theory Probability_Space
imports Lebesgue_Integration Radon_Nikodym
begin
lemma (in measure_space) measure_inter_full_set:
assumes "S \<in> sets M" "T \<in> sets M" and not_\<omega>: "\<mu> (T - S) \<noteq> \<omega>"
assumes T: "\<mu> T = \<mu> (space M)"
shows "\<mu> (S \<inter> T) = \<mu> S"
proof (rule antisym)
show " \<mu> (S \<inter> T) \<le> \<mu> S"
using assms by (auto intro!: measure_mono)
show "\<mu> S \<le> \<mu> (S \<inter> T)"
proof (rule ccontr)
assume contr: "\<not> ?thesis"
have "\<mu> (space M) = \<mu> ((T - S) \<union> (S \<inter> T))"
unfolding T[symmetric] by (auto intro!: arg_cong[where f="\<mu>"])
also have "\<dots> \<le> \<mu> (T - S) + \<mu> (S \<inter> T)"
using assms by (auto intro!: measure_subadditive)
also have "\<dots> < \<mu> (T - S) + \<mu> S"
by (rule pinfreal_less_add[OF not_\<omega>]) (insert contr, auto)
also have "\<dots> = \<mu> (T \<union> S)"
using assms by (subst measure_additive) auto
also have "\<dots> \<le> \<mu> (space M)"
using assms sets_into_space by (auto intro!: measure_mono)
finally show False ..
qed
qed
lemma (in finite_measure) finite_measure_inter_full_set:
assumes "S \<in> sets M" "T \<in> sets M"
assumes T: "\<mu> T = \<mu> (space M)"
shows "\<mu> (S \<inter> T) = \<mu> S"
using measure_inter_full_set[OF assms(1,2) finite_measure assms(3)] assms
by auto
locale prob_space = measure_space +
assumes measure_space_1: "\<mu> (space M) = 1"
sublocale prob_space < finite_measure
proof
from measure_space_1 show "\<mu> (space M) \<noteq> \<omega>" by simp
qed
context prob_space
begin
abbreviation "events \<equiv> sets M"
abbreviation "prob \<equiv> \<lambda>A. real (\<mu> A)"
abbreviation "prob_preserving \<equiv> measure_preserving"
abbreviation "random_variable \<equiv> \<lambda> s X. X \<in> measurable M s"
abbreviation "expectation \<equiv> integral"
definition
"indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B"
definition
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)"
definition
"distribution X = (\<lambda>s. \<mu> ((X -` s) \<inter> (space M)))"
abbreviation
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
lemma prob_space: "prob (space M) = 1"
unfolding measure_space_1 by simp
lemma measure_le_1[simp, intro]:
assumes "A \<in> events" shows "\<mu> A \<le> 1"
proof -
have "\<mu> A \<le> \<mu> (space M)"
using assms sets_into_space by(auto intro!: measure_mono)
also note measure_space_1
finally show ?thesis .
qed
lemma measure_finite[simp, intro]:
assumes "A \<in> events" shows "\<mu> A \<noteq> \<omega>"
using measure_le_1[OF assms] by auto
lemma prob_compl:
assumes "A \<in> events"
shows "prob (space M - A) = 1 - prob A"
using `A \<in> events`[THEN sets_into_space] `A \<in> events` measure_space_1
by (subst real_finite_measure_Diff) auto
lemma indep_space:
assumes "s \<in> events"
shows "indep (space M) s"
using assms prob_space by (simp add: indep_def)
lemma prob_space_increasing: "increasing M prob"
by (auto intro!: real_measure_mono simp: increasing_def)
lemma prob_zero_union:
assumes "s \<in> events" "t \<in> events" "prob t = 0"
shows "prob (s \<union> t) = prob s"
using assms
proof -
have "prob (s \<union> t) \<le> prob s"
using real_finite_measure_subadditive[of s t] assms by auto
moreover have "prob (s \<union> t) \<ge> prob s"
using assms by (blast intro: real_measure_mono)
ultimately show ?thesis by simp
qed
lemma prob_eq_compl:
assumes "s \<in> events" "t \<in> events"
assumes "prob (space M - s) = prob (space M - t)"
shows "prob s = prob t"
using assms prob_compl by auto
lemma prob_one_inter:
assumes events:"s \<in> events" "t \<in> events"
assumes "prob t = 1"
shows "prob (s \<inter> t) = prob s"
proof -
have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)"
using events assms prob_compl[of "t"] by (auto intro!: prob_zero_union)
also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
by blast
finally show "prob (s \<inter> t) = prob s"
using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s])
qed
lemma prob_eq_bigunion_image:
assumes "range f \<subseteq> events" "range g \<subseteq> events"
assumes "disjoint_family f" "disjoint_family g"
assumes "\<And> n :: nat. prob (f n) = prob (g n)"
shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))"
using assms
proof -
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))"
by (rule real_finite_measure_UNION[OF assms(1,3)])
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))"
by (rule real_finite_measure_UNION[OF assms(2,4)])
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
qed
lemma prob_countably_zero:
assumes "range c \<subseteq> events"
assumes "\<And> i. prob (c i) = 0"
shows "prob (\<Union> i :: nat. c i) = 0"
proof (rule antisym)
show "prob (\<Union> i :: nat. c i) \<le> 0"
using real_finite_measurable_countably_subadditive[OF assms(1)]
by (simp add: assms(2) suminf_zero summable_zero)
show "0 \<le> prob (\<Union> i :: nat. c i)" by (rule real_pinfreal_nonneg)
qed
lemma indep_sym:
"indep a b \<Longrightarrow> indep b a"
unfolding indep_def using Int_commute[of a b] by auto
lemma indep_refl:
assumes "a \<in> events"
shows "indep a a = (prob a = 0) \<or> (prob a = 1)"
using assms unfolding indep_def by auto
lemma prob_equiprobable_finite_unions:
assumes "s \<in> events"
assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events"
assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})"
shows "prob s = real (card s) * prob {SOME x. x \<in> s}"
proof (cases "s = {}")
case False hence "\<exists> x. x \<in> s" by blast
from someI_ex[OF this] assms
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
have "prob s = (\<Sum> x \<in> s. prob {x})"
using real_finite_measure_finite_singelton[OF s_finite] by simp
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
using setsum_constant assms by (simp add: real_eq_of_nat)
finally show ?thesis by simp
qed simp
lemma prob_real_sum_image_fn:
assumes "e \<in> events"
assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events"
assumes "finite s"
assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)"
shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))"
proof -
have e: "e = (\<Union> i \<in> s. e \<inter> f i)"
using `e \<in> events` sets_into_space upper by blast
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
proof (rule real_finite_measure_finite_Union)
show "finite s" by fact
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
using disjoint by (auto simp: disjoint_family_on_def)
qed
finally show ?thesis .
qed
lemma distribution_prob_space:
fixes S :: "('c, 'd) algebra_scheme"
assumes "sigma_algebra S" "random_variable S X"
shows "prob_space S (distribution X)"
proof -
interpret S: sigma_algebra S by fact
show ?thesis
proof
show "distribution X {} = 0" unfolding distribution_def by simp
have "X -` space S \<inter> space M = space M"
using `random_variable S X` by (auto simp: measurable_def)
then show "distribution X (space S) = 1" using measure_space_1 by (simp add: distribution_def)
show "countably_additive S (distribution X)"
proof (unfold countably_additive_def, safe)
fix A :: "nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets S" "disjoint_family A"
hence *: "\<And>i. X -` A i \<inter> space M \<in> sets M"
using `random_variable S X` by (auto simp: measurable_def)
moreover hence "\<And>i. \<mu> (X -` A i \<inter> space M) \<noteq> \<omega>"
using finite_measure by auto
moreover have "(\<Union>i. X -` A i \<inter> space M) \<in> sets M"
using * by blast
moreover hence "\<mu> (\<Union>i. X -` A i \<inter> space M) \<noteq> \<omega>"
using finite_measure by auto
moreover have **: "disjoint_family (\<lambda>i. X -` A i \<inter> space M)"
using `disjoint_family A` by (auto simp: disjoint_family_on_def)
ultimately show "(\<Sum>\<^isub>\<infinity> i. distribution X (A i)) = distribution X (\<Union>i. A i)"
using measure_countably_additive[OF _ **]
by (auto simp: distribution_def Real_real comp_def vimage_UN)
qed
qed
qed
lemma distribution_lebesgue_thm1:
assumes "random_variable s X"
assumes "A \<in> sets s"
shows "real (distribution X A) = expectation (indicator (X -` A \<inter> space M))"
unfolding distribution_def
using assms unfolding measurable_def
using integral_indicator by auto
lemma distribution_lebesgue_thm2:
assumes "sigma_algebra S" "random_variable S X" and "A \<in> sets S"
shows "distribution X A =
measure_space.positive_integral S (distribution X) (indicator A)"
(is "_ = measure_space.positive_integral _ ?D _")
proof -
interpret S: prob_space S "distribution X" using assms(1,2) by (rule distribution_prob_space)
show ?thesis
using S.positive_integral_indicator(1)
using assms unfolding distribution_def by auto
qed
lemma finite_expectation1:
assumes "finite (X`space M)" and rv: "random_variable borel_space X"
shows "expectation X = (\<Sum> r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))"
proof (rule integral_on_finite(2)[OF assms(2,1)])
fix x have "X -` {x} \<inter> space M \<in> sets M"
using rv unfolding measurable_def by auto
thus "\<mu> (X -` {x} \<inter> space M) \<noteq> \<omega>" using finite_measure by simp
qed
lemma finite_expectation:
assumes "finite (space M)" "random_variable borel_space X"
shows "expectation X = (\<Sum> r \<in> X ` (space M). r * real (distribution X {r}))"
using assms unfolding distribution_def using finite_expectation1 by auto
lemma prob_x_eq_1_imp_prob_y_eq_0:
assumes "{x} \<in> events"
assumes "prob {x} = 1"
assumes "{y} \<in> events"
assumes "y \<noteq> x"
shows "prob {y} = 0"
using prob_one_inter[of "{y}" "{x}"] assms by auto
lemma distribution_empty[simp]: "distribution X {} = 0"
unfolding distribution_def by simp
lemma distribution_space[simp]: "distribution X (X ` space M) = 1"
proof -
have "X -` X ` space M \<inter> space M = space M" by auto
thus ?thesis unfolding distribution_def by (simp add: measure_space_1)
qed
lemma distribution_one:
assumes "random_variable M X" and "A \<in> events"
shows "distribution X A \<le> 1"
proof -
have "distribution X A \<le> \<mu> (space M)" unfolding distribution_def
using assms[unfolded measurable_def] by (auto intro!: measure_mono)
thus ?thesis by (simp add: measure_space_1)
qed
lemma distribution_finite:
assumes "random_variable M X" and "A \<in> events"
shows "distribution X A \<noteq> \<omega>"
using distribution_one[OF assms] by auto
lemma distribution_x_eq_1_imp_distribution_y_eq_0:
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
(is "random_variable ?S X")
assumes "distribution X {x} = 1"
assumes "y \<noteq> x"
shows "distribution X {y} = 0"
proof -
have "sigma_algebra ?S" by (rule sigma_algebra_Pow)
from distribution_prob_space[OF this X]
interpret S: prob_space ?S "distribution X" by simp
have x: "{x} \<in> sets ?S"
proof (rule ccontr)
assume "{x} \<notin> sets ?S"
hence "X -` {x} \<inter> space M = {}" by auto
thus "False" using assms unfolding distribution_def by auto
qed
have [simp]: "{y} \<inter> {x} = {}" "{x} - {y} = {x}" using `y \<noteq> x` by auto
show ?thesis
proof cases
assume "{y} \<in> sets ?S"
with `{x} \<in> sets ?S` assms show "distribution X {y} = 0"
using S.measure_inter_full_set[of "{y}" "{x}"]
by simp
next
assume "{y} \<notin> sets ?S"
hence "X -` {y} \<inter> space M = {}" by auto
thus "distribution X {y} = 0" unfolding distribution_def by auto
qed
qed
end
locale finite_prob_space = prob_space + finite_measure_space
lemma finite_prob_space_eq:
"finite_prob_space M \<mu> \<longleftrightarrow> finite_measure_space M \<mu> \<and> \<mu> (space M) = 1"
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
by auto
lemma (in prob_space) not_empty: "space M \<noteq> {}"
using prob_space empty_measure by auto
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
using measure_space_1 sum_over_space by simp
lemma (in finite_prob_space) positive_distribution: "0 \<le> distribution X x"
unfolding distribution_def by simp
lemma (in finite_prob_space) joint_distribution_restriction_fst:
"joint_distribution X Y A \<le> distribution X (fst ` A)"
unfolding distribution_def
proof (safe intro!: measure_mono)
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
show "x \<in> X -` fst ` A"
by (auto intro!: image_eqI[OF _ *])
qed (simp_all add: sets_eq_Pow)
lemma (in finite_prob_space) joint_distribution_restriction_snd:
"joint_distribution X Y A \<le> distribution Y (snd ` A)"
unfolding distribution_def
proof (safe intro!: measure_mono)
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
show "x \<in> Y -` snd ` A"
by (auto intro!: image_eqI[OF _ *])
qed (simp_all add: sets_eq_Pow)
lemma (in finite_prob_space) distribution_order:
shows "0 \<le> distribution X x'"
and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')"
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
using positive_distribution[of X x']
positive_distribution[of "\<lambda>x. (X x, Y x)" "{(x, y)}"]
joint_distribution_restriction_fst[of X Y "{(x, y)}"]
joint_distribution_restriction_snd[of X Y "{(x, y)}"]
by auto
lemma (in finite_prob_space) finite_product_measure_space:
assumes "finite s1" "finite s2"
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2)\<rparr> (joint_distribution X Y)"
(is "finite_measure_space ?M ?D")
proof (rule finite_Pow_additivity_sufficient)
show "positive ?D"
unfolding positive_def using assms sets_eq_Pow
by (simp add: distribution_def)
show "additive ?M ?D" unfolding additive_def
proof safe
fix x y
have A: "((\<lambda>x. (X x, Y x)) -` x) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto
have B: "((\<lambda>x. (X x, Y x)) -` y) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto
assume "x \<inter> y = {}"
hence "(\<lambda>x. (X x, Y x)) -` x \<inter> space M \<inter> ((\<lambda>x. (X x, Y x)) -` y \<inter> space M) = {}"
by auto
from additive[unfolded additive_def, rule_format, OF A B] this
finite_measure[OF A] finite_measure[OF B]
show "?D (x \<union> y) = ?D x + ?D y"
apply (simp add: distribution_def)
apply (subst Int_Un_distrib2)
by (auto simp: real_of_pinfreal_add)
qed
show "finite (space ?M)"
using assms by auto
show "sets ?M = Pow (space ?M)"
by simp
{ fix x assume "x \<in> space ?M" thus "?D {x} \<noteq> \<omega>"
unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) }
qed
lemma (in finite_prob_space) finite_product_measure_space_of_images:
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
sets = Pow (X ` space M \<times> Y ` space M) \<rparr>
(joint_distribution X Y)"
using finite_space by (auto intro!: finite_product_measure_space)
lemma (in finite_prob_space) finite_measure_space:
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
(is "finite_measure_space ?S _")
proof (rule finite_Pow_additivity_sufficient, simp_all)
show "finite (X ` space M)" using finite_space by simp
show "positive (distribution X)"
unfolding distribution_def positive_def using sets_eq_Pow by auto
show "additive ?S (distribution X)" unfolding additive_def distribution_def
proof (simp, safe)
fix x y
have x: "(X -` x) \<inter> space M \<in> sets M"
and y: "(X -` y) \<inter> space M \<in> sets M" using sets_eq_Pow by auto
assume "x \<inter> y = {}"
hence "X -` x \<inter> space M \<inter> (X -` y \<inter> space M) = {}" by auto
from additive[unfolded additive_def, rule_format, OF x y] this
finite_measure[OF x] finite_measure[OF y]
have "\<mu> (((X -` x) \<union> (X -` y)) \<inter> space M) =
\<mu> ((X -` x) \<inter> space M) + \<mu> ((X -` y) \<inter> space M)"
by (subst Int_Un_distrib2) auto
thus "\<mu> ((X -` x \<union> X -` y) \<inter> space M) = \<mu> (X -` x \<inter> space M) + \<mu> (X -` y \<inter> space M)"
by auto
qed
{ fix x assume "x \<in> X ` space M" thus "distribution X {x} \<noteq> \<omega>"
unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) }
qed
lemma (in finite_prob_space) finite_prob_space_of_images:
"finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
by (simp add: finite_prob_space_eq finite_measure_space)
lemma (in finite_prob_space) finite_product_prob_space_of_images:
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M)\<rparr>
(joint_distribution X Y)"
(is "finite_prob_space ?S _")
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images)
have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
qed
lemma (in prob_space) prob_space_subalgebra:
assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows "prob_space (M\<lparr> sets := N \<rparr>) \<mu>" sorry
lemma (in measure_space) measure_space_subalgebra:
assumes "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows "measure_space (M\<lparr> sets := N \<rparr>) \<mu>" sorry
lemma pinfreal_0_less_mult_iff[simp]:
fixes x y :: pinfreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y"
by (cases x, cases y) (auto simp: zero_less_mult_iff)
lemma (in sigma_algebra) simple_function_subalgebra:
assumes "sigma_algebra.simple_function (M\<lparr>sets:=N\<rparr>) f"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets:=N\<rparr>)"
shows "simple_function f"
using assms
unfolding simple_function_def
unfolding sigma_algebra.simple_function_def[OF N_subalgebra(2)]
by auto
lemma (in measure_space) simple_integral_subalgebra[simp]:
assumes "measure_space (M\<lparr>sets := N\<rparr>) \<mu>"
shows "measure_space.simple_integral (M\<lparr>sets := N\<rparr>) \<mu> = simple_integral"
unfolding simple_integral_def_raw
unfolding measure_space.simple_integral_def_raw[OF assms] by simp
lemma (in sigma_algebra) borel_measurable_subalgebra:
assumes "N \<subseteq> sets M" "f \<in> borel_measurable (M\<lparr>sets:=N\<rparr>)"
shows "f \<in> borel_measurable M"
using assms unfolding measurable_def by auto
lemma (in measure_space) positive_integral_subalgebra[simp]:
assumes borel: "f \<in> borel_measurable (M\<lparr>sets := N\<rparr>)"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets := N\<rparr>)"
shows "measure_space.positive_integral (M\<lparr>sets := N\<rparr>) \<mu> f = positive_integral f"
proof -
note msN = measure_space_subalgebra[OF N_subalgebra]
then interpret N: measure_space "M\<lparr>sets:=N\<rparr>" \<mu> .
from N.borel_measurable_implies_simple_function_sequence[OF borel]
obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast
then have sf: "\<And>i. simple_function (fs i)"
using simple_function_subalgebra[OF _ N_subalgebra] by blast
from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
show ?thesis unfolding simple_integral_subalgebra[OF msN] isoton_def by simp
qed
section "Conditional Expectation and Probability"
lemma (in prob_space) conditional_expectation_exists:
fixes X :: "'a \<Rightarrow> pinfreal"
assumes borel: "X \<in> borel_measurable M"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows "\<exists>Y\<in>borel_measurable (M\<lparr> sets := N \<rparr>). \<forall>C\<in>N.
positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)"
proof -
interpret P: prob_space "M\<lparr> sets := N \<rparr>" \<mu>
using prob_space_subalgebra[OF N_subalgebra] .
let "?f A" = "\<lambda>x. X x * indicator A x"
let "?Q A" = "positive_integral (?f A)"
from measure_space_density[OF borel]
have Q: "measure_space (M\<lparr> sets := N \<rparr>) ?Q"
by (rule measure_space.measure_space_subalgebra[OF _ N_subalgebra])
then interpret Q: measure_space "M\<lparr> sets := N \<rparr>" ?Q .
have "P.absolutely_continuous ?Q"
unfolding P.absolutely_continuous_def
proof (safe, simp)
fix A assume "A \<in> N" "\<mu> A = 0"
moreover then have f_borel: "?f A \<in> borel_measurable M"
using borel N_subalgebra by (auto intro: borel_measurable_indicator)
moreover have "{x\<in>space M. ?f A x \<noteq> 0} = (?f A -` {0<..} \<inter> space M) \<inter> A"
by (auto simp: indicator_def)
moreover have "\<mu> \<dots> \<le> \<mu> A"
using `A \<in> N` N_subalgebra f_borel
by (auto intro!: measure_mono Int[of _ A] measurable_sets)
ultimately show "?Q A = 0"
by (simp add: positive_integral_0_iff)
qed
from P.Radon_Nikodym[OF Q this]
obtain Y where Y: "Y \<in> borel_measurable (M\<lparr>sets := N\<rparr>)"
"\<And>A. A \<in> sets (M\<lparr>sets:=N\<rparr>) \<Longrightarrow> ?Q A = P.positive_integral (\<lambda>x. Y x * indicator A x)"
by blast
with N_subalgebra show ?thesis
by (auto intro!: bexI[OF _ Y(1)])
qed
definition (in prob_space)
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable (M\<lparr>sets:=N\<rparr>)
\<and> (\<forall>C\<in>N. positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)))"
abbreviation (in prob_space)
"conditional_probabiltiy N A \<equiv> conditional_expectation N (indicator A)"
lemma (in prob_space)
fixes X :: "'a \<Rightarrow> pinfreal"
assumes borel: "X \<in> borel_measurable M"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows borel_measurable_conditional_expectation:
"conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)"
and conditional_expectation: "\<And>C. C \<in> N \<Longrightarrow>
positive_integral (\<lambda>x. conditional_expectation N X x * indicator C x) =
positive_integral (\<lambda>x. X x * indicator C x)"
(is "\<And>C. C \<in> N \<Longrightarrow> ?eq C")
proof -
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
then show "conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)"
unfolding conditional_expectation_def by (rule someI2_ex) blast
from CE show "\<And>C. C\<in>N \<Longrightarrow> ?eq C"
unfolding conditional_expectation_def by (rule someI2_ex) blast
qed
end