author | hoelzl |
Wed, 10 Oct 2012 12:12:24 +0200 | |
changeset 49785 | 0a8adca22974 |
parent 49784 | 5e5b2da42a69 |
child 49789 | e0a4cb91a8a9 |
permissions | -rw-r--r-- |
47694 | 1 |
(* Title: HOL/Probability/Measure_Space.thy |
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Author: Lawrence C Paulson |
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Author: Johannes Hölzl, TU München |
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Author: Armin Heller, TU München |
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*) |
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header {* Measure spaces and their properties *} |
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theory Measure_Space |
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imports |
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Sigma_Algebra |
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"~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits" |
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begin |
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lemma sums_def2: |
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"f sums x \<longleftrightarrow> (\<lambda>n. (\<Sum>i\<le>n. f i)) ----> x" |
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unfolding sums_def |
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apply (subst LIMSEQ_Suc_iff[symmetric]) |
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unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost .. |
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lemma suminf_cmult_indicator: |
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fixes f :: "nat \<Rightarrow> ereal" |
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assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i" |
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shows "(\<Sum>n. f n * indicator (A n) x) = f i" |
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proof - |
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have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)" |
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using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto |
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then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)" |
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by (auto simp: setsum_cases) |
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moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)" |
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proof (rule ereal_SUPI) |
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fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y" |
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from this[of "Suc i"] show "f i \<le> y" by auto |
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qed (insert assms, simp) |
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ultimately show ?thesis using assms |
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by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def) |
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qed |
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lemma suminf_indicator: |
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assumes "disjoint_family A" |
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shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x" |
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proof cases |
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assume *: "x \<in> (\<Union>i. A i)" |
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then obtain i where "x \<in> A i" by auto |
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from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"] |
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show ?thesis using * by simp |
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qed simp |
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text {* |
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The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to |
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represent sigma algebras (with an arbitrary emeasure). |
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*} |
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section "Extend binary sets" |
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lemma LIMSEQ_binaryset: |
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assumes f: "f {} = 0" |
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shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B" |
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proof - |
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have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)" |
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proof |
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fix n |
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show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B" |
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by (induct n) (auto simp add: binaryset_def f) |
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qed |
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moreover |
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have "... ----> f A + f B" by (rule tendsto_const) |
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ultimately |
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have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B" |
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by metis |
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hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B" |
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by simp |
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thus ?thesis by (rule LIMSEQ_offset [where k=2]) |
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qed |
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lemma binaryset_sums: |
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assumes f: "f {} = 0" |
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shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)" |
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by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan) |
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lemma suminf_binaryset_eq: |
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fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}" |
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shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B" |
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by (metis binaryset_sums sums_unique) |
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section {* Properties of a premeasure @{term \<mu>} *} |
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text {* |
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The definitions for @{const positive} and @{const countably_additive} should be here, by they are |
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necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}. |
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*} |
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definition additive where |
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"additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)" |
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definition increasing where |
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"increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)" |
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||
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lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def) |
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lemma positiveD2: "positive M f \<Longrightarrow> A \<in> M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def) |
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lemma positiveD_empty: |
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"positive M f \<Longrightarrow> f {} = 0" |
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by (auto simp add: positive_def) |
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lemma additiveD: |
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"additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y" |
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by (auto simp add: additive_def) |
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lemma increasingD: |
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"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y" |
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by (auto simp add: increasing_def) |
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lemma countably_additiveI: |
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"(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)) |
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\<Longrightarrow> countably_additive M f" |
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by (simp add: countably_additive_def) |
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lemma (in ring_of_sets) disjointed_additive: |
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assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A" |
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shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" |
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proof (induct n) |
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case (Suc n) |
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then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))" |
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by simp |
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also have "\<dots> = f (A n \<union> disjointed A (Suc n))" |
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using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq) |
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also have "A n \<union> disjointed A (Suc n) = A (Suc n)" |
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using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq) |
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finally show ?case . |
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qed simp |
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lemma (in ring_of_sets) additive_sum: |
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fixes A:: "'i \<Rightarrow> 'a set" |
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assumes f: "positive M f" and ad: "additive M f" and "finite S" |
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and A: "A`S \<subseteq> M" |
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and disj: "disjoint_family_on A S" |
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shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)" |
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using `finite S` disj A proof induct |
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case empty show ?case using f by (simp add: positive_def) |
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next |
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case (insert s S) |
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then have "A s \<inter> (\<Union>i\<in>S. A i) = {}" |
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by (auto simp add: disjoint_family_on_def neq_iff) |
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moreover |
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have "A s \<in> M" using insert by blast |
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moreover have "(\<Union>i\<in>S. A i) \<in> M" |
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using insert `finite S` by auto |
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moreover |
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ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)" |
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using ad UNION_in_sets A by (auto simp add: additive_def) |
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with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A] |
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by (auto simp add: additive_def subset_insertI) |
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qed |
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lemma (in ring_of_sets) additive_increasing: |
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assumes posf: "positive M f" and addf: "additive M f" |
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shows "increasing M f" |
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proof (auto simp add: increasing_def) |
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fix x y |
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assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y" |
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then have "y - x \<in> M" by auto |
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then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto |
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then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto |
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also have "... = f (x \<union> (y-x))" using addf |
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by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2)) |
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also have "... = f y" |
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by (metis Un_Diff_cancel Un_absorb1 xy(3)) |
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finally show "f x \<le> f y" by simp |
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qed |
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lemma (in ring_of_sets) countably_additive_additive: |
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assumes posf: "positive M f" and ca: "countably_additive M f" |
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shows "additive M f" |
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proof (auto simp add: additive_def) |
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fix x y |
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assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}" |
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hence "disjoint_family (binaryset x y)" |
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by (auto simp add: disjoint_family_on_def binaryset_def) |
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hence "range (binaryset x y) \<subseteq> M \<longrightarrow> |
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(\<Union>i. binaryset x y i) \<in> M \<longrightarrow> |
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f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))" |
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using ca |
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by (simp add: countably_additive_def) |
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hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow> |
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f (x \<union> y) = (\<Sum>n. f (binaryset x y n))" |
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by (simp add: range_binaryset_eq UN_binaryset_eq) |
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thus "f (x \<union> y) = f x + f y" using posf x y |
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by (auto simp add: Un suminf_binaryset_eq positive_def) |
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qed |
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lemma (in algebra) increasing_additive_bound: |
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fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal" |
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assumes f: "positive M f" and ad: "additive M f" |
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and inc: "increasing M f" |
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and A: "range A \<subseteq> M" |
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and disj: "disjoint_family A" |
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shows "(\<Sum>i. f (A i)) \<le> f \<Omega>" |
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proof (safe intro!: suminf_bound) |
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fix N |
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note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"] |
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have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)" |
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using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N) |
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also have "... \<le> f \<Omega>" using space_closed A |
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by (intro increasingD[OF inc] finite_UN) auto |
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finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp |
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qed (insert f A, auto simp: positive_def) |
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lemma (in ring_of_sets) countably_additiveI_finite: |
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assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>" |
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shows "countably_additive M \<mu>" |
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proof (rule countably_additiveI) |
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fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F" |
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have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto |
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from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto |
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have inj_f: "inj_on f {i. F i \<noteq> {}}" |
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proof (rule inj_onI, simp) |
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fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}" |
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then have "f i \<in> F i" "f j \<in> F j" using f by force+ |
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with disj * show "i = j" by (auto simp: disjoint_family_on_def) |
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qed |
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have "finite (\<Union>i. F i)" |
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by (metis F(2) assms(1) infinite_super sets_into_space) |
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have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}" |
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by (auto simp: positiveD_empty[OF `positive M \<mu>`]) |
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moreover have fin_not_empty: "finite {i. F i \<noteq> {}}" |
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proof (rule finite_imageD) |
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from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto |
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then show "finite (f`{i. F i \<noteq> {}})" |
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by (rule finite_subset) fact |
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qed fact |
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ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}" |
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by (rule finite_subset) |
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have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}" |
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using disj by (auto simp: disjoint_family_on_def) |
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from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))" |
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47761 | 242 |
by (rule suminf_finite) auto |
47694 | 243 |
also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))" |
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using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto |
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also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)" |
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using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto |
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also have "\<dots> = \<mu> (\<Union>i. F i)" |
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by (rule arg_cong[where f=\<mu>]) auto |
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finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" . |
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qed |
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251 |
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49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
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252 |
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below: |
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assumes f: "positive M f" "additive M f" |
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254 |
shows "countably_additive M f \<longleftrightarrow> |
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(\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))" |
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256 |
unfolding countably_additive_def |
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proof safe |
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258 |
assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)" |
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fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M" |
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then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets) |
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with count_sum[THEN spec, of "disjointed A"] A(3) |
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262 |
have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)" |
16907431e477
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by (auto simp: UN_disjointed_eq disjoint_family_disjointed) |
16907431e477
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parents:
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264 |
moreover have "(\<lambda>n. (\<Sum>i=0..<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" |
16907431e477
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parents:
47762
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|
265 |
using f(1)[unfolded positive_def] dA |
16907431e477
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parents:
47762
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266 |
by (auto intro!: summable_sumr_LIMSEQ_suminf summable_ereal_pos) |
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parents:
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|
267 |
from LIMSEQ_Suc[OF this] |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
268 |
have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
269 |
unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost . |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
270 |
moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
271 |
using disjointed_additive[OF f A(1,2)] . |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
272 |
ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
273 |
next |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
274 |
assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
275 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
276 |
have *: "(\<Union>n. (\<Union>i\<le>n. A i)) = (\<Union>i. A i)" by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
277 |
have "(\<lambda>n. f (\<Union>i\<le>n. A i)) ----> f (\<Union>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
278 |
proof (unfold *[symmetric], intro cont[rule_format]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
279 |
show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> M" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
280 |
using A * by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
281 |
qed (force intro!: incseq_SucI) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
282 |
moreover have "\<And>n. f (\<Union>i\<le>n. A i) = (\<Sum>i\<le>n. f (A i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
283 |
using A |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
284 |
by (intro additive_sum[OF f, of _ A, symmetric]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
285 |
(auto intro: disjoint_family_on_mono[where B=UNIV]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
286 |
ultimately |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
287 |
have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
288 |
unfolding sums_def2 by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
289 |
from sums_unique[OF this] |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
290 |
show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
291 |
qed |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
292 |
|
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
293 |
lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous: |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
294 |
assumes f: "positive M f" "additive M f" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
295 |
shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
296 |
\<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
297 |
proof safe |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
298 |
assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
299 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
300 |
with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
301 |
using `positive M f`[unfolded positive_def] by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
302 |
next |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
303 |
assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
304 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
305 |
|
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
306 |
have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
307 |
using additive_increasing[OF f] unfolding increasing_def by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
308 |
|
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
309 |
have decseq_fA: "decseq (\<lambda>i. f (A i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
310 |
using A by (auto simp: decseq_def intro!: f_mono) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
311 |
have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
312 |
using A by (auto simp: decseq_def) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
313 |
then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
314 |
using A unfolding decseq_def by (auto intro!: f_mono Diff) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
315 |
have "f (\<Inter>x. A x) \<le> f (A 0)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
316 |
using A by (auto intro!: f_mono) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
317 |
then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
318 |
using A by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
319 |
{ fix i |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
320 |
have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
321 |
then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
322 |
using A by auto } |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
323 |
note f_fin = this |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
324 |
have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
325 |
proof (intro cont[rule_format, OF _ decseq _ f_fin]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
326 |
show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
327 |
using A by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
328 |
qed |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
329 |
from INF_Lim_ereal[OF decseq_f this] |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
330 |
have "(INF n. f (A n - (\<Inter>i. A i))) = 0" . |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
331 |
moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
332 |
by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
333 |
ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
334 |
using A(4) f_fin f_Int_fin |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
335 |
by (subst INFI_ereal_add) (auto simp: decseq_f) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
336 |
moreover { |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
337 |
fix n |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
338 |
have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
339 |
using A by (subst f(2)[THEN additiveD]) auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
340 |
also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
341 |
by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
342 |
finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . } |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
343 |
ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
344 |
by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
345 |
with LIMSEQ_ereal_INFI[OF decseq_fA] |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
346 |
show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
347 |
qed |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
348 |
|
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
349 |
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below: |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
350 |
assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
351 |
assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
352 |
assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
353 |
shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
354 |
proof - |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
355 |
have "\<forall>A\<in>M. \<exists>x. f A = ereal x" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
356 |
proof |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
357 |
fix A assume "A \<in> M" with f show "\<exists>x. f A = ereal x" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
358 |
unfolding positive_def by (cases "f A") auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
359 |
qed |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
360 |
from bchoice[OF this] guess f' .. note f' = this[rule_format] |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
361 |
from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
362 |
by (intro cont[rule_format]) (auto simp: decseq_def incseq_def) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
363 |
moreover |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
364 |
{ fix i |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
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diff
changeset
|
365 |
have "f ((\<Union>i. A i) - A i) + f (A i) = f ((\<Union>i. A i) - A i \<union> A i)" |
16907431e477
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changeset
|
366 |
using A by (intro f(2)[THEN additiveD, symmetric]) auto |
16907431e477
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diff
changeset
|
367 |
also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)" |
16907431e477
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changeset
|
368 |
by auto |
16907431e477
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diff
changeset
|
369 |
finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)" |
16907431e477
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changeset
|
370 |
using A by (subst (asm) (1 2 3) f') auto |
16907431e477
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changeset
|
371 |
then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))" |
16907431e477
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diff
changeset
|
372 |
using A f' by auto } |
16907431e477
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changeset
|
373 |
ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0" |
16907431e477
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changeset
|
374 |
by (simp add: zero_ereal_def) |
16907431e477
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changeset
|
375 |
then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)" |
16907431e477
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changeset
|
376 |
by (rule LIMSEQ_diff_approach_zero2[OF tendsto_const]) |
16907431e477
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changeset
|
377 |
then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
16907431e477
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changeset
|
378 |
using A by (subst (1 2) f') auto |
16907431e477
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changeset
|
379 |
qed |
16907431e477
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diff
changeset
|
380 |
|
16907431e477
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changeset
|
381 |
lemma (in ring_of_sets) empty_continuous_imp_countably_additive: |
16907431e477
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changeset
|
382 |
assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>" |
16907431e477
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changeset
|
383 |
assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
16907431e477
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changeset
|
384 |
shows "countably_additive M f" |
16907431e477
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changeset
|
385 |
using countably_additive_iff_continuous_from_below[OF f] |
16907431e477
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changeset
|
386 |
using empty_continuous_imp_continuous_from_below[OF f fin] cont |
16907431e477
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|
387 |
by blast |
16907431e477
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|
388 |
|
47694 | 389 |
section {* Properties of @{const emeasure} *} |
390 |
||
391 |
lemma emeasure_positive: "positive (sets M) (emeasure M)" |
|
392 |
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) |
|
393 |
||
394 |
lemma emeasure_empty[simp, intro]: "emeasure M {} = 0" |
|
395 |
using emeasure_positive[of M] by (simp add: positive_def) |
|
396 |
||
397 |
lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A" |
|
398 |
using emeasure_notin_sets[of A M] emeasure_positive[of M] |
|
399 |
by (cases "A \<in> sets M") (auto simp: positive_def) |
|
400 |
||
401 |
lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>" |
|
402 |
using emeasure_nonneg[of M A] by auto |
|
403 |
||
404 |
lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)" |
|
405 |
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) |
|
406 |
||
407 |
lemma suminf_emeasure: |
|
408 |
"range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" |
|
409 |
using countable_UN[of A UNIV M] emeasure_countably_additive[of M] |
|
410 |
by (simp add: countably_additive_def) |
|
411 |
||
412 |
lemma emeasure_additive: "additive (sets M) (emeasure M)" |
|
413 |
by (metis countably_additive_additive emeasure_positive emeasure_countably_additive) |
|
414 |
||
415 |
lemma plus_emeasure: |
|
416 |
"a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)" |
|
417 |
using additiveD[OF emeasure_additive] .. |
|
418 |
||
419 |
lemma setsum_emeasure: |
|
420 |
"F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow> |
|
421 |
(\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)" |
|
422 |
by (metis additive_sum emeasure_positive emeasure_additive) |
|
423 |
||
424 |
lemma emeasure_mono: |
|
425 |
"a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b" |
|
426 |
by (metis additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets |
|
427 |
emeasure_positive increasingD) |
|
428 |
||
429 |
lemma emeasure_space: |
|
430 |
"emeasure M A \<le> emeasure M (space M)" |
|
431 |
by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets_into_space top) |
|
432 |
||
433 |
lemma emeasure_compl: |
|
434 |
assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>" |
|
435 |
shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s" |
|
436 |
proof - |
|
437 |
from s have "0 \<le> emeasure M s" by auto |
|
438 |
have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s |
|
439 |
by (metis Un_Diff_cancel Un_absorb1 s sets_into_space) |
|
440 |
also have "... = emeasure M s + emeasure M (space M - s)" |
|
441 |
by (rule plus_emeasure[symmetric]) (auto simp add: s) |
|
442 |
finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" . |
|
443 |
then show ?thesis |
|
444 |
using fin `0 \<le> emeasure M s` |
|
445 |
unfolding ereal_eq_minus_iff by (auto simp: ac_simps) |
|
446 |
qed |
|
447 |
||
448 |
lemma emeasure_Diff: |
|
449 |
assumes finite: "emeasure M B \<noteq> \<infinity>" |
|
450 |
and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A" |
|
451 |
shows "emeasure M (A - B) = emeasure M A - emeasure M B" |
|
452 |
proof - |
|
453 |
have "0 \<le> emeasure M B" using assms by auto |
|
454 |
have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto |
|
455 |
then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp |
|
456 |
also have "\<dots> = emeasure M (A - B) + emeasure M B" |
|
457 |
using measurable by (subst plus_emeasure[symmetric]) auto |
|
458 |
finally show "emeasure M (A - B) = emeasure M A - emeasure M B" |
|
459 |
unfolding ereal_eq_minus_iff |
|
460 |
using finite `0 \<le> emeasure M B` by auto |
|
461 |
qed |
|
462 |
||
49773
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|
463 |
lemma Lim_emeasure_incseq: |
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changeset
|
464 |
"range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)" |
16907431e477
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diff
changeset
|
465 |
using emeasure_countably_additive |
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diff
changeset
|
466 |
by (auto simp add: countably_additive_iff_continuous_from_below emeasure_positive emeasure_additive) |
47694 | 467 |
|
468 |
lemma incseq_emeasure: |
|
469 |
assumes "range B \<subseteq> sets M" "incseq B" |
|
470 |
shows "incseq (\<lambda>i. emeasure M (B i))" |
|
471 |
using assms by (auto simp: incseq_def intro!: emeasure_mono) |
|
472 |
||
49773
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|
473 |
lemma SUP_emeasure_incseq: |
47694 | 474 |
assumes A: "range A \<subseteq> sets M" "incseq A" |
49773
16907431e477
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changeset
|
475 |
shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)" |
16907431e477
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diff
changeset
|
476 |
using LIMSEQ_ereal_SUPR[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A] |
16907431e477
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diff
changeset
|
477 |
by (simp add: LIMSEQ_unique) |
47694 | 478 |
|
479 |
lemma decseq_emeasure: |
|
480 |
assumes "range B \<subseteq> sets M" "decseq B" |
|
481 |
shows "decseq (\<lambda>i. emeasure M (B i))" |
|
482 |
using assms by (auto simp: decseq_def intro!: emeasure_mono) |
|
483 |
||
484 |
lemma INF_emeasure_decseq: |
|
485 |
assumes A: "range A \<subseteq> sets M" and "decseq A" |
|
486 |
and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" |
|
487 |
shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)" |
|
488 |
proof - |
|
489 |
have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)" |
|
490 |
using A by (auto intro!: emeasure_mono) |
|
491 |
hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto |
|
492 |
||
493 |
have A0: "0 \<le> emeasure M (A 0)" using A by auto |
|
494 |
||
495 |
have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))" |
|
496 |
by (simp add: ereal_SUPR_uminus minus_ereal_def) |
|
497 |
also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))" |
|
498 |
unfolding minus_ereal_def using A0 assms |
|
499 |
by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure) |
|
500 |
also have "\<dots> = (SUP n. emeasure M (A 0 - A n))" |
|
501 |
using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto |
|
502 |
also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)" |
|
503 |
proof (rule SUP_emeasure_incseq) |
|
504 |
show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M" |
|
505 |
using A by auto |
|
506 |
show "incseq (\<lambda>n. A 0 - A n)" |
|
507 |
using `decseq A` by (auto simp add: incseq_def decseq_def) |
|
508 |
qed |
|
509 |
also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)" |
|
510 |
using A finite * by (simp, subst emeasure_Diff) auto |
|
511 |
finally show ?thesis |
|
512 |
unfolding ereal_minus_eq_minus_iff using finite A0 by auto |
|
513 |
qed |
|
514 |
||
515 |
lemma Lim_emeasure_decseq: |
|
516 |
assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" |
|
517 |
shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)" |
|
518 |
using LIMSEQ_ereal_INFI[OF decseq_emeasure, OF A] |
|
519 |
using INF_emeasure_decseq[OF A fin] by simp |
|
520 |
||
521 |
lemma emeasure_subadditive: |
|
522 |
assumes measurable: "A \<in> sets M" "B \<in> sets M" |
|
523 |
shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B" |
|
524 |
proof - |
|
525 |
from plus_emeasure[of A M "B - A"] |
|
526 |
have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)" |
|
527 |
using assms by (simp add: Diff) |
|
528 |
also have "\<dots> \<le> emeasure M A + emeasure M B" |
|
529 |
using assms by (auto intro!: add_left_mono emeasure_mono) |
|
530 |
finally show ?thesis . |
|
531 |
qed |
|
532 |
||
533 |
lemma emeasure_subadditive_finite: |
|
534 |
assumes "finite I" "A ` I \<subseteq> sets M" |
|
535 |
shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))" |
|
536 |
using assms proof induct |
|
537 |
case (insert i I) |
|
538 |
then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))" |
|
539 |
by simp |
|
540 |
also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)" |
|
541 |
using insert by (intro emeasure_subadditive finite_UN) auto |
|
542 |
also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))" |
|
543 |
using insert by (intro add_mono) auto |
|
544 |
also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))" |
|
545 |
using insert by auto |
|
546 |
finally show ?case . |
|
547 |
qed simp |
|
548 |
||
549 |
lemma emeasure_subadditive_countably: |
|
550 |
assumes "range f \<subseteq> sets M" |
|
551 |
shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))" |
|
552 |
proof - |
|
553 |
have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)" |
|
554 |
unfolding UN_disjointed_eq .. |
|
555 |
also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))" |
|
556 |
using range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"] |
|
557 |
by (simp add: disjoint_family_disjointed comp_def) |
|
558 |
also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))" |
|
559 |
using range_disjointed_sets[OF assms] assms |
|
560 |
by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset) |
|
561 |
finally show ?thesis . |
|
562 |
qed |
|
563 |
||
564 |
lemma emeasure_insert: |
|
565 |
assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A" |
|
566 |
shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A" |
|
567 |
proof - |
|
568 |
have "{x} \<inter> A = {}" using `x \<notin> A` by auto |
|
569 |
from plus_emeasure[OF sets this] show ?thesis by simp |
|
570 |
qed |
|
571 |
||
572 |
lemma emeasure_eq_setsum_singleton: |
|
573 |
assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" |
|
574 |
shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})" |
|
575 |
using setsum_emeasure[of "\<lambda>x. {x}" S M] assms |
|
576 |
by (auto simp: disjoint_family_on_def subset_eq) |
|
577 |
||
578 |
lemma setsum_emeasure_cover: |
|
579 |
assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M" |
|
580 |
assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)" |
|
581 |
assumes disj: "disjoint_family_on B S" |
|
582 |
shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))" |
|
583 |
proof - |
|
584 |
have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))" |
|
585 |
proof (rule setsum_emeasure) |
|
586 |
show "disjoint_family_on (\<lambda>i. A \<inter> B i) S" |
|
587 |
using `disjoint_family_on B S` |
|
588 |
unfolding disjoint_family_on_def by auto |
|
589 |
qed (insert assms, auto) |
|
590 |
also have "(\<Union>i\<in>S. A \<inter> (B i)) = A" |
|
591 |
using A by auto |
|
592 |
finally show ?thesis by simp |
|
593 |
qed |
|
594 |
||
595 |
lemma emeasure_eq_0: |
|
596 |
"N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0" |
|
597 |
by (metis emeasure_mono emeasure_nonneg order_eq_iff) |
|
598 |
||
599 |
lemma emeasure_UN_eq_0: |
|
600 |
assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M" |
|
601 |
shows "emeasure M (\<Union> i. N i) = 0" |
|
602 |
proof - |
|
603 |
have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto |
|
604 |
moreover have "emeasure M (\<Union> i. N i) \<le> 0" |
|
605 |
using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp |
|
606 |
ultimately show ?thesis by simp |
|
607 |
qed |
|
608 |
||
609 |
lemma measure_eqI_finite: |
|
610 |
assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A" |
|
611 |
assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" |
|
612 |
shows "M = N" |
|
613 |
proof (rule measure_eqI) |
|
614 |
fix X assume "X \<in> sets M" |
|
615 |
then have X: "X \<subseteq> A" by auto |
|
616 |
then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})" |
|
617 |
using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset) |
|
618 |
also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})" |
|
619 |
using X eq by (auto intro!: setsum_cong) |
|
620 |
also have "\<dots> = emeasure N X" |
|
621 |
using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset) |
|
622 |
finally show "emeasure M X = emeasure N X" . |
|
623 |
qed simp |
|
624 |
||
625 |
lemma measure_eqI_generator_eq: |
|
626 |
fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set" |
|
627 |
assumes "Int_stable E" "E \<subseteq> Pow \<Omega>" |
|
628 |
and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X" |
|
629 |
and M: "sets M = sigma_sets \<Omega> E" |
|
630 |
and N: "sets N = sigma_sets \<Omega> E" |
|
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
631 |
and A: "range A \<subseteq> E" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" |
47694 | 632 |
shows "M = N" |
633 |
proof - |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
634 |
let ?\<mu> = "emeasure M" and ?\<nu> = "emeasure N" |
47694 | 635 |
interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
636 |
{ fix F assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>" |
47694 | 637 |
then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
638 |
let ?D = "{D \<in> sigma_sets \<Omega> E. ?\<mu> (F \<inter> D) = ?\<nu> (F \<inter> D)}" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
639 |
have "?\<nu> F \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` `F \<in> E` eq by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
640 |
interpret D: dynkin_system \<Omega> ?D |
47694 | 641 |
proof (rule dynkin_systemI, simp_all) |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
642 |
fix A assume "A \<in> sigma_sets \<Omega> E \<and> ?\<mu> (F \<inter> A) = ?\<nu> (F \<inter> A)" |
47694 | 643 |
then show "A \<subseteq> \<Omega>" using S.sets_into_space by auto |
644 |
next |
|
645 |
have "F \<inter> \<Omega> = F" using `F \<in> E` `E \<subseteq> Pow \<Omega>` by auto |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
646 |
then show "?\<mu> (F \<inter> \<Omega>) = ?\<nu> (F \<inter> \<Omega>)" |
47694 | 647 |
using `F \<in> E` eq by (auto intro: sigma_sets_top) |
648 |
next |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
649 |
fix A assume *: "A \<in> sigma_sets \<Omega> E \<and> ?\<mu> (F \<inter> A) = ?\<nu> (F \<inter> A)" |
47694 | 650 |
then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)" |
651 |
and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E" |
|
652 |
using `F \<in> E` S.sets_into_space by auto |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
653 |
have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
654 |
then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using `?\<nu> F \<noteq> \<infinity>` by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
655 |
have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
656 |
then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
657 |
then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding ** |
47694 | 658 |
using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N) |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
659 |
also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq `F \<in> E` * by simp |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
660 |
also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding ** |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
661 |
using `F \<inter> A \<in> sigma_sets \<Omega> E` `?\<nu> (F \<inter> A) \<noteq> \<infinity>` |
47694 | 662 |
by (auto intro!: emeasure_Diff[symmetric] simp: M N) |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
663 |
finally show "\<Omega> - A \<in> sigma_sets \<Omega> E \<and> ?\<mu> (F \<inter> (\<Omega> - A)) = ?\<nu> (F \<inter> (\<Omega> - A))" |
47694 | 664 |
using * by auto |
665 |
next |
|
666 |
fix A :: "nat \<Rightarrow> 'a set" |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
667 |
assume "disjoint_family A" and A: "range A \<subseteq> {X \<in> sigma_sets \<Omega> E. ?\<mu> (F \<inter> X) = ?\<nu> (F \<inter> X)}" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
668 |
then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
669 |
by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
670 |
with A show "(\<Union>x. A x) \<in> sigma_sets \<Omega> E \<and> ?\<mu> (F \<inter> (\<Union>x. A x)) = ?\<nu> (F \<inter> (\<Union>x. A x))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
671 |
by auto |
47694 | 672 |
qed |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
673 |
have *: "sigma_sets \<Omega> E = ?D" |
47694 | 674 |
using `F \<in> E` `Int_stable E` |
675 |
by (intro D.dynkin_lemma) (auto simp add: Int_stable_def eq) |
|
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
676 |
have "\<And>D. D \<in> sets M \<Longrightarrow> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
677 |
unfolding M by (subst (asm) *) auto } |
47694 | 678 |
note * = this |
679 |
show "M = N" |
|
680 |
proof (rule measure_eqI) |
|
681 |
show "sets M = sets N" |
|
682 |
using M N by simp |
|
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
683 |
have [simp, intro]: "\<And>i. A i \<in> sets M" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
684 |
using A(1) by (auto simp: subset_eq M) |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
685 |
fix F assume "F \<in> sets M" |
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
686 |
let ?D = "disjointed (\<lambda>i. F \<inter> A i)" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
687 |
have "space M = \<Omega>" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
688 |
using top[of M] space_closed[of M] S.top S.space_closed `sets M = sigma_sets \<Omega> E` by blast |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
689 |
then have F_eq: "F = (\<Union>i. ?D i)" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
690 |
using `F \<in> sets M`[THEN sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq) |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
691 |
have [simp, intro]: "\<And>i. ?D i \<in> sets M" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
692 |
using range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] `F \<in> sets M` |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
693 |
by (auto simp: subset_eq) |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
694 |
have "disjoint_family ?D" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
695 |
by (auto simp: disjoint_family_disjointed) |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
696 |
moreover |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
697 |
{ fix i |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
698 |
have "A i \<inter> ?D i = ?D i" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
699 |
by (auto simp: disjointed_def) |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
700 |
then have "emeasure M (?D i) = emeasure N (?D i)" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
701 |
using *[of "A i" "?D i", OF _ A(3)] A(1) by auto } |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
702 |
ultimately show "emeasure M F = emeasure N F" |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
703 |
using N M |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
704 |
apply (subst (1 2) F_eq) |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
705 |
apply (subst (1 2) suminf_emeasure[symmetric]) |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
706 |
apply auto |
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49773
diff
changeset
|
707 |
done |
47694 | 708 |
qed |
709 |
qed |
|
710 |
||
711 |
lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M" |
|
712 |
proof (intro measure_eqI emeasure_measure_of_sigma) |
|
713 |
show "sigma_algebra (space M) (sets M)" .. |
|
714 |
show "positive (sets M) (emeasure M)" |
|
715 |
by (simp add: positive_def emeasure_nonneg) |
|
716 |
show "countably_additive (sets M) (emeasure M)" |
|
717 |
by (simp add: emeasure_countably_additive) |
|
718 |
qed simp_all |
|
719 |
||
720 |
section "@{text \<mu>}-null sets" |
|
721 |
||
722 |
definition null_sets :: "'a measure \<Rightarrow> 'a set set" where |
|
723 |
"null_sets M = {N\<in>sets M. emeasure M N = 0}" |
|
724 |
||
725 |
lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0" |
|
726 |
by (simp add: null_sets_def) |
|
727 |
||
728 |
lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M" |
|
729 |
unfolding null_sets_def by simp |
|
730 |
||
731 |
lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M" |
|
732 |
unfolding null_sets_def by simp |
|
733 |
||
734 |
interpretation null_sets: ring_of_sets "space M" "null_sets M" for M |
|
47762 | 735 |
proof (rule ring_of_setsI) |
47694 | 736 |
show "null_sets M \<subseteq> Pow (space M)" |
737 |
using sets_into_space by auto |
|
738 |
show "{} \<in> null_sets M" |
|
739 |
by auto |
|
740 |
fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M" |
|
741 |
then have "A \<in> sets M" "B \<in> sets M" |
|
742 |
by auto |
|
743 |
moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B" |
|
744 |
"emeasure M (A - B) \<le> emeasure M A" |
|
745 |
by (auto intro!: emeasure_subadditive emeasure_mono) |
|
746 |
moreover have "emeasure M B = 0" "emeasure M A = 0" |
|
747 |
using sets by auto |
|
748 |
ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M" |
|
749 |
by (auto intro!: antisym) |
|
750 |
qed |
|
751 |
||
752 |
lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))" |
|
753 |
proof - |
|
754 |
have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)" |
|
755 |
unfolding SUP_def image_compose |
|
756 |
unfolding surj_from_nat .. |
|
757 |
then show ?thesis by simp |
|
758 |
qed |
|
759 |
||
760 |
lemma null_sets_UN[intro]: |
|
761 |
assumes "\<And>i::'i::countable. N i \<in> null_sets M" |
|
762 |
shows "(\<Union>i. N i) \<in> null_sets M" |
|
763 |
proof (intro conjI CollectI null_setsI) |
|
764 |
show "(\<Union>i. N i) \<in> sets M" using assms by auto |
|
765 |
have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg) |
|
766 |
moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))" |
|
767 |
unfolding UN_from_nat[of N] |
|
768 |
using assms by (intro emeasure_subadditive_countably) auto |
|
769 |
ultimately show "emeasure M (\<Union>i. N i) = 0" |
|
770 |
using assms by (auto simp: null_setsD1) |
|
771 |
qed |
|
772 |
||
773 |
lemma null_set_Int1: |
|
774 |
assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M" |
|
775 |
proof (intro CollectI conjI null_setsI) |
|
776 |
show "emeasure M (A \<inter> B) = 0" using assms |
|
777 |
by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto |
|
778 |
qed (insert assms, auto) |
|
779 |
||
780 |
lemma null_set_Int2: |
|
781 |
assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M" |
|
782 |
using assms by (subst Int_commute) (rule null_set_Int1) |
|
783 |
||
784 |
lemma emeasure_Diff_null_set: |
|
785 |
assumes "B \<in> null_sets M" "A \<in> sets M" |
|
786 |
shows "emeasure M (A - B) = emeasure M A" |
|
787 |
proof - |
|
788 |
have *: "A - B = (A - (A \<inter> B))" by auto |
|
789 |
have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1) |
|
790 |
then show ?thesis |
|
791 |
unfolding * using assms |
|
792 |
by (subst emeasure_Diff) auto |
|
793 |
qed |
|
794 |
||
795 |
lemma null_set_Diff: |
|
796 |
assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M" |
|
797 |
proof (intro CollectI conjI null_setsI) |
|
798 |
show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto |
|
799 |
qed (insert assms, auto) |
|
800 |
||
801 |
lemma emeasure_Un_null_set: |
|
802 |
assumes "A \<in> sets M" "B \<in> null_sets M" |
|
803 |
shows "emeasure M (A \<union> B) = emeasure M A" |
|
804 |
proof - |
|
805 |
have *: "A \<union> B = A \<union> (B - A)" by auto |
|
806 |
have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff) |
|
807 |
then show ?thesis |
|
808 |
unfolding * using assms |
|
809 |
by (subst plus_emeasure[symmetric]) auto |
|
810 |
qed |
|
811 |
||
812 |
section "Formalize almost everywhere" |
|
813 |
||
814 |
definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where |
|
815 |
"ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)" |
|
816 |
||
817 |
abbreviation |
|
818 |
almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where |
|
819 |
"almost_everywhere M P \<equiv> eventually P (ae_filter M)" |
|
820 |
||
821 |
syntax |
|
822 |
"_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10) |
|
823 |
||
824 |
translations |
|
825 |
"AE x in M. P" == "CONST almost_everywhere M (%x. P)" |
|
826 |
||
827 |
lemma eventually_ae_filter: |
|
828 |
fixes M P |
|
829 |
defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N" |
|
830 |
shows "eventually P (ae_filter M) \<longleftrightarrow> F P" |
|
831 |
unfolding ae_filter_def F_def[symmetric] |
|
832 |
proof (rule eventually_Abs_filter) |
|
833 |
show "is_filter F" |
|
834 |
proof |
|
835 |
fix P Q assume "F P" "F Q" |
|
836 |
then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" |
|
837 |
and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L" |
|
838 |
by auto |
|
839 |
then have "L \<union> N \<in> null_sets M" "{x \<in> space M. \<not> (P x \<and> Q x)} \<subseteq> L \<union> N" by auto |
|
840 |
then show "F (\<lambda>x. P x \<and> Q x)" by auto |
|
841 |
next |
|
842 |
fix P Q assume "F P" |
|
843 |
then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto |
|
844 |
moreover assume "\<forall>x. P x \<longrightarrow> Q x" |
|
845 |
ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto |
|
846 |
then show "F Q" by auto |
|
847 |
qed auto |
|
848 |
qed |
|
849 |
||
850 |
lemma AE_I': |
|
851 |
"N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)" |
|
852 |
unfolding eventually_ae_filter by auto |
|
853 |
||
854 |
lemma AE_iff_null: |
|
855 |
assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M") |
|
856 |
shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M" |
|
857 |
proof |
|
858 |
assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0" |
|
859 |
unfolding eventually_ae_filter by auto |
|
860 |
have "0 \<le> emeasure M ?P" by auto |
|
861 |
moreover have "emeasure M ?P \<le> emeasure M N" |
|
862 |
using assms N(1,2) by (auto intro: emeasure_mono) |
|
863 |
ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto |
|
864 |
then show "?P \<in> null_sets M" using assms by auto |
|
865 |
next |
|
866 |
assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I') |
|
867 |
qed |
|
868 |
||
869 |
lemma AE_iff_null_sets: |
|
870 |
"N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)" |
|
871 |
using Int_absorb1[OF sets_into_space, of N M] |
|
872 |
by (subst AE_iff_null) (auto simp: Int_def[symmetric]) |
|
873 |
||
47761 | 874 |
lemma AE_not_in: |
875 |
"N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" |
|
876 |
by (metis AE_iff_null_sets null_setsD2) |
|
877 |
||
47694 | 878 |
lemma AE_iff_measurable: |
879 |
"N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0" |
|
880 |
using AE_iff_null[of _ P] by auto |
|
881 |
||
882 |
lemma AE_E[consumes 1]: |
|
883 |
assumes "AE x in M. P x" |
|
884 |
obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M" |
|
885 |
using assms unfolding eventually_ae_filter by auto |
|
886 |
||
887 |
lemma AE_E2: |
|
888 |
assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M" |
|
889 |
shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0") |
|
890 |
proof - |
|
891 |
have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto |
|
892 |
with AE_iff_null[of M P] assms show ?thesis by auto |
|
893 |
qed |
|
894 |
||
895 |
lemma AE_I: |
|
896 |
assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M" |
|
897 |
shows "AE x in M. P x" |
|
898 |
using assms unfolding eventually_ae_filter by auto |
|
899 |
||
900 |
lemma AE_mp[elim!]: |
|
901 |
assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x" |
|
902 |
shows "AE x in M. Q x" |
|
903 |
proof - |
|
904 |
from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A" |
|
905 |
and A: "A \<in> sets M" "emeasure M A = 0" |
|
906 |
by (auto elim!: AE_E) |
|
907 |
||
908 |
from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B" |
|
909 |
and B: "B \<in> sets M" "emeasure M B = 0" |
|
910 |
by (auto elim!: AE_E) |
|
911 |
||
912 |
show ?thesis |
|
913 |
proof (intro AE_I) |
|
914 |
have "0 \<le> emeasure M (A \<union> B)" using A B by auto |
|
915 |
moreover have "emeasure M (A \<union> B) \<le> 0" |
|
916 |
using emeasure_subadditive[of A M B] A B by auto |
|
917 |
ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto |
|
918 |
show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B" |
|
919 |
using P imp by auto |
|
920 |
qed |
|
921 |
qed |
|
922 |
||
923 |
(* depricated replace by laws about eventually *) |
|
924 |
lemma |
|
925 |
shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x" |
|
926 |
and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x" |
|
927 |
and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x" |
|
928 |
and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x" |
|
929 |
and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)" |
|
930 |
by auto |
|
931 |
||
932 |
lemma AE_impI: |
|
933 |
"(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x" |
|
934 |
by (cases P) auto |
|
935 |
||
936 |
lemma AE_measure: |
|
937 |
assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M") |
|
938 |
shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)" |
|
939 |
proof - |
|
940 |
from AE_E[OF AE] guess N . note N = this |
|
941 |
with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)" |
|
942 |
by (intro emeasure_mono) auto |
|
943 |
also have "\<dots> \<le> emeasure M ?P + emeasure M N" |
|
944 |
using sets N by (intro emeasure_subadditive) auto |
|
945 |
also have "\<dots> = emeasure M ?P" using N by simp |
|
946 |
finally show "emeasure M ?P = emeasure M (space M)" |
|
947 |
using emeasure_space[of M "?P"] by auto |
|
948 |
qed |
|
949 |
||
950 |
lemma AE_space: "AE x in M. x \<in> space M" |
|
951 |
by (rule AE_I[where N="{}"]) auto |
|
952 |
||
953 |
lemma AE_I2[simp, intro]: |
|
954 |
"(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x" |
|
955 |
using AE_space by force |
|
956 |
||
957 |
lemma AE_Ball_mp: |
|
958 |
"\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x" |
|
959 |
by auto |
|
960 |
||
961 |
lemma AE_cong[cong]: |
|
962 |
"(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)" |
|
963 |
by auto |
|
964 |
||
965 |
lemma AE_all_countable: |
|
966 |
"(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)" |
|
967 |
proof |
|
968 |
assume "\<forall>i. AE x in M. P i x" |
|
969 |
from this[unfolded eventually_ae_filter Bex_def, THEN choice] |
|
970 |
obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto |
|
971 |
have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto |
|
972 |
also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto |
|
973 |
finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" . |
|
974 |
moreover from N have "(\<Union>i. N i) \<in> null_sets M" |
|
975 |
by (intro null_sets_UN) auto |
|
976 |
ultimately show "AE x in M. \<forall>i. P i x" |
|
977 |
unfolding eventually_ae_filter by auto |
|
978 |
qed auto |
|
979 |
||
980 |
lemma AE_finite_all: |
|
981 |
assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)" |
|
982 |
using f by induct auto |
|
983 |
||
984 |
lemma AE_finite_allI: |
|
985 |
assumes "finite S" |
|
986 |
shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x" |
|
987 |
using AE_finite_all[OF `finite S`] by auto |
|
988 |
||
989 |
lemma emeasure_mono_AE: |
|
990 |
assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" |
|
991 |
and B: "B \<in> sets M" |
|
992 |
shows "emeasure M A \<le> emeasure M B" |
|
993 |
proof cases |
|
994 |
assume A: "A \<in> sets M" |
|
995 |
from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M" |
|
996 |
by (auto simp: eventually_ae_filter) |
|
997 |
have "emeasure M A = emeasure M (A - N)" |
|
998 |
using N A by (subst emeasure_Diff_null_set) auto |
|
999 |
also have "emeasure M (A - N) \<le> emeasure M (B - N)" |
|
1000 |
using N A B sets_into_space by (auto intro!: emeasure_mono) |
|
1001 |
also have "emeasure M (B - N) = emeasure M B" |
|
1002 |
using N B by (subst emeasure_Diff_null_set) auto |
|
1003 |
finally show ?thesis . |
|
1004 |
qed (simp add: emeasure_nonneg emeasure_notin_sets) |
|
1005 |
||
1006 |
lemma emeasure_eq_AE: |
|
1007 |
assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" |
|
1008 |
assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
|
1009 |
shows "emeasure M A = emeasure M B" |
|
1010 |
using assms by (safe intro!: antisym emeasure_mono_AE) auto |
|
1011 |
||
1012 |
section {* @{text \<sigma>}-finite Measures *} |
|
1013 |
||
1014 |
locale sigma_finite_measure = |
|
1015 |
fixes M :: "'a measure" |
|
1016 |
assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. |
|
1017 |
range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)" |
|
1018 |
||
1019 |
lemma (in sigma_finite_measure) sigma_finite_disjoint: |
|
1020 |
obtains A :: "nat \<Rightarrow> 'a set" |
|
1021 |
where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A" |
|
1022 |
proof atomize_elim |
|
1023 |
case goal1 |
|
1024 |
obtain A :: "nat \<Rightarrow> 'a set" where |
|
1025 |
range: "range A \<subseteq> sets M" and |
|
1026 |
space: "(\<Union>i. A i) = space M" and |
|
1027 |
measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" |
|
1028 |
using sigma_finite by auto |
|
1029 |
note range' = range_disjointed_sets[OF range] range |
|
1030 |
{ fix i |
|
1031 |
have "emeasure M (disjointed A i) \<le> emeasure M (A i)" |
|
1032 |
using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono) |
|
1033 |
then have "emeasure M (disjointed A i) \<noteq> \<infinity>" |
|
1034 |
using measure[of i] by auto } |
|
1035 |
with disjoint_family_disjointed UN_disjointed_eq[of A] space range' |
|
1036 |
show ?case by (auto intro!: exI[of _ "disjointed A"]) |
|
1037 |
qed |
|
1038 |
||
1039 |
lemma (in sigma_finite_measure) sigma_finite_incseq: |
|
1040 |
obtains A :: "nat \<Rightarrow> 'a set" |
|
1041 |
where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A" |
|
1042 |
proof atomize_elim |
|
1043 |
case goal1 |
|
1044 |
obtain F :: "nat \<Rightarrow> 'a set" where |
|
1045 |
F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>" |
|
1046 |
using sigma_finite by auto |
|
1047 |
then show ?case |
|
1048 |
proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI) |
|
1049 |
from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto |
|
1050 |
then show "(\<Union>n. \<Union> i\<le>n. F i) = space M" |
|
1051 |
using F by fastforce |
|
1052 |
next |
|
1053 |
fix n |
|
1054 |
have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F |
|
1055 |
by (auto intro!: emeasure_subadditive_finite) |
|
1056 |
also have "\<dots> < \<infinity>" |
|
1057 |
using F by (auto simp: setsum_Pinfty) |
|
1058 |
finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp |
|
1059 |
qed (force simp: incseq_def)+ |
|
1060 |
qed |
|
1061 |
||
1062 |
section {* Measure space induced by distribution of @{const measurable}-functions *} |
|
1063 |
||
1064 |
definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where |
|
1065 |
"distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))" |
|
1066 |
||
1067 |
lemma |
|
1068 |
shows sets_distr[simp]: "sets (distr M N f) = sets N" |
|
1069 |
and space_distr[simp]: "space (distr M N f) = space N" |
|
1070 |
by (auto simp: distr_def) |
|
1071 |
||
1072 |
lemma |
|
1073 |
shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'" |
|
1074 |
and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng" |
|
1075 |
by (auto simp: measurable_def) |
|
1076 |
||
1077 |
lemma emeasure_distr: |
|
1078 |
fixes f :: "'a \<Rightarrow> 'b" |
|
1079 |
assumes f: "f \<in> measurable M N" and A: "A \<in> sets N" |
|
1080 |
shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A") |
|
1081 |
unfolding distr_def |
|
1082 |
proof (rule emeasure_measure_of_sigma) |
|
1083 |
show "positive (sets N) ?\<mu>" |
|
1084 |
by (auto simp: positive_def) |
|
1085 |
||
1086 |
show "countably_additive (sets N) ?\<mu>" |
|
1087 |
proof (intro countably_additiveI) |
|
1088 |
fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A" |
|
1089 |
then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto |
|
1090 |
then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M" |
|
1091 |
using f by (auto simp: measurable_def) |
|
1092 |
moreover have "(\<Union>i. f -` A i \<inter> space M) \<in> sets M" |
|
1093 |
using * by blast |
|
1094 |
moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)" |
|
1095 |
using `disjoint_family A` by (auto simp: disjoint_family_on_def) |
|
1096 |
ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" |
|
1097 |
using suminf_emeasure[OF _ **] A f |
|
1098 |
by (auto simp: comp_def vimage_UN) |
|
1099 |
qed |
|
1100 |
show "sigma_algebra (space N) (sets N)" .. |
|
1101 |
qed fact |
|
1102 |
||
1103 |
lemma AE_distrD: |
|
1104 |
assumes f: "f \<in> measurable M M'" |
|
1105 |
and AE: "AE x in distr M M' f. P x" |
|
1106 |
shows "AE x in M. P (f x)" |
|
1107 |
proof - |
|
1108 |
from AE[THEN AE_E] guess N . |
|
1109 |
with f show ?thesis |
|
1110 |
unfolding eventually_ae_filter |
|
1111 |
by (intro bexI[of _ "f -` N \<inter> space M"]) |
|
1112 |
(auto simp: emeasure_distr measurable_def) |
|
1113 |
qed |
|
1114 |
||
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1115 |
lemma AE_distr_iff: |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1116 |
assumes f: "f \<in> measurable M N" and P: "{x \<in> space N. P x} \<in> sets N" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1117 |
shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1118 |
proof (subst (1 2) AE_iff_measurable[OF _ refl]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1119 |
from P show "{x \<in> space (distr M N f). \<not> P x} \<in> sets (distr M N f)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1120 |
by (auto intro!: sets_Collect_neg) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1121 |
moreover |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1122 |
have "f -` {x \<in> space N. P x} \<inter> space M = {x \<in> space M. P (f x)}" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1123 |
using f by (auto dest: measurable_space) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1124 |
then show "{x \<in> space M. \<not> P (f x)} \<in> sets M" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1125 |
using measurable_sets[OF f P] by (auto intro!: sets_Collect_neg) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1126 |
moreover have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1127 |
using f by (auto dest: measurable_space) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1128 |
ultimately show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) = |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1129 |
(emeasure M {x \<in> space M. \<not> P (f x)} = 0)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1130 |
using f by (simp add: emeasure_distr) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1131 |
qed |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1132 |
|
47694 | 1133 |
lemma null_sets_distr_iff: |
1134 |
"f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N" |
|
1135 |
by (auto simp add: null_sets_def emeasure_distr measurable_sets) |
|
1136 |
||
1137 |
lemma distr_distr: |
|
1138 |
assumes f: "g \<in> measurable N L" and g: "f \<in> measurable M N" |
|
1139 |
shows "distr (distr M N f) L g = distr M L (g \<circ> f)" (is "?L = ?R") |
|
1140 |
using measurable_comp[OF g f] f g |
|
1141 |
by (auto simp add: emeasure_distr measurable_sets measurable_space |
|
1142 |
intro!: arg_cong[where f="emeasure M"] measure_eqI) |
|
1143 |
||
1144 |
section {* Real measure values *} |
|
1145 |
||
1146 |
lemma measure_nonneg: "0 \<le> measure M A" |
|
1147 |
using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos) |
|
1148 |
||
1149 |
lemma measure_empty[simp]: "measure M {} = 0" |
|
1150 |
unfolding measure_def by simp |
|
1151 |
||
1152 |
lemma emeasure_eq_ereal_measure: |
|
1153 |
"emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)" |
|
1154 |
using emeasure_nonneg[of M A] |
|
1155 |
by (cases "emeasure M A") (auto simp: measure_def) |
|
1156 |
||
1157 |
lemma measure_Union: |
|
1158 |
assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>" |
|
1159 |
and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}" |
|
1160 |
shows "measure M (A \<union> B) = measure M A + measure M B" |
|
1161 |
unfolding measure_def |
|
1162 |
using plus_emeasure[OF measurable, symmetric] finite |
|
1163 |
by (simp add: emeasure_eq_ereal_measure) |
|
1164 |
||
1165 |
lemma measure_finite_Union: |
|
1166 |
assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S" |
|
1167 |
assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" |
|
1168 |
shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))" |
|
1169 |
unfolding measure_def |
|
1170 |
using setsum_emeasure[OF measurable, symmetric] finite |
|
1171 |
by (simp add: emeasure_eq_ereal_measure) |
|
1172 |
||
1173 |
lemma measure_Diff: |
|
1174 |
assumes finite: "emeasure M A \<noteq> \<infinity>" |
|
1175 |
and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A" |
|
1176 |
shows "measure M (A - B) = measure M A - measure M B" |
|
1177 |
proof - |
|
1178 |
have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A" |
|
1179 |
using measurable by (auto intro!: emeasure_mono) |
|
1180 |
hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B" |
|
1181 |
using measurable finite by (rule_tac measure_Union) auto |
|
1182 |
thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2) |
|
1183 |
qed |
|
1184 |
||
1185 |
lemma measure_UNION: |
|
1186 |
assumes measurable: "range A \<subseteq> sets M" "disjoint_family A" |
|
1187 |
assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" |
|
1188 |
shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))" |
|
1189 |
proof - |
|
1190 |
from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"] |
|
1191 |
suminf_emeasure[OF measurable] emeasure_nonneg[of M] |
|
1192 |
have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp |
|
1193 |
moreover |
|
1194 |
{ fix i |
|
1195 |
have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)" |
|
1196 |
using measurable by (auto intro!: emeasure_mono) |
|
1197 |
then have "emeasure M (A i) = ereal ((measure M (A i)))" |
|
1198 |
using finite by (intro emeasure_eq_ereal_measure) auto } |
|
1199 |
ultimately show ?thesis using finite |
|
1200 |
unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure) |
|
1201 |
qed |
|
1202 |
||
1203 |
lemma measure_subadditive: |
|
1204 |
assumes measurable: "A \<in> sets M" "B \<in> sets M" |
|
1205 |
and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>" |
|
1206 |
shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)" |
|
1207 |
proof - |
|
1208 |
have "emeasure M (A \<union> B) \<noteq> \<infinity>" |
|
1209 |
using emeasure_subadditive[OF measurable] fin by auto |
|
1210 |
then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)" |
|
1211 |
using emeasure_subadditive[OF measurable] fin |
|
1212 |
by (auto simp: emeasure_eq_ereal_measure) |
|
1213 |
qed |
|
1214 |
||
1215 |
lemma measure_subadditive_finite: |
|
1216 |
assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" |
|
1217 |
shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))" |
|
1218 |
proof - |
|
1219 |
{ have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))" |
|
1220 |
using emeasure_subadditive_finite[OF A] . |
|
1221 |
also have "\<dots> < \<infinity>" |
|
1222 |
using fin by (simp add: setsum_Pinfty) |
|
1223 |
finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp } |
|
1224 |
then show ?thesis |
|
1225 |
using emeasure_subadditive_finite[OF A] fin |
|
1226 |
unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg) |
|
1227 |
qed |
|
1228 |
||
1229 |
lemma measure_subadditive_countably: |
|
1230 |
assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>" |
|
1231 |
shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))" |
|
1232 |
proof - |
|
1233 |
from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty) |
|
1234 |
moreover |
|
1235 |
{ have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))" |
|
1236 |
using emeasure_subadditive_countably[OF A] . |
|
1237 |
also have "\<dots> < \<infinity>" |
|
1238 |
using fin by simp |
|
1239 |
finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp } |
|
1240 |
ultimately show ?thesis |
|
1241 |
using emeasure_subadditive_countably[OF A] fin |
|
1242 |
unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg) |
|
1243 |
qed |
|
1244 |
||
1245 |
lemma measure_eq_setsum_singleton: |
|
1246 |
assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" |
|
1247 |
and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>" |
|
1248 |
shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))" |
|
1249 |
unfolding measure_def |
|
1250 |
using emeasure_eq_setsum_singleton[OF S] fin |
|
1251 |
by simp (simp add: emeasure_eq_ereal_measure) |
|
1252 |
||
1253 |
lemma Lim_measure_incseq: |
|
1254 |
assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" |
|
1255 |
shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))" |
|
1256 |
proof - |
|
1257 |
have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)" |
|
1258 |
using fin by (auto simp: emeasure_eq_ereal_measure) |
|
1259 |
then show ?thesis |
|
1260 |
using Lim_emeasure_incseq[OF A] |
|
1261 |
unfolding measure_def |
|
1262 |
by (intro lim_real_of_ereal) simp |
|
1263 |
qed |
|
1264 |
||
1265 |
lemma Lim_measure_decseq: |
|
1266 |
assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" |
|
1267 |
shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)" |
|
1268 |
proof - |
|
1269 |
have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)" |
|
1270 |
using A by (auto intro!: emeasure_mono) |
|
1271 |
also have "\<dots> < \<infinity>" |
|
1272 |
using fin[of 0] by auto |
|
1273 |
finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)" |
|
1274 |
by (auto simp: emeasure_eq_ereal_measure) |
|
1275 |
then show ?thesis |
|
1276 |
unfolding measure_def |
|
1277 |
using Lim_emeasure_decseq[OF A fin] |
|
1278 |
by (intro lim_real_of_ereal) simp |
|
1279 |
qed |
|
1280 |
||
1281 |
section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *} |
|
1282 |
||
1283 |
locale finite_measure = sigma_finite_measure M for M + |
|
1284 |
assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>" |
|
1285 |
||
1286 |
lemma finite_measureI[Pure.intro!]: |
|
1287 |
assumes *: "emeasure M (space M) \<noteq> \<infinity>" |
|
1288 |
shows "finite_measure M" |
|
1289 |
proof |
|
1290 |
show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)" |
|
1291 |
using * by (auto intro!: exI[of _ "\<lambda>_. space M"]) |
|
1292 |
qed fact |
|
1293 |
||
1294 |
lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>" |
|
1295 |
using finite_emeasure_space emeasure_space[of M A] by auto |
|
1296 |
||
1297 |
lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)" |
|
1298 |
unfolding measure_def by (simp add: emeasure_eq_ereal_measure) |
|
1299 |
||
1300 |
lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r" |
|
1301 |
using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto |
|
1302 |
||
1303 |
lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)" |
|
1304 |
using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def) |
|
1305 |
||
1306 |
lemma (in finite_measure) finite_measure_Diff: |
|
1307 |
assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A" |
|
1308 |
shows "measure M (A - B) = measure M A - measure M B" |
|
1309 |
using measure_Diff[OF _ assms] by simp |
|
1310 |
||
1311 |
lemma (in finite_measure) finite_measure_Union: |
|
1312 |
assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}" |
|
1313 |
shows "measure M (A \<union> B) = measure M A + measure M B" |
|
1314 |
using measure_Union[OF _ _ assms] by simp |
|
1315 |
||
1316 |
lemma (in finite_measure) finite_measure_finite_Union: |
|
1317 |
assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S" |
|
1318 |
shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))" |
|
1319 |
using measure_finite_Union[OF assms] by simp |
|
1320 |
||
1321 |
lemma (in finite_measure) finite_measure_UNION: |
|
1322 |
assumes A: "range A \<subseteq> sets M" "disjoint_family A" |
|
1323 |
shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))" |
|
1324 |
using measure_UNION[OF A] by simp |
|
1325 |
||
1326 |
lemma (in finite_measure) finite_measure_mono: |
|
1327 |
assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B" |
|
1328 |
using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def) |
|
1329 |
||
1330 |
lemma (in finite_measure) finite_measure_subadditive: |
|
1331 |
assumes m: "A \<in> sets M" "B \<in> sets M" |
|
1332 |
shows "measure M (A \<union> B) \<le> measure M A + measure M B" |
|
1333 |
using measure_subadditive[OF m] by simp |
|
1334 |
||
1335 |
lemma (in finite_measure) finite_measure_subadditive_finite: |
|
1336 |
assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))" |
|
1337 |
using measure_subadditive_finite[OF assms] by simp |
|
1338 |
||
1339 |
lemma (in finite_measure) finite_measure_subadditive_countably: |
|
1340 |
assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))" |
|
1341 |
shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))" |
|
1342 |
proof - |
|
1343 |
from `summable (\<lambda>i. measure M (A i))` |
|
1344 |
have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))" |
|
1345 |
by (simp add: sums_ereal) (rule summable_sums) |
|
1346 |
from sums_unique[OF this, symmetric] |
|
1347 |
measure_subadditive_countably[OF A] |
|
1348 |
show ?thesis by (simp add: emeasure_eq_measure) |
|
1349 |
qed |
|
1350 |
||
1351 |
lemma (in finite_measure) finite_measure_eq_setsum_singleton: |
|
1352 |
assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" |
|
1353 |
shows "measure M S = (\<Sum>x\<in>S. measure M {x})" |
|
1354 |
using measure_eq_setsum_singleton[OF assms] by simp |
|
1355 |
||
1356 |
lemma (in finite_measure) finite_Lim_measure_incseq: |
|
1357 |
assumes A: "range A \<subseteq> sets M" "incseq A" |
|
1358 |
shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)" |
|
1359 |
using Lim_measure_incseq[OF A] by simp |
|
1360 |
||
1361 |
lemma (in finite_measure) finite_Lim_measure_decseq: |
|
1362 |
assumes A: "range A \<subseteq> sets M" "decseq A" |
|
1363 |
shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)" |
|
1364 |
using Lim_measure_decseq[OF A] by simp |
|
1365 |
||
1366 |
lemma (in finite_measure) finite_measure_compl: |
|
1367 |
assumes S: "S \<in> sets M" |
|
1368 |
shows "measure M (space M - S) = measure M (space M) - measure M S" |
|
1369 |
using measure_Diff[OF _ top S sets_into_space] S by simp |
|
1370 |
||
1371 |
lemma (in finite_measure) finite_measure_mono_AE: |
|
1372 |
assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M" |
|
1373 |
shows "measure M A \<le> measure M B" |
|
1374 |
using assms emeasure_mono_AE[OF imp B] |
|
1375 |
by (simp add: emeasure_eq_measure) |
|
1376 |
||
1377 |
lemma (in finite_measure) finite_measure_eq_AE: |
|
1378 |
assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" |
|
1379 |
assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
|
1380 |
shows "measure M A = measure M B" |
|
1381 |
using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure) |
|
1382 |
||
1383 |
section {* Counting space *} |
|
1384 |
||
1385 |
definition count_space :: "'a set \<Rightarrow> 'a measure" where |
|
1386 |
"count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)" |
|
1387 |
||
1388 |
lemma |
|
1389 |
shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>" |
|
1390 |
and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>" |
|
1391 |
using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>] |
|
1392 |
by (auto simp: count_space_def) |
|
1393 |
||
1394 |
lemma measurable_count_space_eq1[simp]: |
|
1395 |
"f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M" |
|
1396 |
unfolding measurable_def by simp |
|
1397 |
||
1398 |
lemma measurable_count_space_eq2[simp]: |
|
1399 |
assumes "finite A" |
|
1400 |
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))" |
|
1401 |
proof - |
|
1402 |
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A" |
|
1403 |
with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X" |
|
1404 |
by (auto dest: finite_subset) |
|
1405 |
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M" |
|
1406 |
ultimately have "f -` X \<inter> space M \<in> sets M" |
|
1407 |
using `X \<subseteq> A` by (auto intro!: finite_UN simp del: UN_simps) } |
|
1408 |
then show ?thesis |
|
1409 |
unfolding measurable_def by auto |
|
1410 |
qed |
|
1411 |
||
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1412 |
lemma strict_monoI_Suc: |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1413 |
assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1414 |
unfolding strict_mono_def |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1415 |
proof safe |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1416 |
fix n m :: nat assume "n < m" then show "f n < f m" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1417 |
by (induct m) (auto simp: less_Suc_eq intro: less_trans ord) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1418 |
qed |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1419 |
|
47694 | 1420 |
lemma emeasure_count_space: |
1421 |
assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)" |
|
1422 |
(is "_ = ?M X") |
|
1423 |
unfolding count_space_def |
|
1424 |
proof (rule emeasure_measure_of_sigma) |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1425 |
show "X \<in> Pow A" using `X \<subseteq> A` by auto |
47694 | 1426 |
show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow) |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1427 |
show positive: "positive (Pow A) ?M" |
47694 | 1428 |
by (auto simp: positive_def) |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1429 |
have additive: "additive (Pow A) ?M" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1430 |
by (auto simp: card_Un_disjoint additive_def) |
47694 | 1431 |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1432 |
interpret ring_of_sets A "Pow A" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1433 |
by (rule ring_of_setsI) auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1434 |
show "countably_additive (Pow A) ?M" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1435 |
unfolding countably_additive_iff_continuous_from_below[OF positive additive] |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1436 |
proof safe |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1437 |
fix F :: "nat \<Rightarrow> 'a set" assume "incseq F" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1438 |
show "(\<lambda>i. ?M (F i)) ----> ?M (\<Union>i. F i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1439 |
proof cases |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1440 |
assume "\<exists>i. \<forall>j\<ge>i. F i = F j" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1441 |
then guess i .. note i = this |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1442 |
{ fix j from i `incseq F` have "F j \<subseteq> F i" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1443 |
by (cases "i \<le> j") (auto simp: incseq_def) } |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1444 |
then have eq: "(\<Union>i. F i) = F i" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1445 |
by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1446 |
with i show ?thesis |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1447 |
by (auto intro!: Lim_eventually eventually_sequentiallyI[where c=i]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1448 |
next |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1449 |
assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1450 |
then obtain f where "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1451 |
moreover then have "\<And>i. F i \<subseteq> F (f i)" using `incseq F` by (auto simp: incseq_def) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1452 |
ultimately have *: "\<And>i. F i \<subset> F (f i)" by auto |
47694 | 1453 |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1454 |
have "incseq (\<lambda>i. ?M (F i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1455 |
using `incseq F` unfolding incseq_def by (auto simp: card_mono dest: finite_subset) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1456 |
then have "(\<lambda>i. ?M (F i)) ----> (SUP n. ?M (F n))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1457 |
by (rule LIMSEQ_ereal_SUPR) |
47694 | 1458 |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1459 |
moreover have "(SUP n. ?M (F n)) = \<infinity>" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1460 |
proof (rule SUP_PInfty) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1461 |
fix n :: nat show "\<exists>k::nat\<in>UNIV. ereal n \<le> ?M (F k)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1462 |
proof (induct n) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1463 |
case (Suc n) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1464 |
then guess k .. note k = this |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1465 |
moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1466 |
using `F k \<subset> F (f k)` by (simp add: psubset_card_mono) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1467 |
moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1468 |
using `k \<le> f k` `incseq F` by (auto simp: incseq_def dest: finite_subset) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1469 |
ultimately show ?case |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1470 |
by (auto intro!: exI[of _ "f k"]) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1471 |
qed auto |
47694 | 1472 |
qed |
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1473 |
|
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1474 |
moreover |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1475 |
have "inj (\<lambda>n. F ((f ^^ n) 0))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1476 |
by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1477 |
then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1478 |
by (rule range_inj_infinite) |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1479 |
have "infinite (Pow (\<Union>i. F i))" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1480 |
by (rule infinite_super[OF _ 1]) auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1481 |
then have "infinite (\<Union>i. F i)" |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1482 |
by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1483 |
|
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1484 |
ultimately show ?thesis by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1485 |
qed |
47694 | 1486 |
qed |
1487 |
qed |
|
1488 |
||
1489 |
lemma emeasure_count_space_finite[simp]: |
|
1490 |
"X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)" |
|
1491 |
using emeasure_count_space[of X A] by simp |
|
1492 |
||
1493 |
lemma emeasure_count_space_infinite[simp]: |
|
1494 |
"X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>" |
|
1495 |
using emeasure_count_space[of X A] by simp |
|
1496 |
||
1497 |
lemma emeasure_count_space_eq_0: |
|
1498 |
"emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})" |
|
1499 |
proof cases |
|
1500 |
assume X: "X \<subseteq> A" |
|
1501 |
then show ?thesis |
|
1502 |
proof (intro iffI impI) |
|
1503 |
assume "emeasure (count_space A) X = 0" |
|
1504 |
with X show "X = {}" |
|
1505 |
by (subst (asm) emeasure_count_space) (auto split: split_if_asm) |
|
1506 |
qed simp |
|
1507 |
qed (simp add: emeasure_notin_sets) |
|
1508 |
||
1509 |
lemma null_sets_count_space: "null_sets (count_space A) = { {} }" |
|
1510 |
unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0) |
|
1511 |
||
1512 |
lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)" |
|
1513 |
unfolding eventually_ae_filter by (auto simp add: null_sets_count_space) |
|
1514 |
||
1515 |
lemma sigma_finite_measure_count_space: |
|
1516 |
fixes A :: "'a::countable set" |
|
1517 |
shows "sigma_finite_measure (count_space A)" |
|
1518 |
proof |
|
1519 |
show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (count_space A) \<and> (\<Union>i. F i) = space (count_space A) \<and> |
|
1520 |
(\<forall>i. emeasure (count_space A) (F i) \<noteq> \<infinity>)" |
|
1521 |
using surj_from_nat by (intro exI[of _ "\<lambda>i. {from_nat i} \<inter> A"]) (auto simp del: surj_from_nat) |
|
1522 |
qed |
|
1523 |
||
1524 |
lemma finite_measure_count_space: |
|
1525 |
assumes [simp]: "finite A" |
|
1526 |
shows "finite_measure (count_space A)" |
|
1527 |
by rule simp |
|
1528 |
||
1529 |
lemma sigma_finite_measure_count_space_finite: |
|
1530 |
assumes A: "finite A" shows "sigma_finite_measure (count_space A)" |
|
1531 |
proof - |
|
1532 |
interpret finite_measure "count_space A" using A by (rule finite_measure_count_space) |
|
1533 |
show "sigma_finite_measure (count_space A)" .. |
|
1534 |
qed |
|
1535 |
||
1536 |
end |
|
1537 |