author | hoelzl |
Mon, 23 May 2011 19:21:05 +0200 | |
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parent 42902 | e8dbf90a2f3b |
child 42981 | fe7f5a26e4c6 |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Probability_Measure.thy |
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Author: Johannes Hölzl, TU München |
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Author: Armin Heller, TU München |
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*) |
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header {*Probability measure*} |
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theory Probability_Measure |
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imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure Lebesgue_Measure |
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begin |
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locale prob_space = measure_space + |
|
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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assumes measure_space_1: "measure M (space M) = 1" |
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15 |
sublocale prob_space < finite_measure |
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proof |
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from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp |
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qed |
19 |
||
40859 | 20 |
abbreviation (in prob_space) "events \<equiv> sets M" |
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abbreviation (in prob_space) "prob \<equiv> \<mu>'" |
40859 | 22 |
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'" |
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abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M" |
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|
40859 | 25 |
definition (in prob_space) |
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"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B" |
27 |
||
40859 | 28 |
definition (in prob_space) |
35582 | 29 |
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)" |
30 |
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40859 | 31 |
definition (in prob_space) |
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"distribution X A = \<mu>' (X -` A \<inter> space M)" |
35582 | 33 |
|
40859 | 34 |
abbreviation (in prob_space) |
36624 | 35 |
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" |
35582 | 36 |
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declare (in finite_measure) positive_measure'[intro, simp] |
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lemma (in prob_space) distribution_cong: |
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assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" |
|
41 |
shows "distribution X = distribution Y" |
|
39302
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renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
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unfolding distribution_def fun_eq_iff |
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) |
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|
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lemma (in prob_space) joint_distribution_cong: |
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assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" |
|
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assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" |
|
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shows "joint_distribution X Y = joint_distribution X' Y'" |
|
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renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
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unfolding distribution_def fun_eq_iff |
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) |
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|
40859 | 52 |
lemma (in prob_space) distribution_id[simp]: |
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"N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N" |
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by (auto simp: distribution_def intro!: arg_cong[where f=prob]) |
40859 | 55 |
|
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lemma (in prob_space) prob_space: "prob (space M) = 1" |
|
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using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def) |
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58 |
|
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lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1" |
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using bounded_measure[of A] by (simp add: prob_space) |
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61 |
|
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lemma (in prob_space) distribution_positive[simp, intro]: |
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"0 \<le> distribution X A" unfolding distribution_def by auto |
35582 | 64 |
|
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lemma (in prob_space) joint_distribution_remove[simp]: |
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"joint_distribution X X {(x, x)} = distribution X {x}" |
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unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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68 |
|
42950
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move lemmas to Extended_Reals and Extended_Real_Limits
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parents:
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69 |
lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1" |
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move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
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|
70 |
unfolding measure_space_1[symmetric] |
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move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
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using sets_into_space |
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move lemmas to Extended_Reals and Extended_Real_Limits
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parents:
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72 |
by (intro measure_mono) auto |
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move lemmas to Extended_Reals and Extended_Real_Limits
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parents:
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73 |
|
41981
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parents:
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lemma (in prob_space) distribution_1: |
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"distribution X A \<le> 1" |
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unfolding distribution_def by simp |
35582 | 77 |
|
40859 | 78 |
lemma (in prob_space) prob_compl: |
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79 |
assumes A: "A \<in> events" |
38656 | 80 |
shows "prob (space M - A) = 1 - prob A" |
41981
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81 |
using finite_measure_compl[OF A] by (simp add: prob_space) |
35582 | 82 |
|
41981
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lemma (in prob_space) indep_space: "s \<in> events \<Longrightarrow> indep (space M) s" |
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by (simp add: indep_def prob_space) |
35582 | 85 |
|
40859 | 86 |
lemma (in prob_space) prob_space_increasing: "increasing M prob" |
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by (auto intro!: finite_measure_mono simp: increasing_def) |
35582 | 88 |
|
40859 | 89 |
lemma (in prob_space) prob_zero_union: |
35582 | 90 |
assumes "s \<in> events" "t \<in> events" "prob t = 0" |
91 |
shows "prob (s \<union> t) = prob s" |
|
38656 | 92 |
using assms |
35582 | 93 |
proof - |
94 |
have "prob (s \<union> t) \<le> prob s" |
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95 |
using finite_measure_subadditive[of s t] assms by auto |
35582 | 96 |
moreover have "prob (s \<union> t) \<ge> prob s" |
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97 |
using assms by (blast intro: finite_measure_mono) |
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ultimately show ?thesis by simp |
99 |
qed |
|
100 |
||
40859 | 101 |
lemma (in prob_space) prob_eq_compl: |
35582 | 102 |
assumes "s \<in> events" "t \<in> events" |
103 |
assumes "prob (space M - s) = prob (space M - t)" |
|
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shows "prob s = prob t" |
|
38656 | 105 |
using assms prob_compl by auto |
35582 | 106 |
|
40859 | 107 |
lemma (in prob_space) prob_one_inter: |
35582 | 108 |
assumes events:"s \<in> events" "t \<in> events" |
109 |
assumes "prob t = 1" |
|
110 |
shows "prob (s \<inter> t) = prob s" |
|
111 |
proof - |
|
38656 | 112 |
have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)" |
113 |
using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union) |
|
114 |
also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)" |
|
115 |
by blast |
|
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finally show "prob (s \<inter> t) = prob s" |
|
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using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s]) |
|
35582 | 118 |
qed |
119 |
||
40859 | 120 |
lemma (in prob_space) prob_eq_bigunion_image: |
35582 | 121 |
assumes "range f \<subseteq> events" "range g \<subseteq> events" |
122 |
assumes "disjoint_family f" "disjoint_family g" |
|
123 |
assumes "\<And> n :: nat. prob (f n) = prob (g n)" |
|
124 |
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" |
|
125 |
using assms |
|
126 |
proof - |
|
38656 | 127 |
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))" |
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128 |
by (rule finite_measure_UNION[OF assms(1,3)]) |
38656 | 129 |
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" |
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130 |
by (rule finite_measure_UNION[OF assms(2,4)]) |
38656 | 131 |
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp |
35582 | 132 |
qed |
133 |
||
40859 | 134 |
lemma (in prob_space) prob_countably_zero: |
35582 | 135 |
assumes "range c \<subseteq> events" |
136 |
assumes "\<And> i. prob (c i) = 0" |
|
38656 | 137 |
shows "prob (\<Union> i :: nat. c i) = 0" |
138 |
proof (rule antisym) |
|
139 |
show "prob (\<Union> i :: nat. c i) \<le> 0" |
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41981
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140 |
using finite_measure_countably_subadditive[OF assms(1)] |
38656 | 141 |
by (simp add: assms(2) suminf_zero summable_zero) |
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142 |
qed simp |
35582 | 143 |
|
40859 | 144 |
lemma (in prob_space) indep_sym: |
35582 | 145 |
"indep a b \<Longrightarrow> indep b a" |
146 |
unfolding indep_def using Int_commute[of a b] by auto |
|
147 |
||
40859 | 148 |
lemma (in prob_space) indep_refl: |
35582 | 149 |
assumes "a \<in> events" |
150 |
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" |
|
151 |
using assms unfolding indep_def by auto |
|
152 |
||
40859 | 153 |
lemma (in prob_space) prob_equiprobable_finite_unions: |
38656 | 154 |
assumes "s \<in> events" |
155 |
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" |
|
35582 | 156 |
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" |
38656 | 157 |
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" |
35582 | 158 |
proof (cases "s = {}") |
38656 | 159 |
case False hence "\<exists> x. x \<in> s" by blast |
35582 | 160 |
from someI_ex[OF this] assms |
161 |
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast |
|
162 |
have "prob s = (\<Sum> x \<in> s. prob {x})" |
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163 |
using finite_measure_finite_singleton[OF s_finite] by simp |
35582 | 164 |
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto |
38656 | 165 |
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" |
166 |
using setsum_constant assms by (simp add: real_eq_of_nat) |
|
35582 | 167 |
finally show ?thesis by simp |
38656 | 168 |
qed simp |
35582 | 169 |
|
40859 | 170 |
lemma (in prob_space) prob_real_sum_image_fn: |
35582 | 171 |
assumes "e \<in> events" |
172 |
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" |
|
173 |
assumes "finite s" |
|
38656 | 174 |
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" |
175 |
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" |
|
35582 | 176 |
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" |
177 |
proof - |
|
38656 | 178 |
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" |
179 |
using `e \<in> events` sets_into_space upper by blast |
|
180 |
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp |
|
181 |
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" |
|
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parents:
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182 |
proof (rule finite_measure_finite_Union) |
38656 | 183 |
show "finite s" by fact |
184 |
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact |
|
185 |
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" |
|
186 |
using disjoint by (auto simp: disjoint_family_on_def) |
|
187 |
qed |
|
188 |
finally show ?thesis . |
|
35582 | 189 |
qed |
190 |
||
42199 | 191 |
lemma (in prob_space) prob_space_vimage: |
192 |
assumes S: "sigma_algebra S" |
|
193 |
assumes T: "T \<in> measure_preserving M S" |
|
194 |
shows "prob_space S" |
|
35582 | 195 |
proof - |
42199 | 196 |
interpret S: measure_space S |
197 |
using S and T by (rule measure_space_vimage) |
|
38656 | 198 |
show ?thesis |
42199 | 199 |
proof |
200 |
from T[THEN measure_preservingD2] |
|
201 |
have "T -` space S \<inter> space M = space M" |
|
202 |
by (auto simp: measurable_def) |
|
203 |
with T[THEN measure_preservingD, of "space S", symmetric] |
|
204 |
show "measure S (space S) = 1" |
|
205 |
using measure_space_1 by simp |
|
35582 | 206 |
qed |
207 |
qed |
|
208 |
||
42199 | 209 |
lemma (in prob_space) distribution_prob_space: |
210 |
assumes X: "random_variable S X" |
|
211 |
shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" (is "prob_space ?S") |
|
212 |
proof (rule prob_space_vimage) |
|
213 |
show "X \<in> measure_preserving M ?S" |
|
214 |
using X |
|
215 |
unfolding measure_preserving_def distribution_def_raw |
|
216 |
by (auto simp: finite_measure_eq measurable_sets) |
|
217 |
show "sigma_algebra ?S" using X by simp |
|
218 |
qed |
|
219 |
||
40859 | 220 |
lemma (in prob_space) AE_distribution: |
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221 |
assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x" |
40859 | 222 |
shows "AE x. Q (X x)" |
223 |
proof - |
|
41981
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224 |
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space) |
40859 | 225 |
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" |
226 |
using assms unfolding X.almost_everywhere_def by auto |
|
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227 |
from X[unfolded measurable_def] N show "AE x. Q (X x)" |
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228 |
by (intro AE_I'[where N="X -` N \<inter> space M"]) |
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229 |
(auto simp: finite_measure_eq distribution_def measurable_sets) |
40859 | 230 |
qed |
231 |
||
41981
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parents:
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|
232 |
lemma (in prob_space) distribution_eq_integral: |
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parents:
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|
233 |
"random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))" |
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
234 |
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
|
235 |
by (auto simp: measurable_sets distribution_def) |
35582 | 236 |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
237 |
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
|
238 |
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
|
239 |
shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)" |
35582 | 240 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
241 |
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
|
242 |
using assms(1) by (rule distribution_prob_space) |
35582 | 243 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
244 |
using S.positive_integral_indicator(1)[of A] assms by simp |
35582 | 245 |
qed |
246 |
||
40859 | 247 |
lemma (in prob_space) finite_expectation1: |
248 |
assumes f: "finite (X`space M)" and rv: "random_variable borel X" |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
249 |
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:
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changeset
|
250 |
proof (subst integral_on_finite) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
251 |
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
|
252 |
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
|
253 |
"\<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
|
254 |
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto |
38656 | 255 |
qed |
35582 | 256 |
|
40859 | 257 |
lemma (in prob_space) finite_expectation: |
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
|
258 |
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
|
259 |
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" |
38656 | 260 |
using assms unfolding distribution_def using finite_expectation1 by auto |
261 |
||
40859 | 262 |
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: |
35582 | 263 |
assumes "{x} \<in> events" |
38656 | 264 |
assumes "prob {x} = 1" |
35582 | 265 |
assumes "{y} \<in> events" |
266 |
assumes "y \<noteq> x" |
|
267 |
shows "prob {y} = 0" |
|
268 |
using prob_one_inter[of "{y}" "{x}"] assms by auto |
|
269 |
||
40859 | 270 |
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" |
38656 | 271 |
unfolding distribution_def by simp |
272 |
||
40859 | 273 |
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" |
38656 | 274 |
proof - |
275 |
have "X -` X ` space M \<inter> space M = space M" by auto |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
276 |
thus ?thesis unfolding distribution_def by (simp add: prob_space) |
38656 | 277 |
qed |
278 |
||
40859 | 279 |
lemma (in prob_space) distribution_one: |
280 |
assumes "random_variable M' X" and "A \<in> sets M'" |
|
38656 | 281 |
shows "distribution X A \<le> 1" |
282 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
283 |
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
|
284 |
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
|
285 |
thus ?thesis by (simp add: prob_space) |
38656 | 286 |
qed |
287 |
||
40859 | 288 |
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: |
35582 | 289 |
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" |
38656 | 290 |
(is "random_variable ?S X") |
291 |
assumes "distribution X {x} = 1" |
|
35582 | 292 |
assumes "y \<noteq> x" |
293 |
shows "distribution X {y} = 0" |
|
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
|
294 |
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
|
295 |
{ 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
|
296 |
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
|
297 |
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
|
298 |
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
|
299 |
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
|
300 |
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
|
301 |
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
|
302 |
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
|
303 |
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
|
304 |
"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
|
305 |
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
|
306 |
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
|
307 |
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
|
308 |
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
|
309 |
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
|
310 |
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
|
311 |
thus "distribution X {y} = 0" unfolding distribution_def by auto |
35582 | 312 |
qed |
313 |
||
40859 | 314 |
lemma (in prob_space) joint_distribution_Times_le_fst: |
315 |
assumes X: "random_variable MX X" and Y: "random_variable MY Y" |
|
316 |
and A: "A \<in> sets MX" and B: "B \<in> sets MY" |
|
317 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" |
|
318 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
319 |
proof (intro finite_measure_mono) |
40859 | 320 |
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force |
321 |
show "X -` A \<inter> space M \<in> events" |
|
322 |
using X A unfolding measurable_def by simp |
|
323 |
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M = |
|
324 |
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
325 |
qed |
|
326 |
||
327 |
lemma (in prob_space) joint_distribution_commute: |
|
328 |
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
329 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 330 |
|
331 |
lemma (in prob_space) joint_distribution_Times_le_snd: |
|
332 |
assumes X: "random_variable MX X" and Y: "random_variable MY Y" |
|
333 |
and A: "A \<in> sets MX" and B: "B \<in> sets MY" |
|
334 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" |
|
335 |
using assms |
|
336 |
by (subst joint_distribution_commute) |
|
337 |
(simp add: swap_product joint_distribution_Times_le_fst) |
|
338 |
||
339 |
lemma (in prob_space) random_variable_pairI: |
|
340 |
assumes "random_variable MX X" |
|
341 |
assumes "random_variable MY Y" |
|
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
|
342 |
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" |
40859 | 343 |
proof |
344 |
interpret MX: sigma_algebra MX using assms by simp |
|
345 |
interpret MY: sigma_algebra MY using assms by simp |
|
346 |
interpret P: pair_sigma_algebra MX MY by default |
|
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
|
347 |
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default |
40859 | 348 |
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
|
349 |
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" |
41095 | 350 |
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) |
40859 | 351 |
qed |
352 |
||
353 |
lemma (in prob_space) joint_distribution_commute_singleton: |
|
354 |
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
355 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 356 |
|
357 |
lemma (in prob_space) joint_distribution_assoc_singleton: |
|
358 |
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = |
|
359 |
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
360 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 361 |
|
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
|
362 |
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 |
40859 | 363 |
|
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
|
364 |
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
|
365 |
|
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
|
366 |
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
|
367 |
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) |
40859 | 368 |
|
369 |
lemma countably_additiveI[case_names countably]: |
|
370 |
assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
371 |
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" |
40859 | 372 |
shows "countably_additive M \<mu>" |
373 |
using assms unfolding countably_additive_def by auto |
|
374 |
||
375 |
lemma (in prob_space) joint_distribution_prob_space: |
|
376 |
assumes "random_variable MX X" "random_variable MY Y" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
377 |
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
|
378 |
using random_variable_pairI[OF assms] by (rule distribution_prob_space) |
40859 | 379 |
|
380 |
section "Probability spaces on finite sets" |
|
35582 | 381 |
|
35977 | 382 |
locale finite_prob_space = prob_space + finite_measure_space |
383 |
||
40859 | 384 |
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" |
385 |
||
386 |
lemma (in prob_space) finite_random_variableD: |
|
387 |
assumes "finite_random_variable M' X" shows "random_variable M' X" |
|
388 |
proof - |
|
389 |
interpret M': finite_sigma_algebra M' using assms by simp |
|
390 |
then show "random_variable M' X" using assms by simp default |
|
391 |
qed |
|
392 |
||
393 |
lemma (in prob_space) distribution_finite_prob_space: |
|
394 |
assumes "finite_random_variable MX X" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
395 |
shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)" |
40859 | 396 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
397 |
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" |
40859 | 398 |
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) |
399 |
interpret MX: finite_sigma_algebra MX |
|
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
|
400 |
using assms by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
401 |
show ?thesis by default (simp_all add: MX.finite_space) |
40859 | 402 |
qed |
403 |
||
404 |
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]: |
|
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
|
405 |
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
|
406 |
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
|
407 |
(is "finite_random_variable ?X _") |
40859 | 408 |
proof (intro conjI) |
409 |
have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp |
|
41689
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
|
410 |
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
|
411 |
show "finite_sigma_algebra ?X" |
40859 | 412 |
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
|
413 |
show "X \<in> measurable M ?X" |
40859 | 414 |
proof (unfold measurable_def, clarsimp) |
415 |
fix A assume A: "A \<subseteq> X`space M" |
|
416 |
then have "finite A" by (rule finite_subset) simp |
|
417 |
then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" |
|
418 |
unfolding vimage_UN UN_extend_simps |
|
419 |
apply (rule finite_UN) |
|
420 |
using A assms unfolding simple_function_def by auto |
|
421 |
then show "X -` A \<inter> space M \<in> events" by simp |
|
422 |
qed |
|
423 |
qed |
|
424 |
||
425 |
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]: |
|
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
|
426 |
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
|
427 |
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
|
428 |
using simple_function_imp_finite_random_variable[OF assms, of ext] |
40859 | 429 |
by (auto dest!: finite_random_variableD) |
430 |
||
431 |
lemma (in prob_space) sum_over_space_real_distribution: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
432 |
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" |
40859 | 433 |
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
|
434 |
by (subst finite_measure_finite_Union[symmetric]) |
40859 | 435 |
(auto simp add: disjoint_family_on_def simple_function_def |
436 |
intro!: arg_cong[where f=prob]) |
|
437 |
||
438 |
lemma (in prob_space) finite_random_variable_pairI: |
|
439 |
assumes "finite_random_variable MX X" |
|
440 |
assumes "finite_random_variable MY Y" |
|
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
|
441 |
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" |
40859 | 442 |
proof |
443 |
interpret MX: finite_sigma_algebra MX using assms by simp |
|
444 |
interpret MY: finite_sigma_algebra MY using assms by simp |
|
445 |
interpret P: pair_finite_sigma_algebra MX MY by default |
|
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
|
446 |
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default |
40859 | 447 |
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
|
448 |
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" |
41095 | 449 |
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) |
40859 | 450 |
qed |
451 |
||
452 |
lemma (in prob_space) finite_random_variable_imp_sets: |
|
453 |
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" |
|
454 |
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp |
|
455 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
456 |
lemma (in prob_space) finite_random_variable_measurable: |
40859 | 457 |
assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events" |
458 |
proof - |
|
459 |
interpret X: finite_sigma_algebra MX using X by simp |
|
460 |
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and |
|
461 |
"X \<in> space M \<rightarrow> space MX" |
|
462 |
by (auto simp: measurable_def) |
|
463 |
then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M" |
|
464 |
by auto |
|
465 |
show "X -` A \<inter> space M \<in> events" |
|
466 |
unfolding * by (intro vimage) auto |
|
467 |
qed |
|
468 |
||
469 |
lemma (in prob_space) joint_distribution_finite_Times_le_fst: |
|
470 |
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" |
|
471 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" |
|
472 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
473 |
proof (intro finite_measure_mono) |
40859 | 474 |
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force |
475 |
show "X -` A \<inter> space M \<in> events" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
476 |
using finite_random_variable_measurable[OF X] . |
40859 | 477 |
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M = |
478 |
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
479 |
qed |
|
480 |
||
481 |
lemma (in prob_space) joint_distribution_finite_Times_le_snd: |
|
482 |
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" |
|
483 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" |
|
484 |
using assms |
|
485 |
by (subst joint_distribution_commute) |
|
486 |
(simp add: swap_product joint_distribution_finite_Times_le_fst) |
|
487 |
||
488 |
lemma (in prob_space) finite_distribution_order: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
489 |
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" |
40859 | 490 |
assumes "finite_random_variable MX X" "finite_random_variable MY Y" |
491 |
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" |
|
492 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" |
|
493 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" |
|
494 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" |
|
495 |
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
496 |
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
497 |
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] |
|
498 |
using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
499 |
by (auto intro: antisym) |
40859 | 500 |
|
501 |
lemma (in prob_space) setsum_joint_distribution: |
|
502 |
assumes X: "finite_random_variable MX X" |
|
503 |
assumes Y: "random_variable MY Y" "B \<in> sets MY" |
|
504 |
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" |
|
505 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
506 |
proof (subst finite_measure_finite_Union[symmetric]) |
40859 | 507 |
interpret MX: finite_sigma_algebra MX using X by auto |
508 |
show "finite (space MX)" using MX.finite_space . |
|
509 |
let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M" |
|
510 |
{ fix i assume "i \<in> space MX" |
|
511 |
moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
512 |
ultimately show "?d i \<in> events" |
|
513 |
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y |
|
514 |
using MX.sets_eq_Pow by auto } |
|
515 |
show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
516 |
show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
517 |
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 518 |
qed |
519 |
||
520 |
lemma (in prob_space) setsum_joint_distribution_singleton: |
|
521 |
assumes X: "finite_random_variable MX X" |
|
522 |
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" |
|
523 |
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" |
|
524 |
using setsum_joint_distribution[OF X |
|
525 |
finite_random_variableD[OF Y(1)] |
|
526 |
finite_random_variable_imp_sets[OF Y]] by simp |
|
527 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
528 |
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 |
40859 | 529 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
530 |
sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
531 |
sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2 by default |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
532 |
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default |
40859 | 533 |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
534 |
locale product_finite_prob_space = |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
535 |
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
536 |
and I :: "'i set" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
537 |
assumes finite_space: "\<And>i. finite_prob_space (M i)" and finite_index: "finite I" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
538 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
539 |
sublocale product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
540 |
sublocale product_finite_prob_space \<subseteq> finite_product_sigma_finite M I by default (rule finite_index) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
541 |
sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
542 |
proof |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
543 |
show "\<mu> (space P) = 1" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
544 |
using measure_times[OF M.top] M.measure_space_1 |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
545 |
by (simp add: setprod_1 space_product_algebra) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
546 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
547 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
548 |
lemma funset_eq_UN_fun_upd_I: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
549 |
assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
550 |
and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
551 |
and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
552 |
shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
553 |
proof safe |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
554 |
fix f assume f: "f \<in> F (insert a A)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
555 |
show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
556 |
proof (rule UN_I[of "f(a := d)"]) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
557 |
show "f(a := d) \<in> F A" using *[OF f] . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
558 |
show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
559 |
proof (rule image_eqI[of _ _ "f a"]) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
560 |
show "f a \<in> G (f(a := d))" using **[OF f] . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
561 |
qed simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
562 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
563 |
next |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
564 |
fix f x assume "f \<in> F A" "x \<in> G f" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
565 |
from ***[OF this] show "f(a := x) \<in> F (insert a A)" . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
566 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
567 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
568 |
lemma extensional_funcset_insert_eq[simp]: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
569 |
assumes "a \<notin> A" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
570 |
shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
571 |
apply (rule funset_eq_UN_fun_upd_I) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
572 |
using assms |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
573 |
by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
574 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
575 |
lemma finite_extensional_funcset[simp, intro]: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
576 |
assumes "finite A" "finite B" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
577 |
shows "finite (extensional A \<inter> (A \<rightarrow> B))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
578 |
using assms by induct (auto simp: extensional_funcset_insert_eq) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
579 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
580 |
lemma finite_PiE[simp, intro]: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
581 |
assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
582 |
shows "finite (Pi\<^isub>E A B)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
583 |
proof - |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
584 |
have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
585 |
show ?thesis |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
586 |
using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
587 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
588 |
|
42892 | 589 |
lemma (in product_finite_prob_space) singleton_eq_product: |
590 |
assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})" |
|
591 |
proof (safe intro!: ext[of _ x]) |
|
592 |
fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I" |
|
593 |
with x show "y i = x i" |
|
594 |
by (cases "i \<in> I") (auto simp: extensional_def) |
|
595 |
qed (insert x, auto) |
|
596 |
||
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
597 |
sublocale product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
598 |
proof |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
599 |
show "finite (space P)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
600 |
using finite_index M.finite_space by auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
601 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
602 |
{ fix x assume "x \<in> space P" |
42892 | 603 |
with this[THEN singleton_eq_product] |
604 |
have "{x} \<in> sets P" |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
605 |
by (auto intro!: in_P) } |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
606 |
note x_in_P = this |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
607 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
608 |
have "Pow (space P) \<subseteq> sets P" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
609 |
proof |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
610 |
fix X assume "X \<in> Pow (space P)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
611 |
moreover then have "finite X" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
612 |
using `finite (space P)` by (blast intro: finite_subset) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
613 |
ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
614 |
by (intro finite_UN x_in_P) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
615 |
then show "X \<in> sets P" by simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
616 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
617 |
with space_closed show [simp]: "sets P = Pow (space P)" .. |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
618 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
619 |
{ fix x assume "x \<in> space P" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
620 |
from this top have "\<mu> {x} \<le> \<mu> (space P)" by (intro measure_mono) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
621 |
then show "\<mu> {x} \<noteq> \<infinity>" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
622 |
using measure_space_1 by auto } |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
623 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
624 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
625 |
lemma (in product_finite_prob_space) measure_finite_times: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
626 |
"(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
627 |
by (rule measure_times) simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
628 |
|
42892 | 629 |
lemma (in product_finite_prob_space) measure_singleton_times: |
630 |
assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})" |
|
631 |
unfolding singleton_eq_product[OF x] using x |
|
632 |
by (intro measure_finite_times) auto |
|
633 |
||
634 |
lemma (in product_finite_prob_space) prob_finite_times: |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
635 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
636 |
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
637 |
proof - |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
638 |
have "extreal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
639 |
using X by (intro finite_measure_eq[symmetric] in_P) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
640 |
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
641 |
using measure_finite_times X by simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
642 |
also have "\<dots> = extreal (\<Prod>i\<in>I. M.\<mu>' i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
643 |
using X by (simp add: M.finite_measure_eq setprod_extreal) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
644 |
finally show ?thesis by simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
645 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
646 |
|
42892 | 647 |
lemma (in product_finite_prob_space) prob_singleton_times: |
648 |
assumes x: "x \<in> space P" |
|
649 |
shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})" |
|
650 |
unfolding singleton_eq_product[OF x] using x |
|
651 |
by (intro prob_finite_times) auto |
|
652 |
||
653 |
lemma (in product_finite_prob_space) prob_finite_product: |
|
654 |
"A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})" |
|
655 |
by (auto simp add: finite_measure_singleton prob_singleton_times |
|
656 |
simp del: space_product_algebra |
|
657 |
intro!: setsum_cong prob_singleton_times) |
|
658 |
||
40859 | 659 |
lemma (in prob_space) joint_distribution_finite_prob_space: |
660 |
assumes X: "finite_random_variable MX X" |
|
661 |
assumes Y: "finite_random_variable MY Y" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
662 |
shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
663 |
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) |
40859 | 664 |
|
36624 | 665 |
lemma finite_prob_space_eq: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
666 |
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" |
36624 | 667 |
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def |
668 |
by auto |
|
669 |
||
670 |
lemma (in prob_space) not_empty: "space M \<noteq> {}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
671 |
using prob_space empty_measure' by auto |
36624 | 672 |
|
38656 | 673 |
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" |
674 |
using measure_space_1 sum_over_space by simp |
|
36624 | 675 |
|
676 |
lemma (in finite_prob_space) joint_distribution_restriction_fst: |
|
677 |
"joint_distribution X Y A \<le> distribution X (fst ` A)" |
|
678 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
679 |
proof (safe intro!: finite_measure_mono) |
36624 | 680 |
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" |
681 |
show "x \<in> X -` fst ` A" |
|
682 |
by (auto intro!: image_eqI[OF _ *]) |
|
683 |
qed (simp_all add: sets_eq_Pow) |
|
684 |
||
685 |
lemma (in finite_prob_space) joint_distribution_restriction_snd: |
|
686 |
"joint_distribution X Y A \<le> distribution Y (snd ` A)" |
|
687 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
688 |
proof (safe intro!: finite_measure_mono) |
36624 | 689 |
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" |
690 |
show "x \<in> Y -` snd ` A" |
|
691 |
by (auto intro!: image_eqI[OF _ *]) |
|
692 |
qed (simp_all add: sets_eq_Pow) |
|
693 |
||
694 |
lemma (in finite_prob_space) distribution_order: |
|
695 |
shows "0 \<le> distribution X x'" |
|
696 |
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" |
|
697 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" |
|
698 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" |
|
699 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" |
|
700 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" |
|
701 |
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
702 |
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
703 |
using |
36624 | 704 |
joint_distribution_restriction_fst[of X Y "{(x, y)}"] |
705 |
joint_distribution_restriction_snd[of X Y "{(x, y)}"] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
706 |
by (auto intro: antisym) |
36624 | 707 |
|
39097 | 708 |
lemma (in finite_prob_space) distribution_mono: |
709 |
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" |
|
710 |
shows "distribution X x \<le> distribution Y y" |
|
711 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
712 |
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) |
39097 | 713 |
|
714 |
lemma (in finite_prob_space) distribution_mono_gt_0: |
|
715 |
assumes gt_0: "0 < distribution X x" |
|
716 |
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" |
|
717 |
shows "0 < distribution Y y" |
|
718 |
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) |
|
719 |
||
720 |
lemma (in finite_prob_space) sum_over_space_distrib: |
|
721 |
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
722 |
unfolding distribution_def prob_space[symmetric] using finite_space |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
723 |
by (subst finite_measure_finite_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
724 |
(auto simp add: disjoint_family_on_def sets_eq_Pow |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
725 |
intro!: arg_cong[where f=\<mu>']) |
39097 | 726 |
|
727 |
lemma (in finite_prob_space) finite_sum_over_space_eq_1: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
728 |
"(\<Sum>x\<in>space M. prob {x}) = 1" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
729 |
using prob_space finite_space |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
730 |
by (subst (asm) finite_measure_finite_singleton) auto |
39097 | 731 |
|
732 |
lemma (in prob_space) distribution_remove_const: |
|
733 |
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" |
|
734 |
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" |
|
735 |
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" |
|
736 |
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" |
|
737 |
and "distribution (\<lambda>x. ()) {()} = 1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
738 |
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) |
35977 | 739 |
|
39097 | 740 |
lemma (in finite_prob_space) setsum_distribution_gen: |
741 |
assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M" |
|
742 |
and "inj_on f (X`space M)" |
|
743 |
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" |
|
744 |
unfolding distribution_def assms |
|
745 |
using finite_space assms |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
746 |
by (subst finite_measure_finite_Union[symmetric]) |
39097 | 747 |
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def |
748 |
intro!: arg_cong[where f=prob]) |
|
749 |
||
750 |
lemma (in finite_prob_space) setsum_distribution: |
|
751 |
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" |
|
752 |
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" |
|
753 |
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" |
|
754 |
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" |
|
755 |
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" |
|
756 |
by (auto intro!: inj_onI setsum_distribution_gen) |
|
757 |
||
758 |
lemma (in finite_prob_space) uniform_prob: |
|
759 |
assumes "x \<in> space M" |
|
760 |
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
761 |
shows "prob {x} = 1 / card (space M)" |
39097 | 762 |
proof - |
763 |
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" |
|
764 |
using assms(2)[OF _ `x \<in> space M`] by blast |
|
765 |
have "1 = prob (space M)" |
|
766 |
using prob_space by auto |
|
767 |
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
768 |
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] |
39097 | 769 |
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] |
770 |
finite_space unfolding disjoint_family_on_def prob_space[symmetric] |
|
771 |
by (auto simp add:setsum_restrict_set) |
|
772 |
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" |
|
773 |
using prob_x by auto |
|
774 |
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp |
|
775 |
finally have one: "1 = real (card (space M)) * prob {x}" |
|
776 |
using real_eq_of_nat by auto |
|
777 |
hence two: "real (card (space M)) \<noteq> 0" by fastsimp |
|
778 |
from one have three: "prob {x} \<noteq> 0" by fastsimp |
|
779 |
thus ?thesis using one two three divide_cancel_right |
|
780 |
by (auto simp:field_simps) |
|
39092 | 781 |
qed |
35977 | 782 |
|
39092 | 783 |
lemma (in prob_space) prob_space_subalgebra: |
41545 | 784 |
assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
785 |
and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
786 |
shows "prob_space N" |
39092 | 787 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
788 |
interpret N: measure_space N |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
789 |
by (rule measure_space_subalgebra[OF assms]) |
39092 | 790 |
show ?thesis |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
791 |
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) |
35977 | 792 |
qed |
793 |
||
39092 | 794 |
lemma (in prob_space) prob_space_of_restricted_space: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
795 |
assumes "\<mu> A \<noteq> 0" "A \<in> sets M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
796 |
shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
797 |
(is "prob_space ?P") |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
798 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
799 |
interpret A: measure_space "restricted_space A" |
39092 | 800 |
using `A \<in> sets M` by (rule restricted_measure_space) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
801 |
interpret A': sigma_algebra ?P |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
802 |
by (rule A.sigma_algebra_cong) auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
803 |
show "prob_space ?P" |
39092 | 804 |
proof |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
805 |
show "measure ?P (space ?P) = 1" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
806 |
using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
807 |
show "positive ?P (measure ?P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
808 |
proof (simp add: positive_def, safe) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
809 |
show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
810 |
fix B assume "B \<in> events" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
811 |
with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M` |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
812 |
show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
813 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
814 |
show "countably_additive ?P (measure ?P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
815 |
proof (simp add: countably_additive_def, safe) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
816 |
fix B and F :: "nat \<Rightarrow> 'a set" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
817 |
assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
818 |
{ fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
819 |
from F have "F i \<in> op \<inter> A ` events" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
820 |
with `A \<in> events` have "F i \<in> events" by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
821 |
moreover then have "range F \<subseteq> events" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
822 |
moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
823 |
by (simp add: mult_commute divide_extreal_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
824 |
moreover have "0 \<le> inverse (\<mu> A)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
825 |
using real_measure[OF `A \<in> events`] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
826 |
ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
827 |
using measure_countably_additive[of F] F |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
828 |
by (auto simp: suminf_cmult_extreal) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
829 |
qed |
39092 | 830 |
qed |
831 |
qed |
|
832 |
||
833 |
lemma finite_prob_spaceI: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
834 |
assumes "finite (space M)" "sets M = Pow(space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
835 |
and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
836 |
and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
837 |
shows "finite_prob_space M" |
39092 | 838 |
unfolding finite_prob_space_eq |
839 |
proof |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
840 |
show "finite_measure_space M" using assms |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
841 |
by (auto intro!: finite_measure_spaceI) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
842 |
show "measure M (space M) = 1" by fact |
39092 | 843 |
qed |
36624 | 844 |
|
845 |
lemma (in finite_prob_space) finite_measure_space: |
|
39097 | 846 |
fixes X :: "'a \<Rightarrow> 'x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
847 |
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
848 |
(is "finite_measure_space ?S") |
39092 | 849 |
proof (rule finite_measure_spaceI, simp_all) |
36624 | 850 |
show "finite (X ` space M)" using finite_space by simp |
39097 | 851 |
next |
852 |
fix A B :: "'x set" assume "A \<inter> B = {}" |
|
853 |
then show "distribution X (A \<union> B) = distribution X A + distribution X B" |
|
854 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
855 |
by (subst finite_measure_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
856 |
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) |
36624 | 857 |
qed |
858 |
||
39097 | 859 |
lemma (in finite_prob_space) finite_prob_space_of_images: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
860 |
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X \<rparr>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
861 |
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def) |
39097 | 862 |
|
39096 | 863 |
lemma (in finite_prob_space) finite_product_measure_space: |
39097 | 864 |
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" |
39096 | 865 |
assumes "finite s1" "finite s2" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
866 |
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
867 |
(is "finite_measure_space ?M") |
39097 | 868 |
proof (rule finite_measure_spaceI, simp_all) |
869 |
show "finite (s1 \<times> s2)" |
|
39096 | 870 |
using assms by auto |
39097 | 871 |
next |
872 |
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" |
|
873 |
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" |
|
874 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
875 |
by (subst finite_measure_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
876 |
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) |
39096 | 877 |
qed |
878 |
||
39097 | 879 |
lemma (in finite_prob_space) finite_product_measure_space_of_images: |
39096 | 880 |
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M, |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
881 |
sets = Pow (X ` space M \<times> Y ` space M), |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
882 |
measure = extreal \<circ> joint_distribution X Y \<rparr>" |
39096 | 883 |
using finite_space by (auto intro!: finite_product_measure_space) |
884 |
||
40859 | 885 |
lemma (in finite_prob_space) finite_product_prob_space_of_images: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
886 |
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M), |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
887 |
measure = extreal \<circ> joint_distribution X Y \<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
888 |
(is "finite_prob_space ?S") |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
889 |
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def) |
40859 | 890 |
have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto |
891 |
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" |
|
892 |
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) |
|
893 |
qed |
|
894 |
||
39085 | 895 |
section "Conditional Expectation and Probability" |
896 |
||
897 |
lemma (in prob_space) conditional_expectation_exists: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
898 |
fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
899 |
assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
900 |
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
901 |
shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N. |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
902 |
(\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))" |
39083 | 903 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
904 |
note N(4)[simp] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
905 |
interpret P: prob_space N |
41545 | 906 |
using prob_space_subalgebra[OF N] . |
39083 | 907 |
|
908 |
let "?f A" = "\<lambda>x. X x * indicator A x" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
909 |
let "?Q A" = "integral\<^isup>P M (?f A)" |
39083 | 910 |
|
911 |
from measure_space_density[OF borel] |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
912 |
have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
913 |
apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"]) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
914 |
using N by (auto intro!: P.sigma_algebra_cong) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
915 |
then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" . |
39083 | 916 |
|
917 |
have "P.absolutely_continuous ?Q" |
|
918 |
unfolding P.absolutely_continuous_def |
|
41545 | 919 |
proof safe |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
920 |
fix A assume "A \<in> sets N" "P.\<mu> A = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
921 |
then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
922 |
using borel N by (auto intro!: borel_measurable_indicator AE_not_in) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
923 |
then show "?Q A = 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
924 |
by (auto simp add: positive_integral_0_iff_AE) |
39083 | 925 |
qed |
926 |
from P.Radon_Nikodym[OF Q this] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
927 |
obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
928 |
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)" |
39083 | 929 |
by blast |
41545 | 930 |
with N(2) show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
931 |
by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)]) |
39083 | 932 |
qed |
933 |
||
39085 | 934 |
definition (in prob_space) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
935 |
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
936 |
\<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))" |
39085 | 937 |
|
938 |
abbreviation (in prob_space) |
|
39092 | 939 |
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" |
39085 | 940 |
|
941 |
lemma (in prob_space) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
942 |
fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
943 |
assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
944 |
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" |
39085 | 945 |
shows borel_measurable_conditional_expectation: |
41545 | 946 |
"conditional_expectation N X \<in> borel_measurable N" |
947 |
and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow> |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
948 |
(\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) = |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
949 |
(\<integral>\<^isup>+x. X x * indicator C x \<partial>M)" |
41545 | 950 |
(is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C") |
39085 | 951 |
proof - |
952 |
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] |
|
41545 | 953 |
then show "conditional_expectation N X \<in> borel_measurable N" |
39085 | 954 |
unfolding conditional_expectation_def by (rule someI2_ex) blast |
955 |
||
41545 | 956 |
from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C" |
39085 | 957 |
unfolding conditional_expectation_def by (rule someI2_ex) blast |
958 |
qed |
|
959 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
960 |
lemma (in sigma_algebra) factorize_measurable_function_pos: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
961 |
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" |
39091 | 962 |
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
963 |
assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
964 |
shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
965 |
proof - |
39091 | 966 |
interpret M': sigma_algebra M' by fact |
967 |
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto |
|
968 |
from M'.sigma_algebra_vimage[OF this] |
|
969 |
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . |
|
970 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
971 |
from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this |
39091 | 972 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
973 |
have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" |
39091 | 974 |
proof |
975 |
fix i |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
976 |
from f(1)[of i] have "finite (f i`space M)" and B_ex: |
39091 | 977 |
"\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
978 |
unfolding simple_function_def by auto |
39091 | 979 |
from B_ex[THEN bchoice] guess B .. note B = this |
980 |
||
981 |
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" |
|
982 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
983 |
show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" |
39091 | 984 |
proof (intro exI[of _ ?g] conjI ballI) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
985 |
show "simple_function M' ?g" using B by auto |
39091 | 986 |
|
987 |
fix x assume "x \<in> space M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
988 |
then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::extreal)" |
39091 | 989 |
unfolding indicator_def using B by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
990 |
then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i] |
39091 | 991 |
by (subst va.simple_function_indicator_representation) auto |
992 |
qed |
|
993 |
qed |
|
994 |
from choice[OF this] guess g .. note g = this |
|
995 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
996 |
show ?thesis |
39091 | 997 |
proof (intro ballI bexI) |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset
|
998 |
show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'" |
39091 | 999 |
using g by (auto intro: M'.borel_measurable_simple_function) |
1000 |
fix x assume "x \<in> space M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1001 |
have "max 0 (Z x) = (SUP i. f i x)" using f by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1002 |
also have "\<dots> = (SUP i. g i (Y x))" |
39091 | 1003 |
using g `x \<in> space M` by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1004 |
finally show "max 0 (Z x) = (SUP i. g i (Y x))" . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1005 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1006 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1007 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1008 |
lemma (in sigma_algebra) factorize_measurable_function: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1009 |
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1010 |
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1011 |
shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1012 |
\<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1013 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1014 |
interpret M': sigma_algebra M' by fact |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1015 |
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1016 |
from M'.sigma_algebra_vimage[OF this] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1017 |
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1018 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1019 |
{ fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1020 |
with M'.measurable_vimage_algebra[OF Y] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1021 |
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1022 |
by (rule measurable_comp) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1023 |
moreover assume "\<forall>x\<in>space M. Z x = g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1024 |
then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1025 |
g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1026 |
by (auto intro!: measurable_cong) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1027 |
ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1028 |
by simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1029 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1030 |
assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1031 |
with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1032 |
"(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1033 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1034 |
from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1035 |
from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1036 |
let "?g x" = "p x - n x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1037 |
show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1038 |
proof (intro bexI ballI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1039 |
show "?g \<in> borel_measurable M'" using p n by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1040 |
fix x assume "x \<in> space M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1041 |
then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1042 |
using p n by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1043 |
then show "Z x = ?g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1044 |
by (auto split: split_max) |
39091 | 1045 |
qed |
1046 |
qed |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
39089
diff
changeset
|
1047 |
|
42902 | 1048 |
subsection "Borel Measure on {0 .. 1}" |
1049 |
||
1050 |
definition pborel :: "real measure_space" where |
|
1051 |
"pborel = lborel.restricted_space {0 .. 1}" |
|
1052 |
||
1053 |
lemma space_pborel[simp]: |
|
1054 |
"space pborel = {0 .. 1}" |
|
1055 |
unfolding pborel_def by auto |
|
1056 |
||
1057 |
lemma sets_pborel: |
|
1058 |
"A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 .. 1}" |
|
1059 |
unfolding pborel_def by auto |
|
1060 |
||
1061 |
lemma in_pborel[intro, simp]: |
|
1062 |
"A \<subseteq> {0 .. 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel" |
|
1063 |
unfolding pborel_def by auto |
|
1064 |
||
1065 |
interpretation pborel: measure_space pborel |
|
1066 |
using lborel.restricted_measure_space[of "{0 .. 1}"] |
|
1067 |
by (simp add: pborel_def) |
|
1068 |
||
1069 |
interpretation pborel: prob_space pborel |
|
1070 |
by default (simp add: one_extreal_def pborel_def) |
|
1071 |
||
1072 |
lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 .. 1} then real (lborel.\<mu> A) else 0)" |
|
1073 |
unfolding pborel.\<mu>'_def by (auto simp: pborel_def) |
|
1074 |
||
1075 |
lemma pborel_singelton[simp]: "pborel.prob {a} = 0" |
|
1076 |
by (auto simp: pborel_prob) |
|
1077 |
||
1078 |
lemma |
|
1079 |
shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)" |
|
1080 |
and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)" |
|
1081 |
and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)" |
|
1082 |
and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)" |
|
1083 |
unfolding pborel_prob by (auto simp: atLeastLessThan_subseteq_atLeastAtMost_iff |
|
1084 |
greaterThanAtMost_subseteq_atLeastAtMost_iff greaterThanLessThan_subseteq_atLeastAtMost_iff) |
|
1085 |
||
1086 |
lemma pborel_lebesgue_measure: |
|
1087 |
"A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)" |
|
1088 |
by (simp add: sets_pborel pborel_prob) |
|
1089 |
||
1090 |
lemma pborel_alt: |
|
1091 |
"pborel = sigma \<lparr> |
|
1092 |
space = {0..1}, |
|
1093 |
sets = range (\<lambda>(x,y). {x..y} \<inter> {0..1}), |
|
1094 |
measure = measure lborel \<rparr>" (is "_ = ?R") |
|
1095 |
proof - |
|
1096 |
have *: "{0..1::real} \<in> sets borel" by auto |
|
1097 |
have **: "op \<inter> {0..1::real} ` range (\<lambda>(x, y). {x..y}) = range (\<lambda>(x,y). {x..y} \<inter> {0..1})" |
|
1098 |
unfolding image_image by (intro arg_cong[where f=range]) auto |
|
1099 |
have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a .. b :: real}), |
|
1100 |
measure = measure pborel\<rparr>) {0 .. 1}" |
|
1101 |
by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastAtMost lborel_def) |
|
1102 |
also have "\<dots> = ?R" |
|
1103 |
by (subst restricted_sigma) |
|
1104 |
(simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"]) |
|
1105 |
finally show ?thesis . |
|
1106 |
qed |
|
1107 |
||
42860 | 1108 |
subsection "Bernoulli space" |
1109 |
||
1110 |
definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV, |
|
1111 |
measure = extreal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>" |
|
1112 |
||
1113 |
interpretation bernoulli: finite_prob_space "bernoulli_space p" for p |
|
1114 |
by (rule finite_prob_spaceI) |
|
1115 |
(auto simp: bernoulli_space_def UNIV_bool one_extreal_def setsum_Un_disjoint intro!: setsum_nonneg) |
|
1116 |
||
1117 |
lemma bernoulli_measure: |
|
1118 |
"0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)" |
|
1119 |
unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong) |
|
1120 |
||
1121 |
lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p" |
|
1122 |
and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1 - p" |
|
1123 |
unfolding bernoulli_measure by simp_all |
|
1124 |
||
35582 | 1125 |
end |