author | huffman |
Fri, 02 Sep 2011 13:57:12 -0700 | |
changeset 44666 | 8670a39d4420 |
parent 43920 | cedb5cb948fd |
child 44890 | 22f665a2e91c |
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_Measure |
35582 | 10 |
begin |
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12 |
locale prob_space = measure_space + |
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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assumes measure_space_1: "measure M (space M) = 1" |
38656 | 14 |
|
15 |
sublocale prob_space < finite_measure |
|
16 |
proof |
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17 |
from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp |
38656 | 18 |
qed |
19 |
||
40859 | 20 |
abbreviation (in prob_space) "events \<equiv> sets M" |
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abbreviation (in prob_space) "prob \<equiv> \<mu>'" |
40859 | 22 |
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'" |
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abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M" |
35582 | 24 |
|
40859 | 25 |
definition (in prob_space) |
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"distribution X A = \<mu>' (X -` A \<inter> space M)" |
35582 | 27 |
|
40859 | 28 |
abbreviation (in prob_space) |
36624 | 29 |
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" |
35582 | 30 |
|
43339 | 31 |
lemma (in prob_space) prob_space_cong: |
32 |
assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M" |
|
33 |
shows "prob_space N" |
|
34 |
proof - |
|
35 |
interpret N: measure_space N by (intro measure_space_cong assms) |
|
36 |
show ?thesis by default (insert assms measure_space_1, simp) |
|
37 |
qed |
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38 |
||
39097 | 39 |
lemma (in prob_space) distribution_cong: |
40 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" |
|
41 |
shows "distribution X = distribution Y" |
|
39302
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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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|
42 |
unfolding distribution_def fun_eq_iff |
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) |
39097 | 44 |
|
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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" |
|
47 |
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'" |
|
39302
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unfolding distribution_def fun_eq_iff |
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|
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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" |
|
43920 | 57 |
using measure_space_1 unfolding \<mu>'_def by (simp add: one_ereal_def) |
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58 |
|
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parents:
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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 |
|
43339 | 65 |
lemma (in prob_space) not_zero_less_distribution[simp]: |
66 |
"(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0" |
|
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using distribution_positive[of X A] by arith |
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68 |
||
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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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71 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
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72 |
|
43339 | 73 |
lemma (in prob_space) not_empty: "space M \<noteq> {}" |
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using prob_space empty_measure' by auto |
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75 |
||
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move lemmas to Extended_Reals and Extended_Real_Limits
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76 |
lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1" |
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parents:
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changeset
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77 |
unfolding measure_space_1[symmetric] |
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move lemmas to Extended_Reals and Extended_Real_Limits
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78 |
using sets_into_space |
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move lemmas to Extended_Reals and Extended_Real_Limits
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79 |
by (intro measure_mono) auto |
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move lemmas to Extended_Reals and Extended_Real_Limits
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80 |
|
43339 | 81 |
lemma (in prob_space) AE_I_eq_1: |
82 |
assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M" |
|
83 |
shows "AE x. P x" |
|
84 |
proof (rule AE_I) |
|
85 |
show "\<mu> (space M - {x \<in> space M. P x}) = 0" |
|
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using assms measure_space_1 by (simp add: measure_compl) |
|
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qed (insert assms, auto) |
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88 |
||
41981
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hoelzl
parents:
41831
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changeset
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89 |
lemma (in prob_space) distribution_1: |
cdf7693bbe08
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90 |
"distribution X A \<le> 1" |
cdf7693bbe08
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91 |
unfolding distribution_def by simp |
35582 | 92 |
|
40859 | 93 |
lemma (in prob_space) prob_compl: |
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94 |
assumes A: "A \<in> events" |
38656 | 95 |
shows "prob (space M - A) = 1 - prob A" |
41981
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96 |
using finite_measure_compl[OF A] by (simp add: prob_space) |
35582 | 97 |
|
40859 | 98 |
lemma (in prob_space) prob_space_increasing: "increasing M prob" |
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by (auto intro!: finite_measure_mono simp: increasing_def) |
35582 | 100 |
|
40859 | 101 |
lemma (in prob_space) prob_zero_union: |
35582 | 102 |
assumes "s \<in> events" "t \<in> events" "prob t = 0" |
103 |
shows "prob (s \<union> t) = prob s" |
|
38656 | 104 |
using assms |
35582 | 105 |
proof - |
106 |
have "prob (s \<union> t) \<le> prob s" |
|
41981
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107 |
using finite_measure_subadditive[of s t] assms by auto |
35582 | 108 |
moreover have "prob (s \<union> t) \<ge> prob s" |
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|
109 |
using assms by (blast intro: finite_measure_mono) |
35582 | 110 |
ultimately show ?thesis by simp |
111 |
qed |
|
112 |
||
40859 | 113 |
lemma (in prob_space) prob_eq_compl: |
35582 | 114 |
assumes "s \<in> events" "t \<in> events" |
115 |
assumes "prob (space M - s) = prob (space M - t)" |
|
116 |
shows "prob s = prob t" |
|
38656 | 117 |
using assms prob_compl by auto |
35582 | 118 |
|
40859 | 119 |
lemma (in prob_space) prob_one_inter: |
35582 | 120 |
assumes events:"s \<in> events" "t \<in> events" |
121 |
assumes "prob t = 1" |
|
122 |
shows "prob (s \<inter> t) = prob s" |
|
123 |
proof - |
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38656 | 124 |
have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)" |
125 |
using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union) |
|
126 |
also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)" |
|
127 |
by blast |
|
128 |
finally show "prob (s \<inter> t) = prob s" |
|
129 |
using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s]) |
|
35582 | 130 |
qed |
131 |
||
40859 | 132 |
lemma (in prob_space) prob_eq_bigunion_image: |
35582 | 133 |
assumes "range f \<subseteq> events" "range g \<subseteq> events" |
134 |
assumes "disjoint_family f" "disjoint_family g" |
|
135 |
assumes "\<And> n :: nat. prob (f n) = prob (g n)" |
|
136 |
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" |
|
137 |
using assms |
|
138 |
proof - |
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38656 | 139 |
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))" |
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140 |
by (rule finite_measure_UNION[OF assms(1,3)]) |
38656 | 141 |
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" |
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41831
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|
142 |
by (rule finite_measure_UNION[OF assms(2,4)]) |
38656 | 143 |
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp |
35582 | 144 |
qed |
145 |
||
40859 | 146 |
lemma (in prob_space) prob_countably_zero: |
35582 | 147 |
assumes "range c \<subseteq> events" |
148 |
assumes "\<And> i. prob (c i) = 0" |
|
38656 | 149 |
shows "prob (\<Union> i :: nat. c i) = 0" |
150 |
proof (rule antisym) |
|
151 |
show "prob (\<Union> i :: nat. c i) \<le> 0" |
|
41981
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152 |
using finite_measure_countably_subadditive[OF assms(1)] |
38656 | 153 |
by (simp add: assms(2) suminf_zero summable_zero) |
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154 |
qed simp |
35582 | 155 |
|
40859 | 156 |
lemma (in prob_space) prob_equiprobable_finite_unions: |
38656 | 157 |
assumes "s \<in> events" |
158 |
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" |
|
35582 | 159 |
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" |
38656 | 160 |
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" |
35582 | 161 |
proof (cases "s = {}") |
38656 | 162 |
case False hence "\<exists> x. x \<in> s" by blast |
35582 | 163 |
from someI_ex[OF this] assms |
164 |
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast |
|
165 |
have "prob s = (\<Sum> x \<in> s. prob {x})" |
|
41981
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hoelzl
parents:
41831
diff
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|
166 |
using finite_measure_finite_singleton[OF s_finite] by simp |
35582 | 167 |
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto |
38656 | 168 |
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" |
169 |
using setsum_constant assms by (simp add: real_eq_of_nat) |
|
35582 | 170 |
finally show ?thesis by simp |
38656 | 171 |
qed simp |
35582 | 172 |
|
40859 | 173 |
lemma (in prob_space) prob_real_sum_image_fn: |
35582 | 174 |
assumes "e \<in> events" |
175 |
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" |
|
176 |
assumes "finite s" |
|
38656 | 177 |
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" |
178 |
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" |
|
35582 | 179 |
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" |
180 |
proof - |
|
38656 | 181 |
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" |
182 |
using `e \<in> events` sets_into_space upper by blast |
|
183 |
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp |
|
184 |
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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diff
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|
185 |
proof (rule finite_measure_finite_Union) |
38656 | 186 |
show "finite s" by fact |
187 |
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact |
|
188 |
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" |
|
189 |
using disjoint by (auto simp: disjoint_family_on_def) |
|
190 |
qed |
|
191 |
finally show ?thesis . |
|
35582 | 192 |
qed |
193 |
||
42199 | 194 |
lemma (in prob_space) prob_space_vimage: |
195 |
assumes S: "sigma_algebra S" |
|
196 |
assumes T: "T \<in> measure_preserving M S" |
|
197 |
shows "prob_space S" |
|
35582 | 198 |
proof - |
42199 | 199 |
interpret S: measure_space S |
200 |
using S and T by (rule measure_space_vimage) |
|
38656 | 201 |
show ?thesis |
42199 | 202 |
proof |
203 |
from T[THEN measure_preservingD2] |
|
204 |
have "T -` space S \<inter> space M = space M" |
|
205 |
by (auto simp: measurable_def) |
|
206 |
with T[THEN measure_preservingD, of "space S", symmetric] |
|
207 |
show "measure S (space S) = 1" |
|
208 |
using measure_space_1 by simp |
|
35582 | 209 |
qed |
210 |
qed |
|
211 |
||
42988 | 212 |
lemma prob_space_unique_Int_stable: |
213 |
fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set" |
|
214 |
assumes E: "Int_stable E" "space E \<in> sets E" |
|
215 |
and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)" |
|
216 |
and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)" |
|
217 |
and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X" |
|
218 |
assumes "X \<in> sets (sigma E)" |
|
219 |
shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X" |
|
220 |
proof - |
|
221 |
interpret M!: prob_space M by fact |
|
222 |
interpret N!: prob_space N by fact |
|
223 |
have "measure M X = measure N X" |
|
224 |
proof (rule measure_unique_Int_stable[OF `Int_stable E`]) |
|
225 |
show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E" |
|
226 |
using E M N by auto |
|
227 |
show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>" |
|
228 |
using M.measure_space_1 by simp |
|
229 |
show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>" |
|
230 |
using E M N by (auto intro!: M.measure_space_cong) |
|
231 |
show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>" |
|
232 |
using E M N by (auto intro!: N.measure_space_cong) |
|
233 |
{ fix X assume "X \<in> sets E" |
|
234 |
then have "X \<in> sets (sigma E)" |
|
235 |
by (auto simp: sets_sigma sigma_sets.Basic) |
|
236 |
with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X" |
|
237 |
by (simp add: M.finite_measure_eq N.finite_measure_eq) } |
|
238 |
qed fact |
|
239 |
with `X \<in> sets (sigma E)` M N show ?thesis |
|
240 |
by (simp add: M.finite_measure_eq N.finite_measure_eq) |
|
241 |
qed |
|
242 |
||
42199 | 243 |
lemma (in prob_space) distribution_prob_space: |
244 |
assumes X: "random_variable S X" |
|
43920 | 245 |
shows "prob_space (S\<lparr>measure := ereal \<circ> distribution X\<rparr>)" (is "prob_space ?S") |
42199 | 246 |
proof (rule prob_space_vimage) |
247 |
show "X \<in> measure_preserving M ?S" |
|
248 |
using X |
|
249 |
unfolding measure_preserving_def distribution_def_raw |
|
250 |
by (auto simp: finite_measure_eq measurable_sets) |
|
251 |
show "sigma_algebra ?S" using X by simp |
|
252 |
qed |
|
253 |
||
40859 | 254 |
lemma (in prob_space) AE_distribution: |
43920 | 255 |
assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := ereal \<circ> distribution X\<rparr>. Q x" |
40859 | 256 |
shows "AE x. Q (X x)" |
257 |
proof - |
|
43920 | 258 |
interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space) |
40859 | 259 |
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" |
260 |
using assms unfolding X.almost_everywhere_def by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
261 |
from X[unfolded measurable_def] N show "AE x. Q (X x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
262 |
by (intro AE_I'[where N="X -` N \<inter> space M"]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
263 |
(auto simp: finite_measure_eq distribution_def measurable_sets) |
40859 | 264 |
qed |
265 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
266 |
lemma (in prob_space) distribution_eq_integral: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
267 |
"random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
268 |
using finite_measure_eq[of "X -` A \<inter> space M"] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
269 |
by (auto simp: measurable_sets distribution_def) |
35582 | 270 |
|
43339 | 271 |
lemma (in prob_space) expectation_less: |
272 |
assumes [simp]: "integrable M X" |
|
273 |
assumes gt: "\<forall>x\<in>space M. X x < b" |
|
274 |
shows "expectation X < b" |
|
275 |
proof - |
|
276 |
have "expectation X < expectation (\<lambda>x. b)" |
|
277 |
using gt measure_space_1 |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
43339
diff
changeset
|
278 |
by (intro integral_less_AE_space) auto |
43339 | 279 |
then show ?thesis using prob_space by simp |
280 |
qed |
|
281 |
||
282 |
lemma (in prob_space) expectation_greater: |
|
283 |
assumes [simp]: "integrable M X" |
|
284 |
assumes gt: "\<forall>x\<in>space M. a < X x" |
|
285 |
shows "a < expectation X" |
|
286 |
proof - |
|
287 |
have "expectation (\<lambda>x. a) < expectation X" |
|
288 |
using gt measure_space_1 |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
43339
diff
changeset
|
289 |
by (intro integral_less_AE_space) auto |
43339 | 290 |
then show ?thesis using prob_space by simp |
291 |
qed |
|
292 |
||
293 |
lemma convex_le_Inf_differential: |
|
294 |
fixes f :: "real \<Rightarrow> real" |
|
295 |
assumes "convex_on I f" |
|
296 |
assumes "x \<in> interior I" "y \<in> I" |
|
297 |
shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)" |
|
298 |
(is "_ \<ge> _ + Inf (?F x) * (y - x)") |
|
299 |
proof - |
|
300 |
show ?thesis |
|
301 |
proof (cases rule: linorder_cases) |
|
302 |
assume "x < y" |
|
303 |
moreover |
|
304 |
have "open (interior I)" by auto |
|
305 |
from openE[OF this `x \<in> interior I`] guess e . note e = this |
|
306 |
moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)" |
|
307 |
ultimately have "x < t" "t < y" "t \<in> ball x e" |
|
308 |
by (auto simp: mem_ball dist_real_def field_simps split: split_min) |
|
309 |
with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto |
|
310 |
||
311 |
have "open (interior I)" by auto |
|
312 |
from openE[OF this `x \<in> interior I`] guess e . |
|
313 |
moreover def K \<equiv> "x - e / 2" |
|
314 |
with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def) |
|
315 |
ultimately have "K \<in> I" "K < x" "x \<in> I" |
|
316 |
using interior_subset[of I] `x \<in> interior I` by auto |
|
317 |
||
318 |
have "Inf (?F x) \<le> (f x - f y) / (x - y)" |
|
319 |
proof (rule Inf_lower2) |
|
320 |
show "(f x - f t) / (x - t) \<in> ?F x" |
|
321 |
using `t \<in> I` `x < t` by auto |
|
322 |
show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)" |
|
323 |
using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff) |
|
324 |
next |
|
325 |
fix y assume "y \<in> ?F x" |
|
326 |
with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]] |
|
327 |
show "(f K - f x) / (K - x) \<le> y" by auto |
|
328 |
qed |
|
329 |
then show ?thesis |
|
330 |
using `x < y` by (simp add: field_simps) |
|
331 |
next |
|
332 |
assume "y < x" |
|
333 |
moreover |
|
334 |
have "open (interior I)" by auto |
|
335 |
from openE[OF this `x \<in> interior I`] guess e . note e = this |
|
336 |
moreover def t \<equiv> "x + e / 2" |
|
337 |
ultimately have "x < t" "t \<in> ball x e" |
|
338 |
by (auto simp: mem_ball dist_real_def field_simps) |
|
339 |
with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto |
|
340 |
||
341 |
have "(f x - f y) / (x - y) \<le> Inf (?F x)" |
|
342 |
proof (rule Inf_greatest) |
|
343 |
have "(f x - f y) / (x - y) = (f y - f x) / (y - x)" |
|
344 |
using `y < x` by (auto simp: field_simps) |
|
345 |
also |
|
346 |
fix z assume "z \<in> ?F x" |
|
347 |
with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]] |
|
348 |
have "(f y - f x) / (y - x) \<le> z" by auto |
|
349 |
finally show "(f x - f y) / (x - y) \<le> z" . |
|
350 |
next |
|
351 |
have "open (interior I)" by auto |
|
352 |
from openE[OF this `x \<in> interior I`] guess e . note e = this |
|
353 |
then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def) |
|
354 |
with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto |
|
355 |
then show "?F x \<noteq> {}" by blast |
|
356 |
qed |
|
357 |
then show ?thesis |
|
358 |
using `y < x` by (simp add: field_simps) |
|
359 |
qed simp |
|
360 |
qed |
|
361 |
||
362 |
lemma (in prob_space) jensens_inequality: |
|
363 |
fixes a b :: real |
|
364 |
assumes X: "integrable M X" "X ` space M \<subseteq> I" |
|
365 |
assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV" |
|
366 |
assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q" |
|
367 |
shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" |
|
368 |
proof - |
|
369 |
let "?F x" = "Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))" |
|
370 |
from not_empty X(2) have "I \<noteq> {}" by auto |
|
371 |
||
372 |
from I have "open I" by auto |
|
373 |
||
374 |
note I |
|
375 |
moreover |
|
376 |
{ assume "I \<subseteq> {a <..}" |
|
377 |
with X have "a < expectation X" |
|
378 |
by (intro expectation_greater) auto } |
|
379 |
moreover |
|
380 |
{ assume "I \<subseteq> {..< b}" |
|
381 |
with X have "expectation X < b" |
|
382 |
by (intro expectation_less) auto } |
|
383 |
ultimately have "expectation X \<in> I" |
|
384 |
by (elim disjE) (auto simp: subset_eq) |
|
385 |
moreover |
|
386 |
{ fix y assume y: "y \<in> I" |
|
387 |
with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y" |
|
388 |
by (auto intro!: Sup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) } |
|
389 |
ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)" |
|
390 |
by simp |
|
391 |
also have "\<dots> \<le> expectation (\<lambda>w. q (X w))" |
|
392 |
proof (rule Sup_least) |
|
393 |
show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}" |
|
394 |
using `I \<noteq> {}` by auto |
|
395 |
next |
|
396 |
fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I" |
|
397 |
then guess x .. note x = this |
|
398 |
have "q x + ?F x * (expectation X - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))" |
|
399 |
using prob_space |
|
400 |
by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X) |
|
401 |
also have "\<dots> \<le> expectation (\<lambda>w. q (X w))" |
|
402 |
using `x \<in> I` `open I` X(2) |
|
403 |
by (intro integral_mono integral_add integral_cmult integral_diff |
|
404 |
lebesgue_integral_const X q convex_le_Inf_differential) |
|
405 |
(auto simp: interior_open) |
|
406 |
finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto |
|
407 |
qed |
|
408 |
finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" . |
|
409 |
qed |
|
410 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
411 |
lemma (in prob_space) distribution_eq_translated_integral: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
412 |
assumes "random_variable S X" "A \<in> sets S" |
43920 | 413 |
shows "distribution X A = integral\<^isup>P (S\<lparr>measure := ereal \<circ> distribution X\<rparr>) (indicator A)" |
35582 | 414 |
proof - |
43920 | 415 |
interpret S: prob_space "S\<lparr>measure := ereal \<circ> distribution X\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
416 |
using assms(1) by (rule distribution_prob_space) |
35582 | 417 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
418 |
using S.positive_integral_indicator(1)[of A] assms by simp |
35582 | 419 |
qed |
420 |
||
40859 | 421 |
lemma (in prob_space) finite_expectation1: |
422 |
assumes f: "finite (X`space M)" and rv: "random_variable borel X" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
423 |
shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r") |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
424 |
proof (subst integral_on_finite) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
425 |
show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
426 |
show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
427 |
"\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
428 |
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto |
38656 | 429 |
qed |
35582 | 430 |
|
40859 | 431 |
lemma (in prob_space) finite_expectation: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
432 |
assumes "finite (X`space M)" "random_variable borel X" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
433 |
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" |
38656 | 434 |
using assms unfolding distribution_def using finite_expectation1 by auto |
435 |
||
40859 | 436 |
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: |
35582 | 437 |
assumes "{x} \<in> events" |
38656 | 438 |
assumes "prob {x} = 1" |
35582 | 439 |
assumes "{y} \<in> events" |
440 |
assumes "y \<noteq> x" |
|
441 |
shows "prob {y} = 0" |
|
442 |
using prob_one_inter[of "{y}" "{x}"] assms by auto |
|
443 |
||
40859 | 444 |
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" |
38656 | 445 |
unfolding distribution_def by simp |
446 |
||
40859 | 447 |
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" |
38656 | 448 |
proof - |
449 |
have "X -` X ` space M \<inter> space M = space M" by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
450 |
thus ?thesis unfolding distribution_def by (simp add: prob_space) |
38656 | 451 |
qed |
452 |
||
40859 | 453 |
lemma (in prob_space) distribution_one: |
454 |
assumes "random_variable M' X" and "A \<in> sets M'" |
|
38656 | 455 |
shows "distribution X A \<le> 1" |
456 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
457 |
have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
458 |
using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
459 |
thus ?thesis by (simp add: prob_space) |
38656 | 460 |
qed |
461 |
||
40859 | 462 |
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: |
35582 | 463 |
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" |
38656 | 464 |
(is "random_variable ?S X") |
465 |
assumes "distribution X {x} = 1" |
|
35582 | 466 |
assumes "y \<noteq> x" |
467 |
shows "distribution X {y} = 0" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
468 |
proof cases |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
469 |
{ fix x have "X -` {x} \<inter> space M \<in> sets M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
470 |
proof cases |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
471 |
assume "x \<in> X`space M" with X show ?thesis |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
472 |
by (auto simp: measurable_def image_iff) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
473 |
next |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
474 |
assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
475 |
then show ?thesis by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
476 |
qed } note single = this |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
477 |
have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
478 |
"X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
479 |
using `y \<noteq> x` by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
480 |
with finite_measure_inter_full_set[OF single single, of x y] assms(2) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
481 |
show ?thesis by (auto simp: distribution_def prob_space) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
482 |
next |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
483 |
assume "{y} \<notin> sets ?S" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
484 |
then have "X -` {y} \<inter> space M = {}" by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
485 |
thus "distribution X {y} = 0" unfolding distribution_def by auto |
35582 | 486 |
qed |
487 |
||
40859 | 488 |
lemma (in prob_space) joint_distribution_Times_le_fst: |
489 |
assumes X: "random_variable MX X" and Y: "random_variable MY Y" |
|
490 |
and A: "A \<in> sets MX" and B: "B \<in> sets MY" |
|
491 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" |
|
492 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
493 |
proof (intro finite_measure_mono) |
40859 | 494 |
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force |
495 |
show "X -` A \<inter> space M \<in> events" |
|
496 |
using X A unfolding measurable_def by simp |
|
497 |
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M = |
|
498 |
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
499 |
qed |
|
500 |
||
501 |
lemma (in prob_space) joint_distribution_commute: |
|
502 |
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
503 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 504 |
|
505 |
lemma (in prob_space) joint_distribution_Times_le_snd: |
|
506 |
assumes X: "random_variable MX X" and Y: "random_variable MY Y" |
|
507 |
and A: "A \<in> sets MX" and B: "B \<in> sets MY" |
|
508 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" |
|
509 |
using assms |
|
510 |
by (subst joint_distribution_commute) |
|
511 |
(simp add: swap_product joint_distribution_Times_le_fst) |
|
512 |
||
513 |
lemma (in prob_space) random_variable_pairI: |
|
514 |
assumes "random_variable MX X" |
|
515 |
assumes "random_variable MY Y" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
516 |
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" |
40859 | 517 |
proof |
518 |
interpret MX: sigma_algebra MX using assms by simp |
|
519 |
interpret MY: sigma_algebra MY using assms by simp |
|
520 |
interpret P: pair_sigma_algebra MX MY by default |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
521 |
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default |
40859 | 522 |
have sa: "sigma_algebra M" by default |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
523 |
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" |
41095 | 524 |
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) |
40859 | 525 |
qed |
526 |
||
527 |
lemma (in prob_space) joint_distribution_commute_singleton: |
|
528 |
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
529 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 530 |
|
531 |
lemma (in prob_space) joint_distribution_assoc_singleton: |
|
532 |
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = |
|
533 |
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
534 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 535 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
536 |
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 |
40859 | 537 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
538 |
sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
539 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
540 |
sublocale pair_prob_space \<subseteq> P: prob_space P |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
541 |
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) |
40859 | 542 |
|
543 |
lemma countably_additiveI[case_names countably]: |
|
544 |
assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
545 |
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" |
40859 | 546 |
shows "countably_additive M \<mu>" |
547 |
using assms unfolding countably_additive_def by auto |
|
548 |
||
549 |
lemma (in prob_space) joint_distribution_prob_space: |
|
550 |
assumes "random_variable MX X" "random_variable MY Y" |
|
43920 | 551 |
shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
552 |
using random_variable_pairI[OF assms] by (rule distribution_prob_space) |
40859 | 553 |
|
42988 | 554 |
|
555 |
locale finite_product_prob_space = |
|
556 |
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" |
|
557 |
and I :: "'i set" |
|
558 |
assumes prob_space: "\<And>i. prob_space (M i)" and finite_index: "finite I" |
|
559 |
||
560 |
sublocale finite_product_prob_space \<subseteq> M: prob_space "M i" for i |
|
561 |
by (rule prob_space) |
|
562 |
||
563 |
sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite M I |
|
564 |
by default (rule finite_index) |
|
565 |
||
566 |
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i" |
|
567 |
proof qed (simp add: measure_times M.measure_space_1 setprod_1) |
|
568 |
||
569 |
lemma (in finite_product_prob_space) prob_times: |
|
570 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)" |
|
571 |
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))" |
|
572 |
proof - |
|
43920 | 573 |
have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)" |
42988 | 574 |
using X by (intro finite_measure_eq[symmetric] in_P) auto |
575 |
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))" |
|
576 |
using measure_times X by simp |
|
43920 | 577 |
also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))" |
578 |
using X by (simp add: M.finite_measure_eq setprod_ereal) |
|
42988 | 579 |
finally show ?thesis by simp |
580 |
qed |
|
581 |
||
582 |
lemma (in prob_space) random_variable_restrict: |
|
583 |
assumes I: "finite I" |
|
584 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)" |
|
585 |
shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)" |
|
586 |
proof |
|
587 |
{ fix i assume "i \<in> I" |
|
588 |
with X interpret N: sigma_algebra "N i" by simp |
|
589 |
have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) } |
|
590 |
note N_closed = this |
|
591 |
then show "sigma_algebra (Pi\<^isub>M I N)" |
|
592 |
by (simp add: product_algebra_def) |
|
593 |
(intro sigma_algebra_sigma product_algebra_generator_sets_into_space) |
|
594 |
show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)" |
|
595 |
using X by (intro measurable_restrict[OF I N_closed]) auto |
|
596 |
qed |
|
597 |
||
40859 | 598 |
section "Probability spaces on finite sets" |
35582 | 599 |
|
35977 | 600 |
locale finite_prob_space = prob_space + finite_measure_space |
601 |
||
40859 | 602 |
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" |
603 |
||
604 |
lemma (in prob_space) finite_random_variableD: |
|
605 |
assumes "finite_random_variable M' X" shows "random_variable M' X" |
|
606 |
proof - |
|
607 |
interpret M': finite_sigma_algebra M' using assms by simp |
|
608 |
then show "random_variable M' X" using assms by simp default |
|
609 |
qed |
|
610 |
||
611 |
lemma (in prob_space) distribution_finite_prob_space: |
|
612 |
assumes "finite_random_variable MX X" |
|
43920 | 613 |
shows "finite_prob_space (MX\<lparr>measure := ereal \<circ> distribution X\<rparr>)" |
40859 | 614 |
proof - |
43920 | 615 |
interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>" |
40859 | 616 |
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) |
617 |
interpret MX: finite_sigma_algebra MX |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
618 |
using assms by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
619 |
show ?thesis by default (simp_all add: MX.finite_space) |
40859 | 620 |
qed |
621 |
||
622 |
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
623 |
assumes "simple_function M X" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
624 |
shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
625 |
(is "finite_random_variable ?X _") |
40859 | 626 |
proof (intro conjI) |
627 |
have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
628 |
interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
629 |
show "finite_sigma_algebra ?X" |
40859 | 630 |
by default auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
631 |
show "X \<in> measurable M ?X" |
40859 | 632 |
proof (unfold measurable_def, clarsimp) |
633 |
fix A assume A: "A \<subseteq> X`space M" |
|
634 |
then have "finite A" by (rule finite_subset) simp |
|
635 |
then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" |
|
636 |
unfolding vimage_UN UN_extend_simps |
|
637 |
apply (rule finite_UN) |
|
638 |
using A assms unfolding simple_function_def by auto |
|
639 |
then show "X -` A \<inter> space M \<in> events" by simp |
|
640 |
qed |
|
641 |
qed |
|
642 |
||
643 |
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
644 |
assumes "simple_function M X" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
645 |
shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
646 |
using simple_function_imp_finite_random_variable[OF assms, of ext] |
40859 | 647 |
by (auto dest!: finite_random_variableD) |
648 |
||
649 |
lemma (in prob_space) sum_over_space_real_distribution: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
650 |
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" |
40859 | 651 |
unfolding distribution_def prob_space[symmetric] |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
652 |
by (subst finite_measure_finite_Union[symmetric]) |
40859 | 653 |
(auto simp add: disjoint_family_on_def simple_function_def |
654 |
intro!: arg_cong[where f=prob]) |
|
655 |
||
656 |
lemma (in prob_space) finite_random_variable_pairI: |
|
657 |
assumes "finite_random_variable MX X" |
|
658 |
assumes "finite_random_variable MY Y" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
659 |
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" |
40859 | 660 |
proof |
661 |
interpret MX: finite_sigma_algebra MX using assms by simp |
|
662 |
interpret MY: finite_sigma_algebra MY using assms by simp |
|
663 |
interpret P: pair_finite_sigma_algebra MX MY by default |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
664 |
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default |
40859 | 665 |
have sa: "sigma_algebra M" by default |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
666 |
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" |
41095 | 667 |
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) |
40859 | 668 |
qed |
669 |
||
670 |
lemma (in prob_space) finite_random_variable_imp_sets: |
|
671 |
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" |
|
672 |
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp |
|
673 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
674 |
lemma (in prob_space) finite_random_variable_measurable: |
40859 | 675 |
assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events" |
676 |
proof - |
|
677 |
interpret X: finite_sigma_algebra MX using X by simp |
|
678 |
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and |
|
679 |
"X \<in> space M \<rightarrow> space MX" |
|
680 |
by (auto simp: measurable_def) |
|
681 |
then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M" |
|
682 |
by auto |
|
683 |
show "X -` A \<inter> space M \<in> events" |
|
684 |
unfolding * by (intro vimage) auto |
|
685 |
qed |
|
686 |
||
687 |
lemma (in prob_space) joint_distribution_finite_Times_le_fst: |
|
688 |
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" |
|
689 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" |
|
690 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
691 |
proof (intro finite_measure_mono) |
40859 | 692 |
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force |
693 |
show "X -` A \<inter> space M \<in> events" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
694 |
using finite_random_variable_measurable[OF X] . |
40859 | 695 |
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M = |
696 |
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
697 |
qed |
|
698 |
||
699 |
lemma (in prob_space) joint_distribution_finite_Times_le_snd: |
|
700 |
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" |
|
701 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" |
|
702 |
using assms |
|
703 |
by (subst joint_distribution_commute) |
|
704 |
(simp add: swap_product joint_distribution_finite_Times_le_fst) |
|
705 |
||
706 |
lemma (in prob_space) finite_distribution_order: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
707 |
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" |
40859 | 708 |
assumes "finite_random_variable MX X" "finite_random_variable MY Y" |
709 |
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" |
|
710 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" |
|
711 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" |
|
712 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" |
|
713 |
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
714 |
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
715 |
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] |
|
716 |
using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
717 |
by (auto intro: antisym) |
40859 | 718 |
|
719 |
lemma (in prob_space) setsum_joint_distribution: |
|
720 |
assumes X: "finite_random_variable MX X" |
|
721 |
assumes Y: "random_variable MY Y" "B \<in> sets MY" |
|
722 |
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" |
|
723 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
724 |
proof (subst finite_measure_finite_Union[symmetric]) |
40859 | 725 |
interpret MX: finite_sigma_algebra MX using X by auto |
726 |
show "finite (space MX)" using MX.finite_space . |
|
727 |
let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M" |
|
728 |
{ fix i assume "i \<in> space MX" |
|
729 |
moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
730 |
ultimately show "?d i \<in> events" |
|
731 |
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y |
|
732 |
using MX.sets_eq_Pow by auto } |
|
733 |
show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
734 |
show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
735 |
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 736 |
qed |
737 |
||
738 |
lemma (in prob_space) setsum_joint_distribution_singleton: |
|
739 |
assumes X: "finite_random_variable MX X" |
|
740 |
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" |
|
741 |
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" |
|
742 |
using setsum_joint_distribution[OF X |
|
743 |
finite_random_variableD[OF Y(1)] |
|
744 |
finite_random_variable_imp_sets[OF Y]] by simp |
|
745 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
746 |
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 |
40859 | 747 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
748 |
sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
749 |
sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2 by default |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
750 |
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default |
40859 | 751 |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
752 |
locale product_finite_prob_space = |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
753 |
fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
754 |
and I :: "'i set" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
755 |
assumes finite_space: "\<And>i. finite_prob_space (M i)" and finite_index: "finite I" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
756 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
757 |
sublocale product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
758 |
sublocale product_finite_prob_space \<subseteq> finite_product_sigma_finite M I by default (rule finite_index) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
759 |
sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
760 |
proof |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
761 |
show "\<mu> (space P) = 1" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
762 |
using measure_times[OF M.top] M.measure_space_1 |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
763 |
by (simp add: setprod_1 space_product_algebra) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
764 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
765 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
766 |
lemma funset_eq_UN_fun_upd_I: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
767 |
assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
768 |
and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
769 |
and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
770 |
shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
771 |
proof safe |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
772 |
fix f assume f: "f \<in> F (insert a A)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
773 |
show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
774 |
proof (rule UN_I[of "f(a := d)"]) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
775 |
show "f(a := d) \<in> F A" using *[OF f] . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
776 |
show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
777 |
proof (rule image_eqI[of _ _ "f a"]) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
778 |
show "f a \<in> G (f(a := d))" using **[OF f] . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
779 |
qed simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
780 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
781 |
next |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
782 |
fix f x assume "f \<in> F A" "x \<in> G f" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
783 |
from ***[OF this] show "f(a := x) \<in> F (insert a A)" . |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
784 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
785 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
786 |
lemma extensional_funcset_insert_eq[simp]: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
787 |
assumes "a \<notin> A" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
788 |
shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
789 |
apply (rule funset_eq_UN_fun_upd_I) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
790 |
using assms |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
791 |
by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
792 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
793 |
lemma finite_extensional_funcset[simp, intro]: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
794 |
assumes "finite A" "finite B" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
795 |
shows "finite (extensional A \<inter> (A \<rightarrow> B))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
796 |
using assms by induct (auto simp: extensional_funcset_insert_eq) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
797 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
798 |
lemma finite_PiE[simp, intro]: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
799 |
assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
800 |
shows "finite (Pi\<^isub>E A B)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
801 |
proof - |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
802 |
have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
803 |
show ?thesis |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
804 |
using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
805 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
806 |
|
42892 | 807 |
lemma (in product_finite_prob_space) singleton_eq_product: |
808 |
assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})" |
|
809 |
proof (safe intro!: ext[of _ x]) |
|
810 |
fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I" |
|
811 |
with x show "y i = x i" |
|
812 |
by (cases "i \<in> I") (auto simp: extensional_def) |
|
813 |
qed (insert x, auto) |
|
814 |
||
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
815 |
sublocale product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
816 |
proof |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
817 |
show "finite (space P)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
818 |
using finite_index M.finite_space by auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
819 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
820 |
{ fix x assume "x \<in> space P" |
42892 | 821 |
with this[THEN singleton_eq_product] |
822 |
have "{x} \<in> sets P" |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
823 |
by (auto intro!: in_P) } |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
824 |
note x_in_P = this |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
825 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
826 |
have "Pow (space P) \<subseteq> sets P" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
827 |
proof |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
828 |
fix X assume "X \<in> Pow (space P)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
829 |
moreover then have "finite X" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
830 |
using `finite (space P)` by (blast intro: finite_subset) |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
831 |
ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
832 |
by (intro finite_UN x_in_P) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
833 |
then show "X \<in> sets P" by simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
834 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
835 |
with space_closed show [simp]: "sets P = Pow (space P)" .. |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
836 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
837 |
{ fix x assume "x \<in> space P" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
838 |
from this top have "\<mu> {x} \<le> \<mu> (space P)" by (intro measure_mono) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
839 |
then show "\<mu> {x} \<noteq> \<infinity>" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
840 |
using measure_space_1 by auto } |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
841 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
842 |
|
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
843 |
lemma (in product_finite_prob_space) measure_finite_times: |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
844 |
"(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
845 |
by (rule measure_times) simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
846 |
|
42892 | 847 |
lemma (in product_finite_prob_space) measure_singleton_times: |
848 |
assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})" |
|
849 |
unfolding singleton_eq_product[OF x] using x |
|
850 |
by (intro measure_finite_times) auto |
|
851 |
||
852 |
lemma (in product_finite_prob_space) prob_finite_times: |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
853 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
854 |
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
855 |
proof - |
43920 | 856 |
have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)" |
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
857 |
using X by (intro finite_measure_eq[symmetric] in_P) auto |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
858 |
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))" |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
859 |
using measure_finite_times X by simp |
43920 | 860 |
also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))" |
861 |
using X by (simp add: M.finite_measure_eq setprod_ereal) |
|
42859
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
862 |
finally show ?thesis by simp |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
863 |
qed |
d9dfc733f25c
add product of probability spaces with finite cardinality
hoelzl
parents:
42858
diff
changeset
|
864 |
|
42892 | 865 |
lemma (in product_finite_prob_space) prob_singleton_times: |
866 |
assumes x: "x \<in> space P" |
|
867 |
shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})" |
|
868 |
unfolding singleton_eq_product[OF x] using x |
|
869 |
by (intro prob_finite_times) auto |
|
870 |
||
871 |
lemma (in product_finite_prob_space) prob_finite_product: |
|
872 |
"A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})" |
|
873 |
by (auto simp add: finite_measure_singleton prob_singleton_times |
|
874 |
simp del: space_product_algebra |
|
875 |
intro!: setsum_cong prob_singleton_times) |
|
876 |
||
40859 | 877 |
lemma (in prob_space) joint_distribution_finite_prob_space: |
878 |
assumes X: "finite_random_variable MX X" |
|
879 |
assumes Y: "finite_random_variable MY Y" |
|
43920 | 880 |
shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
881 |
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) |
40859 | 882 |
|
36624 | 883 |
lemma finite_prob_space_eq: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
884 |
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" |
36624 | 885 |
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def |
886 |
by auto |
|
887 |
||
38656 | 888 |
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" |
889 |
using measure_space_1 sum_over_space by simp |
|
36624 | 890 |
|
891 |
lemma (in finite_prob_space) joint_distribution_restriction_fst: |
|
892 |
"joint_distribution X Y A \<le> distribution X (fst ` A)" |
|
893 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
894 |
proof (safe intro!: finite_measure_mono) |
36624 | 895 |
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" |
896 |
show "x \<in> X -` fst ` A" |
|
897 |
by (auto intro!: image_eqI[OF _ *]) |
|
898 |
qed (simp_all add: sets_eq_Pow) |
|
899 |
||
900 |
lemma (in finite_prob_space) joint_distribution_restriction_snd: |
|
901 |
"joint_distribution X Y A \<le> distribution Y (snd ` A)" |
|
902 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
903 |
proof (safe intro!: finite_measure_mono) |
36624 | 904 |
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" |
905 |
show "x \<in> Y -` snd ` A" |
|
906 |
by (auto intro!: image_eqI[OF _ *]) |
|
907 |
qed (simp_all add: sets_eq_Pow) |
|
908 |
||
909 |
lemma (in finite_prob_space) distribution_order: |
|
910 |
shows "0 \<le> distribution X x'" |
|
911 |
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" |
|
912 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" |
|
913 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" |
|
914 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" |
|
915 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" |
|
916 |
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
917 |
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
918 |
using |
36624 | 919 |
joint_distribution_restriction_fst[of X Y "{(x, y)}"] |
920 |
joint_distribution_restriction_snd[of X Y "{(x, y)}"] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
921 |
by (auto intro: antisym) |
36624 | 922 |
|
39097 | 923 |
lemma (in finite_prob_space) distribution_mono: |
924 |
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" |
|
925 |
shows "distribution X x \<le> distribution Y y" |
|
926 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
927 |
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) |
39097 | 928 |
|
929 |
lemma (in finite_prob_space) distribution_mono_gt_0: |
|
930 |
assumes gt_0: "0 < distribution X x" |
|
931 |
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" |
|
932 |
shows "0 < distribution Y y" |
|
933 |
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) |
|
934 |
||
935 |
lemma (in finite_prob_space) sum_over_space_distrib: |
|
936 |
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
937 |
unfolding distribution_def prob_space[symmetric] using finite_space |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
938 |
by (subst finite_measure_finite_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
939 |
(auto simp add: disjoint_family_on_def sets_eq_Pow |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
940 |
intro!: arg_cong[where f=\<mu>']) |
39097 | 941 |
|
942 |
lemma (in finite_prob_space) finite_sum_over_space_eq_1: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
943 |
"(\<Sum>x\<in>space M. prob {x}) = 1" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
944 |
using prob_space finite_space |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
945 |
by (subst (asm) finite_measure_finite_singleton) auto |
39097 | 946 |
|
947 |
lemma (in prob_space) distribution_remove_const: |
|
948 |
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" |
|
949 |
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" |
|
950 |
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" |
|
951 |
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" |
|
952 |
and "distribution (\<lambda>x. ()) {()} = 1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
953 |
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) |
35977 | 954 |
|
39097 | 955 |
lemma (in finite_prob_space) setsum_distribution_gen: |
956 |
assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M" |
|
957 |
and "inj_on f (X`space M)" |
|
958 |
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" |
|
959 |
unfolding distribution_def assms |
|
960 |
using finite_space assms |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
961 |
by (subst finite_measure_finite_Union[symmetric]) |
39097 | 962 |
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def |
963 |
intro!: arg_cong[where f=prob]) |
|
964 |
||
965 |
lemma (in finite_prob_space) setsum_distribution: |
|
966 |
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" |
|
967 |
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" |
|
968 |
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" |
|
969 |
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" |
|
970 |
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" |
|
971 |
by (auto intro!: inj_onI setsum_distribution_gen) |
|
972 |
||
973 |
lemma (in finite_prob_space) uniform_prob: |
|
974 |
assumes "x \<in> space M" |
|
975 |
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
976 |
shows "prob {x} = 1 / card (space M)" |
39097 | 977 |
proof - |
978 |
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" |
|
979 |
using assms(2)[OF _ `x \<in> space M`] by blast |
|
980 |
have "1 = prob (space M)" |
|
981 |
using prob_space by auto |
|
982 |
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
983 |
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] |
39097 | 984 |
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] |
985 |
finite_space unfolding disjoint_family_on_def prob_space[symmetric] |
|
986 |
by (auto simp add:setsum_restrict_set) |
|
987 |
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" |
|
988 |
using prob_x by auto |
|
989 |
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp |
|
990 |
finally have one: "1 = real (card (space M)) * prob {x}" |
|
991 |
using real_eq_of_nat by auto |
|
43339 | 992 |
hence two: "real (card (space M)) \<noteq> 0" by fastsimp |
39097 | 993 |
from one have three: "prob {x} \<noteq> 0" by fastsimp |
994 |
thus ?thesis using one two three divide_cancel_right |
|
995 |
by (auto simp:field_simps) |
|
39092 | 996 |
qed |
35977 | 997 |
|
39092 | 998 |
lemma (in prob_space) prob_space_subalgebra: |
41545 | 999 |
assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1000 |
and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1001 |
shows "prob_space N" |
39092 | 1002 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1003 |
interpret N: measure_space N |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1004 |
by (rule measure_space_subalgebra[OF assms]) |
39092 | 1005 |
show ?thesis |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1006 |
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) |
35977 | 1007 |
qed |
1008 |
||
39092 | 1009 |
lemma (in prob_space) prob_space_of_restricted_space: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1010 |
assumes "\<mu> A \<noteq> 0" "A \<in> sets M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1011 |
shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1012 |
(is "prob_space ?P") |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1013 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1014 |
interpret A: measure_space "restricted_space A" |
39092 | 1015 |
using `A \<in> sets M` by (rule restricted_measure_space) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1016 |
interpret A': sigma_algebra ?P |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1017 |
by (rule A.sigma_algebra_cong) auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1018 |
show "prob_space ?P" |
39092 | 1019 |
proof |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1020 |
show "measure ?P (space ?P) = 1" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1021 |
using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1022 |
show "positive ?P (measure ?P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1023 |
proof (simp add: positive_def, safe) |
43920 | 1024 |
show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_ereal_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1025 |
fix B assume "B \<in> events" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1026 |
with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M` |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1027 |
show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1028 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1029 |
show "countably_additive ?P (measure ?P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1030 |
proof (simp add: countably_additive_def, safe) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1031 |
fix B and F :: "nat \<Rightarrow> 'a set" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1032 |
assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1033 |
{ fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1034 |
from F have "F i \<in> op \<inter> A ` events" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1035 |
with `A \<in> events` have "F i \<in> events" by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1036 |
moreover then have "range F \<subseteq> events" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1037 |
moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" |
43920 | 1038 |
by (simp add: mult_commute divide_ereal_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1039 |
moreover have "0 \<le> inverse (\<mu> A)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1040 |
using real_measure[OF `A \<in> events`] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1041 |
ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1042 |
using measure_countably_additive[of F] F |
43920 | 1043 |
by (auto simp: suminf_cmult_ereal) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1044 |
qed |
39092 | 1045 |
qed |
1046 |
qed |
|
1047 |
||
1048 |
lemma finite_prob_spaceI: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1049 |
assumes "finite (space M)" "sets M = Pow(space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1050 |
and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1051 |
and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1052 |
shows "finite_prob_space M" |
39092 | 1053 |
unfolding finite_prob_space_eq |
1054 |
proof |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1055 |
show "finite_measure_space M" using assms |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1056 |
by (auto intro!: finite_measure_spaceI) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1057 |
show "measure M (space M) = 1" by fact |
39092 | 1058 |
qed |
36624 | 1059 |
|
1060 |
lemma (in finite_prob_space) finite_measure_space: |
|
39097 | 1061 |
fixes X :: "'a \<Rightarrow> 'x" |
43920 | 1062 |
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1063 |
(is "finite_measure_space ?S") |
39092 | 1064 |
proof (rule finite_measure_spaceI, simp_all) |
36624 | 1065 |
show "finite (X ` space M)" using finite_space by simp |
39097 | 1066 |
next |
1067 |
fix A B :: "'x set" assume "A \<inter> B = {}" |
|
1068 |
then show "distribution X (A \<union> B) = distribution X A + distribution X B" |
|
1069 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1070 |
by (subst finite_measure_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1071 |
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) |
36624 | 1072 |
qed |
1073 |
||
39097 | 1074 |
lemma (in finite_prob_space) finite_prob_space_of_images: |
43920 | 1075 |
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X \<rparr>" |
1076 |
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_ereal_def) |
|
39097 | 1077 |
|
39096 | 1078 |
lemma (in finite_prob_space) finite_product_measure_space: |
39097 | 1079 |
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" |
39096 | 1080 |
assumes "finite s1" "finite s2" |
43920 | 1081 |
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = ereal \<circ> joint_distribution X Y\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1082 |
(is "finite_measure_space ?M") |
39097 | 1083 |
proof (rule finite_measure_spaceI, simp_all) |
1084 |
show "finite (s1 \<times> s2)" |
|
39096 | 1085 |
using assms by auto |
39097 | 1086 |
next |
1087 |
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" |
|
1088 |
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" |
|
1089 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1090 |
by (subst finite_measure_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1091 |
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) |
39096 | 1092 |
qed |
1093 |
||
39097 | 1094 |
lemma (in finite_prob_space) finite_product_measure_space_of_images: |
39096 | 1095 |
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M, |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1096 |
sets = Pow (X ` space M \<times> Y ` space M), |
43920 | 1097 |
measure = ereal \<circ> joint_distribution X Y \<rparr>" |
39096 | 1098 |
using finite_space by (auto intro!: finite_product_measure_space) |
1099 |
||
40859 | 1100 |
lemma (in finite_prob_space) finite_product_prob_space_of_images: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1101 |
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M), |
43920 | 1102 |
measure = ereal \<circ> joint_distribution X Y \<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1103 |
(is "finite_prob_space ?S") |
43920 | 1104 |
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_ereal_def) |
40859 | 1105 |
have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto |
1106 |
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" |
|
1107 |
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) |
|
1108 |
qed |
|
1109 |
||
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1110 |
subsection "Borel Measure on {0 ..< 1}" |
42902 | 1111 |
|
1112 |
definition pborel :: "real measure_space" where |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1113 |
"pborel = lborel.restricted_space {0 ..< 1}" |
42902 | 1114 |
|
1115 |
lemma space_pborel[simp]: |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1116 |
"space pborel = {0 ..< 1}" |
42902 | 1117 |
unfolding pborel_def by auto |
1118 |
||
1119 |
lemma sets_pborel: |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1120 |
"A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 ..< 1}" |
42902 | 1121 |
unfolding pborel_def by auto |
1122 |
||
1123 |
lemma in_pborel[intro, simp]: |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1124 |
"A \<subseteq> {0 ..< 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel" |
42902 | 1125 |
unfolding pborel_def by auto |
1126 |
||
1127 |
interpretation pborel: measure_space pborel |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1128 |
using lborel.restricted_measure_space[of "{0 ..< 1}"] |
42902 | 1129 |
by (simp add: pborel_def) |
1130 |
||
1131 |
interpretation pborel: prob_space pborel |
|
43920 | 1132 |
by default (simp add: one_ereal_def pborel_def) |
42902 | 1133 |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1134 |
lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 ..< 1} then real (lborel.\<mu> A) else 0)" |
42902 | 1135 |
unfolding pborel.\<mu>'_def by (auto simp: pborel_def) |
1136 |
||
1137 |
lemma pborel_singelton[simp]: "pborel.prob {a} = 0" |
|
1138 |
by (auto simp: pborel_prob) |
|
1139 |
||
1140 |
lemma |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1141 |
shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)" |
42902 | 1142 |
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)" |
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1143 |
and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)" |
42902 | 1144 |
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)" |
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1145 |
unfolding pborel_prob |
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1146 |
by (auto simp: atLeastAtMost_subseteq_atLeastLessThan_iff |
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1147 |
greaterThanAtMost_subseteq_atLeastLessThan_iff greaterThanLessThan_subseteq_atLeastLessThan_iff) |
42902 | 1148 |
|
1149 |
lemma pborel_lebesgue_measure: |
|
1150 |
"A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)" |
|
1151 |
by (simp add: sets_pborel pborel_prob) |
|
1152 |
||
1153 |
lemma pborel_alt: |
|
1154 |
"pborel = sigma \<lparr> |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1155 |
space = {0..<1}, |
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1156 |
sets = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1}), |
42902 | 1157 |
measure = measure lborel \<rparr>" (is "_ = ?R") |
1158 |
proof - |
|
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1159 |
have *: "{0..<1::real} \<in> sets borel" by auto |
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1160 |
have **: "op \<inter> {0..<1::real} ` range (\<lambda>(x, y). {x..<y}) = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1})" |
42902 | 1161 |
unfolding image_image by (intro arg_cong[where f=range]) auto |
43658
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1162 |
have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real}), |
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1163 |
measure = measure pborel\<rparr>) {0 ..< 1}" |
0d96ec6ec33b
the borel probability measure is easier to handle with {0 ..< 1} (coverable by disjoint intervals {_ ..< _})
hoelzl
parents:
43556
diff
changeset
|
1164 |
by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastLessThan lborel_def) |
42902 | 1165 |
also have "\<dots> = ?R" |
1166 |
by (subst restricted_sigma) |
|
1167 |
(simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"]) |
|
1168 |
finally show ?thesis . |
|
1169 |
qed |
|
1170 |
||
42860 | 1171 |
subsection "Bernoulli space" |
1172 |
||
1173 |
definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV, |
|
43920 | 1174 |
measure = ereal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>" |
42860 | 1175 |
|
1176 |
interpretation bernoulli: finite_prob_space "bernoulli_space p" for p |
|
1177 |
by (rule finite_prob_spaceI) |
|
43920 | 1178 |
(auto simp: bernoulli_space_def UNIV_bool one_ereal_def setsum_Un_disjoint intro!: setsum_nonneg) |
42860 | 1179 |
|
1180 |
lemma bernoulli_measure: |
|
1181 |
"0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)" |
|
1182 |
unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong) |
|
1183 |
||
1184 |
lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p" |
|
1185 |
and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1 - p" |
|
1186 |
unfolding bernoulli_measure by simp_all |
|
1187 |
||
35582 | 1188 |
end |