Generalized simple_functionD and less_SUP_iff.
Moved theorems to appropriate places.
(* Title: Sigma_Algebra.thy
Author: Stefan Richter, Markus Wenzel, TU Muenchen
Plus material from the Hurd/Coble measure theory development,
translated by Lawrence Paulson.
*)
header {* Sigma Algebras *}
theory Sigma_Algebra imports Main Countable FuncSet Indicator_Function begin
text {* 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. *}
subsection {* Algebras *}
record 'a algebra =
space :: "'a set"
sets :: "'a set set"
locale algebra =
fixes M :: "'a algebra"
assumes space_closed: "sets M \<subseteq> Pow (space M)"
and empty_sets [iff]: "{} \<in> sets M"
and compl_sets [intro]: "!!a. a \<in> sets M \<Longrightarrow> space M - a \<in> sets M"
and Un [intro]: "!!a b. a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<union> b \<in> sets M"
lemma (in algebra) top [iff]: "space M \<in> sets M"
by (metis Diff_empty compl_sets empty_sets)
lemma (in algebra) sets_into_space: "x \<in> sets M \<Longrightarrow> x \<subseteq> space M"
by (metis PowD contra_subsetD space_closed)
lemma (in algebra) Int [intro]:
assumes a: "a \<in> sets M" and b: "b \<in> sets M" shows "a \<inter> b \<in> sets M"
proof -
have "((space M - a) \<union> (space M - b)) \<in> sets M"
by (metis a b compl_sets Un)
moreover
have "a \<inter> b = space M - ((space M - a) \<union> (space M - b))"
using space_closed a b
by blast
ultimately show ?thesis
by (metis compl_sets)
qed
lemma (in algebra) Diff [intro]:
assumes a: "a \<in> sets M" and b: "b \<in> sets M" shows "a - b \<in> sets M"
proof -
have "(a \<inter> (space M - b)) \<in> sets M"
by (metis a b compl_sets Int)
moreover
have "a - b = (a \<inter> (space M - b))"
by (metis Int_Diff Int_absorb1 Int_commute a sets_into_space)
ultimately show ?thesis
by metis
qed
lemma (in algebra) finite_union [intro]:
"finite X \<Longrightarrow> X \<subseteq> sets M \<Longrightarrow> Union X \<in> sets M"
by (induct set: finite) (auto simp add: Un)
lemma algebra_iff_Int:
"algebra M \<longleftrightarrow>
sets M \<subseteq> Pow (space M) & {} \<in> sets M &
(\<forall>a \<in> sets M. space M - a \<in> sets M) &
(\<forall>a \<in> sets M. \<forall> b \<in> sets M. a \<inter> b \<in> sets M)"
proof (auto simp add: algebra.Int, auto simp add: algebra_def)
fix a b
assume ab: "sets M \<subseteq> Pow (space M)"
"\<forall>a\<in>sets M. space M - a \<in> sets M"
"\<forall>a\<in>sets M. \<forall>b\<in>sets M. a \<inter> b \<in> sets M"
"a \<in> sets M" "b \<in> sets M"
hence "a \<union> b = space M - ((space M - a) \<inter> (space M - b))"
by blast
also have "... \<in> sets M"
by (metis ab)
finally show "a \<union> b \<in> sets M" .
qed
lemma (in algebra) insert_in_sets:
assumes "{x} \<in> sets M" "A \<in> sets M" shows "insert x A \<in> sets M"
proof -
have "{x} \<union> A \<in> sets M" using assms by (rule Un)
thus ?thesis by auto
qed
lemma (in algebra) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"
by (metis Int_absorb1 sets_into_space)
lemma (in algebra) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
by (metis Int_absorb2 sets_into_space)
section {* Restricted algebras *}
abbreviation (in algebra)
"restricted_space A \<equiv> \<lparr> space = A, sets = (\<lambda>S. (A \<inter> S)) ` sets M \<rparr>"
lemma (in algebra) restricted_algebra:
assumes "A \<in> sets M" shows "algebra (restricted_space A)"
using assms by unfold_locales auto
subsection {* Sigma Algebras *}
locale sigma_algebra = algebra +
assumes countable_nat_UN [intro]:
"!!A. range A \<subseteq> sets M \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
lemma countable_UN_eq:
fixes A :: "'i::countable \<Rightarrow> 'a set"
shows "(range A \<subseteq> sets M \<longrightarrow> (\<Union>i. A i) \<in> sets M) \<longleftrightarrow>
(range (A \<circ> from_nat) \<subseteq> sets M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> sets 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_compose intro!: imageI)
show ?thesis unfolding * ** ..
qed
lemma (in sigma_algebra) countable_UN[intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes "A`X \<subseteq> sets M"
shows "(\<Union>x\<in>X. A x) \<in> sets M"
proof -
let "?A i" = "if i \<in> X then A i else {}"
from assms have "range ?A \<subseteq> sets M" by auto
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
have "(\<Union>x. ?A x) \<in> sets M" by auto
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
ultimately show ?thesis by simp
qed
lemma (in sigma_algebra) finite_UN:
assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
shows "(\<Union>i\<in>I. A i) \<in> sets M"
using assms by induct auto
lemma (in sigma_algebra) countable_INT [intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes A: "A`X \<subseteq> sets M" "X \<noteq> {}"
shows "(\<Inter>i\<in>X. A i) \<in> sets M"
proof -
from A have "\<forall>i\<in>X. A i \<in> sets M" by fast
hence "space M - (\<Union>i\<in>X. space M - A i) \<in> sets M" by blast
moreover
have "(\<Inter>i\<in>X. A i) = space M - (\<Union>i\<in>X. space M - A i)" using space_closed A
by blast
ultimately show ?thesis by metis
qed
lemma (in sigma_algebra) finite_INT:
assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
shows "(\<Inter>i\<in>I. A i) \<in> sets M"
using assms by (induct rule: finite_ne_induct) auto
lemma algebra_Pow:
"algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
by (auto simp add: algebra_def)
lemma sigma_algebra_Pow:
"sigma_algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
by (auto simp add: sigma_algebra_def sigma_algebra_axioms_def algebra_Pow)
lemma sigma_algebra_iff:
"sigma_algebra M \<longleftrightarrow>
algebra M \<and> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
subsection {* Binary Unions *}
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
where "binary a b = (\<lambda>\<^isup>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: UNION_eq_Union_image range_binary_eq)
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
by (simp add: INTER_eq_Inter_image range_binary_eq)
lemma sigma_algebra_iff2:
"sigma_algebra M \<longleftrightarrow>
sets M \<subseteq> Pow (space M) \<and>
{} \<in> sets M \<and> (\<forall>s \<in> sets M. space M - s \<in> sets M) \<and>
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
algebra_def Un_range_binary)
subsection {* Initial Sigma Algebra *}
text {*Sigma algebras can naturally be created as the closure of any set of
sets with regard to the properties just postulated. *}
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: "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"
definition
"sigma M = (| space = space M, sets = sigma_sets (space M) (sets M) |)"
lemma sets_sigma: "sets (sigma M) = sigma_sets (space M) (sets M)"
unfolding sigma_def by simp
lemma space_sigma [simp]: "space (sigma M) = space M"
by (simp add: sigma_def)
lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
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_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 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 (in sigma_algebra) sigma_sets_subset:
assumes a: "a \<subseteq> sets M"
shows "sigma_sets (space M) a \<subseteq> sets M"
proof
fix x
assume "x \<in> sigma_sets (space M) a"
from this show "x \<in> sets M"
by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
qed
lemma (in sigma_algebra) sigma_sets_eq:
"sigma_sets (space M) (sets M) = sets M"
proof
show "sets M \<subseteq> sigma_sets (space M) (sets M)"
by (metis Set.subsetI sigma_sets.Basic)
next
show "sigma_sets (space M) (sets M) \<subseteq> sets M"
by (metis sigma_sets_subset subset_refl)
qed
lemma sigma_algebra_sigma_sets:
"a \<subseteq> Pow (space M) \<Longrightarrow> sets M = sigma_sets (space M) a \<Longrightarrow> sigma_algebra M"
apply (auto simp add: sigma_algebra_def sigma_algebra_axioms_def
algebra_def sigma_sets.Empty sigma_sets.Compl sigma_sets_Un)
apply (blast dest: sigma_sets_into_sp)
apply (rule sigma_sets.Union, fast)
done
lemma sigma_algebra_sigma:
"sets M \<subseteq> Pow (space M) \<Longrightarrow> sigma_algebra (sigma M)"
apply (rule sigma_algebra_sigma_sets)
apply (auto simp add: sigma_def)
done
lemma (in sigma_algebra) sigma_subset:
"sets N \<subseteq> sets M \<Longrightarrow> space N = space M \<Longrightarrow> sets (sigma N) \<subseteq> (sets M)"
by (simp add: sigma_def sigma_sets_subset)
lemma sigma_sets_least_sigma_algebra:
assumes "A \<subseteq> Pow S"
shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra \<lparr>space = S, sets = B\<rparr>}"
proof safe
fix B X assume "A \<subseteq> B" and sa: "sigma_algebra \<lparr> space = S, sets = B \<rparr>"
and X: "X \<in> sigma_sets S A"
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
show "X \<in> B" by auto
next
fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra \<lparr>space = S, sets = B\<rparr>}"
then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra \<lparr>space = S, sets = B\<rparr> \<Longrightarrow> X \<in> B"
by simp
have "A \<subseteq> sigma_sets S A" using assms
by (auto intro!: sigma_sets.Basic)
moreover have "sigma_algebra \<lparr>space = S, sets = sigma_sets S A\<rparr>"
using assms by (intro sigma_algebra_sigma_sets[of A]) auto
ultimately show "X \<in> sigma_sets S A" by auto
qed
lemma (in sigma_algebra) restriction_in_sets:
fixes A :: "nat \<Rightarrow> 'a set"
assumes "S \<in> sets M"
and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` sets M" (is "_ \<subseteq> ?r")
shows "range A \<subseteq> sets M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` sets M"
proof -
{ fix i have "A i \<in> ?r" using * by auto
hence "\<exists>B. A i = B \<inter> S \<and> B \<in> sets M" by auto
hence "A i \<subseteq> S" "A i \<in> sets M" using `S \<in> sets M` by auto }
thus "range A \<subseteq> sets M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` sets M"
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
qed
lemma (in sigma_algebra) restricted_sigma_algebra:
assumes "S \<in> sets M"
shows "sigma_algebra (restricted_space S)"
unfolding sigma_algebra_def sigma_algebra_axioms_def
proof safe
show "algebra (restricted_space S)" using restricted_algebra[OF assms] .
next
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets (restricted_space S)"
from restriction_in_sets[OF assms this[simplified]]
show "(\<Union>i. A i) \<in> sets (restricted_space S)" by simp
qed
lemma sigma_sets_Int:
assumes "A \<in> sigma_sets sp st"
shows "op \<inter> A ` sigma_sets sp st = sigma_sets (A \<inter> sp) (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 show "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)
next
fix x assume "x \<in> sigma_sets (A \<inter> sp) (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
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)
qed
lemma sigma_sets_single[simp]: "sigma_sets {X} {{X}} = {{}, {X}}"
proof (intro set_eqI iffI)
fix x assume "x \<in> sigma_sets {X} {{X}}"
from sigma_sets_into_sp[OF _ this]
show "x \<in> {{}, {X}}" by auto
next
fix x assume "x \<in> {{}, {X}}"
then show "x \<in> sigma_sets {X} {{X}}"
by (auto intro: sigma_sets.Empty sigma_sets_top)
qed
lemma (in sigma_algebra) sets_sigma_subset:
assumes "space N = space M"
assumes "sets N \<subseteq> sets M"
shows "sets (sigma N) \<subseteq> sets M"
by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms)
lemma in_sigma[intro, simp]: "A \<in> sets M \<Longrightarrow> A \<in> sets (sigma M)"
unfolding sigma_def by (auto intro!: sigma_sets.Basic)
lemma (in sigma_algebra) sigma_eq[simp]: "sigma M = M"
unfolding sigma_def sigma_sets_eq by simp
section {* Measurable functions *}
definition
"measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
lemma (in sigma_algebra) measurable_sigma:
assumes B: "sets N \<subseteq> Pow (space N)"
and f: "f \<in> space M -> space N"
and ba: "\<And>y. y \<in> sets N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
shows "f \<in> measurable M (sigma N)"
proof -
have "sigma_sets (space N) (sets N) \<subseteq> {y . (f -` y) \<inter> space M \<in> sets M & y \<subseteq> space N}"
proof clarify
fix x
assume "x \<in> sigma_sets (space N) (sets N)"
thus "f -` x \<inter> space M \<in> sets M \<and> x \<subseteq> space N"
proof induct
case (Basic a)
thus ?case
by (auto simp add: ba) (metis B subsetD PowD)
next
case Empty
thus ?case
by auto
next
case (Compl a)
have [simp]: "f -` space N \<inter> space M = space M"
by (auto simp add: funcset_mem [OF f])
thus ?case
by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl)
next
case (Union a)
thus ?case
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
qed
qed
thus ?thesis
by (simp add: measurable_def sigma_algebra_axioms sigma_algebra_sigma B f)
(auto simp add: sigma_def)
qed
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_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 (in sigma_algebra) measurable_subset:
"(\<And>S. S \<in> sets A \<Longrightarrow> S \<subseteq> space A) \<Longrightarrow> measurable M A \<subseteq> measurable M (sigma A)"
by (auto intro: measurable_sigma measurable_sets measurable_space)
lemma measurable_eqI:
"\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
\<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
by (simp add: measurable_def sigma_algebra_iff2)
lemma (in sigma_algebra) measurable_const[intro, simp]:
assumes "c \<in> space M'"
shows "(\<lambda>x. c) \<in> measurable M M'"
using assms by (auto simp add: measurable_def)
lemma (in sigma_algebra) measurable_If:
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
assumes P: "{x\<in>space M. P x} \<in> sets M"
shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
unfolding measurable_def
proof safe
fix x assume "x \<in> space M"
thus "(if P x then f x else g x) \<in> space M'"
using measure unfolding measurable_def by auto
next
fix A assume "A \<in> sets M'"
hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
using measure unfolding measurable_def by (auto split: split_if_asm)
show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
using `A \<in> sets M'` measure P unfolding * measurable_def
by (auto intro!: Un)
qed
lemma (in sigma_algebra) measurable_If_set:
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
assumes P: "A \<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" using `A \<in> sets M` sets_into_space by auto
thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<in> sets M` by auto
qed
lemma (in algebra) measurable_ident[intro, simp]: "id \<in> measurable M M"
by (auto simp add: measurable_def)
lemma measurable_comp[intro]:
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c"
apply (auto simp add: measurable_def vimage_compose)
apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a")
apply force+
done
lemma measurable_strong:
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
assumes f: "f \<in> measurable a b" and g: "g \<in> (space b -> space c)"
and a: "sigma_algebra a" and b: "sigma_algebra b" and c: "sigma_algebra c"
and t: "f ` (space a) \<subseteq> t"
and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
shows "(g o f) \<in> measurable a c"
proof -
have fab: "f \<in> (space a -> space b)"
and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
by (auto simp add: measurable_def)
have eq: "f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
by force
show ?thesis
apply (auto simp add: measurable_def vimage_compose a c)
apply (metis funcset_mem fab g)
apply (subst eq, metis ba cb)
done
qed
lemma measurable_mono1:
"a \<subseteq> b \<Longrightarrow> sigma_algebra \<lparr>space = X, sets = b\<rparr>
\<Longrightarrow> measurable \<lparr>space = X, sets = a\<rparr> c \<subseteq> measurable \<lparr>space = X, sets = b\<rparr> c"
by (auto simp add: measurable_def)
lemma measurable_up_sigma:
"measurable A M \<subseteq> measurable (sigma A) M"
unfolding measurable_def
by (auto simp: sigma_def intro: sigma_sets.Basic)
lemma (in sigma_algebra) measurable_range_reduce:
"\<lbrakk> f \<in> measurable M \<lparr>space = s, sets = Pow s\<rparr> ; s \<noteq> {} \<rbrakk>
\<Longrightarrow> f \<in> measurable M \<lparr>space = s \<inter> (f ` space M), sets = Pow (s \<inter> (f ` space M))\<rparr>"
by (simp add: measurable_def sigma_algebra_Pow del: Pow_Int_eq) blast
lemma (in sigma_algebra) measurable_Pow_to_Pow:
"(sets M = Pow (space M)) \<Longrightarrow> f \<in> measurable M \<lparr>space = UNIV, sets = Pow UNIV\<rparr>"
by (auto simp add: measurable_def sigma_algebra_def sigma_algebra_axioms_def algebra_def)
lemma (in sigma_algebra) measurable_Pow_to_Pow_image:
"sets M = Pow (space M)
\<Longrightarrow> f \<in> measurable M \<lparr>space = f ` space M, sets = Pow (f ` space M)\<rparr>"
by (simp add: measurable_def sigma_algebra_Pow) intro_locales
lemma (in sigma_algebra) measurable_iff_sigma:
assumes "sets E \<subseteq> Pow (space E)" and "f \<in> space M \<rightarrow> space E"
shows "f \<in> measurable M (sigma E) \<longleftrightarrow> (\<forall>A\<in>sets E. f -` A \<inter> space M \<in> sets M)"
using measurable_sigma[OF assms]
by (fastsimp simp: measurable_def sets_sigma intro: sigma_sets.intros)
section "Disjoint families"
definition
disjoint_family_on where
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
abbreviation
"disjoint_family A \<equiv> disjoint_family_on A UNIV"
lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
by blast
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
by blast
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
by blast
lemma disjoint_family_subset:
"disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
by (force simp add: disjoint_family_on_def)
lemma disjoint_family_on_bisimulation:
assumes "disjoint_family_on f S"
and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}"
shows "disjoint_family_on g S"
using assms unfolding disjoint_family_on_def by auto
lemma disjoint_family_on_mono:
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
unfolding disjoint_family_on_def by auto
lemma disjoint_family_Suc:
assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
proof -
{
fix m
have "!!n. A n \<subseteq> A (m+n)"
proof (induct m)
case 0 show ?case by simp
next
case (Suc m) thus ?case
by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans)
qed
}
hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
by (metis add_commute le_add_diff_inverse nat_less_le)
thus ?thesis
by (auto simp add: disjoint_family_on_def)
(metis insert_absorb insert_subset le_SucE le_antisym not_leE)
qed
lemma setsum_indicator_disjoint_family:
fixes f :: "'d \<Rightarrow> 'e::semiring_1"
assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
proof -
have "P \<inter> {i. x \<in> A i} = {j}"
using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
by auto
thus ?thesis
unfolding indicator_def
by (simp add: if_distrib setsum_cases[OF `finite P`])
qed
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
qed
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
apply (rule UN_finite2_eq [where k=0])
apply (simp add: finite_UN_disjointed_eq)
done
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
by (auto simp add: disjointed_def)
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
by (simp add: disjoint_family_on_def)
(metis neq_iff Int_commute less_disjoint_disjointed)
lemma disjointed_subset: "disjointed A n \<subseteq> A n"
by (auto simp add: disjointed_def)
lemma (in algebra) UNION_in_sets:
fixes A:: "nat \<Rightarrow> 'a set"
assumes A: "range A \<subseteq> sets M "
shows "(\<Union>i\<in>{0..<n}. A i) \<in> sets 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 algebra) range_disjointed_sets:
assumes A: "range A \<subseteq> sets M "
shows "range (disjointed A) \<subseteq> sets M"
proof (auto simp add: disjointed_def)
fix n
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> sets M" using UNION_in_sets
by (metis A Diff UNIV_I image_subset_iff)
qed
lemma sigma_algebra_disjoint_iff:
"sigma_algebra M \<longleftrightarrow>
algebra M &
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow>
(\<Union>i::nat. A i) \<in> sets M)"
proof (auto simp add: sigma_algebra_iff)
fix A :: "nat \<Rightarrow> 'a set"
assume M: "algebra M"
and A: "range A \<subseteq> sets M"
and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>
disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
hence "range (disjointed A) \<subseteq> sets M \<longrightarrow>
disjoint_family (disjointed A) \<longrightarrow>
(\<Union>i. disjointed A i) \<in> sets M" by blast
hence "(\<Union>i. disjointed A i) \<in> sets M"
by (simp add: algebra.range_disjointed_sets M A disjoint_family_disjointed)
thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
qed
subsection {* Sigma algebra generated by function preimages *}
definition (in sigma_algebra)
"vimage_algebra S f = \<lparr> space = S, sets = (\<lambda>A. f -` A \<inter> S) ` sets M \<rparr>"
lemma (in sigma_algebra) in_vimage_algebra[simp]:
"A \<in> sets (vimage_algebra S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
by (simp add: vimage_algebra_def image_iff)
lemma (in sigma_algebra) space_vimage_algebra[simp]:
"space (vimage_algebra S f) = S"
by (simp add: vimage_algebra_def)
lemma (in sigma_algebra) sigma_algebra_preimages:
fixes f :: "'x \<Rightarrow> 'a"
assumes "f \<in> A \<rightarrow> space M"
shows "sigma_algebra \<lparr> space = A, sets = (\<lambda>M. f -` M \<inter> A) ` sets M \<rparr>"
(is "sigma_algebra \<lparr> space = _, sets = ?F ` sets M \<rparr>")
proof (simp add: sigma_algebra_iff2, safe)
show "{} \<in> ?F ` sets M" by blast
next
fix S assume "S \<in> sets M"
moreover have "A - ?F S = ?F (space M - S)"
using assms by auto
ultimately show "A - ?F S \<in> ?F ` sets M"
by blast
next
fix S :: "nat \<Rightarrow> 'x set" assume *: "range S \<subseteq> ?F ` sets M"
have "\<forall>i. \<exists>b. b \<in> sets M \<and> S i = ?F b"
proof safe
fix i
have "S i \<in> ?F ` sets M" using * by auto
then show "\<exists>b. b \<in> sets M \<and> S i = ?F b" by auto
qed
from choice[OF this] obtain b where b: "range b \<subseteq> sets M" "\<And>i. S i = ?F (b i)"
by auto
then have "(\<Union>i. S i) = ?F (\<Union>i. b i)" by auto
then show "(\<Union>i. S i) \<in> ?F ` sets M" using b(1) by blast
qed
lemma (in sigma_algebra) sigma_algebra_vimage:
fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
shows "sigma_algebra (vimage_algebra S f)"
proof -
from sigma_algebra_preimages[OF assms]
show ?thesis unfolding vimage_algebra_def by (auto simp: sigma_algebra_iff2)
qed
lemma (in sigma_algebra) measurable_vimage_algebra:
fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
shows "f \<in> measurable (vimage_algebra S f) M"
unfolding measurable_def using assms by force
lemma (in sigma_algebra) measurable_vimage:
fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra S f) M2"
proof -
note measurable_vimage_algebra[OF assms(2)]
from measurable_comp[OF this assms(1)]
show ?thesis by (simp add: comp_def)
qed
lemma (in sigma_algebra) vimage_vimage_inv:
assumes f: "bij_betw f S (space M)"
assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f (g x) = x" and g: "g \<in> space M \<rightarrow> S"
shows "sigma_algebra.vimage_algebra (vimage_algebra S f) (space M) g = M"
proof -
interpret T: sigma_algebra "vimage_algebra S f"
using f by (safe intro!: sigma_algebra_vimage bij_betw_imp_funcset)
have inj: "inj_on f S" and [simp]: "f`S = space M"
using f unfolding bij_betw_def by auto
{ fix A assume A: "A \<in> sets M"
have "g -` f -` A \<inter> g -` S \<inter> space M = (f \<circ> g) -` A \<inter> space M"
using g by auto
also have "\<dots> = A"
using `A \<in> sets M`[THEN sets_into_space] by auto
finally have "g -` f -` A \<inter> g -` S \<inter> space M = A" . }
note X = this
show ?thesis
unfolding T.vimage_algebra_def unfolding vimage_algebra_def
by (simp add: image_compose[symmetric] comp_def X cong: image_cong)
qed
lemma (in sigma_algebra) measurable_vimage_iff:
fixes f :: "'b \<Rightarrow> 'a" assumes f: "bij_betw f S (space M)"
shows "g \<in> measurable M M' \<longleftrightarrow> (g \<circ> f) \<in> measurable (vimage_algebra S f) M'"
proof
assume "g \<in> measurable M M'"
from measurable_vimage[OF this f[THEN bij_betw_imp_funcset]]
show "(g \<circ> f) \<in> measurable (vimage_algebra S f) M'" unfolding comp_def .
next
interpret v: sigma_algebra "vimage_algebra S f"
using f[THEN bij_betw_imp_funcset] by (rule sigma_algebra_vimage)
note f' = f[THEN bij_betw_the_inv_into]
assume "g \<circ> f \<in> measurable (vimage_algebra S f) M'"
from v.measurable_vimage[OF this, unfolded space_vimage_algebra, OF f'[THEN bij_betw_imp_funcset]]
show "g \<in> measurable M M'"
using f f'[THEN bij_betw_imp_funcset] f[unfolded bij_betw_def]
by (subst (asm) vimage_vimage_inv)
(simp_all add: f_the_inv_into_f cong: measurable_cong)
qed
lemma (in sigma_algebra) measurable_vimage_iff_inv:
fixes f :: "'b \<Rightarrow> 'a" assumes f: "bij_betw f S (space M)"
shows "g \<in> measurable (vimage_algebra S f) M' \<longleftrightarrow> (g \<circ> the_inv_into S f) \<in> measurable M M'"
unfolding measurable_vimage_iff[OF f]
using f[unfolded bij_betw_def]
by (auto intro!: measurable_cong simp add: the_inv_into_f_f)
lemma sigma_sets_vimage:
assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
proof (intro set_eqI iffI)
let ?F = "\<lambda>X. f -` X \<inter> S'"
fix X assume "X \<in> sigma_sets S' (?F ` A)"
then show "X \<in> ?F ` sigma_sets S A"
proof induct
case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
by auto
then show ?case by (auto intro!: sigma_sets.Basic)
next
case Empty then show ?case
by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
next
case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
by auto
then have "S - X' \<in> sigma_sets S A"
by (auto intro!: sigma_sets.Compl)
then show ?case
using X assms by (auto intro!: image_eqI[where x="S - X'"])
next
case (Union F)
then have "\<forall>i. \<exists>F'. F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
by (auto simp: image_iff Bex_def)
from choice[OF this] obtain F' where
"\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
by auto
then show ?case
by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
qed
next
let ?F = "\<lambda>X. f -` X \<inter> S'"
fix X assume "X \<in> ?F ` sigma_sets S A"
then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
then show "X \<in> sigma_sets S' (?F ` A)"
proof (induct arbitrary: X)
case (Basic X') then show ?case by (auto intro: sigma_sets.Basic)
next
case Empty then show ?case by (auto intro: sigma_sets.Empty)
next
case (Compl X')
have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
apply (rule sigma_sets.Compl)
using assms by (auto intro!: Compl.hyps simp: Compl.prems)
also have "S' - (S' - X) = X"
using assms Compl by auto
finally show ?case .
next
case (Union F)
have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
by (intro sigma_sets.Union Union.hyps) simp
also have "(\<Union>i. f -` F i \<inter> S') = X"
using assms Union by auto
finally show ?case .
qed
qed
lemma vimage_algebra_sigma:
assumes E: "sets E \<subseteq> Pow (space E)"
and f: "f \<in> space F \<rightarrow> space E"
and "\<And>A. A \<in> sets F \<Longrightarrow> A \<in> (\<lambda>X. f -` X \<inter> space F) ` sets E"
and "\<And>A. A \<in> sets E \<Longrightarrow> f -` A \<inter> space F \<in> sets F"
shows "sigma_algebra.vimage_algebra (sigma E) (space F) f = sigma F"
proof -
interpret sigma_algebra "sigma E"
using assms by (intro sigma_algebra_sigma) auto
have eq: "sets F = (\<lambda>X. f -` X \<inter> space F) ` sets E"
using assms by auto
show "vimage_algebra (space F) f = sigma F"
unfolding vimage_algebra_def using assms
by (simp add: sigma_def eq sigma_sets_vimage)
qed
section {* Conditional space *}
definition (in algebra)
"image_space X = \<lparr> space = X`space M, sets = (\<lambda>S. X`S) ` sets M \<rparr>"
definition (in algebra)
"conditional_space X A = algebra.image_space (restricted_space A) X"
lemma (in algebra) space_conditional_space:
assumes "A \<in> sets M" shows "space (conditional_space X A) = X`A"
proof -
interpret r: algebra "restricted_space A" using assms by (rule restricted_algebra)
show ?thesis unfolding conditional_space_def r.image_space_def
by simp
qed
subsection {* A Two-Element Series *}
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
where "binaryset A B = (\<lambda>\<^isup>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: UNION_eq_Union_image range_binaryset_eq)
section {* Closed CDI *}
definition
closed_cdi where
"closed_cdi M \<longleftrightarrow>
sets M \<subseteq> Pow (space M) &
(\<forall>s \<in> sets M. space M - s \<in> sets M) &
(\<forall>A. (range A \<subseteq> sets M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
(\<Union>i. A i) \<in> sets M) &
(\<forall>A. (range A \<subseteq> sets M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)"
inductive_set
smallest_ccdi_sets :: "('a, 'b) algebra_scheme \<Rightarrow> 'a set set"
for M
where
Basic [intro]:
"a \<in> sets M \<Longrightarrow> a \<in> smallest_ccdi_sets M"
| Compl [intro]:
"a \<in> smallest_ccdi_sets M \<Longrightarrow> space M - a \<in> smallest_ccdi_sets M"
| Inc:
"range A \<in> Pow(smallest_ccdi_sets M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets M"
| Disj:
"range A \<in> Pow(smallest_ccdi_sets M) \<Longrightarrow> disjoint_family A
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets M"
monos Pow_mono
definition
smallest_closed_cdi where
"smallest_closed_cdi M = (|space = space M, sets = smallest_ccdi_sets M|)"
lemma space_smallest_closed_cdi [simp]:
"space (smallest_closed_cdi M) = space M"
by (simp add: smallest_closed_cdi_def)
lemma (in algebra) smallest_closed_cdi1: "sets M \<subseteq> sets (smallest_closed_cdi M)"
by (auto simp add: smallest_closed_cdi_def)
lemma (in algebra) smallest_ccdi_sets:
"smallest_ccdi_sets M \<subseteq> Pow (space M)"
apply (rule subsetI)
apply (erule smallest_ccdi_sets.induct)
apply (auto intro: range_subsetD dest: sets_into_space)
done
lemma (in algebra) smallest_closed_cdi2: "closed_cdi (smallest_closed_cdi M)"
apply (auto simp add: closed_cdi_def smallest_closed_cdi_def smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
done
lemma (in algebra) smallest_closed_cdi3:
"sets (smallest_closed_cdi M) \<subseteq> Pow (space M)"
by (simp add: smallest_closed_cdi_def smallest_ccdi_sets)
lemma closed_cdi_subset: "closed_cdi M \<Longrightarrow> sets M \<subseteq> Pow (space M)"
by (simp add: closed_cdi_def)
lemma closed_cdi_Compl: "closed_cdi M \<Longrightarrow> s \<in> sets M \<Longrightarrow> space M - s \<in> sets M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Inc:
"closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow>
(\<Union>i. A i) \<in> sets M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Disj:
"closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
by (simp add: closed_cdi_def)
lemma closed_cdi_Un:
assumes cdi: "closed_cdi M" and empty: "{} \<in> sets M"
and A: "A \<in> sets M" and B: "B \<in> sets M"
and disj: "A \<inter> B = {}"
shows "A \<union> B \<in> sets M"
proof -
have ra: "range (binaryset A B) \<subseteq> sets 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 M" and B: "B \<in> smallest_ccdi_sets M"
and disj: "A \<inter> B = {}"
shows "A \<union> B \<in> smallest_ccdi_sets M"
proof -
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets 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> sets M"
shows "b \<in> smallest_ccdi_sets M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets 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> (space M - x) = space M - ((space M - a) \<union> (a \<inter> x))"
by blast
also have "... \<in> smallest_ccdi_sets M"
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
Diff_disjoint Int_Diff Int_empty_right Un_commute
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl
smallest_ccdi_sets_Un)
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 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 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 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 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 M"
shows "a \<in> smallest_ccdi_sets M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets 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 "(space M - x) \<inter> b = space M - (x \<inter> b \<union> (space M - b))"
by blast
also have "... \<in> smallest_ccdi_sets 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 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 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 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 M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed
lemma (in algebra) sets_smallest_closed_cdi_Int:
"a \<in> sets (smallest_closed_cdi M) \<Longrightarrow> b \<in> sets (smallest_closed_cdi M)
\<Longrightarrow> a \<inter> b \<in> sets (smallest_closed_cdi M)"
by (simp add: smallest_ccdi_sets_Int smallest_closed_cdi_def)
lemma (in algebra) sigma_property_disjoint_lemma:
assumes sbC: "sets M \<subseteq> C"
and ccdi: "closed_cdi (|space = space M, sets = C|)"
shows "sigma_sets (space M) (sets M) \<subseteq> C"
proof -
have "smallest_ccdi_sets M \<in> {B . sets M \<subseteq> B \<and> sigma_algebra (|space = space M, sets = 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 (space M) (sets M) \<subseteq> smallest_ccdi_sets M"
by clarsimp
(drule sigma_algebra.sigma_sets_subset [where a="sets M"], auto)
also have "... \<subseteq> C"
proof
fix x
assume x: "x \<in> smallest_ccdi_sets 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: "sets M \<subseteq> C"
and compl: "!!s. s \<in> C \<inter> sigma_sets (space M) (sets M) \<Longrightarrow> space M - s \<in> C"
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (space M) (sets 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 (space M) (sets M)
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
shows "sigma_sets (space M) (sets M) \<subseteq> C"
proof -
have "sigma_sets (space M) (sets M) \<subseteq> C \<inter> sigma_sets (space M) (sets M)"
proof (rule sigma_property_disjoint_lemma)
show "sets M \<subseteq> C \<inter> sigma_sets (space M) (sets M)"
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
next
show "closed_cdi \<lparr>space = space M, sets = C \<inter> sigma_sets (space M) (sets M)\<rparr>"
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
section {* Dynkin systems *}
locale dynkin_system =
fixes M :: "'a algebra"
assumes space_closed: "sets M \<subseteq> Pow (space M)"
and space: "space M \<in> sets M"
and compl[intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
lemma (in dynkin_system) sets_into_space: "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
using space_closed by auto
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> sets M"
using space compl[of "space M"] by simp
lemma (in dynkin_system) diff:
assumes sets: "D \<in> sets M" "E \<in> sets M" and "D \<subseteq> E"
shows "E - D \<in> sets M"
proof -
let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then space M - E else {}"
have "range ?f = {D, space M - E, {}}"
by (auto simp: image_iff)
moreover have "D \<union> (space M - E) = (\<Union>i. ?f i)"
by (auto simp: image_iff split: split_if_asm)
moreover
then have "disjoint_family ?f" unfolding disjoint_family_on_def
using `D \<in> sets M`[THEN sets_into_space] `D \<subseteq> E` by auto
ultimately have "space M - (D \<union> (space M - E)) \<in> sets M"
using sets by auto
also have "space M - (D \<union> (space M - E)) = E - D"
using assms sets_into_space by auto
finally show ?thesis .
qed
lemma dynkin_systemI:
assumes "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M" "space M \<in> sets M"
assumes "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
shows "dynkin_system M"
using assms by (auto simp: dynkin_system_def)
lemma dynkin_system_trivial:
shows "dynkin_system \<lparr> space = A, sets = Pow A \<rparr>"
by (rule dynkin_systemI) auto
lemma sigma_algebra_imp_dynkin_system:
assumes "sigma_algebra M" shows "dynkin_system M"
proof -
interpret sigma_algebra M by fact
show ?thesis using sets_into_space by (fastsimp intro!: dynkin_systemI)
qed
subsection "Intersection stable algebras"
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> sets M. \<forall> b \<in> sets M. a \<inter> b \<in> sets M)"
lemma (in algebra) Int_stable: "Int_stable M"
unfolding Int_stable_def by auto
lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
"sigma_algebra M \<longleftrightarrow> Int_stable M"
proof
assume "sigma_algebra M" then show "Int_stable M"
unfolding sigma_algebra_def using algebra.Int_stable by auto
next
assume "Int_stable M"
show "sigma_algebra M"
unfolding sigma_algebra_disjoint_iff algebra_def
proof (intro conjI ballI allI impI)
show "sets M \<subseteq> Pow (space M)" using sets_into_space by auto
next
fix A B assume "A \<in> sets M" "B \<in> sets M"
then have "A \<union> B = space M - ((space M - A) \<inter> (space M - B))"
"space M - A \<in> sets M" "space M - B \<in> sets M"
using sets_into_space by auto
then show "A \<union> B \<in> sets M"
using `Int_stable M` unfolding Int_stable_def by auto
qed auto
qed
subsection "Smallest Dynkin systems"
definition dynkin :: "'a algebra \<Rightarrow> 'a algebra" where
"dynkin M = \<lparr> space = space M,
sets = \<Inter>{D. dynkin_system \<lparr> space = space M, sets = D\<rparr> \<and> sets M \<subseteq> D}\<rparr>"
lemma dynkin_system_dynkin:
fixes M :: "'a algebra"
assumes "sets M \<subseteq> Pow (space M)"
shows "dynkin_system (dynkin M)"
proof (rule dynkin_systemI)
fix A assume "A \<in> sets (dynkin M)"
moreover
{ fix D assume "A \<in> D" and d: "dynkin_system \<lparr> space = space M, sets = D \<rparr>"
from dynkin_system.sets_into_space[OF d] `A \<in> D`
have "A \<subseteq> space M" by auto }
moreover have "{D. dynkin_system \<lparr> space = space M, sets = D\<rparr> \<and> sets M \<subseteq> D} \<noteq> {}"
using assms dynkin_system_trivial by fastsimp
ultimately show "A \<subseteq> space (dynkin M)"
unfolding dynkin_def using assms
by simp (metis dynkin_system.sets_into_space in_mono mem_def)
next
show "space (dynkin M) \<in> sets (dynkin M)"
unfolding dynkin_def using dynkin_system.space by fastsimp
next
fix A assume "A \<in> sets (dynkin M)"
then show "space (dynkin M) - A \<in> sets (dynkin 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> sets (dynkin M)"
show "(\<Union>i. A i) \<in> sets (dynkin M)" unfolding dynkin_def
proof (simp, safe)
fix D assume "dynkin_system \<lparr>space = space M, sets = D\<rparr>" "sets M \<subseteq> D"
with A have "(\<Union>i. A i) \<in> sets \<lparr>space = space M, sets = D\<rparr>"
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> sets M \<Longrightarrow> A \<in> sets (dynkin M)"
unfolding dynkin_def by auto
lemma dynkin_space[simp]:
"space (dynkin M) = space M"
unfolding dynkin_def by auto
lemma (in dynkin_system) restricted_dynkin_system:
assumes "D \<in> sets M"
shows "dynkin_system \<lparr> space = space M,
sets = {Q. Q \<subseteq> space M \<and> Q \<inter> D \<in> sets M} \<rparr>"
proof (rule dynkin_systemI, simp_all)
have "space M \<inter> D = D"
using `D \<in> sets M` sets_into_space by auto
then show "space M \<inter> D \<in> sets M"
using `D \<in> sets M` by auto
next
fix A assume "A \<subseteq> space M \<and> A \<inter> D \<in> sets M"
moreover have "(space M - A) \<inter> D = (space M - (A \<inter> D)) - (space M - D)"
by auto
ultimately show "space M - A \<subseteq> space M \<and> (space M - A) \<inter> D \<in> sets M"
using `D \<in> sets M` by (auto intro: diff)
next
fix A :: "nat \<Rightarrow> 'a set"
assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> space M \<and> Q \<inter> D \<in> sets M}"
then have "\<And>i. A i \<subseteq> space M" "disjoint_family (\<lambda>i. A i \<inter> D)"
"range (\<lambda>i. A i \<inter> D) \<subseteq> sets M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
by ((fastsimp simp: disjoint_family_on_def)+)
then show "(\<Union>x. A x) \<subseteq> space M \<and> (\<Union>x. A x) \<inter> D \<in> sets M"
by (auto simp del: UN_simps)
qed
lemma (in dynkin_system) dynkin_subset:
fixes N :: "'a algebra"
assumes "sets N \<subseteq> sets M"
assumes "space N = space M"
shows "sets (dynkin N) \<subseteq> sets M"
proof -
have *: "\<lparr>space = space N, sets = sets M\<rparr> = M"
unfolding `space N = space M` by simp
have "dynkin_system M" by default
then have "dynkin_system \<lparr>space = space N, sets = sets M\<rparr>"
using assms unfolding * by simp
with `sets N \<subseteq> sets M` show ?thesis by (auto simp add: dynkin_def)
qed
lemma sigma_eq_dynkin:
fixes M :: "'a algebra"
assumes sets: "sets M \<subseteq> Pow (space M)"
assumes "Int_stable M"
shows "sigma M = dynkin M"
proof -
have "sets (dynkin M) \<subseteq> sigma_sets (space M) (sets M)"
using sigma_algebra_imp_dynkin_system
unfolding dynkin_def sigma_def sigma_sets_least_sigma_algebra[OF sets] by auto
moreover
interpret dynkin_system "dynkin M"
using dynkin_system_dynkin[OF sets] .
have "sigma_algebra (dynkin M)"
unfolding sigma_algebra_eq_Int_stable Int_stable_def
proof (intro ballI)
fix A B assume "A \<in> sets (dynkin M)" "B \<in> sets (dynkin M)"
let "?D E" = "\<lparr> space = space M,
sets = {Q. Q \<subseteq> space M \<and> Q \<inter> E \<in> sets (dynkin M)} \<rparr>"
have "sets M \<subseteq> sets (?D B)"
proof
fix E assume "E \<in> sets M"
then have "sets M \<subseteq> sets (?D E)" "E \<in> sets (dynkin M)"
using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
then have "sets (dynkin M) \<subseteq> sets (?D E)"
using restricted_dynkin_system `E \<in> sets (dynkin M)`
by (intro dynkin_system.dynkin_subset) simp_all
then have "B \<in> sets (?D E)"
using `B \<in> sets (dynkin M)` by auto
then have "E \<inter> B \<in> sets (dynkin M)"
by (subst Int_commute) simp
then show "E \<in> sets (?D B)"
using sets `E \<in> sets M` by auto
qed
then have "sets (dynkin M) \<subseteq> sets (?D B)"
using restricted_dynkin_system `B \<in> sets (dynkin M)`
by (intro dynkin_system.dynkin_subset) simp_all
then show "A \<inter> B \<in> sets (dynkin M)"
using `A \<in> sets (dynkin M)` sets_into_space by auto
qed
from sigma_algebra.sigma_sets_subset[OF this, of "sets M"]
have "sigma_sets (space M) (sets M) \<subseteq> sets (dynkin M)" by auto
ultimately have "sigma_sets (space M) (sets M) = sets (dynkin M)" by auto
then show ?thesis
by (intro algebra.equality) (simp_all add: sigma_def)
qed
lemma (in dynkin_system) dynkin_idem:
"dynkin M = M"
proof -
have "sets (dynkin M) = sets M"
proof
show "sets M \<subseteq> sets (dynkin M)"
using dynkin_Basic by auto
show "sets (dynkin M) \<subseteq> sets M"
by (intro dynkin_subset) auto
qed
then show ?thesis
by (auto intro!: algebra.equality)
qed
lemma (in dynkin_system) dynkin_lemma:
fixes E :: "'a algebra"
assumes "Int_stable E" and E: "sets E \<subseteq> sets M" "space E = space M"
and "sets M \<subseteq> sets (sigma E)"
shows "sigma E = M"
proof -
have "sets E \<subseteq> Pow (space E)"
using E sets_into_space by auto
then have "sigma E = dynkin E"
using `Int_stable E` by (rule sigma_eq_dynkin)
moreover then have "sets (dynkin E) = sets M"
using assms dynkin_subset[OF E] by simp
ultimately show ?thesis
using E by simp
qed
locale finite_sigma_algebra = sigma_algebra +
assumes finite_space: "finite (space M)"
and sets_eq_Pow[simp]: "sets M = Pow (space M)"
lemma (in finite_sigma_algebra) sets_image_space_eq_Pow:
"sets (image_space X) = Pow (space (image_space X))"
proof safe
fix x S assume "S \<in> sets (image_space X)" "x \<in> S"
then show "x \<in> space (image_space X)"
using sets_into_space by (auto intro!: imageI simp: image_space_def)
next
fix S assume "S \<subseteq> space (image_space X)"
then obtain S' where "S = X`S'" "S'\<in>sets M"
by (auto simp: subset_image_iff sets_eq_Pow image_space_def)
then show "S \<in> sets (image_space X)"
by (auto simp: image_space_def)
qed
end