(* Title: HOL/Analysis/Sigma_Algebra.thy Author: Stefan Richter, Markus Wenzel, TU München Author: Johannes Hölzl, TU München Plus material from the Hurd/Coble measure theory development, translated by Lawrence Paulson. *) chapter ‹Measure and Integration Theory› theory Sigma_Algebra imports Complex_Main "HOL-Library.Countable_Set" "HOL-Library.FuncSet" "HOL-Library.Indicator_Function" "HOL-Library.Extended_Nonnegative_Real" "HOL-Library.Disjoint_Sets" begin section ‹Sigma Algebra› text ‹Sigma algebras are an elementary concept in measure theory. To measure --- that is to integrate --- functions, we first have to measure sets. Unfortunately, when dealing with a large universe, it is often not possible to consistently assign a measure to every subset. Therefore it is necessary to define the set of measurable subsets of the universe. A sigma algebra is such a set that has three very natural and desirable properties.› subsection ‹Families of sets› locale✐‹tag important› subset_class = fixes Ω :: "'a set" and M :: "'a set set" assumes space_closed: "M ⊆ Pow Ω" lemma (in subset_class) sets_into_space: "x ∈ M ⟹ x ⊆ Ω" by (metis PowD contra_subsetD space_closed) subsubsection ‹Semiring of sets› locale✐‹tag important› semiring_of_sets = subset_class + assumes empty_sets[iff]: "{} ∈ M" assumes Int[intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ∩ b ∈ M" assumes Diff_cover: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ ∃C⊆M. finite C ∧ disjoint C ∧ a - b = ⋃C" lemma (in semiring_of_sets) finite_INT[intro]: assumes "finite I" "I ≠ {}" "⋀i. i ∈ I ⟹ A i ∈ M" shows "(⋂i∈I. A i) ∈ M" using assms by (induct rule: finite_ne_induct) auto lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x ∈ M ⟹ Ω ∩ x = x" by (metis Int_absorb1 sets_into_space) lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x ∈ M ⟹ x ∩ Ω = x" by (metis Int_absorb2 sets_into_space) lemma (in semiring_of_sets) sets_Collect_conj: assumes "{x∈Ω. P x} ∈ M" "{x∈Ω. Q x} ∈ M" shows "{x∈Ω. Q x ∧ P x} ∈ M" proof - have "{x∈Ω. Q x ∧ P x} = {x∈Ω. Q x} ∩ {x∈Ω. P x}" by auto with assms show ?thesis by auto qed lemma (in semiring_of_sets) sets_Collect_finite_All': assumes "⋀i. i ∈ S ⟹ {x∈Ω. P i x} ∈ M" "finite S" "S ≠ {}" shows "{x∈Ω. ∀i∈S. P i x} ∈ M" proof - have "{x∈Ω. ∀i∈S. P i x} = (⋂i∈S. {x∈Ω. P i x})" using ‹S ≠ {}› by auto with assms show ?thesis by auto qed subsubsection ‹Ring of sets› locale✐‹tag important› ring_of_sets = semiring_of_sets + assumes Un [intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ∪ b ∈ M" lemma (in ring_of_sets) finite_Union [intro]: "finite X ⟹ X ⊆ M ⟹ ⋃X ∈ M" by (induct set: finite) (auto simp add: Un) lemma (in ring_of_sets) finite_UN[intro]: assumes "finite I" and "⋀i. i ∈ I ⟹ A i ∈ M" shows "(⋃i∈I. A i) ∈ M" using assms by induct auto lemma (in ring_of_sets) Diff [intro]: assumes "a ∈ M" "b ∈ M" shows "a - b ∈ M" using Diff_cover[OF assms] by auto lemma ring_of_setsI: assumes space_closed: "M ⊆ Pow Ω" assumes empty_sets[iff]: "{} ∈ M" assumes Un[intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ∪ b ∈ M" assumes Diff[intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a - b ∈ M" shows "ring_of_sets Ω M" proof fix a b assume ab: "a ∈ M" "b ∈ M" from ab show "∃C⊆M. finite C ∧ disjoint C ∧ a - b = ⋃C" by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def) have "a ∩ b = a - (a - b)" by auto also have "… ∈ M" using ab by auto finally show "a ∩ b ∈ M" . qed fact+ lemma ring_of_sets_iff: "ring_of_sets Ω M ⟷ M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀a∈M. ∀b∈M. a ∪ b ∈ M) ∧ (∀a∈M. ∀b∈M. a - b ∈ M)" proof assume "ring_of_sets Ω M" then interpret ring_of_sets Ω M . show "M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀a∈M. ∀b∈M. a ∪ b ∈ M) ∧ (∀a∈M. ∀b∈M. a - b ∈ M)" using space_closed by auto qed (auto intro!: ring_of_setsI) lemma (in ring_of_sets) insert_in_sets: assumes "{x} ∈ M" "A ∈ M" shows "insert x A ∈ M" proof - have "{x} ∪ A ∈ M" using assms by (rule Un) thus ?thesis by auto qed lemma (in ring_of_sets) sets_Collect_disj: assumes "{x∈Ω. P x} ∈ M" "{x∈Ω. Q x} ∈ M" shows "{x∈Ω. Q x ∨ P x} ∈ M" proof - have "{x∈Ω. Q x ∨ P x} = {x∈Ω. Q x} ∪ {x∈Ω. P x}" by auto with assms show ?thesis by auto qed lemma (in ring_of_sets) sets_Collect_finite_Ex: assumes "⋀i. i ∈ S ⟹ {x∈Ω. P i x} ∈ M" "finite S" shows "{x∈Ω. ∃i∈S. P i x} ∈ M" proof - have "{x∈Ω. ∃i∈S. P i x} = (⋃i∈S. {x∈Ω. P i x})" by auto with assms show ?thesis by auto qed subsubsection ‹Algebra of sets› locale✐‹tag important› algebra = ring_of_sets + assumes top [iff]: "Ω ∈ M" lemma (in algebra) compl_sets [intro]: "a ∈ M ⟹ Ω - a ∈ M" by auto proposition algebra_iff_Un: "algebra Ω M ⟷ M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀a ∈ M. Ω - a ∈ M) ∧ (∀a ∈ M. ∀ b ∈ M. a ∪ b ∈ M)" (is "_ ⟷ ?Un") proof assume "algebra Ω M" then interpret algebra Ω M . show ?Un using sets_into_space by auto next assume ?Un then have "Ω ∈ M" by auto interpret ring_of_sets Ω M proof (rule ring_of_setsI) show Ω: "M ⊆ Pow Ω" "{} ∈ M" using ‹?Un› by auto fix a b assume a: "a ∈ M" and b: "b ∈ M" then show "a ∪ b ∈ M" using ‹?Un› by auto have "a - b = Ω - ((Ω - a) ∪ b)" using Ω a b by auto then show "a - b ∈ M" using a b ‹?Un› by auto qed show "algebra Ω M" proof qed fact qed proposition algebra_iff_Int: "algebra Ω M ⟷ M ⊆ Pow Ω & {} ∈ M & (∀a ∈ M. Ω - a ∈ M) & (∀a ∈ M. ∀ b ∈ M. a ∩ b ∈ M)" (is "_ ⟷ ?Int") proof assume "algebra Ω M" then interpret algebra Ω M . show ?Int using sets_into_space by auto next assume ?Int show "algebra Ω M" proof (unfold algebra_iff_Un, intro conjI ballI) show Ω: "M ⊆ Pow Ω" "{} ∈ M" using ‹?Int› by auto from ‹?Int› show "⋀a. a ∈ M ⟹ Ω - a ∈ M" by auto fix a b assume M: "a ∈ M" "b ∈ M" hence "a ∪ b = Ω - ((Ω - a) ∩ (Ω - b))" using Ω by blast also have "... ∈ M" using M ‹?Int› by auto finally show "a ∪ b ∈ M" . qed qed lemma (in algebra) sets_Collect_neg: assumes "{x∈Ω. P x} ∈ M" shows "{x∈Ω. ¬ P x} ∈ M" proof - have "{x∈Ω. ¬ P x} = Ω - {x∈Ω. P x}" by auto with assms show ?thesis by auto qed lemma (in algebra) sets_Collect_imp: "{x∈Ω. P x} ∈ M ⟹ {x∈Ω. Q x} ∈ M ⟹ {x∈Ω. Q x ⟶ P x} ∈ M" unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg) lemma (in algebra) sets_Collect_const: "{x∈Ω. P} ∈ M" by (cases P) auto lemma algebra_single_set: "X ⊆ S ⟹ algebra S { {}, X, S - X, S }" by (auto simp: algebra_iff_Int) subsubsection✐‹tag unimportant› ‹Restricted algebras› abbreviation (in algebra) "restricted_space A ≡ ((∩) A) ` M" lemma (in algebra) restricted_algebra: assumes "A ∈ M" shows "algebra A (restricted_space A)" using assms by (auto simp: algebra_iff_Int) subsubsection ‹Sigma Algebras› locale✐‹tag important› sigma_algebra = algebra + assumes countable_nat_UN [intro]: "⋀A. range A ⊆ M ⟹ (⋃i::nat. A i) ∈ M" lemma (in algebra) is_sigma_algebra: assumes "finite M" shows "sigma_algebra Ω M" proof fix A :: "nat ⇒ 'a set" assume "range A ⊆ M" then have "(⋃i. A i) = (⋃s∈M ∩ range A. s)" by auto also have "(⋃s∈M ∩ range A. s) ∈ M" using ‹finite M› by auto finally show "(⋃i. A i) ∈ M" . qed lemma countable_UN_eq: fixes A :: "'i::countable ⇒ 'a set" shows "(range A ⊆ M ⟶ (⋃i. A i) ∈ M) ⟷ (range (A ∘ from_nat) ⊆ M ⟶ (⋃i. (A ∘ from_nat) i) ∈ M)" proof - let ?A' = "A ∘ from_nat" have *: "(⋃i. ?A' i) = (⋃i. A i)" (is "?l = ?r") proof safe fix x i assume "x ∈ A i" thus "x ∈ ?l" by (auto intro!: exI[of _ "to_nat i"]) next fix x i assume "x ∈ ?A' i" thus "x ∈ ?r" by (auto intro!: exI[of _ "from_nat i"]) qed have "A ` range from_nat = range A" using surj_from_nat by simp then have **: "range ?A' = range A" by (simp only: image_comp [symmetric]) show ?thesis unfolding * ** .. qed lemma (in sigma_algebra) countable_Union [intro]: assumes "countable X" "X ⊆ M" shows "⋃X ∈ M" proof cases assume "X ≠ {}" hence "⋃X = (⋃n. from_nat_into X n)" using assms by (auto cong del: SUP_cong) also have "… ∈ M" using assms by (auto intro!: countable_nat_UN) (metis ‹X ≠ {}› from_nat_into subsetD) finally show ?thesis . qed simp lemma (in sigma_algebra) countable_UN[intro]: fixes A :: "'i::countable ⇒ 'a set" assumes "A`X ⊆ M" shows "(⋃x∈X. A x) ∈ M" proof - let ?A = "λi. if i ∈ X then A i else {}" from assms have "range ?A ⊆ M" by auto with countable_nat_UN[of "?A ∘ from_nat"] countable_UN_eq[of ?A M] have "(⋃x. ?A x) ∈ M" by auto moreover have "(⋃x. ?A x) = (⋃x∈X. A x)" by (auto split: if_split_asm) ultimately show ?thesis by simp qed lemma (in sigma_algebra) countable_UN': fixes A :: "'i ⇒ 'a set" assumes X: "countable X" assumes A: "A`X ⊆ M" shows "(⋃x∈X. A x) ∈ M" proof - have "(⋃x∈X. A x) = (⋃i∈to_nat_on X ` X. A (from_nat_into X i))" using X by auto also have "… ∈ M" using A X by (intro countable_UN) auto finally show ?thesis . qed lemma (in sigma_algebra) countable_UN'': "⟦ countable X; ⋀x y. x ∈ X ⟹ A x ∈ M ⟧ ⟹ (⋃x∈X. A x) ∈ M" by(erule countable_UN')(auto) lemma (in sigma_algebra) countable_INT [intro]: fixes A :: "'i::countable ⇒ 'a set" assumes A: "A`X ⊆ M" "X ≠ {}" shows "(⋂i∈X. A i) ∈ M" proof - from A have "∀i∈X. A i ∈ M" by fast hence "Ω - (⋃i∈X. Ω - A i) ∈ M" by blast moreover have "(⋂i∈X. A i) = Ω - (⋃i∈X. Ω - A i)" using space_closed A by blast ultimately show ?thesis by metis qed lemma (in sigma_algebra) countable_INT': fixes A :: "'i ⇒ 'a set" assumes X: "countable X" "X ≠ {}" assumes A: "A`X ⊆ M" shows "(⋂x∈X. A x) ∈ M" proof - have "(⋂x∈X. A x) = (⋂i∈to_nat_on X ` X. A (from_nat_into X i))" using X by auto also have "… ∈ M" using A X by (intro countable_INT) auto finally show ?thesis . qed lemma (in sigma_algebra) countable_INT'': "UNIV ∈ M ⟹ countable I ⟹ (⋀i. i ∈ I ⟹ F i ∈ M) ⟹ (⋂i∈I. F i) ∈ M" by (cases "I = {}") (auto intro: countable_INT') lemma (in sigma_algebra) countable: assumes "⋀a. a ∈ A ⟹ {a} ∈ M" "countable A" shows "A ∈ M" proof - have "(⋃a∈A. {a}) ∈ M" using assms by (intro countable_UN') auto also have "(⋃a∈A. {a}) = A" by auto finally show ?thesis by auto qed lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)" by (auto simp: ring_of_sets_iff) lemma algebra_Pow: "algebra sp (Pow sp)" by (auto simp: algebra_iff_Un) lemma sigma_algebra_iff: "sigma_algebra Ω M ⟷ algebra Ω M ∧ (∀A. range A ⊆ M ⟶ (⋃i::nat. A i) ∈ M)" by (simp add: sigma_algebra_def sigma_algebra_axioms_def) lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)" by (auto simp: sigma_algebra_iff algebra_iff_Int) lemma (in sigma_algebra) sets_Collect_countable_All: assumes "⋀i. {x∈Ω. P i x} ∈ M" shows "{x∈Ω. ∀i::'i::countable. P i x} ∈ M" proof - have "{x∈Ω. ∀i::'i::countable. P i x} = (⋂i. {x∈Ω. P i x})" by auto with assms show ?thesis by auto qed lemma (in sigma_algebra) sets_Collect_countable_Ex: assumes "⋀i. {x∈Ω. P i x} ∈ M" shows "{x∈Ω. ∃i::'i::countable. P i x} ∈ M" proof - have "{x∈Ω. ∃i::'i::countable. P i x} = (⋃i. {x∈Ω. P i x})" by auto with assms show ?thesis by auto qed lemma (in sigma_algebra) sets_Collect_countable_Ex': assumes "⋀i. i ∈ I ⟹ {x∈Ω. P i x} ∈ M" assumes "countable I" shows "{x∈Ω. ∃i∈I. P i x} ∈ M" proof - have "{x∈Ω. ∃i∈I. P i x} = (⋃i∈I. {x∈Ω. P i x})" by auto with assms show ?thesis by (auto intro!: countable_UN') qed lemma (in sigma_algebra) sets_Collect_countable_All': assumes "⋀i. i ∈ I ⟹ {x∈Ω. P i x} ∈ M" assumes "countable I" shows "{x∈Ω. ∀i∈I. P i x} ∈ M" proof - have "{x∈Ω. ∀i∈I. P i x} = (⋂i∈I. {x∈Ω. P i x}) ∩ Ω" by auto with assms show ?thesis by (cases "I = {}") (auto intro!: countable_INT') qed lemma (in sigma_algebra) sets_Collect_countable_Ex1': assumes "⋀i. i ∈ I ⟹ {x∈Ω. P i x} ∈ M" assumes "countable I" shows "{x∈Ω. ∃!i∈I. P i x} ∈ M" proof - have "{x∈Ω. ∃!i∈I. P i x} = {x∈Ω. ∃i∈I. P i x ∧ (∀j∈I. P j x ⟶ i = j)}" by auto with assms show ?thesis by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const) qed lemmas (in sigma_algebra) sets_Collect = sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All lemma (in sigma_algebra) sets_Collect_countable_Ball: assumes "⋀i. {x∈Ω. P i x} ∈ M" shows "{x∈Ω. ∀i::'i::countable∈X. P i x} ∈ M" unfolding Ball_def by (intro sets_Collect assms) lemma (in sigma_algebra) sets_Collect_countable_Bex: assumes "⋀i. {x∈Ω. P i x} ∈ M" shows "{x∈Ω. ∃i::'i::countable∈X. P i x} ∈ M" unfolding Bex_def by (intro sets_Collect assms) lemma sigma_algebra_single_set: assumes "X ⊆ S" shows "sigma_algebra S { {}, X, S - X, S }" using algebra.is_sigma_algebra[OF algebra_single_set[OF ‹X ⊆ S›]] by simp subsubsection✐‹tag unimportant› ‹Binary Unions› definition binary :: "'a ⇒ 'a ⇒ nat ⇒ 'a" where "binary a b = (λx. b)(0 := a)" lemma range_binary_eq: "range(binary a b) = {a,b}" by (auto simp add: binary_def) lemma Un_range_binary: "a ∪ b = (⋃i::nat. binary a b i)" by (simp add: range_binary_eq cong del: SUP_cong_simp) lemma Int_range_binary: "a ∩ b = (⋂i::nat. binary a b i)" by (simp add: range_binary_eq cong del: INF_cong_simp) lemma sigma_algebra_iff2: "sigma_algebra Ω M ⟷ M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀s ∈ M. Ω - s ∈ M) ∧ (∀A. range A ⊆ M ⟶(⋃ i::nat. A i) ∈ M)" (is "?P ⟷ ?R ∧ ?S ∧ ?V ∧ ?W") proof assume ?P then interpret sigma_algebra Ω M . from space_closed show "?R ∧ ?S ∧ ?V ∧ ?W" by auto next assume "?R ∧ ?S ∧ ?V ∧ ?W" then have ?R ?S ?V ?W by simp_all show ?P proof (rule sigma_algebra.intro) show "sigma_algebra_axioms M" by standard (use ‹?W› in simp) from ‹?W› have *: "range (binary a b) ⊆ M ⟹ ⋃ (range (binary a b)) ∈ M" for a b by auto show "algebra Ω M" unfolding algebra_iff_Un using ‹?R› ‹?S› ‹?V› * by (auto simp add: range_binary_eq) qed qed subsubsection ‹Initial Sigma Algebra› text✐‹tag important› ‹Sigma algebras can naturally be created as the closure of any set of M with regard to the properties just postulated.› inductive_set✐‹tag important› sigma_sets :: "'a set ⇒ 'a set set ⇒ 'a set set" for sp :: "'a set" and A :: "'a set set" where Basic[intro, simp]: "a ∈ A ⟹ a ∈ sigma_sets sp A" | Empty: "{} ∈ sigma_sets sp A" | Compl: "a ∈ sigma_sets sp A ⟹ sp - a ∈ sigma_sets sp A" | Union: "(⋀i::nat. a i ∈ sigma_sets sp A) ⟹ (⋃i. a i) ∈ sigma_sets sp A" lemma (in sigma_algebra) sigma_sets_subset: assumes a: "a ⊆ M" shows "sigma_sets Ω a ⊆ M" proof fix x assume "x ∈ sigma_sets Ω a" from this show "x ∈ M" by (induct rule: sigma_sets.induct, auto) (metis a subsetD) qed lemma sigma_sets_into_sp: "A ⊆ Pow sp ⟹ x ∈ sigma_sets sp A ⟹ x ⊆ sp" by (erule sigma_sets.induct, auto) lemma sigma_algebra_sigma_sets: "a ⊆ Pow Ω ⟹ sigma_algebra Ω (sigma_sets Ω a)" by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl) lemma sigma_sets_least_sigma_algebra: assumes "A ⊆ Pow S" shows "sigma_sets S A = ⋂{B. A ⊆ B ∧ sigma_algebra S B}" proof safe fix B X assume "A ⊆ B" and sa: "sigma_algebra S B" and X: "X ∈ sigma_sets S A" from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF ‹A ⊆ B›] X show "X ∈ B" by auto next fix X assume "X ∈ ⋂{B. A ⊆ B ∧ sigma_algebra S B}" then have [intro!]: "⋀B. A ⊆ B ⟹ sigma_algebra S B ⟹ X ∈ B" by simp have "A ⊆ sigma_sets S A" using assms by auto moreover have "sigma_algebra S (sigma_sets S A)" using assms by (intro sigma_algebra_sigma_sets[of A]) auto ultimately show "X ∈ sigma_sets S A" by auto qed lemma sigma_sets_top: "sp ∈ sigma_sets sp A" by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty) lemma binary_in_sigma_sets: "binary a b i ∈ sigma_sets sp A" if "a ∈ sigma_sets sp A" and "b ∈ sigma_sets sp A" using that by (simp add: binary_def) lemma sigma_sets_Un: "a ∪ b ∈ sigma_sets sp A" if "a ∈ sigma_sets sp A" and "b ∈ sigma_sets sp A" using that by (simp add: Un_range_binary binary_in_sigma_sets Union) lemma sigma_sets_Inter: assumes Asb: "A ⊆ Pow sp" shows "(⋀i::nat. a i ∈ sigma_sets sp A) ⟹ (⋂i. a i) ∈ sigma_sets sp A" proof - assume ai: "⋀i::nat. a i ∈ sigma_sets sp A" hence "⋀i::nat. sp-(a i) ∈ sigma_sets sp A" by (rule sigma_sets.Compl) hence "(⋃i. sp-(a i)) ∈ sigma_sets sp A" by (rule sigma_sets.Union) hence "sp-(⋃i. sp-(a i)) ∈ sigma_sets sp A" by (rule sigma_sets.Compl) also have "sp-(⋃i. sp-(a i)) = sp Int (⋂i. a i)" by auto also have "... = (⋂i. a i)" using ai by (blast dest: sigma_sets_into_sp [OF Asb]) finally show ?thesis . qed lemma sigma_sets_INTER: assumes Asb: "A ⊆ Pow sp" and ai: "⋀i::nat. i ∈ S ⟹ a i ∈ sigma_sets sp A" and non: "S ≠ {}" shows "(⋂i∈S. a i) ∈ sigma_sets sp A" proof - from ai have "⋀i. (if i∈S then a i else sp) ∈ sigma_sets sp A" by (simp add: sigma_sets.intros(2-) sigma_sets_top) hence "(⋂i. (if i∈S then a i else sp)) ∈ sigma_sets sp A" by (rule sigma_sets_Inter [OF Asb]) also have "(⋂i. (if i∈S then a i else sp)) = (⋂i∈S. a i)" by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+ finally show ?thesis . qed lemma sigma_sets_UNION: "countable B ⟹ (⋀b. b ∈ B ⟹ b ∈ sigma_sets X A) ⟹ ⋃ B ∈ sigma_sets X A" using from_nat_into [of B] range_from_nat_into [of B] sigma_sets.Union [of "from_nat_into B" X A] by (cases "B = {}") (simp_all add: sigma_sets.Empty cong del: SUP_cong) lemma (in sigma_algebra) sigma_sets_eq: "sigma_sets Ω M = M" proof show "M ⊆ sigma_sets Ω M" by (metis Set.subsetI sigma_sets.Basic) next show "sigma_sets Ω M ⊆ M" by (metis sigma_sets_subset subset_refl) qed lemma sigma_sets_eqI: assumes A: "⋀a. a ∈ A ⟹ a ∈ sigma_sets M B" assumes B: "⋀b. b ∈ B ⟹ b ∈ sigma_sets M A" shows "sigma_sets M A = sigma_sets M B" proof (intro set_eqI iffI) fix a assume "a ∈ sigma_sets M A" from this A show "a ∈ sigma_sets M B" by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) next fix b assume "b ∈ sigma_sets M B" from this B show "b ∈ sigma_sets M A" by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) qed lemma sigma_sets_subseteq: assumes "A ⊆ B" shows "sigma_sets X A ⊆ sigma_sets X B" proof fix x assume "x ∈ sigma_sets X A" then show "x ∈ sigma_sets X B" by induct (insert ‹A ⊆ B›, auto intro: sigma_sets.intros(2-)) qed lemma sigma_sets_mono: assumes "A ⊆ sigma_sets X B" shows "sigma_sets X A ⊆ sigma_sets X B" proof fix x assume "x ∈ sigma_sets X A" then show "x ∈ sigma_sets X B" by induct (insert ‹A ⊆ sigma_sets X B›, auto intro: sigma_sets.intros(2-)) qed lemma sigma_sets_mono': assumes "A ⊆ B" shows "sigma_sets X A ⊆ sigma_sets X B" proof fix x assume "x ∈ sigma_sets X A" then show "x ∈ sigma_sets X B" by induct (insert ‹A ⊆ B›, auto intro: sigma_sets.intros(2-)) qed lemma sigma_sets_superset_generator: "A ⊆ sigma_sets X A" by (auto intro: sigma_sets.Basic) lemma (in sigma_algebra) restriction_in_sets: fixes A :: "nat ⇒ 'a set" assumes "S ∈ M" and *: "range A ⊆ (λA. S ∩ A) ` M" (is "_ ⊆ ?r") shows "range A ⊆ M" "(⋃i. A i) ∈ (λA. S ∩ A) ` M" proof - { fix i have "A i ∈ ?r" using * by auto hence "∃B. A i = B ∩ S ∧ B ∈ M" by auto hence "A i ⊆ S" "A i ∈ M" using ‹S ∈ M› by auto } thus "range A ⊆ M" "(⋃i. A i) ∈ (λA. S ∩ A) ` M" by (auto intro!: image_eqI[of _ _ "(⋃i. A i)"]) qed lemma (in sigma_algebra) restricted_sigma_algebra: assumes "S ∈ M" shows "sigma_algebra S (restricted_space S)" unfolding sigma_algebra_def sigma_algebra_axioms_def proof safe show "algebra S (restricted_space S)" using restricted_algebra[OF assms] . next fix A :: "nat ⇒ 'a set" assume "range A ⊆ restricted_space S" from restriction_in_sets[OF assms this[simplified]] show "(⋃i. A i) ∈ restricted_space S" by simp qed lemma sigma_sets_Int: assumes "A ∈ sigma_sets sp st" "A ⊆ sp" shows "(∩) A ` sigma_sets sp st = sigma_sets A ((∩) A ` st)" proof (intro equalityI subsetI) fix x assume "x ∈ (∩) A ` sigma_sets sp st" then obtain y where "y ∈ sigma_sets sp st" "x = y ∩ A" by auto then have "x ∈ sigma_sets (A ∩ sp) ((∩) A ` st)" proof (induct arbitrary: x) case (Compl a) then show ?case by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps) next case (Union a) then show ?case by (auto intro!: sigma_sets.Union simp add: UN_extend_simps simp del: UN_simps) qed (auto intro!: sigma_sets.intros(2-)) then show "x ∈ sigma_sets A ((∩) A ` st)" using ‹A ⊆ sp› by (simp add: Int_absorb2) next fix x assume "x ∈ sigma_sets A ((∩) A ` st)" then show "x ∈ (∩) A ` sigma_sets sp st" proof induct case (Compl a) then obtain x where "a = A ∩ x" "x ∈ sigma_sets sp st" by auto then show ?case using ‹A ⊆ sp› by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl) next case (Union a) then have "∀i. ∃x. x ∈ sigma_sets sp st ∧ a i = A ∩ x" by (auto simp: image_iff Bex_def) then obtain f where "∀x. f x ∈ sigma_sets sp st ∧ a x = A ∩ f x" by metis then show ?case by (auto intro!: bexI[of _ "(⋃x. f x)"] sigma_sets.Union simp add: image_iff) qed (auto intro!: sigma_sets.intros(2-)) qed lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}" proof (intro set_eqI iffI) fix a assume "a ∈ sigma_sets A {}" then show "a ∈ {{}, A}" by induct blast+ qed (auto intro: sigma_sets.Empty sigma_sets_top) lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}" proof (intro set_eqI iffI) fix x assume "x ∈ sigma_sets A {A}" then show "x ∈ {{}, A}" by induct blast+ next fix x assume "x ∈ {{}, A}" then show "x ∈ sigma_sets A {A}" by (auto intro: sigma_sets.Empty sigma_sets_top) qed lemma sigma_sets_sigma_sets_eq: "M ⊆ Pow S ⟹ sigma_sets S (sigma_sets S M) = sigma_sets S M" by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto lemma sigma_sets_singleton: assumes "X ⊆ S" shows "sigma_sets S { X } = { {}, X, S - X, S }" proof - interpret sigma_algebra S "{ {}, X, S - X, S }" by (rule sigma_algebra_single_set) fact have "sigma_sets S { X } ⊆ sigma_sets S { {}, X, S - X, S }" by (rule sigma_sets_subseteq) simp moreover have "… = { {}, X, S - X, S }" using sigma_sets_eq by simp moreover { fix A assume "A ∈ { {}, X, S - X, S }" then have "A ∈ sigma_sets S { X }" by (auto intro: sigma_sets.intros(2-) sigma_sets_top) } ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }" by (intro antisym) auto with sigma_sets_eq show ?thesis by simp qed lemma restricted_sigma: assumes S: "S ∈ sigma_sets Ω M" and M: "M ⊆ Pow Ω" shows "algebra.restricted_space (sigma_sets Ω M) S = sigma_sets S (algebra.restricted_space M S)" proof - from S sigma_sets_into_sp[OF M] have "S ∈ sigma_sets Ω M" "S ⊆ Ω" by auto from sigma_sets_Int[OF this] show ?thesis by simp qed lemma sigma_sets_vimage_commute: assumes X: "X ∈ Ω → Ω'" shows "{X -` A ∩ Ω |A. A ∈ sigma_sets Ω' M'} = sigma_sets Ω {X -` A ∩ Ω |A. A ∈ M'}" (is "?L = ?R") proof show "?L ⊆ ?R" proof clarify fix A assume "A ∈ sigma_sets Ω' M'" then show "X -` A ∩ Ω ∈ ?R" proof induct case Empty then show ?case by (auto intro!: sigma_sets.Empty) next case (Compl B) have [simp]: "X -` (Ω' - B) ∩ Ω = Ω - (X -` B ∩ Ω)" by (auto simp add: funcset_mem [OF X]) with Compl show ?case by (auto intro!: sigma_sets.Compl) next case (Union F) then show ?case by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps intro!: sigma_sets.Union) qed auto qed show "?R ⊆ ?L" proof clarify fix A assume "A ∈ ?R" then show "∃B. A = X -` B ∩ Ω ∧ B ∈ sigma_sets Ω' M'" proof induct case (Basic B) then show ?case by auto next case Empty then show ?case by (auto intro!: sigma_sets.Empty exI[of _ "{}"]) next case (Compl B) then obtain A where A: "B = X -` A ∩ Ω" "A ∈ sigma_sets Ω' M'" by auto then have [simp]: "Ω - B = X -` (Ω' - A) ∩ Ω" by (auto simp add: funcset_mem [OF X]) with A(2) show ?case by (auto intro: sigma_sets.Compl) next case (Union F) then have "∀i. ∃B. F i = X -` B ∩ Ω ∧ B ∈ sigma_sets Ω' M'" by auto then obtain A where "∀x. F x = X -` A x ∩ Ω ∧ A x ∈ sigma_sets Ω' M'" by metis then show ?case by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union) qed qed qed lemma (in ring_of_sets) UNION_in_sets: fixes A:: "nat ⇒ 'a set" assumes A: "range A ⊆ M" shows "(⋃i∈{0..<n}. A i) ∈ M" proof (induct n) case 0 show ?case by simp next case (Suc n) thus ?case by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff) qed lemma (in ring_of_sets) range_disjointed_sets: assumes A: "range A ⊆ M" shows "range (disjointed A) ⊆ M" proof (auto simp add: disjointed_def) fix n show "A n - (⋃i∈{0..<n}. A i) ∈ M" using UNION_in_sets by (metis A Diff UNIV_I image_subset_iff) qed lemma (in algebra) range_disjointed_sets': "range A ⊆ M ⟹ range (disjointed A) ⊆ M" using range_disjointed_sets . lemma sigma_algebra_disjoint_iff: "sigma_algebra Ω M ⟷ algebra Ω M ∧ (∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i::nat. A i) ∈ M)" proof (auto simp add: sigma_algebra_iff) fix A :: "nat ⇒ 'a set" assume M: "algebra Ω M" and A: "range A ⊆ M" and UnA: "∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i::nat. A i) ∈ M" hence "range (disjointed A) ⊆ M ⟶ disjoint_family (disjointed A) ⟶ (⋃i. disjointed A i) ∈ M" by blast hence "(⋃i. disjointed A i) ∈ M" by (simp add: algebra.range_disjointed_sets'[of Ω] M A disjoint_family_disjointed) thus "(⋃i::nat. A i) ∈ M" by (simp add: UN_disjointed_eq) qed subsubsection✐‹tag unimportant› ‹Ring generated by a semiring› definition (in semiring_of_sets) generated_ring :: "'a set set" where "generated_ring = { ⋃C | C. C ⊆ M ∧ finite C ∧ disjoint C }" lemma (in semiring_of_sets) generated_ringE[elim?]: assumes "a ∈ generated_ring" obtains C where "finite C" "disjoint C" "C ⊆ M" "a = ⋃C" using assms unfolding generated_ring_def by auto lemma (in semiring_of_sets) generated_ringI[intro?]: assumes "finite C" "disjoint C" "C ⊆ M" "a = ⋃C" shows "a ∈ generated_ring" using assms unfolding generated_ring_def by auto lemma (in semiring_of_sets) generated_ringI_Basic: "A ∈ M ⟹ A ∈ generated_ring" by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def) lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]: assumes a: "a ∈ generated_ring" and b: "b ∈ generated_ring" and "a ∩ b = {}" shows "a ∪ b ∈ generated_ring" proof - from a b obtain Ca Cb where Ca: "finite Ca" "disjoint Ca" "Ca ⊆ M" "a = ⋃ Ca" and Cb: "finite Cb" "disjoint Cb" "Cb ⊆ M" "b = ⋃ Cb" using generated_ringE by metis show ?thesis proof from ‹a ∩ b = {}› Ca Cb show "disjoint (Ca ∪ Cb)" by (auto intro!: disjoint_union) qed (use Ca Cb in auto) qed lemma (in semiring_of_sets) generated_ring_empty: "{} ∈ generated_ring" by (auto simp: generated_ring_def disjoint_def) lemma (in semiring_of_sets) generated_ring_disjoint_Union: assumes "finite A" shows "A ⊆ generated_ring ⟹ disjoint A ⟹ ⋃A ∈ generated_ring" using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty) lemma (in semiring_of_sets) generated_ring_disjoint_UNION: "finite I ⟹ disjoint (A ` I) ⟹ (⋀i. i ∈ I ⟹ A i ∈ generated_ring) ⟹ ⋃(A ` I) ∈ generated_ring" by (intro generated_ring_disjoint_Union) auto lemma (in semiring_of_sets) generated_ring_Int: assumes a: "a ∈ generated_ring" and b: "b ∈ generated_ring" shows "a ∩ b ∈ generated_ring" proof - from a b obtain Ca Cb where Ca: "finite Ca" "disjoint Ca" "Ca ⊆ M" "a = ⋃ Ca" and Cb: "finite Cb" "disjoint Cb" "Cb ⊆ M" "b = ⋃ Cb" using generated_ringE by metis define C where "C = (λ(a,b). a ∩ b)` (Ca×Cb)" show ?thesis proof show "disjoint C" proof (simp add: disjoint_def C_def, intro ballI impI) fix a1 b1 a2 b2 assume sets: "a1 ∈ Ca" "b1 ∈ Cb" "a2 ∈ Ca" "b2 ∈ Cb" assume "a1 ∩ b1 ≠ a2 ∩ b2" then have "a1 ≠ a2 ∨ b1 ≠ b2" by auto then show "(a1 ∩ b1) ∩ (a2 ∩ b2) = {}" proof assume "a1 ≠ a2" with sets Ca have "a1 ∩ a2 = {}" by (auto simp: disjoint_def) then show ?thesis by auto next assume "b1 ≠ b2" with sets Cb have "b1 ∩ b2 = {}" by (auto simp: disjoint_def) then show ?thesis by auto qed qed qed (use Ca Cb in ‹auto simp: C_def›) qed lemma (in semiring_of_sets) generated_ring_Inter: assumes "finite A" "A ≠ {}" shows "A ⊆ generated_ring ⟹ ⋂A ∈ generated_ring" using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int) lemma (in semiring_of_sets) generated_ring_INTER: "finite I ⟹ I ≠ {} ⟹ (⋀i. i ∈ I ⟹ A i ∈ generated_ring) ⟹ ⋂(A ` I) ∈ generated_ring" by (intro generated_ring_Inter) auto lemma (in semiring_of_sets) generating_ring: "ring_of_sets Ω generated_ring" proof (rule ring_of_setsI) let ?R = generated_ring show "?R ⊆ Pow Ω" using sets_into_space by (auto simp: generated_ring_def generated_ring_empty) show "{} ∈ ?R" by (rule generated_ring_empty) { fix a b assume "a ∈ ?R" "b ∈ ?R" then obtain Ca Cb where Ca: "finite Ca" "disjoint Ca" "Ca ⊆ M" "a = ⋃ Ca" and Cb: "finite Cb" "disjoint Cb" "Cb ⊆ M" "b = ⋃ Cb" using generated_ringE by metis show "a - b ∈ ?R" proof cases assume "Cb = {}" with Cb ‹a ∈ ?R› show ?thesis by simp next assume "Cb ≠ {}" with Ca Cb have "a - b = (⋃a'∈Ca. ⋂b'∈Cb. a' - b')" by auto also have "… ∈ ?R" proof (intro generated_ring_INTER generated_ring_disjoint_UNION) fix a b assume "a ∈ Ca" "b ∈ Cb" with Ca Cb Diff_cover[of a b] show "a - b ∈ ?R" by (auto simp add: generated_ring_def) (metis DiffI Diff_eq_empty_iff empty_iff) next show "disjoint ((λa'. ⋂b'∈Cb. a' - b')`Ca)" using Ca by (auto simp add: disjoint_def ‹Cb ≠ {}›) next show "finite Ca" "finite Cb" "Cb ≠ {}" by fact+ qed finally show "a - b ∈ ?R" . qed } note Diff = this fix a b assume sets: "a ∈ ?R" "b ∈ ?R" have "a ∪ b = (a - b) ∪ (a ∩ b) ∪ (b - a)" by auto also have "… ∈ ?R" by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto finally show "a ∪ b ∈ ?R" . qed lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets Ω generated_ring = sigma_sets Ω M" proof interpret M: sigma_algebra Ω "sigma_sets Ω M" using space_closed by (rule sigma_algebra_sigma_sets) show "sigma_sets Ω generated_ring ⊆ sigma_sets Ω M" by (blast intro!: sigma_sets_mono elim: generated_ringE) qed (auto intro!: generated_ringI_Basic sigma_sets_mono) subsubsection✐‹tag unimportant› ‹A Two-Element Series› definition binaryset :: "'a set ⇒ 'a set ⇒ nat ⇒ 'a set" where "binaryset A B = (λx. {})(0 := A, Suc 0 := B)" lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}" apply (simp add: binaryset_def) apply (rule set_eqI) apply (auto simp add: image_iff) done lemma UN_binaryset_eq: "(⋃i. binaryset A B i) = A ∪ B" by (simp add: range_binaryset_eq cong del: SUP_cong_simp) subsubsection ‹Closed CDI› definition✐‹tag important› closed_cdi :: "'a set ⇒ 'a set set ⇒ bool" where "closed_cdi Ω M ⟷ M ⊆ Pow Ω & (∀s ∈ M. Ω - s ∈ M) & (∀A. (range A ⊆ M) & (A 0 = {}) & (∀n. A n ⊆ A (Suc n)) ⟶ (⋃i. A i) ∈ M) & (∀A. (range A ⊆ M) & disjoint_family A ⟶ (⋃i::nat. A i) ∈ M)" inductive_set smallest_ccdi_sets :: "'a set ⇒ 'a set set ⇒ 'a set set" for Ω M where Basic [intro]: "a ∈ M ⟹ a ∈ smallest_ccdi_sets Ω M" | Compl [intro]: "a ∈ smallest_ccdi_sets Ω M ⟹ Ω - a ∈ smallest_ccdi_sets Ω M" | Inc: "range A ∈ Pow(smallest_ccdi_sets Ω M) ⟹ A 0 = {} ⟹ (⋀n. A n ⊆ A (Suc n)) ⟹ (⋃i. A i) ∈ smallest_ccdi_sets Ω M" | Disj: "range A ∈ Pow(smallest_ccdi_sets Ω M) ⟹ disjoint_family A ⟹ (⋃i::nat. A i) ∈ smallest_ccdi_sets Ω M" lemma (in subset_class) smallest_closed_cdi1: "M ⊆ smallest_ccdi_sets Ω M" by auto lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets Ω M ⊆ Pow Ω" apply (rule subsetI) apply (erule smallest_ccdi_sets.induct) apply (auto intro: range_subsetD dest: sets_into_space) done lemma (in subset_class) smallest_closed_cdi2: "closed_cdi Ω (smallest_ccdi_sets Ω M)" apply (auto simp add: closed_cdi_def smallest_ccdi_sets) apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) + done lemma closed_cdi_subset: "closed_cdi Ω M ⟹ M ⊆ Pow Ω" by (simp add: closed_cdi_def) lemma closed_cdi_Compl: "closed_cdi Ω M ⟹ s ∈ M ⟹ Ω - s ∈ M" by (simp add: closed_cdi_def) lemma closed_cdi_Inc: "closed_cdi Ω M ⟹ range A ⊆ M ⟹ A 0 = {} ⟹ (!!n. A n ⊆ A (Suc n)) ⟹ (⋃i. A i) ∈ M" by (simp add: closed_cdi_def) lemma closed_cdi_Disj: "closed_cdi Ω M ⟹ range A ⊆ M ⟹ disjoint_family A ⟹ (⋃i::nat. A i) ∈ M" by (simp add: closed_cdi_def) lemma closed_cdi_Un: assumes cdi: "closed_cdi Ω M" and empty: "{} ∈ M" and A: "A ∈ M" and B: "B ∈ M" and disj: "A ∩ B = {}" shows "A ∪ B ∈ M" proof - have ra: "range (binaryset A B) ⊆ M" by (simp add: range_binaryset_eq empty A B) have di: "disjoint_family (binaryset A B)" using disj by (simp add: disjoint_family_on_def binaryset_def Int_commute) from closed_cdi_Disj [OF cdi ra di] show ?thesis by (simp add: UN_binaryset_eq) qed lemma (in algebra) smallest_ccdi_sets_Un: assumes A: "A ∈ smallest_ccdi_sets Ω M" and B: "B ∈ smallest_ccdi_sets Ω M" and disj: "A ∩ B = {}" shows "A ∪ B ∈ smallest_ccdi_sets Ω M" proof - have ra: "range (binaryset A B) ∈ Pow (smallest_ccdi_sets Ω M)" by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic) have di: "disjoint_family (binaryset A B)" using disj by (simp add: disjoint_family_on_def binaryset_def Int_commute) from Disj [OF ra di] show ?thesis by (simp add: UN_binaryset_eq) qed lemma (in algebra) smallest_ccdi_sets_Int1: assumes a: "a ∈ M" shows "b ∈ smallest_ccdi_sets Ω M ⟹ a ∩ b ∈ smallest_ccdi_sets Ω M" proof (induct rule: smallest_ccdi_sets.induct) case (Basic x) thus ?case by (metis a Int smallest_ccdi_sets.Basic) next case (Compl x) have "a ∩ (Ω - x) = Ω - ((Ω - a) ∪ (a ∩ x))" by blast also have "... ∈ smallest_ccdi_sets Ω M" by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2 Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl) finally show ?case . next case (Inc A) have 1: "(⋃i. (λi. a ∩ A i) i) = a ∩ (⋃i. A i)" by blast have "range (λi. a ∩ A i) ∈ Pow(smallest_ccdi_sets Ω M)" using Inc by blast moreover have "(λi. a ∩ A i) 0 = {}" by (simp add: Inc) moreover have "!!n. (λi. a ∩ A i) n ⊆ (λi. a ∩ A i) (Suc n)" using Inc by blast ultimately have 2: "(⋃i. (λi. a ∩ A i) i) ∈ smallest_ccdi_sets Ω M" by (rule smallest_ccdi_sets.Inc) show ?case by (metis 1 2) next case (Disj A) have 1: "(⋃i. (λi. a ∩ A i) i) = a ∩ (⋃i. A i)" by blast have "range (λi. a ∩ A i) ∈ Pow(smallest_ccdi_sets Ω M)" using Disj by blast moreover have "disjoint_family (λi. a ∩ A i)" using Disj by (auto simp add: disjoint_family_on_def) ultimately have 2: "(⋃i. (λi. a ∩ A i) i) ∈ smallest_ccdi_sets Ω M" by (rule smallest_ccdi_sets.Disj) show ?case by (metis 1 2) qed lemma (in algebra) smallest_ccdi_sets_Int: assumes b: "b ∈ smallest_ccdi_sets Ω M" shows "a ∈ smallest_ccdi_sets Ω M ⟹ a ∩ b ∈ smallest_ccdi_sets Ω M" proof (induct rule: smallest_ccdi_sets.induct) case (Basic x) thus ?case by (metis b smallest_ccdi_sets_Int1) next case (Compl x) have "(Ω - x) ∩ b = Ω - (x ∩ b ∪ (Ω - b))" by blast also have "... ∈ smallest_ccdi_sets Ω M" by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b smallest_ccdi_sets.Compl smallest_ccdi_sets_Un) finally show ?case . next case (Inc A) have 1: "(⋃i. (λi. A i ∩ b) i) = (⋃i. A i) ∩ b" by blast have "range (λi. A i ∩ b) ∈ Pow(smallest_ccdi_sets Ω M)" using Inc by blast moreover have "(λi. A i ∩ b) 0 = {}" by (simp add: Inc) moreover have "!!n. (λi. A i ∩ b) n ⊆ (λi. A i ∩ b) (Suc n)" using Inc by blast ultimately have 2: "(⋃i. (λi. A i ∩ b) i) ∈ smallest_ccdi_sets Ω M" by (rule smallest_ccdi_sets.Inc) show ?case by (metis 1 2) next case (Disj A) have 1: "(⋃i. (λi. A i ∩ b) i) = (⋃i. A i) ∩ b" by blast have "range (λi. A i ∩ b) ∈ Pow(smallest_ccdi_sets Ω M)" using Disj by blast moreover have "disjoint_family (λi. A i ∩ b)" using Disj by (auto simp add: disjoint_family_on_def) ultimately have 2: "(⋃i. (λi. A i ∩ b) i) ∈ smallest_ccdi_sets Ω M" by (rule smallest_ccdi_sets.Disj) show ?case by (metis 1 2) qed lemma (in algebra) sigma_property_disjoint_lemma: assumes sbC: "M ⊆ C" and ccdi: "closed_cdi Ω C" shows "sigma_sets Ω M ⊆ C" proof - have "smallest_ccdi_sets Ω M ∈ {B . M ⊆ B ∧ sigma_algebra Ω B}" apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int smallest_ccdi_sets_Int) apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets) apply (blast intro: smallest_ccdi_sets.Disj) done hence "sigma_sets (Ω) (M) ⊆ smallest_ccdi_sets Ω M" by clarsimp (drule sigma_algebra.sigma_sets_subset [where a="M"], auto) also have "... ⊆ C" proof fix x assume x: "x ∈ smallest_ccdi_sets Ω M" thus "x ∈ C" proof (induct rule: smallest_ccdi_sets.induct) case (Basic x) thus ?case by (metis Basic subsetD sbC) next case (Compl x) thus ?case by (blast intro: closed_cdi_Compl [OF ccdi, simplified]) next case (Inc A) thus ?case by (auto intro: closed_cdi_Inc [OF ccdi, simplified]) next case (Disj A) thus ?case by (auto intro: closed_cdi_Disj [OF ccdi, simplified]) qed qed finally show ?thesis . qed lemma (in algebra) sigma_property_disjoint: assumes sbC: "M ⊆ C" and compl: "!!s. s ∈ C ∩ sigma_sets (Ω) (M) ⟹ Ω - s ∈ C" and inc: "!!A. range A ⊆ C ∩ sigma_sets (Ω) (M) ⟹ A 0 = {} ⟹ (!!n. A n ⊆ A (Suc n)) ⟹ (⋃i. A i) ∈ C" and disj: "!!A. range A ⊆ C ∩ sigma_sets (Ω) (M) ⟹ disjoint_family A ⟹ (⋃i::nat. A i) ∈ C" shows "sigma_sets (Ω) (M) ⊆ C" proof - have "sigma_sets (Ω) (M) ⊆ C ∩ sigma_sets (Ω) (M)" proof (rule sigma_property_disjoint_lemma) show "M ⊆ C ∩ sigma_sets (Ω) (M)" by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic) next show "closed_cdi Ω (C ∩ sigma_sets (Ω) (M))" by (simp add: closed_cdi_def compl inc disj) (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed IntE sigma_sets.Compl range_subsetD sigma_sets.Union) qed thus ?thesis by blast qed subsubsection ‹Dynkin systems› locale✐‹tag important› Dynkin_system = subset_class + assumes space: "Ω ∈ M" and compl[intro!]: "⋀A. A ∈ M ⟹ Ω - A ∈ M" and UN[intro!]: "⋀A. disjoint_family A ⟹ range A ⊆ M ⟹ (⋃i::nat. A i) ∈ M" lemma (in Dynkin_system) empty[intro, simp]: "{} ∈ M" using space compl[of "Ω"] by simp lemma (in Dynkin_system) diff: assumes sets: "D ∈ M" "E ∈ M" and "D ⊆ E" shows "E - D ∈ M" proof - let ?f = "λx. if x = 0 then D else if x = Suc 0 then Ω - E else {}" have "range ?f = {D, Ω - E, {}}" by (auto simp: image_iff) moreover have "D ∪ (Ω - E) = (⋃i. ?f i)" by (auto simp: image_iff split: if_split_asm) moreover have "disjoint_family ?f" unfolding disjoint_family_on_def using ‹D ∈ M›[THEN sets_into_space] ‹D ⊆ E› by auto ultimately have "Ω - (D ∪ (Ω - E)) ∈ M" using sets UN by auto fastforce also have "Ω - (D ∪ (Ω - E)) = E - D" using assms sets_into_space by auto finally show ?thesis . qed lemma Dynkin_systemI: assumes "⋀ A. A ∈ M ⟹ A ⊆ Ω" "Ω ∈ M" assumes "⋀ A. A ∈ M ⟹ Ω - A ∈ M" assumes "⋀ A. disjoint_family A ⟹ range A ⊆ M ⟹ (⋃i::nat. A i) ∈ M" shows "Dynkin_system Ω M" using assms by (auto simp: Dynkin_system_def Dynkin_system_axioms_def subset_class_def) lemma Dynkin_systemI': assumes 1: "⋀ A. A ∈ M ⟹ A ⊆ Ω" assumes empty: "{} ∈ M" assumes Diff: "⋀ A. A ∈ M ⟹ Ω - A ∈ M" assumes 2: "⋀ A. disjoint_family A ⟹ range A ⊆ M ⟹ (⋃i::nat. A i) ∈ M" shows "Dynkin_system Ω M" proof - from Diff[OF empty] have "Ω ∈ M" by auto from 1 this Diff 2 show ?thesis by (intro Dynkin_systemI) auto qed lemma Dynkin_system_trivial: shows "Dynkin_system A (Pow A)" by (rule Dynkin_systemI) auto lemma sigma_algebra_imp_Dynkin_system: assumes "sigma_algebra Ω M" shows "Dynkin_system Ω M" proof - interpret sigma_algebra Ω M by fact show ?thesis using sets_into_space by (fastforce intro!: Dynkin_systemI) qed subsubsection "Intersection sets systems" definition✐‹tag important› Int_stable :: "'a set set ⇒ bool" where "Int_stable M ⟷ (∀ a ∈ M. ∀ b ∈ M. a ∩ b ∈ M)" lemma (in algebra) Int_stable: "Int_stable M" unfolding Int_stable_def by auto lemma Int_stableI_image: "(⋀i j. i ∈ I ⟹ j ∈ I ⟹ ∃k∈I. A i ∩ A j = A k) ⟹ Int_stable (A ` I)" by (auto simp: Int_stable_def image_def) lemma Int_stableI: "(⋀a b. a ∈ A ⟹ b ∈ A ⟹ a ∩ b ∈ A) ⟹ Int_stable A" unfolding Int_stable_def by auto lemma Int_stableD: "Int_stable M ⟹ a ∈ M ⟹ b ∈ M ⟹ a ∩ b ∈ M" unfolding Int_stable_def by auto lemma (in Dynkin_system) sigma_algebra_eq_Int_stable: "sigma_algebra Ω M ⟷ Int_stable M" proof assume "sigma_algebra Ω M" then show "Int_stable M" unfolding sigma_algebra_def using algebra.Int_stable by auto next assume "Int_stable M" show "sigma_algebra Ω M" unfolding sigma_algebra_disjoint_iff algebra_iff_Un proof (intro conjI ballI allI impI) show "M ⊆ Pow (Ω)" using sets_into_space by auto next fix A B assume "A ∈ M" "B ∈ M" then have "A ∪ B = Ω - ((Ω - A) ∩ (Ω - B))" "Ω - A ∈ M" "Ω - B ∈ M" using sets_into_space by auto then show "A ∪ B ∈ M" using ‹Int_stable M› unfolding Int_stable_def by auto qed auto qed subsubsection "Smallest Dynkin systems" definition✐‹tag important› Dynkin :: "'a set ⇒ 'a set set ⇒ 'a set set" where "Dynkin Ω M = (⋂{D. Dynkin_system Ω D ∧ M ⊆ D})" lemma Dynkin_system_Dynkin: assumes "M ⊆ Pow (Ω)" shows "Dynkin_system Ω (Dynkin Ω M)" proof (rule Dynkin_systemI) fix A assume "A ∈ Dynkin Ω M" moreover { fix D assume "A ∈ D" and d: "Dynkin_system Ω D" then have "A ⊆ Ω" by (auto simp: Dynkin_system_def subset_class_def) } moreover have "{D. Dynkin_system Ω D ∧ M ⊆ D} ≠ {}" using assms Dynkin_system_trivial by fastforce ultimately show "A ⊆ Ω" unfolding Dynkin_def using assms by auto next show "Ω ∈ Dynkin Ω M" unfolding Dynkin_def using Dynkin_system.space by fastforce next fix A assume "A ∈ Dynkin Ω M" then show "Ω - A ∈ Dynkin Ω M" unfolding Dynkin_def using Dynkin_system.compl by force next fix A :: "nat ⇒ 'a set" assume A: "disjoint_family A" "range A ⊆ Dynkin Ω M" show "(⋃i. A i) ∈ Dynkin Ω M" unfolding Dynkin_def proof (simp, safe) fix D assume "Dynkin_system Ω D" "M ⊆ D" with A have "(⋃i. A i) ∈ D" by (intro Dynkin_system.UN) (auto simp: Dynkin_def) then show "(⋃i. A i) ∈ D" by auto qed qed lemma Dynkin_Basic[intro]: "A ∈ M ⟹ A ∈ Dynkin Ω M" unfolding Dynkin_def by auto lemma (in Dynkin_system) restricted_Dynkin_system: assumes "D ∈ M" shows "Dynkin_system Ω {Q. Q ⊆ Ω ∧ Q ∩ D ∈ M}" proof (rule Dynkin_systemI, simp_all) have "Ω ∩ D = D" using ‹D ∈ M› sets_into_space by auto then show "Ω ∩ D ∈ M" using ‹D ∈ M› by auto next fix A assume "A ⊆ Ω ∧ A ∩ D ∈ M" moreover have "(Ω - A) ∩ D = (Ω - (A ∩ D)) - (Ω - D)" by auto ultimately show "(Ω - A) ∩ D ∈ M" using ‹D ∈ M› by (auto intro: diff) next fix A :: "nat ⇒ 'a set" assume "disjoint_family A" "range A ⊆ {Q. Q ⊆ Ω ∧ Q ∩ D ∈ M}" then have "⋀i. A i ⊆ Ω" "disjoint_family (λi. A i ∩ D)" "range (λi. A i ∩ D) ⊆ M" "(⋃x. A x) ∩ D = (⋃x. A x ∩ D)" by ((fastforce simp: disjoint_family_on_def)+) then show "(⋃x. A x) ⊆ Ω ∧ (⋃x. A x) ∩ D ∈ M" by (auto simp del: UN_simps) qed lemma (in Dynkin_system) Dynkin_subset: assumes "N ⊆ M" shows "Dynkin Ω N ⊆ M" proof - have "Dynkin_system Ω M" .. then have "Dynkin_system Ω M" using assms unfolding Dynkin_system_def Dynkin_system_axioms_def subset_class_def by simp with ‹N ⊆ M› show ?thesis by (auto simp add: Dynkin_def) qed lemma sigma_eq_Dynkin: assumes sets: "M ⊆ Pow Ω" assumes "Int_stable M" shows "sigma_sets Ω M = Dynkin Ω M" proof - have "Dynkin Ω M ⊆ sigma_sets (Ω) (M)" using sigma_algebra_imp_Dynkin_system unfolding Dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto moreover interpret Dynkin_system Ω "Dynkin Ω M" using Dynkin_system_Dynkin[OF sets] . have "sigma_algebra Ω (Dynkin Ω M)" unfolding sigma_algebra_eq_Int_stable Int_stable_def proof (intro ballI) fix A B assume "A ∈ Dynkin Ω M" "B ∈ Dynkin Ω M" let ?D = "λE. {Q. Q ⊆ Ω ∧ Q ∩ E ∈ Dynkin Ω M}" have "M ⊆ ?D B" proof fix E assume "E ∈ M" then have "M ⊆ ?D E" "E ∈ Dynkin Ω M" using sets_into_space ‹Int_stable M› by (auto simp: Int_stable_def) then have "Dynkin Ω M ⊆ ?D E" using restricted_Dynkin_system ‹E ∈ Dynkin Ω M› by (intro Dynkin_system.Dynkin_subset) simp_all then have "B ∈ ?D E" using ‹B ∈ Dynkin Ω M› by auto then have "E ∩ B ∈ Dynkin Ω M" by (subst Int_commute) simp then show "E ∈ ?D B" using sets ‹E ∈ M› by auto qed then have "Dynkin Ω M ⊆ ?D B" using restricted_Dynkin_system ‹B ∈ Dynkin Ω M› by (intro Dynkin_system.Dynkin_subset) simp_all then show "A ∩ B ∈ Dynkin Ω M" using ‹A ∈ Dynkin Ω M› sets_into_space by auto qed from sigma_algebra.sigma_sets_subset[OF this, of "M"] have "sigma_sets (Ω) (M) ⊆ Dynkin Ω M" by auto ultimately have "sigma_sets (Ω) (M) = Dynkin Ω M" by auto then show ?thesis by (auto simp: Dynkin_def) qed lemma (in Dynkin_system) Dynkin_idem: "Dynkin Ω M = M" proof - have "Dynkin Ω M = M" proof show "M ⊆ Dynkin Ω M" using Dynkin_Basic by auto show "Dynkin Ω M ⊆ M" by (intro Dynkin_subset) auto qed then show ?thesis by (auto simp: Dynkin_def) qed lemma (in Dynkin_system) Dynkin_lemma: assumes "Int_stable E" and E: "E ⊆ M" "M ⊆ sigma_sets Ω E" shows "sigma_sets Ω E = M" proof - have "E ⊆ Pow Ω" using E sets_into_space by force then have *: "sigma_sets Ω E = Dynkin Ω E" using ‹Int_stable E› by (rule sigma_eq_Dynkin) then have "Dynkin Ω E = M" using assms Dynkin_subset[OF E(1)] by simp with * show ?thesis using assms by (auto simp: Dynkin_def) qed subsubsection ‹Induction rule for intersection-stable generators› text✐‹tag important› ‹The reason to introduce Dynkin-systems is the following induction rules for ‹σ›-algebras generated by a generator closed under intersection.› proposition sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]: assumes "Int_stable G" and closed: "G ⊆ Pow Ω" and A: "A ∈ sigma_sets Ω G" assumes basic: "⋀A. A ∈ G ⟹ P A" and empty: "P {}" and compl: "⋀A. A ∈ sigma_sets Ω G ⟹ P A ⟹ P (Ω - A)" and union: "⋀A. disjoint_family A ⟹ range A ⊆ sigma_sets Ω G ⟹ (⋀i. P (A i)) ⟹ P (⋃i::nat. A i)" shows "P A" proof - let ?D = "{ A ∈ sigma_sets Ω G. P A }" interpret sigma_algebra Ω "sigma_sets Ω G" using closed by (rule sigma_algebra_sigma_sets) from compl[OF _ empty] closed have space: "P Ω" by simp interpret Dynkin_system Ω ?D by standard (auto dest: sets_into_space intro!: space compl union) have "sigma_sets Ω G = ?D" by (rule Dynkin_lemma) (auto simp: basic ‹Int_stable G›) with A show ?thesis by auto qed subsection ‹Measure type› definition✐‹tag important› positive :: "'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where "positive M μ ⟷ μ {} = 0" definition✐‹tag important› countably_additive :: "'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where "countably_additive M f ⟷ (∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i. A i) ∈ M ⟶ (∑i. f (A i)) = f (⋃i. A i))" definition✐‹tag important› measure_space :: "'a set ⇒ 'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where "measure_space Ω A μ ⟷ sigma_algebra Ω A ∧ positive A μ ∧ countably_additive A μ" typedef✐‹tag important› 'a measure = "{(Ω::'a set, A, μ). (∀a∈-A. μ a = 0) ∧ measure_space Ω A μ }" proof have "sigma_algebra UNIV {{}, UNIV}" by (auto simp: sigma_algebra_iff2) then show "(UNIV, {{}, UNIV}, λA. 0) ∈ {(Ω, A, μ). (∀a∈-A. μ a = 0) ∧ measure_space Ω A μ} " by (auto simp: measure_space_def positive_def countably_additive_def) qed definition✐‹tag important› space :: "'a measure ⇒ 'a set" where "space M = fst (Rep_measure M)" definition✐‹tag important› sets :: "'a measure ⇒ 'a set set" where "sets M = fst (snd (Rep_measure M))" definition✐‹tag important› emeasure :: "'a measure ⇒ 'a set ⇒ ennreal" where "emeasure M = snd (snd (Rep_measure M))" definition✐‹tag important› measure :: "'a measure ⇒ 'a set ⇒ real" where "measure M A = enn2real (emeasure M A)" declare [[coercion sets]] declare [[coercion measure]] declare [[coercion emeasure]] lemma measure_space: "measure_space (space M) (sets M) (emeasure M)" by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse) interpretation sets: sigma_algebra "space M" "sets M" for M :: "'a measure" using measure_space[of M] by (auto simp: measure_space_def) definition✐‹tag important› measure_of :: "'a set ⇒ 'a set set ⇒ ('a set ⇒ ennreal) ⇒ 'a measure" where "measure_of Ω A μ = Abs_measure (Ω, if A ⊆ Pow Ω then sigma_sets Ω A else {{}, Ω}, λa. if a ∈ sigma_sets Ω A ∧ measure_space Ω (sigma_sets Ω A) μ then μ a else 0)" abbreviation "sigma Ω A ≡ measure_of Ω A (λx. 0)" lemma measure_space_0: "A ⊆ Pow Ω ⟹ measure_space Ω (sigma_sets Ω A) (λx. 0)" unfolding measure_space_def by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def) lemma sigma_algebra_trivial: "sigma_algebra Ω {{}, Ω}" by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{Ω}"])+ lemma measure_space_0': "measure_space Ω {{}, Ω} (λx. 0)" by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial) lemma measure_space_closed: assumes "measure_space Ω M μ" shows "M ⊆ Pow Ω" proof - interpret sigma_algebra Ω M using assms by(simp add: measure_space_def) show ?thesis by(rule space_closed) qed lemma (in ring_of_sets) positive_cong_eq: "(⋀a. a ∈ M ⟹ μ' a = μ a) ⟹ positive M μ' = positive M μ" by (auto simp add: positive_def) lemma (in sigma_algebra) countably_additive_eq: "(⋀a. a ∈ M ⟹ μ' a = μ a) ⟹ countably_additive M μ' = countably_additive M μ" unfolding countably_additive_def by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq) lemma measure_space_eq: assumes closed: "A ⊆ Pow Ω" and eq: "⋀a. a ∈ sigma_sets Ω A ⟹ μ a = μ' a" shows "measure_space Ω (sigma_sets Ω A) μ = measure_space Ω (sigma_sets Ω A) μ'" proof - interpret sigma_algebra Ω "sigma_sets Ω A" using closed by (rule sigma_algebra_sigma_sets) from positive_cong_eq[OF eq, of "λi. i"] countably_additive_eq[OF eq, of "λi. i"] show ?thesis by (auto simp: measure_space_def) qed lemma measure_of_eq: assumes closed: "A ⊆ Pow Ω" and eq: "(⋀a. a ∈ sigma_sets Ω A ⟹ μ a = μ' a)" shows "measure_of Ω A μ = measure_of Ω A μ'" proof - have "measure_space Ω (sigma_sets Ω A) μ = measure_space Ω