src/HOL/Probability/Caratheodory.thy
author hoelzl
Thu Sep 02 17:28:00 2010 +0200 (2010-09-02)
changeset 39096 111756225292
parent 38656 d5d342611edb
child 40859 de0b30e6c2d2
permissions -rw-r--r--
merged
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header {*Caratheodory Extension Theorem*}
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theory Caratheodory
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  imports Sigma_Algebra Positive_Infinite_Real
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begin
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text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
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subsection {* Measure Spaces *}
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definition "positive f \<longleftrightarrow> f {} = (0::pinfreal)" -- "Positive is enforced by the type"
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definition
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  additive  where
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  "additive M f \<longleftrightarrow>
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    (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
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    \<longrightarrow> f (x \<union> y) = f x + f y)"
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definition
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  countably_additive  where
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  "countably_additive M f \<longleftrightarrow>
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    (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
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         disjoint_family A \<longrightarrow>
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         (\<Union>i. A i) \<in> sets M \<longrightarrow>
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         (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i))"
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definition
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  increasing  where
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  "increasing M f \<longleftrightarrow> (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
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definition
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  subadditive  where
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  "subadditive M f \<longleftrightarrow>
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    (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
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    \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
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definition
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  countably_subadditive  where
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  "countably_subadditive M f \<longleftrightarrow>
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    (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
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         disjoint_family A \<longrightarrow>
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         (\<Union>i. A i) \<in> sets M \<longrightarrow>
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         f (\<Union>i. A i) \<le> psuminf (\<lambda>n. f (A n)))"
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definition
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  lambda_system where
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  "lambda_system M f =
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    {l. l \<in> sets M & (\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x)}"
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definition
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  outer_measure_space where
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  "outer_measure_space M f  \<longleftrightarrow>
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     positive f \<and> increasing M f \<and> countably_subadditive M f"
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definition
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  measure_set where
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  "measure_set M f X =
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     {r . \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>\<^isub>\<infinity> i. f (A i)) = r}"
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locale measure_space = sigma_algebra +
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  fixes \<mu> :: "'a set \<Rightarrow> pinfreal"
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  assumes empty_measure [simp]: "\<mu> {} = 0"
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      and ca: "countably_additive M \<mu>"
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lemma increasingD:
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     "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
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  by (auto simp add: increasing_def)
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lemma subadditiveD:
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     "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
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      \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
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  by (auto simp add: subadditive_def)
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lemma additiveD:
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     "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
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      \<Longrightarrow> f (x \<union> y) = f x + f y"
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  by (auto simp add: additive_def)
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lemma countably_additiveD:
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  "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
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   \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)"
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  by (simp add: countably_additive_def)
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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 = 0..<n. f (binaryset A B i)) ----> f A + f B"
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proof -
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  have "(\<lambda>n. \<Sum>i = 0..< 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\<Colon>nat = 0\<Colon>nat..<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 LIMSEQ_const)
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  ultimately
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  have "(\<lambda>n. \<Sum>i = 0..< 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 = 0..< 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])
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lemma suminf_binaryset_eq:
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     "f {} = 0 \<Longrightarrow> suminf (\<lambda>n. f (binaryset A B n)) = f A + f B"
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  by (metis binaryset_sums sums_unique)
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lemma binaryset_psuminf:
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  assumes "f {} = 0"
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  shows "(\<Sum>\<^isub>\<infinity> n. f (binaryset A B n)) = f A + f B" (is "?suminf = ?sum")
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proof -
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  have *: "{..<2} = {0, 1::nat}" by auto
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  have "\<forall>n\<ge>2. f (binaryset A B n) = 0"
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    unfolding binaryset_def
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    using assms by auto
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  hence "?suminf = (\<Sum>N<2. f (binaryset A B N))"
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    by (rule psuminf_finite)
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  also have "... = ?sum" unfolding * binaryset_def
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    by simp
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  finally show ?thesis .
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qed
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subsection {* Lambda Systems *}
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lemma (in algebra) lambda_system_eq:
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    "lambda_system M f =
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        {l. l \<in> sets M & (\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x)}"
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proof -
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  have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
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    by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
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  show ?thesis
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    by (auto simp add: lambda_system_def) (metis Int_commute)+
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qed
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lemma (in algebra) lambda_system_empty:
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  "positive f \<Longrightarrow> {} \<in> lambda_system M f"
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  by (auto simp add: positive_def lambda_system_eq)
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lemma lambda_system_sets:
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    "x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
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  by (simp add:  lambda_system_def)
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lemma (in algebra) lambda_system_Compl:
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  fixes f:: "'a set \<Rightarrow> pinfreal"
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  assumes x: "x \<in> lambda_system M f"
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  shows "space M - x \<in> lambda_system M f"
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  proof -
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    have "x \<subseteq> space M"
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      by (metis sets_into_space lambda_system_sets x)
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    hence "space M - (space M - x) = x"
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      by (metis double_diff equalityE)
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    with x show ?thesis
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      by (force simp add: lambda_system_def ac_simps)
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  qed
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lemma (in algebra) lambda_system_Int:
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  fixes f:: "'a set \<Rightarrow> pinfreal"
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  assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
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  shows "x \<inter> y \<in> lambda_system M f"
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  proof -
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    from xl yl show ?thesis
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      proof (auto simp add: positive_def lambda_system_eq Int)
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        fix u
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        assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M"
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           and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z"
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           and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z"
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        have "u - x \<inter> y \<in> sets M"
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          by (metis Diff Diff_Int Un u x y)
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        moreover
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        have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
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        moreover
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        have "u - x \<inter> y - y = u - y" by blast
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        ultimately
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        have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
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          by force
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        have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
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              = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
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          by (simp add: ey ac_simps)
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        also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
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          by (simp add: Int_ac)
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        also have "... = f (u \<inter> y) + f (u - y)"
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          using fx [THEN bspec, of "u \<inter> y"] Int y u
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          by force
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        also have "... = f u"
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          by (metis fy u)
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        finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
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      qed
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  qed
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lemma (in algebra) lambda_system_Un:
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  fixes f:: "'a set \<Rightarrow> pinfreal"
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  assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
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  shows "x \<union> y \<in> lambda_system M f"
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proof -
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  have "(space M - x) \<inter> (space M - y) \<in> sets M"
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    by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
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  moreover
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  have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
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    by auto  (metis subsetD lambda_system_sets sets_into_space xl yl)+
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  ultimately show ?thesis
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    by (metis lambda_system_Compl lambda_system_Int xl yl)
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qed
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lemma (in algebra) lambda_system_algebra:
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  "positive f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))"
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  apply (auto simp add: algebra_def)
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  apply (metis lambda_system_sets set_mp sets_into_space)
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  apply (metis lambda_system_empty)
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  apply (metis lambda_system_Compl)
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  apply (metis lambda_system_Un)
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  done
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lemma (in algebra) lambda_system_strong_additive:
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  assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
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      and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
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  shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
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  proof -
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    have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
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    moreover
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    have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
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    moreover
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    have "(z \<inter> (x \<union> y)) \<in> sets M"
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      by (metis Int Un lambda_system_sets xl yl z)
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    ultimately show ?thesis using xl yl
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      by (simp add: lambda_system_eq)
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  qed
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lemma (in algebra) lambda_system_additive:
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     "additive (M (|sets := lambda_system M f|)) f"
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  proof (auto simp add: additive_def)
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    fix x and y
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    assume disj: "x \<inter> y = {}"
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       and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
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    hence  "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
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    thus "f (x \<union> y) = f x + f y"
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      using lambda_system_strong_additive [OF top disj xl yl]
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      by (simp add: Un)
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  qed
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lemma (in algebra) countably_subadditive_subadditive:
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  assumes f: "positive f" and cs: "countably_subadditive M f"
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  shows  "subadditive M f"
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proof (auto simp add: subadditive_def)
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  fix x y
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  assume x: "x \<in> sets M" and y: "y \<in> sets 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> sets M \<longrightarrow>
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         (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
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         f (\<Union>i. binaryset x y i) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
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    using cs by (simp add: countably_subadditive_def)
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  hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
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         f (x \<union> y) \<le> (\<Sum>\<^isub>\<infinity> 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) \<le>  f x + f y" using f x y
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    by (auto simp add: Un o_def binaryset_psuminf positive_def)
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qed
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lemma (in algebra) additive_sum:
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  fixes A:: "nat \<Rightarrow> 'a set"
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  assumes f: "positive f" and ad: "additive M f"
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      and A: "range A \<subseteq> sets M"
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      and disj: "disjoint_family A"
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  shows  "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
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proof (induct n)
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  case 0 show ?case using f by (simp add: positive_def)
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next
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  case (Suc n)
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  have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
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    by (auto simp add: disjoint_family_on_def neq_iff) blast
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  moreover
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  have "A n \<in> sets M" using A by blast
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  moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
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    by (metis A UNION_in_sets atLeast0LessThan)
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  moreover
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  ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
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    using ad UNION_in_sets A by (auto simp add: additive_def)
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  with Suc.hyps show ?case using ad
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    by (auto simp add: atLeastLessThanSuc additive_def)
paulson@33271
   289
qed
paulson@33271
   290
paulson@33271
   291
paulson@33271
   292
lemma countably_subadditiveD:
paulson@33271
   293
  "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
hoelzl@38656
   294
   (\<Union>i. A i) \<in> sets M \<Longrightarrow> f (\<Union>i. A i) \<le> psuminf (f o A)"
paulson@33271
   295
  by (auto simp add: countably_subadditive_def o_def)
paulson@33271
   296
hoelzl@38656
   297
lemma (in algebra) increasing_additive_bound:
hoelzl@38656
   298
  fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> pinfreal"
hoelzl@38656
   299
  assumes f: "positive f" and ad: "additive M f"
paulson@33271
   300
      and inc: "increasing M f"
paulson@33271
   301
      and A: "range A \<subseteq> sets M"
paulson@33271
   302
      and disj: "disjoint_family A"
hoelzl@38656
   303
  shows  "psuminf (f \<circ> A) \<le> f (space M)"
hoelzl@38656
   304
proof (safe intro!: psuminf_bound)
hoelzl@38656
   305
  fix N
hoelzl@38656
   306
  have "setsum (f \<circ> A) {0..<N} = f (\<Union>i\<in>{0..<N}. A i)"
hoelzl@38656
   307
    by (rule additive_sum [OF f ad A disj])
paulson@33271
   308
  also have "... \<le> f (space M)" using space_closed A
hoelzl@38656
   309
    by (blast intro: increasingD [OF inc] UNION_in_sets top)
hoelzl@38656
   310
  finally show "setsum (f \<circ> A) {..<N} \<le> f (space M)" by (simp add: atLeast0LessThan)
paulson@33271
   311
qed
paulson@33271
   312
paulson@33271
   313
lemma lambda_system_increasing:
paulson@33271
   314
   "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
hoelzl@38656
   315
  by (simp add: increasing_def lambda_system_def)
paulson@33271
   316
paulson@33271
   317
lemma (in algebra) lambda_system_strong_sum:
hoelzl@38656
   318
  fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal"
hoelzl@38656
   319
  assumes f: "positive f" and a: "a \<in> sets M"
paulson@33271
   320
      and A: "range A \<subseteq> lambda_system M f"
paulson@33271
   321
      and disj: "disjoint_family A"
paulson@33271
   322
  shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
paulson@33271
   323
proof (induct n)
hoelzl@38656
   324
  case 0 show ?case using f by (simp add: positive_def)
paulson@33271
   325
next
hoelzl@38656
   326
  case (Suc n)
paulson@33271
   327
  have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
hoelzl@38656
   328
    by (force simp add: disjoint_family_on_def neq_iff)
paulson@33271
   329
  have 3: "A n \<in> lambda_system M f" using A
paulson@33271
   330
    by blast
paulson@33271
   331
  have 4: "UNION {0..<n} A \<in> lambda_system M f"
hoelzl@38656
   332
    using A algebra.UNION_in_sets [OF local.lambda_system_algebra, of f, OF f]
paulson@33271
   333
    by simp
paulson@33271
   334
  from Suc.hyps show ?case
paulson@33271
   335
    by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
paulson@33271
   336
qed
paulson@33271
   337
paulson@33271
   338
paulson@33271
   339
lemma (in sigma_algebra) lambda_system_caratheodory:
paulson@33271
   340
  assumes oms: "outer_measure_space M f"
paulson@33271
   341
      and A: "range A \<subseteq> lambda_system M f"
paulson@33271
   342
      and disj: "disjoint_family A"
hoelzl@38656
   343
  shows  "(\<Union>i. A i) \<in> lambda_system M f \<and> psuminf (f \<circ> A) = f (\<Union>i. A i)"
paulson@33271
   344
proof -
hoelzl@38656
   345
  have pos: "positive f" and inc: "increasing M f"
hoelzl@38656
   346
   and csa: "countably_subadditive M f"
paulson@33271
   347
    by (metis oms outer_measure_space_def)+
paulson@33271
   348
  have sa: "subadditive M f"
hoelzl@38656
   349
    by (metis countably_subadditive_subadditive csa pos)
hoelzl@38656
   350
  have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
paulson@33271
   351
    by simp
paulson@33271
   352
  have alg_ls: "algebra (M(|sets := lambda_system M f|))"
hoelzl@38656
   353
    by (rule lambda_system_algebra) (rule pos)
paulson@33271
   354
  have A'': "range A \<subseteq> sets M"
paulson@33271
   355
     by (metis A image_subset_iff lambda_system_sets)
hoelzl@38656
   356
paulson@33271
   357
  have U_in: "(\<Union>i. A i) \<in> sets M"
huffman@37032
   358
    by (metis A'' countable_UN)
hoelzl@38656
   359
  have U_eq: "f (\<Union>i. A i) = psuminf (f o A)"
paulson@33271
   360
    proof (rule antisym)
hoelzl@38656
   361
      show "f (\<Union>i. A i) \<le> psuminf (f \<circ> A)"
hoelzl@38656
   362
        by (rule countably_subadditiveD [OF csa A'' disj U_in])
hoelzl@38656
   363
      show "psuminf (f \<circ> A) \<le> f (\<Union>i. A i)"
hoelzl@38656
   364
        by (rule psuminf_bound, unfold atLeast0LessThan[symmetric])
paulson@33271
   365
           (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
hoelzl@38656
   366
                  lambda_system_additive subset_Un_eq increasingD [OF inc]
hoelzl@38656
   367
                  A' A'' UNION_in_sets U_in)
paulson@33271
   368
    qed
paulson@33271
   369
  {
hoelzl@38656
   370
    fix a
hoelzl@38656
   371
    assume a [iff]: "a \<in> sets M"
paulson@33271
   372
    have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
paulson@33271
   373
    proof -
paulson@33271
   374
      show ?thesis
paulson@33271
   375
      proof (rule antisym)
wenzelm@33536
   376
        have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
wenzelm@33536
   377
          by blast
hoelzl@38656
   378
        moreover
wenzelm@33536
   379
        have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
hoelzl@38656
   380
          by (auto simp add: disjoint_family_on_def)
hoelzl@38656
   381
        moreover
wenzelm@33536
   382
        have "a \<inter> (\<Union>i. A i) \<in> sets M"
wenzelm@33536
   383
          by (metis Int U_in a)
hoelzl@38656
   384
        ultimately
hoelzl@38656
   385
        have "f (a \<inter> (\<Union>i. A i)) \<le> psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
hoelzl@38656
   386
          using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"]
hoelzl@38656
   387
          by (simp add: o_def)
hoelzl@38656
   388
        hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
hoelzl@38656
   389
            psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i))"
hoelzl@38656
   390
          by (rule add_right_mono)
hoelzl@38656
   391
        moreover
hoelzl@38656
   392
        have "psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i)) \<le> f a"
hoelzl@38656
   393
          proof (safe intro!: psuminf_bound_add)
wenzelm@33536
   394
            fix n
wenzelm@33536
   395
            have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
hoelzl@38656
   396
              by (metis A'' UNION_in_sets)
wenzelm@33536
   397
            have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
huffman@37032
   398
              by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
wenzelm@33536
   399
            have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
hoelzl@38656
   400
              using algebra.UNION_in_sets [OF lambda_system_algebra [of f, OF pos]]
hoelzl@38656
   401
              by (simp add: A)
hoelzl@38656
   402
            hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
huffman@37032
   403
              by (simp add: lambda_system_eq UNION_in)
wenzelm@33536
   404
            have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
hoelzl@38656
   405
              by (blast intro: increasingD [OF inc] UNION_eq_Union_image
huffman@37032
   406
                               UNION_in U_in)
hoelzl@38656
   407
            thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {..<n} + f (a - (\<Union>i. A i)) \<le> f a"
hoelzl@38656
   408
              by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
wenzelm@33536
   409
          qed
hoelzl@38656
   410
        ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
hoelzl@38656
   411
          by (rule order_trans)
paulson@33271
   412
      next
hoelzl@38656
   413
        have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
huffman@37032
   414
          by (blast intro:  increasingD [OF inc] U_in)
wenzelm@33536
   415
        also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
huffman@37032
   416
          by (blast intro: subadditiveD [OF sa] U_in)
wenzelm@33536
   417
        finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
paulson@33271
   418
        qed
paulson@33271
   419
     qed
paulson@33271
   420
  }
paulson@33271
   421
  thus  ?thesis
hoelzl@38656
   422
    by (simp add: lambda_system_eq sums_iff U_eq U_in)
paulson@33271
   423
qed
paulson@33271
   424
paulson@33271
   425
lemma (in sigma_algebra) caratheodory_lemma:
paulson@33271
   426
  assumes oms: "outer_measure_space M f"
hoelzl@38656
   427
  shows "measure_space (|space = space M, sets = lambda_system M f|) f"
paulson@33271
   428
proof -
hoelzl@38656
   429
  have pos: "positive f"
paulson@33271
   430
    by (metis oms outer_measure_space_def)
hoelzl@38656
   431
  have alg: "algebra (|space = space M, sets = lambda_system M f|)"
hoelzl@38656
   432
    using lambda_system_algebra [of f, OF pos]
hoelzl@38656
   433
    by (simp add: algebra_def)
hoelzl@38656
   434
  then moreover
hoelzl@38656
   435
  have "sigma_algebra (|space = space M, sets = lambda_system M f|)"
paulson@33271
   436
    using lambda_system_caratheodory [OF oms]
hoelzl@38656
   437
    by (simp add: sigma_algebra_disjoint_iff)
hoelzl@38656
   438
  moreover
hoelzl@38656
   439
  have "measure_space_axioms (|space = space M, sets = lambda_system M f|) f"
paulson@33271
   440
    using pos lambda_system_caratheodory [OF oms]
hoelzl@38656
   441
    by (simp add: measure_space_axioms_def positive_def lambda_system_sets
hoelzl@38656
   442
                  countably_additive_def o_def)
hoelzl@38656
   443
  ultimately
paulson@33271
   444
  show ?thesis
hoelzl@38656
   445
    by intro_locales (auto simp add: sigma_algebra_def)
paulson@33271
   446
qed
paulson@33271
   447
paulson@33271
   448
lemma (in algebra) additive_increasing:
hoelzl@38656
   449
  assumes posf: "positive f" and addf: "additive M f"
paulson@33271
   450
  shows "increasing M f"
hoelzl@38656
   451
proof (auto simp add: increasing_def)
paulson@33271
   452
  fix x y
paulson@33271
   453
  assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
hoelzl@38656
   454
  have "f x \<le> f x + f (y-x)" ..
paulson@33271
   455
  also have "... = f (x \<union> (y-x))" using addf
huffman@37032
   456
    by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
paulson@33271
   457
  also have "... = f y"
huffman@37032
   458
    by (metis Un_Diff_cancel Un_absorb1 xy(3))
paulson@33271
   459
  finally show "f x \<le> f y" .
paulson@33271
   460
qed
paulson@33271
   461
paulson@33271
   462
lemma (in algebra) countably_additive_additive:
hoelzl@38656
   463
  assumes posf: "positive f" and ca: "countably_additive M f"
paulson@33271
   464
  shows "additive M f"
hoelzl@38656
   465
proof (auto simp add: additive_def)
paulson@33271
   466
  fix x y
paulson@33271
   467
  assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
paulson@33271
   468
  hence "disjoint_family (binaryset x y)"
hoelzl@38656
   469
    by (auto simp add: disjoint_family_on_def binaryset_def)
hoelzl@38656
   470
  hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
hoelzl@38656
   471
         (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
hoelzl@38656
   472
         f (\<Union>i. binaryset x y i) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
paulson@33271
   473
    using ca
hoelzl@38656
   474
    by (simp add: countably_additive_def)
hoelzl@38656
   475
  hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
hoelzl@38656
   476
         f (x \<union> y) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
paulson@33271
   477
    by (simp add: range_binaryset_eq UN_binaryset_eq)
paulson@33271
   478
  thus "f (x \<union> y) = f x + f y" using posf x y
hoelzl@38656
   479
    by (auto simp add: Un binaryset_psuminf positive_def)
hoelzl@38656
   480
qed
hoelzl@38656
   481
hoelzl@39096
   482
lemma inf_measure_nonempty:
hoelzl@39096
   483
  assumes f: "positive f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
hoelzl@39096
   484
  shows "f b \<in> measure_set M f a"
hoelzl@39096
   485
proof -
hoelzl@39096
   486
  have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
hoelzl@39096
   487
    by (rule psuminf_finite) (simp add: f[unfolded positive_def])
hoelzl@39096
   488
  also have "... = f b"
hoelzl@39096
   489
    by simp
hoelzl@39096
   490
  finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
hoelzl@39096
   491
  thus ?thesis using assms
hoelzl@39096
   492
    by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
hoelzl@39096
   493
             simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
hoelzl@39096
   494
qed
hoelzl@39096
   495
paulson@33271
   496
lemma (in algebra) inf_measure_agrees:
hoelzl@38656
   497
  assumes posf: "positive f" and ca: "countably_additive M f"
hoelzl@38656
   498
      and s: "s \<in> sets M"
paulson@33271
   499
  shows "Inf (measure_set M f s) = f s"
hoelzl@38656
   500
  unfolding Inf_pinfreal_def
hoelzl@38656
   501
proof (safe intro!: Greatest_equality)
paulson@33271
   502
  fix z
paulson@33271
   503
  assume z: "z \<in> measure_set M f s"
hoelzl@38656
   504
  from this obtain A where
paulson@33271
   505
    A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
hoelzl@38656
   506
    and "s \<subseteq> (\<Union>x. A x)" and si: "psuminf (f \<circ> A) = z"
hoelzl@38656
   507
    by (auto simp add: measure_set_def comp_def)
paulson@33271
   508
  hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
paulson@33271
   509
  have inc: "increasing M f"
paulson@33271
   510
    by (metis additive_increasing ca countably_additive_additive posf)
hoelzl@38656
   511
  have sums: "psuminf (\<lambda>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
hoelzl@38656
   512
    proof (rule countably_additiveD [OF ca])
paulson@33271
   513
      show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
wenzelm@33536
   514
        by blast
paulson@33271
   515
      show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
hoelzl@35582
   516
        by (auto simp add: disjoint_family_on_def)
paulson@33271
   517
      show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
wenzelm@33536
   518
        by (metis UN_extend_simps(4) s seq)
paulson@33271
   519
    qed
hoelzl@38656
   520
  hence "f s = psuminf (\<lambda>i. f (A i \<inter> s))"
huffman@37032
   521
    using seq [symmetric] by (simp add: sums_iff)
hoelzl@38656
   522
  also have "... \<le> psuminf (f \<circ> A)"
hoelzl@38656
   523
    proof (rule psuminf_le)
hoelzl@38656
   524
      fix n show "f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
hoelzl@38656
   525
        by (force intro: increasingD [OF inc])
paulson@33271
   526
    qed
hoelzl@38656
   527
  also have "... = z" by (rule si)
paulson@33271
   528
  finally show "f s \<le> z" .
paulson@33271
   529
next
paulson@33271
   530
  fix y
hoelzl@38656
   531
  assume y: "\<forall>u \<in> measure_set M f s. y \<le> u"
paulson@33271
   532
  thus "y \<le> f s"
hoelzl@38656
   533
    by (blast intro: inf_measure_nonempty [of f, OF posf s subset_refl])
paulson@33271
   534
qed
paulson@33271
   535
paulson@33271
   536
lemma (in algebra) inf_measure_empty:
hoelzl@39096
   537
  assumes posf: "positive f"  "{} \<in> sets M"
paulson@33271
   538
  shows "Inf (measure_set M f {}) = 0"
paulson@33271
   539
proof (rule antisym)
paulson@33271
   540
  show "Inf (measure_set M f {}) \<le> 0"
hoelzl@39096
   541
    by (metis complete_lattice_class.Inf_lower `{} \<in> sets M` inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
hoelzl@38656
   542
qed simp
paulson@33271
   543
paulson@33271
   544
lemma (in algebra) inf_measure_positive:
hoelzl@38656
   545
  "positive f \<Longrightarrow>
hoelzl@38656
   546
   positive (\<lambda>x. Inf (measure_set M f x))"
hoelzl@38656
   547
  by (simp add: positive_def inf_measure_empty) 
paulson@33271
   548
paulson@33271
   549
lemma (in algebra) inf_measure_increasing:
hoelzl@38656
   550
  assumes posf: "positive f"
paulson@33271
   551
  shows "increasing (| space = space M, sets = Pow (space M) |)
paulson@33271
   552
                    (\<lambda>x. Inf (measure_set M f x))"
hoelzl@38656
   553
apply (auto simp add: increasing_def)
hoelzl@38656
   554
apply (rule complete_lattice_class.Inf_greatest)
hoelzl@38656
   555
apply (rule complete_lattice_class.Inf_lower)
huffman@37032
   556
apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
paulson@33271
   557
done
paulson@33271
   558
paulson@33271
   559
paulson@33271
   560
lemma (in algebra) inf_measure_le:
hoelzl@38656
   561
  assumes posf: "positive f" and inc: "increasing M f"
hoelzl@38656
   562
      and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> psuminf (f \<circ> A) = r}"
paulson@33271
   563
  shows "Inf (measure_set M f s) \<le> x"
paulson@33271
   564
proof -
paulson@33271
   565
  from x
hoelzl@38656
   566
  obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
hoelzl@38656
   567
             and xeq: "psuminf (f \<circ> A) = x"
hoelzl@38656
   568
    by auto
paulson@33271
   569
  have dA: "range (disjointed A) \<subseteq> sets M"
paulson@33271
   570
    by (metis A range_disjointed_sets)
hoelzl@38656
   571
  have "\<forall>n.(f o disjointed A) n \<le> (f \<circ> A) n" unfolding comp_def
hoelzl@38656
   572
    by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
hoelzl@38656
   573
  hence sda: "psuminf (f o disjointed A) \<le> psuminf (f \<circ> A)"
hoelzl@38656
   574
    by (blast intro: psuminf_le)
hoelzl@38656
   575
  hence ley: "psuminf (f o disjointed A) \<le> x"
hoelzl@38656
   576
    by (metis xeq)
hoelzl@38656
   577
  hence y: "psuminf (f o disjointed A) \<in> measure_set M f s"
paulson@33271
   578
    apply (auto simp add: measure_set_def)
hoelzl@38656
   579
    apply (rule_tac x="disjointed A" in exI)
hoelzl@38656
   580
    apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
paulson@33271
   581
    done
paulson@33271
   582
  show ?thesis
hoelzl@38656
   583
    by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower)
paulson@33271
   584
qed
paulson@33271
   585
paulson@33271
   586
lemma (in algebra) inf_measure_close:
hoelzl@38656
   587
  assumes posf: "positive f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
hoelzl@38656
   588
  shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
hoelzl@38656
   589
               psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
hoelzl@38656
   590
proof (cases "Inf (measure_set M f s) = \<omega>")
hoelzl@38656
   591
  case False
hoelzl@38656
   592
  obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
hoelzl@38656
   593
    using Inf_close[OF False e] by auto
hoelzl@38656
   594
  thus ?thesis
hoelzl@38656
   595
    by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
hoelzl@38656
   596
next
hoelzl@38656
   597
  case True
hoelzl@38656
   598
  have "measure_set M f s \<noteq> {}"
hoelzl@39096
   599
    by (metis emptyE ss inf_measure_nonempty [of f, OF posf top _ empty_sets])
hoelzl@38656
   600
  then obtain l where "l \<in> measure_set M f s" by auto
hoelzl@38656
   601
  moreover from True have "l \<le> Inf (measure_set M f s) + e" by simp
hoelzl@38656
   602
  ultimately show ?thesis
hoelzl@38656
   603
    by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
paulson@33271
   604
qed
paulson@33271
   605
paulson@33271
   606
lemma (in algebra) inf_measure_countably_subadditive:
hoelzl@38656
   607
  assumes posf: "positive f" and inc: "increasing M f"
paulson@33271
   608
  shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
paulson@33271
   609
                  (\<lambda>x. Inf (measure_set M f x))"
hoelzl@38656
   610
  unfolding countably_subadditive_def o_def
hoelzl@38656
   611
proof (safe, simp, rule pinfreal_le_epsilon)
hoelzl@38656
   612
  fix A :: "nat \<Rightarrow> 'a set" and e :: pinfreal
hoelzl@38656
   613
hoelzl@38656
   614
  let "?outer n" = "Inf (measure_set M f (A n))"
hoelzl@38656
   615
  assume A: "range A \<subseteq> Pow (space M)"
hoelzl@38656
   616
     and disj: "disjoint_family A"
hoelzl@38656
   617
     and sb: "(\<Union>i. A i) \<subseteq> space M"
hoelzl@38656
   618
     and e: "0 < e"
hoelzl@38656
   619
  hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
hoelzl@38656
   620
                   A n \<subseteq> (\<Union>i. BB n i) \<and>
hoelzl@38656
   621
                   psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
hoelzl@38656
   622
    apply (safe intro!: choice inf_measure_close [of f, OF posf _])
hoelzl@38656
   623
    using e sb by (cases e, auto simp add: not_le mult_pos_pos)
hoelzl@38656
   624
  then obtain BB
hoelzl@38656
   625
    where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
hoelzl@38656
   626
      and disjBB: "\<And>n. disjoint_family (BB n)"
hoelzl@38656
   627
      and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
hoelzl@38656
   628
      and BBle: "\<And>n. psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
hoelzl@38656
   629
    by auto blast
hoelzl@38656
   630
  have sll: "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> psuminf ?outer + e"
hoelzl@38656
   631
    proof -
hoelzl@38656
   632
      have "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> (\<Sum>\<^isub>\<infinity> n. ?outer n + e*(1/2) ^ Suc n)"
hoelzl@38656
   633
        by (rule psuminf_le[OF BBle])
hoelzl@38656
   634
      also have "... = psuminf ?outer + e"
hoelzl@38656
   635
        using psuminf_half_series by simp
hoelzl@38656
   636
      finally show ?thesis .
hoelzl@38656
   637
    qed
hoelzl@38656
   638
  def C \<equiv> "(split BB) o prod_decode"
hoelzl@38656
   639
  have C: "!!n. C n \<in> sets M"
hoelzl@38656
   640
    apply (rule_tac p="prod_decode n" in PairE)
hoelzl@38656
   641
    apply (simp add: C_def)
hoelzl@38656
   642
    apply (metis BB subsetD rangeI)
hoelzl@38656
   643
    done
hoelzl@38656
   644
  have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
hoelzl@38656
   645
    proof (auto simp add: C_def)
hoelzl@38656
   646
      fix x i
hoelzl@38656
   647
      assume x: "x \<in> A i"
hoelzl@38656
   648
      with sbBB [of i] obtain j where "x \<in> BB i j"
hoelzl@38656
   649
        by blast
hoelzl@38656
   650
      thus "\<exists>i. x \<in> split BB (prod_decode i)"
hoelzl@38656
   651
        by (metis prod_encode_inverse prod.cases)
hoelzl@38656
   652
    qed
hoelzl@38656
   653
  have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
hoelzl@38656
   654
    by (rule ext)  (auto simp add: C_def)
hoelzl@38656
   655
  moreover have "psuminf ... = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" using BBle
hoelzl@38656
   656
    by (force intro!: psuminf_2dimen simp: o_def)
hoelzl@38656
   657
  ultimately have Csums: "psuminf (f \<circ> C) = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" by simp
hoelzl@38656
   658
  have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))"
hoelzl@38656
   659
    apply (rule inf_measure_le [OF posf(1) inc], auto)
hoelzl@38656
   660
    apply (rule_tac x="C" in exI)
hoelzl@38656
   661
    apply (auto simp add: C sbC Csums)
hoelzl@38656
   662
    done
hoelzl@38656
   663
  also have "... \<le> (\<Sum>\<^isub>\<infinity>n. Inf (measure_set M f (A n))) + e" using sll
hoelzl@38656
   664
    by blast
hoelzl@38656
   665
  finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> psuminf ?outer + e" .
paulson@33271
   666
qed
paulson@33271
   667
paulson@33271
   668
lemma (in algebra) inf_measure_outer:
hoelzl@38656
   669
  "\<lbrakk> positive f ; increasing M f \<rbrakk>
paulson@33271
   670
   \<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |)
paulson@33271
   671
                          (\<lambda>x. Inf (measure_set M f x))"
hoelzl@38656
   672
  by (simp add: outer_measure_space_def inf_measure_empty
hoelzl@38656
   673
                inf_measure_increasing inf_measure_countably_subadditive positive_def)
paulson@33271
   674
paulson@33271
   675
(*MOVE UP*)
paulson@33271
   676
paulson@33271
   677
lemma (in algebra) algebra_subset_lambda_system:
hoelzl@38656
   678
  assumes posf: "positive f" and inc: "increasing M f"
paulson@33271
   679
      and add: "additive M f"
paulson@33271
   680
  shows "sets M \<subseteq> lambda_system (| space = space M, sets = Pow (space M) |)
paulson@33271
   681
                                (\<lambda>x. Inf (measure_set M f x))"
hoelzl@38656
   682
proof (auto dest: sets_into_space
hoelzl@38656
   683
            simp add: algebra.lambda_system_eq [OF algebra_Pow])
paulson@33271
   684
  fix x s
paulson@33271
   685
  assume x: "x \<in> sets M"
paulson@33271
   686
     and s: "s \<subseteq> space M"
hoelzl@38656
   687
  have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
paulson@33271
   688
    by blast
paulson@33271
   689
  have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
paulson@33271
   690
        \<le> Inf (measure_set M f s)"
hoelzl@38656
   691
    proof (rule pinfreal_le_epsilon)
hoelzl@38656
   692
      fix e :: pinfreal
paulson@33271
   693
      assume e: "0 < e"
hoelzl@38656
   694
      from inf_measure_close [of f, OF posf e s]
hoelzl@38656
   695
      obtain A where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
hoelzl@38656
   696
                 and sUN: "s \<subseteq> (\<Union>i. A i)"
hoelzl@38656
   697
                 and l: "psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
wenzelm@33536
   698
        by auto
paulson@33271
   699
      have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
paulson@33271
   700
                      (f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
wenzelm@33536
   701
        by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD)
paulson@33271
   702
      have  [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)"
wenzelm@33536
   703
        by (subst additiveD [OF add, symmetric])
wenzelm@33536
   704
           (auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint)
paulson@33271
   705
      { fix u
wenzelm@33536
   706
        assume u: "u \<in> sets M"
hoelzl@38656
   707
        have [simp]: "\<And>n. f (A n \<inter> u) \<le> f (A n)"
hoelzl@38656
   708
          by (simp add: increasingD [OF inc] u Int range_subsetD [OF A])
hoelzl@38656
   709
        have 2: "Inf (measure_set M f (s \<inter> u)) \<le> psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A)"
hoelzl@38656
   710
          proof (rule complete_lattice_class.Inf_lower)
hoelzl@38656
   711
            show "psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
hoelzl@38656
   712
              apply (simp add: measure_set_def)
hoelzl@38656
   713
              apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
hoelzl@38656
   714
              apply (auto simp add: disjoint_family_subset [OF disj] o_def)
hoelzl@38656
   715
              apply (blast intro: u range_subsetD [OF A])
paulson@33271
   716
              apply (blast dest: subsetD [OF sUN])
paulson@33271
   717
              done
hoelzl@38656
   718
          qed
paulson@33271
   719
      } note lesum = this
hoelzl@38656
   720
      have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> psuminf (f o (\<lambda>z. z\<inter>x) o A)"
hoelzl@38656
   721
        and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
hoelzl@38656
   722
                   \<le> psuminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
wenzelm@33536
   723
        by (metis Diff lesum top x)+
paulson@33271
   724
      hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
hoelzl@38656
   725
           \<le> psuminf (f o (\<lambda>s. s\<inter>x) o A) + psuminf (f o (\<lambda>s. s-x) o A)"
hoelzl@38656
   726
        by (simp add: x add_mono)
hoelzl@38656
   727
      also have "... \<le> psuminf (f o A)"
hoelzl@38656
   728
        by (simp add: x psuminf_add[symmetric] o_def)
paulson@33271
   729
      also have "... \<le> Inf (measure_set M f s) + e"
hoelzl@38656
   730
        by (rule l)
paulson@33271
   731
      finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
paulson@33271
   732
        \<le> Inf (measure_set M f s) + e" .
paulson@33271
   733
    qed
hoelzl@38656
   734
  moreover
paulson@33271
   735
  have "Inf (measure_set M f s)
paulson@33271
   736
       \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
paulson@33271
   737
    proof -
paulson@33271
   738
    have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
paulson@33271
   739
      by (metis Un_Diff_Int Un_commute)
hoelzl@38656
   740
    also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
hoelzl@38656
   741
      apply (rule subadditiveD)
hoelzl@38656
   742
      apply (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow
wenzelm@33536
   743
               inf_measure_positive inf_measure_countably_subadditive posf inc)
hoelzl@38656
   744
      apply (auto simp add: subsetD [OF s])
paulson@33271
   745
      done
paulson@33271
   746
    finally show ?thesis .
paulson@33271
   747
    qed
hoelzl@38656
   748
  ultimately
paulson@33271
   749
  show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
paulson@33271
   750
        = Inf (measure_set M f s)"
paulson@33271
   751
    by (rule order_antisym)
paulson@33271
   752
qed
paulson@33271
   753
paulson@33271
   754
lemma measure_down:
hoelzl@38656
   755
     "measure_space N \<mu> \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow>
hoelzl@38656
   756
      (\<nu> = \<mu>) \<Longrightarrow> measure_space M \<nu>"
hoelzl@38656
   757
  by (simp add: measure_space_def measure_space_axioms_def positive_def
hoelzl@38656
   758
                countably_additive_def)
paulson@33271
   759
     blast
paulson@33271
   760
paulson@33271
   761
theorem (in algebra) caratheodory:
hoelzl@38656
   762
  assumes posf: "positive f" and ca: "countably_additive M f"
hoelzl@38656
   763
  shows "\<exists>\<mu> :: 'a set \<Rightarrow> pinfreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> measure_space (sigma (space M) (sets M)) \<mu>"
paulson@33271
   764
  proof -
paulson@33271
   765
    have inc: "increasing M f"
hoelzl@38656
   766
      by (metis additive_increasing ca countably_additive_additive posf)
paulson@33271
   767
    let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
paulson@33271
   768
    def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
hoelzl@38656
   769
    have mls: "measure_space \<lparr>space = space M, sets = ls\<rparr> ?infm"
paulson@33271
   770
      using sigma_algebra.caratheodory_lemma
paulson@33271
   771
              [OF sigma_algebra_Pow  inf_measure_outer [OF posf inc]]
paulson@33271
   772
      by (simp add: ls_def)
hoelzl@38656
   773
    hence sls: "sigma_algebra (|space = space M, sets = ls|)"
hoelzl@38656
   774
      by (simp add: measure_space_def)
hoelzl@38656
   775
    have "sets M \<subseteq> ls"
paulson@33271
   776
      by (simp add: ls_def)
paulson@33271
   777
         (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
hoelzl@38656
   778
    hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls"
paulson@33271
   779
      using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"]
paulson@33271
   780
      by simp
hoelzl@38656
   781
    have "measure_space (sigma (space M) (sets M)) ?infm"
hoelzl@38656
   782
      unfolding sigma_def
hoelzl@38656
   783
      by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
paulson@33271
   784
         (simp_all add: sgs_sb space_closed)
hoelzl@38656
   785
    thus ?thesis using inf_measure_agrees [OF posf ca] by (auto intro!: exI[of _ ?infm])
hoelzl@38656
   786
  qed
paulson@33271
   787
paulson@33271
   788
end