author | hoelzl |
Thu, 02 Sep 2010 19:57:16 +0200 | |
changeset 39098 | 21e9bd6cf0a8 |
parent 39092 | 98de40859858 |
child 39198 | f967a16dfcdd |
permissions | -rw-r--r-- |
38656 | 1 |
(* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *) |
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header {*Borel spaces*} |
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33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
5 |
theory Borel |
38656 | 6 |
imports Sigma_Algebra Positive_Infinite_Real Multivariate_Analysis |
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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7 |
begin |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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|
39092 | 9 |
lemma LIMSEQ_max: |
10 |
"u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0" |
|
11 |
by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D) |
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||
38656 | 13 |
section "Generic Borel spaces" |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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|
38656 | 15 |
definition "borel_space = sigma (UNIV::'a::topological_space set) open" |
16 |
abbreviation "borel_measurable M \<equiv> measurable M borel_space" |
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paulson
parents:
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38656 | 18 |
interpretation borel_space: sigma_algebra borel_space |
19 |
using sigma_algebra_sigma by (auto simp: borel_space_def) |
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33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
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|
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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lemma in_borel_measurable: |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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22 |
"f \<in> borel_measurable M \<longleftrightarrow> |
38656 | 23 |
(\<forall>S \<in> sets (sigma UNIV open). |
24 |
f -` S \<inter> space M \<in> sets M)" |
|
25 |
by (auto simp add: measurable_def borel_space_def) |
|
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
38656 | 27 |
lemma in_borel_measurable_borel_space: |
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"f \<in> borel_measurable M \<longleftrightarrow> |
|
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(\<forall>S \<in> sets borel_space. |
|
30 |
f -` S \<inter> space M \<in> sets M)" |
|
31 |
by (auto simp add: measurable_def borel_space_def) |
|
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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32 |
|
38656 | 33 |
lemma space_borel_space[simp]: "space borel_space = UNIV" |
34 |
unfolding borel_space_def by auto |
|
35 |
||
36 |
lemma borel_space_open[simp]: |
|
37 |
assumes "open A" shows "A \<in> sets borel_space" |
|
38 |
proof - |
|
39 |
have "A \<in> open" unfolding mem_def using assms . |
|
40 |
thus ?thesis unfolding borel_space_def sigma_def by (auto intro!: sigma_sets.Basic) |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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qed |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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42 |
|
38656 | 43 |
lemma borel_space_closed[simp]: |
44 |
assumes "closed A" shows "A \<in> sets borel_space" |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
45 |
proof - |
38656 | 46 |
have "space borel_space - (- A) \<in> sets borel_space" |
47 |
using assms unfolding closed_def by (blast intro: borel_space_open) |
|
48 |
thus ?thesis by simp |
|
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
49 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
50 |
|
38656 | 51 |
lemma (in sigma_algebra) borel_measurable_vimage: |
52 |
fixes f :: "'a \<Rightarrow> 'x::t2_space" |
|
53 |
assumes borel: "f \<in> borel_measurable M" |
|
54 |
shows "f -` {x} \<inter> space M \<in> sets M" |
|
55 |
proof (cases "x \<in> f ` space M") |
|
56 |
case True then obtain y where "x = f y" by auto |
|
57 |
from closed_sing[of "f y"] |
|
58 |
have "{f y} \<in> sets borel_space" by (rule borel_space_closed) |
|
59 |
with assms show ?thesis |
|
60 |
unfolding in_borel_measurable_borel_space `x = f y` by auto |
|
61 |
next |
|
62 |
case False hence "f -` {x} \<inter> space M = {}" by auto |
|
63 |
thus ?thesis by auto |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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64 |
qed |
40b44cb20c8c
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paulson
parents:
diff
changeset
|
65 |
|
38656 | 66 |
lemma (in sigma_algebra) borel_measurableI: |
67 |
fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space" |
|
68 |
assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
|
69 |
shows "f \<in> borel_measurable M" |
|
70 |
unfolding borel_space_def |
|
71 |
proof (rule measurable_sigma) |
|
72 |
fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M" |
|
73 |
using assms[of S] by (simp add: mem_def) |
|
74 |
qed simp_all |
|
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paulson
parents:
diff
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|
75 |
|
38656 | 76 |
lemma borel_space_singleton[simp, intro]: |
77 |
fixes x :: "'a::t1_space" |
|
78 |
shows "A \<in> sets borel_space \<Longrightarrow> insert x A \<in> sets borel_space" |
|
79 |
proof (rule borel_space.insert_in_sets) |
|
80 |
show "{x} \<in> sets borel_space" |
|
81 |
using closed_sing[of x] by (rule borel_space_closed) |
|
82 |
qed simp |
|
83 |
||
84 |
lemma (in sigma_algebra) borel_measurable_const[simp, intro]: |
|
85 |
"(\<lambda>x. c) \<in> borel_measurable M" |
|
86 |
by (auto intro!: measurable_const) |
|
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
87 |
|
39083 | 88 |
lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]: |
38656 | 89 |
assumes A: "A \<in> sets M" |
90 |
shows "indicator A \<in> borel_measurable M" |
|
91 |
unfolding indicator_def_raw using A |
|
92 |
by (auto intro!: measurable_If_set borel_measurable_const) |
|
33533
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paulson
parents:
diff
changeset
|
93 |
|
38656 | 94 |
lemma borel_measurable_translate: |
95 |
assumes "A \<in> sets borel_space" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel_space" |
|
96 |
shows "f -` A \<in> sets borel_space" |
|
97 |
proof - |
|
98 |
have "A \<in> sigma_sets UNIV open" using assms |
|
99 |
by (simp add: borel_space_def sigma_def) |
|
100 |
thus ?thesis |
|
101 |
proof (induct rule: sigma_sets.induct) |
|
102 |
case (Basic a) thus ?case using trans[of a] by (simp add: mem_def) |
|
103 |
next |
|
104 |
case (Compl a) |
|
105 |
moreover have "UNIV \<in> sets borel_space" |
|
106 |
by (metis borel_space.top borel_space_def_raw mem_def space_sigma) |
|
107 |
ultimately show ?case |
|
108 |
by (auto simp: vimage_Diff borel_space.Diff) |
|
109 |
qed (auto simp add: vimage_UN) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
110 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
111 |
|
39092 | 112 |
lemma (in sigma_algebra) borel_measurable_restricted: |
113 |
fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" assumes "A \<in> sets M" |
|
114 |
shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow> |
|
115 |
(\<lambda>x. f x * indicator A x) \<in> borel_measurable M" |
|
116 |
(is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M") |
|
117 |
proof - |
|
118 |
interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`]) |
|
119 |
have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R" |
|
120 |
by (auto intro!: measurable_cong) |
|
121 |
show ?thesis unfolding * |
|
122 |
unfolding in_borel_measurable_borel_space |
|
123 |
proof (simp, safe) |
|
124 |
fix S :: "'x set" assume "S \<in> sets borel_space" |
|
125 |
"\<forall>S\<in>sets borel_space. ?f -` S \<inter> A \<in> op \<inter> A ` sets M" |
|
126 |
then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto |
|
127 |
then have f: "?f -` S \<inter> A \<in> sets M" |
|
128 |
using `A \<in> sets M` sets_into_space by fastsimp |
|
129 |
show "?f -` S \<inter> space M \<in> sets M" |
|
130 |
proof cases |
|
131 |
assume "0 \<in> S" |
|
132 |
then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)" |
|
133 |
using `A \<in> sets M` sets_into_space by auto |
|
134 |
then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff) |
|
135 |
next |
|
136 |
assume "0 \<notin> S" |
|
137 |
then have "?f -` S \<inter> space M = ?f -` S \<inter> A" |
|
138 |
using `A \<in> sets M` sets_into_space |
|
139 |
by (auto simp: indicator_def split: split_if_asm) |
|
140 |
then show ?thesis using f by auto |
|
141 |
qed |
|
142 |
next |
|
143 |
fix S :: "'x set" assume "S \<in> sets borel_space" |
|
144 |
"\<forall>S\<in>sets borel_space. ?f -` S \<inter> space M \<in> sets M" |
|
145 |
then have f: "?f -` S \<inter> space M \<in> sets M" by auto |
|
146 |
then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M" |
|
147 |
using `A \<in> sets M` sets_into_space |
|
148 |
apply (simp add: image_iff) |
|
149 |
apply (rule bexI[OF _ f]) |
|
150 |
by auto |
|
151 |
qed |
|
152 |
qed |
|
153 |
||
154 |
lemma (in sigma_algebra) borel_measurable_subalgebra: |
|
155 |
assumes "N \<subseteq> sets M" "f \<in> borel_measurable (M\<lparr>sets:=N\<rparr>)" |
|
156 |
shows "f \<in> borel_measurable M" |
|
157 |
using assms unfolding measurable_def by auto |
|
158 |
||
38656 | 159 |
section "Borel spaces on euclidean spaces" |
160 |
||
161 |
lemma lessThan_borel[simp, intro]: |
|
162 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
163 |
shows "{..< a} \<in> sets borel_space" |
|
164 |
by (blast intro: borel_space_open) |
|
165 |
||
166 |
lemma greaterThan_borel[simp, intro]: |
|
167 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
168 |
shows "{a <..} \<in> sets borel_space" |
|
169 |
by (blast intro: borel_space_open) |
|
170 |
||
171 |
lemma greaterThanLessThan_borel[simp, intro]: |
|
172 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
173 |
shows "{a<..<b} \<in> sets borel_space" |
|
174 |
by (blast intro: borel_space_open) |
|
175 |
||
176 |
lemma atMost_borel[simp, intro]: |
|
177 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
178 |
shows "{..a} \<in> sets borel_space" |
|
179 |
by (blast intro: borel_space_closed) |
|
180 |
||
181 |
lemma atLeast_borel[simp, intro]: |
|
182 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
183 |
shows "{a..} \<in> sets borel_space" |
|
184 |
by (blast intro: borel_space_closed) |
|
185 |
||
186 |
lemma atLeastAtMost_borel[simp, intro]: |
|
187 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
188 |
shows "{a..b} \<in> sets borel_space" |
|
189 |
by (blast intro: borel_space_closed) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
190 |
|
38656 | 191 |
lemma greaterThanAtMost_borel[simp, intro]: |
192 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
193 |
shows "{a<..b} \<in> sets borel_space" |
|
194 |
unfolding greaterThanAtMost_def by blast |
|
195 |
||
196 |
lemma atLeastLessThan_borel[simp, intro]: |
|
197 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
198 |
shows "{a..<b} \<in> sets borel_space" |
|
199 |
unfolding atLeastLessThan_def by blast |
|
200 |
||
201 |
lemma hafspace_less_borel[simp, intro]: |
|
202 |
fixes a :: real |
|
203 |
shows "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel_space" |
|
204 |
by (auto intro!: borel_space_open open_halfspace_component_gt) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
205 |
|
38656 | 206 |
lemma hafspace_greater_borel[simp, intro]: |
207 |
fixes a :: real |
|
208 |
shows "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel_space" |
|
209 |
by (auto intro!: borel_space_open open_halfspace_component_lt) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
210 |
|
38656 | 211 |
lemma hafspace_less_eq_borel[simp, intro]: |
212 |
fixes a :: real |
|
213 |
shows "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel_space" |
|
214 |
by (auto intro!: borel_space_closed closed_halfspace_component_ge) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
215 |
|
38656 | 216 |
lemma hafspace_greater_eq_borel[simp, intro]: |
217 |
fixes a :: real |
|
218 |
shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel_space" |
|
219 |
by (auto intro!: borel_space_closed closed_halfspace_component_le) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
220 |
|
38656 | 221 |
lemma (in sigma_algebra) borel_measurable_less[simp, intro]: |
222 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
223 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
224 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
225 |
shows "{w \<in> space M. f w < g w} \<in> sets M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
226 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
227 |
have "{w \<in> space M. f w < g w} = |
38656 | 228 |
(\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))" |
229 |
using Rats_dense_in_real by (auto simp add: Rats_def) |
|
230 |
then show ?thesis using f g |
|
231 |
by simp (blast intro: measurable_sets) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
232 |
qed |
38656 | 233 |
|
234 |
lemma (in sigma_algebra) borel_measurable_le[simp, intro]: |
|
235 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
236 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
237 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
238 |
shows "{w \<in> space M. f w \<le> g w} \<in> sets M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
239 |
proof - |
38656 | 240 |
have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}" |
241 |
by auto |
|
242 |
thus ?thesis using f g |
|
243 |
by simp blast |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
244 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
245 |
|
38656 | 246 |
lemma (in sigma_algebra) borel_measurable_eq[simp, intro]: |
247 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
248 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
249 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
250 |
shows "{w \<in> space M. f w = g w} \<in> sets M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
251 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
252 |
have "{w \<in> space M. f w = g w} = |
33536 | 253 |
{w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
254 |
by auto |
38656 | 255 |
thus ?thesis using f g by auto |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
256 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
257 |
|
38656 | 258 |
lemma (in sigma_algebra) borel_measurable_neq[simp, intro]: |
259 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
260 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
261 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
262 |
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
263 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
264 |
have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
265 |
by auto |
38656 | 266 |
thus ?thesis using f g by auto |
267 |
qed |
|
268 |
||
269 |
subsection "Borel space equals sigma algebras over intervals" |
|
270 |
||
271 |
lemma rational_boxes: |
|
272 |
fixes x :: "'a\<Colon>ordered_euclidean_space" |
|
273 |
assumes "0 < e" |
|
274 |
shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e" |
|
275 |
proof - |
|
276 |
def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))" |
|
277 |
then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos) |
|
278 |
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i") |
|
279 |
proof |
|
280 |
fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e |
|
281 |
show "?th i" by auto |
|
282 |
qed |
|
283 |
from choice[OF this] guess a .. note a = this |
|
284 |
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i") |
|
285 |
proof |
|
286 |
fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e |
|
287 |
show "?th i" by auto |
|
288 |
qed |
|
289 |
from choice[OF this] guess b .. note b = this |
|
290 |
{ fix y :: 'a assume *: "Chi a < y" "y < Chi b" |
|
291 |
have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)" |
|
292 |
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2) |
|
293 |
also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))" |
|
294 |
proof (rule real_sqrt_less_mono, rule setsum_strict_mono) |
|
295 |
fix i assume i: "i \<in> {..<DIM('a)}" |
|
296 |
have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto |
|
297 |
moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto |
|
298 |
moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto |
|
299 |
ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto |
|
300 |
then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))" |
|
301 |
unfolding e'_def by (auto simp: dist_real_def) |
|
302 |
then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>" |
|
303 |
by (rule power_strict_mono) auto |
|
304 |
then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)" |
|
305 |
by (simp add: power_divide) |
|
306 |
qed auto |
|
307 |
also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive) |
|
308 |
finally have "dist x y < e" . } |
|
309 |
with a b show ?thesis |
|
310 |
apply (rule_tac exI[of _ "Chi a"]) |
|
311 |
apply (rule_tac exI[of _ "Chi b"]) |
|
312 |
using eucl_less[where 'a='a] by auto |
|
313 |
qed |
|
314 |
||
315 |
lemma ex_rat_list: |
|
316 |
fixes x :: "'a\<Colon>ordered_euclidean_space" |
|
317 |
assumes "\<And> i. x $$ i \<in> \<rat>" |
|
318 |
shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)" |
|
319 |
proof - |
|
320 |
have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast |
|
321 |
from choice[OF this] guess r .. |
|
322 |
then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"]) |
|
323 |
qed |
|
324 |
||
325 |
lemma open_UNION: |
|
326 |
fixes M :: "'a\<Colon>ordered_euclidean_space set" |
|
327 |
assumes "open M" |
|
328 |
shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M} |
|
329 |
(\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})" |
|
330 |
(is "M = UNION ?idx ?box") |
|
331 |
proof safe |
|
332 |
fix x assume "x \<in> M" |
|
333 |
obtain e where e: "e > 0" "ball x e \<subseteq> M" |
|
334 |
using openE[OF assms `x \<in> M`] by auto |
|
335 |
then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e" |
|
336 |
using rational_boxes[OF e(1)] by blast |
|
337 |
then obtain p q where pq: "length p = DIM ('a)" |
|
338 |
"length q = DIM ('a)" |
|
339 |
"\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i" |
|
340 |
using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast |
|
341 |
hence p: "Chi (of_rat \<circ> op ! p) = a" |
|
342 |
using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a] |
|
343 |
unfolding o_def by auto |
|
344 |
from pq have q: "Chi (of_rat \<circ> op ! q) = b" |
|
345 |
using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b] |
|
346 |
unfolding o_def by auto |
|
347 |
have "x \<in> ?box (p, q)" |
|
348 |
using p q ab by auto |
|
349 |
thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto |
|
350 |
qed auto |
|
351 |
||
352 |
lemma halfspace_span_open: |
|
353 |
"sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))) |
|
354 |
\<subseteq> sets borel_space" |
|
355 |
by (auto intro!: borel_space.sigma_sets_subset[simplified] borel_space_open |
|
356 |
open_halfspace_component_lt simp: sets_sigma) |
|
357 |
||
358 |
lemma halfspace_lt_in_halfspace: |
|
359 |
"{x\<Colon>'a. x $$ i < a} \<in> sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})))" |
|
360 |
unfolding sets_sigma by (rule sigma_sets.Basic) auto |
|
361 |
||
362 |
lemma halfspace_gt_in_halfspace: |
|
363 |
"{x\<Colon>'a. a < x $$ i} \<in> sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})))" |
|
364 |
(is "?set \<in> sets ?SIGMA") |
|
365 |
proof - |
|
366 |
interpret sigma_algebra ?SIGMA by (rule sigma_algebra_sigma) simp |
|
367 |
have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})" |
|
368 |
proof (safe, simp_all add: not_less) |
|
369 |
fix x assume "a < x $$ i" |
|
370 |
with reals_Archimedean[of "x $$ i - a"] |
|
371 |
obtain n where "a + 1 / real (Suc n) < x $$ i" |
|
372 |
by (auto simp: inverse_eq_divide field_simps) |
|
373 |
then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i" |
|
374 |
by (blast intro: less_imp_le) |
|
375 |
next |
|
376 |
fix x n |
|
377 |
have "a < a + 1 / real (Suc n)" by auto |
|
378 |
also assume "\<dots> \<le> x" |
|
379 |
finally show "a < x" . |
|
380 |
qed |
|
381 |
show "?set \<in> sets ?SIGMA" unfolding * |
|
382 |
by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
383 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
384 |
|
38656 | 385 |
lemma (in sigma_algebra) sets_sigma_subset: |
386 |
assumes "A = space M" |
|
387 |
assumes "B \<subseteq> sets M" |
|
388 |
shows "sets (sigma A B) \<subseteq> sets M" |
|
389 |
by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms) |
|
390 |
||
391 |
lemma open_span_halfspace: |
|
392 |
"sets borel_space \<subseteq> sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i < a})))" |
|
393 |
(is "_ \<subseteq> sets ?SIGMA") |
|
394 |
proof (unfold borel_space_def, rule sigma_algebra.sets_sigma_subset, safe) |
|
395 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp |
|
396 |
then interpret sigma_algebra ?SIGMA . |
|
397 |
fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def . |
|
398 |
from open_UNION[OF this] |
|
399 |
obtain I where *: "S = |
|
400 |
(\<Union>(a, b)\<in>I. |
|
401 |
(\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter> |
|
402 |
(\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))" |
|
403 |
unfolding greaterThanLessThan_def |
|
404 |
unfolding eucl_greaterThan_eq_halfspaces[where 'a='a] |
|
405 |
unfolding eucl_lessThan_eq_halfspaces[where 'a='a] |
|
406 |
by blast |
|
407 |
show "S \<in> sets ?SIGMA" |
|
408 |
unfolding * |
|
409 |
by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) |
|
410 |
qed auto |
|
411 |
||
412 |
lemma halfspace_span_halfspace_le: |
|
413 |
"sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))) \<subseteq> |
|
414 |
sets (sigma UNIV (range (\<lambda> (a, i). {x. x $$ i \<le> a})))" |
|
415 |
(is "_ \<subseteq> sets ?SIGMA") |
|
416 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
417 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
418 |
then interpret sigma_algebra ?SIGMA . |
|
419 |
fix a i |
|
420 |
have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})" |
|
421 |
proof (safe, simp_all) |
|
422 |
fix x::'a assume *: "x$$i < a" |
|
423 |
with reals_Archimedean[of "a - x$$i"] |
|
424 |
obtain n where "x $$ i < a - 1 / (real (Suc n))" |
|
425 |
by (auto simp: field_simps inverse_eq_divide) |
|
426 |
then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))" |
|
427 |
by (blast intro: less_imp_le) |
|
428 |
next |
|
429 |
fix x::'a and n |
|
430 |
assume "x$$i \<le> a - 1 / real (Suc n)" |
|
431 |
also have "\<dots> < a" by auto |
|
432 |
finally show "x$$i < a" . |
|
433 |
qed |
|
434 |
show "{x. x$$i < a} \<in> sets ?SIGMA" unfolding * |
|
435 |
by (safe intro!: countable_UN) |
|
436 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
437 |
qed auto |
|
438 |
||
439 |
lemma halfspace_span_halfspace_ge: |
|
440 |
"sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))) \<subseteq> |
|
441 |
sets (sigma UNIV (range (\<lambda> (a, i). {x. a \<le> x $$ i})))" |
|
442 |
(is "_ \<subseteq> sets ?SIGMA") |
|
443 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
444 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
445 |
then interpret sigma_algebra ?SIGMA . |
|
446 |
fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto |
|
447 |
show "{x. x$$i < a} \<in> sets ?SIGMA" unfolding * |
|
448 |
by (safe intro!: Diff) |
|
449 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
450 |
qed auto |
|
451 |
||
452 |
lemma halfspace_le_span_halfspace_gt: |
|
453 |
"sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a}))) \<subseteq> |
|
454 |
sets (sigma UNIV (range (\<lambda> (a, i). {x. a < x $$ i})))" |
|
455 |
(is "_ \<subseteq> sets ?SIGMA") |
|
456 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
457 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
458 |
then interpret sigma_algebra ?SIGMA . |
|
459 |
fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto |
|
460 |
show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding * |
|
461 |
by (safe intro!: Diff) |
|
462 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
463 |
qed auto |
|
464 |
||
465 |
lemma halfspace_le_span_atMost: |
|
466 |
"sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a}))) \<subseteq> |
|
467 |
sets (sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})))" |
|
468 |
(is "_ \<subseteq> sets ?SIGMA") |
|
469 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
470 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
471 |
then interpret sigma_algebra ?SIGMA . |
|
472 |
fix a i |
|
473 |
show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
|
474 |
proof cases |
|
475 |
assume "i < DIM('a)" |
|
476 |
then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})" |
|
477 |
proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm) |
|
478 |
fix x |
|
479 |
from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat .. |
|
480 |
then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k" |
|
481 |
by (subst (asm) Max_le_iff) auto |
|
482 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k" |
|
483 |
by (auto intro!: exI[of _ k]) |
|
484 |
qed |
|
485 |
show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding * |
|
486 |
by (safe intro!: countable_UN) |
|
487 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
488 |
next |
|
489 |
assume "\<not> i < DIM('a)" |
|
490 |
then show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
|
491 |
using top by auto |
|
492 |
qed |
|
493 |
qed auto |
|
494 |
||
495 |
lemma halfspace_le_span_greaterThan: |
|
496 |
"sets (sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a}))) \<subseteq> |
|
497 |
sets (sigma UNIV (range (\<lambda>a. {a<..})))" |
|
498 |
(is "_ \<subseteq> sets ?SIGMA") |
|
499 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
500 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
501 |
then interpret sigma_algebra ?SIGMA . |
|
502 |
fix a i |
|
503 |
show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
|
504 |
proof cases |
|
505 |
assume "i < DIM('a)" |
|
506 |
have "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto |
|
507 |
also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)` |
|
508 |
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm) |
|
509 |
fix x |
|
510 |
from real_arch_lt[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"] |
|
511 |
guess k::nat .. note k = this |
|
512 |
{ fix i assume "i < DIM('a)" |
|
513 |
then have "-x$$i < real k" |
|
514 |
using k by (subst (asm) Max_less_iff) auto |
|
515 |
then have "- real k < x$$i" by simp } |
|
516 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia" |
|
517 |
by (auto intro!: exI[of _ k]) |
|
518 |
qed |
|
519 |
finally show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
|
520 |
apply (simp only:) |
|
521 |
apply (safe intro!: countable_UN Diff) |
|
522 |
by (auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
523 |
next |
|
524 |
assume "\<not> i < DIM('a)" |
|
525 |
then show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
|
526 |
using top by auto |
|
527 |
qed |
|
528 |
qed auto |
|
529 |
||
530 |
lemma atMost_span_atLeastAtMost: |
|
531 |
"sets (sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))) \<subseteq> |
|
532 |
sets (sigma UNIV (range (\<lambda>(a,b). {a..b})))" |
|
533 |
(is "_ \<subseteq> sets ?SIGMA") |
|
534 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
535 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
536 |
then interpret sigma_algebra ?SIGMA . |
|
537 |
fix a::'a |
|
538 |
have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})" |
|
539 |
proof (safe, simp_all add: eucl_le[where 'a='a]) |
|
540 |
fix x |
|
541 |
from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"] |
|
542 |
guess k::nat .. note k = this |
|
543 |
{ fix i assume "i < DIM('a)" |
|
544 |
with k have "- x$$i \<le> real k" |
|
545 |
by (subst (asm) Max_le_iff) (auto simp: field_simps) |
|
546 |
then have "- real k \<le> x$$i" by simp } |
|
547 |
then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i" |
|
548 |
by (auto intro!: exI[of _ k]) |
|
549 |
qed |
|
550 |
show "{..a} \<in> sets ?SIGMA" unfolding * |
|
551 |
by (safe intro!: countable_UN) |
|
552 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
553 |
qed auto |
|
554 |
||
555 |
lemma borel_space_eq_greaterThanLessThan: |
|
556 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})))" |
|
557 |
(is "_ = sets ?SIGMA") |
|
558 |
proof (rule antisym) |
|
559 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
560 |
by (rule borel_space.sets_sigma_subset) auto |
|
561 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
562 |
unfolding borel_space_def |
|
563 |
proof (rule sigma_algebra.sets_sigma_subset, safe) |
|
564 |
show "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
565 |
then interpret sigma_algebra ?SIGMA . |
|
566 |
fix M :: "'a set" assume "M \<in> open" |
|
567 |
then have "open M" by (simp add: mem_def) |
|
568 |
show "M \<in> sets ?SIGMA" |
|
569 |
apply (subst open_UNION[OF `open M`]) |
|
570 |
apply (safe intro!: countable_UN) |
|
571 |
by (auto simp add: sigma_def intro!: sigma_sets.Basic) |
|
572 |
qed auto |
|
573 |
qed |
|
574 |
||
575 |
lemma borel_space_eq_atMost: |
|
576 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})))" |
|
577 |
(is "_ = sets ?SIGMA") |
|
578 |
proof (rule antisym) |
|
579 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
580 |
using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace |
|
581 |
by auto |
|
582 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
583 |
by (rule borel_space.sets_sigma_subset) auto |
|
584 |
qed |
|
585 |
||
586 |
lemma borel_space_eq_atLeastAtMost: |
|
587 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})))" |
|
588 |
(is "_ = sets ?SIGMA") |
|
589 |
proof (rule antisym) |
|
590 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
591 |
using atMost_span_atLeastAtMost halfspace_le_span_atMost |
|
592 |
halfspace_span_halfspace_le open_span_halfspace |
|
593 |
by auto |
|
594 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
595 |
by (rule borel_space.sets_sigma_subset) auto |
|
596 |
qed |
|
597 |
||
598 |
lemma borel_space_eq_greaterThan: |
|
599 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})))" |
|
600 |
(is "_ = sets ?SIGMA") |
|
601 |
proof (rule antisym) |
|
602 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
603 |
using halfspace_le_span_greaterThan |
|
604 |
halfspace_span_halfspace_le open_span_halfspace |
|
605 |
by auto |
|
606 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
607 |
by (rule borel_space.sets_sigma_subset) auto |
|
608 |
qed |
|
609 |
||
610 |
lemma borel_space_eq_halfspace_le: |
|
611 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i \<le> a})))" |
|
612 |
(is "_ = sets ?SIGMA") |
|
613 |
proof (rule antisym) |
|
614 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
615 |
using open_span_halfspace halfspace_span_halfspace_le by auto |
|
616 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
617 |
by (rule borel_space.sets_sigma_subset) auto |
|
618 |
qed |
|
619 |
||
620 |
lemma borel_space_eq_halfspace_less: |
|
621 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})))" |
|
622 |
(is "_ = sets ?SIGMA") |
|
623 |
proof (rule antisym) |
|
624 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
625 |
using open_span_halfspace . |
|
626 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
627 |
by (rule borel_space.sets_sigma_subset) auto |
|
628 |
qed |
|
629 |
||
630 |
lemma borel_space_eq_halfspace_gt: |
|
631 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})))" |
|
632 |
(is "_ = sets ?SIGMA") |
|
633 |
proof (rule antisym) |
|
634 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
635 |
using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto |
|
636 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
637 |
by (rule borel_space.sets_sigma_subset) auto |
|
638 |
qed |
|
639 |
||
640 |
lemma borel_space_eq_halfspace_ge: |
|
641 |
"sets borel_space = sets (sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x$$i})))" |
|
642 |
(is "_ = sets ?SIGMA") |
|
643 |
proof (rule antisym) |
|
644 |
show "sets borel_space \<subseteq> sets ?SIGMA" |
|
645 |
using halfspace_span_halfspace_ge open_span_halfspace by auto |
|
646 |
show "sets ?SIGMA \<subseteq> sets borel_space" |
|
647 |
by (rule borel_space.sets_sigma_subset) auto |
|
648 |
qed |
|
649 |
||
650 |
lemma (in sigma_algebra) borel_measurable_halfspacesI: |
|
651 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
652 |
assumes "sets borel_space = sets (sigma UNIV (range F))" |
|
653 |
and "\<And>a i. S a i = f -` F (a,i) \<inter> space M" |
|
654 |
and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M" |
|
655 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)" |
|
656 |
proof safe |
|
657 |
fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M" |
|
658 |
then show "S a i \<in> sets M" unfolding assms |
|
659 |
by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def) |
|
660 |
next |
|
661 |
assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M" |
|
662 |
{ fix a i have "S a i \<in> sets M" |
|
663 |
proof cases |
|
664 |
assume "i < DIM('c)" |
|
665 |
with a show ?thesis unfolding assms(2) by simp |
|
666 |
next |
|
667 |
assume "\<not> i < DIM('c)" |
|
668 |
from assms(3)[OF this] show ?thesis . |
|
669 |
qed } |
|
670 |
then have "f \<in> measurable M (sigma UNIV (range F))" |
|
671 |
by (auto intro!: measurable_sigma simp: assms(2)) |
|
672 |
then show "f \<in> borel_measurable M" unfolding measurable_def |
|
673 |
unfolding assms(1) by simp |
|
674 |
qed |
|
675 |
||
676 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_le: |
|
677 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
678 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)" |
|
679 |
by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_le]) auto |
|
680 |
||
681 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_less: |
|
682 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
683 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)" |
|
684 |
by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_less]) auto |
|
685 |
||
686 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge: |
|
687 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
688 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)" |
|
689 |
by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_ge]) auto |
|
690 |
||
691 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater: |
|
692 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
693 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)" |
|
694 |
by (rule borel_measurable_halfspacesI[OF borel_space_eq_halfspace_gt]) auto |
|
695 |
||
696 |
lemma (in sigma_algebra) borel_measurable_iff_le: |
|
697 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)" |
|
698 |
using borel_measurable_iff_halfspace_le[where 'c=real] by simp |
|
699 |
||
700 |
lemma (in sigma_algebra) borel_measurable_iff_less: |
|
701 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)" |
|
702 |
using borel_measurable_iff_halfspace_less[where 'c=real] by simp |
|
703 |
||
704 |
lemma (in sigma_algebra) borel_measurable_iff_ge: |
|
705 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)" |
|
706 |
using borel_measurable_iff_halfspace_ge[where 'c=real] by simp |
|
707 |
||
708 |
lemma (in sigma_algebra) borel_measurable_iff_greater: |
|
709 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)" |
|
710 |
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp |
|
711 |
||
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
712 |
lemma borel_measureable_euclidean_component: |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
713 |
"(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel_space" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
714 |
unfolding borel_space_def[where 'a=real] |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
715 |
proof (rule borel_space.measurable_sigma) |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
716 |
fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def . |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
717 |
from open_vimage_euclidean_component[OF this] |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
718 |
show "(\<lambda>x. x $$ i) -` S \<inter> space borel_space \<in> sets borel_space" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
719 |
by (auto intro: borel_space_open) |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
720 |
qed auto |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
721 |
|
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
722 |
lemma (in sigma_algebra) borel_measureable_euclidean_space: |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
723 |
fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
724 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
725 |
proof safe |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
726 |
fix i assume "f \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
727 |
then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
728 |
using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def] |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
729 |
by (auto intro: borel_measureable_euclidean_component) |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
730 |
next |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
731 |
assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
732 |
then show "f \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
733 |
unfolding borel_measurable_iff_halfspace_le by auto |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
734 |
qed |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
735 |
|
38656 | 736 |
subsection "Borel measurable operators" |
737 |
||
738 |
lemma (in sigma_algebra) affine_borel_measurable_vector: |
|
739 |
fixes f :: "'a \<Rightarrow> 'x::real_normed_vector" |
|
740 |
assumes "f \<in> borel_measurable M" |
|
741 |
shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M" |
|
742 |
proof (rule borel_measurableI) |
|
743 |
fix S :: "'x set" assume "open S" |
|
744 |
show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M" |
|
745 |
proof cases |
|
746 |
assume "b \<noteq> 0" |
|
747 |
with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open") |
|
748 |
by (auto intro!: open_affinity simp: scaleR.add_right mem_def) |
|
749 |
hence "?S \<in> sets borel_space" |
|
750 |
unfolding borel_space_def by (auto simp: sigma_def intro!: sigma_sets.Basic) |
|
751 |
moreover |
|
752 |
from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S" |
|
753 |
apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all) |
|
754 |
ultimately show ?thesis using assms unfolding in_borel_measurable_borel_space |
|
755 |
by auto |
|
756 |
qed simp |
|
757 |
qed |
|
758 |
||
759 |
lemma (in sigma_algebra) affine_borel_measurable: |
|
760 |
fixes g :: "'a \<Rightarrow> real" |
|
761 |
assumes g: "g \<in> borel_measurable M" |
|
762 |
shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M" |
|
763 |
using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute) |
|
764 |
||
765 |
lemma (in sigma_algebra) borel_measurable_add[simp, intro]: |
|
766 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
767 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
768 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
769 |
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
770 |
proof - |
38656 | 771 |
have 1: "\<And>a. {w\<in>space M. a \<le> f w + g w} = {w \<in> space M. a + g w * -1 \<le> f w}" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
772 |
by auto |
38656 | 773 |
have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M" |
774 |
by (rule affine_borel_measurable [OF g]) |
|
775 |
then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f |
|
776 |
by auto |
|
777 |
then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M" |
|
778 |
by (simp add: 1) |
|
779 |
then show ?thesis |
|
780 |
by (simp add: borel_measurable_iff_ge) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
781 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
782 |
|
38656 | 783 |
lemma (in sigma_algebra) borel_measurable_square: |
784 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
785 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
786 |
shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
787 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
788 |
{ |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
789 |
fix a |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
790 |
have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
791 |
proof (cases rule: linorder_cases [of a 0]) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
792 |
case less |
38656 | 793 |
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
794 |
by auto (metis less order_le_less_trans power2_less_0) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
795 |
also have "... \<in> sets M" |
38656 | 796 |
by (rule empty_sets) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
797 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
798 |
next |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
799 |
case equal |
38656 | 800 |
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
801 |
{w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
802 |
by auto |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
803 |
also have "... \<in> sets M" |
38656 | 804 |
apply (insert f) |
805 |
apply (rule Int) |
|
806 |
apply (simp add: borel_measurable_iff_le) |
|
807 |
apply (simp add: borel_measurable_iff_ge) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
808 |
done |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
809 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
810 |
next |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
811 |
case greater |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
812 |
have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a \<le> f x & f x \<le> sqrt a)" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
813 |
by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
814 |
real_sqrt_le_iff real_sqrt_power) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
815 |
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = |
38656 | 816 |
{w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
817 |
using greater by auto |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
818 |
also have "... \<in> sets M" |
38656 | 819 |
apply (insert f) |
820 |
apply (rule Int) |
|
821 |
apply (simp add: borel_measurable_iff_ge) |
|
822 |
apply (simp add: borel_measurable_iff_le) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
823 |
done |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
824 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
825 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
826 |
} |
38656 | 827 |
thus ?thesis by (auto simp add: borel_measurable_iff_le) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
828 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
829 |
|
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
830 |
lemma times_eq_sum_squares: |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
831 |
fixes x::real |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
832 |
shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4" |
38656 | 833 |
by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric]) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
834 |
|
38656 | 835 |
lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]: |
836 |
fixes g :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
837 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
838 |
shows "(\<lambda>x. - g x) \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
839 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
840 |
have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
841 |
by simp |
38656 | 842 |
also have "... \<in> borel_measurable M" |
843 |
by (fast intro: affine_borel_measurable g) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
844 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
845 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
846 |
|
38656 | 847 |
lemma (in sigma_algebra) borel_measurable_times[simp, intro]: |
848 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
849 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
850 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
851 |
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
852 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
853 |
have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M" |
38656 | 854 |
using assms by (fast intro: affine_borel_measurable borel_measurable_square) |
855 |
have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) = |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
856 |
(\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))" |
35582 | 857 |
by (simp add: minus_divide_right) |
38656 | 858 |
also have "... \<in> borel_measurable M" |
859 |
using f g by (fast intro: affine_borel_measurable borel_measurable_square f g) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
860 |
finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
861 |
show ?thesis |
38656 | 862 |
apply (simp add: times_eq_sum_squares diff_minus) |
863 |
using 1 2 by simp |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
864 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
865 |
|
38656 | 866 |
lemma (in sigma_algebra) borel_measurable_diff[simp, intro]: |
867 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
868 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
869 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
870 |
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
38656 | 871 |
unfolding diff_minus using assms by fast |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
872 |
|
38656 | 873 |
lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]: |
874 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real" |
|
875 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
|
876 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
|
877 |
proof cases |
|
878 |
assume "finite S" |
|
879 |
thus ?thesis using assms by induct auto |
|
880 |
qed simp |
|
35692 | 881 |
|
38656 | 882 |
lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]: |
883 |
fixes f :: "'a \<Rightarrow> real" |
|
35692 | 884 |
assumes "f \<in> borel_measurable M" |
885 |
shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M" |
|
38656 | 886 |
unfolding borel_measurable_iff_ge unfolding inverse_eq_divide |
887 |
proof safe |
|
888 |
fix a :: real |
|
889 |
have *: "{w \<in> space M. a \<le> 1 / f w} = |
|
890 |
({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union> |
|
891 |
({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union> |
|
892 |
({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq) |
|
893 |
show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding * |
|
894 |
by (auto intro!: Int Un) |
|
35692 | 895 |
qed |
896 |
||
38656 | 897 |
lemma (in sigma_algebra) borel_measurable_divide[simp, intro]: |
898 |
fixes f :: "'a \<Rightarrow> real" |
|
35692 | 899 |
assumes "f \<in> borel_measurable M" |
900 |
and "g \<in> borel_measurable M" |
|
901 |
shows "(\<lambda>x. f x / g x) \<in> borel_measurable M" |
|
902 |
unfolding field_divide_inverse |
|
38656 | 903 |
by (rule borel_measurable_inverse borel_measurable_times assms)+ |
904 |
||
905 |
lemma (in sigma_algebra) borel_measurable_max[intro, simp]: |
|
906 |
fixes f g :: "'a \<Rightarrow> real" |
|
907 |
assumes "f \<in> borel_measurable M" |
|
908 |
assumes "g \<in> borel_measurable M" |
|
909 |
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M" |
|
910 |
unfolding borel_measurable_iff_le |
|
911 |
proof safe |
|
912 |
fix a |
|
913 |
have "{x \<in> space M. max (g x) (f x) \<le> a} = |
|
914 |
{x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto |
|
915 |
thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M" |
|
916 |
using assms unfolding borel_measurable_iff_le |
|
917 |
by (auto intro!: Int) |
|
918 |
qed |
|
919 |
||
920 |
lemma (in sigma_algebra) borel_measurable_min[intro, simp]: |
|
921 |
fixes f g :: "'a \<Rightarrow> real" |
|
922 |
assumes "f \<in> borel_measurable M" |
|
923 |
assumes "g \<in> borel_measurable M" |
|
924 |
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M" |
|
925 |
unfolding borel_measurable_iff_ge |
|
926 |
proof safe |
|
927 |
fix a |
|
928 |
have "{x \<in> space M. a \<le> min (g x) (f x)} = |
|
929 |
{x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto |
|
930 |
thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M" |
|
931 |
using assms unfolding borel_measurable_iff_ge |
|
932 |
by (auto intro!: Int) |
|
933 |
qed |
|
934 |
||
935 |
lemma (in sigma_algebra) borel_measurable_abs[simp, intro]: |
|
936 |
assumes "f \<in> borel_measurable M" |
|
937 |
shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M" |
|
938 |
proof - |
|
939 |
have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def) |
|
940 |
show ?thesis unfolding * using assms by auto |
|
941 |
qed |
|
942 |
||
943 |
section "Borel space over the real line with infinity" |
|
35692 | 944 |
|
38656 | 945 |
lemma borel_space_Real_measurable: |
946 |
"A \<in> sets borel_space \<Longrightarrow> Real -` A \<in> sets borel_space" |
|
947 |
proof (rule borel_measurable_translate) |
|
948 |
fix B :: "pinfreal set" assume "open B" |
|
949 |
then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and |
|
950 |
x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B" |
|
951 |
unfolding open_pinfreal_def by blast |
|
952 |
||
953 |
have "Real -` B = Real -` (B - {\<omega>})" by auto |
|
954 |
also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp |
|
955 |
also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})" |
|
956 |
apply (auto simp add: Real_eq_Real image_iff) |
|
957 |
apply (rule_tac x="max 0 x" in bexI) |
|
958 |
by (auto simp: max_def) |
|
959 |
finally show "Real -` B \<in> sets borel_space" |
|
960 |
using `open T` by auto |
|
961 |
qed simp |
|
962 |
||
963 |
lemma borel_space_real_measurable: |
|
964 |
"A \<in> sets borel_space \<Longrightarrow> (real -` A :: pinfreal set) \<in> sets borel_space" |
|
965 |
proof (rule borel_measurable_translate) |
|
966 |
fix B :: "real set" assume "open B" |
|
967 |
{ fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto } |
|
968 |
note Ex_less_real = this |
|
969 |
have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))" |
|
970 |
by (force simp: Ex_less_real) |
|
971 |
||
972 |
have "open (real -` (B \<inter> {0 <..}) :: pinfreal set)" |
|
973 |
unfolding open_pinfreal_def using `open B` |
|
974 |
by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real) |
|
975 |
then show "(real -` B :: pinfreal set) \<in> sets borel_space" unfolding * by auto |
|
976 |
qed simp |
|
977 |
||
978 |
lemma (in sigma_algebra) borel_measurable_Real[intro, simp]: |
|
979 |
assumes "f \<in> borel_measurable M" |
|
980 |
shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M" |
|
981 |
unfolding in_borel_measurable_borel_space |
|
982 |
proof safe |
|
983 |
fix S :: "pinfreal set" assume "S \<in> sets borel_space" |
|
984 |
from borel_space_Real_measurable[OF this] |
|
985 |
have "(Real \<circ> f) -` S \<inter> space M \<in> sets M" |
|
986 |
using assms |
|
987 |
unfolding vimage_compose in_borel_measurable_borel_space |
|
988 |
by auto |
|
989 |
thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def) |
|
35748 | 990 |
qed |
35692 | 991 |
|
38656 | 992 |
lemma (in sigma_algebra) borel_measurable_real[intro, simp]: |
993 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
994 |
assumes "f \<in> borel_measurable M" |
|
995 |
shows "(\<lambda>x. real (f x)) \<in> borel_measurable M" |
|
996 |
unfolding in_borel_measurable_borel_space |
|
997 |
proof safe |
|
998 |
fix S :: "real set" assume "S \<in> sets borel_space" |
|
999 |
from borel_space_real_measurable[OF this] |
|
1000 |
have "(real \<circ> f) -` S \<inter> space M \<in> sets M" |
|
1001 |
using assms |
|
1002 |
unfolding vimage_compose in_borel_measurable_borel_space |
|
1003 |
by auto |
|
1004 |
thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def) |
|
1005 |
qed |
|
35692 | 1006 |
|
38656 | 1007 |
lemma (in sigma_algebra) borel_measurable_Real_eq: |
1008 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" |
|
1009 |
shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" |
|
1010 |
proof |
|
1011 |
have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}" |
|
1012 |
by auto |
|
1013 |
assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M" |
|
1014 |
hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M" |
|
1015 |
by (rule borel_measurable_real) |
|
1016 |
moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x" |
|
1017 |
using assms by auto |
|
1018 |
ultimately show "f \<in> borel_measurable M" |
|
1019 |
by (simp cong: measurable_cong) |
|
1020 |
qed auto |
|
35692 | 1021 |
|
38656 | 1022 |
lemma (in sigma_algebra) borel_measurable_pinfreal_eq_real: |
1023 |
"f \<in> borel_measurable M \<longleftrightarrow> |
|
1024 |
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)" |
|
1025 |
proof safe |
|
1026 |
assume "f \<in> borel_measurable M" |
|
1027 |
then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M" |
|
1028 |
by (auto intro: borel_measurable_vimage borel_measurable_real) |
|
1029 |
next |
|
1030 |
assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M" |
|
1031 |
have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto |
|
1032 |
with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp |
|
1033 |
have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))" |
|
1034 |
by (simp add: expand_fun_eq Real_real) |
|
1035 |
show "f \<in> borel_measurable M" |
|
1036 |
apply (subst f) |
|
1037 |
apply (rule measurable_If) |
|
1038 |
using * ** by auto |
|
1039 |
qed |
|
1040 |
||
1041 |
lemma (in sigma_algebra) less_eq_ge_measurable: |
|
1042 |
fixes f :: "'a \<Rightarrow> 'c::linorder" |
|
1043 |
shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M" |
|
1044 |
proof |
|
1045 |
assume "{x\<in>space M. f x \<le> a} \<in> sets M" |
|
1046 |
moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto |
|
1047 |
ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto |
|
1048 |
next |
|
1049 |
assume "{x\<in>space M. a < f x} \<in> sets M" |
|
1050 |
moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto |
|
1051 |
ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto |
|
1052 |
qed |
|
35692 | 1053 |
|
38656 | 1054 |
lemma (in sigma_algebra) greater_eq_le_measurable: |
1055 |
fixes f :: "'a \<Rightarrow> 'c::linorder" |
|
1056 |
shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M" |
|
1057 |
proof |
|
1058 |
assume "{x\<in>space M. a \<le> f x} \<in> sets M" |
|
1059 |
moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto |
|
1060 |
ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto |
|
1061 |
next |
|
1062 |
assume "{x\<in>space M. f x < a} \<in> sets M" |
|
1063 |
moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto |
|
1064 |
ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto |
|
1065 |
qed |
|
1066 |
||
1067 |
lemma (in sigma_algebra) less_eq_le_pinfreal_measurable: |
|
1068 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
1069 |
shows "(\<forall>a. {x\<in>space M. a < f x} \<in> sets M) \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)" |
|
1070 |
proof |
|
1071 |
assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M" |
|
1072 |
show "\<forall>a. {x \<in> space M. a < f x} \<in> sets M" |
|
1073 |
proof |
|
1074 |
fix a show "{x \<in> space M. a < f x} \<in> sets M" |
|
1075 |
proof (cases a) |
|
1076 |
case (preal r) |
|
1077 |
have "{x\<in>space M. a < f x} = (\<Union>i. {x\<in>space M. a + inverse (of_nat (Suc i)) \<le> f x})" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
1078 |
proof safe |
38656 | 1079 |
fix x assume "a < f x" and [simp]: "x \<in> space M" |
1080 |
with ex_pinfreal_inverse_of_nat_Suc_less[of "f x - a"] |
|
1081 |
obtain n where "a + inverse (of_nat (Suc n)) < f x" |
|
1082 |
by (cases "f x", auto simp: pinfreal_minus_order) |
|
1083 |
then have "a + inverse (of_nat (Suc n)) \<le> f x" by simp |
|
1084 |
then show "x \<in> (\<Union>i. {x \<in> space M. a + inverse (of_nat (Suc i)) \<le> f x})" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
1085 |
by auto |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
1086 |
next |
38656 | 1087 |
fix i x assume [simp]: "x \<in> space M" |
1088 |
have "a < a + inverse (of_nat (Suc i))" using preal by auto |
|
1089 |
also assume "a + inverse (of_nat (Suc i)) \<le> f x" |
|
1090 |
finally show "a < f x" . |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
1091 |
qed |
38656 | 1092 |
with a show ?thesis by auto |
1093 |
qed simp |
|
35582 | 1094 |
qed |
1095 |
next |
|
38656 | 1096 |
assume a': "\<forall>a. {x \<in> space M. a < f x} \<in> sets M" |
1097 |
then have a: "\<forall>a. {x \<in> space M. f x \<le> a} \<in> sets M" unfolding less_eq_ge_measurable . |
|
1098 |
show "\<forall>a. {x \<in> space M. a \<le> f x} \<in> sets M" unfolding greater_eq_le_measurable[symmetric] |
|
1099 |
proof |
|
1100 |
fix a show "{x \<in> space M. f x < a} \<in> sets M" |
|
1101 |
proof (cases a) |
|
1102 |
case (preal r) |
|
1103 |
show ?thesis |
|
1104 |
proof cases |
|
1105 |
assume "a = 0" then show ?thesis by simp |
|
1106 |
next |
|
1107 |
assume "a \<noteq> 0" |
|
1108 |
have "{x\<in>space M. f x < a} = (\<Union>i. {x\<in>space M. f x \<le> a - inverse (of_nat (Suc i))})" |
|
1109 |
proof safe |
|
1110 |
fix x assume "f x < a" and [simp]: "x \<in> space M" |
|
1111 |
with ex_pinfreal_inverse_of_nat_Suc_less[of "a - f x"] |
|
1112 |
obtain n where "inverse (of_nat (Suc n)) < a - f x" |
|
1113 |
using preal by (cases "f x") auto |
|
1114 |
then have "f x \<le> a - inverse (of_nat (Suc n)) " |
|
1115 |
using preal by (cases "f x") (auto split: split_if_asm) |
|
1116 |
then show "x \<in> (\<Union>i. {x \<in> space M. f x \<le> a - inverse (of_nat (Suc i))})" |
|
1117 |
by auto |
|
1118 |
next |
|
1119 |
fix i x assume [simp]: "x \<in> space M" |
|
1120 |
assume "f x \<le> a - inverse (of_nat (Suc i))" |
|
1121 |
also have "\<dots> < a" using `a \<noteq> 0` preal by auto |
|
1122 |
finally show "f x < a" . |
|
1123 |
qed |
|
1124 |
with a show ?thesis by auto |
|
1125 |
qed |
|
1126 |
next |
|
1127 |
case infinite |
|
1128 |
have "f -` {\<omega>} \<inter> space M = (\<Inter>n. {x\<in>space M. of_nat n < f x})" |
|
1129 |
proof (safe, simp_all, safe) |
|
1130 |
fix x assume *: "\<forall>n::nat. Real (real n) < f x" |
|
1131 |
show "f x = \<omega>" proof (rule ccontr) |
|
1132 |
assume "f x \<noteq> \<omega>" |
|
1133 |
with real_arch_lt[of "real (f x)"] obtain n where "f x < of_nat n" |
|
1134 |
by (auto simp: pinfreal_noteq_omega_Ex) |
|
1135 |
with *[THEN spec, of n] show False by auto |
|
1136 |
qed |
|
1137 |
qed |
|
1138 |
with a' have \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" by auto |
|
1139 |
moreover have "{x \<in> space M. f x < a} = space M - f -` {\<omega>} \<inter> space M" |
|
1140 |
using infinite by auto |
|
1141 |
ultimately show ?thesis by auto |
|
1142 |
qed |
|
35582 | 1143 |
qed |
1144 |
qed |
|
1145 |
||
38656 | 1146 |
lemma (in sigma_algebra) borel_measurable_pinfreal_iff_greater: |
1147 |
"(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)" |
|
1148 |
proof safe |
|
1149 |
fix a assume f: "f \<in> borel_measurable M" |
|
1150 |
have "{x\<in>space M. a < f x} = f -` {a <..} \<inter> space M" by auto |
|
1151 |
with f show "{x\<in>space M. a < f x} \<in> sets M" |
|
1152 |
by (auto intro!: measurable_sets) |
|
1153 |
next |
|
1154 |
assume *: "\<forall>a. {x\<in>space M. a < f x} \<in> sets M" |
|
1155 |
hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M" |
|
1156 |
unfolding less_eq_le_pinfreal_measurable |
|
1157 |
unfolding greater_eq_le_measurable . |
|
35582 | 1158 |
|
38656 | 1159 |
show "f \<in> borel_measurable M" unfolding borel_measurable_pinfreal_eq_real borel_measurable_iff_greater |
1160 |
proof safe |
|
1161 |
have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto |
|
1162 |
then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto |
|
35582 | 1163 |
|
38656 | 1164 |
fix a |
1165 |
have "{w \<in> space M. a < real (f w)} = |
|
1166 |
(if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)" |
|
1167 |
proof (split split_if, safe del: notI) |
|
1168 |
fix x assume "0 \<le> a" |
|
1169 |
{ assume "a < real (f x)" then show "Real a < f x" "x \<notin> f -` {\<omega>} \<inter> space M" |
|
1170 |
using `0 \<le> a` by (cases "f x", auto) } |
|
1171 |
{ assume "Real a < f x" "x \<notin> f -` {\<omega>}" then show "a < real (f x)" |
|
1172 |
using `0 \<le> a` by (cases "f x", auto) } |
|
1173 |
next |
|
1174 |
fix x assume "\<not> 0 \<le> a" then show "a < real (f x)" by (cases "f x") auto |
|
1175 |
qed |
|
1176 |
then show "{w \<in> space M. a < real (f w)} \<in> sets M" |
|
1177 |
using \<omega> * by (auto intro!: Diff) |
|
35582 | 1178 |
qed |
1179 |
qed |
|
1180 |
||
38656 | 1181 |
lemma (in sigma_algebra) borel_measurable_pinfreal_iff_less: |
1182 |
"(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)" |
|
1183 |
using borel_measurable_pinfreal_iff_greater unfolding less_eq_le_pinfreal_measurable greater_eq_le_measurable . |
|
1184 |
||
1185 |
lemma (in sigma_algebra) borel_measurable_pinfreal_iff_le: |
|
1186 |
"(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)" |
|
1187 |
using borel_measurable_pinfreal_iff_greater unfolding less_eq_ge_measurable . |
|
1188 |
||
1189 |
lemma (in sigma_algebra) borel_measurable_pinfreal_iff_ge: |
|
1190 |
"(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)" |
|
1191 |
using borel_measurable_pinfreal_iff_greater unfolding less_eq_le_pinfreal_measurable . |
|
1192 |
||
1193 |
lemma (in sigma_algebra) borel_measurable_pinfreal_eq_const: |
|
1194 |
fixes f :: "'a \<Rightarrow> pinfreal" assumes "f \<in> borel_measurable M" |
|
1195 |
shows "{x\<in>space M. f x = c} \<in> sets M" |
|
1196 |
proof - |
|
1197 |
have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto |
|
1198 |
then show ?thesis using assms by (auto intro!: measurable_sets) |
|
1199 |
qed |
|
1200 |
||
1201 |
lemma (in sigma_algebra) borel_measurable_pinfreal_neq_const: |
|
1202 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
1203 |
assumes "f \<in> borel_measurable M" |
|
1204 |
shows "{x\<in>space M. f x \<noteq> c} \<in> sets M" |
|
1205 |
proof - |
|
1206 |
have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto |
|
1207 |
then show ?thesis using assms by (auto intro!: measurable_sets) |
|
1208 |
qed |
|
1209 |
||
1210 |
lemma (in sigma_algebra) borel_measurable_pinfreal_less[intro,simp]: |
|
1211 |
fixes f g :: "'a \<Rightarrow> pinfreal" |
|
1212 |
assumes f: "f \<in> borel_measurable M" |
|
1213 |
assumes g: "g \<in> borel_measurable M" |
|
1214 |
shows "{x \<in> space M. f x < g x} \<in> sets M" |
|
1215 |
proof - |
|
1216 |
have "(\<lambda>x. real (f x)) \<in> borel_measurable M" |
|
1217 |
"(\<lambda>x. real (g x)) \<in> borel_measurable M" |
|
1218 |
using assms by (auto intro!: borel_measurable_real) |
|
1219 |
from borel_measurable_less[OF this] |
|
1220 |
have "{x \<in> space M. real (f x) < real (g x)} \<in> sets M" . |
|
1221 |
moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pinfreal_neq_const) |
|
1222 |
moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pinfreal_eq_const) |
|
1223 |
moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pinfreal_neq_const) |
|
1224 |
moreover have "{x \<in> space M. f x < g x} = ({x \<in> space M. g x = \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>}) \<union> |
|
1225 |
({x \<in> space M. g x \<noteq> \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>} \<inter> {x \<in> space M. real (f x) < real (g x)})" |
|
1226 |
by (auto simp: real_of_pinfreal_strict_mono_iff) |
|
1227 |
ultimately show ?thesis by auto |
|
1228 |
qed |
|
1229 |
||
1230 |
lemma (in sigma_algebra) borel_measurable_pinfreal_le[intro,simp]: |
|
1231 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
1232 |
assumes f: "f \<in> borel_measurable M" |
|
1233 |
assumes g: "g \<in> borel_measurable M" |
|
1234 |
shows "{x \<in> space M. f x \<le> g x} \<in> sets M" |
|
1235 |
proof - |
|
1236 |
have "{x \<in> space M. f x \<le> g x} = space M - {x \<in> space M. g x < f x}" by auto |
|
1237 |
then show ?thesis using g f by auto |
|
1238 |
qed |
|
1239 |
||
1240 |
lemma (in sigma_algebra) borel_measurable_pinfreal_eq[intro,simp]: |
|
1241 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
1242 |
assumes f: "f \<in> borel_measurable M" |
|
1243 |
assumes g: "g \<in> borel_measurable M" |
|
1244 |
shows "{w \<in> space M. f w = g w} \<in> sets M" |
|
1245 |
proof - |
|
1246 |
have "{x \<in> space M. f x = g x} = {x \<in> space M. g x \<le> f x} \<inter> {x \<in> space M. f x \<le> g x}" by auto |
|
1247 |
then show ?thesis using g f by auto |
|
1248 |
qed |
|
1249 |
||
1250 |
lemma (in sigma_algebra) borel_measurable_pinfreal_neq[intro,simp]: |
|
1251 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
1252 |
assumes f: "f \<in> borel_measurable M" |
|
1253 |
assumes g: "g \<in> borel_measurable M" |
|
1254 |
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
|
35692 | 1255 |
proof - |
38656 | 1256 |
have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto |
1257 |
thus ?thesis using f g by auto |
|
1258 |
qed |
|
1259 |
||
1260 |
lemma (in sigma_algebra) borel_measurable_pinfreal_add[intro, simp]: |
|
1261 |
fixes f :: "'a \<Rightarrow> pinfreal" |
|
1262 |
assumes measure: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
1263 |
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
|
1264 |
proof - |
|
1265 |
have *: "(\<lambda>x. f x + g x) = |
|
1266 |
(\<lambda>x. if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))" |
|
1267 |
by (auto simp: expand_fun_eq pinfreal_noteq_omega_Ex) |
|
1268 |
show ?thesis using assms unfolding * |
|
1269 |
by (auto intro!: measurable_If) |
|
1270 |
qed |
|
1271 |
||
1272 |
lemma (in sigma_algebra) borel_measurable_pinfreal_times[intro, simp]: |
|
1273 |
fixes f :: "'a \<Rightarrow> pinfreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
1274 |
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
|
1275 |
proof - |
|
1276 |
have *: "(\<lambda>x. f x * g x) = |
|
1277 |
(\<lambda>x. if f x = 0 then 0 else if g x = 0 then 0 else if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else |
|
1278 |
Real (real (f x) * real (g x)))" |
|
1279 |
by (auto simp: expand_fun_eq pinfreal_noteq_omega_Ex) |
|
1280 |
show ?thesis using assms unfolding * |
|
1281 |
by (auto intro!: measurable_If) |
|
1282 |
qed |
|
1283 |
||
1284 |
lemma (in sigma_algebra) borel_measurable_pinfreal_setsum[simp, intro]: |
|
1285 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pinfreal" |
|
1286 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
|
1287 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
|
1288 |
proof cases |
|
1289 |
assume "finite S" |
|
1290 |
thus ?thesis using assms |
|
1291 |
by induct auto |
|
1292 |
qed (simp add: borel_measurable_const) |
|
1293 |
||
1294 |
lemma (in sigma_algebra) borel_measurable_pinfreal_min[intro, simp]: |
|
1295 |
fixes f g :: "'a \<Rightarrow> pinfreal" |
|
1296 |
assumes "f \<in> borel_measurable M" |
|
1297 |
assumes "g \<in> borel_measurable M" |
|
1298 |
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M" |
|
1299 |
using assms unfolding min_def by (auto intro!: measurable_If) |
|
1300 |
||
1301 |
lemma (in sigma_algebra) borel_measurable_pinfreal_max[intro]: |
|
1302 |
fixes f g :: "'a \<Rightarrow> pinfreal" |
|
1303 |
assumes "f \<in> borel_measurable M" |
|
1304 |
and "g \<in> borel_measurable M" |
|
1305 |
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M" |
|
1306 |
using assms unfolding max_def by (auto intro!: measurable_If) |
|
1307 |
||
1308 |
lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]: |
|
1309 |
fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pinfreal" |
|
1310 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M" |
|
1311 |
shows "(SUP i : A. f i) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M") |
|
1312 |
unfolding borel_measurable_pinfreal_iff_greater |
|
1313 |
proof safe |
|
1314 |
fix a |
|
1315 |
have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})" |
|
38705 | 1316 |
by (auto simp: less_Sup_iff SUPR_def[where 'a=pinfreal] SUPR_fun_expand[where 'c=pinfreal]) |
38656 | 1317 |
then show "{x\<in>space M. a < ?sup x} \<in> sets M" |
1318 |
using assms by auto |
|
1319 |
qed |
|
1320 |
||
1321 |
lemma (in sigma_algebra) borel_measurable_INF[simp, intro]: |
|
1322 |
fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pinfreal" |
|
1323 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M" |
|
1324 |
shows "(INF i : A. f i) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M") |
|
1325 |
unfolding borel_measurable_pinfreal_iff_less |
|
1326 |
proof safe |
|
1327 |
fix a |
|
1328 |
have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})" |
|
1329 |
by (auto simp: Inf_less_iff INFI_def[where 'a=pinfreal] INFI_fun_expand) |
|
1330 |
then show "{x\<in>space M. ?inf x < a} \<in> sets M" |
|
1331 |
using assms by auto |
|
1332 |
qed |
|
1333 |
||
1334 |
lemma (in sigma_algebra) borel_measurable_pinfreal_diff: |
|
1335 |
fixes f g :: "'a \<Rightarrow> pinfreal" |
|
1336 |
assumes "f \<in> borel_measurable M" |
|
1337 |
assumes "g \<in> borel_measurable M" |
|
1338 |
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
|
1339 |
unfolding borel_measurable_pinfreal_iff_greater |
|
1340 |
proof safe |
|
1341 |
fix a |
|
1342 |
have "{x \<in> space M. a < f x - g x} = {x \<in> space M. g x + a < f x}" |
|
1343 |
by (simp add: pinfreal_less_minus_iff) |
|
1344 |
then show "{x \<in> space M. a < f x - g x} \<in> sets M" |
|
1345 |
using assms by auto |
|
35692 | 1346 |
qed |
1347 |
||
39092 | 1348 |
lemma (in sigma_algebra) borel_measurable_psuminf: |
1349 |
assumes "\<And>i. f i \<in> borel_measurable M" |
|
1350 |
shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M" |
|
1351 |
using assms unfolding psuminf_def |
|
1352 |
by (auto intro!: borel_measurable_SUP[unfolded SUPR_fun_expand]) |
|
1353 |
||
1354 |
section "LIMSEQ is borel measurable" |
|
1355 |
||
1356 |
lemma (in sigma_algebra) borel_measurable_LIMSEQ: |
|
1357 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" |
|
1358 |
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x" |
|
1359 |
and u: "\<And>i. u i \<in> borel_measurable M" |
|
1360 |
shows "u' \<in> borel_measurable M" |
|
1361 |
proof - |
|
1362 |
let "?pu x i" = "max (u i x) 0" |
|
1363 |
let "?nu x i" = "max (- u i x) 0" |
|
1364 |
||
1365 |
{ fix x assume x: "x \<in> space M" |
|
1366 |
have "(?pu x) ----> max (u' x) 0" |
|
1367 |
"(?nu x) ----> max (- u' x) 0" |
|
1368 |
using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus) |
|
1369 |
from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)] |
|
1370 |
have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)" |
|
1371 |
"(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)" |
|
1372 |
by (simp_all add: Real_max'[symmetric]) } |
|
1373 |
note eq = this |
|
1374 |
||
1375 |
have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x" |
|
1376 |
by auto |
|
1377 |
||
1378 |
have "(SUP n. INF m. (\<lambda>x. Real (u (n + m) x))) \<in> borel_measurable M" |
|
1379 |
"(SUP n. INF m. (\<lambda>x. Real (- u (n + m) x))) \<in> borel_measurable M" |
|
1380 |
using u by (auto intro: borel_measurable_SUP borel_measurable_INF borel_measurable_Real) |
|
1381 |
with eq[THEN measurable_cong, of M "\<lambda>x. x" borel_space] |
|
1382 |
have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M" |
|
1383 |
"(\<lambda>x. Real (- u' x)) \<in> borel_measurable M" |
|
1384 |
unfolding SUPR_fun_expand INFI_fun_expand by auto |
|
1385 |
note this[THEN borel_measurable_real] |
|
1386 |
from borel_measurable_diff[OF this] |
|
1387 |
show ?thesis unfolding * . |
|
1388 |
qed |
|
1389 |
||
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
1390 |
end |