author | hoelzl |
Fri, 01 Apr 2011 17:20:33 +0200 | |
changeset 42199 | aded34119213 |
parent 42148 | d596e7bb251f |
child 42200 | 8df8e5cc3119 |
permissions | -rw-r--r-- |
42148 | 1 |
(* 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 |
42146
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split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure |
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begin |
11 |
||
41981
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lemma real_of_extreal_inverse[simp]: |
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fixes X :: extreal |
40859 | 14 |
shows "real (inverse X) = 1 / real X" |
15 |
by (cases X) (auto simp: inverse_eq_divide) |
|
16 |
||
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
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17 |
lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)" |
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by (cases X) auto |
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reworked Probability theory: measures are not type restricted to positive extended reals
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19 |
|
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reworked Probability theory: measures are not type restricted to positive extended reals
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lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>" |
40859 | 21 |
by (cases X) auto |
22 |
||
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23 |
lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)" |
40859 | 24 |
by (cases X) auto |
25 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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lemma real_of_extreal_le_1: fixes X :: extreal shows "X \<le> 1 \<Longrightarrow> real X \<le> 1" |
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27 |
by (cases X) (auto simp: one_extreal_def) |
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|
35582 | 29 |
locale prob_space = measure_space + |
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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30 |
assumes measure_space_1: "measure M (space M) = 1" |
38656 | 31 |
|
32 |
sublocale prob_space < finite_measure |
|
33 |
proof |
|
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from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp |
38656 | 35 |
qed |
36 |
||
40859 | 37 |
abbreviation (in prob_space) "events \<equiv> sets M" |
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abbreviation (in prob_space) "prob \<equiv> \<mu>'" |
40859 | 39 |
abbreviation (in prob_space) "prob_preserving \<equiv> measure_preserving" |
40 |
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'" |
|
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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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abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M" |
35582 | 42 |
|
40859 | 43 |
definition (in prob_space) |
35582 | 44 |
"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B" |
45 |
||
40859 | 46 |
definition (in prob_space) |
35582 | 47 |
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)" |
48 |
||
40859 | 49 |
definition (in prob_space) |
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"distribution X A = \<mu>' (X -` A \<inter> space M)" |
35582 | 51 |
|
40859 | 52 |
abbreviation (in prob_space) |
36624 | 53 |
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))" |
35582 | 54 |
|
41981
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55 |
declare (in finite_measure) positive_measure'[intro, simp] |
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56 |
|
39097 | 57 |
lemma (in prob_space) distribution_cong: |
58 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" |
|
59 |
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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60 |
unfolding distribution_def fun_eq_iff |
41981
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61 |
using assms by (auto intro!: arg_cong[where f="\<mu>'"]) |
39097 | 62 |
|
63 |
lemma (in prob_space) joint_distribution_cong: |
|
64 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" |
|
65 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" |
|
66 |
shows "joint_distribution X Y = joint_distribution X' 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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|
67 |
unfolding distribution_def fun_eq_iff |
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using assms by (auto intro!: arg_cong[where f="\<mu>'"]) |
39097 | 69 |
|
40859 | 70 |
lemma (in prob_space) distribution_id[simp]: |
41981
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71 |
"N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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72 |
by (auto simp: distribution_def intro!: arg_cong[where f=prob]) |
40859 | 73 |
|
74 |
lemma (in prob_space) prob_space: "prob (space M) = 1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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75 |
using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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76 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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77 |
lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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78 |
using bounded_measure[of A] by (simp add: prob_space) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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79 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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80 |
lemma (in prob_space) distribution_positive[simp, intro]: |
cdf7693bbe08
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81 |
"0 \<le> distribution X A" unfolding distribution_def by auto |
35582 | 82 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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83 |
lemma (in prob_space) joint_distribution_remove[simp]: |
cdf7693bbe08
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parents:
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84 |
"joint_distribution X X {(x, x)} = distribution X {x}" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
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changeset
|
85 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
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86 |
|
cdf7693bbe08
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87 |
lemma (in prob_space) distribution_1: |
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88 |
"distribution X A \<le> 1" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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89 |
unfolding distribution_def by simp |
35582 | 90 |
|
40859 | 91 |
lemma (in prob_space) prob_compl: |
41981
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92 |
assumes A: "A \<in> events" |
38656 | 93 |
shows "prob (space M - A) = 1 - prob A" |
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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|
94 |
using finite_measure_compl[OF A] by (simp add: prob_space) |
35582 | 95 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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96 |
lemma (in prob_space) indep_space: "s \<in> events \<Longrightarrow> indep (space M) s" |
cdf7693bbe08
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|
97 |
by (simp add: indep_def prob_space) |
35582 | 98 |
|
40859 | 99 |
lemma (in prob_space) prob_space_increasing: "increasing M prob" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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|
100 |
by (auto intro!: finite_measure_mono simp: increasing_def) |
35582 | 101 |
|
40859 | 102 |
lemma (in prob_space) prob_zero_union: |
35582 | 103 |
assumes "s \<in> events" "t \<in> events" "prob t = 0" |
104 |
shows "prob (s \<union> t) = prob s" |
|
38656 | 105 |
using assms |
35582 | 106 |
proof - |
107 |
have "prob (s \<union> t) \<le> prob s" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
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|
108 |
using finite_measure_subadditive[of s t] assms by auto |
35582 | 109 |
moreover have "prob (s \<union> t) \<ge> prob s" |
41981
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|
110 |
using assms by (blast intro: finite_measure_mono) |
35582 | 111 |
ultimately show ?thesis by simp |
112 |
qed |
|
113 |
||
40859 | 114 |
lemma (in prob_space) prob_eq_compl: |
35582 | 115 |
assumes "s \<in> events" "t \<in> events" |
116 |
assumes "prob (space M - s) = prob (space M - t)" |
|
117 |
shows "prob s = prob t" |
|
38656 | 118 |
using assms prob_compl by auto |
35582 | 119 |
|
40859 | 120 |
lemma (in prob_space) prob_one_inter: |
35582 | 121 |
assumes events:"s \<in> events" "t \<in> events" |
122 |
assumes "prob t = 1" |
|
123 |
shows "prob (s \<inter> t) = prob s" |
|
124 |
proof - |
|
38656 | 125 |
have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)" |
126 |
using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union) |
|
127 |
also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)" |
|
128 |
by blast |
|
129 |
finally show "prob (s \<inter> t) = prob s" |
|
130 |
using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s]) |
|
35582 | 131 |
qed |
132 |
||
40859 | 133 |
lemma (in prob_space) prob_eq_bigunion_image: |
35582 | 134 |
assumes "range f \<subseteq> events" "range g \<subseteq> events" |
135 |
assumes "disjoint_family f" "disjoint_family g" |
|
136 |
assumes "\<And> n :: nat. prob (f n) = prob (g n)" |
|
137 |
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))" |
|
138 |
using assms |
|
139 |
proof - |
|
38656 | 140 |
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))" |
41981
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141 |
by (rule finite_measure_UNION[OF assms(1,3)]) |
38656 | 142 |
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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|
143 |
by (rule finite_measure_UNION[OF assms(2,4)]) |
38656 | 144 |
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp |
35582 | 145 |
qed |
146 |
||
40859 | 147 |
lemma (in prob_space) prob_countably_zero: |
35582 | 148 |
assumes "range c \<subseteq> events" |
149 |
assumes "\<And> i. prob (c i) = 0" |
|
38656 | 150 |
shows "prob (\<Union> i :: nat. c i) = 0" |
151 |
proof (rule antisym) |
|
152 |
show "prob (\<Union> i :: nat. c i) \<le> 0" |
|
41981
cdf7693bbe08
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hoelzl
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41831
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|
153 |
using finite_measure_countably_subadditive[OF assms(1)] |
38656 | 154 |
by (simp add: assms(2) suminf_zero summable_zero) |
41981
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155 |
qed simp |
35582 | 156 |
|
40859 | 157 |
lemma (in prob_space) indep_sym: |
35582 | 158 |
"indep a b \<Longrightarrow> indep b a" |
159 |
unfolding indep_def using Int_commute[of a b] by auto |
|
160 |
||
40859 | 161 |
lemma (in prob_space) indep_refl: |
35582 | 162 |
assumes "a \<in> events" |
163 |
shows "indep a a = (prob a = 0) \<or> (prob a = 1)" |
|
164 |
using assms unfolding indep_def by auto |
|
165 |
||
40859 | 166 |
lemma (in prob_space) prob_equiprobable_finite_unions: |
38656 | 167 |
assumes "s \<in> events" |
168 |
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events" |
|
35582 | 169 |
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})" |
38656 | 170 |
shows "prob s = real (card s) * prob {SOME x. x \<in> s}" |
35582 | 171 |
proof (cases "s = {}") |
38656 | 172 |
case False hence "\<exists> x. x \<in> s" by blast |
35582 | 173 |
from someI_ex[OF this] assms |
174 |
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast |
|
175 |
have "prob s = (\<Sum> x \<in> s. prob {x})" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
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|
176 |
using finite_measure_finite_singleton[OF s_finite] by simp |
35582 | 177 |
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto |
38656 | 178 |
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}" |
179 |
using setsum_constant assms by (simp add: real_eq_of_nat) |
|
35582 | 180 |
finally show ?thesis by simp |
38656 | 181 |
qed simp |
35582 | 182 |
|
40859 | 183 |
lemma (in prob_space) prob_real_sum_image_fn: |
35582 | 184 |
assumes "e \<in> events" |
185 |
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events" |
|
186 |
assumes "finite s" |
|
38656 | 187 |
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" |
188 |
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" |
|
35582 | 189 |
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))" |
190 |
proof - |
|
38656 | 191 |
have e: "e = (\<Union> i \<in> s. e \<inter> f i)" |
192 |
using `e \<in> events` sets_into_space upper by blast |
|
193 |
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp |
|
194 |
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))" |
|
41981
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parents:
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|
195 |
proof (rule finite_measure_finite_Union) |
38656 | 196 |
show "finite s" by fact |
197 |
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact |
|
198 |
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" |
|
199 |
using disjoint by (auto simp: disjoint_family_on_def) |
|
200 |
qed |
|
201 |
finally show ?thesis . |
|
35582 | 202 |
qed |
203 |
||
42199 | 204 |
lemma (in prob_space) prob_space_vimage: |
205 |
assumes S: "sigma_algebra S" |
|
206 |
assumes T: "T \<in> measure_preserving M S" |
|
207 |
shows "prob_space S" |
|
35582 | 208 |
proof - |
42199 | 209 |
interpret S: measure_space S |
210 |
using S and T by (rule measure_space_vimage) |
|
38656 | 211 |
show ?thesis |
42199 | 212 |
proof |
213 |
from T[THEN measure_preservingD2] |
|
214 |
have "T -` space S \<inter> space M = space M" |
|
215 |
by (auto simp: measurable_def) |
|
216 |
with T[THEN measure_preservingD, of "space S", symmetric] |
|
217 |
show "measure S (space S) = 1" |
|
218 |
using measure_space_1 by simp |
|
35582 | 219 |
qed |
220 |
qed |
|
221 |
||
42199 | 222 |
lemma (in prob_space) distribution_prob_space: |
223 |
assumes X: "random_variable S X" |
|
224 |
shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" (is "prob_space ?S") |
|
225 |
proof (rule prob_space_vimage) |
|
226 |
show "X \<in> measure_preserving M ?S" |
|
227 |
using X |
|
228 |
unfolding measure_preserving_def distribution_def_raw |
|
229 |
by (auto simp: finite_measure_eq measurable_sets) |
|
230 |
show "sigma_algebra ?S" using X by simp |
|
231 |
qed |
|
232 |
||
40859 | 233 |
lemma (in prob_space) AE_distribution: |
41981
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hoelzl
parents:
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|
234 |
assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x" |
40859 | 235 |
shows "AE x. Q (X x)" |
236 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
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changeset
|
237 |
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space) |
40859 | 238 |
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N" |
239 |
using assms unfolding X.almost_everywhere_def by auto |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
240 |
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
|
241 |
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
|
242 |
(auto simp: finite_measure_eq distribution_def measurable_sets) |
40859 | 243 |
qed |
244 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
245 |
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
|
246 |
"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
|
247 |
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:
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diff
changeset
|
248 |
by (auto simp: measurable_sets distribution_def) |
35582 | 249 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
250 |
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
|
251 |
assumes "random_variable S X" "A \<in> sets S" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
252 |
shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)" |
35582 | 253 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
254 |
interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
255 |
using assms(1) by (rule distribution_prob_space) |
35582 | 256 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
257 |
using S.positive_integral_indicator(1)[of A] assms by simp |
35582 | 258 |
qed |
259 |
||
40859 | 260 |
lemma (in prob_space) finite_expectation1: |
261 |
assumes f: "finite (X`space M)" and rv: "random_variable borel X" |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
262 |
shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r") |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
263 |
proof (subst integral_on_finite) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
264 |
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
|
265 |
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
|
266 |
"\<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
|
267 |
using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto |
38656 | 268 |
qed |
35582 | 269 |
|
40859 | 270 |
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
|
271 |
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
|
272 |
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})" |
38656 | 273 |
using assms unfolding distribution_def using finite_expectation1 by auto |
274 |
||
40859 | 275 |
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0: |
35582 | 276 |
assumes "{x} \<in> events" |
38656 | 277 |
assumes "prob {x} = 1" |
35582 | 278 |
assumes "{y} \<in> events" |
279 |
assumes "y \<noteq> x" |
|
280 |
shows "prob {y} = 0" |
|
281 |
using prob_one_inter[of "{y}" "{x}"] assms by auto |
|
282 |
||
40859 | 283 |
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0" |
38656 | 284 |
unfolding distribution_def by simp |
285 |
||
40859 | 286 |
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1" |
38656 | 287 |
proof - |
288 |
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
|
289 |
thus ?thesis unfolding distribution_def by (simp add: prob_space) |
38656 | 290 |
qed |
291 |
||
40859 | 292 |
lemma (in prob_space) distribution_one: |
293 |
assumes "random_variable M' X" and "A \<in> sets M'" |
|
38656 | 294 |
shows "distribution X A \<le> 1" |
295 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
296 |
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
|
297 |
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
|
298 |
thus ?thesis by (simp add: prob_space) |
38656 | 299 |
qed |
300 |
||
40859 | 301 |
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0: |
35582 | 302 |
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X" |
38656 | 303 |
(is "random_variable ?S X") |
304 |
assumes "distribution X {x} = 1" |
|
35582 | 305 |
assumes "y \<noteq> x" |
306 |
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
|
307 |
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
|
308 |
{ 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
|
309 |
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
|
310 |
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
|
311 |
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
|
312 |
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
|
313 |
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
|
314 |
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
|
315 |
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
|
316 |
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
|
317 |
"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
|
318 |
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
|
319 |
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
|
320 |
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
|
321 |
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
|
322 |
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
|
323 |
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
|
324 |
thus "distribution X {y} = 0" unfolding distribution_def by auto |
35582 | 325 |
qed |
326 |
||
40859 | 327 |
lemma (in prob_space) joint_distribution_Times_le_fst: |
328 |
assumes X: "random_variable MX X" and Y: "random_variable MY Y" |
|
329 |
and A: "A \<in> sets MX" and B: "B \<in> sets MY" |
|
330 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" |
|
331 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
332 |
proof (intro finite_measure_mono) |
40859 | 333 |
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force |
334 |
show "X -` A \<inter> space M \<in> events" |
|
335 |
using X A unfolding measurable_def by simp |
|
336 |
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M = |
|
337 |
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
338 |
qed |
|
339 |
||
340 |
lemma (in prob_space) joint_distribution_commute: |
|
341 |
"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
|
342 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 343 |
|
344 |
lemma (in prob_space) joint_distribution_Times_le_snd: |
|
345 |
assumes X: "random_variable MX X" and Y: "random_variable MY Y" |
|
346 |
and A: "A \<in> sets MX" and B: "B \<in> sets MY" |
|
347 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" |
|
348 |
using assms |
|
349 |
by (subst joint_distribution_commute) |
|
350 |
(simp add: swap_product joint_distribution_Times_le_fst) |
|
351 |
||
352 |
lemma (in prob_space) random_variable_pairI: |
|
353 |
assumes "random_variable MX X" |
|
354 |
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
|
355 |
shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" |
40859 | 356 |
proof |
357 |
interpret MX: sigma_algebra MX using assms by simp |
|
358 |
interpret MY: sigma_algebra MY using assms by simp |
|
359 |
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
|
360 |
show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default |
40859 | 361 |
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
|
362 |
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" |
41095 | 363 |
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) |
40859 | 364 |
qed |
365 |
||
366 |
lemma (in prob_space) joint_distribution_commute_singleton: |
|
367 |
"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
|
368 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 369 |
|
370 |
lemma (in prob_space) joint_distribution_assoc_singleton: |
|
371 |
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = |
|
372 |
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
|
373 |
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 374 |
|
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
|
375 |
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2 |
40859 | 376 |
|
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
|
377 |
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
|
378 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
379 |
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
|
380 |
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure) |
40859 | 381 |
|
382 |
lemma countably_additiveI[case_names countably]: |
|
383 |
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
|
384 |
(\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)" |
40859 | 385 |
shows "countably_additive M \<mu>" |
386 |
using assms unfolding countably_additive_def by auto |
|
387 |
||
388 |
lemma (in prob_space) joint_distribution_prob_space: |
|
389 |
assumes "random_variable MX X" "random_variable MY Y" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
390 |
shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
391 |
using random_variable_pairI[OF assms] by (rule distribution_prob_space) |
40859 | 392 |
|
393 |
section "Probability spaces on finite sets" |
|
35582 | 394 |
|
35977 | 395 |
locale finite_prob_space = prob_space + finite_measure_space |
396 |
||
40859 | 397 |
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'" |
398 |
||
399 |
lemma (in prob_space) finite_random_variableD: |
|
400 |
assumes "finite_random_variable M' X" shows "random_variable M' X" |
|
401 |
proof - |
|
402 |
interpret M': finite_sigma_algebra M' using assms by simp |
|
403 |
then show "random_variable M' X" using assms by simp default |
|
404 |
qed |
|
405 |
||
406 |
lemma (in prob_space) distribution_finite_prob_space: |
|
407 |
assumes "finite_random_variable MX X" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
408 |
shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)" |
40859 | 409 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
410 |
interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" |
40859 | 411 |
using assms[THEN finite_random_variableD] by (rule distribution_prob_space) |
412 |
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
|
413 |
using assms by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
414 |
show ?thesis by default (simp_all add: MX.finite_space) |
40859 | 415 |
qed |
416 |
||
417 |
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
|
418 |
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
|
419 |
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
|
420 |
(is "finite_random_variable ?X _") |
40859 | 421 |
proof (intro conjI) |
422 |
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
|
423 |
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
|
424 |
show "finite_sigma_algebra ?X" |
40859 | 425 |
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
|
426 |
show "X \<in> measurable M ?X" |
40859 | 427 |
proof (unfold measurable_def, clarsimp) |
428 |
fix A assume A: "A \<subseteq> X`space M" |
|
429 |
then have "finite A" by (rule finite_subset) simp |
|
430 |
then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events" |
|
431 |
unfolding vimage_UN UN_extend_simps |
|
432 |
apply (rule finite_UN) |
|
433 |
using A assms unfolding simple_function_def by auto |
|
434 |
then show "X -` A \<inter> space M \<in> events" by simp |
|
435 |
qed |
|
436 |
qed |
|
437 |
||
438 |
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
|
439 |
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
|
440 |
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
|
441 |
using simple_function_imp_finite_random_variable[OF assms, of ext] |
40859 | 442 |
by (auto dest!: finite_random_variableD) |
443 |
||
444 |
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
|
445 |
"simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1" |
40859 | 446 |
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
|
447 |
by (subst finite_measure_finite_Union[symmetric]) |
40859 | 448 |
(auto simp add: disjoint_family_on_def simple_function_def |
449 |
intro!: arg_cong[where f=prob]) |
|
450 |
||
451 |
lemma (in prob_space) finite_random_variable_pairI: |
|
452 |
assumes "finite_random_variable MX X" |
|
453 |
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
|
454 |
shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))" |
40859 | 455 |
proof |
456 |
interpret MX: finite_sigma_algebra MX using assms by simp |
|
457 |
interpret MY: finite_sigma_algebra MY using assms by simp |
|
458 |
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
|
459 |
show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default |
40859 | 460 |
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
|
461 |
show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)" |
41095 | 462 |
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def) |
40859 | 463 |
qed |
464 |
||
465 |
lemma (in prob_space) finite_random_variable_imp_sets: |
|
466 |
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX" |
|
467 |
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp |
|
468 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
469 |
lemma (in prob_space) finite_random_variable_measurable: |
40859 | 470 |
assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events" |
471 |
proof - |
|
472 |
interpret X: finite_sigma_algebra MX using X by simp |
|
473 |
from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and |
|
474 |
"X \<in> space M \<rightarrow> space MX" |
|
475 |
by (auto simp: measurable_def) |
|
476 |
then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M" |
|
477 |
by auto |
|
478 |
show "X -` A \<inter> space M \<in> events" |
|
479 |
unfolding * by (intro vimage) auto |
|
480 |
qed |
|
481 |
||
482 |
lemma (in prob_space) joint_distribution_finite_Times_le_fst: |
|
483 |
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" |
|
484 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A" |
|
485 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
486 |
proof (intro finite_measure_mono) |
40859 | 487 |
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force |
488 |
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
|
489 |
using finite_random_variable_measurable[OF X] . |
40859 | 490 |
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M = |
491 |
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
492 |
qed |
|
493 |
||
494 |
lemma (in prob_space) joint_distribution_finite_Times_le_snd: |
|
495 |
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" |
|
496 |
shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B" |
|
497 |
using assms |
|
498 |
by (subst joint_distribution_commute) |
|
499 |
(simp add: swap_product joint_distribution_finite_Times_le_fst) |
|
500 |
||
501 |
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
|
502 |
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme" |
40859 | 503 |
assumes "finite_random_variable MX X" "finite_random_variable MY Y" |
504 |
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" |
|
505 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" |
|
506 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" |
|
507 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" |
|
508 |
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
509 |
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
510 |
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"] |
|
511 |
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
|
512 |
by (auto intro: antisym) |
40859 | 513 |
|
514 |
lemma (in prob_space) setsum_joint_distribution: |
|
515 |
assumes X: "finite_random_variable MX X" |
|
516 |
assumes Y: "random_variable MY Y" "B \<in> sets MY" |
|
517 |
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B" |
|
518 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
519 |
proof (subst finite_measure_finite_Union[symmetric]) |
40859 | 520 |
interpret MX: finite_sigma_algebra MX using X by auto |
521 |
show "finite (space MX)" using MX.finite_space . |
|
522 |
let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M" |
|
523 |
{ fix i assume "i \<in> space MX" |
|
524 |
moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto |
|
525 |
ultimately show "?d i \<in> events" |
|
526 |
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y |
|
527 |
using MX.sets_eq_Pow by auto } |
|
528 |
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
|
529 |
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
|
530 |
using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>']) |
40859 | 531 |
qed |
532 |
||
533 |
lemma (in prob_space) setsum_joint_distribution_singleton: |
|
534 |
assumes X: "finite_random_variable MX X" |
|
535 |
assumes Y: "finite_random_variable MY Y" "b \<in> space MY" |
|
536 |
shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}" |
|
537 |
using setsum_joint_distribution[OF X |
|
538 |
finite_random_variableD[OF Y(1)] |
|
539 |
finite_random_variable_imp_sets[OF Y]] by simp |
|
540 |
||
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
|
541 |
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2 |
40859 | 542 |
|
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
|
543 |
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
|
544 |
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
|
545 |
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default |
40859 | 546 |
|
547 |
lemma (in prob_space) joint_distribution_finite_prob_space: |
|
548 |
assumes X: "finite_random_variable MX X" |
|
549 |
assumes Y: "finite_random_variable MY Y" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
550 |
shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
551 |
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y) |
40859 | 552 |
|
36624 | 553 |
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
|
554 |
"finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1" |
36624 | 555 |
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def |
556 |
by auto |
|
557 |
||
558 |
lemma (in prob_space) not_empty: "space M \<noteq> {}" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
559 |
using prob_space empty_measure' by auto |
36624 | 560 |
|
38656 | 561 |
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1" |
562 |
using measure_space_1 sum_over_space by simp |
|
36624 | 563 |
|
564 |
lemma (in finite_prob_space) joint_distribution_restriction_fst: |
|
565 |
"joint_distribution X Y A \<le> distribution X (fst ` A)" |
|
566 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
567 |
proof (safe intro!: finite_measure_mono) |
36624 | 568 |
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" |
569 |
show "x \<in> X -` fst ` A" |
|
570 |
by (auto intro!: image_eqI[OF _ *]) |
|
571 |
qed (simp_all add: sets_eq_Pow) |
|
572 |
||
573 |
lemma (in finite_prob_space) joint_distribution_restriction_snd: |
|
574 |
"joint_distribution X Y A \<le> distribution Y (snd ` A)" |
|
575 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
576 |
proof (safe intro!: finite_measure_mono) |
36624 | 577 |
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A" |
578 |
show "x \<in> Y -` snd ` A" |
|
579 |
by (auto intro!: image_eqI[OF _ *]) |
|
580 |
qed (simp_all add: sets_eq_Pow) |
|
581 |
||
582 |
lemma (in finite_prob_space) distribution_order: |
|
583 |
shows "0 \<le> distribution X x'" |
|
584 |
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')" |
|
585 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}" |
|
586 |
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}" |
|
587 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}" |
|
588 |
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}" |
|
589 |
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0" |
|
590 |
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
|
591 |
using |
36624 | 592 |
joint_distribution_restriction_fst[of X Y "{(x, y)}"] |
593 |
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
|
594 |
by (auto intro: antisym) |
36624 | 595 |
|
39097 | 596 |
lemma (in finite_prob_space) distribution_mono: |
597 |
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" |
|
598 |
shows "distribution X x \<le> distribution Y y" |
|
599 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
600 |
using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono) |
39097 | 601 |
|
602 |
lemma (in finite_prob_space) distribution_mono_gt_0: |
|
603 |
assumes gt_0: "0 < distribution X x" |
|
604 |
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" |
|
605 |
shows "0 < distribution Y y" |
|
606 |
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) |
|
607 |
||
608 |
lemma (in finite_prob_space) sum_over_space_distrib: |
|
609 |
"(\<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
|
610 |
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
|
611 |
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
|
612 |
(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
|
613 |
intro!: arg_cong[where f=\<mu>']) |
39097 | 614 |
|
615 |
lemma (in finite_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
|
616 |
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" |
39097 | 617 |
unfolding distribution_def prob_space[symmetric] using finite_space |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
618 |
by (subst finite_measure_finite_Union[symmetric]) |
39097 | 619 |
(auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob]) |
620 |
||
621 |
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
|
622 |
"(\<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
|
623 |
using prob_space finite_space |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
624 |
by (subst (asm) finite_measure_finite_singleton) auto |
39097 | 625 |
|
626 |
lemma (in prob_space) distribution_remove_const: |
|
627 |
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" |
|
628 |
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" |
|
629 |
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" |
|
630 |
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" |
|
631 |
and "distribution (\<lambda>x. ()) {()} = 1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
632 |
by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric]) |
35977 | 633 |
|
39097 | 634 |
lemma (in finite_prob_space) setsum_distribution_gen: |
635 |
assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M" |
|
636 |
and "inj_on f (X`space M)" |
|
637 |
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" |
|
638 |
unfolding distribution_def assms |
|
639 |
using finite_space assms |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
640 |
by (subst finite_measure_finite_Union[symmetric]) |
39097 | 641 |
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def |
642 |
intro!: arg_cong[where f=prob]) |
|
643 |
||
644 |
lemma (in finite_prob_space) setsum_distribution: |
|
645 |
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" |
|
646 |
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" |
|
647 |
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" |
|
648 |
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" |
|
649 |
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" |
|
650 |
by (auto intro!: inj_onI setsum_distribution_gen) |
|
651 |
||
652 |
lemma (in finite_prob_space) uniform_prob: |
|
653 |
assumes "x \<in> space M" |
|
654 |
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
|
655 |
shows "prob {x} = 1 / card (space M)" |
39097 | 656 |
proof - |
657 |
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" |
|
658 |
using assms(2)[OF _ `x \<in> space M`] by blast |
|
659 |
have "1 = prob (space M)" |
|
660 |
using prob_space by auto |
|
661 |
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
|
662 |
using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified] |
39097 | 663 |
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] |
664 |
finite_space unfolding disjoint_family_on_def prob_space[symmetric] |
|
665 |
by (auto simp add:setsum_restrict_set) |
|
666 |
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" |
|
667 |
using prob_x by auto |
|
668 |
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp |
|
669 |
finally have one: "1 = real (card (space M)) * prob {x}" |
|
670 |
using real_eq_of_nat by auto |
|
671 |
hence two: "real (card (space M)) \<noteq> 0" by fastsimp |
|
672 |
from one have three: "prob {x} \<noteq> 0" by fastsimp |
|
673 |
thus ?thesis using one two three divide_cancel_right |
|
674 |
by (auto simp:field_simps) |
|
39092 | 675 |
qed |
35977 | 676 |
|
39092 | 677 |
lemma (in prob_space) prob_space_subalgebra: |
41545 | 678 |
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
|
679 |
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
|
680 |
shows "prob_space N" |
39092 | 681 |
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
|
682 |
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
|
683 |
by (rule measure_space_subalgebra[OF assms]) |
39092 | 684 |
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
|
685 |
proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1) |
35977 | 686 |
qed |
687 |
||
39092 | 688 |
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
|
689 |
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
|
690 |
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
|
691 |
(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
|
692 |
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
|
693 |
interpret A: measure_space "restricted_space A" |
39092 | 694 |
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
|
695 |
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
|
696 |
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
|
697 |
show "prob_space ?P" |
39092 | 698 |
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
|
699 |
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
|
700 |
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
|
701 |
show "positive ?P (measure ?P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
702 |
proof (simp add: positive_def, safe) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
703 |
show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
704 |
fix B assume "B \<in> events" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
705 |
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
|
706 |
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
|
707 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
708 |
show "countably_additive ?P (measure ?P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
709 |
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
|
710 |
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
|
711 |
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
|
712 |
{ fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
713 |
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
|
714 |
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
|
715 |
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
|
716 |
moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
717 |
by (simp add: mult_commute divide_extreal_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
718 |
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
|
719 |
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
|
720 |
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
|
721 |
using measure_countably_additive[of F] F |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
722 |
by (auto simp: suminf_cmult_extreal) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
723 |
qed |
39092 | 724 |
qed |
725 |
qed |
|
726 |
||
727 |
lemma finite_prob_spaceI: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
728 |
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
|
729 |
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
|
730 |
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
|
731 |
shows "finite_prob_space M" |
39092 | 732 |
unfolding finite_prob_space_eq |
733 |
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
|
734 |
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
|
735 |
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
|
736 |
show "measure M (space M) = 1" by fact |
39092 | 737 |
qed |
36624 | 738 |
|
739 |
lemma (in finite_prob_space) finite_measure_space: |
|
39097 | 740 |
fixes X :: "'a \<Rightarrow> 'x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
741 |
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
742 |
(is "finite_measure_space ?S") |
39092 | 743 |
proof (rule finite_measure_spaceI, simp_all) |
36624 | 744 |
show "finite (X ` space M)" using finite_space by simp |
39097 | 745 |
next |
746 |
fix A B :: "'x set" assume "A \<inter> B = {}" |
|
747 |
then show "distribution X (A \<union> B) = distribution X A + distribution X B" |
|
748 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
749 |
by (subst finite_measure_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
750 |
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) |
36624 | 751 |
qed |
752 |
||
39097 | 753 |
lemma (in finite_prob_space) finite_prob_space_of_images: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
754 |
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X \<rparr>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
755 |
by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def) |
39097 | 756 |
|
39096 | 757 |
lemma (in finite_prob_space) finite_product_measure_space: |
39097 | 758 |
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y" |
39096 | 759 |
assumes "finite s1" "finite s2" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
760 |
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
761 |
(is "finite_measure_space ?M") |
39097 | 762 |
proof (rule finite_measure_spaceI, simp_all) |
763 |
show "finite (s1 \<times> s2)" |
|
39096 | 764 |
using assms by auto |
39097 | 765 |
next |
766 |
fix A B :: "('x*'y) set" assume "A \<inter> B = {}" |
|
767 |
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B" |
|
768 |
unfolding distribution_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
769 |
by (subst finite_measure_Union[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
770 |
(auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow) |
39096 | 771 |
qed |
772 |
||
39097 | 773 |
lemma (in finite_prob_space) finite_product_measure_space_of_images: |
39096 | 774 |
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
|
775 |
sets = Pow (X ` space M \<times> Y ` space M), |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
776 |
measure = extreal \<circ> joint_distribution X Y \<rparr>" |
39096 | 777 |
using finite_space by (auto intro!: finite_product_measure_space) |
778 |
||
40859 | 779 |
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
|
780 |
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M), |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
781 |
measure = extreal \<circ> joint_distribution X Y \<rparr>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
782 |
(is "finite_prob_space ?S") |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
783 |
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def) |
40859 | 784 |
have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto |
785 |
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1" |
|
786 |
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1) |
|
787 |
qed |
|
788 |
||
39085 | 789 |
section "Conditional Expectation and Probability" |
790 |
||
791 |
lemma (in prob_space) conditional_expectation_exists: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
792 |
fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
793 |
assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
794 |
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
795 |
shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N. |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
796 |
(\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))" |
39083 | 797 |
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
|
798 |
note N(4)[simp] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
799 |
interpret P: prob_space N |
41545 | 800 |
using prob_space_subalgebra[OF N] . |
39083 | 801 |
|
802 |
let "?f A" = "\<lambda>x. X x * indicator A x" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
803 |
let "?Q A" = "integral\<^isup>P M (?f A)" |
39083 | 804 |
|
805 |
from measure_space_density[OF borel] |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
806 |
have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
807 |
apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"]) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
808 |
using N by (auto intro!: P.sigma_algebra_cong) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
809 |
then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" . |
39083 | 810 |
|
811 |
have "P.absolutely_continuous ?Q" |
|
812 |
unfolding P.absolutely_continuous_def |
|
41545 | 813 |
proof safe |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
814 |
fix A assume "A \<in> sets N" "P.\<mu> A = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
815 |
then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
816 |
using borel N by (auto intro!: borel_measurable_indicator AE_not_in) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
817 |
then show "?Q A = 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
818 |
by (auto simp add: positive_integral_0_iff_AE) |
39083 | 819 |
qed |
820 |
from P.Radon_Nikodym[OF Q this] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
821 |
obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
822 |
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)" |
39083 | 823 |
by blast |
41545 | 824 |
with N(2) show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
825 |
by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)]) |
39083 | 826 |
qed |
827 |
||
39085 | 828 |
definition (in prob_space) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
829 |
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
830 |
\<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))" |
39085 | 831 |
|
832 |
abbreviation (in prob_space) |
|
39092 | 833 |
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)" |
39085 | 834 |
|
835 |
lemma (in prob_space) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
836 |
fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
837 |
assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
838 |
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A" |
39085 | 839 |
shows borel_measurable_conditional_expectation: |
41545 | 840 |
"conditional_expectation N X \<in> borel_measurable N" |
841 |
and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow> |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
842 |
(\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) = |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
843 |
(\<integral>\<^isup>+x. X x * indicator C x \<partial>M)" |
41545 | 844 |
(is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C") |
39085 | 845 |
proof - |
846 |
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def] |
|
41545 | 847 |
then show "conditional_expectation N X \<in> borel_measurable N" |
39085 | 848 |
unfolding conditional_expectation_def by (rule someI2_ex) blast |
849 |
||
41545 | 850 |
from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C" |
39085 | 851 |
unfolding conditional_expectation_def by (rule someI2_ex) blast |
852 |
qed |
|
853 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
854 |
lemma (in sigma_algebra) factorize_measurable_function_pos: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
855 |
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" |
39091 | 856 |
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
857 |
assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
858 |
shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
859 |
proof - |
39091 | 860 |
interpret M': sigma_algebra M' by fact |
861 |
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto |
|
862 |
from M'.sigma_algebra_vimage[OF this] |
|
863 |
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . |
|
864 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
865 |
from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this |
39091 | 866 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
867 |
have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" |
39091 | 868 |
proof |
869 |
fix i |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
870 |
from f(1)[of i] have "finite (f i`space M)" and B_ex: |
39091 | 871 |
"\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
872 |
unfolding simple_function_def by auto |
39091 | 873 |
from B_ex[THEN bchoice] guess B .. note B = this |
874 |
||
875 |
let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x" |
|
876 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
877 |
show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))" |
39091 | 878 |
proof (intro exI[of _ ?g] conjI ballI) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
879 |
show "simple_function M' ?g" using B by auto |
39091 | 880 |
|
881 |
fix x assume "x \<in> space M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
882 |
then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::extreal)" |
39091 | 883 |
unfolding indicator_def using B by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
884 |
then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i] |
39091 | 885 |
by (subst va.simple_function_indicator_representation) auto |
886 |
qed |
|
887 |
qed |
|
888 |
from choice[OF this] guess g .. note g = this |
|
889 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
890 |
show ?thesis |
39091 | 891 |
proof (intro ballI bexI) |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset
|
892 |
show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'" |
39091 | 893 |
using g by (auto intro: M'.borel_measurable_simple_function) |
894 |
fix x assume "x \<in> space M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
895 |
have "max 0 (Z x) = (SUP i. f i x)" using f by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
896 |
also have "\<dots> = (SUP i. g i (Y x))" |
39091 | 897 |
using g `x \<in> space M` by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
898 |
finally show "max 0 (Z x) = (SUP i. g i (Y x))" . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
899 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
900 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
901 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
902 |
lemma extreal_0_le_iff_le_0[simp]: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
903 |
fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
904 |
by (cases rule: extreal2_cases[of a]) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
905 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
906 |
lemma (in sigma_algebra) factorize_measurable_function: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
907 |
fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
908 |
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
909 |
shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
910 |
\<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
911 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
912 |
interpret M': sigma_algebra M' by fact |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
913 |
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
914 |
from M'.sigma_algebra_vimage[OF this] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
915 |
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
916 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
917 |
{ fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
918 |
with M'.measurable_vimage_algebra[OF Y] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
919 |
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
920 |
by (rule measurable_comp) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
921 |
moreover assume "\<forall>x\<in>space M. Z x = g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
922 |
then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
923 |
g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
924 |
by (auto intro!: measurable_cong) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
925 |
ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
926 |
by simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
927 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
928 |
assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
929 |
with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
930 |
"(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
931 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
932 |
from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
933 |
from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
934 |
let "?g x" = "p x - n x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
935 |
show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
936 |
proof (intro bexI ballI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
937 |
show "?g \<in> borel_measurable M'" using p n by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
938 |
fix x assume "x \<in> space M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
939 |
then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
940 |
using p n by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
941 |
then show "Z x = ?g (Y x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
942 |
by (auto split: split_max) |
39091 | 943 |
qed |
944 |
qed |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
39089
diff
changeset
|
945 |
|
35582 | 946 |
end |