author  hoelzl 
Mon, 14 Mar 2011 14:37:49 +0100  
changeset 41981  cdf7693bbe08 
parent 41831  91a2b435dd7a 
child 42067  66c8281349ec 
permissions  rwrr 
35582  1 
theory Probability_Space 
40859  2 
imports Lebesgue_Integration Radon_Nikodym Product_Measure 
35582  3 
begin 
4 

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lemma real_of_extreal_inverse[simp]: 
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fixes X :: extreal 
40859  7 
shows "real (inverse X) = 1 / real X" 
8 
by (cases X) (auto simp: inverse_eq_divide) 

9 

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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  14 
by (cases X) auto 
15 

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lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)" 
40859  17 
by (cases X) auto 
18 

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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  22 
locale prob_space = measure_space + 
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assumes measure_space_1: "measure M (space M) = 1" 
38656  24 

25 
sublocale prob_space < finite_measure 

26 
proof 

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from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp 
38656  28 
qed 
29 

40859  30 
abbreviation (in prob_space) "events \<equiv> sets M" 
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abbreviation (in prob_space) "prob \<equiv> \<mu>'" 
40859  32 
abbreviation (in prob_space) "prob_preserving \<equiv> measure_preserving" 
33 
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  35 

40859  36 
definition (in prob_space) 
35582  37 
"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B" 
38 

40859  39 
definition (in prob_space) 
35582  40 
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)" 
41 

40859  42 
definition (in prob_space) 
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"distribution X A = \<mu>' (X ` A \<inter> space M)" 
35582  44 

40859  45 
abbreviation (in prob_space) 
36624  46 
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" 
35582  47 

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declare (in finite_measure) positive_measure'[intro, simp] 
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39097  50 
lemma (in prob_space) distribution_cong: 
51 
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" 

52 
shows "distribution X = distribution Y" 

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53 
unfolding distribution_def fun_eq_iff 
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) 
39097  55 

56 
lemma (in prob_space) joint_distribution_cong: 

57 
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" 

58 
assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" 

59 
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  62 

40859  63 
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  66 

67 
lemma (in prob_space) prob_space: "prob (space M) = 1" 

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68 
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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71 
using bounded_measure[of A] by (simp add: prob_space) 
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72 

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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  75 

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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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79 

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lemma (in prob_space) distribution_1: 
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81 
"distribution X A \<le> 1" 
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82 
unfolding distribution_def by simp 
35582  83 

40859  84 
lemma (in prob_space) prob_compl: 
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assumes A: "A \<in> events" 
38656  86 
shows "prob (space M  A) = 1  prob A" 
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87 
using finite_measure_compl[OF A] by (simp add: prob_space) 
35582  88 

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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  91 

40859  92 
lemma (in prob_space) prob_space_increasing: "increasing M prob" 
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by (auto intro!: finite_measure_mono simp: increasing_def) 
35582  94 

40859  95 
lemma (in prob_space) prob_zero_union: 
35582  96 
assumes "s \<in> events" "t \<in> events" "prob t = 0" 
97 
shows "prob (s \<union> t) = prob s" 

38656  98 
using assms 
35582  99 
proof  
100 
have "prob (s \<union> t) \<le> prob s" 

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101 
using finite_measure_subadditive[of s t] assms by auto 
35582  102 
moreover have "prob (s \<union> t) \<ge> prob s" 
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103 
using assms by (blast intro: finite_measure_mono) 
35582  104 
ultimately show ?thesis by simp 
105 
qed 

106 

40859  107 
lemma (in prob_space) prob_eq_compl: 
35582  108 
assumes "s \<in> events" "t \<in> events" 
109 
assumes "prob (space M  s) = prob (space M  t)" 

110 
shows "prob s = prob t" 

38656  111 
using assms prob_compl by auto 
35582  112 

40859  113 
lemma (in prob_space) prob_one_inter: 
35582  114 
assumes events:"s \<in> events" "t \<in> events" 
115 
assumes "prob t = 1" 

116 
shows "prob (s \<inter> t) = prob s" 

117 
proof  

38656  118 
have "prob ((space M  s) \<union> (space M  t)) = prob (space M  s)" 
119 
using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union) 

120 
also have "(space M  s) \<union> (space M  t) = space M  (s \<inter> t)" 

121 
by blast 

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

123 
using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s]) 

35582  124 
qed 
125 

40859  126 
lemma (in prob_space) prob_eq_bigunion_image: 
35582  127 
assumes "range f \<subseteq> events" "range g \<subseteq> events" 
128 
assumes "disjoint_family f" "disjoint_family g" 

129 
assumes "\<And> n :: nat. prob (f n) = prob (g n)" 

130 
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" 

131 
using assms 

132 
proof  

38656  133 
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  135 
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" 
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136 
by (rule finite_measure_UNION[OF assms(2,4)]) 
38656  137 
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp 
35582  138 
qed 
139 

40859  140 
lemma (in prob_space) prob_countably_zero: 
35582  141 
assumes "range c \<subseteq> events" 
142 
assumes "\<And> i. prob (c i) = 0" 

38656  143 
shows "prob (\<Union> i :: nat. c i) = 0" 
144 
proof (rule antisym) 

145 
show "prob (\<Union> i :: nat. c i) \<le> 0" 

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146 
using finite_measure_countably_subadditive[OF assms(1)] 
38656  147 
by (simp add: assms(2) suminf_zero summable_zero) 
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148 
qed simp 
35582  149 

40859  150 
lemma (in prob_space) indep_sym: 
35582  151 
"indep a b \<Longrightarrow> indep b a" 
152 
unfolding indep_def using Int_commute[of a b] by auto 

153 

40859  154 
lemma (in prob_space) indep_refl: 
35582  155 
assumes "a \<in> events" 
156 
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" 

157 
using assms unfolding indep_def by auto 

158 

40859  159 
lemma (in prob_space) prob_equiprobable_finite_unions: 
38656  160 
assumes "s \<in> events" 
161 
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" 

35582  162 
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" 
38656  163 
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" 
35582  164 
proof (cases "s = {}") 
38656  165 
case False hence "\<exists> x. x \<in> s" by blast 
35582  166 
from someI_ex[OF this] assms 
167 
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast 

168 
have "prob s = (\<Sum> x \<in> s. prob {x})" 

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169 
using finite_measure_finite_singleton[OF s_finite] by simp 
35582  170 
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto 
38656  171 
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" 
172 
using setsum_constant assms by (simp add: real_eq_of_nat) 

35582  173 
finally show ?thesis by simp 
38656  174 
qed simp 
35582  175 

40859  176 
lemma (in prob_space) prob_real_sum_image_fn: 
35582  177 
assumes "e \<in> events" 
178 
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" 

179 
assumes "finite s" 

38656  180 
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" 
181 
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" 

35582  182 
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" 
183 
proof  

38656  184 
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" 
185 
using `e \<in> events` sets_into_space upper by blast 

186 
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp 

187 
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" 

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188 
proof (rule finite_measure_finite_Union) 
38656  189 
show "finite s" by fact 
190 
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact 

191 
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" 

192 
using disjoint by (auto simp: disjoint_family_on_def) 

193 
qed 

194 
finally show ?thesis . 

35582  195 
qed 
196 

40859  197 
lemma (in prob_space) distribution_prob_space: 
198 
assumes "random_variable S X" 

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199 
shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" 
35582  200 
proof  
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201 
interpret S: measure_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
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202 
proof (rule measure_space.measure_space_cong) 
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203 
show "measure_space (S\<lparr> measure := \<lambda>A. \<mu> (X ` A \<inter> space M) \<rparr>)" 
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204 
using assms by (auto intro!: measure_space_vimage simp: measure_preserving_def) 
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205 
qed (insert assms, auto simp add: finite_measure_eq distribution_def measurable_sets) 
38656  206 
show ?thesis 
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207 
proof (default, simp) 
38656  208 
have "X ` space S \<inter> space M = space M" 
209 
using `random_variable S X` by (auto simp: measurable_def) 

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210 
then show "extreal (distribution X (space S)) = 1" 
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211 
by (simp add: distribution_def one_extreal_def prob_space) 
35582  212 
qed 
213 
qed 

214 

40859  215 
lemma (in prob_space) AE_distribution: 
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216 
assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x" 
40859  217 
shows "AE x. Q (X x)" 
218 
proof  

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219 
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space) 
40859  220 
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" 
221 
using assms unfolding X.almost_everywhere_def by auto 

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222 
from X[unfolded measurable_def] N show "AE x. Q (X x)" 
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223 
by (intro AE_I'[where N="X ` N \<inter> space M"]) 
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224 
(auto simp: finite_measure_eq distribution_def measurable_sets) 
40859  225 
qed 
226 

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227 
lemma (in prob_space) distribution_eq_integral: 
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228 
"random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X ` A \<inter> space M))" 
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229 
using finite_measure_eq[of "X ` A \<inter> space M"] 
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230 
by (auto simp: measurable_sets distribution_def) 
35582  231 

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232 
lemma (in prob_space) distribution_eq_translated_integral: 
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233 
assumes "random_variable S X" "A \<in> sets S" 
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234 
shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)" 
35582  235 
proof  
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236 
interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
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237 
using assms(1) by (rule distribution_prob_space) 
35582  238 
show ?thesis 
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239 
using S.positive_integral_indicator(1)[of A] assms by simp 
35582  240 
qed 
241 

40859  242 
lemma (in prob_space) finite_expectation1: 
243 
assumes f: "finite (X`space M)" and rv: "random_variable borel X" 

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244 
shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X ` {r} \<inter> space M))" (is "_ = ?r") 
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245 
proof (subst integral_on_finite) 
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246 
show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto 
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247 
show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X ` {r} \<inter> space M))) = ?r" 
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248 
"\<And>x. \<mu> (X ` {x} \<inter> space M) \<noteq> \<infinity>" 
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249 
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto 
38656  250 
qed 
35582  251 

40859  252 
lemma (in prob_space) finite_expectation: 
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253 
assumes "finite (X`space M)" "random_variable borel X" 
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254 
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" 
38656  255 
using assms unfolding distribution_def using finite_expectation1 by auto 
256 

40859  257 
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: 
35582  258 
assumes "{x} \<in> events" 
38656  259 
assumes "prob {x} = 1" 
35582  260 
assumes "{y} \<in> events" 
261 
assumes "y \<noteq> x" 

262 
shows "prob {y} = 0" 

263 
using prob_one_inter[of "{y}" "{x}"] assms by auto 

264 

40859  265 
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" 
38656  266 
unfolding distribution_def by simp 
267 

40859  268 
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" 
38656  269 
proof  
270 
have "X ` X ` space M \<inter> space M = space M" by auto 

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271 
thus ?thesis unfolding distribution_def by (simp add: prob_space) 
38656  272 
qed 
273 

40859  274 
lemma (in prob_space) distribution_one: 
275 
assumes "random_variable M' X" and "A \<in> sets M'" 

38656  276 
shows "distribution X A \<le> 1" 
277 
proof  

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278 
have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def 
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279 
using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono) 
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280 
thus ?thesis by (simp add: prob_space) 
38656  281 
qed 
282 

40859  283 
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: 
35582  284 
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" 
38656  285 
(is "random_variable ?S X") 
286 
assumes "distribution X {x} = 1" 

35582  287 
assumes "y \<noteq> x" 
288 
shows "distribution X {y} = 0" 

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289 
proof cases 
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290 
{ fix x have "X ` {x} \<inter> space M \<in> sets M" 
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291 
proof cases 
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292 
assume "x \<in> X`space M" with X show ?thesis 
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293 
by (auto simp: measurable_def image_iff) 
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294 
next 
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295 
assume "x \<notin> X`space M" then have "X ` {x} \<inter> space M = {}" by auto 
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296 
then show ?thesis by auto 
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297 
qed } note single = this 
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298 
have "X ` {x} \<inter> space M  X ` {y} \<inter> space M = X ` {x} \<inter> space M" 
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299 
"X ` {y} \<inter> space M \<inter> (X ` {x} \<inter> space M) = {}" 
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300 
using `y \<noteq> x` by auto 
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301 
with finite_measure_inter_full_set[OF single single, of x y] assms(2) 
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302 
show ?thesis by (auto simp: distribution_def prob_space) 
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303 
next 
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304 
assume "{y} \<notin> sets ?S" 
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305 
then have "X ` {y} \<inter> space M = {}" by auto 
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306 
thus "distribution X {y} = 0" unfolding distribution_def by auto 
35582  307 
qed 
308 

40859  309 
lemma (in prob_space) joint_distribution_Times_le_fst: 
310 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

311 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

312 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

313 
unfolding distribution_def 

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314 
proof (intro finite_measure_mono) 
40859  315 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
316 
show "X ` A \<inter> space M \<in> events" 

317 
using X A unfolding measurable_def by simp 

318 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 

319 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

320 
qed 

321 

322 
lemma (in prob_space) joint_distribution_commute: 

323 
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)" 

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324 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  325 

326 
lemma (in prob_space) joint_distribution_Times_le_snd: 

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 Y B" 

330 
using assms 

331 
by (subst joint_distribution_commute) 

332 
(simp add: swap_product joint_distribution_Times_le_fst) 

333 

334 
lemma (in prob_space) random_variable_pairI: 

335 
assumes "random_variable MX X" 

336 
assumes "random_variable MY Y" 

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337 
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  338 
proof 
339 
interpret MX: sigma_algebra MX using assms by simp 

340 
interpret MY: sigma_algebra MY using assms by simp 

341 
interpret P: pair_sigma_algebra MX MY by default 

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342 
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  343 
have sa: "sigma_algebra M" by default 
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344 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  345 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  346 
qed 
347 

348 
lemma (in prob_space) joint_distribution_commute_singleton: 

349 
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}" 

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350 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  351 

352 
lemma (in prob_space) joint_distribution_assoc_singleton: 

353 
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 

354 
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}" 

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

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357 
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 
40859  358 

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359 
sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default 
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changeset

360 

3e39b0e730d6
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361 
sublocale pair_prob_space \<subseteq> P: prob_space P 
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362 
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) 
40859  363 

364 
lemma countably_additiveI[case_names countably]: 

365 
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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366 
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" 
40859  367 
shows "countably_additive M \<mu>" 
368 
using assms unfolding countably_additive_def by auto 

369 

370 
lemma (in prob_space) joint_distribution_prob_space: 

371 
assumes "random_variable MX X" "random_variable MY Y" 

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372 
shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" 
41689
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373 
using random_variable_pairI[OF assms] by (rule distribution_prob_space) 
40859  374 

375 
section "Probability spaces on finite sets" 

35582  376 

35977  377 
locale finite_prob_space = prob_space + finite_measure_space 
378 

40859  379 
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" 
380 

381 
lemma (in prob_space) finite_random_variableD: 

382 
assumes "finite_random_variable M' X" shows "random_variable M' X" 

383 
proof  

384 
interpret M': finite_sigma_algebra M' using assms by simp 

385 
then show "random_variable M' X" using assms by simp default 

386 
qed 

387 

388 
lemma (in prob_space) distribution_finite_prob_space: 

389 
assumes "finite_random_variable MX X" 

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390 
shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)" 
40859  391 
proof  
41981
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changeset

392 
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
40859  393 
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) 
394 
interpret MX: finite_sigma_algebra MX 

41689
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395 
using assms by auto 
41981
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396 
show ?thesis by default (simp_all add: MX.finite_space) 
40859  397 
qed 
398 

399 
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]: 

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400 
assumes "simple_function M X" 
3e39b0e730d6
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changeset

401 
shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X" 
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changeset

402 
(is "finite_random_variable ?X _") 
40859  403 
proof (intro conjI) 
404 
have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp 

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changeset

405 
interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow) 
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hoelzl
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changeset

406 
show "finite_sigma_algebra ?X" 
40859  407 
by default auto 
41689
3e39b0e730d6
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hoelzl
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changeset

408 
show "X \<in> measurable M ?X" 
40859  409 
proof (unfold measurable_def, clarsimp) 
410 
fix A assume A: "A \<subseteq> X`space M" 

411 
then have "finite A" by (rule finite_subset) simp 

412 
then have "X ` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" 

413 
unfolding vimage_UN UN_extend_simps 

414 
apply (rule finite_UN) 

415 
using A assms unfolding simple_function_def by auto 

416 
then show "X ` A \<inter> space M \<in> events" by simp 

417 
qed 

418 
qed 

419 

420 
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]: 

41689
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421 
assumes "simple_function M X" 
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changeset

422 
shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X" 
3e39b0e730d6
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hoelzl
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changeset

423 
using simple_function_imp_finite_random_variable[OF assms, of ext] 
40859  424 
by (auto dest!: finite_random_variableD) 
425 

426 
lemma (in prob_space) sum_over_space_real_distribution: 

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427 
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" 
40859  428 
unfolding distribution_def prob_space[symmetric] 
41981
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429 
by (subst finite_measure_finite_Union[symmetric]) 
40859  430 
(auto simp add: disjoint_family_on_def simple_function_def 
431 
intro!: arg_cong[where f=prob]) 

432 

433 
lemma (in prob_space) finite_random_variable_pairI: 

434 
assumes "finite_random_variable MX X" 

435 
assumes "finite_random_variable MY Y" 

41689
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hoelzl
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diff
changeset

436 
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  437 
proof 
438 
interpret MX: finite_sigma_algebra MX using assms by simp 

439 
interpret MY: finite_sigma_algebra MY using assms by simp 

440 
interpret P: pair_finite_sigma_algebra MX MY by default 

41689
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hoelzl
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changeset

441 
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  442 
have sa: "sigma_algebra M" by default 
41689
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hoelzl
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diff
changeset

443 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  444 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  445 
qed 
446 

447 
lemma (in prob_space) finite_random_variable_imp_sets: 

448 
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" 

449 
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp 

450 

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451 
lemma (in prob_space) finite_random_variable_measurable: 
40859  452 
assumes X: "finite_random_variable MX X" shows "X ` A \<inter> space M \<in> events" 
453 
proof  

454 
interpret X: finite_sigma_algebra MX using X by simp 

455 
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X ` A \<inter> space M \<in> events" and 

456 
"X \<in> space M \<rightarrow> space MX" 

457 
by (auto simp: measurable_def) 

458 
then have *: "X ` A \<inter> space M = X ` (A \<inter> space MX) \<inter> space M" 

459 
by auto 

460 
show "X ` A \<inter> space M \<in> events" 

461 
unfolding * by (intro vimage) auto 

462 
qed 

463 

464 
lemma (in prob_space) joint_distribution_finite_Times_le_fst: 

465 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

466 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

467 
unfolding distribution_def 

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468 
proof (intro finite_measure_mono) 
40859  469 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
470 
show "X ` A \<inter> space M \<in> events" 

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471 
using finite_random_variable_measurable[OF X] . 
40859  472 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 
473 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

474 
qed 

475 

476 
lemma (in prob_space) joint_distribution_finite_Times_le_snd: 

477 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

478 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

479 
using assms 

480 
by (subst joint_distribution_commute) 

481 
(simp add: swap_product joint_distribution_finite_Times_le_fst) 

482 

483 
lemma (in prob_space) finite_distribution_order: 

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484 
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" 
40859  485 
assumes "finite_random_variable MX X" "finite_random_variable MY Y" 
486 
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

487 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

488 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

489 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

490 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

491 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

492 
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] 

493 
using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"] 

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494 
by (auto intro: antisym) 
40859  495 

496 
lemma (in prob_space) setsum_joint_distribution: 

497 
assumes X: "finite_random_variable MX X" 

498 
assumes Y: "random_variable MY Y" "B \<in> sets MY" 

499 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" 

500 
unfolding distribution_def 

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501 
proof (subst finite_measure_finite_Union[symmetric]) 
40859  502 
interpret MX: finite_sigma_algebra MX using X by auto 
503 
show "finite (space MX)" using MX.finite_space . 

504 
let "?d i" = "(\<lambda>x. (X x, Y x)) ` ({i} \<times> B) \<inter> space M" 

505 
{ fix i assume "i \<in> space MX" 

506 
moreover have "?d i = (X ` {i} \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

507 
ultimately show "?d i \<in> events" 

508 
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y 

509 
using MX.sets_eq_Pow by auto } 

510 
show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def) 

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511 
show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y ` B \<inter> space M)" 
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512 
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) 
40859  513 
qed 
514 

515 
lemma (in prob_space) setsum_joint_distribution_singleton: 

516 
assumes X: "finite_random_variable MX X" 

517 
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" 

518 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" 

519 
using setsum_joint_distribution[OF X 

520 
finite_random_variableD[OF Y(1)] 

521 
finite_random_variable_imp_sets[OF Y]] by simp 

522 

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523 
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 
40859  524 

41689
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525 
sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default 
3e39b0e730d6
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changeset

526 
sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2 by default 
3e39b0e730d6
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changeset

527 
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default 
40859  528 

529 
lemma (in prob_space) joint_distribution_finite_prob_space: 

530 
assumes X: "finite_random_variable MX X" 

531 
assumes Y: "finite_random_variable MY Y" 

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changeset

532 
shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" 
41689
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changeset

533 
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) 
40859  534 

36624  535 
lemma finite_prob_space_eq: 
41689
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hoelzl
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536 
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" 
36624  537 
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def 
538 
by auto 

539 

540 
lemma (in prob_space) not_empty: "space M \<noteq> {}" 

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changeset

541 
using prob_space empty_measure' by auto 
36624  542 

38656  543 
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" 
544 
using measure_space_1 sum_over_space by simp 

36624  545 

546 
lemma (in finite_prob_space) joint_distribution_restriction_fst: 

547 
"joint_distribution X Y A \<le> distribution X (fst ` A)" 

548 
unfolding distribution_def 

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549 
proof (safe intro!: finite_measure_mono) 
36624  550 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
551 
show "x \<in> X ` fst ` A" 

552 
by (auto intro!: image_eqI[OF _ *]) 

553 
qed (simp_all add: sets_eq_Pow) 

554 

555 
lemma (in finite_prob_space) joint_distribution_restriction_snd: 

556 
"joint_distribution X Y A \<le> distribution Y (snd ` A)" 

557 
unfolding distribution_def 

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558 
proof (safe intro!: finite_measure_mono) 
36624  559 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
560 
show "x \<in> Y ` snd ` A" 

561 
by (auto intro!: image_eqI[OF _ *]) 

562 
qed (simp_all add: sets_eq_Pow) 

563 

564 
lemma (in finite_prob_space) distribution_order: 

565 
shows "0 \<le> distribution X x'" 

566 
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" 

567 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

568 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

569 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

570 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

571 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

572 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

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changeset

573 
using 
36624  574 
joint_distribution_restriction_fst[of X Y "{(x, y)}"] 
575 
joint_distribution_restriction_snd[of X Y "{(x, y)}"] 

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changeset

576 
by (auto intro: antisym) 
36624  577 

39097  578 
lemma (in finite_prob_space) distribution_mono: 
579 
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

580 
shows "distribution X x \<le> distribution Y y" 

581 
unfolding distribution_def 

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582 
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) 
39097  583 

584 
lemma (in finite_prob_space) distribution_mono_gt_0: 

585 
assumes gt_0: "0 < distribution X x" 

586 
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

587 
shows "0 < distribution Y y" 

588 
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) 

589 

590 
lemma (in finite_prob_space) sum_over_space_distrib: 

591 
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" 

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changeset

592 
unfolding distribution_def prob_space[symmetric] using finite_space 
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changeset

593 
by (subst finite_measure_finite_Union[symmetric]) 
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diff
changeset

594 
(auto simp add: disjoint_family_on_def sets_eq_Pow 
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changeset

595 
intro!: arg_cong[where f=\<mu>']) 
39097  596 

597 
lemma (in finite_prob_space) sum_over_space_real_distribution: 

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changeset

598 
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" 
39097  599 
unfolding distribution_def prob_space[symmetric] using finite_space 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

600 
by (subst finite_measure_finite_Union[symmetric]) 
39097  601 
(auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob]) 
602 

603 
lemma (in finite_prob_space) finite_sum_over_space_eq_1: 

41981
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diff
changeset

604 
"(\<Sum>x\<in>space M. prob {x}) = 1" 
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hoelzl
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diff
changeset

605 
using prob_space finite_space 
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diff
changeset

606 
by (subst (asm) finite_measure_finite_singleton) auto 
39097  607 

608 
lemma (in prob_space) distribution_remove_const: 

609 
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" 

610 
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" 

611 
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" 

612 
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" 

613 
and "distribution (\<lambda>x. ()) {()} = 1" 

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changeset

614 
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) 
35977  615 

39097  616 
lemma (in finite_prob_space) setsum_distribution_gen: 
617 
assumes "Z ` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y ` {f x}) \<inter> space M" 

618 
and "inj_on f (X`space M)" 

619 
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" 

620 
unfolding distribution_def assms 

621 
using finite_space assms 

41981
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changeset

622 
by (subst finite_measure_finite_Union[symmetric]) 
39097  623 
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def 
624 
intro!: arg_cong[where f=prob]) 

625 

626 
lemma (in finite_prob_space) setsum_distribution: 

627 
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" 

628 
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" 

629 
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" 

630 
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" 

631 
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" 

632 
by (auto intro!: inj_onI setsum_distribution_gen) 

633 

634 
lemma (in finite_prob_space) uniform_prob: 

635 
assumes "x \<in> space M" 

636 
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}" 

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changeset

637 
shows "prob {x} = 1 / card (space M)" 
39097  638 
proof  
639 
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" 

640 
using assms(2)[OF _ `x \<in> space M`] by blast 

641 
have "1 = prob (space M)" 

642 
using prob_space by auto 

643 
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})" 

41981
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changeset

644 
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] 
39097  645 
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] 
646 
finite_space unfolding disjoint_family_on_def prob_space[symmetric] 

647 
by (auto simp add:setsum_restrict_set) 

648 
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" 

649 
using prob_x by auto 

650 
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp 

651 
finally have one: "1 = real (card (space M)) * prob {x}" 

652 
using real_eq_of_nat by auto 

653 
hence two: "real (card (space M)) \<noteq> 0" by fastsimp 

654 
from one have three: "prob {x} \<noteq> 0" by fastsimp 

655 
thus ?thesis using one two three divide_cancel_right 

656 
by (auto simp:field_simps) 

39092  657 
qed 
35977  658 

39092  659 
lemma (in prob_space) prob_space_subalgebra: 
41545  660 
assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" 
41689
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41661
diff
changeset

661 
and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" 
3e39b0e730d6
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hoelzl
parents:
41661
diff
changeset

662 
shows "prob_space N" 
39092  663 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset

664 
interpret N: measure_space N 
3e39b0e730d6
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hoelzl
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diff
changeset

665 
by (rule measure_space_subalgebra[OF assms]) 
39092  666 
show ?thesis 
41689
3e39b0e730d6
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hoelzl
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diff
changeset

667 
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) 
35977  668 
qed 
669 

39092  670 
lemma (in prob_space) prob_space_of_restricted_space: 
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hoelzl
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changeset

671 
assumes "\<mu> A \<noteq> 0" "A \<in> sets M" 
41689
3e39b0e730d6
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hoelzl
parents:
41661
diff
changeset

672 
shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)" 
3e39b0e730d6
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hoelzl
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diff
changeset

673 
(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

674 
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

675 
interpret A: measure_space "restricted_space A" 
39092  676 
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:
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diff
changeset

677 
interpret A': sigma_algebra ?P 
3e39b0e730d6
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hoelzl
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41661
diff
changeset

678 
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

679 
show "prob_space ?P" 
39092  680 
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

681 
show "measure ?P (space ?P) = 1" 
41981
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changeset

682 
using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto 
cdf7693bbe08
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hoelzl
parents:
41831
diff
changeset

683 
show "positive ?P (measure ?P)" 
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parents:
41831
diff
changeset

684 
proof (simp add: positive_def, safe) 
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parents:
41831
diff
changeset

685 
show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def) 
cdf7693bbe08
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changeset

686 
fix B assume "B \<in> events" 
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41831
diff
changeset

687 
with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M` 
cdf7693bbe08
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hoelzl
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688 
show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int) 
cdf7693bbe08
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hoelzl
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changeset

689 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

690 
show "countably_additive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

691 
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

692 
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

693 
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

694 
{ fix i 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

695 
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

696 
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

697 
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

698 
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

699 
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

700 
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

701 
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

702 
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

703 
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

704 
by (auto simp: suminf_cmult_extreal) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

705 
qed 
39092  706 
qed 
707 
qed 

708 

709 
lemma finite_prob_spaceI: 

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

710 
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

711 
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

712 
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

713 
shows "finite_prob_space M" 
39092  714 
unfolding finite_prob_space_eq 
715 
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

716 
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

717 
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

718 
show "measure M (space M) = 1" by fact 
39092  719 
qed 
36624  720 

721 
lemma (in finite_prob_space) finite_measure_space: 

39097  722 
fixes X :: "'a \<Rightarrow> 'x" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

723 
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

724 
(is "finite_measure_space ?S") 
39092  725 
proof (rule finite_measure_spaceI, simp_all) 
36624  726 
show "finite (X ` space M)" using finite_space by simp 
39097  727 
next 
728 
fix A B :: "'x set" assume "A \<inter> B = {}" 

729 
then show "distribution X (A \<union> B) = distribution X A + distribution X B" 

730 
unfolding distribution_def 

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

731 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

732 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
36624  733 
qed 
734 

39097  735 
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

736 
"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

737 
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def) 
39097  738 

39096  739 
lemma (in finite_prob_space) finite_product_measure_space: 
39097  740 
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" 
39096  741 
assumes "finite s1" "finite s2" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

742 
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

743 
(is "finite_measure_space ?M") 
39097  744 
proof (rule finite_measure_spaceI, simp_all) 
745 
show "finite (s1 \<times> s2)" 

39096  746 
using assms by auto 
39097  747 
next 
748 
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" 

749 
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" 

750 
unfolding distribution_def 

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

751 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

752 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
39096  753 
qed 
754 

39097  755 
lemma (in finite_prob_space) finite_product_measure_space_of_images: 
39096  756 
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

757 
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

758 
measure = extreal \<circ> joint_distribution X Y \<rparr>" 
39096  759 
using finite_space by (auto intro!: finite_product_measure_space) 
760 

40859  761 
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

762 
"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

763 
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

764 
(is "finite_prob_space ?S") 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

765 
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def) 
40859  766 
have "X ` X ` space M \<inter> Y ` Y ` space M \<inter> space M = space M" by auto 
767 
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" 

768 
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) 

769 
qed 

770 

39085  771 
section "Conditional Expectation and Probability" 
772 

773 
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

774 
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

775 
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

776 
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

777 
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

778 
(\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))" 
39083  779 
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

780 
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

781 
interpret P: prob_space N 
41545  782 
using prob_space_subalgebra[OF N] . 
39083  783 

784 
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

785 
let "?Q A" = "integral\<^isup>P M (?f A)" 
39083  786 

787 
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

788 
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

789 
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

790 
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

791 
then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" . 
39083  792 

793 
have "P.absolutely_continuous ?Q" 

794 
unfolding P.absolutely_continuous_def 

41545  795 
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

796 
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

797 
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

798 
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

799 
then show "?Q A = 0" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

800 
by (auto simp add: positive_integral_0_iff_AE) 
39083  801 
qed 
802 
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

803 
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

804 
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)" 
39083  805 
by blast 
41545  806 
with N(2) show ?thesis 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

807 
by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)]) 
39083  808 
qed 
809 

39085  810 
definition (in prob_space) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

811 
"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

812 
\<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  813 

814 
abbreviation (in prob_space) 

39092  815 
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" 
39085  816 

817 
lemma (in prob_space) 

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

818 
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

819 
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

820 
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" 
39085  821 
shows borel_measurable_conditional_expectation: 
41545  822 
"conditional_expectation N X \<in> borel_measurable N" 
823 
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

824 
(\<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

825 
(\<integral>\<^isup>+x. X x * indicator C x \<partial>M)" 
41545  826 
(is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C") 
39085  827 
proof  
828 
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] 

41545  829 
then show "conditional_expectation N X \<in> borel_measurable N" 
39085  830 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
831 

41545  832 
from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C" 
39085  833 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
834 
qed 

835 

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

836 
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

837 
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" 
39091  838 
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

839 
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

840 
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

841 
proof  
39091  842 
interpret M': sigma_algebra M' by fact 
843 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 

844 
from M'.sigma_algebra_vimage[OF this] 

845 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 

846 

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

847 
from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this 
39091  848 

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

849 
have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 
39091  850 
proof 
851 
fix i 

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

852 
from f(1)[of i] have "finite (f i`space M)" and B_ex: 
39091  853 
"\<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

854 
unfolding simple_function_def by auto 
39091  855 
from B_ex[THEN bchoice] guess B .. note B = this 
856 

857 
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" 

858 

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

859 
show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" 
39091  860 
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

861 
show "simple_function M' ?g" using B by auto 
39091  862 

863 
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

864 
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  865 
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

866 
then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i] 
39091  867 
by (subst va.simple_function_indicator_representation) auto 
868 
qed 

869 
qed 

870 
from choice[OF this] guess g .. note g = this 

871 

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

872 
show ?thesis 
39091  873 
proof (intro ballI bexI) 
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

874 
show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'" 
39091  875 
using g by (auto intro: M'.borel_measurable_simple_function) 
876 
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

877 
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

878 
also have "\<dots> = (SUP i. g i (Y x))" 
39091  879 
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

880 
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

881 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

882 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

883 

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

884 
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

885 
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

886 
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

887 

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

888 
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

889 
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

890 
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

891 
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

892 
\<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

893 
proof safe 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

894 
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

895 
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

896 
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

897 
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

898 

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

899 
{ fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

900 
with M'.measurable_vimage_algebra[OF Y] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

901 
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

902 
by (rule measurable_comp) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

903 
moreover assume "\<forall>x\<in>space M. Z x = g (Y x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

904 
then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

905 
g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

906 
by (auto intro!: measurable_cong) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

907 
ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

908 
by simp } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

909 

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

910 
assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

911 
with assms have "(\<lambda>x.  Z x) \<in> borel_measurable M" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

912 
"(\<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:
41831
diff
changeset

913 
by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

914 
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
hoelzl
parents:
41831
diff
changeset

915 
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:
41831
diff
changeset

916 
let "?g x" = "p x  n x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

917 
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:
41831
diff
changeset

918 
proof (intro bexI ballI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

919 
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:
41831
diff
changeset

920 
fix x assume "x \<in> space M" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

921 
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:
41831
diff
changeset

922 
using p n by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

923 
then show "Z x = ?g (Y x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

924 
by (auto split: split_max) 
39091  925 
qed 
926 
qed 

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

927 

35582  928 
end 