author | hoelzl |
Thu, 02 Sep 2010 17:28:00 +0200 | |
changeset 39096 | 111756225292 |
parent 38656 | d5d342611edb |
child 40859 | de0b30e6c2d2 |
permissions | -rw-r--r-- |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1 |
header {*Caratheodory Extension Theorem*} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
2 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
3 |
theory Caratheodory |
38656 | 4 |
imports Sigma_Algebra Positive_Infinite_Real |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
5 |
begin |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
6 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
7 |
text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
8 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
9 |
subsection {* Measure Spaces *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
10 |
|
38656 | 11 |
definition "positive f \<longleftrightarrow> f {} = (0::pinfreal)" -- "Positive is enforced by the type" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
12 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
13 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
14 |
additive where |
38656 | 15 |
"additive M f \<longleftrightarrow> |
16 |
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
17 |
\<longrightarrow> f (x \<union> y) = f x + f y)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
18 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
19 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
20 |
countably_additive where |
38656 | 21 |
"countably_additive M f \<longleftrightarrow> |
22 |
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
23 |
disjoint_family A \<longrightarrow> |
38656 | 24 |
(\<Union>i. A i) \<in> sets M \<longrightarrow> |
25 |
(\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i))" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
26 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
27 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
28 |
increasing where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
29 |
"increasing M f \<longleftrightarrow> (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
30 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
31 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
32 |
subadditive where |
38656 | 33 |
"subadditive M f \<longleftrightarrow> |
34 |
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
35 |
\<longrightarrow> f (x \<union> y) \<le> f x + f y)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
36 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
37 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
38 |
countably_subadditive where |
38656 | 39 |
"countably_subadditive M f \<longleftrightarrow> |
40 |
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
41 |
disjoint_family A \<longrightarrow> |
38656 | 42 |
(\<Union>i. A i) \<in> sets M \<longrightarrow> |
43 |
f (\<Union>i. A i) \<le> psuminf (\<lambda>n. f (A n)))" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
44 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
45 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
46 |
lambda_system where |
38656 | 47 |
"lambda_system M f = |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
48 |
{l. l \<in> sets M & (\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x)}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
49 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
50 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
51 |
outer_measure_space where |
38656 | 52 |
"outer_measure_space M f \<longleftrightarrow> |
53 |
positive f \<and> increasing M f \<and> countably_subadditive M f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
54 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
55 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
56 |
measure_set where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
57 |
"measure_set M f X = |
38656 | 58 |
{r . \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>\<^isub>\<infinity> i. f (A i)) = r}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
59 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
60 |
locale measure_space = sigma_algebra + |
38656 | 61 |
fixes \<mu> :: "'a set \<Rightarrow> pinfreal" |
62 |
assumes empty_measure [simp]: "\<mu> {} = 0" |
|
63 |
and ca: "countably_additive M \<mu>" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
64 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
65 |
lemma increasingD: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
66 |
"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
67 |
by (auto simp add: increasing_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
68 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
69 |
lemma subadditiveD: |
38656 | 70 |
"subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
71 |
\<Longrightarrow> f (x \<union> y) \<le> f x + f y" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
72 |
by (auto simp add: subadditive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
73 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
74 |
lemma additiveD: |
38656 | 75 |
"additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
76 |
\<Longrightarrow> f (x \<union> y) = f x + f y" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
77 |
by (auto simp add: additive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
78 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
79 |
lemma countably_additiveD: |
35582 | 80 |
"countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A |
38656 | 81 |
\<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)" |
35582 | 82 |
by (simp add: countably_additive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
83 |
|
38656 | 84 |
section "Extend binary sets" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
85 |
|
35582 | 86 |
lemma LIMSEQ_binaryset: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
87 |
assumes f: "f {} = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
88 |
shows "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
89 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
90 |
have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)" |
35582 | 91 |
proof |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
92 |
fix n |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
93 |
show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<Suc (Suc n). f (binaryset A B i)) = f A + f B" |
35582 | 94 |
by (induct n) (auto simp add: binaryset_def f) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
95 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
96 |
moreover |
35582 | 97 |
have "... ----> f A + f B" by (rule LIMSEQ_const) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
98 |
ultimately |
35582 | 99 |
have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
100 |
by metis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
101 |
hence "(\<lambda>n. \<Sum>i = 0..< n+2. f (binaryset A B i)) ----> f A + f B" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
102 |
by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
103 |
thus ?thesis by (rule LIMSEQ_offset [where k=2]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
104 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
105 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
106 |
lemma binaryset_sums: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
107 |
assumes f: "f {} = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
108 |
shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)" |
38656 | 109 |
by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
110 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
111 |
lemma suminf_binaryset_eq: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
112 |
"f {} = 0 \<Longrightarrow> suminf (\<lambda>n. f (binaryset A B n)) = f A + f B" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
113 |
by (metis binaryset_sums sums_unique) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
114 |
|
38656 | 115 |
lemma binaryset_psuminf: |
116 |
assumes "f {} = 0" |
|
117 |
shows "(\<Sum>\<^isub>\<infinity> n. f (binaryset A B n)) = f A + f B" (is "?suminf = ?sum") |
|
118 |
proof - |
|
119 |
have *: "{..<2} = {0, 1::nat}" by auto |
|
120 |
have "\<forall>n\<ge>2. f (binaryset A B n) = 0" |
|
121 |
unfolding binaryset_def |
|
122 |
using assms by auto |
|
123 |
hence "?suminf = (\<Sum>N<2. f (binaryset A B N))" |
|
124 |
by (rule psuminf_finite) |
|
125 |
also have "... = ?sum" unfolding * binaryset_def |
|
126 |
by simp |
|
127 |
finally show ?thesis . |
|
128 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
129 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
130 |
subsection {* Lambda Systems *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
131 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
132 |
lemma (in algebra) lambda_system_eq: |
38656 | 133 |
"lambda_system M f = |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
134 |
{l. l \<in> sets M & (\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x)}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
135 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
136 |
have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l" |
37032 | 137 |
by (metis Int_Diff Int_absorb1 Int_commute sets_into_space) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
138 |
show ?thesis |
37032 | 139 |
by (auto simp add: lambda_system_def) (metis Int_commute)+ |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
140 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
141 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
142 |
lemma (in algebra) lambda_system_empty: |
38656 | 143 |
"positive f \<Longrightarrow> {} \<in> lambda_system M f" |
144 |
by (auto simp add: positive_def lambda_system_eq) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
145 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
146 |
lemma lambda_system_sets: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
147 |
"x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
148 |
by (simp add: lambda_system_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
149 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
150 |
lemma (in algebra) lambda_system_Compl: |
38656 | 151 |
fixes f:: "'a set \<Rightarrow> pinfreal" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
152 |
assumes x: "x \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
153 |
shows "space M - x \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
154 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
155 |
have "x \<subseteq> space M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
156 |
by (metis sets_into_space lambda_system_sets x) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
157 |
hence "space M - (space M - x) = x" |
38656 | 158 |
by (metis double_diff equalityE) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
159 |
with x show ?thesis |
38656 | 160 |
by (force simp add: lambda_system_def ac_simps) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
161 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
162 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
163 |
lemma (in algebra) lambda_system_Int: |
38656 | 164 |
fixes f:: "'a set \<Rightarrow> pinfreal" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
165 |
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
166 |
shows "x \<inter> y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
167 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
168 |
from xl yl show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
169 |
proof (auto simp add: positive_def lambda_system_eq Int) |
33536 | 170 |
fix u |
171 |
assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
172 |
and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
173 |
and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z" |
33536 | 174 |
have "u - x \<inter> y \<in> sets M" |
175 |
by (metis Diff Diff_Int Un u x y) |
|
176 |
moreover |
|
177 |
have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast |
|
178 |
moreover |
|
179 |
have "u - x \<inter> y - y = u - y" by blast |
|
180 |
ultimately |
|
181 |
have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy |
|
182 |
by force |
|
38656 | 183 |
have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
184 |
= (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)" |
38656 | 185 |
by (simp add: ey ac_simps) |
33536 | 186 |
also have "... = (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)" |
38656 | 187 |
by (simp add: Int_ac) |
33536 | 188 |
also have "... = f (u \<inter> y) + f (u - y)" |
189 |
using fx [THEN bspec, of "u \<inter> y"] Int y u |
|
190 |
by force |
|
191 |
also have "... = f u" |
|
38656 | 192 |
by (metis fy u) |
33536 | 193 |
finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" . |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
194 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
195 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
196 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
197 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
198 |
lemma (in algebra) lambda_system_Un: |
38656 | 199 |
fixes f:: "'a set \<Rightarrow> pinfreal" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
200 |
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
201 |
shows "x \<union> y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
202 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
203 |
have "(space M - x) \<inter> (space M - y) \<in> sets M" |
38656 | 204 |
by (metis Diff_Un Un compl_sets lambda_system_sets xl yl) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
205 |
moreover |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
206 |
have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
207 |
by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
208 |
ultimately show ?thesis |
38656 | 209 |
by (metis lambda_system_Compl lambda_system_Int xl yl) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
210 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
211 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
212 |
lemma (in algebra) lambda_system_algebra: |
38656 | 213 |
"positive f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))" |
214 |
apply (auto simp add: algebra_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
215 |
apply (metis lambda_system_sets set_mp sets_into_space) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
216 |
apply (metis lambda_system_empty) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
217 |
apply (metis lambda_system_Compl) |
38656 | 218 |
apply (metis lambda_system_Un) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
219 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
220 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
221 |
lemma (in algebra) lambda_system_strong_additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
222 |
assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
223 |
and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
224 |
shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
225 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
226 |
have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
227 |
moreover |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
228 |
have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast |
38656 | 229 |
moreover |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
230 |
have "(z \<inter> (x \<union> y)) \<in> sets M" |
38656 | 231 |
by (metis Int Un lambda_system_sets xl yl z) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
232 |
ultimately show ?thesis using xl yl |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
233 |
by (simp add: lambda_system_eq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
234 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
235 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
236 |
lemma (in algebra) lambda_system_additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
237 |
"additive (M (|sets := lambda_system M f|)) f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
238 |
proof (auto simp add: additive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
239 |
fix x and y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
240 |
assume disj: "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
241 |
and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
242 |
hence "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+ |
38656 | 243 |
thus "f (x \<union> y) = f x + f y" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
244 |
using lambda_system_strong_additive [OF top disj xl yl] |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
245 |
by (simp add: Un) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
246 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
247 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
248 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
249 |
lemma (in algebra) countably_subadditive_subadditive: |
38656 | 250 |
assumes f: "positive f" and cs: "countably_subadditive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
251 |
shows "subadditive M f" |
35582 | 252 |
proof (auto simp add: subadditive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
253 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
254 |
assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
255 |
hence "disjoint_family (binaryset x y)" |
35582 | 256 |
by (auto simp add: disjoint_family_on_def binaryset_def) |
257 |
hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow> |
|
258 |
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow> |
|
38656 | 259 |
f (\<Union>i. binaryset x y i) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))" |
35582 | 260 |
using cs by (simp add: countably_subadditive_def) |
261 |
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow> |
|
38656 | 262 |
f (x \<union> y) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
263 |
by (simp add: range_binaryset_eq UN_binaryset_eq) |
38656 | 264 |
thus "f (x \<union> y) \<le> f x + f y" using f x y |
265 |
by (auto simp add: Un o_def binaryset_psuminf positive_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
266 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
267 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
268 |
lemma (in algebra) additive_sum: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
269 |
fixes A:: "nat \<Rightarrow> 'a set" |
38656 | 270 |
assumes f: "positive f" and ad: "additive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
271 |
and A: "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
272 |
and disj: "disjoint_family A" |
38656 | 273 |
shows "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
274 |
proof (induct n) |
38656 | 275 |
case 0 show ?case using f by (simp add: positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
276 |
next |
38656 | 277 |
case (Suc n) |
278 |
have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj |
|
35582 | 279 |
by (auto simp add: disjoint_family_on_def neq_iff) blast |
38656 | 280 |
moreover |
281 |
have "A n \<in> sets M" using A by blast |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
282 |
moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
283 |
by (metis A UNION_in_sets atLeast0LessThan) |
38656 | 284 |
moreover |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
285 |
ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)" |
38656 | 286 |
using ad UNION_in_sets A by (auto simp add: additive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
287 |
with Suc.hyps show ?case using ad |
38656 | 288 |
by (auto simp add: atLeastLessThanSuc additive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
289 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
290 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
291 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
292 |
lemma countably_subadditiveD: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
293 |
"countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> |
38656 | 294 |
(\<Union>i. A i) \<in> sets M \<Longrightarrow> f (\<Union>i. A i) \<le> psuminf (f o A)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
295 |
by (auto simp add: countably_subadditive_def o_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
296 |
|
38656 | 297 |
lemma (in algebra) increasing_additive_bound: |
298 |
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal" |
|
299 |
assumes f: "positive f" and ad: "additive M f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
300 |
and inc: "increasing M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
301 |
and A: "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
302 |
and disj: "disjoint_family A" |
38656 | 303 |
shows "psuminf (f \<circ> A) \<le> f (space M)" |
304 |
proof (safe intro!: psuminf_bound) |
|
305 |
fix N |
|
306 |
have "setsum (f \<circ> A) {0..<N} = f (\<Union>i\<in>{0..<N}. A i)" |
|
307 |
by (rule additive_sum [OF f ad A disj]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
308 |
also have "... \<le> f (space M)" using space_closed A |
38656 | 309 |
by (blast intro: increasingD [OF inc] UNION_in_sets top) |
310 |
finally show "setsum (f \<circ> A) {..<N} \<le> f (space M)" by (simp add: atLeast0LessThan) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
311 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
312 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
313 |
lemma lambda_system_increasing: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
314 |
"increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f" |
38656 | 315 |
by (simp add: increasing_def lambda_system_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
316 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
317 |
lemma (in algebra) lambda_system_strong_sum: |
38656 | 318 |
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal" |
319 |
assumes f: "positive f" and a: "a \<in> sets M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
320 |
and A: "range A \<subseteq> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
321 |
and disj: "disjoint_family A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
322 |
shows "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
323 |
proof (induct n) |
38656 | 324 |
case 0 show ?case using f by (simp add: positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
325 |
next |
38656 | 326 |
case (Suc n) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
327 |
have 2: "A n \<inter> UNION {0..<n} A = {}" using disj |
38656 | 328 |
by (force simp add: disjoint_family_on_def neq_iff) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
329 |
have 3: "A n \<in> lambda_system M f" using A |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
330 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
331 |
have 4: "UNION {0..<n} A \<in> lambda_system M f" |
38656 | 332 |
using A algebra.UNION_in_sets [OF local.lambda_system_algebra, of f, OF f] |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
333 |
by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
334 |
from Suc.hyps show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
335 |
by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
336 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
337 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
338 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
339 |
lemma (in sigma_algebra) lambda_system_caratheodory: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
340 |
assumes oms: "outer_measure_space M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
341 |
and A: "range A \<subseteq> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
342 |
and disj: "disjoint_family A" |
38656 | 343 |
shows "(\<Union>i. A i) \<in> lambda_system M f \<and> psuminf (f \<circ> A) = f (\<Union>i. A i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
344 |
proof - |
38656 | 345 |
have pos: "positive f" and inc: "increasing M f" |
346 |
and csa: "countably_subadditive M f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
347 |
by (metis oms outer_measure_space_def)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
348 |
have sa: "subadditive M f" |
38656 | 349 |
by (metis countably_subadditive_subadditive csa pos) |
350 |
have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
351 |
by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
352 |
have alg_ls: "algebra (M(|sets := lambda_system M f|))" |
38656 | 353 |
by (rule lambda_system_algebra) (rule pos) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
354 |
have A'': "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
355 |
by (metis A image_subset_iff lambda_system_sets) |
38656 | 356 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
357 |
have U_in: "(\<Union>i. A i) \<in> sets M" |
37032 | 358 |
by (metis A'' countable_UN) |
38656 | 359 |
have U_eq: "f (\<Union>i. A i) = psuminf (f o A)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
360 |
proof (rule antisym) |
38656 | 361 |
show "f (\<Union>i. A i) \<le> psuminf (f \<circ> A)" |
362 |
by (rule countably_subadditiveD [OF csa A'' disj U_in]) |
|
363 |
show "psuminf (f \<circ> A) \<le> f (\<Union>i. A i)" |
|
364 |
by (rule psuminf_bound, unfold atLeast0LessThan[symmetric]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
365 |
(metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right |
38656 | 366 |
lambda_system_additive subset_Un_eq increasingD [OF inc] |
367 |
A' A'' UNION_in_sets U_in) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
368 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
369 |
{ |
38656 | 370 |
fix a |
371 |
assume a [iff]: "a \<in> sets M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
372 |
have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
373 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
374 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
375 |
proof (rule antisym) |
33536 | 376 |
have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A'' |
377 |
by blast |
|
38656 | 378 |
moreover |
33536 | 379 |
have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj |
38656 | 380 |
by (auto simp add: disjoint_family_on_def) |
381 |
moreover |
|
33536 | 382 |
have "a \<inter> (\<Union>i. A i) \<in> sets M" |
383 |
by (metis Int U_in a) |
|
38656 | 384 |
ultimately |
385 |
have "f (a \<inter> (\<Union>i. A i)) \<le> psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)" |
|
386 |
using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"] |
|
387 |
by (simp add: o_def) |
|
388 |
hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> |
|
389 |
psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i))" |
|
390 |
by (rule add_right_mono) |
|
391 |
moreover |
|
392 |
have "psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i)) \<le> f a" |
|
393 |
proof (safe intro!: psuminf_bound_add) |
|
33536 | 394 |
fix n |
395 |
have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M" |
|
38656 | 396 |
by (metis A'' UNION_in_sets) |
33536 | 397 |
have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A'' |
37032 | 398 |
by (blast intro: increasingD [OF inc] A'' UNION_in_sets) |
33536 | 399 |
have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f" |
38656 | 400 |
using algebra.UNION_in_sets [OF lambda_system_algebra [of f, OF pos]] |
401 |
by (simp add: A) |
|
402 |
hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))" |
|
37032 | 403 |
by (simp add: lambda_system_eq UNION_in) |
33536 | 404 |
have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))" |
38656 | 405 |
by (blast intro: increasingD [OF inc] UNION_eq_Union_image |
37032 | 406 |
UNION_in U_in) |
38656 | 407 |
thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {..<n} + f (a - (\<Union>i. A i)) \<le> f a" |
408 |
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric]) |
|
33536 | 409 |
qed |
38656 | 410 |
ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
411 |
by (rule order_trans) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
412 |
next |
38656 | 413 |
have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))" |
37032 | 414 |
by (blast intro: increasingD [OF inc] U_in) |
33536 | 415 |
also have "... \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" |
37032 | 416 |
by (blast intro: subadditiveD [OF sa] U_in) |
33536 | 417 |
finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" . |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
418 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
419 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
420 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
421 |
thus ?thesis |
38656 | 422 |
by (simp add: lambda_system_eq sums_iff U_eq U_in) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
423 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
424 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
425 |
lemma (in sigma_algebra) caratheodory_lemma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
426 |
assumes oms: "outer_measure_space M f" |
38656 | 427 |
shows "measure_space (|space = space M, sets = lambda_system M f|) f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
428 |
proof - |
38656 | 429 |
have pos: "positive f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
430 |
by (metis oms outer_measure_space_def) |
38656 | 431 |
have alg: "algebra (|space = space M, sets = lambda_system M f|)" |
432 |
using lambda_system_algebra [of f, OF pos] |
|
433 |
by (simp add: algebra_def) |
|
434 |
then moreover |
|
435 |
have "sigma_algebra (|space = space M, sets = lambda_system M f|)" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
436 |
using lambda_system_caratheodory [OF oms] |
38656 | 437 |
by (simp add: sigma_algebra_disjoint_iff) |
438 |
moreover |
|
439 |
have "measure_space_axioms (|space = space M, sets = lambda_system M f|) f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
440 |
using pos lambda_system_caratheodory [OF oms] |
38656 | 441 |
by (simp add: measure_space_axioms_def positive_def lambda_system_sets |
442 |
countably_additive_def o_def) |
|
443 |
ultimately |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
444 |
show ?thesis |
38656 | 445 |
by intro_locales (auto simp add: sigma_algebra_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
446 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
447 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
448 |
lemma (in algebra) additive_increasing: |
38656 | 449 |
assumes posf: "positive f" and addf: "additive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
450 |
shows "increasing M f" |
38656 | 451 |
proof (auto simp add: increasing_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
452 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
453 |
assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y" |
38656 | 454 |
have "f x \<le> f x + f (y-x)" .. |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
455 |
also have "... = f (x \<union> (y-x))" using addf |
37032 | 456 |
by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2)) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
457 |
also have "... = f y" |
37032 | 458 |
by (metis Un_Diff_cancel Un_absorb1 xy(3)) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
459 |
finally show "f x \<le> f y" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
460 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
461 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
462 |
lemma (in algebra) countably_additive_additive: |
38656 | 463 |
assumes posf: "positive f" and ca: "countably_additive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
464 |
shows "additive M f" |
38656 | 465 |
proof (auto simp add: additive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
466 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
467 |
assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
468 |
hence "disjoint_family (binaryset x y)" |
38656 | 469 |
by (auto simp add: disjoint_family_on_def binaryset_def) |
470 |
hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow> |
|
471 |
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow> |
|
472 |
f (\<Union>i. binaryset x y i) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
473 |
using ca |
38656 | 474 |
by (simp add: countably_additive_def) |
475 |
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow> |
|
476 |
f (x \<union> y) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
477 |
by (simp add: range_binaryset_eq UN_binaryset_eq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
478 |
thus "f (x \<union> y) = f x + f y" using posf x y |
38656 | 479 |
by (auto simp add: Un binaryset_psuminf positive_def) |
480 |
qed |
|
481 |
||
39096 | 482 |
lemma inf_measure_nonempty: |
483 |
assumes f: "positive f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M" |
|
484 |
shows "f b \<in> measure_set M f a" |
|
485 |
proof - |
|
486 |
have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}" |
|
487 |
by (rule psuminf_finite) (simp add: f[unfolded positive_def]) |
|
488 |
also have "... = f b" |
|
489 |
by simp |
|
490 |
finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" . |
|
491 |
thus ?thesis using assms |
|
492 |
by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"] |
|
493 |
simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def) |
|
494 |
qed |
|
495 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
496 |
lemma (in algebra) inf_measure_agrees: |
38656 | 497 |
assumes posf: "positive f" and ca: "countably_additive M f" |
498 |
and s: "s \<in> sets M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
499 |
shows "Inf (measure_set M f s) = f s" |
38656 | 500 |
unfolding Inf_pinfreal_def |
501 |
proof (safe intro!: Greatest_equality) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
502 |
fix z |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
503 |
assume z: "z \<in> measure_set M f s" |
38656 | 504 |
from this obtain A where |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
505 |
A: "range A \<subseteq> sets M" and disj: "disjoint_family A" |
38656 | 506 |
and "s \<subseteq> (\<Union>x. A x)" and si: "psuminf (f \<circ> A) = z" |
507 |
by (auto simp add: measure_set_def comp_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
508 |
hence seq: "s = (\<Union>i. A i \<inter> s)" by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
509 |
have inc: "increasing M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
510 |
by (metis additive_increasing ca countably_additive_additive posf) |
38656 | 511 |
have sums: "psuminf (\<lambda>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)" |
512 |
proof (rule countably_additiveD [OF ca]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
513 |
show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s |
33536 | 514 |
by blast |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
515 |
show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj |
35582 | 516 |
by (auto simp add: disjoint_family_on_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
517 |
show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s |
33536 | 518 |
by (metis UN_extend_simps(4) s seq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
519 |
qed |
38656 | 520 |
hence "f s = psuminf (\<lambda>i. f (A i \<inter> s))" |
37032 | 521 |
using seq [symmetric] by (simp add: sums_iff) |
38656 | 522 |
also have "... \<le> psuminf (f \<circ> A)" |
523 |
proof (rule psuminf_le) |
|
524 |
fix n show "f (A n \<inter> s) \<le> (f \<circ> A) n" using A s |
|
525 |
by (force intro: increasingD [OF inc]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
526 |
qed |
38656 | 527 |
also have "... = z" by (rule si) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
528 |
finally show "f s \<le> z" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
529 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
530 |
fix y |
38656 | 531 |
assume y: "\<forall>u \<in> measure_set M f s. y \<le> u" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
532 |
thus "y \<le> f s" |
38656 | 533 |
by (blast intro: inf_measure_nonempty [of f, OF posf s subset_refl]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
534 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
535 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
536 |
lemma (in algebra) inf_measure_empty: |
39096 | 537 |
assumes posf: "positive f" "{} \<in> sets M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
538 |
shows "Inf (measure_set M f {}) = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
539 |
proof (rule antisym) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
540 |
show "Inf (measure_set M f {}) \<le> 0" |
39096 | 541 |
by (metis complete_lattice_class.Inf_lower `{} \<in> sets M` inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def]) |
38656 | 542 |
qed simp |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
543 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
544 |
lemma (in algebra) inf_measure_positive: |
38656 | 545 |
"positive f \<Longrightarrow> |
546 |
positive (\<lambda>x. Inf (measure_set M f x))" |
|
547 |
by (simp add: positive_def inf_measure_empty) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
548 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
549 |
lemma (in algebra) inf_measure_increasing: |
38656 | 550 |
assumes posf: "positive f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
551 |
shows "increasing (| space = space M, sets = Pow (space M) |) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
552 |
(\<lambda>x. Inf (measure_set M f x))" |
38656 | 553 |
apply (auto simp add: increasing_def) |
554 |
apply (rule complete_lattice_class.Inf_greatest) |
|
555 |
apply (rule complete_lattice_class.Inf_lower) |
|
37032 | 556 |
apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
557 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
558 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
559 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
560 |
lemma (in algebra) inf_measure_le: |
38656 | 561 |
assumes posf: "positive f" and inc: "increasing M f" |
562 |
and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> psuminf (f \<circ> A) = r}" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
563 |
shows "Inf (measure_set M f s) \<le> x" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
564 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
565 |
from x |
38656 | 566 |
obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)" |
567 |
and xeq: "psuminf (f \<circ> A) = x" |
|
568 |
by auto |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
569 |
have dA: "range (disjointed A) \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
570 |
by (metis A range_disjointed_sets) |
38656 | 571 |
have "\<forall>n.(f o disjointed A) n \<le> (f \<circ> A) n" unfolding comp_def |
572 |
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def) |
|
573 |
hence sda: "psuminf (f o disjointed A) \<le> psuminf (f \<circ> A)" |
|
574 |
by (blast intro: psuminf_le) |
|
575 |
hence ley: "psuminf (f o disjointed A) \<le> x" |
|
576 |
by (metis xeq) |
|
577 |
hence y: "psuminf (f o disjointed A) \<in> measure_set M f s" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
578 |
apply (auto simp add: measure_set_def) |
38656 | 579 |
apply (rule_tac x="disjointed A" in exI) |
580 |
apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
581 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
582 |
show ?thesis |
38656 | 583 |
by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
584 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
585 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
586 |
lemma (in algebra) inf_measure_close: |
38656 | 587 |
assumes posf: "positive f" and e: "0 < e" and ss: "s \<subseteq> (space M)" |
588 |
shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and> |
|
589 |
psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e" |
|
590 |
proof (cases "Inf (measure_set M f s) = \<omega>") |
|
591 |
case False |
|
592 |
obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e" |
|
593 |
using Inf_close[OF False e] by auto |
|
594 |
thus ?thesis |
|
595 |
by (auto intro!: exI[of _ l] simp: measure_set_def comp_def) |
|
596 |
next |
|
597 |
case True |
|
598 |
have "measure_set M f s \<noteq> {}" |
|
39096 | 599 |
by (metis emptyE ss inf_measure_nonempty [of f, OF posf top _ empty_sets]) |
38656 | 600 |
then obtain l where "l \<in> measure_set M f s" by auto |
601 |
moreover from True have "l \<le> Inf (measure_set M f s) + e" by simp |
|
602 |
ultimately show ?thesis |
|
603 |
by (auto intro!: exI[of _ l] simp: measure_set_def comp_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
604 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
605 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
606 |
lemma (in algebra) inf_measure_countably_subadditive: |
38656 | 607 |
assumes posf: "positive f" and inc: "increasing M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
608 |
shows "countably_subadditive (| space = space M, sets = Pow (space M) |) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
609 |
(\<lambda>x. Inf (measure_set M f x))" |
38656 | 610 |
unfolding countably_subadditive_def o_def |
611 |
proof (safe, simp, rule pinfreal_le_epsilon) |
|
612 |
fix A :: "nat \<Rightarrow> 'a set" and e :: pinfreal |
|
613 |
||
614 |
let "?outer n" = "Inf (measure_set M f (A n))" |
|
615 |
assume A: "range A \<subseteq> Pow (space M)" |
|
616 |
and disj: "disjoint_family A" |
|
617 |
and sb: "(\<Union>i. A i) \<subseteq> space M" |
|
618 |
and e: "0 < e" |
|
619 |
hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and> |
|
620 |
A n \<subseteq> (\<Union>i. BB n i) \<and> |
|
621 |
psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)" |
|
622 |
apply (safe intro!: choice inf_measure_close [of f, OF posf _]) |
|
623 |
using e sb by (cases e, auto simp add: not_le mult_pos_pos) |
|
624 |
then obtain BB |
|
625 |
where BB: "\<And>n. (range (BB n) \<subseteq> sets M)" |
|
626 |
and disjBB: "\<And>n. disjoint_family (BB n)" |
|
627 |
and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)" |
|
628 |
and BBle: "\<And>n. psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)" |
|
629 |
by auto blast |
|
630 |
have sll: "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> psuminf ?outer + e" |
|
631 |
proof - |
|
632 |
have "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> (\<Sum>\<^isub>\<infinity> n. ?outer n + e*(1/2) ^ Suc n)" |
|
633 |
by (rule psuminf_le[OF BBle]) |
|
634 |
also have "... = psuminf ?outer + e" |
|
635 |
using psuminf_half_series by simp |
|
636 |
finally show ?thesis . |
|
637 |
qed |
|
638 |
def C \<equiv> "(split BB) o prod_decode" |
|
639 |
have C: "!!n. C n \<in> sets M" |
|
640 |
apply (rule_tac p="prod_decode n" in PairE) |
|
641 |
apply (simp add: C_def) |
|
642 |
apply (metis BB subsetD rangeI) |
|
643 |
done |
|
644 |
have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)" |
|
645 |
proof (auto simp add: C_def) |
|
646 |
fix x i |
|
647 |
assume x: "x \<in> A i" |
|
648 |
with sbBB [of i] obtain j where "x \<in> BB i j" |
|
649 |
by blast |
|
650 |
thus "\<exists>i. x \<in> split BB (prod_decode i)" |
|
651 |
by (metis prod_encode_inverse prod.cases) |
|
652 |
qed |
|
653 |
have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode" |
|
654 |
by (rule ext) (auto simp add: C_def) |
|
655 |
moreover have "psuminf ... = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" using BBle |
|
656 |
by (force intro!: psuminf_2dimen simp: o_def) |
|
657 |
ultimately have Csums: "psuminf (f \<circ> C) = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" by simp |
|
658 |
have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" |
|
659 |
apply (rule inf_measure_le [OF posf(1) inc], auto) |
|
660 |
apply (rule_tac x="C" in exI) |
|
661 |
apply (auto simp add: C sbC Csums) |
|
662 |
done |
|
663 |
also have "... \<le> (\<Sum>\<^isub>\<infinity>n. Inf (measure_set M f (A n))) + e" using sll |
|
664 |
by blast |
|
665 |
finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> psuminf ?outer + e" . |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
666 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
667 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
668 |
lemma (in algebra) inf_measure_outer: |
38656 | 669 |
"\<lbrakk> positive f ; increasing M f \<rbrakk> |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
670 |
\<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
671 |
(\<lambda>x. Inf (measure_set M f x))" |
38656 | 672 |
by (simp add: outer_measure_space_def inf_measure_empty |
673 |
inf_measure_increasing inf_measure_countably_subadditive positive_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
674 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
675 |
(*MOVE UP*) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
676 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
677 |
lemma (in algebra) algebra_subset_lambda_system: |
38656 | 678 |
assumes posf: "positive f" and inc: "increasing M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
679 |
and add: "additive M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
680 |
shows "sets M \<subseteq> lambda_system (| space = space M, sets = Pow (space M) |) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
681 |
(\<lambda>x. Inf (measure_set M f x))" |
38656 | 682 |
proof (auto dest: sets_into_space |
683 |
simp add: algebra.lambda_system_eq [OF algebra_Pow]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
684 |
fix x s |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
685 |
assume x: "x \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
686 |
and s: "s \<subseteq> space M" |
38656 | 687 |
have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
688 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
689 |
have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
690 |
\<le> Inf (measure_set M f s)" |
38656 | 691 |
proof (rule pinfreal_le_epsilon) |
692 |
fix e :: pinfreal |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
693 |
assume e: "0 < e" |
38656 | 694 |
from inf_measure_close [of f, OF posf e s] |
695 |
obtain A where A: "range A \<subseteq> sets M" and disj: "disjoint_family A" |
|
696 |
and sUN: "s \<subseteq> (\<Union>i. A i)" |
|
697 |
and l: "psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e" |
|
33536 | 698 |
by auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
699 |
have [simp]: "!!x. x \<in> sets M \<Longrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
700 |
(f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)" |
33536 | 701 |
by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
702 |
have [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)" |
33536 | 703 |
by (subst additiveD [OF add, symmetric]) |
704 |
(auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
705 |
{ fix u |
33536 | 706 |
assume u: "u \<in> sets M" |
38656 | 707 |
have [simp]: "\<And>n. f (A n \<inter> u) \<le> f (A n)" |
708 |
by (simp add: increasingD [OF inc] u Int range_subsetD [OF A]) |
|
709 |
have 2: "Inf (measure_set M f (s \<inter> u)) \<le> psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A)" |
|
710 |
proof (rule complete_lattice_class.Inf_lower) |
|
711 |
show "psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)" |
|
712 |
apply (simp add: measure_set_def) |
|
713 |
apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI) |
|
714 |
apply (auto simp add: disjoint_family_subset [OF disj] o_def) |
|
715 |
apply (blast intro: u range_subsetD [OF A]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
716 |
apply (blast dest: subsetD [OF sUN]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
717 |
done |
38656 | 718 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
719 |
} note lesum = this |
38656 | 720 |
have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> psuminf (f o (\<lambda>z. z\<inter>x) o A)" |
721 |
and inf2: "Inf (measure_set M f (s \<inter> (space M - x))) |
|
722 |
\<le> psuminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)" |
|
33536 | 723 |
by (metis Diff lesum top x)+ |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
724 |
hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x)) |
38656 | 725 |
\<le> psuminf (f o (\<lambda>s. s\<inter>x) o A) + psuminf (f o (\<lambda>s. s-x) o A)" |
726 |
by (simp add: x add_mono) |
|
727 |
also have "... \<le> psuminf (f o A)" |
|
728 |
by (simp add: x psuminf_add[symmetric] o_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
729 |
also have "... \<le> Inf (measure_set M f s) + e" |
38656 | 730 |
by (rule l) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
731 |
finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
732 |
\<le> Inf (measure_set M f s) + e" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
733 |
qed |
38656 | 734 |
moreover |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
735 |
have "Inf (measure_set M f s) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
736 |
\<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
737 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
738 |
have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
739 |
by (metis Un_Diff_Int Un_commute) |
38656 | 740 |
also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" |
741 |
apply (rule subadditiveD) |
|
742 |
apply (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow |
|
33536 | 743 |
inf_measure_positive inf_measure_countably_subadditive posf inc) |
38656 | 744 |
apply (auto simp add: subsetD [OF s]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
745 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
746 |
finally show ?thesis . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
747 |
qed |
38656 | 748 |
ultimately |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
749 |
show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
750 |
= Inf (measure_set M f s)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
751 |
by (rule order_antisym) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
752 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
753 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
754 |
lemma measure_down: |
38656 | 755 |
"measure_space N \<mu> \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow> |
756 |
(\<nu> = \<mu>) \<Longrightarrow> measure_space M \<nu>" |
|
757 |
by (simp add: measure_space_def measure_space_axioms_def positive_def |
|
758 |
countably_additive_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
759 |
blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
760 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
761 |
theorem (in algebra) caratheodory: |
38656 | 762 |
assumes posf: "positive f" and ca: "countably_additive M f" |
763 |
shows "\<exists>\<mu> :: 'a set \<Rightarrow> pinfreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> measure_space (sigma (space M) (sets M)) \<mu>" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
764 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
765 |
have inc: "increasing M f" |
38656 | 766 |
by (metis additive_increasing ca countably_additive_additive posf) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
767 |
let ?infm = "(\<lambda>x. Inf (measure_set M f x))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
768 |
def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm" |
38656 | 769 |
have mls: "measure_space \<lparr>space = space M, sets = ls\<rparr> ?infm" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
770 |
using sigma_algebra.caratheodory_lemma |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
771 |
[OF sigma_algebra_Pow inf_measure_outer [OF posf inc]] |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
772 |
by (simp add: ls_def) |
38656 | 773 |
hence sls: "sigma_algebra (|space = space M, sets = ls|)" |
774 |
by (simp add: measure_space_def) |
|
775 |
have "sets M \<subseteq> ls" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
776 |
by (simp add: ls_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
777 |
(metis ca posf inc countably_additive_additive algebra_subset_lambda_system) |
38656 | 778 |
hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
779 |
using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"] |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
780 |
by simp |
38656 | 781 |
have "measure_space (sigma (space M) (sets M)) ?infm" |
782 |
unfolding sigma_def |
|
783 |
by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
784 |
(simp_all add: sgs_sb space_closed) |
38656 | 785 |
thus ?thesis using inf_measure_agrees [OF posf ca] by (auto intro!: exI[of _ ?infm]) |
786 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
787 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
788 |
end |