author | hoelzl |
Tue, 07 Oct 2014 14:02:24 +0200 | |
changeset 58608 | 5b7f0b5da884 |
parent 58606 | 9c66f7c541fb |
child 58876 | 1888e3cb8048 |
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
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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
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5 |
translated by Lawrence Paulson. |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
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6 |
*) |
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New theory Probability, which contains a development of measure theory
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parents:
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|
56994 | 8 |
header {* Describing measurable sets *} |
33271
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New theory Probability, which contains a development of measure theory
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9 |
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41413
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explicit file specifications -- avoid secondary load path;
wenzelm
parents:
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theory Sigma_Algebra |
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explicit file specifications -- avoid secondary load path;
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11 |
imports |
42145 | 12 |
Complex_Main |
50245
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based countable topological basis on Countable_Set
immler
parents:
50244
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"~~/src/HOL/Library/Countable_Set" |
41413
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"~~/src/HOL/Library/FuncSet" |
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15 |
"~~/src/HOL/Library/Indicator_Function" |
47694 | 16 |
"~~/src/HOL/Library/Extended_Real" |
41413
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17 |
begin |
33271
7be66dee1a5a
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18 |
|
7be66dee1a5a
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text {* Sigma algebras are an elementary concept in measure |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
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parents:
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theory. To measure --- that is to integrate --- functions, we first have |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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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:
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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:
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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:
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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:
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25 |
three very natural and desirable properties. *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
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parents:
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26 |
|
47762 | 27 |
subsection {* Families of sets *} |
33271
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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
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New theory Probability, which contains a development of measure theory
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parents:
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32 |
|
47694 | 33 |
lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>" |
33271
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New theory Probability, which contains a development of measure theory
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34 |
by (metis PowD contra_subsetD space_closed) |
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New theory Probability, which contains a development of measure theory
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parents:
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35 |
|
56994 | 36 |
subsubsection {* Semiring of sets *} |
47762 | 37 |
|
38 |
definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})" |
|
39 |
||
40 |
lemma disjointI: |
|
41 |
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A" |
|
42 |
unfolding disjoint_def by auto |
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43 |
||
44 |
lemma disjointD: |
|
45 |
"disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}" |
|
46 |
unfolding disjoint_def by auto |
|
47 |
||
48 |
lemma disjoint_empty[iff]: "disjoint {}" |
|
49 |
by (auto simp: disjoint_def) |
|
42065
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hoelzl
parents:
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50 |
|
47762 | 51 |
lemma disjoint_union: |
52 |
assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}" |
|
53 |
shows "disjoint (C \<union> B)" |
|
54 |
proof (rule disjointI) |
|
55 |
fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d" |
|
56 |
show "c \<inter> d = {}" |
|
57 |
proof cases |
|
58 |
assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)" |
|
59 |
then show ?thesis |
|
60 |
proof |
|
61 |
assume "c \<in> C \<and> d \<in> C" with `c \<noteq> d` C show "c \<inter> d = {}" |
|
62 |
by (auto simp: disjoint_def) |
|
63 |
next |
|
64 |
assume "c \<in> B \<and> d \<in> B" with `c \<noteq> d` B show "c \<inter> d = {}" |
|
65 |
by (auto simp: disjoint_def) |
|
66 |
qed |
|
67 |
next |
|
68 |
assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))" |
|
69 |
with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)" |
|
70 |
by auto |
|
71 |
with disj show "c \<inter> d = {}" by auto |
|
72 |
qed |
|
73 |
qed |
|
74 |
||
53816 | 75 |
lemma disjoint_singleton [simp]: "disjoint {A}" |
76 |
by(simp add: disjoint_def) |
|
77 |
||
47762 | 78 |
locale semiring_of_sets = subset_class + |
79 |
assumes empty_sets[iff]: "{} \<in> M" |
|
80 |
assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M" |
|
81 |
assumes Diff_cover: |
|
82 |
"\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> \<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C" |
|
83 |
||
84 |
lemma (in semiring_of_sets) finite_INT[intro]: |
|
85 |
assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" |
|
86 |
shows "(\<Inter>i\<in>I. A i) \<in> M" |
|
87 |
using assms by (induct rule: finite_ne_induct) auto |
|
88 |
||
89 |
lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x" |
|
90 |
by (metis Int_absorb1 sets_into_space) |
|
91 |
||
92 |
lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x" |
|
93 |
by (metis Int_absorb2 sets_into_space) |
|
94 |
||
95 |
lemma (in semiring_of_sets) sets_Collect_conj: |
|
96 |
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M" |
|
97 |
shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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|
98 |
proof - |
47762 | 99 |
have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}" |
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100 |
by auto |
47762 | 101 |
with assms show ?thesis by auto |
33271
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102 |
qed |
7be66dee1a5a
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paulson
parents:
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103 |
|
50002
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add measurability prover; add support for Borel sets
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parents:
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104 |
lemma (in semiring_of_sets) sets_Collect_finite_All': |
47762 | 105 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}" |
106 |
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M" |
|
107 |
proof - |
|
108 |
have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})" |
|
109 |
using `S \<noteq> {}` by auto |
|
110 |
with assms show ?thesis by auto |
|
111 |
qed |
|
112 |
||
113 |
locale ring_of_sets = semiring_of_sets + |
|
114 |
assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M" |
|
115 |
||
42065
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41983
diff
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116 |
lemma (in ring_of_sets) finite_Union [intro]: |
47694 | 117 |
"finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M" |
38656 | 118 |
by (induct set: finite) (auto simp add: Un) |
33271
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paulson
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119 |
|
42065
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120 |
lemma (in ring_of_sets) finite_UN[intro]: |
47694 | 121 |
assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" |
122 |
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
|
123 |
using assms by induct auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
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124 |
|
47762 | 125 |
lemma (in ring_of_sets) Diff [intro]: |
126 |
assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M" |
|
127 |
using Diff_cover[OF assms] by auto |
|
128 |
||
129 |
lemma ring_of_setsI: |
|
130 |
assumes space_closed: "M \<subseteq> Pow \<Omega>" |
|
131 |
assumes empty_sets[iff]: "{} \<in> M" |
|
132 |
assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M" |
|
133 |
assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M" |
|
134 |
shows "ring_of_sets \<Omega> M" |
|
135 |
proof |
|
136 |
fix a b assume ab: "a \<in> M" "b \<in> M" |
|
137 |
from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C" |
|
138 |
by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def) |
|
139 |
have "a \<inter> b = a - (a - b)" by auto |
|
140 |
also have "\<dots> \<in> M" using ab by auto |
|
141 |
finally show "a \<inter> b \<in> M" . |
|
142 |
qed fact+ |
|
143 |
||
144 |
lemma ring_of_sets_iff: "ring_of_sets \<Omega> M \<longleftrightarrow> M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)" |
|
145 |
proof |
|
146 |
assume "ring_of_sets \<Omega> M" |
|
147 |
then interpret ring_of_sets \<Omega> M . |
|
148 |
show "M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)" |
|
149 |
using space_closed by auto |
|
150 |
qed (auto intro!: ring_of_setsI) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
151 |
|
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152 |
lemma (in ring_of_sets) insert_in_sets: |
47694 | 153 |
assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M" |
38656 | 154 |
proof - |
47694 | 155 |
have "{x} \<union> A \<in> M" using assms by (rule Un) |
38656 | 156 |
thus ?thesis by auto |
157 |
qed |
|
158 |
||
42867 | 159 |
lemma (in ring_of_sets) sets_Collect_disj: |
47694 | 160 |
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M" |
161 |
shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M" |
|
42867 | 162 |
proof - |
47694 | 163 |
have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}" |
42867 | 164 |
by auto |
165 |
with assms show ?thesis by auto |
|
166 |
qed |
|
167 |
||
168 |
lemma (in ring_of_sets) sets_Collect_finite_Ex: |
|
47694 | 169 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" |
170 |
shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M" |
|
42867 | 171 |
proof - |
47694 | 172 |
have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})" |
42867 | 173 |
by auto |
174 |
with assms show ?thesis by auto |
|
175 |
qed |
|
176 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
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41983
diff
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177 |
locale algebra = ring_of_sets + |
47694 | 178 |
assumes top [iff]: "\<Omega> \<in> M" |
42065
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179 |
|
2b98b4c2e2f1
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|
180 |
lemma (in algebra) compl_sets [intro]: |
47694 | 181 |
"a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" |
42065
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diff
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|
182 |
by auto |
2b98b4c2e2f1
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parents:
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diff
changeset
|
183 |
|
2b98b4c2e2f1
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parents:
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diff
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|
184 |
lemma algebra_iff_Un: |
47694 | 185 |
"algebra \<Omega> M \<longleftrightarrow> |
186 |
M \<subseteq> Pow \<Omega> \<and> |
|
187 |
{} \<in> M \<and> |
|
188 |
(\<forall>a \<in> M. \<Omega> - a \<in> M) \<and> |
|
189 |
(\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un") |
|
42065
2b98b4c2e2f1
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hoelzl
parents:
41983
diff
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|
190 |
proof |
47694 | 191 |
assume "algebra \<Omega> M" |
192 |
then interpret algebra \<Omega> M . |
|
42065
2b98b4c2e2f1
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diff
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|
193 |
show ?Un using sets_into_space by auto |
2b98b4c2e2f1
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41983
diff
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|
194 |
next |
2b98b4c2e2f1
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parents:
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diff
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|
195 |
assume ?Un |
47762 | 196 |
then have "\<Omega> \<in> M" by auto |
197 |
interpret ring_of_sets \<Omega> M |
|
198 |
proof (rule ring_of_setsI) |
|
199 |
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
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41983
diff
changeset
|
200 |
using `?Un` by auto |
47694 | 201 |
fix a b assume a: "a \<in> M" and b: "b \<in> M" |
202 |
then show "a \<union> b \<in> M" using `?Un` by auto |
|
203 |
have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)" |
|
204 |
using \<Omega> a b by auto |
|
205 |
then show "a - b \<in> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
206 |
using a b `?Un` by auto |
2b98b4c2e2f1
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hoelzl
parents:
41983
diff
changeset
|
207 |
qed |
47762 | 208 |
show "algebra \<Omega> M" proof qed fact |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
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diff
changeset
|
209 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
210 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
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diff
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|
211 |
lemma algebra_iff_Int: |
47694 | 212 |
"algebra \<Omega> M \<longleftrightarrow> |
213 |
M \<subseteq> Pow \<Omega> & {} \<in> M & |
|
214 |
(\<forall>a \<in> M. \<Omega> - a \<in> M) & |
|
215 |
(\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int") |
|
42065
2b98b4c2e2f1
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hoelzl
parents:
41983
diff
changeset
|
216 |
proof |
47694 | 217 |
assume "algebra \<Omega> M" |
218 |
then interpret algebra \<Omega> M . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
219 |
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
|
220 |
next |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
221 |
assume ?Int |
47694 | 222 |
show "algebra \<Omega> M" |
42065
2b98b4c2e2f1
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hoelzl
parents:
41983
diff
changeset
|
223 |
proof (unfold algebra_iff_Un, intro conjI ballI) |
47694 | 224 |
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
|
225 |
using `?Int` by auto |
47694 | 226 |
from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto |
227 |
fix a b assume M: "a \<in> M" "b \<in> M" |
|
228 |
hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))" |
|
229 |
using \<Omega> by blast |
|
230 |
also have "... \<in> M" |
|
231 |
using M `?Int` by auto |
|
232 |
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
|
233 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
234 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
235 |
|
42867 | 236 |
lemma (in algebra) sets_Collect_neg: |
47694 | 237 |
assumes "{x\<in>\<Omega>. P x} \<in> M" |
238 |
shows "{x\<in>\<Omega>. \<not> P x} \<in> M" |
|
42867 | 239 |
proof - |
47694 | 240 |
have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto |
42867 | 241 |
with assms show ?thesis by auto |
242 |
qed |
|
243 |
||
244 |
lemma (in algebra) sets_Collect_imp: |
|
47694 | 245 |
"{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 | 246 |
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg) |
247 |
||
248 |
lemma (in algebra) sets_Collect_const: |
|
47694 | 249 |
"{x\<in>\<Omega>. P} \<in> M" |
42867 | 250 |
by (cases P) auto |
251 |
||
42984 | 252 |
lemma algebra_single_set: |
47762 | 253 |
"X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }" |
254 |
by (auto simp: algebra_iff_Int) |
|
42984 | 255 |
|
56994 | 256 |
subsubsection {* Restricted algebras *} |
39092 | 257 |
|
258 |
abbreviation (in algebra) |
|
47694 | 259 |
"restricted_space A \<equiv> (op \<inter> A) ` M" |
39092 | 260 |
|
38656 | 261 |
lemma (in algebra) restricted_algebra: |
47694 | 262 |
assumes "A \<in> M" shows "algebra A (restricted_space A)" |
47762 | 263 |
using assms by (auto simp: algebra_iff_Int) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
264 |
|
56994 | 265 |
subsubsection {* Sigma Algebras *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
266 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
267 |
locale sigma_algebra = algebra + |
47694 | 268 |
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
|
269 |
|
42984 | 270 |
lemma (in algebra) is_sigma_algebra: |
47694 | 271 |
assumes "finite M" |
272 |
shows "sigma_algebra \<Omega> M" |
|
42984 | 273 |
proof |
47694 | 274 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M" |
275 |
then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)" |
|
42984 | 276 |
by auto |
47694 | 277 |
also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M" |
278 |
using `finite M` by auto |
|
279 |
finally show "(\<Union>i. A i) \<in> M" . |
|
42984 | 280 |
qed |
281 |
||
38656 | 282 |
lemma countable_UN_eq: |
283 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
|
47694 | 284 |
shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow> |
285 |
(range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)" |
|
38656 | 286 |
proof - |
287 |
let ?A' = "A \<circ> from_nat" |
|
288 |
have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r") |
|
289 |
proof safe |
|
290 |
fix x i assume "x \<in> A i" thus "x \<in> ?l" |
|
291 |
by (auto intro!: exI[of _ "to_nat i"]) |
|
292 |
next |
|
293 |
fix x i assume "x \<in> ?A' i" thus "x \<in> ?r" |
|
294 |
by (auto intro!: exI[of _ "from_nat i"]) |
|
295 |
qed |
|
296 |
have **: "range ?A' = range A" |
|
40702 | 297 |
using surj_from_nat |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
54420
diff
changeset
|
298 |
by (auto simp: image_comp [symmetric] intro!: imageI) |
38656 | 299 |
show ?thesis unfolding * ** .. |
300 |
qed |
|
301 |
||
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
302 |
lemma (in sigma_algebra) countable_Union [intro]: |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
303 |
assumes "countable X" "X \<subseteq> M" shows "Union X \<in> M" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
304 |
proof cases |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
305 |
assume "X \<noteq> {}" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
306 |
hence "\<Union>X = (\<Union>n. from_nat_into X n)" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
307 |
using assms by (auto intro: from_nat_into) (metis from_nat_into_surj) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
308 |
also have "\<dots> \<in> M" using assms |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
309 |
by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
310 |
finally show ?thesis . |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
311 |
qed simp |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
312 |
|
38656 | 313 |
lemma (in sigma_algebra) countable_UN[intro]: |
314 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
|
47694 | 315 |
assumes "A`X \<subseteq> M" |
316 |
shows "(\<Union>x\<in>X. A x) \<in> M" |
|
38656 | 317 |
proof - |
46731 | 318 |
let ?A = "\<lambda>i. if i \<in> X then A i else {}" |
47694 | 319 |
from assms have "range ?A \<subseteq> M" by auto |
38656 | 320 |
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M] |
47694 | 321 |
have "(\<Union>x. ?A x) \<in> M" by auto |
38656 | 322 |
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm) |
323 |
ultimately show ?thesis by simp |
|
324 |
qed |
|
325 |
||
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
326 |
lemma (in sigma_algebra) countable_UN': |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
327 |
fixes A :: "'i \<Rightarrow> 'a set" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
328 |
assumes X: "countable X" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
329 |
assumes A: "A`X \<subseteq> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
330 |
shows "(\<Union>x\<in>X. A x) \<in> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
331 |
proof - |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
332 |
have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
333 |
using X by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
334 |
also have "\<dots> \<in> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
335 |
using A X |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
336 |
by (intro countable_UN) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
337 |
finally show ?thesis . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
338 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
339 |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33271
diff
changeset
|
340 |
lemma (in sigma_algebra) countable_INT [intro]: |
38656 | 341 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
47694 | 342 |
assumes A: "A`X \<subseteq> M" "X \<noteq> {}" |
343 |
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
|
344 |
proof - |
47694 | 345 |
from A have "\<forall>i\<in>X. A i \<in> M" by fast |
346 |
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
|
347 |
moreover |
47694 | 348 |
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
|
349 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
350 |
ultimately show ?thesis by metis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
351 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
352 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
353 |
lemma (in sigma_algebra) countable_INT': |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
354 |
fixes A :: "'i \<Rightarrow> 'a set" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
355 |
assumes X: "countable X" "X \<noteq> {}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
356 |
assumes A: "A`X \<subseteq> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
357 |
shows "(\<Inter>x\<in>X. A x) \<in> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
358 |
proof - |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
359 |
have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
360 |
using X by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
361 |
also have "\<dots> \<in> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
362 |
using A X |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
363 |
by (intro countable_INT) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
364 |
finally show ?thesis . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
365 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
366 |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
367 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
368 |
lemma (in sigma_algebra) countable: |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
369 |
assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> M" "countable A" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
370 |
shows "A \<in> M" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
371 |
proof - |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
372 |
have "(\<Union>a\<in>A. {a}) \<in> M" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
373 |
using assms by (intro countable_UN') auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
374 |
also have "(\<Union>a\<in>A. {a}) = A" by auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
375 |
finally show ?thesis by auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
376 |
qed |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
377 |
|
47694 | 378 |
lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)" |
47762 | 379 |
by (auto simp: ring_of_sets_iff) |
42145 | 380 |
|
47694 | 381 |
lemma algebra_Pow: "algebra sp (Pow sp)" |
47762 | 382 |
by (auto simp: algebra_iff_Un) |
38656 | 383 |
|
384 |
lemma sigma_algebra_iff: |
|
47694 | 385 |
"sigma_algebra \<Omega> M \<longleftrightarrow> |
386 |
algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 387 |
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
|
388 |
|
47762 | 389 |
lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)" |
390 |
by (auto simp: sigma_algebra_iff algebra_iff_Int) |
|
391 |
||
42867 | 392 |
lemma (in sigma_algebra) sets_Collect_countable_All: |
47694 | 393 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
394 |
shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M" |
|
42867 | 395 |
proof - |
47694 | 396 |
have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto |
42867 | 397 |
with assms show ?thesis by auto |
398 |
qed |
|
399 |
||
400 |
lemma (in sigma_algebra) sets_Collect_countable_Ex: |
|
47694 | 401 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
402 |
shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M" |
|
42867 | 403 |
proof - |
47694 | 404 |
have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto |
42867 | 405 |
with assms show ?thesis by auto |
406 |
qed |
|
407 |
||
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
408 |
lemma (in sigma_algebra) sets_Collect_countable_Ex': |
54418 | 409 |
assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
410 |
assumes "countable I" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
411 |
shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
412 |
proof - |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
413 |
have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
414 |
with assms show ?thesis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
415 |
by (auto intro!: countable_UN') |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
416 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50387
diff
changeset
|
417 |
|
54418 | 418 |
lemma (in sigma_algebra) sets_Collect_countable_All': |
419 |
assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" |
|
420 |
assumes "countable I" |
|
421 |
shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M" |
|
422 |
proof - |
|
423 |
have "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} = (\<Inter>i\<in>I. {x\<in>\<Omega>. P i x}) \<inter> \<Omega>" by auto |
|
424 |
with assms show ?thesis |
|
425 |
by (cases "I = {}") (auto intro!: countable_INT') |
|
426 |
qed |
|
427 |
||
428 |
lemma (in sigma_algebra) sets_Collect_countable_Ex1': |
|
429 |
assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" |
|
430 |
assumes "countable I" |
|
431 |
shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M" |
|
432 |
proof - |
|
433 |
have "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} = {x\<in>\<Omega>. \<exists>i\<in>I. P i x \<and> (\<forall>j\<in>I. P j x \<longrightarrow> i = j)}" |
|
434 |
by auto |
|
435 |
with assms show ?thesis |
|
436 |
by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const) |
|
437 |
qed |
|
438 |
||
42867 | 439 |
lemmas (in sigma_algebra) sets_Collect = |
440 |
sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const |
|
441 |
sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All |
|
442 |
||
47694 | 443 |
lemma (in sigma_algebra) sets_Collect_countable_Ball: |
444 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
|
445 |
shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M" |
|
446 |
unfolding Ball_def by (intro sets_Collect assms) |
|
447 |
||
448 |
lemma (in sigma_algebra) sets_Collect_countable_Bex: |
|
449 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
|
450 |
shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M" |
|
451 |
unfolding Bex_def by (intro sets_Collect assms) |
|
452 |
||
42984 | 453 |
lemma sigma_algebra_single_set: |
454 |
assumes "X \<subseteq> S" |
|
47694 | 455 |
shows "sigma_algebra S { {}, X, S - X, S }" |
42984 | 456 |
using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp |
457 |
||
56994 | 458 |
subsubsection {* Binary Unions *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
459 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
460 |
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a" |
50252 | 461 |
where "binary a b = (\<lambda>x. b)(0 := a)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
462 |
|
38656 | 463 |
lemma range_binary_eq: "range(binary a b) = {a,b}" |
464 |
by (auto simp add: binary_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
465 |
|
38656 | 466 |
lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)" |
44106 | 467 |
by (simp add: SUP_def range_binary_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
468 |
|
38656 | 469 |
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)" |
44106 | 470 |
by (simp add: INF_def range_binary_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
471 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
472 |
lemma sigma_algebra_iff2: |
47694 | 473 |
"sigma_algebra \<Omega> M \<longleftrightarrow> |
474 |
M \<subseteq> Pow \<Omega> \<and> |
|
475 |
{} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and> |
|
476 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 477 |
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
|
478 |
algebra_iff_Un Un_range_binary) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
479 |
|
56994 | 480 |
subsubsection {* Initial Sigma Algebra *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
481 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
482 |
text {*Sigma algebras can naturally be created as the closure of any set of |
47694 | 483 |
M with regard to the properties just postulated. *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
484 |
|
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
485 |
inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
486 |
for sp :: "'a set" and A :: "'a set set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
487 |
where |
47694 | 488 |
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
|
489 |
| Empty: "{} \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
490 |
| 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
|
491 |
| 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
|
492 |
|
41543 | 493 |
lemma (in sigma_algebra) sigma_sets_subset: |
47694 | 494 |
assumes a: "a \<subseteq> M" |
495 |
shows "sigma_sets \<Omega> a \<subseteq> M" |
|
41543 | 496 |
proof |
497 |
fix x |
|
47694 | 498 |
assume "x \<in> sigma_sets \<Omega> a" |
499 |
from this show "x \<in> M" |
|
41543 | 500 |
by (induct rule: sigma_sets.induct, auto) (metis a subsetD) |
501 |
qed |
|
502 |
||
503 |
lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp" |
|
504 |
by (erule sigma_sets.induct, auto) |
|
505 |
||
506 |
lemma sigma_algebra_sigma_sets: |
|
47694 | 507 |
"a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)" |
41543 | 508 |
by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp |
509 |
intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl) |
|
510 |
||
511 |
lemma sigma_sets_least_sigma_algebra: |
|
512 |
assumes "A \<subseteq> Pow S" |
|
47694 | 513 |
shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}" |
41543 | 514 |
proof safe |
47694 | 515 |
fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B" |
41543 | 516 |
and X: "X \<in> sigma_sets S A" |
517 |
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X |
|
518 |
show "X \<in> B" by auto |
|
519 |
next |
|
47694 | 520 |
fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}" |
521 |
then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B" |
|
41543 | 522 |
by simp |
47694 | 523 |
have "A \<subseteq> sigma_sets S A" using assms by auto |
524 |
moreover have "sigma_algebra S (sigma_sets S A)" |
|
41543 | 525 |
using assms by (intro sigma_algebra_sigma_sets[of A]) auto |
526 |
ultimately show "X \<in> sigma_sets S A" by auto |
|
527 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
528 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
529 |
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
|
530 |
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
|
531 |
|
38656 | 532 |
lemma sigma_sets_Un: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
533 |
"a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A" |
38656 | 534 |
apply (simp add: Un_range_binary range_binary_eq) |
40859 | 535 |
apply (rule Union, simp add: binary_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
536 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
537 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
538 |
lemma sigma_sets_Inter: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
539 |
assumes Asb: "A \<subseteq> Pow sp" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
540 |
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
|
541 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
542 |
assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A" |
38656 | 543 |
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
|
544 |
by (rule sigma_sets.Compl) |
38656 | 545 |
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
|
546 |
by (rule sigma_sets.Union) |
38656 | 547 |
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
|
548 |
by (rule sigma_sets.Compl) |
38656 | 549 |
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
|
550 |
by auto |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
551 |
also have "... = (\<Inter>i. a i)" using ai |
38656 | 552 |
by (blast dest: sigma_sets_into_sp [OF Asb]) |
553 |
finally show ?thesis . |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
554 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
555 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
556 |
lemma sigma_sets_INTER: |
38656 | 557 |
assumes Asb: "A \<subseteq> Pow sp" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
558 |
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
|
559 |
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
|
560 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
561 |
from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A" |
47694 | 562 |
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
|
563 |
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
|
564 |
by (rule sigma_sets_Inter [OF Asb]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
565 |
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
|
566 |
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
|
567 |
finally show ?thesis . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
568 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
569 |
|
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
570 |
lemma sigma_sets_UNION: "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
571 |
using from_nat_into[of B] range_from_nat_into[of B] sigma_sets.Union[of "from_nat_into B" X A] |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
572 |
apply (cases "B = {}") |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
573 |
apply (simp add: sigma_sets.Empty) |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
574 |
apply (simp del: Union_image_eq add: Union_image_eq[symmetric]) |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
575 |
done |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
50526
diff
changeset
|
576 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
577 |
lemma (in sigma_algebra) sigma_sets_eq: |
47694 | 578 |
"sigma_sets \<Omega> M = M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
579 |
proof |
47694 | 580 |
show "M \<subseteq> sigma_sets \<Omega> M" |
37032 | 581 |
by (metis Set.subsetI sigma_sets.Basic) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
582 |
next |
47694 | 583 |
show "sigma_sets \<Omega> M \<subseteq> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
584 |
by (metis sigma_sets_subset subset_refl) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
585 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
586 |
|
42981 | 587 |
lemma sigma_sets_eqI: |
588 |
assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B" |
|
589 |
assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A" |
|
590 |
shows "sigma_sets M A = sigma_sets M B" |
|
591 |
proof (intro set_eqI iffI) |
|
592 |
fix a assume "a \<in> sigma_sets M A" |
|
593 |
from this A show "a \<in> sigma_sets M B" |
|
47694 | 594 |
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) |
42981 | 595 |
next |
596 |
fix b assume "b \<in> sigma_sets M B" |
|
597 |
from this B show "b \<in> sigma_sets M A" |
|
47694 | 598 |
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) |
42981 | 599 |
qed |
600 |
||
42984 | 601 |
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
602 |
proof |
|
603 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
47694 | 604 |
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-)) |
42984 | 605 |
qed |
606 |
||
47762 | 607 |
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
608 |
proof |
|
609 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
610 |
by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-)) |
|
611 |
qed |
|
612 |
||
613 |
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
|
614 |
proof |
|
615 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
616 |
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-)) |
|
617 |
qed |
|
618 |
||
619 |
lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A" |
|
620 |
by (auto intro: sigma_sets.Basic) |
|
621 |
||
38656 | 622 |
lemma (in sigma_algebra) restriction_in_sets: |
623 |
fixes A :: "nat \<Rightarrow> 'a set" |
|
47694 | 624 |
assumes "S \<in> M" |
625 |
and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r") |
|
626 |
shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M" |
|
38656 | 627 |
proof - |
628 |
{ fix i have "A i \<in> ?r" using * by auto |
|
47694 | 629 |
hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto |
630 |
hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto } |
|
631 |
thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M" |
|
38656 | 632 |
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"]) |
633 |
qed |
|
634 |
||
635 |
lemma (in sigma_algebra) restricted_sigma_algebra: |
|
47694 | 636 |
assumes "S \<in> M" |
637 |
shows "sigma_algebra S (restricted_space S)" |
|
38656 | 638 |
unfolding sigma_algebra_def sigma_algebra_axioms_def |
639 |
proof safe |
|
47694 | 640 |
show "algebra S (restricted_space S)" using restricted_algebra[OF assms] . |
38656 | 641 |
next |
47694 | 642 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S" |
38656 | 643 |
from restriction_in_sets[OF assms this[simplified]] |
47694 | 644 |
show "(\<Union>i. A i) \<in> restricted_space S" by simp |
38656 | 645 |
qed |
646 |
||
40859 | 647 |
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
|
648 |
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
|
649 |
shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)" |
40859 | 650 |
proof (intro equalityI subsetI) |
651 |
fix x assume "x \<in> op \<inter> A ` sigma_sets sp st" |
|
652 |
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
|
653 |
then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)" |
40859 | 654 |
proof (induct arbitrary: x) |
655 |
case (Compl a) |
|
656 |
then show ?case |
|
657 |
by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps) |
|
658 |
next |
|
659 |
case (Union a) |
|
660 |
then show ?case |
|
661 |
by (auto intro!: sigma_sets.Union |
|
662 |
simp add: UN_extend_simps simp del: UN_simps) |
|
47694 | 663 |
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
|
664 |
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
|
665 |
using `A \<subseteq> sp` by (simp add: Int_absorb2) |
40859 | 666 |
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
|
667 |
fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)" |
40859 | 668 |
then show "x \<in> op \<inter> A ` sigma_sets sp st" |
669 |
proof induct |
|
670 |
case (Compl a) |
|
671 |
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
|
672 |
then show ?case using `A \<subseteq> sp` |
40859 | 673 |
by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl) |
674 |
next |
|
675 |
case (Union a) |
|
676 |
then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x" |
|
677 |
by (auto simp: image_iff Bex_def) |
|
678 |
from choice[OF this] guess f .. |
|
679 |
then show ?case |
|
680 |
by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union |
|
681 |
simp add: image_iff) |
|
47694 | 682 |
qed (auto intro!: sigma_sets.intros(2-)) |
40859 | 683 |
qed |
684 |
||
47694 | 685 |
lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}" |
40859 | 686 |
proof (intro set_eqI iffI) |
47694 | 687 |
fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}" |
688 |
by induct blast+ |
|
689 |
qed (auto intro: sigma_sets.Empty sigma_sets_top) |
|
690 |
||
691 |
lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}" |
|
692 |
proof (intro set_eqI iffI) |
|
693 |
fix x assume "x \<in> sigma_sets A {A}" |
|
694 |
then show "x \<in> {{}, A}" |
|
695 |
by induct blast+ |
|
40859 | 696 |
next |
47694 | 697 |
fix x assume "x \<in> {{}, A}" |
698 |
then show "x \<in> sigma_sets A {A}" |
|
40859 | 699 |
by (auto intro: sigma_sets.Empty sigma_sets_top) |
700 |
qed |
|
701 |
||
42987 | 702 |
lemma sigma_sets_sigma_sets_eq: |
703 |
"M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M" |
|
47694 | 704 |
by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto |
42987 | 705 |
|
42984 | 706 |
lemma sigma_sets_singleton: |
707 |
assumes "X \<subseteq> S" |
|
708 |
shows "sigma_sets S { X } = { {}, X, S - X, S }" |
|
709 |
proof - |
|
47694 | 710 |
interpret sigma_algebra S "{ {}, X, S - X, S }" |
42984 | 711 |
by (rule sigma_algebra_single_set) fact |
712 |
have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }" |
|
713 |
by (rule sigma_sets_subseteq) simp |
|
714 |
moreover have "\<dots> = { {}, X, S - X, S }" |
|
47694 | 715 |
using sigma_sets_eq by simp |
42984 | 716 |
moreover |
717 |
{ fix A assume "A \<in> { {}, X, S - X, S }" |
|
718 |
then have "A \<in> sigma_sets S { X }" |
|
47694 | 719 |
by (auto intro: sigma_sets.intros(2-) sigma_sets_top) } |
42984 | 720 |
ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }" |
721 |
by (intro antisym) auto |
|
47694 | 722 |
with sigma_sets_eq show ?thesis by simp |
42984 | 723 |
qed |
724 |
||
42863 | 725 |
lemma restricted_sigma: |
47694 | 726 |
assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>" |
727 |
shows "algebra.restricted_space (sigma_sets \<Omega> M) S = |
|
728 |
sigma_sets S (algebra.restricted_space M S)" |
|
42863 | 729 |
proof - |
730 |
from S sigma_sets_into_sp[OF M] |
|
47694 | 731 |
have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto |
42863 | 732 |
from sigma_sets_Int[OF this] |
47694 | 733 |
show ?thesis by simp |
42863 | 734 |
qed |
735 |
||
42987 | 736 |
lemma sigma_sets_vimage_commute: |
47694 | 737 |
assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'" |
738 |
shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'} |
|
739 |
= sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R") |
|
42987 | 740 |
proof |
741 |
show "?L \<subseteq> ?R" |
|
742 |
proof clarify |
|
47694 | 743 |
fix A assume "A \<in> sigma_sets \<Omega>' M'" |
744 |
then show "X -` A \<inter> \<Omega> \<in> ?R" |
|
42987 | 745 |
proof induct |
746 |
case Empty then show ?case |
|
747 |
by (auto intro!: sigma_sets.Empty) |
|
748 |
next |
|
749 |
case (Compl B) |
|
47694 | 750 |
have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)" |
42987 | 751 |
by (auto simp add: funcset_mem [OF X]) |
752 |
with Compl show ?case |
|
753 |
by (auto intro!: sigma_sets.Compl) |
|
754 |
next |
|
755 |
case (Union F) |
|
756 |
then show ?case |
|
757 |
by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps |
|
758 |
intro!: sigma_sets.Union) |
|
47694 | 759 |
qed auto |
42987 | 760 |
qed |
761 |
show "?R \<subseteq> ?L" |
|
762 |
proof clarify |
|
763 |
fix A assume "A \<in> ?R" |
|
47694 | 764 |
then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" |
42987 | 765 |
proof induct |
766 |
case (Basic B) then show ?case by auto |
|
767 |
next |
|
768 |
case Empty then show ?case |
|
47694 | 769 |
by (auto intro!: sigma_sets.Empty exI[of _ "{}"]) |
42987 | 770 |
next |
771 |
case (Compl B) |
|
47694 | 772 |
then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto |
773 |
then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>" |
|
42987 | 774 |
by (auto simp add: funcset_mem [OF X]) |
775 |
with A(2) show ?case |
|
47694 | 776 |
by (auto intro: sigma_sets.Compl) |
42987 | 777 |
next |
778 |
case (Union F) |
|
47694 | 779 |
then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto |
42987 | 780 |
from choice[OF this] guess A .. note A = this |
781 |
with A show ?case |
|
47694 | 782 |
by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union) |
42987 | 783 |
qed |
784 |
qed |
|
785 |
qed |
|
786 |
||
56994 | 787 |
subsubsection "Disjoint families" |
38656 | 788 |
|
789 |
definition |
|
790 |
disjoint_family_on where |
|
791 |
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})" |
|
792 |
||
793 |
abbreviation |
|
794 |
"disjoint_family A \<equiv> disjoint_family_on A UNIV" |
|
795 |
||
796 |
lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B" |
|
797 |
by blast |
|
798 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
799 |
lemma disjoint_family_onD: "disjoint_family_on A I \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
800 |
by (auto simp: disjoint_family_on_def) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
801 |
|
38656 | 802 |
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}" |
803 |
by blast |
|
804 |
||
805 |
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A" |
|
806 |
by blast |
|
807 |
||
808 |
lemma disjoint_family_subset: |
|
809 |
"disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B" |
|
810 |
by (force simp add: disjoint_family_on_def) |
|
811 |
||
40859 | 812 |
lemma disjoint_family_on_bisimulation: |
813 |
assumes "disjoint_family_on f S" |
|
814 |
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 = {}" |
|
815 |
shows "disjoint_family_on g S" |
|
816 |
using assms unfolding disjoint_family_on_def by auto |
|
817 |
||
38656 | 818 |
lemma disjoint_family_on_mono: |
819 |
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A" |
|
820 |
unfolding disjoint_family_on_def by auto |
|
821 |
||
822 |
lemma disjoint_family_Suc: |
|
823 |
assumes Suc: "!!n. A n \<subseteq> A (Suc n)" |
|
824 |
shows "disjoint_family (\<lambda>i. A (Suc i) - A i)" |
|
825 |
proof - |
|
826 |
{ |
|
827 |
fix m |
|
828 |
have "!!n. A n \<subseteq> A (m+n)" |
|
829 |
proof (induct m) |
|
830 |
case 0 show ?case by simp |
|
831 |
next |
|
832 |
case (Suc m) thus ?case |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
833 |
by (metis Suc_eq_plus1 assms add.commute add.left_commute subset_trans) |
38656 | 834 |
qed |
835 |
} |
|
836 |
hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
837 |
by (metis add.commute le_add_diff_inverse nat_less_le) |
38656 | 838 |
thus ?thesis |
839 |
by (auto simp add: disjoint_family_on_def) |
|
840 |
(metis insert_absorb insert_subset le_SucE le_antisym not_leE) |
|
841 |
qed |
|
842 |
||
39092 | 843 |
lemma setsum_indicator_disjoint_family: |
844 |
fixes f :: "'d \<Rightarrow> 'e::semiring_1" |
|
845 |
assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P" |
|
846 |
shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j" |
|
847 |
proof - |
|
848 |
have "P \<inter> {i. x \<in> A i} = {j}" |
|
849 |
using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def |
|
850 |
by auto |
|
851 |
thus ?thesis |
|
852 |
unfolding indicator_def |
|
57418 | 853 |
by (simp add: if_distrib setsum.If_cases[OF `finite P`]) |
39092 | 854 |
qed |
855 |
||
38656 | 856 |
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set " |
857 |
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)" |
|
858 |
||
859 |
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)" |
|
860 |
proof (induct n) |
|
861 |
case 0 show ?case by simp |
|
862 |
next |
|
863 |
case (Suc n) |
|
864 |
thus ?case by (simp add: atLeastLessThanSuc disjointed_def) |
|
865 |
qed |
|
866 |
||
867 |
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)" |
|
868 |
apply (rule UN_finite2_eq [where k=0]) |
|
869 |
apply (simp add: finite_UN_disjointed_eq) |
|
870 |
done |
|
871 |
||
872 |
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}" |
|
873 |
by (auto simp add: disjointed_def) |
|
874 |
||
875 |
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)" |
|
876 |
by (simp add: disjoint_family_on_def) |
|
877 |
(metis neq_iff Int_commute less_disjoint_disjointed) |
|
878 |
||
879 |
lemma disjointed_subset: "disjointed A n \<subseteq> A n" |
|
880 |
by (auto simp add: disjointed_def) |
|
881 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
882 |
lemma (in ring_of_sets) UNION_in_sets: |
38656 | 883 |
fixes A:: "nat \<Rightarrow> 'a set" |
47694 | 884 |
assumes A: "range A \<subseteq> M" |
885 |
shows "(\<Union>i\<in>{0..<n}. A i) \<in> M" |
|
38656 | 886 |
proof (induct n) |
887 |
case 0 show ?case by simp |
|
888 |
next |
|
889 |
case (Suc n) |
|
890 |
thus ?case |
|
891 |
by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff) |
|
892 |
qed |
|
893 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
894 |
lemma (in ring_of_sets) range_disjointed_sets: |
47694 | 895 |
assumes A: "range A \<subseteq> M" |
896 |
shows "range (disjointed A) \<subseteq> M" |
|
38656 | 897 |
proof (auto simp add: disjointed_def) |
898 |
fix n |
|
47694 | 899 |
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets |
38656 | 900 |
by (metis A Diff UNIV_I image_subset_iff) |
901 |
qed |
|
902 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
903 |
lemma (in algebra) range_disjointed_sets': |
47694 | 904 |
"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
|
905 |
using range_disjointed_sets . |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
906 |
|
42145 | 907 |
lemma disjointed_0[simp]: "disjointed A 0 = A 0" |
908 |
by (simp add: disjointed_def) |
|
909 |
||
910 |
lemma incseq_Un: |
|
911 |
"incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n" |
|
912 |
unfolding incseq_def by auto |
|
913 |
||
914 |
lemma disjointed_incseq: |
|
915 |
"incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n" |
|
916 |
using incseq_Un[of A] |
|
917 |
by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost) |
|
918 |
||
38656 | 919 |
lemma sigma_algebra_disjoint_iff: |
47694 | 920 |
"sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and> |
921 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 922 |
proof (auto simp add: sigma_algebra_iff) |
923 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 924 |
assume M: "algebra \<Omega> M" |
925 |
and A: "range A \<subseteq> M" |
|
926 |
and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
927 |
hence "range (disjointed A) \<subseteq> M \<longrightarrow> |
|
38656 | 928 |
disjoint_family (disjointed A) \<longrightarrow> |
47694 | 929 |
(\<Union>i. disjointed A i) \<in> M" by blast |
930 |
hence "(\<Union>i. disjointed A i) \<in> M" |
|
931 |
by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed) |
|
932 |
thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq) |
|
933 |
qed |
|
934 |
||
47762 | 935 |
lemma disjoint_family_on_disjoint_image: |
936 |
"disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)" |
|
937 |
unfolding disjoint_family_on_def disjoint_def by force |
|
938 |
||
939 |
lemma disjoint_image_disjoint_family_on: |
|
940 |
assumes d: "disjoint (A ` I)" and i: "inj_on A I" |
|
941 |
shows "disjoint_family_on A I" |
|
942 |
unfolding disjoint_family_on_def |
|
943 |
proof (intro ballI impI) |
|
944 |
fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m" |
|
945 |
with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}" |
|
946 |
by (intro disjointD[OF d]) auto |
|
947 |
qed |
|
948 |
||
56994 | 949 |
subsubsection {* Ring generated by a semiring *} |
47762 | 950 |
|
951 |
definition (in semiring_of_sets) |
|
952 |
"generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }" |
|
953 |
||
954 |
lemma (in semiring_of_sets) generated_ringE[elim?]: |
|
955 |
assumes "a \<in> generated_ring" |
|
956 |
obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C" |
|
957 |
using assms unfolding generated_ring_def by auto |
|
958 |
||
959 |
lemma (in semiring_of_sets) generated_ringI[intro?]: |
|
960 |
assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C" |
|
961 |
shows "a \<in> generated_ring" |
|
962 |
using assms unfolding generated_ring_def by auto |
|
963 |
||
964 |
lemma (in semiring_of_sets) generated_ringI_Basic: |
|
965 |
"A \<in> M \<Longrightarrow> A \<in> generated_ring" |
|
966 |
by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def) |
|
967 |
||
968 |
lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]: |
|
969 |
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring" |
|
970 |
and "a \<inter> b = {}" |
|
971 |
shows "a \<union> b \<in> generated_ring" |
|
972 |
proof - |
|
973 |
from a guess Ca .. note Ca = this |
|
974 |
from b guess Cb .. note Cb = this |
|
975 |
show ?thesis |
|
976 |
proof |
|
977 |
show "disjoint (Ca \<union> Cb)" |
|
978 |
using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union) |
|
979 |
qed (insert Ca Cb, auto) |
|
980 |
qed |
|
981 |
||
982 |
lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring" |
|
983 |
by (auto simp: generated_ring_def disjoint_def) |
|
984 |
||
985 |
lemma (in semiring_of_sets) generated_ring_disjoint_Union: |
|
986 |
assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring" |
|
987 |
using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty) |
|
988 |
||
989 |
lemma (in semiring_of_sets) generated_ring_disjoint_UNION: |
|
990 |
"finite I \<Longrightarrow> disjoint (A ` I) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> UNION I A \<in> generated_ring" |
|
991 |
unfolding SUP_def by (intro generated_ring_disjoint_Union) auto |
|
992 |
||
993 |
lemma (in semiring_of_sets) generated_ring_Int: |
|
994 |
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring" |
|
995 |
shows "a \<inter> b \<in> generated_ring" |
|
996 |
proof - |
|
997 |
from a guess Ca .. note Ca = this |
|
998 |
from b guess Cb .. note Cb = this |
|
999 |
def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)" |
|
1000 |
show ?thesis |
|
1001 |
proof |
|
1002 |
show "disjoint C" |
|
1003 |
proof (simp add: disjoint_def C_def, intro ballI impI) |
|
1004 |
fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb" |
|
1005 |
assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2" |
|
1006 |
then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto |
|
1007 |
then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}" |
|
1008 |
proof |
|
1009 |
assume "a1 \<noteq> a2" |
|
1010 |
with sets Ca have "a1 \<inter> a2 = {}" |
|
1011 |
by (auto simp: disjoint_def) |
|
1012 |
then show ?thesis by auto |
|
1013 |
next |
|
1014 |
assume "b1 \<noteq> b2" |
|
1015 |
with sets Cb have "b1 \<inter> b2 = {}" |
|
1016 |
by (auto simp: disjoint_def) |
|
1017 |
then show ?thesis by auto |
|
1018 |
qed |
|
1019 |
qed |
|
1020 |
qed (insert Ca Cb, auto simp: C_def) |
|
1021 |
qed |
|
1022 |
||
1023 |
lemma (in semiring_of_sets) generated_ring_Inter: |
|
1024 |
assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring" |
|
1025 |
using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int) |
|
1026 |
||
1027 |
lemma (in semiring_of_sets) generated_ring_INTER: |
|
1028 |
"finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring" |
|
1029 |
unfolding INF_def by (intro generated_ring_Inter) auto |
|
1030 |
||
1031 |
lemma (in semiring_of_sets) generating_ring: |
|
1032 |
"ring_of_sets \<Omega> generated_ring" |
|
1033 |
proof (rule ring_of_setsI) |
|
1034 |
let ?R = generated_ring |
|
1035 |
show "?R \<subseteq> Pow \<Omega>" |
|
1036 |
using sets_into_space by (auto simp: generated_ring_def generated_ring_empty) |
|
1037 |
show "{} \<in> ?R" by (rule generated_ring_empty) |
|
1038 |
||
1039 |
{ fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this |
|
1040 |
fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this |
|
1041 |
||
1042 |
show "a - b \<in> ?R" |
|
1043 |
proof cases |
|
1044 |
assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis |
|
1045 |
by simp |
|
1046 |
next |
|
1047 |
assume "Cb \<noteq> {}" |
|
1048 |
with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto |
|
1049 |
also have "\<dots> \<in> ?R" |
|
1050 |
proof (intro generated_ring_INTER generated_ring_disjoint_UNION) |
|
1051 |
fix a b assume "a \<in> Ca" "b \<in> Cb" |
|
1052 |
with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R" |
|
1053 |
by (auto simp add: generated_ring_def) |
|
1054 |
next |
|
1055 |
show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)" |
|
1056 |
using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`) |
|
1057 |
next |
|
1058 |
show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+ |
|
1059 |
qed |
|
1060 |
finally show "a - b \<in> ?R" . |
|
1061 |
qed } |
|
1062 |
note Diff = this |
|
1063 |
||
1064 |
fix a b assume sets: "a \<in> ?R" "b \<in> ?R" |
|
1065 |
have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto |
|
1066 |
also have "\<dots> \<in> ?R" |
|
1067 |
by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto |
|
1068 |
finally show "a \<union> b \<in> ?R" . |
|
1069 |
qed |
|
1070 |
||
1071 |
lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M" |
|
1072 |
proof |
|
1073 |
interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M" |
|
1074 |
using space_closed by (rule sigma_algebra_sigma_sets) |
|
1075 |
show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M" |
|
1076 |
by (blast intro!: sigma_sets_mono elim: generated_ringE) |
|
1077 |
qed (auto intro!: generated_ringI_Basic sigma_sets_mono) |
|
1078 |
||
56994 | 1079 |
subsubsection {* A Two-Element Series *} |
38656 | 1080 |
|
1081 |
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set " |
|
50252 | 1082 |
where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)" |
38656 | 1083 |
|
1084 |
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}" |
|
1085 |
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
|
1086 |
apply (rule set_eqI) |
38656 | 1087 |
apply (auto simp add: image_iff) |
1088 |
done |
|
1089 |
||
1090 |
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B" |
|
44106 | 1091 |
by (simp add: SUP_def range_binaryset_eq) |
38656 | 1092 |
|
56994 | 1093 |
subsubsection {* Closed CDI *} |
38656 | 1094 |
|
47694 | 1095 |
definition closed_cdi where |
1096 |
"closed_cdi \<Omega> M \<longleftrightarrow> |
|
1097 |
M \<subseteq> Pow \<Omega> & |
|
1098 |
(\<forall>s \<in> M. \<Omega> - s \<in> M) & |
|
1099 |
(\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow> |
|
1100 |
(\<Union>i. A i) \<in> M) & |
|
1101 |
(\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 1102 |
|
1103 |
inductive_set |
|
47694 | 1104 |
smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" |
1105 |
for \<Omega> M |
|
38656 | 1106 |
where |
1107 |
Basic [intro]: |
|
47694 | 1108 |
"a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1109 |
| Compl [intro]: |
47694 | 1110 |
"a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1111 |
| Inc: |
47694 | 1112 |
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n)) |
1113 |
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1114 |
| Disj: |
47694 | 1115 |
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A |
1116 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1117 |
|
47694 | 1118 |
lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M" |
1119 |
by auto |
|
38656 | 1120 |
|
47694 | 1121 |
lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>" |
38656 | 1122 |
apply (rule subsetI) |
1123 |
apply (erule smallest_ccdi_sets.induct) |
|
1124 |
apply (auto intro: range_subsetD dest: sets_into_space) |
|
1125 |
done |
|
1126 |
||
47694 | 1127 |
lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)" |
1128 |
apply (auto simp add: closed_cdi_def smallest_ccdi_sets) |
|
38656 | 1129 |
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) + |
1130 |
done |
|
1131 |
||
47694 | 1132 |
lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>" |
38656 | 1133 |
by (simp add: closed_cdi_def) |
1134 |
||
47694 | 1135 |
lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M" |
38656 | 1136 |
by (simp add: closed_cdi_def) |
1137 |
||
1138 |
lemma closed_cdi_Inc: |
|
47694 | 1139 |
"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 | 1140 |
by (simp add: closed_cdi_def) |
1141 |
||
1142 |
lemma closed_cdi_Disj: |
|
47694 | 1143 |
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
38656 | 1144 |
by (simp add: closed_cdi_def) |
1145 |
||
1146 |
lemma closed_cdi_Un: |
|
47694 | 1147 |
assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M" |
1148 |
and A: "A \<in> M" and B: "B \<in> M" |
|
38656 | 1149 |
and disj: "A \<inter> B = {}" |
47694 | 1150 |
shows "A \<union> B \<in> M" |
38656 | 1151 |
proof - |
47694 | 1152 |
have ra: "range (binaryset A B) \<subseteq> M" |
38656 | 1153 |
by (simp add: range_binaryset_eq empty A B) |
1154 |
have di: "disjoint_family (binaryset A B)" using disj |
|
1155 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
|
1156 |
from closed_cdi_Disj [OF cdi ra di] |
|
1157 |
show ?thesis |
|
1158 |
by (simp add: UN_binaryset_eq) |
|
1159 |
qed |
|
1160 |
||
1161 |
lemma (in algebra) smallest_ccdi_sets_Un: |
|
47694 | 1162 |
assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1163 |
and disj: "A \<inter> B = {}" |
47694 | 1164 |
shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1165 |
proof - |
47694 | 1166 |
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)" |
38656 | 1167 |
by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic) |
1168 |
have di: "disjoint_family (binaryset A B)" using disj |
|
1169 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
|
1170 |
from Disj [OF ra di] |
|
1171 |
show ?thesis |
|
1172 |
by (simp add: UN_binaryset_eq) |
|
1173 |
qed |
|
1174 |
||
1175 |
lemma (in algebra) smallest_ccdi_sets_Int1: |
|
47694 | 1176 |
assumes a: "a \<in> M" |
1177 |
shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1178 |
proof (induct rule: smallest_ccdi_sets.induct) |
1179 |
case (Basic x) |
|
1180 |
thus ?case |
|
1181 |
by (metis a Int smallest_ccdi_sets.Basic) |
|
1182 |
next |
|
1183 |
case (Compl x) |
|
47694 | 1184 |
have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))" |
38656 | 1185 |
by blast |
47694 | 1186 |
also have "... \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1187 |
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2 |
47694 | 1188 |
Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un |
1189 |
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl) |
|
38656 | 1190 |
finally show ?case . |
1191 |
next |
|
1192 |
case (Inc A) |
|
1193 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
|
1194 |
by blast |
|
47694 | 1195 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc |
38656 | 1196 |
by blast |
1197 |
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}" |
|
1198 |
by (simp add: Inc) |
|
1199 |
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc |
|
1200 |
by blast |
|
47694 | 1201 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1202 |
by (rule smallest_ccdi_sets.Inc) |
1203 |
show ?case |
|
1204 |
by (metis 1 2) |
|
1205 |
next |
|
1206 |
case (Disj A) |
|
1207 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
|
1208 |
by blast |
|
47694 | 1209 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj |
38656 | 1210 |
by blast |
1211 |
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj |
|
1212 |
by (auto simp add: disjoint_family_on_def) |
|
47694 | 1213 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1214 |
by (rule smallest_ccdi_sets.Disj) |
1215 |
show ?case |
|
1216 |
by (metis 1 2) |
|
1217 |
qed |
|
1218 |
||
1219 |
||
1220 |
lemma (in algebra) smallest_ccdi_sets_Int: |
|
47694 | 1221 |
assumes b: "b \<in> smallest_ccdi_sets \<Omega> M" |
1222 |
shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 1223 |
proof (induct rule: smallest_ccdi_sets.induct) |
1224 |
case (Basic x) |
|
1225 |
thus ?case |
|
1226 |
by (metis b smallest_ccdi_sets_Int1) |
|
1227 |
next |
|
1228 |
case (Compl x) |
|
47694 | 1229 |
have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))" |
38656 | 1230 |
by blast |
47694 | 1231 |
also have "... \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1232 |
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b |
1233 |
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un) |
|
1234 |
finally show ?case . |
|
1235 |
next |
|
1236 |
case (Inc A) |
|
1237 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
|
1238 |
by blast |
|
47694 | 1239 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc |
38656 | 1240 |
by blast |
1241 |
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}" |
|
1242 |
by (simp add: Inc) |
|
1243 |
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc |
|
1244 |
by blast |
|
47694 | 1245 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1246 |
by (rule smallest_ccdi_sets.Inc) |
1247 |
show ?case |
|
1248 |
by (metis 1 2) |
|
1249 |
next |
|
1250 |
case (Disj A) |
|
1251 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
|
1252 |
by blast |
|
47694 | 1253 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj |
38656 | 1254 |
by blast |
1255 |
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj |
|
1256 |
by (auto simp add: disjoint_family_on_def) |
|
47694 | 1257 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1258 |
by (rule smallest_ccdi_sets.Disj) |
1259 |
show ?case |
|
1260 |
by (metis 1 2) |
|
1261 |
qed |
|
1262 |
||
1263 |
lemma (in algebra) sigma_property_disjoint_lemma: |
|
47694 | 1264 |
assumes sbC: "M \<subseteq> C" |
1265 |
and ccdi: "closed_cdi \<Omega> C" |
|
1266 |
shows "sigma_sets \<Omega> M \<subseteq> C" |
|
38656 | 1267 |
proof - |
47694 | 1268 |
have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}" |
38656 | 1269 |
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int |
1270 |
smallest_ccdi_sets_Int) |
|
1271 |
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets) |
|
1272 |
apply (blast intro: smallest_ccdi_sets.Disj) |
|
1273 |
done |
|
47694 | 1274 |
hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M" |
38656 | 1275 |
by clarsimp |
47694 | 1276 |
(drule sigma_algebra.sigma_sets_subset [where a="M"], auto) |
38656 | 1277 |
also have "... \<subseteq> C" |
1278 |
proof |
|
1279 |
fix x |
|
47694 | 1280 |
assume x: "x \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 1281 |
thus "x \<in> C" |
1282 |
proof (induct rule: smallest_ccdi_sets.induct) |
|
1283 |
case (Basic x) |
|
1284 |
thus ?case |
|
1285 |
by (metis Basic subsetD sbC) |
|
1286 |
next |
|
1287 |
case (Compl x) |
|
1288 |
thus ?case |
|
1289 |
by (blast intro: closed_cdi_Compl [OF ccdi, simplified]) |
|
1290 |
next |
|
1291 |
case (Inc A) |
|
1292 |
thus ?case |
|
1293 |
by (auto intro: closed_cdi_Inc [OF ccdi, simplified]) |
|
1294 |
next |
|
1295 |
case (Disj A) |
|
1296 |
thus ?case |
|
1297 |
by (auto intro: closed_cdi_Disj [OF ccdi, simplified]) |
|
1298 |
qed |
|
1299 |
qed |
|
1300 |
finally show ?thesis . |
|
1301 |
qed |
|
1302 |
||
1303 |
lemma (in algebra) sigma_property_disjoint: |
|
47694 | 1304 |
assumes sbC: "M \<subseteq> C" |
1305 |
and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C" |
|
1306 |
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M) |
|
38656 | 1307 |
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) |
1308 |
\<Longrightarrow> (\<Union>i. A i) \<in> C" |
|
47694 | 1309 |
and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M) |
38656 | 1310 |
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C" |
47694 | 1311 |
shows "sigma_sets (\<Omega>) (M) \<subseteq> C" |
38656 | 1312 |
proof - |
47694 | 1313 |
have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)" |
38656 | 1314 |
proof (rule sigma_property_disjoint_lemma) |
47694 | 1315 |
show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)" |
38656 | 1316 |
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic) |
1317 |
next |
|
47694 | 1318 |
show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))" |
38656 | 1319 |
by (simp add: closed_cdi_def compl inc disj) |
1320 |
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed |
|
1321 |
IntE sigma_sets.Compl range_subsetD sigma_sets.Union) |
|
1322 |
qed |
|
1323 |
thus ?thesis |
|
1324 |
by blast |
|
1325 |
qed |
|
1326 |
||
56994 | 1327 |
subsubsection {* Dynkin systems *} |
40859 | 1328 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1329 |
locale dynkin_system = subset_class + |
47694 | 1330 |
assumes space: "\<Omega> \<in> M" |
1331 |
and compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
1332 |
and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
1333 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
40859 | 1334 |
|
47694 | 1335 |
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M" |
1336 |
using space compl[of "\<Omega>"] by simp |
|
40859 | 1337 |
|
1338 |
lemma (in dynkin_system) diff: |
|
47694 | 1339 |
assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E" |
1340 |
shows "E - D \<in> M" |
|
40859 | 1341 |
proof - |
47694 | 1342 |
let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}" |
1343 |
have "range ?f = {D, \<Omega> - E, {}}" |
|
40859 | 1344 |
by (auto simp: image_iff) |
47694 | 1345 |
moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)" |
40859 | 1346 |
by (auto simp: image_iff split: split_if_asm) |
1347 |
moreover |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
51683
diff
changeset
|
1348 |
have "disjoint_family ?f" unfolding disjoint_family_on_def |
47694 | 1349 |
using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto |
1350 |
ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M" |
|
40859 | 1351 |
using sets by auto |
47694 | 1352 |
also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D" |
40859 | 1353 |
using assms sets_into_space by auto |
1354 |
finally show ?thesis . |
|
1355 |
qed |
|
1356 |
||
1357 |
lemma dynkin_systemI: |
|
47694 | 1358 |
assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M" |
1359 |
assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
1360 |
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
1361 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
1362 |
shows "dynkin_system \<Omega> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1363 |
using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def) |
40859 | 1364 |
|
42988 | 1365 |
lemma dynkin_systemI': |
47694 | 1366 |
assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" |
1367 |
assumes empty: "{} \<in> M" |
|
1368 |
assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
1369 |
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
1370 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
1371 |
shows "dynkin_system \<Omega> M" |
|
42988 | 1372 |
proof - |
47694 | 1373 |
from Diff[OF empty] have "\<Omega> \<in> M" by auto |
42988 | 1374 |
from 1 this Diff 2 show ?thesis |
1375 |
by (intro dynkin_systemI) auto |
|
1376 |
qed |
|
1377 |
||
40859 | 1378 |
lemma dynkin_system_trivial: |
47694 | 1379 |
shows "dynkin_system A (Pow A)" |
40859 | 1380 |
by (rule dynkin_systemI) auto |
1381 |
||
1382 |
lemma sigma_algebra_imp_dynkin_system: |
|
47694 | 1383 |
assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M" |
40859 | 1384 |
proof - |
47694 | 1385 |
interpret sigma_algebra \<Omega> M by fact |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
1386 |
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI) |
40859 | 1387 |
qed |
1388 |
||
56994 | 1389 |
subsubsection "Intersection sets systems" |
40859 | 1390 |
|
47694 | 1391 |
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" |
40859 | 1392 |
|
1393 |
lemma (in algebra) Int_stable: "Int_stable M" |
|
1394 |
unfolding Int_stable_def by auto |
|
1395 |
||
42981 | 1396 |
lemma Int_stableI: |
47694 | 1397 |
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A" |
42981 | 1398 |
unfolding Int_stable_def by auto |
1399 |
||
1400 |
lemma Int_stableD: |
|
47694 | 1401 |
"Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M" |
42981 | 1402 |
unfolding Int_stable_def by auto |
1403 |
||
40859 | 1404 |
lemma (in dynkin_system) sigma_algebra_eq_Int_stable: |
47694 | 1405 |
"sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M" |
40859 | 1406 |
proof |
47694 | 1407 |
assume "sigma_algebra \<Omega> M" then show "Int_stable M" |
40859 | 1408 |
unfolding sigma_algebra_def using algebra.Int_stable by auto |
1409 |
next |
|
1410 |
assume "Int_stable M" |
|
47694 | 1411 |
show "sigma_algebra \<Omega> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1412 |
unfolding sigma_algebra_disjoint_iff algebra_iff_Un |
40859 | 1413 |
proof (intro conjI ballI allI impI) |
47694 | 1414 |
show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto |
40859 | 1415 |
next |
47694 | 1416 |
fix A B assume "A \<in> M" "B \<in> M" |
1417 |
then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))" |
|
1418 |
"\<Omega> - A \<in> M" "\<Omega> - B \<in> M" |
|
40859 | 1419 |
using sets_into_space by auto |
47694 | 1420 |
then show "A \<union> B \<in> M" |
40859 | 1421 |
using `Int_stable M` unfolding Int_stable_def by auto |
1422 |
qed auto |
|
1423 |
qed |
|
1424 |
||
56994 | 1425 |
subsubsection "Smallest Dynkin systems" |
40859 | 1426 |
|
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
|
1427 |
definition dynkin where |
47694 | 1428 |
"dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})" |
40859 | 1429 |
|
1430 |
lemma dynkin_system_dynkin: |
|
47694 | 1431 |
assumes "M \<subseteq> Pow (\<Omega>)" |
1432 |
shows "dynkin_system \<Omega> (dynkin \<Omega> M)" |
|
40859 | 1433 |
proof (rule dynkin_systemI) |
47694 | 1434 |
fix A assume "A \<in> dynkin \<Omega> M" |
40859 | 1435 |
moreover |
47694 | 1436 |
{ fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D" |
1437 |
then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) } |
|
1438 |
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
|
1439 |
using assms dynkin_system_trivial by fastforce |
47694 | 1440 |
ultimately show "A \<subseteq> \<Omega>" |
40859 | 1441 |
unfolding dynkin_def using assms |
47694 | 1442 |
by auto |
40859 | 1443 |
next |
47694 | 1444 |
show "\<Omega> \<in> dynkin \<Omega> M" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
1445 |
unfolding dynkin_def using dynkin_system.space by fastforce |
40859 | 1446 |
next |
47694 | 1447 |
fix A assume "A \<in> dynkin \<Omega> M" |
1448 |
then show "\<Omega> - A \<in> dynkin \<Omega> M" |
|
40859 | 1449 |
unfolding dynkin_def using dynkin_system.compl by force |
1450 |
next |
|
1451 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 1452 |
assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M" |
1453 |
show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def |
|
40859 | 1454 |
proof (simp, safe) |
47694 | 1455 |
fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D" |
1456 |
with A have "(\<Union>i. A i) \<in> D" |
|
40859 | 1457 |
by (intro dynkin_system.UN) (auto simp: dynkin_def) |
1458 |
then show "(\<Union>i. A i) \<in> D" by auto |
|
1459 |
qed |
|
1460 |
qed |
|
1461 |
||
47694 | 1462 |
lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M" |
40859 | 1463 |
unfolding dynkin_def by auto |
1464 |
||
1465 |
lemma (in dynkin_system) restricted_dynkin_system: |
|
47694 | 1466 |
assumes "D \<in> M" |
1467 |
shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}" |
|
40859 | 1468 |
proof (rule dynkin_systemI, simp_all) |
47694 | 1469 |
have "\<Omega> \<inter> D = D" |
1470 |
using `D \<in> M` sets_into_space by auto |
|
1471 |
then show "\<Omega> \<inter> D \<in> M" |
|
1472 |
using `D \<in> M` by auto |
|
40859 | 1473 |
next |
47694 | 1474 |
fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M" |
1475 |
moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)" |
|
40859 | 1476 |
by auto |
47694 | 1477 |
ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M" |
1478 |
using `D \<in> M` by (auto intro: diff) |
|
40859 | 1479 |
next |
1480 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 1481 |
assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}" |
1482 |
then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)" |
|
1483 |
"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
|
1484 |
by ((fastforce simp: disjoint_family_on_def)+) |
47694 | 1485 |
then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M" |
40859 | 1486 |
by (auto simp del: UN_simps) |
1487 |
qed |
|
1488 |
||
1489 |
lemma (in dynkin_system) dynkin_subset: |
|
47694 | 1490 |
assumes "N \<subseteq> M" |
1491 |
shows "dynkin \<Omega> N \<subseteq> M" |
|
40859 | 1492 |
proof - |
47694 | 1493 |
have "dynkin_system \<Omega> M" by default |
1494 |
then have "dynkin_system \<Omega> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
1495 |
using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp |
47694 | 1496 |
with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def) |
40859 | 1497 |
qed |
1498 |
||
1499 |
lemma sigma_eq_dynkin: |
|
47694 | 1500 |
assumes sets: "M \<subseteq> Pow \<Omega>" |
40859 | 1501 |
assumes "Int_stable M" |
47694 | 1502 |
shows "sigma_sets \<Omega> M = dynkin \<Omega> M" |
40859 | 1503 |
proof - |
47694 | 1504 |
have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)" |
40859 | 1505 |
using sigma_algebra_imp_dynkin_system |
47694 | 1506 |
unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto |
40859 | 1507 |
moreover |
47694 | 1508 |
interpret dynkin_system \<Omega> "dynkin \<Omega> M" |
40859 | 1509 |
using dynkin_system_dynkin[OF sets] . |
47694 | 1510 |
have "sigma_algebra \<Omega> (dynkin \<Omega> M)" |
40859 | 1511 |
unfolding sigma_algebra_eq_Int_stable Int_stable_def |
1512 |
proof (intro ballI) |
|
47694 | 1513 |
fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M" |
1514 |
let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}" |
|
1515 |
have "M \<subseteq> ?D B" |
|
40859 | 1516 |
proof |
47694 | 1517 |
fix E assume "E \<in> M" |
1518 |
then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M" |
|
40859 | 1519 |
using sets_into_space `Int_stable M` by (auto simp: Int_stable_def) |
47694 | 1520 |
then have "dynkin \<Omega> M \<subseteq> ?D E" |
1521 |
using restricted_dynkin_system `E \<in> dynkin \<Omega> M` |
|
40859 | 1522 |
by (intro dynkin_system.dynkin_subset) simp_all |
47694 | 1523 |
then have "B \<in> ?D E" |
1524 |
using `B \<in> dynkin \<Omega> M` by auto |
|
1525 |
then have "E \<inter> B \<in> dynkin \<Omega> M" |
|
40859 | 1526 |
by (subst Int_commute) simp |
47694 | 1527 |
then show "E \<in> ?D B" |
1528 |
using sets `E \<in> M` by auto |
|
40859 | 1529 |
qed |
47694 | 1530 |
then have "dynkin \<Omega> M \<subseteq> ?D B" |
1531 |
using restricted_dynkin_system `B \<in> dynkin \<Omega> M` |
|
40859 | 1532 |
by (intro dynkin_system.dynkin_subset) simp_all |
47694 | 1533 |
then show "A \<inter> B \<in> dynkin \<Omega> M" |
1534 |
using `A \<in> dynkin \<Omega> M` sets_into_space by auto |
|
40859 | 1535 |
qed |
47694 | 1536 |
from sigma_algebra.sigma_sets_subset[OF this, of "M"] |
1537 |
have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto |
|
1538 |
ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto |
|
40859 | 1539 |
then show ?thesis |
47694 | 1540 |
by (auto simp: dynkin_def) |
40859 | 1541 |
qed |
1542 |
||
1543 |
lemma (in dynkin_system) dynkin_idem: |
|
47694 | 1544 |
"dynkin \<Omega> M = M" |
40859 | 1545 |
proof - |
47694 | 1546 |
have "dynkin \<Omega> M = M" |
40859 | 1547 |
proof |
47694 | 1548 |
show "M \<subseteq> dynkin \<Omega> M" |
40859 | 1549 |
using dynkin_Basic by auto |
47694 | 1550 |
show "dynkin \<Omega> M \<subseteq> M" |
40859 | 1551 |
by (intro dynkin_subset) auto |
1552 |
qed |
|
1553 |
then show ?thesis |
|
47694 | 1554 |
by (auto simp: dynkin_def) |
40859 | 1555 |
qed |
1556 |
||
1557 |
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
|
1558 |
assumes "Int_stable E" |
47694 | 1559 |
and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E" |
1560 |
shows "sigma_sets \<Omega> E = M" |
|
40859 | 1561 |
proof - |
47694 | 1562 |
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
|
1563 |
using E sets_into_space by force |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
51683
diff
changeset
|
1564 |
then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E" |
40859 | 1565 |
using `Int_stable E` by (rule sigma_eq_dynkin) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
51683
diff
changeset
|
1566 |
then have "dynkin \<Omega> E = M" |
47694 | 1567 |
using assms dynkin_subset[OF E(1)] by simp |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
51683
diff
changeset
|
1568 |
with * show ?thesis |
47694 | 1569 |
using assms by (auto simp: dynkin_def) |
42864 | 1570 |
qed |
1571 |
||
56994 | 1572 |
subsubsection {* Induction rule for intersection-stable generators *} |
1573 |
||
1574 |
text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras |
|
1575 |
generated by a generator closed under intersection. *} |
|
1576 |
||
49789
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1577 |
lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]: |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1578 |
assumes "Int_stable G" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1579 |
and closed: "G \<subseteq> Pow \<Omega>" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1580 |
and A: "A \<in> sigma_sets \<Omega> G" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1581 |
assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1582 |
and empty: "P {}" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1583 |
and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1584 |
and union: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sigma_sets \<Omega> G \<Longrightarrow> (\<And>i. P (A i)) \<Longrightarrow> P (\<Union>i::nat. A i)" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1585 |
shows "P A" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1586 |
proof - |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1587 |
let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1588 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1589 |
using closed by (rule sigma_algebra_sigma_sets) |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1590 |
from compl[OF _ empty] closed have space: "P \<Omega>" by simp |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1591 |
interpret dynkin_system \<Omega> ?D |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1592 |
by default (auto dest: sets_into_space intro!: space compl union) |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1593 |
have "sigma_sets \<Omega> G = ?D" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1594 |
by (rule dynkin_lemma) (auto simp: basic `Int_stable G`) |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1595 |
with A show ?thesis by auto |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1596 |
qed |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
1597 |
|
56994 | 1598 |
subsection {* Measure type *} |
1599 |
||
1600 |
definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
1601 |
"positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)" |
|
1602 |
||
1603 |
definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
1604 |
"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> |
|
1605 |
(\<Sum>i. f (A i)) = f (\<Union>i. A i))" |
|
1606 |
||
1607 |
definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
1608 |
"measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>" |
|
1609 |
||
1610 |
typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }" |
|
1611 |
proof |
|
1612 |
have "sigma_algebra UNIV {{}, UNIV}" |
|
1613 |
by (auto simp: sigma_algebra_iff2) |
|
1614 |
then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} " |
|
1615 |
by (auto simp: measure_space_def positive_def countably_additive_def) |
|
1616 |
qed |
|
1617 |
||
1618 |
definition space :: "'a measure \<Rightarrow> 'a set" where |
|
1619 |
"space M = fst (Rep_measure M)" |
|
1620 |
||
1621 |
definition sets :: "'a measure \<Rightarrow> 'a set set" where |
|
1622 |
"sets M = fst (snd (Rep_measure M))" |
|
1623 |
||
1624 |
definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where |
|
1625 |
"emeasure M = snd (snd (Rep_measure M))" |
|
1626 |
||
1627 |
definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where |
|
1628 |
"measure M A = real (emeasure M A)" |
|
1629 |
||
1630 |
declare [[coercion sets]] |
|
1631 |
||
1632 |
declare [[coercion measure]] |
|
1633 |
||
1634 |
declare [[coercion emeasure]] |
|
1635 |
||
1636 |
lemma measure_space: "measure_space (space M) (sets M) (emeasure M)" |
|
1637 |
by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse) |
|
1638 |
||
1639 |
interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure" |
|
1640 |
using measure_space[of M] by (auto simp: measure_space_def) |
|
1641 |
||
1642 |
definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
1643 |
"measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>}, |
|
1644 |
\<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)" |
|
1645 |
||
1646 |
abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)" |
|
1647 |
||
1648 |
lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)" |
|
1649 |
unfolding measure_space_def |
|
1650 |
by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def) |
|
1651 |
||
1652 |
lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}" |
|
1653 |
by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+ |
|
1654 |
||
1655 |
lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)" |
|
1656 |
by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial) |
|
1657 |
||
1658 |
lemma measure_space_closed: |
|
1659 |
assumes "measure_space \<Omega> M \<mu>" |
|
1660 |
shows "M \<subseteq> Pow \<Omega>" |
|
1661 |
proof - |
|
1662 |
interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def) |
|
1663 |
show ?thesis by(rule space_closed) |
|
1664 |
qed |
|
1665 |
||
1666 |
lemma (in ring_of_sets) positive_cong_eq: |
|
1667 |
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>" |
|
1668 |
by (auto simp add: positive_def) |
|
1669 |
||
1670 |
lemma (in sigma_algebra) countably_additive_eq: |
|
1671 |
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>" |
|
1672 |
unfolding countably_additive_def |
|
1673 |
by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq) |
|
1674 |
||
1675 |
lemma measure_space_eq: |
|
1676 |
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a" |
|
1677 |
shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'" |
|
1678 |
proof - |
|
1679 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets) |
|
1680 |
from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis |
|
1681 |
by (auto simp: measure_space_def) |
|
1682 |
qed |
|
1683 |
||
1684 |
lemma measure_of_eq: |
|
1685 |
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)" |
|
1686 |
shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'" |
|
1687 |
proof - |
|
1688 |
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'" |
|
1689 |
using assms by (rule measure_space_eq) |
|
1690 |
with eq show ?thesis |
|
1691 |
by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure]) |
|
1692 |
qed |
|
1693 |
||
1694 |
lemma |
|
1695 |
shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space) |
|
1696 |
and sets_measure_of_conv: |
|
1697 |
"sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets) |
|
1698 |
and emeasure_measure_of_conv: |
|
1699 |
"emeasure (measure_of \<Omega> A \<mu>) = |
|
1700 |
(\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)" (is ?emeasure) |
|
1701 |
proof - |
|
1702 |
have "?space \<and> ?sets \<and> ?emeasure" |
|
1703 |
proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>") |
|
1704 |
case True |
|
1705 |
from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>] |
|
1706 |
have "A \<subseteq> Pow \<Omega>" by simp |
|
1707 |
hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) |
|
1708 |
(\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)" |
|
1709 |
by(rule measure_space_eq) auto |
|
1710 |
with True `A \<subseteq> Pow \<Omega>` show ?thesis |
|
1711 |
by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse) |
|
1712 |
next |
|
1713 |
case False thus ?thesis |
|
1714 |
by(cases "A \<subseteq> Pow \<Omega>")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0') |
|
1715 |
qed |
|
1716 |
thus ?space ?sets ?emeasure by simp_all |
|
1717 |
qed |
|
1718 |
||
1719 |
lemma [simp]: |
|
1720 |
assumes A: "A \<subseteq> Pow \<Omega>" |
|
1721 |
shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" |
|
1722 |
and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>" |
|
1723 |
using assms |
|
1724 |
by(simp_all add: sets_measure_of_conv space_measure_of_conv) |
|
1725 |
||
1726 |
lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M" |
|
1727 |
using space_closed by (auto intro!: sigma_sets_eq) |
|
1728 |
||
1729 |
lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>" |
|
1730 |
by (rule space_measure_of_conv) |
|
1731 |
||
1732 |
lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')" |
|
1733 |
by (auto intro!: sigma_sets_subseteq) |
|
1734 |
||
1735 |
lemma sigma_sets_mono'': |
|
1736 |
assumes "A \<in> sigma_sets C D" |
|
1737 |
assumes "B \<subseteq> D" |
|
1738 |
assumes "D \<subseteq> Pow C" |
|
1739 |
shows "sigma_sets A B \<subseteq> sigma_sets C D" |
|
1740 |
proof |
|
1741 |
fix x assume "x \<in> sigma_sets A B" |
|
1742 |
thus "x \<in> sigma_sets C D" |
|
1743 |
proof induct |
|
1744 |
case (Basic a) with assms have "a \<in> D" by auto |
|
1745 |
thus ?case .. |
|
1746 |
next |
|
1747 |
case Empty show ?case by (rule sigma_sets.Empty) |
|
1748 |
next |
|
1749 |
from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`]) |
|
1750 |
moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`]) |
|
1751 |
ultimately have "A - a \<in> sets (sigma C D)" .. |
|
1752 |
thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`]) |
|
1753 |
next |
|
1754 |
case (Union a) |
|
1755 |
thus ?case by (intro sigma_sets.Union) |
|
1756 |
qed |
|
1757 |
qed |
|
1758 |
||
1759 |
lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)" |
|
1760 |
by auto |
|
1761 |
||
58606 | 1762 |
lemma space_empty_iff: "space N = {} \<longleftrightarrow> sets N = {{}}" |
1763 |
by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff |
|
1764 |
sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD) |
|
1765 |
||
56994 | 1766 |
subsubsection {* Constructing simple @{typ "'a measure"} *} |
1767 |
||
1768 |
lemma emeasure_measure_of: |
|
1769 |
assumes M: "M = measure_of \<Omega> A \<mu>" |
|
1770 |
assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>" |
|
1771 |
assumes X: "X \<in> sets M" |
|
1772 |
shows "emeasure M X = \<mu> X" |
|
1773 |
proof - |
|
1774 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact |
|
1775 |
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
1776 |
using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets) |
|
1777 |
thus ?thesis using X ms |
|
1778 |
by(simp add: M emeasure_measure_of_conv sets_measure_of_conv) |
|
1779 |
qed |
|
1780 |
||
1781 |
lemma emeasure_measure_of_sigma: |
|
1782 |
assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>" |
|
1783 |
assumes A: "A \<in> M" |
|
1784 |
shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A" |
|
1785 |
proof - |
|
1786 |
interpret sigma_algebra \<Omega> M by fact |
|
1787 |
have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
|
1788 |
using ms sigma_sets_eq by (simp add: measure_space_def) |
|
1789 |
thus ?thesis by(simp add: emeasure_measure_of_conv A) |
|
1790 |
qed |
|
1791 |
||
1792 |
lemma measure_cases[cases type: measure]: |
|
1793 |
obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>" |
|
1794 |
by atomize_elim (cases x, auto) |
|
1795 |
||
1796 |
lemma sets_eq_imp_space_eq: |
|
1797 |
"sets M = sets M' \<Longrightarrow> space M = space M'" |
|
1798 |
using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M'] |
|
1799 |
by blast |
|
1800 |
||
1801 |
lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0" |
|
1802 |
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) |
|
1803 |
||
1804 |
lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M" |
|
1805 |
using emeasure_notin_sets[of A M] by blast |
|
1806 |
||
1807 |
lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0" |
|
1808 |
by (simp add: measure_def emeasure_notin_sets) |
|
1809 |
||
1810 |
lemma measure_eqI: |
|
1811 |
fixes M N :: "'a measure" |
|
1812 |
assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" |
|
1813 |
shows "M = N" |
|
1814 |
proof (cases M N rule: measure_cases[case_product measure_cases]) |
|
1815 |
case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>') |
|
1816 |
interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def) |
|
1817 |
interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def) |
|
1818 |
have "A = sets M" "A' = sets N" |
|
1819 |
using measure_measure by (simp_all add: sets_def Abs_measure_inverse) |
|
1820 |
with `sets M = sets N` have AA': "A = A'" by simp |
|
1821 |
moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto |
|
1822 |
moreover { fix B have "\<mu> B = \<mu>' B" |
|
1823 |
proof cases |
|
1824 |
assume "B \<in> A" |
|
1825 |
with eq `A = sets M` have "emeasure M B = emeasure N B" by simp |
|
1826 |
with measure_measure show "\<mu> B = \<mu>' B" |
|
1827 |
by (simp add: emeasure_def Abs_measure_inverse) |
|
1828 |
next |
|
1829 |
assume "B \<notin> A" |
|
1830 |
with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N" |
|
1831 |
by auto |
|
1832 |
then have "emeasure M B = 0" "emeasure N B = 0" |
|
1833 |
by (simp_all add: emeasure_notin_sets) |
|
1834 |
with measure_measure show "\<mu> B = \<mu>' B" |
|
1835 |
by (simp add: emeasure_def Abs_measure_inverse) |
|
1836 |
qed } |
|
1837 |
then have "\<mu> = \<mu>'" by auto |
|
1838 |
ultimately show "M = N" |
|
1839 |
by (simp add: measure_measure) |
|
1840 |
qed |
|
1841 |
||
1842 |
lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)" |
|
1843 |
using measure_space_0[of A \<Omega>] |
|
1844 |
by (simp add: measure_of_def emeasure_def Abs_measure_inverse) |
|
1845 |
||
1846 |
lemma sigma_eqI: |
|
1847 |
assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N" |
|
1848 |
shows "sigma \<Omega> M = sigma \<Omega> N" |
|
1849 |
by (rule measure_eqI) (simp_all add: emeasure_sigma) |
|
1850 |
||
1851 |
subsubsection {* Measurable functions *} |
|
1852 |
||
1853 |
definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where |
|
1854 |
"measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}" |
|
1855 |
||
1856 |
lemma measurable_space: |
|
1857 |
"f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A" |
|
1858 |
unfolding measurable_def by auto |
|
1859 |
||
1860 |
lemma measurable_sets: |
|
1861 |
"f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
|
1862 |
unfolding measurable_def by auto |
|
1863 |
||
1864 |
lemma measurable_sets_Collect: |
|
1865 |
assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M" |
|
1866 |
proof - |
|
1867 |
have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}" |
|
1868 |
using measurable_space[OF f] by auto |
|
1869 |
with measurable_sets[OF f P] show ?thesis |
|
1870 |
by simp |
|
1871 |
qed |
|
1872 |
||
1873 |
lemma measurable_sigma_sets: |
|
1874 |
assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>" |
|
1875 |
and f: "f \<in> space M \<rightarrow> \<Omega>" |
|
1876 |
and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M" |
|
1877 |
shows "f \<in> measurable M N" |
|
1878 |
proof - |
|
1879 |
interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets) |
|
1880 |
from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force |
|
1881 |
||
1882 |
{ fix X assume "X \<in> sigma_sets \<Omega> A" |
|
1883 |
then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>" |
|
1884 |
proof induct |
|
1885 |
case (Basic a) then show ?case |
|
1886 |
by (auto simp add: ba) (metis B(2) subsetD PowD) |
|
1887 |
next |
|
1888 |
case (Compl a) |
|
1889 |
have [simp]: "f -` \<Omega> \<inter> space M = space M" |
|
1890 |
by (auto simp add: funcset_mem [OF f]) |
|
1891 |
then show ?case |
|
1892 |
by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl) |
|
1893 |
next |
|
1894 |
case (Union a) |
|
1895 |
then show ?case |
|
1896 |
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast |
|
1897 |
qed auto } |
|
1898 |
with f show ?thesis |
|
1899 |
by (auto simp add: measurable_def B \<Omega>) |
|
1900 |
qed |
|
1901 |
||
1902 |
lemma measurable_measure_of: |
|
1903 |
assumes B: "N \<subseteq> Pow \<Omega>" |
|
1904 |
and f: "f \<in> space M \<rightarrow> \<Omega>" |
|
1905 |
and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M" |
|
1906 |
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)" |
|
1907 |
proof - |
|
1908 |
have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N" |
|
1909 |
using B by (rule sets_measure_of) |
|
1910 |
from this assms show ?thesis by (rule measurable_sigma_sets) |
|
1911 |
qed |
|
1912 |
||
1913 |
lemma measurable_iff_measure_of: |
|
1914 |
assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>" |
|
1915 |
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)" |
|
1916 |
by (metis assms in_measure_of measurable_measure_of assms measurable_sets) |
|
1917 |
||
1918 |
lemma measurable_cong_sets: |
|
1919 |
assumes sets: "sets M = sets M'" "sets N = sets N'" |
|
1920 |
shows "measurable M N = measurable M' N'" |
|
1921 |
using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def) |
|
1922 |
||
1923 |
lemma measurable_cong: |
|
1924 |
assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w" |
|
1925 |
shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'" |
|
1926 |
unfolding measurable_def using assms |
|
1927 |
by (simp cong: vimage_inter_cong Pi_cong) |
|
1928 |
||
1929 |
lemma measurable_cong_strong: |
|
1930 |
"M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow> |
|
1931 |
f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'" |
|
1932 |
by (metis measurable_cong) |
|
1933 |
||
1934 |
lemma measurable_eqI: |
|
1935 |
"\<lbrakk> space m1 = space m1' ; space m2 = space m2' ; |
|
1936 |
sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk> |
|
1937 |
\<Longrightarrow> measurable m1 m2 = measurable m1' m2'" |
|
1938 |
by (simp add: measurable_def sigma_algebra_iff2) |
|
1939 |
||
1940 |
lemma measurable_compose: |
|
1941 |
assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L" |
|
1942 |
shows "(\<lambda>x. g (f x)) \<in> measurable M L" |
|
1943 |
proof - |
|
1944 |
have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M" |
|
1945 |
using measurable_space[OF f] by auto |
|
1946 |
with measurable_space[OF f] measurable_space[OF g] show ?thesis |
|
1947 |
by (auto intro: measurable_sets[OF f] measurable_sets[OF g] |
|
1948 |
simp del: vimage_Int simp add: measurable_def) |
|
1949 |
qed |
|
1950 |
||
1951 |
lemma measurable_comp: |
|
1952 |
"f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L" |
|
1953 |
using measurable_compose[of f M N g L] by (simp add: comp_def) |
|
1954 |
||
1955 |
lemma measurable_const: |
|
1956 |
"c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'" |
|
1957 |
by (auto simp add: measurable_def) |
|
1958 |
||
1959 |
lemma measurable_If: |
|
1960 |
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'" |
|
1961 |
assumes P: "{x\<in>space M. P x} \<in> sets M" |
|
1962 |
shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'" |
|
1963 |
unfolding measurable_def |
|
1964 |
proof safe |
|
1965 |
fix x assume "x \<in> space M" |
|
1966 |
thus "(if P x then f x else g x) \<in> space M'" |
|
1967 |
using measure unfolding measurable_def by auto |
|
1968 |
next |
|
1969 |
fix A assume "A \<in> sets M'" |
|
1970 |
hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M = |
|
1971 |
((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union> |
|
1972 |
((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))" |
|
1973 |
using measure unfolding measurable_def by (auto split: split_if_asm) |
|
1974 |
show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M" |
|
1975 |
using `A \<in> sets M'` measure P unfolding * measurable_def |
|
1976 |
by (auto intro!: sets.Un) |
|
1977 |
qed |
|
1978 |
||
1979 |
lemma measurable_If_set: |
|
1980 |
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'" |
|
1981 |
assumes P: "A \<inter> space M \<in> sets M" |
|
1982 |
shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'" |
|
1983 |
proof (rule measurable_If[OF measure]) |
|
1984 |
have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto |
|
1985 |
thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto |
|
1986 |
qed |
|
1987 |
||
1988 |
lemma measurable_ident: "id \<in> measurable M M" |
|
1989 |
by (auto simp add: measurable_def) |
|
1990 |
||
1991 |
lemma measurable_ident_sets: |
|
1992 |
assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'" |
|
1993 |
using measurable_ident[of M] |
|
1994 |
unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] . |
|
1995 |
||
1996 |
lemma sets_Least: |
|
1997 |
assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M" |
|
1998 |
shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M" |
|
1999 |
proof - |
|
2000 |
{ fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M" |
|
2001 |
proof cases |
|
2002 |
assume i: "(LEAST j. False) = i" |
|
2003 |
have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M = |
|
2004 |
{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}))" |
|
2005 |
by (simp add: set_eq_iff, safe) |
|
2006 |
(insert i, auto dest: Least_le intro: LeastI intro!: Least_equality) |
|
2007 |
with meas show ?thesis |
|
2008 |
by (auto intro!: sets.Int) |
|
2009 |
next |
|
2010 |
assume i: "(LEAST j. False) \<noteq> i" |
|
2011 |
then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M = |
|
2012 |
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))" |
|
2013 |
proof (simp add: set_eq_iff, safe) |
|
2014 |
fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)" |
|
2015 |
have "\<exists>j. P j x" |
|
2016 |
by (rule ccontr) (insert neq, auto) |
|
2017 |
then show "P (LEAST j. P j x) x" by (rule LeastI_ex) |
|
2018 |
qed (auto dest: Least_le intro!: Least_equality) |
|
2019 |
with meas show ?thesis |
|
2020 |
by auto |
|
2021 |
qed } |
|
2022 |
then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M" |
|
2023 |
by (intro sets.countable_UN) auto |
|
2024 |
moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) = |
|
2025 |
(\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto |
|
2026 |
ultimately show ?thesis by auto |
|
2027 |
qed |
|
2028 |
||
2029 |
lemma measurable_strong: |
|
2030 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
|
2031 |
assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c" |
|
2032 |
and t: "f ` (space a) \<subseteq> t" |
|
2033 |
and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b" |
|
2034 |
shows "(g o f) \<in> measurable a c" |
|
2035 |
proof - |
|
2036 |
have fab: "f \<in> (space a -> space b)" |
|
2037 |
and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f |
|
2038 |
by (auto simp add: measurable_def) |
|
2039 |
have eq: "\<And>y. (g \<circ> f) -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t |
|
2040 |
by force |
|
2041 |
show ?thesis |
|
2042 |
apply (auto simp add: measurable_def vimage_comp) |
|
2043 |
apply (metis funcset_mem fab g) |
|
2044 |
apply (subst eq) |
|
2045 |
apply (metis ba cb) |
|
2046 |
done |
|
2047 |
qed |
|
2048 |
||
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2049 |
lemma measurable_discrete_difference: |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2050 |
assumes f: "f \<in> measurable M N" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2051 |
assumes X: "countable X" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2052 |
assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2053 |
assumes space: "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2054 |
assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2055 |
shows "g \<in> measurable M N" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2056 |
unfolding measurable_def |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2057 |
proof safe |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2058 |
fix x assume "x \<in> space M" then show "g x \<in> space N" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2059 |
using measurable_space[OF f, of x] eq[of x] space[of x] by (cases "x \<in> X") auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2060 |
next |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2061 |
fix S assume S: "S \<in> sets N" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2062 |
have "g -` S \<inter> space M = (f -` S \<inter> space M) - (\<Union>x\<in>X. {x}) \<union> (\<Union>x\<in>{x\<in>X. g x \<in> S}. {x})" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2063 |
using sets.sets_into_space[OF sets] eq by auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2064 |
also have "\<dots> \<in> sets M" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2065 |
by (safe intro!: sets.Diff sets.Un measurable_sets[OF f] S sets.countable_UN' X countable_Collect sets) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2066 |
finally show "g -` S \<inter> space M \<in> sets M" . |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2067 |
qed |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57138
diff
changeset
|
2068 |
|
56994 | 2069 |
lemma measurable_mono1: |
2070 |
"M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow> |
|
2071 |
measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N" |
|
2072 |
using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def) |
|
2073 |
||
2074 |
subsubsection {* Counting space *} |
|
2075 |
||
2076 |
definition count_space :: "'a set \<Rightarrow> 'a measure" where |
|
2077 |
"count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)" |
|
2078 |
||
2079 |
lemma |
|
2080 |
shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>" |
|
2081 |
and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>" |
|
2082 |
using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>] |
|
2083 |
by (auto simp: count_space_def) |
|
2084 |
||
2085 |
lemma measurable_count_space_eq1[simp]: |
|
2086 |
"f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M" |
|
2087 |
unfolding measurable_def by simp |
|
2088 |
||
2089 |
lemma measurable_count_space_eq2: |
|
2090 |
assumes "finite A" |
|
2091 |
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))" |
|
2092 |
proof - |
|
2093 |
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A" |
|
2094 |
with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X" |
|
2095 |
by (auto dest: finite_subset) |
|
2096 |
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M" |
|
2097 |
ultimately have "f -` X \<inter> space M \<in> sets M" |
|
2098 |
using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) } |
|
2099 |
then show ?thesis |
|
2100 |
unfolding measurable_def by auto |
|
2101 |
qed |
|
2102 |
||
2103 |
lemma measurable_count_space_eq2_countable: |
|
2104 |
fixes f :: "'a => 'c::countable" |
|
2105 |
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))" |
|
2106 |
proof - |
|
2107 |
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A" |
|
2108 |
assume *: "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M" |
|
2109 |
have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" |
|
2110 |
by auto |
|
2111 |
also have "\<dots> \<in> sets M" |
|
2112 |
using * `X \<subseteq> A` by (intro sets.countable_UN) auto |
|
2113 |
finally have "f -` X \<inter> space M \<in> sets M" . } |
|
2114 |
then show ?thesis |
|
2115 |
unfolding measurable_def by auto |
|
2116 |
qed |
|
2117 |
||
2118 |
lemma measurable_compose_countable: |
|
2119 |
assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)" |
|
2120 |
shows "(\<lambda>x. f (g x) x) \<in> measurable M N" |
|
2121 |
unfolding measurable_def |
|
2122 |
proof safe |
|
2123 |
fix x assume "x \<in> space M" then show "f (g x) x \<in> space N" |
|
2124 |
using f[THEN measurable_space] g[THEN measurable_space] by auto |
|
2125 |
next |
|
2126 |
fix A assume A: "A \<in> sets N" |
|
2127 |
have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))" |
|
2128 |
by auto |
|
2129 |
also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets] |
|
2130 |
by (auto intro!: sets.countable_UN measurable_sets) |
|
2131 |
finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" . |
|
2132 |
qed |
|
2133 |
||
2134 |
lemma measurable_count_space_const: |
|
2135 |
"(\<lambda>x. c) \<in> measurable M (count_space UNIV)" |
|
2136 |
by (simp add: measurable_const) |
|
2137 |
||
2138 |
lemma measurable_count_space: |
|
2139 |
"f \<in> measurable (count_space A) (count_space UNIV)" |
|
2140 |
by simp |
|
2141 |
||
2142 |
lemma measurable_compose_rev: |
|
2143 |
assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L" |
|
2144 |
shows "(\<lambda>x. f (g x)) \<in> measurable M N" |
|
2145 |
using measurable_compose[OF g f] . |
|
2146 |
||
2147 |
lemma measurable_count_space_eq_countable: |
|
2148 |
assumes "countable A" |
|
2149 |
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))" |
|
2150 |
proof - |
|
2151 |
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A" |
|
2152 |
with `countable A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X" |
|
2153 |
by (auto dest: countable_subset) |
|
2154 |
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M" |
|
2155 |
ultimately have "f -` X \<inter> space M \<in> sets M" |
|
2156 |
using `X \<subseteq> A` by (auto intro!: sets.countable_UN' simp del: UN_simps) } |
|
2157 |
then show ?thesis |
|
2158 |
unfolding measurable_def by auto |
|
2159 |
qed |
|
2160 |
||
58606 | 2161 |
lemma measurable_empty_iff: |
2162 |
"space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}" |
|
2163 |
by (auto simp add: measurable_def Pi_iff) |
|
2164 |
||
56994 | 2165 |
subsubsection {* Extend measure *} |
2166 |
||
2167 |
definition "extend_measure \<Omega> I G \<mu> = |
|
2168 |
(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) |
|
2169 |
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>') |
|
2170 |
else measure_of \<Omega> (G`I) (\<lambda>_. 0))" |
|
2171 |
||
2172 |
lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>" |
|
2173 |
unfolding extend_measure_def by simp |
|
2174 |
||
2175 |
lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)" |
|
2176 |
unfolding extend_measure_def by simp |
|
2177 |
||
2178 |
lemma emeasure_extend_measure: |
|
2179 |
assumes M: "M = extend_measure \<Omega> I G \<mu>" |
|
2180 |
and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i" |
|
2181 |
and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'" |
|
2182 |
and "i \<in> I" |
|
2183 |
shows "emeasure M (G i) = \<mu> i" |
|
2184 |
proof cases |
|
2185 |
assume *: "(\<forall>i\<in>I. \<mu> i = 0)" |
|
2186 |
with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)" |
|
2187 |
by (simp add: extend_measure_def) |
|
2188 |
from measure_space_0[OF ms(1)] ms `i\<in>I` |
|
2189 |
have "emeasure M (G i) = 0" |
|
2190 |
by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure) |
|
2191 |
with `i\<in>I` * show ?thesis |
|
2192 |
by simp |
|
2193 |
next |
|
2194 |
def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'" |
|
2195 |
assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)" |
|
2196 |
moreover |
|
2197 |
have "measure_space (space M) (sets M) \<mu>'" |
|
2198 |
using ms unfolding measure_space_def by auto default |
|
2199 |
with ms eq have "\<exists>\<mu>'. P \<mu>'" |
|
2200 |
unfolding P_def |
|
2201 |
by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure) |
|
2202 |
ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)" |
|
2203 |
by (simp add: M extend_measure_def P_def[symmetric]) |
|
2204 |
||
2205 |
from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex) |
|
2206 |
show "emeasure M (G i) = \<mu> i" |
|
2207 |
proof (subst emeasure_measure_of[OF M_eq]) |
|
2208 |
have sets_M: "sets M = sigma_sets \<Omega> (G`I)" |
|
2209 |
using M_eq ms by (auto simp: sets_extend_measure) |
|
2210 |
then show "G i \<in> sets M" using `i \<in> I` by auto |
|
2211 |
show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i" |
|
2212 |
using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def) |
|
2213 |
qed fact |
|
2214 |
qed |
|
2215 |
||
2216 |
lemma emeasure_extend_measure_Pair: |
|
2217 |
assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)" |
|
2218 |
and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j" |
|
2219 |
and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'" |
|
2220 |
and "I i j" |
|
2221 |
shows "emeasure M (G i j) = \<mu> i j" |
|
2222 |
using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j` |
|
2223 |
by (auto simp: subset_eq) |
|
2224 |
||
58608 | 2225 |
subsubsection {* Supremum of a set of $\sigma$-algebras *} |
58588 | 2226 |
|
2227 |
definition "Sup_sigma M = sigma (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)" |
|
2228 |
||
2229 |
syntax |
|
2230 |
"_SUP_sigma" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>\<^sub>\<sigma> _\<in>_./ _)" [0, 0, 10] 10) |
|
56994 | 2231 |
|
58588 | 2232 |
translations |
2233 |
"\<Squnion>\<^sub>\<sigma> x\<in>A. B" == "CONST Sup_sigma ((\<lambda>x. B) ` A)" |
|
2234 |
||
2235 |
lemma space_Sup_sigma: "space (Sup_sigma M) = (\<Union>x\<in>M. space x)" |
|
2236 |
unfolding Sup_sigma_def by (rule space_measure_of) (auto dest: sets.sets_into_space) |
|
2237 |
||
2238 |
lemma sets_Sup_sigma: "sets (Sup_sigma M) = sigma_sets (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)" |
|
2239 |
unfolding Sup_sigma_def by (rule sets_measure_of) (auto dest: sets.sets_into_space) |
|
2240 |
||
2241 |
lemma in_Sup_sigma: "m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup_sigma M)" |
|
2242 |
unfolding sets_Sup_sigma by auto |
|
56994 | 2243 |
|
58588 | 2244 |
lemma sets_Sup_in_sets: |
2245 |
assumes "M \<noteq> {}" |
|
2246 |
assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N" |
|
2247 |
assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N" |
|
2248 |
shows "sets (Sup_sigma M) \<subseteq> sets N" |
|
2249 |
proof - |
|
2250 |
have *: "UNION M space = space N" |
|
56994 | 2251 |
using assms by auto |
58588 | 2252 |
show ?thesis |
2253 |
unfolding sets_Sup_sigma * using assms by (auto intro!: sets.sigma_sets_subset) |
|
2254 |
qed |
|
2255 |
||
2256 |
lemma measurable_Sup_sigma1: |
|
2257 |
assumes m: "m \<in> M" and f: "f \<in> measurable m N" |
|
2258 |
and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n" |
|
2259 |
shows "f \<in> measurable (Sup_sigma M) N" |
|
2260 |
proof - |
|
2261 |
have "space (Sup_sigma M) = space m" |
|
2262 |
using m by (auto simp add: space_Sup_sigma dest: const_space) |
|
2263 |
then show ?thesis |
|
2264 |
using m f unfolding measurable_def by (auto intro: in_Sup_sigma) |
|
56994 | 2265 |
qed |
2266 |
||
58588 | 2267 |
lemma measurable_Sup_sigma2: |
2268 |
assumes M: "M \<noteq> {}" |
|
2269 |
assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m" |
|
2270 |
shows "f \<in> measurable N (Sup_sigma M)" |
|
2271 |
unfolding Sup_sigma_def |
|
2272 |
proof (rule measurable_measure_of) |
|
2273 |
show "f \<in> space N \<rightarrow> UNION M space" |
|
2274 |
using measurable_space[OF f] M by auto |
|
2275 |
qed (auto intro: measurable_sets f dest: sets.sets_into_space) |
|
56994 | 2276 |
|
58608 | 2277 |
subsection {* The smallest $\sigma$-algebra regarding a function *} |
56994 | 2278 |
|
58588 | 2279 |
definition |
2280 |
"vimage_algebra X f M = sigma X {f -` A \<inter> X | A. A \<in> sets M}" |
|
2281 |
||
2282 |
lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X" |
|
2283 |
unfolding vimage_algebra_def by (rule space_measure_of) auto |
|
56994 | 2284 |
|
58588 | 2285 |
lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f -` A \<inter> X | A. A \<in> sets M}" |
2286 |
unfolding vimage_algebra_def by (rule sets_measure_of) auto |
|
2287 |
||
2288 |
lemma sets_vimage_algebra2: |
|
2289 |
"f \<in> X \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra X f M) = {f -` A \<inter> X | A. A \<in> sets M}" |
|
2290 |
using sigma_sets_vimage_commute[of f X "space M" "sets M"] |
|
2291 |
unfolding sets_vimage_algebra sets.sigma_sets_eq by simp |
|
56994 | 2292 |
|
58588 | 2293 |
lemma in_vimage_algebra: "A \<in> sets M \<Longrightarrow> f -` A \<inter> X \<in> sets (vimage_algebra X f M)" |
2294 |
by (auto simp: vimage_algebra_def) |
|
2295 |
||
2296 |
lemma sets_image_in_sets: |
|
2297 |
assumes N: "space N = X" |
|
2298 |
assumes f: "f \<in> measurable N M" |
|
2299 |
shows "sets (vimage_algebra X f M) \<subseteq> sets N" |
|
2300 |
unfolding sets_vimage_algebra N[symmetric] |
|
2301 |
by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f) |
|
2302 |
||
2303 |
lemma measurable_vimage_algebra1: "f \<in> X \<rightarrow> space M \<Longrightarrow> f \<in> measurable (vimage_algebra X f M) M" |
|
2304 |
unfolding measurable_def by (auto intro: in_vimage_algebra) |
|
2305 |
||
2306 |
lemma measurable_vimage_algebra2: |
|
2307 |
assumes g: "g \<in> space N \<rightarrow> X" and f: "(\<lambda>x. f (g x)) \<in> measurable N M" |
|
2308 |
shows "g \<in> measurable N (vimage_algebra X f M)" |
|
2309 |
unfolding vimage_algebra_def |
|
2310 |
proof (rule measurable_measure_of) |
|
2311 |
fix A assume "A \<in> {f -` A \<inter> X | A. A \<in> sets M}" |
|
2312 |
then obtain Y where Y: "Y \<in> sets M" and A: "A = f -` Y \<inter> X" |
|
2313 |
by auto |
|
2314 |
then have "g -` A \<inter> space N = (\<lambda>x. f (g x)) -` Y \<inter> space N" |
|
2315 |
using g by auto |
|
2316 |
also have "\<dots> \<in> sets N" |
|
2317 |
using f Y by (rule measurable_sets) |
|
2318 |
finally show "g -` A \<inter> space N \<in> sets N" . |
|
2319 |
qed (insert g, auto) |
|
56994 | 2320 |
|
2321 |
subsubsection {* Restricted Space Sigma Algebra *} |
|
2322 |
||
57025 | 2323 |
definition restrict_space where |
2324 |
"restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>) ` sets M) (emeasure M)" |
|
56994 | 2325 |
|
57025 | 2326 |
lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M" |
2327 |
using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto |
|
2328 |
||
2329 |
lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>" |
|
2330 |
by (simp add: space_restrict_space sets.sets_into_space) |
|
56994 | 2331 |
|
57025 | 2332 |
lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M" |
58588 | 2333 |
unfolding restrict_space_def |
2334 |
proof (subst sets_measure_of) |
|
2335 |
show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" |
|
2336 |
by (auto dest: sets.sets_into_space) |
|
2337 |
have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = |
|
57025 | 2338 |
(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M" |
58588 | 2339 |
by (subst sigma_sets_vimage_commute[symmetric, where \<Omega>' = "space M"]) |
2340 |
(auto simp add: sets.sigma_sets_eq) |
|
2341 |
moreover have "{((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M" |
|
2342 |
by auto |
|
2343 |
moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M = (op \<inter> \<Omega>) ` sets M" |
|
2344 |
by (intro image_cong) (auto dest: sets.sets_into_space) |
|
2345 |
ultimately show "sigma_sets (\<Omega> \<inter> space M) (op \<inter> \<Omega> ` sets M) = op \<inter> \<Omega> ` sets M" |
|
2346 |
by simp |
|
57025 | 2347 |
qed |
56994 | 2348 |
|
2349 |
lemma sets_restrict_space_iff: |
|
57025 | 2350 |
"\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)" |
2351 |
proof (subst sets_restrict_space, safe) |
|
2352 |
fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M" |
|
2353 |
then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M" |
|
2354 |
by rule |
|
2355 |
also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A" |
|
2356 |
using sets.sets_into_space[OF A] by auto |
|
2357 |
finally show "\<Omega> \<inter> A \<in> sets M" |
|
2358 |
by auto |
|
2359 |
qed auto |
|
56994 | 2360 |
|
2361 |
lemma measurable_restrict_space1: |
|
57025 | 2362 |
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M" and f: "f \<in> measurable M N" |
2363 |
shows "f \<in> measurable (restrict_space M \<Omega>) N" |
|
56994 | 2364 |
unfolding measurable_def |
2365 |
proof (intro CollectI conjI ballI) |
|
2366 |
show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N" |
|
2367 |
using measurable_space[OF f] sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space) |
|
2368 |
||
2369 |
fix A assume "A \<in> sets N" |
|
57025 | 2370 |
have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> (\<Omega> \<inter> space M)" |
56994 | 2371 |
using sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space) |
2372 |
also have "\<dots> \<in> sets (restrict_space M \<Omega>)" |
|
2373 |
unfolding sets_restrict_space_iff[OF \<Omega>] |
|
2374 |
using measurable_sets[OF f `A \<in> sets N`] \<Omega> by blast |
|
2375 |
finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" . |
|
2376 |
qed |
|
2377 |
||
2378 |
lemma measurable_restrict_space2: |
|
57137 | 2379 |
"\<Omega> \<inter> space N \<in> sets N \<Longrightarrow> f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> |
57025 | 2380 |
f \<in> measurable M (restrict_space N \<Omega>)" |
57137 | 2381 |
by (simp add: measurable_def space_restrict_space sets_restrict_space_iff Pi_Int[symmetric]) |
56994 | 2382 |
|
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2383 |
lemma measurable_restrict_space_iff: |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2384 |
assumes \<Omega>[simp, intro]: "\<Omega> \<inter> space M \<in> sets M" "c \<in> space N" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2385 |
shows "f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow> |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2386 |
(\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N" (is "f \<in> measurable ?R N \<longleftrightarrow> ?f \<in> measurable M N") |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2387 |
unfolding measurable_def |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2388 |
proof safe |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2389 |
fix x assume "f \<in> space ?R \<rightarrow> space N" "x \<in> space M" then show "?f x \<in> space N" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2390 |
using `c\<in>space N` by (auto simp: space_restrict_space) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2391 |
next |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2392 |
fix x assume "?f \<in> space M \<rightarrow> space N" "x \<in> space ?R" then show "f x \<in> space N" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2393 |
using `c\<in>space N` by (auto simp: space_restrict_space Pi_iff) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2394 |
next |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2395 |
fix X assume X: "X \<in> sets N" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2396 |
assume *[THEN bspec]: "\<forall>y\<in>sets N. f -` y \<inter> space ?R \<in> sets ?R" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2397 |
have "?f -` X \<inter> space M = (f -` X \<inter> (\<Omega> \<inter> space M)) \<union> (if c \<in> X then (space M - (\<Omega> \<inter> space M)) else {})" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2398 |
by (auto split: split_if_asm) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2399 |
also have "\<dots> \<in> sets M" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2400 |
using *[OF X] by (auto simp add: space_restrict_space sets_restrict_space_iff) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2401 |
finally show "?f -` X \<inter> space M \<in> sets M" . |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2402 |
next |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2403 |
assume *[THEN bspec]: "\<forall>y\<in>sets N. ?f -` y \<inter> space M \<in> sets M" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2404 |
fix X :: "'b set" assume X: "X \<in> sets N" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2405 |
have "f -` X \<inter> (\<Omega> \<inter> space M) = (?f -` X \<inter> space M) \<inter> (\<Omega> \<inter> space M)" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2406 |
by (auto simp: space_restrict_space) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2407 |
also have "\<dots> \<in> sets M" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2408 |
using *[OF X] by auto |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2409 |
finally show "f -` X \<inter> space ?R \<in> sets ?R" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2410 |
by (auto simp add: sets_restrict_space_iff space_restrict_space) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2411 |
qed |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
2412 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
2413 |
end |
57025 | 2414 |