(* Title: HOL/Analysis/Measure_Space.thy Author: Lawrence C Paulson Author: Johannes Hölzl, TU München Author: Armin Heller, TU München *) section ‹Measure spaces and their properties› theory Measure_Space imports Measurable "HOL-Library.Extended_Nonnegative_Real" begin subsection%unimportant "Relate extended reals and the indicator function" lemma suminf_cmult_indicator: fixes f :: "nat ⇒ ennreal" assumes "disjoint_family A" "x ∈ A i" shows "(∑n. f n * indicator (A n) x) = f i" proof - have **: "⋀n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)" using ‹x ∈ A i› assms unfolding disjoint_family_on_def indicator_def by auto then have "⋀n. (∑j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ennreal)" by (auto simp: sum.If_cases) moreover have "(SUP n. if i < n then f i else 0) = (f i :: ennreal)" proof (rule SUP_eqI) fix y :: ennreal assume "⋀n. n ∈ UNIV ⟹ (if i < n then f i else 0) ≤ y" from this[of "Suc i"] show "f i ≤ y" by auto qed (insert assms, simp) ultimately show ?thesis using assms by (subst suminf_eq_SUP) (auto simp: indicator_def) qed lemma suminf_indicator: assumes "disjoint_family A" shows "(∑n. indicator (A n) x :: ennreal) = indicator (⋃i. A i) x" proof cases assume *: "x ∈ (⋃i. A i)" then obtain i where "x ∈ A i" by auto from suminf_cmult_indicator[OF assms(1), OF ‹x ∈ A i›, of "λk. 1"] show ?thesis using * by simp qed simp lemma sum_indicator_disjoint_family: fixes f :: "'d ⇒ 'e::semiring_1" assumes d: "disjoint_family_on A P" and "x ∈ A j" and "finite P" and "j ∈ P" shows "(∑i∈P. f i * indicator (A i) x) = f j" proof - have "P ∩ {i. x ∈ A i} = {j}" using d ‹x ∈ A j› ‹j ∈ P› unfolding disjoint_family_on_def by auto thus ?thesis unfolding indicator_def by (simp add: if_distrib sum.If_cases[OF ‹finite P›]) qed text ‹ The type for emeasure spaces is already defined in @{theory "HOL-Analysis.Sigma_Algebra"}, as it is also used to represent sigma algebras (with an arbitrary emeasure). › subsection%unimportant "Extend binary sets" lemma LIMSEQ_binaryset: assumes f: "f {} = 0" shows "(λn. ∑i<n. f (binaryset A B i)) ⇢ f A + f B" proof - have "(λn. ∑i < Suc (Suc n). f (binaryset A B i)) = (λn. f A + f B)" proof fix n show "(∑i < Suc (Suc n). f (binaryset A B i)) = f A + f B" by (induct n) (auto simp add: binaryset_def f) qed moreover have "... ⇢ f A + f B" by (rule tendsto_const) ultimately have "(λn. ∑i< Suc (Suc n). f (binaryset A B i)) ⇢ f A + f B" by metis hence "(λn. ∑i< n+2. f (binaryset A B i)) ⇢ f A + f B" by simp thus ?thesis by (rule LIMSEQ_offset [where k=2]) qed lemma binaryset_sums: assumes f: "f {} = 0" shows "(λn. f (binaryset A B n)) sums (f A + f B)" by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan) lemma suminf_binaryset_eq: fixes f :: "'a set ⇒ 'b::{comm_monoid_add, t2_space}" shows "f {} = 0 ⟹ (∑n. f (binaryset A B n)) = f A + f B" by (metis binaryset_sums sums_unique) subsection%unimportant ‹Properties of a premeasure @{term μ}› text ‹ The definitions for @{const positive} and @{const countably_additive} should be here, by they are necessary to define @{typ "'a measure"} in @{theory "HOL-Analysis.Sigma_Algebra"}. › definition subadditive where "subadditive M f ⟷ (∀x∈M. ∀y∈M. x ∩ y = {} ⟶ f (x ∪ y) ≤ f x + f y)" lemma subadditiveD: "subadditive M f ⟹ x ∩ y = {} ⟹ x ∈ M ⟹ y ∈ M ⟹ f (x ∪ y) ≤ f x + f y" by (auto simp add: subadditive_def) definition countably_subadditive where "countably_subadditive M f ⟷ (∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i. A i) ∈ M ⟶ (f (⋃i. A i) ≤ (∑i. f (A i))))" lemma (in ring_of_sets) countably_subadditive_subadditive: fixes f :: "'a set ⇒ ennreal" assumes f: "positive M f" and cs: "countably_subadditive M f" shows "subadditive M f" proof (auto simp add: subadditive_def) fix x y assume x: "x ∈ M" and y: "y ∈ M" and "x ∩ y = {}" hence "disjoint_family (binaryset x y)" by (auto simp add: disjoint_family_on_def binaryset_def) hence "range (binaryset x y) ⊆ M ⟶ (⋃i. binaryset x y i) ∈ M ⟶ f (⋃i. binaryset x y i) ≤ (∑ n. f (binaryset x y n))" using cs by (auto simp add: countably_subadditive_def) hence "{x,y,{}} ⊆ M ⟶ x ∪ y ∈ M ⟶ f (x ∪ y) ≤ (∑ n. f (binaryset x y n))" by (simp add: range_binaryset_eq UN_binaryset_eq) thus "f (x ∪ y) ≤ f x + f y" using f x y by (auto simp add: Un o_def suminf_binaryset_eq positive_def) qed definition additive where "additive M μ ⟷ (∀x∈M. ∀y∈M. x ∩ y = {} ⟶ μ (x ∪ y) = μ x + μ y)" definition increasing where "increasing M μ ⟷ (∀x∈M. ∀y∈M. x ⊆ y ⟶ μ x ≤ μ y)" lemma positiveD1: "positive M f ⟹ f {} = 0" by (auto simp: positive_def) lemma positiveD_empty: "positive M f ⟹ f {} = 0" by (auto simp add: positive_def) lemma additiveD: "additive M f ⟹ x ∩ y = {} ⟹ x ∈ M ⟹ y ∈ M ⟹ f (x ∪ y) = f x + f y" by (auto simp add: additive_def) lemma increasingD: "increasing M f ⟹ x ⊆ y ⟹ x∈M ⟹ y∈M ⟹ f x ≤ f y" by (auto simp add: increasing_def) lemma countably_additiveI[case_names countably]: "(⋀A. range A ⊆ M ⟹ disjoint_family A ⟹ (⋃i. A i) ∈ M ⟹ (∑i. f (A i)) = f (⋃i. A i)) ⟹ countably_additive M f" by (simp add: countably_additive_def) lemma (in ring_of_sets) disjointed_additive: assumes f: "positive M f" "additive M f" and A: "range A ⊆ M" "incseq A" shows "(∑i≤n. f (disjointed A i)) = f (A n)" proof (induct n) case (Suc n) then have "(∑i≤Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))" by simp also have "… = f (A n ∪ disjointed A (Suc n))" using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_mono) also have "A n ∪ disjointed A (Suc n) = A (Suc n)" using ‹incseq A› by (auto dest: incseq_SucD simp: disjointed_mono) finally show ?case . qed simp lemma (in ring_of_sets) additive_sum: fixes A:: "'i ⇒ 'a set" assumes f: "positive M f" and ad: "additive M f" and "finite S" and A: "A`S ⊆ M" and disj: "disjoint_family_on A S" shows "(∑i∈S. f (A i)) = f (⋃i∈S. A i)" using ‹finite S› disj A proof induct case empty show ?case using f by (simp add: positive_def) next case (insert s S) then have "A s ∩ (⋃i∈S. A i) = {}" by (auto simp add: disjoint_family_on_def neq_iff) moreover have "A s ∈ M" using insert by blast moreover have "(⋃i∈S. A i) ∈ M" using insert ‹finite S› by auto ultimately have "f (A s ∪ (⋃i∈S. A i)) = f (A s) + f(⋃i∈S. A i)" using ad UNION_in_sets A by (auto simp add: additive_def) with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A] by (auto simp add: additive_def subset_insertI) qed lemma (in ring_of_sets) additive_increasing: fixes f :: "'a set ⇒ ennreal" assumes posf: "positive M f" and addf: "additive M f" shows "increasing M f" proof (auto simp add: increasing_def) fix x y assume xy: "x ∈ M" "y ∈ M" "x ⊆ y" then have "y - x ∈ M" by auto then have "f x + 0 ≤ f x + f (y-x)" by (intro add_left_mono zero_le) also have "... = f (x ∪ (y-x))" using addf by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2)) also have "... = f y" by (metis Un_Diff_cancel Un_absorb1 xy(3)) finally show "f x ≤ f y" by simp qed lemma (in ring_of_sets) subadditive: fixes f :: "'a set ⇒ ennreal" assumes f: "positive M f" "additive M f" and A: "A`S ⊆ M" and S: "finite S" shows "f (⋃i∈S. A i) ≤ (∑i∈S. f (A i))" using S A proof (induct S) case empty thus ?case using f by (auto simp: positive_def) next case (insert x F) hence in_M: "A x ∈ M" "(⋃i∈F. A i) ∈ M" "(⋃i∈F. A i) - A x ∈ M" using A by force+ have subs: "(⋃i∈F. A i) - A x ⊆ (⋃i∈F. A i)" by auto have "(⋃i∈(insert x F). A i) = A x ∪ ((⋃i∈F. A i) - A x)" by auto hence "f (⋃i∈(insert x F). A i) = f (A x ∪ ((⋃i∈F. A i) - A x))" by simp also have "… = f (A x) + f ((⋃i∈F. A i) - A x)" using f(2) by (rule additiveD) (insert in_M, auto) also have "… ≤ f (A x) + f (⋃i∈F. A i)" using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono) also have "… ≤ f (A x) + (∑i∈F. f (A i))" using insert by (auto intro: add_left_mono) finally show "f (⋃i∈(insert x F). A i) ≤ (∑i∈(insert x F). f (A i))" using insert by simp qed lemma (in ring_of_sets) countably_additive_additive: fixes f :: "'a set ⇒ ennreal" assumes posf: "positive M f" and ca: "countably_additive M f" shows "additive M f" proof (auto simp add: additive_def) fix x y assume x: "x ∈ M" and y: "y ∈ M" and "x ∩ y = {}" hence "disjoint_family (binaryset x y)" by (auto simp add: disjoint_family_on_def binaryset_def) hence "range (binaryset x y) ⊆ M ⟶ (⋃i. binaryset x y i) ∈ M ⟶ f (⋃i. binaryset x y i) = (∑ n. f (binaryset x y n))" using ca by (simp add: countably_additive_def) hence "{x,y,{}} ⊆ M ⟶ x ∪ y ∈ M ⟶ f (x ∪ y) = (∑n. f (binaryset x y n))" by (simp add: range_binaryset_eq UN_binaryset_eq) thus "f (x ∪ y) = f x + f y" using posf x y by (auto simp add: Un suminf_binaryset_eq positive_def) qed lemma (in algebra) increasing_additive_bound: fixes A:: "nat ⇒ 'a set" and f :: "'a set ⇒ ennreal" assumes f: "positive M f" and ad: "additive M f" and inc: "increasing M f" and A: "range A ⊆ M" and disj: "disjoint_family A" shows "(∑i. f (A i)) ≤ f Ω" proof (safe intro!: suminf_le_const) fix N note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"] have "(∑i<N. f (A i)) = f (⋃i∈{..<N}. A i)" using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N) also have "... ≤ f Ω" using space_closed A by (intro increasingD[OF inc] finite_UN) auto finally show "(∑i<N. f (A i)) ≤ f Ω" by simp qed (insert f A, auto simp: positive_def) lemma (in ring_of_sets) countably_additiveI_finite: fixes μ :: "'a set ⇒ ennreal" assumes "finite Ω" "positive M μ" "additive M μ" shows "countably_additive M μ" proof (rule countably_additiveI) fix F :: "nat ⇒ 'a set" assume F: "range F ⊆ M" "(⋃i. F i) ∈ M" and disj: "disjoint_family F" have "∀i∈{i. F i ≠ {}}. ∃x. x ∈ F i" by auto from bchoice[OF this] obtain f where f: "⋀i. F i ≠ {} ⟹ f i ∈ F i" by auto have inj_f: "inj_on f {i. F i ≠ {}}" proof (rule inj_onI, simp) fix i j a b assume *: "f i = f j" "F i ≠ {}" "F j ≠ {}" then have "f i ∈ F i" "f j ∈ F j" using f by force+ with disj * show "i = j" by (auto simp: disjoint_family_on_def) qed have "finite (⋃i. F i)" by (metis F(2) assms(1) infinite_super sets_into_space) have F_subset: "{i. μ (F i) ≠ 0} ⊆ {i. F i ≠ {}}" by (auto simp: positiveD_empty[OF ‹positive M μ›]) moreover have fin_not_empty: "finite {i. F i ≠ {}}" proof (rule finite_imageD) from f have "f`{i. F i ≠ {}} ⊆ (⋃i. F i)" by auto then show "finite (f`{i. F i ≠ {}})" by (rule finite_subset) fact qed fact ultimately have fin_not_0: "finite {i. μ (F i) ≠ 0}" by (rule finite_subset) have disj_not_empty: "disjoint_family_on F {i. F i ≠ {}}" using disj by (auto simp: disjoint_family_on_def) from fin_not_0 have "(∑i. μ (F i)) = (∑i | μ (F i) ≠ 0. μ (F i))" by (rule suminf_finite) auto also have "… = (∑i | F i ≠ {}. μ (F i))" using fin_not_empty F_subset by (rule sum.mono_neutral_left) auto also have "… = μ (⋃i∈{i. F i ≠ {}}. F i)" using ‹positive M μ› ‹additive M μ› fin_not_empty disj_not_empty F by (intro additive_sum) auto also have "… = μ (⋃i. F i)" by (rule arg_cong[where f=μ]) auto finally show "(∑i. μ (F i)) = μ (⋃i. F i)" . qed lemma (in ring_of_sets) countably_additive_iff_continuous_from_below: fixes f :: "'a set ⇒ ennreal" assumes f: "positive M f" "additive M f" shows "countably_additive M f ⟷ (∀A. range A ⊆ M ⟶ incseq A ⟶ (⋃i. A i) ∈ M ⟶ (λi. f (A i)) ⇢ f (⋃i. A i))" unfolding countably_additive_def proof safe assume count_sum: "∀A. range A ⊆ M ⟶ disjoint_family A ⟶ UNION UNIV A ∈ M ⟶ (∑i. f (A i)) = f (UNION UNIV A)" fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ M" "incseq A" "(⋃i. A i) ∈ M" then have dA: "range (disjointed A) ⊆ M" by (auto simp: range_disjointed_sets) with count_sum[THEN spec, of "disjointed A"] A(3) have f_UN: "(∑i. f (disjointed A i)) = f (⋃i. A i)" by (auto simp: UN_disjointed_eq disjoint_family_disjointed) moreover have "(λn. (∑i<n. f (disjointed A i))) ⇢ (∑i. f (disjointed A i))" using f(1)[unfolded positive_def] dA by (auto intro!: summable_LIMSEQ) from LIMSEQ_Suc[OF this] have "(λn. (∑i≤n. f (disjointed A i))) ⇢ (∑i. f (disjointed A i))" unfolding lessThan_Suc_atMost . moreover have "⋀n. (∑i≤n. f (disjointed A i)) = f (A n)" using disjointed_additive[OF f A(1,2)] . ultimately show "(λi. f (A i)) ⇢ f (⋃i. A i)" by simp next assume cont: "∀A. range A ⊆ M ⟶ incseq A ⟶ (⋃i. A i) ∈ M ⟶ (λi. f (A i)) ⇢ f (⋃i. A i)" fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ M" "disjoint_family A" "(⋃i. A i) ∈ M" have *: "(⋃n. (⋃i<n. A i)) = (⋃i. A i)" by auto have "(λn. f (⋃i<n. A i)) ⇢ f (⋃i. A i)" proof (unfold *[symmetric], intro cont[rule_format]) show "range (λi. ⋃i<i. A i) ⊆ M" "(⋃i. ⋃i<i. A i) ∈ M" using A * by auto qed (force intro!: incseq_SucI) moreover have "⋀n. f (⋃i<n. A i) = (∑i<n. f (A i))" using A by (intro additive_sum[OF f, of _ A, symmetric]) (auto intro: disjoint_family_on_mono[where B=UNIV]) ultimately have "(λi. f (A i)) sums f (⋃i. A i)" unfolding sums_def by simp from sums_unique[OF this] show "(∑i. f (A i)) = f (⋃i. A i)" by simp qed lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous: fixes f :: "'a set ⇒ ennreal" assumes f: "positive M f" "additive M f" shows "(∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) ∈ M ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ f (⋂i. A i)) ⟷ (∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) = {} ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ 0)" proof safe assume cont: "(∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) ∈ M ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ f (⋂i. A i))" fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ M" "decseq A" "(⋂i. A i) = {}" "∀i. f (A i) ≠ ∞" with cont[THEN spec, of A] show "(λi. f (A i)) ⇢ 0" using ‹positive M f›[unfolded positive_def] by auto next assume cont: "∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) = {} ⟶ (∀i. f (A i) ≠ ∞) ⟶ (λi. f (A i)) ⇢ 0" fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ M" "decseq A" "(⋂i. A i) ∈ M" "∀i. f (A i) ≠ ∞" have f_mono: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ⊆ b ⟹ f a ≤ f b" using additive_increasing[OF f] unfolding increasing_def by simp have decseq_fA: "decseq (λi. f (A i))" using A by (auto simp: decseq_def intro!: f_mono) have decseq: "decseq (λi. A i - (⋂i. A i))" using A by (auto simp: decseq_def) then have decseq_f: "decseq (λi. f (A i - (⋂i. A i)))" using A unfolding decseq_def by (auto intro!: f_mono Diff) have "f (⋂x. A x) ≤ f (A 0)" using A by (auto intro!: f_mono) then have f_Int_fin: "f (⋂x. A x) ≠ ∞" using A by (auto simp: top_unique) { fix i have "f (A i - (⋂i. A i)) ≤ f (A i)" using A by (auto intro!: f_mono) then have "f (A i - (⋂i. A i)) ≠ ∞" using A by (auto simp: top_unique) } note f_fin = this have "(λi. f (A i - (⋂i. A i))) ⇢ 0" proof (intro cont[rule_format, OF _ decseq _ f_fin]) show "range (λi. A i - (⋂i. A i)) ⊆ M" "(⋂i. A i - (⋂i. A i)) = {}" using A by auto qed from INF_Lim[OF decseq_f this] have "(INF n. f (A n - (⋂i. A i))) = 0" . moreover have "(INF n. f (⋂i. A i)) = f (⋂i. A i)" by auto ultimately have "(INF n. f (A n - (⋂i. A i)) + f (⋂i. A i)) = 0 + f (⋂i. A i)" using A(4) f_fin f_Int_fin by (subst INF_ennreal_add_const) (auto simp: decseq_f) moreover { fix n have "f (A n - (⋂i. A i)) + f (⋂i. A i) = f ((A n - (⋂i. A i)) ∪ (⋂i. A i))" using A by (subst f(2)[THEN additiveD]) auto also have "(A n - (⋂i. A i)) ∪ (⋂i. A i) = A n" by auto finally have "f (A n - (⋂i. A i)) + f (⋂i. A i) = f (A n)" . } ultimately have "(INF n. f (A n)) = f (⋂i. A i)" by simp with LIMSEQ_INF[OF decseq_fA] show "(λi. f (A i)) ⇢ f (⋂i. A i)" by simp qed lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below: fixes f :: "'a set ⇒ ennreal" assumes f: "positive M f" "additive M f" "∀A∈M. f A ≠ ∞" assumes cont: "∀A. range A ⊆ M ⟶ decseq A ⟶ (⋂i. A i) = {} ⟶ (λi. f (A i)) ⇢ 0" assumes A: "range A ⊆ M" "incseq A" "(⋃i. A i) ∈ M" shows "(λi. f (A i)) ⇢ f (⋃i. A i)" proof - from A have "(λi. f ((⋃i. A i) - A i)) ⇢ 0" by (intro cont[rule_format]) (auto simp: decseq_def incseq_def) moreover { fix i have "f ((⋃i. A i) - A i ∪ A i) = f ((⋃i. A i) - A i) + f (A i)" using A by (intro f(2)[THEN additiveD]) auto also have "((⋃i. A i) - A i) ∪ A i = (⋃i. A i)" by auto finally have "f ((⋃i. A i) - A i) = f (⋃i. A i) - f (A i)" using f(3)[rule_format, of "A i"] A by (auto simp: ennreal_add_diff_cancel subset_eq) } moreover have "∀⇩_{F}i in sequentially. f (A i) ≤ f (⋃i. A i)" using increasingD[OF additive_increasing[OF f(1, 2)], of "A _" "⋃i. A i"] A by (auto intro!: always_eventually simp: subset_eq) ultimately show "(λi. f (A i)) ⇢ f (⋃i. A i)" by (auto intro: ennreal_tendsto_const_minus) qed lemma (in ring_of_sets) empty_continuous_imp_countably_additive: fixes f :: "'a set ⇒ ennreal" assumes f: "positive M f" "additive M f" and fin: "∀A∈M. f A ≠ ∞" assumes cont: "⋀A. range A ⊆ M ⟹ decseq A ⟹ (⋂i. A i) = {} ⟹ (λi. f (A i)) ⇢ 0" shows "countably_additive M f" using countably_additive_iff_continuous_from_below[OF f] using empty_continuous_imp_continuous_from_below[OF f fin] cont by blast subsection%unimportant ‹Properties of @{const emeasure}› lemma emeasure_positive: "positive (sets M) (emeasure M)" by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) lemma emeasure_empty[simp, intro]: "emeasure M {} = 0" using emeasure_positive[of M] by (simp add: positive_def) lemma emeasure_single_in_space: "emeasure M {x} ≠ 0 ⟹ x ∈ space M" using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space zero_less_iff_neq_zero[THEN iffD2]) lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)" by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) lemma suminf_emeasure: "range A ⊆ sets M ⟹ disjoint_family A ⟹ (∑i. emeasure M (A i)) = emeasure M (⋃i. A i)" using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M] by (simp add: countably_additive_def) lemma sums_emeasure: "disjoint_family F ⟹ (⋀i. F i ∈ sets M) ⟹ (λi. emeasure M (F i)) sums emeasure M (⋃i. F i)" unfolding sums_iff by (intro conjI suminf_emeasure) auto lemma emeasure_additive: "additive (sets M) (emeasure M)" by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive) lemma plus_emeasure: "a ∈ sets M ⟹ b ∈ sets M ⟹ a ∩ b = {} ⟹ emeasure M a + emeasure M b = emeasure M (a ∪ b)" using additiveD[OF emeasure_additive] .. lemma emeasure_Union: "A ∈ sets M ⟹ B ∈ sets M ⟹ emeasure M (A ∪ B) = emeasure M A + emeasure M (B - A)" using plus_emeasure[of A M "B - A"] by auto lemma sum_emeasure: "F`I ⊆ sets M ⟹ disjoint_family_on F I ⟹ finite I ⟹ (∑i∈I. emeasure M (F i)) = emeasure M (⋃i∈I. F i)" by (metis sets.additive_sum emeasure_positive emeasure_additive) lemma emeasure_mono: "a ⊆ b ⟹ b ∈ sets M ⟹ emeasure M a ≤ emeasure M b" by (metis zero_le sets.additive_increasing emeasure_additive emeasure_notin_sets emeasure_positive increasingD) lemma emeasure_space: "emeasure M A ≤ emeasure M (space M)" by (metis emeasure_mono emeasure_notin_sets sets.sets_into_space sets.top zero_le) lemma emeasure_Diff: assumes finite: "emeasure M B ≠ ∞" and [measurable]: "A ∈ sets M" "B ∈ sets M" and "B ⊆ A" shows "emeasure M (A - B) = emeasure M A - emeasure M B" proof - have "(A - B) ∪ B = A" using ‹B ⊆ A› by auto then have "emeasure M A = emeasure M ((A - B) ∪ B)" by simp also have "… = emeasure M (A - B) + emeasure M B" by (subst plus_emeasure[symmetric]) auto finally show "emeasure M (A - B) = emeasure M A - emeasure M B" using finite by simp qed lemma emeasure_compl: "s ∈ sets M ⟹ emeasure M s ≠ ∞ ⟹ emeasure M (space M - s) = emeasure M (space M) - emeasure M s" by (rule emeasure_Diff) (auto dest: sets.sets_into_space) lemma Lim_emeasure_incseq: "range A ⊆ sets M ⟹ incseq A ⟹ (λi. (emeasure M (A i))) ⇢ emeasure M (⋃i. A i)" using emeasure_countably_additive by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive emeasure_additive) lemma incseq_emeasure: assumes "range B ⊆ sets M" "incseq B" shows "incseq (λi. emeasure M (B i))" using assms by (auto simp: incseq_def intro!: emeasure_mono) lemma SUP_emeasure_incseq: assumes A: "range A ⊆ sets M" "incseq A" shows "(SUP n. emeasure M (A n)) = emeasure M (⋃i. A i)" using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A] by (simp add: LIMSEQ_unique) lemma decseq_emeasure: assumes "range B ⊆ sets M" "decseq B" shows "decseq (λi. emeasure M (B i))" using assms by (auto simp: decseq_def intro!: emeasure_mono) lemma INF_emeasure_decseq: assumes A: "range A ⊆ sets M" and "decseq A" and finite: "⋀i. emeasure M (A i) ≠ ∞" shows "(INF n. emeasure M (A n)) = emeasure M (⋂i. A i)" proof - have le_MI: "emeasure M (⋂i. A i) ≤ emeasure M (A 0)" using A by (auto intro!: emeasure_mono) hence *: "emeasure M (⋂i. A i) ≠ ∞" using finite[of 0] by (auto simp: top_unique) have "emeasure M (A 0) - (INF n. emeasure M (A n)) = (SUP n. emeasure M (A 0) - emeasure M (A n))" by (simp add: ennreal_INF_const_minus) also have "… = (SUP n. emeasure M (A 0 - A n))" using A finite ‹decseq A›[unfolded decseq_def] by (subst emeasure_Diff) auto also have "… = emeasure M (⋃i. A 0 - A i)" proof (rule SUP_emeasure_incseq) show "range (λn. A 0 - A n) ⊆ sets M" using A by auto show "incseq (λn. A 0 - A n)" using ‹decseq A› by (auto simp add: incseq_def decseq_def) qed also have "… = emeasure M (A 0) - emeasure M (⋂i. A i)" using A finite * by (simp, subst emeasure_Diff) auto finally show ?thesis by (rule ennreal_minus_cancel[rotated 3]) (insert finite A, auto intro: INF_lower emeasure_mono) qed lemma INF_emeasure_decseq': assumes A: "⋀i. A i ∈ sets M" and "decseq A" and finite: "∃i. emeasure M (A i) ≠ ∞" shows "(INF n. emeasure M (A n)) = emeasure M (⋂i. A i)" proof - from finite obtain i where i: "emeasure M (A i) < ∞" by (auto simp: less_top) have fin: "i ≤ j ⟹ emeasure M (A j) < ∞" for j by (rule le_less_trans[OF emeasure_mono i]) (auto intro!: decseqD[OF ‹decseq A›] A) have "(INF n. emeasure M (A n)) = (INF n. emeasure M (A (n + i)))" proof (rule INF_eq) show "∃j∈UNIV. emeasure M (A (j + i)) ≤ emeasure M (A i')" for i' by (intro bexI[of _ i'] emeasure_mono decseqD[OF ‹decseq A›] A) auto qed auto also have "… = emeasure M (INF n. (A (n + i)))" using A ‹decseq A› fin by (intro INF_emeasure_decseq) (auto simp: decseq_def less_top) also have "(INF n. (A (n + i))) = (INF n. A n)" by (meson INF_eq UNIV_I assms(2) decseqD le_add1) finally show ?thesis . qed lemma emeasure_INT_decseq_subset: fixes F :: "nat ⇒ 'a set" assumes I: "I ≠ {}" and F: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ i ≤ j ⟹ F j ⊆ F i" assumes F_sets[measurable]: "⋀i. i ∈ I ⟹ F i ∈ sets M" and fin: "⋀i. i ∈ I ⟹ emeasure M (F i) ≠ ∞" shows "emeasure M (⋂i∈I. F i) = (INF i:I. emeasure M (F i))" proof cases assume "finite I" have "(⋂i∈I. F i) = F (Max I)" using I ‹finite I› by (intro antisym INF_lower INF_greatest F) auto moreover have "(INF i:I. emeasure M (F i)) = emeasure M (F (Max I))" using I ‹finite I› by (intro antisym INF_lower INF_greatest F emeasure_mono) auto ultimately show ?thesis by simp next assume "infinite I" define L where "L n = (LEAST i. i ∈ I ∧ i ≥ n)" for n have L: "L n ∈ I ∧ n ≤ L n" for n unfolding L_def proof (rule LeastI_ex) show "∃x. x ∈ I ∧ n ≤ x" using ‹infinite I› finite_subset[of I "{..< n}"] by (rule_tac ccontr) (auto simp: not_le) qed have L_eq[simp]: "i ∈ I ⟹ L i = i" for i unfolding L_def by (intro Least_equality) auto have L_mono: "i ≤ j ⟹ L i ≤ L j" for i j using L[of j] unfolding L_def by (intro Least_le) (auto simp: L_def) have "emeasure M (⋂i. F (L i)) = (INF i. emeasure M (F (L i)))" proof (intro INF_emeasure_decseq[symmetric]) show "decseq (λi. F (L i))" using L by (intro antimonoI F L_mono) auto qed (insert L fin, auto) also have "… = (INF i:I. emeasure M (F i))" proof (intro antisym INF_greatest) show "i ∈ I ⟹ (INF i. emeasure M (F (L i))) ≤ emeasure M (F i)" for i by (intro INF_lower2[of i]) auto qed (insert L, auto intro: INF_lower) also have "(⋂i. F (L i)) = (⋂i∈I. F i)" proof (intro antisym INF_greatest) show "i ∈ I ⟹ (⋂i. F (L i)) ⊆ F i" for i by (intro INF_lower2[of i]) auto qed (insert L, auto) finally show ?thesis . qed lemma Lim_emeasure_decseq: assumes A: "range A ⊆ sets M" "decseq A" and fin: "⋀i. emeasure M (A i) ≠ ∞" shows "(λi. emeasure M (A i)) ⇢ emeasure M (⋂i. A i)" using LIMSEQ_INF[OF decseq_emeasure, OF A] using INF_emeasure_decseq[OF A fin] by simp lemma emeasure_lfp'[consumes 1, case_names cont measurable]: assumes "P M" assumes cont: "sup_continuous F" assumes *: "⋀M A. P M ⟹ (⋀N. P N ⟹ Measurable.pred N A) ⟹ Measurable.pred M (F A)" shows "emeasure M {x∈space M. lfp F x} = (SUP i. emeasure M {x∈space M. (F ^^ i) (λx. False) x})" proof - have "emeasure M {x∈space M. lfp F x} = emeasure M (⋃i. {x∈space M. (F ^^ i) (λx. False) x})" using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure]) moreover { fix i from ‹P M› have "{x∈space M. (F ^^ i) (λx. False) x} ∈ sets M" by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) } moreover have "incseq (λi. {x∈space M. (F ^^ i) (λx. False) x})" proof (rule incseq_SucI) fix i have "(F ^^ i) (λx. False) ≤ (F ^^ (Suc i)) (λx. False)" proof (induct i) case 0 show ?case by (simp add: le_fun_def) next case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto qed then show "{x ∈ space M. (F ^^ i) (λx. False) x} ⊆ {x ∈ space M. (F ^^ Suc i) (λx. False) x}" by auto qed ultimately show ?thesis by (subst SUP_emeasure_incseq) auto qed lemma emeasure_lfp: assumes [simp]: "⋀s. sets (M s) = sets N" assumes cont: "sup_continuous F" "sup_continuous f" assumes meas: "⋀P. Measurable.pred N P ⟹ Measurable.pred N (F P)" assumes iter: "⋀P s. Measurable.pred N P ⟹ P ≤ lfp F ⟹ emeasure (M s) {x∈space N. F P x} = f (λs. emeasure (M s) {x∈space N. P x}) s" shows "emeasure (M s) {x∈space N. lfp F x} = lfp f s" proof (subst lfp_transfer_bounded[where α="λF s. emeasure (M s) {x∈space N. F x}" and g=f and f=F and P="Measurable.pred N", symmetric]) fix C assume "incseq C" "⋀i. Measurable.pred N (C i)" then show "(λs. emeasure (M s) {x ∈ space N. (SUP i. C i) x}) = (SUP i. (λs. emeasure (M s) {x ∈ space N. C i x}))" unfolding SUP_apply[abs_def] by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure]) qed (auto simp add: iter le_fun_def SUP_apply[abs_def] intro!: meas cont) lemma emeasure_subadditive_finite: "finite I ⟹ A ` I ⊆ sets M ⟹ emeasure M (⋃i∈I. A i) ≤ (∑i∈I. emeasure M (A i))" by (rule sets.subadditive[OF emeasure_positive emeasure_additive]) auto lemma emeasure_subadditive: "A ∈ sets M ⟹ B ∈ sets M ⟹ emeasure M (A ∪ B) ≤ emeasure M A + emeasure M B" using emeasure_subadditive_finite[of "{True, False}" "λTrue ⇒ A | False ⇒ B" M] by simp lemma emeasure_subadditive_countably: assumes "range f ⊆ sets M" shows "emeasure M (⋃i. f i) ≤ (∑i. emeasure M (f i))" proof - have "emeasure M (⋃i. f i) = emeasure M (⋃i. disjointed f i)" unfolding UN_disjointed_eq .. also have "… = (∑i. emeasure M (disjointed f i))" using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"] by (simp add: disjoint_family_disjointed comp_def) also have "… ≤ (∑i. emeasure M (f i))" using sets.range_disjointed_sets[OF assms] assms by (auto intro!: suminf_le emeasure_mono disjointed_subset) finally show ?thesis . qed lemma emeasure_insert: assumes sets: "{x} ∈ sets M" "A ∈ sets M" and "x ∉ A" shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A" proof - have "{x} ∩ A = {}" using ‹x ∉ A› by auto from plus_emeasure[OF sets this] show ?thesis by simp qed lemma emeasure_insert_ne: "A ≠ {} ⟹ {x} ∈ sets M ⟹ A ∈ sets M ⟹ x ∉ A ⟹ emeasure M (insert x A) = emeasure M {x} + emeasure M A" by (rule emeasure_insert) lemma emeasure_eq_sum_singleton: assumes "finite S" "⋀x. x ∈ S ⟹ {x} ∈ sets M" shows "emeasure M S = (∑x∈S. emeasure M {x})" using sum_emeasure[of "λx. {x}" S M] assms by (auto simp: disjoint_family_on_def subset_eq) lemma sum_emeasure_cover: assumes "finite S" and "A ∈ sets M" and br_in_M: "B ` S ⊆ sets M" assumes A: "A ⊆ (⋃i∈S. B i)" assumes disj: "disjoint_family_on B S" shows "emeasure M A = (∑i∈S. emeasure M (A ∩ (B i)))" proof - have "(∑i∈S. emeasure M (A ∩ (B i))) = emeasure M (⋃i∈S. A ∩ (B i))" proof (rule sum_emeasure) show "disjoint_family_on (λi. A ∩ B i) S" using ‹disjoint_family_on B S› unfolding disjoint_family_on_def by auto qed (insert assms, auto) also have "(⋃i∈S. A ∩ (B i)) = A" using A by auto finally show ?thesis by simp qed lemma emeasure_eq_0: "N ∈ sets M ⟹ emeasure M N = 0 ⟹ K ⊆ N ⟹ emeasure M K = 0" by (metis emeasure_mono order_eq_iff zero_le) lemma emeasure_UN_eq_0: assumes "⋀i::nat. emeasure M (N i) = 0" and "range N ⊆ sets M" shows "emeasure M (⋃i. N i) = 0" proof - have "emeasure M (⋃i. N i) ≤ 0" using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp then show ?thesis by (auto intro: antisym zero_le) qed lemma measure_eqI_finite: assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A" assumes eq: "⋀a. a ∈ A ⟹ emeasure M {a} = emeasure N {a}" shows "M = N" proof (rule measure_eqI) fix X assume "X ∈ sets M" then have X: "X ⊆ A" by auto then have "emeasure M X = (∑a∈X. emeasure M {a})" using ‹finite A› by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset) also have "… = (∑a∈X. emeasure N {a})" using X eq by (auto intro!: sum.cong) also have "… = emeasure N X" using X ‹finite A› by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset) finally show "emeasure M X = emeasure N X" . qed simp lemma measure_eqI_generator_eq: fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat ⇒ 'a set" assumes "Int_stable E" "E ⊆ Pow Ω" and eq: "⋀X. X ∈ E ⟹ emeasure M X = emeasure N X" and M: "sets M = sigma_sets Ω E" and N: "sets N = sigma_sets Ω E" and A: "range A ⊆ E" "(⋃i. A i) = Ω" "⋀i. emeasure M (A i) ≠ ∞" shows "M = N" proof - let ?μ = "emeasure M" and ?ν = "emeasure N" interpret S: sigma_algebra Ω "sigma_sets Ω E" by (rule sigma_algebra_sigma_sets) fact have "space M = Ω" using sets.top[of M] sets.space_closed[of M] S.top S.space_closed ‹sets M = sigma_sets Ω E› by blast { fix F D assume "F ∈ E" and "?μ F ≠ ∞" then have [intro]: "F ∈ sigma_sets Ω E" by auto have "?ν F ≠ ∞" using ‹?μ F ≠ ∞› ‹F ∈ E› eq by simp assume "D ∈ sets M" with ‹Int_stable E› ‹E ⊆ Pow Ω› have "emeasure M (F ∩ D) = emeasure N (F ∩ D)" unfolding M proof (induct rule: sigma_sets_induct_disjoint) case (basic A) then have "F ∩ A ∈ E" using ‹Int_stable E› ‹F ∈ E› by (auto simp: Int_stable_def) then show ?case using eq by auto next case empty then show ?case by simp next case (compl A) then have **: "F ∩ (Ω - A) = F - (F ∩ A)" and [intro]: "F ∩ A ∈ sigma_sets Ω E" using ‹F ∈ E› S.sets_into_space by (auto simp: M) have "?ν (F ∩ A) ≤ ?ν F" by (auto intro!: emeasure_mono simp: M N) then have "?ν (F ∩ A) ≠ ∞" using ‹?ν F ≠ ∞› by (auto simp: top_unique) have "?μ (F ∩ A) ≤ ?μ F" by (auto intro!: emeasure_mono simp: M N) then have "?μ (F ∩ A) ≠ ∞" using ‹?μ F ≠ ∞› by (auto simp: top_unique) then have "?μ (F ∩ (Ω - A)) = ?μ F - ?μ (F ∩ A)" unfolding ** using ‹F ∩ A ∈ sigma_sets Ω E› by (auto intro!: emeasure_Diff simp: M N) also have "… = ?ν F - ?ν (F ∩ A)" using eq ‹F ∈ E› compl by simp also have "… = ?ν (F ∩ (Ω - A))" unfolding ** using ‹F ∩ A ∈ sigma_sets Ω E› ‹?ν (F ∩ A) ≠ ∞› by (auto intro!: emeasure_Diff[symmetric] simp: M N) finally show ?case using ‹space M = Ω› by auto next case (union A) then have "?μ (⋃x. F ∩ A x) = ?ν (⋃x. F ∩ A x)" by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N) with A show ?case by auto qed } note * = this show "M = N" proof (rule measure_eqI) show "sets M = sets N" using M N by simp have [simp, intro]: "⋀i. A i ∈ sets M" using A(1) by (auto simp: subset_eq M) fix F assume "F ∈ sets M" let ?D = "disjointed (λi. F ∩ A i)" from ‹space M = Ω› have F_eq: "F = (⋃i. ?D i)" using ‹F ∈ sets M›[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq) have [simp, intro]: "⋀i. ?D i ∈ sets M" using sets.range_disjointed_sets[of "λi. F ∩ A i" M] ‹F ∈ sets M› by (auto simp: subset_eq) have "disjoint_family ?D" by (auto simp: disjoint_family_disjointed) moreover have "(∑i. emeasure M (?D i)) = (∑i. emeasure N (?D i))" proof (intro arg_cong[where f=suminf] ext) fix i have "A i ∩ ?D i = ?D i" by (auto simp: disjointed_def) then show "emeasure M (?D i) = emeasure N (?D i)" using *[of "A i" "?D i", OF _ A(3)] A(1) by auto qed ultimately show "emeasure M F = emeasure N F" by (simp add: image_subset_iff ‹sets M = sets N›[symmetric] F_eq[symmetric] suminf_emeasure) qed qed lemma space_empty: "space M = {} ⟹ M = count_space {}" by (rule measure_eqI) (simp_all add: space_empty_iff) lemma measure_eqI_generator_eq_countable: fixes M N :: "'a measure" and E :: "'a set set" and A :: "'a set set" assumes E: "Int_stable E" "E ⊆ Pow Ω" "⋀X. X ∈ E ⟹ emeasure M X = emeasure N X" and sets: "sets M = sigma_sets Ω E" "sets N = sigma_sets Ω E" and A: "A ⊆ E" "(⋃A) = Ω" "countable A" "⋀a. a ∈ A ⟹ emeasure M a ≠ ∞" shows "M = N" proof cases assume "Ω = {}" have *: "sigma_sets Ω E = sets (sigma Ω E)" using E(2) by simp have "space M = Ω" "space N = Ω" using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of) then show "M = N" unfolding ‹Ω = {}› by (auto dest: space_empty) next assume "Ω ≠ {}" with ‹⋃A = Ω› have "A ≠ {}" by auto from this ‹countable A› have rng: "range (from_nat_into A) = A" by (rule range_from_nat_into) show "M = N" proof (rule measure_eqI_generator_eq[OF E sets]) show "range (from_nat_into A) ⊆ E" unfolding rng using ‹A ⊆ E› . show "(⋃i. from_nat_into A i) = Ω" unfolding rng using ‹⋃A = Ω› . show "emeasure M (from_nat_into A i) ≠ ∞" for i using rng by (intro A) auto qed qed lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M" proof (intro measure_eqI emeasure_measure_of_sigma) show "sigma_algebra (space M) (sets M)" .. show "positive (sets M) (emeasure M)" by (simp add: positive_def) show "countably_additive (sets M) (emeasure M)" by (simp add: emeasure_countably_additive) qed simp_all subsection ‹‹μ›-null sets› definition%important null_sets :: "'a measure ⇒ 'a set set" where "null_sets M = {N∈sets M. emeasure M N = 0}" lemma null_setsD1[dest]: "A ∈ null_sets M ⟹ emeasure M A = 0" by (simp add: null_sets_def) lemma null_setsD2[dest]: "A ∈ null_sets M ⟹ A ∈ sets M" unfolding null_sets_def by simp lemma null_setsI[intro]: "emeasure M A = 0 ⟹ A ∈ sets M ⟹ A ∈ null_sets M" unfolding null_sets_def by simp interpretation null_sets: ring_of_sets "space M" "null_sets M" for M proof (rule ring_of_setsI) show "null_sets M ⊆ Pow (space M)" using sets.sets_into_space by auto show "{} ∈ null_sets M" by auto fix A B assume null_sets: "A ∈ null_sets M" "B ∈ null_sets M" then have sets: "A ∈ sets M" "B ∈ sets M" by auto then have *: "emeasure M (A ∪ B) ≤ emeasure M A + emeasure M B" "emeasure M (A - B) ≤ emeasure M A" by (auto intro!: emeasure_subadditive emeasure_mono) then have "emeasure M B = 0" "emeasure M A = 0" using null_sets by auto with sets * show "A - B ∈ null_sets M" "A ∪ B ∈ null_sets M" by (auto intro!: antisym zero_le) qed lemma UN_from_nat_into: assumes I: "countable I" "I ≠ {}" shows "(⋃i∈I. N i) = (⋃i. N (from_nat_into I i))" proof - have "(⋃i∈I. N i) = ⋃(N ` range (from_nat_into I))" using I by simp also have "… = (⋃i. (N ∘ from_nat_into I) i)" by simp finally show ?thesis by simp qed lemma null_sets_UN': assumes "countable I" assumes "⋀i. i ∈ I ⟹ N i ∈ null_sets M" shows "(⋃i∈I. N i) ∈ null_sets M" proof cases assume "I = {}" then show ?thesis by simp next assume "I ≠ {}" show ?thesis proof (intro conjI CollectI null_setsI) show "(⋃i∈I. N i) ∈ sets M" using assms by (intro sets.countable_UN') auto have "emeasure M (⋃i∈I. N i) ≤ (∑n. emeasure M (N (from_nat_into I n)))" unfolding UN_from_nat_into[OF ‹countable I› ‹I ≠ {}›] using assms ‹I ≠ {}› by (intro emeasure_subadditive_countably) (auto intro: from_nat_into) also have "(λn. emeasure M (N (from_nat_into I n))) = (λ_. 0)" using assms ‹I ≠ {}› by (auto intro: from_nat_into) finally show "emeasure M (⋃i∈I. N i) = 0" by (intro antisym zero_le) simp qed qed lemma null_sets_UN[intro]: "(⋀i::'i::countable. N i ∈ null_sets M) ⟹ (⋃i. N i) ∈ null_sets M" by (rule null_sets_UN') auto lemma null_set_Int1: assumes "B ∈ null_sets M" "A ∈ sets M" shows "A ∩ B ∈ null_sets M" proof (intro CollectI conjI null_setsI) show "emeasure M (A ∩ B) = 0" using assms by (intro emeasure_eq_0[of B _ "A ∩ B"]) auto qed (insert assms, auto) lemma null_set_Int2: assumes "B ∈ null_sets M" "A ∈ sets M" shows "B ∩ A ∈ null_sets M" using assms by (subst Int_commute) (rule null_set_Int1) lemma emeasure_Diff_null_set: assumes "B ∈ null_sets M" "A ∈ sets M" shows "emeasure M (A - B) = emeasure M A" proof - have *: "A - B = (A - (A ∩ B))" by auto have "A ∩ B ∈ null_sets M" using assms by (rule null_set_Int1) then show ?thesis unfolding * using assms by (subst emeasure_Diff) auto qed lemma null_set_Diff: assumes "B ∈ null_sets M" "A ∈ sets M" shows "B - A ∈ null_sets M" proof (intro CollectI conjI null_setsI) show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto qed (insert assms, auto) lemma emeasure_Un_null_set: assumes "A ∈ sets M" "B ∈ null_sets M" shows "emeasure M (A ∪ B) = emeasure M A" proof - have *: "A ∪ B = A ∪ (B - A)" by auto have "B - A ∈ null_sets M" using assms(2,1) by (rule null_set_Diff) then show ?thesis unfolding * using assms by (subst plus_emeasure[symmetric]) auto qed subsection ‹The almost everywhere filter (i.e.\ quantifier)› definition%important ae_filter :: "'a measure ⇒ 'a filter" where "ae_filter M = (INF N:null_sets M. principal (space M - N))" abbreviation almost_everywhere :: "'a measure ⇒ ('a ⇒ bool) ⇒ bool" where "almost_everywhere M P ≡ eventually P (ae_filter M)" syntax "_almost_everywhere" :: "pttrn ⇒ 'a ⇒ bool ⇒ bool" ("AE _ in _. _" [0,0,10] 10) translations "AE x in M. P" ⇌ "CONST almost_everywhere M (λx. P)" abbreviation "set_almost_everywhere A M P ≡ AE x in M. x ∈ A ⟶ P x" syntax "_set_almost_everywhere" :: "pttrn ⇒ 'a set ⇒ 'a ⇒ bool ⇒ bool" ("AE _∈_ in _./ _" [0,0,0,10] 10) translations "AE x∈A in M. P" ⇌ "CONST set_almost_everywhere A M (λx. P)" lemma eventually_ae_filter: "eventually P (ae_filter M) ⟷ (∃N∈null_sets M. {x ∈ space M. ¬ P x} ⊆ N)" unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq) lemma AE_I': "N ∈ null_sets M ⟹ {x∈space M. ¬ P x} ⊆ N ⟹ (AE x in M. P x)" unfolding eventually_ae_filter by auto lemma AE_iff_null: assumes "{x∈space M. ¬ P x} ∈ sets M" (is "?P ∈ sets M") shows "(AE x in M. P x) ⟷ {x∈space M. ¬ P x} ∈ null_sets M" proof assume "AE x in M. P x" then obtain N where N: "N ∈ sets M" "?P ⊆ N" "emeasure M N = 0" unfolding eventually_ae_filter by auto have "emeasure M ?P ≤ emeasure M N" using assms N(1,2) by (auto intro: emeasure_mono) then have "emeasure M ?P = 0" unfolding ‹emeasure M N = 0› by auto then show "?P ∈ null_sets M" using assms by auto next assume "?P ∈ null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I') qed lemma AE_iff_null_sets: "N ∈ sets M ⟹ N ∈ null_sets M ⟷ (AE x in M. x ∉ N)" using Int_absorb1[OF sets.sets_into_space, of N M] by (subst AE_iff_null) (auto simp: Int_def[symmetric]) lemma AE_not_in: "N ∈ null_sets M ⟹ AE x in M. x ∉ N" by (metis AE_iff_null_sets null_setsD2) lemma AE_iff_measurable: "N ∈ sets M ⟹ {x∈space M. ¬ P x} = N ⟹ (AE x in M. P x) ⟷ emeasure M N = 0" using AE_iff_null[of _ P] by auto lemma AE_E[consumes 1]: assumes "AE x in M. P x" obtains N where "{x ∈ space M. ¬ P x} ⊆ N" "emeasure M N = 0" "N ∈ sets M" using assms unfolding eventually_ae_filter by auto lemma AE_E2: assumes "AE x in M. P x" "{x∈space M. P x} ∈ sets M" shows "emeasure M {x∈space M. ¬ P x} = 0" (is "emeasure M ?P = 0") proof - have "{x∈space M. ¬ P x} = space M - {x∈space M. P x}" by auto with AE_iff_null[of M P] assms show ?thesis by auto qed lemma AE_E3: assumes "AE x in M. P x" obtains N where "⋀x. x ∈ space M - N ⟹ P x" "N ∈ null_sets M" using assms unfolding eventually_ae_filter by auto lemma AE_I: assumes "{x ∈ space M. ¬ P x} ⊆ N" "emeasure M N = 0" "N ∈ sets M" shows "AE x in M. P x" using assms unfolding eventually_ae_filter by auto lemma AE_mp[elim!]: assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x ⟶ Q x" shows "AE x in M. Q x" proof - from AE_P obtain A where P: "{x∈space M. ¬ P x} ⊆ A" and A: "A ∈ sets M" "emeasure M A = 0" by (auto elim!: AE_E) from AE_imp obtain B where imp: "{x∈space M. P x ∧ ¬ Q x} ⊆ B" and B: "B ∈ sets M" "emeasure M B = 0" by (auto elim!: AE_E) show ?thesis proof (intro AE_I) have "emeasure M (A ∪ B) ≤ 0" using emeasure_subadditive[of A M B] A B by auto then show "A ∪ B ∈ sets M" "emeasure M (A ∪ B) = 0" using A B by auto show "{x∈space M. ¬ Q x} ⊆ A ∪ B" using P imp by auto qed qed text ‹The next lemma is convenient to combine with a lemma whose conclusion is of the form ‹AE x in M. P x = Q x›: for such a lemma, there is no \verb+[symmetric]+ variant, but using ‹AE_symmetric[OF...]› will replace it.› (* depricated replace by laws about eventually *) lemma shows AE_iffI: "AE x in M. P x ⟹ AE x in M. P x ⟷ Q x ⟹ AE x in M. Q x" and AE_disjI1: "AE x in M. P x ⟹ AE x in M. P x ∨ Q x" and AE_disjI2: "AE x in M. Q x ⟹ AE x in M. P x ∨ Q x" and AE_conjI: "AE x in M. P x ⟹ AE x in M. Q x ⟹ AE x in M. P x ∧ Q x" and AE_conj_iff[simp]: "(AE x in M. P x ∧ Q x) ⟷ (AE x in M. P x) ∧ (AE x in M. Q x)" by auto lemma AE_symmetric: assumes "AE x in M. P x = Q x" shows "AE x in M. Q x = P x" using assms by auto lemma AE_impI: "(P ⟹ AE x in M. Q x) ⟹ AE x in M. P ⟶ Q x" by (cases P) auto 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] guess N . note N = this with sets have "emeasure M (space M) ≤ emeasure M (?P ∪ N)" by (intro emeasure_mono) auto also have "… ≤ emeasure M ?P + emeasure M N" using sets N by (intro emeasure_subadditive) auto also have "… = emeasure M ?P" using N by simp finally show "emeasure M ?P = emeasure M (space M)" using emeasure_space[of M "?P"] by auto qed lemma AE_space: "AE x in M. x ∈ space M" by (rule AE_I[where N="{}"]) auto lemma AE_I2[simp, intro]: "(⋀x. x ∈ space M ⟹ P x) ⟹ AE x in M. P x" using AE_space by force lemma AE_Ball_mp: "∀x∈space M. P x ⟹ AE x in M. P x ⟶ Q x ⟹ AE x in M. Q x" by auto lemma AE_cong[cong]: "(⋀x. x ∈ space M ⟹ P x ⟷ Q x) ⟹ (AE x in M. P x) ⟷ (AE x in M. Q x)" by auto lemma AE_cong_strong: "M = N ⟹ (⋀x. x ∈ space N =simp=> P x = Q x) ⟹ (AE x in M. P x) ⟷ (AE x in N. Q x)" by (auto simp: simp_implies_def) lemma AE_all_countable: "(AE x in M. ∀i. P i x) ⟷ (∀i::'i::countable. AE x in M. P i x)" proof assume "∀i. AE x in M. P i x" from this[unfolded eventually_ae_filter Bex_def, THEN choice] obtain N where N: "⋀i. N i ∈ null_sets M" "⋀i. {x∈space M. ¬ P i x} ⊆ N i" by auto have "{x∈space M. ¬ (∀i. P i x)} ⊆ (⋃i. {x∈space M. ¬ P i x})" by auto also have "… ⊆ (⋃i. N i)" using N by auto finally have "{x∈space M. ¬ (∀i. P i x)} ⊆ (⋃i. N i)" . moreover from N have "(⋃i. N i) ∈ null_sets M" by (intro null_sets_UN) auto ultimately show "AE x in M. ∀i. P i x" unfolding eventually_ae_filter by auto qed auto lemma AE_ball_countable: assumes [intro]: "countable X" shows "(AE x in M. ∀y∈X. P x y) ⟷ (∀y∈X. AE x in M. P x y)" proof assume "∀y∈X. AE x in M. P x y" from this[unfolded eventually_ae_filter Bex_def, THEN bchoice] obtain N where N: "⋀y. y ∈ X ⟹ N y ∈ null_sets M" "⋀y. y ∈ X ⟹ {x∈space M. ¬ P x y} ⊆ N y" by auto have "{x∈space M. ¬ (∀y∈X. P x y)} ⊆ (⋃y∈X. {x∈space M. ¬ P x y})" by auto also have "… ⊆ (⋃y∈X. N y)" using N by auto finally have "{x∈space M. ¬ (∀y∈X. P x y)} ⊆ (⋃y∈X. N y)" . moreover from N have "(⋃y∈X. N y) ∈ null_sets M" by (intro null_sets_UN') auto ultimately show "AE x in M. ∀y∈X. P x y" unfolding eventually_ae_filter by auto qed auto lemma AE_ball_countable': "(⋀N. N ∈ I ⟹ AE x in M. P N x) ⟹ countable I ⟹ AE x in M. ∀N ∈ I. P N x" unfolding AE_ball_countable by simp lemma pairwise_alt: "pairwise R S ⟷ (∀x∈S. ∀y∈S-{x}. R x y)" by (auto simp add: pairwise_def) lemma AE_pairwise: "countable F ⟹ pairwise (λA B. AE x in M. R x A B) F ⟷ (AE x in M. pairwise (R x) F)" unfolding pairwise_alt by (simp add: AE_ball_countable) lemma AE_discrete_difference: assumes X: "countable X" assumes null: "⋀x. x ∈ X ⟹ emeasure M {x} = 0" assumes sets: "⋀x. x ∈ X ⟹ {x} ∈ sets M" shows "AE x in M. x ∉ X" proof - have "(⋃x∈X. {x}) ∈ null_sets M" using assms by (intro null_sets_UN') auto from AE_not_in[OF this] show "AE x in M. x ∉ X" by auto qed lemma AE_finite_all: assumes f: "finite S" shows "(AE x in M. ∀i∈S. P i x) ⟷ (∀i∈S. AE x in M. P i x)" using f by induct auto lemma AE_finite_allI: assumes "finite S" shows "(⋀s. s ∈ S ⟹ AE x in M. Q s x) ⟹ AE x in M. ∀s∈S. Q s x" using AE_finite_all[OF ‹finite S›] by auto lemma emeasure_mono_AE: assumes imp: "AE x in M. x ∈ A ⟶ x ∈ B" and B: "B ∈ sets M" shows "emeasure M A ≤ emeasure M B" proof cases assume A: "A ∈ sets M" from imp obtain N where N: "{x∈space M. ¬ (x ∈ A ⟶ x ∈ B)} ⊆ N" "N ∈ null_sets M" by (auto simp: eventually_ae_filter) have "emeasure M A = emeasure M (A - N)" using N A by (subst emeasure_Diff_null_set) auto also have "emeasure M (A - N) ≤ emeasure M (B - N)" using N A B sets.sets_into_space by (auto intro!: emeasure_mono) also have "emeasure M (B - N) = emeasure M B" using N B by (subst emeasure_Diff_null_set) auto finally show ?thesis . qed (simp add: emeasure_notin_sets) lemma emeasure_eq_AE: assumes iff: "AE x in M. x ∈ A ⟷ x ∈ B" assumes A: "A ∈ sets M" and B: "B ∈ sets M" shows "emeasure M A = emeasure M B" using assms by (safe intro!: antisym emeasure_mono_AE) auto lemma emeasure_Collect_eq_AE: "AE x in M. P x ⟷ Q x ⟹ Measurable.pred M Q ⟹ Measurable.pred M P ⟹ emeasure M {x∈space M. P x} = emeasure M {x∈space M. Q x}" by (intro emeasure_eq_AE) auto lemma emeasure_eq_0_AE: "AE x in M. ¬ P x ⟹ emeasure M {x∈space M. P x} = 0" using AE_iff_measurable[OF _ refl, of M "λx. ¬ P x"] by (cases "{x∈space M. P x} ∈ sets M") (simp_all add: emeasure_notin_sets) lemma emeasure_0_AE: assumes "emeasure M (space M) = 0" shows "AE x in M. P x" using eventually_ae_filter assms by blast lemma emeasure_add_AE: assumes [measurable]: "A ∈ sets M" "B ∈ sets M" "C ∈ sets M" assumes 1: "AE x in M. x ∈ C ⟷ x ∈ A ∨ x ∈ B" assumes 2: "AE x in M. ¬ (x ∈ A ∧ x ∈ B)" shows "emeasure M C = emeasure M A + emeasure M B" proof - have "emeasure M C = emeasure M (A ∪ B)" by (rule emeasure_eq_AE) (insert 1, auto) also have "… = emeasure M A + emeasure M (B - A)" by (subst plus_emeasure) auto also have "emeasure M (B - A) = emeasure M B" by (rule emeasure_eq_AE) (insert 2, auto) finally show ?thesis . qed subsection ‹‹σ›-finite Measures› locale%important sigma_finite_measure = fixes M :: "'a measure" assumes sigma_finite_countable: "∃A::'a set set. countable A ∧ A ⊆ sets M ∧ (⋃A) = space M ∧ (∀a∈A. emeasure M a ≠ ∞)" lemma (in sigma_finite_measure) sigma_finite: obtains A :: "nat ⇒ 'a set" where "range A ⊆ sets M" "(⋃i. A i) = space M" "⋀i. emeasure M (A i) ≠ ∞" proof - obtain A :: "'a set set" where [simp]: "countable A" and A: "A ⊆ sets M" "(⋃A) = space M" "⋀a. a ∈ A ⟹ emeasure M a ≠ ∞" using sigma_finite_countable by metis show thesis proof cases assume "A = {}" with ‹(⋃A) = space M› show thesis by (intro that[of "λ_. {}"]) auto next assume "A ≠ {}" show thesis proof show "range (from_nat_into A) ⊆ sets M" using ‹A ≠ {}› A by auto have "(⋃i. from_nat_into A i) = ⋃A" using range_from_nat_into[OF ‹A ≠ {}› ‹countable A›] by auto with A show "(⋃i. from_nat_into A i) = space M" by auto qed (intro A from_nat_into ‹A ≠ {}›) qed qed lemma (in sigma_finite_measure) sigma_finite_disjoint: obtains A :: "nat ⇒ 'a set" where "range A ⊆ sets M" "(⋃i. A i) = space M" "⋀i. emeasure M (A i) ≠ ∞" "disjoint_family A" proof - obtain A :: "nat ⇒ 'a set" where range: "range A ⊆ sets M" and space: "(⋃i. A i) = space M" and measure: "⋀i. emeasure M (A i) ≠ ∞" using sigma_finite by blast show thesis proof (rule that[of "disjointed A"]) show "range (disjointed A) ⊆ sets M" by (rule sets.range_disjointed_sets[OF range]) show "(⋃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) ≠ ∞" for i proof - have "emeasure M (disjointed A i) ≤ emeasure M (A i)" using range disjointed_subset[of A i] by (auto intro!: emeasure_mono) then show ?thesis using measure[of i] by (auto simp: top_unique) qed qed qed lemma (in sigma_finite_measure) sigma_finite_incseq: obtains A :: "nat ⇒ 'a set" where "range A ⊆ sets M" "(⋃i. A i) = space M" "⋀i. emeasure M (A i) ≠ ∞" "incseq A" proof - obtain F :: "nat ⇒ 'a set" where F: "range F ⊆ sets M" "(⋃i. F i) = space M" "⋀i. emeasure M (F i) ≠ ∞" using sigma_finite by blast show thesis proof (rule that[of "λn. ⋃i≤n. F i"]) show "range (λn. ⋃i≤n. F i) ⊆ sets M" using F by (force simp: incseq_def) show "(⋃n. ⋃i≤n. F i) = space M" proof - from F have "⋀x. x ∈ space M ⟹ ∃i. x ∈ F i" by auto with F show ?thesis by fastforce qed show "emeasure M (⋃i≤n. F i) ≠ ∞" for n proof - have "emeasure M (⋃i≤n. F i) ≤ (∑i≤n. emeasure M (F i))" using F by (auto intro!: emeasure_subadditive_finite) also have "… < ∞" using F by (auto simp: sum_Pinfty less_top) finally show ?thesis by simp qed show "incseq (λn. ⋃i≤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: "W ∈ sets M" and W_inf: "emeasure M W = ∞" obtains Z where "Z ∈ sets M" "Z ⊆ W" "emeasure M Z < ∞" "emeasure M Z > C" proof - obtain A :: "nat ⇒ 'a set" where A: "range A ⊆ sets M" "(⋃i. A i) = space M" "⋀i. emeasure M (A i) ≠ ∞" "incseq A" using sigma_finite_incseq by blast define B where "B = (λi. W ∩ A i)" have B_meas: "⋀i. B i ∈ sets M" using W_meas ‹range A ⊆ sets M› B_def by blast have b: "⋀i. B i ⊆ W" using B_def by blast { fix i have "emeasure M (B i) ≤ emeasure M (A i)" using A by (intro emeasure_mono) (auto simp: B_def) also have "emeasure M (A i) < ∞" using ‹⋀i. emeasure M (A i) ≠ ∞› by (simp add: less_top) finally have "emeasure M (B i) < ∞" . } note c = this have "W = (⋃i. B i)" using B_def ‹(⋃i. A i) = space M› W_meas by auto moreover have "incseq B" using B_def ‹incseq A› by (simp add: incseq_def subset_eq) ultimately have "(λi. emeasure M (B i)) ⇢ emeasure M W" using W_meas B_meas by (simp add: B_meas Lim_emeasure_incseq image_subset_iff) then have "(λi. emeasure M (B i)) ⇢ ∞" using W_inf by simp from order_tendstoD(1)[OF this, of C] obtain i where d: "emeasure M (B i) > C" by (auto simp: eventually_sequentially) have "B i ∈ sets M" "B i ⊆ W" "emeasure M (B i) < ∞" "emeasure M (B i) > C" using B_meas b c d by auto then show ?thesis using that by blast qed subsection ‹Measure space induced by distribution of @{const measurable}-functions› definition%important distr :: "'a measure ⇒ 'b measure ⇒ ('a ⇒ 'b) ⇒ 'b measure" where "distr M N f = measure_of (space N) (sets N) (λA. emeasure M (f -` A ∩ space M))" lemma shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N" and space_distr[simp]: "space (distr M N f) = 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 ⟹ sets N = sets L ⟹ (⋀x. x ∈ space M ⟹ f x = g x) ⟹ 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 f :: "'a ⇒ 'b" assumes f: "f ∈ measurable M N" and A: "A ∈ sets N" shows "emeasure (distr M N f) A = emeasure M (f -` A ∩ space M)" (is "_ = ?μ A") unfolding distr_def proof (rule emeasure_measure_of_sigma) show "positive (sets N) ?μ" by (auto simp: positive_def) show "countably_additive (sets N) ?μ" proof (intro countably_additiveI) fix A :: "nat ⇒ 'b set" assume "range A ⊆ sets N" "disjoint_family A" then have A: "⋀i. A i ∈ sets N" "(⋃i. A i) ∈ sets N" by auto then have *: "range (λi. f -` (A i) ∩ space M) ⊆ sets M" using f by (auto simp: measurable_def) moreover have "(⋃i. f -` A i ∩ space M) ∈ sets M" using * by blast moreover have **: "disjoint_family (λi. f -` A i ∩ space M)" using ‹disjoint_family A› by (auto simp: disjoint_family_on_def) ultimately show "(∑i. ?μ (A i)) = ?μ (⋃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 ∈ measurable M N" "Measurable.pred N P" shows "emeasure (distr M N X) {x∈space N. P x} = emeasure M {x∈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: "⋀M. P M ⟹ f ∈ measurable M' M" 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 (f x)} = (SUP i. emeasure M' {x∈space M'. (F ^^ i) (λx. False) (f x)})" proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f]) show "f ∈ measurable M' M" "f ∈ measurable M' M" using f[OF ‹P M›] by auto { fix i show "Measurable.pred M ((F ^^ i) (λx. False))" using ‹P M› by (induction i arbitrary: M) (auto intro!: *) } show "Measurable.pred M (lfp F)" using ‹P M› cont * by (rule measurable_lfp_coinduct[of P]) have "emeasure (distr M' M f) {x ∈ space (distr M' M f). lfp F x} = (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 N M (λx. x) = N" 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 add: emeasure_distr intro!: emeasure_eq_AE measurable_sets) qed (insert 1, simp) lemma AE_distrD: assumes f: "f ∈ measurable M M'" and AE: "AE x in distr M M' f. P x" shows "AE x in M. P (f x)" proof - from AE[THEN AE_E] guess N . with f show ?thesis unfolding eventually_ae_filter by (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 add: null_sets_def emeasure_distr) proposition distr_distr: "g ∈ measurable N L ⟹ f ∈ measurable M N ⟹ distr (distr M N f) L g = distr M L (g ∘ f)" by (auto simp add: emeasure_distr measurable_space intro!: arg_cong[where f="emeasure M"] measure_eqI) subsection%unimportant ‹Real measure values› lemma ring_of_finite_sets: "ring_of_sets (space M) {A∈sets M. emeasure M A ≠ top}" proof (rule ring_of_setsI) show "a ∈ {A ∈ sets M. emeasure M A ≠ top} ⟹ b ∈ {A ∈ sets M. emeasure M A ≠ top} ⟹ a ∪ b ∈ {A ∈ sets M. emeasure M A ≠ top}" for a b using emeasure_subadditive[of a M b] by (auto simp: top_unique) show "a ∈ {A ∈ sets M. emeasure M A ≠ top} ⟹ b ∈ {A ∈ sets M. emeasure M A ≠ top} ⟹ a - b ∈ {A ∈ sets M. emeasure M A ≠ top}" for a b using emeasure_mono[of "a - b" a M] by (auto simp: Diff_subset top_unique) qed (auto dest: sets.sets_into_space) 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 M A ⟷ measure M A ≠ 0" using measure_nonneg[of M A] by (auto simp add: 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 enn2ereal_top) 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 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 disjoint_family_on_insert: "i ∉ I ⟹ disjoint_family_on A (insert i I) ⟷ A i ∩ (⋃i∈I. A i) = {} ∧ disjoint_family_on A I" by (fastforce simp: disjoint_family_on_def) 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) moreover { fix i have "emeasure M (A i) ≤ emeasure M (⋃i. A i)" using measurable by (auto intro!: emeasure_mono) then have "emeasure M (A i) = ennreal ((measure M (A i)))" using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) } ultimately show ?thesis using finite by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all 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)" using emeasure_subadditive[OF measurable] fin apply simp apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure) apply (auto simp flip: ennreal_plus) done 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) ≤ (∑i∈I. emeasure M (A i))" using emeasure_subadditive_finite[OF A] . also have "… < ∞" using fin by (simp add: less_top A) finally have "emeasure M (⋃i∈I. A i) ≠ top" by simp } note * = this 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 - from fin have **: "⋀i. emeasure M (A i) ≠ top" using ennreal_suminf_lessD[of "λi. emeasure M (A i)"] by (simp add: less_top) { have "emeasure M (⋃i. A i) ≤ (∑i. emeasure M (A i))" using emeasure_subadditive_countably[OF A] . also have "… < ∞" using fin by (simp add: less_top) finally have "emeasure M (⋃i. A i) ≠ top" by simp } then have "ennreal (measure M (⋃i. A i)) = emeasure M (⋃i. A i)" 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 (subst suminf_ennreal) (auto simp: **) finally show ?thesis apply (rule ennreal_le_iff[THEN iffD1, rotated]) apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top) using fin apply (simp add: emeasure_eq_ennreal_measure[OF **]) done 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" by (simp add: measure_def emeasure_Diff_null_set) 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%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 Diff_subset) qed subsection‹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" "⋀i. i ∈ I ⟹ F i ⊆ A" "⋀i. i ∈ I ⟹ F i ∈ sets M" "A ∈ fmeasurable M" 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∈I. F i) ∈ sets M" 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 add: 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 ∈ fmeasurable M" 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" "S ∈ fmeasurable M" and sub: "S ⊆ T" shows "T ∈ fmeasurable M" 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 - A)" 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 ∉ A ∨ x ∉ B ⟹ A ∈ fmeasurable M ⟹ B ∈ fmeasurable M ⟹ 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 ∈ I ⟹ F i ∈ fmeasurable M) ⟹ pairwise (λi j. AE x in M. x ∉ F i ∨ x ∉ F j) 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 ∪ UNION I F) = measure M (F x) + measure M (UNION I F)" 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 assume "¬ (A ∈ fmeasurable M ∧ B ∈ fmeasurable M)" 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 (⋃i∈I. F i) ≤ (∑i∈I. measure M (F i))" proof (induction I rule: finite_induct) case (insert i I) then have "measure M (⋃i∈insert i I. F i) ≤ measure M (F i) + measure M (⋃i∈I. F i)" by (auto intro!: measure_Un_le) 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 ⟹ (⋀S. S ∈ F ⟹ S ∈ sets M) ⟹ measure M (⋃F) ≤ (∑S∈F. measure M S)" using measure_UNION_le[of F "λx. x" M] by simp text‹Version for indexed union over a countable set› lemma assumes "countable I" and I: "⋀i. i ∈ I ⟹ A i ∈ fmeasurable M" 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 then have "emeasure M (⋃i∈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 (simp add: 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 then obtain D :: "nat ⇒ 'a set" where D: "𝒟 = range 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 by (intro conjI fmeasurable_UN_bound [OF _ 1 2] measure_UN_bound [OF _ 1 2]) auto 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 (⋃n. S n) ≤ B" 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 add: dist_norm suminf_exist_split[OF _ assms]) lemma emeasure_union_summable: assumes [measurable]: "⋀n. A n ∈ sets M" 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" for N unfolding B_def by auto have "(λN. emeasure M (B N)) ⇢ emeasure M (⋃N. B N)" apply (rule Lim_emeasure_incseq) unfolding B_def by (auto simp add: SUP_subset_mono incseq_def) 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 assms(3) measure_nonneg sum_le_suminf by blast have "emeasure M (B N) ≤ (∑n∈{..<N}. emeasure M (A n))" unfolding B_def by (rule emeasure_subadditive_finite, auto) also have "... = (∑n∈{..<N}. ennreal(measure M (A n)))" using assms(2) by (simp add: emeasure_eq_ennreal_measure 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 ?thesis by simp 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) < ∞" "summable (λn. measure 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 assms(3)]) 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 add: limsup_INF_SUP) also have "... = emeasure M (⋃k. A (k+n))" using I by auto also have "... ≤ (∑k. measure M (A (k+n)))" apply (rule emeasure_union_summable) using assms summable_ignore_initial_segment[OF assms(3), 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 { fix x assume "x ∉ limsup A" then obtain N where "x ∉ (⋃n∈{N..}. A n)" unfolding limsup_INF_SUP by blast then have "eventually (λn. x ∉ A n) sequentially" using eventually_sequentially by auto } ultimately show ?thesis by auto qed subsection ‹Measure spaces with @{term "emeasure M (space M) < ∞"}› locale%important finite_measure = sigma_finite_measure M for M + assumes finite_emeasure_space: "emeasure M (space M) ≠ top" lemma finite_measureI[Pure.intro!]: "emeasure M (space M) ≠ ∞ ⟹ finite_measure M" proof qed (auto intro!: exI[of _ "{space M}"]) lemma (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) lemma (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M" by (auto simp: fmeasurable_def less_top[symmetric]) lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)" by (intro emeasure_eq_ennreal_measure) simp lemma (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 lemma (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) lemma (in finite_measure) finite_measure_Diff: assumes sets: "A ∈ sets M" "B ∈ sets M" and "B ⊆ A" shows "measure M (A - B) = measure M A - measure M B" using measure_Diff[OF _ assms] by simp lemma (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 lemma (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 lemma (in finite_measure) finite_measure_UNION: assumes A: "range A ⊆ sets M" "disjoint_family A" shows "(λi. measure M (A i)) sums (measure M (⋃i. A i))" using measure_UNION[OF A] by simp lemma (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) lemma (in finite_measure) finite_measure_subadditive: assumes m: "A ∈ sets M" "B ∈ sets M" shows "measure M (A ∪ B) ≤ measure M A + measure M B" using measure_subadditive[OF m] by simp lemma (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 lemma (in finite_measure) finite_measure_subadditive_countably: "range A ⊆ sets M ⟹ summable (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) lemma (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 lemma (in finite_measure) finite_Lim_measure_incseq: assumes A: "range A ⊆ sets M" "incseq A" shows "(λi. measure M (A i)) ⇢ measure M (⋃i. A i)" using Lim_measure_incseq[OF A] by simp lemma (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 lemma (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_Diff[OF _ sets.top S sets.sets_into_space] S by simp lemma (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) lemma (in finite_measure) finite_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: emeasure_eq_measure) lemma (in finite_measure) measure_increasing: "increasing M (measure M)" by (auto intro!: finite_measure_mono simp: increasing_def) lemma (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" using assms proof - have "measure M (s ∪ t) ≤ measure M s" using finite_measure_subadditive[of s t] assms by auto moreover have "measure M (s ∪ t) ≥ measure M s" using assms by (blast intro: finite_measure_mono) ultimately show ?thesis by simp qed lemma (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 lemma (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)" using assms proof - 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 qed lemma (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" proof (rule antisym) show "measure M (⋃i :: nat. c i) ≤ 0" using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2)) qed simp lemma (in finite_measure) measure_space_inter: assumes events:"s ∈ sets M" "t ∈ sets M" assumes "measure M t = measure M (space M)" shows "measure M (s ∩ t) = measure M s" proof - 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 ∩ t" s]) qed lemma (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}" proof 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 qed simp lemma (in finite_measure) measure_real_sum_image_fn: assumes "e ∈ sets M" assumes "⋀ x. x ∈ s ⟹ e ∩ f x ∈ sets M" assumes "finite s" assumes disjoint: "⋀ x y. ⟦x ∈ s ; y ∈ s ; x ≠ y⟧ ⟹ f x ∩ f y = {}" assumes upper: "space M ⊆ (⋃i ∈ s. f i)" shows "measure M e = (∑ x ∈ s. measure M (e ∩ f x))" proof - have "e ⊆ (⋃i∈s. f i)" using ‹e ∈ sets M› sets.sets_into_space upper by blast then have e: "e = (⋃i ∈ s. e ∩ f i)" by auto hence "measure M e = measure M (⋃i ∈ s. e ∩ f i)" by simp 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 ⊆ sets M" using assms(2) by auto show "disjoint_family_on (λi. e ∩ f i) s" using disjoint by (auto simp: disjoint_family_on_def) qed finally show ?thesis . qed lemma (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) lemma (in finite_measure) finite_measure_distr: assumes f: "f ∈ measurable M M'" shows "finite_measure (distr M M' f)" proof (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) qed lemma 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 ⟹ emeasure (M s) {x∈space N. F P x} = f (λs. emeasure (M s) {x∈space N. P x}) s" 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" proof (subst gfp_transfer_bounded[where α="λF s. emeasure (M s) {x∈space N. F x}" and g=f and f=F 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 ∈ space N. C i x}))" unfolding INF_apply[abs_def] by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure]) next 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) qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont) subsection%unimportant ‹Counting space› lemma strict_monoI_Suc: assumes ord [simp]: "(⋀n. f n < f (Suc n))" shows "strict_mono f" unfolding strict_mono_def proof safe fix n m :: nat assume "n < m" then show "f n < f m" by (induct m) (auto simp: less_Suc_eq intro: less_trans ord) qed lemma 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 proof (rule emeasure_measure_of_sigma) 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" then guess i .. note i = this { fix j from i ‹incseq F› have "F j ⊆ F i" 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 metis 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 guess k .. note k = this 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 "finite (F (f k)) ⟹ finite (F k)" using ‹k ≤ f k› ‹incseq F› by (auto simp: incseq_def dest: finite_subset) 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 auto 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: subset_inj_on) 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 emeasure_count_space[of X A] by simp 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 add: 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 add: null_sets_count_space) lemma sigma_finite_measure_count_space_countable: assumes A: "countable A" shows "sigma_finite_measure (count_space A)" proof qed (insert A, 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)" by (rule sigma_finite_measure_count_space_countable) auto 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: "finite A" shows "sigma_finite_measure (count_space A)" proof - interpret finite_measure "count_space A" using A by (rule finite_measure_count_space) show "sigma_finite_measure (count_space A)" .. qed subsection%unimportant ‹Measure restricted to space› lemma emeasure_restrict_space: assumes "Ω ∩ space M ∈ sets M" "A ⊆ Ω" shows "emeasure (restrict_space M Ω) A = emeasure M A" proof (cases "A ∈ sets M") case True show ?thesis proof (rule emeasure_measure_of[OF restrict_space_def]) 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 ⊆ sets (restrict_space M Ω)" "disjoint_family 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 (subst suminf_emeasure) (auto simp: disjoint_family_subset) qed qed next case False with assms have "A ∉ sets (restrict_space M Ω)" by (simp add: sets_restrict_space_iff) with False show ?thesis by (simp add: emeasure_notin_sets) qed lemma measure_restrict_space: assumes "Ω ∩ space M ∈ sets M" "A ⊆ Ω" 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 { fix X assume X: "X ∈ sets M" "emeasure M X = 0" then have "emeasure M (Ω ∩ space M ∩ X) ≤ emeasure M X" by (intro emeasure_mono) auto then have "emeasure M (Ω ∩ space M ∩ X) = 0" using X by (auto intro!: antisym) } with assms show ?thesis unfolding eventually_ae_filter by (auto simp add: 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 add: 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 add: 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 add: emeasure_restrict_space intro: emeasure_mono) also have "… < ∞" using C(4)[rule_format, of a'] ‹a' ∈ C› by (simp add: less_top) finally have "emeasure (restrict_space M A) a ≠ ∞" by simp } ultimately show ?thesis by unfold_locales (rule exI conjI|assumption|blast)+ qed lemma finite_measure_restrict_space: assumes "finite_measure M" and A: "A ∈ sets M" shows "finite_measure (restrict_space M A)" proof - interpret finite_measure M by fact show ?thesis by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space) qed 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 (subst emeasure_distr) (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr intro!: measurable_restrict_space2) 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 add: 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 ∈ E" then show "emeasure (restrict_space M Ω) X = emeasure (restrict_space N Ω) X" using E Ω by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq]) next show "range (from_nat_into A) ⊆ E" "(⋃i. from_nat_into A i) = Ω" using A by (auto cong del: strong_SUP_cong) 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 add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE) finally show "emeasure M X = emeasure N X" . qed fact subsection%unimportant ‹Null measure› definition "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) (auto simp: positive_def countably_additive_def emeasure_notin_sets null_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" by(rule measure_eqI) simp_all subsection ‹Scaling a measure› definition%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, measurable_cong]: "sets (scale_measure r M) = sets M" by (simp add: scale_measure_def) lemma emeasure_scale_measure [simp]: "emeasure (scale_measure r M) A = r * emeasure M A" (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 add: max_def mult.assoc) lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M" by (rule measure_eqI) simp_all subsection ‹Complete lattice structure on measures› lemma (in finite_measure) finite_measure_Diff': "A ∈ sets M ⟹ B ∈ sets M ⟹ measure M (A - B) = measure M A - measure M (A ∩ B)" using finite_measure_Diff[of A "A ∩ B"] by (auto simp: Diff_Int) lemma (in finite_measure) finite_measure_Union': "A ∈ sets M ⟹ B ∈ sets M ⟹ measure M (A ∪ 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" shows "∃Y∈sets M. (∀X∈sets M. X ⊆ Y ⟶ N X ≤ M X) ∧ (∀X∈sets M. X ∩ Y = {} ⟶ M X ≤ N X)" 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:sets M. d X)" have le_γ[intro]: "X ∈ sets M ⟹ d X ≤ γ" for X by (auto simp: γ_def intro!: cSUP_upper) have "∃f. ∀n. f n ∈ sets M ∧ d (f n) > γ - 1 / 2^n" proof (intro choice_iff[THEN iffD1] allI) fix n have "∃X∈sets M. γ - 1 / 2^n < d X" unfolding γ_def by (intro less_cSUP_iff[THEN iffD1]) auto then show "∃y. y ∈ sets M ∧ γ - 1 / 2 ^ n < d y" by auto qed then obtain E where [measurable]: "E n ∈ sets M" and E: "d (E n) > γ - 1 / 2^n" for n 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 ≤ d (F m n)" if "m ≤ n" for m n 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 ≤ i› by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union') 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)" proof (rule LIMSEQ_le) show "(λn. γ - 2 / 2 ^ n) ⇢ γ - 0" by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto have "incseq F'" by (auto simp: incseq_def F'_def) then show "(λm. d (F' m)) ⇢ d (⋃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 "(λn. d (F m 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 "(λn. γ - 2 / 2 ^ m + 1 / 2 ^ n) ⇢ γ - 2 / 2 ^ m + 0" 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 qed proposition unsigned_Hahn_decomposition: assumes [simp]: "sets N = sets M" and [measurable]: "A ∈ sets M" and [simp]: "emeasure M A ≠ top" "emeasure N A ≠ top" 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)" proof - have "∃Y∈sets (restrict_space M A). (∀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) then guess Y .. then show ?thesis apply (intro bexI[of _ Y] conjI ballI conjI) apply (simp_all add: sets_restrict_space emeasure_restrict_space) apply safe subgoal for X Z by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1) subgoal for X Z by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1 ac_simps) apply auto done qed text%important ‹ Define a lexicographical order on @{type measure}, in the order space, sets and measure. The parts of the lexicographical order are point-wise ordered. › instantiation measure :: (type) order_bot begin inductive less_eq_measure :: "'a measure ⇒ 'a measure ⇒ bool" where "space M ⊂ space N ⟹ less_eq_measure M N" | "space M = space N ⟹ sets M ⊂ sets N ⟹ less_eq_measure M N" | "space M = space N ⟹ sets M = sets N ⟹ emeasure M ≤ emeasure N ⟹ less_eq_measure M N" lemma le_measure_iff: "M ≤ N ⟷ (if space M = space N then 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) definition%important less_measure :: "'a measure ⇒ 'a measure ⇒ bool" where "less_measure M N ⟷ (M ≤ N ∧ ¬ N ≤ M)" definition%important bot_measure :: "'a measure" where "bot_measure = sigma {} {}" lemma shows space_bot[simp]: "space bot = {}" 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 :: "'a measure" by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def) 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)" apply - apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq) subgoal for X by (cases "X ∈ sets M") (auto simp: emeasure_notin_sets) done definition%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 = "λX Y. emeasure A (X ∩ Y) + emeasure B (X ∩ - Y)" 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" by auto 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 (X i) = top ∨ emeasure B (X i) = top" then show ?thesis proof assume "emeasure A (X i) = top" then show ?thesis by (intro bexI[of _ "X i"]) auto next assume "emeasure B (X i) = top" then show ?thesis by (intro bexI[of _ "{}"]) auto qed next assume finite: "¬ (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[of _ Y] 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 proof safe fix x j assume "x ∈ X i" "x ∈ C j" moreover have "i = j ∨ X i ∩ X j = {}" using ‹disjoint_family X› by (auto simp: disjoint_family_on_def) ultimately show "x ∈ C i" using ‹C i ⊆ X i› ‹C j ⊆ X j› by auto qed auto have **: "X i ∩ - (⋃i. C i) = X i ∩ - C i" for i proof safe fix x j assume "x ∈ X i" "x ∉ C i" "x ∈ C j" moreover have "i = j ∨ X i ∩ X j = {}" using ‹disjoint_family X› by (auto simp: disjoint_family_on_def) ultimately show False using ‹C i ⊆ X i› ‹C j ⊆ X j› by auto qed auto show "∃c∈sets A. ∀i∈J. ?d (X i) a ≤ ?d (X i) c ∧ ?d (X i) b ≤ ?d (X i) c" apply (intro bexI[of _ "⋃i. C i"]) unfolding * ** apply (intro C ballI conjI) apply auto done qed also have "… = ?S (⋃i. X i)" unfolding UN_extend_simps(4) by (auto simp add: suminf_add[symmetric] Diff_eq[symmetric] simp del: UN_simps intro!: SUP_cong arg_cong2[where f="(+)"] suminf_emeasure disjoint_family_on_bisimulation[OF ‹disjoint_family X›]) 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 ⟹ emeasure B Y ≤ emeasure C Y" shows "emeasure (sup_measure' A B) X ≤ emeasure C X" proof (subst emeasure_sup_measure') show "(SUP Y:sets A. emeasure A (X ∩ Y) + emeasure B (X ∩ - Y)) ≤ emeasure C X" 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%important sup_lexord :: "'a ⇒ 'a ⇒ ('a ⇒ 'b::order) ⇒ 'a ⇒ 'a ⇒ 'a" where "sup_lexord A B k s c = (if k A = k B then c else if ¬ k A ≤ k B ∧ ¬ k B ≤ k A then s else if k B ≤ k A then A else B)" lemma sup_lexord: "(k A < k B ⟹ P B) ⟹ (k B < k A ⟹ P A) ⟹ (k A = k B ⟹ P c) ⟹ (¬ k B ≤ k A ⟹ ¬ k A ≤ k B ⟹ P s) ⟹ P (sup_lexord A B k s c)" 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" by (simp add: sup_lexord_def) 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 x] by auto lemma sigma_le_iff: "𝒜 ⊆ Pow Ω ⟹ sigma Ω 𝒜 ≤ x ⟷ (Ω ⊆ space x ∧ (space x = Ω ⟶ 𝒜 ⊆ sets x))" by (cases "Ω = space x") (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def sigma_sets_superset_generator sigma_sets_le_sets_iff) instantiation measure :: (type) semilattice_sup begin definition%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 ≤ 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 x)" using sets.space_closed by auto assume "¬ sets y ⊆ sets x" "¬ sets x ⊆ sets y" 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)" by (simp add: space_measure_of[OF *] less_eq_measure.intros(2)) next assume "¬ space y ⊆ space x" "¬ space x ⊆ space y" then show "x ≤ sigma (space x ∪ space y) {}" by (intro less_eq_measure.intros) auto next assume "sets x = sets y" then show "x ≤ sup_measure' x y" 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)" by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2)) 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 "x ≤ y ⟹ z ≤ y ⟹ sup x z ≤ y" 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 add: 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: le_measure_iff split: if_split_asm) 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 add: 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: "UNION S sets ⊆ Pow (UNION S space)" using sets.space_closed by auto definition Sup_lexord :: "('a ⇒ 'b::complete_lattice) ⇒ ('a set ⇒ 'a) ⇒ ('a set ⇒ 'a) ⇒ 'a set ⇒ 'a" where "Sup_lexord k c s A = (let U = (SUP a:A. k a) in if ∃a∈A. k a = U then c {a∈A. k a = U} else s A)" lemma Sup_lexord: "(⋀a S. a ∈ A ⟹ k a = (SUP a:A. k a) ⟹ S = {a'∈A. k a' = k a} ⟹ P (c S)) ⟹ ((⋀a. a ∈ A ⟹ k a ≠ (SUP a:A. k a)) ⟹ P (s A)) ⟹ P (Sup_lexord k c s A)" by (auto simp: Sup_lexord_def Let_def) lemma Sup_lexord1: assumes A: "A ≠ {}" "(⋀a. a ∈ A ⟹ k a = (⋃a∈A. k a))" "P (c A)" shows "P (Sup_lexord k c s A)" unfolding Sup_lexord_def Let_def proof (clarsimp, safe) show "∀a∈A. k a ≠ (⋃x∈A. k x) ⟹ P (s A)" by (metis assms(1,2) ex_in_conv) next fix a assume "a ∈ A" "k a = (⋃x∈A. k x)" then have "{a ∈ A. k a = (⋃x∈A. k x)} = {a ∈ A. k a = k a}" by (metis A(2)[symmetric]) then show "P (c {a ∈ A. k a = (⋃x∈A. k x)})" by (simp add: A(3)) 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 ⟹ finite I ⟹ sup_measure.F id I ≤ sup_measure.F id (I ∪ J)" proof (induction J rule: finite_induct) case empty then show ?case by simp next case (insert i J) show ?case proof cases assume "i ∈ I" with insert show ?thesis by (auto simp: insert_absorb) next assume "i ∉ I" have "sup_measure.F id I ≤ sup_measure.F id (I ∪ J)" by (intro insert) also have "… ≤ sup_measure.F id (insert i (I ∪ J))" using insert ‹i ∉ I› by (subst sup_measure.insert) auto finally show ?thesis by auto qed qed lemma sup_measure_F_mono: "finite I ⟹ J ⊆ I ⟹ sup_measure.F id J ≤ 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 ⟹ I ≠ {} ⟹ (⋀i. i ∈ I ⟹ sets i = sets M) ⟹ sets (sup_measure.F id I) = sets M" by (induction I rule: finite_ne_induct) (simp_all add: sets_sup) definition%important Sup_measure' :: "'a measure set ⇒ 'a measure" where "Sup_measure' M = measure_of (⋃a∈M. space a) (⋃a∈M. sets a) (λX. (SUP P:{P. finite P ∧ P ⊆ M }. sup_measure.F id P X))" lemma space_Sup_measure'2: "space (Sup_measure' M) = (⋃m∈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 (⋃m∈M. space m) (⋃m∈M. sets m)" unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed]) lemma sets_Sup_measure': assumes sets_eq[simp]: "⋀m. m ∈ M ⟹ sets m = sets A" and "M ≠ {}" shows "sets (Sup_measure' M) = sets A" using sets_eq[THEN sets_eq_imp_space_eq, simp] ‹M ≠ {}› by (simp add: Sup_measure'_def) lemma space_Sup_measure': assumes sets_eq[simp]: "⋀m. m ∈ M ⟹ sets m = sets A" and "M ≠ {}" shows "space (Sup_measure' M) = space A" using sets_eq[THEN sets_eq_imp_space_eq, simp] ‹M ≠ {}› by (simp add: Sup_measure'_def ) lemma emeasure_Sup_measure': assumes sets_eq[simp]: "⋀m. m ∈ M ⟹ sets m = sets A" and "X ∈ sets A" "M ≠ {}" shows "emeasure (Sup_measure' M) X = (SUP P:{P. finite P ∧ P ⊆ 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] have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A" using ‹M ≠ {}› by (simp_all add: Sup_measure'_def) let ?μ = "sup_measure.F id" show "countably_additive (sets (Sup_measure' M)) ?S" proof (rule countably_additiveI, goal_cases) case (1 F) then have **: "range F ⊆ sets A" by (auto simp: *) show "(∑i. ?S (F i)) = ?S (⋃i. F i)" proof (subst ennreal_suminf_SUP_eq_directed) fix i j and N :: "nat set" assume ij: "i ∈ {P. finite P ∧ P ⊆ M}" "j ∈ {P. finite P ∧ P ⊆ M}" have "(i ≠ {} ⟶ sets (?μ i) = sets A) ∧ (j ≠ {} ⟶ sets (?μ j) = sets A) ∧ (i ≠ {} ∨ j ≠ {} ⟶ sets (?μ (i ∪ j)) = sets A)" using ij by (intro impI sets_sup_measure_F conjI) auto then have "?μ j (F n) ≤ ?μ (i ∪ j) (F n) ∧ ?μ i (F n) ≤ ?μ (i ∪ 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 "∃k∈{P. finite P ∧ P ⊆ M}. ∀n∈N. ?μ i (F n) ≤ ?μ k (F n) ∧ ?μ j (F n) ≤ ?μ k (F n)" by (safe intro!: bexI[of _ "i ∪ j"]) auto next show "(SUP P : {P. finite P ∧ P ⊆ M}. ∑n. ?μ P (F n)) = (SUP P : {P. finite P ∧ P ⊆ M}. ?μ P (UNION UNIV F))" proof (intro SUP_cong refl) fix i assume i: "i ∈ {P. finite P ∧ P ⊆ M}" show "(∑n. ?μ i (F n)) = ?μ i (UNION UNIV F)" proof cases assume "i ≠ {}" with i ** show ?thesis apply (intro suminf_emeasure ‹disjoint_family F›) apply (subst sets_sup_measure_F[OF _ _ sets_eq]) apply auto done qed simp qed qed qed show "positive (sets (Sup_measure' M)) ?S" by (auto simp: positive_def bot_ennreal[symmetric]) show "X ∈ sets (Sup_measure' M)" using assms * by auto qed (rule UN_space_closed) definition%important Sup_measure :: "'a measure set ⇒ 'a measure" where "Sup_measure = Sup_lexord space (Sup_lexord sets Sup_measure' (λU. sigma (⋃u∈U. space u) (⋃u∈U. sets u))) (λU. sigma (⋃u∈U. space u) {})" definition%important Inf_measure :: "'a measure set ⇒ 'a measure" where "Inf_measure A = Sup {x. ∀a∈A. x ≤ a}" definition%important inf_measure :: "'a measure ⇒ 'a measure ⇒ 'a measure" where "inf_measure a b = Inf {a, b}" definition%important top_measure :: "'a measure" where "top_measure = Inf {}" instance proof note UN_space_closed [simp] show upper: "x ≤ Sup A" if x: "x ∈ A" for x :: "'a measure" and A unfolding Sup_measure_def proof (intro Sup_lexord[where P="λy. x ≤ y"]) assume "⋀a. a ∈ A ⟹ space a ≠ (⋃a∈A. space a)" from this[OF ‹x ∈ A›] ‹x ∈ A› show "x ≤ sigma (⋃a∈A. space a) {}" by (intro less_eq_measure.intros) auto next fix a S assume "a ∈ A" and a: "space a = (⋃a∈A. space a)" and S: "S = {a' ∈ A. space a' = space a}" and neq: "⋀aa. aa ∈ S ⟹ sets aa ≠ (⋃a∈S. sets a)" have sp_a: "space a = (UNION S space)" using ‹a∈A› by (auto simp: S) show "x ≤ sigma (UNION S space) (UNION S sets)" proof cases assume [simp]: "space x = space a" have "sets x ⊂ (⋃a∈S. sets a)" using ‹x∈A› neq[of x] by (auto simp: S) also have "… ⊆ sigma_sets (⋃x∈S. space x) (⋃x∈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 ≠ space a" moreover have "space x ≤ space a" unfolding a using ‹x∈A› 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 ∈ A" and a: "space a = (⋃a∈A. space a)" and S: "S = {a' ∈ A. space a' = space a}" and "b ∈ S" and b: "sets b = (⋃a∈S. sets a)" and S': "S' = {a' ∈ S. sets a' = sets b}" then have "S' ≠ {}" "space b = space a" by auto have sets_eq: "⋀x. x ∈ S' ⟹ 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 ‹S' ≠ {}› by (simp_all add: Sup_measure'_def sets_eq) show "x ≤ Sup_measure' S'" proof cases assume "x ∈ S" with ‹b ∈ S› have "space x = space b" by (simp add: S) show ?thesis proof cases assume "x ∈ S'" show "x ≤ Sup_measure' S'" proof (intro le_measure[THEN iffD2] ballI) show "sets x = sets (Sup_measure' S')" using ‹x∈S'› * by (simp add: S') fix X assume "X ∈ sets x" show "emeasure x X ≤ emeasure (Sup_measure' S') X" proof (subst emeasure_Sup_measure'[OF _ ‹X ∈ sets x›]) show "emeasure x X ≤ (SUP P : {P. finite P ∧ P ⊆ S'}. emeasure (sup_measure.F id P) X)" using ‹x∈S'› by (intro SUP_upper2[where i="{x}"]) auto qed (insert ‹x∈S'› S', auto) qed next assume "x ∉ S'" then have "sets x ≠ sets b" using ‹x∈S› by (auto simp: S') moreover have "sets x ≤ sets b" using ‹x∈S› unfolding b by auto ultimately show ?thesis using * ‹x ∈ S› by (intro less_eq_measure.intros(2)) (simp_all add: * ‹space x = space b› less_le) qed next assume "x ∉ S" with ‹x∈A› ‹x ∉ S› ‹space b = space a› show ?thesis by (intro less_eq_measure.intros) (simp_all add: * less_le a SUP_upper S) qed qed show least: "Sup A ≤ x" if x: "⋀z. z ∈ A ⟹ z ≤ x" for x :: "'a measure" and A unfolding Sup_measure_def proof (intro Sup_lexord[where P="λy. y ≤ x"]) assume "⋀a. a ∈ A ⟹ space a ≠ (⋃a∈A. space a)" show "sigma (UNION A space) {} ≤ x" using x[THEN le_measureD1] by (subst sigma_le_iff) auto next fix a S assume "a ∈ A" "space a = (⋃a∈A. space a)" and S: "S = {a' ∈ A. space a' = space a}" "⋀a. a ∈ S ⟹ sets a ≠ (⋃a∈S. sets a)" have "UNION S space ⊆ space x" using S le_measureD1[OF x] by auto moreover have "UNION S space = space a" using ‹a∈A› S by auto then have "space x = UNION S space ⟹ UNION S sets ⊆ sets x" using ‹a ∈ A› le_measureD2[OF x] by (auto simp: S) ultimately show "sigma (UNION S space) (UNION S sets) ≤ x" by (subst sigma_le_iff) simp_all next fix a b S S' assume "a ∈ A" and a: "space a = (⋃a∈A. space a)" and S: "S = {a' ∈ A. space a' = space a}" and "b ∈ S" and b: "sets b = (⋃a∈S. sets a)" and S': "S' = {a' ∈ S. sets a' = sets b}" then have "S' ≠ {}" "space b = space a" by auto have sets_eq: "⋀x. x ∈ S' ⟹ 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 ‹S' ≠ {}› by (simp_all add: Sup_measure'_def sets_eq) show "Sup_measure' S' ≤ 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" by (simp add: * **) fix X assume "X ∈ sets (Sup_measure' S')" show "emeasure (Sup_measure' S') X ≤ emeasure x X" unfolding *** proof (subst emeasure_Sup_measure'[OF _ ‹X ∈ sets (Sup_measure' S')›]) show "(SUP P : {P. finite P ∧ P ⊆ S'}. emeasure (sup_measure.F id P) X) ≤ emeasure x X" proof (safe intro!: SUP_least) fix P assume P: "finite P" "P ⊆ S'" show "emeasure (sup_measure.F id P) X ≤ emeasure x X" proof cases assume "P = {}" then show ?thesis by auto next assume "P ≠ {}" from P have "finite P" "P ⊆ A" unfolding S' S by (simp_all add: subset_eq) then have "sup_measure.F id P ≤ x" by (induction P) (auto simp: x) moreover have "sets (sup_measure.F id P) = sets x" using ‹finite P› ‹P ≠ {}› ‹P ⊆ S'› ‹sets x = sets b› by (intro sets_sup_measure_F) (auto simp: S') ultimately show "emeasure (sup_measure.F id P) X ≤ emeasure x X" by (rule le_measureD3) qed qed show "m ∈ S' ⟹ sets m = sets (Sup_measure' S')" for m unfolding * by (simp add: S') qed fact qed next assume "sets x ≠ sets b" moreover have "sets b ≤ sets x" unfolding b S using x[THEN le_measureD2] ‹space x = space a› by auto ultimately show "Sup_measure' S' ≤ x" using ‹space x = space a› ‹b ∈ S› by (intro less_eq_measure.intros(2)) (simp_all add: * S) qed next assume "space x ≠ space a" then have "space a < space x" using le_measureD1[OF x[OF ‹a∈A›]] by auto then show "Sup_measure' S' ≤ x" by (intro less_eq_measure.intros) (simp add: * ‹space b = space a›) qed qed show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)" by (auto intro!: antisym least simp: top_measure_def) show lower: "x ∈ A ⟹ Inf A ≤ x" for x :: "'a measure" and A unfolding Inf_measure_def by (intro least) auto show greatest: "(⋀z. z ∈ A ⟹ x ≤ z) ⟹ x ≤ Inf A" for x :: "'a measure" and A unfolding Inf_measure_def by (intro upper) auto show "inf x y ≤ x" "inf x y ≤ y" "x ≤ y ⟹ x ≤ z ⟹ x ≤ inf y z" for x y z :: "'a measure" by (auto simp: inf_measure_def intro!: lower greatest) qed end lemma sets_SUP: assumes "⋀x. x ∈ I ⟹ sets (M x) = sets N" shows "I ≠ {} ⟹ sets (SUP i: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="λx. sets x = sets N"]) auto lemma emeasure_SUP: assumes sets: "⋀i. i ∈ I ⟹ sets (M i) = sets N" "X ∈ sets N" "I ≠ {}" shows "emeasure (SUP i:I. M i) X = (SUP J:{J. J ≠ {} ∧ finite J ∧ J ⊆ I}. emeasure (SUP i:J. M i) X)" proof - interpret sup_measure: comm_monoid_set sup "bot :: 'b measure" by standard (auto intro!: antisym) have eq: "finite J ⟹ sup_measure.F id J = (SUP i:J. i)" for J :: "'b measure set" by (induction J rule: finite_induct) auto have 1: "J ≠ {} ⟹ J ⊆ I ⟹ sets (SUP x:J. M x) = sets N" for J by (intro sets_SUP sets) (auto ) from ‹I ≠ {}› obtain i where "i∈I" by auto have "Sup_measure' (M`I) X = (SUP P:{P. finite P ∧ P ⊆ M`I}. sup_measure.F id P X)" using sets by (intro emeasure_Sup_measure') auto also have "Sup_measure' (M`I) = (SUP i:I. M i)" unfolding Sup_measure_def using ‹I ≠ {}› sets sets(1)[THEN sets_eq_imp_space_eq] by (intro Sup_lexord1[where P="λx. _ = x"]) auto also have "(SUP P:{P. finite P ∧ P ⊆ M`I}. sup_measure.F id P X) = (SUP J:{J. J ≠ {} ∧ finite J ∧ J ⊆ I}. (SUP i:J. M i) X)" proof (intro SUP_eq) fix J assume "J ∈ {P. finite P ∧ P ⊆ M`I}" then obtain J' where J': "J' ⊆ I" "finite J'" and J: "J = M`J'" and "finite J" using finite_subset_image[of J M I] by auto show "∃j∈{J. J ≠ {} ∧ finite J ∧ J ⊆ I}. sup_measure.F id J X ≤ (SUP i:j. M i) X" proof cases assume "J' = {}" with ‹i ∈ I› show ?thesis by (auto simp add: J) next assume "J' ≠ {}" with J J' show ?thesis by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply) qed next fix J assume J: "J ∈ {P. P ≠ {} ∧ finite P ∧ P ⊆ I}" show "∃J'∈{J. finite J ∧ J ⊆ M`I}. (SUP i:J. M i) X ≤ sup_measure.F id J' X" using J by (intro bexI[of _ "M`J"]) (auto simp add: eq simp del: id_apply) qed finally show ?thesis . qed lemma emeasure_SUP_chain: assumes sets: "⋀i. i ∈ A ⟹ sets (M i) = sets N" "X ∈ sets N" assumes ch: "Complete_Partial_Order.chain (≤) (M ` A)" and "A ≠ {}" shows "emeasure (SUP i:A. M i) X = (SUP i:A. emeasure (M i) X)" proof (subst emeasure_SUP[OF sets ‹A ≠ {}›]) show "(SUP J:{J. J ≠ {} ∧ finite J ∧ J ⊆ A}. emeasure (SUPREMUM J M) X) = (SUP i:A. emeasure (M i) X)" proof (rule SUP_eq) fix J assume "J ∈ {J. J ≠ {} ∧ finite J ∧ J ⊆ A}" then have J: "Complete_Partial_Order.chain (≤) (M ` J)" "finite J" "J ≠ {}" and "J ⊆ A" using ch[THEN chain_subset, of "M`J"] by auto with in_chain_finite[OF J(1)] obtain j where "j ∈ J" "(SUP j:J. M j) = M j" by auto with ‹J ⊆ A› show "∃j∈A. emeasure (SUPREMUM J M) X ≤ emeasure (M j) X" by auto next fix j assume "j∈A" then show "∃i∈{J. J ≠ {} ∧ finite J ∧ J ⊆ A}. emeasure (M j) X ≤ emeasure (SUPREMUM i M) X" by (intro bexI[of _ "{j}"]) auto qed qed subsubsection%unimportant ‹Supremum of a set of $\sigma$-algebras› lemma space_Sup_eq_UN: "space (Sup M) = (⋃x∈M. space x)" unfolding Sup_measure_def apply (intro Sup_lexord[where P="λx. space x = _"]) apply (subst space_Sup_measure'2) apply auto [] apply (subst space_measure_of[OF UN_space_closed]) apply auto done lemma sets_Sup_eq: assumes *: "⋀m. m ∈ M ⟹ space m = X" and "M ≠ {}" shows "sets (Sup M) = sigma_sets X (⋃x∈M. sets x)" unfolding Sup_measure_def apply (rule Sup_lexord1) apply fact apply (simp add: assms) apply (rule Sup_lexord) subgoal premises that for a S unfolding that(3) that(2)[symmetric] using that(1) apply (subst sets_Sup_measure'2) apply (intro arg_cong2[where f=sigma_sets]) apply (auto simp: *) done apply (subst sets_measure_of[OF UN_space_closed]) apply (simp add: assms) done lemma in_sets_Sup: "(⋀m. m ∈ M ⟹ space m = X) ⟹ m ∈ M ⟹ A ∈ sets m ⟹ A ∈ sets (Sup M)" by (subst sets_Sup_eq[where X=X]) auto lemma Sup_lexord_rel: assumes "⋀i. i ∈ I ⟹ k (A i) = k (B i)" "R (c (A ` {a ∈ I. k (B a) = (SUP x:I. k (B x))})) (c (B ` {a ∈ I. k (B a) = (SUP x: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 ∈ I. k (B a) = (SUP x:I. k (B x))} = {a ∈ A ` I. k a = (SUP x:I. k (B x))}" using assms(1) by auto moreover have "B ` {a ∈ I. k (B a) = (SUP x:I. k (B x))} = {a ∈ B ` I. k a = (SUP x:I. k (B x))}" by auto ultimately show ?thesis using assms by (auto simp: Sup_lexord_def Let_def) qed lemma sets_SUP_cong: assumes eq: "⋀i. i ∈ I ⟹ sets (M i) = sets (N i)" shows "sets (SUP i:I. M i) = sets (SUP i:I. N i)" unfolding Sup_measure_def using eq eq[THEN sets_eq_imp_space_eq] apply (intro Sup_lexord_rel[where R="λx y. sets x = sets y"]) apply simp apply simp apply (simp add: sets_Sup_measure'2) apply (intro arg_cong2[where f="λx y. sets (sigma x y)"]) apply auto done lemma sets_Sup_in_sets: assumes "M ≠ {}" assumes "⋀m. m ∈ M ⟹ space m = space N" assumes "⋀m. m ∈ M ⟹ sets m ⊆ sets N" shows "sets (Sup M) ⊆ sets N" proof - have *: "UNION M space = 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 ∈ M" and f: "f ∈ measurable m N" and const_space: "⋀m n. m ∈ M ⟹ n ∈ M ⟹ space m = space n" shows "f ∈ measurable (Sup M) N" proof - have "space (Sup M) = space m" using m by (auto simp add: 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 ≠ {}" assumes f: "⋀m. m ∈ M ⟹ f ∈ measurable N m" and const_space: "⋀m n. m ∈ M ⟹ n ∈ M ⟹ space m = space n" shows "f ∈ measurable N (Sup M)" proof - from M obtain m where "m ∈ M" by auto have space_eq: "⋀n. n ∈ M ⟹ space n = space m" by (intro const_space ‹m ∈ M›) have "f ∈ measurable N (sigma (⋃m∈M. space m) (⋃m∈M. sets m))" proof (rule measurable_measure_of) show "f ∈ space N → UNION M space" using measurable_space[OF f] M by auto qed (auto intro: measurable_sets f dest: sets.sets_into_space) also have "measurable N (sigma (⋃m∈M. space m) (⋃m∈M. sets m)) = measurable N (Sup M)" apply (intro measurable_cong_sets refl) apply (subst sets_Sup_eq[OF space_eq M]) apply simp apply (subst sets_measure_of[OF UN_space_closed]) apply (simp add: space_eq M) done finally show ?thesis . qed lemma measurable_SUP2: "I ≠ {} ⟹ (⋀i. i ∈ I ⟹ f ∈ measurable N (M i)) ⟹ (⋀i j. i ∈ I ⟹ j ∈ I ⟹ space (M i) = space (M j)) ⟹ f ∈ measurable N (SUP i:I. M i)" by (auto intro!: measurable_Sup2) lemma sets_Sup_sigma: assumes [simp]: "M ≠ {}" and M: "⋀m. m ∈ M ⟹ m ⊆ Pow Ω" shows "sets (SUP m:M. sigma Ω m) = sets (sigma Ω (⋃M))" proof - { fix a m assume "a ∈ sigma_sets Ω m" "m ∈ M" then have "a ∈ sigma_sets Ω (⋃M)" by induction (auto intro: sigma_sets.intros(2-)) } then show "sets (SUP m:M. sigma Ω m) = sets (sigma Ω (⋃M))" apply (subst sets_Sup_eq[where X="Ω"]) apply (auto simp add: M) [] apply auto [] apply (simp add: space_measure_of_conv M Union_least) apply (rule sigma_sets_eqI) apply auto done qed lemma Sup_sigma: assumes [simp]: "M ≠ {}" and M: "⋀m. m ∈ M ⟹ m ⊆ Pow Ω" shows "(SUP m:M. sigma Ω m) = (sigma Ω (⋃M))" proof (intro antisym SUP_least) have *: "⋃M ⊆ Pow Ω" using M by auto show "sigma Ω (⋃M) ≤ (SUP m:M. sigma Ω m)" proof (intro less_eq_measure.intros(3)) show "space (sigma Ω (⋃M)) = space (SUP m:M. sigma Ω m)" "sets (sigma Ω (⋃M)) = sets (SUP m:M. sigma Ω m)" using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq] by auto qed (simp add: emeasure_sigma le_fun_def) fix m assume "m ∈ M" then show "sigma Ω m ≤ sigma Ω (⋃M)" by (subst sigma_le_iff) (auto simp add: M *) qed lemma SUP_sigma_sigma: "M ≠ {} ⟹ (⋀m. m ∈ M ⟹ f m ⊆ Pow Ω) ⟹ (SUP m:M. sigma Ω (f m)) = sigma Ω (⋃m∈M. f m)" using Sup_sigma[of "f`M" Ω] by auto lemma sets_vimage_Sup_eq: assumes *: "M ≠ {}" "f ∈ X → Y" "⋀m. m ∈ M ⟹ space m = Y" shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m : M. vimage_algebra X f m)" (is "?IS = ?SI") proof show "?IS ⊆ ?SI" apply (intro sets_image_in_sets measurable_Sup2) apply (simp add: space_Sup_eq_UN *) apply (simp add: *) apply