
diff -r 5001ed24e129 -r d5d342611edb src/HOL/Probability/Sigma_Algebra.thy
--- 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




