--- a/src/HOL/Analysis/Sigma_Algebra.thy Mon Dec 31 13:05:15 2018 +0100
+++ b/src/HOL/Analysis/Sigma_Algebra.thy Mon Dec 31 13:24:20 2018 +0100
@@ -1181,16 +1181,16 @@
subsubsection \<open>Dynkin systems\<close>
-locale%important dynkin_system = subset_class +
+locale%important Dynkin_system = subset_class +
assumes space: "\<Omega> \<in> M"
and compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
-lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
+lemma (in Dynkin_system) empty[intro, simp]: "{} \<in> M"
using space compl[of "\<Omega>"] by simp
-lemma (in dynkin_system) diff:
+lemma (in Dynkin_system) diff:
assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
shows "E - D \<in> M"
proof -
@@ -1209,36 +1209,36 @@
finally show ?thesis .
qed
-lemma dynkin_systemI:
+lemma Dynkin_systemI:
assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
- shows "dynkin_system \<Omega> M"
- using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
+ shows "Dynkin_system \<Omega> M"
+ using assms by (auto simp: Dynkin_system_def Dynkin_system_axioms_def subset_class_def)
-lemma dynkin_systemI':
+lemma Dynkin_systemI':
assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
assumes empty: "{} \<in> M"
assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
- shows "dynkin_system \<Omega> M"
+ shows "Dynkin_system \<Omega> M"
proof -
from Diff[OF empty] have "\<Omega> \<in> M" by auto
from 1 this Diff 2 show ?thesis
- by (intro dynkin_systemI) auto
+ by (intro Dynkin_systemI) auto
qed
-lemma dynkin_system_trivial:
- shows "dynkin_system A (Pow A)"
- by (rule dynkin_systemI) auto
+lemma Dynkin_system_trivial:
+ shows "Dynkin_system A (Pow A)"
+ by (rule Dynkin_systemI) auto
-lemma sigma_algebra_imp_dynkin_system:
- assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
+lemma sigma_algebra_imp_Dynkin_system:
+ assumes "sigma_algebra \<Omega> M" shows "Dynkin_system \<Omega> M"
proof -
interpret sigma_algebra \<Omega> M by fact
- show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
+ show ?thesis using sets_into_space by (fastforce intro!: Dynkin_systemI)
qed
subsubsection "Intersection sets systems"
@@ -1261,7 +1261,7 @@
"Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
unfolding Int_stable_def by auto
-lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
+lemma (in Dynkin_system) sigma_algebra_eq_Int_stable:
"sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
proof
assume "sigma_algebra \<Omega> M" then show "Int_stable M"
@@ -1284,48 +1284,48 @@
subsubsection "Smallest Dynkin systems"
-definition%important dynkin :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" where
- "dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
+definition%important Dynkin :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" where
+ "Dynkin \<Omega> M = (\<Inter>{D. Dynkin_system \<Omega> D \<and> M \<subseteq> D})"
-lemma dynkin_system_dynkin:
+lemma Dynkin_system_Dynkin:
assumes "M \<subseteq> Pow (\<Omega>)"
- shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
-proof (rule dynkin_systemI)
- fix A assume "A \<in> dynkin \<Omega> M"
+ shows "Dynkin_system \<Omega> (Dynkin \<Omega> M)"
+proof (rule Dynkin_systemI)
+ fix A assume "A \<in> Dynkin \<Omega> M"
moreover
- { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
- then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
- moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
- using assms dynkin_system_trivial by fastforce
+ { fix D assume "A \<in> D" and d: "Dynkin_system \<Omega> D"
+ then have "A \<subseteq> \<Omega>" by (auto simp: Dynkin_system_def subset_class_def) }
+ moreover have "{D. Dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
+ using assms Dynkin_system_trivial by fastforce
ultimately show "A \<subseteq> \<Omega>"
- unfolding dynkin_def using assms
+ unfolding Dynkin_def using assms
by auto
next
- show "\<Omega> \<in> dynkin \<Omega> M"
- unfolding dynkin_def using dynkin_system.space by fastforce
+ show "\<Omega> \<in> Dynkin \<Omega> M"
+ unfolding Dynkin_def using Dynkin_system.space by fastforce
next
- fix A assume "A \<in> dynkin \<Omega> M"
- then show "\<Omega> - A \<in> dynkin \<Omega> M"
- unfolding dynkin_def using dynkin_system.compl by force
+ fix A assume "A \<in> Dynkin \<Omega> M"
+ then show "\<Omega> - A \<in> Dynkin \<Omega> M"
+ unfolding Dynkin_def using Dynkin_system.compl by force
next
fix A :: "nat \<Rightarrow> 'a set"
- assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
- show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
+ assume A: "disjoint_family A" "range A \<subseteq> Dynkin \<Omega> M"
+ show "(\<Union>i. A i) \<in> Dynkin \<Omega> M" unfolding Dynkin_def
proof (simp, safe)
- fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
+ fix D assume "Dynkin_system \<Omega> D" "M \<subseteq> D"
with A have "(\<Union>i. A i) \<in> D"
- by (intro dynkin_system.UN) (auto simp: dynkin_def)
+ by (intro Dynkin_system.UN) (auto simp: Dynkin_def)
then show "(\<Union>i. A i) \<in> D" by auto
qed
qed
-lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
- unfolding dynkin_def by auto
+lemma Dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> Dynkin \<Omega> M"
+ unfolding Dynkin_def by auto
-lemma (in dynkin_system) restricted_dynkin_system:
+lemma (in Dynkin_system) restricted_Dynkin_system:
assumes "D \<in> M"
- shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
-proof (rule dynkin_systemI, simp_all)
+ shows "Dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
+proof (rule Dynkin_systemI, simp_all)
have "\<Omega> \<inter> D = D"
using \<open>D \<in> M\<close> sets_into_space by auto
then show "\<Omega> \<inter> D \<in> M"
@@ -1346,87 +1346,87 @@
by (auto simp del: UN_simps)
qed
-lemma (in dynkin_system) dynkin_subset:
+lemma (in Dynkin_system) Dynkin_subset:
assumes "N \<subseteq> M"
- shows "dynkin \<Omega> N \<subseteq> M"
+ shows "Dynkin \<Omega> N \<subseteq> M"
proof -
- have "dynkin_system \<Omega> M" ..
- then have "dynkin_system \<Omega> M"
- using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
- with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: dynkin_def)
+ have "Dynkin_system \<Omega> M" ..
+ then have "Dynkin_system \<Omega> M"
+ using assms unfolding Dynkin_system_def Dynkin_system_axioms_def subset_class_def by simp
+ with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: Dynkin_def)
qed
-lemma sigma_eq_dynkin:
+lemma sigma_eq_Dynkin:
assumes sets: "M \<subseteq> Pow \<Omega>"
assumes "Int_stable M"
- shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
+ shows "sigma_sets \<Omega> M = Dynkin \<Omega> M"
proof -
- have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
- using sigma_algebra_imp_dynkin_system
- unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
+ have "Dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
+ using sigma_algebra_imp_Dynkin_system
+ unfolding Dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
moreover
- interpret dynkin_system \<Omega> "dynkin \<Omega> M"
- using dynkin_system_dynkin[OF sets] .
- have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
+ interpret Dynkin_system \<Omega> "Dynkin \<Omega> M"
+ using Dynkin_system_Dynkin[OF sets] .
+ have "sigma_algebra \<Omega> (Dynkin \<Omega> M)"
unfolding sigma_algebra_eq_Int_stable Int_stable_def
proof (intro ballI)
- fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
- let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
+ fix A B assume "A \<in> Dynkin \<Omega> M" "B \<in> Dynkin \<Omega> M"
+ let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> Dynkin \<Omega> M}"
have "M \<subseteq> ?D B"
proof
fix E assume "E \<in> M"
- then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
+ then have "M \<subseteq> ?D E" "E \<in> Dynkin \<Omega> M"
using sets_into_space \<open>Int_stable M\<close> by (auto simp: Int_stable_def)
- then have "dynkin \<Omega> M \<subseteq> ?D E"
- using restricted_dynkin_system \<open>E \<in> dynkin \<Omega> M\<close>
- by (intro dynkin_system.dynkin_subset) simp_all
+ then have "Dynkin \<Omega> M \<subseteq> ?D E"
+ using restricted_Dynkin_system \<open>E \<in> Dynkin \<Omega> M\<close>
+ by (intro Dynkin_system.Dynkin_subset) simp_all
then have "B \<in> ?D E"
- using \<open>B \<in> dynkin \<Omega> M\<close> by auto
- then have "E \<inter> B \<in> dynkin \<Omega> M"
+ using \<open>B \<in> Dynkin \<Omega> M\<close> by auto
+ then have "E \<inter> B \<in> Dynkin \<Omega> M"
by (subst Int_commute) simp
then show "E \<in> ?D B"
using sets \<open>E \<in> M\<close> by auto
qed
- then have "dynkin \<Omega> M \<subseteq> ?D B"
- using restricted_dynkin_system \<open>B \<in> dynkin \<Omega> M\<close>
- by (intro dynkin_system.dynkin_subset) simp_all
- then show "A \<inter> B \<in> dynkin \<Omega> M"
- using \<open>A \<in> dynkin \<Omega> M\<close> sets_into_space by auto
+ then have "Dynkin \<Omega> M \<subseteq> ?D B"
+ using restricted_Dynkin_system \<open>B \<in> Dynkin \<Omega> M\<close>
+ by (intro Dynkin_system.Dynkin_subset) simp_all
+ then show "A \<inter> B \<in> Dynkin \<Omega> M"
+ using \<open>A \<in> Dynkin \<Omega> M\<close> sets_into_space by auto
qed
from sigma_algebra.sigma_sets_subset[OF this, of "M"]
- have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
- ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
+ have "sigma_sets (\<Omega>) (M) \<subseteq> Dynkin \<Omega> M" by auto
+ ultimately have "sigma_sets (\<Omega>) (M) = Dynkin \<Omega> M" by auto
then show ?thesis
- by (auto simp: dynkin_def)
+ by (auto simp: Dynkin_def)
qed
-lemma (in dynkin_system) dynkin_idem:
- "dynkin \<Omega> M = M"
+lemma (in Dynkin_system) Dynkin_idem:
+ "Dynkin \<Omega> M = M"
proof -
- have "dynkin \<Omega> M = M"
+ have "Dynkin \<Omega> M = M"
proof
- show "M \<subseteq> dynkin \<Omega> M"
- using dynkin_Basic by auto
- show "dynkin \<Omega> M \<subseteq> M"
- by (intro dynkin_subset) auto
+ show "M \<subseteq> Dynkin \<Omega> M"
+ using Dynkin_Basic by auto
+ show "Dynkin \<Omega> M \<subseteq> M"
+ by (intro Dynkin_subset) auto
qed
then show ?thesis
- by (auto simp: dynkin_def)
+ by (auto simp: Dynkin_def)
qed
-lemma (in dynkin_system) dynkin_lemma:
+lemma (in Dynkin_system) Dynkin_lemma:
assumes "Int_stable E"
and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
shows "sigma_sets \<Omega> E = M"
proof -
have "E \<subseteq> Pow \<Omega>"
using E sets_into_space by force
- then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
- using \<open>Int_stable E\<close> by (rule sigma_eq_dynkin)
- then have "dynkin \<Omega> E = M"
- using assms dynkin_subset[OF E(1)] by simp
+ then have *: "sigma_sets \<Omega> E = Dynkin \<Omega> E"
+ using \<open>Int_stable E\<close> by (rule sigma_eq_Dynkin)
+ then have "Dynkin \<Omega> E = M"
+ using assms Dynkin_subset[OF E(1)] by simp
with * show ?thesis
- using assms by (auto simp: dynkin_def)
+ using assms by (auto simp: Dynkin_def)
qed
subsubsection \<open>Induction rule for intersection-stable generators\<close>
@@ -1448,10 +1448,10 @@
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
using closed by (rule sigma_algebra_sigma_sets)
from compl[OF _ empty] closed have space: "P \<Omega>" by simp
- interpret dynkin_system \<Omega> ?D
+ interpret Dynkin_system \<Omega> ?D
by standard (auto dest: sets_into_space intro!: space compl union)
have "sigma_sets \<Omega> G = ?D"
- by (rule dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)
+ by (rule Dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)
with A show ?thesis by auto
qed