session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
(* Title: HOL/Analysis/Sigma_Algebra.thy
Author: Stefan Richter, Markus Wenzel, TU München
Author: Johannes Hölzl, TU München
Plus material from the Hurd/Coble measure theory development,
translated by Lawrence Paulson.
*)
section \<open>Describing measurable sets\<close>
theory Sigma_Algebra
imports
Complex_Main
"HOL-Library.Countable_Set"
"HOL-Library.FuncSet"
"HOL-Library.Indicator_Function"
"HOL-Library.Extended_Nonnegative_Real"
"HOL-Library.Disjoint_Sets"
begin
text \<open>Sigma algebras are an elementary concept in measure
theory. To measure --- that is to integrate --- functions, we first have
to measure sets. Unfortunately, when dealing with a large universe,
it is often not possible to consistently assign a measure to every
subset. Therefore it is necessary to define the set of measurable
subsets of the universe. A sigma algebra is such a set that has
three very natural and desirable properties.\<close>
subsection \<open>Families of sets\<close>
locale subset_class =
fixes \<Omega> :: "'a set" and M :: "'a set set"
assumes space_closed: "M \<subseteq> Pow \<Omega>"
lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
by (metis PowD contra_subsetD space_closed)
subsubsection \<open>Semiring of sets\<close>
locale semiring_of_sets = subset_class +
assumes empty_sets[iff]: "{} \<in> M"
assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
assumes Diff_cover:
"\<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"
lemma (in semiring_of_sets) finite_INT[intro]:
assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
shows "(\<Inter>i\<in>I. A i) \<in> M"
using assms by (induct rule: finite_ne_induct) auto
lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
by (metis Int_absorb1 sets_into_space)
lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
by (metis Int_absorb2 sets_into_space)
lemma (in semiring_of_sets) sets_Collect_conj:
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
proof -
have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
by auto
with assms show ?thesis by auto
qed
lemma (in semiring_of_sets) sets_Collect_finite_All':
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
using \<open>S \<noteq> {}\<close> by auto
with assms show ?thesis by auto
qed
locale ring_of_sets = semiring_of_sets +
assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
lemma (in ring_of_sets) finite_Union [intro]:
"finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> \<Union>X \<in> M"
by (induct set: finite) (auto simp add: Un)
lemma (in ring_of_sets) finite_UN[intro]:
assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
shows "(\<Union>i\<in>I. A i) \<in> M"
using assms by induct auto
lemma (in ring_of_sets) Diff [intro]:
assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"
using Diff_cover[OF assms] by auto
lemma ring_of_setsI:
assumes space_closed: "M \<subseteq> Pow \<Omega>"
assumes empty_sets[iff]: "{} \<in> M"
assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
shows "ring_of_sets \<Omega> M"
proof
fix a b assume ab: "a \<in> M" "b \<in> M"
from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
have "a \<inter> b = a - (a - b)" by auto
also have "\<dots> \<in> M" using ab by auto
finally show "a \<inter> b \<in> M" .
qed fact+
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)"
proof
assume "ring_of_sets \<Omega> M"
then interpret ring_of_sets \<Omega> M .
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)"
using space_closed by auto
qed (auto intro!: ring_of_setsI)
lemma (in ring_of_sets) insert_in_sets:
assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
proof -
have "{x} \<union> A \<in> M" using assms by (rule Un)
thus ?thesis by auto
qed
lemma (in ring_of_sets) sets_Collect_disj:
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
proof -
have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
by auto
with assms show ?thesis by auto
qed
lemma (in ring_of_sets) sets_Collect_finite_Ex:
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
by auto
with assms show ?thesis by auto
qed
locale algebra = ring_of_sets +
assumes top [iff]: "\<Omega> \<in> M"
lemma (in algebra) compl_sets [intro]:
"a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
by auto
lemma algebra_iff_Un:
"algebra \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> \<and>
{} \<in> M \<and>
(\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
(\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
proof
assume "algebra \<Omega> M"
then interpret algebra \<Omega> M .
show ?Un using sets_into_space by auto
next
assume ?Un
then have "\<Omega> \<in> M" by auto
interpret ring_of_sets \<Omega> M
proof (rule ring_of_setsI)
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
using \<open>?Un\<close> by auto
fix a b assume a: "a \<in> M" and b: "b \<in> M"
then show "a \<union> b \<in> M" using \<open>?Un\<close> by auto
have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
using \<Omega> a b by auto
then show "a - b \<in> M"
using a b \<open>?Un\<close> by auto
qed
show "algebra \<Omega> M" proof qed fact
qed
lemma algebra_iff_Int:
"algebra \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> & {} \<in> M &
(\<forall>a \<in> M. \<Omega> - a \<in> M) &
(\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
proof
assume "algebra \<Omega> M"
then interpret algebra \<Omega> M .
show ?Int using sets_into_space by auto
next
assume ?Int
show "algebra \<Omega> M"
proof (unfold algebra_iff_Un, intro conjI ballI)
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
using \<open>?Int\<close> by auto
from \<open>?Int\<close> show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
fix a b assume M: "a \<in> M" "b \<in> M"
hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
using \<Omega> by blast
also have "... \<in> M"
using M \<open>?Int\<close> by auto
finally show "a \<union> b \<in> M" .
qed
qed
lemma (in algebra) sets_Collect_neg:
assumes "{x\<in>\<Omega>. P x} \<in> M"
shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
with assms show ?thesis by auto
qed
lemma (in algebra) sets_Collect_imp:
"{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"
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)
lemma (in algebra) sets_Collect_const:
"{x\<in>\<Omega>. P} \<in> M"
by (cases P) auto
lemma algebra_single_set:
"X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
by (auto simp: algebra_iff_Int)
subsubsection \<open>Restricted algebras\<close>
abbreviation (in algebra)
"restricted_space A \<equiv> (op \<inter> A) ` M"
lemma (in algebra) restricted_algebra:
assumes "A \<in> M" shows "algebra A (restricted_space A)"
using assms by (auto simp: algebra_iff_Int)
subsubsection \<open>Sigma Algebras\<close>
locale sigma_algebra = algebra +
assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
lemma (in algebra) is_sigma_algebra:
assumes "finite M"
shows "sigma_algebra \<Omega> M"
proof
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
by auto
also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
using \<open>finite M\<close> by auto
finally show "(\<Union>i. A i) \<in> M" .
qed
lemma countable_UN_eq:
fixes A :: "'i::countable \<Rightarrow> 'a set"
shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
(range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
proof -
let ?A' = "A \<circ> from_nat"
have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
proof safe
fix x i assume "x \<in> A i" thus "x \<in> ?l"
by (auto intro!: exI[of _ "to_nat i"])
next
fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"
by (auto intro!: exI[of _ "from_nat i"])
qed
have **: "range ?A' = range A"
using surj_from_nat
by (auto simp: image_comp [symmetric] intro!: imageI)
show ?thesis unfolding * ** ..
qed
lemma (in sigma_algebra) countable_Union [intro]:
assumes "countable X" "X \<subseteq> M" shows "\<Union>X \<in> M"
proof cases
assume "X \<noteq> {}"
hence "\<Union>X = (\<Union>n. from_nat_into X n)"
using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
also have "\<dots> \<in> M" using assms
by (auto intro!: countable_nat_UN) (metis \<open>X \<noteq> {}\<close> from_nat_into set_mp)
finally show ?thesis .
qed simp
lemma (in sigma_algebra) countable_UN[intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes "A`X \<subseteq> M"
shows "(\<Union>x\<in>X. A x) \<in> M"
proof -
let ?A = "\<lambda>i. if i \<in> X then A i else {}"
from assms have "range ?A \<subseteq> M" by auto
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
have "(\<Union>x. ?A x) \<in> M" by auto
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: if_split_asm)
ultimately show ?thesis by simp
qed
lemma (in sigma_algebra) countable_UN':
fixes A :: "'i \<Rightarrow> 'a set"
assumes X: "countable X"
assumes A: "A`X \<subseteq> M"
shows "(\<Union>x\<in>X. A x) \<in> M"
proof -
have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
using X by auto
also have "\<dots> \<in> M"
using A X
by (intro countable_UN) auto
finally show ?thesis .
qed
lemma (in sigma_algebra) countable_UN'':
"\<lbrakk> countable X; \<And>x y. x \<in> X \<Longrightarrow> A x \<in> M \<rbrakk> \<Longrightarrow> (\<Union>x\<in>X. A x) \<in> M"
by(erule countable_UN')(auto)
lemma (in sigma_algebra) countable_INT [intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
shows "(\<Inter>i\<in>X. A i) \<in> M"
proof -
from A have "\<forall>i\<in>X. A i \<in> M" by fast
hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
moreover
have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
by blast
ultimately show ?thesis by metis
qed
lemma (in sigma_algebra) countable_INT':
fixes A :: "'i \<Rightarrow> 'a set"
assumes X: "countable X" "X \<noteq> {}"
assumes A: "A`X \<subseteq> M"
shows "(\<Inter>x\<in>X. A x) \<in> M"
proof -
have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
using X by auto
also have "\<dots> \<in> M"
using A X
by (intro countable_INT) auto
finally show ?thesis .
qed
lemma (in sigma_algebra) countable_INT'':
"UNIV \<in> M \<Longrightarrow> countable I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> M) \<Longrightarrow> (\<Inter>i\<in>I. F i) \<in> M"
by (cases "I = {}") (auto intro: countable_INT')
lemma (in sigma_algebra) countable:
assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> M" "countable A"
shows "A \<in> M"
proof -
have "(\<Union>a\<in>A. {a}) \<in> M"
using assms by (intro countable_UN') auto
also have "(\<Union>a\<in>A. {a}) = A" by auto
finally show ?thesis by auto
qed
lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
by (auto simp: ring_of_sets_iff)
lemma algebra_Pow: "algebra sp (Pow sp)"
by (auto simp: algebra_iff_Un)
lemma sigma_algebra_iff:
"sigma_algebra \<Omega> M \<longleftrightarrow>
algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
by (auto simp: sigma_algebra_iff algebra_iff_Int)
lemma (in sigma_algebra) sets_Collect_countable_All:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis by auto
qed
lemma (in sigma_algebra) sets_Collect_countable_Ex:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis by auto
qed
lemma (in sigma_algebra) sets_Collect_countable_Ex':
assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
assumes "countable I"
shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis
by (auto intro!: countable_UN')
qed
lemma (in sigma_algebra) sets_Collect_countable_All':
assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
assumes "countable I"
shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M"
proof -
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
with assms show ?thesis
by (cases "I = {}") (auto intro!: countable_INT')
qed
lemma (in sigma_algebra) sets_Collect_countable_Ex1':
assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
assumes "countable I"
shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M"
proof -
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)}"
by auto
with assms show ?thesis
by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)
qed
lemmas (in sigma_algebra) sets_Collect =
sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
lemma (in sigma_algebra) sets_Collect_countable_Ball:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
unfolding Ball_def by (intro sets_Collect assms)
lemma (in sigma_algebra) sets_Collect_countable_Bex:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
unfolding Bex_def by (intro sets_Collect assms)
lemma sigma_algebra_single_set:
assumes "X \<subseteq> S"
shows "sigma_algebra S { {}, X, S - X, S }"
using algebra.is_sigma_algebra[OF algebra_single_set[OF \<open>X \<subseteq> S\<close>]] by simp
subsubsection \<open>Binary Unions\<close>
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
where "binary a b = (\<lambda>x. b)(0 := a)"
lemma range_binary_eq: "range(binary a b) = {a,b}"
by (auto simp add: binary_def)
lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
by (simp add: range_binary_eq cong del: strong_SUP_cong)
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
by (simp add: range_binary_eq cong del: strong_INF_cong)
lemma sigma_algebra_iff2:
"sigma_algebra \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> \<and>
{} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
(\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
algebra_iff_Un Un_range_binary)
subsubsection \<open>Initial Sigma Algebra\<close>
text \<open>Sigma algebras can naturally be created as the closure of any set of
M with regard to the properties just postulated.\<close>
inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
for sp :: "'a set" and A :: "'a set set"
where
Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
| Empty: "{} \<in> sigma_sets sp A"
| Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
| Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
lemma (in sigma_algebra) sigma_sets_subset:
assumes a: "a \<subseteq> M"
shows "sigma_sets \<Omega> a \<subseteq> M"
proof
fix x
assume "x \<in> sigma_sets \<Omega> a"
from this show "x \<in> M"
by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
qed
lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
by (erule sigma_sets.induct, auto)
lemma sigma_algebra_sigma_sets:
"a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
lemma sigma_sets_least_sigma_algebra:
assumes "A \<subseteq> Pow S"
shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
proof safe
fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
and X: "X \<in> sigma_sets S A"
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF \<open>A \<subseteq> B\<close>] X
show "X \<in> B" by auto
next
fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
by simp
have "A \<subseteq> sigma_sets S A" using assms by auto
moreover have "sigma_algebra S (sigma_sets S A)"
using assms by (intro sigma_algebra_sigma_sets[of A]) auto
ultimately show "X \<in> sigma_sets S A" by auto
qed
lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
lemma sigma_sets_Un:
"a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
apply (simp add: Un_range_binary range_binary_eq)
apply (rule Union, simp add: binary_def)
done
lemma sigma_sets_Inter:
assumes Asb: "A \<subseteq> Pow sp"
shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
proof -
assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
by (rule sigma_sets.Compl)
hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
by (rule sigma_sets.Union)
hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
by (rule sigma_sets.Compl)
also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
by auto
also have "... = (\<Inter>i. a i)" using ai
by (blast dest: sigma_sets_into_sp [OF Asb])
finally show ?thesis .
qed
lemma sigma_sets_INTER:
assumes Asb: "A \<subseteq> Pow sp"
and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
proof -
from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
by (simp add: sigma_sets.intros(2-) sigma_sets_top)
hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
by (rule sigma_sets_Inter [OF Asb])
also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
finally show ?thesis .
qed
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"
apply (cases "B = {}")
apply (simp add: sigma_sets.Empty)
using from_nat_into [of B] range_from_nat_into [of B] sigma_sets.Union [of "from_nat_into B" X A]
apply simp
apply auto
apply (metis Sup_bot_conv(1) Union_empty \<open>\<lbrakk>B \<noteq> {}; countable B\<rbrakk> \<Longrightarrow> range (from_nat_into B) = B\<close>)
done
lemma (in sigma_algebra) sigma_sets_eq:
"sigma_sets \<Omega> M = M"
proof
show "M \<subseteq> sigma_sets \<Omega> M"
by (metis Set.subsetI sigma_sets.Basic)
next
show "sigma_sets \<Omega> M \<subseteq> M"
by (metis sigma_sets_subset subset_refl)
qed
lemma sigma_sets_eqI:
assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
shows "sigma_sets M A = sigma_sets M B"
proof (intro set_eqI iffI)
fix a assume "a \<in> sigma_sets M A"
from this A show "a \<in> sigma_sets M B"
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
next
fix b assume "b \<in> sigma_sets M B"
from this B show "b \<in> sigma_sets M A"
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
qed
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))
qed
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert \<open>A \<subseteq> sigma_sets X B\<close>, auto intro: sigma_sets.intros(2-))
qed
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))
qed
lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
by (auto intro: sigma_sets.Basic)
lemma (in sigma_algebra) restriction_in_sets:
fixes A :: "nat \<Rightarrow> 'a set"
assumes "S \<in> M"
and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
proof -
{ fix i have "A i \<in> ?r" using * by auto
hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
hence "A i \<subseteq> S" "A i \<in> M" using \<open>S \<in> M\<close> by auto }
thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
qed
lemma (in sigma_algebra) restricted_sigma_algebra:
assumes "S \<in> M"
shows "sigma_algebra S (restricted_space S)"
unfolding sigma_algebra_def sigma_algebra_axioms_def
proof safe
show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
next
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
from restriction_in_sets[OF assms this[simplified]]
show "(\<Union>i. A i) \<in> restricted_space S" by simp
qed
lemma sigma_sets_Int:
assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
proof (intro equalityI subsetI)
fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
proof (induct arbitrary: x)
case (Compl a)
then show ?case
by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
next
case (Union a)
then show ?case
by (auto intro!: sigma_sets.Union
simp add: UN_extend_simps simp del: UN_simps)
qed (auto intro!: sigma_sets.intros(2-))
then show "x \<in> sigma_sets A (op \<inter> A ` st)"
using \<open>A \<subseteq> sp\<close> by (simp add: Int_absorb2)
next
fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
then show "x \<in> op \<inter> A ` sigma_sets sp st"
proof induct
case (Compl a)
then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
then show ?case using \<open>A \<subseteq> sp\<close>
by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
next
case (Union a)
then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
by (auto simp: image_iff Bex_def)
from choice[OF this] guess f ..
then show ?case
by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
simp add: image_iff)
qed (auto intro!: sigma_sets.intros(2-))
qed
lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
proof (intro set_eqI iffI)
fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
by induct blast+
qed (auto intro: sigma_sets.Empty sigma_sets_top)
lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
proof (intro set_eqI iffI)
fix x assume "x \<in> sigma_sets A {A}"
then show "x \<in> {{}, A}"
by induct blast+
next
fix x assume "x \<in> {{}, A}"
then show "x \<in> sigma_sets A {A}"
by (auto intro: sigma_sets.Empty sigma_sets_top)
qed
lemma sigma_sets_sigma_sets_eq:
"M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
lemma sigma_sets_singleton:
assumes "X \<subseteq> S"
shows "sigma_sets S { X } = { {}, X, S - X, S }"
proof -
interpret sigma_algebra S "{ {}, X, S - X, S }"
by (rule sigma_algebra_single_set) fact
have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
by (rule sigma_sets_subseteq) simp
moreover have "\<dots> = { {}, X, S - X, S }"
using sigma_sets_eq by simp
moreover
{ fix A assume "A \<in> { {}, X, S - X, S }"
then have "A \<in> sigma_sets S { X }"
by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
by (intro antisym) auto
with sigma_sets_eq show ?thesis by simp
qed
lemma restricted_sigma:
assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
sigma_sets S (algebra.restricted_space M S)"
proof -
from S sigma_sets_into_sp[OF M]
have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
from sigma_sets_Int[OF this]
show ?thesis by simp
qed
lemma sigma_sets_vimage_commute:
assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
= sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
proof
show "?L \<subseteq> ?R"
proof clarify
fix A assume "A \<in> sigma_sets \<Omega>' M'"
then show "X -` A \<inter> \<Omega> \<in> ?R"
proof induct
case Empty then show ?case
by (auto intro!: sigma_sets.Empty)
next
case (Compl B)
have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
by (auto simp add: funcset_mem [OF X])
with Compl show ?case
by (auto intro!: sigma_sets.Compl)
next
case (Union F)
then show ?case
by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
intro!: sigma_sets.Union)
qed auto
qed
show "?R \<subseteq> ?L"
proof clarify
fix A assume "A \<in> ?R"
then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
proof induct
case (Basic B) then show ?case by auto
next
case Empty then show ?case
by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
next
case (Compl B)
then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
by (auto simp add: funcset_mem [OF X])
with A(2) show ?case
by (auto intro: sigma_sets.Compl)
next
case (Union F)
then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
from choice[OF this] guess A .. note A = this
with A show ?case
by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
qed
qed
qed
lemma (in ring_of_sets) UNION_in_sets:
fixes A:: "nat \<Rightarrow> 'a set"
assumes A: "range A \<subseteq> M"
shows "(\<Union>i\<in>{0..<n}. A i) \<in> M"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case
by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
qed
lemma (in ring_of_sets) range_disjointed_sets:
assumes A: "range A \<subseteq> M"
shows "range (disjointed A) \<subseteq> M"
proof (auto simp add: disjointed_def)
fix n
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
by (metis A Diff UNIV_I image_subset_iff)
qed
lemma (in algebra) range_disjointed_sets':
"range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
using range_disjointed_sets .
lemma sigma_algebra_disjoint_iff:
"sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
proof (auto simp add: sigma_algebra_iff)
fix A :: "nat \<Rightarrow> 'a set"
assume M: "algebra \<Omega> M"
and A: "range A \<subseteq> M"
and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
hence "range (disjointed A) \<subseteq> M \<longrightarrow>
disjoint_family (disjointed A) \<longrightarrow>
(\<Union>i. disjointed A i) \<in> M" by blast
hence "(\<Union>i. disjointed A i) \<in> M"
by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
qed
subsubsection \<open>Ring generated by a semiring\<close>
definition (in semiring_of_sets)
"generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
lemma (in semiring_of_sets) generated_ringE[elim?]:
assumes "a \<in> generated_ring"
obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
using assms unfolding generated_ring_def by auto
lemma (in semiring_of_sets) generated_ringI[intro?]:
assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
shows "a \<in> generated_ring"
using assms unfolding generated_ring_def by auto
lemma (in semiring_of_sets) generated_ringI_Basic:
"A \<in> M \<Longrightarrow> A \<in> generated_ring"
by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
and "a \<inter> b = {}"
shows "a \<union> b \<in> generated_ring"
proof -
from a guess Ca .. note Ca = this
from b guess Cb .. note Cb = this
show ?thesis
proof
show "disjoint (Ca \<union> Cb)"
using \<open>a \<inter> b = {}\<close> Ca Cb by (auto intro!: disjoint_union)
qed (insert Ca Cb, auto)
qed
lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
by (auto simp: generated_ring_def disjoint_def)
lemma (in semiring_of_sets) generated_ring_disjoint_Union:
assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
"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"
by (intro generated_ring_disjoint_Union) auto
lemma (in semiring_of_sets) generated_ring_Int:
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
shows "a \<inter> b \<in> generated_ring"
proof -
from a guess Ca .. note Ca = this
from b guess Cb .. note Cb = this
define C where "C = (\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
show ?thesis
proof
show "disjoint C"
proof (simp add: disjoint_def C_def, intro ballI impI)
fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
proof
assume "a1 \<noteq> a2"
with sets Ca have "a1 \<inter> a2 = {}"
by (auto simp: disjoint_def)
then show ?thesis by auto
next
assume "b1 \<noteq> b2"
with sets Cb have "b1 \<inter> b2 = {}"
by (auto simp: disjoint_def)
then show ?thesis by auto
qed
qed
qed (insert Ca Cb, auto simp: C_def)
qed
lemma (in semiring_of_sets) generated_ring_Inter:
assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
lemma (in semiring_of_sets) generated_ring_INTER:
"finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
by (intro generated_ring_Inter) auto
lemma (in semiring_of_sets) generating_ring:
"ring_of_sets \<Omega> generated_ring"
proof (rule ring_of_setsI)
let ?R = generated_ring
show "?R \<subseteq> Pow \<Omega>"
using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
show "{} \<in> ?R" by (rule generated_ring_empty)
{ fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
show "a - b \<in> ?R"
proof cases
assume "Cb = {}" with Cb \<open>a \<in> ?R\<close> show ?thesis
by simp
next
assume "Cb \<noteq> {}"
with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
also have "\<dots> \<in> ?R"
proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
fix a b assume "a \<in> Ca" "b \<in> Cb"
with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
by (auto simp add: generated_ring_def)
(metis DiffI Diff_eq_empty_iff empty_iff)
next
show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
using Ca by (auto simp add: disjoint_def \<open>Cb \<noteq> {}\<close>)
next
show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
qed
finally show "a - b \<in> ?R" .
qed }
note Diff = this
fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
also have "\<dots> \<in> ?R"
by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
finally show "a \<union> b \<in> ?R" .
qed
lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
proof
interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
using space_closed by (rule sigma_algebra_sigma_sets)
show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
by (blast intro!: sigma_sets_mono elim: generated_ringE)
qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
subsubsection \<open>A Two-Element Series\<close>
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set"
where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
apply (simp add: binaryset_def)
apply (rule set_eqI)
apply (auto simp add: image_iff)
done
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
by (simp add: range_binaryset_eq cong del: strong_SUP_cong)
subsubsection \<open>Closed CDI\<close>
definition closed_cdi where
"closed_cdi \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> &
(\<forall>s \<in> M. \<Omega> - s \<in> M) &
(\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
(\<Union>i. A i) \<in> M) &
(\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
inductive_set
smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
for \<Omega> M
where
Basic [intro]:
"a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
| Compl [intro]:
"a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
| Inc:
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
| Disj:
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
by auto
lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
apply (rule subsetI)
apply (erule smallest_ccdi_sets.induct)
apply (auto intro: range_subsetD dest: sets_into_space)
done
lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
done
lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
by (simp add: closed_cdi_def)
lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Inc:
"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"
by (simp add: closed_cdi_def)
lemma closed_cdi_Disj:
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Un:
assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
and A: "A \<in> M" and B: "B \<in> M"
and disj: "A \<inter> B = {}"
shows "A \<union> B \<in> M"
proof -
have ra: "range (binaryset A B) \<subseteq> M"
by (simp add: range_binaryset_eq empty A B)
have di: "disjoint_family (binaryset A B)" using disj
by (simp add: disjoint_family_on_def binaryset_def Int_commute)
from closed_cdi_Disj [OF cdi ra di]
show ?thesis
by (simp add: UN_binaryset_eq)
qed
lemma (in algebra) smallest_ccdi_sets_Un:
assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
and disj: "A \<inter> B = {}"
shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
proof -
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic)
have di: "disjoint_family (binaryset A B)" using disj
by (simp add: disjoint_family_on_def binaryset_def Int_commute)
from Disj [OF ra di]
show ?thesis
by (simp add: UN_binaryset_eq)
qed
lemma (in algebra) smallest_ccdi_sets_Int1:
assumes a: "a \<in> M"
shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis a Int smallest_ccdi_sets.Basic)
next
case (Compl x)
have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
by blast
also have "... \<in> smallest_ccdi_sets \<Omega> M"
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
finally show ?case .
next
case (Inc A)
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
by blast
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
by blast
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
by (simp add: Inc)
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
by blast
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
next
case (Disj A)
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
by blast
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
by blast
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
by (auto simp add: disjoint_family_on_def)
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed
lemma (in algebra) smallest_ccdi_sets_Int:
assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis b smallest_ccdi_sets_Int1)
next
case (Compl x)
have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
by blast
also have "... \<in> smallest_ccdi_sets \<Omega> M"
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
finally show ?case .
next
case (Inc A)
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
by blast
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
by blast
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
by (simp add: Inc)
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
by blast
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
next
case (Disj A)
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
by blast
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
by blast
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
by (auto simp add: disjoint_family_on_def)
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed
lemma (in algebra) sigma_property_disjoint_lemma:
assumes sbC: "M \<subseteq> C"
and ccdi: "closed_cdi \<Omega> C"
shows "sigma_sets \<Omega> M \<subseteq> C"
proof -
have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
smallest_ccdi_sets_Int)
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Disj)
done
hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
by clarsimp
(drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
also have "... \<subseteq> C"
proof
fix x
assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
thus "x \<in> C"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis Basic subsetD sbC)
next
case (Compl x)
thus ?case
by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
next
case (Inc A)
thus ?case
by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
next
case (Disj A)
thus ?case
by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
qed
qed
finally show ?thesis .
qed
lemma (in algebra) sigma_property_disjoint:
assumes sbC: "M \<subseteq> C"
and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
\<Longrightarrow> (\<Union>i. A i) \<in> C"
and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
proof -
have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
proof (rule sigma_property_disjoint_lemma)
show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
next
show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
by (simp add: closed_cdi_def compl inc disj)
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
qed
thus ?thesis
by blast
qed
subsubsection \<open>Dynkin systems\<close>
locale dynkin_system = subset_class +
assumes space: "\<Omega> \<in> M"
and compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
using space compl[of "\<Omega>"] by simp
lemma (in dynkin_system) diff:
assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
shows "E - D \<in> M"
proof -
let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
have "range ?f = {D, \<Omega> - E, {}}"
by (auto simp: image_iff)
moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
by (auto simp: image_iff split: if_split_asm)
moreover
have "disjoint_family ?f" unfolding disjoint_family_on_def
using \<open>D \<in> M\<close>[THEN sets_into_space] \<open>D \<subseteq> E\<close> by auto
ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
using sets by auto
also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
using assms sets_into_space by auto
finally show ?thesis .
qed
lemma dynkin_systemI:
assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
shows "dynkin_system \<Omega> M"
using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
lemma dynkin_systemI':
assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
assumes empty: "{} \<in> M"
assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
shows "dynkin_system \<Omega> M"
proof -
from Diff[OF empty] have "\<Omega> \<in> M" by auto
from 1 this Diff 2 show ?thesis
by (intro dynkin_systemI) auto
qed
lemma dynkin_system_trivial:
shows "dynkin_system A (Pow A)"
by (rule dynkin_systemI) auto
lemma sigma_algebra_imp_dynkin_system:
assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
proof -
interpret sigma_algebra \<Omega> M by fact
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
qed
subsubsection "Intersection sets systems"
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
lemma (in algebra) Int_stable: "Int_stable M"
unfolding Int_stable_def by auto
lemma Int_stableI_image:
"(\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. A i \<inter> A j = A k) \<Longrightarrow> Int_stable (A ` I)"
by (auto simp: Int_stable_def image_def)
lemma Int_stableI:
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
unfolding Int_stable_def by auto
lemma Int_stableD:
"Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
unfolding Int_stable_def by auto
lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
"sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
proof
assume "sigma_algebra \<Omega> M" then show "Int_stable M"
unfolding sigma_algebra_def using algebra.Int_stable by auto
next
assume "Int_stable M"
show "sigma_algebra \<Omega> M"
unfolding sigma_algebra_disjoint_iff algebra_iff_Un
proof (intro conjI ballI allI impI)
show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
next
fix A B assume "A \<in> M" "B \<in> M"
then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
"\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
using sets_into_space by auto
then show "A \<union> B \<in> M"
using \<open>Int_stable M\<close> unfolding Int_stable_def by auto
qed auto
qed
subsubsection "Smallest Dynkin systems"
definition dynkin where
"dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
lemma dynkin_system_dynkin:
assumes "M \<subseteq> Pow (\<Omega>)"
shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
proof (rule dynkin_systemI)
fix A assume "A \<in> dynkin \<Omega> M"
moreover
{ fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
using assms dynkin_system_trivial by fastforce
ultimately show "A \<subseteq> \<Omega>"
unfolding dynkin_def using assms
by auto
next
show "\<Omega> \<in> dynkin \<Omega> M"
unfolding dynkin_def using dynkin_system.space by fastforce
next
fix A assume "A \<in> dynkin \<Omega> M"
then show "\<Omega> - A \<in> dynkin \<Omega> M"
unfolding dynkin_def using dynkin_system.compl by force
next
fix A :: "nat \<Rightarrow> 'a set"
assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
proof (simp, safe)
fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
with A have "(\<Union>i. A i) \<in> D"
by (intro dynkin_system.UN) (auto simp: dynkin_def)
then show "(\<Union>i. A i) \<in> D" by auto
qed
qed
lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
unfolding dynkin_def by auto
lemma (in dynkin_system) restricted_dynkin_system:
assumes "D \<in> M"
shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
proof (rule dynkin_systemI, simp_all)
have "\<Omega> \<inter> D = D"
using \<open>D \<in> M\<close> sets_into_space by auto
then show "\<Omega> \<inter> D \<in> M"
using \<open>D \<in> M\<close> by auto
next
fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
by auto
ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
using \<open>D \<in> M\<close> by (auto intro: diff)
next
fix A :: "nat \<Rightarrow> 'a set"
assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
"range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
by ((fastforce simp: disjoint_family_on_def)+)
then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
by (auto simp del: UN_simps)
qed
lemma (in dynkin_system) dynkin_subset:
assumes "N \<subseteq> M"
shows "dynkin \<Omega> N \<subseteq> M"
proof -
have "dynkin_system \<Omega> M" ..
then have "dynkin_system \<Omega> M"
using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: dynkin_def)
qed
lemma sigma_eq_dynkin:
assumes sets: "M \<subseteq> Pow \<Omega>"
assumes "Int_stable M"
shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
proof -
have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
using sigma_algebra_imp_dynkin_system
unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
moreover
interpret dynkin_system \<Omega> "dynkin \<Omega> M"
using dynkin_system_dynkin[OF sets] .
have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
unfolding sigma_algebra_eq_Int_stable Int_stable_def
proof (intro ballI)
fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
have "M \<subseteq> ?D B"
proof
fix E assume "E \<in> M"
then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
using sets_into_space \<open>Int_stable M\<close> by (auto simp: Int_stable_def)
then have "dynkin \<Omega> M \<subseteq> ?D E"
using restricted_dynkin_system \<open>E \<in> dynkin \<Omega> M\<close>
by (intro dynkin_system.dynkin_subset) simp_all
then have "B \<in> ?D E"
using \<open>B \<in> dynkin \<Omega> M\<close> by auto
then have "E \<inter> B \<in> dynkin \<Omega> M"
by (subst Int_commute) simp
then show "E \<in> ?D B"
using sets \<open>E \<in> M\<close> by auto
qed
then have "dynkin \<Omega> M \<subseteq> ?D B"
using restricted_dynkin_system \<open>B \<in> dynkin \<Omega> M\<close>
by (intro dynkin_system.dynkin_subset) simp_all
then show "A \<inter> B \<in> dynkin \<Omega> M"
using \<open>A \<in> dynkin \<Omega> M\<close> sets_into_space by auto
qed
from sigma_algebra.sigma_sets_subset[OF this, of "M"]
have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
then show ?thesis
by (auto simp: dynkin_def)
qed
lemma (in dynkin_system) dynkin_idem:
"dynkin \<Omega> M = M"
proof -
have "dynkin \<Omega> M = M"
proof
show "M \<subseteq> dynkin \<Omega> M"
using dynkin_Basic by auto
show "dynkin \<Omega> M \<subseteq> M"
by (intro dynkin_subset) auto
qed
then show ?thesis
by (auto simp: dynkin_def)
qed
lemma (in dynkin_system) dynkin_lemma:
assumes "Int_stable E"
and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
shows "sigma_sets \<Omega> E = M"
proof -
have "E \<subseteq> Pow \<Omega>"
using E sets_into_space by force
then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
using \<open>Int_stable E\<close> by (rule sigma_eq_dynkin)
then have "dynkin \<Omega> E = M"
using assms dynkin_subset[OF E(1)] by simp
with * show ?thesis
using assms by (auto simp: dynkin_def)
qed
subsubsection \<open>Induction rule for intersection-stable generators\<close>
text \<open>The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
generated by a generator closed under intersection.\<close>
lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
assumes "Int_stable G"
and closed: "G \<subseteq> Pow \<Omega>"
and A: "A \<in> sigma_sets \<Omega> G"
assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
and empty: "P {}"
and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
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)"
shows "P A"
proof -
let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
using closed by (rule sigma_algebra_sigma_sets)
from compl[OF _ empty] closed have space: "P \<Omega>" by simp
interpret dynkin_system \<Omega> ?D
by standard (auto dest: sets_into_space intro!: space compl union)
have "sigma_sets \<Omega> G = ?D"
by (rule dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)
with A show ?thesis by auto
qed
subsection \<open>Measure type\<close>
definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
"positive M \<mu> \<longleftrightarrow> \<mu> {} = 0"
definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
(\<Sum>i. f (A i)) = f (\<Union>i. A i))"
definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
"measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
proof
have "sigma_algebra UNIV {{}, UNIV}"
by (auto simp: sigma_algebra_iff2)
then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
by (auto simp: measure_space_def positive_def countably_additive_def)
qed
definition space :: "'a measure \<Rightarrow> 'a set" where
"space M = fst (Rep_measure M)"
definition sets :: "'a measure \<Rightarrow> 'a set set" where
"sets M = fst (snd (Rep_measure M))"
definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ennreal" where
"emeasure M = snd (snd (Rep_measure M))"
definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
"measure M A = enn2real (emeasure M A)"
declare [[coercion sets]]
declare [[coercion measure]]
declare [[coercion emeasure]]
lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
interpretation sets: sigma_algebra "space M" "sets M" for M :: "'a measure"
using measure_space[of M] by (auto simp: measure_space_def)
definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a measure" where
"measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
\<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
unfolding measure_space_def
by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
lemma measure_space_closed:
assumes "measure_space \<Omega> M \<mu>"
shows "M \<subseteq> Pow \<Omega>"
proof -
interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
show ?thesis by(rule space_closed)
qed
lemma (in ring_of_sets) positive_cong_eq:
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
by (auto simp add: positive_def)
lemma (in sigma_algebra) countably_additive_eq:
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
unfolding countably_additive_def
by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
lemma measure_space_eq:
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
proof -
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
by (auto simp: measure_space_def)
qed
lemma measure_of_eq:
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
proof -
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
using assms by (rule measure_space_eq)
with eq show ?thesis
by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
qed
lemma
shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
and sets_measure_of_conv:
"sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
and emeasure_measure_of_conv:
"emeasure (measure_of \<Omega> A \<mu>) =
(\<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)
proof -
have "?space \<and> ?sets \<and> ?emeasure"
proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
case True
from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
have "A \<subseteq> Pow \<Omega>" by simp
hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
(\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
by(rule measure_space_eq) auto
with True \<open>A \<subseteq> Pow \<Omega>\<close> show ?thesis
by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
next
case False thus ?thesis
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')
qed
thus ?space ?sets ?emeasure by simp_all
qed
lemma [simp]:
assumes A: "A \<subseteq> Pow \<Omega>"
shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
using assms
by(simp_all add: sets_measure_of_conv space_measure_of_conv)
lemma space_in_measure_of[simp]: "\<Omega> \<in> sets (measure_of \<Omega> M \<mu>)"
by (subst sets_measure_of_conv) (auto simp: sigma_sets_top)
lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
using space_closed by (auto intro!: sigma_sets_eq)
lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
by (rule space_measure_of_conv)
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>')"
by (auto intro!: sigma_sets_subseteq)
lemma emeasure_sigma: "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"
unfolding measure_of_def emeasure_def
by (subst Abs_measure_inverse)
(auto simp: measure_space_def positive_def countably_additive_def
intro!: sigma_algebra_sigma_sets sigma_algebra_trivial)
lemma sigma_sets_mono'':
assumes "A \<in> sigma_sets C D"
assumes "B \<subseteq> D"
assumes "D \<subseteq> Pow C"
shows "sigma_sets A B \<subseteq> sigma_sets C D"
proof
fix x assume "x \<in> sigma_sets A B"
thus "x \<in> sigma_sets C D"
proof induct
case (Basic a) with assms have "a \<in> D" by auto
thus ?case ..
next
case Empty show ?case by (rule sigma_sets.Empty)
next
from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
ultimately have "A - a \<in> sets (sigma C D)" ..
thus ?case by (subst (asm) sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
next
case (Union a)
thus ?case by (intro sigma_sets.Union)
qed
qed
lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
by auto
lemma space_empty_iff: "space N = {} \<longleftrightarrow> sets N = {{}}"
by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff
sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)
subsubsection \<open>Constructing simple @{typ "'a measure"}\<close>
lemma emeasure_measure_of:
assumes M: "M = measure_of \<Omega> A \<mu>"
assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
assumes X: "X \<in> sets M"
shows "emeasure M X = \<mu> X"
proof -
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
thus ?thesis using X ms
by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
qed
lemma emeasure_measure_of_sigma:
assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
assumes A: "A \<in> M"
shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
proof -
interpret sigma_algebra \<Omega> M by fact
have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
using ms sigma_sets_eq by (simp add: measure_space_def)
thus ?thesis by(simp add: emeasure_measure_of_conv A)
qed
lemma measure_cases[cases type: measure]:
obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
by atomize_elim (cases x, auto)
lemma sets_le_imp_space_le: "sets A \<subseteq> sets B \<Longrightarrow> space A \<subseteq> space B"
by (auto dest: sets.sets_into_space)
lemma sets_eq_imp_space_eq: "sets M = sets M' \<Longrightarrow> space M = space M'"
by (auto intro!: antisym sets_le_imp_space_le)
lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
using emeasure_notin_sets[of A M] by blast
lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
by (simp add: measure_def emeasure_notin_sets zero_ennreal.rep_eq)
lemma measure_eqI:
fixes M N :: "'a measure"
assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
shows "M = N"
proof (cases M N rule: measure_cases[case_product measure_cases])
case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
have "A = sets M" "A' = sets N"
using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
with \<open>sets M = sets N\<close> have AA': "A = A'" by simp
moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
moreover { fix B have "\<mu> B = \<mu>' B"
proof cases
assume "B \<in> A"
with eq \<open>A = sets M\<close> have "emeasure M B = emeasure N B" by simp
with measure_measure show "\<mu> B = \<mu>' B"
by (simp add: emeasure_def Abs_measure_inverse)
next
assume "B \<notin> A"
with \<open>A = sets M\<close> \<open>A' = sets N\<close> \<open>A = A'\<close> have "B \<notin> sets M" "B \<notin> sets N"
by auto
then have "emeasure M B = 0" "emeasure N B = 0"
by (simp_all add: emeasure_notin_sets)
with measure_measure show "\<mu> B = \<mu>' B"
by (simp add: emeasure_def Abs_measure_inverse)
qed }
then have "\<mu> = \<mu>'" by auto
ultimately show "M = N"
by (simp add: measure_measure)
qed
lemma sigma_eqI:
assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
shows "sigma \<Omega> M = sigma \<Omega> N"
by (rule measure_eqI) (simp_all add: emeasure_sigma)
subsubsection \<open>Measurable functions\<close>
definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>M" 60) where
"measurable A B = {f \<in> space A \<rightarrow> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
lemma measurableI:
"(\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N) \<Longrightarrow> (\<And>A. A \<in> sets N \<Longrightarrow> f -` A \<inter> space M \<in> sets M) \<Longrightarrow>
f \<in> measurable M N"
by (auto simp: measurable_def)
lemma measurable_space:
"f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
unfolding measurable_def by auto
lemma measurable_sets:
"f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
unfolding measurable_def by auto
lemma measurable_sets_Collect:
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"
proof -
have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
using measurable_space[OF f] by auto
with measurable_sets[OF f P] show ?thesis
by simp
qed
lemma measurable_sigma_sets:
assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
and f: "f \<in> space M \<rightarrow> \<Omega>"
and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
shows "f \<in> measurable M N"
proof -
interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
{ fix X assume "X \<in> sigma_sets \<Omega> A"
then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
proof induct
case (Basic a) then show ?case
by (auto simp add: ba) (metis B(2) subsetD PowD)
next
case (Compl a)
have [simp]: "f -` \<Omega> \<inter> space M = space M"
by (auto simp add: funcset_mem [OF f])
then show ?case
by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
next
case (Union a)
then show ?case
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
qed auto }
with f show ?thesis
by (auto simp add: measurable_def B \<Omega>)
qed
lemma measurable_measure_of:
assumes B: "N \<subseteq> Pow \<Omega>"
and f: "f \<in> space M \<rightarrow> \<Omega>"
and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
proof -
have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
using B by (rule sets_measure_of)
from this assms show ?thesis by (rule measurable_sigma_sets)
qed
lemma measurable_iff_measure_of:
assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
lemma measurable_cong_sets:
assumes sets: "sets M = sets M'" "sets N = sets N'"
shows "measurable M N = measurable M' N'"
using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
lemma measurable_cong:
assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w"
shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
unfolding measurable_def using assms
by (simp cong: vimage_inter_cong Pi_cong)
lemma measurable_cong':
assumes "\<And>w. w \<in> space M =simp=> f w = g w"
shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
unfolding measurable_def using assms
by (simp cong: vimage_inter_cong Pi_cong add: simp_implies_def)
lemma measurable_cong_strong:
"M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
by (metis measurable_cong)
lemma measurable_compose:
assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
shows "(\<lambda>x. g (f x)) \<in> measurable M L"
proof -
have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
using measurable_space[OF f] by auto
with measurable_space[OF f] measurable_space[OF g] show ?thesis
by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
simp del: vimage_Int simp add: measurable_def)
qed
lemma measurable_comp:
"f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
using measurable_compose[of f M N g L] by (simp add: comp_def)
lemma measurable_const:
"c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
by (auto simp add: measurable_def)
lemma measurable_ident: "id \<in> measurable M M"
by (auto simp add: measurable_def)
lemma measurable_id: "(\<lambda>x. x) \<in> measurable M M"
by (simp add: measurable_def)
lemma measurable_ident_sets:
assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
using measurable_ident[of M]
unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
lemma sets_Least:
assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
proof -
{ fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
proof cases
assume i: "(LEAST j. False) = i"
have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
{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}))"
by (simp add: set_eq_iff, safe)
(insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
with meas show ?thesis
by (auto intro!: sets.Int)
next
assume i: "(LEAST j. False) \<noteq> i"
then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
proof (simp add: set_eq_iff, safe)
fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
have "\<exists>j. P j x"
by (rule ccontr) (insert neq, auto)
then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
qed (auto dest: Least_le intro!: Least_equality)
with meas show ?thesis
by auto
qed }
then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
by (intro sets.countable_UN) auto
moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
(\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
ultimately show ?thesis by auto
qed
lemma measurable_mono1:
"M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
subsubsection \<open>Counting space\<close>
definition count_space :: "'a set \<Rightarrow> 'a measure" where
"count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then of_nat (card A) else \<infinity>)"
lemma
shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
by (auto simp: count_space_def)
lemma measurable_count_space_eq1[simp]:
"f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
unfolding measurable_def by simp
lemma measurable_compose_countable':
assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f i x) \<in> measurable M N"
and g: "g \<in> measurable M (count_space I)" and I: "countable I"
shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
unfolding measurable_def
proof safe
fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
using measurable_space[OF f] g[THEN measurable_space] by auto
next
fix A assume A: "A \<in> sets N"
have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i\<in>I. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))"
using measurable_space[OF g] by auto
also have "\<dots> \<in> sets M"
using f[THEN measurable_sets, OF _ A] g[THEN measurable_sets]
by (auto intro!: sets.countable_UN' I intro: sets.Int[OF measurable_sets measurable_sets])
finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
qed
lemma measurable_count_space_eq_countable:
assumes "countable A"
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
proof -
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
with \<open>countable A\<close> have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
by (auto dest: countable_subset)
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
ultimately have "f -` X \<inter> space M \<in> sets M"
using \<open>X \<subseteq> A\<close> by (auto intro!: sets.countable_UN' simp del: UN_simps) }
then show ?thesis
unfolding measurable_def by auto
qed
lemma measurable_count_space_eq2:
"finite A \<Longrightarrow> f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
by (intro measurable_count_space_eq_countable countable_finite)
lemma measurable_count_space_eq2_countable:
fixes f :: "'a => 'c::countable"
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
by (intro measurable_count_space_eq_countable countableI_type)
lemma measurable_compose_countable:
assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
by (rule measurable_compose_countable'[OF assms]) auto
lemma measurable_count_space_const:
"(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
by (simp add: measurable_const)
lemma measurable_count_space:
"f \<in> measurable (count_space A) (count_space UNIV)"
by simp
lemma measurable_compose_rev:
assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
shows "(\<lambda>x. f (g x)) \<in> measurable M N"
using measurable_compose[OF g f] .
lemma measurable_empty_iff:
"space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}"
by (auto simp add: measurable_def Pi_iff)
subsubsection \<open>Extend measure\<close>
definition "extend_measure \<Omega> I G \<mu> =
(if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>')
else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
unfolding extend_measure_def by simp
lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
unfolding extend_measure_def by simp
lemma emeasure_extend_measure:
assumes M: "M = extend_measure \<Omega> I G \<mu>"
and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
and "i \<in> I"
shows "emeasure M (G i) = \<mu> i"
proof cases
assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
by (simp add: extend_measure_def)
from measure_space_0[OF ms(1)] ms \<open>i\<in>I\<close>
have "emeasure M (G i) = 0"
by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
with \<open>i\<in>I\<close> * show ?thesis
by simp
next
define P where "P \<mu>' \<longleftrightarrow> (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'" for \<mu>'
assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
moreover
have "measure_space (space M) (sets M) \<mu>'"
using ms unfolding measure_space_def by auto standard
with ms eq have "\<exists>\<mu>'. P \<mu>'"
unfolding P_def
by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
by (simp add: M extend_measure_def P_def[symmetric])
from \<open>\<exists>\<mu>'. P \<mu>'\<close> have P: "P (Eps P)" by (rule someI_ex)
show "emeasure M (G i) = \<mu> i"
proof (subst emeasure_measure_of[OF M_eq])
have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
using M_eq ms by (auto simp: sets_extend_measure)
then show "G i \<in> sets M" using \<open>i \<in> I\<close> by auto
show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
using P \<open>i\<in>I\<close> by (auto simp add: sets_M measure_space_def P_def)
qed fact
qed
lemma emeasure_extend_measure_Pair:
assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
and "I i j"
shows "emeasure M (G i j) = \<mu> i j"
using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) \<open>I i j\<close>
by (auto simp: subset_eq)
subsection \<open>The smallest $\sigma$-algebra regarding a function\<close>
definition
"vimage_algebra X f M = sigma X {f -` A \<inter> X | A. A \<in> sets M}"
lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X"
unfolding vimage_algebra_def by (rule space_measure_of) auto
lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f -` A \<inter> X | A. A \<in> sets M}"
unfolding vimage_algebra_def by (rule sets_measure_of) auto
lemma sets_vimage_algebra2:
"f \<in> X \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra X f M) = {f -` A \<inter> X | A. A \<in> sets M}"
using sigma_sets_vimage_commute[of f X "space M" "sets M"]
unfolding sets_vimage_algebra sets.sigma_sets_eq by simp
lemma sets_vimage_algebra_cong: "sets M = sets N \<Longrightarrow> sets (vimage_algebra X f M) = sets (vimage_algebra X f N)"
by (simp add: sets_vimage_algebra)
lemma vimage_algebra_cong:
assumes "X = Y"
assumes "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"
assumes "sets M = sets N"
shows "vimage_algebra X f M = vimage_algebra Y g N"
by (auto simp: vimage_algebra_def assms intro!: arg_cong2[where f=sigma])
lemma in_vimage_algebra: "A \<in> sets M \<Longrightarrow> f -` A \<inter> X \<in> sets (vimage_algebra X f M)"
by (auto simp: vimage_algebra_def)
lemma sets_image_in_sets:
assumes N: "space N = X"
assumes f: "f \<in> measurable N M"
shows "sets (vimage_algebra X f M) \<subseteq> sets N"
unfolding sets_vimage_algebra N[symmetric]
by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f)
lemma measurable_vimage_algebra1: "f \<in> X \<rightarrow> space M \<Longrightarrow> f \<in> measurable (vimage_algebra X f M) M"
unfolding measurable_def by (auto intro: in_vimage_algebra)
lemma measurable_vimage_algebra2:
assumes g: "g \<in> space N \<rightarrow> X" and f: "(\<lambda>x. f (g x)) \<in> measurable N M"
shows "g \<in> measurable N (vimage_algebra X f M)"
unfolding vimage_algebra_def
proof (rule measurable_measure_of)
fix A assume "A \<in> {f -` A \<inter> X | A. A \<in> sets M}"
then obtain Y where Y: "Y \<in> sets M" and A: "A = f -` Y \<inter> X"
by auto
then have "g -` A \<inter> space N = (\<lambda>x. f (g x)) -` Y \<inter> space N"
using g by auto
also have "\<dots> \<in> sets N"
using f Y by (rule measurable_sets)
finally show "g -` A \<inter> space N \<in> sets N" .
qed (insert g, auto)
lemma vimage_algebra_sigma:
assumes X: "X \<subseteq> Pow \<Omega>'" and f: "f \<in> \<Omega> \<rightarrow> \<Omega>'"
shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f -` A \<inter> \<Omega> | A. A \<in> X }" (is "?V = ?S")
proof (rule measure_eqI)
have \<Omega>: "{f -` A \<inter> \<Omega> |A. A \<in> X} \<subseteq> Pow \<Omega>" by auto
show "sets ?V = sets ?S"
using sigma_sets_vimage_commute[OF f, of X]
by (simp add: space_measure_of_conv f sets_vimage_algebra2 \<Omega> X)
qed (simp add: vimage_algebra_def emeasure_sigma)
lemma vimage_algebra_vimage_algebra_eq:
assumes *: "f \<in> X \<rightarrow> Y" "g \<in> Y \<rightarrow> space M"
shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"
(is "?VV = ?V")
proof (rule measure_eqI)
have "(\<lambda>x. g (f x)) \<in> X \<rightarrow> space M" "\<And>A. A \<inter> f -` Y \<inter> X = A \<inter> X"
using * by auto
with * show "sets ?VV = sets ?V"
by (simp add: sets_vimage_algebra2 ex_simps[symmetric] vimage_comp comp_def del: ex_simps)
qed (simp add: vimage_algebra_def emeasure_sigma)
subsubsection \<open>Restricted Space Sigma Algebra\<close>
definition restrict_space where
"restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>) ` sets M) (emeasure M)"
lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto
lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"
by (simp add: space_restrict_space sets.sets_into_space)
lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
unfolding restrict_space_def
proof (subst sets_measure_of)
show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)"
by (auto dest: sets.sets_into_space)
have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} =
(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M"
by (subst sigma_sets_vimage_commute[symmetric, where \<Omega>' = "space M"])
(auto simp add: sets.sigma_sets_eq)
moreover have "{((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M"
by auto
moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M = (op \<inter> \<Omega>) ` sets M"
by (intro image_cong) (auto dest: sets.sets_into_space)
ultimately show "sigma_sets (\<Omega> \<inter> space M) (op \<inter> \<Omega> ` sets M) = op \<inter> \<Omega> ` sets M"
by simp
qed
lemma restrict_space_sets_cong:
"A = B \<Longrightarrow> sets M = sets N \<Longrightarrow> sets (restrict_space M A) = sets (restrict_space N B)"
by (auto simp: sets_restrict_space)
lemma sets_restrict_space_count_space :
"sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"
by(auto simp add: sets_restrict_space)
lemma sets_restrict_UNIV[simp]: "sets (restrict_space M UNIV) = sets M"
by (auto simp add: sets_restrict_space)
lemma sets_restrict_restrict_space:
"sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"
unfolding sets_restrict_space image_comp by (intro image_cong) auto
lemma sets_restrict_space_iff:
"\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
proof (subst sets_restrict_space, safe)
fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"
then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"
by rule
also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"
using sets.sets_into_space[OF A] by auto
finally show "\<Omega> \<inter> A \<in> sets M"
by auto
qed auto
lemma sets_restrict_space_cong: "sets M = sets N \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"
by (simp add: sets_restrict_space)
lemma restrict_space_eq_vimage_algebra:
"\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (vimage_algebra \<Omega> (\<lambda>x. x) M)"
unfolding restrict_space_def
apply (subst sets_measure_of)
apply (auto simp add: image_subset_iff dest: sets.sets_into_space) []
apply (auto simp add: sets_vimage_algebra intro!: arg_cong2[where f=sigma_sets])
done
lemma sets_Collect_restrict_space_iff:
assumes "S \<in> sets M"
shows "{x\<in>space (restrict_space M S). P x} \<in> sets (restrict_space M S) \<longleftrightarrow> {x\<in>space M. x \<in> S \<and> P x} \<in> sets M"
proof -
have "{x\<in>S. P x} = {x\<in>space M. x \<in> S \<and> P x}"
using sets.sets_into_space[OF assms] by auto
then show ?thesis
by (subst sets_restrict_space_iff) (auto simp add: space_restrict_space assms)
qed
lemma measurable_restrict_space1:
assumes f: "f \<in> measurable M N"
shows "f \<in> measurable (restrict_space M \<Omega>) N"
unfolding measurable_def
proof (intro CollectI conjI ballI)
show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"
using measurable_space[OF f] by (auto simp: space_restrict_space)
fix A assume "A \<in> sets N"
have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"
by (auto simp: space_restrict_space)
also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
unfolding sets_restrict_space
using measurable_sets[OF f \<open>A \<in> sets N\<close>] by blast
finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
qed
lemma measurable_restrict_space2_iff:
"f \<in> measurable M (restrict_space N \<Omega>) \<longleftrightarrow> (f \<in> measurable M N \<and> f \<in> space M \<rightarrow> \<Omega>)"
proof -
have "\<And>A. f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f -` \<Omega> \<inter> f -` A \<inter> space M = f -` A \<inter> space M"
by auto
then show ?thesis
by (auto simp: measurable_def space_restrict_space Pi_Int[symmetric] sets_restrict_space)
qed
lemma measurable_restrict_space2:
"f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"
by (simp add: measurable_restrict_space2_iff)
lemma measurable_piecewise_restrict:
assumes I: "countable C"
and X: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M" "space M \<subseteq> \<Union>C"
and f: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> measurable (restrict_space M \<Omega>) N"
shows "f \<in> measurable M N"
proof (rule measurableI)
fix x assume "x \<in> space M"
with X obtain \<Omega> where "\<Omega> \<in> C" "x \<in> \<Omega>" "x \<in> space M" by auto
then show "f x \<in> space N"
by (auto simp: space_restrict_space intro: f measurable_space)
next
fix A assume A: "A \<in> sets N"
have "f -` A \<inter> space M = (\<Union>\<Omega>\<in>C. (f -` A \<inter> (\<Omega> \<inter> space M)))"
using X by (auto simp: subset_eq)
also have "\<dots> \<in> sets M"
using measurable_sets[OF f A] X I
by (intro sets.countable_UN') (auto simp: sets_restrict_space_iff space_restrict_space)
finally show "f -` A \<inter> space M \<in> sets M" .
qed
lemma measurable_piecewise_restrict_iff:
"countable C \<Longrightarrow> (\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M) \<Longrightarrow> space M \<subseteq> (\<Union>C) \<Longrightarrow>
f \<in> measurable M N \<longleftrightarrow> (\<forall>\<Omega>\<in>C. f \<in> measurable (restrict_space M \<Omega>) N)"
by (auto intro: measurable_piecewise_restrict measurable_restrict_space1)
lemma measurable_If_restrict_space_iff:
"{x\<in>space M. P x} \<in> sets M \<Longrightarrow>
(\<lambda>x. if P x then f x else g x) \<in> measurable M N \<longleftrightarrow>
(f \<in> measurable (restrict_space M {x. P x}) N \<and> g \<in> measurable (restrict_space M {x. \<not> P x}) N)"
by (subst measurable_piecewise_restrict_iff[where C="{{x. P x}, {x. \<not> P x}}"])
(auto simp: Int_def sets.sets_Collect_neg space_restrict_space conj_commute[of _ "x \<in> space M" for x]
cong: measurable_cong')
lemma measurable_If:
"f \<in> measurable M M' \<Longrightarrow> g \<in> measurable M M' \<Longrightarrow> {x\<in>space M. P x} \<in> sets M \<Longrightarrow>
(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
unfolding measurable_If_restrict_space_iff by (auto intro: measurable_restrict_space1)
lemma measurable_If_set:
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
assumes P: "A \<inter> space M \<in> sets M"
shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
proof (rule measurable_If[OF measure])
have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
thus "{x \<in> space M. x \<in> A} \<in> sets M" using \<open>A \<inter> space M \<in> sets M\<close> by auto
qed
lemma measurable_restrict_space_iff:
"\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> c \<in> space N \<Longrightarrow>
f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow> (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N"
by (subst measurable_If_restrict_space_iff)
(simp_all add: Int_def conj_commute measurable_const)
lemma restrict_space_singleton: "{x} \<in> sets M \<Longrightarrow> sets (restrict_space M {x}) = sets (count_space {x})"
using sets_restrict_space_iff[of "{x}" M]
by (auto simp add: sets_restrict_space_iff dest!: subset_singletonD)
lemma measurable_restrict_countable:
assumes X[intro]: "countable X"
assumes sets[simp]: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
assumes space[simp]: "\<And>x. x \<in> X \<Longrightarrow> f x \<in> space N"
assumes f: "f \<in> measurable (restrict_space M (- X)) N"
shows "f \<in> measurable M N"
using f sets.countable[OF sets X]
by (intro measurable_piecewise_restrict[where M=M and C="{- X} \<union> ((\<lambda>x. {x}) ` X)"])
(auto simp: Diff_Int_distrib2 Compl_eq_Diff_UNIV Int_insert_left sets.Diff restrict_space_singleton
simp del: sets_count_space cong: measurable_cong_sets)
lemma measurable_discrete_difference:
assumes f: "f \<in> measurable M N"
assumes X: "countable X" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"
assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
shows "g \<in> measurable M N"
by (rule measurable_restrict_countable[OF X])
(auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)
lemma measurable_count_space_extend: "A \<subseteq> B \<Longrightarrow> f \<in> space M \<rightarrow> A \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M count_space B \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M count_space A"
by (auto simp: measurable_def)
end