author | bulwahn |
Sun, 16 Sep 2012 06:51:36 +0200 | |
changeset 49394 | 52e636ace94e |
parent 47762 | d31085f07f60 |
child 49773 | 16907431e477 |
permissions | -rw-r--r-- |
42067 | 1 |
(* Title: HOL/Probability/Caratheodory.thy |
2 |
Author: Lawrence C Paulson |
|
3 |
Author: Johannes Hölzl, TU München |
|
4 |
*) |
|
5 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
6 |
header {*Caratheodory Extension Theorem*} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
7 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
8 |
theory Caratheodory |
47694 | 9 |
imports Measure_Space |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
10 |
begin |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
11 |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42145
diff
changeset
|
12 |
lemma sums_def2: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42145
diff
changeset
|
13 |
"f sums x \<longleftrightarrow> (\<lambda>n. (\<Sum>i\<le>n. f i)) ----> x" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42145
diff
changeset
|
14 |
unfolding sums_def |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42145
diff
changeset
|
15 |
apply (subst LIMSEQ_Suc_iff[symmetric]) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42145
diff
changeset
|
16 |
unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost .. |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42145
diff
changeset
|
17 |
|
42067 | 18 |
text {* |
19 |
Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson. |
|
20 |
*} |
|
21 |
||
43920 | 22 |
lemma suminf_ereal_2dimen: |
23 |
fixes f:: "nat \<times> nat \<Rightarrow> ereal" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
24 |
assumes pos: "\<And>p. 0 \<le> f p" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
25 |
assumes "\<And>m. g m = (\<Sum>n. f (m,n))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
26 |
shows "(\<Sum>i. f (prod_decode i)) = suminf g" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
27 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
28 |
have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
29 |
using assms by (simp add: fun_eq_iff) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
30 |
have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
31 |
by (simp add: setsum_reindex[OF inj_prod_decode] comp_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
32 |
{ fix n |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
33 |
let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
34 |
{ fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
35 |
then have "a < ?M fst" "b < ?M snd" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
36 |
by (auto intro!: Max_ge le_imp_less_Suc image_eqI) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
37 |
then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
38 |
by (auto intro!: setsum_mono3 simp: pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
39 |
then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
40 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
41 |
{ fix a b |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
42 |
let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
43 |
{ fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
44 |
by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
45 |
then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
46 |
by (auto intro!: setsum_mono3 simp: pos) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
47 |
ultimately |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
48 |
show ?thesis unfolding g_def using pos |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset
|
49 |
by (auto intro!: SUPR_eq simp: setsum_cartesian_product reindex SUP_upper2 |
43920 | 50 |
setsum_nonneg suminf_ereal_eq_SUPR SUPR_pair |
51 |
SUPR_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
52 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
53 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
54 |
subsection {* Measure Spaces *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
55 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
56 |
definition subadditive where "subadditive M f \<longleftrightarrow> |
47694 | 57 |
(\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
58 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
59 |
definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow> |
47694 | 60 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
61 |
(f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
62 |
|
47694 | 63 |
definition lambda_system where "lambda_system \<Omega> M f = {l \<in> M. |
64 |
\<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
65 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
66 |
definition outer_measure_space where "outer_measure_space M f \<longleftrightarrow> |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
67 |
positive M f \<and> increasing M f \<and> countably_subadditive M f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
68 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
69 |
definition measure_set where "measure_set M f X = {r. |
47694 | 70 |
\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
71 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
72 |
lemma subadditiveD: |
47694 | 73 |
"subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
74 |
by (auto simp add: subadditive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
75 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
76 |
subsection {* Lambda Systems *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
77 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
78 |
lemma (in algebra) lambda_system_eq: |
47694 | 79 |
shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
80 |
proof - |
47694 | 81 |
have [simp]: "!!l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l" |
37032 | 82 |
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
|
83 |
show ?thesis |
37032 | 84 |
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
|
85 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
86 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
87 |
lemma (in algebra) lambda_system_empty: |
47694 | 88 |
"positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
89 |
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
|
90 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
91 |
lemma lambda_system_sets: |
47694 | 92 |
"x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
93 |
by (simp add: lambda_system_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
94 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
95 |
lemma (in algebra) lambda_system_Compl: |
43920 | 96 |
fixes f:: "'a set \<Rightarrow> ereal" |
47694 | 97 |
assumes x: "x \<in> lambda_system \<Omega> M f" |
98 |
shows "\<Omega> - x \<in> lambda_system \<Omega> M f" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
99 |
proof - |
47694 | 100 |
have "x \<subseteq> \<Omega>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
101 |
by (metis sets_into_space lambda_system_sets x) |
47694 | 102 |
hence "\<Omega> - (\<Omega> - x) = x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
103 |
by (metis double_diff equalityE) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
104 |
with x show ?thesis |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
105 |
by (force simp add: lambda_system_def ac_simps) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
106 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
107 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
108 |
lemma (in algebra) lambda_system_Int: |
43920 | 109 |
fixes f:: "'a set \<Rightarrow> ereal" |
47694 | 110 |
assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f" |
111 |
shows "x \<inter> y \<in> lambda_system \<Omega> M f" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
112 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
113 |
from xl yl show ?thesis |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
114 |
proof (auto simp add: positive_def lambda_system_eq Int) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
115 |
fix u |
47694 | 116 |
assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M" |
117 |
and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z" |
|
118 |
and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z" |
|
119 |
have "u - x \<inter> y \<in> M" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
120 |
by (metis Diff Diff_Int Un u x y) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
121 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
122 |
have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
123 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
124 |
have "u - x \<inter> y - y = u - y" by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
125 |
ultimately |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
126 |
have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
127 |
by force |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
128 |
have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
129 |
= (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
130 |
by (simp add: ey ac_simps) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
131 |
also have "... = (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
132 |
by (simp add: Int_ac) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
133 |
also have "... = f (u \<inter> y) + f (u - y)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
134 |
using fx [THEN bspec, of "u \<inter> y"] Int y u |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
135 |
by force |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
136 |
also have "... = f u" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
137 |
by (metis fy u) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
138 |
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
|
139 |
qed |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
140 |
qed |
33271
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_Un: |
43920 | 143 |
fixes f:: "'a set \<Rightarrow> ereal" |
47694 | 144 |
assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f" |
145 |
shows "x \<union> y \<in> lambda_system \<Omega> M f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
146 |
proof - |
47694 | 147 |
have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M" |
38656 | 148 |
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
|
149 |
moreover |
47694 | 150 |
have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - y))" |
46731 | 151 |
by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+ |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
152 |
ultimately show ?thesis |
38656 | 153 |
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
|
154 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
155 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
156 |
lemma (in algebra) lambda_system_algebra: |
47694 | 157 |
"positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
158 |
apply (auto simp add: algebra_iff_Un) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
159 |
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
|
160 |
apply (metis lambda_system_empty) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
161 |
apply (metis lambda_system_Compl) |
38656 | 162 |
apply (metis lambda_system_Un) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
163 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
164 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
165 |
lemma (in algebra) lambda_system_strong_additive: |
47694 | 166 |
assumes z: "z \<in> M" and disj: "x \<inter> y = {}" |
167 |
and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
168 |
shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
169 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
170 |
have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
171 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
172 |
have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
173 |
moreover |
47694 | 174 |
have "(z \<inter> (x \<union> y)) \<in> M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
175 |
by (metis Int Un lambda_system_sets xl yl z) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
176 |
ultimately show ?thesis using xl yl |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
177 |
by (simp add: lambda_system_eq) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
178 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
179 |
|
47694 | 180 |
lemma (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
181 |
proof (auto simp add: additive_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
182 |
fix x and y |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
183 |
assume disj: "x \<inter> y = {}" |
47694 | 184 |
and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f" |
185 |
hence "x \<in> M" "y \<in> M" by (blast intro: lambda_system_sets)+ |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
186 |
thus "f (x \<union> y) = f x + f y" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
187 |
using lambda_system_strong_additive [OF top disj xl yl] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
188 |
by (simp add: Un) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
189 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
190 |
|
42145 | 191 |
lemma (in ring_of_sets) countably_subadditive_subadditive: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
192 |
assumes f: "positive M f" and cs: "countably_subadditive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
193 |
shows "subadditive M f" |
35582 | 194 |
proof (auto simp add: subadditive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
195 |
fix x y |
47694 | 196 |
assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
197 |
hence "disjoint_family (binaryset x y)" |
35582 | 198 |
by (auto simp add: disjoint_family_on_def binaryset_def) |
47694 | 199 |
hence "range (binaryset x y) \<subseteq> M \<longrightarrow> |
200 |
(\<Union>i. binaryset x y i) \<in> M \<longrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
201 |
f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
202 |
using cs by (auto simp add: countably_subadditive_def) |
47694 | 203 |
hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
204 |
f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
205 |
by (simp add: range_binaryset_eq UN_binaryset_eq) |
38656 | 206 |
thus "f (x \<union> y) \<le> f x + f y" using f x y |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
207 |
by (auto simp add: Un o_def suminf_binaryset_eq positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
208 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
209 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
210 |
lemma lambda_system_increasing: |
47694 | 211 |
"increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f" |
38656 | 212 |
by (simp add: increasing_def lambda_system_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
213 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
214 |
lemma lambda_system_positive: |
47694 | 215 |
"positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
216 |
by (simp add: positive_def lambda_system_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
217 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
218 |
lemma (in algebra) lambda_system_strong_sum: |
43920 | 219 |
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal" |
47694 | 220 |
assumes f: "positive M f" and a: "a \<in> M" |
221 |
and A: "range A \<subseteq> lambda_system \<Omega> M f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
222 |
and disj: "disjoint_family A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
223 |
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
|
224 |
proof (induct n) |
38656 | 225 |
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
|
226 |
next |
38656 | 227 |
case (Suc n) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
228 |
have 2: "A n \<inter> UNION {0..<n} A = {}" using disj |
38656 | 229 |
by (force simp add: disjoint_family_on_def neq_iff) |
47694 | 230 |
have 3: "A n \<in> lambda_system \<Omega> M f" using A |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
231 |
by blast |
47694 | 232 |
interpret l: algebra \<Omega> "lambda_system \<Omega> M f" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
233 |
using f by (rule lambda_system_algebra) |
47694 | 234 |
have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> M f" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
235 |
using A l.UNION_in_sets by simp |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
236 |
from Suc.hyps show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
237 |
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
|
238 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
239 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
240 |
lemma (in sigma_algebra) lambda_system_caratheodory: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
241 |
assumes oms: "outer_measure_space M f" |
47694 | 242 |
and A: "range A \<subseteq> lambda_system \<Omega> M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
243 |
and disj: "disjoint_family A" |
47694 | 244 |
shows "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
245 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
246 |
have pos: "positive M f" and inc: "increasing M f" |
38656 | 247 |
and csa: "countably_subadditive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
248 |
by (metis oms outer_measure_space_def)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
249 |
have sa: "subadditive M f" |
38656 | 250 |
by (metis countably_subadditive_subadditive csa pos) |
47694 | 251 |
have A': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A |
252 |
by auto |
|
253 |
interpret ls: algebra \<Omega> "lambda_system \<Omega> M f" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
254 |
using pos by (rule lambda_system_algebra) |
47694 | 255 |
have A'': "range A \<subseteq> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
256 |
by (metis A image_subset_iff lambda_system_sets) |
38656 | 257 |
|
47694 | 258 |
have U_in: "(\<Union>i. A i) \<in> M" |
37032 | 259 |
by (metis A'' countable_UN) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
260 |
have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
261 |
proof (rule antisym) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
262 |
show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
263 |
using csa[unfolded countably_subadditive_def] A'' disj U_in by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
264 |
have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
265 |
have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
266 |
show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
267 |
using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis] |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
268 |
using A'' |
47694 | 269 |
by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] countable_UN) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
270 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
271 |
{ |
38656 | 272 |
fix a |
47694 | 273 |
assume a [iff]: "a \<in> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
274 |
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
|
275 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
276 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
277 |
proof (rule antisym) |
47694 | 278 |
have "range (\<lambda>i. a \<inter> A i) \<subseteq> M" using A'' |
33536 | 279 |
by blast |
38656 | 280 |
moreover |
33536 | 281 |
have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj |
38656 | 282 |
by (auto simp add: disjoint_family_on_def) |
283 |
moreover |
|
47694 | 284 |
have "a \<inter> (\<Union>i. A i) \<in> M" |
33536 | 285 |
by (metis Int U_in a) |
38656 | 286 |
ultimately |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
287 |
have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
288 |
using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"] |
38656 | 289 |
by (simp add: o_def) |
290 |
hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
291 |
(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))" |
38656 | 292 |
by (rule add_right_mono) |
293 |
moreover |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
294 |
have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
295 |
proof (intro suminf_bound_add allI) |
33536 | 296 |
fix n |
47694 | 297 |
have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M" |
38656 | 298 |
by (metis A'' UNION_in_sets) |
33536 | 299 |
have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A'' |
37032 | 300 |
by (blast intro: increasingD [OF inc] A'' UNION_in_sets) |
47694 | 301 |
have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system \<Omega> M f" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
302 |
using ls.UNION_in_sets by (simp add: A) |
38656 | 303 |
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 | 304 |
by (simp add: lambda_system_eq UNION_in) |
33536 | 305 |
have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))" |
44106 | 306 |
by (blast intro: increasingD [OF inc] UNION_in U_in) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
307 |
thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
38656 | 308 |
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
309 |
next |
47694 | 310 |
have "\<And>i. a \<inter> A i \<in> M" using A'' by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
311 |
then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto |
47694 | 312 |
have "\<And>i. a - (\<Union>i. A i) \<in> M" using A'' by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
313 |
then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
314 |
then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto |
33536 | 315 |
qed |
38656 | 316 |
ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
317 |
by (rule order_trans) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
318 |
next |
38656 | 319 |
have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))" |
37032 | 320 |
by (blast intro: increasingD [OF inc] U_in) |
33536 | 321 |
also have "... \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" |
37032 | 322 |
by (blast intro: subadditiveD [OF sa] U_in) |
33536 | 323 |
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
|
324 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
325 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
326 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
327 |
thus ?thesis |
38656 | 328 |
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
|
329 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
330 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
331 |
lemma (in sigma_algebra) caratheodory_lemma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
332 |
assumes oms: "outer_measure_space M f" |
47694 | 333 |
defines "L \<equiv> lambda_system \<Omega> M f" |
334 |
shows "measure_space \<Omega> L f" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
335 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
336 |
have pos: "positive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
337 |
by (metis oms outer_measure_space_def) |
47694 | 338 |
have alg: "algebra \<Omega> L" |
38656 | 339 |
using lambda_system_algebra [of f, OF pos] |
47694 | 340 |
by (simp add: algebra_iff_Un L_def) |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
341 |
then |
47694 | 342 |
have "sigma_algebra \<Omega> L" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
343 |
using lambda_system_caratheodory [OF oms] |
47694 | 344 |
by (simp add: sigma_algebra_disjoint_iff L_def) |
38656 | 345 |
moreover |
47694 | 346 |
have "countably_additive L f" "positive L f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
347 |
using pos lambda_system_caratheodory [OF oms] |
47694 | 348 |
by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def) |
38656 | 349 |
ultimately |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
350 |
show ?thesis |
47694 | 351 |
using pos by (simp add: measure_space_def) |
38656 | 352 |
qed |
353 |
||
39096 | 354 |
lemma inf_measure_nonempty: |
47694 | 355 |
assumes f: "positive M f" and b: "b \<in> M" and a: "a \<subseteq> b" "{} \<in> M" |
39096 | 356 |
shows "f b \<in> measure_set M f a" |
357 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
358 |
let ?A = "\<lambda>i::nat. (if i = 0 then b else {})" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
359 |
have "(\<Sum>i. f (?A i)) = (\<Sum>i<1::nat. f (?A i))" |
47761 | 360 |
by (rule suminf_finite) (simp_all add: f[unfolded positive_def]) |
39096 | 361 |
also have "... = f b" |
362 |
by simp |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
363 |
finally show ?thesis using assms |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
364 |
by (auto intro!: exI [of _ ?A] |
39096 | 365 |
simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def) |
366 |
qed |
|
367 |
||
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
368 |
lemma (in ring_of_sets) inf_measure_agrees: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
369 |
assumes posf: "positive M f" and ca: "countably_additive M f" |
47694 | 370 |
and s: "s \<in> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
371 |
shows "Inf (measure_set M f s) = f s" |
43920 | 372 |
unfolding Inf_ereal_def |
38656 | 373 |
proof (safe intro!: Greatest_equality) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
374 |
fix z |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
375 |
assume z: "z \<in> measure_set M f s" |
38656 | 376 |
from this obtain A where |
47694 | 377 |
A: "range A \<subseteq> M" and disj: "disjoint_family A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
378 |
and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z" |
38656 | 379 |
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
|
380 |
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
|
381 |
have inc: "increasing M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
382 |
by (metis additive_increasing ca countably_additive_additive posf) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
383 |
have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
384 |
proof (rule ca[unfolded countably_additive_def, rule_format]) |
47694 | 385 |
show "range (\<lambda>n. A n \<inter> s) \<subseteq> M" using A s |
33536 | 386 |
by blast |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
387 |
show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj |
35582 | 388 |
by (auto simp add: disjoint_family_on_def) |
47694 | 389 |
show "(\<Union>i. A i \<inter> s) \<in> M" using A s |
33536 | 390 |
by (metis UN_extend_simps(4) s seq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
391 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
392 |
hence "f s = (\<Sum>i. f (A i \<inter> s))" |
37032 | 393 |
using seq [symmetric] by (simp add: sums_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
394 |
also have "... \<le> (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
395 |
proof (rule suminf_le_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
396 |
fix n show "f (A n \<inter> s) \<le> f (A n)" using A s |
38656 | 397 |
by (force intro: increasingD [OF inc]) |
47694 | 398 |
fix N have "A N \<inter> s \<in> M" using A s by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
399 |
then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
400 |
qed |
38656 | 401 |
also have "... = z" by (rule si) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
402 |
finally show "f s \<le> z" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
403 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
404 |
fix y |
38656 | 405 |
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
|
406 |
thus "y \<le> f s" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
407 |
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
|
408 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
409 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
410 |
lemma measure_set_pos: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
411 |
assumes posf: "positive M f" "r \<in> measure_set M f X" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
412 |
shows "0 \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
413 |
proof - |
47694 | 414 |
obtain A where "range A \<subseteq> M" and r: "r = (\<Sum>i. f (A i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
415 |
using `r \<in> measure_set M f X` unfolding measure_set_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
416 |
then show "0 \<le> r" using posf unfolding r positive_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
417 |
by (intro suminf_0_le) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
418 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
419 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
420 |
lemma inf_measure_pos: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
421 |
assumes posf: "positive M f" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
422 |
shows "0 \<le> Inf (measure_set M f X)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
423 |
proof (rule complete_lattice_class.Inf_greatest) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
424 |
fix r assume "r \<in> measure_set M f X" with posf show "0 \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
425 |
by (rule measure_set_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
426 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
427 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
428 |
lemma inf_measure_empty: |
47694 | 429 |
assumes posf: "positive M f" and "{} \<in> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
430 |
shows "Inf (measure_set M f {}) = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
431 |
proof (rule antisym) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
432 |
show "Inf (measure_set M f {}) \<le> 0" |
47694 | 433 |
by (metis complete_lattice_class.Inf_lower `{} \<in> M` |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
434 |
inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
435 |
qed (rule inf_measure_pos[OF posf]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
436 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
437 |
lemma (in ring_of_sets) inf_measure_positive: |
47694 | 438 |
assumes p: "positive M f" and "{} \<in> M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
439 |
shows "positive M (\<lambda>x. Inf (measure_set M f x))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
440 |
proof (unfold positive_def, intro conjI ballI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
441 |
show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto |
47694 | 442 |
fix A assume "A \<in> M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
443 |
qed (rule inf_measure_pos[OF p]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
444 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
445 |
lemma (in ring_of_sets) inf_measure_increasing: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
446 |
assumes posf: "positive M f" |
47694 | 447 |
shows "increasing (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))" |
44918 | 448 |
apply (clarsimp simp add: increasing_def) |
38656 | 449 |
apply (rule complete_lattice_class.Inf_greatest) |
450 |
apply (rule complete_lattice_class.Inf_lower) |
|
37032 | 451 |
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
|
452 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
453 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
454 |
lemma (in ring_of_sets) inf_measure_le: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
455 |
assumes posf: "positive M f" and inc: "increasing M f" |
47694 | 456 |
and x: "x \<in> {r . \<exists>A. range A \<subseteq> M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
457 |
shows "Inf (measure_set M f s) \<le> x" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
458 |
proof - |
47694 | 459 |
obtain A where A: "range A \<subseteq> M" and ss: "s \<subseteq> (\<Union>i. A i)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
460 |
and xeq: "(\<Sum>i. f (A i)) = x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
461 |
using x by auto |
47694 | 462 |
have dA: "range (disjointed A) \<subseteq> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
463 |
by (metis A range_disjointed_sets) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
464 |
have "\<forall>n. f (disjointed A n) \<le> f (A n)" |
38656 | 465 |
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
466 |
moreover have "\<forall>i. 0 \<le> f (disjointed A i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
467 |
using posf dA unfolding positive_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
468 |
ultimately have sda: "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
469 |
by (blast intro!: suminf_le_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
470 |
hence ley: "(\<Sum>i. f (disjointed A i)) \<le> x" |
38656 | 471 |
by (metis xeq) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
472 |
hence y: "(\<Sum>i. f (disjointed A i)) \<in> measure_set M f s" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
473 |
apply (auto simp add: measure_set_def) |
38656 | 474 |
apply (rule_tac x="disjointed A" in exI) |
475 |
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
|
476 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
477 |
show ?thesis |
38656 | 478 |
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
|
479 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
480 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
481 |
lemma (in ring_of_sets) inf_measure_close: |
43920 | 482 |
fixes e :: ereal |
47694 | 483 |
assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (\<Omega>)" and "Inf (measure_set M f s) \<noteq> \<infinity>" |
484 |
shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
485 |
(\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
486 |
proof - |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
487 |
from `Inf (measure_set M f s) \<noteq> \<infinity>` have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
488 |
using inf_measure_pos[OF posf, of s] by auto |
38656 | 489 |
obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e" |
43920 | 490 |
using Inf_ereal_close[OF fin e] by auto |
38656 | 491 |
thus ?thesis |
492 |
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
|
493 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
494 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
495 |
lemma (in ring_of_sets) inf_measure_countably_subadditive: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
496 |
assumes posf: "positive M f" and inc: "increasing M f" |
47694 | 497 |
shows "countably_subadditive (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
498 |
proof (simp add: countably_subadditive_def, safe) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
499 |
fix A :: "nat \<Rightarrow> 'a set" |
46731 | 500 |
let ?outer = "\<lambda>B. Inf (measure_set M f B)" |
47694 | 501 |
assume A: "range A \<subseteq> Pow (\<Omega>)" |
38656 | 502 |
and disj: "disjoint_family A" |
47694 | 503 |
and sb: "(\<Union>i. A i) \<subseteq> \<Omega>" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
504 |
|
43920 | 505 |
{ fix e :: ereal assume e: "0 < e" and "\<forall>i. ?outer (A i) \<noteq> \<infinity>" |
47694 | 506 |
hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> M \<and> disjoint_family (BB n) \<and> |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
507 |
A n \<subseteq> (\<Union>i. BB n i) \<and> (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
508 |
apply (safe intro!: choice inf_measure_close [of f, OF posf]) |
43920 | 509 |
using e sb by (auto simp: ereal_zero_less_0_iff one_ereal_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
510 |
then obtain BB |
47694 | 511 |
where BB: "\<And>n. (range (BB n) \<subseteq> M)" |
38656 | 512 |
and disjBB: "\<And>n. disjoint_family (BB n)" |
513 |
and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)" |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
514 |
and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
515 |
by auto blast |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
516 |
have sll: "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n)) + e" |
38656 | 517 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
518 |
have sum_eq_1: "(\<Sum>n. e*(1/2) ^ Suc n) = e" |
43920 | 519 |
using suminf_half_series_ereal e |
520 |
by (simp add: ereal_zero_le_0_iff zero_le_divide_ereal suminf_cmult_ereal) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
521 |
have "\<And>n i. 0 \<le> f (BB n i)" using posf[unfolded positive_def] BB by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
522 |
then have "\<And>n. 0 \<le> (\<Sum>i. f (BB n i))" by (rule suminf_0_le) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
523 |
then have "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n) + e*(1/2) ^ Suc n)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
524 |
by (rule suminf_le_pos[OF BBle]) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
525 |
also have "... = (\<Sum>n. ?outer (A n)) + e" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
526 |
using sum_eq_1 inf_measure_pos[OF posf] e |
43920 | 527 |
by (subst suminf_add_ereal) (auto simp add: ereal_zero_le_0_iff) |
38656 | 528 |
finally show ?thesis . |
529 |
qed |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
530 |
def C \<equiv> "(split BB) o prod_decode" |
47694 | 531 |
have C: "!!n. C n \<in> M" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
532 |
apply (rule_tac p="prod_decode n" in PairE) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
533 |
apply (simp add: C_def) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
534 |
apply (metis BB subsetD rangeI) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
535 |
done |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
536 |
have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)" |
38656 | 537 |
proof (auto simp add: C_def) |
538 |
fix x i |
|
539 |
assume x: "x \<in> A i" |
|
540 |
with sbBB [of i] obtain j where "x \<in> BB i j" |
|
541 |
by blast |
|
542 |
thus "\<exists>i. x \<in> split BB (prod_decode i)" |
|
543 |
by (metis prod_encode_inverse prod.cases) |
|
544 |
qed |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
545 |
have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
546 |
by (rule ext) (auto simp add: C_def) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
547 |
moreover have "suminf ... = (\<Sum>n. \<Sum>i. f (BB n i))" using BBle |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
548 |
using BB posf[unfolded positive_def] |
43920 | 549 |
by (force intro!: suminf_ereal_2dimen simp: o_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
550 |
ultimately have Csums: "(\<Sum>i. f (C i)) = (\<Sum>n. \<Sum>i. f (BB n i))" by (simp add: o_def) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
551 |
have "?outer (\<Union>i. A i) \<le> (\<Sum>n. \<Sum>i. f (BB n i))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
552 |
apply (rule inf_measure_le [OF posf(1) inc], auto) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
553 |
apply (rule_tac x="C" in exI) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
554 |
apply (auto simp add: C sbC Csums) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
555 |
done |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
556 |
also have "... \<le> (\<Sum>n. ?outer (A n)) + e" using sll |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
557 |
by blast |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
558 |
finally have "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n)) + e" . } |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
559 |
note for_finite_Inf = this |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
560 |
|
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
561 |
show "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
562 |
proof cases |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
563 |
assume "\<forall>i. ?outer (A i) \<noteq> \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
564 |
with for_finite_Inf show ?thesis |
43920 | 565 |
by (intro ereal_le_epsilon) auto |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
566 |
next |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
567 |
assume "\<not> (\<forall>i. ?outer (A i) \<noteq> \<infinity>)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
568 |
then have "\<exists>i. ?outer (A i) = \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
569 |
by auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
570 |
then have "(\<Sum>n. ?outer (A n)) = \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
571 |
using suminf_PInfty[OF inf_measure_pos, OF posf] |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
572 |
by metis |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
573 |
then show ?thesis by simp |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
574 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
575 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
576 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
577 |
lemma (in ring_of_sets) inf_measure_outer: |
47694 | 578 |
"\<lbrakk> positive M f ; increasing M f \<rbrakk> \<Longrightarrow> |
579 |
outer_measure_space (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
580 |
using inf_measure_pos[of M f] |
38656 | 581 |
by (simp add: outer_measure_space_def inf_measure_empty |
582 |
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
|
583 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
584 |
lemma (in ring_of_sets) algebra_subset_lambda_system: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
585 |
assumes posf: "positive M f" and inc: "increasing M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
586 |
and add: "additive M f" |
47694 | 587 |
shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))" |
38656 | 588 |
proof (auto dest: sets_into_space |
589 |
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
|
590 |
fix x s |
47694 | 591 |
assume x: "x \<in> M" |
592 |
and s: "s \<subseteq> \<Omega>" |
|
593 |
have [simp]: "!!x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - x) = s-x" using s |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
594 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
595 |
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
|
596 |
\<le> Inf (measure_set M f s)" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
597 |
proof cases |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
598 |
assume "Inf (measure_set M f s) = \<infinity>" then show ?thesis by simp |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
599 |
next |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
600 |
assume fin: "Inf (measure_set M f s) \<noteq> \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
601 |
then have "measure_set M f s \<noteq> {}" |
43920 | 602 |
by (auto simp: top_ereal_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
603 |
show ?thesis |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
604 |
proof (rule complete_lattice_class.Inf_greatest) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
605 |
fix r assume "r \<in> measure_set M f s" |
47694 | 606 |
then obtain A where A: "disjoint_family A" "range A \<subseteq> M" "s \<subseteq> (\<Union>i. A i)" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
607 |
and r: "r = (\<Sum>i. f (A i))" unfolding measure_set_def by auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
608 |
have "Inf (measure_set M f (s \<inter> x)) \<le> (\<Sum>i. f (A i \<inter> x))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
609 |
unfolding measure_set_def |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
610 |
proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i \<inter> x"]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
611 |
from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
612 |
by (rule disjoint_family_on_bisimulation) auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
613 |
qed (insert x A, auto) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
614 |
moreover |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
615 |
have "Inf (measure_set M f (s - x)) \<le> (\<Sum>i. f (A i - x))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
616 |
unfolding measure_set_def |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
617 |
proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i - x"]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
618 |
from A(1) show "disjoint_family (\<lambda>i. A i - x)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
619 |
by (rule disjoint_family_on_bisimulation) auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
620 |
qed (insert x A, auto) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
621 |
ultimately have "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le> |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
622 |
(\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
623 |
also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))" |
43920 | 624 |
using A(2) x posf by (subst suminf_add_ereal) (auto simp: positive_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
625 |
also have "\<dots> = (\<Sum>i. f (A i))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
626 |
using A x |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
627 |
by (subst add[THEN additiveD, symmetric]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
628 |
(auto intro!: arg_cong[where f=suminf] arg_cong[where f=f]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
629 |
finally show "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le> r" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
630 |
using r by simp |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
631 |
qed |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
632 |
qed |
38656 | 633 |
moreover |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
634 |
have "Inf (measure_set M f s) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
635 |
\<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" |
42145 | 636 |
proof - |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
637 |
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
|
638 |
by (metis Un_Diff_Int Un_commute) |
38656 | 639 |
also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" |
640 |
apply (rule subadditiveD) |
|
42145 | 641 |
apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
642 |
apply (simp add: positive_def inf_measure_empty[OF posf] inf_measure_pos[OF posf]) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
643 |
apply (rule inf_measure_countably_subadditive) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
644 |
using s by (auto intro!: posf inc) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
645 |
finally show ?thesis . |
42145 | 646 |
qed |
38656 | 647 |
ultimately |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
648 |
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
|
649 |
= Inf (measure_set M f s)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
650 |
by (rule order_antisym) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
651 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
652 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
653 |
lemma measure_down: |
47694 | 654 |
"measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>" |
655 |
by (simp add: measure_space_def positive_def countably_additive_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
656 |
blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
657 |
|
47762 | 658 |
theorem (in ring_of_sets) caratheodory': |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
659 |
assumes posf: "positive M f" and ca: "countably_additive M f" |
47694 | 660 |
shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
661 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
662 |
have inc: "increasing M f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
663 |
by (metis additive_increasing ca countably_additive_additive posf) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
664 |
let ?infm = "(\<lambda>x. Inf (measure_set M f x))" |
47694 | 665 |
def ls \<equiv> "lambda_system \<Omega> (Pow \<Omega>) ?infm" |
666 |
have mls: "measure_space \<Omega> ls ?infm" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
667 |
using sigma_algebra.caratheodory_lemma |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
668 |
[OF sigma_algebra_Pow inf_measure_outer [OF posf inc]] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
669 |
by (simp add: ls_def) |
47694 | 670 |
hence sls: "sigma_algebra \<Omega> ls" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
671 |
by (simp add: measure_space_def) |
47694 | 672 |
have "M \<subseteq> ls" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
673 |
by (simp add: ls_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
674 |
(metis ca posf inc countably_additive_additive algebra_subset_lambda_system) |
47694 | 675 |
hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls" |
676 |
using sigma_algebra.sigma_sets_subset [OF sls, of "M"] |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
677 |
by simp |
47694 | 678 |
have "measure_space \<Omega> (sigma_sets \<Omega> M) ?infm" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
679 |
by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
680 |
(simp_all add: sgs_sb space_closed) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
681 |
thus ?thesis using inf_measure_agrees [OF posf ca] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
682 |
by (intro exI[of _ ?infm]) auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
683 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
684 |
|
42145 | 685 |
subsubsection {*Alternative instances of caratheodory*} |
686 |
||
687 |
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below: |
|
688 |
assumes f: "positive M f" "additive M f" |
|
689 |
shows "countably_additive M f \<longleftrightarrow> |
|
47694 | 690 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))" |
42145 | 691 |
unfolding countably_additive_def |
692 |
proof safe |
|
47694 | 693 |
assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)" |
694 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M" |
|
695 |
then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets) |
|
42145 | 696 |
with count_sum[THEN spec, of "disjointed A"] A(3) |
697 |
have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)" |
|
698 |
by (auto simp: UN_disjointed_eq disjoint_family_disjointed) |
|
699 |
moreover have "(\<lambda>n. (\<Sum>i=0..<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" |
|
700 |
using f(1)[unfolded positive_def] dA |
|
43920 | 701 |
by (auto intro!: summable_sumr_LIMSEQ_suminf summable_ereal_pos) |
42145 | 702 |
from LIMSEQ_Suc[OF this] |
703 |
have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" |
|
704 |
unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost . |
|
705 |
moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" |
|
706 |
using disjointed_additive[OF f A(1,2)] . |
|
707 |
ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp |
|
708 |
next |
|
47694 | 709 |
assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
710 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M" |
|
42145 | 711 |
have *: "(\<Union>n. (\<Union>i\<le>n. A i)) = (\<Union>i. A i)" by auto |
712 |
have "(\<lambda>n. f (\<Union>i\<le>n. A i)) ----> f (\<Union>i. A i)" |
|
713 |
proof (unfold *[symmetric], intro cont[rule_format]) |
|
47694 | 714 |
show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> M" |
42145 | 715 |
using A * by auto |
716 |
qed (force intro!: incseq_SucI) |
|
717 |
moreover have "\<And>n. f (\<Union>i\<le>n. A i) = (\<Sum>i\<le>n. f (A i))" |
|
718 |
using A |
|
719 |
by (intro additive_sum[OF f, of _ A, symmetric]) |
|
720 |
(auto intro: disjoint_family_on_mono[where B=UNIV]) |
|
721 |
ultimately |
|
722 |
have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)" |
|
723 |
unfolding sums_def2 by simp |
|
724 |
from sums_unique[OF this] |
|
725 |
show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp |
|
726 |
qed |
|
727 |
||
728 |
lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous: |
|
729 |
assumes f: "positive M f" "additive M f" |
|
47694 | 730 |
shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)) |
731 |
\<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)" |
|
42145 | 732 |
proof safe |
47694 | 733 |
assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))" |
734 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>" |
|
42145 | 735 |
with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0" |
736 |
using `positive M f`[unfolded positive_def] by auto |
|
737 |
next |
|
47694 | 738 |
assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
739 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>" |
|
42145 | 740 |
|
47694 | 741 |
have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b" |
42145 | 742 |
using additive_increasing[OF f] unfolding increasing_def by simp |
743 |
||
744 |
have decseq_fA: "decseq (\<lambda>i. f (A i))" |
|
745 |
using A by (auto simp: decseq_def intro!: f_mono) |
|
746 |
have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))" |
|
747 |
using A by (auto simp: decseq_def) |
|
748 |
then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))" |
|
749 |
using A unfolding decseq_def by (auto intro!: f_mono Diff) |
|
750 |
have "f (\<Inter>x. A x) \<le> f (A 0)" |
|
751 |
using A by (auto intro!: f_mono) |
|
752 |
then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>" |
|
753 |
using A by auto |
|
754 |
{ fix i |
|
755 |
have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono) |
|
756 |
then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>" |
|
757 |
using A by auto } |
|
758 |
note f_fin = this |
|
759 |
have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0" |
|
760 |
proof (intro cont[rule_format, OF _ decseq _ f_fin]) |
|
47694 | 761 |
show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}" |
42145 | 762 |
using A by auto |
763 |
qed |
|
43920 | 764 |
from INF_Lim_ereal[OF decseq_f this] |
42145 | 765 |
have "(INF n. f (A n - (\<Inter>i. A i))) = 0" . |
766 |
moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)" |
|
767 |
by auto |
|
768 |
ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)" |
|
769 |
using A(4) f_fin f_Int_fin |
|
43920 | 770 |
by (subst INFI_ereal_add) (auto simp: decseq_f) |
42145 | 771 |
moreover { |
772 |
fix n |
|
773 |
have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))" |
|
774 |
using A by (subst f(2)[THEN additiveD]) auto |
|
775 |
also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n" |
|
776 |
by auto |
|
777 |
finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . } |
|
778 |
ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)" |
|
779 |
by simp |
|
43920 | 780 |
with LIMSEQ_ereal_INFI[OF decseq_fA] |
42145 | 781 |
show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp |
782 |
qed |
|
783 |
||
784 |
lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def) |
|
47694 | 785 |
lemma positiveD2: "positive M f \<Longrightarrow> A \<in> M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def) |
42145 | 786 |
|
787 |
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below: |
|
47694 | 788 |
assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>" |
789 |
assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
790 |
assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M" |
|
42145 | 791 |
shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
792 |
proof - |
|
47694 | 793 |
have "\<forall>A\<in>M. \<exists>x. f A = ereal x" |
42145 | 794 |
proof |
47694 | 795 |
fix A assume "A \<in> M" with f show "\<exists>x. f A = ereal x" |
42145 | 796 |
unfolding positive_def by (cases "f A") auto |
797 |
qed |
|
798 |
from bchoice[OF this] guess f' .. note f' = this[rule_format] |
|
799 |
from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0" |
|
800 |
by (intro cont[rule_format]) (auto simp: decseq_def incseq_def) |
|
801 |
moreover |
|
802 |
{ fix i |
|
803 |
have "f ((\<Union>i. A i) - A i) + f (A i) = f ((\<Union>i. A i) - A i \<union> A i)" |
|
804 |
using A by (intro f(2)[THEN additiveD, symmetric]) auto |
|
805 |
also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)" |
|
806 |
by auto |
|
807 |
finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)" |
|
808 |
using A by (subst (asm) (1 2 3) f') auto |
|
43920 | 809 |
then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))" |
42145 | 810 |
using A f' by auto } |
811 |
ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0" |
|
43920 | 812 |
by (simp add: zero_ereal_def) |
42145 | 813 |
then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)" |
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44106
diff
changeset
|
814 |
by (rule LIMSEQ_diff_approach_zero2[OF tendsto_const]) |
42145 | 815 |
then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
816 |
using A by (subst (1 2) f') auto |
|
817 |
qed |
|
818 |
||
819 |
lemma (in ring_of_sets) empty_continuous_imp_countably_additive: |
|
47694 | 820 |
assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>" |
821 |
assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
42145 | 822 |
shows "countably_additive M f" |
823 |
using countably_additive_iff_continuous_from_below[OF f] |
|
824 |
using empty_continuous_imp_continuous_from_below[OF f fin] cont |
|
825 |
by blast |
|
826 |
||
827 |
lemma (in ring_of_sets) caratheodory_empty_continuous: |
|
47694 | 828 |
assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>" |
829 |
assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
830 |
shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
|
47762 | 831 |
proof (intro caratheodory' empty_continuous_imp_countably_additive f) |
47694 | 832 |
show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto |
42145 | 833 |
qed (rule cont) |
834 |
||
47762 | 835 |
section {* Volumes *} |
836 |
||
837 |
definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
838 |
"volume M f \<longleftrightarrow> |
|
839 |
(f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and> |
|
840 |
(\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))" |
|
841 |
||
842 |
lemma volumeI: |
|
843 |
assumes "f {} = 0" |
|
844 |
assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a" |
|
845 |
assumes "\<And>C. C \<subseteq> M \<Longrightarrow> disjoint C \<Longrightarrow> finite C \<Longrightarrow> \<Union>C \<in> M \<Longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c)" |
|
846 |
shows "volume M f" |
|
847 |
using assms by (auto simp: volume_def) |
|
848 |
||
849 |
lemma volume_positive: |
|
850 |
"volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a" |
|
851 |
by (auto simp: volume_def) |
|
852 |
||
853 |
lemma volume_empty: |
|
854 |
"volume M f \<Longrightarrow> f {} = 0" |
|
855 |
by (auto simp: volume_def) |
|
856 |
||
857 |
lemma volume_finite_additive: |
|
858 |
assumes "volume M f" |
|
859 |
assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M" |
|
860 |
shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))" |
|
861 |
proof - |
|
862 |
have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>A`I \<in> M" |
|
863 |
using A unfolding SUP_def by (auto simp: disjoint_family_on_disjoint_image) |
|
864 |
with `volume M f` have "f (\<Union>A`I) = (\<Sum>a\<in>A`I. f a)" |
|
865 |
unfolding volume_def by blast |
|
866 |
also have "\<dots> = (\<Sum>i\<in>I. f (A i))" |
|
867 |
proof (subst setsum_reindex_nonzero) |
|
868 |
fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j" |
|
869 |
with `disjoint_family_on A I` have "A i = {}" |
|
870 |
by (auto simp: disjoint_family_on_def) |
|
871 |
then show "f (A i) = 0" |
|
872 |
using volume_empty[OF `volume M f`] by simp |
|
873 |
qed (auto intro: `finite I`) |
|
874 |
finally show "f (UNION I A) = (\<Sum>i\<in>I. f (A i))" |
|
875 |
by simp |
|
876 |
qed |
|
877 |
||
878 |
lemma (in ring_of_sets) volume_additiveI: |
|
879 |
assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a" |
|
880 |
assumes [simp]: "\<mu> {} = 0" |
|
881 |
assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b" |
|
882 |
shows "volume M \<mu>" |
|
883 |
proof (unfold volume_def, safe) |
|
884 |
fix C assume "finite C" "C \<subseteq> M" "disjoint C" |
|
885 |
then show "\<mu> (\<Union>C) = setsum \<mu> C" |
|
886 |
proof (induct C) |
|
887 |
case (insert c C) |
|
888 |
from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)" |
|
889 |
by (auto intro!: add simp: disjoint_def) |
|
890 |
with insert show ?case |
|
891 |
by (simp add: disjoint_def) |
|
892 |
qed simp |
|
893 |
qed fact+ |
|
894 |
||
895 |
lemma (in semiring_of_sets) extend_volume: |
|
896 |
assumes "volume M \<mu>" |
|
897 |
shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)" |
|
898 |
proof - |
|
899 |
let ?R = generated_ring |
|
900 |
have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)" |
|
901 |
by (auto simp: generated_ring_def) |
|
902 |
from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this |
|
903 |
||
904 |
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C" |
|
905 |
fix D assume D: "D \<subseteq> M" "finite D" "disjoint D" |
|
906 |
assume "\<Union>C = \<Union>D" |
|
907 |
have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))" |
|
908 |
proof (intro setsum_cong refl) |
|
909 |
fix d assume "d \<in> D" |
|
910 |
have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d" |
|
911 |
using `d \<in> D` `\<Union>C = \<Union>D` by auto |
|
912 |
moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" |
|
913 |
proof (rule volume_finite_additive) |
|
914 |
{ fix c assume "c \<in> C" then show "c \<inter> d \<in> M" |
|
915 |
using C D `d \<in> D` by auto } |
|
916 |
show "(\<Union>a\<in>C. a \<inter> d) \<in> M" |
|
917 |
unfolding Un_eq_d using `d \<in> D` D by auto |
|
918 |
show "disjoint_family_on (\<lambda>a. a \<inter> d) C" |
|
919 |
using `disjoint C` by (auto simp: disjoint_family_on_def disjoint_def) |
|
920 |
qed fact+ |
|
921 |
ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp |
|
922 |
qed } |
|
923 |
note split_sum = this |
|
924 |
||
925 |
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C" |
|
926 |
fix D assume D: "D \<subseteq> M" "finite D" "disjoint D" |
|
927 |
assume "\<Union>C = \<Union>D" |
|
928 |
with split_sum[OF C D] split_sum[OF D C] |
|
929 |
have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)" |
|
930 |
by (simp, subst setsum_commute, simp add: ac_simps) } |
|
931 |
note sum_eq = this |
|
932 |
||
933 |
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C" |
|
934 |
then have "\<Union>C \<in> ?R" by (auto simp: generated_ring_def) |
|
935 |
with \<mu>'_spec[THEN bspec, of "\<Union>C"] |
|
936 |
obtain D where |
|
937 |
D: "D \<subseteq> M" "finite D" "disjoint D" "\<Union>C = \<Union>D" and "\<mu>' (\<Union>C) = (\<Sum>d\<in>D. \<mu> d)" |
|
938 |
by blast |
|
939 |
with sum_eq[OF C D] have "\<mu>' (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" by simp } |
|
940 |
note \<mu>' = this |
|
941 |
||
942 |
show ?thesis |
|
943 |
proof (intro exI conjI ring_of_sets.volume_additiveI[OF generating_ring] ballI) |
|
944 |
fix a assume "a \<in> M" with \<mu>'[of "{a}"] show "\<mu>' a = \<mu> a" |
|
945 |
by (simp add: disjoint_def) |
|
946 |
next |
|
947 |
fix a assume "a \<in> ?R" then guess Ca .. note Ca = this |
|
948 |
with \<mu>'[of Ca] `volume M \<mu>`[THEN volume_positive] |
|
949 |
show "0 \<le> \<mu>' a" |
|
950 |
by (auto intro!: setsum_nonneg) |
|
951 |
next |
|
952 |
show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto |
|
953 |
next |
|
954 |
fix a assume "a \<in> ?R" then guess Ca .. note Ca = this |
|
955 |
fix b assume "b \<in> ?R" then guess Cb .. note Cb = this |
|
956 |
assume "a \<inter> b = {}" |
|
957 |
with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto |
|
958 |
then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto |
|
959 |
||
960 |
from `a \<inter> b = {}` have "\<mu>' (\<Union> (Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)" |
|
961 |
using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union) |
|
962 |
also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)" |
|
963 |
using C_Int_cases volume_empty[OF `volume M \<mu>`] by (elim disjE) simp_all |
|
964 |
also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)" |
|
965 |
using Ca Cb by (simp add: setsum_Un_Int) |
|
966 |
also have "\<dots> = \<mu>' a + \<mu>' b" |
|
967 |
using Ca Cb by (simp add: \<mu>') |
|
968 |
finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b" |
|
969 |
using Ca Cb by simp |
|
970 |
qed |
|
971 |
qed |
|
972 |
||
973 |
section {* Caratheodory on semirings *} |
|
974 |
||
975 |
theorem (in semiring_of_sets) caratheodory: |
|
976 |
assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>" |
|
977 |
shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'" |
|
978 |
proof - |
|
979 |
have "volume M \<mu>" |
|
980 |
proof (rule volumeI) |
|
981 |
{ fix a assume "a \<in> M" then show "0 \<le> \<mu> a" |
|
982 |
using pos unfolding positive_def by auto } |
|
983 |
note p = this |
|
984 |
||
985 |
fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C" |
|
986 |
have "\<exists>F'. bij_betw F' {..<card C} C" |
|
987 |
by (rule finite_same_card_bij[OF _ `finite C`]) auto |
|
988 |
then guess F' .. note F' = this |
|
989 |
then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}" |
|
990 |
by (auto simp: bij_betw_def) |
|
991 |
{ fix i j assume *: "i < card C" "j < card C" "i \<noteq> j" |
|
992 |
with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j" |
|
993 |
unfolding inj_on_def by auto |
|
994 |
with `disjoint C`[THEN disjointD] |
|
995 |
have "F' i \<inter> F' j = {}" |
|
996 |
by auto } |
|
997 |
note F'_disj = this |
|
998 |
def F \<equiv> "\<lambda>i. if i < card C then F' i else {}" |
|
999 |
then have "disjoint_family F" |
|
1000 |
using F'_disj by (auto simp: disjoint_family_on_def) |
|
1001 |
moreover from F' have "(\<Union>i. F i) = \<Union>C" |
|
1002 |
by (auto simp: F_def set_eq_iff split: split_if_asm) |
|
1003 |
moreover have sets_F: "\<And>i. F i \<in> M" |
|
1004 |
using F' sets_C by (auto simp: F_def) |
|
1005 |
moreover note sets_C |
|
1006 |
ultimately have "\<mu> (\<Union>C) = (\<Sum>i. \<mu> (F i))" |
|
1007 |
using ca[unfolded countably_additive_def, THEN spec, of F] by auto |
|
1008 |
also have "\<dots> = (\<Sum>i<card C. \<mu> (F' i))" |
|
1009 |
proof - |
|
1010 |
have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) sums (\<Sum>i<card C. \<mu> (F' i))" |
|
1011 |
by (rule sums_If_finite_set) auto |
|
1012 |
also have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) = (\<lambda>i. \<mu> (F i))" |
|
1013 |
using pos by (auto simp: positive_def F_def) |
|
1014 |
finally show "(\<Sum>i. \<mu> (F i)) = (\<Sum>i<card C. \<mu> (F' i))" |
|
1015 |
by (simp add: sums_iff) |
|
1016 |
qed |
|
1017 |
also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)" |
|
1018 |
using F'(2) by (subst (2) F') (simp add: setsum_reindex) |
|
1019 |
finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" . |
|
1020 |
next |
|
1021 |
show "\<mu> {} = 0" |
|
1022 |
using `positive M \<mu>` by (rule positiveD1) |
|
1023 |
qed |
|
1024 |
from extend_volume[OF this] obtain \<mu>_r where |
|
1025 |
V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a" |
|
1026 |
by auto |
|
1027 |
||
1028 |
interpret G: ring_of_sets \<Omega> generated_ring |
|
1029 |
by (rule generating_ring) |
|
1030 |
||
1031 |
have pos: "positive generated_ring \<mu>_r" |
|
1032 |
using V unfolding positive_def by (auto simp: positive_def intro!: volume_positive volume_empty) |
|
1033 |
||
1034 |
have "countably_additive generated_ring \<mu>_r" |
|
1035 |
proof (rule countably_additiveI) |
|
1036 |
fix A' :: "nat \<Rightarrow> 'a set" assume A': "range A' \<subseteq> generated_ring" "disjoint_family A'" |
|
1037 |
and Un_A: "(\<Union>i. A' i) \<in> generated_ring" |
|
1038 |
||
1039 |
from generated_ringE[OF Un_A] guess C' . note C' = this |
|
1040 |
||
1041 |
{ fix c assume "c \<in> C'" |
|
1042 |
moreover def A \<equiv> "\<lambda>i. A' i \<inter> c" |
|
1043 |
ultimately have A: "range A \<subseteq> generated_ring" "disjoint_family A" |
|
1044 |
and Un_A: "(\<Union>i. A i) \<in> generated_ring" |
|
1045 |
using A' C' |
|
1046 |
by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def) |
|
1047 |
from A C' `c \<in> C'` have UN_eq: "(\<Union>i. A i) = c" |
|
1048 |
by (auto simp: A_def) |
|
1049 |
||
1050 |
have "\<forall>i::nat. \<exists>f::nat \<Rightarrow> 'a set. \<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j)) \<and> disjoint_family f \<and> \<Union>range f = A i \<and> (\<forall>j. f j \<in> M)" |
|
1051 |
(is "\<forall>i. ?P i") |
|
1052 |
proof |
|
1053 |
fix i |
|
1054 |
from A have Ai: "A i \<in> generated_ring" by auto |
|
1055 |
from generated_ringE[OF this] guess C . note C = this |
|
1056 |
||
1057 |
have "\<exists>F'. bij_betw F' {..<card C} C" |
|
1058 |
by (rule finite_same_card_bij[OF _ `finite C`]) auto |
|
1059 |
then guess F .. note F = this |
|
1060 |
def f \<equiv> "\<lambda>i. if i < card C then F i else {}" |
|
1061 |
then have f: "bij_betw f {..< card C} C" |
|
1062 |
by (intro bij_betw_cong[THEN iffD1, OF _ F]) auto |
|
1063 |
with C have "\<forall>j. f j \<in> M" |
|
1064 |
by (auto simp: Pi_iff f_def dest!: bij_betw_imp_funcset) |
|
1065 |
moreover |
|
1066 |
from f C have d_f: "disjoint_family_on f {..<card C}" |
|
1067 |
by (intro disjoint_image_disjoint_family_on) (auto simp: bij_betw_def) |
|
1068 |
then have "disjoint_family f" |
|
1069 |
by (auto simp: disjoint_family_on_def f_def) |
|
1070 |
moreover |
|
1071 |
have Ai_eq: "A i = (\<Union> x<card C. f x)" |
|
1072 |
using f C Ai unfolding bij_betw_def by (simp add: Union_image_eq[symmetric]) |
|
1073 |
then have "\<Union>range f = A i" |
|
1074 |
using f C Ai unfolding bij_betw_def by (auto simp: f_def) |
|
1075 |
moreover |
|
1076 |
{ have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)" |
|
1077 |
using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def) |
|
1078 |
also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))" |
|
1079 |
by (rule sums_If_finite_set[THEN sums_unique, symmetric]) simp |
|
1080 |
also have "\<dots> = \<mu>_r (A i)" |
|
1081 |
using C f[THEN bij_betw_imp_funcset] unfolding Ai_eq |
|
1082 |
by (intro volume_finite_additive[OF V(1) _ d_f, symmetric]) |
|
1083 |
(auto simp: Pi_iff Ai_eq intro: generated_ringI_Basic) |
|
1084 |
finally have "\<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j))" .. } |
|
1085 |
ultimately show "?P i" |
|
1086 |
by blast |
|
1087 |
qed |
|
1088 |
from choice[OF this] guess f .. note f = this |
|
1089 |
then have UN_f_eq: "(\<Union>i. split f (prod_decode i)) = (\<Union>i. A i)" |
|
1090 |
unfolding UN_extend_simps surj_prod_decode by (auto simp: set_eq_iff) |
|
1091 |
||
1092 |
have d: "disjoint_family (\<lambda>i. split f (prod_decode i))" |
|
1093 |
unfolding disjoint_family_on_def |
|
1094 |
proof (intro ballI impI) |
|
1095 |
fix m n :: nat assume "m \<noteq> n" |
|
1096 |
then have neq: "prod_decode m \<noteq> prod_decode n" |
|
1097 |
using inj_prod_decode[of UNIV] by (auto simp: inj_on_def) |
|
1098 |
show "split f (prod_decode m) \<inter> split f (prod_decode n) = {}" |
|
1099 |
proof cases |
|
1100 |
assume "fst (prod_decode m) = fst (prod_decode n)" |
|
1101 |
then show ?thesis |
|
1102 |
using neq f by (fastforce simp: disjoint_family_on_def) |
|
1103 |
next |
|
1104 |
assume neq: "fst (prod_decode m) \<noteq> fst (prod_decode n)" |
|
1105 |
have "split f (prod_decode m) \<subseteq> A (fst (prod_decode m))" |
|
1106 |
"split f (prod_decode n) \<subseteq> A (fst (prod_decode n))" |
|
1107 |
using f[THEN spec, of "fst (prod_decode m)"] |
|
1108 |
using f[THEN spec, of "fst (prod_decode n)"] |
|
1109 |
by (auto simp: set_eq_iff) |
|
1110 |
with f A neq show ?thesis |
|
1111 |
by (fastforce simp: disjoint_family_on_def subset_eq set_eq_iff) |
|
1112 |
qed |
|
1113 |
qed |
|
1114 |
from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (split f (prod_decode n)))" |
|
1115 |
by (intro suminf_ereal_2dimen[symmetric] positiveD2[OF pos] generated_ringI_Basic) |
|
1116 |
(auto split: prod.split) |
|
1117 |
also have "\<dots> = (\<Sum>n. \<mu> (split f (prod_decode n)))" |
|
1118 |
using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split) |
|
1119 |
also have "\<dots> = \<mu> (\<Union>i. split f (prod_decode i))" |
|
1120 |
using f `c \<in> C'` C' |
|
1121 |
by (intro ca[unfolded countably_additive_def, rule_format]) |
|
1122 |
(auto split: prod.split simp: UN_f_eq d UN_eq) |
|
1123 |
finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c" |
|
1124 |
using UN_f_eq UN_eq by (simp add: A_def) } |
|
1125 |
note eq = this |
|
1126 |
||
1127 |
have "(\<Sum>n. \<mu>_r (A' n)) = (\<Sum>n. \<Sum>c\<in>C'. \<mu>_r (A' n \<inter> c))" |
|
49394
52e636ace94e
removing find_theorems commands that were left in the developments accidently
bulwahn
parents:
47762
diff
changeset
|
1128 |
using C' A' |
47762 | 1129 |
by (subst volume_finite_additive[symmetric, OF V(1)]) |
1130 |
(auto simp: disjoint_def disjoint_family_on_def Union_image_eq[symmetric] simp del: Union_image_eq |
|
1131 |
intro!: G.Int G.finite_Union arg_cong[where f="\<lambda>X. suminf (\<lambda>i. \<mu>_r (X i))"] ext |
|
1132 |
intro: generated_ringI_Basic) |
|
1133 |
also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))" |
|
1134 |
using C' A' |
|
1135 |
by (intro suminf_setsum_ereal positiveD2[OF pos] G.Int G.finite_Union) |
|
1136 |
(auto intro: generated_ringI_Basic) |
|
1137 |
also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)" |
|
1138 |
using eq V C' by (auto intro!: setsum_cong) |
|
1139 |
also have "\<dots> = \<mu>_r (\<Union>C')" |
|
1140 |
using C' Un_A |
|
1141 |
by (subst volume_finite_additive[symmetric, OF V(1)]) |
|
1142 |
(auto simp: disjoint_family_on_def disjoint_def Union_image_eq[symmetric] simp del: Union_image_eq |
|
1143 |
intro: generated_ringI_Basic) |
|
1144 |
finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)" |
|
1145 |
using C' by simp |
|
1146 |
qed |
|
1147 |
from G.caratheodory'[OF `positive generated_ring \<mu>_r` `countably_additive generated_ring \<mu>_r`] |
|
1148 |
guess \<mu>' .. |
|
1149 |
with V show ?thesis |
|
1150 |
unfolding sigma_sets_generated_ring_eq |
|
1151 |
by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic) |
|
1152 |
qed |
|
1153 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1154 |
end |