author | hoelzl |
Tue, 19 Jul 2011 14:36:12 +0200 | |
changeset 43920 | cedb5cb948fd |
parent 42990 | 3706951a6421 |
child 43923 | ab93d0190a5d |
permissions | -rw-r--r-- |
42150 | 1 |
(* Title: HOL/Probability/Borel_Space.thy |
42067 | 2 |
Author: Johannes Hölzl, TU München |
3 |
Author: Armin Heller, TU München |
|
4 |
*) |
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38656 | 5 |
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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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|
40859 | 8 |
theory Borel_Space |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
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|
9 |
imports Sigma_Algebra Multivariate_Analysis |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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10 |
begin |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
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11 |
|
38656 | 12 |
section "Generic Borel spaces" |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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13 |
|
40859 | 14 |
definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>" |
15 |
abbreviation "borel_measurable M \<equiv> measurable M borel" |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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16 |
|
40859 | 17 |
interpretation borel: sigma_algebra borel |
18 |
by (auto simp: borel_def intro!: sigma_algebra_sigma) |
|
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
19 |
|
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
20 |
lemma in_borel_measurable: |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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21 |
"f \<in> borel_measurable M \<longleftrightarrow> |
40859 | 22 |
(\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = open\<rparr>). |
38656 | 23 |
f -` S \<inter> space M \<in> sets M)" |
40859 | 24 |
by (auto simp add: measurable_def borel_def) |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
25 |
|
40859 | 26 |
lemma in_borel_measurable_borel: |
38656 | 27 |
"f \<in> borel_measurable M \<longleftrightarrow> |
40859 | 28 |
(\<forall>S \<in> sets borel. |
38656 | 29 |
f -` S \<inter> space M \<in> sets M)" |
40859 | 30 |
by (auto simp add: measurable_def borel_def) |
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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31 |
|
40859 | 32 |
lemma space_borel[simp]: "space borel = UNIV" |
33 |
unfolding borel_def by auto |
|
38656 | 34 |
|
40859 | 35 |
lemma borel_open[simp]: |
36 |
assumes "open A" shows "A \<in> sets borel" |
|
38656 | 37 |
proof - |
38 |
have "A \<in> open" unfolding mem_def using assms . |
|
40859 | 39 |
thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic) |
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
40 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
41 |
|
40859 | 42 |
lemma borel_closed[simp]: |
43 |
assumes "closed A" shows "A \<in> sets borel" |
|
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
44 |
proof - |
40859 | 45 |
have "space borel - (- A) \<in> sets borel" |
46 |
using assms unfolding closed_def by (blast intro: borel_open) |
|
38656 | 47 |
thus ?thesis by simp |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
48 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
49 |
|
41830 | 50 |
lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel" |
51 |
unfolding Compl_eq_Diff_UNIV by (intro borel.Diff) auto |
|
52 |
||
38656 | 53 |
lemma (in sigma_algebra) borel_measurable_vimage: |
54 |
fixes f :: "'a \<Rightarrow> 'x::t2_space" |
|
55 |
assumes borel: "f \<in> borel_measurable M" |
|
56 |
shows "f -` {x} \<inter> space M \<in> sets M" |
|
57 |
proof (cases "x \<in> f ` space M") |
|
58 |
case True then obtain y where "x = f y" by auto |
|
41969 | 59 |
from closed_singleton[of "f y"] |
40859 | 60 |
have "{f y} \<in> sets borel" by (rule borel_closed) |
38656 | 61 |
with assms show ?thesis |
40859 | 62 |
unfolding in_borel_measurable_borel `x = f y` by auto |
38656 | 63 |
next |
64 |
case False hence "f -` {x} \<inter> space M = {}" by auto |
|
65 |
thus ?thesis by auto |
|
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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|
66 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
67 |
|
38656 | 68 |
lemma (in sigma_algebra) borel_measurableI: |
69 |
fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space" |
|
70 |
assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
|
71 |
shows "f \<in> borel_measurable M" |
|
40859 | 72 |
unfolding borel_def |
73 |
proof (rule measurable_sigma, simp_all) |
|
38656 | 74 |
fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M" |
75 |
using assms[of S] by (simp add: mem_def) |
|
40859 | 76 |
qed |
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
77 |
|
40859 | 78 |
lemma borel_singleton[simp, intro]: |
38656 | 79 |
fixes x :: "'a::t1_space" |
40859 | 80 |
shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel" |
81 |
proof (rule borel.insert_in_sets) |
|
82 |
show "{x} \<in> sets borel" |
|
41969 | 83 |
using closed_singleton[of x] by (rule borel_closed) |
38656 | 84 |
qed simp |
85 |
||
86 |
lemma (in sigma_algebra) borel_measurable_const[simp, intro]: |
|
87 |
"(\<lambda>x. c) \<in> borel_measurable M" |
|
88 |
by (auto intro!: measurable_const) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
89 |
|
39083 | 90 |
lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]: |
38656 | 91 |
assumes A: "A \<in> sets M" |
92 |
shows "indicator A \<in> borel_measurable M" |
|
93 |
unfolding indicator_def_raw using A |
|
94 |
by (auto intro!: measurable_If_set borel_measurable_const) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
95 |
|
40859 | 96 |
lemma (in sigma_algebra) borel_measurable_indicator_iff: |
97 |
"(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M" |
|
98 |
(is "?I \<in> borel_measurable M \<longleftrightarrow> _") |
|
99 |
proof |
|
100 |
assume "?I \<in> borel_measurable M" |
|
101 |
then have "?I -` {1} \<inter> space M \<in> sets M" |
|
102 |
unfolding measurable_def by auto |
|
103 |
also have "?I -` {1} \<inter> space M = A \<inter> space M" |
|
104 |
unfolding indicator_def_raw by auto |
|
105 |
finally show "A \<inter> space M \<in> sets M" . |
|
106 |
next |
|
107 |
assume "A \<inter> space M \<in> sets M" |
|
108 |
moreover have "?I \<in> borel_measurable M \<longleftrightarrow> |
|
109 |
(indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M" |
|
110 |
by (intro measurable_cong) (auto simp: indicator_def) |
|
111 |
ultimately show "?I \<in> borel_measurable M" by auto |
|
112 |
qed |
|
113 |
||
39092 | 114 |
lemma (in sigma_algebra) borel_measurable_restricted: |
43920 | 115 |
fixes f :: "'a \<Rightarrow> ereal" assumes "A \<in> sets M" |
39092 | 116 |
shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow> |
117 |
(\<lambda>x. f x * indicator A x) \<in> borel_measurable M" |
|
118 |
(is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M") |
|
119 |
proof - |
|
120 |
interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`]) |
|
121 |
have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R" |
|
122 |
by (auto intro!: measurable_cong) |
|
123 |
show ?thesis unfolding * |
|
40859 | 124 |
unfolding in_borel_measurable_borel |
39092 | 125 |
proof (simp, safe) |
43920 | 126 |
fix S :: "ereal set" assume "S \<in> sets borel" |
40859 | 127 |
"\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M" |
39092 | 128 |
then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto |
129 |
then have f: "?f -` S \<inter> A \<in> sets M" |
|
130 |
using `A \<in> sets M` sets_into_space by fastsimp |
|
131 |
show "?f -` S \<inter> space M \<in> sets M" |
|
132 |
proof cases |
|
133 |
assume "0 \<in> S" |
|
134 |
then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)" |
|
135 |
using `A \<in> sets M` sets_into_space by auto |
|
136 |
then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff) |
|
137 |
next |
|
138 |
assume "0 \<notin> S" |
|
139 |
then have "?f -` S \<inter> space M = ?f -` S \<inter> A" |
|
140 |
using `A \<in> sets M` sets_into_space |
|
141 |
by (auto simp: indicator_def split: split_if_asm) |
|
142 |
then show ?thesis using f by auto |
|
143 |
qed |
|
144 |
next |
|
43920 | 145 |
fix S :: "ereal set" assume "S \<in> sets borel" |
40859 | 146 |
"\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M" |
39092 | 147 |
then have f: "?f -` S \<inter> space M \<in> sets M" by auto |
148 |
then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M" |
|
149 |
using `A \<in> sets M` sets_into_space |
|
150 |
apply (simp add: image_iff) |
|
151 |
apply (rule bexI[OF _ f]) |
|
152 |
by auto |
|
153 |
qed |
|
154 |
qed |
|
155 |
||
156 |
lemma (in sigma_algebra) borel_measurable_subalgebra: |
|
41545 | 157 |
assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N" |
39092 | 158 |
shows "f \<in> borel_measurable M" |
159 |
using assms unfolding measurable_def by auto |
|
160 |
||
38656 | 161 |
section "Borel spaces on euclidean spaces" |
162 |
||
163 |
lemma lessThan_borel[simp, intro]: |
|
164 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 165 |
shows "{..< a} \<in> sets borel" |
166 |
by (blast intro: borel_open) |
|
38656 | 167 |
|
168 |
lemma greaterThan_borel[simp, intro]: |
|
169 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 170 |
shows "{a <..} \<in> sets borel" |
171 |
by (blast intro: borel_open) |
|
38656 | 172 |
|
173 |
lemma greaterThanLessThan_borel[simp, intro]: |
|
174 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 175 |
shows "{a<..<b} \<in> sets borel" |
176 |
by (blast intro: borel_open) |
|
38656 | 177 |
|
178 |
lemma atMost_borel[simp, intro]: |
|
179 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 180 |
shows "{..a} \<in> sets borel" |
181 |
by (blast intro: borel_closed) |
|
38656 | 182 |
|
183 |
lemma atLeast_borel[simp, intro]: |
|
184 |
fixes a :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 185 |
shows "{a..} \<in> sets borel" |
186 |
by (blast intro: borel_closed) |
|
38656 | 187 |
|
188 |
lemma atLeastAtMost_borel[simp, intro]: |
|
189 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 190 |
shows "{a..b} \<in> sets borel" |
191 |
by (blast intro: borel_closed) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
192 |
|
38656 | 193 |
lemma greaterThanAtMost_borel[simp, intro]: |
194 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 195 |
shows "{a<..b} \<in> sets borel" |
38656 | 196 |
unfolding greaterThanAtMost_def by blast |
197 |
||
198 |
lemma atLeastLessThan_borel[simp, intro]: |
|
199 |
fixes a b :: "'a\<Colon>ordered_euclidean_space" |
|
40859 | 200 |
shows "{a..<b} \<in> sets borel" |
38656 | 201 |
unfolding atLeastLessThan_def by blast |
202 |
||
203 |
lemma hafspace_less_borel[simp, intro]: |
|
204 |
fixes a :: real |
|
40859 | 205 |
shows "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel" |
206 |
by (auto intro!: borel_open open_halfspace_component_gt) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
207 |
|
38656 | 208 |
lemma hafspace_greater_borel[simp, intro]: |
209 |
fixes a :: real |
|
40859 | 210 |
shows "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel" |
211 |
by (auto intro!: borel_open open_halfspace_component_lt) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
212 |
|
38656 | 213 |
lemma hafspace_less_eq_borel[simp, intro]: |
214 |
fixes a :: real |
|
40859 | 215 |
shows "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel" |
216 |
by (auto intro!: borel_closed closed_halfspace_component_ge) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
217 |
|
38656 | 218 |
lemma hafspace_greater_eq_borel[simp, intro]: |
219 |
fixes a :: real |
|
40859 | 220 |
shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel" |
221 |
by (auto intro!: borel_closed closed_halfspace_component_le) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
222 |
|
38656 | 223 |
lemma (in sigma_algebra) borel_measurable_less[simp, intro]: |
224 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
225 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
226 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
227 |
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
|
228 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
229 |
have "{w \<in> space M. f w < g w} = |
38656 | 230 |
(\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))" |
231 |
using Rats_dense_in_real by (auto simp add: Rats_def) |
|
232 |
then show ?thesis using f g |
|
233 |
by simp (blast intro: measurable_sets) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
234 |
qed |
38656 | 235 |
|
236 |
lemma (in sigma_algebra) borel_measurable_le[simp, intro]: |
|
237 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
238 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
239 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
240 |
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
|
241 |
proof - |
38656 | 242 |
have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}" |
243 |
by auto |
|
244 |
thus ?thesis using f g |
|
245 |
by simp blast |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
246 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
247 |
|
38656 | 248 |
lemma (in sigma_algebra) borel_measurable_eq[simp, intro]: |
249 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
250 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
251 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
252 |
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
|
253 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
254 |
have "{w \<in> space M. f w = g w} = |
33536 | 255 |
{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
|
256 |
by auto |
38656 | 257 |
thus ?thesis using f g by auto |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
258 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
259 |
|
38656 | 260 |
lemma (in sigma_algebra) borel_measurable_neq[simp, intro]: |
261 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
262 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
263 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
264 |
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
|
265 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
266 |
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
|
267 |
by auto |
38656 | 268 |
thus ?thesis using f g by auto |
269 |
qed |
|
270 |
||
271 |
subsection "Borel space equals sigma algebras over intervals" |
|
272 |
||
273 |
lemma rational_boxes: |
|
274 |
fixes x :: "'a\<Colon>ordered_euclidean_space" |
|
275 |
assumes "0 < e" |
|
276 |
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" |
|
277 |
proof - |
|
278 |
def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))" |
|
279 |
then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos) |
|
280 |
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i") |
|
281 |
proof |
|
282 |
fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e |
|
283 |
show "?th i" by auto |
|
284 |
qed |
|
285 |
from choice[OF this] guess a .. note a = this |
|
286 |
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i") |
|
287 |
proof |
|
288 |
fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e |
|
289 |
show "?th i" by auto |
|
290 |
qed |
|
291 |
from choice[OF this] guess b .. note b = this |
|
292 |
{ fix y :: 'a assume *: "Chi a < y" "y < Chi b" |
|
293 |
have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)" |
|
294 |
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2) |
|
295 |
also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))" |
|
296 |
proof (rule real_sqrt_less_mono, rule setsum_strict_mono) |
|
297 |
fix i assume i: "i \<in> {..<DIM('a)}" |
|
298 |
have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto |
|
299 |
moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto |
|
300 |
moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto |
|
301 |
ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto |
|
302 |
then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))" |
|
303 |
unfolding e'_def by (auto simp: dist_real_def) |
|
304 |
then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>" |
|
305 |
by (rule power_strict_mono) auto |
|
306 |
then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)" |
|
307 |
by (simp add: power_divide) |
|
308 |
qed auto |
|
309 |
also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive) |
|
310 |
finally have "dist x y < e" . } |
|
311 |
with a b show ?thesis |
|
312 |
apply (rule_tac exI[of _ "Chi a"]) |
|
313 |
apply (rule_tac exI[of _ "Chi b"]) |
|
314 |
using eucl_less[where 'a='a] by auto |
|
315 |
qed |
|
316 |
||
317 |
lemma ex_rat_list: |
|
318 |
fixes x :: "'a\<Colon>ordered_euclidean_space" |
|
319 |
assumes "\<And> i. x $$ i \<in> \<rat>" |
|
320 |
shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)" |
|
321 |
proof - |
|
322 |
have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast |
|
323 |
from choice[OF this] guess r .. |
|
324 |
then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"]) |
|
325 |
qed |
|
326 |
||
327 |
lemma open_UNION: |
|
328 |
fixes M :: "'a\<Colon>ordered_euclidean_space set" |
|
329 |
assumes "open M" |
|
330 |
shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M} |
|
331 |
(\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})" |
|
332 |
(is "M = UNION ?idx ?box") |
|
333 |
proof safe |
|
334 |
fix x assume "x \<in> M" |
|
335 |
obtain e where e: "e > 0" "ball x e \<subseteq> M" |
|
336 |
using openE[OF assms `x \<in> M`] by auto |
|
337 |
then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e" |
|
338 |
using rational_boxes[OF e(1)] by blast |
|
339 |
then obtain p q where pq: "length p = DIM ('a)" |
|
340 |
"length q = DIM ('a)" |
|
341 |
"\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i" |
|
342 |
using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast |
|
343 |
hence p: "Chi (of_rat \<circ> op ! p) = a" |
|
344 |
using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a] |
|
345 |
unfolding o_def by auto |
|
346 |
from pq have q: "Chi (of_rat \<circ> op ! q) = b" |
|
347 |
using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b] |
|
348 |
unfolding o_def by auto |
|
349 |
have "x \<in> ?box (p, q)" |
|
350 |
using p q ab by auto |
|
351 |
thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto |
|
352 |
qed auto |
|
353 |
||
354 |
lemma halfspace_span_open: |
|
40859 | 355 |
"sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})) |
356 |
\<subseteq> sets borel" |
|
357 |
by (auto intro!: borel.sigma_sets_subset[simplified] borel_open |
|
358 |
open_halfspace_component_lt) |
|
38656 | 359 |
|
360 |
lemma halfspace_lt_in_halfspace: |
|
40859 | 361 |
"{x\<Colon>'a. x $$ i < a} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>)" |
362 |
by (auto intro!: sigma_sets.Basic simp: sets_sigma) |
|
38656 | 363 |
|
364 |
lemma halfspace_gt_in_halfspace: |
|
40859 | 365 |
"{x\<Colon>'a. a < x $$ i} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>)" |
366 |
(is "?set \<in> sets ?SIGMA") |
|
38656 | 367 |
proof - |
40859 | 368 |
interpret sigma_algebra "?SIGMA" |
369 |
by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma) |
|
38656 | 370 |
have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})" |
371 |
proof (safe, simp_all add: not_less) |
|
372 |
fix x assume "a < x $$ i" |
|
373 |
with reals_Archimedean[of "x $$ i - a"] |
|
374 |
obtain n where "a + 1 / real (Suc n) < x $$ i" |
|
375 |
by (auto simp: inverse_eq_divide field_simps) |
|
376 |
then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i" |
|
377 |
by (blast intro: less_imp_le) |
|
378 |
next |
|
379 |
fix x n |
|
380 |
have "a < a + 1 / real (Suc n)" by auto |
|
381 |
also assume "\<dots> \<le> x" |
|
382 |
finally show "a < x" . |
|
383 |
qed |
|
384 |
show "?set \<in> sets ?SIGMA" unfolding * |
|
385 |
by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
386 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
387 |
|
38656 | 388 |
lemma open_span_halfspace: |
40859 | 389 |
"sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i < a})\<rparr>)" |
38656 | 390 |
(is "_ \<subseteq> sets ?SIGMA") |
40859 | 391 |
proof - |
392 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp |
|
38656 | 393 |
then interpret sigma_algebra ?SIGMA . |
40859 | 394 |
{ fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def . |
395 |
from open_UNION[OF this] |
|
396 |
obtain I where *: "S = |
|
397 |
(\<Union>(a, b)\<in>I. |
|
398 |
(\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter> |
|
399 |
(\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))" |
|
400 |
unfolding greaterThanLessThan_def |
|
401 |
unfolding eucl_greaterThan_eq_halfspaces[where 'a='a] |
|
402 |
unfolding eucl_lessThan_eq_halfspaces[where 'a='a] |
|
403 |
by blast |
|
404 |
have "S \<in> sets ?SIGMA" |
|
405 |
unfolding * |
|
406 |
by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) } |
|
407 |
then show ?thesis unfolding borel_def |
|
408 |
by (intro sets_sigma_subset) auto |
|
409 |
qed |
|
38656 | 410 |
|
411 |
lemma halfspace_span_halfspace_le: |
|
40859 | 412 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq> |
413 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x $$ i \<le> a})\<rparr>)" |
|
38656 | 414 |
(is "_ \<subseteq> sets ?SIGMA") |
40859 | 415 |
proof - |
416 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
38656 | 417 |
then interpret sigma_algebra ?SIGMA . |
40859 | 418 |
{ fix a i |
419 |
have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})" |
|
420 |
proof (safe, simp_all) |
|
421 |
fix x::'a assume *: "x$$i < a" |
|
422 |
with reals_Archimedean[of "a - x$$i"] |
|
423 |
obtain n where "x $$ i < a - 1 / (real (Suc n))" |
|
424 |
by (auto simp: field_simps inverse_eq_divide) |
|
425 |
then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))" |
|
426 |
by (blast intro: less_imp_le) |
|
427 |
next |
|
428 |
fix x::'a and n |
|
429 |
assume "x$$i \<le> a - 1 / real (Suc n)" |
|
430 |
also have "\<dots> < a" by auto |
|
431 |
finally show "x$$i < a" . |
|
432 |
qed |
|
433 |
have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding * |
|
434 |
by (safe intro!: countable_UN) |
|
435 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) } |
|
436 |
then show ?thesis by (intro sets_sigma_subset) auto |
|
437 |
qed |
|
38656 | 438 |
|
439 |
lemma halfspace_span_halfspace_ge: |
|
40859 | 440 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq> |
441 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x $$ i})\<rparr>)" |
|
38656 | 442 |
(is "_ \<subseteq> sets ?SIGMA") |
40859 | 443 |
proof - |
444 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
38656 | 445 |
then interpret sigma_algebra ?SIGMA . |
40859 | 446 |
{ fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto |
447 |
have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding * |
|
448 |
by (safe intro!: Diff) |
|
449 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) } |
|
450 |
then show ?thesis by (intro sets_sigma_subset) auto |
|
451 |
qed |
|
38656 | 452 |
|
453 |
lemma halfspace_le_span_halfspace_gt: |
|
40859 | 454 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq> |
455 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x $$ i})\<rparr>)" |
|
38656 | 456 |
(is "_ \<subseteq> sets ?SIGMA") |
40859 | 457 |
proof - |
458 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
38656 | 459 |
then interpret sigma_algebra ?SIGMA . |
40859 | 460 |
{ fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto |
461 |
have "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding * |
|
462 |
by (safe intro!: Diff) |
|
463 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) } |
|
464 |
then show ?thesis by (intro sets_sigma_subset) auto |
|
465 |
qed |
|
38656 | 466 |
|
467 |
lemma halfspace_le_span_atMost: |
|
40859 | 468 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq> |
469 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)" |
|
38656 | 470 |
(is "_ \<subseteq> sets ?SIGMA") |
40859 | 471 |
proof - |
472 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
38656 | 473 |
then interpret sigma_algebra ?SIGMA . |
40859 | 474 |
have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA" |
38656 | 475 |
proof cases |
40859 | 476 |
fix a i assume "i < DIM('a)" |
38656 | 477 |
then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})" |
478 |
proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm) |
|
479 |
fix x |
|
480 |
from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat .. |
|
481 |
then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k" |
|
482 |
by (subst (asm) Max_le_iff) auto |
|
483 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k" |
|
484 |
by (auto intro!: exI[of _ k]) |
|
485 |
qed |
|
486 |
show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding * |
|
487 |
by (safe intro!: countable_UN) |
|
488 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
489 |
next |
|
40859 | 490 |
fix a i assume "\<not> i < DIM('a)" |
38656 | 491 |
then show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
492 |
using top by auto |
|
493 |
qed |
|
40859 | 494 |
then show ?thesis by (intro sets_sigma_subset) auto |
495 |
qed |
|
38656 | 496 |
|
497 |
lemma halfspace_le_span_greaterThan: |
|
40859 | 498 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq> |
499 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)" |
|
38656 | 500 |
(is "_ \<subseteq> sets ?SIGMA") |
40859 | 501 |
proof - |
502 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
38656 | 503 |
then interpret sigma_algebra ?SIGMA . |
40859 | 504 |
have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA" |
38656 | 505 |
proof cases |
40859 | 506 |
fix a i assume "i < DIM('a)" |
38656 | 507 |
have "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto |
508 |
also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)` |
|
509 |
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm) |
|
510 |
fix x |
|
511 |
from real_arch_lt[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"] |
|
512 |
guess k::nat .. note k = this |
|
513 |
{ fix i assume "i < DIM('a)" |
|
514 |
then have "-x$$i < real k" |
|
515 |
using k by (subst (asm) Max_less_iff) auto |
|
516 |
then have "- real k < x$$i" by simp } |
|
517 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia" |
|
518 |
by (auto intro!: exI[of _ k]) |
|
519 |
qed |
|
520 |
finally show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
|
521 |
apply (simp only:) |
|
522 |
apply (safe intro!: countable_UN Diff) |
|
523 |
by (auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
524 |
next |
|
40859 | 525 |
fix a i assume "\<not> i < DIM('a)" |
38656 | 526 |
then show "{x. x$$i \<le> a} \<in> sets ?SIGMA" |
527 |
using top by auto |
|
528 |
qed |
|
40859 | 529 |
then show ?thesis by (intro sets_sigma_subset) auto |
530 |
qed |
|
531 |
||
532 |
lemma halfspace_le_span_lessThan: |
|
533 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i})\<rparr>) \<subseteq> |
|
534 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)" |
|
535 |
(is "_ \<subseteq> sets ?SIGMA") |
|
536 |
proof - |
|
537 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
538 |
then interpret sigma_algebra ?SIGMA . |
|
539 |
have "\<And>a i. {x. a \<le> x$$i} \<in> sets ?SIGMA" |
|
540 |
proof cases |
|
541 |
fix a i assume "i < DIM('a)" |
|
542 |
have "{x::'a. a \<le> x$$i} = space ?SIGMA - {x::'a. x$$i < a}" by auto |
|
543 |
also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)` |
|
544 |
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm) |
|
545 |
fix x |
|
546 |
from real_arch_lt[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] |
|
547 |
guess k::nat .. note k = this |
|
548 |
{ fix i assume "i < DIM('a)" |
|
549 |
then have "x$$i < real k" |
|
550 |
using k by (subst (asm) Max_less_iff) auto |
|
551 |
then have "x$$i < real k" by simp } |
|
552 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k" |
|
553 |
by (auto intro!: exI[of _ k]) |
|
554 |
qed |
|
555 |
finally show "{x. a \<le> x$$i} \<in> sets ?SIGMA" |
|
556 |
apply (simp only:) |
|
557 |
apply (safe intro!: countable_UN Diff) |
|
558 |
by (auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
559 |
next |
|
560 |
fix a i assume "\<not> i < DIM('a)" |
|
561 |
then show "{x. a \<le> x$$i} \<in> sets ?SIGMA" |
|
562 |
using top by auto |
|
563 |
qed |
|
564 |
then show ?thesis by (intro sets_sigma_subset) auto |
|
565 |
qed |
|
566 |
||
567 |
lemma atMost_span_atLeastAtMost: |
|
568 |
"sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq> |
|
569 |
sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)" |
|
570 |
(is "_ \<subseteq> sets ?SIGMA") |
|
571 |
proof - |
|
572 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
573 |
then interpret sigma_algebra ?SIGMA . |
|
574 |
{ fix a::'a |
|
575 |
have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})" |
|
576 |
proof (safe, simp_all add: eucl_le[where 'a='a]) |
|
577 |
fix x |
|
578 |
from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"] |
|
579 |
guess k::nat .. note k = this |
|
580 |
{ fix i assume "i < DIM('a)" |
|
581 |
with k have "- x$$i \<le> real k" |
|
582 |
by (subst (asm) Max_le_iff) (auto simp: field_simps) |
|
583 |
then have "- real k \<le> x$$i" by simp } |
|
584 |
then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i" |
|
585 |
by (auto intro!: exI[of _ k]) |
|
586 |
qed |
|
587 |
have "{..a} \<in> sets ?SIGMA" unfolding * |
|
588 |
by (safe intro!: countable_UN) |
|
589 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) } |
|
590 |
then show ?thesis by (intro sets_sigma_subset) auto |
|
591 |
qed |
|
592 |
||
593 |
lemma borel_eq_atMost: |
|
594 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)" |
|
595 |
(is "_ = ?SIGMA") |
|
40869 | 596 |
proof (intro algebra.equality antisym) |
40859 | 597 |
show "sets borel \<subseteq> sets ?SIGMA" |
598 |
using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace |
|
599 |
by auto |
|
600 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
601 |
by (rule borel.sets_sigma_subset) auto |
|
602 |
qed auto |
|
603 |
||
604 |
lemma borel_eq_atLeastAtMost: |
|
605 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)" |
|
606 |
(is "_ = ?SIGMA") |
|
40869 | 607 |
proof (intro algebra.equality antisym) |
40859 | 608 |
show "sets borel \<subseteq> sets ?SIGMA" |
609 |
using atMost_span_atLeastAtMost halfspace_le_span_atMost |
|
610 |
halfspace_span_halfspace_le open_span_halfspace |
|
611 |
by auto |
|
612 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
613 |
by (rule borel.sets_sigma_subset) auto |
|
614 |
qed auto |
|
615 |
||
616 |
lemma borel_eq_greaterThan: |
|
617 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)" |
|
618 |
(is "_ = ?SIGMA") |
|
40869 | 619 |
proof (intro algebra.equality antisym) |
40859 | 620 |
show "sets borel \<subseteq> sets ?SIGMA" |
621 |
using halfspace_le_span_greaterThan |
|
622 |
halfspace_span_halfspace_le open_span_halfspace |
|
623 |
by auto |
|
624 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
625 |
by (rule borel.sets_sigma_subset) auto |
|
626 |
qed auto |
|
627 |
||
628 |
lemma borel_eq_lessThan: |
|
629 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)" |
|
630 |
(is "_ = ?SIGMA") |
|
40869 | 631 |
proof (intro algebra.equality antisym) |
40859 | 632 |
show "sets borel \<subseteq> sets ?SIGMA" |
633 |
using halfspace_le_span_lessThan |
|
634 |
halfspace_span_halfspace_ge open_span_halfspace |
|
635 |
by auto |
|
636 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
637 |
by (rule borel.sets_sigma_subset) auto |
|
638 |
qed auto |
|
639 |
||
640 |
lemma borel_eq_greaterThanLessThan: |
|
641 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)" |
|
642 |
(is "_ = ?SIGMA") |
|
40869 | 643 |
proof (intro algebra.equality antisym) |
40859 | 644 |
show "sets ?SIGMA \<subseteq> sets borel" |
645 |
by (rule borel.sets_sigma_subset) auto |
|
646 |
show "sets borel \<subseteq> sets ?SIGMA" |
|
647 |
proof - |
|
648 |
have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto |
|
649 |
then interpret sigma_algebra ?SIGMA . |
|
650 |
{ fix M :: "'a set" assume "M \<in> open" |
|
651 |
then have "open M" by (simp add: mem_def) |
|
652 |
have "M \<in> sets ?SIGMA" |
|
653 |
apply (subst open_UNION[OF `open M`]) |
|
654 |
apply (safe intro!: countable_UN) |
|
655 |
by (auto simp add: sigma_def intro!: sigma_sets.Basic) } |
|
656 |
then show ?thesis |
|
657 |
unfolding borel_def by (intro sets_sigma_subset) auto |
|
658 |
qed |
|
38656 | 659 |
qed auto |
660 |
||
42862 | 661 |
lemma borel_eq_atLeastLessThan: |
662 |
"borel = sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real})\<rparr>" (is "_ = ?S") |
|
663 |
proof (intro algebra.equality antisym) |
|
664 |
interpret sigma_algebra ?S |
|
665 |
by (rule sigma_algebra_sigma) auto |
|
666 |
show "sets borel \<subseteq> sets ?S" |
|
667 |
unfolding borel_eq_lessThan |
|
668 |
proof (intro sets_sigma_subset subsetI) |
|
669 |
have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto |
|
670 |
fix A :: "real set" assume "A \<in> sets \<lparr>space = UNIV, sets = range lessThan\<rparr>" |
|
671 |
then obtain x where "A = {..< x}" by auto |
|
672 |
then have "A = (\<Union>i::nat. {-real i ..< x})" |
|
673 |
by (auto simp: move_uminus real_arch_simple) |
|
674 |
then show "A \<in> sets ?S" |
|
675 |
by (auto simp: sets_sigma intro!: sigma_sets.intros) |
|
676 |
qed simp |
|
677 |
show "sets ?S \<subseteq> sets borel" |
|
678 |
by (intro borel.sets_sigma_subset) auto |
|
679 |
qed simp_all |
|
680 |
||
40859 | 681 |
lemma borel_eq_halfspace_le: |
682 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i \<le> a})\<rparr>)" |
|
683 |
(is "_ = ?SIGMA") |
|
40869 | 684 |
proof (intro algebra.equality antisym) |
40859 | 685 |
show "sets borel \<subseteq> sets ?SIGMA" |
686 |
using open_span_halfspace halfspace_span_halfspace_le by auto |
|
687 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
688 |
by (rule borel.sets_sigma_subset) auto |
|
689 |
qed auto |
|
690 |
||
691 |
lemma borel_eq_halfspace_less: |
|
692 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})\<rparr>)" |
|
693 |
(is "_ = ?SIGMA") |
|
40869 | 694 |
proof (intro algebra.equality antisym) |
40859 | 695 |
show "sets borel \<subseteq> sets ?SIGMA" |
696 |
using open_span_halfspace . |
|
697 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
698 |
by (rule borel.sets_sigma_subset) auto |
|
38656 | 699 |
qed auto |
700 |
||
40859 | 701 |
lemma borel_eq_halfspace_gt: |
702 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})\<rparr>)" |
|
703 |
(is "_ = ?SIGMA") |
|
40869 | 704 |
proof (intro algebra.equality antisym) |
40859 | 705 |
show "sets borel \<subseteq> sets ?SIGMA" |
706 |
using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto |
|
707 |
show "sets ?SIGMA \<subseteq> sets borel" |
|
708 |
by (rule borel.sets_sigma_subset) auto |
|
709 |
qed auto |
|
38656 | 710 |
|
40859 | 711 |
lemma borel_eq_halfspace_ge: |
712 |
"borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x$$i})\<rparr>)" |
|
713 |
(is "_ = ?SIGMA") |
|
40869 | 714 |
proof (intro algebra.equality antisym) |
40859 | 715 |
show "sets borel \<subseteq> sets ?SIGMA" |
38656 | 716 |
using halfspace_span_halfspace_ge open_span_halfspace by auto |
40859 | 717 |
show "sets ?SIGMA \<subseteq> sets borel" |
718 |
by (rule borel.sets_sigma_subset) auto |
|
719 |
qed auto |
|
38656 | 720 |
|
721 |
lemma (in sigma_algebra) borel_measurable_halfspacesI: |
|
722 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
40859 | 723 |
assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)" |
38656 | 724 |
and "\<And>a i. S a i = f -` F (a,i) \<inter> space M" |
725 |
and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M" |
|
726 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)" |
|
727 |
proof safe |
|
728 |
fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M" |
|
729 |
then show "S a i \<in> sets M" unfolding assms |
|
730 |
by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def) |
|
731 |
next |
|
732 |
assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M" |
|
733 |
{ fix a i have "S a i \<in> sets M" |
|
734 |
proof cases |
|
735 |
assume "i < DIM('c)" |
|
736 |
with a show ?thesis unfolding assms(2) by simp |
|
737 |
next |
|
738 |
assume "\<not> i < DIM('c)" |
|
739 |
from assms(3)[OF this] show ?thesis . |
|
740 |
qed } |
|
40859 | 741 |
then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)" |
38656 | 742 |
by (auto intro!: measurable_sigma simp: assms(2)) |
743 |
then show "f \<in> borel_measurable M" unfolding measurable_def |
|
744 |
unfolding assms(1) by simp |
|
745 |
qed |
|
746 |
||
747 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_le: |
|
748 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
749 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)" |
|
40859 | 750 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto |
38656 | 751 |
|
752 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_less: |
|
753 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
754 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)" |
|
40859 | 755 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto |
38656 | 756 |
|
757 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge: |
|
758 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
759 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)" |
|
40859 | 760 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto |
38656 | 761 |
|
762 |
lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater: |
|
763 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
|
764 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)" |
|
40859 | 765 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto |
38656 | 766 |
|
767 |
lemma (in sigma_algebra) borel_measurable_iff_le: |
|
768 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)" |
|
769 |
using borel_measurable_iff_halfspace_le[where 'c=real] by simp |
|
770 |
||
771 |
lemma (in sigma_algebra) borel_measurable_iff_less: |
|
772 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)" |
|
773 |
using borel_measurable_iff_halfspace_less[where 'c=real] by simp |
|
774 |
||
775 |
lemma (in sigma_algebra) borel_measurable_iff_ge: |
|
776 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)" |
|
777 |
using borel_measurable_iff_halfspace_ge[where 'c=real] by simp |
|
778 |
||
779 |
lemma (in sigma_algebra) borel_measurable_iff_greater: |
|
780 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)" |
|
781 |
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp |
|
782 |
||
41025 | 783 |
lemma borel_measurable_euclidean_component: |
40859 | 784 |
"(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel" |
785 |
unfolding borel_def[where 'a=real] |
|
786 |
proof (rule borel.measurable_sigma, simp_all) |
|
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
787 |
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
|
788 |
from open_vimage_euclidean_component[OF this] |
40859 | 789 |
show "(\<lambda>x. x $$ i) -` S \<in> sets borel" |
790 |
by (auto intro: borel_open) |
|
791 |
qed |
|
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
792 |
|
41025 | 793 |
lemma (in sigma_algebra) borel_measurable_euclidean_space: |
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
794 |
fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
795 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
796 |
proof safe |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
797 |
fix i assume "f \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
798 |
then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
799 |
using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def] |
41025 | 800 |
by (auto intro: borel_measurable_euclidean_component) |
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
801 |
next |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
802 |
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
|
803 |
then show "f \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
804 |
unfolding borel_measurable_iff_halfspace_le by auto |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
805 |
qed |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
806 |
|
38656 | 807 |
subsection "Borel measurable operators" |
808 |
||
809 |
lemma (in sigma_algebra) affine_borel_measurable_vector: |
|
810 |
fixes f :: "'a \<Rightarrow> 'x::real_normed_vector" |
|
811 |
assumes "f \<in> borel_measurable M" |
|
812 |
shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M" |
|
813 |
proof (rule borel_measurableI) |
|
814 |
fix S :: "'x set" assume "open S" |
|
815 |
show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M" |
|
816 |
proof cases |
|
817 |
assume "b \<noteq> 0" |
|
818 |
with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open") |
|
819 |
by (auto intro!: open_affinity simp: scaleR.add_right mem_def) |
|
40859 | 820 |
hence "?S \<in> sets borel" |
821 |
unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic) |
|
38656 | 822 |
moreover |
823 |
from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S" |
|
824 |
apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all) |
|
40859 | 825 |
ultimately show ?thesis using assms unfolding in_borel_measurable_borel |
38656 | 826 |
by auto |
827 |
qed simp |
|
828 |
qed |
|
829 |
||
830 |
lemma (in sigma_algebra) affine_borel_measurable: |
|
831 |
fixes g :: "'a \<Rightarrow> real" |
|
832 |
assumes g: "g \<in> borel_measurable M" |
|
833 |
shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M" |
|
834 |
using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute) |
|
835 |
||
836 |
lemma (in sigma_algebra) borel_measurable_add[simp, intro]: |
|
837 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
838 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
839 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
840 |
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
|
841 |
proof - |
38656 | 842 |
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
|
843 |
by auto |
38656 | 844 |
have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M" |
845 |
by (rule affine_borel_measurable [OF g]) |
|
846 |
then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f |
|
847 |
by auto |
|
848 |
then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M" |
|
849 |
by (simp add: 1) |
|
850 |
then show ?thesis |
|
851 |
by (simp add: borel_measurable_iff_ge) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
852 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
853 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
854 |
lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
855 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
856 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
857 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
858 |
proof cases |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
859 |
assume "finite S" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
860 |
thus ?thesis using assms by induct auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
861 |
qed simp |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
862 |
|
38656 | 863 |
lemma (in sigma_algebra) borel_measurable_square: |
864 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
865 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
866 |
shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
867 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
868 |
{ |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
869 |
fix a |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
870 |
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
|
871 |
proof (cases rule: linorder_cases [of a 0]) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
872 |
case less |
38656 | 873 |
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
|
874 |
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
|
875 |
also have "... \<in> sets M" |
38656 | 876 |
by (rule empty_sets) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
877 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
878 |
next |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
879 |
case equal |
38656 | 880 |
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
|
881 |
{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
|
882 |
by auto |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
883 |
also have "... \<in> sets M" |
38656 | 884 |
apply (insert f) |
885 |
apply (rule Int) |
|
886 |
apply (simp add: borel_measurable_iff_le) |
|
887 |
apply (simp add: borel_measurable_iff_ge) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
888 |
done |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
889 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
890 |
next |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
891 |
case greater |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
892 |
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
|
893 |
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
|
894 |
real_sqrt_le_iff real_sqrt_power) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
895 |
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = |
38656 | 896 |
{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
|
897 |
using greater by auto |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
898 |
also have "... \<in> sets M" |
38656 | 899 |
apply (insert f) |
900 |
apply (rule Int) |
|
901 |
apply (simp add: borel_measurable_iff_ge) |
|
902 |
apply (simp add: borel_measurable_iff_le) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
903 |
done |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
904 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
905 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
906 |
} |
38656 | 907 |
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
|
908 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
909 |
|
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
910 |
lemma times_eq_sum_squares: |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
911 |
fixes x::real |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
912 |
shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4" |
38656 | 913 |
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
|
914 |
|
38656 | 915 |
lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]: |
916 |
fixes g :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
917 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
918 |
shows "(\<lambda>x. - g x) \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
919 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
920 |
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
|
921 |
by simp |
38656 | 922 |
also have "... \<in> borel_measurable M" |
923 |
by (fast intro: affine_borel_measurable g) |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
924 |
finally show ?thesis . |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
925 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
926 |
|
38656 | 927 |
lemma (in sigma_algebra) borel_measurable_times[simp, intro]: |
928 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
929 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
930 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
931 |
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
|
932 |
proof - |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
933 |
have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M" |
38656 | 934 |
using assms by (fast intro: affine_borel_measurable borel_measurable_square) |
935 |
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
|
936 |
(\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))" |
35582 | 937 |
by (simp add: minus_divide_right) |
38656 | 938 |
also have "... \<in> borel_measurable M" |
939 |
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
|
940 |
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
|
941 |
show ?thesis |
38656 | 942 |
apply (simp add: times_eq_sum_squares diff_minus) |
943 |
using 1 2 by simp |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
944 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
945 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
946 |
lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
947 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
948 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
949 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
950 |
proof cases |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
951 |
assume "finite S" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
952 |
thus ?thesis using assms by induct auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
953 |
qed simp |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
954 |
|
38656 | 955 |
lemma (in sigma_algebra) borel_measurable_diff[simp, intro]: |
956 |
fixes f :: "'a \<Rightarrow> real" |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
957 |
assumes f: "f \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
958 |
assumes g: "g \<in> borel_measurable M" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
959 |
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
38656 | 960 |
unfolding diff_minus using assms by fast |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
961 |
|
38656 | 962 |
lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]: |
963 |
fixes f :: "'a \<Rightarrow> real" |
|
35692 | 964 |
assumes "f \<in> borel_measurable M" |
965 |
shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M" |
|
38656 | 966 |
unfolding borel_measurable_iff_ge unfolding inverse_eq_divide |
967 |
proof safe |
|
968 |
fix a :: real |
|
969 |
have *: "{w \<in> space M. a \<le> 1 / f w} = |
|
970 |
({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union> |
|
971 |
({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union> |
|
972 |
({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq) |
|
973 |
show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding * |
|
974 |
by (auto intro!: Int Un) |
|
35692 | 975 |
qed |
976 |
||
38656 | 977 |
lemma (in sigma_algebra) borel_measurable_divide[simp, intro]: |
978 |
fixes f :: "'a \<Rightarrow> real" |
|
35692 | 979 |
assumes "f \<in> borel_measurable M" |
980 |
and "g \<in> borel_measurable M" |
|
981 |
shows "(\<lambda>x. f x / g x) \<in> borel_measurable M" |
|
982 |
unfolding field_divide_inverse |
|
38656 | 983 |
by (rule borel_measurable_inverse borel_measurable_times assms)+ |
984 |
||
985 |
lemma (in sigma_algebra) borel_measurable_max[intro, simp]: |
|
986 |
fixes f g :: "'a \<Rightarrow> real" |
|
987 |
assumes "f \<in> borel_measurable M" |
|
988 |
assumes "g \<in> borel_measurable M" |
|
989 |
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M" |
|
990 |
unfolding borel_measurable_iff_le |
|
991 |
proof safe |
|
992 |
fix a |
|
993 |
have "{x \<in> space M. max (g x) (f x) \<le> a} = |
|
994 |
{x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto |
|
995 |
thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M" |
|
996 |
using assms unfolding borel_measurable_iff_le |
|
997 |
by (auto intro!: Int) |
|
998 |
qed |
|
999 |
||
1000 |
lemma (in sigma_algebra) borel_measurable_min[intro, simp]: |
|
1001 |
fixes f g :: "'a \<Rightarrow> real" |
|
1002 |
assumes "f \<in> borel_measurable M" |
|
1003 |
assumes "g \<in> borel_measurable M" |
|
1004 |
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M" |
|
1005 |
unfolding borel_measurable_iff_ge |
|
1006 |
proof safe |
|
1007 |
fix a |
|
1008 |
have "{x \<in> space M. a \<le> min (g x) (f x)} = |
|
1009 |
{x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto |
|
1010 |
thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M" |
|
1011 |
using assms unfolding borel_measurable_iff_ge |
|
1012 |
by (auto intro!: Int) |
|
1013 |
qed |
|
1014 |
||
1015 |
lemma (in sigma_algebra) borel_measurable_abs[simp, intro]: |
|
1016 |
assumes "f \<in> borel_measurable M" |
|
1017 |
shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M" |
|
1018 |
proof - |
|
1019 |
have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def) |
|
1020 |
show ?thesis unfolding * using assms by auto |
|
1021 |
qed |
|
1022 |
||
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1023 |
lemma borel_measurable_nth[simp, intro]: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1024 |
"(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1025 |
using borel_measurable_euclidean_component |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1026 |
unfolding nth_conv_component by auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1027 |
|
41830 | 1028 |
lemma borel_measurable_continuous_on1: |
1029 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space" |
|
1030 |
assumes "continuous_on UNIV f" |
|
1031 |
shows "f \<in> borel_measurable borel" |
|
1032 |
apply(rule borel.borel_measurableI) |
|
1033 |
using continuous_open_preimage[OF assms] unfolding vimage_def by auto |
|
1034 |
||
1035 |
lemma borel_measurable_continuous_on: |
|
1036 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space" |
|
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1037 |
assumes cont: "continuous_on A f" "open A" |
41830 | 1038 |
shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _") |
1039 |
proof (rule borel.borel_measurableI) |
|
1040 |
fix S :: "'b set" assume "open S" |
|
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1041 |
then have "open {x\<in>A. f x \<in> S}" |
41830 | 1042 |
by (intro continuous_open_preimage[OF cont]) auto |
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1043 |
then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1044 |
have "?f -` S \<inter> space borel = |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1045 |
{x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1046 |
by (auto split: split_if_asm) |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1047 |
also have "\<dots> \<in> sets borel" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1048 |
using * `open A` by (auto simp del: space_borel intro!: borel.Un) |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1049 |
finally show "?f -` S \<inter> space borel \<in> sets borel" . |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1050 |
qed |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1051 |
|
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1052 |
lemma (in sigma_algebra) convex_measurable: |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1053 |
fixes a b :: real |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1054 |
assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1055 |
assumes q: "convex_on { a <..< b} q" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1056 |
shows "q \<circ> X \<in> borel_measurable M" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1057 |
proof - |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1058 |
have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<in> borel_measurable borel" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1059 |
proof (rule borel_measurable_continuous_on) |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1060 |
show "open {a<..<b}" by auto |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1061 |
from this q show "continuous_on {a<..<b} q" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1062 |
by (rule convex_on_continuous) |
41830 | 1063 |
qed |
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1064 |
then have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<circ> X \<in> borel_measurable M" (is ?qX) |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1065 |
using X by (intro measurable_comp) auto |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1066 |
moreover have "?qX \<longleftrightarrow> q \<circ> X \<in> borel_measurable M" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1067 |
using X by (intro measurable_cong) auto |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
1068 |
ultimately show ?thesis by simp |
41830 | 1069 |
qed |
1070 |
||
1071 |
lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel" |
|
1072 |
proof - |
|
1073 |
{ fix x :: real assume x: "x \<le> 0" |
|
1074 |
{ fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto } |
|
1075 |
from this[of x] x this[of 0] have "log b 0 = log b x" |
|
1076 |
by (auto simp: ln_def log_def) } |
|
1077 |
note log_imp = this |
|
1078 |
have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel" |
|
1079 |
proof (rule borel_measurable_continuous_on) |
|
1080 |
show "continuous_on {0<..} (log b)" |
|
1081 |
by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont |
|
1082 |
simp: continuous_isCont[symmetric]) |
|
1083 |
show "open ({0<..}::real set)" by auto |
|
1084 |
qed |
|
1085 |
also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b" |
|
1086 |
by (simp add: fun_eq_iff not_less log_imp) |
|
1087 |
finally show ?thesis . |
|
1088 |
qed |
|
1089 |
||
1090 |
lemma (in sigma_algebra) borel_measurable_log[simp,intro]: |
|
1091 |
assumes f: "f \<in> borel_measurable M" and "1 < b" |
|
1092 |
shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M" |
|
1093 |
using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]] |
|
1094 |
by (simp add: comp_def) |
|
1095 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1096 |
subsection "Borel space on the extended reals" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1097 |
|
43920 | 1098 |
lemma borel_measurable_ereal_borel: |
1099 |
"ereal \<in> borel_measurable borel" |
|
1100 |
unfolding borel_def[where 'a=ereal] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1101 |
proof (rule borel.measurable_sigma) |
43920 | 1102 |
fix X :: "ereal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open \<rparr>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1103 |
then have "open X" by (auto simp: mem_def) |
43920 | 1104 |
then have "open (ereal -` X \<inter> space borel)" |
1105 |
by (simp add: open_ereal_vimage) |
|
1106 |
then show "ereal -` X \<inter> space borel \<in> sets borel" by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1107 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1108 |
|
43920 | 1109 |
lemma (in sigma_algebra) borel_measurable_ereal[simp, intro]: |
1110 |
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M" |
|
1111 |
using measurable_comp[OF f borel_measurable_ereal_borel] unfolding comp_def . |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1112 |
|
43920 | 1113 |
lemma borel_measurable_real_of_ereal_borel: |
1114 |
"(real :: ereal \<Rightarrow> real) \<in> borel_measurable borel" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1115 |
unfolding borel_def[where 'a=real] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1116 |
proof (rule borel.measurable_sigma) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1117 |
fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = open \<rparr>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1118 |
then have "open B" by (auto simp: mem_def) |
43920 | 1119 |
have *: "ereal -` real -` (B - {0}) = B - {0}" by auto |
1120 |
have open_real: "open (real -` (B - {0}) :: ereal set)" |
|
1121 |
unfolding open_ereal_def * using `open B` by auto |
|
1122 |
show "(real -` B \<inter> space borel :: ereal set) \<in> sets borel" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1123 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1124 |
assume "0 \<in> B" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1125 |
then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}" |
43920 | 1126 |
by (auto simp add: real_of_ereal_eq_0) |
1127 |
then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1128 |
using open_real by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1129 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1130 |
assume "0 \<notin> B" |
43920 | 1131 |
then have *: "(real -` B :: ereal set) = real -` (B - {0})" |
1132 |
by (auto simp add: real_of_ereal_eq_0) |
|
1133 |
then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1134 |
using open_real by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1135 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1136 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1137 |
|
43920 | 1138 |
lemma (in sigma_algebra) borel_measurable_real_of_ereal[simp, intro]: |
1139 |
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: ereal)) \<in> borel_measurable M" |
|
1140 |
using measurable_comp[OF f borel_measurable_real_of_ereal_borel] unfolding comp_def . |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1141 |
|
43920 | 1142 |
lemma (in sigma_algebra) borel_measurable_ereal_iff: |
1143 |
shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1144 |
proof |
43920 | 1145 |
assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M" |
1146 |
from borel_measurable_real_of_ereal[OF this] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1147 |
show "f \<in> borel_measurable M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1148 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1149 |
|
43920 | 1150 |
lemma (in sigma_algebra) borel_measurable_ereal_iff_real: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1151 |
"f \<in> borel_measurable M \<longleftrightarrow> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1152 |
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1153 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1154 |
assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1155 |
have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1156 |
with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all |
43920 | 1157 |
let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1158 |
have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto |
43920 | 1159 |
also have "?f = f" by (auto simp: fun_eq_iff ereal_real) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1160 |
finally show "f \<in> borel_measurable M" . |
43920 | 1161 |
qed (auto intro: measurable_sets borel_measurable_real_of_ereal) |
41830 | 1162 |
|
38656 | 1163 |
lemma (in sigma_algebra) less_eq_ge_measurable: |
1164 |
fixes f :: "'a \<Rightarrow> 'c::linorder" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1165 |
shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M" |
38656 | 1166 |
proof |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1167 |
assume "f -` {a <..} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1168 |
moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1169 |
ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto |
38656 | 1170 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1171 |
assume "f -` {..a} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1172 |
moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1173 |
ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto |
38656 | 1174 |
qed |
35692 | 1175 |
|
38656 | 1176 |
lemma (in sigma_algebra) greater_eq_le_measurable: |
1177 |
fixes f :: "'a \<Rightarrow> 'c::linorder" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1178 |
shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M" |
38656 | 1179 |
proof |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1180 |
assume "f -` {a ..} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1181 |
moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1182 |
ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto |
38656 | 1183 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1184 |
assume "f -` {..< a} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1185 |
moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1186 |
ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto |
38656 | 1187 |
qed |
1188 |
||
43920 | 1189 |
lemma (in sigma_algebra) borel_measurable_uminus_borel_ereal: |
1190 |
"(uminus :: ereal \<Rightarrow> ereal) \<in> borel_measurable borel" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1191 |
proof (subst borel_def, rule borel.measurable_sigma) |
43920 | 1192 |
fix X :: "ereal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open\<rparr>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1193 |
then have "open X" by (simp add: mem_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1194 |
have "uminus -` X = uminus ` X" by (force simp: image_iff) |
43920 | 1195 |
then have "open (uminus -` X)" using `open X` ereal_open_uminus by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1196 |
then show "uminus -` X \<inter> space borel \<in> sets borel" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1197 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1198 |
|
43920 | 1199 |
lemma (in sigma_algebra) borel_measurable_uminus_ereal[intro]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1200 |
assumes "f \<in> borel_measurable M" |
43920 | 1201 |
shows "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M" |
1202 |
using measurable_comp[OF assms borel_measurable_uminus_borel_ereal] by (simp add: comp_def) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1203 |
|
43920 | 1204 |
lemma (in sigma_algebra) borel_measurable_uminus_eq_ereal[simp]: |
1205 |
"(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r") |
|
38656 | 1206 |
proof |
43920 | 1207 |
assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1208 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1209 |
|
43920 | 1210 |
lemma (in sigma_algebra) borel_measurable_eq_atMost_ereal: |
1211 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1212 |
proof (intro iffI allI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1213 |
assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1214 |
show "f \<in> borel_measurable M" |
43920 | 1215 |
unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1216 |
proof (intro conjI allI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1217 |
fix a :: real |
43920 | 1218 |
{ fix x :: ereal assume *: "\<forall>i::nat. real i < x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1219 |
have "x = \<infinity>" |
43920 | 1220 |
proof (rule ereal_top) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1221 |
fix B from real_arch_lt[of B] guess n .. |
43920 | 1222 |
then have "ereal B < real n" by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1223 |
with * show "B \<le> x" by (metis less_trans less_imp_le) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1224 |
qed } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1225 |
then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1226 |
by (auto simp: not_le) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1227 |
then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1228 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1229 |
have "{-\<infinity>} = {..-\<infinity>}" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1230 |
then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto |
43920 | 1231 |
moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M" |
1232 |
using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1233 |
moreover have "{w \<in> space M. real (f w) \<le> a} = |
43920 | 1234 |
(if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M |
1235 |
else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1236 |
proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1237 |
ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto |
35582 | 1238 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1239 |
qed (simp add: measurable_sets) |
35582 | 1240 |
|
43920 | 1241 |
lemma (in sigma_algebra) borel_measurable_eq_atLeast_ereal: |
1242 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1243 |
proof |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1244 |
assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1245 |
moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}" |
43920 | 1246 |
by (auto simp: ereal_uminus_le_reorder) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1247 |
ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M" |
43920 | 1248 |
unfolding borel_measurable_eq_atMost_ereal by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1249 |
then show "f \<in> borel_measurable M" by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1250 |
qed (simp add: measurable_sets) |
35582 | 1251 |
|
43920 | 1252 |
lemma (in sigma_algebra) borel_measurable_ereal_iff_less: |
1253 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)" |
|
1254 |
unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable .. |
|
38656 | 1255 |
|
43920 | 1256 |
lemma (in sigma_algebra) borel_measurable_ereal_iff_ge: |
1257 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)" |
|
1258 |
unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable .. |
|
38656 | 1259 |
|
43920 | 1260 |
lemma (in sigma_algebra) borel_measurable_ereal_eq_const: |
1261 |
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" |
|
38656 | 1262 |
shows "{x\<in>space M. f x = c} \<in> sets M" |
1263 |
proof - |
|
1264 |
have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto |
|
1265 |
then show ?thesis using assms by (auto intro!: measurable_sets) |
|
1266 |
qed |
|
1267 |
||
43920 | 1268 |
lemma (in sigma_algebra) borel_measurable_ereal_neq_const: |
1269 |
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" |
|
38656 | 1270 |
shows "{x\<in>space M. f x \<noteq> c} \<in> sets M" |
1271 |
proof - |
|
1272 |
have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto |
|
1273 |
then show ?thesis using assms by (auto intro!: measurable_sets) |
|
1274 |
qed |
|
1275 |
||
43920 | 1276 |
lemma (in sigma_algebra) borel_measurable_ereal_le[intro,simp]: |
1277 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1278 |
assumes f: "f \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1279 |
assumes g: "g \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1280 |
shows "{x \<in> space M. f x \<le> g x} \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1281 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1282 |
have "{x \<in> space M. f x \<le> g x} = |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1283 |
{x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1284 |
f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r") |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1285 |
proof (intro set_eqI) |
43920 | 1286 |
fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: ereal2_cases[of "f x" "g x"]) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1287 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1288 |
with f g show ?thesis by (auto intro!: Un simp: measurable_sets) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1289 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1290 |
|
43920 | 1291 |
lemma (in sigma_algebra) borel_measurable_ereal_less[intro,simp]: |
1292 |
fixes f :: "'a \<Rightarrow> ereal" |
|
38656 | 1293 |
assumes f: "f \<in> borel_measurable M" |
1294 |
assumes g: "g \<in> borel_measurable M" |
|
1295 |
shows "{x \<in> space M. f x < g x} \<in> sets M" |
|
1296 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1297 |
have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto |
38656 | 1298 |
then show ?thesis using g f by auto |
1299 |
qed |
|
1300 |
||
43920 | 1301 |
lemma (in sigma_algebra) borel_measurable_ereal_eq[intro,simp]: |
1302 |
fixes f :: "'a \<Rightarrow> ereal" |
|
38656 | 1303 |
assumes f: "f \<in> borel_measurable M" |
1304 |
assumes g: "g \<in> borel_measurable M" |
|
1305 |
shows "{w \<in> space M. f w = g w} \<in> sets M" |
|
1306 |
proof - |
|
1307 |
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 |
|
1308 |
then show ?thesis using g f by auto |
|
1309 |
qed |
|
1310 |
||
43920 | 1311 |
lemma (in sigma_algebra) borel_measurable_ereal_neq[intro,simp]: |
1312 |
fixes f :: "'a \<Rightarrow> ereal" |
|
38656 | 1313 |
assumes f: "f \<in> borel_measurable M" |
1314 |
assumes g: "g \<in> borel_measurable M" |
|
1315 |
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
|
35692 | 1316 |
proof - |
38656 | 1317 |
have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto |
1318 |
thus ?thesis using f g by auto |
|
1319 |
qed |
|
1320 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1321 |
lemma (in sigma_algebra) split_sets: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1322 |
"{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1323 |
"{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1324 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1325 |
|
43920 | 1326 |
lemma (in sigma_algebra) borel_measurable_ereal_add[intro, simp]: |
1327 |
fixes f :: "'a \<Rightarrow> ereal" |
|
41025 | 1328 |
assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
38656 | 1329 |
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
1330 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1331 |
{ fix x assume "x \<in> space M" then have "f x + g x = |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1332 |
(if f x = \<infinity> \<or> g x = \<infinity> then \<infinity> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1333 |
else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity> |
43920 | 1334 |
else ereal (real (f x) + real (g x)))" |
1335 |
by (cases rule: ereal2_cases[of "f x" "g x"]) auto } |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1336 |
with assms show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1337 |
by (auto cong: measurable_cong simp: split_sets |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1338 |
intro!: Un measurable_If measurable_sets) |
38656 | 1339 |
qed |
1340 |
||
43920 | 1341 |
lemma (in sigma_algebra) borel_measurable_ereal_setsum[simp, intro]: |
1342 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal" |
|
41096 | 1343 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
1344 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
|
1345 |
proof cases |
|
1346 |
assume "finite S" |
|
1347 |
thus ?thesis using assms |
|
1348 |
by induct auto |
|
1349 |
qed (simp add: borel_measurable_const) |
|
1350 |
||
43920 | 1351 |
lemma (in sigma_algebra) borel_measurable_ereal_abs[intro, simp]: |
1352 |
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1353 |
shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1354 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1355 |
{ fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1356 |
then show ?thesis using assms by (auto intro!: measurable_If) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1357 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1358 |
|
43920 | 1359 |
lemma (in sigma_algebra) borel_measurable_ereal_times[intro, simp]: |
1360 |
fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
38656 | 1361 |
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
1362 |
proof - |
|
43920 | 1363 |
{ fix f g :: "'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1364 |
assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1365 |
and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1366 |
{ fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0 |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1367 |
else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity> |
43920 | 1368 |
else ereal (real (f x) * real (g x)))" |
1369 |
apply (cases rule: ereal2_cases[of "f x" "g x"]) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1370 |
using pos[of x] by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1371 |
with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1372 |
by (auto cong: measurable_cong simp: split_sets |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1373 |
intro!: Un measurable_If measurable_sets) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1374 |
note pos_times = this |
38656 | 1375 |
have *: "(\<lambda>x. f x * g x) = |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1376 |
(\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1377 |
by (auto simp: fun_eq_iff) |
38656 | 1378 |
show ?thesis using assms unfolding * |
43920 | 1379 |
by (intro measurable_If pos_times borel_measurable_uminus_ereal) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1380 |
(auto simp: split_sets intro!: Int) |
38656 | 1381 |
qed |
1382 |
||
43920 | 1383 |
lemma (in sigma_algebra) borel_measurable_ereal_setprod[simp, intro]: |
1384 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal" |
|
38656 | 1385 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
41096 | 1386 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
38656 | 1387 |
proof cases |
1388 |
assume "finite S" |
|
41096 | 1389 |
thus ?thesis using assms by induct auto |
1390 |
qed simp |
|
38656 | 1391 |
|
43920 | 1392 |
lemma (in sigma_algebra) borel_measurable_ereal_min[simp, intro]: |
1393 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
38656 | 1394 |
assumes "f \<in> borel_measurable M" |
1395 |
assumes "g \<in> borel_measurable M" |
|
1396 |
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M" |
|
1397 |
using assms unfolding min_def by (auto intro!: measurable_If) |
|
1398 |
||
43920 | 1399 |
lemma (in sigma_algebra) borel_measurable_ereal_max[simp, intro]: |
1400 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
38656 | 1401 |
assumes "f \<in> borel_measurable M" |
1402 |
and "g \<in> borel_measurable M" |
|
1403 |
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M" |
|
1404 |
using assms unfolding max_def by (auto intro!: measurable_If) |
|
1405 |
||
1406 |
lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]: |
|
43920 | 1407 |
fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal" |
38656 | 1408 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M" |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
1409 |
shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M") |
43920 | 1410 |
unfolding borel_measurable_ereal_iff_ge |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1411 |
proof |
38656 | 1412 |
fix a |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1413 |
have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})" |
41083
c987429a1298
work around problems with eta-expansion of equations
haftmann
parents:
41080
diff
changeset
|
1414 |
by (auto simp: less_SUP_iff SUPR_apply) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1415 |
then show "?sup -` {a<..} \<inter> space M \<in> sets M" |
38656 | 1416 |
using assms by auto |
1417 |
qed |
|
1418 |
||
1419 |
lemma (in sigma_algebra) borel_measurable_INF[simp, intro]: |
|
43920 | 1420 |
fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal" |
38656 | 1421 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M" |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
1422 |
shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M") |
43920 | 1423 |
unfolding borel_measurable_ereal_iff_less |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1424 |
proof |
38656 | 1425 |
fix a |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1426 |
have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})" |
41083
c987429a1298
work around problems with eta-expansion of equations
haftmann
parents:
41080
diff
changeset
|
1427 |
by (auto simp: INF_less_iff INFI_apply) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1428 |
then show "?inf -` {..<a} \<inter> space M \<in> sets M" |
38656 | 1429 |
using assms by auto |
1430 |
qed |
|
1431 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1432 |
lemma (in sigma_algebra) borel_measurable_liminf[simp, intro]: |
43920 | 1433 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1434 |
assumes "\<And>i. f i \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1435 |
shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1436 |
unfolding liminf_SUPR_INFI using assms by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1437 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1438 |
lemma (in sigma_algebra) borel_measurable_limsup[simp, intro]: |
43920 | 1439 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1440 |
assumes "\<And>i. f i \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1441 |
shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1442 |
unfolding limsup_INFI_SUPR using assms by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1443 |
|
43920 | 1444 |
lemma (in sigma_algebra) borel_measurable_ereal_diff[simp, intro]: |
1445 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
38656 | 1446 |
assumes "f \<in> borel_measurable M" |
1447 |
assumes "g \<in> borel_measurable M" |
|
1448 |
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
|
43920 | 1449 |
unfolding minus_ereal_def using assms by auto |
35692 | 1450 |
|
40870
94427db32392
Tuned setup for borel_measurable with min, max and psuminf.
hoelzl
parents:
40869
diff
changeset
|
1451 |
lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]: |
43920 | 1452 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1453 |
assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1454 |
shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1455 |
apply (subst measurable_cong) |
43920 | 1456 |
apply (subst suminf_ereal_eq_SUPR) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1457 |
apply (rule pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1458 |
using assms by auto |
39092 | 1459 |
|
1460 |
section "LIMSEQ is borel measurable" |
|
1461 |
||
1462 |
lemma (in sigma_algebra) borel_measurable_LIMSEQ: |
|
1463 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" |
|
1464 |
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x" |
|
1465 |
and u: "\<And>i. u i \<in> borel_measurable M" |
|
1466 |
shows "u' \<in> borel_measurable M" |
|
1467 |
proof - |
|
43920 | 1468 |
have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)" |
1469 |
using u' by (simp add: lim_imp_Liminf trivial_limit_sequentially lim_ereal) |
|
1470 |
moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M" |
|
39092 | 1471 |
by auto |
43920 | 1472 |
ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff) |
39092 | 1473 |
qed |
1474 |
||
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
1475 |
end |