author  hoelzl 
Fri, 27 Aug 2010 16:23:51 +0200  
changeset 39091  11314c196e11 
parent 39090  a2d38b8b693e 
child 39092  98de40859858 
permissions  rwrr 
35582  1 
theory Probability_Space 
39083  2 
imports Lebesgue_Integration Radon_Nikodym 
35582  3 
begin 
4 

38656  5 
lemma (in measure_space) measure_inter_full_set: 
6 
assumes "S \<in> sets M" "T \<in> sets M" and not_\<omega>: "\<mu> (T  S) \<noteq> \<omega>" 

7 
assumes T: "\<mu> T = \<mu> (space M)" 

8 
shows "\<mu> (S \<inter> T) = \<mu> S" 

9 
proof (rule antisym) 

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show " \<mu> (S \<inter> T) \<le> \<mu> S" 

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using assms by (auto intro!: measure_mono) 

12 

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show "\<mu> S \<le> \<mu> (S \<inter> T)" 

14 
proof (rule ccontr) 

15 
assume contr: "\<not> ?thesis" 

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have "\<mu> (space M) = \<mu> ((T  S) \<union> (S \<inter> T))" 

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unfolding T[symmetric] by (auto intro!: arg_cong[where f="\<mu>"]) 

18 
also have "\<dots> \<le> \<mu> (T  S) + \<mu> (S \<inter> T)" 

19 
using assms by (auto intro!: measure_subadditive) 

20 
also have "\<dots> < \<mu> (T  S) + \<mu> S" 

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by (rule pinfreal_less_add[OF not_\<omega>]) (insert contr, auto) 

22 
also have "\<dots> = \<mu> (T \<union> S)" 

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using assms by (subst measure_additive) auto 

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also have "\<dots> \<le> \<mu> (space M)" 

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using assms sets_into_space by (auto intro!: measure_mono) 

26 
finally show False .. 

27 
qed 

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qed 

29 

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lemma (in finite_measure) finite_measure_inter_full_set: 

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assumes "S \<in> sets M" "T \<in> sets M" 

32 
assumes T: "\<mu> T = \<mu> (space M)" 

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shows "\<mu> (S \<inter> T) = \<mu> S" 

34 
using measure_inter_full_set[OF assms(1,2) finite_measure assms(3)] assms 

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by auto 

36 

35582  37 
locale prob_space = measure_space + 
38656  38 
assumes measure_space_1: "\<mu> (space M) = 1" 
39 

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sublocale prob_space < finite_measure 

41 
proof 

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from measure_space_1 show "\<mu> (space M) \<noteq> \<omega>" by simp 

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qed 

44 

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context prob_space 

35582  46 
begin 
47 

48 
abbreviation "events \<equiv> sets M" 

38656  49 
abbreviation "prob \<equiv> \<lambda>A. real (\<mu> A)" 
35582  50 
abbreviation "prob_preserving \<equiv> measure_preserving" 
51 
abbreviation "random_variable \<equiv> \<lambda> s X. X \<in> measurable M s" 

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abbreviation "expectation \<equiv> integral" 

53 

54 
definition 

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"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B" 

56 

57 
definition 

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"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)" 

59 

60 
definition 

38656  61 
"distribution X = (\<lambda>s. \<mu> ((X ` s) \<inter> (space M)))" 
35582  62 

36624  63 
abbreviation 
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"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" 

35582  65 

38656  66 
lemma prob_space: "prob (space M) = 1" 
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unfolding measure_space_1 by simp 

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38656  69 
lemma measure_le_1[simp, intro]: 
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assumes "A \<in> events" shows "\<mu> A \<le> 1" 

71 
proof  

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have "\<mu> A \<le> \<mu> (space M)" 

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using assms sets_into_space by(auto intro!: measure_mono) 

74 
also note measure_space_1 

75 
finally show ?thesis . 

76 
qed 

35582  77 

38656  78 
lemma measure_finite[simp, intro]: 
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assumes "A \<in> events" shows "\<mu> A \<noteq> \<omega>" 

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using measure_le_1[OF assms] by auto 

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lemma prob_compl: 

38656  83 
assumes "A \<in> events" 
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shows "prob (space M  A) = 1  prob A" 

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using `A \<in> events`[THEN sets_into_space] `A \<in> events` measure_space_1 

86 
by (subst real_finite_measure_Diff) auto 

35582  87 

88 
lemma indep_space: 

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assumes "s \<in> events" 

90 
shows "indep (space M) s" 

38656  91 
using assms prob_space by (simp add: indep_def) 
35582  92 

38656  93 
lemma prob_space_increasing: "increasing M prob" 
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by (auto intro!: real_measure_mono simp: increasing_def) 

35582  95 

96 
lemma prob_zero_union: 

97 
assumes "s \<in> events" "t \<in> events" "prob t = 0" 

98 
shows "prob (s \<union> t) = prob s" 

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using assms 
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proof  
101 
have "prob (s \<union> t) \<le> prob s" 

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using real_finite_measure_subadditive[of s t] assms by auto 
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moreover have "prob (s \<union> t) \<ge> prob s" 
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using assms by (blast intro: real_measure_mono) 
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ultimately show ?thesis by simp 
106 
qed 

107 

108 
lemma prob_eq_compl: 

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assumes "s \<in> events" "t \<in> events" 

110 
assumes "prob (space M  s) = prob (space M  t)" 

111 
shows "prob s = prob t" 

38656  112 
using assms prob_compl by auto 
35582  113 

114 
lemma prob_one_inter: 

115 
assumes events:"s \<in> events" "t \<in> events" 

116 
assumes "prob t = 1" 

117 
shows "prob (s \<inter> t) = prob s" 

118 
proof  

38656  119 
have "prob ((space M  s) \<union> (space M  t)) = prob (space M  s)" 
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using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union) 

121 
also have "(space M  s) \<union> (space M  t) = space M  (s \<inter> t)" 

122 
by blast 

123 
finally show "prob (s \<inter> t) = prob s" 

124 
using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s]) 

35582  125 
qed 
126 

127 
lemma prob_eq_bigunion_image: 

128 
assumes "range f \<subseteq> events" "range g \<subseteq> events" 

129 
assumes "disjoint_family f" "disjoint_family g" 

130 
assumes "\<And> n :: nat. prob (f n) = prob (g n)" 

131 
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" 

132 
using assms 

133 
proof  

38656  134 
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))" 
135 
by (rule real_finite_measure_UNION[OF assms(1,3)]) 

136 
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" 

137 
by (rule real_finite_measure_UNION[OF assms(2,4)]) 

138 
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp 

35582  139 
qed 
140 

141 
lemma prob_countably_zero: 

142 
assumes "range c \<subseteq> events" 

143 
assumes "\<And> i. prob (c i) = 0" 

38656  144 
shows "prob (\<Union> i :: nat. c i) = 0" 
145 
proof (rule antisym) 

146 
show "prob (\<Union> i :: nat. c i) \<le> 0" 

147 
using real_finite_measurable_countably_subadditive[OF assms(1)] 

148 
by (simp add: assms(2) suminf_zero summable_zero) 

149 
show "0 \<le> prob (\<Union> i :: nat. c i)" by (rule real_pinfreal_nonneg) 

35582  150 
qed 
151 

152 
lemma indep_sym: 

153 
"indep a b \<Longrightarrow> indep b a" 

154 
unfolding indep_def using Int_commute[of a b] by auto 

155 

156 
lemma indep_refl: 

157 
assumes "a \<in> events" 

158 
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" 

159 
using assms unfolding indep_def by auto 

160 

161 
lemma prob_equiprobable_finite_unions: 

38656  162 
assumes "s \<in> events" 
163 
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" 

35582  164 
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" 
38656  165 
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" 
35582  166 
proof (cases "s = {}") 
38656  167 
case False hence "\<exists> x. x \<in> s" by blast 
35582  168 
from someI_ex[OF this] assms 
169 
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast 

170 
have "prob s = (\<Sum> x \<in> s. prob {x})" 

38656  171 
using real_finite_measure_finite_singelton[OF s_finite] by simp 
35582  172 
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto 
38656  173 
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" 
174 
using setsum_constant assms by (simp add: real_eq_of_nat) 

35582  175 
finally show ?thesis by simp 
38656  176 
qed simp 
35582  177 

178 
lemma prob_real_sum_image_fn: 

179 
assumes "e \<in> events" 

180 
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" 

181 
assumes "finite s" 

38656  182 
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" 
183 
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" 

35582  184 
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" 
185 
proof  

38656  186 
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" 
187 
using `e \<in> events` sets_into_space upper by blast 

188 
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp 

189 
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" 

190 
proof (rule real_finite_measure_finite_Union) 

191 
show "finite s" by fact 

192 
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact 

193 
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" 

194 
using disjoint by (auto simp: disjoint_family_on_def) 

195 
qed 

196 
finally show ?thesis . 

35582  197 
qed 
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199 
lemma distribution_prob_space: 

39089  200 
assumes S: "sigma_algebra S" "random_variable S X" 
38656  201 
shows "prob_space S (distribution X)" 
35582  202 
proof  
39089  203 
interpret S: measure_space S "distribution X" 
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using measure_space_vimage[OF S(2,1)] unfolding distribution_def . 

38656  205 
show ?thesis 
206 
proof 

207 
have "X ` space S \<inter> space M = space M" 

208 
using `random_variable S X` by (auto simp: measurable_def) 

39089  209 
then show "distribution X (space S) = 1" 
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using measure_space_1 by (simp add: distribution_def) 

35582  211 
qed 
212 
qed 

213 

214 
lemma distribution_lebesgue_thm1: 

215 
assumes "random_variable s X" 

216 
assumes "A \<in> sets s" 

38656  217 
shows "real (distribution X A) = expectation (indicator (X ` A \<inter> space M))" 
35582  218 
unfolding distribution_def 
219 
using assms unfolding measurable_def 

38656  220 
using integral_indicator by auto 
35582  221 

222 
lemma distribution_lebesgue_thm2: 

38656  223 
assumes "sigma_algebra S" "random_variable S X" and "A \<in> sets S" 
224 
shows "distribution X A = 

225 
measure_space.positive_integral S (distribution X) (indicator A)" 

226 
(is "_ = measure_space.positive_integral _ ?D _") 

35582  227 
proof  
38656  228 
interpret S: prob_space S "distribution X" using assms(1,2) by (rule distribution_prob_space) 
35582  229 

230 
show ?thesis 

38656  231 
using S.positive_integral_indicator(1) 
35582  232 
using assms unfolding distribution_def by auto 
233 
qed 

234 

235 
lemma finite_expectation1: 

38656  236 
assumes "finite (X`space M)" and rv: "random_variable borel_space X" 
35582  237 
shows "expectation X = (\<Sum> r \<in> X ` space M. r * prob (X ` {r} \<inter> space M))" 
38656  238 
proof (rule integral_on_finite(2)[OF assms(2,1)]) 
239 
fix x have "X ` {x} \<inter> space M \<in> sets M" 

240 
using rv unfolding measurable_def by auto 

241 
thus "\<mu> (X ` {x} \<inter> space M) \<noteq> \<omega>" using finite_measure by simp 

242 
qed 

35582  243 

244 
lemma finite_expectation: 

38656  245 
assumes "finite (space M)" "random_variable borel_space X" 
246 
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * real (distribution X {r}))" 

247 
using assms unfolding distribution_def using finite_expectation1 by auto 

248 

35582  249 
lemma prob_x_eq_1_imp_prob_y_eq_0: 
250 
assumes "{x} \<in> events" 

38656  251 
assumes "prob {x} = 1" 
35582  252 
assumes "{y} \<in> events" 
253 
assumes "y \<noteq> x" 

254 
shows "prob {y} = 0" 

255 
using prob_one_inter[of "{y}" "{x}"] assms by auto 

256 

38656  257 
lemma distribution_empty[simp]: "distribution X {} = 0" 
258 
unfolding distribution_def by simp 

259 

260 
lemma distribution_space[simp]: "distribution X (X ` space M) = 1" 

261 
proof  

262 
have "X ` X ` space M \<inter> space M = space M" by auto 

263 
thus ?thesis unfolding distribution_def by (simp add: measure_space_1) 

264 
qed 

265 

266 
lemma distribution_one: 

267 
assumes "random_variable M X" and "A \<in> events" 

268 
shows "distribution X A \<le> 1" 

269 
proof  

270 
have "distribution X A \<le> \<mu> (space M)" unfolding distribution_def 

271 
using assms[unfolded measurable_def] by (auto intro!: measure_mono) 

272 
thus ?thesis by (simp add: measure_space_1) 

273 
qed 

274 

275 
lemma distribution_finite: 

276 
assumes "random_variable M X" and "A \<in> events" 

277 
shows "distribution X A \<noteq> \<omega>" 

278 
using distribution_one[OF assms] by auto 

279 

35582  280 
lemma distribution_x_eq_1_imp_distribution_y_eq_0: 
281 
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" 

38656  282 
(is "random_variable ?S X") 
283 
assumes "distribution X {x} = 1" 

35582  284 
assumes "y \<noteq> x" 
285 
shows "distribution X {y} = 0" 

286 
proof  

38656  287 
have "sigma_algebra ?S" by (rule sigma_algebra_Pow) 
288 
from distribution_prob_space[OF this X] 

289 
interpret S: prob_space ?S "distribution X" by simp 

290 

291 
have x: "{x} \<in> sets ?S" 

292 
proof (rule ccontr) 

293 
assume "{x} \<notin> sets ?S" 

35582  294 
hence "X ` {x} \<inter> space M = {}" by auto 
38656  295 
thus "False" using assms unfolding distribution_def by auto 
296 
qed 

297 

298 
have [simp]: "{y} \<inter> {x} = {}" "{x}  {y} = {x}" using `y \<noteq> x` by auto 

299 

300 
show ?thesis 

301 
proof cases 

302 
assume "{y} \<in> sets ?S" 

303 
with `{x} \<in> sets ?S` assms show "distribution X {y} = 0" 

304 
using S.measure_inter_full_set[of "{y}" "{x}"] 

305 
by simp 

306 
next 

307 
assume "{y} \<notin> sets ?S" 

35582  308 
hence "X ` {y} \<inter> space M = {}" by auto 
38656  309 
thus "distribution X {y} = 0" unfolding distribution_def by auto 
310 
qed 

35582  311 
qed 
312 

313 
end 

314 

35977  315 
locale finite_prob_space = prob_space + finite_measure_space 
316 

36624  317 
lemma finite_prob_space_eq: 
38656  318 
"finite_prob_space M \<mu> \<longleftrightarrow> finite_measure_space M \<mu> \<and> \<mu> (space M) = 1" 
36624  319 
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def 
320 
by auto 

321 

322 
lemma (in prob_space) not_empty: "space M \<noteq> {}" 

323 
using prob_space empty_measure by auto 

324 

38656  325 
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" 
326 
using measure_space_1 sum_over_space by simp 

36624  327 

328 
lemma (in finite_prob_space) positive_distribution: "0 \<le> distribution X x" 

38656  329 
unfolding distribution_def by simp 
36624  330 

331 
lemma (in finite_prob_space) joint_distribution_restriction_fst: 

332 
"joint_distribution X Y A \<le> distribution X (fst ` A)" 

333 
unfolding distribution_def 

334 
proof (safe intro!: measure_mono) 

335 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 

336 
show "x \<in> X ` fst ` A" 

337 
by (auto intro!: image_eqI[OF _ *]) 

338 
qed (simp_all add: sets_eq_Pow) 

339 

340 
lemma (in finite_prob_space) joint_distribution_restriction_snd: 

341 
"joint_distribution X Y A \<le> distribution Y (snd ` A)" 

342 
unfolding distribution_def 

343 
proof (safe intro!: measure_mono) 

344 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 

345 
show "x \<in> Y ` snd ` A" 

346 
by (auto intro!: image_eqI[OF _ *]) 

347 
qed (simp_all add: sets_eq_Pow) 

348 

349 
lemma (in finite_prob_space) distribution_order: 

350 
shows "0 \<le> distribution X x'" 

351 
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" 

352 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

353 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

354 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

355 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

356 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

357 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

358 
using positive_distribution[of X x'] 

359 
positive_distribution[of "\<lambda>x. (X x, Y x)" "{(x, y)}"] 

360 
joint_distribution_restriction_fst[of X Y "{(x, y)}"] 

361 
joint_distribution_restriction_snd[of X Y "{(x, y)}"] 

362 
by auto 

363 

364 
lemma (in finite_prob_space) finite_product_measure_space: 

35977  365 
assumes "finite s1" "finite s2" 
38656  366 
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2)\<rparr> (joint_distribution X Y)" 
367 
(is "finite_measure_space ?M ?D") 

35977  368 
proof (rule finite_Pow_additivity_sufficient) 
38656  369 
show "positive ?D" 
370 
unfolding positive_def using assms sets_eq_Pow 

36624  371 
by (simp add: distribution_def) 
35977  372 

38656  373 
show "additive ?M ?D" unfolding additive_def 
35977  374 
proof safe 
375 
fix x y 

376 
have A: "((\<lambda>x. (X x, Y x)) ` x) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto 

377 
have B: "((\<lambda>x. (X x, Y x)) ` y) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto 

378 
assume "x \<inter> y = {}" 

38656  379 
hence "(\<lambda>x. (X x, Y x)) ` x \<inter> space M \<inter> ((\<lambda>x. (X x, Y x)) ` y \<inter> space M) = {}" 
380 
by auto 

35977  381 
from additive[unfolded additive_def, rule_format, OF A B] this 
38656  382 
finite_measure[OF A] finite_measure[OF B] 
383 
show "?D (x \<union> y) = ?D x + ?D y" 

36624  384 
apply (simp add: distribution_def) 
35977  385 
apply (subst Int_Un_distrib2) 
38656  386 
by (auto simp: real_of_pinfreal_add) 
35977  387 
qed 
388 

389 
show "finite (space ?M)" 

390 
using assms by auto 

391 

392 
show "sets ?M = Pow (space ?M)" 

393 
by simp 

38656  394 

395 
{ fix x assume "x \<in> space ?M" thus "?D {x} \<noteq> \<omega>" 

396 
unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) } 

35977  397 
qed 
398 

36624  399 
lemma (in finite_prob_space) finite_product_measure_space_of_images: 
35977  400 
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M, 
38656  401 
sets = Pow (X ` space M \<times> Y ` space M) \<rparr> 
402 
(joint_distribution X Y)" 

36624  403 
using finite_space by (auto intro!: finite_product_measure_space) 
404 

405 
lemma (in finite_prob_space) finite_measure_space: 

38656  406 
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)" 
407 
(is "finite_measure_space ?S _") 

36624  408 
proof (rule finite_Pow_additivity_sufficient, simp_all) 
409 
show "finite (X ` space M)" using finite_space by simp 

410 

38656  411 
show "positive (distribution X)" 
412 
unfolding distribution_def positive_def using sets_eq_Pow by auto 

36624  413 

414 
show "additive ?S (distribution X)" unfolding additive_def distribution_def 

415 
proof (simp, safe) 

416 
fix x y 

417 
have x: "(X ` x) \<inter> space M \<in> sets M" 

418 
and y: "(X ` y) \<inter> space M \<in> sets M" using sets_eq_Pow by auto 

419 
assume "x \<inter> y = {}" 

38656  420 
hence "X ` x \<inter> space M \<inter> (X ` y \<inter> space M) = {}" by auto 
36624  421 
from additive[unfolded additive_def, rule_format, OF x y] this 
38656  422 
finite_measure[OF x] finite_measure[OF y] 
423 
have "\<mu> (((X ` x) \<union> (X ` y)) \<inter> space M) = 

424 
\<mu> ((X ` x) \<inter> space M) + \<mu> ((X ` y) \<inter> space M)" 

425 
by (subst Int_Un_distrib2) auto 

426 
thus "\<mu> ((X ` x \<union> X ` y) \<inter> space M) = \<mu> (X ` x \<inter> space M) + \<mu> (X ` y \<inter> space M)" 

36624  427 
by auto 
428 
qed 

38656  429 

430 
{ fix x assume "x \<in> X ` space M" thus "distribution X {x} \<noteq> \<omega>" 

431 
unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) } 

36624  432 
qed 
433 

434 
lemma (in finite_prob_space) finite_prob_space_of_images: 

38656  435 
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)" 
436 
by (simp add: finite_prob_space_eq finite_measure_space) 

36624  437 

438 
lemma (in finite_prob_space) finite_product_prob_space_of_images: 

38656  439 
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M)\<rparr> 
440 
(joint_distribution X Y)" 

441 
(is "finite_prob_space ?S _") 

442 
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images) 

36624  443 
have "X ` X ` space M \<inter> Y ` Y ` space M \<inter> space M = space M" by auto 
444 
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" 

38656  445 
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) 
36624  446 
qed 
35977  447 

39083  448 
lemma (in prob_space) prob_space_subalgebra: 
449 
assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

450 
shows "prob_space (M\<lparr> sets := N \<rparr>) \<mu>" sorry 

451 

452 
lemma (in measure_space) measure_space_subalgebra: 

453 
assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

454 
shows "measure_space (M\<lparr> sets := N \<rparr>) \<mu>" sorry 

455 

456 
lemma pinfreal_0_less_mult_iff[simp]: 

457 
fixes x y :: pinfreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y" 

458 
by (cases x, cases y) (auto simp: zero_less_mult_iff) 

459 

460 
lemma (in sigma_algebra) simple_function_subalgebra: 

461 
assumes "sigma_algebra.simple_function (M\<lparr>sets:=N\<rparr>) f" 

462 
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets:=N\<rparr>)" 

463 
shows "simple_function f" 

464 
using assms 

465 
unfolding simple_function_def 

466 
unfolding sigma_algebra.simple_function_def[OF N_subalgebra(2)] 

467 
by auto 

468 

469 
lemma (in measure_space) simple_integral_subalgebra[simp]: 

470 
assumes "measure_space (M\<lparr>sets := N\<rparr>) \<mu>" 

471 
shows "measure_space.simple_integral (M\<lparr>sets := N\<rparr>) \<mu> = simple_integral" 

472 
unfolding simple_integral_def_raw 

473 
unfolding measure_space.simple_integral_def_raw[OF assms] by simp 

474 

475 
lemma (in sigma_algebra) borel_measurable_subalgebra: 

476 
assumes "N \<subseteq> sets M" "f \<in> borel_measurable (M\<lparr>sets:=N\<rparr>)" 

477 
shows "f \<in> borel_measurable M" 

478 
using assms unfolding measurable_def by auto 

479 

480 
lemma (in measure_space) positive_integral_subalgebra[simp]: 

481 
assumes borel: "f \<in> borel_measurable (M\<lparr>sets := N\<rparr>)" 

482 
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets := N\<rparr>)" 

483 
shows "measure_space.positive_integral (M\<lparr>sets := N\<rparr>) \<mu> f = positive_integral f" 

484 
proof  

485 
note msN = measure_space_subalgebra[OF N_subalgebra] 

486 
then interpret N: measure_space "M\<lparr>sets:=N\<rparr>" \<mu> . 

487 

488 
from N.borel_measurable_implies_simple_function_sequence[OF borel] 

489 
obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast 

490 
then have sf: "\<And>i. simple_function (fs i)" 

491 
using simple_function_subalgebra[OF _ N_subalgebra] by blast 

492 

493 
from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf] 

494 
show ?thesis unfolding simple_integral_subalgebra[OF msN] isoton_def by simp 

495 
qed 

39084  496 

39085  497 
section "Conditional Expectation and Probability" 
498 

499 
lemma (in prob_space) conditional_expectation_exists: 

39083  500 
fixes X :: "'a \<Rightarrow> pinfreal" 
501 
assumes borel: "X \<in> borel_measurable M" 

502 
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

503 
shows "\<exists>Y\<in>borel_measurable (M\<lparr> sets := N \<rparr>). \<forall>C\<in>N. 

504 
positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)" 

505 
proof  

506 
interpret P: prob_space "M\<lparr> sets := N \<rparr>" \<mu> 

507 
using prob_space_subalgebra[OF N_subalgebra] . 

508 

509 
let "?f A" = "\<lambda>x. X x * indicator A x" 

510 
let "?Q A" = "positive_integral (?f A)" 

511 

512 
from measure_space_density[OF borel] 

513 
have Q: "measure_space (M\<lparr> sets := N \<rparr>) ?Q" 

514 
by (rule measure_space.measure_space_subalgebra[OF _ N_subalgebra]) 

515 
then interpret Q: measure_space "M\<lparr> sets := N \<rparr>" ?Q . 

516 

517 
have "P.absolutely_continuous ?Q" 

518 
unfolding P.absolutely_continuous_def 

519 
proof (safe, simp) 

520 
fix A assume "A \<in> N" "\<mu> A = 0" 

521 
moreover then have f_borel: "?f A \<in> borel_measurable M" 

522 
using borel N_subalgebra by (auto intro: borel_measurable_indicator) 

523 
moreover have "{x\<in>space M. ?f A x \<noteq> 0} = (?f A ` {0<..} \<inter> space M) \<inter> A" 

524 
by (auto simp: indicator_def) 

525 
moreover have "\<mu> \<dots> \<le> \<mu> A" 

526 
using `A \<in> N` N_subalgebra f_borel 

527 
by (auto intro!: measure_mono Int[of _ A] measurable_sets) 

528 
ultimately show "?Q A = 0" 

529 
by (simp add: positive_integral_0_iff) 

530 
qed 

531 
from P.Radon_Nikodym[OF Q this] 

532 
obtain Y where Y: "Y \<in> borel_measurable (M\<lparr>sets := N\<rparr>)" 

533 
"\<And>A. A \<in> sets (M\<lparr>sets:=N\<rparr>) \<Longrightarrow> ?Q A = P.positive_integral (\<lambda>x. Y x * indicator A x)" 

534 
by blast 

39084  535 
with N_subalgebra show ?thesis 
536 
by (auto intro!: bexI[OF _ Y(1)]) 

39083  537 
qed 
538 

39085  539 
definition (in prob_space) 
540 
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable (M\<lparr>sets:=N\<rparr>) 

541 
\<and> (\<forall>C\<in>N. positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)))" 

542 

543 
abbreviation (in prob_space) 

544 
"conditional_probabiltiy N A \<equiv> conditional_expectation N (indicator A)" 

545 

546 
lemma (in prob_space) 

547 
fixes X :: "'a \<Rightarrow> pinfreal" 

548 
assumes borel: "X \<in> borel_measurable M" 

549 
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

550 
shows borel_measurable_conditional_expectation: 

551 
"conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)" 

552 
and conditional_expectation: "\<And>C. C \<in> N \<Longrightarrow> 

553 
positive_integral (\<lambda>x. conditional_expectation N X x * indicator C x) = 

554 
positive_integral (\<lambda>x. X x * indicator C x)" 

555 
(is "\<And>C. C \<in> N \<Longrightarrow> ?eq C") 

556 
proof  

557 
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] 

558 
then show "conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)" 

559 
unfolding conditional_expectation_def by (rule someI2_ex) blast 

560 

561 
from CE show "\<And>C. C\<in>N \<Longrightarrow> ?eq C" 

562 
unfolding conditional_expectation_def by (rule someI2_ex) blast 

563 
qed 

564 

39091  565 
lemma (in sigma_algebra) factorize_measurable_function: 
566 
fixes Z :: "'a \<Rightarrow> pinfreal" and Y :: "'a \<Rightarrow> 'c" 

567 
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" 

568 
shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y) 

569 
\<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))" 

570 
proof safe 

571 
interpret M': sigma_algebra M' by fact 

572 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 

573 
from M'.sigma_algebra_vimage[OF this] 

574 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 

575 

576 
{ fix g :: "'c \<Rightarrow> pinfreal" assume "g \<in> borel_measurable M'" 

577 
with M'.measurable_vimage_algebra[OF Y] 

578 
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

579 
by (rule measurable_comp) 

580 
moreover assume "\<forall>x\<in>space M. Z x = g (Y x)" 

581 
then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow> 

582 
g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

583 
by (auto intro!: measurable_cong) 

584 
ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

585 
by simp } 

586 

587 
assume "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

588 
from va.borel_measurable_implies_simple_function_sequence[OF this] 

589 
obtain f where f: "\<And>i. va.simple_function (f i)" and "f \<up> Z" by blast 

590 

591 
have "\<forall>i. \<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 

592 
proof 

593 
fix i 

594 
from f[of i] have "finite (f i`space M)" and B_ex: 

595 
"\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) ` {z} \<inter> space M = Y ` B \<inter> space M" 

596 
unfolding va.simple_function_def by auto 

597 
from B_ex[THEN bchoice] guess B .. note B = this 

598 

599 
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" 

600 

601 
show "\<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 

602 
proof (intro exI[of _ ?g] conjI ballI) 

603 
show "M'.simple_function ?g" using B by auto 

604 

605 
fix x assume "x \<in> space M" 

606 
then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i ` {z} \<inter> space M) x::pinfreal)" 

607 
unfolding indicator_def using B by auto 

608 
then show "f i x = ?g (Y x)" using `x \<in> space M` f[of i] 

609 
by (subst va.simple_function_indicator_representation) auto 

610 
qed 

611 
qed 

612 
from choice[OF this] guess g .. note g = this 

613 

614 
show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)" 

615 
proof (intro ballI bexI) 

616 
show "(SUP i. g i) \<in> borel_measurable M'" 

617 
using g by (auto intro: M'.borel_measurable_simple_function) 

618 
fix x assume "x \<in> space M" 

619 
have "Z x = (SUP i. f i) x" using `f \<up> Z` unfolding isoton_def by simp 

620 
also have "\<dots> = (SUP i. g i) (Y x)" unfolding SUPR_fun_expand 

621 
using g `x \<in> space M` by simp 

622 
finally show "Z x = (SUP i. g i) (Y x)" . 

623 
qed 

624 
qed 

39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
39089
diff
changeset

625 

35582  626 
end 