
  apply (metis (mono_tags, lifting) "*" empty_iff mem_Collect_eq sets.sigma_sets_eq sets_Sup_measure')
  by (metis "*" SUP_eq_const UN_space_closed assms(2) sets_measure_of)
qed (use * in blast)


lemma in_sets_Sup: "(\<And>m. m \<in> M \<Longrightarrow> space m = X) \<Longrightarrow> m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup M)"
  by (subst sets_Sup_eq[where X=X]) auto

lemma Sup_lexord_rel:
  assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"
    "R (c (A ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))})) (c (B ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))}))"
    "R (s (A`I)) (s (B`I))"
  shows "R (Sup_lexord k c s (A`I)) (Sup_lexord k c s (B`I))"
proof -
  have "A ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))} =  {a \<in> A ` I. k a = (SUP x\<in>I. k (B x))}"
    using assms(1) by auto
  moreover have "B ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))} =  {a \<in> B ` I. k a = (SUP x\<in>I. k (B x))}"
    by auto
  ultimately show ?thesis
    using assms by (auto simp: Sup_lexord_def Let_def image_comp)
qed

lemma sets_SUP_cong:
  assumes eq: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" 
  shows "sets (SUP i\<in>I. M i) = sets (SUP i\<in>I. N i)"
  unfolding Sup_measure_def
  using eq eq[THEN sets_eq_imp_space_eq]
  by (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"], simp_all add: sets_Sup_measure'2)

lemma sets_Sup_in_sets:
  assumes "M \<noteq> {}"
  assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"
  assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"
  shows "sets (Sup M) \<subseteq> sets N"
proof -
  have *: "\<Union>(space ` M) = space N"
    using assms by auto
  show ?thesis
    unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)
qed

lemma measurable_Sup1:
  assumes m: "m \<in> M" and f: "f \<in> measurable m N"
    and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
  shows "f \<in> measurable (Sup M) N"
proof -
  have "space (Sup M) = space m"
    using m by (auto simp: space_Sup_eq_UN dest: const_space)
  then show ?thesis
    using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])
qed

lemma measurable_Sup2:
  assumes M: "M \<noteq> {}"
  assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"
    and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
  shows "f \<in> measurable N (Sup M)"
proof -
  from M obtain m where "m \<in> M" by auto
  have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"
    by (intro const_space \<open>m \<in> M\<close>)
  have eq: "sets (sigma (\<Union> (space ` M)) (\<Union> (sets ` M))) = sets (Sup M)"
    by (metis M SUP_eq_const UN_space_closed sets_Sup_eq sets_measure_of space_eq)
  have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"
  proof (rule measurable_measure_of)
    show "f \<in> space N \<rightarrow> \<Union>(space ` M)"
      using measurable_space[OF f] M by auto
  qed (auto intro: measurable_sets f dest: sets.sets_into_space)
  also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"
    using eq measurable_cong_sets by blast
  finally show ?thesis .
qed

lemma measurable_SUP2:
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f \<in> measurable N (M i)) \<Longrightarrow>
    (\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> space (M i) = space (M j)) \<Longrightarrow> f \<in> measurable N (SUP i\<in>I. M i)"
  by (auto intro!: measurable_Sup2)

lemma sets_Sup_sigma:
  assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
  shows "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
proof -
  have "a \<in> sigma_sets \<Omega> (\<Union>M)"
    if "a \<in> sigma_sets \<Omega> m" "m \<in> M" for a m
    using that by induction (auto intro: sigma_sets.intros) 
  then have "sigma_sets \<Omega> (\<Union> (sigma_sets \<Omega> ` M)) = sigma_sets \<Omega> (\<Union> M)"
    by (smt (verit, best) UN_iff Union_iff sigma_sets.Basic sigma_sets_eqI)
  then show "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
    by (subst sets_Sup_eq) (fastforce simp: M Union_least)+
qed

lemma Sup_sigma:
  assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
  shows "(SUP m\<in>M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"
proof (intro antisym SUP_least)
  have *: "\<Union>M \<subseteq> Pow \<Omega>"
    using M by auto
  show "sigma \<Omega> (\<Union>M) \<le> (SUP m\<in>M. sigma \<Omega> m)"
  proof (intro less_eq_measure.intros(3))
    show "space (sigma \<Omega> (\<Union>M)) = space (SUP m\<in>M. sigma \<Omega> m)"
         "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m\<in>M. sigma \<Omega> m)"
      by (auto simp: M sets_Sup_sigma sets_eq_imp_space_eq space_measure_of_conv)
  qed (simp add: emeasure_sigma le_fun_def)
  fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"
    by (subst sigma_le_iff) (auto simp: M *)
qed

lemma SUP_sigma_sigma:
  "M \<noteq> {} \<Longrightarrow> (\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>) \<Longrightarrow> (SUP m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"
  using Sup_sigma[of "f`M" \<Omega>] by (auto simp: image_comp)

lemma sets_vimage_Sup_eq:
  assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"
  shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m \<in> M. vimage_algebra X f m)"
  (is "?L = ?R")
proof
  { fix m
    assume "m \<in> M"
    then have "f \<in> vimage_algebra X f m \<rightarrow>\<^sub>M m"
      by (simp add: assms measurable_vimage_algebra1)
    then have "f \<in> Sup (vimage_algebra X f ` M) \<rightarrow>\<^sub>M m"
      using \<open>m \<in> M\<close> by (force simp: intro: measurable_Sup1)
  }
  then show "?L \<subseteq> ?R"
    by (intro sets_image_in_sets measurable_Sup2) (simp_all add: space_Sup_eq_UN *)
  show "?R \<subseteq> ?L"
    apply (intro sets_Sup_in_sets)
    apply (force simp: * space_Sup_eq_UN sets_vimage_algebra2 intro: in_sets_Sup)+
    done
qed

lemma restrict_space_eq_vimage_algebra':
  "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"
  by (metis Int_assoc image_cong inf_le2 restrict_space_eq_vimage_algebra
      sets.Int_space_eq1 sets_restrict_space)

lemma sigma_le_sets:
  assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"
proof
  have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"
    by (auto intro: sigma_sets_top)
  moreover assume "sets (sigma X A) \<subseteq> sets N"
  ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"
    by auto
next
  assume *: "X \<in> sets N \<and> A \<subseteq> sets N"
  { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"
      by induction auto }
  then show "sets (sigma X A) \<subseteq> sets N"
    by auto
qed

lemma measurable_iff_sets:
  "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"
  (is "?L = ?R")
proof
  show "?L \<Longrightarrow> ?R"
    by (simp add: measurable_space sets_image_in_sets)
  show "?R \<Longrightarrow> ?L"
    by (simp add: in_vimage_algebra measurable_def subset_eq)
qed

lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"
  using sets.top[of "vimage_algebra X f M"] by simp

lemma measurable_mono:
  assumes N: "sets N' \<le> sets N" "space N = space N'"
  assumes M: "sets M \<le> sets M'" "space M = space M'"
  shows "measurable M N \<subseteq> measurable M' N'"
  unfolding measurable_def
proof safe
  fix f A 
  assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'" 
         "\<forall>y\<in>sets N. f -` y \<inter> space M \<in> sets M" 
  then show "f -` A \<inter> space M' \<in> sets M'"
    using assms by (metis subset_eq)
qed (use N M in auto)

lemma measurable_Sup_measurable:
  assumes f: "f \<in> space N \<rightarrow> A"
  shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"
proof (rule measurable_Sup2)
  have "f \<in> N \<rightarrow>\<^sub>M sigma A {}"
    by (meson empty_subsetI equals0D f measurable_measure_of)
  then show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"
    by fastforce
qed auto

lemma (in sigma_algebra) sigma_sets_subset':
  assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"
  shows "sigma_sets \<Omega>' a \<subseteq> M"
proof
  show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x
    using x by (induct rule: sigma_sets.induct) (use a in auto)
qed

lemma in_sets_SUP: "i \<in> I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> space (M i) = Y) \<Longrightarrow> X \<in> sets (M i) \<Longrightarrow> X \<in> sets (SUP i\<in>I. M i)"
  by (intro in_sets_Sup[where X=Y]) auto

lemma measurable_SUP1:
  "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<And>m n. m \<in> I \<Longrightarrow> n \<in> I \<Longrightarrow> space (M m) = space (M n)) \<Longrightarrow>
    f \<in> measurable (SUP i\<in>I. M i) N"
  by (auto intro: measurable_Sup1)

lemma sets_image_in_sets':
  assumes X: "X \<in> sets N"
  assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f -` A \<inter> X \<in> sets N"
  shows "sets (vimage_algebra X f M) \<subseteq> sets N"
  unfolding sets_vimage_algebra
  by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)

lemma mono_vimage_algebra:
  "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"
  by (simp add: in_vimage_algebra sets_image_in_sets' sets_vimage_algebra_space subsetD)

lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"
  unfolding sets_restrict_space by (rule image_mono)

lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"
  by (metis measure_eqI emeasure_empty sets_bot singletonD)

lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"
  using sets_eq_bot[of M] by blast


lemma (in finite_measure) countable_support:
  "countable {x. measure M {x} \<noteq> 0}"
proof cases
  assume "measure M (space M) = 0"
  then show ?thesis 
    by (metis (mono_tags, lifting) bounded_measure measure_le_0_iff Collect_empty_eq countable_empty) 
next
  let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
  assume "?M \<noteq> 0"
  then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
    using reals_Archimedean[of "?m x / ?M" for x]
    by (auto simp: field_simps not_le[symmetric] divide_le_0_iff measure_le_0_iff)
  have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
  proof (rule ccontr)
    fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
    then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
      by (metis infinite_arbitrarily_large)
    then have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
      by auto
    { fix x assume "x \<in> X"
      from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
      then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
    note singleton_sets = this
    have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
      using \<open>?M \<noteq> 0\<close>
      by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)
    also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
      by (rule sum_mono) fact
    also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
      using singleton_sets \<open>finite X\<close>
      by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
    finally have "?M < measure M (\<Union>x\<in>X. {x})" .
    moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
      using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
    ultimately show False by simp
  qed
  show ?thesis
    unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
qed

end