src/HOL/Probability/Sigma_Algebra.thy
author haftmann
Sat Jul 05 11:01:53 2014 +0200 (2014-07-05)
changeset 57514 bdc2c6b40bf2
parent 57512 cc97b347b301
child 58588 93d87fd1583d
permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
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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@57275
   367
hoelzl@57275
   368
lemma (in sigma_algebra) countable:
hoelzl@57275
   369
  assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> M" "countable A"
hoelzl@57275
   370
  shows "A \<in> M"
hoelzl@57275
   371
proof -
hoelzl@57275
   372
  have "(\<Union>a\<in>A. {a}) \<in> M"
hoelzl@57275
   373
    using assms by (intro countable_UN') auto
hoelzl@57275
   374
  also have "(\<Union>a\<in>A. {a}) = A" by auto
hoelzl@57275
   375
  finally show ?thesis by auto
hoelzl@57275
   376
qed
hoelzl@57275
   377
hoelzl@47694
   378
lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
hoelzl@47762
   379
  by (auto simp: ring_of_sets_iff)
hoelzl@42145
   380
hoelzl@47694
   381
lemma algebra_Pow: "algebra sp (Pow sp)"
hoelzl@47762
   382
  by (auto simp: algebra_iff_Un)
hoelzl@38656
   383
hoelzl@38656
   384
lemma sigma_algebra_iff:
hoelzl@47694
   385
  "sigma_algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694
   386
    algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
   387
  by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
paulson@33271
   388
hoelzl@47762
   389
lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
hoelzl@47762
   390
  by (auto simp: sigma_algebra_iff algebra_iff_Int)
hoelzl@47762
   391
hoelzl@42867
   392
lemma (in sigma_algebra) sets_Collect_countable_All:
hoelzl@47694
   393
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   394
  shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
hoelzl@42867
   395
proof -
hoelzl@47694
   396
  have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
hoelzl@42867
   397
  with assms show ?thesis by auto
hoelzl@42867
   398
qed
hoelzl@42867
   399
hoelzl@42867
   400
lemma (in sigma_algebra) sets_Collect_countable_Ex:
hoelzl@47694
   401
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   402
  shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
hoelzl@42867
   403
proof -
hoelzl@47694
   404
  have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
hoelzl@42867
   405
  with assms show ?thesis by auto
hoelzl@42867
   406
qed
hoelzl@42867
   407
hoelzl@50526
   408
lemma (in sigma_algebra) sets_Collect_countable_Ex':
hoelzl@54418
   409
  assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@50526
   410
  assumes "countable I"
hoelzl@50526
   411
  shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
hoelzl@50526
   412
proof -
hoelzl@50526
   413
  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
   414
  with assms show ?thesis 
hoelzl@50526
   415
    by (auto intro!: countable_UN')
hoelzl@50526
   416
qed
hoelzl@50526
   417
hoelzl@54418
   418
lemma (in sigma_algebra) sets_Collect_countable_All':
hoelzl@54418
   419
  assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@54418
   420
  assumes "countable I"
hoelzl@54418
   421
  shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M"
hoelzl@54418
   422
proof -
hoelzl@54418
   423
  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
   424
  with assms show ?thesis 
hoelzl@54418
   425
    by (cases "I = {}") (auto intro!: countable_INT')
hoelzl@54418
   426
qed
hoelzl@54418
   427
hoelzl@54418
   428
lemma (in sigma_algebra) sets_Collect_countable_Ex1':
hoelzl@54418
   429
  assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@54418
   430
  assumes "countable I"
hoelzl@54418
   431
  shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M"
hoelzl@54418
   432
proof -
hoelzl@54418
   433
  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
   434
    by auto
hoelzl@54418
   435
  with assms show ?thesis 
hoelzl@54418
   436
    by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)
hoelzl@54418
   437
qed
hoelzl@54418
   438
hoelzl@42867
   439
lemmas (in sigma_algebra) sets_Collect =
hoelzl@42867
   440
  sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
hoelzl@42867
   441
  sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
hoelzl@42867
   442
hoelzl@47694
   443
lemma (in sigma_algebra) sets_Collect_countable_Ball:
hoelzl@47694
   444
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   445
  shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
hoelzl@47694
   446
  unfolding Ball_def by (intro sets_Collect assms)
hoelzl@47694
   447
hoelzl@47694
   448
lemma (in sigma_algebra) sets_Collect_countable_Bex:
hoelzl@47694
   449
  assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694
   450
  shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
hoelzl@47694
   451
  unfolding Bex_def by (intro sets_Collect assms)
hoelzl@47694
   452
hoelzl@42984
   453
lemma sigma_algebra_single_set:
hoelzl@42984
   454
  assumes "X \<subseteq> S"
hoelzl@47694
   455
  shows "sigma_algebra S { {}, X, S - X, S }"
hoelzl@42984
   456
  using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
hoelzl@42984
   457
hoelzl@56994
   458
subsubsection {* Binary Unions *}
paulson@33271
   459
paulson@33271
   460
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
wenzelm@50252
   461
  where "binary a b =  (\<lambda>x. b)(0 := a)"
paulson@33271
   462
hoelzl@38656
   463
lemma range_binary_eq: "range(binary a b) = {a,b}"
hoelzl@38656
   464
  by (auto simp add: binary_def)
paulson@33271
   465
hoelzl@38656
   466
lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
haftmann@44106
   467
  by (simp add: SUP_def range_binary_eq)
paulson@33271
   468
hoelzl@38656
   469
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
haftmann@44106
   470
  by (simp add: INF_def range_binary_eq)
paulson@33271
   471
paulson@33271
   472
lemma sigma_algebra_iff2:
hoelzl@47694
   473
     "sigma_algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694
   474
       M \<subseteq> Pow \<Omega> \<and>
hoelzl@47694
   475
       {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
hoelzl@47694
   476
       (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
   477
  by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
hoelzl@42065
   478
         algebra_iff_Un Un_range_binary)
paulson@33271
   479
hoelzl@56994
   480
subsubsection {* Initial Sigma Algebra *}
paulson@33271
   481
paulson@33271
   482
text {*Sigma algebras can naturally be created as the closure of any set of
hoelzl@47694
   483
  M with regard to the properties just postulated.  *}
paulson@33271
   484
hoelzl@51683
   485
inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
paulson@33271
   486
  for sp :: "'a set" and A :: "'a set set"
paulson@33271
   487
  where
hoelzl@47694
   488
    Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
paulson@33271
   489
  | Empty: "{} \<in> sigma_sets sp A"
paulson@33271
   490
  | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
paulson@33271
   491
  | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
paulson@33271
   492
hoelzl@41543
   493
lemma (in sigma_algebra) sigma_sets_subset:
hoelzl@47694
   494
  assumes a: "a \<subseteq> M"
hoelzl@47694
   495
  shows "sigma_sets \<Omega> a \<subseteq> M"
hoelzl@41543
   496
proof
hoelzl@41543
   497
  fix x
hoelzl@47694
   498
  assume "x \<in> sigma_sets \<Omega> a"
hoelzl@47694
   499
  from this show "x \<in> M"
hoelzl@41543
   500
    by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
hoelzl@41543
   501
qed
hoelzl@41543
   502
hoelzl@41543
   503
lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
hoelzl@41543
   504
  by (erule sigma_sets.induct, auto)
hoelzl@41543
   505
hoelzl@41543
   506
lemma sigma_algebra_sigma_sets:
hoelzl@47694
   507
     "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
hoelzl@41543
   508
  by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
hoelzl@41543
   509
           intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
hoelzl@41543
   510
hoelzl@41543
   511
lemma sigma_sets_least_sigma_algebra:
hoelzl@41543
   512
  assumes "A \<subseteq> Pow S"
hoelzl@47694
   513
  shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
hoelzl@41543
   514
proof safe
hoelzl@47694
   515
  fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
hoelzl@41543
   516
    and X: "X \<in> sigma_sets S A"
hoelzl@41543
   517
  from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
hoelzl@41543
   518
  show "X \<in> B" by auto
hoelzl@41543
   519
next
hoelzl@47694
   520
  fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
hoelzl@47694
   521
  then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
hoelzl@41543
   522
     by simp
hoelzl@47694
   523
  have "A \<subseteq> sigma_sets S A" using assms by auto
hoelzl@47694
   524
  moreover have "sigma_algebra S (sigma_sets S A)"
hoelzl@41543
   525
    using assms by (intro sigma_algebra_sigma_sets[of A]) auto
hoelzl@41543
   526
  ultimately show "X \<in> sigma_sets S A" by auto
hoelzl@41543
   527
qed
paulson@33271
   528
paulson@33271
   529
lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
paulson@33271
   530
  by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
paulson@33271
   531
hoelzl@38656
   532
lemma sigma_sets_Un:
paulson@33271
   533
  "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
hoelzl@38656
   534
apply (simp add: Un_range_binary range_binary_eq)
hoelzl@40859
   535
apply (rule Union, simp add: binary_def)
paulson@33271
   536
done
paulson@33271
   537
paulson@33271
   538
lemma sigma_sets_Inter:
paulson@33271
   539
  assumes Asb: "A \<subseteq> Pow sp"
paulson@33271
   540
  shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
paulson@33271
   541
proof -
paulson@33271
   542
  assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
hoelzl@38656
   543
  hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
paulson@33271
   544
    by (rule sigma_sets.Compl)
hoelzl@38656
   545
  hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
paulson@33271
   546
    by (rule sigma_sets.Union)
hoelzl@38656
   547
  hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
paulson@33271
   548
    by (rule sigma_sets.Compl)
hoelzl@38656
   549
  also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
paulson@33271
   550
    by auto
paulson@33271
   551
  also have "... = (\<Inter>i. a i)" using ai
hoelzl@38656
   552
    by (blast dest: sigma_sets_into_sp [OF Asb])
hoelzl@38656
   553
  finally show ?thesis .
paulson@33271
   554
qed
paulson@33271
   555
paulson@33271
   556
lemma sigma_sets_INTER:
hoelzl@38656
   557
  assumes Asb: "A \<subseteq> Pow sp"
paulson@33271
   558
      and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
paulson@33271
   559
  shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
paulson@33271
   560
proof -
paulson@33271
   561
  from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
hoelzl@47694
   562
    by (simp add: sigma_sets.intros(2-) sigma_sets_top)
paulson@33271
   563
  hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
paulson@33271
   564
    by (rule sigma_sets_Inter [OF Asb])
paulson@33271
   565
  also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
paulson@33271
   566
    by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
paulson@33271
   567
  finally show ?thesis .
paulson@33271
   568
qed
paulson@33271
   569
hoelzl@51683
   570
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
   571
  using from_nat_into[of B] range_from_nat_into[of B] sigma_sets.Union[of "from_nat_into B" X A]
hoelzl@51683
   572
  apply (cases "B = {}")
hoelzl@51683
   573
  apply (simp add: sigma_sets.Empty)
hoelzl@51683
   574
  apply (simp del: Union_image_eq add: Union_image_eq[symmetric])
hoelzl@51683
   575
  done
hoelzl@51683
   576
paulson@33271
   577
lemma (in sigma_algebra) sigma_sets_eq:
hoelzl@47694
   578
     "sigma_sets \<Omega> M = M"
paulson@33271
   579
proof
hoelzl@47694
   580
  show "M \<subseteq> sigma_sets \<Omega> M"
huffman@37032
   581
    by (metis Set.subsetI sigma_sets.Basic)
paulson@33271
   582
  next
hoelzl@47694
   583
  show "sigma_sets \<Omega> M \<subseteq> M"
paulson@33271
   584
    by (metis sigma_sets_subset subset_refl)
paulson@33271
   585
qed
paulson@33271
   586
hoelzl@42981
   587
lemma sigma_sets_eqI:
hoelzl@42981
   588
  assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
hoelzl@42981
   589
  assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
hoelzl@42981
   590
  shows "sigma_sets M A = sigma_sets M B"
hoelzl@42981
   591
proof (intro set_eqI iffI)
hoelzl@42981
   592
  fix a assume "a \<in> sigma_sets M A"
hoelzl@42981
   593
  from this A show "a \<in> sigma_sets M B"
hoelzl@47694
   594
    by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
hoelzl@42981
   595
next
hoelzl@42981
   596
  fix b assume "b \<in> sigma_sets M B"
hoelzl@42981
   597
  from this B show "b \<in> sigma_sets M A"
hoelzl@47694
   598
    by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
hoelzl@42981
   599
qed
hoelzl@42981
   600
hoelzl@42984
   601
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@42984
   602
proof
hoelzl@42984
   603
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47694
   604
    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
hoelzl@42984
   605
qed
hoelzl@42984
   606
hoelzl@47762
   607
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@47762
   608
proof
hoelzl@47762
   609
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47762
   610
    by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
hoelzl@47762
   611
qed
hoelzl@47762
   612
hoelzl@47762
   613
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@47762
   614
proof
hoelzl@47762
   615
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47762
   616
    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
hoelzl@47762
   617
qed
hoelzl@47762
   618
hoelzl@47762
   619
lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
hoelzl@47762
   620
  by (auto intro: sigma_sets.Basic)
hoelzl@47762
   621
hoelzl@38656
   622
lemma (in sigma_algebra) restriction_in_sets:
hoelzl@38656
   623
  fixes A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
   624
  assumes "S \<in> M"
hoelzl@47694
   625
  and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
hoelzl@47694
   626
  shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
hoelzl@38656
   627
proof -
hoelzl@38656
   628
  { fix i have "A i \<in> ?r" using * by auto
hoelzl@47694
   629
    hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
hoelzl@47694
   630
    hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
hoelzl@47694
   631
  thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
hoelzl@38656
   632
    by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
hoelzl@38656
   633
qed
hoelzl@38656
   634
hoelzl@38656
   635
lemma (in sigma_algebra) restricted_sigma_algebra:
hoelzl@47694
   636
  assumes "S \<in> M"
hoelzl@47694
   637
  shows "sigma_algebra S (restricted_space S)"
hoelzl@38656
   638
  unfolding sigma_algebra_def sigma_algebra_axioms_def
hoelzl@38656
   639
proof safe
hoelzl@47694
   640
  show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
hoelzl@38656
   641
next
hoelzl@47694
   642
  fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
hoelzl@38656
   643
  from restriction_in_sets[OF assms this[simplified]]
hoelzl@47694
   644
  show "(\<Union>i. A i) \<in> restricted_space S" by simp
hoelzl@38656
   645
qed
hoelzl@38656
   646
hoelzl@40859
   647
lemma sigma_sets_Int:
hoelzl@41689
   648
  assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
hoelzl@41689
   649
  shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
hoelzl@40859
   650
proof (intro equalityI subsetI)
hoelzl@40859
   651
  fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
hoelzl@40859
   652
  then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
hoelzl@41689
   653
  then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
hoelzl@40859
   654
  proof (induct arbitrary: x)
hoelzl@40859
   655
    case (Compl a)
hoelzl@40859
   656
    then show ?case
hoelzl@40859
   657
      by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
hoelzl@40859
   658
  next
hoelzl@40859
   659
    case (Union a)
hoelzl@40859
   660
    then show ?case
hoelzl@40859
   661
      by (auto intro!: sigma_sets.Union
hoelzl@40859
   662
               simp add: UN_extend_simps simp del: UN_simps)
hoelzl@47694
   663
  qed (auto intro!: sigma_sets.intros(2-))
hoelzl@41689
   664
  then show "x \<in> sigma_sets A (op \<inter> A ` st)"
hoelzl@41689
   665
    using `A \<subseteq> sp` by (simp add: Int_absorb2)
hoelzl@40859
   666
next
hoelzl@41689
   667
  fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
hoelzl@40859
   668
  then show "x \<in> op \<inter> A ` sigma_sets sp st"
hoelzl@40859
   669
  proof induct
hoelzl@40859
   670
    case (Compl a)
hoelzl@40859
   671
    then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
hoelzl@41689
   672
    then show ?case using `A \<subseteq> sp`
hoelzl@40859
   673
      by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
hoelzl@40859
   674
  next
hoelzl@40859
   675
    case (Union a)
hoelzl@40859
   676
    then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
hoelzl@40859
   677
      by (auto simp: image_iff Bex_def)
hoelzl@40859
   678
    from choice[OF this] guess f ..
hoelzl@40859
   679
    then show ?case
hoelzl@40859
   680
      by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
hoelzl@40859
   681
               simp add: image_iff)
hoelzl@47694
   682
  qed (auto intro!: sigma_sets.intros(2-))
hoelzl@40859
   683
qed
hoelzl@40859
   684
hoelzl@47694
   685
lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
hoelzl@40859
   686
proof (intro set_eqI iffI)
hoelzl@47694
   687
  fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
hoelzl@47694
   688
    by induct blast+
hoelzl@47694
   689
qed (auto intro: sigma_sets.Empty sigma_sets_top)
hoelzl@47694
   690
hoelzl@47694
   691
lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
hoelzl@47694
   692
proof (intro set_eqI iffI)
hoelzl@47694
   693
  fix x assume "x \<in> sigma_sets A {A}"
hoelzl@47694
   694
  then show "x \<in> {{}, A}"
hoelzl@47694
   695
    by induct blast+
hoelzl@40859
   696
next
hoelzl@47694
   697
  fix x assume "x \<in> {{}, A}"
hoelzl@47694
   698
  then show "x \<in> sigma_sets A {A}"
hoelzl@40859
   699
    by (auto intro: sigma_sets.Empty sigma_sets_top)
hoelzl@40859
   700
qed
hoelzl@40859
   701
hoelzl@42987
   702
lemma sigma_sets_sigma_sets_eq:
hoelzl@42987
   703
  "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
hoelzl@47694
   704
  by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
hoelzl@42987
   705
hoelzl@42984
   706
lemma sigma_sets_singleton:
hoelzl@42984
   707
  assumes "X \<subseteq> S"
hoelzl@42984
   708
  shows "sigma_sets S { X } = { {}, X, S - X, S }"
hoelzl@42984
   709
proof -
hoelzl@47694
   710
  interpret sigma_algebra S "{ {}, X, S - X, S }"
hoelzl@42984
   711
    by (rule sigma_algebra_single_set) fact
hoelzl@42984
   712
  have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
hoelzl@42984
   713
    by (rule sigma_sets_subseteq) simp
hoelzl@42984
   714
  moreover have "\<dots> = { {}, X, S - X, S }"
hoelzl@47694
   715
    using sigma_sets_eq by simp
hoelzl@42984
   716
  moreover
hoelzl@42984
   717
  { fix A assume "A \<in> { {}, X, S - X, S }"
hoelzl@42984
   718
    then have "A \<in> sigma_sets S { X }"
hoelzl@47694
   719
      by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
hoelzl@42984
   720
  ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
hoelzl@42984
   721
    by (intro antisym) auto
hoelzl@47694
   722
  with sigma_sets_eq show ?thesis by simp
hoelzl@42984
   723
qed
hoelzl@42984
   724
hoelzl@42863
   725
lemma restricted_sigma:
hoelzl@47694
   726
  assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
hoelzl@47694
   727
  shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
hoelzl@47694
   728
    sigma_sets S (algebra.restricted_space M S)"
hoelzl@42863
   729
proof -
hoelzl@42863
   730
  from S sigma_sets_into_sp[OF M]
hoelzl@47694
   731
  have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
hoelzl@42863
   732
  from sigma_sets_Int[OF this]
hoelzl@47694
   733
  show ?thesis by simp
hoelzl@42863
   734
qed
hoelzl@42863
   735
hoelzl@42987
   736
lemma sigma_sets_vimage_commute:
hoelzl@47694
   737
  assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
hoelzl@47694
   738
  shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
hoelzl@47694
   739
       = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
hoelzl@42987
   740
proof
hoelzl@42987
   741
  show "?L \<subseteq> ?R"
hoelzl@42987
   742
  proof clarify
hoelzl@47694
   743
    fix A assume "A \<in> sigma_sets \<Omega>' M'"
hoelzl@47694
   744
    then show "X -` A \<inter> \<Omega> \<in> ?R"
hoelzl@42987
   745
    proof induct
hoelzl@42987
   746
      case Empty then show ?case
hoelzl@42987
   747
        by (auto intro!: sigma_sets.Empty)
hoelzl@42987
   748
    next
hoelzl@42987
   749
      case (Compl B)
hoelzl@47694
   750
      have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
hoelzl@42987
   751
        by (auto simp add: funcset_mem [OF X])
hoelzl@42987
   752
      with Compl show ?case
hoelzl@42987
   753
        by (auto intro!: sigma_sets.Compl)
hoelzl@42987
   754
    next
hoelzl@42987
   755
      case (Union F)
hoelzl@42987
   756
      then show ?case
hoelzl@42987
   757
        by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
hoelzl@42987
   758
                 intro!: sigma_sets.Union)
hoelzl@47694
   759
    qed auto
hoelzl@42987
   760
  qed
hoelzl@42987
   761
  show "?R \<subseteq> ?L"
hoelzl@42987
   762
  proof clarify
hoelzl@42987
   763
    fix A assume "A \<in> ?R"
hoelzl@47694
   764
    then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
hoelzl@42987
   765
    proof induct
hoelzl@42987
   766
      case (Basic B) then show ?case by auto
hoelzl@42987
   767
    next
hoelzl@42987
   768
      case Empty then show ?case
hoelzl@47694
   769
        by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
hoelzl@42987
   770
    next
hoelzl@42987
   771
      case (Compl B)
hoelzl@47694
   772
      then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
hoelzl@47694
   773
      then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
hoelzl@42987
   774
        by (auto simp add: funcset_mem [OF X])
hoelzl@42987
   775
      with A(2) show ?case
hoelzl@47694
   776
        by (auto intro: sigma_sets.Compl)
hoelzl@42987
   777
    next
hoelzl@42987
   778
      case (Union F)
hoelzl@47694
   779
      then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
hoelzl@42987
   780
      from choice[OF this] guess A .. note A = this
hoelzl@42987
   781
      with A show ?case
hoelzl@47694
   782
        by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
hoelzl@42987
   783
    qed
hoelzl@42987
   784
  qed
hoelzl@42987
   785
qed
hoelzl@42987
   786
hoelzl@56994
   787
subsubsection "Disjoint families"
hoelzl@38656
   788
hoelzl@38656
   789
definition
hoelzl@38656
   790
  disjoint_family_on  where
hoelzl@38656
   791
  "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
   792
hoelzl@38656
   793
abbreviation
hoelzl@38656
   794
  "disjoint_family A \<equiv> disjoint_family_on A UNIV"
hoelzl@38656
   795
hoelzl@38656
   796
lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
hoelzl@38656
   797
  by blast
hoelzl@38656
   798
hoelzl@57447
   799
lemma disjoint_family_onD: "disjoint_family_on A I \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
hoelzl@57447
   800
  by (auto simp: disjoint_family_on_def)
hoelzl@57447
   801
hoelzl@38656
   802
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
hoelzl@38656
   803
  by blast
hoelzl@38656
   804
hoelzl@38656
   805
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
hoelzl@38656
   806
  by blast
hoelzl@38656
   807
hoelzl@38656
   808
lemma disjoint_family_subset:
hoelzl@38656
   809
     "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
hoelzl@38656
   810
  by (force simp add: disjoint_family_on_def)
hoelzl@38656
   811
hoelzl@40859
   812
lemma disjoint_family_on_bisimulation:
hoelzl@40859
   813
  assumes "disjoint_family_on f S"
hoelzl@40859
   814
  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
   815
  shows "disjoint_family_on g S"
hoelzl@40859
   816
  using assms unfolding disjoint_family_on_def by auto
hoelzl@40859
   817
hoelzl@38656
   818
lemma disjoint_family_on_mono:
hoelzl@38656
   819
  "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
hoelzl@38656
   820
  unfolding disjoint_family_on_def by auto
hoelzl@38656
   821
hoelzl@38656
   822
lemma disjoint_family_Suc:
hoelzl@38656
   823
  assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
hoelzl@38656
   824
  shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
hoelzl@38656
   825
proof -
hoelzl@38656
   826
  {
hoelzl@38656
   827
    fix m
hoelzl@38656
   828
    have "!!n. A n \<subseteq> A (m+n)"
hoelzl@38656
   829
    proof (induct m)
hoelzl@38656
   830
      case 0 show ?case by simp
hoelzl@38656
   831
    next
hoelzl@38656
   832
      case (Suc m) thus ?case
haftmann@57512
   833
        by (metis Suc_eq_plus1 assms add.commute add.left_commute subset_trans)
hoelzl@38656
   834
    qed
hoelzl@38656
   835
  }
hoelzl@38656
   836
  hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
haftmann@57512
   837
    by (metis add.commute le_add_diff_inverse nat_less_le)
hoelzl@38656
   838
  thus ?thesis
hoelzl@38656
   839
    by (auto simp add: disjoint_family_on_def)
hoelzl@38656
   840
      (metis insert_absorb insert_subset le_SucE le_antisym not_leE)
hoelzl@38656
   841
qed
hoelzl@38656
   842
hoelzl@39092
   843
lemma setsum_indicator_disjoint_family:
hoelzl@39092
   844
  fixes f :: "'d \<Rightarrow> 'e::semiring_1"
hoelzl@39092
   845
  assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
hoelzl@39092
   846
  shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
hoelzl@39092
   847
proof -
hoelzl@39092
   848
  have "P \<inter> {i. x \<in> A i} = {j}"
hoelzl@39092
   849
    using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
hoelzl@39092
   850
    by auto
hoelzl@39092
   851
  thus ?thesis
hoelzl@39092
   852
    unfolding indicator_def
haftmann@57418
   853
    by (simp add: if_distrib setsum.If_cases[OF `finite P`])
hoelzl@39092
   854
qed
hoelzl@39092
   855
hoelzl@38656
   856
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
hoelzl@38656
   857
  where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
hoelzl@38656
   858
hoelzl@38656
   859
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
hoelzl@38656
   860
proof (induct n)
hoelzl@38656
   861
  case 0 show ?case by simp
hoelzl@38656
   862
next
hoelzl@38656
   863
  case (Suc n)
hoelzl@38656
   864
  thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
hoelzl@38656
   865
qed
hoelzl@38656
   866
hoelzl@38656
   867
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
hoelzl@38656
   868
  apply (rule UN_finite2_eq [where k=0])
hoelzl@38656
   869
  apply (simp add: finite_UN_disjointed_eq)
hoelzl@38656
   870
  done
hoelzl@38656
   871
hoelzl@38656
   872
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
hoelzl@38656
   873
  by (auto simp add: disjointed_def)
hoelzl@38656
   874
hoelzl@38656
   875
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
hoelzl@38656
   876
  by (simp add: disjoint_family_on_def)
hoelzl@38656
   877
     (metis neq_iff Int_commute less_disjoint_disjointed)
hoelzl@38656
   878
hoelzl@38656
   879
lemma disjointed_subset: "disjointed A n \<subseteq> A n"
hoelzl@38656
   880
  by (auto simp add: disjointed_def)
hoelzl@38656
   881
hoelzl@42065
   882
lemma (in ring_of_sets) UNION_in_sets:
hoelzl@38656
   883
  fixes A:: "nat \<Rightarrow> 'a set"
hoelzl@47694
   884
  assumes A: "range A \<subseteq> M"
hoelzl@47694
   885
  shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
hoelzl@38656
   886
proof (induct n)
hoelzl@38656
   887
  case 0 show ?case by simp
hoelzl@38656
   888
next
hoelzl@38656
   889
  case (Suc n)
hoelzl@38656
   890
  thus ?case
hoelzl@38656
   891
    by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
hoelzl@38656
   892
qed
hoelzl@38656
   893
hoelzl@42065
   894
lemma (in ring_of_sets) range_disjointed_sets:
hoelzl@47694
   895
  assumes A: "range A \<subseteq> M"
hoelzl@47694
   896
  shows  "range (disjointed A) \<subseteq> M"
hoelzl@38656
   897
proof (auto simp add: disjointed_def)
hoelzl@38656
   898
  fix n
hoelzl@47694
   899
  show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
hoelzl@38656
   900
    by (metis A Diff UNIV_I image_subset_iff)
hoelzl@38656
   901
qed
hoelzl@38656
   902
hoelzl@42065
   903
lemma (in algebra) range_disjointed_sets':
hoelzl@47694
   904
  "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
hoelzl@42065
   905
  using range_disjointed_sets .
hoelzl@42065
   906
hoelzl@42145
   907
lemma disjointed_0[simp]: "disjointed A 0 = A 0"
hoelzl@42145
   908
  by (simp add: disjointed_def)
hoelzl@42145
   909
hoelzl@42145
   910
lemma incseq_Un:
hoelzl@42145
   911
  "incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
hoelzl@42145
   912
  unfolding incseq_def by auto
hoelzl@42145
   913
hoelzl@42145
   914
lemma disjointed_incseq:
hoelzl@42145
   915
  "incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
hoelzl@42145
   916
  using incseq_Un[of A]
hoelzl@42145
   917
  by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
hoelzl@42145
   918
hoelzl@38656
   919
lemma sigma_algebra_disjoint_iff:
hoelzl@47694
   920
  "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
hoelzl@47694
   921
    (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
   922
proof (auto simp add: sigma_algebra_iff)
hoelzl@38656
   923
  fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
   924
  assume M: "algebra \<Omega> M"
hoelzl@47694
   925
     and A: "range A \<subseteq> M"
hoelzl@47694
   926
     and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694
   927
  hence "range (disjointed A) \<subseteq> M \<longrightarrow>
hoelzl@38656
   928
         disjoint_family (disjointed A) \<longrightarrow>
hoelzl@47694
   929
         (\<Union>i. disjointed A i) \<in> M" by blast
hoelzl@47694
   930
  hence "(\<Union>i. disjointed A i) \<in> M"
hoelzl@47694
   931
    by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
hoelzl@47694
   932
  thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
hoelzl@47694
   933
qed
hoelzl@47694
   934
hoelzl@47762
   935
lemma disjoint_family_on_disjoint_image:
hoelzl@47762
   936
  "disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
hoelzl@47762
   937
  unfolding disjoint_family_on_def disjoint_def by force
hoelzl@47762
   938
hoelzl@47762
   939
lemma disjoint_image_disjoint_family_on:
hoelzl@47762
   940
  assumes d: "disjoint (A ` I)" and i: "inj_on A I"
hoelzl@47762
   941
  shows "disjoint_family_on A I"
hoelzl@47762
   942
  unfolding disjoint_family_on_def
hoelzl@47762
   943
proof (intro ballI impI)
hoelzl@47762
   944
  fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
hoelzl@47762
   945
  with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
hoelzl@47762
   946
    by (intro disjointD[OF d]) auto
hoelzl@47762
   947
qed
hoelzl@47762
   948
hoelzl@56994
   949
subsubsection {* Ring generated by a semiring *}
hoelzl@47762
   950
hoelzl@47762
   951
definition (in semiring_of_sets)
hoelzl@47762
   952
  "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
hoelzl@47762
   953
hoelzl@47762
   954
lemma (in semiring_of_sets) generated_ringE[elim?]:
hoelzl@47762
   955
  assumes "a \<in> generated_ring"
hoelzl@47762
   956
  obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
hoelzl@47762
   957
  using assms unfolding generated_ring_def by auto
hoelzl@47762
   958
hoelzl@47762
   959
lemma (in semiring_of_sets) generated_ringI[intro?]:
hoelzl@47762
   960
  assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
hoelzl@47762
   961
  shows "a \<in> generated_ring"
hoelzl@47762
   962
  using assms unfolding generated_ring_def by auto
hoelzl@47762
   963
hoelzl@47762
   964
lemma (in semiring_of_sets) generated_ringI_Basic:
hoelzl@47762
   965
  "A \<in> M \<Longrightarrow> A \<in> generated_ring"
hoelzl@47762
   966
  by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
hoelzl@47762
   967
hoelzl@47762
   968
lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
hoelzl@47762
   969
  assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
hoelzl@47762
   970
  and "a \<inter> b = {}"
hoelzl@47762
   971
  shows "a \<union> b \<in> generated_ring"
hoelzl@47762
   972
proof -
hoelzl@47762
   973
  from a guess Ca .. note Ca = this
hoelzl@47762
   974
  from b guess Cb .. note Cb = this
hoelzl@47762
   975
  show ?thesis
hoelzl@47762
   976
  proof
hoelzl@47762
   977
    show "disjoint (Ca \<union> Cb)"
hoelzl@47762
   978
      using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
hoelzl@47762
   979
  qed (insert Ca Cb, auto)
hoelzl@47762
   980
qed
hoelzl@47762
   981
hoelzl@47762
   982
lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
hoelzl@47762
   983
  by (auto simp: generated_ring_def disjoint_def)
hoelzl@47762
   984
hoelzl@47762
   985
lemma (in semiring_of_sets) generated_ring_disjoint_Union:
hoelzl@47762
   986
  assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
hoelzl@47762
   987
  using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
hoelzl@47762
   988
hoelzl@47762
   989
lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
hoelzl@47762
   990
  "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
   991
  unfolding SUP_def by (intro generated_ring_disjoint_Union) auto
hoelzl@47762
   992
hoelzl@47762
   993
lemma (in semiring_of_sets) generated_ring_Int:
hoelzl@47762
   994
  assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
hoelzl@47762
   995
  shows "a \<inter> b \<in> generated_ring"
hoelzl@47762
   996
proof -
hoelzl@47762
   997
  from a guess Ca .. note Ca = this
hoelzl@47762
   998
  from b guess Cb .. note Cb = this
hoelzl@47762
   999
  def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
hoelzl@47762
  1000
  show ?thesis
hoelzl@47762
  1001
  proof
hoelzl@47762
  1002
    show "disjoint C"
hoelzl@47762
  1003
    proof (simp add: disjoint_def C_def, intro ballI impI)
hoelzl@47762
  1004
      fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
hoelzl@47762
  1005
      assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
hoelzl@47762
  1006
      then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
hoelzl@47762
  1007
      then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
hoelzl@47762
  1008
      proof
hoelzl@47762
  1009
        assume "a1 \<noteq> a2"
hoelzl@47762
  1010
        with sets Ca have "a1 \<inter> a2 = {}"
hoelzl@47762
  1011
          by (auto simp: disjoint_def)
hoelzl@47762
  1012
        then show ?thesis by auto
hoelzl@47762
  1013
      next
hoelzl@47762
  1014
        assume "b1 \<noteq> b2"
hoelzl@47762
  1015
        with sets Cb have "b1 \<inter> b2 = {}"
hoelzl@47762
  1016
          by (auto simp: disjoint_def)
hoelzl@47762
  1017
        then show ?thesis by auto
hoelzl@47762
  1018
      qed
hoelzl@47762
  1019
    qed
hoelzl@47762
  1020
  qed (insert Ca Cb, auto simp: C_def)
hoelzl@47762
  1021
qed
hoelzl@47762
  1022
hoelzl@47762
  1023
lemma (in semiring_of_sets) generated_ring_Inter:
hoelzl@47762
  1024
  assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
hoelzl@47762
  1025
  using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
hoelzl@47762
  1026
hoelzl@47762
  1027
lemma (in semiring_of_sets) generated_ring_INTER:
hoelzl@47762
  1028
  "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
  1029
  unfolding INF_def by (intro generated_ring_Inter) auto
hoelzl@47762
  1030
hoelzl@47762
  1031
lemma (in semiring_of_sets) generating_ring:
hoelzl@47762
  1032
  "ring_of_sets \<Omega> generated_ring"
hoelzl@47762
  1033
proof (rule ring_of_setsI)
hoelzl@47762
  1034
  let ?R = generated_ring
hoelzl@47762
  1035
  show "?R \<subseteq> Pow \<Omega>"
hoelzl@47762
  1036
    using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
hoelzl@47762
  1037
  show "{} \<in> ?R" by (rule generated_ring_empty)
hoelzl@47762
  1038
hoelzl@47762
  1039
  { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
hoelzl@47762
  1040
    fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
hoelzl@47762
  1041
  
hoelzl@47762
  1042
    show "a - b \<in> ?R"
hoelzl@47762
  1043
    proof cases
hoelzl@47762
  1044
      assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
hoelzl@47762
  1045
        by simp
hoelzl@47762
  1046
    next
hoelzl@47762
  1047
      assume "Cb \<noteq> {}"
hoelzl@47762
  1048
      with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
hoelzl@47762
  1049
      also have "\<dots> \<in> ?R"
hoelzl@47762
  1050
      proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
hoelzl@47762
  1051
        fix a b assume "a \<in> Ca" "b \<in> Cb"
hoelzl@47762
  1052
        with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
hoelzl@47762
  1053
          by (auto simp add: generated_ring_def)
hoelzl@47762
  1054
      next
hoelzl@47762
  1055
        show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
hoelzl@47762
  1056
          using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
hoelzl@47762
  1057
      next
hoelzl@47762
  1058
        show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
hoelzl@47762
  1059
      qed
hoelzl@47762
  1060
      finally show "a - b \<in> ?R" .
hoelzl@47762
  1061
    qed }
hoelzl@47762
  1062
  note Diff = this
hoelzl@47762
  1063
hoelzl@47762
  1064
  fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
hoelzl@47762
  1065
  have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
hoelzl@47762
  1066
  also have "\<dots> \<in> ?R"
hoelzl@47762
  1067
    by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
hoelzl@47762
  1068
  finally show "a \<union> b \<in> ?R" .
hoelzl@47762
  1069
qed
hoelzl@47762
  1070
hoelzl@47762
  1071
lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
hoelzl@47762
  1072
proof
hoelzl@47762
  1073
  interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
hoelzl@47762
  1074
    using space_closed by (rule sigma_algebra_sigma_sets)
hoelzl@47762
  1075
  show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
hoelzl@47762
  1076
    by (blast intro!: sigma_sets_mono elim: generated_ringE)
hoelzl@47762
  1077
qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
hoelzl@47762
  1078
hoelzl@56994
  1079
subsubsection {* A Two-Element Series *}
hoelzl@38656
  1080
hoelzl@38656
  1081
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
wenzelm@50252
  1082
  where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
hoelzl@38656
  1083
hoelzl@38656
  1084
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
hoelzl@38656
  1085
  apply (simp add: binaryset_def)
nipkow@39302
  1086
  apply (rule set_eqI)
hoelzl@38656
  1087
  apply (auto simp add: image_iff)
hoelzl@38656
  1088
  done
hoelzl@38656
  1089
hoelzl@38656
  1090
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
haftmann@44106
  1091
  by (simp add: SUP_def range_binaryset_eq)
hoelzl@38656
  1092
hoelzl@56994
  1093
subsubsection {* Closed CDI *}
hoelzl@38656
  1094
hoelzl@47694
  1095
definition closed_cdi where
hoelzl@47694
  1096
  "closed_cdi \<Omega> M \<longleftrightarrow>
hoelzl@47694
  1097
   M \<subseteq> Pow \<Omega> &
hoelzl@47694
  1098
   (\<forall>s \<in> M. \<Omega> - s \<in> M) &
hoelzl@47694
  1099
   (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
hoelzl@47694
  1100
        (\<Union>i. A i) \<in> M) &
hoelzl@47694
  1101
   (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656
  1102
hoelzl@38656
  1103
inductive_set
hoelzl@47694
  1104
  smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
hoelzl@47694
  1105
  for \<Omega> M
hoelzl@38656
  1106
  where
hoelzl@38656
  1107
    Basic [intro]:
hoelzl@47694
  1108
      "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1109
  | Compl [intro]:
hoelzl@47694
  1110
      "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1111
  | Inc:
hoelzl@47694
  1112
      "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
hoelzl@47694
  1113
       \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1114
  | Disj:
hoelzl@47694
  1115
      "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
hoelzl@47694
  1116
       \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1117
hoelzl@47694
  1118
lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
hoelzl@47694
  1119
  by auto
hoelzl@38656
  1120
hoelzl@47694
  1121
lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
hoelzl@38656
  1122
  apply (rule subsetI)
hoelzl@38656
  1123
  apply (erule smallest_ccdi_sets.induct)
hoelzl@38656
  1124
  apply (auto intro: range_subsetD dest: sets_into_space)
hoelzl@38656
  1125
  done
hoelzl@38656
  1126
hoelzl@47694
  1127
lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
hoelzl@47694
  1128
  apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
hoelzl@38656
  1129
  apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
hoelzl@38656
  1130
  done
hoelzl@38656
  1131
hoelzl@47694
  1132
lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
hoelzl@38656
  1133
  by (simp add: closed_cdi_def)
hoelzl@38656
  1134
hoelzl@47694
  1135
lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
hoelzl@38656
  1136
  by (simp add: closed_cdi_def)
hoelzl@38656
  1137
hoelzl@38656
  1138
lemma closed_cdi_Inc:
hoelzl@47694
  1139
  "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
  1140
  by (simp add: closed_cdi_def)
hoelzl@38656
  1141
hoelzl@38656
  1142
lemma closed_cdi_Disj:
hoelzl@47694
  1143
  "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@38656
  1144
  by (simp add: closed_cdi_def)
hoelzl@38656
  1145
hoelzl@38656
  1146
lemma closed_cdi_Un:
hoelzl@47694
  1147
  assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
hoelzl@47694
  1148
      and A: "A \<in> M" and B: "B \<in> M"
hoelzl@38656
  1149
      and disj: "A \<inter> B = {}"
hoelzl@47694
  1150
    shows "A \<union> B \<in> M"
hoelzl@38656
  1151
proof -
hoelzl@47694
  1152
  have ra: "range (binaryset A B) \<subseteq> M"
hoelzl@38656
  1153
   by (simp add: range_binaryset_eq empty A B)
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 closed_cdi_Disj [OF cdi 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_Un:
hoelzl@47694
  1162
  assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1163
      and disj: "A \<inter> B = {}"
hoelzl@47694
  1164
    shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1165
proof -
hoelzl@47694
  1166
  have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
hoelzl@38656
  1167
    by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
hoelzl@38656
  1168
  have di:  "disjoint_family (binaryset A B)" using disj
hoelzl@38656
  1169
    by (simp add: disjoint_family_on_def binaryset_def Int_commute)
hoelzl@38656
  1170
  from Disj [OF ra di]
hoelzl@38656
  1171
  show ?thesis
hoelzl@38656
  1172
    by (simp add: UN_binaryset_eq)
hoelzl@38656
  1173
qed
hoelzl@38656
  1174
hoelzl@38656
  1175
lemma (in algebra) smallest_ccdi_sets_Int1:
hoelzl@47694
  1176
  assumes a: "a \<in> M"
hoelzl@47694
  1177
  shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1178
proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656
  1179
  case (Basic x)
hoelzl@38656
  1180
  thus ?case
hoelzl@38656
  1181
    by (metis a Int smallest_ccdi_sets.Basic)
hoelzl@38656
  1182
next
hoelzl@38656
  1183
  case (Compl x)
hoelzl@47694
  1184
  have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
hoelzl@38656
  1185
    by blast
hoelzl@47694
  1186
  also have "... \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1187
    by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
hoelzl@47694
  1188
           Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
hoelzl@47694
  1189
           smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
hoelzl@38656
  1190
  finally show ?case .
hoelzl@38656
  1191
next
hoelzl@38656
  1192
  case (Inc 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 Inc
hoelzl@38656
  1196
    by blast
hoelzl@38656
  1197
  moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
hoelzl@38656
  1198
    by (simp add: Inc)
hoelzl@38656
  1199
  moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
hoelzl@38656
  1200
    by blast
hoelzl@47694
  1201
  ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1202
    by (rule smallest_ccdi_sets.Inc)
hoelzl@38656
  1203
  show ?case
hoelzl@38656
  1204
    by (metis 1 2)
hoelzl@38656
  1205
next
hoelzl@38656
  1206
  case (Disj A)
hoelzl@38656
  1207
  have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
hoelzl@38656
  1208
    by blast
hoelzl@47694
  1209
  have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
hoelzl@38656
  1210
    by blast
hoelzl@38656
  1211
  moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
hoelzl@38656
  1212
    by (auto simp add: disjoint_family_on_def)
hoelzl@47694
  1213
  ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1214
    by (rule smallest_ccdi_sets.Disj)
hoelzl@38656
  1215
  show ?case
hoelzl@38656
  1216
    by (metis 1 2)
hoelzl@38656
  1217
qed
hoelzl@38656
  1218
hoelzl@38656
  1219
hoelzl@38656
  1220
lemma (in algebra) smallest_ccdi_sets_Int:
hoelzl@47694
  1221
  assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@47694
  1222
  shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1223
proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656
  1224
  case (Basic x)
hoelzl@38656
  1225
  thus ?case
hoelzl@38656
  1226
    by (metis b smallest_ccdi_sets_Int1)
hoelzl@38656
  1227
next
hoelzl@38656
  1228
  case (Compl x)
hoelzl@47694
  1229
  have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
hoelzl@38656
  1230
    by blast
hoelzl@47694
  1231
  also have "... \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1232
    by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
hoelzl@38656
  1233
           smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
hoelzl@38656
  1234
  finally show ?case .
hoelzl@38656
  1235
next
hoelzl@38656
  1236
  case (Inc 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 Inc
hoelzl@38656
  1240
    by blast
hoelzl@38656
  1241
  moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
hoelzl@38656
  1242
    by (simp add: Inc)
hoelzl@38656
  1243
  moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
hoelzl@38656
  1244
    by blast
hoelzl@47694
  1245
  ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1246
    by (rule smallest_ccdi_sets.Inc)
hoelzl@38656
  1247
  show ?case
hoelzl@38656
  1248
    by (metis 1 2)
hoelzl@38656
  1249
next
hoelzl@38656
  1250
  case (Disj A)
hoelzl@38656
  1251
  have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
hoelzl@38656
  1252
    by blast
hoelzl@47694
  1253
  have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
hoelzl@38656
  1254
    by blast
hoelzl@38656
  1255
  moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
hoelzl@38656
  1256
    by (auto simp add: disjoint_family_on_def)
hoelzl@47694
  1257
  ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1258
    by (rule smallest_ccdi_sets.Disj)
hoelzl@38656
  1259
  show ?case
hoelzl@38656
  1260
    by (metis 1 2)
hoelzl@38656
  1261
qed
hoelzl@38656
  1262
hoelzl@38656
  1263
lemma (in algebra) sigma_property_disjoint_lemma:
hoelzl@47694
  1264
  assumes sbC: "M \<subseteq> C"
hoelzl@47694
  1265
      and ccdi: "closed_cdi \<Omega> C"
hoelzl@47694
  1266
  shows "sigma_sets \<Omega> M \<subseteq> C"
hoelzl@38656
  1267
proof -
hoelzl@47694
  1268
  have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
hoelzl@38656
  1269
    apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
hoelzl@38656
  1270
            smallest_ccdi_sets_Int)
hoelzl@38656
  1271
    apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
hoelzl@38656
  1272
    apply (blast intro: smallest_ccdi_sets.Disj)
hoelzl@38656
  1273
    done
hoelzl@47694
  1274
  hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1275
    by clarsimp
hoelzl@47694
  1276
       (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
hoelzl@38656
  1277
  also have "...  \<subseteq> C"
hoelzl@38656
  1278
    proof
hoelzl@38656
  1279
      fix x
hoelzl@47694
  1280
      assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656
  1281
      thus "x \<in> C"
hoelzl@38656
  1282
        proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656
  1283
          case (Basic x)
hoelzl@38656
  1284
          thus ?case
hoelzl@38656
  1285
            by (metis Basic subsetD sbC)
hoelzl@38656
  1286
        next
hoelzl@38656
  1287
          case (Compl x)
hoelzl@38656
  1288
          thus ?case
hoelzl@38656
  1289
            by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
hoelzl@38656
  1290
        next
hoelzl@38656
  1291
          case (Inc A)
hoelzl@38656
  1292
          thus ?case
hoelzl@38656
  1293
               by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
hoelzl@38656
  1294
        next
hoelzl@38656
  1295
          case (Disj A)
hoelzl@38656
  1296
          thus ?case
hoelzl@38656
  1297
               by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
hoelzl@38656
  1298
        qed
hoelzl@38656
  1299
    qed
hoelzl@38656
  1300
  finally show ?thesis .
hoelzl@38656
  1301
qed
hoelzl@38656
  1302
hoelzl@38656
  1303
lemma (in algebra) sigma_property_disjoint:
hoelzl@47694
  1304
  assumes sbC: "M \<subseteq> C"
hoelzl@47694
  1305
      and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
hoelzl@47694
  1306
      and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
hoelzl@38656
  1307
                     \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
hoelzl@38656
  1308
                     \<Longrightarrow> (\<Union>i. A i) \<in> C"
hoelzl@47694
  1309
      and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
hoelzl@38656
  1310
                      \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
hoelzl@47694
  1311
  shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
hoelzl@38656
  1312
proof -
hoelzl@47694
  1313
  have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
hoelzl@38656
  1314
    proof (rule sigma_property_disjoint_lemma)
hoelzl@47694
  1315
      show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
hoelzl@38656
  1316
        by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
hoelzl@38656
  1317
    next
hoelzl@47694
  1318
      show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
hoelzl@38656
  1319
        by (simp add: closed_cdi_def compl inc disj)
hoelzl@38656
  1320
           (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
hoelzl@38656
  1321
             IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
hoelzl@38656
  1322
    qed
hoelzl@38656
  1323
  thus ?thesis
hoelzl@38656
  1324
    by blast
hoelzl@38656
  1325
qed
hoelzl@38656
  1326
hoelzl@56994
  1327
subsubsection {* Dynkin systems *}
hoelzl@40859
  1328
hoelzl@42065
  1329
locale dynkin_system = subset_class +
hoelzl@47694
  1330
  assumes space: "\<Omega> \<in> M"
hoelzl@47694
  1331
    and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694
  1332
    and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694
  1333
                           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@40859
  1334
hoelzl@47694
  1335
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
hoelzl@47694
  1336
  using space compl[of "\<Omega>"] by simp
hoelzl@40859
  1337
hoelzl@40859
  1338
lemma (in dynkin_system) diff:
hoelzl@47694
  1339
  assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
hoelzl@47694
  1340
  shows "E - D \<in> M"
hoelzl@40859
  1341
proof -
hoelzl@47694
  1342
  let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
hoelzl@47694
  1343
  have "range ?f = {D, \<Omega> - E, {}}"
hoelzl@40859
  1344
    by (auto simp: image_iff)
hoelzl@47694
  1345
  moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
hoelzl@40859
  1346
    by (auto simp: image_iff split: split_if_asm)
hoelzl@40859
  1347
  moreover
wenzelm@53374
  1348
  have "disjoint_family ?f" unfolding disjoint_family_on_def
hoelzl@47694
  1349
    using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
hoelzl@47694
  1350
  ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
hoelzl@40859
  1351
    using sets by auto
hoelzl@47694
  1352
  also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
hoelzl@40859
  1353
    using assms sets_into_space by auto
hoelzl@40859
  1354
  finally show ?thesis .
hoelzl@40859
  1355
qed
hoelzl@40859
  1356
hoelzl@40859
  1357
lemma dynkin_systemI:
hoelzl@47694
  1358
  assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
hoelzl@47694
  1359
  assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694
  1360
  assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694
  1361
          \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694
  1362
  shows "dynkin_system \<Omega> M"
hoelzl@42065
  1363
  using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
hoelzl@40859
  1364
hoelzl@42988
  1365
lemma dynkin_systemI':
hoelzl@47694
  1366
  assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
hoelzl@47694
  1367
  assumes empty: "{} \<in> M"
hoelzl@47694
  1368
  assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694
  1369
  assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694
  1370
          \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694
  1371
  shows "dynkin_system \<Omega> M"
hoelzl@42988
  1372
proof -
hoelzl@47694
  1373
  from Diff[OF empty] have "\<Omega> \<in> M" by auto
hoelzl@42988
  1374
  from 1 this Diff 2 show ?thesis
hoelzl@42988
  1375
    by (intro dynkin_systemI) auto
hoelzl@42988
  1376
qed
hoelzl@42988
  1377
hoelzl@40859
  1378
lemma dynkin_system_trivial:
hoelzl@47694
  1379
  shows "dynkin_system A (Pow A)"
hoelzl@40859
  1380
  by (rule dynkin_systemI) auto
hoelzl@40859
  1381
hoelzl@40859
  1382
lemma sigma_algebra_imp_dynkin_system:
hoelzl@47694
  1383
  assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
hoelzl@40859
  1384
proof -
hoelzl@47694
  1385
  interpret sigma_algebra \<Omega> M by fact
nipkow@44890
  1386
  show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
hoelzl@40859
  1387
qed
hoelzl@40859
  1388
hoelzl@56994
  1389
subsubsection "Intersection sets systems"
hoelzl@40859
  1390
hoelzl@47694
  1391
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
hoelzl@40859
  1392
hoelzl@40859
  1393
lemma (in algebra) Int_stable: "Int_stable M"
hoelzl@40859
  1394
  unfolding Int_stable_def by auto
hoelzl@40859
  1395
hoelzl@42981
  1396
lemma Int_stableI:
hoelzl@47694
  1397
  "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
hoelzl@42981
  1398
  unfolding Int_stable_def by auto
hoelzl@42981
  1399
hoelzl@42981
  1400
lemma Int_stableD:
hoelzl@47694
  1401
  "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
hoelzl@42981
  1402
  unfolding Int_stable_def by auto
hoelzl@42981
  1403
hoelzl@40859
  1404
lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
hoelzl@47694
  1405
  "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
hoelzl@40859
  1406
proof
hoelzl@47694
  1407
  assume "sigma_algebra \<Omega> M" then show "Int_stable M"
hoelzl@40859
  1408
    unfolding sigma_algebra_def using algebra.Int_stable by auto
hoelzl@40859
  1409
next
hoelzl@40859
  1410
  assume "Int_stable M"
hoelzl@47694
  1411
  show "sigma_algebra \<Omega> M"
hoelzl@42065
  1412
    unfolding sigma_algebra_disjoint_iff algebra_iff_Un
hoelzl@40859
  1413
  proof (intro conjI ballI allI impI)
hoelzl@47694
  1414
    show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
hoelzl@40859
  1415
  next
hoelzl@47694
  1416
    fix A B assume "A \<in> M" "B \<in> M"
hoelzl@47694
  1417
    then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
hoelzl@47694
  1418
              "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
hoelzl@40859
  1419
      using sets_into_space by auto
hoelzl@47694
  1420
    then show "A \<union> B \<in> M"
hoelzl@40859
  1421
      using `Int_stable M` unfolding Int_stable_def by auto
hoelzl@40859
  1422
  qed auto
hoelzl@40859
  1423
qed
hoelzl@40859
  1424
hoelzl@56994
  1425
subsubsection "Smallest Dynkin systems"
hoelzl@40859
  1426
hoelzl@41689
  1427
definition dynkin where
hoelzl@47694
  1428
  "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
hoelzl@40859
  1429
hoelzl@40859
  1430
lemma dynkin_system_dynkin:
hoelzl@47694
  1431
  assumes "M \<subseteq> Pow (\<Omega>)"
hoelzl@47694
  1432
  shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
hoelzl@40859
  1433
proof (rule dynkin_systemI)
hoelzl@47694
  1434
  fix A assume "A \<in> dynkin \<Omega> M"
hoelzl@40859
  1435
  moreover
hoelzl@47694
  1436
  { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
hoelzl@47694
  1437
    then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
hoelzl@47694
  1438
  moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
nipkow@44890
  1439
    using assms dynkin_system_trivial by fastforce
hoelzl@47694
  1440
  ultimately show "A \<subseteq> \<Omega>"
hoelzl@40859
  1441
    unfolding dynkin_def using assms
hoelzl@47694
  1442
    by auto
hoelzl@40859
  1443
next
hoelzl@47694
  1444
  show "\<Omega> \<in> dynkin \<Omega> M"
nipkow@44890
  1445
    unfolding dynkin_def using dynkin_system.space by fastforce
hoelzl@40859
  1446
next
hoelzl@47694
  1447
  fix A assume "A \<in> dynkin \<Omega> M"
hoelzl@47694
  1448
  then show "\<Omega> - A \<in> dynkin \<Omega> M"
hoelzl@40859
  1449
    unfolding dynkin_def using dynkin_system.compl by force
hoelzl@40859
  1450
next
hoelzl@40859
  1451
  fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
  1452
  assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
hoelzl@47694
  1453
  show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
hoelzl@40859
  1454
  proof (simp, safe)
hoelzl@47694
  1455
    fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
hoelzl@47694
  1456
    with A have "(\<Union>i. A i) \<in> D"
hoelzl@40859
  1457
      by (intro dynkin_system.UN) (auto simp: dynkin_def)
hoelzl@40859
  1458
    then show "(\<Union>i. A i) \<in> D" by auto
hoelzl@40859
  1459
  qed
hoelzl@40859
  1460
qed
hoelzl@40859
  1461
hoelzl@47694
  1462
lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
hoelzl@40859
  1463
  unfolding dynkin_def by auto
hoelzl@40859
  1464
hoelzl@40859
  1465
lemma (in dynkin_system) restricted_dynkin_system:
hoelzl@47694
  1466
  assumes "D \<in> M"
hoelzl@47694
  1467
  shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
hoelzl@40859
  1468
proof (rule dynkin_systemI, simp_all)
hoelzl@47694
  1469
  have "\<Omega> \<inter> D = D"
hoelzl@47694
  1470
    using `D \<in> M` sets_into_space by auto
hoelzl@47694
  1471
  then show "\<Omega> \<inter> D \<in> M"
hoelzl@47694
  1472
    using `D \<in> M` by auto
hoelzl@40859
  1473
next
hoelzl@47694
  1474
  fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
hoelzl@47694
  1475
  moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
hoelzl@40859
  1476
    by auto
hoelzl@47694
  1477
  ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
hoelzl@47694
  1478
    using  `D \<in> M` by (auto intro: diff)
hoelzl@40859
  1479
next
hoelzl@40859
  1480
  fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694
  1481
  assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
hoelzl@47694
  1482
  then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
hoelzl@47694
  1483
    "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
nipkow@44890
  1484
    by ((fastforce simp: disjoint_family_on_def)+)
hoelzl@47694
  1485
  then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
hoelzl@40859
  1486
    by (auto simp del: UN_simps)
hoelzl@40859
  1487
qed
hoelzl@40859
  1488
hoelzl@40859
  1489
lemma (in dynkin_system) dynkin_subset:
hoelzl@47694
  1490
  assumes "N \<subseteq> M"
hoelzl@47694
  1491
  shows "dynkin \<Omega> N \<subseteq> M"
hoelzl@40859
  1492
proof -
hoelzl@47694
  1493
  have "dynkin_system \<Omega> M" by default
hoelzl@47694
  1494
  then have "dynkin_system \<Omega> M"
hoelzl@42065
  1495
    using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
hoelzl@47694
  1496
  with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
hoelzl@40859
  1497
qed
hoelzl@40859
  1498
hoelzl@40859
  1499
lemma sigma_eq_dynkin:
hoelzl@47694
  1500
  assumes sets: "M \<subseteq> Pow \<Omega>"
hoelzl@40859
  1501
  assumes "Int_stable M"
hoelzl@47694
  1502
  shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
hoelzl@40859
  1503
proof -
hoelzl@47694
  1504
  have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
hoelzl@40859
  1505
    using sigma_algebra_imp_dynkin_system
hoelzl@47694
  1506
    unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
hoelzl@40859
  1507
  moreover
hoelzl@47694
  1508
  interpret dynkin_system \<Omega> "dynkin \<Omega> M"
hoelzl@40859
  1509
    using dynkin_system_dynkin[OF sets] .
hoelzl@47694
  1510
  have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
hoelzl@40859
  1511
    unfolding sigma_algebra_eq_Int_stable Int_stable_def
hoelzl@40859
  1512
  proof (intro ballI)
hoelzl@47694
  1513
    fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
hoelzl@47694
  1514
    let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
hoelzl@47694
  1515
    have "M \<subseteq> ?D B"
hoelzl@40859
  1516
    proof
hoelzl@47694
  1517
      fix E assume "E \<in> M"
hoelzl@47694
  1518
      then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
hoelzl@40859
  1519
        using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
hoelzl@47694
  1520
      then have "dynkin \<Omega> M \<subseteq> ?D E"
hoelzl@47694
  1521
        using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
hoelzl@40859
  1522
        by (intro dynkin_system.dynkin_subset) simp_all
hoelzl@47694
  1523
      then have "B \<in> ?D E"
hoelzl@47694
  1524
        using `B \<in> dynkin \<Omega> M` by auto
hoelzl@47694
  1525
      then have "E \<inter> B \<in> dynkin \<Omega> M"
hoelzl@40859
  1526
        by (subst Int_commute) simp
hoelzl@47694
  1527
      then show "E \<in> ?D B"
hoelzl@47694
  1528
        using sets `E \<in> M` by auto
hoelzl@40859
  1529
    qed
hoelzl@47694
  1530
    then have "dynkin \<Omega> M \<subseteq> ?D B"
hoelzl@47694
  1531
      using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
hoelzl@40859
  1532
      by (intro dynkin_system.dynkin_subset) simp_all
hoelzl@47694
  1533
    then show "A \<inter> B \<in> dynkin \<Omega> M"
hoelzl@47694
  1534
      using `A \<in> dynkin \<Omega> M` sets_into_space by auto
hoelzl@40859
  1535
  qed
hoelzl@47694
  1536
  from sigma_algebra.sigma_sets_subset[OF this, of "M"]
hoelzl@47694
  1537
  have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
hoelzl@47694
  1538
  ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
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_idem:
hoelzl@47694
  1544
  "dynkin \<Omega> M = M"
hoelzl@40859
  1545
proof -
hoelzl@47694
  1546
  have "dynkin \<Omega> M = M"
hoelzl@40859
  1547
  proof
hoelzl@47694
  1548
    show "M \<subseteq> dynkin \<Omega> M"
hoelzl@40859
  1549
      using dynkin_Basic by auto
hoelzl@47694
  1550
    show "dynkin \<Omega> M \<subseteq> M"
hoelzl@40859
  1551
      by (intro dynkin_subset) auto
hoelzl@40859
  1552
  qed
hoelzl@40859
  1553
  then show ?thesis
hoelzl@47694
  1554
    by (auto simp: dynkin_def)
hoelzl@40859
  1555
qed
hoelzl@40859
  1556
hoelzl@40859
  1557
lemma (in dynkin_system) dynkin_lemma:
hoelzl@41689
  1558
  assumes "Int_stable E"
hoelzl@47694
  1559
  and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
hoelzl@47694
  1560
  shows "sigma_sets \<Omega> E = M"
hoelzl@40859
  1561
proof -
hoelzl@47694
  1562
  have "E \<subseteq> Pow \<Omega>"
hoelzl@41689
  1563
    using E sets_into_space by force
wenzelm@53374
  1564
  then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
hoelzl@40859
  1565
    using `Int_stable E` by (rule sigma_eq_dynkin)
wenzelm@53374
  1566
  then have "dynkin \<Omega> E = M"
hoelzl@47694
  1567
    using assms dynkin_subset[OF E(1)] by simp
wenzelm@53374
  1568
  with * show ?thesis
hoelzl@47694
  1569
    using assms by (auto simp: dynkin_def)
hoelzl@42864
  1570
qed
hoelzl@42864
  1571
hoelzl@56994
  1572
subsubsection {* Induction rule for intersection-stable generators *}
hoelzl@56994
  1573
hoelzl@56994
  1574
text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
hoelzl@56994
  1575
generated by a generator closed under intersection. *}
hoelzl@56994
  1576
hoelzl@49789
  1577
lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
hoelzl@49789
  1578
  assumes "Int_stable G"
hoelzl@49789
  1579
    and closed: "G \<subseteq> Pow \<Omega>"
hoelzl@49789
  1580
    and A: "A \<in> sigma_sets \<Omega> G"
hoelzl@49789
  1581
  assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
hoelzl@49789
  1582
    and empty: "P {}"
hoelzl@49789
  1583
    and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
hoelzl@49789
  1584
    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
  1585
  shows "P A"
hoelzl@49789
  1586
proof -
hoelzl@49789
  1587
  let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
hoelzl@49789
  1588
  interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
hoelzl@49789
  1589
    using closed by (rule sigma_algebra_sigma_sets)
hoelzl@49789
  1590
  from compl[OF _ empty] closed have space: "P \<Omega>" by simp
hoelzl@49789
  1591
  interpret dynkin_system \<Omega> ?D
hoelzl@49789
  1592
    by default (auto dest: sets_into_space intro!: space compl union)
hoelzl@49789
  1593
  have "sigma_sets \<Omega> G = ?D"
hoelzl@49789
  1594
    by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
hoelzl@49789
  1595
  with A show ?thesis by auto
hoelzl@49789
  1596
qed
hoelzl@49789
  1597
hoelzl@56994
  1598
subsection {* Measure type *}
hoelzl@56994
  1599
hoelzl@56994
  1600
definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994
  1601
  "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
hoelzl@56994
  1602
hoelzl@56994
  1603
definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994
  1604
  "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
  1605
    (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
hoelzl@56994
  1606
hoelzl@56994
  1607
definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994
  1608
  "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
hoelzl@56994
  1609
hoelzl@56994
  1610
typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
hoelzl@56994
  1611
proof
hoelzl@56994
  1612
  have "sigma_algebra UNIV {{}, UNIV}"
hoelzl@56994
  1613
    by (auto simp: sigma_algebra_iff2)
hoelzl@56994
  1614
  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
  1615
    by (auto simp: measure_space_def positive_def countably_additive_def)
hoelzl@56994
  1616
qed
hoelzl@56994
  1617
hoelzl@56994
  1618
definition space :: "'a measure \<Rightarrow> 'a set" where
hoelzl@56994
  1619
  "space M = fst (Rep_measure M)"
hoelzl@56994
  1620
hoelzl@56994
  1621
definition sets :: "'a measure \<Rightarrow> 'a set set" where
hoelzl@56994
  1622
  "sets M = fst (snd (Rep_measure M))"
hoelzl@56994
  1623
hoelzl@56994
  1624
definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
hoelzl@56994
  1625
  "emeasure M = snd (snd (Rep_measure M))"
hoelzl@56994
  1626
hoelzl@56994
  1627
definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
hoelzl@56994
  1628
  "measure M A = real (emeasure M A)"
hoelzl@56994
  1629
hoelzl@56994
  1630
declare [[coercion sets]]
hoelzl@56994
  1631
hoelzl@56994
  1632
declare [[coercion measure]]
hoelzl@56994
  1633
hoelzl@56994
  1634
declare [[coercion emeasure]]
hoelzl@56994
  1635
hoelzl@56994
  1636
lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
hoelzl@56994
  1637
  by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
hoelzl@56994
  1638
hoelzl@56994
  1639
interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
hoelzl@56994
  1640
  using measure_space[of M] by (auto simp: measure_space_def)
hoelzl@56994
  1641
hoelzl@56994
  1642
definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
hoelzl@56994
  1643
  "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
hoelzl@56994
  1644
    \<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
  1645
hoelzl@56994
  1646
abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
hoelzl@56994
  1647
hoelzl@56994
  1648
lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
hoelzl@56994
  1649
  unfolding measure_space_def
hoelzl@56994
  1650
  by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
hoelzl@56994
  1651
hoelzl@56994
  1652
lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
hoelzl@56994
  1653
by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
hoelzl@56994
  1654
hoelzl@56994
  1655
lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
hoelzl@56994
  1656
by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
hoelzl@56994
  1657
hoelzl@56994
  1658
lemma measure_space_closed:
hoelzl@56994
  1659
  assumes "measure_space \<Omega> M \<mu>"
hoelzl@56994
  1660
  shows "M \<subseteq> Pow \<Omega>"
hoelzl@56994
  1661
proof -
hoelzl@56994
  1662
  interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
hoelzl@56994
  1663
  show ?thesis by(rule space_closed)
hoelzl@56994
  1664
qed
hoelzl@56994
  1665
hoelzl@56994
  1666
lemma (in ring_of_sets) positive_cong_eq:
hoelzl@56994
  1667
  "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
hoelzl@56994
  1668
  by (auto simp add: positive_def)
hoelzl@56994
  1669
hoelzl@56994
  1670
lemma (in sigma_algebra) countably_additive_eq:
hoelzl@56994
  1671
  "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
hoelzl@56994
  1672
  unfolding countably_additive_def
hoelzl@56994
  1673
  by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
hoelzl@56994
  1674
hoelzl@56994
  1675
lemma measure_space_eq:
hoelzl@56994
  1676
  assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
hoelzl@56994
  1677
  shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
hoelzl@56994
  1678
proof -
hoelzl@56994
  1679
  interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
hoelzl@56994
  1680
  from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
hoelzl@56994
  1681
    by (auto simp: measure_space_def)
hoelzl@56994
  1682
qed
hoelzl@56994
  1683
hoelzl@56994
  1684
lemma measure_of_eq:
hoelzl@56994
  1685
  assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
hoelzl@56994
  1686
  shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
hoelzl@56994
  1687
proof -
hoelzl@56994
  1688
  have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
hoelzl@56994
  1689
    using assms by (rule measure_space_eq)
hoelzl@56994
  1690
  with eq show ?thesis
hoelzl@56994
  1691
    by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
hoelzl@56994
  1692
qed
hoelzl@56994
  1693
hoelzl@56994
  1694
lemma
hoelzl@56994
  1695
  shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
hoelzl@56994
  1696
  and sets_measure_of_conv:
hoelzl@56994
  1697
  "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
hoelzl@56994
  1698
  and emeasure_measure_of_conv: 
hoelzl@56994
  1699
  "emeasure (measure_of \<Omega> A \<mu>) = 
hoelzl@56994
  1700
  (\<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
  1701
proof -
hoelzl@56994
  1702
  have "?space \<and> ?sets \<and> ?emeasure"
hoelzl@56994
  1703
  proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
hoelzl@56994
  1704
    case True
hoelzl@56994
  1705
    from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
hoelzl@56994
  1706
    have "A \<subseteq> Pow \<Omega>" by simp
hoelzl@56994
  1707
    hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
hoelzl@56994
  1708
      (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
hoelzl@56994
  1709
      by(rule measure_space_eq) auto
hoelzl@56994
  1710
    with True `A \<subseteq> Pow \<Omega>` show ?thesis
hoelzl@56994
  1711
      by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
hoelzl@56994
  1712
  next
hoelzl@56994
  1713
    case False thus ?thesis
hoelzl@56994
  1714
      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
  1715
  qed
hoelzl@56994
  1716
  thus ?space ?sets ?emeasure by simp_all
hoelzl@56994
  1717
qed
hoelzl@56994
  1718
hoelzl@56994
  1719
lemma [simp]:
hoelzl@56994
  1720
  assumes A: "A \<subseteq> Pow \<Omega>"
hoelzl@56994
  1721
  shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
hoelzl@56994
  1722
    and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
hoelzl@56994
  1723
using assms
hoelzl@56994
  1724
by(simp_all add: sets_measure_of_conv space_measure_of_conv)
hoelzl@56994
  1725
hoelzl@56994
  1726
lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
hoelzl@56994
  1727
  using space_closed by (auto intro!: sigma_sets_eq)
hoelzl@56994
  1728
hoelzl@56994
  1729
lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
hoelzl@56994
  1730
  by (rule space_measure_of_conv)
hoelzl@56994
  1731
hoelzl@56994
  1732
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
  1733
  by (auto intro!: sigma_sets_subseteq)
hoelzl@56994
  1734
hoelzl@56994
  1735
lemma sigma_sets_mono'':
hoelzl@56994
  1736
  assumes "A \<in> sigma_sets C D"
hoelzl@56994
  1737
  assumes "B \<subseteq> D"
hoelzl@56994
  1738
  assumes "D \<subseteq> Pow C"
hoelzl@56994
  1739
  shows "sigma_sets A B \<subseteq> sigma_sets C D"
hoelzl@56994
  1740
proof
hoelzl@56994
  1741
  fix x assume "x \<in> sigma_sets A B"
hoelzl@56994
  1742
  thus "x \<in> sigma_sets C D"
hoelzl@56994
  1743
  proof induct
hoelzl@56994
  1744
    case (Basic a) with assms have "a \<in> D" by auto
hoelzl@56994
  1745
    thus ?case ..
hoelzl@56994
  1746
  next
hoelzl@56994
  1747
    case Empty show ?case by (rule sigma_sets.Empty)
hoelzl@56994
  1748
  next
hoelzl@56994
  1749
    from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994
  1750
    moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994
  1751
    ultimately have "A - a \<in> sets (sigma C D)" ..
hoelzl@56994
  1752
    thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994
  1753
  next
hoelzl@56994
  1754
    case (Union a)
hoelzl@56994
  1755
    thus ?case by (intro sigma_sets.Union)
hoelzl@56994
  1756
  qed
hoelzl@56994
  1757
qed
hoelzl@56994
  1758
hoelzl@56994
  1759
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
  1760
  by auto
hoelzl@56994
  1761
hoelzl@56994
  1762
subsubsection {* Constructing simple @{typ "'a measure"} *}
hoelzl@56994
  1763
hoelzl@56994
  1764
lemma emeasure_measure_of:
hoelzl@56994
  1765
  assumes M: "M = measure_of \<Omega> A \<mu>"
hoelzl@56994
  1766
  assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
hoelzl@56994
  1767
  assumes X: "X \<in> sets M"
hoelzl@56994
  1768
  shows "emeasure M X = \<mu> X"
hoelzl@56994
  1769
proof -
hoelzl@56994
  1770
  interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
hoelzl@56994
  1771
  have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
hoelzl@56994
  1772
    using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
hoelzl@56994
  1773
  thus ?thesis using X ms
hoelzl@56994
  1774
    by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
hoelzl@56994
  1775
qed
hoelzl@56994
  1776
hoelzl@56994
  1777
lemma emeasure_measure_of_sigma:
hoelzl@56994
  1778
  assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
hoelzl@56994
  1779
  assumes A: "A \<in> M"
hoelzl@56994
  1780
  shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
hoelzl@56994
  1781
proof -
hoelzl@56994
  1782
  interpret sigma_algebra \<Omega> M by fact
hoelzl@56994
  1783
  have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
hoelzl@56994
  1784
    using ms sigma_sets_eq by (simp add: measure_space_def)
hoelzl@56994
  1785
  thus ?thesis by(simp add: emeasure_measure_of_conv A)
hoelzl@56994
  1786
qed
hoelzl@56994
  1787
hoelzl@56994
  1788
lemma measure_cases[cases type: measure]:
hoelzl@56994
  1789
  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
  1790
  by atomize_elim (cases x, auto)
hoelzl@56994
  1791
hoelzl@56994
  1792
lemma sets_eq_imp_space_eq:
hoelzl@56994
  1793
  "sets M = sets M' \<Longrightarrow> space M = space M'"
hoelzl@56994
  1794
  using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
hoelzl@56994
  1795
  by blast
hoelzl@56994
  1796
hoelzl@56994
  1797
lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
hoelzl@56994
  1798
  by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
hoelzl@56994
  1799
hoelzl@56994
  1800
lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
hoelzl@56994
  1801
  using emeasure_notin_sets[of A M] by blast
hoelzl@56994
  1802
hoelzl@56994
  1803
lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
hoelzl@56994
  1804
  by (simp add: measure_def emeasure_notin_sets)
hoelzl@56994
  1805
hoelzl@56994
  1806
lemma measure_eqI:
hoelzl@56994
  1807
  fixes M N :: "'a measure"
hoelzl@56994
  1808
  assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
hoelzl@56994
  1809
  shows "M = N"
hoelzl@56994
  1810
proof (cases M N rule: measure_cases[case_product measure_cases])
hoelzl@56994
  1811
  case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
hoelzl@56994
  1812
  interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
hoelzl@56994
  1813
  interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
hoelzl@56994
  1814
  have "A = sets M" "A' = sets N"
hoelzl@56994
  1815
    using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
hoelzl@56994
  1816
  with `sets M = sets N` have AA': "A = A'" by simp
hoelzl@56994
  1817
  moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
hoelzl@56994
  1818
  moreover { fix B have "\<mu> B = \<mu>' B"
hoelzl@56994
  1819
    proof cases
hoelzl@56994
  1820
      assume "B \<in> A"
hoelzl@56994
  1821
      with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
hoelzl@56994
  1822
      with measure_measure show "\<mu> B = \<mu>' B"
hoelzl@56994
  1823
        by (simp add: emeasure_def Abs_measure_inverse)
hoelzl@56994
  1824
    next
hoelzl@56994
  1825
      assume "B \<notin> A"
hoelzl@56994
  1826
      with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
hoelzl@56994
  1827
        by auto
hoelzl@56994
  1828
      then have "emeasure M B = 0" "emeasure N B = 0"
hoelzl@56994
  1829
        by (simp_all add: emeasure_notin_sets)
hoelzl@56994
  1830
      with measure_measure show "\<mu> B = \<mu>' B"
hoelzl@56994
  1831
        by (simp add: emeasure_def Abs_measure_inverse)
hoelzl@56994
  1832
    qed }
hoelzl@56994
  1833
  then have "\<mu> = \<mu>'" by auto
hoelzl@56994
  1834
  ultimately show "M = N"
hoelzl@56994
  1835
    by (simp add: measure_measure)
hoelzl@56994
  1836
qed
hoelzl@56994
  1837
hoelzl@56994
  1838
lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
hoelzl@56994
  1839
  using measure_space_0[of A \<Omega>]
hoelzl@56994
  1840
  by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
hoelzl@56994
  1841
hoelzl@56994
  1842
lemma sigma_eqI:
hoelzl@56994
  1843
  assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
hoelzl@56994
  1844
  shows "sigma \<Omega> M = sigma \<Omega> N"
hoelzl@56994
  1845
  by (rule measure_eqI) (simp_all add: emeasure_sigma)
hoelzl@56994
  1846
hoelzl@56994
  1847
subsubsection {* Measurable functions *}
hoelzl@56994
  1848
hoelzl@56994
  1849
definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
hoelzl@56994
  1850
  "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
hoelzl@56994
  1851
hoelzl@56994
  1852
lemma measurable_space:
hoelzl@56994
  1853
  "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
hoelzl@56994
  1854
   unfolding measurable_def by auto
hoelzl@56994
  1855
hoelzl@56994
  1856
lemma measurable_sets:
hoelzl@56994
  1857
  "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
hoelzl@56994
  1858
   unfolding measurable_def by auto
hoelzl@56994
  1859
hoelzl@56994
  1860
lemma measurable_sets_Collect:
hoelzl@56994
  1861
  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
  1862
proof -
hoelzl@56994
  1863
  have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
hoelzl@56994
  1864
    using measurable_space[OF f] by auto
hoelzl@56994
  1865
  with measurable_sets[OF f P] show ?thesis
hoelzl@56994
  1866
    by simp
hoelzl@56994
  1867
qed
hoelzl@56994
  1868
hoelzl@56994
  1869
lemma measurable_sigma_sets:
hoelzl@56994
  1870
  assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
hoelzl@56994
  1871
      and f: "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994
  1872
      and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
hoelzl@56994
  1873
  shows "f \<in> measurable M N"
hoelzl@56994
  1874
proof -
hoelzl@56994
  1875
  interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
hoelzl@56994
  1876
  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
  1877
  
hoelzl@56994
  1878
  { fix X assume "X \<in> sigma_sets \<Omega> A"
hoelzl@56994
  1879
    then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
hoelzl@56994
  1880
      proof induct
hoelzl@56994
  1881
        case (Basic a) then show ?case
hoelzl@56994
  1882
          by (auto simp add: ba) (metis B(2) subsetD PowD)
hoelzl@56994
  1883
      next
hoelzl@56994
  1884
        case (Compl a)
hoelzl@56994
  1885
        have [simp]: "f -` \<Omega> \<inter> space M = space M"
hoelzl@56994
  1886
          by (auto simp add: funcset_mem [OF f])
hoelzl@56994
  1887
        then show ?case
hoelzl@56994
  1888
          by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
hoelzl@56994
  1889
      next
hoelzl@56994
  1890
        case (Union a)
hoelzl@56994
  1891
        then show ?case
hoelzl@56994
  1892
          by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
hoelzl@56994
  1893
      qed auto }
hoelzl@56994
  1894
  with f show ?thesis
hoelzl@56994
  1895
    by (auto simp add: measurable_def B \<Omega>)
hoelzl@56994
  1896
qed
hoelzl@56994
  1897
hoelzl@56994
  1898
lemma measurable_measure_of:
hoelzl@56994
  1899
  assumes B: "N \<subseteq> Pow \<Omega>"
hoelzl@56994
  1900
      and f: "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994
  1901
      and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
hoelzl@56994
  1902
  shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
hoelzl@56994
  1903
proof -
hoelzl@56994
  1904
  have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
hoelzl@56994
  1905
    using B by (rule sets_measure_of)
hoelzl@56994
  1906
  from this assms show ?thesis by (rule measurable_sigma_sets)
hoelzl@56994
  1907
qed
hoelzl@56994
  1908
hoelzl@56994
  1909
lemma measurable_iff_measure_of:
hoelzl@56994
  1910
  assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994
  1911
  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
  1912
  by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
hoelzl@56994
  1913
hoelzl@56994
  1914
lemma measurable_cong_sets:
hoelzl@56994
  1915
  assumes sets: "sets M = sets M'" "sets N = sets N'"
hoelzl@56994
  1916
  shows "measurable M N = measurable M' N'"
hoelzl@56994
  1917
  using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
hoelzl@56994
  1918
hoelzl@56994
  1919
lemma measurable_cong:
hoelzl@56994
  1920
  assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
hoelzl@56994
  1921
  shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
hoelzl@56994
  1922
  unfolding measurable_def using assms
hoelzl@56994
  1923
  by (simp cong: vimage_inter_cong Pi_cong)
hoelzl@56994
  1924
hoelzl@56994
  1925
lemma measurable_cong_strong:
hoelzl@56994
  1926
  "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
hoelzl@56994
  1927
    f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
hoelzl@56994
  1928
  by (metis measurable_cong)
hoelzl@56994
  1929
hoelzl@56994
  1930
lemma measurable_eqI:
hoelzl@56994
  1931
     "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
hoelzl@56994
  1932
        sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
hoelzl@56994
  1933
      \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
hoelzl@56994
  1934
  by (simp add: measurable_def sigma_algebra_iff2)
hoelzl@56994
  1935
hoelzl@56994
  1936
lemma measurable_compose:
hoelzl@56994
  1937
  assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
hoelzl@56994
  1938
  shows "(\<lambda>x. g (f x)) \<in> measurable M L"
hoelzl@56994
  1939
proof -
hoelzl@56994
  1940
  have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
hoelzl@56994
  1941
    using measurable_space[OF f] by auto
hoelzl@56994
  1942
  with measurable_space[OF f] measurable_space[OF g] show ?thesis
hoelzl@56994
  1943
    by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
hoelzl@56994
  1944
             simp del: vimage_Int simp add: measurable_def)
hoelzl@56994
  1945
qed
hoelzl@56994
  1946
hoelzl@56994
  1947
lemma measurable_comp:
hoelzl@56994
  1948
  "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
hoelzl@56994
  1949
  using measurable_compose[of f M N g L] by (simp add: comp_def)
hoelzl@56994
  1950
hoelzl@56994
  1951
lemma measurable_const:
hoelzl@56994
  1952
  "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
hoelzl@56994
  1953
  by (auto simp add: measurable_def)
hoelzl@56994
  1954
hoelzl@56994
  1955
lemma measurable_If:
hoelzl@56994
  1956
  assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
hoelzl@56994
  1957
  assumes P: "{x\<in>space M. P x} \<in> sets M"
hoelzl@56994
  1958
  shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
hoelzl@56994
  1959
  unfolding measurable_def
hoelzl@56994
  1960
proof safe
hoelzl@56994
  1961
  fix x assume "x \<in> space M"
hoelzl@56994
  1962
  thus "(if P x then f x else g x) \<in> space M'"
hoelzl@56994
  1963
    using measure unfolding measurable_def by auto
hoelzl@56994
  1964
next
hoelzl@56994
  1965
  fix A assume "A \<in> sets M'"
hoelzl@56994
  1966
  hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
hoelzl@56994
  1967
    ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
hoelzl@56994
  1968
    ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
hoelzl@56994
  1969
    using measure unfolding measurable_def by (auto split: split_if_asm)
hoelzl@56994
  1970
  show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
hoelzl@56994
  1971
    using `A \<in> sets M'` measure P unfolding * measurable_def
hoelzl@56994
  1972
    by (auto intro!: sets.Un)
hoelzl@56994
  1973
qed
hoelzl@56994
  1974
hoelzl@56994
  1975
lemma measurable_If_set:
hoelzl@56994
  1976
  assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
hoelzl@56994
  1977
  assumes P: "A \<inter> space M \<in> sets M"
hoelzl@56994
  1978
  shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
hoelzl@56994
  1979
proof (rule measurable_If[OF measure])
hoelzl@56994
  1980
  have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
hoelzl@56994
  1981
  thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
hoelzl@56994
  1982
qed
hoelzl@56994
  1983
hoelzl@56994
  1984
lemma measurable_ident: "id \<in> measurable M M"
hoelzl@56994
  1985
  by (auto simp add: measurable_def)
hoelzl@56994
  1986
hoelzl@56994
  1987
lemma measurable_ident_sets:
hoelzl@56994
  1988
  assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
hoelzl@56994
  1989
  using measurable_ident[of M]
hoelzl@56994
  1990
  unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
hoelzl@56994
  1991
hoelzl@56994
  1992
lemma sets_Least:
hoelzl@56994
  1993
  assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
hoelzl@56994
  1994
  shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
hoelzl@56994
  1995
proof -
hoelzl@56994
  1996
  { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
hoelzl@56994
  1997
    proof cases
hoelzl@56994
  1998
      assume i: "(LEAST j. False) = i"
hoelzl@56994
  1999
      have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
hoelzl@56994
  2000
        {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
  2001
        by (simp add: set_eq_iff, safe)
hoelzl@56994
  2002
           (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
hoelzl@56994
  2003
      with meas show ?thesis
hoelzl@56994
  2004
        by (auto intro!: sets.Int)
hoelzl@56994
  2005
    next
hoelzl@56994
  2006
      assume i: "(LEAST j. False) \<noteq> i"
hoelzl@56994
  2007
      then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
hoelzl@56994
  2008
        {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
hoelzl@56994
  2009
      proof (simp add: set_eq_iff, safe)
hoelzl@56994
  2010
        fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
hoelzl@56994
  2011
        have "\<exists>j. P j x"
hoelzl@56994
  2012
          by (rule ccontr) (insert neq, auto)
hoelzl@56994
  2013
        then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
hoelzl@56994
  2014
      qed (auto dest: Least_le intro!: Least_equality)
hoelzl@56994
  2015
      with meas show ?thesis
hoelzl@56994
  2016
        by auto
hoelzl@56994
  2017
    qed }
hoelzl@56994
  2018
  then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
hoelzl@56994
  2019
    by (intro sets.countable_UN) auto
hoelzl@56994
  2020
  moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
hoelzl@56994
  2021
    (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
hoelzl@56994
  2022
  ultimately show ?thesis by auto
hoelzl@56994
  2023
qed
hoelzl@56994
  2024
hoelzl@56994
  2025
lemma measurable_strong:
hoelzl@56994
  2026
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
hoelzl@56994
  2027
  assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
hoelzl@56994
  2028
      and t: "f ` (space a) \<subseteq> t"
hoelzl@56994
  2029
      and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
hoelzl@56994
  2030
  shows "(g o f) \<in> measurable a c"
hoelzl@56994
  2031
proof -
hoelzl@56994
  2032
  have fab: "f \<in> (space a -> space b)"
hoelzl@56994
  2033
   and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
hoelzl@56994
  2034
     by (auto simp add: measurable_def)
hoelzl@56994
  2035
  have eq: "\<And>y. (g \<circ> f) -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
hoelzl@56994
  2036
    by force
hoelzl@56994
  2037
  show ?thesis
hoelzl@56994
  2038
    apply (auto simp add: measurable_def vimage_comp)
hoelzl@56994
  2039
    apply (metis funcset_mem fab g)
hoelzl@56994
  2040
    apply (subst eq)
hoelzl@56994
  2041
    apply (metis ba cb)
hoelzl@56994
  2042
    done
hoelzl@56994
  2043
qed
hoelzl@56994
  2044
hoelzl@57275
  2045
lemma measurable_discrete_difference: