author | hoelzl |
Wed, 25 Apr 2012 15:06:59 +0200 | |
changeset 47756 | 7b2836a43cc9 |
parent 47694 | 05663f75964c |
child 47762 | d31085f07f60 |
permissions | -rw-r--r-- |
41983 | 1 |
(* Title: HOL/Probability/Sigma_Algebra.thy |
42067 | 2 |
Author: Stefan Richter, Markus Wenzel, TU München |
3 |
Author: Johannes Hölzl, TU München |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
4 |
Plus material from the Hurd/Coble measure theory development, |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
5 |
translated by Lawrence Paulson. |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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6 |
*) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
7 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
8 |
header {* Sigma Algebras *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
9 |
|
41413
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
diff
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|
10 |
theory Sigma_Algebra |
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
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11 |
imports |
42145 | 12 |
Complex_Main |
41413
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explicit file specifications -- avoid secondary load path;
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parents:
41095
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13 |
"~~/src/HOL/Library/Countable" |
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explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
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|
14 |
"~~/src/HOL/Library/FuncSet" |
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explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
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15 |
"~~/src/HOL/Library/Indicator_Function" |
47694 | 16 |
"~~/src/HOL/Library/Extended_Real" |
41413
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
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parents:
41095
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17 |
begin |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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18 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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19 |
text {* Sigma algebras are an elementary concept in measure |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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20 |
theory. To measure --- that is to integrate --- functions, we first have |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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21 |
to measure sets. Unfortunately, when dealing with a large universe, |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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22 |
it is often not possible to consistently assign a measure to every |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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23 |
subset. Therefore it is necessary to define the set of measurable |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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|
24 |
subsets of the universe. A sigma algebra is such a set that has |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
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25 |
three very natural and desirable properties. *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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26 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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|
27 |
subsection {* Algebras *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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28 |
|
47694 | 29 |
locale subset_class = |
30 |
fixes \<Omega> :: "'a set" and M :: "'a set set" |
|
31 |
assumes space_closed: "M \<subseteq> Pow \<Omega>" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
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parents:
diff
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32 |
|
47694 | 33 |
lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
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34 |
by (metis PowD contra_subsetD space_closed) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
35 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
36 |
locale ring_of_sets = subset_class + |
47694 | 37 |
assumes empty_sets [iff]: "{} \<in> M" |
38 |
and Diff [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M" |
|
39 |
and Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
40 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
41 |
lemma (in ring_of_sets) Int [intro]: |
47694 | 42 |
assumes a: "a \<in> M" and b: "b \<in> M" shows "a \<inter> b \<in> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
43 |
proof - |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
44 |
have "a \<inter> b = a - (a - b)" |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
45 |
by auto |
47694 | 46 |
then show "a \<inter> b \<in> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
47 |
using a b by auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
48 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
49 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
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50 |
lemma (in ring_of_sets) finite_Union [intro]: |
47694 | 51 |
"finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M" |
38656 | 52 |
by (induct set: finite) (auto simp add: Un) |
33271
7be66dee1a5a
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paulson
parents:
diff
changeset
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53 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
54 |
lemma (in ring_of_sets) finite_UN[intro]: |
47694 | 55 |
assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" |
56 |
shows "(\<Union>i\<in>I. A i) \<in> M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
57 |
using assms by induct auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
58 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
59 |
lemma (in ring_of_sets) finite_INT[intro]: |
47694 | 60 |
assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" |
61 |
shows "(\<Inter>i\<in>I. A i) \<in> M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
62 |
using assms by (induct rule: finite_ne_induct) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
63 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
64 |
lemma (in ring_of_sets) insert_in_sets: |
47694 | 65 |
assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M" |
38656 | 66 |
proof - |
47694 | 67 |
have "{x} \<union> A \<in> M" using assms by (rule Un) |
38656 | 68 |
thus ?thesis by auto |
69 |
qed |
|
70 |
||
47694 | 71 |
lemma (in ring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x" |
38656 | 72 |
by (metis Int_absorb1 sets_into_space) |
73 |
||
47694 | 74 |
lemma (in ring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x" |
38656 | 75 |
by (metis Int_absorb2 sets_into_space) |
76 |
||
42867 | 77 |
lemma (in ring_of_sets) sets_Collect_conj: |
47694 | 78 |
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M" |
79 |
shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M" |
|
42867 | 80 |
proof - |
47694 | 81 |
have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}" |
42867 | 82 |
by auto |
83 |
with assms show ?thesis by auto |
|
84 |
qed |
|
85 |
||
86 |
lemma (in ring_of_sets) sets_Collect_disj: |
|
47694 | 87 |
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M" |
88 |
shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M" |
|
42867 | 89 |
proof - |
47694 | 90 |
have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}" |
42867 | 91 |
by auto |
92 |
with assms show ?thesis by auto |
|
93 |
qed |
|
94 |
||
95 |
lemma (in ring_of_sets) sets_Collect_finite_All: |
|
47694 | 96 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}" |
97 |
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M" |
|
42867 | 98 |
proof - |
47694 | 99 |
have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})" |
42867 | 100 |
using `S \<noteq> {}` by auto |
101 |
with assms show ?thesis by auto |
|
102 |
qed |
|
103 |
||
104 |
lemma (in ring_of_sets) sets_Collect_finite_Ex: |
|
47694 | 105 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" |
106 |
shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M" |
|
42867 | 107 |
proof - |
47694 | 108 |
have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})" |
42867 | 109 |
by auto |
110 |
with assms show ?thesis by auto |
|
111 |
qed |
|
112 |
||
42065
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parents:
41983
diff
changeset
|
113 |
locale algebra = ring_of_sets + |
47694 | 114 |
assumes top [iff]: "\<Omega> \<in> M" |
42065
2b98b4c2e2f1
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hoelzl
parents:
41983
diff
changeset
|
115 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
116 |
lemma (in algebra) compl_sets [intro]: |
47694 | 117 |
"a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" |
42065
2b98b4c2e2f1
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hoelzl
parents:
41983
diff
changeset
|
118 |
by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
119 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
120 |
lemma algebra_iff_Un: |
47694 | 121 |
"algebra \<Omega> M \<longleftrightarrow> |
122 |
M \<subseteq> Pow \<Omega> \<and> |
|
123 |
{} \<in> M \<and> |
|
124 |
(\<forall>a \<in> M. \<Omega> - a \<in> M) \<and> |
|
125 |
(\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un") |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
126 |
proof |
47694 | 127 |
assume "algebra \<Omega> M" |
128 |
then interpret algebra \<Omega> M . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
129 |
show ?Un using sets_into_space by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
130 |
next |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
131 |
assume ?Un |
47694 | 132 |
show "algebra \<Omega> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
133 |
proof |
47694 | 134 |
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M" "\<Omega> \<in> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
135 |
using `?Un` by auto |
47694 | 136 |
fix a b assume a: "a \<in> M" and b: "b \<in> M" |
137 |
then show "a \<union> b \<in> M" using `?Un` by auto |
|
138 |
have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)" |
|
139 |
using \<Omega> a b by auto |
|
140 |
then show "a - b \<in> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
141 |
using a b `?Un` by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
142 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
143 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
144 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
145 |
lemma algebra_iff_Int: |
47694 | 146 |
"algebra \<Omega> M \<longleftrightarrow> |
147 |
M \<subseteq> Pow \<Omega> & {} \<in> M & |
|
148 |
(\<forall>a \<in> M. \<Omega> - a \<in> M) & |
|
149 |
(\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int") |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
150 |
proof |
47694 | 151 |
assume "algebra \<Omega> M" |
152 |
then interpret algebra \<Omega> M . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
153 |
show ?Int using sets_into_space by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
154 |
next |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
155 |
assume ?Int |
47694 | 156 |
show "algebra \<Omega> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
157 |
proof (unfold algebra_iff_Un, intro conjI ballI) |
47694 | 158 |
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
159 |
using `?Int` by auto |
47694 | 160 |
from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto |
161 |
fix a b assume M: "a \<in> M" "b \<in> M" |
|
162 |
hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))" |
|
163 |
using \<Omega> by blast |
|
164 |
also have "... \<in> M" |
|
165 |
using M `?Int` by auto |
|
166 |
finally show "a \<union> b \<in> M" . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
167 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
168 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
169 |
|
42867 | 170 |
lemma (in algebra) sets_Collect_neg: |
47694 | 171 |
assumes "{x\<in>\<Omega>. P x} \<in> M" |
172 |
shows "{x\<in>\<Omega>. \<not> P x} \<in> M" |
|
42867 | 173 |
proof - |
47694 | 174 |
have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto |
42867 | 175 |
with assms show ?thesis by auto |
176 |
qed |
|
177 |
||
178 |
lemma (in algebra) sets_Collect_imp: |
|
47694 | 179 |
"{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M" |
42867 | 180 |
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg) |
181 |
||
182 |
lemma (in algebra) sets_Collect_const: |
|
47694 | 183 |
"{x\<in>\<Omega>. P} \<in> M" |
42867 | 184 |
by (cases P) auto |
185 |
||
42984 | 186 |
lemma algebra_single_set: |
187 |
assumes "X \<subseteq> S" |
|
47694 | 188 |
shows "algebra S { {}, X, S - X, S }" |
42984 | 189 |
by default (insert `X \<subseteq> S`, auto) |
190 |
||
39092 | 191 |
section {* Restricted algebras *} |
192 |
||
193 |
abbreviation (in algebra) |
|
47694 | 194 |
"restricted_space A \<equiv> (op \<inter> A) ` M" |
39092 | 195 |
|
38656 | 196 |
lemma (in algebra) restricted_algebra: |
47694 | 197 |
assumes "A \<in> M" shows "algebra A (restricted_space A)" |
38656 | 198 |
using assms by unfold_locales auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
199 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
200 |
subsection {* Sigma Algebras *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
201 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
202 |
locale sigma_algebra = algebra + |
47694 | 203 |
assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
204 |
|
42984 | 205 |
lemma (in algebra) is_sigma_algebra: |
47694 | 206 |
assumes "finite M" |
207 |
shows "sigma_algebra \<Omega> M" |
|
42984 | 208 |
proof |
47694 | 209 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M" |
210 |
then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)" |
|
42984 | 211 |
by auto |
47694 | 212 |
also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M" |
213 |
using `finite M` by auto |
|
214 |
finally show "(\<Union>i. A i) \<in> M" . |
|
42984 | 215 |
qed |
216 |
||
38656 | 217 |
lemma countable_UN_eq: |
218 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
|
47694 | 219 |
shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow> |
220 |
(range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)" |
|
38656 | 221 |
proof - |
222 |
let ?A' = "A \<circ> from_nat" |
|
223 |
have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r") |
|
224 |
proof safe |
|
225 |
fix x i assume "x \<in> A i" thus "x \<in> ?l" |
|
226 |
by (auto intro!: exI[of _ "to_nat i"]) |
|
227 |
next |
|
228 |
fix x i assume "x \<in> ?A' i" thus "x \<in> ?r" |
|
229 |
by (auto intro!: exI[of _ "from_nat i"]) |
|
230 |
qed |
|
231 |
have **: "range ?A' = range A" |
|
40702 | 232 |
using surj_from_nat |
38656 | 233 |
by (auto simp: image_compose intro!: imageI) |
234 |
show ?thesis unfolding * ** .. |
|
235 |
qed |
|
236 |
||
237 |
lemma (in sigma_algebra) countable_UN[intro]: |
|
238 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
|
47694 | 239 |
assumes "A`X \<subseteq> M" |
240 |
shows "(\<Union>x\<in>X. A x) \<in> M" |
|
38656 | 241 |
proof - |
46731 | 242 |
let ?A = "\<lambda>i. if i \<in> X then A i else {}" |
47694 | 243 |
from assms have "range ?A \<subseteq> M" by auto |
38656 | 244 |
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M] |
47694 | 245 |
have "(\<Union>x. ?A x) \<in> M" by auto |
38656 | 246 |
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm) |
247 |
ultimately show ?thesis by simp |
|
248 |
qed |
|
249 |
||
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33271
diff
changeset
|
250 |
lemma (in sigma_algebra) countable_INT [intro]: |
38656 | 251 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
47694 | 252 |
assumes A: "A`X \<subseteq> M" "X \<noteq> {}" |
253 |
shows "(\<Inter>i\<in>X. A i) \<in> M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
254 |
proof - |
47694 | 255 |
from A have "\<forall>i\<in>X. A i \<in> M" by fast |
256 |
hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
257 |
moreover |
47694 | 258 |
have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
259 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
260 |
ultimately show ?thesis by metis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
261 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
262 |
|
47694 | 263 |
lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)" |
42145 | 264 |
by default auto |
265 |
||
47694 | 266 |
lemma algebra_Pow: "algebra sp (Pow sp)" |
42145 | 267 |
by default auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
268 |
|
47694 | 269 |
lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)" |
42145 | 270 |
by default auto |
38656 | 271 |
|
272 |
lemma sigma_algebra_iff: |
|
47694 | 273 |
"sigma_algebra \<Omega> M \<longleftrightarrow> |
274 |
algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 275 |
by (simp add: sigma_algebra_def sigma_algebra_axioms_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
276 |
|
42867 | 277 |
lemma (in sigma_algebra) sets_Collect_countable_All: |
47694 | 278 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
279 |
shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M" |
|
42867 | 280 |
proof - |
47694 | 281 |
have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto |
42867 | 282 |
with assms show ?thesis by auto |
283 |
qed |
|
284 |
||
285 |
lemma (in sigma_algebra) sets_Collect_countable_Ex: |
|
47694 | 286 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
287 |
shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M" |
|
42867 | 288 |
proof - |
47694 | 289 |
have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto |
42867 | 290 |
with assms show ?thesis by auto |
291 |
qed |
|
292 |
||
293 |
lemmas (in sigma_algebra) sets_Collect = |
|
294 |
sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const |
|
295 |
sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All |
|
296 |
||
47694 | 297 |
lemma (in sigma_algebra) sets_Collect_countable_Ball: |
298 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
|
299 |
shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M" |
|
300 |
unfolding Ball_def by (intro sets_Collect assms) |
|
301 |
||
302 |
lemma (in sigma_algebra) sets_Collect_countable_Bex: |
|
303 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
|
304 |
shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M" |
|
305 |
unfolding Bex_def by (intro sets_Collect assms) |
|
306 |
||
42984 | 307 |
lemma sigma_algebra_single_set: |
308 |
assumes "X \<subseteq> S" |
|
47694 | 309 |
shows "sigma_algebra S { {}, X, S - X, S }" |
42984 | 310 |
using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp |
311 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
312 |
subsection {* Binary Unions *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
313 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
314 |
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
315 |
where "binary a b = (\<lambda>\<^isup>x. b)(0 := a)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
316 |
|
38656 | 317 |
lemma range_binary_eq: "range(binary a b) = {a,b}" |
318 |
by (auto simp add: binary_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
319 |
|
38656 | 320 |
lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)" |
44106 | 321 |
by (simp add: SUP_def range_binary_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
322 |
|
38656 | 323 |
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)" |
44106 | 324 |
by (simp add: INF_def range_binary_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
325 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
326 |
lemma sigma_algebra_iff2: |
47694 | 327 |
"sigma_algebra \<Omega> M \<longleftrightarrow> |
328 |
M \<subseteq> Pow \<Omega> \<and> |
|
329 |
{} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and> |
|
330 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 331 |
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
332 |
algebra_iff_Un Un_range_binary) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
333 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
334 |
subsection {* Initial Sigma Algebra *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
335 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
336 |
text {*Sigma algebras can naturally be created as the closure of any set of |
47694 | 337 |
M with regard to the properties just postulated. *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
338 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
339 |
inductive_set |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
340 |
sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
341 |
for sp :: "'a set" and A :: "'a set set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
342 |
where |
47694 | 343 |
Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
344 |
| Empty: "{} \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
345 |
| Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
346 |
| Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
347 |
|
41543 | 348 |
lemma (in sigma_algebra) sigma_sets_subset: |
47694 | 349 |
assumes a: "a \<subseteq> M" |
350 |
shows "sigma_sets \<Omega> a \<subseteq> M" |
|
41543 | 351 |
proof |
352 |
fix x |
|
47694 | 353 |
assume "x \<in> sigma_sets \<Omega> a" |
354 |
from this show "x \<in> M" |
|
41543 | 355 |
by (induct rule: sigma_sets.induct, auto) (metis a subsetD) |
356 |
qed |
|
357 |
||
358 |
lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp" |
|
359 |
by (erule sigma_sets.induct, auto) |
|
360 |
||
361 |
lemma sigma_algebra_sigma_sets: |
|
47694 | 362 |
"a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)" |
41543 | 363 |
by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp |
364 |
intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl) |
|
365 |
||
366 |
lemma sigma_sets_least_sigma_algebra: |
|
367 |
assumes "A \<subseteq> Pow S" |
|
47694 | 368 |
shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}" |
41543 | 369 |
proof safe |
47694 | 370 |
fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B" |
41543 | 371 |
and X: "X \<in> sigma_sets S A" |
372 |
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X |
|
373 |
show "X \<in> B" by auto |
|
374 |
next |
|
47694 | 375 |
fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}" |
376 |
then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B" |
|
41543 | 377 |
by simp |
47694 | 378 |
have "A \<subseteq> sigma_sets S A" using assms by auto |
379 |
moreover have "sigma_algebra S (sigma_sets S A)" |
|
41543 | 380 |
using assms by (intro sigma_algebra_sigma_sets[of A]) auto |
381 |
ultimately show "X \<in> sigma_sets S A" by auto |
|
382 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
383 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
384 |
lemma sigma_sets_top: "sp \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
385 |
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
386 |
|
38656 | 387 |
lemma sigma_sets_Un: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
388 |
"a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A" |
38656 | 389 |
apply (simp add: Un_range_binary range_binary_eq) |
40859 | 390 |
apply (rule Union, simp add: binary_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
391 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
392 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
393 |
lemma sigma_sets_Inter: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
394 |
assumes Asb: "A \<subseteq> Pow sp" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
395 |
shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
396 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
397 |
assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A" |
38656 | 398 |
hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
399 |
by (rule sigma_sets.Compl) |
38656 | 400 |
hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
401 |
by (rule sigma_sets.Union) |
38656 | 402 |
hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
403 |
by (rule sigma_sets.Compl) |
38656 | 404 |
also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
405 |
by auto |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
406 |
also have "... = (\<Inter>i. a i)" using ai |
38656 | 407 |
by (blast dest: sigma_sets_into_sp [OF Asb]) |
408 |
finally show ?thesis . |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
409 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
410 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
411 |
lemma sigma_sets_INTER: |
38656 | 412 |
assumes Asb: "A \<subseteq> Pow sp" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
413 |
and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
414 |
shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
415 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
416 |
from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A" |
47694 | 417 |
by (simp add: sigma_sets.intros(2-) sigma_sets_top) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
418 |
hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
419 |
by (rule sigma_sets_Inter [OF Asb]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
420 |
also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
421 |
by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
422 |
finally show ?thesis . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
423 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
424 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
425 |
lemma (in sigma_algebra) sigma_sets_eq: |
47694 | 426 |
"sigma_sets \<Omega> M = M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
427 |
proof |
47694 | 428 |
show "M \<subseteq> sigma_sets \<Omega> M" |
37032 | 429 |
by (metis Set.subsetI sigma_sets.Basic) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
430 |
next |
47694 | 431 |
show "sigma_sets \<Omega> M \<subseteq> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
432 |
by (metis sigma_sets_subset subset_refl) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
433 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
434 |
|
42981 | 435 |
lemma sigma_sets_eqI: |
436 |
assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B" |
|
437 |
assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A" |
|
438 |
shows "sigma_sets M A = sigma_sets M B" |
|
439 |
proof (intro set_eqI iffI) |
|
440 |
fix a assume "a \<in> sigma_sets M A" |
|
441 |
from this A show "a \<in> sigma_sets M B" |
|
47694 | 442 |
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) |
42981 | 443 |
next |
444 |
fix b assume "b \<in> sigma_sets M B" |
|
445 |
from this B show "b \<in> sigma_sets M A" |
|
47694 | 446 |
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) |
42981 | 447 |
qed |
448 |
||
42984 | 449 |
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
450 |
proof |
|
451 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
47694 | 452 |
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-)) |
42984 | 453 |
qed |
454 |
||
38656 | 455 |
lemma (in sigma_algebra) restriction_in_sets: |
456 |
fixes A :: "nat \<Rightarrow> 'a set" |
|
47694 | 457 |
assumes "S \<in> M" |
458 |
and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r") |
|
459 |
shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M" |
|
38656 | 460 |
proof - |
461 |
{ fix i have "A i \<in> ?r" using * by auto |
|
47694 | 462 |
hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto |
463 |
hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto } |
|
464 |
thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M" |
|
38656 | 465 |
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"]) |
466 |
qed |
|
467 |
||
468 |
lemma (in sigma_algebra) restricted_sigma_algebra: |
|
47694 | 469 |
assumes "S \<in> M" |
470 |
shows "sigma_algebra S (restricted_space S)" |
|
38656 | 471 |
unfolding sigma_algebra_def sigma_algebra_axioms_def |
472 |
proof safe |
|
47694 | 473 |
show "algebra S (restricted_space S)" using restricted_algebra[OF assms] . |
38656 | 474 |
next |
47694 | 475 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S" |
38656 | 476 |
from restriction_in_sets[OF assms this[simplified]] |
47694 | 477 |
show "(\<Union>i. A i) \<in> restricted_space S" by simp |
38656 | 478 |
qed |
479 |
||
40859 | 480 |
lemma sigma_sets_Int: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
481 |
assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
482 |
shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)" |
40859 | 483 |
proof (intro equalityI subsetI) |
484 |
fix x assume "x \<in> op \<inter> A ` sigma_sets sp st" |
|
485 |
then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
486 |
then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)" |
40859 | 487 |
proof (induct arbitrary: x) |
488 |
case (Compl a) |
|
489 |
then show ?case |
|
490 |
by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps) |
|
491 |
next |
|
492 |
case (Union a) |
|
493 |
then show ?case |
|
494 |
by (auto intro!: sigma_sets.Union |
|
495 |
simp add: UN_extend_simps simp del: UN_simps) |
|
47694 | 496 |
qed (auto intro!: sigma_sets.intros(2-)) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
497 |
then show "x \<in> sigma_sets A (op \<inter> A ` st)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
498 |
using `A \<subseteq> sp` by (simp add: Int_absorb2) |
40859 | 499 |
next |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
500 |
fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)" |
40859 | 501 |
then show "x \<in> op \<inter> A ` sigma_sets sp st" |
502 |
proof induct |
|
503 |
case (Compl a) |
|
504 |
then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
505 |
then show ?case using `A \<subseteq> sp` |
40859 | 506 |
by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl) |
507 |
next |
|
508 |
case (Union a) |
|
509 |
then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x" |
|
510 |
by (auto simp: image_iff Bex_def) |
|
511 |
from choice[OF this] guess f .. |
|
512 |
then show ?case |
|
513 |
by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union |
|
514 |
simp add: image_iff) |
|
47694 | 515 |
qed (auto intro!: sigma_sets.intros(2-)) |
40859 | 516 |
qed |
517 |
||
47694 | 518 |
lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}" |
40859 | 519 |
proof (intro set_eqI iffI) |
47694 | 520 |
fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}" |
521 |
by induct blast+ |
|
522 |
qed (auto intro: sigma_sets.Empty sigma_sets_top) |
|
523 |
||
524 |
lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}" |
|
525 |
proof (intro set_eqI iffI) |
|
526 |
fix x assume "x \<in> sigma_sets A {A}" |
|
527 |
then show "x \<in> {{}, A}" |
|
528 |
by induct blast+ |
|
40859 | 529 |
next |
47694 | 530 |
fix x assume "x \<in> {{}, A}" |
531 |
then show "x \<in> sigma_sets A {A}" |
|
40859 | 532 |
by (auto intro: sigma_sets.Empty sigma_sets_top) |
533 |
qed |
|
534 |
||
42987 | 535 |
lemma sigma_sets_sigma_sets_eq: |
536 |
"M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M" |
|
47694 | 537 |
by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto |
42987 | 538 |
|
42984 | 539 |
lemma sigma_sets_singleton: |
540 |
assumes "X \<subseteq> S" |
|
541 |
shows "sigma_sets S { X } = { {}, X, S - X, S }" |
|
542 |
proof - |
|
47694 | 543 |
interpret sigma_algebra S "{ {}, X, S - X, S }" |
42984 | 544 |
by (rule sigma_algebra_single_set) fact |
545 |
have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }" |
|
546 |
by (rule sigma_sets_subseteq) simp |
|
547 |
moreover have "\<dots> = { {}, X, S - X, S }" |
|
47694 | 548 |
using sigma_sets_eq by simp |
42984 | 549 |
moreover |
550 |
{ fix A assume "A \<in> { {}, X, S - X, S }" |
|
551 |
then have "A \<in> sigma_sets S { X }" |
|
47694 | 552 |
by (auto intro: sigma_sets.intros(2-) sigma_sets_top) } |
42984 | 553 |
ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }" |
554 |
by (intro antisym) auto |
|
47694 | 555 |
with sigma_sets_eq show ?thesis by simp |
42984 | 556 |
qed |
557 |
||
42863 | 558 |
lemma restricted_sigma: |
47694 | 559 |
assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>" |
560 |
shows "algebra.restricted_space (sigma_sets \<Omega> M) S = |
|
561 |
sigma_sets S (algebra.restricted_space M S)" |
|
42863 | 562 |
proof - |
563 |
from S sigma_sets_into_sp[OF M] |
|
47694 | 564 |
have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto |
42863 | 565 |
from sigma_sets_Int[OF this] |
47694 | 566 |
show ?thesis by simp |
42863 | 567 |
qed |
568 |
||
42987 | 569 |
lemma sigma_sets_vimage_commute: |
47694 | 570 |
assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'" |
571 |
shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'} |
|
572 |
= sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R") |
|
42987 | 573 |
proof |
574 |
show "?L \<subseteq> ?R" |
|
575 |
proof clarify |
|
47694 | 576 |
fix A assume "A \<in> sigma_sets \<Omega>' M'" |
577 |
then show "X -` A \<inter> \<Omega> \<in> ?R" |
|
42987 | 578 |
proof induct |
579 |
case Empty then show ?case |
|
580 |
by (auto intro!: sigma_sets.Empty) |
|
581 |
next |
|
582 |
case (Compl B) |
|
47694 | 583 |
have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)" |
42987 | 584 |
by (auto simp add: funcset_mem [OF X]) |
585 |
with Compl show ?case |
|
586 |
by (auto intro!: sigma_sets.Compl) |
|
587 |
next |
|
588 |
case (Union F) |
|
589 |
then show ?case |
|
590 |
by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps |
|
591 |
intro!: sigma_sets.Union) |
|
47694 | 592 |
qed auto |
42987 | 593 |
qed |
594 |
show "?R \<subseteq> ?L" |
|
595 |
proof clarify |
|
596 |
fix A assume "A \<in> ?R" |
|
47694 | 597 |
then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" |
42987 | 598 |
proof induct |
599 |
case (Basic B) then show ?case by auto |
|
600 |
next |
|
601 |
case Empty then show ?case |
|
47694 | 602 |
by (auto intro!: sigma_sets.Empty exI[of _ "{}"]) |
42987 | 603 |
next |
604 |
case (Compl B) |
|
47694 | 605 |
then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto |
606 |
then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>" |
|
42987 | 607 |
by (auto simp add: funcset_mem [OF X]) |
608 |
with A(2) show ?case |
|
47694 | 609 |
by (auto intro: sigma_sets.Compl) |
42987 | 610 |
next |
611 |
case (Union F) |
|
47694 | 612 |
then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto |
42987 | 613 |
from choice[OF this] guess A .. note A = this |
614 |
with A show ?case |
|
47694 | 615 |
by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union) |
42987 | 616 |
qed |
617 |
qed |
|
618 |
qed |
|
619 |
||
38656 | 620 |
section "Disjoint families" |
621 |
||
622 |
definition |
|
623 |
disjoint_family_on where |
|
624 |
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})" |
|
625 |
||
626 |
abbreviation |
|
627 |
"disjoint_family A \<equiv> disjoint_family_on A UNIV" |
|
628 |
||
629 |
lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B" |
|
630 |
by blast |
|
631 |
||
632 |
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}" |
|
633 |
by blast |
|
634 |
||
635 |
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A" |
|
636 |
by blast |
|
637 |
||
638 |
lemma disjoint_family_subset: |
|
639 |
"disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B" |
|
640 |
by (force simp add: disjoint_family_on_def) |
|
641 |
||
40859 | 642 |
lemma disjoint_family_on_bisimulation: |
643 |
assumes "disjoint_family_on f S" |
|
644 |
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 = {}" |
|
645 |
shows "disjoint_family_on g S" |
|
646 |
using assms unfolding disjoint_family_on_def by auto |
|
647 |
||
38656 | 648 |
lemma disjoint_family_on_mono: |
649 |
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A" |
|
650 |
unfolding disjoint_family_on_def by auto |
|
651 |
||
652 |
lemma disjoint_family_Suc: |
|
653 |
assumes Suc: "!!n. A n \<subseteq> A (Suc n)" |
|
654 |
shows "disjoint_family (\<lambda>i. A (Suc i) - A i)" |
|
655 |
proof - |
|
656 |
{ |
|
657 |
fix m |
|
658 |
have "!!n. A n \<subseteq> A (m+n)" |
|
659 |
proof (induct m) |
|
660 |
case 0 show ?case by simp |
|
661 |
next |
|
662 |
case (Suc m) thus ?case |
|
663 |
by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans) |
|
664 |
qed |
|
665 |
} |
|
666 |
hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n" |
|
667 |
by (metis add_commute le_add_diff_inverse nat_less_le) |
|
668 |
thus ?thesis |
|
669 |
by (auto simp add: disjoint_family_on_def) |
|
670 |
(metis insert_absorb insert_subset le_SucE le_antisym not_leE) |
|
671 |
qed |
|
672 |
||
39092 | 673 |
lemma setsum_indicator_disjoint_family: |
674 |
fixes f :: "'d \<Rightarrow> 'e::semiring_1" |
|
675 |
assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P" |
|
676 |
shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j" |
|
677 |
proof - |
|
678 |
have "P \<inter> {i. x \<in> A i} = {j}" |
|
679 |
using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def |
|
680 |
by auto |
|
681 |
thus ?thesis |
|
682 |
unfolding indicator_def |
|
683 |
by (simp add: if_distrib setsum_cases[OF `finite P`]) |
|
684 |
qed |
|
685 |
||
38656 | 686 |
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set " |
687 |
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)" |
|
688 |
||
689 |
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)" |
|
690 |
proof (induct n) |
|
691 |
case 0 show ?case by simp |
|
692 |
next |
|
693 |
case (Suc n) |
|
694 |
thus ?case by (simp add: atLeastLessThanSuc disjointed_def) |
|
695 |
qed |
|
696 |
||
697 |
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)" |
|
698 |
apply (rule UN_finite2_eq [where k=0]) |
|
699 |
apply (simp add: finite_UN_disjointed_eq) |
|
700 |
done |
|
701 |
||
702 |
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}" |
|
703 |
by (auto simp add: disjointed_def) |
|
704 |
||
705 |
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)" |
|
706 |
by (simp add: disjoint_family_on_def) |
|
707 |
(metis neq_iff Int_commute less_disjoint_disjointed) |
|
708 |
||
709 |
lemma disjointed_subset: "disjointed A n \<subseteq> A n" |
|
710 |
by (auto simp add: disjointed_def) |
|
711 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
712 |
lemma (in ring_of_sets) UNION_in_sets: |
38656 | 713 |
fixes A:: "nat \<Rightarrow> 'a set" |
47694 | 714 |
assumes A: "range A \<subseteq> M" |
715 |
shows "(\<Union>i\<in>{0..<n}. A i) \<in> M" |
|
38656 | 716 |
proof (induct n) |
717 |
case 0 show ?case by simp |
|
718 |
next |
|
719 |
case (Suc n) |
|
720 |
thus ?case |
|
721 |
by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff) |
|
722 |
qed |
|
723 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
724 |
lemma (in ring_of_sets) range_disjointed_sets: |
47694 | 725 |
assumes A: "range A \<subseteq> M" |
726 |
shows "range (disjointed A) \<subseteq> M" |
|
38656 | 727 |
proof (auto simp add: disjointed_def) |
728 |
fix n |
|
47694 | 729 |
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets |
38656 | 730 |
by (metis A Diff UNIV_I image_subset_iff) |
731 |
qed |
|
732 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
733 |
lemma (in algebra) range_disjointed_sets': |
47694 | 734 |
"range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
735 |
using range_disjointed_sets . |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
736 |
|
42145 | 737 |
lemma disjointed_0[simp]: "disjointed A 0 = A 0" |
738 |
by (simp add: disjointed_def) |
|
739 |
||
740 |
lemma incseq_Un: |
|
741 |
"incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n" |
|
742 |
unfolding incseq_def by auto |
|
743 |
||
744 |
lemma disjointed_incseq: |
|
745 |
"incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n" |
|
746 |
using incseq_Un[of A] |
|
747 |
by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost) |
|
748 |
||
38656 | 749 |
lemma sigma_algebra_disjoint_iff: |
47694 | 750 |
"sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and> |
751 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 752 |
proof (auto simp add: sigma_algebra_iff) |
753 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 754 |
assume M: "algebra \<Omega> M" |
755 |
and A: "range A \<subseteq> M" |
|
756 |
and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
757 |
hence "range (disjointed A) \<subseteq> M \<longrightarrow> |
|
38656 | 758 |
disjoint_family (disjointed A) \<longrightarrow> |
47694 | 759 |
(\<Union>i. disjointed A i) \<in> M" by blast |
760 |
hence "(\<Union>i. disjointed A i) \<in> M" |
|
761 |
by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed) |
|
762 |
thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq) |
|
763 |
qed |
|
764 |
||
765 |
section {* Measure type *} |
|
766 |
||
767 |
definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
768 |
"positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)" |
|
769 |
||
770 |
definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
771 |
"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> |
|
772 |
(\<Sum>i. f (A i)) = f (\<Union>i. A i))" |
|
773 |
||
774 |
definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
775 |
"measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>" |
|
776 |
||
777 |
typedef (open) 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }" |
|
778 |
proof |
|
779 |
have "sigma_algebra UNIV {{}, UNIV}" |
|
780 |
proof qed auto |
|
781 |
then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} " |
|
782 |
by (auto simp: measure_space_def positive_def countably_additive_def) |
|
783 |
qed |
|
784 |
||
785 |
definition space :: "'a measure \<Rightarrow> 'a set" where |
|
786 |
"space M = fst (Rep_measure M)" |
|
787 |
||
788 |
definition sets :: "'a measure \<Rightarrow> 'a set set" where |
|
789 |
"sets M = fst (snd (Rep_measure M))" |
|
790 |
||
791 |
definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where |
|
792 |
"emeasure M = snd (snd (Rep_measure M))" |
|
793 |
||
794 |
definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where |
|
795 |
"measure M A = real (emeasure M A)" |
|
796 |
||
797 |
declare [[coercion sets]] |
|
798 |
||
799 |
declare [[coercion measure]] |
|
800 |
||
801 |
declare [[coercion emeasure]] |
|
802 |
||
803 |
lemma measure_space: "measure_space (space M) (sets M) (emeasure M)" |
|
804 |
by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse) |
|
805 |
||
806 |
interpretation sigma_algebra "space M" "sets M" for M :: "'a measure" |
|
807 |
using measure_space[of M] by (auto simp: measure_space_def) |
|
808 |
||
809 |
definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
810 |
"measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, sigma_sets \<Omega> A, |
|
811 |
\<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)" |
|
812 |
||
813 |
abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)" |
|
814 |
||
815 |
lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)" |
|
816 |
unfolding measure_space_def |
|
817 |
by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def) |
|
818 |
||
819 |
lemma (in ring_of_sets) positive_cong_eq: |
|
820 |
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>" |
|
821 |
by (auto simp add: positive_def) |
|
822 |
||
823 |
lemma (in sigma_algebra) countably_additive_eq: |
|
824 |
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>" |
|
825 |
unfolding countably_additive_def |
|
826 |
by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq) |
|
827 |
||
828 |
lemma measure_space_eq: |
|
829 |
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a" |
|
830 |
shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'" |
|
831 |
proof - |
|
832 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets) |
|
833 |
from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis |
|
834 |
by (auto simp: measure_space_def) |
|
835 |
qed |
|
836 |
||
837 |
lemma measure_of_eq: |
|
838 |
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)" |
|
839 |
shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'" |
|
840 |
proof - |
|
841 |
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'" |
|
842 |
using assms by (rule measure_space_eq) |
|
843 |
with eq show ?thesis |
|
844 |
by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure]) |
|
845 |
qed |
|
846 |
||
847 |
lemma |
|
848 |
assumes A: "A \<subseteq> Pow \<Omega>" |
|
849 |
shows sets_measure_of[simp]: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" (is ?sets) |
|
850 |
and space_measure_of[simp]: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space) |
|
851 |
proof - |
|
852 |
have "?sets \<and> ?space" |
|
853 |
proof cases |
|
854 |
assume "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
855 |
moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) |
|
856 |
(\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)" |
|
857 |
using A by (rule measure_space_eq) auto |
|
858 |
ultimately show "?sets \<and> ?space" |
|
859 |
by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def) |
|
860 |
next |
|
861 |
assume "\<not> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
862 |
with A show "?sets \<and> ?space" |
|
863 |
by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def measure_space_0) |
|
864 |
qed |
|
865 |
then show ?sets ?space by auto |
|
866 |
qed |
|
867 |
||
868 |
lemma (in sigma_algebra) sets_measure_of_eq[simp]: |
|
869 |
"sets (measure_of \<Omega> M \<mu>) = M" |
|
870 |
using space_closed by (auto intro!: sigma_sets_eq) |
|
871 |
||
872 |
lemma (in sigma_algebra) space_measure_of_eq[simp]: |
|
873 |
"space (measure_of \<Omega> M \<mu>) = \<Omega>" |
|
874 |
using space_closed by (auto intro!: sigma_sets_eq) |
|
875 |
||
876 |
lemma measure_of_subset: |
|
877 |
"M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')" |
|
878 |
by (auto intro!: sigma_sets_subseteq) |
|
879 |
||
47756 | 880 |
lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)" |
47694 | 881 |
by auto |
882 |
||
883 |
section {* Constructing simple @{typ "'a measure"} *} |
|
884 |
||
885 |
lemma emeasure_measure_of: |
|
886 |
assumes M: "M = measure_of \<Omega> A \<mu>" |
|
887 |
assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>" |
|
888 |
assumes X: "X \<in> sets M" |
|
889 |
shows "emeasure M X = \<mu> X" |
|
890 |
proof - |
|
891 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact |
|
892 |
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
893 |
using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets) |
|
894 |
moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0) |
|
895 |
= measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
896 |
using ms(1) by (rule measure_space_eq) auto |
|
897 |
moreover have "X \<in> sigma_sets \<Omega> A" |
|
898 |
using X M ms by simp |
|
899 |
ultimately show ?thesis |
|
900 |
unfolding emeasure_def measure_of_def M |
|
901 |
by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq) |
|
902 |
qed |
|
903 |
||
904 |
lemma emeasure_measure_of_sigma: |
|
905 |
assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>" |
|
906 |
assumes A: "A \<in> M" |
|
907 |
shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A" |
|
908 |
proof - |
|
909 |
interpret sigma_algebra \<Omega> M by fact |
|
910 |
have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
|
911 |
using ms sigma_sets_eq by (simp add: measure_space_def) |
|
912 |
moreover have "measure_space \<Omega> (sigma_sets \<Omega> M) (\<lambda>a. if a \<in> sigma_sets \<Omega> M then \<mu> a else 0) |
|
913 |
= measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
|
914 |
using space_closed by (rule measure_space_eq) auto |
|
915 |
ultimately show ?thesis using A |
|
916 |
unfolding emeasure_def measure_of_def |
|
917 |
by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq) |
|
918 |
qed |
|
919 |
||
920 |
lemma measure_cases[cases type: measure]: |
|
921 |
obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>" |
|
922 |
by atomize_elim (cases x, auto) |
|
923 |
||
924 |
lemma sets_eq_imp_space_eq: |
|
925 |
"sets M = sets M' \<Longrightarrow> space M = space M'" |
|
926 |
using top[of M] top[of M'] space_closed[of M] space_closed[of M'] |
|
927 |
by blast |
|
928 |
||
929 |
lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0" |
|
930 |
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) |
|
931 |
||
932 |
lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0" |
|
933 |
by (simp add: measure_def emeasure_notin_sets) |
|
934 |
||
935 |
lemma measure_eqI: |
|
936 |
fixes M N :: "'a measure" |
|
937 |
assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" |
|
938 |
shows "M = N" |
|
939 |
proof (cases M N rule: measure_cases[case_product measure_cases]) |
|
940 |
case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>') |
|
941 |
interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def) |
|
942 |
interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def) |
|
943 |
have "A = sets M" "A' = sets N" |
|
944 |
using measure_measure by (simp_all add: sets_def Abs_measure_inverse) |
|
945 |
with `sets M = sets N` have "A = A'" by simp |
|
946 |
moreover with M.top N.top M.space_closed N.space_closed have "\<Omega> = \<Omega>'" by auto |
|
947 |
moreover { fix B have "\<mu> B = \<mu>' B" |
|
948 |
proof cases |
|
949 |
assume "B \<in> A" |
|
950 |
with eq `A = sets M` have "emeasure M B = emeasure N B" by simp |
|
951 |
with measure_measure show "\<mu> B = \<mu>' B" |
|
952 |
by (simp add: emeasure_def Abs_measure_inverse) |
|
953 |
next |
|
954 |
assume "B \<notin> A" |
|
955 |
with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N" |
|
956 |
by auto |
|
957 |
then have "emeasure M B = 0" "emeasure N B = 0" |
|
958 |
by (simp_all add: emeasure_notin_sets) |
|
959 |
with measure_measure show "\<mu> B = \<mu>' B" |
|
960 |
by (simp add: emeasure_def Abs_measure_inverse) |
|
961 |
qed } |
|
962 |
then have "\<mu> = \<mu>'" by auto |
|
963 |
ultimately show "M = N" |
|
964 |
by (simp add: measure_measure) |
|
38656 | 965 |
qed |
966 |
||
47694 | 967 |
lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)" |
968 |
using measure_space_0[of A \<Omega>] |
|
969 |
by (simp add: measure_of_def emeasure_def Abs_measure_inverse) |
|
970 |
||
971 |
lemma sigma_eqI: |
|
972 |
assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N" |
|
973 |
shows "sigma \<Omega> M = sigma \<Omega> N" |
|
974 |
by (rule measure_eqI) (simp_all add: emeasure_sigma) |
|
975 |
||
976 |
section {* Measurable functions *} |
|
977 |
||
978 |
definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where |
|
979 |
"measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}" |
|
980 |
||
981 |
lemma measurable_space: |
|
982 |
"f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A" |
|
983 |
unfolding measurable_def by auto |
|
984 |
||
985 |
lemma measurable_sets: |
|
986 |
"f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
|
987 |
unfolding measurable_def by auto |
|
988 |
||
989 |
lemma measurable_sigma_sets: |
|
990 |
assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>" |
|
991 |
and f: "f \<in> space M \<rightarrow> \<Omega>" |
|
992 |
and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M" |
|
993 |
shows "f \<in> measurable M N" |
|
994 |
proof - |
|
995 |
interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets) |
|
996 |
from B top[of N] A.top space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force |
|
997 |
||
998 |
{ fix X assume "X \<in> sigma_sets \<Omega> A" |
|
999 |
then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>" |
|
1000 |
proof induct |
|
1001 |
case (Basic a) then show ?case |
|
1002 |
by (auto simp add: ba) (metis B(2) subsetD PowD) |
|
1003 |
next |
|
1004 |
case (Compl a) |
|
1005 |
have [simp]: "f -` \<Omega> \<inter> space M = space M" |
|
1006 |
by (auto simp add: funcset_mem [OF f]) |
|
1007 |
then show ?case |
|
1008 |
by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl) |
|
1009 |
next |
|
1010 |
case (Union a) |
|
1011 |
then show ?case |
|
1012 |
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast |
|
1013 |
qed auto } |
|
1014 |
with f show ?thesis |
|
1015 |
by (auto simp add: measurable_def B \<Omega>) |
|
1016 |
qed |
|
1017 |
||
1018 |
lemma measurable_measure_of: |
|
1019 |
assumes B: "N \<subseteq> Pow \<Omega>" |
|
1020 |
and f: "f \<in> space M \<rightarrow> \<Omega>" |
|
1021 |
and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M" |
|
1022 |
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)" |
|
1023 |
proof - |
|
1024 |
have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N" |
|
1025 |
using B by (rule sets_measure_of) |
|
1026 |
from this assms show ?thesis by (rule measurable_sigma_sets) |
|
1027 |
qed |
|
1028 |
||
1029 |
lemma measurable_iff_measure_of: |
|
1030 |
assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>" |
|
1031 |
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)" |
|
47756 | 1032 |
by (metis assms in_measure_of measurable_measure_of assms measurable_sets) |
47694 | 1033 |
|
1034 |
lemma measurable_cong: |
|
1035 |
assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w" |
|
1036 |
shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'" |
|
1037 |
unfolding measurable_def using assms |
|
1038 |
by (simp cong: vimage_inter_cong Pi_cong) |
|
1039 |
||
1040 |
lemma measurable_eqI: |
|
1041 |
"\<lbrakk> space m1 = space m1' ; space m2 = space m2' ; |
|
1042 |
sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk> |
|
1043 |
\<Longrightarrow> measurable m1 m2 = measurable m1' m2'" |
|
1044 |
by (simp add: measurable_def sigma_algebra_iff2) |
|
1045 |
||
1046 |
lemma measurable_const[intro, simp]: |
|
1047 |
"c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'" |
|
1048 |
by (auto simp add: measurable_def) |
|
1049 |
||
1050 |
lemma measurable_If: |
|
1051 |
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'" |
|
1052 |
assumes P: "{x\<in>space M. P x} \<in> sets M" |
|
1053 |
shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'" |
|
1054 |
unfolding measurable_def |
|
1055 |
proof safe |
|
1056 |
fix x assume "x \<in> space M" |
|
1057 |
thus "(if P x then f x else g x) \<in> space M'" |
|
1058 |
using measure unfolding measurable_def by auto |
|
1059 |
next |
|
1060 |
fix A assume "A \<in> sets M'" |
|
1061 |
hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M = |
|
1062 |
((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union> |
|
1063 |
((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))" |
|
1064 |
using measure unfolding measurable_def by (auto split: split_if_asm) |
|
1065 |
show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M" |
|
1066 |
using `A \<in> sets M'` measure P unfolding * measurable_def |
|
1067 |
by (auto intro!: Un) |
|
1068 |
qed |
|
1069 |
||
1070 |
lemma measurable_If_set: |
|
1071 |
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'" |
|
1072 |
assumes P: "A \<in> sets M" |
|
1073 |
shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'" |
|
1074 |
proof (rule measurable_If[OF measure]) |
|
1075 |
have "{x \<in> space M. x \<in> A} = A" using `A \<in> sets M` sets_into_space by auto |
|
1076 |
thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<in> sets M` by auto |
|
1077 |
qed |
|
1078 |
||
1079 |
lemma measurable_ident[intro, simp]: "id \<in> measurable M M" |
|
1080 |
by (auto simp add: measurable_def) |
|
1081 |
||
1082 |
lemma measurable_comp[intro]: |
|
1083 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
|
1084 |
shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c" |
|
1085 |
apply (auto simp add: measurable_def vimage_compose) |
|
1086 |
apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a") |
|
1087 |
apply force+ |
|
1088 |
done |
|
1089 |
||
1090 |
lemma measurable_Least: |
|
1091 |
assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M" |
|
1092 |
shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M" |
|
1093 |
proof - |
|
1094 |
{ fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M" |
|
1095 |
proof cases |
|
1096 |
assume i: "(LEAST j. False) = i" |
|
1097 |
have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M = |
|
1098 |
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))" |
|
1099 |
by (simp add: set_eq_iff, safe) |
|
1100 |
(insert i, auto dest: Least_le intro: LeastI intro!: Least_equality) |
|
1101 |
with meas show ?thesis |
|
1102 |
by (auto intro!: Int) |
|
1103 |
next |
|
1104 |
assume i: "(LEAST j. False) \<noteq> i" |
|
1105 |
then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M = |
|
1106 |
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))" |
|
1107 |
proof (simp add: set_eq_iff, safe) |
|
1108 |
fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)" |
|
1109 |
have "\<exists>j. P j x" |
|
1110 |
by (rule ccontr) (insert neq, auto) |
|
1111 |
then show "P (LEAST j. P j x) x" by (rule LeastI_ex) |
|
1112 |
qed (auto dest: Least_le intro!: Least_equality) |
|
1113 |
with meas show ?thesis |
|
1114 |
by auto |
|
1115 |
qed } |
|
1116 |
then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M" |
|
1117 |
by (intro countable_UN) auto |
|
1118 |
moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) = |
|
1119 |
(\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto |
|
1120 |
ultimately show ?thesis by auto |
|
1121 |
qed |
|
1122 |
||
1123 |
lemma measurable_strong: |
|
1124 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
|
1125 |
assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c" |
|
1126 |
and t: "f ` (space a) \<subseteq> t" |
|
1127 |
and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b" |
|
1128 |
shows "(g o f) \<in> measurable a c" |
|
1129 |
proof - |
|
1130 |
have fab: "f \<in> (space a -> space b)" |
|
1131 |
and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f |
|
1132 |
by (auto simp add: measurable_def) |
|
1133 |
have eq: "\<And>y. f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t |
|
1134 |
by force |
|
1135 |
show ?thesis |
|
1136 |
apply (auto simp add: measurable_def vimage_compose) |
|
1137 |
apply (metis funcset_mem fab g) |
|
1138 |
apply (subst eq, metis ba cb) |
|
1139 |
done |
|
1140 |
qed |
|
1141 |
||
1142 |
lemma measurable_mono1: |
|
1143 |
"M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow> |
|
1144 |
measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N" |
|
1145 |
using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def) |
|
1146 |
||
1147 |
subsection {* Extend measure *} |
|
1148 |
||
1149 |
definition "extend_measure \<Omega> I G \<mu> = |
|
1150 |
(if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0) |
|
1151 |
then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') |
|
1152 |
else measure_of \<Omega> (G`I) (\<lambda>_. 0))" |
|
1153 |
||
1154 |
lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>" |
|
1155 |
unfolding extend_measure_def by simp |
|
1156 |
||
1157 |
lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)" |
|
1158 |
unfolding extend_measure_def by simp |
|
1159 |
||
1160 |
lemma emeasure_extend_measure: |
|
1161 |
assumes M: "M = extend_measure \<Omega> I G \<mu>" |
|
1162 |
and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i" |
|
1163 |
and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'" |
|
1164 |
and "i \<in> I" |
|
1165 |
shows "emeasure M (G i) = \<mu> i" |
|
1166 |
proof cases |
|
1167 |
assume *: "(\<forall>i\<in>I. \<mu> i = 0)" |
|
1168 |
with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)" |
|
1169 |
by (simp add: extend_measure_def) |
|
1170 |
from measure_space_0[OF ms(1)] ms `i\<in>I` |
|
1171 |
have "emeasure M (G i) = 0" |
|
1172 |
by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure) |
|
1173 |
with `i\<in>I` * show ?thesis |
|
1174 |
by simp |
|
1175 |
next |
|
1176 |
def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'" |
|
1177 |
assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)" |
|
1178 |
moreover |
|
1179 |
have "measure_space (space M) (sets M) \<mu>'" |
|
1180 |
using ms unfolding measure_space_def by auto default |
|
1181 |
with ms eq have "\<exists>\<mu>'. P \<mu>'" |
|
1182 |
unfolding P_def |
|
1183 |
by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure) |
|
1184 |
ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)" |
|
1185 |
by (simp add: M extend_measure_def P_def[symmetric]) |
|
1186 |
||
1187 |
from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex) |
|
1188 |
show "emeasure M (G i) = \<mu> i" |
|
1189 |
proof (subst emeasure_measure_of[OF M_eq]) |
|
1190 |
have sets_M: "sets M = sigma_sets \<Omega> (G`I)" |
|
1191 |
using M_eq ms by (auto simp: sets_extend_measure) |
|
1192 |
then show "G i \<in> sets M" using `i \<in> I` by auto |
|
1193 |
show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i" |
|
1194 |
using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def) |
|
1195 |
qed fact |
|
1196 |
qed |
|
1197 |
||
1198 |
lemma emeasure_extend_measure_Pair: |
|
1199 |
assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)" |
|
1200 |
and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j" |
|
1201 |
and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'" |
|
1202 |
and "I i j" |
|
1203 |
shows "emeasure M (G i j) = \<mu> i j" |
|
1204 |
using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j` |
|
1205 |
by (auto simp: subset_eq) |
|
1206 |
||
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
1207 |
subsection {* Sigma algebra generated by function preimages *} |
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
1208 |
|
47694 | 1209 |
definition |
1210 |
"vimage_algebra M S f = sigma S ((\<lambda>A. f -` A \<inter> S) ` sets M)" |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
1211 |
|
47694 | 1212 |
lemma sigma_algebra_preimages: |
40859 | 1213 |
fixes f :: "'x \<Rightarrow> 'a" |
47694 | 1214 |
assumes "f \<in> S \<rightarrow> space M" |
1215 |
shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)" |
|
1216 |
(is "sigma_algebra _ (?F ` sets M)") |
|
40859 | 1217 |
proof (simp add: sigma_algebra_iff2, safe) |
1218 |
show "{} \<in> ?F ` sets M" by blast |
|
1219 |
next |
|
47694 | 1220 |
fix A assume "A \<in> sets M" |
1221 |
moreover have "S - ?F A = ?F (space M - A)" |
|
40859 | 1222 |
using assms by auto |
47694 | 1223 |
ultimately show "S - ?F A \<in> ?F ` sets M" |
40859 | 1224 |
by blast |
1225 |
next |
|
47694 | 1226 |
fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M" |
1227 |
have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b" |
|
40859 | 1228 |
proof safe |
1229 |
fix i |
|
47694 | 1230 |
have "A i \<in> ?F ` M" using * by auto |
1231 |
then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto |
|
40859 | 1232 |
qed |
47694 | 1233 |
from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)" |
40859 | 1234 |
by auto |
47694 | 1235 |
then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto |
1236 |
then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast |
|
40859 | 1237 |
qed |
1238 |
||
47694 | 1239 |
lemma sets_vimage_algebra[simp]: |
1240 |
"f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M" |
|
1241 |
using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M] |
|
1242 |
by (simp add: vimage_algebra_def) |
|
1243 |
||
1244 |
lemma space_vimage_algebra[simp]: |
|
1245 |
"f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S" |
|
1246 |
using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M] |
|
1247 |
by (simp add: vimage_algebra_def) |
|
1248 |
||
1249 |
lemma in_vimage_algebra[simp]: |
|
1250 |
"f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)" |
|
1251 |
by (simp add: image_iff) |
|
1252 |
||
1253 |
lemma measurable_vimage_algebra: |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
1254 |
fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M" |
47694 | 1255 |
shows "f \<in> measurable (vimage_algebra M S f) M" |
1256 |
unfolding measurable_def using assms by force |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
1257 |
|
47694 | 1258 |
lemma measurable_vimage: |
40859 | 1259 |
fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a" |
1260 |
assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M" |
|
47694 | 1261 |
shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2" |
40859 | 1262 |
proof - |
1263 |
note measurable_vimage_algebra[OF assms(2)] |
|
1264 |
from measurable_comp[OF this assms(1)] |
|
1265 |
show ?thesis by (simp add: comp_def) |
|
1266 |
qed |
|
1267 |
||
1268 |
lemma sigma_sets_vimage: |
|
1269 |
assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S" |
|
1270 |
shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A" |
|
1271 |
proof (intro set_eqI iffI) |
|
1272 |
let ?F = "\<lambda>X. f -` X \<inter> S'" |
|
1273 |
fix X assume "X \<in> sigma_sets S' (?F ` A)" |
|
1274 |
then show "X \<in> ?F ` sigma_sets S A" |
|
1275 |
proof induct |
|
1276 |
case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A" |
|
1277 |
by auto |
|
47694 | 1278 |
then show ?case by auto |
40859 | 1279 |
next |
1280 |
case Empty then show ?case |
|
1281 |
by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty) |
|
1282 |
next |
|
1283 |
case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A" |
|
1284 |
by auto |
|
1285 |
then have "S - X' \<in> sigma_sets S A" |
|
1286 |
by (auto intro!: sigma_sets.Compl) |
|
1287 |
then show ?case |
|
1288 |
using X assms by (auto intro!: image_eqI[where x="S - X'"]) |
|
1289 |
next |
|
1290 |
case (Union F) |
|
1291 |
then have "\<forall>i. \<exists>F'. F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'" |
|
1292 |
by (auto simp: image_iff Bex_def) |
|
1293 |
from choice[OF this] obtain F' where |
|
1294 |
"\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'" |
|
1295 |
by auto |
|
1296 |
then show ?case |
|
1297 |
by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"]) |
|
1298 |
qed |
|
1299 |
next |
|
1300 |
let ?F = "\<lambda>X. f -` X \<inter> S'" |
|
1301 |
fix X assume "X \<in> ?F ` sigma_sets S A" |
|
1302 |
then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto |
|
1303 |
then show "X \<in> sigma_sets S' (?F ` A)" |
|
1304 |
proof (induct arbitrary: X) |
|
1305 |
case Empty then show ?case by (auto intro: sigma_sets.Empty) |
|
1306 |
next |
|
1307 |
case (Compl X') |
|
1308 |
have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)" |
|
1309 |
apply (rule sigma_sets.Compl) |
|
1310 |
using assms by (auto intro!: Compl.hyps simp: Compl.prems) |
|
1311 |
also have "S' - (S' - X) = X" |
|
1312 |
using assms Compl by auto |
|
1313 |
finally show ?case . |
|
1314 |
next |
|
1315 |
case (Union F) |
|
1316 |
have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)" |
|
1317 |
by (intro sigma_sets.Union Union.hyps) simp |
|
1318 |
also have "(\<Union>i. f -` F i \<inter> S') = X" |
|
1319 |
using assms Union by auto |
|
1320 |
finally show ?case . |
|
47694 | 1321 |
qed auto |
39092 | 1322 |
qed |
1323 |
||
38656 | 1324 |
subsection {* A Two-Element Series *} |
1325 |
||
1326 |
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set " |
|
1327 |
where "binaryset A B = (\<lambda>\<^isup>x. {})(0 := A, Suc 0 := B)" |
|
1328 |
||
1329 |
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}" |
|
1330 |
apply (simp add: binaryset_def) |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39092
diff
changeset
|
1331 |
apply (rule set_eqI) |
38656 | 1332 |
apply (auto simp add: image_iff) |
1333 |
done |
|
1334 |
||
1335 |
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B" |
|
44106 | 1336 |
by (simp add: SUP_def range_binaryset_eq) |
38656 | 1337 |
|
1338 |
section {* Closed CDI *} |
|
1339 |
||
47694 | 1340 |
definition closed_cdi where |
1341 |
"closed_cdi \<Omega> M \<longleftrightarrow> |
|
1342 |
M \<subseteq> Pow \<Omega> & |
|
1343 |
(\<forall>s \<in> M. \<Omega> - s \<in> M) & |
|
1344 |
(\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow> |
|
1345 |
(\<Union>i. A i) \<in> M) & |
|
1346 |
(\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 1347 |
|
1348 |
inductive_set |
|
47694 | 1349 |
smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" |
1350 |
for \<Omega> M |
|
38656 | 1351 |
where |
1352 |
Basic [intro]: |
|
47694 | 1353 |
"a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1354 |
| Compl [intro]: |
47694 | 1355 |
"a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1356 |
| Inc: |
47694 | 1357 |
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n)) |
1358 |
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1359 |
| Disj: |
47694 | 1360 |
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A |
1361 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1362 |
|
47694 | 1363 |
lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M" |
1364 |
by auto |
|
38656 | 1365 |
|
47694 | 1366 |
lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>" |
38656 | 1367 |
apply (rule subsetI) |
1368 |
apply (erule smallest_ccdi_sets.induct) |
|
1369 |
apply (auto intro: range_subsetD dest: sets_into_space) |
|
1370 |
done |
|
1371 |
||
47694 | 1372 |
lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)" |
1373 |
apply (auto simp add: closed_cdi_def smallest_ccdi_sets) |
|
38656 | 1374 |
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) + |
1375 |
done |
|
1376 |
||
47694 | 1377 |
lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>" |
38656 | 1378 |
by (simp add: closed_cdi_def) |
1379 |
||
47694 | 1380 |
lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M" |
38656 | 1381 |
by (simp add: closed_cdi_def) |
1382 |
||
1383 |
lemma closed_cdi_Inc: |
|
47694 | 1384 |
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M" |
38656 | 1385 |
by (simp add: closed_cdi_def) |
1386 |
||
1387 |
lemma closed_cdi_Disj: |
|
47694 | 1388 |
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
38656 | 1389 |
by (simp add: closed_cdi_def) |
1390 |
||
1391 |
lemma closed_cdi_Un: |
|
47694 | 1392 |
assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M" |
1393 |
and A: "A \<in> M" and B: "B \<in> M" |
|
38656 | 1394 |
and disj: "A \<inter> B = {}" |
47694 | 1395 |
shows "A \<union> B \<in> M" |
38656 | 1396 |
proof - |
47694 | 1397 |
have ra: "range (binaryset A B) \<subseteq> M" |
38656 | 1398 |
by (simp add: range_binaryset_eq empty A B) |
1399 |
have di: "disjoint_family (binaryset A B)" using disj |
|
1400 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
|
1401 |
from closed_cdi_Disj [OF cdi ra di] |
|
1402 |
show ?thesis |
|
1403 |
by (simp add: UN_binaryset_eq) |
|
1404 |
qed |
|
1405 |
||
1406 |
lemma (in algebra) smallest_ccdi_sets_Un: |
|
47694 | 1407 |
assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1408 |
and disj: "A \<inter> B = {}" |
47694 | 1409 |
shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1410 |
proof - |
47694 | 1411 |
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)" |
38656 | 1412 |
by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic) |
1413 |
have di: "disjoint_family (binaryset A B)" using disj |
|
1414 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
|
1415 |
from Disj [OF ra di] |
|
1416 |
show ?thesis |
|
1417 |
by (simp add: UN_binaryset_eq) |
|
1418 |
qed |
|
1419 |
||
1420 |
lemma (in algebra) smallest_ccdi_sets_Int1: |
|
47694 | 1421 |
assumes a: "a \<in> M" |
1422 |
shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1423 |
proof (induct rule: smallest_ccdi_sets.induct) |
1424 |
case (Basic x) |
|
1425 |
thus ?case |
|
1426 |
by (metis a Int smallest_ccdi_sets.Basic) |
|
1427 |
next |
|
1428 |
case (Compl x) |
|
47694 | 1429 |
have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))" |
38656 | 1430 |
by blast |
47694 | 1431 |
also have "... \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1432 |
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2 |
47694 | 1433 |
Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un |
1434 |
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl) |
|
38656 | 1435 |
finally show ?case . |
1436 |
next |
|
1437 |
case (Inc A) |
|
1438 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
|
1439 |
by blast |
|
47694 | 1440 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc |
38656 | 1441 |
by blast |
1442 |
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}" |
|
1443 |
by (simp add: Inc) |
|
1444 |
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc |
|
1445 |
by blast |
|
47694 | 1446 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1447 |
by (rule smallest_ccdi_sets.Inc) |
1448 |
show ?case |
|
1449 |
by (metis 1 2) |
|
1450 |
next |
|
1451 |
case (Disj A) |
|
1452 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
|
1453 |
by blast |
|
47694 | 1454 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj |
38656 | 1455 |
by blast |
1456 |
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj |
|
1457 |
by (auto simp add: disjoint_family_on_def) |
|
47694 | 1458 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1459 |
by (rule smallest_ccdi_sets.Disj) |
1460 |
show ?case |
|
1461 |
by (metis 1 2) |
|
1462 |
qed |
|
1463 |
||
1464 |
||
1465 |
lemma (in algebra) smallest_ccdi_sets_Int: |
|
47694 | 1466 |
assumes b: "b \<in> smallest_ccdi_sets \<Omega> M" |
1467 |
shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1468 |
proof (induct rule: smallest_ccdi_sets.induct) |
1469 |
case (Basic x) |
|
1470 |
thus ?case |
|
1471 |
by (metis b smallest_ccdi_sets_Int1) |
|
1472 |
next |
|
1473 |
case (Compl x) |
|
47694 | 1474 |
have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))" |
38656 | 1475 |
by blast |
47694 | 1476 |
also have "... \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1477 |
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b |
1478 |
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un) |
|
1479 |
finally show ?case . |
|
1480 |
next |
|
1481 |
case (Inc A) |
|
1482 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
|
1483 |
by blast |
|
47694 | 1484 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc |
38656 | 1485 |
by blast |
1486 |
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}" |
|
1487 |
by (simp add: Inc) |
|
1488 |
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc |
|
1489 |
by blast |
|
47694 | 1490 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1491 |
by (rule smallest_ccdi_sets.Inc) |
1492 |
show ?case |
|
1493 |
by (metis 1 2) |
|
1494 |
next |
|
1495 |
case (Disj A) |
|
1496 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
|
1497 |
by blast |
|
47694 | 1498 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj |
38656 | 1499 |
by blast |
1500 |
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj |
|
1501 |
by (auto simp add: disjoint_family_on_def) |
|
47694 | 1502 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1503 |
by (rule smallest_ccdi_sets.Disj) |
1504 |
show ?case |
|
1505 |
by (metis 1 2) |
|
1506 |
qed |
|
1507 |
||
1508 |
lemma (in algebra) sigma_property_disjoint_lemma: |
|
47694 | 1509 |
assumes sbC: "M \<subseteq> C" |
1510 |
and ccdi: "closed_cdi \<Omega> C" |
|
1511 |
shows "sigma_sets \<Omega> M \<subseteq> C" |
|
38656 | 1512 |
proof - |
47694 | 1513 |
have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}" |
38656 | 1514 |
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int |
1515 |
smallest_ccdi_sets_Int) |
|
1516 |
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets) |
|
1517 |
apply (blast intro: smallest_ccdi_sets.Disj) |
|
1518 |
done |
|
47694 | 1519 |
hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M" |
38656 | 1520 |
by clarsimp |
47694 | 1521 |
(drule sigma_algebra.sigma_sets_subset [where a="M"], auto) |
38656 | 1522 |
also have "... \<subseteq> C" |
1523 |
proof |
|
1524 |
fix x |
|
47694 | 1525 |
assume x: "x \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1526 |
thus "x \<in> C" |
1527 |
proof (induct rule: smallest_ccdi_sets.induct) |
|
1528 |
case (Basic x) |
|
1529 |
thus ?case |
|
1530 |
by (metis Basic subsetD sbC) |
|
1531 |
next |
|
1532 |
case (Compl x) |
|
1533 |
thus ?case |
|
1534 |
by (blast intro: closed_cdi_Compl [OF ccdi, simplified]) |
|
1535 |
next |
|
1536 |
case (Inc A) |
|
1537 |
thus ?case |
|
1538 |
by (auto intro: closed_cdi_Inc [OF ccdi, simplified]) |
|
1539 |
next |
|
1540 |
case (Disj A) |
|
1541 |
thus ?case |
|
1542 |
by (auto intro: closed_cdi_Disj [OF ccdi, simplified]) |
|
1543 |
qed |
|
1544 |
qed |
|
1545 |
finally show ?thesis . |
|
1546 |
qed |
|
1547 |
||
1548 |
lemma (in algebra) sigma_property_disjoint: |
|
47694 | 1549 |
assumes sbC: "M \<subseteq> C" |
1550 |
and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C" |
|
1551 |
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M) |
|
38656 | 1552 |
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) |
1553 |
\<Longrightarrow> (\<Union>i. A i) \<in> C" |
|
47694 | 1554 |
and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M) |
38656 | 1555 |
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C" |
47694 | 1556 |
shows "sigma_sets (\<Omega>) (M) \<subseteq> C" |
38656 | 1557 |
proof - |
47694 | 1558 |
have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)" |
38656 | 1559 |
proof (rule sigma_property_disjoint_lemma) |
47694 | 1560 |
show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)" |
38656 | 1561 |
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic) |
1562 |
next |
|
47694 | 1563 |
show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))" |
38656 | 1564 |
by (simp add: closed_cdi_def compl inc disj) |
1565 |
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed |
|
1566 |
IntE sigma_sets.Compl range_subsetD sigma_sets.Union) |
|
1567 |
qed |
|
1568 |
thus ?thesis |
|
1569 |
by blast |
|
1570 |
qed |
|
1571 |
||
40859 | 1572 |
section {* Dynkin systems *} |
1573 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1574 |
locale dynkin_system = subset_class + |
47694 | 1575 |
assumes space: "\<Omega> \<in> M" |
1576 |
and compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
1577 |
and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
1578 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
40859 | 1579 |
|
47694 | 1580 |
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M" |
1581 |
using space compl[of "\<Omega>"] by simp |
|
40859 | 1582 |
|
1583 |
lemma (in dynkin_system) diff: |
|
47694 | 1584 |
assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E" |
1585 |
shows "E - D \<in> M" |
|
40859 | 1586 |
proof - |
47694 | 1587 |
let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}" |
1588 |
have "range ?f = {D, \<Omega> - E, {}}" |
|
40859 | 1589 |
by (auto simp: image_iff) |
47694 | 1590 |
moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)" |
40859 | 1591 |
by (auto simp: image_iff split: split_if_asm) |
1592 |
moreover |
|
1593 |
then have "disjoint_family ?f" unfolding disjoint_family_on_def |
|
47694 | 1594 |
using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto |
1595 |
ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M" |
|
40859 | 1596 |
using sets by auto |
47694 | 1597 |
also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D" |
40859 | 1598 |
using assms sets_into_space by auto |
1599 |
finally show ?thesis . |
|
1600 |
qed |
|
1601 |
||
1602 |
lemma dynkin_systemI: |
|
47694 | 1603 |
assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M" |
1604 |
assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
1605 |
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
1606 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
1607 |
shows "dynkin_system \<Omega> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1608 |
using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def) |
40859 | 1609 |
|
42988 | 1610 |
lemma dynkin_systemI': |
47694 | 1611 |
assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" |
1612 |
assumes empty: "{} \<in> M" |
|
1613 |
assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
1614 |
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
1615 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
1616 |
shows "dynkin_system \<Omega> M" |
|
42988 | 1617 |
proof - |
47694 | 1618 |
from Diff[OF empty] have "\<Omega> \<in> M" by auto |
42988 | 1619 |
from 1 this Diff 2 show ?thesis |
1620 |
by (intro dynkin_systemI) auto |
|
1621 |
qed |
|
1622 |
||
40859 | 1623 |
lemma dynkin_system_trivial: |
47694 | 1624 |
shows "dynkin_system A (Pow A)" |
40859 | 1625 |
by (rule dynkin_systemI) auto |
1626 |
||
1627 |
lemma sigma_algebra_imp_dynkin_system: |
|
47694 | 1628 |
assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M" |
40859 | 1629 |
proof - |
47694 | 1630 |
interpret sigma_algebra \<Omega> M by fact |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
1631 |
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI) |
40859 | 1632 |
qed |
1633 |
||
1634 |
subsection "Intersection stable algebras" |
|
1635 |
||
47694 | 1636 |
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" |
40859 | 1637 |
|
1638 |
lemma (in algebra) Int_stable: "Int_stable M" |
|
1639 |
unfolding Int_stable_def by auto |
|
1640 |
||
42981 | 1641 |
lemma Int_stableI: |
47694 | 1642 |
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A" |
42981 | 1643 |
unfolding Int_stable_def by auto |
1644 |
||
1645 |
lemma Int_stableD: |
|
47694 | 1646 |
"Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M" |
42981 | 1647 |
unfolding Int_stable_def by auto |
1648 |
||
40859 | 1649 |
lemma (in dynkin_system) sigma_algebra_eq_Int_stable: |
47694 | 1650 |
"sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M" |
40859 | 1651 |
proof |
47694 | 1652 |
assume "sigma_algebra \<Omega> M" then show "Int_stable M" |
40859 | 1653 |
unfolding sigma_algebra_def using algebra.Int_stable by auto |
1654 |
next |
|
1655 |
assume "Int_stable M" |
|
47694 | 1656 |
show "sigma_algebra \<Omega> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1657 |
unfolding sigma_algebra_disjoint_iff algebra_iff_Un |
40859 | 1658 |
proof (intro conjI ballI allI impI) |
47694 | 1659 |
show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto |
40859 | 1660 |
next |
47694 | 1661 |
fix A B assume "A \<in> M" "B \<in> M" |
1662 |
then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))" |
|
1663 |
"\<Omega> - A \<in> M" "\<Omega> - B \<in> M" |
|
40859 | 1664 |
using sets_into_space by auto |
47694 | 1665 |
then show "A \<union> B \<in> M" |
40859 | 1666 |
using `Int_stable M` unfolding Int_stable_def by auto |
1667 |
qed auto |
|
1668 |
qed |
|
1669 |
||
1670 |
subsection "Smallest Dynkin systems" |
|
1671 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
1672 |
definition dynkin where |
47694 | 1673 |
"dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})" |
40859 | 1674 |
|
1675 |
lemma dynkin_system_dynkin: |
|
47694 | 1676 |
assumes "M \<subseteq> Pow (\<Omega>)" |
1677 |
shows "dynkin_system \<Omega> (dynkin \<Omega> M)" |
|
40859 | 1678 |
proof (rule dynkin_systemI) |
47694 | 1679 |
fix A assume "A \<in> dynkin \<Omega> M" |
40859 | 1680 |
moreover |
47694 | 1681 |
{ fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D" |
1682 |
then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) } |
|
1683 |
moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
1684 |
using assms dynkin_system_trivial by fastforce |
47694 | 1685 |
ultimately show "A \<subseteq> \<Omega>" |
40859 | 1686 |
unfolding dynkin_def using assms |
47694 | 1687 |
by auto |
40859 | 1688 |
next |
47694 | 1689 |
show "\<Omega> \<in> dynkin \<Omega> M" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
1690 |
unfolding dynkin_def using dynkin_system.space by fastforce |
40859 | 1691 |
next |
47694 | 1692 |
fix A assume "A \<in> dynkin \<Omega> M" |
1693 |
then show "\<Omega> - A \<in> dynkin \<Omega> M" |
|
40859 | 1694 |
unfolding dynkin_def using dynkin_system.compl by force |
1695 |
next |
|
1696 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 1697 |
assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M" |
1698 |
show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def |
|
40859 | 1699 |
proof (simp, safe) |
47694 | 1700 |
fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D" |
1701 |
with A have "(\<Union>i. A i) \<in> D" |
|
40859 | 1702 |
by (intro dynkin_system.UN) (auto simp: dynkin_def) |
1703 |
then show "(\<Union>i. A i) \<in> D" by auto |
|
1704 |
qed |
|
1705 |
qed |
|
1706 |
||
47694 | 1707 |
lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M" |
40859 | 1708 |
unfolding dynkin_def by auto |
1709 |
||
1710 |
lemma (in dynkin_system) restricted_dynkin_system: |
|
47694 | 1711 |
assumes "D \<in> M" |
1712 |
shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}" |
|
40859 | 1713 |
proof (rule dynkin_systemI, simp_all) |
47694 | 1714 |
have "\<Omega> \<inter> D = D" |
1715 |
using `D \<in> M` sets_into_space by auto |
|
1716 |
then show "\<Omega> \<inter> D \<in> M" |
|
1717 |
using `D \<in> M` by auto |
|
40859 | 1718 |
next |
47694 | 1719 |
fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M" |
1720 |
moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)" |
|
40859 | 1721 |
by auto |
47694 | 1722 |
ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M" |
1723 |
using `D \<in> M` by (auto intro: diff) |
|
40859 | 1724 |
next |
1725 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 1726 |
assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}" |
1727 |
then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)" |
|
1728 |
"range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
1729 |
by ((fastforce simp: disjoint_family_on_def)+) |
47694 | 1730 |
then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M" |
40859 | 1731 |
by (auto simp del: UN_simps) |
1732 |
qed |
|
1733 |
||
1734 |
lemma (in dynkin_system) dynkin_subset: |
|
47694 | 1735 |
assumes "N \<subseteq> M" |
1736 |
shows "dynkin \<Omega> N \<subseteq> M" |
|
40859 | 1737 |
proof - |
47694 | 1738 |
have "dynkin_system \<Omega> M" by default |
1739 |
then have "dynkin_system \<Omega> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1740 |
using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp |
47694 | 1741 |
with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def) |
40859 | 1742 |
qed |
1743 |
||
1744 |
lemma sigma_eq_dynkin: |
|
47694 | 1745 |
assumes sets: "M \<subseteq> Pow \<Omega>" |
40859 | 1746 |
assumes "Int_stable M" |
47694 | 1747 |
shows "sigma_sets \<Omega> M = dynkin \<Omega> M" |
40859 | 1748 |
proof - |
47694 | 1749 |
have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)" |
40859 | 1750 |
using sigma_algebra_imp_dynkin_system |
47694 | 1751 |
unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto |
40859 | 1752 |
moreover |
47694 | 1753 |
interpret dynkin_system \<Omega> "dynkin \<Omega> M" |
40859 | 1754 |
using dynkin_system_dynkin[OF sets] . |
47694 | 1755 |
have "sigma_algebra \<Omega> (dynkin \<Omega> M)" |
40859 | 1756 |
unfolding sigma_algebra_eq_Int_stable Int_stable_def |
1757 |
proof (intro ballI) |
|
47694 | 1758 |
fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M" |
1759 |
let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}" |
|
1760 |
have "M \<subseteq> ?D B" |
|
40859 | 1761 |
proof |
47694 | 1762 |
fix E assume "E \<in> M" |
1763 |
then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M" |
|
40859 | 1764 |
using sets_into_space `Int_stable M` by (auto simp: Int_stable_def) |
47694 | 1765 |
then have "dynkin \<Omega> M \<subseteq> ?D E" |
1766 |
using restricted_dynkin_system `E \<in> dynkin \<Omega> M` |
|
40859 | 1767 |
by (intro dynkin_system.dynkin_subset) simp_all |
47694 | 1768 |
then have "B \<in> ?D E" |
1769 |
using `B \<in> dynkin \<Omega> M` by auto |
|
1770 |
then have "E \<inter> B \<in> dynkin \<Omega> M" |
|
40859 | 1771 |
by (subst Int_commute) simp |
47694 | 1772 |
then show "E \<in> ?D B" |
1773 |
using sets `E \<in> M` by auto |
|
40859 | 1774 |
qed |
47694 | 1775 |
then have "dynkin \<Omega> M \<subseteq> ?D B" |
1776 |
using restricted_dynkin_system `B \<in> dynkin \<Omega> M` |
|
40859 | 1777 |
by (intro dynkin_system.dynkin_subset) simp_all |
47694 | 1778 |
then show "A \<inter> B \<in> dynkin \<Omega> M" |
1779 |
using `A \<in> dynkin \<Omega> M` sets_into_space by auto |
|
40859 | 1780 |
qed |
47694 | 1781 |
from sigma_algebra.sigma_sets_subset[OF this, of "M"] |
1782 |
have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto |
|
1783 |
ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto |
|
40859 | 1784 |
then show ?thesis |
47694 | 1785 |
by (auto simp: dynkin_def) |
40859 | 1786 |
qed |
1787 |
||
1788 |
lemma (in dynkin_system) dynkin_idem: |
|
47694 | 1789 |
"dynkin \<Omega> M = M" |
40859 | 1790 |
proof - |
47694 | 1791 |
have "dynkin \<Omega> M = M" |
40859 | 1792 |
proof |
47694 | 1793 |
show "M \<subseteq> dynkin \<Omega> M" |
40859 | 1794 |
using dynkin_Basic by auto |
47694 | 1795 |
show "dynkin \<Omega> M \<subseteq> M" |
40859 | 1796 |
by (intro dynkin_subset) auto |
1797 |
qed |
|
1798 |
then show ?thesis |
|
47694 | 1799 |
by (auto simp: dynkin_def) |
40859 | 1800 |
qed |
1801 |
||
1802 |
lemma (in dynkin_system) dynkin_lemma: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
1803 |
assumes "Int_stable E" |
47694 | 1804 |
and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E" |
1805 |
shows "sigma_sets \<Omega> E = M" |
|
40859 | 1806 |
proof - |
47694 | 1807 |
have "E \<subseteq> Pow \<Omega>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
1808 |
using E sets_into_space by force |
47694 | 1809 |
then have "sigma_sets \<Omega> E = dynkin \<Omega> E" |
40859 | 1810 |
using `Int_stable E` by (rule sigma_eq_dynkin) |
47694 | 1811 |
moreover then have "dynkin \<Omega> E = M" |
1812 |
using assms dynkin_subset[OF E(1)] by simp |
|
40859 | 1813 |
ultimately show ?thesis |
47694 | 1814 |
using assms by (auto simp: dynkin_def) |
42864 | 1815 |
qed |
1816 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1817 |
end |