author  hoelzl 
Thu, 02 Sep 2010 17:12:40 +0200  
changeset 39092  98de40859858 
parent 39091  11314c196e11 
child 39096  111756225292 
permissions  rwrr 
35582  1 
theory Probability_Space 
39083  2 
imports Lebesgue_Integration Radon_Nikodym 
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begin 
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locale prob_space = measure_space + 

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assumes measure_space_1: "\<mu> (space M) = 1" 
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sublocale prob_space < finite_measure 

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proof 

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

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qed 

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context prob_space 

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begin 
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abbreviation "events \<equiv> sets M" 

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

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abbreviation "expectation \<equiv> integral" 

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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" 

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definition 

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

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definition 

38656  29 
"distribution X = (\<lambda>s. \<mu> ((X ` s) \<inter> (space M)))" 
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36624  31 
abbreviation 
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"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" 

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38656  34 
lemma prob_space: "prob (space M) = 1" 
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unfolding measure_space_1 by simp 

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38656  37 
lemma measure_le_1[simp, intro]: 
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assumes "A \<in> events" shows "\<mu> A \<le> 1" 

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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) 

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also note measure_space_1 

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finally show ?thesis . 

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qed 

35582  45 

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lemma prob_compl: 

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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 

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by (subst real_finite_measure_Diff) auto 

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lemma indep_space: 

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assumes "s \<in> events" 

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shows "indep (space M) s" 

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using assms prob_space by (simp add: indep_def) 
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38656  57 
lemma prob_space_increasing: "increasing M prob" 
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by (auto intro!: real_measure_mono simp: increasing_def) 

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lemma prob_zero_union: 

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

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

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using assms 
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proof  
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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 
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qed 

71 

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lemma prob_eq_compl: 

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

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

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shows "prob s = prob t" 

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using assms prob_compl by auto 
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lemma prob_one_inter: 

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

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assumes "prob t = 1" 

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

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proof  

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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) 

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

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by blast 

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

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

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qed 
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lemma prob_eq_bigunion_image: 

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

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assumes "disjoint_family f" "disjoint_family g" 

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

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

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using assms 

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proof  

38656  98 
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))" 
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by (rule real_finite_measure_UNION[OF assms(1,3)]) 

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

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

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

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qed 
104 

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lemma prob_countably_zero: 

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assumes "range c \<subseteq> events" 

107 
assumes "\<And> i. prob (c i) = 0" 

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

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

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using real_finite_measurable_countably_subadditive[OF assms(1)] 

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

113 
show "0 \<le> prob (\<Union> i :: nat. c i)" by (rule real_pinfreal_nonneg) 

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qed 
115 

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lemma indep_sym: 

117 
"indep a b \<Longrightarrow> indep b a" 

118 
unfolding indep_def using Int_commute[of a b] by auto 

119 

120 
lemma indep_refl: 

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assumes "a \<in> events" 

122 
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" 

123 
using assms unfolding indep_def by auto 

124 

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lemma prob_equiprobable_finite_unions: 

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assumes "s \<in> events" 
127 
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" 

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assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" 
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shows "prob s = real (card s) * prob {SOME x. x \<in> s}" 
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proof (cases "s = {}") 
38656  131 
case False hence "\<exists> x. x \<in> s" by blast 
35582  132 
from someI_ex[OF this] assms 
133 
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast 

134 
have "prob s = (\<Sum> x \<in> s. prob {x})" 

38656  135 
using real_finite_measure_finite_singelton[OF s_finite] by simp 
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also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto 
38656  137 
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" 
138 
using setsum_constant assms by (simp add: real_eq_of_nat) 

35582  139 
finally show ?thesis by simp 
38656  140 
qed simp 
35582  141 

142 
lemma prob_real_sum_image_fn: 

143 
assumes "e \<in> events" 

144 
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" 

145 
assumes "finite s" 

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assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" 
147 
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" 

35582  148 
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" 
149 
proof  

38656  150 
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" 
151 
using `e \<in> events` sets_into_space upper by blast 

152 
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp 

153 
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" 

154 
proof (rule real_finite_measure_finite_Union) 

155 
show "finite s" by fact 

156 
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact 

157 
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" 

158 
using disjoint by (auto simp: disjoint_family_on_def) 

159 
qed 

160 
finally show ?thesis . 

35582  161 
qed 
162 

163 
lemma distribution_prob_space: 

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assumes S: "sigma_algebra S" "random_variable S X" 
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shows "prob_space S (distribution X)" 
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proof  
39089  167 
interpret S: measure_space S "distribution X" 
168 
using measure_space_vimage[OF S(2,1)] unfolding distribution_def . 

38656  169 
show ?thesis 
170 
proof 

171 
have "X ` space S \<inter> space M = space M" 

172 
using `random_variable S X` by (auto simp: measurable_def) 

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then show "distribution X (space S) = 1" 
174 
using measure_space_1 by (simp add: distribution_def) 

35582  175 
qed 
176 
qed 

177 

178 
lemma distribution_lebesgue_thm1: 

179 
assumes "random_variable s X" 

180 
assumes "A \<in> sets s" 

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shows "real (distribution X A) = expectation (indicator (X ` A \<inter> space M))" 
35582  182 
unfolding distribution_def 
183 
using assms unfolding measurable_def 

38656  184 
using integral_indicator by auto 
35582  185 

186 
lemma distribution_lebesgue_thm2: 

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assumes "sigma_algebra S" "random_variable S X" and "A \<in> sets S" 
188 
shows "distribution X A = 

189 
measure_space.positive_integral S (distribution X) (indicator A)" 

190 
(is "_ = measure_space.positive_integral _ ?D _") 

35582  191 
proof  
38656  192 
interpret S: prob_space S "distribution X" using assms(1,2) by (rule distribution_prob_space) 
35582  193 

194 
show ?thesis 

38656  195 
using S.positive_integral_indicator(1) 
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using assms unfolding distribution_def by auto 
197 
qed 

198 

199 
lemma finite_expectation1: 

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assumes "finite (X`space M)" and rv: "random_variable borel_space X" 
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shows "expectation X = (\<Sum> r \<in> X ` space M. r * prob (X ` {r} \<inter> space M))" 
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proof (rule integral_on_finite(2)[OF assms(2,1)]) 
203 
fix x have "X ` {x} \<inter> space M \<in> sets M" 

204 
using rv unfolding measurable_def by auto 

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

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qed 

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lemma finite_expectation: 

38656  209 
assumes "finite (space M)" "random_variable borel_space X" 
210 
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * real (distribution X {r}))" 

211 
using assms unfolding distribution_def using finite_expectation1 by auto 

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

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assumes "prob {x} = 1" 
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assumes "{y} \<in> events" 
217 
assumes "y \<noteq> x" 

218 
shows "prob {y} = 0" 

219 
using prob_one_inter[of "{y}" "{x}"] assms by auto 

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

223 

224 
lemma distribution_space[simp]: "distribution X (X ` space M) = 1" 

225 
proof  

226 
have "X ` X ` space M \<inter> space M = space M" by auto 

227 
thus ?thesis unfolding distribution_def by (simp add: measure_space_1) 

228 
qed 

229 

230 
lemma distribution_one: 

231 
assumes "random_variable M X" and "A \<in> events" 

232 
shows "distribution X A \<le> 1" 

233 
proof  

234 
have "distribution X A \<le> \<mu> (space M)" unfolding distribution_def 

235 
using assms[unfolded measurable_def] by (auto intro!: measure_mono) 

236 
thus ?thesis by (simp add: measure_space_1) 

237 
qed 

238 

239 
lemma distribution_finite: 

240 
assumes "random_variable M X" and "A \<in> events" 

241 
shows "distribution X A \<noteq> \<omega>" 

242 
using distribution_one[OF assms] by auto 

243 

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lemma distribution_x_eq_1_imp_distribution_y_eq_0: 
245 
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" 

38656  246 
(is "random_variable ?S X") 
247 
assumes "distribution X {x} = 1" 

35582  248 
assumes "y \<noteq> x" 
249 
shows "distribution X {y} = 0" 

250 
proof  

38656  251 
have "sigma_algebra ?S" by (rule sigma_algebra_Pow) 
252 
from distribution_prob_space[OF this X] 

253 
interpret S: prob_space ?S "distribution X" by simp 

254 

255 
have x: "{x} \<in> sets ?S" 

256 
proof (rule ccontr) 

257 
assume "{x} \<notin> sets ?S" 

35582  258 
hence "X ` {x} \<inter> space M = {}" by auto 
38656  259 
thus "False" using assms unfolding distribution_def by auto 
260 
qed 

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262 
have [simp]: "{y} \<inter> {x} = {}" "{x}  {y} = {x}" using `y \<noteq> x` by auto 

263 

264 
show ?thesis 

265 
proof cases 

266 
assume "{y} \<in> sets ?S" 

267 
with `{x} \<in> sets ?S` assms show "distribution X {y} = 0" 

268 
using S.measure_inter_full_set[of "{y}" "{x}"] 

269 
by simp 

270 
next 

271 
assume "{y} \<notin> sets ?S" 

35582  272 
hence "X ` {y} \<inter> space M = {}" by auto 
38656  273 
thus "distribution X {y} = 0" unfolding distribution_def by auto 
274 
qed 

35582  275 
qed 
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277 
end 

278 

35977  279 
locale finite_prob_space = prob_space + finite_measure_space 
280 

36624  281 
lemma finite_prob_space_eq: 
38656  282 
"finite_prob_space M \<mu> \<longleftrightarrow> finite_measure_space M \<mu> \<and> \<mu> (space M) = 1" 
36624  283 
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def 
284 
by auto 

285 

286 
lemma (in prob_space) not_empty: "space M \<noteq> {}" 

287 
using prob_space empty_measure by auto 

288 

38656  289 
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" 
290 
using measure_space_1 sum_over_space by simp 

36624  291 

292 
lemma (in finite_prob_space) positive_distribution: "0 \<le> distribution X x" 

38656  293 
unfolding distribution_def by simp 
36624  294 

295 
lemma (in finite_prob_space) joint_distribution_restriction_fst: 

296 
"joint_distribution X Y A \<le> distribution X (fst ` A)" 

297 
unfolding distribution_def 

298 
proof (safe intro!: measure_mono) 

299 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 

300 
show "x \<in> X ` fst ` A" 

301 
by (auto intro!: image_eqI[OF _ *]) 

302 
qed (simp_all add: sets_eq_Pow) 

303 

304 
lemma (in finite_prob_space) joint_distribution_restriction_snd: 

305 
"joint_distribution X Y A \<le> distribution Y (snd ` A)" 

306 
unfolding distribution_def 

307 
proof (safe intro!: measure_mono) 

308 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 

309 
show "x \<in> Y ` snd ` A" 

310 
by (auto intro!: image_eqI[OF _ *]) 

311 
qed (simp_all add: sets_eq_Pow) 

312 

313 
lemma (in finite_prob_space) distribution_order: 

314 
shows "0 \<le> distribution X x'" 

315 
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" 

316 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

317 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

318 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

319 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

320 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

321 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

322 
using positive_distribution[of X x'] 

323 
positive_distribution[of "\<lambda>x. (X x, Y x)" "{(x, y)}"] 

324 
joint_distribution_restriction_fst[of X Y "{(x, y)}"] 

325 
joint_distribution_restriction_snd[of X Y "{(x, y)}"] 

326 
by auto 

327 

39092  328 
lemma (in finite_prob_space) finite_prob_space_of_images: 
329 
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)" 

330 
by (simp add: finite_prob_space_eq finite_measure_space) 

35977  331 

39092  332 
lemma (in finite_prob_space) finite_product_prob_space_of_images: 
333 
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M)\<rparr> 

334 
(joint_distribution X Y)" 

335 
(is "finite_prob_space ?S _") 

336 
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images) 

337 
have "X ` X ` space M \<inter> Y ` Y ` space M \<inter> space M = space M" by auto 

338 
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" 

339 
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) 

340 
qed 

35977  341 

39092  342 
lemma (in prob_space) prob_space_subalgebra: 
343 
assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

344 
shows "prob_space (M\<lparr> sets := N \<rparr>) \<mu>" 

345 
proof  

346 
interpret N: measure_space "M\<lparr> sets := N \<rparr>" \<mu> 

347 
using measure_space_subalgebra[OF assms] . 

348 
show ?thesis 

349 
proof qed (simp add: measure_space_1) 

35977  350 
qed 
351 

39092  352 
lemma (in prob_space) prob_space_of_restricted_space: 
353 
assumes "\<mu> A \<noteq> 0" "\<mu> A \<noteq> \<omega>" "A \<in> sets M" 

354 
shows "prob_space (restricted_space A) (\<lambda>S. \<mu> S / \<mu> A)" 

355 
unfolding prob_space_def prob_space_axioms_def 

356 
proof 

357 
show "\<mu> (space (restricted_space A)) / \<mu> A = 1" 

358 
using `\<mu> A \<noteq> 0` `\<mu> A \<noteq> \<omega>` by (auto simp: pinfreal_noteq_omega_Ex) 

359 
have *: "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" by (simp add: mult_commute) 

360 
interpret A: measure_space "restricted_space A" \<mu> 

361 
using `A \<in> sets M` by (rule restricted_measure_space) 

362 
show "measure_space (restricted_space A) (\<lambda>S. \<mu> S / \<mu> A)" 

363 
proof 

364 
show "\<mu> {} / \<mu> A = 0" by auto 

365 
show "countably_additive (restricted_space A) (\<lambda>S. \<mu> S / \<mu> A)" 

366 
unfolding countably_additive_def psuminf_cmult_right * 

367 
using A.measure_countably_additive by auto 

368 
qed 

369 
qed 

370 

371 
lemma finite_prob_spaceI: 

372 
assumes "finite (space M)" "sets M = Pow(space M)" "\<mu> (space M) = 1" "\<mu> {} = 0" 

373 
and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> \<mu> (A \<union> B) = \<mu> A + \<mu> B" 

374 
shows "finite_prob_space M \<mu>" 

375 
unfolding finite_prob_space_eq 

376 
proof 

377 
show "finite_measure_space M \<mu>" using assms 

378 
by (auto intro!: finite_measure_spaceI) 

379 
show "\<mu> (space M) = 1" by fact 

380 
qed 

36624  381 

382 
lemma (in finite_prob_space) finite_measure_space: 

38656  383 
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)" 
384 
(is "finite_measure_space ?S _") 

39092  385 
proof (rule finite_measure_spaceI, simp_all) 
36624  386 
show "finite (X ` space M)" using finite_space by simp 
387 

38656  388 
show "positive (distribution X)" 
389 
unfolding distribution_def positive_def using sets_eq_Pow by auto 

36624  390 

391 
show "additive ?S (distribution X)" unfolding additive_def distribution_def 

392 
proof (simp, safe) 

393 
fix x y 

394 
have x: "(X ` x) \<inter> space M \<in> sets M" 

395 
and y: "(X ` y) \<inter> space M \<in> sets M" using sets_eq_Pow by auto 

396 
assume "x \<inter> y = {}" 

38656  397 
hence "X ` x \<inter> space M \<inter> (X ` y \<inter> space M) = {}" by auto 
36624  398 
from additive[unfolded additive_def, rule_format, OF x y] this 
38656  399 
finite_measure[OF x] finite_measure[OF y] 
400 
have "\<mu> (((X ` x) \<union> (X ` y)) \<inter> space M) = 

401 
\<mu> ((X ` x) \<inter> space M) + \<mu> ((X ` y) \<inter> space M)" 

402 
by (subst Int_Un_distrib2) auto 

403 
thus "\<mu> ((X ` x \<union> X ` y) \<inter> space M) = \<mu> (X ` x \<inter> space M) + \<mu> (X ` y \<inter> space M)" 

36624  404 
by auto 
405 
qed 

38656  406 

407 
{ fix x assume "x \<in> X ` space M" thus "distribution X {x} \<noteq> \<omega>" 

408 
unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) } 

36624  409 
qed 
410 

39085  411 
section "Conditional Expectation and Probability" 
412 

413 
lemma (in prob_space) conditional_expectation_exists: 

39083  414 
fixes X :: "'a \<Rightarrow> pinfreal" 
415 
assumes borel: "X \<in> borel_measurable M" 

416 
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

417 
shows "\<exists>Y\<in>borel_measurable (M\<lparr> sets := N \<rparr>). \<forall>C\<in>N. 

418 
positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)" 

419 
proof  

420 
interpret P: prob_space "M\<lparr> sets := N \<rparr>" \<mu> 

421 
using prob_space_subalgebra[OF N_subalgebra] . 

422 

423 
let "?f A" = "\<lambda>x. X x * indicator A x" 

424 
let "?Q A" = "positive_integral (?f A)" 

425 

426 
from measure_space_density[OF borel] 

427 
have Q: "measure_space (M\<lparr> sets := N \<rparr>) ?Q" 

428 
by (rule measure_space.measure_space_subalgebra[OF _ N_subalgebra]) 

429 
then interpret Q: measure_space "M\<lparr> sets := N \<rparr>" ?Q . 

430 

431 
have "P.absolutely_continuous ?Q" 

432 
unfolding P.absolutely_continuous_def 

433 
proof (safe, simp) 

434 
fix A assume "A \<in> N" "\<mu> A = 0" 

435 
moreover then have f_borel: "?f A \<in> borel_measurable M" 

436 
using borel N_subalgebra by (auto intro: borel_measurable_indicator) 

437 
moreover have "{x\<in>space M. ?f A x \<noteq> 0} = (?f A ` {0<..} \<inter> space M) \<inter> A" 

438 
by (auto simp: indicator_def) 

439 
moreover have "\<mu> \<dots> \<le> \<mu> A" 

440 
using `A \<in> N` N_subalgebra f_borel 

441 
by (auto intro!: measure_mono Int[of _ A] measurable_sets) 

442 
ultimately show "?Q A = 0" 

443 
by (simp add: positive_integral_0_iff) 

444 
qed 

445 
from P.Radon_Nikodym[OF Q this] 

446 
obtain Y where Y: "Y \<in> borel_measurable (M\<lparr>sets := N\<rparr>)" 

447 
"\<And>A. A \<in> sets (M\<lparr>sets:=N\<rparr>) \<Longrightarrow> ?Q A = P.positive_integral (\<lambda>x. Y x * indicator A x)" 

448 
by blast 

39084  449 
with N_subalgebra show ?thesis 
450 
by (auto intro!: bexI[OF _ Y(1)]) 

39083  451 
qed 
452 

39085  453 
definition (in prob_space) 
454 
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable (M\<lparr>sets:=N\<rparr>) 

455 
\<and> (\<forall>C\<in>N. positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)))" 

456 

457 
abbreviation (in prob_space) 

39092  458 
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" 
39085  459 

460 
lemma (in prob_space) 

461 
fixes X :: "'a \<Rightarrow> pinfreal" 

462 
assumes borel: "X \<in> borel_measurable M" 

463 
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)" 

464 
shows borel_measurable_conditional_expectation: 

465 
"conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)" 

466 
and conditional_expectation: "\<And>C. C \<in> N \<Longrightarrow> 

467 
positive_integral (\<lambda>x. conditional_expectation N X x * indicator C x) = 

468 
positive_integral (\<lambda>x. X x * indicator C x)" 

469 
(is "\<And>C. C \<in> N \<Longrightarrow> ?eq C") 

470 
proof  

471 
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] 

472 
then show "conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)" 

473 
unfolding conditional_expectation_def by (rule someI2_ex) blast 

474 

475 
from CE show "\<And>C. C\<in>N \<Longrightarrow> ?eq C" 

476 
unfolding conditional_expectation_def by (rule someI2_ex) blast 

477 
qed 

478 

39091  479 
lemma (in sigma_algebra) factorize_measurable_function: 
480 
fixes Z :: "'a \<Rightarrow> pinfreal" and Y :: "'a \<Rightarrow> 'c" 

481 
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" 

482 
shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y) 

483 
\<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))" 

484 
proof safe 

485 
interpret M': sigma_algebra M' by fact 

486 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 

487 
from M'.sigma_algebra_vimage[OF this] 

488 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 

489 

490 
{ fix g :: "'c \<Rightarrow> pinfreal" assume "g \<in> borel_measurable M'" 

491 
with M'.measurable_vimage_algebra[OF Y] 

492 
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

493 
by (rule measurable_comp) 

494 
moreover assume "\<forall>x\<in>space M. Z x = g (Y x)" 

495 
then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow> 

496 
g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

497 
by (auto intro!: measurable_cong) 

498 
ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

499 
by simp } 

500 

501 
assume "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 

502 
from va.borel_measurable_implies_simple_function_sequence[OF this] 

503 
obtain f where f: "\<And>i. va.simple_function (f i)" and "f \<up> Z" by blast 

504 

505 
have "\<forall>i. \<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 

506 
proof 

507 
fix i 

508 
from f[of i] have "finite (f i`space M)" and B_ex: 

509 
"\<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" 

510 
unfolding va.simple_function_def by auto 

511 
from B_ex[THEN bchoice] guess B .. note B = this 

512 

513 
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" 

514 

515 
show "\<exists>g. M'.simple_function g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 

516 
proof (intro exI[of _ ?g] conjI ballI) 

517 
show "M'.simple_function ?g" using B by auto 

518 

519 
fix x assume "x \<in> space M" 

520 
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)" 

521 
unfolding indicator_def using B by auto 

522 
then show "f i x = ?g (Y x)" using `x \<in> space M` f[of i] 

523 
by (subst va.simple_function_indicator_representation) auto 

524 
qed 

525 
qed 

526 
from choice[OF this] guess g .. note g = this 

527 

528 
show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)" 

529 
proof (intro ballI bexI) 

530 
show "(SUP i. g i) \<in> borel_measurable M'" 

531 
using g by (auto intro: M'.borel_measurable_simple_function) 

532 
fix x assume "x \<in> space M" 

533 
have "Z x = (SUP i. f i) x" using `f \<up> Z` unfolding isoton_def by simp 

534 
also have "\<dots> = (SUP i. g i) (Y x)" unfolding SUPR_fun_expand 

535 
using g `x \<in> space M` by simp 

536 
finally show "Z x = (SUP i. g i) (Y x)" . 

537 
qed 

538 
qed 

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

539 

35582  540 
end 