src/HOL/Probability/Measure_Space.thy
author hoelzl
Wed, 10 Oct 2012 12:12:23 +0200
changeset 49784 5e5b2da42a69
parent 49773 16907431e477
child 49789 e0a4cb91a8a9
permissions -rw-r--r--
remove incseq assumption from measure_eqI_generator_eq
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  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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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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   164
  then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   165
  also have "... = f (x \<union> (y-x))" using addf
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   166
    by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   167
  also have "... = f y"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   168
    by (metis Un_Diff_cancel Un_absorb1 xy(3))
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   169
  finally show "f x \<le> f y" by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   170
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   171
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   172
lemma (in ring_of_sets) countably_additive_additive:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   173
  assumes posf: "positive M f" and ca: "countably_additive M f"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   174
  shows "additive M f"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   175
proof (auto simp add: additive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   176
  fix x y
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   177
  assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   178
  hence "disjoint_family (binaryset x y)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   179
    by (auto simp add: disjoint_family_on_def binaryset_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   180
  hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   181
         (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   182
         f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   183
    using ca
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   184
    by (simp add: countably_additive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   185
  hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   186
         f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   187
    by (simp add: range_binaryset_eq UN_binaryset_eq)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   188
  thus "f (x \<union> y) = f x + f y" using posf x y
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   189
    by (auto simp add: Un suminf_binaryset_eq positive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   190
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   191
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   192
lemma (in algebra) increasing_additive_bound:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   193
  fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> ereal"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   194
  assumes f: "positive M f" and ad: "additive M f"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   195
      and inc: "increasing M f"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   196
      and A: "range A \<subseteq> M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   197
      and disj: "disjoint_family A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   198
  shows  "(\<Sum>i. f (A i)) \<le> f \<Omega>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   199
proof (safe intro!: suminf_bound)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   200
  fix N
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   201
  note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   202
  have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   203
    using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   204
  also have "... \<le> f \<Omega>" using space_closed A
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   205
    by (intro increasingD[OF inc] finite_UN) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   206
  finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   207
qed (insert f A, auto simp: positive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   208
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   209
lemma (in ring_of_sets) countably_additiveI_finite:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   210
  assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   211
  shows "countably_additive M \<mu>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   212
proof (rule countably_additiveI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   213
  fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   214
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   215
  have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   216
  from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   217
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   218
  have inj_f: "inj_on f {i. F i \<noteq> {}}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   219
  proof (rule inj_onI, simp)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   220
    fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   221
    then have "f i \<in> F i" "f j \<in> F j" using f by force+
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   222
    with disj * show "i = j" by (auto simp: disjoint_family_on_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   223
  qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   224
  have "finite (\<Union>i. F i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   225
    by (metis F(2) assms(1) infinite_super sets_into_space)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   226
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   227
  have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   228
    by (auto simp: positiveD_empty[OF `positive M \<mu>`])
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   229
  moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   230
  proof (rule finite_imageD)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   231
    from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   232
    then show "finite (f`{i. F i \<noteq> {}})"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   233
      by (rule finite_subset) fact
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   234
  qed fact
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   235
  ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   236
    by (rule finite_subset)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   237
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   238
  have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   239
    using disj by (auto simp: disjoint_family_on_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   240
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   241
  from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   242
    by (rule suminf_finite) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   243
  also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   244
    using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   245
  also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   246
    using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   247
  also have "\<dots> = \<mu> (\<Union>i. F i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   248
    by (rule arg_cong[where f=\<mu>]) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   249
  finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   250
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   251
49773
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   252
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   253
  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
   254
  shows "countably_additive M f \<longleftrightarrow>
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   255
    (\<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
   256
  unfolding countably_additive_def
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   257
proof safe
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   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)"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   259
  fix A :: "nat \<Rightarrow> 'a set" assume 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
   260
  then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets)
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   261
  with count_sum[THEN spec, of "disjointed A"] A(3)
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   262
  have f_UN: "(\<Sum>i. f (disjointed 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
   263
    by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   264
  moreover have "(\<lambda>n. (\<Sum>i=0..<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
   265
    using f(1)[unfolded positive_def] dA
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   266
    by (auto intro!: summable_sumr_LIMSEQ_suminf summable_ereal_pos)
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   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
hoelzl
parents: 47762
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 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   366
      using A by (intro f(2)[THEN additiveD, symmetric]) auto
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   367
    also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   368
      by auto
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   369
    finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   370
      using A by (subst (asm) (1 2 3) f') auto
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   371
    then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   372
      using A f' by auto }
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   373
  ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   374
    by (simp add: zero_ereal_def)
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   375
  then have "(\<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
   376
    by (rule LIMSEQ_diff_approach_zero2[OF tendsto_const])
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   377
  then show "(\<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
   378
    using A by (subst (1 2) f') auto
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   379
qed
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   380
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   381
lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   382
  assumes f: "positive M f" "additive M f" and fin: "\<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
   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 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   384
  shows "countably_additive M f"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   385
  using countably_additive_iff_continuous_from_below[OF f]
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   386
  using empty_continuous_imp_continuous_from_below[OF f fin] cont
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   387
  by blast
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   388
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   389
section {* Properties of @{const emeasure} *}
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   390
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   391
lemma emeasure_positive: "positive (sets M) (emeasure M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   392
  by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   393
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   394
lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   395
  using emeasure_positive[of M] by (simp add: positive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   396
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   397
lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   398
  using emeasure_notin_sets[of A M] emeasure_positive[of M]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   399
  by (cases "A \<in> sets M") (auto simp: positive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   400
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   401
lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   402
  using emeasure_nonneg[of M A] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   403
  
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   404
lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   405
  by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   406
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   407
lemma suminf_emeasure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   408
  "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   409
  using countable_UN[of A UNIV M] emeasure_countably_additive[of M]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   410
  by (simp add: countably_additive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   411
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   412
lemma emeasure_additive: "additive (sets M) (emeasure M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   413
  by (metis countably_additive_additive emeasure_positive emeasure_countably_additive)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   414
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   415
lemma plus_emeasure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   417
  using additiveD[OF emeasure_additive] ..
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   418
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   419
lemma setsum_emeasure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   420
  "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   421
    (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   422
  by (metis additive_sum emeasure_positive emeasure_additive)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   423
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   424
lemma emeasure_mono:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   425
  "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   426
  by (metis additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   427
            emeasure_positive increasingD)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   428
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   429
lemma emeasure_space:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   430
  "emeasure M A \<le> emeasure M (space M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   431
  by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets_into_space top)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   432
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   433
lemma emeasure_compl:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   434
  assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   435
  shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   436
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   437
  from s have "0 \<le> emeasure M s" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   438
  have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   439
    by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   440
  also have "... = emeasure M s + emeasure M (space M - s)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   441
    by (rule plus_emeasure[symmetric]) (auto simp add: s)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   442
  finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   443
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   444
    using fin `0 \<le> emeasure M s`
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   445
    unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   446
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   447
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   448
lemma emeasure_Diff:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   449
  assumes finite: "emeasure M B \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   450
  and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   451
  shows "emeasure M (A - B) = emeasure M A - emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   452
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   453
  have "0 \<le> emeasure M B" using assms by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   454
  have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   455
  then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   456
  also have "\<dots> = emeasure M (A - B) + emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   457
    using measurable by (subst plus_emeasure[symmetric]) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   458
  finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   459
    unfolding ereal_eq_minus_iff
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   460
    using finite `0 \<le> emeasure M B` by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   461
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   462
49773
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   463
lemma Lim_emeasure_incseq:
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   464
  "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   465
  using emeasure_countably_additive
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   466
  by (auto simp add: countably_additive_iff_continuous_from_below emeasure_positive emeasure_additive)
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   467
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   468
lemma incseq_emeasure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   469
  assumes "range B \<subseteq> sets M" "incseq B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   470
  shows "incseq (\<lambda>i. emeasure M (B i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   471
  using assms by (auto simp: incseq_def intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   472
49773
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   473
lemma SUP_emeasure_incseq:
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   474
  assumes A: "range A \<subseteq> sets M" "incseq A"
49773
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   475
  shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   476
  using LIMSEQ_ereal_SUPR[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]
16907431e477 tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents: 47762
diff changeset
   477
  by (simp add: LIMSEQ_unique)
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   478
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   479
lemma decseq_emeasure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   480
  assumes "range B \<subseteq> sets M" "decseq B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   481
  shows "decseq (\<lambda>i. emeasure M (B i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   482
  using assms by (auto simp: decseq_def intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   483
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   484
lemma INF_emeasure_decseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   485
  assumes A: "range A \<subseteq> sets M" and "decseq A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   486
  and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   487
  shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   488
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   489
  have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   490
    using A by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   491
  hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   492
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   493
  have A0: "0 \<le> emeasure M (A 0)" using A by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   494
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   495
  have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   496
    by (simp add: ereal_SUPR_uminus minus_ereal_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   497
  also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   498
    unfolding minus_ereal_def using A0 assms
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   499
    by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   500
  also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   501
    using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   502
  also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   503
  proof (rule SUP_emeasure_incseq)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   504
    show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   505
      using A by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   506
    show "incseq (\<lambda>n. A 0 - A n)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   507
      using `decseq A` by (auto simp add: incseq_def decseq_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   508
  qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   509
  also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   510
    using A finite * by (simp, subst emeasure_Diff) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   511
  finally show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   512
    unfolding ereal_minus_eq_minus_iff using finite A0 by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   513
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   514
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   515
lemma Lim_emeasure_decseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   516
  assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   517
  shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   518
  using LIMSEQ_ereal_INFI[OF decseq_emeasure, OF A]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   519
  using INF_emeasure_decseq[OF A fin] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   520
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   521
lemma emeasure_subadditive:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   522
  assumes measurable: "A \<in> sets M" "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   523
  shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   524
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   525
  from plus_emeasure[of A M "B - A"]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   526
  have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   527
    using assms by (simp add: Diff)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   528
  also have "\<dots> \<le> emeasure M A + emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   529
    using assms by (auto intro!: add_left_mono emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   530
  finally show ?thesis .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   531
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   532
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   533
lemma emeasure_subadditive_finite:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   534
  assumes "finite I" "A ` I \<subseteq> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   535
  shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   536
using assms proof induct
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   537
  case (insert i I)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   538
  then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   539
    by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   540
  also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   541
    using insert by (intro emeasure_subadditive finite_UN) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   542
  also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   543
    using insert by (intro add_mono) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   544
  also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   545
    using insert by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   546
  finally show ?case .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   547
qed simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   548
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   549
lemma emeasure_subadditive_countably:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   550
  assumes "range f \<subseteq> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   551
  shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   552
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   553
  have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   554
    unfolding UN_disjointed_eq ..
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   555
  also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   556
    using range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   557
    by (simp add:  disjoint_family_disjointed comp_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   558
  also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   559
    using range_disjointed_sets[OF assms] assms
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   560
    by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   561
  finally show ?thesis .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   562
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   563
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   564
lemma emeasure_insert:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   565
  assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   566
  shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   567
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   568
  have "{x} \<inter> A = {}" using `x \<notin> A` by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   569
  from plus_emeasure[OF sets this] show ?thesis by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   570
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   571
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   572
lemma emeasure_eq_setsum_singleton:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   573
  assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   574
  shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   575
  using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   576
  by (auto simp: disjoint_family_on_def subset_eq)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   577
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   578
lemma setsum_emeasure_cover:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   579
  assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   580
  assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   581
  assumes disj: "disjoint_family_on B S"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   582
  shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   583
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   584
  have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   585
  proof (rule setsum_emeasure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   586
    show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   587
      using `disjoint_family_on B S`
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   588
      unfolding disjoint_family_on_def by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   589
  qed (insert assms, auto)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   590
  also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   591
    using A by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   592
  finally show ?thesis by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   593
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   594
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   595
lemma emeasure_eq_0:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   596
  "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   597
  by (metis emeasure_mono emeasure_nonneg order_eq_iff)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   598
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   599
lemma emeasure_UN_eq_0:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   600
  assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   601
  shows "emeasure M (\<Union> i. N i) = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   602
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   603
  have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   604
  moreover have "emeasure M (\<Union> i. N i) \<le> 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   605
    using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   606
  ultimately show ?thesis by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   607
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   608
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   609
lemma measure_eqI_finite:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   610
  assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   611
  assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   612
  shows "M = N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   613
proof (rule measure_eqI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   614
  fix X assume "X \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   615
  then have X: "X \<subseteq> A" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   616
  then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   617
    using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   618
  also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   619
    using X eq by (auto intro!: setsum_cong)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   620
  also have "\<dots> = emeasure N X"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   621
    using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   622
  finally show "emeasure M X = emeasure N X" .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   623
qed simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   624
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   625
lemma measure_eqI_generator_eq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   626
  fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   627
  assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   628
  and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   629
  and M: "sets M = sigma_sets \<Omega> E"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   632
  shows "M = N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   643
      then show "A \<subseteq> \<Omega>" using S.sets_into_space by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   644
    next
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   647
        using `F \<in> E` eq by (auto intro: sigma_sets_top)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   650
      then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   651
        and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   664
        using * by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   665
    next
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   674
      using `F \<in> E` `Int_stable E`
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   678
  note * = this
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   679
  show "M = N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   680
  proof (rule measure_eqI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   681
    show "sets M = sets N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   708
  qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   709
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   710
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   711
lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   712
proof (intro measure_eqI emeasure_measure_of_sigma)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   713
  show "sigma_algebra (space M) (sets M)" ..
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   714
  show "positive (sets M) (emeasure M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   715
    by (simp add: positive_def emeasure_nonneg)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   716
  show "countably_additive (sets M) (emeasure M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   717
    by (simp add: emeasure_countably_additive)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   718
qed simp_all
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   719
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   720
section "@{text \<mu>}-null sets"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   721
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   722
definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   723
  "null_sets M = {N\<in>sets M. emeasure M N = 0}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   724
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   725
lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   726
  by (simp add: null_sets_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   727
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   728
lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   729
  unfolding null_sets_def by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   730
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   731
lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   732
  unfolding null_sets_def by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   733
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   734
interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
47762
d31085f07f60 add Caratheodories theorem for semi-rings of sets
hoelzl
parents: 47761
diff changeset
   735
proof (rule ring_of_setsI)
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   736
  show "null_sets M \<subseteq> Pow (space M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   737
    using sets_into_space by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   738
  show "{} \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   739
    by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   740
  fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   741
  then have "A \<in> sets M" "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   742
    by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   743
  moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   744
    "emeasure M (A - B) \<le> emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   745
    by (auto intro!: emeasure_subadditive emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   746
  moreover have "emeasure M B = 0" "emeasure M A = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   747
    using sets by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   748
  ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   749
    by (auto intro!: antisym)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   750
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   751
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   752
lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   753
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   754
  have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   755
    unfolding SUP_def image_compose
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   756
    unfolding surj_from_nat ..
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   757
  then show ?thesis by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   758
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   759
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   760
lemma null_sets_UN[intro]:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   761
  assumes "\<And>i::'i::countable. N i \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   762
  shows "(\<Union>i. N i) \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   763
proof (intro conjI CollectI null_setsI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   764
  show "(\<Union>i. N i) \<in> sets M" using assms by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   765
  have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   766
  moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   767
    unfolding UN_from_nat[of N]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   768
    using assms by (intro emeasure_subadditive_countably) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   769
  ultimately show "emeasure M (\<Union>i. N i) = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   770
    using assms by (auto simp: null_setsD1)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   771
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   772
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   773
lemma null_set_Int1:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   774
  assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   775
proof (intro CollectI conjI null_setsI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   776
  show "emeasure M (A \<inter> B) = 0" using assms
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   777
    by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   778
qed (insert assms, auto)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   779
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   780
lemma null_set_Int2:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   781
  assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   782
  using assms by (subst Int_commute) (rule null_set_Int1)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   783
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   784
lemma emeasure_Diff_null_set:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   785
  assumes "B \<in> null_sets M" "A \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   786
  shows "emeasure M (A - B) = emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   787
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   788
  have *: "A - B = (A - (A \<inter> B))" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   789
  have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   790
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   791
    unfolding * using assms
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   792
    by (subst emeasure_Diff) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   793
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   794
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   795
lemma null_set_Diff:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   796
  assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   797
proof (intro CollectI conjI null_setsI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   798
  show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   799
qed (insert assms, auto)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   800
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   801
lemma emeasure_Un_null_set:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   802
  assumes "A \<in> sets M" "B \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   803
  shows "emeasure M (A \<union> B) = emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   804
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   805
  have *: "A \<union> B = A \<union> (B - A)" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   806
  have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   807
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   808
    unfolding * using assms
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   809
    by (subst plus_emeasure[symmetric]) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   810
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   811
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   812
section "Formalize almost everywhere"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   813
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   814
definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   815
  "ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   816
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   817
abbreviation
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   818
  almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   819
  "almost_everywhere M P \<equiv> eventually P (ae_filter M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   820
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   821
syntax
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   822
  "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   823
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   824
translations
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   825
  "AE x in M. P" == "CONST almost_everywhere M (%x. P)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   826
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   827
lemma eventually_ae_filter:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   828
  fixes M P
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   829
  defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N" 
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   830
  shows "eventually P (ae_filter M) \<longleftrightarrow> F P"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   831
  unfolding ae_filter_def F_def[symmetric]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   832
proof (rule eventually_Abs_filter)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   833
  show "is_filter F"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   834
  proof
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   835
    fix P Q assume "F P" "F Q"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   836
    then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   837
      and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   838
      by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   840
    then show "F (\<lambda>x. P x \<and> Q x)" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   841
  next
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   842
    fix P Q assume "F P"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   843
    then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   844
    moreover assume "\<forall>x. P x \<longrightarrow> Q x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   845
    ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   846
    then show "F Q" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   847
  qed auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   848
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   849
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   850
lemma AE_I':
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   851
  "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   852
  unfolding eventually_ae_filter by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   853
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   854
lemma AE_iff_null:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   855
  assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   856
  shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   857
proof
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   858
  assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   859
    unfolding eventually_ae_filter by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   860
  have "0 \<le> emeasure M ?P" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   861
  moreover have "emeasure M ?P \<le> emeasure M N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   862
    using assms N(1,2) by (auto intro: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   863
  ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   864
  then show "?P \<in> null_sets M" using assms by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   865
next
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   866
  assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   867
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   868
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   869
lemma AE_iff_null_sets:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   870
  "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   871
  using Int_absorb1[OF sets_into_space, of N M]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   872
  by (subst AE_iff_null) (auto simp: Int_def[symmetric])
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   873
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   874
lemma AE_not_in:
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   875
  "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   876
  by (metis AE_iff_null_sets null_setsD2)
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   877
47694
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   878
lemma AE_iff_measurable:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   880
  using AE_iff_null[of _ P] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   881
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   882
lemma AE_E[consumes 1]:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   883
  assumes "AE x in M. P x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   884
  obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   885
  using assms unfolding eventually_ae_filter by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   886
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   887
lemma AE_E2:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   888
  assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   889
  shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   890
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   891
  have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   892
  with AE_iff_null[of M P] assms show ?thesis by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   893
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   894
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   895
lemma AE_I:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   896
  assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   897
  shows "AE x in M. P x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   898
  using assms unfolding eventually_ae_filter by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   899
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   900
lemma AE_mp[elim!]:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   901
  assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   902
  shows "AE x in M. Q x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   903
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   904
  from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   905
    and A: "A \<in> sets M" "emeasure M A = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   906
    by (auto elim!: AE_E)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   907
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   908
  from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   909
    and B: "B \<in> sets M" "emeasure M B = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   910
    by (auto elim!: AE_E)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   911
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   912
  show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   913
  proof (intro AE_I)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   914
    have "0 \<le> emeasure M (A \<union> B)" using A B by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   915
    moreover have "emeasure M (A \<union> B) \<le> 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   916
      using emeasure_subadditive[of A M B] A B by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   917
    ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   918
    show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   919
      using P imp by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   920
  qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   921
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   922
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   923
(* depricated replace by laws about eventually *)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   924
lemma
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   926
    and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   927
    and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   930
  by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   931
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   932
lemma AE_impI:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   933
  "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   934
  by (cases P) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   935
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   936
lemma AE_measure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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")
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   938
  shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   939
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   940
  from AE_E[OF AE] guess N . note N = this
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   941
  with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   942
    by (intro emeasure_mono) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   943
  also have "\<dots> \<le> emeasure M ?P + emeasure M N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   944
    using sets N by (intro emeasure_subadditive) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   945
  also have "\<dots> = emeasure M ?P" using N by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   946
  finally show "emeasure M ?P = emeasure M (space M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   947
    using emeasure_space[of M "?P"] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   948
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   949
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   950
lemma AE_space: "AE x in M. x \<in> space M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   951
  by (rule AE_I[where N="{}"]) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   952
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   953
lemma AE_I2[simp, intro]:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   954
  "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   955
  using AE_space by force
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   956
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   957
lemma AE_Ball_mp:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   959
  by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   960
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   961
lemma AE_cong[cong]:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   963
  by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   964
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   965
lemma AE_all_countable:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   966
  "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   967
proof
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   968
  assume "\<forall>i. AE x in M. P i x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   969
  from this[unfolded eventually_ae_filter Bex_def, THEN choice]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   972
  also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   973
  finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   974
  moreover from N have "(\<Union>i. N i) \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   975
    by (intro null_sets_UN) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   976
  ultimately show "AE x in M. \<forall>i. P i x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   977
    unfolding eventually_ae_filter by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   978
qed auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   979
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   980
lemma AE_finite_all:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   982
  using f by induct auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   983
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   984
lemma AE_finite_allI:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   985
  assumes "finite S"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   987
  using AE_finite_all[OF `finite S`] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   988
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   989
lemma emeasure_mono_AE:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   990
  assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   991
    and B: "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   992
  shows "emeasure M A \<le> emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   993
proof cases
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   994
  assume A: "A \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   996
    by (auto simp: eventually_ae_filter)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   997
  have "emeasure M A = emeasure M (A - N)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   998
    using N A by (subst emeasure_Diff_null_set) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
   999
  also have "emeasure M (A - N) \<le> emeasure M (B - N)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1000
    using N A B sets_into_space by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1001
  also have "emeasure M (B - N) = emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1002
    using N B by (subst emeasure_Diff_null_set) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1003
  finally show ?thesis .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1004
qed (simp add: emeasure_nonneg emeasure_notin_sets)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1005
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1006
lemma emeasure_eq_AE:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1007
  assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1008
  assumes A: "A \<in> sets M" and B: "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1009
  shows "emeasure M A = emeasure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1010
  using assms by (safe intro!: antisym emeasure_mono_AE) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1011
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1012
section {* @{text \<sigma>}-finite Measures *}
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1013
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1014
locale sigma_finite_measure =
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1015
  fixes M :: "'a measure"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1016
  assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set.
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1017
    range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1018
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1019
lemma (in sigma_finite_measure) sigma_finite_disjoint:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1020
  obtains A :: "nat \<Rightarrow> 'a set"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1021
  where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1022
proof atomize_elim
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1023
  case goal1
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1024
  obtain A :: "nat \<Rightarrow> 'a set" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1025
    range: "range A \<subseteq> sets M" and
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1026
    space: "(\<Union>i. A i) = space M" and
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1027
    measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1028
    using sigma_finite by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1029
  note range' = range_disjointed_sets[OF range] range
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1030
  { fix i
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1031
    have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1032
      using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1033
    then have "emeasure M (disjointed A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1034
      using measure[of i] by auto }
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1035
  with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1036
  show ?case by (auto intro!: exI[of _ "disjointed A"])
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1037
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1038
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1039
lemma (in sigma_finite_measure) sigma_finite_incseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1040
  obtains A :: "nat \<Rightarrow> 'a set"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1041
  where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1042
proof atomize_elim
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1043
  case goal1
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1044
  obtain F :: "nat \<Rightarrow> 'a set" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1045
    F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1046
    using sigma_finite by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1047
  then show ?case
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1048
  proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1049
    from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1050
    then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1051
      using F by fastforce
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1052
  next
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1053
    fix n
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1054
    have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1055
      by (auto intro!: emeasure_subadditive_finite)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1056
    also have "\<dots> < \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1057
      using F by (auto simp: setsum_Pinfty)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1058
    finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1059
  qed (force simp: incseq_def)+
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1060
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1061
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1062
section {* Measure space induced by distribution of @{const measurable}-functions *}
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1063
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1064
definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1065
  "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1066
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1067
lemma
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1068
  shows sets_distr[simp]: "sets (distr M N f) = sets N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1069
    and space_distr[simp]: "space (distr M N f) = space N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1070
  by (auto simp: distr_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1071
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1072
lemma
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1073
  shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1074
    and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1075
  by (auto simp: measurable_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1076
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1077
lemma emeasure_distr:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1078
  fixes f :: "'a \<Rightarrow> 'b"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1079
  assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1080
  shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1081
  unfolding distr_def
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1082
proof (rule emeasure_measure_of_sigma)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1083
  show "positive (sets N) ?\<mu>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1084
    by (auto simp: positive_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1085
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1086
  show "countably_additive (sets N) ?\<mu>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1087
  proof (intro countably_additiveI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1088
    fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1089
    then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1090
    then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1091
      using f by (auto simp: measurable_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1092
    moreover have "(\<Union>i. f -`  A i \<inter> space M) \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1093
      using * by blast
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1094
    moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1095
      using `disjoint_family A` by (auto simp: disjoint_family_on_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1096
    ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1097
      using suminf_emeasure[OF _ **] A f
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1098
      by (auto simp: comp_def vimage_UN)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1099
  qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1100
  show "sigma_algebra (space N) (sets N)" ..
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1101
qed fact
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1102
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1103
lemma AE_distrD:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1104
  assumes f: "f \<in> measurable M M'"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1105
    and AE: "AE x in distr M M' f. P x"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1106
  shows "AE x in M. P (f x)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1107
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1108
  from AE[THEN AE_E] guess N .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1109
  with f show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1110
    unfolding eventually_ae_filter
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1111
    by (intro bexI[of _ "f -` N \<inter> space M"])
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1112
       (auto simp: emeasure_distr measurable_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1113
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  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
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1133
lemma null_sets_distr_iff:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  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"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1135
  by (auto simp add: null_sets_def emeasure_distr measurable_sets)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1136
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1137
lemma distr_distr:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1138
  assumes f: "g \<in> measurable N L" and g: "f \<in> measurable M N"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1139
  shows "distr (distr M N f) L g = distr M L (g \<circ> f)" (is "?L = ?R")
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1140
  using measurable_comp[OF g f] f g
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1141
  by (auto simp add: emeasure_distr measurable_sets measurable_space
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1142
           intro!: arg_cong[where f="emeasure M"] measure_eqI)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1143
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1144
section {* Real measure values *}
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1145
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1146
lemma measure_nonneg: "0 \<le> measure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1147
  using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1148
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1149
lemma measure_empty[simp]: "measure M {} = 0"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1150
  unfolding measure_def by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1151
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1152
lemma emeasure_eq_ereal_measure:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1153
  "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1154
  using emeasure_nonneg[of M A]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1155
  by (cases "emeasure M A") (auto simp: measure_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1156
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1157
lemma measure_Union:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1158
  assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1159
  and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1160
  shows "measure M (A \<union> B) = measure M A + measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1161
  unfolding measure_def
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1162
  using plus_emeasure[OF measurable, symmetric] finite
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1163
  by (simp add: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1164
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1165
lemma measure_finite_Union:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1166
  assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1167
  assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1168
  shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1169
  unfolding measure_def
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1170
  using setsum_emeasure[OF measurable, symmetric] finite
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1171
  by (simp add: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1172
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1173
lemma measure_Diff:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1174
  assumes finite: "emeasure M A \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1175
  and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1176
  shows "measure M (A - B) = measure M A - measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1177
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1178
  have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1179
    using measurable by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1180
  hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1181
    using measurable finite by (rule_tac measure_Union) auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1182
  thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1183
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1184
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1185
lemma measure_UNION:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1186
  assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1187
  assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1188
  shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1189
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1190
  from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1191
       suminf_emeasure[OF measurable] emeasure_nonneg[of M]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1192
  have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1193
  moreover
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1194
  { fix i
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1195
    have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1196
      using measurable by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1197
    then have "emeasure M (A i) = ereal ((measure M (A i)))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1198
      using finite by (intro emeasure_eq_ereal_measure) auto }
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1199
  ultimately show ?thesis using finite
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1200
    unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1201
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1202
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1203
lemma measure_subadditive:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1204
  assumes measurable: "A \<in> sets M" "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1205
  and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1206
  shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1207
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1208
  have "emeasure M (A \<union> B) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1209
    using emeasure_subadditive[OF measurable] fin by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1210
  then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1211
    using emeasure_subadditive[OF measurable] fin
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1212
    by (auto simp: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1213
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1214
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1215
lemma measure_subadditive_finite:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1216
  assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1217
  shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1218
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1219
  { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1220
      using emeasure_subadditive_finite[OF A] .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1221
    also have "\<dots> < \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1222
      using fin by (simp add: setsum_Pinfty)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1223
    finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp }
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1224
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1225
    using emeasure_subadditive_finite[OF A] fin
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1226
    unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1227
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1228
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1229
lemma measure_subadditive_countably:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1230
  assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1231
  shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1232
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1233
  from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1234
  moreover
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1235
  { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1236
      using emeasure_subadditive_countably[OF A] .
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1237
    also have "\<dots> < \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1238
      using fin by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1239
    finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp }
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1240
  ultimately  show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1241
    using emeasure_subadditive_countably[OF A] fin
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1242
    unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1243
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1244
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1245
lemma measure_eq_setsum_singleton:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1246
  assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1247
  and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1248
  shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1249
  unfolding measure_def
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1250
  using emeasure_eq_setsum_singleton[OF S] fin
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1251
  by simp (simp add: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1252
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1253
lemma Lim_measure_incseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1254
  assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1255
  shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1256
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1257
  have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1258
    using fin by (auto simp: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1259
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1260
    using Lim_emeasure_incseq[OF A]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1261
    unfolding measure_def
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1262
    by (intro lim_real_of_ereal) simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1263
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1264
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1265
lemma Lim_measure_decseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1266
  assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1267
  shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1268
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1269
  have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1270
    using A by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1271
  also have "\<dots> < \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1272
    using fin[of 0] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1273
  finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1274
    by (auto simp: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1275
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1276
    unfolding measure_def
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1277
    using Lim_emeasure_decseq[OF A fin]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1278
    by (intro lim_real_of_ereal) simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1279
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1280
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1281
section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1282
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1283
locale finite_measure = sigma_finite_measure M for M +
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1284
  assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1285
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1286
lemma finite_measureI[Pure.intro!]:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1287
  assumes *: "emeasure M (space M) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1288
  shows "finite_measure M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1289
proof
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  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>)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1291
    using * by (auto intro!: exI[of _ "\<lambda>_. space M"])
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1292
qed fact
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1293
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1294
lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1295
  using finite_emeasure_space emeasure_space[of M A] by auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1296
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1297
lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1298
  unfolding measure_def by (simp add: emeasure_eq_ereal_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1299
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1300
lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1301
  using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1302
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1303
lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1304
  using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1305
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1306
lemma (in finite_measure) finite_measure_Diff:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1307
  assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1308
  shows "measure M (A - B) = measure M A - measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1309
  using measure_Diff[OF _ assms] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1310
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1311
lemma (in finite_measure) finite_measure_Union:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1312
  assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1313
  shows "measure M (A \<union> B) = measure M A + measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1314
  using measure_Union[OF _ _ assms] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1315
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1316
lemma (in finite_measure) finite_measure_finite_Union:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1317
  assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1318
  shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1319
  using measure_finite_Union[OF assms] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1320
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1321
lemma (in finite_measure) finite_measure_UNION:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1322
  assumes A: "range A \<subseteq> sets M" "disjoint_family A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1323
  shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1324
  using measure_UNION[OF A] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1325
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1326
lemma (in finite_measure) finite_measure_mono:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1327
  assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1328
  using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1329
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1330
lemma (in finite_measure) finite_measure_subadditive:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1331
  assumes m: "A \<in> sets M" "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1332
  shows "measure M (A \<union> B) \<le> measure M A + measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1333
  using measure_subadditive[OF m] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1334
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1335
lemma (in finite_measure) finite_measure_subadditive_finite:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  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))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1337
  using measure_subadditive_finite[OF assms] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1338
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1339
lemma (in finite_measure) finite_measure_subadditive_countably:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1340
  assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1341
  shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1342
proof -
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1343
  from `summable (\<lambda>i. measure M (A i))`
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1344
  have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1345
    by (simp add: sums_ereal) (rule summable_sums)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1346
  from sums_unique[OF this, symmetric]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1347
       measure_subadditive_countably[OF A]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1348
  show ?thesis by (simp add: emeasure_eq_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1349
qed
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1350
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1351
lemma (in finite_measure) finite_measure_eq_setsum_singleton:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1352
  assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1353
  shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1354
  using measure_eq_setsum_singleton[OF assms] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1355
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1356
lemma (in finite_measure) finite_Lim_measure_incseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1357
  assumes A: "range A \<subseteq> sets M" "incseq A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1358
  shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1359
  using Lim_measure_incseq[OF A] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1360
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1361
lemma (in finite_measure) finite_Lim_measure_decseq:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1362
  assumes A: "range A \<subseteq> sets M" "decseq A"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1363
  shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1364
  using Lim_measure_decseq[OF A] by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1365
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1366
lemma (in finite_measure) finite_measure_compl:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1367
  assumes S: "S \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1368
  shows "measure M (space M - S) = measure M (space M) - measure M S"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1369
  using measure_Diff[OF _ top S sets_into_space] S by simp
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1370
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1371
lemma (in finite_measure) finite_measure_mono_AE:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1372
  assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1373
  shows "measure M A \<le> measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1374
  using assms emeasure_mono_AE[OF imp B]
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1375
  by (simp add: emeasure_eq_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1376
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1377
lemma (in finite_measure) finite_measure_eq_AE:
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1378
  assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1379
  assumes A: "A \<in> sets M" and B: "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1380
  shows "measure M A = measure M B"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1381
  using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1382
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1383
section {* Counting space *}
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1384
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1385
definition count_space :: "'a set \<Rightarrow> 'a measure" where
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1386
  "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1387
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1388
lemma 
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1389
  shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1390
    and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
05663f75964c reworked Probability theory
hoelzl
parents:
diff changeset
  1391
  using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
05663f75964c reworked Probability theory
h