--- a/src/HOL/Probability/Sigma_Algebra.thy Mon Aug 23 17:46:13 2010 +0200
+++ b/src/HOL/Probability/Sigma_Algebra.thy Mon Aug 23 19:35:57 2010 +0200
@@ -1,12 +1,12 @@
(* Title: Sigma_Algebra.thy
Author: Stefan Richter, Markus Wenzel, TU Muenchen
- Plus material from the Hurd/Coble measure theory development,
+ Plus material from the Hurd/Coble measure theory development,
translated by Lawrence Paulson.
*)
header {* Sigma Algebras *}
-theory Sigma_Algebra imports Complex_Main begin
+theory Sigma_Algebra imports Main Countable FuncSet begin
text {* Sigma algebras are an elementary concept in measure
theory. To measure --- that is to integrate --- functions, we first have
@@ -18,8 +18,8 @@
subsection {* Algebras *}
-record 'a algebra =
- space :: "'a set"
+record 'a algebra =
+ space :: "'a set"
sets :: "'a set set"
locale algebra =
@@ -35,20 +35,20 @@
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]:
+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"
+ 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))"
+ 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]:
+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"
@@ -60,74 +60,143 @@
by metis
qed
-lemma (in algebra) finite_union [intro]:
+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)
+ 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)
+
+lemma (in algebra) restricted_algebra:
+ assumes "S \<in> sets M"
+ shows "algebra (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>)"
+ (is "algebra ?r")
+ using assms by unfold_locales auto
subsection {* Sigma Algebras *}
locale sigma_algebra = algebra +
- assumes countable_UN [intro]:
+ 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_range[OF 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]:
- assumes a: "range a \<subseteq> sets M"
- shows "(\<Inter>i::nat. a i) \<in> sets M"
+ 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. a i \<in> sets M" by fast
- hence "space M - (\<Union>i. space M - a i) \<in> sets M" by blast
+ 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. a i) = space M - (\<Union>i. space M - a i)" using space_closed a
+ 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) gen_countable_UN:
- fixes f :: "nat \<Rightarrow> 'c"
- shows "I = range f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> (\<Union>x\<in>I. A x) \<in> sets M"
-by auto
-
-lemma (in sigma_algebra) gen_countable_INT:
- fixes f :: "nat \<Rightarrow> 'c"
- shows "I = range f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> (\<Inter>x\<in>I. A x) \<in> sets M"
-by auto
+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 (| space = sp, sets = Pow sp |)"
- by (auto simp add: algebra_def)
+ "algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
+ by (auto simp add: algebra_def)
lemma sigma_algebra_Pow:
- "sigma_algebra (| space = sp, sets = Pow sp |)"
- by (auto simp add: sigma_algebra_def sigma_algebra_axioms_def 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 range_binary_eq: "range(binary a b) = {a,b}"
+ by (auto simp add: binary_def)
-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 Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
+ by (simp add: UNION_eq_Union_image range_binary_eq)
-lemma sigma_algebra_iff:
- "sigma_algebra M \<longleftrightarrow>
- algebra M & (\<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)
+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) &
- {} \<in> sets M & (\<forall>s \<in> sets M. space M - s \<in> sets M) &
+ 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 (force simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
- algebra_def Un_range_binary)
-
+ by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
+ algebra_def Un_range_binary)
subsection {* Initial Sigma Algebra *}
@@ -148,19 +217,21 @@
sigma where
"sigma sp A = (| space = sp, sets = sigma_sets sp A |)"
+lemma sets_sigma: "sets (sigma A B) = sigma_sets A B"
+ unfolding sigma_def by simp
lemma space_sigma [simp]: "space (sigma X B) = X"
- by (simp add: sigma_def)
+ 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)
+ by (erule sigma_sets.induct, auto)
-lemma sigma_sets_Un:
+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 (simp add: Un_range_binary range_binary_eq)
apply (rule Union, simp add: binary_def COMBK_def fun_upd_apply)
done
@@ -169,21 +240,21 @@
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"
+ 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"
+ 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"
+ 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)"
+ 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 .
+ by (blast dest: sigma_sets_into_sp [OF Asb])
+ finally show ?thesis .
qed
lemma sigma_sets_INTER:
- assumes Asb: "A \<subseteq> Pow sp"
+ 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 -
@@ -197,13 +268,13 @@
qed
lemma (in sigma_algebra) sigma_sets_subset:
- assumes a: "a \<subseteq> sets M"
+ 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)
+ by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
qed
lemma (in sigma_algebra) sigma_sets_eq:
@@ -219,19 +290,612 @@
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)
+ 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:
"a \<subseteq> Pow X \<Longrightarrow> sigma_algebra (sigma X a)"
- apply (rule sigma_algebra_sigma_sets)
- apply (auto simp add: sigma_def)
+ apply (rule sigma_algebra_sigma_sets)
+ apply (auto simp add: sigma_def)
done
lemma (in sigma_algebra) sigma_subset:
"a \<subseteq> sets M ==> sets (sigma (space M) a) \<subseteq> (sets M)"
by (simp add: sigma_def sigma_sets_subset)
+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 (M\<lparr> space := S, sets := (\<lambda>A. S \<inter> A) ` sets M \<rparr>)"
+ (is "sigma_algebra ?r")
+ unfolding sigma_algebra_def sigma_algebra_axioms_def
+proof safe
+ show "algebra ?r" using restricted_algebra[OF assms] .
+next
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets ?r"
+ from restriction_in_sets[OF assms this[simplified]]
+ show "(\<Union>i. A i) \<in> sets ?r" by simp
+qed
+
+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: "B \<subseteq> Pow X"
+ and f: "f \<in> space M -> X"
+ and ba: "\<And>y. y \<in> B \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
+ shows "f \<in> measurable M (sigma X B)"
+proof -
+ have "sigma_sets X B \<subseteq> {y . (f -` y) \<inter> space M \<in> sets M & y \<subseteq> X}"
+ proof clarify
+ fix x
+ assume "x \<in> sigma_sets X B"
+ thus "f -` x \<inter> space M \<in> sets M \<and> x \<subseteq> X"
+ 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 -` X \<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 intro: countable_UN)
+ 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 (space A) (sets 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 (space A) (sets 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) 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
+
+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_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
+
+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 {* 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_ext)
+ 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
+
end