src/HOL/Probability/Sigma_Algebra.thy
author hoelzl
Tue May 20 19:24:39 2014 +0200 (2014-05-20)
changeset 57025 e7fd64f82876
parent 56994 8d5e5ec1cac3
child 57137 f174712d0a84
permissions -rw-r--r--
add various lemmas
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(*  Title:      HOL/Probability/Sigma_Algebra.thy
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    Author:     Stefan Richter, Markus Wenzel, TU München
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    Author:     Johannes Hölzl, TU München
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    Plus material from the Hurd/Coble measure theory development,
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    translated by Lawrence Paulson.
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*)
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header {* Describing measurable sets *}
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theory Sigma_Algebra
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imports
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  Complex_Main
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  "~~/src/HOL/Library/Countable_Set"
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  "~~/src/HOL/Library/FuncSet"
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  "~~/src/HOL/Library/Indicator_Function"
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  "~~/src/HOL/Library/Extended_Real"
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begin
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text {* Sigma algebras are an elementary concept in measure
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  theory. To measure --- that is to integrate --- functions, we first have
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  to measure sets. Unfortunately, when dealing with a large universe,
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  it is often not possible to consistently assign a measure to every
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  subset. Therefore it is necessary to define the set of measurable
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  subsets of the universe. A sigma algebra is such a set that has
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  three very natural and desirable properties. *}
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subsection {* Families of sets *}
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locale subset_class =
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  fixes \<Omega> :: "'a set" and M :: "'a set set"
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  assumes space_closed: "M \<subseteq> Pow \<Omega>"
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lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
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  by (metis PowD contra_subsetD space_closed)
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subsubsection {* Semiring of sets *}
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definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
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lemma disjointI:
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  "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"
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  unfolding disjoint_def by auto
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lemma disjointD:
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  "disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"
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  unfolding disjoint_def by auto
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lemma disjoint_empty[iff]: "disjoint {}"
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  by (auto simp: disjoint_def)
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lemma disjoint_union: 
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  assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}"
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  shows "disjoint (C \<union> B)"
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proof (rule disjointI)
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  fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d"
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  show "c \<inter> d = {}"
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  proof cases
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    assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)"
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    then show ?thesis
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    proof 
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      assume "c \<in> C \<and> d \<in> C" with `c \<noteq> d` C show "c \<inter> d = {}"
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        by (auto simp: disjoint_def)
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    next
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      assume "c \<in> B \<and> d \<in> B" with `c \<noteq> d` B show "c \<inter> d = {}"
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        by (auto simp: disjoint_def)
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    qed
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  next
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    assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))"
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    with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)"
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      by auto
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    with disj show "c \<inter> d = {}" by auto
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  qed
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qed
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lemma disjoint_singleton [simp]: "disjoint {A}"
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by(simp add: disjoint_def)
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locale semiring_of_sets = subset_class +
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  assumes empty_sets[iff]: "{} \<in> M"
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  assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
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  assumes Diff_cover:
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    "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> \<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
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lemma (in semiring_of_sets) finite_INT[intro]:
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  assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
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  shows "(\<Inter>i\<in>I. A i) \<in> M"
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  using assms by (induct rule: finite_ne_induct) auto
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lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
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  by (metis Int_absorb1 sets_into_space)
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lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
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  by (metis Int_absorb2 sets_into_space)
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lemma (in semiring_of_sets) sets_Collect_conj:
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  assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
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  shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
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proof -
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  have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
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    by auto
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  with assms show ?thesis by auto
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qed
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lemma (in semiring_of_sets) sets_Collect_finite_All':
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  assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
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  shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
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proof -
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  have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
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    using `S \<noteq> {}` by auto
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  with assms show ?thesis by auto
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qed
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locale ring_of_sets = semiring_of_sets +
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  assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
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lemma (in ring_of_sets) finite_Union [intro]:
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  "finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M"
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  by (induct set: finite) (auto simp add: Un)
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lemma (in ring_of_sets) finite_UN[intro]:
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  assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
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  shows "(\<Union>i\<in>I. A i) \<in> M"
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  using assms by induct auto
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lemma (in ring_of_sets) Diff [intro]:
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  assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"
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  using Diff_cover[OF assms] by auto
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lemma ring_of_setsI:
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  assumes space_closed: "M \<subseteq> Pow \<Omega>"
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  assumes empty_sets[iff]: "{} \<in> M"
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  assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
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  assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
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  shows "ring_of_sets \<Omega> M"
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proof
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  fix a b assume ab: "a \<in> M" "b \<in> M"
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  from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
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    by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
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  have "a \<inter> b = a - (a - b)" by auto
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  also have "\<dots> \<in> M" using ab by auto
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  finally show "a \<inter> b \<in> M" .
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qed fact+
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lemma ring_of_sets_iff: "ring_of_sets \<Omega> M \<longleftrightarrow> M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"
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proof
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  assume "ring_of_sets \<Omega> M"
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  then interpret ring_of_sets \<Omega> M .
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  show "M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"
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    using space_closed by auto
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qed (auto intro!: ring_of_setsI)
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lemma (in ring_of_sets) insert_in_sets:
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  assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
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proof -
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  have "{x} \<union> A \<in> M" using assms by (rule Un)
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  thus ?thesis by auto
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qed
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lemma (in ring_of_sets) sets_Collect_disj:
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  assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
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  shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
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proof -
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  have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
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    by auto
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  with assms show ?thesis by auto
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qed
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lemma (in ring_of_sets) sets_Collect_finite_Ex:
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  assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
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  shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
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proof -
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  have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
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    by auto
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  with assms show ?thesis by auto
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qed
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locale algebra = ring_of_sets +
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  assumes top [iff]: "\<Omega> \<in> M"
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lemma (in algebra) compl_sets [intro]:
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  "a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
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  by auto
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lemma algebra_iff_Un:
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  "algebra \<Omega> M \<longleftrightarrow>
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    M \<subseteq> Pow \<Omega> \<and>
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    {} \<in> M \<and>
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    (\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
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    (\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
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proof
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  assume "algebra \<Omega> M"
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  then interpret algebra \<Omega> M .
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  show ?Un using sets_into_space by auto
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next
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  assume ?Un
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  then have "\<Omega> \<in> M" by auto
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  interpret ring_of_sets \<Omega> M
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  proof (rule ring_of_setsI)
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    show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
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      using `?Un` by auto
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    fix a b assume a: "a \<in> M" and b: "b \<in> M"
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    then show "a \<union> b \<in> M" using `?Un` by auto
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    have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
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      using \<Omega> a b by auto
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    then show "a - b \<in> M"
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      using a b  `?Un` by auto
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  qed
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  show "algebra \<Omega> M" proof qed fact
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qed
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lemma algebra_iff_Int:
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     "algebra \<Omega> M \<longleftrightarrow>
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       M \<subseteq> Pow \<Omega> & {} \<in> M &
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       (\<forall>a \<in> M. \<Omega> - a \<in> M) &
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       (\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
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proof
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  assume "algebra \<Omega> M"
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  then interpret algebra \<Omega> M .
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  show ?Int using sets_into_space by auto
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next
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  assume ?Int
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  show "algebra \<Omega> M"
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  proof (unfold algebra_iff_Un, intro conjI ballI)
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    show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
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      using `?Int` by auto
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    from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
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    fix a b assume M: "a \<in> M" "b \<in> M"
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    hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
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      using \<Omega> by blast
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    also have "... \<in> M"
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      using M `?Int` by auto
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    finally show "a \<union> b \<in> M" .
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  qed
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qed
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lemma (in algebra) sets_Collect_neg:
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  assumes "{x\<in>\<Omega>. P x} \<in> M"
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  shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
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proof -
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  have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
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  with assms show ?thesis by auto
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qed
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lemma (in algebra) sets_Collect_imp:
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  "{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M"
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  unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)
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lemma (in algebra) sets_Collect_const:
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  "{x\<in>\<Omega>. P} \<in> M"
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  by (cases P) auto
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lemma algebra_single_set:
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  "X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
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  by (auto simp: algebra_iff_Int)
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subsubsection {* Restricted algebras *}
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abbreviation (in algebra)
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  "restricted_space A \<equiv> (op \<inter> A) ` M"
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lemma (in algebra) restricted_algebra:
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  assumes "A \<in> M" shows "algebra A (restricted_space A)"
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  using assms by (auto simp: algebra_iff_Int)
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subsubsection {* Sigma Algebras *}
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locale sigma_algebra = algebra +
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  assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
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lemma (in algebra) is_sigma_algebra:
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  assumes "finite M"
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  shows "sigma_algebra \<Omega> M"
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proof
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  fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
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  then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
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    by auto
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  also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
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    using `finite M` by auto
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  finally show "(\<Union>i. A i) \<in> M" .
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qed
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lemma countable_UN_eq:
hoelzl@38656
   283
  fixes A :: "'i::countable \<Rightarrow> 'a set"
hoelzl@47694
   284
  shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
hoelzl@47694
   285
    (range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
hoelzl@38656
   286
proof -
hoelzl@38656
   287
  let ?A' = "A \<circ> from_nat"
hoelzl@38656
   288
  have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
hoelzl@38656
   289
  proof safe
hoelzl@38656
   290
    fix x i assume "x \<in> A i" thus "x \<in> ?l"
hoelzl@38656
   291
      by (auto intro!: exI[of _ "to_nat i"])
hoelzl@38656
   292
  next
hoelzl@38656
   293
    fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"
hoelzl@38656
   294
      by (auto intro!: exI[of _ "from_nat i"])
hoelzl@38656
   295
  qed
hoelzl@38656
   296
  have **: "range ?A' = range A"
hoelzl@40702
   297
    using surj_from_nat
haftmann@56154
   298
    by (auto simp: image_comp [symmetric] intro!: imageI)
hoelzl@38656
   299
  show ?thesis unfolding * ** ..
hoelzl@38656
   300
qed
hoelzl@38656
   301
immler@50245
   302
lemma (in sigma_algebra) countable_Union [intro]:
immler@50245
   303
  assumes "countable X" "X \<subseteq> M" shows "Union X \<in> M"
immler@50245
   304
proof cases
immler@50245
   305
  assume "X \<noteq> {}"
immler@50245
   306
  hence "\<Union>X = (\<Union>n. from_nat_into X n)"
immler@50245
   307
    using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
immler@50245
   308
  also have "\<dots> \<in> M" using assms
immler@50245
   309
    by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp)
immler@50245
   310
  finally show ?thesis .
immler@50245
   311
qed simp
immler@50245
   312
hoelzl@38656
   313
lemma (in sigma_algebra) countable_UN[intro]:
hoelzl@38656
   314
  fixes A :: "'i::countable \<Rightarrow> 'a set"
hoelzl@47694
   315
  assumes "A`X \<subseteq> M"
hoelzl@47694
   316
  shows  "(\<Union>x\<in>X. A x) \<in> M"
hoelzl@38656
   317
proof -
wenzelm@46731
   318
  let ?A = "\<lambda>i. if i \<in> X then A i else {}"
hoelzl@47694
   319
  from assms have "range ?A \<subseteq> M" by auto
hoelzl@38656
   320
  with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
hoelzl@47694
   321
  have "(\<Union>x. ?A x) \<in> M" by auto
hoelzl@38656
   322
  moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
hoelzl@38656
   323
  ultimately show ?thesis by simp
hoelzl@38656
   324
qed
hoelzl@38656
   325
hoelzl@50526
   326
lemma (in sigma_algebra) countable_UN':
hoelzl@50526
   327
  fixes A :: "'i \<Rightarrow> 'a set"
hoelzl@50526
   328
  assumes X: "countable X"
hoelzl@50526
   329
  assumes A: "A`X \<subseteq> M"
hoelzl@50526
   330
  shows  "(\<Union>x\<in>X. A x) \<in> M"
hoelzl@50526
   331
proof -
hoelzl@50526
   332
  have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
hoelzl@50526
   333
    using X by auto
hoelzl@50526
   334
  also have "\<dots> \<in> M"
hoelzl@50526
   335
    using A X
hoelzl@50526
   336
    by (intro countable_UN) auto
hoelzl@50526
   337
  finally show ?thesis .
hoelzl@50526
   338
qed
hoelzl@50526
   339
paulson@33533
   340
lemma (in sigma_algebra) countable_INT [intro]:
hoelzl@38656
   341
  fixes A :: "'i::countable \<Rightarrow> 'a set"
hoelzl@47694
   342
  assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
hoelzl@47694
   343
  shows "(\<Inter>i\<in>X. A i) \<in> M"
paulson@33271
   344
proof -
hoelzl@47694
   345
  from A have "\<forall>i\<in>X. A i \<in> M" by fast
hoelzl@47694
   346
  hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
paulson@33271
   347
  moreover
hoelzl@47694
   348
  have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
paulson@33271
   349
    by blast
paulson@33271
   350
  ultimately show ?thesis by metis
paulson@33271
   351
qed
paulson@33271
   352
hoelzl@50526
   353
lemma (in sigma_algebra) countable_INT':
hoelzl@50526
   354
  fixes A :: "'i \<Rightarrow> 'a set"
hoelzl@50526
   355
  assumes X: "countable X" "X \<noteq> {}"
hoelzl@50526
   356
  assumes A: "A`X \<subseteq> M"
hoelzl@50526
   357
  shows  "(\<Inter>x\<in>X. A x) \<in> M"
hoelzl@50526
   358
proof -
hoelzl@50526
   359
  have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
hoelzl@50526
   360
    using X by auto
hoelzl@50526
   361
  also have "\<dots> \<in> M"
hoelzl@50526
   362
    using A X
hoelzl@50526
   363
    by (intro countable_INT) auto
hoelzl@50526
   364
  finally show ?thesis .
hoelzl@50526
   365
qed
hoelzl@50526
   366
hoelzl@47694
   367
lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
hoelzl@47762
   368
  by (auto simp: ring_of_sets_iff)
hoelzl@42145
   369
hoelzl@47694
   370
lemma algebra_Pow: "algebra sp (Pow sp)"
hoelzl@47762
   371
  by (auto simp: algebra_iff_Un)
hoelzl@38656
   372
hoelzl@38656
   373
lemma sigma_algebra_iff:
hoelzl@47694
   374
  "sigma_algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694
   375
    algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
   376
  by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
paulson@33271
   377
hoelzl@47762
   378
lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
hoelzl@47762
   379
  by (auto simp: sigma_algebra_iff algebra_iff_Int)
hoelzl@47762
   380
hoelzl@42867
   381
lemma (in sigma_algebra) sets_Collect_countable_All:
hoelzl@47694
   382
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   383
  shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
hoelzl@42867
   384
proof -
hoelzl@47694
   385
  have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
hoelzl@42867
   386
  with assms show ?thesis by auto
hoelzl@42867
   387
qed
hoelzl@42867
   388
hoelzl@42867
   389
lemma (in sigma_algebra) sets_Collect_countable_Ex:
hoelzl@47694
   390
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   391
  shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
hoelzl@42867
   392
proof -
hoelzl@47694
   393
  have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
hoelzl@42867
   394
  with assms show ?thesis by auto
hoelzl@42867
   395
qed
hoelzl@42867
   396
hoelzl@50526
   397
lemma (in sigma_algebra) sets_Collect_countable_Ex':
hoelzl@54418
   398
  assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@50526
   399
  assumes "countable I"
hoelzl@50526
   400
  shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
hoelzl@50526
   401
proof -
hoelzl@50526
   402
  have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto
hoelzl@50526
   403
  with assms show ?thesis 
hoelzl@50526
   404
    by (auto intro!: countable_UN')
hoelzl@50526
   405
qed
hoelzl@50526
   406
hoelzl@54418
   407
lemma (in sigma_algebra) sets_Collect_countable_All':
hoelzl@54418
   408
  assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@54418
   409
  assumes "countable I"
hoelzl@54418
   410
  shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M"
hoelzl@54418
   411
proof -
hoelzl@54418
   412
  have "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} = (\<Inter>i\<in>I. {x\<in>\<Omega>. P i x}) \<inter> \<Omega>" by auto
hoelzl@54418
   413
  with assms show ?thesis 
hoelzl@54418
   414
    by (cases "I = {}") (auto intro!: countable_INT')
hoelzl@54418
   415
qed
hoelzl@54418
   416
hoelzl@54418
   417
lemma (in sigma_algebra) sets_Collect_countable_Ex1':
hoelzl@54418
   418
  assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@54418
   419
  assumes "countable I"
hoelzl@54418
   420
  shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M"
hoelzl@54418
   421
proof -
hoelzl@54418
   422
  have "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} = {x\<in>\<Omega>. \<exists>i\<in>I. P i x \<and> (\<forall>j\<in>I. P j x \<longrightarrow> i = j)}"
hoelzl@54418
   423
    by auto
hoelzl@54418
   424
  with assms show ?thesis 
hoelzl@54418
   425
    by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)
hoelzl@54418
   426
qed
hoelzl@54418
   427
hoelzl@42867
   428
lemmas (in sigma_algebra) sets_Collect =
hoelzl@42867
   429
  sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
hoelzl@42867
   430
  sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
hoelzl@42867
   431
hoelzl@47694
   432
lemma (in sigma_algebra) sets_Collect_countable_Ball:
hoelzl@47694
   433
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   434
  shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
hoelzl@47694
   435
  unfolding Ball_def by (intro sets_Collect assms)
hoelzl@47694
   436
hoelzl@47694
   437
lemma (in sigma_algebra) sets_Collect_countable_Bex:
hoelzl@47694
   438
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   439
  shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
hoelzl@47694
   440
  unfolding Bex_def by (intro sets_Collect assms)
hoelzl@47694
   441
hoelzl@42984
   442
lemma sigma_algebra_single_set:
hoelzl@42984
   443
  assumes "X \<subseteq> S"
hoelzl@47694
   444
  shows "sigma_algebra S { {}, X, S - X, S }"
hoelzl@42984
   445
  using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
hoelzl@42984
   446
hoelzl@56994
   447
subsubsection {* Binary Unions *}
paulson@33271
   448
paulson@33271
   449
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
wenzelm@50252
   450
  where "binary a b =  (\<lambda>x. b)(0 := a)"
paulson@33271
   451
hoelzl@38656
   452
lemma range_binary_eq: "range(binary a b) = {a,b}"
hoelzl@38656
   453
  by (auto simp add: binary_def)
paulson@33271
   454
hoelzl@38656
   455
lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
haftmann@44106
   456
  by (simp add: SUP_def range_binary_eq)
paulson@33271
   457
hoelzl@38656
   458
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
haftmann@44106
   459
  by (simp add: INF_def range_binary_eq)
paulson@33271
   460
paulson@33271
   461
lemma sigma_algebra_iff2:
hoelzl@47694
   462
     "sigma_algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694
   463
       M \<subseteq> Pow \<Omega> \<and>
hoelzl@47694
   464
       {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
hoelzl@47694
   465
       (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
   466
  by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
hoelzl@42065
   467
         algebra_iff_Un Un_range_binary)
paulson@33271
   468
hoelzl@56994
   469
subsubsection {* Initial Sigma Algebra *}
paulson@33271
   470
paulson@33271
   471
text {*Sigma algebras can naturally be created as the closure of any set of
hoelzl@47694
   472
  M with regard to the properties just postulated.  *}
paulson@33271
   473
hoelzl@51683
   474
inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
paulson@33271
   475
  for sp :: "'a set" and A :: "'a set set"
paulson@33271
   476
  where
hoelzl@47694
   477
    Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
paulson@33271
   478
  | Empty: "{} \<in> sigma_sets sp A"
paulson@33271
   479
  | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
paulson@33271
   480
  | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
paulson@33271
   481
hoelzl@41543
   482
lemma (in sigma_algebra) sigma_sets_subset:
hoelzl@47694
   483
  assumes a: "a \<subseteq> M"
hoelzl@47694
   484
  shows "sigma_sets \<Omega> a \<subseteq> M"
hoelzl@41543
   485
proof
hoelzl@41543
   486
  fix x
hoelzl@47694
   487
  assume "x \<in> sigma_sets \<Omega> a"
hoelzl@47694
   488
  from this show "x \<in> M"
hoelzl@41543
   489
    by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
hoelzl@41543
   490
qed
hoelzl@41543
   491
hoelzl@41543
   492
lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
hoelzl@41543
   493
  by (erule sigma_sets.induct, auto)
hoelzl@41543
   494
hoelzl@41543
   495
lemma sigma_algebra_sigma_sets:
hoelzl@47694
   496
     "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
hoelzl@41543
   497
  by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
hoelzl@41543
   498
           intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
hoelzl@41543
   499
hoelzl@41543
   500
lemma sigma_sets_least_sigma_algebra:
hoelzl@41543
   501
  assumes "A \<subseteq> Pow S"
hoelzl@47694
   502
  shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
hoelzl@41543
   503
proof safe
hoelzl@47694
   504
  fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
hoelzl@41543
   505
    and X: "X \<in> sigma_sets S A"
hoelzl@41543
   506
  from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
hoelzl@41543
   507
  show "X \<in> B" by auto
hoelzl@41543
   508
next
hoelzl@47694
   509
  fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
hoelzl@47694
   510
  then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
hoelzl@41543
   511
     by simp
hoelzl@47694
   512
  have "A \<subseteq> sigma_sets S A" using assms by auto
hoelzl@47694
   513
  moreover have "sigma_algebra S (sigma_sets S A)"
hoelzl@41543
   514
    using assms by (intro sigma_algebra_sigma_sets[of A]) auto
hoelzl@41543
   515
  ultimately show "X \<in> sigma_sets S A" by auto
hoelzl@41543
   516
qed
paulson@33271
   517
paulson@33271
   518
lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
paulson@33271
   519
  by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
paulson@33271
   520
hoelzl@38656
   521
lemma sigma_sets_Un:
paulson@33271
   522
  "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
hoelzl@38656
   523
apply (simp add: Un_range_binary range_binary_eq)
hoelzl@40859
   524
apply (rule Union, simp add: binary_def)
paulson@33271
   525
done
paulson@33271
   526
paulson@33271
   527
lemma sigma_sets_Inter:
paulson@33271
   528
  assumes Asb: "A \<subseteq> Pow sp"
paulson@33271
   529
  shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
paulson@33271
   530
proof -
paulson@33271
   531
  assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
hoelzl@38656
   532
  hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
paulson@33271
   533
    by (rule sigma_sets.Compl)
hoelzl@38656
   534
  hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
paulson@33271
   535
    by (rule sigma_sets.Union)
hoelzl@38656
   536
  hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
paulson@33271
   537
    by (rule sigma_sets.Compl)
hoelzl@38656
   538
  also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
paulson@33271
   539
    by auto
paulson@33271
   540
  also have "... = (\<Inter>i. a i)" using ai
hoelzl@38656
   541
    by (blast dest: sigma_sets_into_sp [OF Asb])
hoelzl@38656
   542
  finally show ?thesis .
paulson@33271
   543
qed
paulson@33271
   544
paulson@33271
   545
lemma sigma_sets_INTER:
hoelzl@38656
   546
  assumes Asb: "A \<subseteq> Pow sp"
paulson@33271
   547
      and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
paulson@33271
   548
  shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
paulson@33271
   549
proof -
paulson@33271
   550
  from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
hoelzl@47694
   551
    by (simp add: sigma_sets.intros(2-) sigma_sets_top)
paulson@33271
   552
  hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
paulson@33271
   553
    by (rule sigma_sets_Inter [OF Asb])
paulson@33271
   554
  also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
paulson@33271
   555
    by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
paulson@33271
   556
  finally show ?thesis .
paulson@33271
   557
qed
paulson@33271
   558
hoelzl@51683
   559
lemma sigma_sets_UNION: "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"
hoelzl@51683
   560
  using from_nat_into[of B] range_from_nat_into[of B] sigma_sets.Union[of "from_nat_into B" X A]
hoelzl@51683
   561
  apply (cases "B = {}")
hoelzl@51683
   562
  apply (simp add: sigma_sets.Empty)
hoelzl@51683
   563
  apply (simp del: Union_image_eq add: Union_image_eq[symmetric])
hoelzl@51683
   564
  done
hoelzl@51683
   565
paulson@33271
   566
lemma (in sigma_algebra) sigma_sets_eq:
hoelzl@47694
   567
     "sigma_sets \<Omega> M = M"
paulson@33271
   568
proof
hoelzl@47694
   569
  show "M \<subseteq> sigma_sets \<Omega> M"
huffman@37032
   570
    by (metis Set.subsetI sigma_sets.Basic)
paulson@33271
   571
  next
hoelzl@47694
   572
  show "sigma_sets \<Omega> M \<subseteq> M"
paulson@33271
   573
    by (metis sigma_sets_subset subset_refl)
paulson@33271
   574
qed
paulson@33271
   575
hoelzl@42981
   576
lemma sigma_sets_eqI:
hoelzl@42981
   577
  assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
hoelzl@42981
   578
  assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
hoelzl@42981
   579
  shows "sigma_sets M A = sigma_sets M B"
hoelzl@42981
   580
proof (intro set_eqI iffI)
hoelzl@42981
   581
  fix a assume "a \<in> sigma_sets M A"
hoelzl@42981
   582
  from this A show "a \<in> sigma_sets M B"
hoelzl@47694
   583
    by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
hoelzl@42981
   584
next
hoelzl@42981
   585
  fix b assume "b \<in> sigma_sets M B"
hoelzl@42981
   586
  from this B show "b \<in> sigma_sets M A"
hoelzl@47694
   587
    by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
hoelzl@42981
   588
qed
hoelzl@42981
   589
hoelzl@42984
   590
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@42984
   591
proof
hoelzl@42984
   592
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47694
   593
    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
hoelzl@42984
   594
qed
hoelzl@42984
   595
hoelzl@47762
   596
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@47762
   597
proof
hoelzl@47762
   598
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47762
   599
    by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
hoelzl@47762
   600
qed
hoelzl@47762
   601
hoelzl@47762
   602
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@47762
   603
proof
hoelzl@47762
   604
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47762
   605
    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
hoelzl@47762
   606
qed
hoelzl@47762
   607
hoelzl@47762
   608
lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
hoelzl@47762
   609
  by (auto intro: sigma_sets.Basic)
hoelzl@47762
   610
hoelzl@38656
   611
lemma (in sigma_algebra) restriction_in_sets:
hoelzl@38656
   612
  fixes A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
   613
  assumes "S \<in> M"
hoelzl@47694
   614
  and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
hoelzl@47694
   615
  shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
hoelzl@38656
   616
proof -
hoelzl@38656
   617
  { fix i have "A i \<in> ?r" using * by auto
hoelzl@47694
   618
    hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
hoelzl@47694
   619
    hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
hoelzl@47694
   620
  thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
hoelzl@38656
   621
    by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
hoelzl@38656
   622
qed
hoelzl@38656
   623
hoelzl@38656
   624
lemma (in sigma_algebra) restricted_sigma_algebra:
hoelzl@47694
   625
  assumes "S \<in> M"
hoelzl@47694
   626
  shows "sigma_algebra S (restricted_space S)"
hoelzl@38656
   627
  unfolding sigma_algebra_def sigma_algebra_axioms_def
hoelzl@38656
   628
proof safe
hoelzl@47694
   629
  show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
hoelzl@38656
   630
next
hoelzl@47694
   631
  fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
hoelzl@38656
   632
  from restriction_in_sets[OF assms this[simplified]]
hoelzl@47694
   633
  show "(\<Union>i. A i) \<in> restricted_space S" by simp
hoelzl@38656
   634
qed
hoelzl@38656
   635
hoelzl@40859
   636
lemma sigma_sets_Int:
hoelzl@41689
   637
  assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
hoelzl@41689
   638
  shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
hoelzl@40859
   639
proof (intro equalityI subsetI)
hoelzl@40859
   640
  fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
hoelzl@40859
   641
  then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
hoelzl@41689
   642
  then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
hoelzl@40859
   643
  proof (induct arbitrary: x)
hoelzl@40859
   644
    case (Compl a)
hoelzl@40859
   645
    then show ?case
hoelzl@40859
   646
      by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
hoelzl@40859
   647
  next
hoelzl@40859
   648
    case (Union a)
hoelzl@40859
   649
    then show ?case
hoelzl@40859
   650
      by (auto intro!: sigma_sets.Union
hoelzl@40859
   651
               simp add: UN_extend_simps simp del: UN_simps)
hoelzl@47694
   652
  qed (auto intro!: sigma_sets.intros(2-))
hoelzl@41689
   653
  then show "x \<in> sigma_sets A (op \<inter> A ` st)"
hoelzl@41689
   654
    using `A \<subseteq> sp` by (simp add: Int_absorb2)
hoelzl@40859
   655
next
hoelzl@41689
   656
  fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
hoelzl@40859
   657
  then show "x \<in> op \<inter> A ` sigma_sets sp st"
hoelzl@40859
   658
  proof induct
hoelzl@40859
   659
    case (Compl a)
hoelzl@40859
   660
    then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
hoelzl@41689
   661
    then show ?case using `A \<subseteq> sp`
hoelzl@40859
   662
      by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
hoelzl@40859
   663
  next
hoelzl@40859
   664
    case (Union a)
hoelzl@40859
   665
    then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
hoelzl@40859
   666
      by (auto simp: image_iff Bex_def)
hoelzl@40859
   667
    from choice[OF this] guess f ..
hoelzl@40859
   668
    then show ?case
hoelzl@40859
   669
      by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
hoelzl@40859
   670
               simp add: image_iff)
hoelzl@47694
   671
  qed (auto intro!: sigma_sets.intros(2-))
hoelzl@40859
   672
qed
hoelzl@40859
   673
hoelzl@47694
   674
lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
hoelzl@40859
   675
proof (intro set_eqI iffI)
hoelzl@47694
   676
  fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
hoelzl@47694
   677
    by induct blast+
hoelzl@47694
   678
qed (auto intro: sigma_sets.Empty sigma_sets_top)
hoelzl@47694
   679
hoelzl@47694
   680
lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
hoelzl@47694
   681
proof (intro set_eqI iffI)
hoelzl@47694
   682
  fix x assume "x \<in> sigma_sets A {A}"
hoelzl@47694
   683
  then show "x \<in> {{}, A}"
hoelzl@47694
   684
    by induct blast+
hoelzl@40859
   685
next
hoelzl@47694
   686
  fix x assume "x \<in> {{}, A}"
hoelzl@47694
   687
  then show "x \<in> sigma_sets A {A}"
hoelzl@40859
   688
    by (auto intro: sigma_sets.Empty sigma_sets_top)
hoelzl@40859
   689
qed
hoelzl@40859
   690
hoelzl@42987
   691
lemma sigma_sets_sigma_sets_eq:
hoelzl@42987
   692
  "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
hoelzl@47694
   693
  by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
hoelzl@42987
   694
hoelzl@42984
   695
lemma sigma_sets_singleton:
hoelzl@42984
   696
  assumes "X \<subseteq> S"
hoelzl@42984
   697
  shows "sigma_sets S { X } = { {}, X, S - X, S }"
hoelzl@42984
   698
proof -
hoelzl@47694
   699
  interpret sigma_algebra S "{ {}, X, S - X, S }"
hoelzl@42984
   700
    by (rule sigma_algebra_single_set) fact
hoelzl@42984
   701
  have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
hoelzl@42984
   702
    by (rule sigma_sets_subseteq) simp
hoelzl@42984
   703
  moreover have "\<dots> = { {}, X, S - X, S }"
hoelzl@47694
   704
    using sigma_sets_eq by simp
hoelzl@42984
   705
  moreover
hoelzl@42984
   706
  { fix A assume "A \<in> { {}, X, S - X, S }"
hoelzl@42984
   707
    then have "A \<in> sigma_sets S { X }"
hoelzl@47694
   708
      by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
hoelzl@42984
   709
  ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
hoelzl@42984
   710
    by (intro antisym) auto
hoelzl@47694
   711
  with sigma_sets_eq show ?thesis by simp
hoelzl@42984
   712
qed
hoelzl@42984
   713
hoelzl@42863
   714
lemma restricted_sigma:
hoelzl@47694
   715
  assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
hoelzl@47694
   716
  shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
hoelzl@47694
   717
    sigma_sets S (algebra.restricted_space M S)"
hoelzl@42863
   718
proof -
hoelzl@42863
   719
  from S sigma_sets_into_sp[OF M]
hoelzl@47694
   720
  have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
hoelzl@42863
   721
  from sigma_sets_Int[OF this]
hoelzl@47694
   722
  show ?thesis by simp
hoelzl@42863
   723
qed
hoelzl@42863
   724
hoelzl@42987
   725
lemma sigma_sets_vimage_commute:
hoelzl@47694
   726
  assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
hoelzl@47694
   727
  shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
hoelzl@47694
   728
       = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
hoelzl@42987
   729
proof
hoelzl@42987
   730
  show "?L \<subseteq> ?R"
hoelzl@42987
   731
  proof clarify
hoelzl@47694
   732
    fix A assume "A \<in> sigma_sets \<Omega>' M'"
hoelzl@47694
   733
    then show "X -` A \<inter> \<Omega> \<in> ?R"
hoelzl@42987
   734
    proof induct
hoelzl@42987
   735
      case Empty then show ?case
hoelzl@42987
   736
        by (auto intro!: sigma_sets.Empty)
hoelzl@42987
   737
    next
hoelzl@42987
   738
      case (Compl B)
hoelzl@47694
   739
      have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
hoelzl@42987
   740
        by (auto simp add: funcset_mem [OF X])
hoelzl@42987
   741
      with Compl show ?case
hoelzl@42987
   742
        by (auto intro!: sigma_sets.Compl)
hoelzl@42987
   743
    next
hoelzl@42987
   744
      case (Union F)
hoelzl@42987
   745
      then show ?case
hoelzl@42987
   746
        by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
hoelzl@42987
   747
                 intro!: sigma_sets.Union)
hoelzl@47694
   748
    qed auto
hoelzl@42987
   749
  qed
hoelzl@42987
   750
  show "?R \<subseteq> ?L"
hoelzl@42987
   751
  proof clarify
hoelzl@42987
   752
    fix A assume "A \<in> ?R"
hoelzl@47694
   753
    then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
hoelzl@42987
   754
    proof induct
hoelzl@42987
   755
      case (Basic B) then show ?case by auto
hoelzl@42987
   756
    next
hoelzl@42987
   757
      case Empty then show ?case
hoelzl@47694
   758
        by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
hoelzl@42987
   759
    next
hoelzl@42987
   760
      case (Compl B)
hoelzl@47694
   761
      then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
hoelzl@47694
   762
      then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
hoelzl@42987
   763
        by (auto simp add: funcset_mem [OF X])
hoelzl@42987
   764
      with A(2) show ?case
hoelzl@47694
   765
        by (auto intro: sigma_sets.Compl)
hoelzl@42987
   766
    next
hoelzl@42987
   767
      case (Union F)
hoelzl@47694
   768
      then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
hoelzl@42987
   769
      from choice[OF this] guess A .. note A = this
hoelzl@42987
   770
      with A show ?case
hoelzl@47694
   771
        by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
hoelzl@42987
   772
    qed
hoelzl@42987
   773
  qed
hoelzl@42987
   774
qed
hoelzl@42987
   775
hoelzl@56994
   776
subsubsection "Disjoint families"
hoelzl@38656
   777
hoelzl@38656
   778
definition
hoelzl@38656
   779
  disjoint_family_on  where
hoelzl@38656
   780
  "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
hoelzl@38656
   781
hoelzl@38656
   782
abbreviation
hoelzl@38656
   783
  "disjoint_family A \<equiv> disjoint_family_on A UNIV"
hoelzl@38656
   784
hoelzl@38656
   785
lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
hoelzl@38656
   786
  by blast
hoelzl@38656
   787
hoelzl@38656
   788
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
hoelzl@38656
   789
  by blast
hoelzl@38656
   790
hoelzl@38656
   791
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
hoelzl@38656
   792
  by blast
hoelzl@38656
   793
hoelzl@38656
   794
lemma disjoint_family_subset:
hoelzl@38656
   795
     "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
hoelzl@38656
   796
  by (force simp add: disjoint_family_on_def)
hoelzl@38656
   797
hoelzl@40859
   798
lemma disjoint_family_on_bisimulation:
hoelzl@40859
   799
  assumes "disjoint_family_on f S"
hoelzl@40859
   800
  and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}"
hoelzl@40859
   801
  shows "disjoint_family_on g S"
hoelzl@40859
   802
  using assms unfolding disjoint_family_on_def by auto
hoelzl@40859
   803
hoelzl@38656
   804
lemma disjoint_family_on_mono:
hoelzl@38656
   805
  "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
hoelzl@38656
   806
  unfolding disjoint_family_on_def by auto
hoelzl@38656
   807
hoelzl@38656
   808
lemma disjoint_family_Suc:
hoelzl@38656
   809
  assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
hoelzl@38656
   810
  shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
hoelzl@38656
   811
proof -
hoelzl@38656
   812
  {
hoelzl@38656
   813
    fix m
hoelzl@38656
   814
    have "!!n. A n \<subseteq> A (m+n)"
hoelzl@38656
   815
    proof (induct m)
hoelzl@38656
   816
      case 0 show ?case by simp
hoelzl@38656
   817
    next
hoelzl@38656
   818
      case (Suc m) thus ?case
hoelzl@38656
   819
        by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans)
hoelzl@38656
   820
    qed
hoelzl@38656
   821
  }
hoelzl@38656
   822
  hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
hoelzl@38656
   823
    by (metis add_commute le_add_diff_inverse nat_less_le)
hoelzl@38656
   824
  thus ?thesis
hoelzl@38656
   825
    by (auto simp add: disjoint_family_on_def)
hoelzl@38656
   826
      (metis insert_absorb insert_subset le_SucE le_antisym not_leE)
hoelzl@38656
   827
qed
hoelzl@38656
   828
hoelzl@39092
   829
lemma setsum_indicator_disjoint_family:
hoelzl@39092
   830
  fixes f :: "'d \<Rightarrow> 'e::semiring_1"
hoelzl@39092
   831
  assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
hoelzl@39092
   832
  shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
hoelzl@39092
   833
proof -
hoelzl@39092
   834
  have "P \<inter> {i. x \<in> A i} = {j}"
hoelzl@39092
   835
    using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
hoelzl@39092
   836
    by auto
hoelzl@39092
   837
  thus ?thesis
hoelzl@39092
   838
    unfolding indicator_def
hoelzl@39092
   839
    by (simp add: if_distrib setsum_cases[OF `finite P`])
hoelzl@39092
   840
qed
hoelzl@39092
   841
hoelzl@38656
   842
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
hoelzl@38656
   843
  where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
hoelzl@38656
   844
hoelzl@38656
   845
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
hoelzl@38656
   846
proof (induct n)
hoelzl@38656
   847
  case 0 show ?case by simp
hoelzl@38656
   848
next
hoelzl@38656
   849
  case (Suc n)
hoelzl@38656
   850
  thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
hoelzl@38656
   851
qed
hoelzl@38656
   852
hoelzl@38656
   853
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
hoelzl@38656
   854
  apply (rule UN_finite2_eq [where k=0])
hoelzl@38656
   855
  apply (simp add: finite_UN_disjointed_eq)
hoelzl@38656
   856
  done
hoelzl@38656
   857
hoelzl@38656
   858
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
hoelzl@38656
   859
  by (auto simp add: disjointed_def)
hoelzl@38656
   860
hoelzl@38656
   861
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
hoelzl@38656
   862
  by (simp add: disjoint_family_on_def)
hoelzl@38656
   863
     (metis neq_iff Int_commute less_disjoint_disjointed)
hoelzl@38656
   864
hoelzl@38656
   865
lemma disjointed_subset: "disjointed A n \<subseteq> A n"
hoelzl@38656
   866
  by (auto simp add: disjointed_def)
hoelzl@38656
   867
hoelzl@42065
   868
lemma (in ring_of_sets) UNION_in_sets:
hoelzl@38656
   869
  fixes A:: "nat \<Rightarrow> 'a set"
hoelzl@47694
   870
  assumes A: "range A \<subseteq> M"
hoelzl@47694
   871
  shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
hoelzl@38656
   872
proof (induct n)
hoelzl@38656
   873
  case 0 show ?case by simp
hoelzl@38656
   874
next
hoelzl@38656
   875
  case (Suc n)
hoelzl@38656
   876
  thus ?case
hoelzl@38656
   877
    by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
hoelzl@38656
   878
qed
hoelzl@38656
   879
hoelzl@42065
   880
lemma (in ring_of_sets) range_disjointed_sets:
hoelzl@47694
   881
  assumes A: "range A \<subseteq> M"
hoelzl@47694
   882
  shows  "range (disjointed A) \<subseteq> M"
hoelzl@38656
   883
proof (auto simp add: disjointed_def)
hoelzl@38656
   884
  fix n
hoelzl@47694
   885
  show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
hoelzl@38656
   886
    by (metis A Diff UNIV_I image_subset_iff)
hoelzl@38656
   887
qed
hoelzl@38656
   888
hoelzl@42065
   889
lemma (in algebra) range_disjointed_sets':
hoelzl@47694
   890
  "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
hoelzl@42065
   891
  using range_disjointed_sets .
hoelzl@42065
   892
hoelzl@42145
   893
lemma disjointed_0[simp]: "disjointed A 0 = A 0"
hoelzl@42145
   894
  by (simp add: disjointed_def)
hoelzl@42145
   895
hoelzl@42145
   896
lemma incseq_Un:
hoelzl@42145
   897
  "incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
hoelzl@42145
   898
  unfolding incseq_def by auto
hoelzl@42145
   899
hoelzl@42145
   900
lemma disjointed_incseq:
hoelzl@42145
   901
  "incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
hoelzl@42145
   902
  using incseq_Un[of A]
hoelzl@42145
   903
  by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
hoelzl@42145
   904
hoelzl@38656
   905
lemma sigma_algebra_disjoint_iff:
hoelzl@47694
   906
  "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
hoelzl@47694
   907
    (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
   908
proof (auto simp add: sigma_algebra_iff)
hoelzl@38656
   909
  fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
   910
  assume M: "algebra \<Omega> M"
hoelzl@47694
   911
     and A: "range A \<subseteq> M"
hoelzl@47694
   912
     and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694
   913
  hence "range (disjointed A) \<subseteq> M \<longrightarrow>
hoelzl@38656
   914
         disjoint_family (disjointed A) \<longrightarrow>
hoelzl@47694
   915
         (\<Union>i. disjointed A i) \<in> M" by blast
hoelzl@47694
   916
  hence "(\<Union>i. disjointed A i) \<in> M"
hoelzl@47694
   917
    by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
hoelzl@47694
   918
  thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
hoelzl@47694
   919
qed
hoelzl@47694
   920
hoelzl@47762
   921
lemma disjoint_family_on_disjoint_image:
hoelzl@47762
   922
  "disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
hoelzl@47762
   923
  unfolding disjoint_family_on_def disjoint_def by force
hoelzl@47762
   924
hoelzl@47762
   925
lemma disjoint_image_disjoint_family_on:
hoelzl@47762
   926
  assumes d: "disjoint (A ` I)" and i: "inj_on A I"
hoelzl@47762
   927
  shows "disjoint_family_on A I"
hoelzl@47762
   928
  unfolding disjoint_family_on_def
hoelzl@47762
   929
proof (intro ballI impI)
hoelzl@47762
   930
  fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
hoelzl@47762
   931
  with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
hoelzl@47762
   932
    by (intro disjointD[OF d]) auto
hoelzl@47762
   933
qed
hoelzl@47762
   934
hoelzl@56994
   935
subsubsection {* Ring generated by a semiring *}
hoelzl@47762
   936
hoelzl@47762
   937
definition (in semiring_of_sets)
hoelzl@47762
   938
  "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
hoelzl@47762
   939
hoelzl@47762
   940
lemma (in semiring_of_sets) generated_ringE[elim?]:
hoelzl@47762
   941
  assumes "a \<in> generated_ring"
hoelzl@47762
   942
  obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
hoelzl@47762
   943
  using assms unfolding generated_ring_def by auto
hoelzl@47762
   944
hoelzl@47762
   945
lemma (in semiring_of_sets) generated_ringI[intro?]:
hoelzl@47762
   946
  assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
hoelzl@47762
   947
  shows "a \<in> generated_ring"
hoelzl@47762
   948
  using assms unfolding generated_ring_def by auto
hoelzl@47762
   949
hoelzl@47762
   950
lemma (in semiring_of_sets) generated_ringI_Basic:
hoelzl@47762
   951
  "A \<in> M \<Longrightarrow> A \<in> generated_ring"
hoelzl@47762
   952
  by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
hoelzl@47762
   953
hoelzl@47762
   954
lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
hoelzl@47762
   955
  assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
hoelzl@47762
   956
  and "a \<inter> b = {}"
hoelzl@47762
   957
  shows "a \<union> b \<in> generated_ring"
hoelzl@47762
   958
proof -
hoelzl@47762
   959
  from a guess Ca .. note Ca = this
hoelzl@47762
   960
  from b guess Cb .. note Cb = this
hoelzl@47762
   961
  show ?thesis
hoelzl@47762
   962
  proof
hoelzl@47762
   963
    show "disjoint (Ca \<union> Cb)"
hoelzl@47762
   964
      using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
hoelzl@47762
   965
  qed (insert Ca Cb, auto)
hoelzl@47762
   966
qed
hoelzl@47762
   967
hoelzl@47762
   968
lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
hoelzl@47762
   969
  by (auto simp: generated_ring_def disjoint_def)
hoelzl@47762
   970
hoelzl@47762
   971
lemma (in semiring_of_sets) generated_ring_disjoint_Union:
hoelzl@47762
   972
  assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
hoelzl@47762
   973
  using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
hoelzl@47762
   974
hoelzl@47762
   975
lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
hoelzl@47762
   976
  "finite I \<Longrightarrow> disjoint (A ` I) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> UNION I A \<in> generated_ring"
hoelzl@47762
   977
  unfolding SUP_def by (intro generated_ring_disjoint_Union) auto
hoelzl@47762
   978
hoelzl@47762
   979
lemma (in semiring_of_sets) generated_ring_Int:
hoelzl@47762
   980
  assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
hoelzl@47762
   981
  shows "a \<inter> b \<in> generated_ring"
hoelzl@47762
   982
proof -
hoelzl@47762
   983
  from a guess Ca .. note Ca = this
hoelzl@47762
   984
  from b guess Cb .. note Cb = this
hoelzl@47762
   985
  def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
hoelzl@47762
   986
  show ?thesis
hoelzl@47762
   987
  proof
hoelzl@47762
   988
    show "disjoint C"
hoelzl@47762
   989
    proof (simp add: disjoint_def C_def, intro ballI impI)
hoelzl@47762
   990
      fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
hoelzl@47762
   991
      assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
hoelzl@47762
   992
      then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
hoelzl@47762
   993
      then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
hoelzl@47762
   994
      proof
hoelzl@47762
   995
        assume "a1 \<noteq> a2"
hoelzl@47762
   996
        with sets Ca have "a1 \<inter> a2 = {}"
hoelzl@47762
   997
          by (auto simp: disjoint_def)
hoelzl@47762
   998
        then show ?thesis by auto
hoelzl@47762
   999
      next
hoelzl@47762
  1000
        assume "b1 \<noteq> b2"
hoelzl@47762
  1001
        with sets Cb have "b1 \<inter> b2 = {}"
hoelzl@47762
  1002
          by (auto simp: disjoint_def)
hoelzl@47762
  1003
        then show ?thesis by auto
hoelzl@47762
  1004
      qed
hoelzl@47762
  1005
    qed
hoelzl@47762
  1006
  qed (insert Ca Cb, auto simp: C_def)
hoelzl@47762
  1007
qed
hoelzl@47762
  1008
hoelzl@47762
  1009
lemma (in semiring_of_sets) generated_ring_Inter:
hoelzl@47762
  1010
  assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
hoelzl@47762
  1011
  using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
hoelzl@47762
  1012
hoelzl@47762
  1013
lemma (in semiring_of_sets) generated_ring_INTER:
hoelzl@47762
  1014
  "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
hoelzl@47762
  1015
  unfolding INF_def by (intro generated_ring_Inter) auto
hoelzl@47762
  1016
hoelzl@47762
  1017
lemma (in semiring_of_sets) generating_ring:
hoelzl@47762
  1018
  "ring_of_sets \<Omega> generated_ring"
hoelzl@47762
  1019
proof (rule ring_of_setsI)
hoelzl@47762
  1020
  let ?R = generated_ring
hoelzl@47762
  1021
  show "?R \<subseteq> Pow \<Omega>"
hoelzl@47762
  1022
    using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
hoelzl@47762
  1023
  show "{} \<in> ?R" by (rule generated_ring_empty)
hoelzl@47762
  1024
hoelzl@47762
  1025
  { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
hoelzl@47762
  1026
    fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
hoelzl@47762
  1027
  
hoelzl@47762
  1028
    show "a - b \<in> ?R"
hoelzl@47762
  1029
    proof cases
hoelzl@47762
  1030
      assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
hoelzl@47762
  1031
        by simp
hoelzl@47762
  1032
    next
hoelzl@47762
  1033
      assume "Cb \<noteq> {}"
hoelzl@47762
  1034
      with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
hoelzl@47762
  1035
      also have "\<dots> \<in> ?R"
hoelzl@47762
  1036
      proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
hoelzl@47762
  1037
        fix a b assume "a \<in> Ca" "b \<in> Cb"
hoelzl@47762
  1038
        with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
hoelzl@47762
  1039
          by (auto simp add: generated_ring_def)
hoelzl@47762
  1040
      next
hoelzl@47762
  1041
        show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
hoelzl@47762
  1042
          using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
hoelzl@47762
  1043
      next
hoelzl@47762
  1044
        show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
hoelzl@47762
  1045
      qed
hoelzl@47762
  1046
      finally show "a - b \<in> ?R" .
hoelzl@47762
  1047
    qed }
hoelzl@47762
  1048
  note Diff = this
hoelzl@47762
  1049
hoelzl@47762
  1050
  fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
hoelzl@47762
  1051
  have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
hoelzl@47762
  1052
  also have "\<dots> \<in> ?R"
hoelzl@47762
  1053
    by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
hoelzl@47762
  1054
  finally show "a \<union> b \<in> ?R" .
hoelzl@47762
  1055
qed
hoelzl@47762
  1056
hoelzl@47762
  1057
lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
hoelzl@47762
  1058
proof
hoelzl@47762
  1059
  interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
hoelzl@47762
  1060
    using space_closed by (rule sigma_algebra_sigma_sets)
hoelzl@47762
  1061
  show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
hoelzl@47762
  1062
    by (blast intro!: sigma_sets_mono elim: generated_ringE)
hoelzl@47762
  1063
qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
hoelzl@47762
  1064
hoelzl@56994
  1065
subsubsection {* A Two-Element Series *}
hoelzl@38656
  1066
hoelzl@38656
  1067
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
wenzelm@50252
  1068
  where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
hoelzl@38656
  1069
hoelzl@38656
  1070
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
hoelzl@38656
  1071
  apply (simp add: binaryset_def)
nipkow@39302
  1072
  apply (rule set_eqI)
hoelzl@38656
  1073
  apply (auto simp add: image_iff)
hoelzl@38656
  1074
  done
hoelzl@38656
  1075
hoelzl@38656
  1076
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
haftmann@44106
  1077
  by (simp add: SUP_def range_binaryset_eq)
hoelzl@38656
  1078
hoelzl@56994
  1079
subsubsection {* Closed CDI *}
hoelzl@38656
  1080
hoelzl@47694
  1081
definition closed_cdi where
hoelzl@47694
  1082
  "closed_cdi \<Omega> M \<longleftrightarrow>
hoelzl@47694
  1083
   M \<subseteq> Pow \<Omega> &
hoelzl@47694
  1084
   (\<forall>s \<in> M. \<Omega> - s \<in> M) &
hoelzl@47694
  1085
   (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
hoelzl@47694
  1086
        (\<Union>i. A i) \<in> M) &
hoelzl@47694
  1087
   (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
  1088
hoelzl@38656
  1089
inductive_set
hoelzl@47694
  1090
  smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
hoelzl@47694
  1091
  for \<Omega> M
hoelzl@38656
  1092
  where
hoelzl@38656
  1093
    Basic [intro]:
hoelzl@47694
  1094
      "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1095
  | Compl [intro]:
hoelzl@47694
  1096
      "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1097
  | Inc:
hoelzl@47694
  1098
      "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
hoelzl@47694
  1099
       \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1100
  | Disj:
hoelzl@47694
  1101
      "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
hoelzl@47694
  1102
       \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1103
hoelzl@47694
  1104
lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
hoelzl@47694
  1105
  by auto
hoelzl@38656
  1106
hoelzl@47694
  1107
lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
hoelzl@38656
  1108
  apply (rule subsetI)
hoelzl@38656
  1109
  apply (erule smallest_ccdi_sets.induct)
hoelzl@38656
  1110
  apply (auto intro: range_subsetD dest: sets_into_space)
hoelzl@38656
  1111
  done
hoelzl@38656
  1112
hoelzl@47694
  1113
lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
hoelzl@47694
  1114
  apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
hoelzl@38656
  1115
  apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
hoelzl@38656
  1116
  done
hoelzl@38656
  1117
hoelzl@47694
  1118
lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
hoelzl@38656
  1119
  by (simp add: closed_cdi_def)
hoelzl@38656
  1120
hoelzl@47694
  1121
lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
hoelzl@38656
  1122
  by (simp add: closed_cdi_def)
hoelzl@38656
  1123
hoelzl@38656
  1124
lemma closed_cdi_Inc:
hoelzl@47694
  1125
  "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M"
hoelzl@38656
  1126
  by (simp add: closed_cdi_def)
hoelzl@38656
  1127
hoelzl@38656
  1128
lemma closed_cdi_Disj:
hoelzl@47694
  1129
  "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@38656
  1130
  by (simp add: closed_cdi_def)
hoelzl@38656
  1131
hoelzl@38656
  1132
lemma closed_cdi_Un:
hoelzl@47694
  1133
  assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
hoelzl@47694
  1134
      and A: "A \<in> M" and B: "B \<in> M"
hoelzl@38656
  1135
      and disj: "A \<inter> B = {}"
hoelzl@47694
  1136
    shows "A \<union> B \<in> M"
hoelzl@38656
  1137
proof -
hoelzl@47694
  1138
  have ra: "range (binaryset A B) \<subseteq> M"
hoelzl@38656
  1139
   by (simp add: range_binaryset_eq empty A B)
hoelzl@38656
  1140
 have di:  "disjoint_family (binaryset A B)" using disj
hoelzl@38656
  1141
   by (simp add: disjoint_family_on_def binaryset_def Int_commute)
hoelzl@38656
  1142
 from closed_cdi_Disj [OF cdi ra di]
hoelzl@38656
  1143
 show ?thesis
hoelzl@38656
  1144
   by (simp add: UN_binaryset_eq)
hoelzl@38656
  1145
qed
hoelzl@38656
  1146
hoelzl@38656
  1147
lemma (in algebra) smallest_ccdi_sets_Un:
hoelzl@47694
  1148
  assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1149
      and disj: "A \<inter> B = {}"
hoelzl@47694
  1150
    shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1151
proof -
hoelzl@47694
  1152
  have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
hoelzl@38656
  1153
    by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
hoelzl@38656
  1154
  have di:  "disjoint_family (binaryset A B)" using disj
hoelzl@38656
  1155
    by (simp add: disjoint_family_on_def binaryset_def Int_commute)
hoelzl@38656
  1156
  from Disj [OF ra di]
hoelzl@38656
  1157
  show ?thesis
hoelzl@38656
  1158
    by (simp add: UN_binaryset_eq)
hoelzl@38656
  1159
qed
hoelzl@38656
  1160
hoelzl@38656
  1161
lemma (in algebra) smallest_ccdi_sets_Int1:
hoelzl@47694
  1162
  assumes a: "a \<in> M"
hoelzl@47694
  1163
  shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1164
proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656
  1165
  case (Basic x)
hoelzl@38656
  1166
  thus ?case
hoelzl@38656
  1167
    by (metis a Int smallest_ccdi_sets.Basic)
hoelzl@38656
  1168
next
hoelzl@38656
  1169
  case (Compl x)
hoelzl@47694
  1170
  have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
hoelzl@38656
  1171
    by blast
hoelzl@47694
  1172
  also have "... \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1173
    by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
hoelzl@47694
  1174
           Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
hoelzl@47694
  1175
           smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
hoelzl@38656
  1176
  finally show ?case .
hoelzl@38656
  1177
next
hoelzl@38656
  1178
  case (Inc A)
hoelzl@38656
  1179
  have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
hoelzl@38656
  1180
    by blast
hoelzl@47694
  1181
  have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
hoelzl@38656
  1182
    by blast
hoelzl@38656
  1183
  moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
hoelzl@38656
  1184
    by (simp add: Inc)
hoelzl@38656
  1185
  moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
hoelzl@38656
  1186
    by blast
hoelzl@47694
  1187
  ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1188
    by (rule smallest_ccdi_sets.Inc)
hoelzl@38656
  1189
  show ?case
hoelzl@38656
  1190
    by (metis 1 2)
hoelzl@38656
  1191
next
hoelzl@38656
  1192
  case (Disj A)
hoelzl@38656
  1193
  have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
hoelzl@38656
  1194
    by blast
hoelzl@47694
  1195
  have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
hoelzl@38656
  1196
    by blast
hoelzl@38656
  1197
  moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
hoelzl@38656
  1198
    by (auto simp add: disjoint_family_on_def)
hoelzl@47694
  1199
  ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1200
    by (rule smallest_ccdi_sets.Disj)
hoelzl@38656
  1201
  show ?case
hoelzl@38656
  1202
    by (metis 1 2)
hoelzl@38656
  1203
qed
hoelzl@38656
  1204
hoelzl@38656
  1205
hoelzl@38656
  1206
lemma (in algebra) smallest_ccdi_sets_Int:
hoelzl@47694
  1207
  assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@47694
  1208
  shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1209
proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656
  1210
  case (Basic x)
hoelzl@38656
  1211
  thus ?case
hoelzl@38656
  1212
    by (metis b smallest_ccdi_sets_Int1)
hoelzl@38656
  1213
next
hoelzl@38656
  1214
  case (Compl x)
hoelzl@47694
  1215
  have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
hoelzl@38656
  1216
    by blast
hoelzl@47694
  1217
  also have "... \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1218
    by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
hoelzl@38656
  1219
           smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
hoelzl@38656
  1220
  finally show ?case .
hoelzl@38656
  1221
next
hoelzl@38656
  1222
  case (Inc A)
hoelzl@38656
  1223
  have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
hoelzl@38656
  1224
    by blast
hoelzl@47694
  1225
  have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
hoelzl@38656
  1226
    by blast
hoelzl@38656
  1227
  moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
hoelzl@38656
  1228
    by (simp add: Inc)
hoelzl@38656
  1229
  moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
hoelzl@38656
  1230
    by blast
hoelzl@47694
  1231
  ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1232
    by (rule smallest_ccdi_sets.Inc)
hoelzl@38656
  1233
  show ?case
hoelzl@38656
  1234
    by (metis 1 2)
hoelzl@38656
  1235
next
hoelzl@38656
  1236
  case (Disj A)
hoelzl@38656
  1237
  have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
hoelzl@38656
  1238
    by blast
hoelzl@47694
  1239
  have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
hoelzl@38656
  1240
    by blast
hoelzl@38656
  1241
  moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
hoelzl@38656
  1242
    by (auto simp add: disjoint_family_on_def)
hoelzl@47694
  1243
  ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1244
    by (rule smallest_ccdi_sets.Disj)
hoelzl@38656
  1245
  show ?case
hoelzl@38656
  1246
    by (metis 1 2)
hoelzl@38656
  1247
qed
hoelzl@38656
  1248
hoelzl@38656
  1249
lemma (in algebra) sigma_property_disjoint_lemma:
hoelzl@47694
  1250
  assumes sbC: "M \<subseteq> C"
hoelzl@47694
  1251
      and ccdi: "closed_cdi \<Omega> C"
hoelzl@47694
  1252
  shows "sigma_sets \<Omega> M \<subseteq> C"
hoelzl@38656
  1253
proof -
hoelzl@47694
  1254
  have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
hoelzl@38656
  1255
    apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
hoelzl@38656
  1256
            smallest_ccdi_sets_Int)
hoelzl@38656
  1257
    apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
hoelzl@38656
  1258
    apply (blast intro: smallest_ccdi_sets.Disj)
hoelzl@38656
  1259
    done
hoelzl@47694
  1260
  hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1261
    by clarsimp
hoelzl@47694
  1262
       (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
hoelzl@38656
  1263
  also have "...  \<subseteq> C"
hoelzl@38656
  1264
    proof
hoelzl@38656
  1265
      fix x
hoelzl@47694
  1266
      assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1267
      thus "x \<in> C"
hoelzl@38656
  1268
        proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656
  1269
          case (Basic x)
hoelzl@38656
  1270
          thus ?case
hoelzl@38656
  1271
            by (metis Basic subsetD sbC)
hoelzl@38656
  1272
        next
hoelzl@38656
  1273
          case (Compl x)
hoelzl@38656
  1274
          thus ?case
hoelzl@38656
  1275
            by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
hoelzl@38656
  1276
        next
hoelzl@38656
  1277
          case (Inc A)
hoelzl@38656
  1278
          thus ?case
hoelzl@38656
  1279
               by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
hoelzl@38656
  1280
        next
hoelzl@38656
  1281
          case (Disj A)
hoelzl@38656
  1282
          thus ?case
hoelzl@38656
  1283
               by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
hoelzl@38656
  1284
        qed
hoelzl@38656
  1285
    qed
hoelzl@38656
  1286
  finally show ?thesis .
hoelzl@38656
  1287
qed
hoelzl@38656
  1288
hoelzl@38656
  1289
lemma (in algebra) sigma_property_disjoint:
hoelzl@47694
  1290
  assumes sbC: "M \<subseteq> C"
hoelzl@47694
  1291
      and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
hoelzl@47694
  1292
      and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
hoelzl@38656
  1293
                     \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
hoelzl@38656
  1294
                     \<Longrightarrow> (\<Union>i. A i) \<in> C"
hoelzl@47694
  1295
      and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
hoelzl@38656
  1296
                      \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
hoelzl@47694
  1297
  shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
hoelzl@38656
  1298
proof -
hoelzl@47694
  1299
  have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
hoelzl@38656
  1300
    proof (rule sigma_property_disjoint_lemma)
hoelzl@47694
  1301
      show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
hoelzl@38656
  1302
        by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
hoelzl@38656
  1303
    next
hoelzl@47694
  1304
      show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
hoelzl@38656
  1305
        by (simp add: closed_cdi_def compl inc disj)
hoelzl@38656
  1306
           (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
hoelzl@38656
  1307
             IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
hoelzl@38656
  1308
    qed
hoelzl@38656
  1309
  thus ?thesis
hoelzl@38656
  1310
    by blast
hoelzl@38656
  1311
qed
hoelzl@38656
  1312
hoelzl@56994
  1313
subsubsection {* Dynkin systems *}
hoelzl@40859
  1314
hoelzl@42065
  1315
locale dynkin_system = subset_class +
hoelzl@47694
  1316
  assumes space: "\<Omega> \<in> M"
hoelzl@47694
  1317
    and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694
  1318
    and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694
  1319
                           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@40859
  1320
hoelzl@47694
  1321
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
hoelzl@47694
  1322
  using space compl[of "\<Omega>"] by simp
hoelzl@40859
  1323
hoelzl@40859
  1324
lemma (in dynkin_system) diff:
hoelzl@47694
  1325
  assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
hoelzl@47694
  1326
  shows "E - D \<in> M"
hoelzl@40859
  1327
proof -
hoelzl@47694
  1328
  let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
hoelzl@47694
  1329
  have "range ?f = {D, \<Omega> - E, {}}"
hoelzl@40859
  1330
    by (auto simp: image_iff)
hoelzl@47694
  1331
  moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
hoelzl@40859
  1332
    by (auto simp: image_iff split: split_if_asm)
hoelzl@40859
  1333
  moreover
wenzelm@53374
  1334
  have "disjoint_family ?f" unfolding disjoint_family_on_def
hoelzl@47694
  1335
    using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
hoelzl@47694
  1336
  ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
hoelzl@40859
  1337
    using sets by auto
hoelzl@47694
  1338
  also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
hoelzl@40859
  1339
    using assms sets_into_space by auto
hoelzl@40859
  1340
  finally show ?thesis .
hoelzl@40859
  1341
qed
hoelzl@40859
  1342
hoelzl@40859
  1343
lemma dynkin_systemI:
hoelzl@47694
  1344
  assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
hoelzl@47694
  1345
  assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694
  1346
  assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694
  1347
          \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694
  1348
  shows "dynkin_system \<Omega> M"
hoelzl@42065
  1349
  using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
hoelzl@40859
  1350
hoelzl@42988
  1351
lemma dynkin_systemI':
hoelzl@47694
  1352
  assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
hoelzl@47694
  1353
  assumes empty: "{} \<in> M"
hoelzl@47694
  1354
  assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694
  1355
  assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694
  1356
          \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694
  1357
  shows "dynkin_system \<Omega> M"
hoelzl@42988
  1358
proof -
hoelzl@47694
  1359
  from Diff[OF empty] have "\<Omega> \<in> M" by auto
hoelzl@42988
  1360
  from 1 this Diff 2 show ?thesis
hoelzl@42988
  1361
    by (intro dynkin_systemI) auto
hoelzl@42988
  1362
qed
hoelzl@42988
  1363
hoelzl@40859
  1364
lemma dynkin_system_trivial:
hoelzl@47694
  1365
  shows "dynkin_system A (Pow A)"
hoelzl@40859
  1366
  by (rule dynkin_systemI) auto
hoelzl@40859
  1367
hoelzl@40859
  1368
lemma sigma_algebra_imp_dynkin_system:
hoelzl@47694
  1369
  assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
hoelzl@40859
  1370
proof -
hoelzl@47694
  1371
  interpret sigma_algebra \<Omega> M by fact
nipkow@44890
  1372
  show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
hoelzl@40859
  1373
qed
hoelzl@40859
  1374
hoelzl@56994
  1375
subsubsection "Intersection sets systems"
hoelzl@40859
  1376
hoelzl@47694
  1377
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
hoelzl@40859
  1378
hoelzl@40859
  1379
lemma (in algebra) Int_stable: "Int_stable M"
hoelzl@40859
  1380
  unfolding Int_stable_def by auto
hoelzl@40859
  1381
hoelzl@42981
  1382
lemma Int_stableI:
hoelzl@47694
  1383
  "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
hoelzl@42981
  1384
  unfolding Int_stable_def by auto
hoelzl@42981
  1385
hoelzl@42981
  1386
lemma Int_stableD:
hoelzl@47694
  1387
  "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
hoelzl@42981
  1388
  unfolding Int_stable_def by auto
hoelzl@42981
  1389
hoelzl@40859
  1390
lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
hoelzl@47694
  1391
  "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
hoelzl@40859
  1392
proof
hoelzl@47694
  1393
  assume "sigma_algebra \<Omega> M" then show "Int_stable M"
hoelzl@40859
  1394
    unfolding sigma_algebra_def using algebra.Int_stable by auto
hoelzl@40859
  1395
next
hoelzl@40859
  1396
  assume "Int_stable M"
hoelzl@47694
  1397
  show "sigma_algebra \<Omega> M"
hoelzl@42065
  1398
    unfolding sigma_algebra_disjoint_iff algebra_iff_Un
hoelzl@40859
  1399
  proof (intro conjI ballI allI impI)
hoelzl@47694
  1400
    show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
hoelzl@40859
  1401
  next
hoelzl@47694
  1402
    fix A B assume "A \<in> M" "B \<in> M"
hoelzl@47694
  1403
    then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
hoelzl@47694
  1404
              "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
hoelzl@40859
  1405
      using sets_into_space by auto
hoelzl@47694
  1406
    then show "A \<union> B \<in> M"
hoelzl@40859
  1407
      using `Int_stable M` unfolding Int_stable_def by auto
hoelzl@40859
  1408
  qed auto
hoelzl@40859
  1409
qed
hoelzl@40859
  1410
hoelzl@56994
  1411
subsubsection "Smallest Dynkin systems"
hoelzl@40859
  1412
hoelzl@41689
  1413
definition dynkin where
hoelzl@47694
  1414
  "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
hoelzl@40859
  1415
hoelzl@40859
  1416
lemma dynkin_system_dynkin:
hoelzl@47694
  1417
  assumes "M \<subseteq> Pow (\<Omega>)"
hoelzl@47694
  1418
  shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
hoelzl@40859
  1419
proof (rule dynkin_systemI)
hoelzl@47694
  1420
  fix A assume "A \<in> dynkin \<Omega> M"
hoelzl@40859
  1421
  moreover
hoelzl@47694
  1422
  { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
hoelzl@47694
  1423
    then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
hoelzl@47694
  1424
  moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
nipkow@44890
  1425
    using assms dynkin_system_trivial by fastforce
hoelzl@47694
  1426
  ultimately show "A \<subseteq> \<Omega>"
hoelzl@40859
  1427
    unfolding dynkin_def using assms
hoelzl@47694
  1428
    by auto
hoelzl@40859
  1429
next
hoelzl@47694
  1430
  show "\<Omega> \<in> dynkin \<Omega> M"
nipkow@44890
  1431
    unfolding dynkin_def using dynkin_system.space by fastforce
hoelzl@40859
  1432
next
hoelzl@47694
  1433
  fix A assume "A \<in> dynkin \<Omega> M"
hoelzl@47694
  1434
  then show "\<Omega> - A \<in> dynkin \<Omega> M"
hoelzl@40859
  1435
    unfolding dynkin_def using dynkin_system.compl by force
hoelzl@40859
  1436
next
hoelzl@40859
  1437
  fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
  1438
  assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
hoelzl@47694
  1439
  show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
hoelzl@40859
  1440
  proof (simp, safe)
hoelzl@47694
  1441
    fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
hoelzl@47694
  1442
    with A have "(\<Union>i. A i) \<in> D"
hoelzl@40859
  1443
      by (intro dynkin_system.UN) (auto simp: dynkin_def)
hoelzl@40859
  1444
    then show "(\<Union>i. A i) \<in> D" by auto
hoelzl@40859
  1445
  qed
hoelzl@40859
  1446
qed
hoelzl@40859
  1447
hoelzl@47694
  1448
lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
hoelzl@40859
  1449
  unfolding dynkin_def by auto
hoelzl@40859
  1450
hoelzl@40859
  1451
lemma (in dynkin_system) restricted_dynkin_system:
hoelzl@47694
  1452
  assumes "D \<in> M"
hoelzl@47694
  1453
  shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
hoelzl@40859
  1454
proof (rule dynkin_systemI, simp_all)
hoelzl@47694
  1455
  have "\<Omega> \<inter> D = D"
hoelzl@47694
  1456
    using `D \<in> M` sets_into_space by auto
hoelzl@47694
  1457
  then show "\<Omega> \<inter> D \<in> M"
hoelzl@47694
  1458
    using `D \<in> M` by auto
hoelzl@40859
  1459
next
hoelzl@47694
  1460
  fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
hoelzl@47694
  1461
  moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
hoelzl@40859
  1462
    by auto
hoelzl@47694
  1463
  ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
hoelzl@47694
  1464
    using  `D \<in> M` by (auto intro: diff)
hoelzl@40859
  1465
next
hoelzl@40859
  1466
  fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
  1467
  assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
hoelzl@47694
  1468
  then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
hoelzl@47694
  1469
    "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
nipkow@44890
  1470
    by ((fastforce simp: disjoint_family_on_def)+)
hoelzl@47694
  1471
  then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
hoelzl@40859
  1472
    by (auto simp del: UN_simps)
hoelzl@40859
  1473
qed
hoelzl@40859
  1474
hoelzl@40859
  1475
lemma (in dynkin_system) dynkin_subset:
hoelzl@47694
  1476
  assumes "N \<subseteq> M"
hoelzl@47694
  1477
  shows "dynkin \<Omega> N \<subseteq> M"
hoelzl@40859
  1478
proof -
hoelzl@47694
  1479
  have "dynkin_system \<Omega> M" by default
hoelzl@47694
  1480
  then have "dynkin_system \<Omega> M"
hoelzl@42065
  1481
    using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
hoelzl@47694
  1482
  with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
hoelzl@40859
  1483
qed
hoelzl@40859
  1484
hoelzl@40859
  1485
lemma sigma_eq_dynkin:
hoelzl@47694
  1486
  assumes sets: "M \<subseteq> Pow \<Omega>"
hoelzl@40859
  1487
  assumes "Int_stable M"
hoelzl@47694
  1488
  shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
hoelzl@40859
  1489
proof -
hoelzl@47694
  1490
  have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
hoelzl@40859
  1491
    using sigma_algebra_imp_dynkin_system
hoelzl@47694
  1492
    unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
hoelzl@40859
  1493
  moreover
hoelzl@47694
  1494
  interpret dynkin_system \<Omega> "dynkin \<Omega> M"
hoelzl@40859
  1495
    using dynkin_system_dynkin[OF sets] .
hoelzl@47694
  1496
  have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
hoelzl@40859
  1497
    unfolding sigma_algebra_eq_Int_stable Int_stable_def
hoelzl@40859
  1498
  proof (intro ballI)
hoelzl@47694
  1499
    fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
hoelzl@47694
  1500
    let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
hoelzl@47694
  1501
    have "M \<subseteq> ?D B"
hoelzl@40859
  1502
    proof
hoelzl@47694
  1503
      fix E assume "E \<in> M"
hoelzl@47694
  1504
      then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
hoelzl@40859
  1505
        using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
hoelzl@47694
  1506
      then have "dynkin \<Omega> M \<subseteq> ?D E"
hoelzl@47694
  1507
        using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
hoelzl@40859
  1508
        by (intro dynkin_system.dynkin_subset) simp_all
hoelzl@47694
  1509
      then have "B \<in> ?D E"
hoelzl@47694
  1510
        using `B \<in> dynkin \<Omega> M` by auto
hoelzl@47694
  1511
      then have "E \<inter> B \<in> dynkin \<Omega> M"
hoelzl@40859
  1512
        by (subst Int_commute) simp
hoelzl@47694
  1513
      then show "E \<in> ?D B"
hoelzl@47694
  1514
        using sets `E \<in> M` by auto
hoelzl@40859
  1515
    qed
hoelzl@47694
  1516
    then have "dynkin \<Omega> M \<subseteq> ?D B"
hoelzl@47694
  1517
      using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
hoelzl@40859
  1518
      by (intro dynkin_system.dynkin_subset) simp_all
hoelzl@47694
  1519
    then show "A \<inter> B \<in> dynkin \<Omega> M"
hoelzl@47694
  1520
      using `A \<in> dynkin \<Omega> M` sets_into_space by auto
hoelzl@40859
  1521
  qed
hoelzl@47694
  1522
  from sigma_algebra.sigma_sets_subset[OF this, of "M"]
hoelzl@47694
  1523
  have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
hoelzl@47694
  1524
  ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
hoelzl@40859
  1525
  then show ?thesis
hoelzl@47694
  1526
    by (auto simp: dynkin_def)
hoelzl@40859
  1527
qed
hoelzl@40859
  1528
hoelzl@40859
  1529
lemma (in dynkin_system) dynkin_idem:
hoelzl@47694
  1530
  "dynkin \<Omega> M = M"
hoelzl@40859
  1531
proof -
hoelzl@47694
  1532
  have "dynkin \<Omega> M = M"
hoelzl@40859
  1533
  proof
hoelzl@47694
  1534
    show "M \<subseteq> dynkin \<Omega> M"
hoelzl@40859
  1535
      using dynkin_Basic by auto
hoelzl@47694
  1536
    show "dynkin \<Omega> M \<subseteq> M"
hoelzl@40859
  1537
      by (intro dynkin_subset) auto
hoelzl@40859
  1538
  qed
hoelzl@40859
  1539
  then show ?thesis
hoelzl@47694
  1540
    by (auto simp: dynkin_def)
hoelzl@40859
  1541
qed
hoelzl@40859
  1542
hoelzl@40859
  1543
lemma (in dynkin_system) dynkin_lemma:
hoelzl@41689
  1544
  assumes "Int_stable E"
hoelzl@47694
  1545
  and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
hoelzl@47694
  1546
  shows "sigma_sets \<Omega> E = M"
hoelzl@40859
  1547
proof -
hoelzl@47694
  1548
  have "E \<subseteq> Pow \<Omega>"
hoelzl@41689
  1549
    using E sets_into_space by force
wenzelm@53374
  1550
  then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
hoelzl@40859
  1551
    using `Int_stable E` by (rule sigma_eq_dynkin)
wenzelm@53374
  1552
  then have "dynkin \<Omega> E = M"
hoelzl@47694
  1553
    using assms dynkin_subset[OF E(1)] by simp
wenzelm@53374
  1554
  with * show ?thesis
hoelzl@47694
  1555
    using assms by (auto simp: dynkin_def)
hoelzl@42864
  1556
qed
hoelzl@42864
  1557
hoelzl@56994
  1558
subsubsection {* Induction rule for intersection-stable generators *}
hoelzl@56994
  1559
hoelzl@56994
  1560
text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
hoelzl@56994
  1561
generated by a generator closed under intersection. *}
hoelzl@56994
  1562
hoelzl@49789
  1563
lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
hoelzl@49789
  1564
  assumes "Int_stable G"
hoelzl@49789
  1565
    and closed: "G \<subseteq> Pow \<Omega>"
hoelzl@49789
  1566
    and A: "A \<in> sigma_sets \<Omega> G"
hoelzl@49789
  1567
  assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
hoelzl@49789
  1568
    and empty: "P {}"
hoelzl@49789
  1569
    and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
hoelzl@49789
  1570
    and union: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sigma_sets \<Omega> G \<Longrightarrow> (\<And>i. P (A i)) \<Longrightarrow> P (\<Union>i::nat. A i)"
hoelzl@49789
  1571
  shows "P A"
hoelzl@49789
  1572
proof -
hoelzl@49789
  1573
  let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
hoelzl@49789
  1574
  interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
hoelzl@49789
  1575
    using closed by (rule sigma_algebra_sigma_sets)
hoelzl@49789
  1576
  from compl[OF _ empty] closed have space: "P \<Omega>" by simp
hoelzl@49789
  1577
  interpret dynkin_system \<Omega> ?D
hoelzl@49789
  1578
    by default (auto dest: sets_into_space intro!: space compl union)
hoelzl@49789
  1579
  have "sigma_sets \<Omega> G = ?D"
hoelzl@49789
  1580
    by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
hoelzl@49789
  1581
  with A show ?thesis by auto
hoelzl@49789
  1582
qed
hoelzl@49789
  1583
hoelzl@56994
  1584
subsection {* Measure type *}
hoelzl@56994
  1585
hoelzl@56994
  1586
definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994
  1587
  "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
hoelzl@56994
  1588
hoelzl@56994
  1589
definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994
  1590
  "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
hoelzl@56994
  1591
    (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
hoelzl@56994
  1592
hoelzl@56994
  1593
definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994
  1594
  "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
hoelzl@56994
  1595
hoelzl@56994
  1596
typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
hoelzl@56994
  1597
proof
hoelzl@56994
  1598
  have "sigma_algebra UNIV {{}, UNIV}"
hoelzl@56994
  1599
    by (auto simp: sigma_algebra_iff2)
hoelzl@56994
  1600
  then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
hoelzl@56994
  1601
    by (auto simp: measure_space_def positive_def countably_additive_def)
hoelzl@56994
  1602
qed
hoelzl@56994
  1603
hoelzl@56994
  1604
definition space :: "'a measure \<Rightarrow> 'a set" where
hoelzl@56994
  1605
  "space M = fst (Rep_measure M)"
hoelzl@56994
  1606
hoelzl@56994
  1607
definition sets :: "'a measure \<Rightarrow> 'a set set" where
hoelzl@56994
  1608
  "sets M = fst (snd (Rep_measure M))"
hoelzl@56994
  1609
hoelzl@56994
  1610
definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
hoelzl@56994
  1611
  "emeasure M = snd (snd (Rep_measure M))"
hoelzl@56994
  1612
hoelzl@56994
  1613
definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
hoelzl@56994
  1614
  "measure M A = real (emeasure M A)"
hoelzl@56994
  1615
hoelzl@56994
  1616
declare [[coercion sets]]
hoelzl@56994
  1617
hoelzl@56994
  1618
declare [[coercion measure]]
hoelzl@56994
  1619
hoelzl@56994
  1620
declare [[coercion emeasure]]
hoelzl@56994
  1621
hoelzl@56994
  1622
lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
hoelzl@56994
  1623
  by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
hoelzl@56994
  1624
hoelzl@56994
  1625
interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
hoelzl@56994
  1626
  using measure_space[of M] by (auto simp: measure_space_def)
hoelzl@56994
  1627
hoelzl@56994
  1628
definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
hoelzl@56994
  1629
  "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
hoelzl@56994
  1630
    \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
hoelzl@56994
  1631
hoelzl@56994
  1632
abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
hoelzl@56994
  1633
hoelzl@56994
  1634
lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
hoelzl@56994
  1635
  unfolding measure_space_def
hoelzl@56994
  1636
  by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
hoelzl@56994
  1637
hoelzl@56994
  1638
lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
hoelzl@56994
  1639
by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
hoelzl@56994
  1640
hoelzl@56994
  1641
lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
hoelzl@56994
  1642
by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
hoelzl@56994
  1643
hoelzl@56994
  1644
lemma measure_space_closed:
hoelzl@56994
  1645
  assumes "measure_space \<Omega> M \<mu>"
hoelzl@56994
  1646
  shows "M \<subseteq> Pow \<Omega>"
hoelzl@56994
  1647
proof -
hoelzl@56994
  1648
  interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
hoelzl@56994
  1649
  show ?thesis by(rule space_closed)
hoelzl@56994
  1650
qed
hoelzl@56994
  1651
hoelzl@56994
  1652
lemma (in ring_of_sets) positive_cong_eq:
hoelzl@56994
  1653
  "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
hoelzl@56994
  1654
  by (auto simp add: positive_def)
hoelzl@56994
  1655
hoelzl@56994
  1656
lemma (in sigma_algebra) countably_additive_eq:
hoelzl@56994
  1657
  "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
hoelzl@56994
  1658
  unfolding countably_additive_def
hoelzl@56994
  1659
  by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
hoelzl@56994
  1660
hoelzl@56994
  1661
lemma measure_space_eq:
hoelzl@56994
  1662
  assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
hoelzl@56994
  1663
  shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
hoelzl@56994
  1664
proof -
hoelzl@56994
  1665
  interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
hoelzl@56994
  1666
  from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
hoelzl@56994
  1667
    by (auto simp: measure_space_def)
hoelzl@56994
  1668
qed
hoelzl@56994
  1669
hoelzl@56994
  1670
lemma measure_of_eq:
hoelzl@56994
  1671
  assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
hoelzl@56994
  1672
  shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
hoelzl@56994
  1673
proof -
hoelzl@56994
  1674
  have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
hoelzl@56994
  1675
    using assms by (rule measure_space_eq)
hoelzl@56994
  1676
  with eq show ?thesis
hoelzl@56994
  1677
    by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
hoelzl@56994
  1678
qed
hoelzl@56994
  1679
hoelzl@56994
  1680
lemma
hoelzl@56994
  1681
  shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
hoelzl@56994
  1682
  and sets_measure_of_conv:
hoelzl@56994
  1683
  "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
hoelzl@56994
  1684
  and emeasure_measure_of_conv: 
hoelzl@56994
  1685
  "emeasure (measure_of \<Omega> A \<mu>) = 
hoelzl@56994
  1686
  (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)" (is ?emeasure)
hoelzl@56994
  1687
proof -
hoelzl@56994
  1688
  have "?space \<and> ?sets \<and> ?emeasure"
hoelzl@56994
  1689
  proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
hoelzl@56994
  1690
    case True
hoelzl@56994
  1691
    from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
hoelzl@56994
  1692
    have "A \<subseteq> Pow \<Omega>" by simp
hoelzl@56994
  1693
    hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
hoelzl@56994
  1694
      (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
hoelzl@56994
  1695
      by(rule measure_space_eq) auto
hoelzl@56994
  1696
    with True `A \<subseteq> Pow \<Omega>` show ?thesis
hoelzl@56994
  1697
      by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
hoelzl@56994
  1698
  next
hoelzl@56994
  1699
    case False thus ?thesis
hoelzl@56994
  1700
      by(cases "A \<subseteq> Pow \<Omega>")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0')
hoelzl@56994
  1701
  qed
hoelzl@56994
  1702
  thus ?space ?sets ?emeasure by simp_all
hoelzl@56994
  1703
qed
hoelzl@56994
  1704
hoelzl@56994
  1705
lemma [simp]:
hoelzl@56994
  1706
  assumes A: "A \<subseteq> Pow \<Omega>"
hoelzl@56994
  1707
  shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
hoelzl@56994
  1708
    and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
hoelzl@56994
  1709
using assms
hoelzl@56994
  1710
by(simp_all add: sets_measure_of_conv space_measure_of_conv)
hoelzl@56994
  1711
hoelzl@56994
  1712
lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
hoelzl@56994
  1713
  using space_closed by (auto intro!: sigma_sets_eq)
hoelzl@56994
  1714
hoelzl@56994
  1715
lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
hoelzl@56994
  1716
  by (rule space_measure_of_conv)
hoelzl@56994
  1717
hoelzl@56994
  1718
lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
hoelzl@56994
  1719
  by (auto intro!: sigma_sets_subseteq)
hoelzl@56994
  1720
hoelzl@56994
  1721
lemma sigma_sets_mono'':
hoelzl@56994
  1722
  assumes "A \<in> sigma_sets C D"
hoelzl@56994
  1723
  assumes "B \<subseteq> D"
hoelzl@56994
  1724
  assumes "D \<subseteq> Pow C"
hoelzl@56994
  1725
  shows "sigma_sets A B \<subseteq> sigma_sets C D"
hoelzl@56994
  1726
proof
hoelzl@56994
  1727
  fix x assume "x \<in> sigma_sets A B"
hoelzl@56994
  1728
  thus "x \<in> sigma_sets C D"
hoelzl@56994
  1729
  proof induct
hoelzl@56994
  1730
    case (Basic a) with assms have "a \<in> D" by auto
hoelzl@56994
  1731
    thus ?case ..
hoelzl@56994
  1732
  next
hoelzl@56994
  1733
    case Empty show ?case by (rule sigma_sets.Empty)
hoelzl@56994
  1734
  next
hoelzl@56994
  1735
    from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994
  1736
    moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994
  1737
    ultimately have "A - a \<in> sets (sigma C D)" ..
hoelzl@56994
  1738
    thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994
  1739
  next
hoelzl@56994
  1740
    case (Union a)
hoelzl@56994
  1741
    thus ?case by (intro sigma_sets.Union)
hoelzl@56994
  1742
  qed
hoelzl@56994
  1743
qed
hoelzl@56994
  1744
hoelzl@56994
  1745
lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
hoelzl@56994
  1746
  by auto
hoelzl@56994
  1747
hoelzl@56994
  1748
subsubsection {* Constructing simple @{typ "'a measure"} *}
hoelzl@56994
  1749
hoelzl@56994
  1750
lemma emeasure_measure_of:
hoelzl@56994
  1751
  assumes M: "M = measure_of \<Omega> A \<mu>"
hoelzl@56994
  1752
  assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
hoelzl@56994
  1753
  assumes X: "X \<in> sets M"
hoelzl@56994
  1754
  shows "emeasure M X = \<mu> X"
hoelzl@56994
  1755
proof -
hoelzl@56994
  1756
  interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
hoelzl@56994
  1757
  have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
hoelzl@56994
  1758
    using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
hoelzl@56994
  1759
  thus ?thesis using X ms
hoelzl@56994
  1760
    by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
hoelzl@56994
  1761
qed
hoelzl@56994
  1762
hoelzl@56994
  1763
lemma emeasure_measure_of_sigma:
hoelzl@56994
  1764
  assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
hoelzl@56994
  1765
  assumes A: "A \<in> M"
hoelzl@56994
  1766
  shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
hoelzl@56994
  1767
proof -
hoelzl@56994
  1768
  interpret sigma_algebra \<Omega> M by fact
hoelzl@56994
  1769
  have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
hoelzl@56994
  1770
    using ms sigma_sets_eq by (simp add: measure_space_def)
hoelzl@56994
  1771
  thus ?thesis by(simp add: emeasure_measure_of_conv A)
hoelzl@56994
  1772
qed
hoelzl@56994
  1773
hoelzl@56994
  1774
lemma measure_cases[cases type: measure]:
hoelzl@56994
  1775
  obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
hoelzl@56994
  1776
  by atomize_elim (cases x, auto)
hoelzl@56994
  1777
hoelzl@56994
  1778
lemma sets_eq_imp_space_eq:
hoelzl@56994
  1779
  "sets M = sets M' \<Longrightarrow> space M = space M'"
hoelzl@56994
  1780
  using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
hoelzl@56994
  1781
  by blast
hoelzl@56994
  1782
hoelzl@56994
  1783
lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
hoelzl@56994
  1784
  by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
hoelzl@56994
  1785
hoelzl@56994
  1786
lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
hoelzl@56994
  1787
  using emeasure_notin_sets[of A M] by blast
hoelzl@56994
  1788
hoelzl@56994
  1789
lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
hoelzl@56994
  1790
  by (simp add: measure_def emeasure_notin_sets)
hoelzl@56994
  1791
hoelzl@56994
  1792
lemma measure_eqI:
hoelzl@56994
  1793
  fixes M N :: "'a measure"
hoelzl@56994
  1794
  assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
hoelzl@56994
  1795
  shows "M = N"
hoelzl@56994
  1796
proof (cases M N rule: measure_cases[case_product measure_cases])
hoelzl@56994
  1797
  case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
hoelzl@56994
  1798
  interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
hoelzl@56994
  1799
  interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
hoelzl@56994
  1800
  have "A = sets M" "A' = sets N"
hoelzl@56994
  1801
    using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
hoelzl@56994
  1802
  with `sets M = sets N` have AA': "A = A'" by simp
hoelzl@56994
  1803
  moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
hoelzl@56994
  1804
  moreover { fix B have "\<mu> B = \<mu>' B"
hoelzl@56994
  1805
    proof cases
hoelzl@56994
  1806
      assume "B \<in> A"
hoelzl@56994
  1807
      with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
hoelzl@56994
  1808
      with measure_measure show "\<mu> B = \<mu>' B"
hoelzl@56994
  1809
        by (simp add: emeasure_def Abs_measure_inverse)
hoelzl@56994
  1810
    next
hoelzl@56994
  1811
      assume "B \<notin> A"
hoelzl@56994
  1812
      with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
hoelzl@56994
  1813
        by auto
hoelzl@56994
  1814
      then have "emeasure M B = 0" "emeasure N B = 0"
hoelzl@56994
  1815
        by (simp_all add: emeasure_notin_sets)
hoelzl@56994
  1816
      with measure_measure show "\<mu> B = \<mu>' B"
hoelzl@56994
  1817
        by (simp add: emeasure_def Abs_measure_inverse)
hoelzl@56994
  1818
    qed }
hoelzl@56994
  1819
  then have "\<mu> = \<mu>'" by auto
hoelzl@56994
  1820
  ultimately show "M = N"
hoelzl@56994
  1821
    by (simp add: measure_measure)
hoelzl@56994
  1822
qed
hoelzl@56994
  1823
hoelzl@56994
  1824
lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
hoelzl@56994
  1825
  using measure_space_0[of A \<Omega>]
hoelzl@56994
  1826
  by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
hoelzl@56994
  1827
hoelzl@56994
  1828
lemma sigma_eqI:
hoelzl@56994
  1829
  assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
hoelzl@56994
  1830
  shows "sigma \<Omega> M = sigma \<Omega> N"
hoelzl@56994
  1831
  by (rule measure_eqI) (simp_all add: emeasure_sigma)
hoelzl@56994
  1832
hoelzl@56994
  1833
subsubsection {* Measurable functions *}
hoelzl@56994
  1834
hoelzl@56994
  1835
definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
hoelzl@56994
  1836
  "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
hoelzl@56994
  1837
hoelzl@56994
  1838
lemma measurable_space:
hoelzl@56994
  1839
  "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
hoelzl@56994
  1840
   unfolding measurable_def by auto
hoelzl@56994
  1841
hoelzl@56994
  1842
lemma measurable_sets:
hoelzl@56994
  1843
  "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
hoelzl@56994
  1844
   unfolding measurable_def by auto
hoelzl@56994
  1845
hoelzl@56994
  1846
lemma measurable_sets_Collect:
hoelzl@56994
  1847
  assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M"
hoelzl@56994
  1848
proof -
hoelzl@56994
  1849
  have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
hoelzl@56994
  1850
    using measurable_space[OF f] by auto
hoelzl@56994
  1851
  with measurable_sets[OF f P] show ?thesis
hoelzl@56994
  1852
    by simp
hoelzl@56994
  1853
qed
hoelzl@56994
  1854
hoelzl@56994
  1855
lemma measurable_sigma_sets:
hoelzl@56994
  1856
  assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
hoelzl@56994
  1857
      and f: "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994
  1858
      and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
hoelzl@56994
  1859
  shows "f \<in> measurable M N"
hoelzl@56994
  1860
proof -
hoelzl@56994
  1861
  interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
hoelzl@56994
  1862
  from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
hoelzl@56994
  1863
  
hoelzl@56994
  1864
  { fix X assume "X \<in> sigma_sets \<Omega> A"
hoelzl@56994
  1865
    then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
hoelzl@56994
  1866
      proof induct
hoelzl@56994
  1867
        case (Basic a) then show ?case
hoelzl@56994
  1868
          by (auto simp add: ba) (metis B(2) subsetD PowD)
hoelzl@56994
  1869
      next
hoelzl@56994
  1870
        case (Compl a)
hoelzl@56994
  1871
        have [simp]: "f -` \<Omega> \<inter> space M = space M"
hoelzl@56994
  1872
          by (auto simp add: funcset_mem [OF f])
hoelzl@56994
  1873
        then show ?case
hoelzl@56994
  1874
          by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
hoelzl@56994
  1875
      next
hoelzl@56994
  1876
        case (Union a)
hoelzl@56994
  1877
        then show ?case
hoelzl@56994
  1878
          by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
hoelzl@56994
  1879
      qed auto }
hoelzl@56994
  1880
  with f show ?thesis
hoelzl@56994
  1881
    by (auto simp add: measurable_def B \<Omega>)
hoelzl@56994
  1882
qed
hoelzl@56994
  1883
hoelzl@56994
  1884
lemma measurable_measure_of:
hoelzl@56994
  1885
  assumes B: "N \<subseteq> Pow \<Omega>"
hoelzl@56994
  1886
      and f: "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994
  1887
      and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
hoelzl@56994
  1888
  shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
hoelzl@56994
  1889
proof -
hoelzl@56994
  1890
  have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
hoelzl@56994
  1891
    using B by (rule sets_measure_of)
hoelzl@56994
  1892
  from this assms show ?thesis by (rule measurable_sigma_sets)
hoelzl@56994
  1893
qed
hoelzl@56994
  1894
hoelzl@56994
  1895
lemma measurable_iff_measure_of:
hoelzl@56994
  1896
  assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994
  1897
  shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
hoelzl@56994
  1898
  by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
hoelzl@56994
  1899
hoelzl@56994
  1900
lemma measurable_cong_sets:
hoelzl@56994
  1901
  assumes sets: "sets M = sets M'" "sets N = sets N'"
hoelzl@56994
  1902
  shows "measurable M N = measurable M' N'"
hoelzl@56994
  1903
  using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
hoelzl@56994
  1904
hoelzl@56994
  1905
lemma measurable_cong:
hoelzl@56994
  1906
  assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
hoelzl@56994
  1907
  shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
hoelzl@56994
  1908
  unfolding measurable_def using assms
hoelzl@56994
  1909
  by (simp cong: vimage_inter_cong Pi_cong)
hoelzl@56994
  1910
hoelzl@56994
  1911
lemma measurable_cong_strong:
hoelzl@56994
  1912
  "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
hoelzl@56994
  1913
    f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
hoelzl@56994
  1914
  by (metis measurable_cong)
hoelzl@56994
  1915
hoelzl@56994
  1916
lemma measurable_eqI:
hoelzl@56994
  1917
     "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
hoelzl@56994
  1918
        sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
hoelzl@56994
  1919
      \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
hoelzl@56994
  1920
  by (simp add: measurable_def sigma_algebra_iff2)
hoelzl@56994
  1921
hoelzl@56994
  1922
lemma measurable_compose:
hoelzl@56994
  1923
  assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
hoelzl@56994
  1924
  shows "(\<lambda>x. g (f x)) \<in> measurable M L"
hoelzl@56994
  1925
proof -
hoelzl@56994
  1926
  have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
hoelzl@56994
  1927
    using measurable_space[OF f] by auto
hoelzl@56994
  1928
  with measurable_space[OF f] measurable_space[OF g] show ?thesis
hoelzl@56994
  1929
    by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
hoelzl@56994
  1930
             simp del: vimage_Int simp add: measurable_def)
hoelzl@56994
  1931
qed
hoelzl@56994
  1932
hoelzl@56994
  1933
lemma measurable_comp:
hoelzl@56994
  1934
  "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
hoelzl@56994
  1935
  using measurable_compose[of f M N g L] by (simp add: comp_def)
hoelzl@56994
  1936
hoelzl@56994
  1937
lemma measurable_const:
hoelzl@56994
  1938
  "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
hoelzl@56994
  1939
  by (auto simp add: measurable_def)
hoelzl@56994
  1940
hoelzl@56994
  1941
lemma measurable_If:
hoelzl@56994
  1942
  assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
hoelzl@56994
  1943
  assumes P: "{x\<in>space M. P x} \<in> sets M"
hoelzl@56994
  1944
  shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
hoelzl@56994
  1945
  unfolding measurable_def
hoelzl@56994
  1946
proof safe
hoelzl@56994
  1947
  fix x assume "x \<in> space M"
hoelzl@56994
  1948
  thus "(if P x then f x else g x) \<in> space M'"
hoelzl@56994
  1949
    using measure unfolding measurable_def by auto
hoelzl@56994
  1950
next
hoelzl@56994
  1951
  fix A assume "A \<in> sets M'"
hoelzl@56994
  1952
  hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
hoelzl@56994
  1953
    ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
hoelzl@56994
  1954
    ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
hoelzl@56994
  1955
    using measure unfolding measurable_def by (auto split: split_if_asm)
hoelzl@56994
  1956
  show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
hoelzl@56994
  1957
    using `A \<in> sets M'` measure P unfolding * measurable_def
hoelzl@56994
  1958
    by (auto intro!: sets.Un)
hoelzl@56994
  1959
qed
hoelzl@56994
  1960
hoelzl@56994
  1961
lemma measurable_If_set:
hoelzl@56994
  1962
  assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
hoelzl@56994
  1963
  assumes P: "A \<inter> space M \<in> sets M"
hoelzl@56994
  1964
  shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
hoelzl@56994
  1965
proof (rule measurable_If[OF measure])
hoelzl@56994
  1966
  have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
hoelzl@56994
  1967
  thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
hoelzl@56994
  1968
qed
hoelzl@56994
  1969
hoelzl@56994
  1970
lemma measurable_ident: "id \<in> measurable M M"
hoelzl@56994
  1971
  by (auto simp add: measurable_def)
hoelzl@56994
  1972
hoelzl@56994
  1973
lemma measurable_ident_sets:
hoelzl@56994
  1974
  assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
hoelzl@56994
  1975
  using measurable_ident[of M]
hoelzl@56994
  1976
  unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
hoelzl@56994
  1977
hoelzl@56994
  1978
lemma sets_Least:
hoelzl@56994
  1979
  assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
hoelzl@56994
  1980
  shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
hoelzl@56994
  1981
proof -
hoelzl@56994
  1982
  { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
hoelzl@56994
  1983
    proof cases
hoelzl@56994
  1984
      assume i: "(LEAST j. False) = i"
hoelzl@56994
  1985
      have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
hoelzl@56994
  1986
        {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
hoelzl@56994
  1987
        by (simp add: set_eq_iff, safe)
hoelzl@56994
  1988
           (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
hoelzl@56994
  1989
      with meas show ?thesis
hoelzl@56994
  1990
        by (auto intro!: sets.Int)
hoelzl@56994
  1991
    next
hoelzl@56994
  1992
      assume i: "(LEAST j. False) \<noteq> i"
hoelzl@56994
  1993
      then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
hoelzl@56994
  1994
        {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
hoelzl@56994
  1995
      proof (simp add: set_eq_iff, safe)
hoelzl@56994
  1996
        fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
hoelzl@56994
  1997
        have "\<exists>j. P j x"
hoelzl@56994
  1998
          by (rule ccontr) (insert neq, auto)
hoelzl@56994
  1999
        then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
hoelzl@56994
  2000
      qed (auto dest: Least_le intro!: Least_equality)
hoelzl@56994
  2001
      with meas show ?thesis
hoelzl@56994
  2002
        by auto
hoelzl@56994
  2003
    qed }
hoelzl@56994
  2004
  then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
hoelzl@56994
  2005
    by (intro sets.countable_UN) auto
hoelzl@56994
  2006
  moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
hoelzl@56994
  2007
    (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
hoelzl@56994
  2008
  ultimately show ?thesis by auto
hoelzl@56994
  2009
qed
hoelzl@56994
  2010
hoelzl@56994
  2011
lemma measurable_strong:
hoelzl@56994
  2012
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
hoelzl@56994
  2013
  assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
hoelzl@56994
  2014
      and t: "f ` (space a) \<subseteq> t"
hoelzl@56994
  2015
      and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
hoelzl@56994
  2016
  shows "(g o f) \<in> measurable a c"
hoelzl@56994
  2017
proof -
hoelzl@56994
  2018
  have fab: "f \<in> (space a -> space b)"
hoelzl@56994
  2019
   and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
hoelzl@56994
  2020
     by (auto simp add: measurable_def)
hoelzl@56994
  2021
  have eq: "\<And>y. (g \<circ> f) -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
hoelzl@56994
  2022
    by force
hoelzl@56994
  2023
  show ?thesis
hoelzl@56994
  2024
    apply (auto simp add: measurable_def vimage_comp)
hoelzl@56994
  2025
    apply (metis funcset_mem fab g)
hoelzl@56994
  2026
    apply (subst eq)
hoelzl@56994
  2027
    apply (metis ba cb)
hoelzl@56994
  2028
    done
hoelzl@56994
  2029
qed
hoelzl@56994
  2030
hoelzl@56994
  2031
lemma measurable_mono1:
hoelzl@56994
  2032
  "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
hoelzl@56994
  2033
    measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
hoelzl@56994
  2034
  using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
hoelzl@56994
  2035
hoelzl@56994
  2036
subsubsection {* Counting space *}
hoelzl@56994
  2037
hoelzl@56994
  2038
definition count_space :: "'a set \<Rightarrow> 'a measure" where
hoelzl@56994
  2039
  "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
hoelzl@56994
  2040
hoelzl@56994
  2041
lemma 
hoelzl@56994
  2042
  shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
hoelzl@56994
  2043
    and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
hoelzl@56994
  2044
  using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
hoelzl@56994
  2045
  by (auto simp: count_space_def)