author  hoelzl 
Mon, 23 May 2011 19:21:05 +0200  
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parent 42902  e8dbf90a2f3b 
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permissions  rwrr 
42148  1 
(* Title: HOL/Probability/Probability_Measure.thy 
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Author: Johannes Hölzl, TU München 
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Author: Armin Heller, TU München 

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*) 

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42148  6 
header {*Probability measure*} 
42067  7 

42148  8 
theory Probability_Measure 
42902  9 
imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure Lebesgue_Measure 
35582  10 
begin 
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12 
locale prob_space = measure_space + 

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assumes measure_space_1: "measure M (space M) = 1" 
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15 
sublocale prob_space < finite_measure 

16 
proof 

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from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp 
38656  18 
qed 
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40859  20 
abbreviation (in prob_space) "events \<equiv> sets M" 
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abbreviation (in prob_space) "prob \<equiv> \<mu>'" 
40859  22 
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'" 
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abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M" 
35582  24 

40859  25 
definition (in prob_space) 
35582  26 
"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B" 
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40859  28 
definition (in prob_space) 
35582  29 
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)" 
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40859  31 
definition (in prob_space) 
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"distribution X A = \<mu>' (X ` A \<inter> space M)" 
35582  33 

40859  34 
abbreviation (in prob_space) 
36624  35 
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" 
35582  36 

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declare (in finite_measure) positive_measure'[intro, simp] 
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39097  39 
lemma (in prob_space) distribution_cong: 
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assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" 

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shows "distribution X = distribution Y" 

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unfolding distribution_def fun_eq_iff 
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) 
39097  44 

45 
lemma (in prob_space) joint_distribution_cong: 

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assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" 

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assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" 

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shows "joint_distribution X Y = joint_distribution X' Y'" 

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unfolding distribution_def fun_eq_iff 
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) 
39097  51 

40859  52 
lemma (in prob_space) distribution_id[simp]: 
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"N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N" 
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by (auto simp: distribution_def intro!: arg_cong[where f=prob]) 
40859  55 

56 
lemma (in prob_space) prob_space: "prob (space M) = 1" 

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using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def) 
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lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1" 
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using bounded_measure[of A] by (simp add: prob_space) 
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lemma (in prob_space) distribution_positive[simp, intro]: 
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"0 \<le> distribution X A" unfolding distribution_def by auto 
35582  64 

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lemma (in prob_space) joint_distribution_remove[simp]: 
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"joint_distribution X X {(x, x)} = distribution X {x}" 
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unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
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lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1" 
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unfolding measure_space_1[symmetric] 
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using sets_into_space 
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by (intro measure_mono) auto 
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lemma (in prob_space) distribution_1: 
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"distribution X A \<le> 1" 
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unfolding distribution_def by simp 
35582  77 

40859  78 
lemma (in prob_space) prob_compl: 
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assumes A: "A \<in> events" 
38656  80 
shows "prob (space M  A) = 1  prob A" 
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using finite_measure_compl[OF A] by (simp add: prob_space) 
35582  82 

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lemma (in prob_space) indep_space: "s \<in> events \<Longrightarrow> indep (space M) s" 
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by (simp add: indep_def prob_space) 
35582  85 

40859  86 
lemma (in prob_space) prob_space_increasing: "increasing M prob" 
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by (auto intro!: finite_measure_mono simp: increasing_def) 
35582  88 

40859  89 
lemma (in prob_space) prob_zero_union: 
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assumes "s \<in> events" "t \<in> events" "prob t = 0" 
91 
shows "prob (s \<union> t) = prob s" 

38656  92 
using assms 
35582  93 
proof  
94 
have "prob (s \<union> t) \<le> prob s" 

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using finite_measure_subadditive[of s t] assms by auto 
35582  96 
moreover have "prob (s \<union> t) \<ge> prob s" 
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using assms by (blast intro: finite_measure_mono) 
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ultimately show ?thesis by simp 
99 
qed 

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40859  101 
lemma (in prob_space) prob_eq_compl: 
35582  102 
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" 

38656  105 
using assms prob_compl by auto 
35582  106 

40859  107 
lemma (in prob_space) prob_one_inter: 
35582  108 
assumes events:"s \<in> events" "t \<in> events" 
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assumes "prob t = 1" 

110 
shows "prob (s \<inter> t) = prob s" 

111 
proof  

38656  112 
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) 

114 
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]) 

35582  118 
qed 
119 

40859  120 
lemma (in prob_space) prob_eq_bigunion_image: 
35582  121 
assumes "range f \<subseteq> events" "range g \<subseteq> events" 
122 
assumes "disjoint_family f" "disjoint_family g" 

123 
assumes "\<And> n :: nat. prob (f n) = prob (g n)" 

124 
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" 

125 
using assms 

126 
proof  

38656  127 
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))" 
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by (rule finite_measure_UNION[OF assms(1,3)]) 
38656  129 
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" 
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by (rule finite_measure_UNION[OF assms(2,4)]) 
38656  131 
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp 
35582  132 
qed 
133 

40859  134 
lemma (in prob_space) prob_countably_zero: 
35582  135 
assumes "range c \<subseteq> events" 
136 
assumes "\<And> i. prob (c i) = 0" 

38656  137 
shows "prob (\<Union> i :: nat. c i) = 0" 
138 
proof (rule antisym) 

139 
show "prob (\<Union> i :: nat. c i) \<le> 0" 

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using finite_measure_countably_subadditive[OF assms(1)] 
38656  141 
by (simp add: assms(2) suminf_zero summable_zero) 
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qed simp 
35582  143 

40859  144 
lemma (in prob_space) indep_sym: 
35582  145 
"indep a b \<Longrightarrow> indep b a" 
146 
unfolding indep_def using Int_commute[of a b] by auto 

147 

40859  148 
lemma (in prob_space) indep_refl: 
35582  149 
assumes "a \<in> events" 
150 
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" 

151 
using assms unfolding indep_def by auto 

152 

40859  153 
lemma (in prob_space) prob_equiprobable_finite_unions: 
38656  154 
assumes "s \<in> events" 
155 
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" 

35582  156 
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" 
38656  157 
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" 
35582  158 
proof (cases "s = {}") 
38656  159 
case False hence "\<exists> x. x \<in> s" by blast 
35582  160 
from someI_ex[OF this] assms 
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have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast 

162 
have "prob s = (\<Sum> x \<in> s. prob {x})" 

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using finite_measure_finite_singleton[OF s_finite] by simp 
35582  164 
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto 
38656  165 
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" 
166 
using setsum_constant assms by (simp add: real_eq_of_nat) 

35582  167 
finally show ?thesis by simp 
38656  168 
qed simp 
35582  169 

40859  170 
lemma (in prob_space) prob_real_sum_image_fn: 
35582  171 
assumes "e \<in> events" 
172 
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" 

173 
assumes "finite s" 

38656  174 
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" 
175 
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" 

35582  176 
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" 
177 
proof  

38656  178 
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" 
179 
using `e \<in> events` sets_into_space upper by blast 

180 
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp 

181 
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" 

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proof (rule finite_measure_finite_Union) 
38656  183 
show "finite s" by fact 
184 
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact 

185 
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" 

186 
using disjoint by (auto simp: disjoint_family_on_def) 

187 
qed 

188 
finally show ?thesis . 

35582  189 
qed 
190 

42199  191 
lemma (in prob_space) prob_space_vimage: 
192 
assumes S: "sigma_algebra S" 

193 
assumes T: "T \<in> measure_preserving M S" 

194 
shows "prob_space S" 

35582  195 
proof  
42199  196 
interpret S: measure_space S 
197 
using S and T by (rule measure_space_vimage) 

38656  198 
show ?thesis 
42199  199 
proof 
200 
from T[THEN measure_preservingD2] 

201 
have "T ` space S \<inter> space M = space M" 

202 
by (auto simp: measurable_def) 

203 
with T[THEN measure_preservingD, of "space S", symmetric] 

204 
show "measure S (space S) = 1" 

205 
using measure_space_1 by simp 

35582  206 
qed 
207 
qed 

208 

42199  209 
lemma (in prob_space) distribution_prob_space: 
210 
assumes X: "random_variable S X" 

211 
shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" (is "prob_space ?S") 

212 
proof (rule prob_space_vimage) 

213 
show "X \<in> measure_preserving M ?S" 

214 
using X 

215 
unfolding measure_preserving_def distribution_def_raw 

216 
by (auto simp: finite_measure_eq measurable_sets) 

217 
show "sigma_algebra ?S" using X by simp 

218 
qed 

219 

40859  220 
lemma (in prob_space) AE_distribution: 
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assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x" 
40859  222 
shows "AE x. Q (X x)" 
223 
proof  

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interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space) 
40859  225 
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" 
226 
using assms unfolding X.almost_everywhere_def by auto 

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from X[unfolded measurable_def] N show "AE x. Q (X x)" 
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by (intro AE_I'[where N="X ` N \<inter> space M"]) 
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(auto simp: finite_measure_eq distribution_def measurable_sets) 
40859  230 
qed 
231 

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lemma (in prob_space) distribution_eq_integral: 
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"random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X ` A \<inter> space M))" 
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using finite_measure_eq[of "X ` A \<inter> space M"] 
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by (auto simp: measurable_sets distribution_def) 
35582  236 

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lemma (in prob_space) distribution_eq_translated_integral: 
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assumes "random_variable S X" "A \<in> sets S" 
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shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)" 
35582  240 
proof  
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interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
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using assms(1) by (rule distribution_prob_space) 
35582  243 
show ?thesis 
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using S.positive_integral_indicator(1)[of A] assms by simp 
35582  245 
qed 
246 

40859  247 
lemma (in prob_space) finite_expectation1: 
248 
assumes f: "finite (X`space M)" and rv: "random_variable borel X" 

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shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X ` {r} \<inter> space M))" (is "_ = ?r") 
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proof (subst integral_on_finite) 
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show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto 
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252 
show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X ` {r} \<inter> space M))) = ?r" 
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253 
"\<And>x. \<mu> (X ` {x} \<inter> space M) \<noteq> \<infinity>" 
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254 
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto 
38656  255 
qed 
35582  256 

40859  257 
lemma (in prob_space) finite_expectation: 
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258 
assumes "finite (X`space M)" "random_variable borel X" 
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259 
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" 
38656  260 
using assms unfolding distribution_def using finite_expectation1 by auto 
261 

40859  262 
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: 
35582  263 
assumes "{x} \<in> events" 
38656  264 
assumes "prob {x} = 1" 
35582  265 
assumes "{y} \<in> events" 
266 
assumes "y \<noteq> x" 

267 
shows "prob {y} = 0" 

268 
using prob_one_inter[of "{y}" "{x}"] assms by auto 

269 

40859  270 
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" 
38656  271 
unfolding distribution_def by simp 
272 

40859  273 
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" 
38656  274 
proof  
275 
have "X ` X ` space M \<inter> space M = space M" by auto 

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276 
thus ?thesis unfolding distribution_def by (simp add: prob_space) 
38656  277 
qed 
278 

40859  279 
lemma (in prob_space) distribution_one: 
280 
assumes "random_variable M' X" and "A \<in> sets M'" 

38656  281 
shows "distribution X A \<le> 1" 
282 
proof  

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283 
have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def 
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284 
using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono) 
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285 
thus ?thesis by (simp add: prob_space) 
38656  286 
qed 
287 

40859  288 
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: 
35582  289 
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" 
38656  290 
(is "random_variable ?S X") 
291 
assumes "distribution X {x} = 1" 

35582  292 
assumes "y \<noteq> x" 
293 
shows "distribution X {y} = 0" 

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294 
proof cases 
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295 
{ fix x have "X ` {x} \<inter> space M \<in> sets M" 
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296 
proof cases 
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297 
assume "x \<in> X`space M" with X show ?thesis 
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298 
by (auto simp: measurable_def image_iff) 
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299 
next 
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300 
assume "x \<notin> X`space M" then have "X ` {x} \<inter> space M = {}" by auto 
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301 
then show ?thesis by auto 
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302 
qed } note single = this 
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303 
have "X ` {x} \<inter> space M  X ` {y} \<inter> space M = X ` {x} \<inter> space M" 
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304 
"X ` {y} \<inter> space M \<inter> (X ` {x} \<inter> space M) = {}" 
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305 
using `y \<noteq> x` by auto 
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306 
with finite_measure_inter_full_set[OF single single, of x y] assms(2) 
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307 
show ?thesis by (auto simp: distribution_def prob_space) 
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308 
next 
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309 
assume "{y} \<notin> sets ?S" 
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310 
then have "X ` {y} \<inter> space M = {}" by auto 
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311 
thus "distribution X {y} = 0" unfolding distribution_def by auto 
35582  312 
qed 
313 

40859  314 
lemma (in prob_space) joint_distribution_Times_le_fst: 
315 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

316 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

317 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

318 
unfolding distribution_def 

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319 
proof (intro finite_measure_mono) 
40859  320 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
321 
show "X ` A \<inter> space M \<in> events" 

322 
using X A unfolding measurable_def by simp 

323 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 

324 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

325 
qed 

326 

327 
lemma (in prob_space) joint_distribution_commute: 

328 
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)" 

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329 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  330 

331 
lemma (in prob_space) joint_distribution_Times_le_snd: 

332 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

333 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

334 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

335 
using assms 

336 
by (subst joint_distribution_commute) 

337 
(simp add: swap_product joint_distribution_Times_le_fst) 

338 

339 
lemma (in prob_space) random_variable_pairI: 

340 
assumes "random_variable MX X" 

341 
assumes "random_variable MY Y" 

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342 
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  343 
proof 
344 
interpret MX: sigma_algebra MX using assms by simp 

345 
interpret MY: sigma_algebra MY using assms by simp 

346 
interpret P: pair_sigma_algebra MX MY by default 

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347 
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  348 
have sa: "sigma_algebra M" by default 
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349 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  350 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  351 
qed 
352 

353 
lemma (in prob_space) joint_distribution_commute_singleton: 

354 
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}" 

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355 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  356 

357 
lemma (in prob_space) joint_distribution_assoc_singleton: 

358 
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 

359 
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}" 

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360 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  361 

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362 
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 
40859  363 

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364 
sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default 
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365 

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366 
sublocale pair_prob_space \<subseteq> P: prob_space P 
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367 
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) 
40859  368 

369 
lemma countably_additiveI[case_names countably]: 

370 
assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow> 

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371 
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" 
40859  372 
shows "countably_additive M \<mu>" 
373 
using assms unfolding countably_additive_def by auto 

374 

375 
lemma (in prob_space) joint_distribution_prob_space: 

376 
assumes "random_variable MX X" "random_variable MY Y" 

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377 
shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" 
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378 
using random_variable_pairI[OF assms] by (rule distribution_prob_space) 
40859  379 

380 
section "Probability spaces on finite sets" 

35582  381 

35977  382 
locale finite_prob_space = prob_space + finite_measure_space 
383 

40859  384 
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" 
385 

386 
lemma (in prob_space) finite_random_variableD: 

387 
assumes "finite_random_variable M' X" shows "random_variable M' X" 

388 
proof  

389 
interpret M': finite_sigma_algebra M' using assms by simp 

390 
then show "random_variable M' X" using assms by simp default 

391 
qed 

392 

393 
lemma (in prob_space) distribution_finite_prob_space: 

394 
assumes "finite_random_variable MX X" 

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395 
shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)" 
40859  396 
proof  
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397 
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
40859  398 
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) 
399 
interpret MX: finite_sigma_algebra MX 

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400 
using assms by auto 
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401 
show ?thesis by default (simp_all add: MX.finite_space) 
40859  402 
qed 
403 

404 
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]: 

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405 
assumes "simple_function M X" 
3e39b0e730d6
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406 
shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X" 
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407 
(is "finite_random_variable ?X _") 
40859  408 
proof (intro conjI) 
409 
have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp 

41689
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changeset

410 
interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow) 
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411 
show "finite_sigma_algebra ?X" 
40859  412 
by default auto 
41689
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hoelzl
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413 
show "X \<in> measurable M ?X" 
40859  414 
proof (unfold measurable_def, clarsimp) 
415 
fix A assume A: "A \<subseteq> X`space M" 

416 
then have "finite A" by (rule finite_subset) simp 

417 
then have "X ` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" 

418 
unfolding vimage_UN UN_extend_simps 

419 
apply (rule finite_UN) 

420 
using A assms unfolding simple_function_def by auto 

421 
then show "X ` A \<inter> space M \<in> events" by simp 

422 
qed 

423 
qed 

424 

425 
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]: 

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426 
assumes "simple_function M X" 
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diff
changeset

427 
shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X" 
3e39b0e730d6
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changeset

428 
using simple_function_imp_finite_random_variable[OF assms, of ext] 
40859  429 
by (auto dest!: finite_random_variableD) 
430 

431 
lemma (in prob_space) sum_over_space_real_distribution: 

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432 
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" 
40859  433 
unfolding distribution_def prob_space[symmetric] 
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434 
by (subst finite_measure_finite_Union[symmetric]) 
40859  435 
(auto simp add: disjoint_family_on_def simple_function_def 
436 
intro!: arg_cong[where f=prob]) 

437 

438 
lemma (in prob_space) finite_random_variable_pairI: 

439 
assumes "finite_random_variable MX X" 

440 
assumes "finite_random_variable MY Y" 

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hoelzl
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diff
changeset

441 
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  442 
proof 
443 
interpret MX: finite_sigma_algebra MX using assms by simp 

444 
interpret MY: finite_sigma_algebra MY using assms by simp 

445 
interpret P: pair_finite_sigma_algebra MX MY by default 

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446 
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  447 
have sa: "sigma_algebra M" by default 
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changeset

448 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  449 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  450 
qed 
451 

452 
lemma (in prob_space) finite_random_variable_imp_sets: 

453 
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" 

454 
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp 

455 

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456 
lemma (in prob_space) finite_random_variable_measurable: 
40859  457 
assumes X: "finite_random_variable MX X" shows "X ` A \<inter> space M \<in> events" 
458 
proof  

459 
interpret X: finite_sigma_algebra MX using X by simp 

460 
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X ` A \<inter> space M \<in> events" and 

461 
"X \<in> space M \<rightarrow> space MX" 

462 
by (auto simp: measurable_def) 

463 
then have *: "X ` A \<inter> space M = X ` (A \<inter> space MX) \<inter> space M" 

464 
by auto 

465 
show "X ` A \<inter> space M \<in> events" 

466 
unfolding * by (intro vimage) auto 

467 
qed 

468 

469 
lemma (in prob_space) joint_distribution_finite_Times_le_fst: 

470 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

471 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

472 
unfolding distribution_def 

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473 
proof (intro finite_measure_mono) 
40859  474 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
475 
show "X ` A \<inter> space M \<in> events" 

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476 
using finite_random_variable_measurable[OF X] . 
40859  477 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 
478 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

479 
qed 

480 

481 
lemma (in prob_space) joint_distribution_finite_Times_le_snd: 

482 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

483 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

484 
using assms 

485 
by (subst joint_distribution_commute) 

486 
(simp add: swap_product joint_distribution_finite_Times_le_fst) 

487 

488 
lemma (in prob_space) finite_distribution_order: 

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hoelzl
parents:
41831
diff
changeset

489 
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" 
40859  490 
assumes "finite_random_variable MX X" "finite_random_variable MY Y" 
491 
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

492 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

493 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

494 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

495 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

496 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

497 
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] 

498 
using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"] 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

499 
by (auto intro: antisym) 
40859  500 

501 
lemma (in prob_space) setsum_joint_distribution: 

502 
assumes X: "finite_random_variable MX X" 

503 
assumes Y: "random_variable MY Y" "B \<in> sets MY" 

504 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" 

505 
unfolding distribution_def 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

506 
proof (subst finite_measure_finite_Union[symmetric]) 
40859  507 
interpret MX: finite_sigma_algebra MX using X by auto 
508 
show "finite (space MX)" using MX.finite_space . 

509 
let "?d i" = "(\<lambda>x. (X x, Y x)) ` ({i} \<times> B) \<inter> space M" 

510 
{ fix i assume "i \<in> space MX" 

511 
moreover have "?d i = (X ` {i} \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

512 
ultimately show "?d i \<in> events" 

513 
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y 

514 
using MX.sets_eq_Pow by auto } 

515 
show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

516 
show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y ` B \<inter> space M)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

517 
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) 
40859  518 
qed 
519 

520 
lemma (in prob_space) setsum_joint_distribution_singleton: 

521 
assumes X: "finite_random_variable MX X" 

522 
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" 

523 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" 

524 
using setsum_joint_distribution[OF X 

525 
finite_random_variableD[OF Y(1)] 

526 
finite_random_variable_imp_sets[OF Y]] by simp 

527 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

528 
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 
40859  529 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

530 
sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

531 
sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2 by default 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

532 
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default 
40859  533 

42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

534 
locale product_finite_prob_space = 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

535 
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

536 
and I :: "'i set" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

537 
assumes finite_space: "\<And>i. finite_prob_space (M i)" and finite_index: "finite I" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

538 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

539 
sublocale product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

540 
sublocale product_finite_prob_space \<subseteq> finite_product_sigma_finite M I by default (rule finite_index) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

541 
sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

542 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

543 
show "\<mu> (space P) = 1" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

544 
using measure_times[OF M.top] M.measure_space_1 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

545 
by (simp add: setprod_1 space_product_algebra) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

546 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

547 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

548 
lemma funset_eq_UN_fun_upd_I: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

549 
assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

550 
and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

551 
and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

552 
shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

553 
proof safe 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

554 
fix f assume f: "f \<in> F (insert a A)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

555 
show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

556 
proof (rule UN_I[of "f(a := d)"]) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

557 
show "f(a := d) \<in> F A" using *[OF f] . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

558 
show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

559 
proof (rule image_eqI[of _ _ "f a"]) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

560 
show "f a \<in> G (f(a := d))" using **[OF f] . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

561 
qed simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

562 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

563 
next 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

564 
fix f x assume "f \<in> F A" "x \<in> G f" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

565 
from ***[OF this] show "f(a := x) \<in> F (insert a A)" . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

566 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

567 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

568 
lemma extensional_funcset_insert_eq[simp]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

569 
assumes "a \<notin> A" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

570 
shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

571 
apply (rule funset_eq_UN_fun_upd_I) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

572 
using assms 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

573 
by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

574 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

575 
lemma finite_extensional_funcset[simp, intro]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

576 
assumes "finite A" "finite B" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

577 
shows "finite (extensional A \<inter> (A \<rightarrow> B))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

578 
using assms by induct (auto simp: extensional_funcset_insert_eq) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

579 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

580 
lemma finite_PiE[simp, intro]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

581 
assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

582 
shows "finite (Pi\<^isub>E A B)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

583 
proof  
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

584 
have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

585 
show ?thesis 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

586 
using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

587 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

588 

42892  589 
lemma (in product_finite_prob_space) singleton_eq_product: 
590 
assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})" 

591 
proof (safe intro!: ext[of _ x]) 

592 
fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I" 

593 
with x show "y i = x i" 

594 
by (cases "i \<in> I") (auto simp: extensional_def) 

595 
qed (insert x, auto) 

596 

42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

597 
sublocale product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

598 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

599 
show "finite (space P)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

600 
using finite_index M.finite_space by auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

601 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

602 
{ fix x assume "x \<in> space P" 
42892  603 
with this[THEN singleton_eq_product] 
604 
have "{x} \<in> sets P" 

42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

605 
by (auto intro!: in_P) } 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

606 
note x_in_P = this 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

607 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

608 
have "Pow (space P) \<subseteq> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

609 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

610 
fix X assume "X \<in> Pow (space P)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

611 
moreover then have "finite X" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

612 
using `finite (space P)` by (blast intro: finite_subset) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

613 
ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

614 
by (intro finite_UN x_in_P) auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

615 
then show "X \<in> sets P" by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

616 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

617 
with space_closed show [simp]: "sets P = Pow (space P)" .. 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

618 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

619 
{ fix x assume "x \<in> space P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

620 
from this top have "\<mu> {x} \<le> \<mu> (space P)" by (intro measure_mono) auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

621 
then show "\<mu> {x} \<noteq> \<infinity>" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

622 
using measure_space_1 by auto } 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

623 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

624 

d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

625 
lemma (in product_finite_prob_space) measure_finite_times: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

626 
"(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

627 
by (rule measure_times) simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

628 

42892  629 
lemma (in product_finite_prob_space) measure_singleton_times: 
630 
assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})" 

631 
unfolding singleton_eq_product[OF x] using x 

632 
by (intro measure_finite_times) auto 

633 

634 
lemma (in product_finite_prob_space) prob_finite_times: 

42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

635 
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

636 
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

637 
proof  
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

638 
have "extreal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

639 
using X by (intro finite_measure_eq[symmetric] in_P) auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

640 
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

641 
using measure_finite_times X by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

642 
also have "\<dots> = extreal (\<Prod>i\<in>I. M.\<mu>' i (X i))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

643 
using X by (simp add: M.finite_measure_eq setprod_extreal) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

644 
finally show ?thesis by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

645 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

646 

42892  647 
lemma (in product_finite_prob_space) prob_singleton_times: 
648 
assumes x: "x \<in> space P" 

649 
shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})" 

650 
unfolding singleton_eq_product[OF x] using x 

651 
by (intro prob_finite_times) auto 

652 

653 
lemma (in product_finite_prob_space) prob_finite_product: 

654 
"A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})" 

655 
by (auto simp add: finite_measure_singleton prob_singleton_times 

656 
simp del: space_product_algebra 

657 
intro!: setsum_cong prob_singleton_times) 

658 

40859  659 
lemma (in prob_space) joint_distribution_finite_prob_space: 
660 
assumes X: "finite_random_variable MX X" 

661 
assumes Y: "finite_random_variable MY Y" 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

662 
shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

663 
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) 
40859  664 

36624  665 
lemma finite_prob_space_eq: 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

666 
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" 
36624  667 
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def 
668 
by auto 

669 

670 
lemma (in prob_space) not_empty: "space M \<noteq> {}" 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

671 
using prob_space empty_measure' by auto 
36624  672 

38656  673 
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" 
674 
using measure_space_1 sum_over_space by simp 

36624  675 

676 
lemma (in finite_prob_space) joint_distribution_restriction_fst: 

677 
"joint_distribution X Y A \<le> distribution X (fst ` A)" 

678 
unfolding distribution_def 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

679 
proof (safe intro!: finite_measure_mono) 
36624  680 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
681 
show "x \<in> X ` fst ` A" 

682 
by (auto intro!: image_eqI[OF _ *]) 

683 
qed (simp_all add: sets_eq_Pow) 

684 

685 
lemma (in finite_prob_space) joint_distribution_restriction_snd: 

686 
"joint_distribution X Y A \<le> distribution Y (snd ` A)" 

687 
unfolding distribution_def 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
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diff
changeset

688 
proof (safe intro!: finite_measure_mono) 
36624  689 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
690 
show "x \<in> Y ` snd ` A" 

691 
by (auto intro!: image_eqI[OF _ *]) 

692 
qed (simp_all add: sets_eq_Pow) 

693 

694 
lemma (in finite_prob_space) distribution_order: 

695 
shows "0 \<le> distribution X x'" 

696 
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" 

697 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

698 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

699 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

700 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

701 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

702 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

703 
using 
36624  704 
joint_distribution_restriction_fst[of X Y "{(x, y)}"] 
705 
joint_distribution_restriction_snd[of X Y "{(x, y)}"] 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

706 
by (auto intro: antisym) 
36624  707 

39097  708 
lemma (in finite_prob_space) distribution_mono: 
709 
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

710 
shows "distribution X x \<le> distribution Y y" 

711 
unfolding distribution_def 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

712 
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) 
39097  713 

714 
lemma (in finite_prob_space) distribution_mono_gt_0: 

715 
assumes gt_0: "0 < distribution X x" 

716 
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

717 
shows "0 < distribution Y y" 

718 
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) 

719 

720 
lemma (in finite_prob_space) sum_over_space_distrib: 

721 
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

722 
unfolding distribution_def prob_space[symmetric] using finite_space 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

723 
by (subst finite_measure_finite_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

724 
(auto simp add: disjoint_family_on_def sets_eq_Pow 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

725 
intro!: arg_cong[where f=\<mu>']) 
39097  726 

727 
lemma (in finite_prob_space) finite_sum_over_space_eq_1: 

41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

728 
"(\<Sum>x\<in>space M. prob {x}) = 1" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

729 
using prob_space finite_space 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

730 
by (subst (asm) finite_measure_finite_singleton) auto 
39097  731 

732 
lemma (in prob_space) distribution_remove_const: 

733 
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" 

734 
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" 

735 
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" 

736 
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" 

737 
and "distribution (\<lambda>x. ()) {()} = 1" 

41981
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hoelzl
parents:
41831
diff
changeset

738 
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) 
35977  739 

39097  740 
lemma (in finite_prob_space) setsum_distribution_gen: 
741 
assumes "Z ` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y ` {f x}) \<inter> space M" 

742 
and "inj_on f (X`space M)" 

743 
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" 

744 
unfolding distribution_def assms 

745 
using finite_space assms 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

746 
by (subst finite_measure_finite_Union[symmetric]) 
39097  747 
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def 
748 
intro!: arg_cong[where f=prob]) 

749 

750 
lemma (in finite_prob_space) setsum_distribution: 

751 
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" 

752 
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" 

753 
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" 

754 
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" 

755 
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" 

756 
by (auto intro!: inj_onI setsum_distribution_gen) 

757 

758 
lemma (in finite_prob_space) uniform_prob: 

759 
assumes "x \<in> space M" 

760 
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

761 
shows "prob {x} = 1 / card (space M)" 
39097  762 
proof  
763 
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" 

764 
using assms(2)[OF _ `x \<in> space M`] by blast 

765 
have "1 = prob (space M)" 

766 
using prob_space by auto 

767 
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

768 
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] 
39097  769 
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] 
770 
finite_space unfolding disjoint_family_on_def prob_space[symmetric] 

771 
by (auto simp add:setsum_restrict_set) 

772 
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" 

773 
using prob_x by auto 

774 
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp 

775 
finally have one: "1 = real (card (space M)) * prob {x}" 

776 
using real_eq_of_nat by auto 

777 
hence two: "real (card (space M)) \<noteq> 0" by fastsimp 

778 
from one have three: "prob {x} \<noteq> 0" by fastsimp 

779 
thus ?thesis using one two three divide_cancel_right 

780 
by (auto simp:field_simps) 

39092  781 
qed 
35977  782 

39092  783 
lemma (in prob_space) prob_space_subalgebra: 
41545  784 
assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

785 
and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

786 
shows "prob_space N" 
39092  787 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

788 
interpret N: measure_space N 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

789 
by (rule measure_space_subalgebra[OF assms]) 
39092  790 
show ?thesis 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

791 
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) 
35977  792 
qed 
793 

39092  794 
lemma (in prob_space) prob_space_of_restricted_space: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

795 
assumes "\<mu> A \<noteq> 0" "A \<in> sets M" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

796 
shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

797 
(is "prob_space ?P") 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

798 
proof  
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

799 
interpret A: measure_space "restricted_space A" 
39092  800 
using `A \<in> sets M` by (rule restricted_measure_space) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

801 
interpret A': sigma_algebra ?P 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

802 
by (rule A.sigma_algebra_cong) auto 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

803 
show "prob_space ?P" 
39092  804 
proof 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

805 
show "measure ?P (space ?P) = 1" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

806 
using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

807 
show "positive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

808 
proof (simp add: positive_def, safe) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

809 
show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

810 
fix B assume "B \<in> events" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

811 
with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M` 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

812 
show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

813 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

814 
show "countably_additive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

815 
proof (simp add: countably_additive_def, safe) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

816 
fix B and F :: "nat \<Rightarrow> 'a set" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

817 
assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

818 
{ fix i 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

819 
from F have "F i \<in> op \<inter> A ` events" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

820 
with `A \<in> events` have "F i \<in> events" by auto } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

821 
moreover then have "range F \<subseteq> events" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

822 
moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

823 
by (simp add: mult_commute divide_extreal_def) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

824 
moreover have "0 \<le> inverse (\<mu> A)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

825 
using real_measure[OF `A \<in> events`] by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

826 
ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

827 
using measure_countably_additive[of F] F 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

828 
by (auto simp: suminf_cmult_extreal) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

829 
qed 
39092  830 
qed 
831 
qed 

832 

833 
lemma finite_prob_spaceI: 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

834 
assumes "finite (space M)" "sets M = Pow(space M)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

835 
and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

836 
and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

837 
shows "finite_prob_space M" 
39092  838 
unfolding finite_prob_space_eq 
839 
proof 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

840 
show "finite_measure_space M" using assms 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

841 
by (auto intro!: finite_measure_spaceI) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

842 
show "measure M (space M) = 1" by fact 
39092  843 
qed 
36624  844 

845 
lemma (in finite_prob_space) finite_measure_space: 

39097  846 
fixes X :: "'a \<Rightarrow> 'x" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

847 
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

848 
(is "finite_measure_space ?S") 
39092  849 
proof (rule finite_measure_spaceI, simp_all) 
36624  850 
show "finite (X ` space M)" using finite_space by simp 
39097  851 
next 
852 
fix A B :: "'x set" assume "A \<inter> B = {}" 

853 
then show "distribution X (A \<union> B) = distribution X A + distribution X B" 

854 
unfolding distribution_def 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

855 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

856 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
36624  857 
qed 
858 

39097  859 
lemma (in finite_prob_space) finite_prob_space_of_images: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

860 
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X \<rparr>" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

861 
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def) 
39097  862 

39096  863 
lemma (in finite_prob_space) finite_product_measure_space: 
39097  864 
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" 
39096  865 
assumes "finite s1" "finite s2" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

866 
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

867 
(is "finite_measure_space ?M") 
39097  868 
proof (rule finite_measure_spaceI, simp_all) 
869 
show "finite (s1 \<times> s2)" 

39096  870 
using assms by auto 
39097  871 
next 
872 
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" 

873 
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" 

874 
unfolding distribution_def 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

875 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

876 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
39096  877 
qed 
878 

39097  879 
lemma (in finite_prob_space) finite_product_measure_space_of_images: 
39096  880 
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M, 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

881 
sets = Pow (X ` space M \<times> Y ` space M), 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

882 
measure = extreal \<circ> joint_distribution X Y \<rparr>" 
39096  883 
using finite_space by (auto intro!: finite_product_measure_space) 
884 

40859  885 
lemma (in finite_prob_space) finite_product_prob_space_of_images: 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

886 
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M), 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

887 
measure = extreal \<circ> joint_distribution X Y \<rparr>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

888 
(is "finite_prob_space ?S") 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

889 
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def) 
40859  890 
have "X ` X ` space M \<inter> Y ` Y ` space M \<inter> space M = space M" by auto 
891 
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" 

892 
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) 

893 
qed 

894 

39085  895 
section "Conditional Expectation and Probability" 
896 

897 
lemma (in prob_space) conditional_expectation_exists: 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

898 
fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

899 
assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

900 
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

901 
shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N. 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

902 
(\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))" 
39083  903 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

904 
note N(4)[simp] 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

905 
interpret P: prob_space N 
41545  906 
using prob_space_subalgebra[OF N] . 
39083  907 

908 
let "?f A" = "\<lambda>x. X x * indicator A x" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

909 
let "?Q A" = "integral\<^isup>P M (?f A)" 
39083  910 

911 
from measure_space_density[OF borel] 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

912 
have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

913 
apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"]) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

914 
using N by (auto intro!: P.sigma_algebra_cong) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

915 
then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" . 
39083  916 

917 
have "P.absolutely_continuous ?Q" 

918 
unfolding P.absolutely_continuous_def 

41545  919 
proof safe 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

920 
fix A assume "A \<in> sets N" "P.\<mu> A = 0" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

921 
then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

922 
using borel N by (auto intro!: borel_measurable_indicator AE_not_in) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

923 
then show "?Q A = 0" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

924 
by (auto simp add: positive_integral_0_iff_AE) 
39083  925 
qed 
926 
from P.Radon_Nikodym[OF Q this] 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

927 
obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

928 
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)" 
39083  929 
by blast 
41545  930 
with N(2) show ?thesis 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

931 
by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)]) 
39083  932 
qed 
933 

39085  934 
definition (in prob_space) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

935 
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

936 
\<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))" 
39085  937 

938 
abbreviation (in prob_space) 

39092  939 
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" 
39085  940 

941 
lemma (in prob_space) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

942 
fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

943 
assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

944 
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" 
39085  945 
shows borel_measurable_conditional_expectation: 
41545  946 
"conditional_expectation N X \<in> borel_measurable N" 
947 
and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow> 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

948 
(\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) = 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

949 
(\<integral>\<^isup>+x. X x * indicator C x \<partial>M)" 
41545  950 
(is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C") 
39085  951 
proof  
952 
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] 

41545  953 
then show "conditional_expectation N X \<in> borel_measurable N" 
39085  954 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
955 

41545  956 
from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C" 
39085  957 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
958 
qed 

959 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

960 
lemma (in sigma_algebra) factorize_measurable_function_pos: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

961 
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" 
39091  962 
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

963 
assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

964 
shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

965 
proof  
39091  966 
interpret M': sigma_algebra M' by fact 
967 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 

968 
from M'.sigma_algebra_vimage[OF this] 

969 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 

970 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

971 
from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this 
39091  972 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

973 
have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 
39091  974 
proof 
975 
fix i 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

976 
from f(1)[of i] have "finite (f i`space M)" and B_ex: 
39091  977 
"\<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" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

978 
unfolding simple_function_def by auto 
39091  979 
from B_ex[THEN bchoice] guess B .. note B = this 
980 

981 
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" 

982 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

983 
show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 
39091  984 
proof (intro exI[of _ ?g] conjI ballI) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

985 
show "simple_function M' ?g" using B by auto 
39091  986 

987 
fix x assume "x \<in> space M" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

988 
then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i ` {z} \<inter> space M) x::extreal)" 
39091  989 
unfolding indicator_def using B by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

990 
then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i] 
39091  991 
by (subst va.simple_function_indicator_representation) auto 
992 
qed 

993 
qed 

994 
from choice[OF this] guess g .. note g = this 

995 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

996 
show ?thesis 
39091  997 
proof (intro ballI bexI) 
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

998 
show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'" 
39091  999 
using g by (auto intro: M'.borel_measurable_simple_function) 
1000 
fix x assume "x \<in> space M" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1001 
have "max 0 (Z x) = (SUP i. f i x)" using f by simp 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1002 
also have "\<dots> = (SUP i. g i (Y x))" 
39091  1003 
using g `x \<in> space M` by simp 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1004 
finally show "max 0 (Z x) = (SUP i. g i (Y x))" . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1005 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1006 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1007 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1008 
lemma (in sigma_algebra) factorize_measurable_function: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1009 
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1010 
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1011 
shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1012 
\<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1013 
proof safe 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1014 
interpret M': sigma_algebra M' by fact 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1015 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1016 
from M'.sigma_algebra_vimage[OF this] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1017 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1018 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1019 
{ fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1020 
with M'.measurable_vimage_algebra[OF Y] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1021 
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1022 
by (rule measurable_comp) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1023 
moreover assume "\<forall>x\<in>space M. Z x = g (Y x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1024 
then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1025 
g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1026 
by (auto intro!: measurable_cong) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1027 
ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1028 
by simp } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1029 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1030 
assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1031 
with assms have "(\<lambda>x.  Z x) \<in> borel_measurable M" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1032 
"(\<lambda>x.  Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
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1033 
by auto 
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1034 
from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this 
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1035 
from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this 
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reworked Probability theory: measures are not type restricted to positive extended reals
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1036 
let "?g x" = "p x  n x" 
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reworked Probability theory: measures are not type restricted to positive extended reals
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1037 
show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)" 
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1038 
proof (intro bexI ballI) 
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changeset

1039 
show "?g \<in> borel_measurable M'" using p n by auto 
cdf7693bbe08
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diff
changeset

1040 
fix x assume "x \<in> space M" 
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diff
changeset

1041 
then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 ( Z x)" 
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1042 
using p n by auto 
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1043 
then show "Z x = ?g (Y x)" 
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1044 
by (auto split: split_max) 
39091  1045 
qed 
1046 
qed 

39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
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diff
changeset

1047 

42902  1048 
subsection "Borel Measure on {0 .. 1}" 
1049 

1050 
definition pborel :: "real measure_space" where 

1051 
"pborel = lborel.restricted_space {0 .. 1}" 

1052 

1053 
lemma space_pborel[simp]: 

1054 
"space pborel = {0 .. 1}" 

1055 
unfolding pborel_def by auto 

1056 

1057 
lemma sets_pborel: 

1058 
"A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 .. 1}" 

1059 
unfolding pborel_def by auto 

1060 

1061 
lemma in_pborel[intro, simp]: 

1062 
"A \<subseteq> {0 .. 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel" 

1063 
unfolding pborel_def by auto 

1064 

1065 
interpretation pborel: measure_space pborel 

1066 
using lborel.restricted_measure_space[of "{0 .. 1}"] 

1067 
by (simp add: pborel_def) 

1068 

1069 
interpretation pborel: prob_space pborel 

1070 
by default (simp add: one_extreal_def pborel_def) 

1071 

1072 
lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 .. 1} then real (lborel.\<mu> A) else 0)" 

1073 
unfolding pborel.\<mu>'_def by (auto simp: pborel_def) 

1074 

1075 
lemma pborel_singelton[simp]: "pborel.prob {a} = 0" 

1076 
by (auto simp: pborel_prob) 

1077 

1078 
lemma 

1079 
shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b  a else 0)" 

1080 
and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b  a else 0)" 

1081 
and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b  a else 0)" 

1082 
and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b  a else 0)" 

1083 
unfolding pborel_prob by (auto simp: atLeastLessThan_subseteq_atLeastAtMost_iff 

1084 
greaterThanAtMost_subseteq_atLeastAtMost_iff greaterThanLessThan_subseteq_atLeastAtMost_iff) 

1085 

1086 
lemma pborel_lebesgue_measure: 

1087 
"A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)" 

1088 
by (simp add: sets_pborel pborel_prob) 

1089 

1090 
lemma pborel_alt: 

1091 
"pborel = sigma \<lparr> 

1092 
space = {0..1}, 

1093 
sets = range (\<lambda>(x,y). {x..y} \<inter> {0..1}), 

1094 
measure = measure lborel \<rparr>" (is "_ = ?R") 

1095 
proof  

1096 
have *: "{0..1::real} \<in> sets borel" by auto 

1097 
have **: "op \<inter> {0..1::real} ` range (\<lambda>(x, y). {x..y}) = range (\<lambda>(x,y). {x..y} \<inter> {0..1})" 

1098 
unfolding image_image by (intro arg_cong[where f=range]) auto 

1099 
have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a .. b :: real}), 

1100 
measure = measure pborel\<rparr>) {0 .. 1}" 

1101 
by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastAtMost lborel_def) 

1102 
also have "\<dots> = ?R" 

1103 
by (subst restricted_sigma) 

1104 
(simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"]) 

1105 
finally show ?thesis . 

1106 
qed 

1107 

42860  1108 
subsection "Bernoulli space" 
1109 

1110 
definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV, 

1111 
measure = extreal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1  min 1 (max 0 p)) \<rparr>" 

1112 

1113 
interpretation bernoulli: finite_prob_space "bernoulli_space p" for p 

1114 
by (rule finite_prob_spaceI) 

1115 
(auto simp: bernoulli_space_def UNIV_bool one_extreal_def setsum_Un_disjoint intro!: setsum_nonneg) 

1116 

1117 
lemma bernoulli_measure: 

1118 
"0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1  p)" 

1119 
unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong) 

1120 

1121 
lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p" 

1122 
and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1  p" 

1123 
unfolding bernoulli_measure by simp_all 

1124 

35582  1125 
end 