author  hoelzl 
Thu, 19 May 2011 19:57:59 +0200  
changeset 42859  d9dfc733f25c 
parent 42858  348fa5df7d3f 
child 42860  b02349e70d5a 
permissions  rwrr 
42148  1 
(* Title: HOL/Probability/Probability_Measure.thy 
42067  2 
Author: Johannes Hölzl, TU München 
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Author: Armin Heller, TU München 

4 
*) 

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42148  6 
header {*Probability measure*} 
42067  7 

42148  8 
theory Probability_Measure 
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split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure 
35582  10 
begin 
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lemma real_of_extreal_inverse[simp]: 
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fixes X :: extreal 
40859  14 
shows "real (inverse X) = 1 / real X" 
15 
by (cases X) (auto simp: inverse_eq_divide) 

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lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)" 
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by (cases X) auto 
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lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>" 
40859  21 
by (cases X) auto 
22 

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lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)" 
40859  24 
by (cases X) auto 
25 

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lemma real_of_extreal_le_1: fixes X :: extreal shows "X \<le> 1 \<Longrightarrow> real X \<le> 1" 
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by (cases X) (auto simp: one_extreal_def) 
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35582  29 
locale prob_space = measure_space + 
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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assumes measure_space_1: "measure M (space M) = 1" 
38656  31 

32 
sublocale prob_space < finite_measure 

33 
proof 

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from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp 
38656  35 
qed 
36 

40859  37 
abbreviation (in prob_space) "events \<equiv> sets M" 
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abbreviation (in prob_space) "prob \<equiv> \<mu>'" 
40859  39 
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  41 

40859  42 
definition (in prob_space) 
35582  43 
"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B" 
44 

40859  45 
definition (in prob_space) 
35582  46 
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)" 
47 

40859  48 
definition (in prob_space) 
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"distribution X A = \<mu>' (X ` A \<inter> space M)" 
35582  50 

40859  51 
abbreviation (in prob_space) 
36624  52 
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" 
35582  53 

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declare (in finite_measure) positive_measure'[intro, simp] 
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39097  56 
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  61 

62 
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  68 

40859  69 
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  72 

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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  81 

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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) distribution_1: 
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"distribution X A \<le> 1" 
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unfolding distribution_def by simp 
35582  89 

40859  90 
lemma (in prob_space) prob_compl: 
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assumes A: "A \<in> events" 
38656  92 
shows "prob (space M  A) = 1  prob A" 
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using finite_measure_compl[OF A] by (simp add: prob_space) 
35582  94 

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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  97 

40859  98 
lemma (in prob_space) prob_space_increasing: "increasing M prob" 
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by (auto intro!: finite_measure_mono simp: increasing_def) 
35582  100 

40859  101 
lemma (in prob_space) prob_zero_union: 
35582  102 
assumes "s \<in> events" "t \<in> events" "prob t = 0" 
103 
shows "prob (s \<union> t) = prob s" 

38656  104 
using assms 
35582  105 
proof  
106 
have "prob (s \<union> t) \<le> prob s" 

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using finite_measure_subadditive[of s t] assms by auto 
35582  108 
moreover have "prob (s \<union> t) \<ge> prob s" 
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using assms by (blast intro: finite_measure_mono) 
35582  110 
ultimately show ?thesis by simp 
111 
qed 

112 

40859  113 
lemma (in prob_space) prob_eq_compl: 
35582  114 
assumes "s \<in> events" "t \<in> events" 
115 
assumes "prob (space M  s) = prob (space M  t)" 

116 
shows "prob s = prob t" 

38656  117 
using assms prob_compl by auto 
35582  118 

40859  119 
lemma (in prob_space) prob_one_inter: 
35582  120 
assumes events:"s \<in> events" "t \<in> events" 
121 
assumes "prob t = 1" 

122 
shows "prob (s \<inter> t) = prob s" 

123 
proof  

38656  124 
have "prob ((space M  s) \<union> (space M  t)) = prob (space M  s)" 
125 
using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union) 

126 
also have "(space M  s) \<union> (space M  t) = space M  (s \<inter> t)" 

127 
by blast 

128 
finally show "prob (s \<inter> t) = prob s" 

129 
using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s]) 

35582  130 
qed 
131 

40859  132 
lemma (in prob_space) prob_eq_bigunion_image: 
35582  133 
assumes "range f \<subseteq> events" "range g \<subseteq> events" 
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assumes "disjoint_family f" "disjoint_family g" 

135 
assumes "\<And> n :: nat. prob (f n) = prob (g n)" 

136 
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" 

137 
using assms 

138 
proof  

38656  139 
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  141 
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  143 
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp 
35582  144 
qed 
145 

40859  146 
lemma (in prob_space) prob_countably_zero: 
35582  147 
assumes "range c \<subseteq> events" 
148 
assumes "\<And> i. prob (c i) = 0" 

38656  149 
shows "prob (\<Union> i :: nat. c i) = 0" 
150 
proof (rule antisym) 

151 
show "prob (\<Union> i :: nat. c i) \<le> 0" 

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using finite_measure_countably_subadditive[OF assms(1)] 
38656  153 
by (simp add: assms(2) suminf_zero summable_zero) 
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qed simp 
35582  155 

40859  156 
lemma (in prob_space) indep_sym: 
35582  157 
"indep a b \<Longrightarrow> indep b a" 
158 
unfolding indep_def using Int_commute[of a b] by auto 

159 

40859  160 
lemma (in prob_space) indep_refl: 
35582  161 
assumes "a \<in> events" 
162 
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" 

163 
using assms unfolding indep_def by auto 

164 

40859  165 
lemma (in prob_space) prob_equiprobable_finite_unions: 
38656  166 
assumes "s \<in> events" 
167 
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" 

35582  168 
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" 
38656  169 
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" 
35582  170 
proof (cases "s = {}") 
38656  171 
case False hence "\<exists> x. x \<in> s" by blast 
35582  172 
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 

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

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using finite_measure_finite_singleton[OF s_finite] by simp 
35582  176 
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto 
38656  177 
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" 
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using setsum_constant assms by (simp add: real_eq_of_nat) 

35582  179 
finally show ?thesis by simp 
38656  180 
qed simp 
35582  181 

40859  182 
lemma (in prob_space) prob_real_sum_image_fn: 
35582  183 
assumes "e \<in> events" 
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assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" 

185 
assumes "finite s" 

38656  186 
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" 
187 
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" 

35582  188 
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" 
189 
proof  

38656  190 
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" 
191 
using `e \<in> events` sets_into_space upper by blast 

192 
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp 

193 
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" 

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proof (rule finite_measure_finite_Union) 
38656  195 
show "finite s" by fact 
196 
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact 

197 
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" 

198 
using disjoint by (auto simp: disjoint_family_on_def) 

199 
qed 

200 
finally show ?thesis . 

35582  201 
qed 
202 

42199  203 
lemma (in prob_space) prob_space_vimage: 
204 
assumes S: "sigma_algebra S" 

205 
assumes T: "T \<in> measure_preserving M S" 

206 
shows "prob_space S" 

35582  207 
proof  
42199  208 
interpret S: measure_space S 
209 
using S and T by (rule measure_space_vimage) 

38656  210 
show ?thesis 
42199  211 
proof 
212 
from T[THEN measure_preservingD2] 

213 
have "T ` space S \<inter> space M = space M" 

214 
by (auto simp: measurable_def) 

215 
with T[THEN measure_preservingD, of "space S", symmetric] 

216 
show "measure S (space S) = 1" 

217 
using measure_space_1 by simp 

35582  218 
qed 
219 
qed 

220 

42199  221 
lemma (in prob_space) distribution_prob_space: 
222 
assumes X: "random_variable S X" 

223 
shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" (is "prob_space ?S") 

224 
proof (rule prob_space_vimage) 

225 
show "X \<in> measure_preserving M ?S" 

226 
using X 

227 
unfolding measure_preserving_def distribution_def_raw 

228 
by (auto simp: finite_measure_eq measurable_sets) 

229 
show "sigma_algebra ?S" using X by simp 

230 
qed 

231 

40859  232 
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  234 
shows "AE x. Q (X x)" 
235 
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  237 
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" 
238 
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  242 
qed 
243 

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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  248 

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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  252 
proof  
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interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
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254 
using assms(1) by (rule distribution_prob_space) 
35582  255 
show ?thesis 
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256 
using S.positive_integral_indicator(1)[of A] assms by simp 
35582  257 
qed 
258 

40859  259 
lemma (in prob_space) finite_expectation1: 
260 
assumes f: "finite (X`space M)" and rv: "random_variable borel X" 

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261 
shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X ` {r} \<inter> space M))" (is "_ = ?r") 
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262 
proof (subst integral_on_finite) 
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263 
show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto 
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264 
show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X ` {r} \<inter> space M))) = ?r" 
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265 
"\<And>x. \<mu> (X ` {x} \<inter> space M) \<noteq> \<infinity>" 
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266 
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto 
38656  267 
qed 
35582  268 

40859  269 
lemma (in prob_space) finite_expectation: 
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270 
assumes "finite (X`space M)" "random_variable borel X" 
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271 
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" 
38656  272 
using assms unfolding distribution_def using finite_expectation1 by auto 
273 

40859  274 
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: 
35582  275 
assumes "{x} \<in> events" 
38656  276 
assumes "prob {x} = 1" 
35582  277 
assumes "{y} \<in> events" 
278 
assumes "y \<noteq> x" 

279 
shows "prob {y} = 0" 

280 
using prob_one_inter[of "{y}" "{x}"] assms by auto 

281 

40859  282 
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" 
38656  283 
unfolding distribution_def by simp 
284 

40859  285 
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" 
38656  286 
proof  
287 
have "X ` X ` space M \<inter> space M = space M" by auto 

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288 
thus ?thesis unfolding distribution_def by (simp add: prob_space) 
38656  289 
qed 
290 

40859  291 
lemma (in prob_space) distribution_one: 
292 
assumes "random_variable M' X" and "A \<in> sets M'" 

38656  293 
shows "distribution X A \<le> 1" 
294 
proof  

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295 
have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def 
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296 
using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono) 
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297 
thus ?thesis by (simp add: prob_space) 
38656  298 
qed 
299 

40859  300 
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: 
35582  301 
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" 
38656  302 
(is "random_variable ?S X") 
303 
assumes "distribution X {x} = 1" 

35582  304 
assumes "y \<noteq> x" 
305 
shows "distribution X {y} = 0" 

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306 
proof cases 
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307 
{ fix x have "X ` {x} \<inter> space M \<in> sets M" 
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308 
proof cases 
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309 
assume "x \<in> X`space M" with X show ?thesis 
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310 
by (auto simp: measurable_def image_iff) 
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311 
next 
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312 
assume "x \<notin> X`space M" then have "X ` {x} \<inter> space M = {}" by auto 
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313 
then show ?thesis by auto 
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314 
qed } note single = this 
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315 
have "X ` {x} \<inter> space M  X ` {y} \<inter> space M = X ` {x} \<inter> space M" 
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316 
"X ` {y} \<inter> space M \<inter> (X ` {x} \<inter> space M) = {}" 
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317 
using `y \<noteq> x` by auto 
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318 
with finite_measure_inter_full_set[OF single single, of x y] assms(2) 
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319 
show ?thesis by (auto simp: distribution_def prob_space) 
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320 
next 
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321 
assume "{y} \<notin> sets ?S" 
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322 
then have "X ` {y} \<inter> space M = {}" by auto 
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323 
thus "distribution X {y} = 0" unfolding distribution_def by auto 
35582  324 
qed 
325 

40859  326 
lemma (in prob_space) joint_distribution_Times_le_fst: 
327 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

328 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

329 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

330 
unfolding distribution_def 

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331 
proof (intro finite_measure_mono) 
40859  332 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
333 
show "X ` A \<inter> space M \<in> events" 

334 
using X A unfolding measurable_def by simp 

335 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 

336 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

337 
qed 

338 

339 
lemma (in prob_space) joint_distribution_commute: 

340 
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)" 

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341 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  342 

343 
lemma (in prob_space) joint_distribution_Times_le_snd: 

344 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

345 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

346 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

347 
using assms 

348 
by (subst joint_distribution_commute) 

349 
(simp add: swap_product joint_distribution_Times_le_fst) 

350 

351 
lemma (in prob_space) random_variable_pairI: 

352 
assumes "random_variable MX X" 

353 
assumes "random_variable MY Y" 

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354 
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  355 
proof 
356 
interpret MX: sigma_algebra MX using assms by simp 

357 
interpret MY: sigma_algebra MY using assms by simp 

358 
interpret P: pair_sigma_algebra MX MY by default 

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359 
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  360 
have sa: "sigma_algebra M" by default 
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361 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  362 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  363 
qed 
364 

365 
lemma (in prob_space) joint_distribution_commute_singleton: 

366 
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}" 

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367 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  368 

369 
lemma (in prob_space) joint_distribution_assoc_singleton: 

370 
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 

371 
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}" 

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372 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  373 

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374 
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 
40859  375 

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376 
sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default 
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377 

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378 
sublocale pair_prob_space \<subseteq> P: prob_space P 
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379 
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) 
40859  380 

381 
lemma countably_additiveI[case_names countably]: 

382 
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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383 
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" 
40859  384 
shows "countably_additive M \<mu>" 
385 
using assms unfolding countably_additive_def by auto 

386 

387 
lemma (in prob_space) joint_distribution_prob_space: 

388 
assumes "random_variable MX X" "random_variable MY Y" 

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389 
shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" 
41689
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390 
using random_variable_pairI[OF assms] by (rule distribution_prob_space) 
40859  391 

392 
section "Probability spaces on finite sets" 

35582  393 

35977  394 
locale finite_prob_space = prob_space + finite_measure_space 
395 

40859  396 
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" 
397 

398 
lemma (in prob_space) finite_random_variableD: 

399 
assumes "finite_random_variable M' X" shows "random_variable M' X" 

400 
proof  

401 
interpret M': finite_sigma_algebra M' using assms by simp 

402 
then show "random_variable M' X" using assms by simp default 

403 
qed 

404 

405 
lemma (in prob_space) distribution_finite_prob_space: 

406 
assumes "finite_random_variable MX X" 

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407 
shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)" 
40859  408 
proof  
41981
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409 
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
40859  410 
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) 
411 
interpret MX: finite_sigma_algebra MX 

41689
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412 
using assms by auto 
41981
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413 
show ?thesis by default (simp_all add: MX.finite_space) 
40859  414 
qed 
415 

416 
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]: 

41689
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417 
assumes "simple_function M X" 
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418 
shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X" 
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419 
(is "finite_random_variable ?X _") 
40859  420 
proof (intro conjI) 
421 
have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp 

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422 
interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow) 
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423 
show "finite_sigma_algebra ?X" 
40859  424 
by default auto 
41689
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425 
show "X \<in> measurable M ?X" 
40859  426 
proof (unfold measurable_def, clarsimp) 
427 
fix A assume A: "A \<subseteq> X`space M" 

428 
then have "finite A" by (rule finite_subset) simp 

429 
then have "X ` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" 

430 
unfolding vimage_UN UN_extend_simps 

431 
apply (rule finite_UN) 

432 
using A assms unfolding simple_function_def by auto 

433 
then show "X ` A \<inter> space M \<in> events" by simp 

434 
qed 

435 
qed 

436 

437 
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]: 

41689
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438 
assumes "simple_function M X" 
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diff
changeset

439 
shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X" 
3e39b0e730d6
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changeset

440 
using simple_function_imp_finite_random_variable[OF assms, of ext] 
40859  441 
by (auto dest!: finite_random_variableD) 
442 

443 
lemma (in prob_space) sum_over_space_real_distribution: 

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444 
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" 
40859  445 
unfolding distribution_def prob_space[symmetric] 
41981
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446 
by (subst finite_measure_finite_Union[symmetric]) 
40859  447 
(auto simp add: disjoint_family_on_def simple_function_def 
448 
intro!: arg_cong[where f=prob]) 

449 

450 
lemma (in prob_space) finite_random_variable_pairI: 

451 
assumes "finite_random_variable MX X" 

452 
assumes "finite_random_variable MY Y" 

41689
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hoelzl
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changeset

453 
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  454 
proof 
455 
interpret MX: finite_sigma_algebra MX using assms by simp 

456 
interpret MY: finite_sigma_algebra MY using assms by simp 

457 
interpret P: pair_finite_sigma_algebra MX MY by default 

41689
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458 
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  459 
have sa: "sigma_algebra M" by default 
41689
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changeset

460 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  461 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  462 
qed 
463 

464 
lemma (in prob_space) finite_random_variable_imp_sets: 

465 
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" 

466 
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp 

467 

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468 
lemma (in prob_space) finite_random_variable_measurable: 
40859  469 
assumes X: "finite_random_variable MX X" shows "X ` A \<inter> space M \<in> events" 
470 
proof  

471 
interpret X: finite_sigma_algebra MX using X by simp 

472 
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X ` A \<inter> space M \<in> events" and 

473 
"X \<in> space M \<rightarrow> space MX" 

474 
by (auto simp: measurable_def) 

475 
then have *: "X ` A \<inter> space M = X ` (A \<inter> space MX) \<inter> space M" 

476 
by auto 

477 
show "X ` A \<inter> space M \<in> events" 

478 
unfolding * by (intro vimage) auto 

479 
qed 

480 

481 
lemma (in prob_space) joint_distribution_finite_Times_le_fst: 

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 X A" 

484 
unfolding distribution_def 

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

485 
proof (intro finite_measure_mono) 
40859  486 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
487 
show "X ` A \<inter> space M \<in> events" 

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

488 
using finite_random_variable_measurable[OF X] . 
40859  489 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 
490 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

491 
qed 

492 

493 
lemma (in prob_space) joint_distribution_finite_Times_le_snd: 

494 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

495 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

496 
using assms 

497 
by (subst joint_distribution_commute) 

498 
(simp add: swap_product joint_distribution_finite_Times_le_fst) 

499 

500 
lemma (in prob_space) finite_distribution_order: 

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

501 
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" 
40859  502 
assumes "finite_random_variable MX X" "finite_random_variable MY Y" 
503 
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

504 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

505 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

506 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

507 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

508 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

509 
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] 

510 
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:
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diff
changeset

511 
by (auto intro: antisym) 
40859  512 

513 
lemma (in prob_space) setsum_joint_distribution: 

514 
assumes X: "finite_random_variable MX X" 

515 
assumes Y: "random_variable MY Y" "B \<in> sets MY" 

516 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" 

517 
unfolding distribution_def 

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

518 
proof (subst finite_measure_finite_Union[symmetric]) 
40859  519 
interpret MX: finite_sigma_algebra MX using X by auto 
520 
show "finite (space MX)" using MX.finite_space . 

521 
let "?d i" = "(\<lambda>x. (X x, Y x)) ` ({i} \<times> B) \<inter> space M" 

522 
{ fix i assume "i \<in> space MX" 

523 
moreover have "?d i = (X ` {i} \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

524 
ultimately show "?d i \<in> events" 

525 
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y 

526 
using MX.sets_eq_Pow by auto } 

527 
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

528 
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

529 
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) 
40859  530 
qed 
531 

532 
lemma (in prob_space) setsum_joint_distribution_singleton: 

533 
assumes X: "finite_random_variable MX X" 

534 
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" 

535 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" 

536 
using setsum_joint_distribution[OF X 

537 
finite_random_variableD[OF Y(1)] 

538 
finite_random_variable_imp_sets[OF Y]] by simp 

539 

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

540 
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 
40859  541 

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

542 
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

543 
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

544 
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default 
40859  545 

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

546 
locale product_finite_prob_space = 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

547 
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

548 
and I :: "'i set" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

549 
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

550 

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

551 
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:
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diff
changeset

552 
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:
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diff
changeset

553 
sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

554 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

555 
show "\<mu> (space P) = 1" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

556 
using measure_times[OF M.top] M.measure_space_1 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

557 
by (simp add: setprod_1 space_product_algebra) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

558 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

559 

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

560 
lemma funset_eq_UN_fun_upd_I: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

561 
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

562 
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:
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diff
changeset

563 
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

564 
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

565 
proof safe 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

566 
fix f assume f: "f \<in> F (insert a A)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

567 
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

568 
proof (rule UN_I[of "f(a := d)"]) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

569 
show "f(a := d) \<in> F A" using *[OF f] . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

570 
show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

571 
proof (rule image_eqI[of _ _ "f a"]) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

572 
show "f a \<in> G (f(a := d))" using **[OF f] . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
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diff
changeset

573 
qed simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

574 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

575 
next 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

576 
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

577 
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

578 
qed 
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 extensional_funcset_insert_eq[simp]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

581 
assumes "a \<notin> A" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

582 
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

583 
apply (rule funset_eq_UN_fun_upd_I) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

584 
using assms 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

585 
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

586 

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

587 
lemma finite_extensional_funcset[simp, intro]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

588 
assumes "finite A" "finite B" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

589 
shows "finite (extensional A \<inter> (A \<rightarrow> B))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

590 
using assms by induct (auto simp: extensional_funcset_insert_eq) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

591 

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

592 
lemma finite_PiE[simp, intro]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

593 
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

594 
shows "finite (Pi\<^isub>E A B)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

595 
proof  
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

596 
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

597 
show ?thesis 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

598 
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

599 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

600 

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

601 
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

602 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

603 
show "finite (space P)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

604 
using finite_index M.finite_space by auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

605 

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

606 
{ fix x assume "x \<in> space P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

607 
then have x: "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

608 
proof safe 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

609 
fix y assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

610 
show "y = x" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

611 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

612 
fix i show "y i = x i" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

613 
using * `x \<in> space P` by (cases "i \<in> I") (auto simp: extensional_def) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

614 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

615 
qed auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

616 
with `x \<in> space P` have "{x} \<in> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

617 
by (auto intro!: in_P) } 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

618 
note x_in_P = this 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

619 

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

620 
have "Pow (space P) \<subseteq> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

621 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

622 
fix X assume "X \<in> Pow (space P)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

623 
moreover then have "finite X" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

624 
using `finite (space P)` by (blast intro: finite_subset) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

625 
ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

626 
by (intro finite_UN x_in_P) auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

627 
then show "X \<in> sets P" by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

628 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

629 
with space_closed show [simp]: "sets P = Pow (space P)" .. 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

630 

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

631 

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

632 
{ fix x assume "x \<in> space P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

633 
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

634 
then show "\<mu> {x} \<noteq> \<infinity>" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

635 
using measure_space_1 by auto } 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

636 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

637 

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

638 
lemma (in product_finite_prob_space) measure_finite_times: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

639 
"(\<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

640 
by (rule measure_times) simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

641 

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

642 
lemma (in product_finite_prob_space) prob_times: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

643 
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

644 
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

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

646 
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

647 
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

648 
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

649 
using measure_finite_times X by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

650 
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

651 
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

652 
finally show ?thesis by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

653 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

654 

40859  655 
lemma (in prob_space) joint_distribution_finite_prob_space: 
656 
assumes X: "finite_random_variable MX X" 

657 
assumes Y: "finite_random_variable MY Y" 

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

658 
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

659 
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) 
40859  660 

36624  661 
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

662 
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" 
36624  663 
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def 
664 
by auto 

665 

666 
lemma (in prob_space) not_empty: "space M \<noteq> {}" 

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

667 
using prob_space empty_measure' by auto 
36624  668 

38656  669 
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" 
670 
using measure_space_1 sum_over_space by simp 

36624  671 

672 
lemma (in finite_prob_space) joint_distribution_restriction_fst: 

673 
"joint_distribution X Y A \<le> distribution X (fst ` A)" 

674 
unfolding distribution_def 

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

675 
proof (safe intro!: finite_measure_mono) 
36624  676 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
677 
show "x \<in> X ` fst ` A" 

678 
by (auto intro!: image_eqI[OF _ *]) 

679 
qed (simp_all add: sets_eq_Pow) 

680 

681 
lemma (in finite_prob_space) joint_distribution_restriction_snd: 

682 
"joint_distribution X Y A \<le> distribution Y (snd ` A)" 

683 
unfolding distribution_def 

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

684 
proof (safe intro!: finite_measure_mono) 
36624  685 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
686 
show "x \<in> Y ` snd ` A" 

687 
by (auto intro!: image_eqI[OF _ *]) 

688 
qed (simp_all add: sets_eq_Pow) 

689 

690 
lemma (in finite_prob_space) distribution_order: 

691 
shows "0 \<le> distribution X x'" 

692 
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" 

693 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

694 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

695 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

696 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

697 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

698 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

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

699 
using 
36624  700 
joint_distribution_restriction_fst[of X Y "{(x, y)}"] 
701 
joint_distribution_restriction_snd[of X Y "{(x, y)}"] 

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

702 
by (auto intro: antisym) 
36624  703 

39097  704 
lemma (in finite_prob_space) distribution_mono: 
705 
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

706 
shows "distribution X x \<le> distribution Y y" 

707 
unfolding distribution_def 

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

708 
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) 
39097  709 

710 
lemma (in finite_prob_space) distribution_mono_gt_0: 

711 
assumes gt_0: "0 < distribution X x" 

712 
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

713 
shows "0 < distribution Y y" 

714 
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) 

715 

716 
lemma (in finite_prob_space) sum_over_space_distrib: 

717 
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" 

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

718 
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

719 
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

720 
(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

721 
intro!: arg_cong[where f=\<mu>']) 
39097  722 

723 
lemma (in finite_prob_space) finite_sum_over_space_eq_1: 

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

724 
"(\<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

725 
using prob_space finite_space 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

726 
by (subst (asm) finite_measure_finite_singleton) auto 
39097  727 

728 
lemma (in prob_space) distribution_remove_const: 

729 
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" 

730 
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" 

731 
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" 

732 
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" 

733 
and "distribution (\<lambda>x. ()) {()} = 1" 

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

734 
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) 
35977  735 

39097  736 
lemma (in finite_prob_space) setsum_distribution_gen: 
737 
assumes "Z ` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y ` {f x}) \<inter> space M" 

738 
and "inj_on f (X`space M)" 

739 
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" 

740 
unfolding distribution_def assms 

741 
using finite_space assms 

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

742 
by (subst finite_measure_finite_Union[symmetric]) 
39097  743 
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def 
744 
intro!: arg_cong[where f=prob]) 

745 

746 
lemma (in finite_prob_space) setsum_distribution: 

747 
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" 

748 
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" 

749 
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" 

750 
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" 

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

752 
by (auto intro!: inj_onI setsum_distribution_gen) 

753 

754 
lemma (in finite_prob_space) uniform_prob: 

755 
assumes "x \<in> space M" 

756 
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

757 
shows "prob {x} = 1 / card (space M)" 
39097  758 
proof  
759 
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" 

760 
using assms(2)[OF _ `x \<in> space M`] by blast 

761 
have "1 = prob (space M)" 

762 
using prob_space by auto 

763 
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

764 
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] 
39097  765 
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] 
766 
finite_space unfolding disjoint_family_on_def prob_space[symmetric] 

767 
by (auto simp add:setsum_restrict_set) 

768 
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" 

769 
using prob_x by auto 

770 
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp 

771 
finally have one: "1 = real (card (space M)) * prob {x}" 

772 
using real_eq_of_nat by auto 

773 
hence two: "real (card (space M)) \<noteq> 0" by fastsimp 

774 
from one have three: "prob {x} \<noteq> 0" by fastsimp 

775 
thus ?thesis using one two three divide_cancel_right 

776 
by (auto simp:field_simps) 

39092  777 
qed 
35977  778 

39092  779 
lemma (in prob_space) prob_space_subalgebra: 
41545  780 
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

781 
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

782 
shows "prob_space N" 
39092  783 
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

784 
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

785 
by (rule measure_space_subalgebra[OF assms]) 
39092  786 
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

787 
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) 
35977  788 
qed 
789 

39092  790 
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

791 
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

792 
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

793 
(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

794 
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

795 
interpret A: measure_space "restricted_space A" 
39092  796 
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

797 
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

798 
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

799 
show "prob_space ?P" 
39092  800 
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

801 
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

802 
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

803 
show "positive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

804 
proof (simp add: positive_def, safe) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

805 
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

806 
fix B assume "B \<in> events" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

807 
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

808 
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

809 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

810 
show "countably_additive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

811 
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

812 
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

813 
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

814 
{ fix i 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

815 
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

816 
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

817 
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

818 
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

819 
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

820 
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

821 
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

822 
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

823 
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

824 
by (auto simp: suminf_cmult_extreal) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

825 
qed 
39092  826 
qed 
827 
qed 

828 

829 
lemma finite_prob_spaceI: 

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

830 
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

831 
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

832 
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

833 
shows "finite_prob_space M" 
39092  834 
unfolding finite_prob_space_eq 
835 
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

836 
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

837 
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

838 
show "measure M (space M) = 1" by fact 
39092  839 
qed 
36624  840 

841 
lemma (in finite_prob_space) finite_measure_space: 

39097  842 
fixes X :: "'a \<Rightarrow> 'x" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

843 
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

844 
(is "finite_measure_space ?S") 
39092  845 
proof (rule finite_measure_spaceI, simp_all) 
36624  846 
show "finite (X ` space M)" using finite_space by simp 
39097  847 
next 
848 
fix A B :: "'x set" assume "A \<inter> B = {}" 

849 
then show "distribution X (A \<union> B) = distribution X A + distribution X B" 

850 
unfolding distribution_def 

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

851 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

852 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
36624  853 
qed 
854 

39097  855 
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

856 
"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

857 
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def) 
39097  858 

39096  859 
lemma (in finite_prob_space) finite_product_measure_space: 
39097  860 
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" 
39096  861 
assumes "finite s1" "finite s2" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

862 
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

863 
(is "finite_measure_space ?M") 
39097  864 
proof (rule finite_measure_spaceI, simp_all) 
865 
show "finite (s1 \<times> s2)" 

39096  866 
using assms by auto 
39097  867 
next 
868 
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" 

869 
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" 

870 
unfolding distribution_def 

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

871 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

872 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
39096  873 
qed 
874 

39097  875 
lemma (in finite_prob_space) finite_product_measure_space_of_images: 
39096  876 
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

877 
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

878 
measure = extreal \<circ> joint_distribution X Y \<rparr>" 
39096  879 
using finite_space by (auto intro!: finite_product_measure_space) 
880 

40859  881 
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

882 
"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

883 
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

884 
(is "finite_prob_space ?S") 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

885 
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def) 
40859  886 
have "X ` X ` space M \<inter> Y ` Y ` space M \<inter> space M = space M" by auto 
887 
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" 

888 
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) 

889 
qed 

890 

39085  891 
section "Conditional Expectation and Probability" 
892 

893 
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

894 
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

895 
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

896 
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

897 
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

898 
(\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))" 
39083  899 
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

900 
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

901 
interpret P: prob_space N 
41545  902 
using prob_space_subalgebra[OF N] . 
39083  903 

904 
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

905 
let "?Q A" = "integral\<^isup>P M (?f A)" 
39083  906 

907 
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

908 
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

909 
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

910 
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

911 
then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" . 
39083  912 

913 
have "P.absolutely_continuous ?Q" 

914 
unfolding P.absolutely_continuous_def 

41545  915 
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

916 
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

917 
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

918 
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

919 
then show "?Q A = 0" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

920 
by (auto simp add: positive_integral_0_iff_AE) 
39083  921 
qed 
922 
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

923 
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

924 
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)" 
39083  925 
by blast 
41545  926 
with N(2) show ?thesis 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

927 
by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)]) 
39083  928 
qed 
929 

39085  930 
definition (in prob_space) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

931 
"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

932 
\<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  933 

934 
abbreviation (in prob_space) 

39092  935 
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" 
39085  936 

937 
lemma (in prob_space) 

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

938 
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

939 
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

940 
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" 
39085  941 
shows borel_measurable_conditional_expectation: 
41545  942 
"conditional_expectation N X \<in> borel_measurable N" 
943 
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

944 
(\<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

945 
(\<integral>\<^isup>+x. X x * indicator C x \<partial>M)" 
41545  946 
(is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C") 
39085  947 
proof  
948 
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] 

41545  949 
then show "conditional_expectation N X \<in> borel_measurable N" 
39085  950 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
951 

41545  952 
from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C" 
39085  953 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
954 
qed 

955 

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

956 
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

957 
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" 
39091  958 
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

959 
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

960 
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

961 
proof  
39091  962 
interpret M': sigma_algebra M' by fact 
963 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 

964 
from M'.sigma_algebra_vimage[OF this] 

965 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 

966 

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

967 
from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this 
39091  968 

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

969 
have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 
39091  970 
proof 
971 
fix i 

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

972 
from f(1)[of i] have "finite (f i`space M)" and B_ex: 
39091  973 
"\<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

974 
unfolding simple_function_def by auto 
39091  975 
from B_ex[THEN bchoice] guess B .. note B = this 
976 

977 
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" 

978 

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

979 
show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 
39091  980 
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

981 
show "simple_function M' ?g" using B by auto 
39091  982 

983 
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

984 
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  985 
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

986 
then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i] 
39091  987 
by (subst va.simple_function_indicator_representation) auto 
988 
qed 

989 
qed 

990 
from choice[OF this] guess g .. note g = this 

991 

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

992 
show ?thesis 
39091  993 
proof (intro ballI bexI) 
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

994 
show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'" 
39091  995 
using g by (auto intro: M'.borel_measurable_simple_function) 
996 
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

997 
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

998 
also have "\<dots> = (SUP i. g i (Y x))" 
39091  999 
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

1000 
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

1001 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1002 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1003 

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

1004 
lemma extreal_0_le_iff_le_0[simp]: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1005 
fixes a :: extreal shows "0 \<le> a \<longleftrightarrow> a \<le> 0" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1006 
by (cases rule: extreal2_cases[of a]) auto 
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
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changeset

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

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

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

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

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

1029 

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

1032 
"(\<lambda>x.  Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1033 
by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1034 
from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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diff
changeset

1035 
from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1036 
let "?g x" = "p x  n x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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diff
changeset

1037 
show "\<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:
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diff
changeset

1038 
proof (intro bexI ballI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1039 
show "?g \<in> borel_measurable M'" using p n by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1040 
fix x assume "x \<in> space M" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1041 
then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 ( Z x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1042 
using p n by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1043 
then show "Z x = ?g (Y x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

1044 
by (auto split: split_max) 
39091  1045 
qed 
1046 
qed 

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

1047 

35582  1048 
end 