author  hoelzl 
Thu, 09 Jun 2011 13:55:11 +0200  
changeset 43339  9ba256ad6781 
parent 42991  3fa22920bf86 
child 43340  60e181c4eae4 
permissions  rwrr 
42148  1 
(* Title: HOL/Probability/Probability_Measure.thy 
42067  2 
Author: Johannes Hölzl, TU München 
3 
Author: Armin Heller, TU München 

4 
*) 

5 

42148  6 
header {*Probability measure*} 
42067  7 

42148  8 
theory Probability_Measure 
42902  9 
imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure Lebesgue_Measure 
35582  10 
begin 
11 

12 
locale prob_space = 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

13 
assumes measure_space_1: "measure M (space M) = 1" 
38656  14 

15 
sublocale prob_space < finite_measure 

16 
proof 

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

17 
from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp 
38656  18 
qed 
19 

40859  20 
abbreviation (in prob_space) "events \<equiv> sets M" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

21 
abbreviation (in prob_space) "prob \<equiv> \<mu>'" 
40859  22 
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'" 
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

23 
abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M" 
35582  24 

40859  25 
definition (in prob_space) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

26 
"distribution X A = \<mu>' (X ` A \<inter> space M)" 
35582  27 

40859  28 
abbreviation (in prob_space) 
36624  29 
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" 
35582  30 

43339  31 
lemma (in prob_space) prob_space_cong: 
32 
assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M" 

33 
shows "prob_space N" 

34 
proof  

35 
interpret N: measure_space N by (intro measure_space_cong assms) 

36 
show ?thesis by default (insert assms measure_space_1, simp) 

37 
qed 

38 

39097  39 
lemma (in prob_space) distribution_cong: 
40 
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" 

41 
shows "distribution X = distribution Y" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

42 
unfolding distribution_def fun_eq_iff 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

43 
using assms by (auto intro!: arg_cong[where f="\<mu>'"]) 
39097  44 

45 
lemma (in prob_space) joint_distribution_cong: 

46 
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" 

47 
assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" 

48 
shows "joint_distribution X Y = joint_distribution X' Y'" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

49 
unfolding distribution_def fun_eq_iff 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

50 
using assms by (auto intro!: arg_cong[where f="\<mu>'"]) 
39097  51 

40859  52 
lemma (in prob_space) distribution_id[simp]: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

53 
"N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

54 
by (auto simp: distribution_def intro!: arg_cong[where f=prob]) 
40859  55 

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

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

57 
using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

58 

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

59 
lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

60 
using bounded_measure[of A] by (simp add: prob_space) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

61 

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

62 
lemma (in prob_space) distribution_positive[simp, intro]: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

63 
"0 \<le> distribution X A" unfolding distribution_def by auto 
35582  64 

43339  65 
lemma (in prob_space) not_zero_less_distribution[simp]: 
66 
"(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0" 

67 
using distribution_positive[of X A] by arith 

68 

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

69 
lemma (in prob_space) joint_distribution_remove[simp]: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

70 
"joint_distribution X X {(x, x)} = distribution X {x}" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

71 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

72 

43339  73 
lemma (in prob_space) not_empty: "space M \<noteq> {}" 
74 
using prob_space empty_measure' by auto 

75 

42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42902
diff
changeset

76 
lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1" 
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42902
diff
changeset

77 
unfolding measure_space_1[symmetric] 
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42902
diff
changeset

78 
using sets_into_space 
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42902
diff
changeset

79 
by (intro measure_mono) auto 
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42902
diff
changeset

80 

43339  81 
lemma (in prob_space) AE_I_eq_1: 
82 
assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M" 

83 
shows "AE x. P x" 

84 
proof (rule AE_I) 

85 
show "\<mu> (space M  {x \<in> space M. P x}) = 0" 

86 
using assms measure_space_1 by (simp add: measure_compl) 

87 
qed (insert assms, auto) 

88 

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

89 
lemma (in prob_space) distribution_1: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

90 
"distribution X A \<le> 1" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

91 
unfolding distribution_def by simp 
35582  92 

40859  93 
lemma (in prob_space) prob_compl: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

94 
assumes A: "A \<in> events" 
38656  95 
shows "prob (space M  A) = 1  prob A" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

96 
using finite_measure_compl[OF A] by (simp add: prob_space) 
35582  97 

40859  98 
lemma (in prob_space) prob_space_increasing: "increasing M prob" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

99 
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" 

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

107 
using finite_measure_subadditive[of s t] assms by auto 
35582  108 
moreover have "prob (s \<union> t) \<ge> prob s" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

109 
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" 
134 
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))" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

140 
by (rule finite_measure_UNION[OF assms(1,3)]) 
38656  141 
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

142 
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" 

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

152 
using finite_measure_countably_subadditive[OF assms(1)] 
38656  153 
by (simp add: assms(2) suminf_zero summable_zero) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

154 
qed simp 
35582  155 

40859  156 
lemma (in prob_space) prob_equiprobable_finite_unions: 
38656  157 
assumes "s \<in> events" 
158 
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" 

35582  159 
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" 
38656  160 
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" 
35582  161 
proof (cases "s = {}") 
38656  162 
case False hence "\<exists> x. x \<in> s" by blast 
35582  163 
from someI_ex[OF this] assms 
164 
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast 

165 
have "prob s = (\<Sum> x \<in> s. prob {x})" 

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

166 
using finite_measure_finite_singleton[OF s_finite] by simp 
35582  167 
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto 
38656  168 
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" 
169 
using setsum_constant assms by (simp add: real_eq_of_nat) 

35582  170 
finally show ?thesis by simp 
38656  171 
qed simp 
35582  172 

40859  173 
lemma (in prob_space) prob_real_sum_image_fn: 
35582  174 
assumes "e \<in> events" 
175 
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" 

176 
assumes "finite s" 

38656  177 
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" 
178 
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" 

35582  179 
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" 
180 
proof  

38656  181 
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" 
182 
using `e \<in> events` sets_into_space upper by blast 

183 
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp 

184 
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" 

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

185 
proof (rule finite_measure_finite_Union) 
38656  186 
show "finite s" by fact 
187 
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact 

188 
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" 

189 
using disjoint by (auto simp: disjoint_family_on_def) 

190 
qed 

191 
finally show ?thesis . 

35582  192 
qed 
193 

42199  194 
lemma (in prob_space) prob_space_vimage: 
195 
assumes S: "sigma_algebra S" 

196 
assumes T: "T \<in> measure_preserving M S" 

197 
shows "prob_space S" 

35582  198 
proof  
42199  199 
interpret S: measure_space S 
200 
using S and T by (rule measure_space_vimage) 

38656  201 
show ?thesis 
42199  202 
proof 
203 
from T[THEN measure_preservingD2] 

204 
have "T ` space S \<inter> space M = space M" 

205 
by (auto simp: measurable_def) 

206 
with T[THEN measure_preservingD, of "space S", symmetric] 

207 
show "measure S (space S) = 1" 

208 
using measure_space_1 by simp 

35582  209 
qed 
210 
qed 

211 

42988  212 
lemma prob_space_unique_Int_stable: 
213 
fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set" 

214 
assumes E: "Int_stable E" "space E \<in> sets E" 

215 
and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)" 

216 
and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)" 

217 
and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X" 

218 
assumes "X \<in> sets (sigma E)" 

219 
shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X" 

220 
proof  

221 
interpret M!: prob_space M by fact 

222 
interpret N!: prob_space N by fact 

223 
have "measure M X = measure N X" 

224 
proof (rule measure_unique_Int_stable[OF `Int_stable E`]) 

225 
show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E" 

226 
using E M N by auto 

227 
show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>" 

228 
using M.measure_space_1 by simp 

229 
show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>" 

230 
using E M N by (auto intro!: M.measure_space_cong) 

231 
show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>" 

232 
using E M N by (auto intro!: N.measure_space_cong) 

233 
{ fix X assume "X \<in> sets E" 

234 
then have "X \<in> sets (sigma E)" 

235 
by (auto simp: sets_sigma sigma_sets.Basic) 

236 
with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X" 

237 
by (simp add: M.finite_measure_eq N.finite_measure_eq) } 

238 
qed fact 

239 
with `X \<in> sets (sigma E)` M N show ?thesis 

240 
by (simp add: M.finite_measure_eq N.finite_measure_eq) 

241 
qed 

242 

42199  243 
lemma (in prob_space) distribution_prob_space: 
244 
assumes X: "random_variable S X" 

245 
shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" (is "prob_space ?S") 

246 
proof (rule prob_space_vimage) 

247 
show "X \<in> measure_preserving M ?S" 

248 
using X 

249 
unfolding measure_preserving_def distribution_def_raw 

250 
by (auto simp: finite_measure_eq measurable_sets) 

251 
show "sigma_algebra ?S" using X by simp 

252 
qed 

253 

40859  254 
lemma (in prob_space) AE_distribution: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

255 
assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x" 
40859  256 
shows "AE x. Q (X x)" 
257 
proof  

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

258 
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space) 
40859  259 
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" 
260 
using assms unfolding X.almost_everywhere_def by auto 

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

261 
from X[unfolded measurable_def] N show "AE x. Q (X x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

262 
by (intro AE_I'[where N="X ` N \<inter> space M"]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

263 
(auto simp: finite_measure_eq distribution_def measurable_sets) 
40859  264 
qed 
265 

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

266 
lemma (in prob_space) distribution_eq_integral: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

267 
"random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X ` A \<inter> space M))" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

268 
using finite_measure_eq[of "X ` A \<inter> space M"] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

269 
by (auto simp: measurable_sets distribution_def) 
35582  270 

43339  271 
lemma (in prob_space) expectation_less: 
272 
assumes [simp]: "integrable M X" 

273 
assumes gt: "\<forall>x\<in>space M. X x < b" 

274 
shows "expectation X < b" 

275 
proof  

276 
have "expectation X < expectation (\<lambda>x. b)" 

277 
using gt measure_space_1 

278 
by (intro integral_less_AE) auto 

279 
then show ?thesis using prob_space by simp 

280 
qed 

281 

282 
lemma (in prob_space) expectation_greater: 

283 
assumes [simp]: "integrable M X" 

284 
assumes gt: "\<forall>x\<in>space M. a < X x" 

285 
shows "a < expectation X" 

286 
proof  

287 
have "expectation (\<lambda>x. a) < expectation X" 

288 
using gt measure_space_1 

289 
by (intro integral_less_AE) auto 

290 
then show ?thesis using prob_space by simp 

291 
qed 

292 

293 
lemma convex_le_Inf_differential: 

294 
fixes f :: "real \<Rightarrow> real" 

295 
assumes "convex_on I f" 

296 
assumes "x \<in> interior I" "y \<in> I" 

297 
shows "f y \<ge> f x + Inf ((\<lambda>t. (f x  f t) / (x  t)) ` ({x<..} \<inter> I)) * (y  x)" 

298 
(is "_ \<ge> _ + Inf (?F x) * (y  x)") 

299 
proof  

300 
show ?thesis 

301 
proof (cases rule: linorder_cases) 

302 
assume "x < y" 

303 
moreover 

304 
have "open (interior I)" by auto 

305 
from openE[OF this `x \<in> interior I`] guess e . note e = this 

306 
moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)" 

307 
ultimately have "x < t" "t < y" "t \<in> ball x e" 

308 
by (auto simp: mem_ball dist_real_def field_simps split: split_min) 

309 
with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto 

310 

311 
have "open (interior I)" by auto 

312 
from openE[OF this `x \<in> interior I`] guess e . 

313 
moreover def K \<equiv> "x  e / 2" 

314 
with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def) 

315 
ultimately have "K \<in> I" "K < x" "x \<in> I" 

316 
using interior_subset[of I] `x \<in> interior I` by auto 

317 

318 
have "Inf (?F x) \<le> (f x  f y) / (x  y)" 

319 
proof (rule Inf_lower2) 

320 
show "(f x  f t) / (x  t) \<in> ?F x" 

321 
using `t \<in> I` `x < t` by auto 

322 
show "(f x  f t) / (x  t) \<le> (f x  f y) / (x  y)" 

323 
using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff) 

324 
next 

325 
fix y assume "y \<in> ?F x" 

326 
with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]] 

327 
show "(f K  f x) / (K  x) \<le> y" by auto 

328 
qed 

329 
then show ?thesis 

330 
using `x < y` by (simp add: field_simps) 

331 
next 

332 
assume "y < x" 

333 
moreover 

334 
have "open (interior I)" by auto 

335 
from openE[OF this `x \<in> interior I`] guess e . note e = this 

336 
moreover def t \<equiv> "x + e / 2" 

337 
ultimately have "x < t" "t \<in> ball x e" 

338 
by (auto simp: mem_ball dist_real_def field_simps) 

339 
with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto 

340 

341 
have "(f x  f y) / (x  y) \<le> Inf (?F x)" 

342 
proof (rule Inf_greatest) 

343 
have "(f x  f y) / (x  y) = (f y  f x) / (y  x)" 

344 
using `y < x` by (auto simp: field_simps) 

345 
also 

346 
fix z assume "z \<in> ?F x" 

347 
with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]] 

348 
have "(f y  f x) / (y  x) \<le> z" by auto 

349 
finally show "(f x  f y) / (x  y) \<le> z" . 

350 
next 

351 
have "open (interior I)" by auto 

352 
from openE[OF this `x \<in> interior I`] guess e . note e = this 

353 
then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def) 

354 
with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto 

355 
then show "?F x \<noteq> {}" by blast 

356 
qed 

357 
then show ?thesis 

358 
using `y < x` by (simp add: field_simps) 

359 
qed simp 

360 
qed 

361 

362 
lemma (in prob_space) jensens_inequality: 

363 
fixes a b :: real 

364 
assumes X: "integrable M X" "X ` space M \<subseteq> I" 

365 
assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV" 

366 
assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q" 

367 
shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" 

368 
proof  

369 
let "?F x" = "Inf ((\<lambda>t. (q x  q t) / (x  t)) ` ({x<..} \<inter> I))" 

370 
from not_empty X(2) have "I \<noteq> {}" by auto 

371 

372 
from I have "open I" by auto 

373 

374 
note I 

375 
moreover 

376 
{ assume "I \<subseteq> {a <..}" 

377 
with X have "a < expectation X" 

378 
by (intro expectation_greater) auto } 

379 
moreover 

380 
{ assume "I \<subseteq> {..< b}" 

381 
with X have "expectation X < b" 

382 
by (intro expectation_less) auto } 

383 
ultimately have "expectation X \<in> I" 

384 
by (elim disjE) (auto simp: subset_eq) 

385 
moreover 

386 
{ fix y assume y: "y \<in> I" 

387 
with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y  x)) ` I) = q y" 

388 
by (auto intro!: Sup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) } 

389 
ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X  x)) ` I)" 

390 
by simp 

391 
also have "\<dots> \<le> expectation (\<lambda>w. q (X w))" 

392 
proof (rule Sup_least) 

393 
show "(\<lambda>x. q x + ?F x * (expectation X  x)) ` I \<noteq> {}" 

394 
using `I \<noteq> {}` by auto 

395 
next 

396 
fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X  x)) ` I" 

397 
then guess x .. note x = this 

398 
have "q x + ?F x * (expectation X  x) = expectation (\<lambda>w. q x + ?F x * (X w  x))" 

399 
using prob_space 

400 
by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X) 

401 
also have "\<dots> \<le> expectation (\<lambda>w. q (X w))" 

402 
using `x \<in> I` `open I` X(2) 

403 
by (intro integral_mono integral_add integral_cmult integral_diff 

404 
lebesgue_integral_const X q convex_le_Inf_differential) 

405 
(auto simp: interior_open) 

406 
finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto 

407 
qed 

408 
finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" . 

409 
qed 

410 

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

411 
lemma (in prob_space) distribution_eq_translated_integral: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

412 
assumes "random_variable S X" "A \<in> sets S" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

413 
shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)" 
35582  414 
proof  
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

415 
interpret S: prob_space "S\<lparr>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

416 
using assms(1) by (rule distribution_prob_space) 
35582  417 
show ?thesis 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

418 
using S.positive_integral_indicator(1)[of A] assms by simp 
35582  419 
qed 
420 

40859  421 
lemma (in prob_space) finite_expectation1: 
422 
assumes f: "finite (X`space M)" and rv: "random_variable borel X" 

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

423 
shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X ` {r} \<inter> space M))" (is "_ = ?r") 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

424 
proof (subst integral_on_finite) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

425 
show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

426 
show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X ` {r} \<inter> space M))) = ?r" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

427 
"\<And>x. \<mu> (X ` {x} \<inter> space M) \<noteq> \<infinity>" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

428 
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto 
38656  429 
qed 
35582  430 

40859  431 
lemma (in prob_space) finite_expectation: 
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

432 
assumes "finite (X`space M)" "random_variable borel X" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

433 
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" 
38656  434 
using assms unfolding distribution_def using finite_expectation1 by auto 
435 

40859  436 
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: 
35582  437 
assumes "{x} \<in> events" 
38656  438 
assumes "prob {x} = 1" 
35582  439 
assumes "{y} \<in> events" 
440 
assumes "y \<noteq> x" 

441 
shows "prob {y} = 0" 

442 
using prob_one_inter[of "{y}" "{x}"] assms by auto 

443 

40859  444 
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" 
38656  445 
unfolding distribution_def by simp 
446 

40859  447 
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" 
38656  448 
proof  
449 
have "X ` X ` space M \<inter> space M = space M" by auto 

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

450 
thus ?thesis unfolding distribution_def by (simp add: prob_space) 
38656  451 
qed 
452 

40859  453 
lemma (in prob_space) distribution_one: 
454 
assumes "random_variable M' X" and "A \<in> sets M'" 

38656  455 
shows "distribution X A \<le> 1" 
456 
proof  

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

457 
have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

458 
using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

459 
thus ?thesis by (simp add: prob_space) 
38656  460 
qed 
461 

40859  462 
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: 
35582  463 
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" 
38656  464 
(is "random_variable ?S X") 
465 
assumes "distribution X {x} = 1" 

35582  466 
assumes "y \<noteq> x" 
467 
shows "distribution X {y} = 0" 

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

468 
proof cases 
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

469 
{ fix x have "X ` {x} \<inter> space M \<in> sets 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

470 
proof cases 
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

471 
assume "x \<in> X`space M" with X show ?thesis 
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

472 
by (auto simp: measurable_def image_iff) 
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

473 
next 
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

474 
assume "x \<notin> X`space M" then have "X ` {x} \<inter> space M = {}" by 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

475 
then show ?thesis by 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

476 
qed } note single = this 
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

477 
have "X ` {x} \<inter> space M  X ` {y} \<inter> space M = X ` {x} \<inter> space 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

478 
"X ` {y} \<inter> space M \<inter> (X ` {x} \<inter> space 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

479 
using `y \<noteq> x` by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

480 
with finite_measure_inter_full_set[OF single single, of x y] assms(2) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

481 
show ?thesis by (auto simp: distribution_def prob_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

482 
next 
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

483 
assume "{y} \<notin> sets ?S" 
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

484 
then have "X ` {y} \<inter> space M = {}" by 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

485 
thus "distribution X {y} = 0" unfolding distribution_def by auto 
35582  486 
qed 
487 

40859  488 
lemma (in prob_space) joint_distribution_Times_le_fst: 
489 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

490 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

491 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

492 
unfolding distribution_def 

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

493 
proof (intro finite_measure_mono) 
40859  494 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
495 
show "X ` A \<inter> space M \<in> events" 

496 
using X A unfolding measurable_def by simp 

497 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 

498 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

499 
qed 

500 

501 
lemma (in prob_space) joint_distribution_commute: 

502 
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)" 

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

503 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  504 

505 
lemma (in prob_space) joint_distribution_Times_le_snd: 

506 
assumes X: "random_variable MX X" and Y: "random_variable MY Y" 

507 
and A: "A \<in> sets MX" and B: "B \<in> sets MY" 

508 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

509 
using assms 

510 
by (subst joint_distribution_commute) 

511 
(simp add: swap_product joint_distribution_Times_le_fst) 

512 

513 
lemma (in prob_space) random_variable_pairI: 

514 
assumes "random_variable MX X" 

515 
assumes "random_variable MY Y" 

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

516 
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  517 
proof 
518 
interpret MX: sigma_algebra MX using assms by simp 

519 
interpret MY: sigma_algebra MY using assms by simp 

520 
interpret P: pair_sigma_algebra MX MY by default 

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

521 
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  522 
have sa: "sigma_algebra M" by default 
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

523 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  524 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  525 
qed 
526 

527 
lemma (in prob_space) joint_distribution_commute_singleton: 

528 
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}" 

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

529 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  530 

531 
lemma (in prob_space) joint_distribution_assoc_singleton: 

532 
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 

533 
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}" 

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

534 
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) 
40859  535 

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

536 
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 
40859  537 

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

538 
sublocale pair_prob_space \<subseteq> pair_sigma_finite 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

539 

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 
sublocale pair_prob_space \<subseteq> P: 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

541 
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) 
40859  542 

543 
lemma countably_additiveI[case_names countably]: 

544 
assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow> 

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

545 
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" 
40859  546 
shows "countably_additive M \<mu>" 
547 
using assms unfolding countably_additive_def by auto 

548 

549 
lemma (in prob_space) joint_distribution_prob_space: 

550 
assumes "random_variable MX X" "random_variable MY Y" 

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

551 
shows "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

552 
using random_variable_pairI[OF assms] by (rule distribution_prob_space) 
40859  553 

42988  554 

555 
locale finite_product_prob_space = 

556 
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" 

557 
and I :: "'i set" 

558 
assumes prob_space: "\<And>i. prob_space (M i)" and finite_index: "finite I" 

559 

560 
sublocale finite_product_prob_space \<subseteq> M: prob_space "M i" for i 

561 
by (rule prob_space) 

562 

563 
sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite M I 

564 
by default (rule finite_index) 

565 

566 
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i" 

567 
proof qed (simp add: measure_times M.measure_space_1 setprod_1) 

568 

569 
lemma (in finite_product_prob_space) prob_times: 

570 
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)" 

571 
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))" 

572 
proof  

573 
have "extreal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)" 

574 
using X by (intro finite_measure_eq[symmetric] in_P) auto 

575 
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))" 

576 
using measure_times X by simp 

577 
also have "\<dots> = extreal (\<Prod>i\<in>I. M.\<mu>' i (X i))" 

578 
using X by (simp add: M.finite_measure_eq setprod_extreal) 

579 
finally show ?thesis by simp 

580 
qed 

581 

582 
lemma (in prob_space) random_variable_restrict: 

583 
assumes I: "finite I" 

584 
assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)" 

585 
shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)" 

586 
proof 

587 
{ fix i assume "i \<in> I" 

588 
with X interpret N: sigma_algebra "N i" by simp 

589 
have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) } 

590 
note N_closed = this 

591 
then show "sigma_algebra (Pi\<^isub>M I N)" 

592 
by (simp add: product_algebra_def) 

593 
(intro sigma_algebra_sigma product_algebra_generator_sets_into_space) 

594 
show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)" 

595 
using X by (intro measurable_restrict[OF I N_closed]) auto 

596 
qed 

597 

40859  598 
section "Probability spaces on finite sets" 
35582  599 

35977  600 
locale finite_prob_space = prob_space + finite_measure_space 
601 

40859  602 
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" 
603 

604 
lemma (in prob_space) finite_random_variableD: 

605 
assumes "finite_random_variable M' X" shows "random_variable M' X" 

606 
proof  

607 
interpret M': finite_sigma_algebra M' using assms by simp 

608 
then show "random_variable M' X" using assms by simp default 

609 
qed 

610 

611 
lemma (in prob_space) distribution_finite_prob_space: 

612 
assumes "finite_random_variable MX X" 

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

613 
shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)" 
40859  614 
proof  
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

615 
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" 
40859  616 
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) 
617 
interpret MX: finite_sigma_algebra MX 

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

618 
using assms by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

619 
show ?thesis by default (simp_all add: MX.finite_space) 
40859  620 
qed 
621 

622 
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]: 

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

623 
assumes "simple_function M X" 
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

624 
shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X" 
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

625 
(is "finite_random_variable ?X _") 
40859  626 
proof (intro conjI) 
627 
have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp 

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

628 
interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow) 
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

629 
show "finite_sigma_algebra ?X" 
40859  630 
by default auto 
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

631 
show "X \<in> measurable M ?X" 
40859  632 
proof (unfold measurable_def, clarsimp) 
633 
fix A assume A: "A \<subseteq> X`space M" 

634 
then have "finite A" by (rule finite_subset) simp 

635 
then have "X ` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" 

636 
unfolding vimage_UN UN_extend_simps 

637 
apply (rule finite_UN) 

638 
using A assms unfolding simple_function_def by auto 

639 
then show "X ` A \<inter> space M \<in> events" by simp 

640 
qed 

641 
qed 

642 

643 
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]: 

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

644 
assumes "simple_function M X" 
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

645 
shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X" 
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

646 
using simple_function_imp_finite_random_variable[OF assms, of ext] 
40859  647 
by (auto dest!: finite_random_variableD) 
648 

649 
lemma (in prob_space) sum_over_space_real_distribution: 

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

650 
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" 
40859  651 
unfolding distribution_def prob_space[symmetric] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

652 
by (subst finite_measure_finite_Union[symmetric]) 
40859  653 
(auto simp add: disjoint_family_on_def simple_function_def 
654 
intro!: arg_cong[where f=prob]) 

655 

656 
lemma (in prob_space) finite_random_variable_pairI: 

657 
assumes "finite_random_variable MX X" 

658 
assumes "finite_random_variable MY Y" 

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 
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" 
40859  660 
proof 
661 
interpret MX: finite_sigma_algebra MX using assms by simp 

662 
interpret MY: finite_sigma_algebra MY using assms by simp 

663 
interpret P: pair_finite_sigma_algebra MX MY by default 

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

664 
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default 
40859  665 
have sa: "sigma_algebra M" by default 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

666 
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" 
41095  667 
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) 
40859  668 
qed 
669 

670 
lemma (in prob_space) finite_random_variable_imp_sets: 

671 
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" 

672 
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp 

673 

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

674 
lemma (in prob_space) finite_random_variable_measurable: 
40859  675 
assumes X: "finite_random_variable MX X" shows "X ` A \<inter> space M \<in> events" 
676 
proof  

677 
interpret X: finite_sigma_algebra MX using X by simp 

678 
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X ` A \<inter> space M \<in> events" and 

679 
"X \<in> space M \<rightarrow> space MX" 

680 
by (auto simp: measurable_def) 

681 
then have *: "X ` A \<inter> space M = X ` (A \<inter> space MX) \<inter> space M" 

682 
by auto 

683 
show "X ` A \<inter> space M \<in> events" 

684 
unfolding * by (intro vimage) auto 

685 
qed 

686 

687 
lemma (in prob_space) joint_distribution_finite_Times_le_fst: 

688 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

689 
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" 

690 
unfolding distribution_def 

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

691 
proof (intro finite_measure_mono) 
40859  692 
show "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M \<subseteq> X ` A \<inter> space M" by force 
693 
show "X ` A \<inter> space M \<in> events" 

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

694 
using finite_random_variable_measurable[OF X] . 
40859  695 
have *: "(\<lambda>x. (X x, Y x)) ` (A \<times> B) \<inter> space M = 
696 
(X ` A \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

697 
qed 

698 

699 
lemma (in prob_space) joint_distribution_finite_Times_le_snd: 

700 
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" 

701 
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" 

702 
using assms 

703 
by (subst joint_distribution_commute) 

704 
(simp add: swap_product joint_distribution_finite_Times_le_fst) 

705 

706 
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

707 
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" 
40859  708 
assumes "finite_random_variable MX X" "finite_random_variable MY Y" 
709 
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

710 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

711 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

712 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

713 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

714 
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

715 
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] 

716 
using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"] 

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

717 
by (auto intro: antisym) 
40859  718 

719 
lemma (in prob_space) setsum_joint_distribution: 

720 
assumes X: "finite_random_variable MX X" 

721 
assumes Y: "random_variable MY Y" "B \<in> sets MY" 

722 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" 

723 
unfolding distribution_def 

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

724 
proof (subst finite_measure_finite_Union[symmetric]) 
40859  725 
interpret MX: finite_sigma_algebra MX using X by auto 
726 
show "finite (space MX)" using MX.finite_space . 

727 
let "?d i" = "(\<lambda>x. (X x, Y x)) ` ({i} \<times> B) \<inter> space M" 

728 
{ fix i assume "i \<in> space MX" 

729 
moreover have "?d i = (X ` {i} \<inter> space M) \<inter> (Y ` B \<inter> space M)" by auto 

730 
ultimately show "?d i \<in> events" 

731 
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y 

732 
using MX.sets_eq_Pow by auto } 

733 
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

734 
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

735 
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) 
40859  736 
qed 
737 

738 
lemma (in prob_space) setsum_joint_distribution_singleton: 

739 
assumes X: "finite_random_variable MX X" 

740 
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" 

741 
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" 

742 
using setsum_joint_distribution[OF X 

743 
finite_random_variableD[OF Y(1)] 

744 
finite_random_variable_imp_sets[OF Y]] by simp 

745 

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

746 
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 
40859  747 

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

748 
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

749 
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

750 
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default 
40859  751 

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

752 
locale product_finite_prob_space = 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

753 
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

754 
and I :: "'i set" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

755 
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

756 

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

757 
sublocale product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

758 
sublocale product_finite_prob_space \<subseteq> finite_product_sigma_finite M I by default (rule finite_index) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

759 
sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

760 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

761 
show "\<mu> (space P) = 1" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

762 
using measure_times[OF M.top] M.measure_space_1 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

763 
by (simp add: setprod_1 space_product_algebra) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

764 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

765 

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

766 
lemma funset_eq_UN_fun_upd_I: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

767 
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

768 
and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

769 
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

770 
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

771 
proof safe 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

772 
fix f assume f: "f \<in> F (insert a A)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

773 
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

774 
proof (rule UN_I[of "f(a := d)"]) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

775 
show "f(a := d) \<in> F A" using *[OF f] . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

776 
show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

777 
proof (rule image_eqI[of _ _ "f a"]) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

778 
show "f a \<in> G (f(a := d))" using **[OF f] . 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

779 
qed simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

780 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

781 
next 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

782 
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

783 
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

784 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

785 

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

786 
lemma extensional_funcset_insert_eq[simp]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

787 
assumes "a \<notin> A" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

788 
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

789 
apply (rule funset_eq_UN_fun_upd_I) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

790 
using assms 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

791 
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

792 

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

793 
lemma finite_extensional_funcset[simp, intro]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

794 
assumes "finite A" "finite B" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

795 
shows "finite (extensional A \<inter> (A \<rightarrow> B))" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

796 
using assms by induct (auto simp: extensional_funcset_insert_eq) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

797 

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

798 
lemma finite_PiE[simp, intro]: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

799 
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

800 
shows "finite (Pi\<^isub>E A B)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

801 
proof  
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

802 
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

803 
show ?thesis 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

804 
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

805 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

806 

42892  807 
lemma (in product_finite_prob_space) singleton_eq_product: 
808 
assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})" 

809 
proof (safe intro!: ext[of _ x]) 

810 
fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I" 

811 
with x show "y i = x i" 

812 
by (cases "i \<in> I") (auto simp: extensional_def) 

813 
qed (insert x, auto) 

814 

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

815 
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

816 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

817 
show "finite (space P)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

818 
using finite_index M.finite_space by auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

819 

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

820 
{ fix x assume "x \<in> space P" 
42892  821 
with this[THEN singleton_eq_product] 
822 
have "{x} \<in> sets P" 

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

823 
by (auto intro!: in_P) } 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

824 
note x_in_P = this 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

825 

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

826 
have "Pow (space P) \<subseteq> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

827 
proof 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

828 
fix X assume "X \<in> Pow (space P)" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

829 
moreover then have "finite X" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

830 
using `finite (space P)` by (blast intro: finite_subset) 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

831 
ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

832 
by (intro finite_UN x_in_P) auto 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

833 
then show "X \<in> sets P" by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

834 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

835 
with space_closed show [simp]: "sets P = Pow (space P)" .. 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

836 

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

837 
{ fix x assume "x \<in> space P" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

838 
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

839 
then show "\<mu> {x} \<noteq> \<infinity>" 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

840 
using measure_space_1 by auto } 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

841 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

842 

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

843 
lemma (in product_finite_prob_space) measure_finite_times: 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

844 
"(\<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

845 
by (rule measure_times) simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

846 

42892  847 
lemma (in product_finite_prob_space) measure_singleton_times: 
848 
assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})" 

849 
unfolding singleton_eq_product[OF x] using x 

850 
by (intro measure_finite_times) auto 

851 

852 
lemma (in product_finite_prob_space) prob_finite_times: 

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

853 
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

854 
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

855 
proof  
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

856 
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

857 
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

858 
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

859 
using measure_finite_times X by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

860 
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

861 
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

862 
finally show ?thesis by simp 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

863 
qed 
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset

864 

42892  865 
lemma (in product_finite_prob_space) prob_singleton_times: 
866 
assumes x: "x \<in> space P" 

867 
shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})" 

868 
unfolding singleton_eq_product[OF x] using x 

869 
by (intro prob_finite_times) auto 

870 

871 
lemma (in product_finite_prob_space) prob_finite_product: 

872 
"A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})" 

873 
by (auto simp add: finite_measure_singleton prob_singleton_times 

874 
simp del: space_product_algebra 

875 
intro!: setsum_cong prob_singleton_times) 

876 

40859  877 
lemma (in prob_space) joint_distribution_finite_prob_space: 
878 
assumes X: "finite_random_variable MX X" 

879 
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

880 
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

881 
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) 
40859  882 

36624  883 
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

884 
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" 
36624  885 
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def 
886 
by auto 

887 

38656  888 
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" 
889 
using measure_space_1 sum_over_space by simp 

36624  890 

891 
lemma (in finite_prob_space) joint_distribution_restriction_fst: 

892 
"joint_distribution X Y A \<le> distribution X (fst ` A)" 

893 
unfolding distribution_def 

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

894 
proof (safe intro!: finite_measure_mono) 
36624  895 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
896 
show "x \<in> X ` fst ` A" 

897 
by (auto intro!: image_eqI[OF _ *]) 

898 
qed (simp_all add: sets_eq_Pow) 

899 

900 
lemma (in finite_prob_space) joint_distribution_restriction_snd: 

901 
"joint_distribution X Y A \<le> distribution Y (snd ` A)" 

902 
unfolding distribution_def 

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

903 
proof (safe intro!: finite_measure_mono) 
36624  904 
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" 
905 
show "x \<in> Y ` snd ` A" 

906 
by (auto intro!: image_eqI[OF _ *]) 

907 
qed (simp_all add: sets_eq_Pow) 

908 

909 
lemma (in finite_prob_space) distribution_order: 

910 
shows "0 \<le> distribution X x'" 

911 
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" 

912 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" 

913 
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" 

914 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" 

915 
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" 

916 
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" 

917 
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

918 
using 
36624  919 
joint_distribution_restriction_fst[of X Y "{(x, y)}"] 
920 
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

921 
by (auto intro: antisym) 
36624  922 

39097  923 
lemma (in finite_prob_space) distribution_mono: 
924 
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

925 
shows "distribution X x \<le> distribution Y y" 

926 
unfolding distribution_def 

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

927 
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) 
39097  928 

929 
lemma (in finite_prob_space) distribution_mono_gt_0: 

930 
assumes gt_0: "0 < distribution X x" 

931 
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

932 
shows "0 < distribution Y y" 

933 
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) 

934 

935 
lemma (in finite_prob_space) sum_over_space_distrib: 

936 
"(\<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

937 
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

938 
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

939 
(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

940 
intro!: arg_cong[where f=\<mu>']) 
39097  941 

942 
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

943 
"(\<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

944 
using prob_space finite_space 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

945 
by (subst (asm) finite_measure_finite_singleton) auto 
39097  946 

947 
lemma (in prob_space) distribution_remove_const: 

948 
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" 

949 
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" 

950 
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" 

951 
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" 

952 
and "distribution (\<lambda>x. ()) {()} = 1" 

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

953 
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) 
35977  954 

39097  955 
lemma (in finite_prob_space) setsum_distribution_gen: 
956 
assumes "Z ` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y ` {f x}) \<inter> space M" 

957 
and "inj_on f (X`space M)" 

958 
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" 

959 
unfolding distribution_def assms 

960 
using finite_space assms 

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

961 
by (subst finite_measure_finite_Union[symmetric]) 
39097  962 
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def 
963 
intro!: arg_cong[where f=prob]) 

964 

965 
lemma (in finite_prob_space) setsum_distribution: 

966 
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" 

967 
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" 

968 
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" 

969 
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" 

970 
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" 

971 
by (auto intro!: inj_onI setsum_distribution_gen) 

972 

973 
lemma (in finite_prob_space) uniform_prob: 

974 
assumes "x \<in> space M" 

975 
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

976 
shows "prob {x} = 1 / card (space M)" 
39097  977 
proof  
978 
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" 

979 
using assms(2)[OF _ `x \<in> space M`] by blast 

980 
have "1 = prob (space M)" 

981 
using prob_space by auto 

982 
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

983 
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] 
39097  984 
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] 
985 
finite_space unfolding disjoint_family_on_def prob_space[symmetric] 

986 
by (auto simp add:setsum_restrict_set) 

987 
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" 

988 
using prob_x by auto 

989 
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp 

990 
finally have one: "1 = real (card (space M)) * prob {x}" 

991 
using real_eq_of_nat by auto 

43339  992 
hence two: "real (card (space M)) \<noteq> 0" by fastsimp 
39097  993 
from one have three: "prob {x} \<noteq> 0" by fastsimp 
994 
thus ?thesis using one two three divide_cancel_right 

995 
by (auto simp:field_simps) 

39092  996 
qed 
35977  997 

39092  998 
lemma (in prob_space) prob_space_subalgebra: 
41545  999 
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

1000 
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

1001 
shows "prob_space N" 
39092  1002 
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

1003 
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

1004 
by (rule measure_space_subalgebra[OF assms]) 
39092  1005 
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

1006 
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) 
35977  1007 
qed 
1008 

39092  1009 
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

1010 
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

1011 
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

1012 
(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

1013 
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

1014 
interpret A: measure_space "restricted_space A" 
39092  1015 
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

1016 
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

1017 
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

1018 
show "prob_space ?P" 
39092  1019 
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

1020 
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

1021 
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

1022 
show "positive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1023 
proof (simp add: positive_def, safe) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1024 
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

1025 
fix B assume "B \<in> events" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1026 
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

1027 
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

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

1029 
show "countably_additive ?P (measure ?P)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1030 
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

1031 
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

1032 
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

1033 
{ fix i 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1034 
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

1035 
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

1036 
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

1037 
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

1038 
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

1039 
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

1040 
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

1041 
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

1042 
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

1043 
by (auto simp: suminf_cmult_extreal) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1044 
qed 
39092  1045 
qed 
1046 
qed 

1047 

1048 
lemma finite_prob_spaceI: 

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

1049 
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

1050 
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

1051 
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

1052 
shows "finite_prob_space M" 
39092  1053 
unfolding finite_prob_space_eq 
1054 
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

1055 
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

1056 
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

1057 
show "measure M (space M) = 1" by fact 
39092  1058 
qed 
36624  1059 

1060 
lemma (in finite_prob_space) finite_measure_space: 

39097  1061 
fixes X :: "'a \<Rightarrow> 'x" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1062 
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

1063 
(is "finite_measure_space ?S") 
39092  1064 
proof (rule finite_measure_spaceI, simp_all) 
36624  1065 
show "finite (X ` space M)" using finite_space by simp 
39097  1066 
next 
1067 
fix A B :: "'x set" assume "A \<inter> B = {}" 

1068 
then show "distribution X (A \<union> B) = distribution X A + distribution X B" 

1069 
unfolding distribution_def 

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

1070 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1071 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
36624  1072 
qed 
1073 

39097  1074 
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

1075 
"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

1076 
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def) 
39097  1077 

39096  1078 
lemma (in finite_prob_space) finite_product_measure_space: 
39097  1079 
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" 
39096  1080 
assumes "finite s1" "finite s2" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1081 
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

1082 
(is "finite_measure_space ?M") 
39097  1083 
proof (rule finite_measure_spaceI, simp_all) 
1084 
show "finite (s1 \<times> s2)" 

39096  1085 
using assms by auto 
39097  1086 
next 
1087 
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" 

1088 
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" 

1089 
unfolding distribution_def 

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

1090 
by (subst finite_measure_Union[symmetric]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1091 
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) 
39096  1092 
qed 
1093 

39097  1094 
lemma (in finite_prob_space) finite_product_measure_space_of_images: 
39096  1095 
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

1096 
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

1097 
measure = extreal \<circ> joint_distribution X Y \<rparr>" 
39096  1098 
using finite_space by (auto intro!: finite_product_measure_space) 
1099 

40859  1100 
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

1101 
"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

1102 
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

1103 
(is "finite_prob_space ?S") 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1104 
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def) 
40859  1105 
have "X ` X ` space M \<inter> Y ` Y ` space M \<inter> space M = space M" by auto 
1106 
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" 

1107 
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) 

1108 
qed 

1109 

39085  1110 
section "Conditional Expectation and Probability" 
1111 

1112 
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

1113 
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

1114 
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

1115 
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

1116 
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

1117 
(\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))" 
39083  1118 
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

1119 
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

1120 
interpret P: prob_space N 
41545  1121 
using prob_space_subalgebra[OF N] . 
39083  1122 

1123 
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

1124 
let "?Q A" = "integral\<^isup>P M (?f A)" 
39083  1125 

1126 
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

1127 
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

1128 
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

1129 
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

1130 
then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" . 
39083  1131 

1132 
have "P.absolutely_continuous ?Q" 

1133 
unfolding P.absolutely_continuous_def 

41545  1134 
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

1135 
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

1136 
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

1137 
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

1138 
then show "?Q A = 0" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1139 
by (auto simp add: positive_integral_0_iff_AE) 
39083  1140 
qed 
1141 
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

1142 
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

1143 
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)" 
39083  1144 
by blast 
41545  1145 
with N(2) show ?thesis 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1146 
by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)]) 
39083  1147 
qed 
1148 

39085  1149 
definition (in prob_space) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

1150 
"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

1151 
\<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  1152 

1153 
abbreviation (in prob_space) 

39092  1154 
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" 
39085  1155 

1156 
lemma (in prob_space) 

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

1157 
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

1158 
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

1159 
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" 
39085  1160 
shows borel_measurable_conditional_expectation: 
41545  1161 
"conditional_expectation N X \<in> borel_measurable N" 
1162 
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

1163 
(\<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

1164 
(\<integral>\<^isup>+x. X x * indicator C x \<partial>M)" 
41545  1165 
(is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C") 
39085  1166 
proof  
1167 
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] 

41545  1168 
then show "conditional_expectation N X \<in> borel_measurable N" 
39085  1169 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
1170 

41545  1171 
from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C" 
39085  1172 
unfolding conditional_expectation_def by (rule someI2_ex) blast 
1173 
qed 

1174 

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

1175 
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

1176 
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" 
39091  1177 
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

1178 
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

1179 
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

1180 
proof  
39091  1181 
interpret M': sigma_algebra M' by fact 
1182 
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto 

1183 
from M'.sigma_algebra_vimage[OF this] 

1184 
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . 

1185 

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

1186 
from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this 
39091  1187 

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