author | haftmann |
Tue, 20 Jul 2010 06:35:29 +0200 | |
changeset 37891 | c26f9d06e82c |
parent 37032 | 58a0757031dd |
child 38656 | d5d342611edb |
permissions | -rw-r--r-- |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1 |
header {*Measures*} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
2 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
3 |
theory Measure |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
4 |
imports Caratheodory FuncSet |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
5 |
begin |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
6 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
7 |
text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
8 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
9 |
definition prod_sets where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
10 |
"prod_sets A B = {z. \<exists>x \<in> A. \<exists>y \<in> B. z = x \<times> y}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
11 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
12 |
lemma prod_setsI: "x \<in> A \<Longrightarrow> y \<in> B \<Longrightarrow> (x \<times> y) \<in> prod_sets A B" |
35977 | 13 |
by (auto simp add: prod_sets_def) |
14 |
||
15 |
lemma sigma_prod_sets_finite: |
|
16 |
assumes "finite A" and "finite B" |
|
17 |
shows "sets (sigma (A \<times> B) (prod_sets (Pow A) (Pow B))) = Pow (A \<times> B)" |
|
18 |
proof (simp add: sigma_def, safe) |
|
19 |
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) |
|
20 |
||
21 |
fix x assume subset: "x \<subseteq> A \<times> B" |
|
22 |
hence "finite x" using fin by (rule finite_subset) |
|
23 |
from this subset show "x \<in> sigma_sets (A\<times>B) (prod_sets (Pow A) (Pow B))" |
|
24 |
(is "x \<in> sigma_sets ?prod ?sets") |
|
25 |
proof (induct x) |
|
26 |
case empty show ?case by (rule sigma_sets.Empty) |
|
27 |
next |
|
28 |
case (insert a x) |
|
29 |
hence "{a} \<in> sigma_sets ?prod ?sets" by (auto simp: prod_sets_def intro!: sigma_sets.Basic) |
|
30 |
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto |
|
31 |
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) |
|
32 |
qed |
|
33 |
next |
|
34 |
fix x a b |
|
35 |
assume "x \<in> sigma_sets (A\<times>B) (prod_sets (Pow A) (Pow B))" and "(a, b) \<in> x" |
|
36 |
from sigma_sets_into_sp[OF _ this(1)] this(2) |
|
37 |
show "a \<in> A" and "b \<in> B" |
|
38 |
by (auto simp: prod_sets_def) |
|
39 |
qed |
|
40 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
41 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
42 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
43 |
closed_cdi where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
44 |
"closed_cdi M \<longleftrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
45 |
sets M \<subseteq> Pow (space M) & |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
46 |
(\<forall>s \<in> sets M. space M - s \<in> sets M) & |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
47 |
(\<forall>A. (range A \<subseteq> sets M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
48 |
(\<Union>i. A i) \<in> sets M) & |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
49 |
(\<forall>A. (range A \<subseteq> sets M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
50 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
51 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
52 |
inductive_set |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
53 |
smallest_ccdi_sets :: "('a, 'b) algebra_scheme \<Rightarrow> 'a set set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
54 |
for M |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
55 |
where |
35977 | 56 |
Basic [intro]: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
57 |
"a \<in> sets M \<Longrightarrow> a \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
58 |
| Compl [intro]: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
59 |
"a \<in> smallest_ccdi_sets M \<Longrightarrow> space M - a \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
60 |
| Inc: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
61 |
"range A \<in> Pow(smallest_ccdi_sets M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
62 |
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
63 |
| Disj: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
64 |
"range A \<in> Pow(smallest_ccdi_sets M) \<Longrightarrow> disjoint_family A |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
65 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
66 |
monos Pow_mono |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
67 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
68 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
69 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
70 |
smallest_closed_cdi where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
71 |
"smallest_closed_cdi M = (|space = space M, sets = smallest_ccdi_sets M|)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
72 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
73 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
74 |
measurable where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
75 |
"measurable a b = {f . sigma_algebra a & sigma_algebra b & |
33536 | 76 |
f \<in> (space a -> space b) & |
77 |
(\<forall>y \<in> sets b. (f -` y) \<inter> (space a) \<in> sets a)}" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
78 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
79 |
definition |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
80 |
measure_preserving where |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
81 |
"measure_preserving m1 m2 = |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
82 |
measurable m1 m2 \<inter> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
83 |
{f . measure_space m1 & measure_space m2 & |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
84 |
(\<forall>y \<in> sets m2. measure m1 ((f -` y) \<inter> (space m1)) = measure m2 y)}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
85 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
86 |
lemma space_smallest_closed_cdi [simp]: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
87 |
"space (smallest_closed_cdi M) = space M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
88 |
by (simp add: smallest_closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
89 |
|
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 (in algebra) smallest_closed_cdi1: "sets M \<subseteq> sets (smallest_closed_cdi M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
92 |
by (auto simp add: smallest_closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
93 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
94 |
lemma (in algebra) smallest_ccdi_sets: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
95 |
"smallest_ccdi_sets M \<subseteq> Pow (space M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
96 |
apply (rule subsetI) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
97 |
apply (erule smallest_ccdi_sets.induct) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
98 |
apply (auto intro: range_subsetD dest: sets_into_space) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
99 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
100 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
101 |
lemma (in algebra) smallest_closed_cdi2: "closed_cdi (smallest_closed_cdi M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
102 |
apply (auto simp add: closed_cdi_def smallest_closed_cdi_def smallest_ccdi_sets) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
103 |
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) + |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
104 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
105 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
106 |
lemma (in algebra) smallest_closed_cdi3: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
107 |
"sets (smallest_closed_cdi M) \<subseteq> Pow (space M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
108 |
by (simp add: smallest_closed_cdi_def smallest_ccdi_sets) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
109 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
110 |
lemma closed_cdi_subset: "closed_cdi M \<Longrightarrow> sets M \<subseteq> Pow (space M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
111 |
by (simp add: closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
112 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
113 |
lemma closed_cdi_Compl: "closed_cdi M \<Longrightarrow> s \<in> sets M \<Longrightarrow> space M - s \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
114 |
by (simp add: closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
115 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
116 |
lemma closed_cdi_Inc: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
117 |
"closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
118 |
(\<Union>i. A i) \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
119 |
by (simp add: closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
120 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
121 |
lemma closed_cdi_Disj: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
122 |
"closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
123 |
by (simp add: closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
124 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
125 |
lemma closed_cdi_Un: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
126 |
assumes cdi: "closed_cdi M" and empty: "{} \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
127 |
and A: "A \<in> sets M" and B: "B \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
128 |
and disj: "A \<inter> B = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
129 |
shows "A \<union> B \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
130 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
131 |
have ra: "range (binaryset A B) \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
132 |
by (simp add: range_binaryset_eq empty A B) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
133 |
have di: "disjoint_family (binaryset A B)" using disj |
35582 | 134 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
135 |
from closed_cdi_Disj [OF cdi ra di] |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
136 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
137 |
by (simp add: UN_binaryset_eq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
138 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
139 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
140 |
lemma (in algebra) smallest_ccdi_sets_Un: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
141 |
assumes A: "A \<in> smallest_ccdi_sets M" and B: "B \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
142 |
and disj: "A \<inter> B = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
143 |
shows "A \<union> B \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
144 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
145 |
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets M)" |
37032 | 146 |
by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
147 |
have di: "disjoint_family (binaryset A B)" using disj |
35582 | 148 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
149 |
from Disj [OF ra di] |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
150 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
151 |
by (simp add: UN_binaryset_eq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
152 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
153 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
154 |
|
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) smallest_ccdi_sets_Int1: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
157 |
assumes a: "a \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
158 |
shows "b \<in> smallest_ccdi_sets M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
159 |
proof (induct rule: smallest_ccdi_sets.induct) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
160 |
case (Basic x) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
161 |
thus ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
162 |
by (metis a Int smallest_ccdi_sets.Basic) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
163 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
164 |
case (Compl x) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
165 |
have "a \<inter> (space M - x) = space M - ((space M - a) \<union> (a \<inter> x))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
166 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
167 |
also have "... \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
168 |
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2 |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
169 |
Diff_disjoint Int_Diff Int_empty_right Un_commute |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
170 |
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
171 |
smallest_ccdi_sets_Un) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
172 |
finally show ?case . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
173 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
174 |
case (Inc A) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
175 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
176 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
177 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets M)" using Inc |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
178 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
179 |
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
180 |
by (simp add: Inc) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
181 |
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
182 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
183 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
184 |
by (rule smallest_ccdi_sets.Inc) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
185 |
show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
186 |
by (metis 1 2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
187 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
188 |
case (Disj A) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
189 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
190 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
191 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets M)" using Disj |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
192 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
193 |
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj |
35582 | 194 |
by (auto simp add: disjoint_family_on_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
195 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
196 |
by (rule smallest_ccdi_sets.Disj) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
197 |
show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
198 |
by (metis 1 2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
199 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
200 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
201 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
202 |
lemma (in algebra) smallest_ccdi_sets_Int: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
203 |
assumes b: "b \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
204 |
shows "a \<in> smallest_ccdi_sets M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
205 |
proof (induct rule: smallest_ccdi_sets.induct) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
206 |
case (Basic x) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
207 |
thus ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
208 |
by (metis b smallest_ccdi_sets_Int1) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
209 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
210 |
case (Compl x) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
211 |
have "(space M - x) \<inter> b = space M - (x \<inter> b \<union> (space M - b))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
212 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
213 |
also have "... \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
214 |
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
215 |
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
216 |
finally show ?case . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
217 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
218 |
case (Inc A) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
219 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
220 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
221 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets M)" using Inc |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
222 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
223 |
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
224 |
by (simp add: Inc) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
225 |
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
226 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
227 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
228 |
by (rule smallest_ccdi_sets.Inc) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
229 |
show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
230 |
by (metis 1 2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
231 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
232 |
case (Disj A) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
233 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
234 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
235 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets M)" using Disj |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
236 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
237 |
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj |
35582 | 238 |
by (auto simp add: disjoint_family_on_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
239 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
240 |
by (rule smallest_ccdi_sets.Disj) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
241 |
show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
242 |
by (metis 1 2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
243 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
244 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
245 |
lemma (in algebra) sets_smallest_closed_cdi_Int: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
246 |
"a \<in> sets (smallest_closed_cdi M) \<Longrightarrow> b \<in> sets (smallest_closed_cdi M) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
247 |
\<Longrightarrow> a \<inter> b \<in> sets (smallest_closed_cdi M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
248 |
by (simp add: smallest_ccdi_sets_Int smallest_closed_cdi_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
249 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
250 |
lemma algebra_iff_Int: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
251 |
"algebra M \<longleftrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
252 |
sets M \<subseteq> Pow (space M) & {} \<in> sets M & |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
253 |
(\<forall>a \<in> sets M. space M - a \<in> sets M) & |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
254 |
(\<forall>a \<in> sets M. \<forall> b \<in> sets M. a \<inter> b \<in> sets M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
255 |
proof (auto simp add: algebra.Int, auto simp add: algebra_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
256 |
fix a b |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
257 |
assume ab: "sets M \<subseteq> Pow (space M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
258 |
"\<forall>a\<in>sets M. space M - a \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
259 |
"\<forall>a\<in>sets M. \<forall>b\<in>sets M. a \<inter> b \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
260 |
"a \<in> sets M" "b \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
261 |
hence "a \<union> b = space M - ((space M - a) \<inter> (space M - b))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
262 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
263 |
also have "... \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
264 |
by (metis ab) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
265 |
finally show "a \<union> b \<in> sets M" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
266 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
267 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
268 |
lemma (in algebra) sigma_property_disjoint_lemma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
269 |
assumes sbC: "sets M \<subseteq> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
270 |
and ccdi: "closed_cdi (|space = space M, sets = C|)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
271 |
shows "sigma_sets (space M) (sets M) \<subseteq> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
272 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
273 |
have "smallest_ccdi_sets M \<in> {B . sets M \<subseteq> B \<and> sigma_algebra (|space = space M, sets = B|)}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
274 |
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
275 |
smallest_ccdi_sets_Int) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
276 |
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
277 |
apply (blast intro: smallest_ccdi_sets.Disj) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
278 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
279 |
hence "sigma_sets (space M) (sets M) \<subseteq> smallest_ccdi_sets M" |
37032 | 280 |
by clarsimp |
281 |
(drule sigma_algebra.sigma_sets_subset [where a="sets M"], auto) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
282 |
also have "... \<subseteq> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
283 |
proof |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
284 |
fix x |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
285 |
assume x: "x \<in> smallest_ccdi_sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
286 |
thus "x \<in> C" |
33536 | 287 |
proof (induct rule: smallest_ccdi_sets.induct) |
288 |
case (Basic x) |
|
289 |
thus ?case |
|
290 |
by (metis Basic subsetD sbC) |
|
291 |
next |
|
292 |
case (Compl x) |
|
293 |
thus ?case |
|
294 |
by (blast intro: closed_cdi_Compl [OF ccdi, simplified]) |
|
295 |
next |
|
296 |
case (Inc A) |
|
297 |
thus ?case |
|
298 |
by (auto intro: closed_cdi_Inc [OF ccdi, simplified]) |
|
299 |
next |
|
300 |
case (Disj A) |
|
301 |
thus ?case |
|
302 |
by (auto intro: closed_cdi_Disj [OF ccdi, simplified]) |
|
303 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
304 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
305 |
finally show ?thesis . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
306 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
307 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
308 |
lemma (in algebra) sigma_property_disjoint: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
309 |
assumes sbC: "sets M \<subseteq> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
310 |
and compl: "!!s. s \<in> C \<inter> sigma_sets (space M) (sets M) \<Longrightarrow> space M - s \<in> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
311 |
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (space M) (sets M) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
312 |
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
313 |
\<Longrightarrow> (\<Union>i. A i) \<in> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
314 |
and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (space M) (sets M) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
315 |
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
316 |
shows "sigma_sets (space M) (sets M) \<subseteq> C" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
317 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
318 |
have "sigma_sets (space M) (sets M) \<subseteq> C \<inter> sigma_sets (space M) (sets M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
319 |
proof (rule sigma_property_disjoint_lemma) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
320 |
show "sets M \<subseteq> C \<inter> sigma_sets (space M) (sets M)" |
33536 | 321 |
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
322 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
323 |
show "closed_cdi \<lparr>space = space M, sets = C \<inter> sigma_sets (space M) (sets M)\<rparr>" |
33536 | 324 |
by (simp add: closed_cdi_def compl inc disj) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
325 |
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed |
33536 | 326 |
IntE sigma_sets.Compl range_subsetD sigma_sets.Union) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
327 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
328 |
thus ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
329 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
330 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
331 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
332 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
333 |
(* or just countably_additiveD [OF measure_space.ca] *) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
334 |
lemma (in measure_space) measure_countably_additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
335 |
"range A \<subseteq> sets M |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
336 |
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
337 |
\<Longrightarrow> (measure M o A) sums measure M (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
338 |
by (insert ca) (simp add: countably_additive_def o_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
339 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
340 |
lemma (in measure_space) additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
341 |
"additive M (measure M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
342 |
proof (rule algebra.countably_additive_additive [OF _ _ ca]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
343 |
show "algebra M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
344 |
by (blast intro: sigma_algebra.axioms local.sigma_algebra_axioms) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
345 |
show "positive M (measure M)" |
37032 | 346 |
by (simp add: positive_def positive) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
347 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
348 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
349 |
lemma (in measure_space) measure_additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
350 |
"a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
351 |
\<Longrightarrow> measure M a + measure M b = measure M (a \<union> b)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
352 |
by (metis additiveD additive) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
353 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
354 |
lemma (in measure_space) measure_compl: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
355 |
assumes s: "s \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
356 |
shows "measure M (space M - s) = measure M (space M) - measure M s" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
357 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
358 |
have "measure M (space M) = measure M (s \<union> (space M - s))" using s |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
359 |
by (metis Un_Diff_cancel Un_absorb1 s sets_into_space) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
360 |
also have "... = measure M s + measure M (space M - s)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
361 |
by (rule additiveD [OF additive]) (auto simp add: s) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
362 |
finally have "measure M (space M) = measure M s + measure M (space M - s)" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
363 |
thus ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
364 |
by arith |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
365 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
366 |
|
36624 | 367 |
lemma (in measure_space) measure_mono: |
368 |
assumes "a \<subseteq> b" "a \<in> sets M" "b \<in> sets M" |
|
369 |
shows "measure M a \<le> measure M b" |
|
370 |
proof - |
|
371 |
have "b = a \<union> (b - a)" using assms by auto |
|
372 |
moreover have "{} = a \<inter> (b - a)" by auto |
|
373 |
ultimately have "measure M b = measure M a + measure M (b - a)" |
|
374 |
using measure_additive[of a "b - a"] local.Diff[of b a] assms by auto |
|
375 |
moreover have "measure M (b - a) \<ge> 0" using positive assms by auto |
|
376 |
ultimately show "measure M a \<le> measure M b" by auto |
|
377 |
qed |
|
378 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
379 |
lemma disjoint_family_Suc: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
380 |
assumes Suc: "!!n. A n \<subseteq> A (Suc n)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
381 |
shows "disjoint_family (\<lambda>i. A (Suc i) - A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
382 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
383 |
{ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
384 |
fix m |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
385 |
have "!!n. A n \<subseteq> A (m+n)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
386 |
proof (induct m) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
387 |
case 0 show ?case by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
388 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
389 |
case (Suc m) thus ?case |
33536 | 390 |
by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
391 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
392 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
393 |
hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
394 |
by (metis add_commute le_add_diff_inverse nat_less_le) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
395 |
thus ?thesis |
35582 | 396 |
by (auto simp add: disjoint_family_on_def) |
33657 | 397 |
(metis insert_absorb insert_subset le_SucE le_antisym not_leE) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
398 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
399 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
400 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
401 |
lemma (in measure_space) measure_countable_increasing: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
402 |
assumes A: "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
403 |
and A0: "A 0 = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
404 |
and ASuc: "!!n. A n \<subseteq> A (Suc n)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
405 |
shows "(measure M o A) ----> measure M (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
406 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
407 |
{ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
408 |
fix n |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
409 |
have "(measure M \<circ> A) n = |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
410 |
setsum (measure M \<circ> (\<lambda>i. A (Suc i) - A i)) {0..<n}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
411 |
proof (induct n) |
37032 | 412 |
case 0 thus ?case by (auto simp add: A0) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
413 |
next |
33536 | 414 |
case (Suc m) |
415 |
have "A (Suc m) = A m \<union> (A (Suc m) - A m)" |
|
416 |
by (metis ASuc Un_Diff_cancel Un_absorb1) |
|
417 |
hence "measure M (A (Suc m)) = |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
418 |
measure M (A m) + measure M (A (Suc m) - A m)" |
33536 | 419 |
by (subst measure_additive) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
420 |
(auto simp add: measure_additive range_subsetD [OF A]) |
33536 | 421 |
with Suc show ?case |
422 |
by simp |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
423 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
424 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
425 |
hence Meq: "measure M o A = (\<lambda>n. setsum (measure M o (\<lambda>i. A(Suc i) - A i)) {0..<n})" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
426 |
by (blast intro: ext) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
427 |
have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
428 |
proof (rule UN_finite2_eq [where k=1], simp) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
429 |
fix i |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
430 |
show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)" |
33536 | 431 |
proof (induct i) |
432 |
case 0 thus ?case by (simp add: A0) |
|
433 |
next |
|
434 |
case (Suc i) |
|
435 |
thus ?case |
|
436 |
by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc]) |
|
437 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
438 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
439 |
have A1: "\<And>i. A (Suc i) - A i \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
440 |
by (metis A Diff range_subsetD) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
441 |
have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M" |
37032 | 442 |
by (blast intro: range_subsetD [OF A]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
443 |
have "(measure M o (\<lambda>i. A (Suc i) - A i)) sums measure M (\<Union>i. A (Suc i) - A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
444 |
by (rule measure_countably_additive) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
445 |
(auto simp add: disjoint_family_Suc ASuc A1 A2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
446 |
also have "... = measure M (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
447 |
by (simp add: Aeq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
448 |
finally have "(measure M o (\<lambda>i. A (Suc i) - A i)) sums measure M (\<Union>i. A i)" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
449 |
thus ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
450 |
by (auto simp add: sums_iff Meq dest: summable_sumr_LIMSEQ_suminf) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
451 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
452 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
453 |
lemma (in measure_space) monotone_convergence: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
454 |
assumes A: "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
455 |
and ASuc: "!!n. A n \<subseteq> A (Suc n)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
456 |
shows "(measure M \<circ> A) ----> measure M (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
457 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
458 |
have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
459 |
by (auto simp add: split: nat.splits) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
460 |
have meq: "measure M \<circ> A = (\<lambda>n. (measure M \<circ> nat_case {} A) (Suc n))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
461 |
by (rule ext) simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
462 |
have "(measure M \<circ> nat_case {} A) ----> measure M (\<Union>i. nat_case {} A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
463 |
by (rule measure_countable_increasing) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
464 |
(auto simp add: range_subsetD [OF A] subsetD [OF ASuc] split: nat.splits) |
36670 | 465 |
also have "measure M (\<Union>i. nat_case {} A i) = measure M (\<Union>i. A i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
466 |
by (simp add: ueq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
467 |
finally have "(measure M \<circ> nat_case {} A) ----> measure M (\<Union>i. A i)" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
468 |
thus ?thesis using meq |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
469 |
by (metis LIMSEQ_Suc) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
470 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
471 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
472 |
lemma measurable_sigma_preimages: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
473 |
assumes a: "sigma_algebra a" and b: "sigma_algebra b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
474 |
and f: "f \<in> space a -> space b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
475 |
and ba: "!!y. y \<in> sets b ==> f -` y \<in> sets a" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
476 |
shows "sigma_algebra (|space = space a, sets = (vimage f) ` (sets b) |)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
477 |
proof (simp add: sigma_algebra_iff2, intro conjI) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
478 |
show "op -` f ` sets b \<subseteq> Pow (space a)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
479 |
by auto (metis IntE a algebra.Int_space_eq1 ba sigma_algebra_iff vimageI) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
480 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
481 |
show "{} \<in> op -` f ` sets b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
482 |
by (metis algebra.empty_sets b image_iff sigma_algebra_iff vimage_empty) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
483 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
484 |
{ fix y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
485 |
assume y: "y \<in> sets b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
486 |
with a b f |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
487 |
have "space a - f -` y = f -` (space b - y)" |
37032 | 488 |
by (auto, clarsimp simp add: sigma_algebra_iff2) |
489 |
(drule ba, blast intro: ba) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
490 |
hence "space a - (f -` y) \<in> (vimage f) ` sets b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
491 |
by auto |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
492 |
(metis b y subsetD equalityE imageI sigma_algebra.sigma_sets_eq |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
493 |
sigma_sets.Compl) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
494 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
495 |
thus "\<forall>s\<in>sets b. space a - f -` s \<in> (vimage f) ` sets b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
496 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
497 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
498 |
{ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
499 |
fix A:: "nat \<Rightarrow> 'a set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
500 |
assume A: "range A \<subseteq> (vimage f) ` (sets b)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
501 |
have "(\<Union>i. A i) \<in> (vimage f) ` (sets b)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
502 |
proof - |
33536 | 503 |
def B \<equiv> "\<lambda>i. @v. v \<in> sets b \<and> f -` v = A i" |
504 |
{ |
|
505 |
fix i |
|
506 |
have "A i \<in> (vimage f) ` (sets b)" using A |
|
507 |
by blast |
|
508 |
hence "\<exists>v. v \<in> sets b \<and> f -` v = A i" |
|
509 |
by blast |
|
510 |
hence "B i \<in> sets b \<and> f -` B i = A i" |
|
511 |
by (simp add: B_def) (erule someI_ex) |
|
512 |
} note B = this |
|
513 |
show ?thesis |
|
514 |
proof |
|
515 |
show "(\<Union>i. A i) = f -` (\<Union>i. B i)" |
|
516 |
by (simp add: vimage_UN B) |
|
517 |
show "(\<Union>i. B i) \<in> sets b" using B |
|
518 |
by (auto intro: sigma_algebra.countable_UN [OF b]) |
|
519 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
520 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
521 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
522 |
thus "\<forall>A::nat \<Rightarrow> 'a set. |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
523 |
range A \<subseteq> (vimage f) ` sets b \<longrightarrow> (\<Union>i. A i) \<in> (vimage f) ` sets b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
524 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
525 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
526 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
527 |
lemma (in sigma_algebra) measurable_sigma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
528 |
assumes B: "B \<subseteq> Pow X" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
529 |
and f: "f \<in> space M -> X" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
530 |
and ba: "!!y. y \<in> B ==> (f -` y) \<inter> space M \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
531 |
shows "f \<in> measurable M (sigma X B)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
532 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
533 |
have "sigma_sets X B \<subseteq> {y . (f -` y) \<inter> space M \<in> sets M & y \<subseteq> X}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
534 |
proof clarify |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
535 |
fix x |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
536 |
assume "x \<in> sigma_sets X B" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
537 |
thus "f -` x \<inter> space M \<in> sets M \<and> x \<subseteq> X" |
33536 | 538 |
proof induct |
539 |
case (Basic a) |
|
540 |
thus ?case |
|
541 |
by (auto simp add: ba) (metis B subsetD PowD) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
542 |
next |
33536 | 543 |
case Empty |
544 |
thus ?case |
|
545 |
by auto |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
546 |
next |
33536 | 547 |
case (Compl a) |
548 |
have [simp]: "f -` X \<inter> space M = space M" |
|
549 |
by (auto simp add: funcset_mem [OF f]) |
|
550 |
thus ?case |
|
551 |
by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl) |
|
552 |
next |
|
553 |
case (Union a) |
|
554 |
thus ?case |
|
555 |
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) |
|
556 |
(blast intro: countable_UN) |
|
557 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
558 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
559 |
thus ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
560 |
by (simp add: measurable_def sigma_algebra_axioms sigma_algebra_sigma B f) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
561 |
(auto simp add: sigma_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
562 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
563 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
564 |
lemma measurable_subset: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
565 |
"measurable a b \<subseteq> measurable a (sigma (space b) (sets b))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
566 |
apply clarify |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
567 |
apply (rule sigma_algebra.measurable_sigma) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
568 |
apply (auto simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
569 |
apply (metis algebra.sets_into_space subsetD sigma_algebra_iff) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
570 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
571 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
572 |
lemma measurable_eqI: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
573 |
"space m1 = space m1' \<Longrightarrow> space m2 = space m2' |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
574 |
\<Longrightarrow> sets m1 = sets m1' \<Longrightarrow> sets m2 = sets m2' |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
575 |
\<Longrightarrow> measurable m1 m2 = measurable m1' m2'" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
576 |
by (simp add: measurable_def sigma_algebra_iff2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
577 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
578 |
lemma measure_preserving_lift: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
579 |
fixes f :: "'a1 \<Rightarrow> 'a2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
580 |
and m1 :: "('a1, 'b1) measure_space_scheme" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
581 |
and m2 :: "('a2, 'b2) measure_space_scheme" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
582 |
assumes m1: "measure_space m1" and m2: "measure_space m2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
583 |
defines "m x \<equiv> (|space = space m2, sets = x, measure = measure m2, ... = more m2|)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
584 |
assumes setsm2: "sets m2 = sigma_sets (space m2) a" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
585 |
and fmp: "f \<in> measure_preserving m1 (m a)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
586 |
shows "f \<in> measure_preserving m1 m2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
587 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
588 |
have [simp]: "!!x. space (m x) = space m2 & sets (m x) = x & measure (m x) = measure m2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
589 |
by (simp add: m_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
590 |
have sa1: "sigma_algebra m1" using m1 |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
591 |
by (simp add: measure_space_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
592 |
show ?thesis using fmp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
593 |
proof (clarsimp simp add: measure_preserving_def m1 m2) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
594 |
assume fm: "f \<in> measurable m1 (m a)" |
33536 | 595 |
and mam: "measure_space (m a)" |
596 |
and meq: "\<forall>y\<in>a. measure m1 (f -` y \<inter> space m1) = measure m2 y" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
597 |
have "f \<in> measurable m1 (sigma (space (m a)) (sets (m a)))" |
33536 | 598 |
by (rule subsetD [OF measurable_subset fm]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
599 |
also have "... = measurable m1 m2" |
33536 | 600 |
by (rule measurable_eqI) (simp_all add: m_def setsm2 sigma_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
601 |
finally have f12: "f \<in> measurable m1 m2" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
602 |
hence ffn: "f \<in> space m1 \<rightarrow> space m2" |
33536 | 603 |
by (simp add: measurable_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
604 |
have "space m1 \<subseteq> f -` (space m2)" |
33536 | 605 |
by auto (metis PiE ffn) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
606 |
hence fveq [simp]: "(f -` (space m2)) \<inter> space m1 = space m1" |
33536 | 607 |
by blast |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
608 |
{ |
33536 | 609 |
fix y |
610 |
assume y: "y \<in> sets m2" |
|
611 |
have ama: "algebra (m a)" using mam |
|
612 |
by (simp add: measure_space_def sigma_algebra_iff) |
|
613 |
have spa: "space m2 \<in> a" using algebra.top [OF ama] |
|
614 |
by (simp add: m_def) |
|
615 |
have "sigma_sets (space m2) a = sigma_sets (space (m a)) (sets (m a))" |
|
616 |
by (simp add: m_def) |
|
617 |
also have "... \<subseteq> {s . measure m1 ((f -` s) \<inter> space m1) = measure m2 s}" |
|
618 |
proof (rule algebra.sigma_property_disjoint, auto simp add: ama) |
|
619 |
fix x |
|
620 |
assume x: "x \<in> a" |
|
621 |
thus "measure m1 (f -` x \<inter> space m1) = measure m2 x" |
|
622 |
by (simp add: meq) |
|
623 |
next |
|
624 |
fix s |
|
625 |
assume eq: "measure m1 (f -` s \<inter> space m1) = measure m2 s" |
|
626 |
and s: "s \<in> sigma_sets (space m2) a" |
|
627 |
hence s2: "s \<in> sets m2" |
|
628 |
by (simp add: setsm2) |
|
629 |
thus "measure m1 (f -` (space m2 - s) \<inter> space m1) = |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
630 |
measure m2 (space m2 - s)" using f12 |
33536 | 631 |
by (simp add: vimage_Diff Diff_Int_distrib2 eq m1 m2 |
632 |
measure_space.measure_compl measurable_def) |
|
633 |
(metis fveq meq spa) |
|
634 |
next |
|
635 |
fix A |
|
636 |
assume A0: "A 0 = {}" |
|
637 |
and ASuc: "!!n. A n \<subseteq> A (Suc n)" |
|
638 |
and rA1: "range A \<subseteq> |
|
639 |
{s. measure m1 (f -` s \<inter> space m1) = measure m2 s}" |
|
640 |
and "range A \<subseteq> sigma_sets (space m2) a" |
|
641 |
hence rA2: "range A \<subseteq> sets m2" by (metis setsm2) |
|
642 |
have eq1: "!!i. measure m1 (f -` A i \<inter> space m1) = measure m2 (A i)" |
|
643 |
using rA1 by blast |
|
644 |
have "(measure m2 \<circ> A) = measure m1 o (\<lambda>i. (f -` A i \<inter> space m1))" |
|
645 |
by (simp add: o_def eq1) |
|
646 |
also have "... ----> measure m1 (\<Union>i. f -` A i \<inter> space m1)" |
|
647 |
proof (rule measure_space.measure_countable_increasing [OF m1]) |
|
648 |
show "range (\<lambda>i. f -` A i \<inter> space m1) \<subseteq> sets m1" |
|
649 |
using f12 rA2 by (auto simp add: measurable_def) |
|
650 |
show "f -` A 0 \<inter> space m1 = {}" using A0 |
|
651 |
by blast |
|
652 |
show "\<And>n. f -` A n \<inter> space m1 \<subseteq> f -` A (Suc n) \<inter> space m1" |
|
653 |
using ASuc by auto |
|
654 |
qed |
|
36670 | 655 |
also have "measure m1 (\<Union>i. f -` A i \<inter> space m1) = measure m1 (f -` (\<Union>i. A i) \<inter> space m1)" |
33536 | 656 |
by (simp add: vimage_UN) |
657 |
finally have "(measure m2 \<circ> A) ----> measure m1 (f -` (\<Union>i. A i) \<inter> space m1)" . |
|
658 |
moreover |
|
659 |
have "(measure m2 \<circ> A) ----> measure m2 (\<Union>i. A i)" |
|
660 |
by (rule measure_space.measure_countable_increasing |
|
661 |
[OF m2 rA2, OF A0 ASuc]) |
|
662 |
ultimately |
|
663 |
show "measure m1 (f -` (\<Union>i. A i) \<inter> space m1) = |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
664 |
measure m2 (\<Union>i. A i)" |
33536 | 665 |
by (rule LIMSEQ_unique) |
666 |
next |
|
667 |
fix A :: "nat => 'a2 set" |
|
668 |
assume dA: "disjoint_family A" |
|
669 |
and rA1: "range A \<subseteq> |
|
670 |
{s. measure m1 (f -` s \<inter> space m1) = measure m2 s}" |
|
671 |
and "range A \<subseteq> sigma_sets (space m2) a" |
|
672 |
hence rA2: "range A \<subseteq> sets m2" by (metis setsm2) |
|
673 |
hence Um2: "(\<Union>i. A i) \<in> sets m2" |
|
674 |
by (metis range_subsetD setsm2 sigma_sets.Union) |
|
675 |
have "!!i. measure m1 (f -` A i \<inter> space m1) = measure m2 (A i)" |
|
676 |
using rA1 by blast |
|
677 |
hence "measure m2 o A = measure m1 o (\<lambda>i. f -` A i \<inter> space m1)" |
|
678 |
by (simp add: o_def) |
|
679 |
also have "... sums measure m1 (\<Union>i. f -` A i \<inter> space m1)" |
|
680 |
proof (rule measure_space.measure_countably_additive [OF m1] ) |
|
681 |
show "range (\<lambda>i. f -` A i \<inter> space m1) \<subseteq> sets m1" |
|
682 |
using f12 rA2 by (auto simp add: measurable_def) |
|
683 |
show "disjoint_family (\<lambda>i. f -` A i \<inter> space m1)" using dA |
|
35582 | 684 |
by (auto simp add: disjoint_family_on_def) |
33536 | 685 |
show "(\<Union>i. f -` A i \<inter> space m1) \<in> sets m1" |
686 |
proof (rule sigma_algebra.countable_UN [OF sa1]) |
|
687 |
show "range (\<lambda>i. f -` A i \<inter> space m1) \<subseteq> sets m1" using f12 rA2 |
|
688 |
by (auto simp add: measurable_def) |
|
689 |
qed |
|
690 |
qed |
|
691 |
finally have "(measure m2 o A) sums measure m1 (\<Union>i. f -` A i \<inter> space m1)" . |
|
692 |
with measure_space.measure_countably_additive [OF m2 rA2 dA Um2] |
|
693 |
have "measure m1 (\<Union>i. f -` A i \<inter> space m1) = measure m2 (\<Union>i. A i)" |
|
694 |
by (metis sums_unique) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
695 |
|
33536 | 696 |
moreover have "measure m1 (f -` (\<Union>i. A i) \<inter> space m1) = measure m1 (\<Union>i. f -` A i \<inter> space m1)" |
697 |
by (simp add: vimage_UN) |
|
698 |
ultimately show "measure m1 (f -` (\<Union>i. A i) \<inter> space m1) = |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
699 |
measure m2 (\<Union>i. A i)" |
33536 | 700 |
by metis |
701 |
qed |
|
702 |
finally have "sigma_sets (space m2) a |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
703 |
\<subseteq> {s . measure m1 ((f -` s) \<inter> space m1) = measure m2 s}" . |
33536 | 704 |
hence "measure m1 (f -` y \<inter> space m1) = measure m2 y" using y |
705 |
by (force simp add: setsm2) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
706 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
707 |
thus "f \<in> measurable m1 m2 \<and> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
708 |
(\<forall>y\<in>sets m2. |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
709 |
measure m1 (f -` y \<inter> space m1) = measure m2 y)" |
33536 | 710 |
by (blast intro: f12) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
711 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
712 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
713 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
714 |
lemma measurable_ident: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
715 |
"sigma_algebra M ==> id \<in> measurable M M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
716 |
apply (simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
717 |
apply (auto simp add: sigma_algebra_def algebra.Int algebra.top) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
718 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
719 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
720 |
lemma measurable_comp: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
721 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
722 |
shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
723 |
apply (auto simp add: measurable_def vimage_compose) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
724 |
apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a") |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
725 |
apply force+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
726 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
727 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
728 |
lemma measurable_strong: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
729 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
730 |
assumes f: "f \<in> measurable a b" and g: "g \<in> (space b -> space c)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
731 |
and c: "sigma_algebra c" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
732 |
and t: "f ` (space a) \<subseteq> t" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
733 |
and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
734 |
shows "(g o f) \<in> measurable a c" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
735 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
736 |
have a: "sigma_algebra a" and b: "sigma_algebra b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
737 |
and fab: "f \<in> (space a -> space b)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
738 |
and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
739 |
by (auto simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
740 |
have eq: "f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
741 |
by force |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
742 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
743 |
apply (auto simp add: measurable_def vimage_compose a c) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
744 |
apply (metis funcset_mem fab g) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
745 |
apply (subst eq, metis ba cb) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
746 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
747 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
748 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
749 |
lemma measurable_mono1: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
750 |
"a \<subseteq> b \<Longrightarrow> sigma_algebra \<lparr>space = X, sets = b\<rparr> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
751 |
\<Longrightarrow> measurable \<lparr>space = X, sets = a\<rparr> c \<subseteq> measurable \<lparr>space = X, sets = b\<rparr> c" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
752 |
by (auto simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
753 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
754 |
lemma measurable_up_sigma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
755 |
"measurable a b \<subseteq> measurable (sigma (space a) (sets a)) b" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
756 |
apply (auto simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
757 |
apply (metis sigma_algebra_iff2 sigma_algebra_sigma) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
758 |
apply (metis algebra.simps(2) sigma_algebra.sigma_sets_eq sigma_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
759 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
760 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
761 |
lemma measure_preserving_up: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
762 |
"f \<in> measure_preserving \<lparr>space = space m1, sets = a, measure = measure m1\<rparr> m2 \<Longrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
763 |
measure_space m1 \<Longrightarrow> sigma_algebra m1 \<Longrightarrow> a \<subseteq> sets m1 |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
764 |
\<Longrightarrow> f \<in> measure_preserving m1 m2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
765 |
by (auto simp add: measure_preserving_def measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
766 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
767 |
lemma measure_preserving_up_sigma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
768 |
"measure_space m1 \<Longrightarrow> (sets m1 = sets (sigma (space m1) a)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
769 |
\<Longrightarrow> measure_preserving \<lparr>space = space m1, sets = a, measure = measure m1\<rparr> m2 |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
770 |
\<subseteq> measure_preserving m1 m2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
771 |
apply (rule subsetI) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
772 |
apply (rule measure_preserving_up) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
773 |
apply (simp_all add: measure_space_def sigma_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
774 |
apply (metis sigma_algebra.sigma_sets_eq subsetI sigma_sets.Basic) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
775 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
776 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
777 |
lemma (in sigma_algebra) measurable_prod_sigma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
778 |
assumes 1: "(fst o f) \<in> measurable M a1" and 2: "(snd o f) \<in> measurable M a2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
779 |
shows "f \<in> measurable M (sigma ((space a1) \<times> (space a2)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
780 |
(prod_sets (sets a1) (sets a2)))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
781 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
782 |
from 1 have sa1: "sigma_algebra a1" and fn1: "fst \<circ> f \<in> space M \<rightarrow> space a1" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
783 |
and q1: "\<forall>y\<in>sets a1. (fst \<circ> f) -` y \<inter> space M \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
784 |
by (auto simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
785 |
from 2 have sa2: "sigma_algebra a2" and fn2: "snd \<circ> f \<in> space M \<rightarrow> space a2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
786 |
and q2: "\<forall>y\<in>sets a2. (snd \<circ> f) -` y \<inter> space M \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
787 |
by (auto simp add: measurable_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
788 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
789 |
proof (rule measurable_sigma) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
790 |
show "prod_sets (sets a1) (sets a2) \<subseteq> Pow (space a1 \<times> space a2)" using sa1 sa2 |
37032 | 791 |
by (auto simp add: prod_sets_def sigma_algebra_iff dest: algebra.space_closed) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
792 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
793 |
show "f \<in> space M \<rightarrow> space a1 \<times> space a2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
794 |
by (rule prod_final [OF fn1 fn2]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
795 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
796 |
fix z |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
797 |
assume z: "z \<in> prod_sets (sets a1) (sets a2)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
798 |
thus "f -` z \<inter> space M \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
799 |
proof (auto simp add: prod_sets_def vimage_Times) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
800 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
801 |
assume x: "x \<in> sets a1" and y: "y \<in> sets a2" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
802 |
have "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M = |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
803 |
((fst \<circ> f) -` x \<inter> space M) \<inter> ((snd \<circ> f) -` y \<inter> space M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
804 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
805 |
also have "... \<in> sets M" using x y q1 q2 |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
806 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
807 |
finally show "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M \<in> sets M" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
808 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
809 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
810 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
811 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
812 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
813 |
lemma (in measure_space) measurable_range_reduce: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
814 |
"f \<in> measurable M \<lparr>space = s, sets = Pow s\<rparr> \<Longrightarrow> |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
815 |
s \<noteq> {} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
816 |
\<Longrightarrow> f \<in> measurable M \<lparr>space = s \<inter> (f ` space M), sets = Pow (s \<inter> (f ` space M))\<rparr>" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
817 |
by (simp add: measurable_def sigma_algebra_Pow del: Pow_Int_eq) blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
818 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
819 |
lemma (in measure_space) measurable_Pow_to_Pow: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
820 |
"(sets M = Pow (space M)) \<Longrightarrow> f \<in> measurable M \<lparr>space = UNIV, sets = Pow UNIV\<rparr>" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
821 |
by (auto simp add: measurable_def sigma_algebra_def sigma_algebra_axioms_def algebra_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
822 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
823 |
lemma (in measure_space) measurable_Pow_to_Pow_image: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
824 |
"sets M = Pow (space M) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
825 |
\<Longrightarrow> f \<in> measurable M \<lparr>space = f ` space M, sets = Pow (f ` space M)\<rparr>" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
826 |
by (simp add: measurable_def sigma_algebra_Pow) intro_locales |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
827 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
828 |
lemma (in measure_space) measure_real_sum_image: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
829 |
assumes s: "s \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
830 |
and ssets: "\<And>x. x \<in> s ==> {x} \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
831 |
and fin: "finite s" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
832 |
shows "measure M s = (\<Sum>x\<in>s. measure M {x})" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
833 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
834 |
{ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
835 |
fix u |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
836 |
assume u: "u \<subseteq> s & u \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
837 |
hence "finite u" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
838 |
by (metis fin finite_subset) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
839 |
hence "measure M u = (\<Sum>x\<in>u. measure M {x})" using u |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
840 |
proof (induct u) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
841 |
case empty |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
842 |
thus ?case by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
843 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
844 |
case (insert x t) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
845 |
hence x: "x \<in> s" and t: "t \<subseteq> s" |
37032 | 846 |
by simp_all |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
847 |
hence ts: "t \<in> sets M" using insert |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
848 |
by (metis Diff_insert_absorb Diff ssets) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
849 |
have "measure M (insert x t) = measure M ({x} \<union> t)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
850 |
by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
851 |
also have "... = measure M {x} + measure M t" |
37032 | 852 |
by (simp add: measure_additive insert ssets x ts |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
853 |
del: Un_insert_left) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
854 |
also have "... = (\<Sum>x\<in>insert x t. measure M {x})" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
855 |
by (simp add: x t ts insert) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
856 |
finally show ?case . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
857 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
858 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
859 |
thus ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
860 |
by (metis subset_refl s) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
861 |
qed |
35582 | 862 |
|
863 |
lemma (in measure_space) measure_finitely_additive': |
|
864 |
assumes "f \<in> ({0 :: nat ..< n} \<rightarrow> sets M)" |
|
865 |
assumes "\<And> a b. \<lbrakk>a < n ; b < n ; a \<noteq> b\<rbrakk> \<Longrightarrow> f a \<inter> f b = {}" |
|
866 |
assumes "s = \<Union> (f ` {0 ..< n})" |
|
867 |
shows "(\<Sum> i \<in> {0 ..< n}. (measure M \<circ> f) i) = measure M s" |
|
868 |
proof - |
|
869 |
def f' == "\<lambda> i. (if i < n then f i else {})" |
|
870 |
have rf: "range f' \<subseteq> sets M" unfolding f'_def |
|
871 |
using assms empty_sets by auto |
|
872 |
have df: "disjoint_family f'" unfolding f'_def disjoint_family_on_def |
|
873 |
using assms by simp |
|
874 |
have "\<Union> range f' = (\<Union> i \<in> {0 ..< n}. f i)" |
|
875 |
unfolding f'_def by auto |
|
876 |
then have "measure M s = measure M (\<Union> range f')" |
|
877 |
using assms by simp |
|
878 |
then have part1: "(\<lambda> i. measure M (f' i)) sums measure M s" |
|
879 |
using df rf ca[unfolded countably_additive_def, rule_format, of f'] |
|
880 |
by auto |
|
881 |
||
882 |
have "(\<lambda> i. measure M (f' i)) sums (\<Sum> i \<in> {0 ..< n}. measure M (f' i))" |
|
883 |
using series_zero[of "n" "\<lambda> i. measure M (f' i)", rule_format] |
|
884 |
unfolding f'_def by simp |
|
885 |
also have "\<dots> = (\<Sum> i \<in> {0 ..< n}. measure M (f i))" |
|
886 |
unfolding f'_def by auto |
|
887 |
finally have part2: "(\<lambda> i. measure M (f' i)) sums (\<Sum> i \<in> {0 ..< n}. measure M (f i))" by simp |
|
888 |
show ?thesis |
|
889 |
using sums_unique[OF part1] |
|
890 |
sums_unique[OF part2] by auto |
|
891 |
qed |
|
892 |
||
893 |
||
894 |
lemma (in measure_space) measure_finitely_additive: |
|
895 |
assumes "finite r" |
|
896 |
assumes "r \<subseteq> sets M" |
|
897 |
assumes d: "\<And> a b. \<lbrakk>a \<in> r ; b \<in> r ; a \<noteq> b\<rbrakk> \<Longrightarrow> a \<inter> b = {}" |
|
898 |
shows "(\<Sum> i \<in> r. measure M i) = measure M (\<Union> r)" |
|
899 |
using assms |
|
900 |
proof - |
|
901 |
(* counting the elements in r is enough *) |
|
902 |
let ?R = "{0 ..< card r}" |
|
903 |
obtain f where f: "f ` ?R = r" "inj_on f ?R" |
|
904 |
using ex_bij_betw_nat_finite[unfolded bij_betw_def, OF `finite r`] |
|
905 |
by auto |
|
906 |
hence f_into_sets: "f \<in> ?R \<rightarrow> sets M" using assms by auto |
|
907 |
have disj: "\<And> a b. \<lbrakk>a \<in> ?R ; b \<in> ?R ; a \<noteq> b\<rbrakk> \<Longrightarrow> f a \<inter> f b = {}" |
|
908 |
proof - |
|
909 |
fix a b assume asm: "a \<in> ?R" "b \<in> ?R" "a \<noteq> b" |
|
910 |
hence neq: "f a \<noteq> f b" using f[unfolded inj_on_def, rule_format] by blast |
|
911 |
from asm have "f a \<in> r" "f b \<in> r" using f by auto |
|
912 |
thus "f a \<inter> f b = {}" using d[of "f a" "f b"] f using neq by auto |
|
913 |
qed |
|
914 |
have "(\<Union> r) = (\<Union> i \<in> ?R. f i)" |
|
915 |
using f by auto |
|
916 |
hence "measure M (\<Union> r)= measure M (\<Union> i \<in> ?R. f i)" by simp |
|
917 |
also have "\<dots> = (\<Sum> i \<in> ?R. measure M (f i))" |
|
918 |
using measure_finitely_additive'[OF f_into_sets disj] by simp |
|
919 |
also have "\<dots> = (\<Sum> a \<in> r. measure M a)" |
|
920 |
using f[rule_format] setsum_reindex[of f ?R "\<lambda> a. measure M a"] by auto |
|
921 |
finally show ?thesis by simp |
|
922 |
qed |
|
923 |
||
924 |
lemma (in measure_space) measure_finitely_additive'': |
|
925 |
assumes "finite s" |
|
926 |
assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<in> sets M" |
|
927 |
assumes d: "disjoint_family_on a s" |
|
928 |
shows "(\<Sum> i \<in> s. measure M (a i)) = measure M (\<Union> i \<in> s. a i)" |
|
929 |
using assms |
|
930 |
proof - |
|
931 |
(* counting the elements in s is enough *) |
|
932 |
let ?R = "{0 ..< card s}" |
|
933 |
obtain f where f: "f ` ?R = s" "inj_on f ?R" |
|
934 |
using ex_bij_betw_nat_finite[unfolded bij_betw_def, OF `finite s`] |
|
935 |
by auto |
|
936 |
hence f_into_sets: "a \<circ> f \<in> ?R \<rightarrow> sets M" using assms unfolding o_def by auto |
|
937 |
have disj: "\<And> i j. \<lbrakk>i \<in> ?R ; j \<in> ?R ; i \<noteq> j\<rbrakk> \<Longrightarrow> (a \<circ> f) i \<inter> (a \<circ> f) j = {}" |
|
938 |
proof - |
|
939 |
fix i j assume asm: "i \<in> ?R" "j \<in> ?R" "i \<noteq> j" |
|
940 |
hence neq: "f i \<noteq> f j" using f[unfolded inj_on_def, rule_format] by blast |
|
941 |
from asm have "f i \<in> s" "f j \<in> s" using f by auto |
|
942 |
thus "(a \<circ> f) i \<inter> (a \<circ> f) j = {}" |
|
943 |
using d f neq unfolding disjoint_family_on_def by auto |
|
944 |
qed |
|
945 |
have "(\<Union> i \<in> s. a i) = (\<Union> i \<in> f ` ?R. a i)" using f by auto |
|
946 |
hence "(\<Union> i \<in> s. a i) = (\<Union> i \<in> ?R. a (f i))" by auto |
|
947 |
hence "measure M (\<Union> i \<in> s. a i) = (\<Sum> i \<in> ?R. measure M (a (f i)))" |
|
948 |
using measure_finitely_additive'[OF f_into_sets disj] by simp |
|
949 |
also have "\<dots> = (\<Sum> i \<in> s. measure M (a i))" |
|
950 |
using f[rule_format] setsum_reindex[of f ?R "\<lambda> i. measure M (a i)"] by auto |
|
951 |
finally show ?thesis by simp |
|
952 |
qed |
|
953 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
954 |
lemma (in sigma_algebra) sigma_algebra_extend: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
955 |
"sigma_algebra \<lparr>space = space M, sets = sets M, measure_space.measure = f\<rparr>" |
33273 | 956 |
by unfold_locales (auto intro!: space_closed) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
957 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
958 |
lemma (in sigma_algebra) finite_additivity_sufficient: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
959 |
assumes fin: "finite (space M)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
960 |
and posf: "positive M f" and addf: "additive M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
961 |
defines "Mf \<equiv> \<lparr>space = space M, sets = sets M, measure = f\<rparr>" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
962 |
shows "measure_space Mf" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
963 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
964 |
have [simp]: "f {} = 0" using posf |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
965 |
by (simp add: positive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
966 |
have "countably_additive Mf f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
967 |
proof (auto simp add: countably_additive_def positive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
968 |
fix A :: "nat \<Rightarrow> 'a set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
969 |
assume A: "range A \<subseteq> sets Mf" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
970 |
and disj: "disjoint_family A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
971 |
and UnA: "(\<Union>i. A i) \<in> sets Mf" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
972 |
def I \<equiv> "{i. A i \<noteq> {}}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
973 |
have "Union (A ` I) \<subseteq> space M" using A |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
974 |
by (auto simp add: Mf_def) (metis range_subsetD subsetD sets_into_space) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
975 |
hence "finite (A ` I)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
976 |
by (metis finite_UnionD finite_subset fin) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
977 |
moreover have "inj_on A I" using disj |
35582 | 978 |
by (auto simp add: I_def disjoint_family_on_def inj_on_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
979 |
ultimately have finI: "finite I" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
980 |
by (metis finite_imageD) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
981 |
hence "\<exists>N. \<forall>m\<ge>N. A m = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
982 |
proof (cases "I = {}") |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
983 |
case True thus ?thesis by (simp add: I_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
984 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
985 |
case False |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
986 |
hence "\<forall>i\<in>I. i < Suc(Max I)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
987 |
by (simp add: Max_less_iff [symmetric] finI) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
988 |
hence "\<forall>m \<ge> Suc(Max I). A m = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
989 |
by (simp add: I_def) (metis less_le_not_le) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
990 |
thus ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
991 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
992 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
993 |
then obtain N where N: "\<forall>m\<ge>N. A m = {}" by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
994 |
hence "\<forall>m\<ge>N. (f o A) m = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
995 |
by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
996 |
hence "(\<lambda>n. f (A n)) sums setsum (f o A) {0..<N}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
997 |
by (simp add: series_zero o_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
998 |
also have "... = f (\<Union>i<N. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
999 |
proof (induct N) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1000 |
case 0 thus ?case by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1001 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1002 |
case (Suc n) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1003 |
have "f (A n \<union> (\<Union> x<n. A x)) = f (A n) + f (\<Union> i<n. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1004 |
proof (rule Caratheodory.additiveD [OF addf]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1005 |
show "A n \<inter> (\<Union> x<n. A x) = {}" using disj |
35582 | 1006 |
by (auto simp add: disjoint_family_on_def nat_less_le) blast |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1007 |
show "A n \<in> sets M" using A |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1008 |
by (force simp add: Mf_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1009 |
show "(\<Union>i<n. A i) \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1010 |
proof (induct n) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1011 |
case 0 thus ?case by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1012 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1013 |
case (Suc n) thus ?case using A |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1014 |
by (simp add: lessThan_Suc Mf_def Un range_subsetD) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1015 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1016 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1017 |
thus ?case using Suc |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1018 |
by (simp add: lessThan_Suc) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1019 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1020 |
also have "... = f (\<Union>i. A i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1021 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1022 |
have "(\<Union> i<N. A i) = (\<Union>i. A i)" using N |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1023 |
by auto (metis Int_absorb N disjoint_iff_not_equal lessThan_iff not_leE) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1024 |
thus ?thesis by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1025 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1026 |
finally show "(\<lambda>n. f (A n)) sums f (\<Union>i. A i)" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1027 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1028 |
thus ?thesis using posf |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1029 |
by (simp add: Mf_def measure_space_def measure_space_axioms_def sigma_algebra_extend positive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1030 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1031 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1032 |
lemma finite_additivity_sufficient: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1033 |
"sigma_algebra M |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1034 |
\<Longrightarrow> finite (space M) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1035 |
\<Longrightarrow> positive M (measure M) \<Longrightarrow> additive M (measure M) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1036 |
\<Longrightarrow> measure_space M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1037 |
by (rule measure_down [OF sigma_algebra.finite_additivity_sufficient], auto) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1038 |
|
35692 | 1039 |
lemma (in measure_space) measure_setsum_split: |
1040 |
assumes "finite r" and "a \<in> sets M" and br_in_M: "b ` r \<subseteq> sets M" |
|
1041 |
assumes "(\<Union>i \<in> r. b i) = space M" |
|
1042 |
assumes "disjoint_family_on b r" |
|
1043 |
shows "(measure M) a = (\<Sum> i \<in> r. measure M (a \<inter> (b i)))" |
|
1044 |
proof - |
|
1045 |
have *: "measure M a = measure M (\<Union>i \<in> r. a \<inter> b i)" |
|
1046 |
using assms by auto |
|
1047 |
show ?thesis unfolding * |
|
1048 |
proof (rule measure_finitely_additive''[symmetric]) |
|
1049 |
show "finite r" using `finite r` by auto |
|
1050 |
{ fix i assume "i \<in> r" |
|
1051 |
hence "b i \<in> sets M" using br_in_M by auto |
|
1052 |
thus "a \<inter> b i \<in> sets M" using `a \<in> sets M` by auto |
|
1053 |
} |
|
1054 |
show "disjoint_family_on (\<lambda>i. a \<inter> b i) r" |
|
1055 |
using `disjoint_family_on b r` |
|
1056 |
unfolding disjoint_family_on_def by auto |
|
1057 |
qed |
|
1058 |
qed |
|
1059 |
||
36624 | 1060 |
locale finite_measure_space = measure_space + |
1061 |
assumes finite_space: "finite (space M)" |
|
1062 |
and sets_eq_Pow: "sets M = Pow (space M)" |
|
1063 |
||
1064 |
lemma (in finite_measure_space) sum_over_space: "(\<Sum>x\<in>space M. measure M {x}) = measure M (space M)" |
|
1065 |
using measure_finitely_additive''[of "space M" "\<lambda>i. {i}"] |
|
1066 |
by (simp add: sets_eq_Pow disjoint_family_on_def finite_space) |
|
1067 |
||
35582 | 1068 |
end |