--- a/src/HOL/IsaMakefile Tue Mar 29 14:27:41 2011 +0200
+++ b/src/HOL/IsaMakefile Tue Mar 29 14:27:42 2011 +0200
@@ -1193,7 +1193,7 @@
Probability/Finite_Product_Measure.thy \
Probability/Infinite_Product_Measure.thy Probability/Information.thy \
Probability/Lebesgue_Integration.thy Probability/Lebesgue_Measure.thy \
- Probability/Measure.thy Probability/Probability_Space.thy \
+ Probability/Measure.thy Probability/Probability_Measure.thy \
Probability/Probability.thy Probability/Radon_Nikodym.thy \
Probability/ROOT.ML Probability/Sigma_Algebra.thy \
Library/Countable.thy Library/FuncSet.thy Library/Nat_Bijection.thy
--- a/src/HOL/Probability/Infinite_Product_Measure.thy Tue Mar 29 14:27:41 2011 +0200
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy Tue Mar 29 14:27:42 2011 +0200
@@ -5,7 +5,7 @@
header {*Infinite Product Measure*}
theory Infinite_Product_Measure
- imports Probability_Space
+ imports Probability_Measure
begin
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
--- a/src/HOL/Probability/Information.thy Tue Mar 29 14:27:41 2011 +0200
+++ b/src/HOL/Probability/Information.thy Tue Mar 29 14:27:42 2011 +0200
@@ -7,7 +7,7 @@
theory Information
imports
- Probability_Space
+ Probability_Measure
"~~/src/HOL/Library/Convex"
begin
--- a/src/HOL/Probability/Probability.thy Tue Mar 29 14:27:41 2011 +0200
+++ b/src/HOL/Probability/Probability.thy Tue Mar 29 14:27:42 2011 +0200
@@ -2,7 +2,7 @@
imports
Complete_Measure
Lebesgue_Measure
- Probability
+ Probability_Measure
Infinite_Product_Measure
Information
"ex/Dining_Cryptographers"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Probability_Measure.thy Tue Mar 29 14:27:42 2011 +0200
@@ -0,0 +1,935 @@
+(* Title: HOL/Probability/Probability_Measure.thy
+ Author: Johannes Hölzl, TU München
+ Author: Armin Heller, TU München
+*)
+
+header {*Probability measure*}
+
+theory Probability_Measure
+imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure
+begin
+
+lemma real_of_extreal_inverse[simp]:
+ fixes X :: extreal
+ shows "real (inverse X) = 1 / real X"
+ by (cases X) (auto simp: inverse_eq_divide)
+
+lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)"
+ by (cases X) auto
+
+lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>"
+ by (cases X) auto
+
+lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)"
+ by (cases X) auto
+
+lemma real_of_extreal_le_1: fixes X :: extreal shows "X \<le> 1 \<Longrightarrow> real X \<le> 1"
+ by (cases X) (auto simp: one_extreal_def)
+
+locale prob_space = measure_space +
+ assumes measure_space_1: "measure M (space M) = 1"
+
+sublocale prob_space < finite_measure
+proof
+ from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp
+qed
+
+abbreviation (in prob_space) "events \<equiv> sets M"
+abbreviation (in prob_space) "prob \<equiv> \<mu>'"
+abbreviation (in prob_space) "prob_preserving \<equiv> measure_preserving"
+abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
+abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
+
+definition (in prob_space)
+ "indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B"
+
+definition (in prob_space)
+ "indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)"
+
+definition (in prob_space)
+ "distribution X A = \<mu>' (X -` A \<inter> space M)"
+
+abbreviation (in prob_space)
+ "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
+
+declare (in finite_measure) positive_measure'[intro, simp]
+
+lemma (in prob_space) distribution_cong:
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
+ shows "distribution X = distribution Y"
+ unfolding distribution_def fun_eq_iff
+ using assms by (auto intro!: arg_cong[where f="\<mu>'"])
+
+lemma (in prob_space) joint_distribution_cong:
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
+ shows "joint_distribution X Y = joint_distribution X' Y'"
+ unfolding distribution_def fun_eq_iff
+ using assms by (auto intro!: arg_cong[where f="\<mu>'"])
+
+lemma (in prob_space) distribution_id[simp]:
+ "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
+ by (auto simp: distribution_def intro!: arg_cong[where f=prob])
+
+lemma (in prob_space) prob_space: "prob (space M) = 1"
+ using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def)
+
+lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
+ using bounded_measure[of A] by (simp add: prob_space)
+
+lemma (in prob_space) distribution_positive[simp, intro]:
+ "0 \<le> distribution X A" unfolding distribution_def by auto
+
+lemma (in prob_space) joint_distribution_remove[simp]:
+ "joint_distribution X X {(x, x)} = distribution X {x}"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+
+lemma (in prob_space) distribution_1:
+ "distribution X A \<le> 1"
+ unfolding distribution_def by simp
+
+lemma (in prob_space) prob_compl:
+ assumes A: "A \<in> events"
+ shows "prob (space M - A) = 1 - prob A"
+ using finite_measure_compl[OF A] by (simp add: prob_space)
+
+lemma (in prob_space) indep_space: "s \<in> events \<Longrightarrow> indep (space M) s"
+ by (simp add: indep_def prob_space)
+
+lemma (in prob_space) prob_space_increasing: "increasing M prob"
+ by (auto intro!: finite_measure_mono simp: increasing_def)
+
+lemma (in prob_space) 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 finite_measure_subadditive[of s t] assms by auto
+ moreover have "prob (s \<union> t) \<ge> prob s"
+ using assms by (blast intro: finite_measure_mono)
+ ultimately show ?thesis by simp
+qed
+
+lemma (in prob_space) 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 (in prob_space) 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 (in prob_space) 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 finite_measure_UNION[OF assms(1,3)])
+ have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))"
+ by (rule finite_measure_UNION[OF assms(2,4)])
+ show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
+qed
+
+lemma (in prob_space) 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 finite_measure_countably_subadditive[OF assms(1)]
+ by (simp add: assms(2) suminf_zero summable_zero)
+qed simp
+
+lemma (in prob_space) indep_sym:
+ "indep a b \<Longrightarrow> indep b a"
+unfolding indep_def using Int_commute[of a b] by auto
+
+lemma (in prob_space) 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 (in prob_space) 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 finite_measure_finite_singleton[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 (in prob_space) 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 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 (in prob_space) distribution_prob_space:
+ assumes "random_variable S X"
+ shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
+proof -
+ interpret S: measure_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
+ proof (rule measure_space.measure_space_cong)
+ show "measure_space (S\<lparr> measure := \<lambda>A. \<mu> (X -` A \<inter> space M) \<rparr>)"
+ using assms by (auto intro!: measure_space_vimage simp: measure_preserving_def)
+ qed (insert assms, auto simp add: finite_measure_eq distribution_def measurable_sets)
+ show ?thesis
+ proof (default, simp)
+ have "X -` space S \<inter> space M = space M"
+ using `random_variable S X` by (auto simp: measurable_def)
+ then show "extreal (distribution X (space S)) = 1"
+ by (simp add: distribution_def one_extreal_def prob_space)
+ qed
+qed
+
+lemma (in prob_space) AE_distribution:
+ assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x"
+ shows "AE x. Q (X x)"
+proof -
+ interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
+ obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
+ using assms unfolding X.almost_everywhere_def by auto
+ from X[unfolded measurable_def] N show "AE x. Q (X x)"
+ by (intro AE_I'[where N="X -` N \<inter> space M"])
+ (auto simp: finite_measure_eq distribution_def measurable_sets)
+qed
+
+lemma (in prob_space) distribution_eq_integral:
+ "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
+ using finite_measure_eq[of "X -` A \<inter> space M"]
+ by (auto simp: measurable_sets distribution_def)
+
+lemma (in prob_space) distribution_eq_translated_integral:
+ assumes "random_variable S X" "A \<in> sets S"
+ shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)"
+proof -
+ interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
+ using assms(1) by (rule distribution_prob_space)
+ show ?thesis
+ using S.positive_integral_indicator(1)[of A] assms by simp
+qed
+
+lemma (in prob_space) finite_expectation1:
+ assumes f: "finite (X`space M)" and rv: "random_variable borel X"
+ shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
+proof (subst integral_on_finite)
+ show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
+ show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
+ "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
+ using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
+qed
+
+lemma (in prob_space) finite_expectation:
+ assumes "finite (X`space M)" "random_variable borel X"
+ shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
+ using assms unfolding distribution_def using finite_expectation1 by auto
+
+lemma (in prob_space) 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 (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
+ unfolding distribution_def by simp
+
+lemma (in prob_space) 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: prob_space)
+qed
+
+lemma (in prob_space) distribution_one:
+ assumes "random_variable M' X" and "A \<in> sets M'"
+ 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!: finite_measure_mono)
+ thus ?thesis by (simp add: prob_space)
+qed
+
+lemma (in prob_space) 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 cases
+ { fix x have "X -` {x} \<inter> space M \<in> sets M"
+ proof cases
+ assume "x \<in> X`space M" with X show ?thesis
+ by (auto simp: measurable_def image_iff)
+ next
+ assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
+ then show ?thesis by auto
+ qed } note single = this
+ have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
+ "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
+ using `y \<noteq> x` by auto
+ with finite_measure_inter_full_set[OF single single, of x y] assms(2)
+ show ?thesis by (auto simp: distribution_def prob_space)
+next
+ assume "{y} \<notin> sets ?S"
+ then have "X -` {y} \<inter> space M = {}" by auto
+ thus "distribution X {y} = 0" unfolding distribution_def by auto
+qed
+
+lemma (in prob_space) joint_distribution_Times_le_fst:
+ assumes X: "random_variable MX X" and Y: "random_variable MY Y"
+ and A: "A \<in> sets MX" and B: "B \<in> sets MY"
+ shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
+ unfolding distribution_def
+proof (intro finite_measure_mono)
+ show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
+ show "X -` A \<inter> space M \<in> events"
+ using X A unfolding measurable_def by simp
+ have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
+ (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
+qed
+
+lemma (in prob_space) joint_distribution_commute:
+ "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+
+lemma (in prob_space) joint_distribution_Times_le_snd:
+ assumes X: "random_variable MX X" and Y: "random_variable MY Y"
+ and A: "A \<in> sets MX" and B: "B \<in> sets MY"
+ shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
+ using assms
+ by (subst joint_distribution_commute)
+ (simp add: swap_product joint_distribution_Times_le_fst)
+
+lemma (in prob_space) random_variable_pairI:
+ assumes "random_variable MX X"
+ assumes "random_variable MY Y"
+ shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
+proof
+ interpret MX: sigma_algebra MX using assms by simp
+ interpret MY: sigma_algebra MY using assms by simp
+ interpret P: pair_sigma_algebra MX MY by default
+ show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
+ have sa: "sigma_algebra M" by default
+ show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
+ unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
+qed
+
+lemma (in prob_space) joint_distribution_commute_singleton:
+ "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+
+lemma (in prob_space) joint_distribution_assoc_singleton:
+ "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
+ joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+
+locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2
+
+sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default
+
+sublocale pair_prob_space \<subseteq> P: prob_space P
+by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
+
+lemma countably_additiveI[case_names countably]:
+ assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
+ (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
+ shows "countably_additive M \<mu>"
+ using assms unfolding countably_additive_def by auto
+
+lemma (in prob_space) joint_distribution_prob_space:
+ assumes "random_variable MX X" "random_variable MY Y"
+ shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
+ using random_variable_pairI[OF assms] by (rule distribution_prob_space)
+
+section "Probability spaces on finite sets"
+
+locale finite_prob_space = prob_space + finite_measure_space
+
+abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
+
+lemma (in prob_space) finite_random_variableD:
+ assumes "finite_random_variable M' X" shows "random_variable M' X"
+proof -
+ interpret M': finite_sigma_algebra M' using assms by simp
+ then show "random_variable M' X" using assms by simp default
+qed
+
+lemma (in prob_space) distribution_finite_prob_space:
+ assumes "finite_random_variable MX X"
+ shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
+proof -
+ interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>"
+ using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
+ interpret MX: finite_sigma_algebra MX
+ using assms by auto
+ show ?thesis by default (simp_all add: MX.finite_space)
+qed
+
+lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
+ assumes "simple_function M X"
+ shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
+ (is "finite_random_variable ?X _")
+proof (intro conjI)
+ have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp
+ interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
+ show "finite_sigma_algebra ?X"
+ by default auto
+ show "X \<in> measurable M ?X"
+ proof (unfold measurable_def, clarsimp)
+ fix A assume A: "A \<subseteq> X`space M"
+ then have "finite A" by (rule finite_subset) simp
+ then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
+ unfolding vimage_UN UN_extend_simps
+ apply (rule finite_UN)
+ using A assms unfolding simple_function_def by auto
+ then show "X -` A \<inter> space M \<in> events" by simp
+ qed
+qed
+
+lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
+ assumes "simple_function M X"
+ shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
+ using simple_function_imp_finite_random_variable[OF assms, of ext]
+ by (auto dest!: finite_random_variableD)
+
+lemma (in prob_space) sum_over_space_real_distribution:
+ "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
+ unfolding distribution_def prob_space[symmetric]
+ by (subst finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def simple_function_def
+ intro!: arg_cong[where f=prob])
+
+lemma (in prob_space) finite_random_variable_pairI:
+ assumes "finite_random_variable MX X"
+ assumes "finite_random_variable MY Y"
+ shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
+proof
+ interpret MX: finite_sigma_algebra MX using assms by simp
+ interpret MY: finite_sigma_algebra MY using assms by simp
+ interpret P: pair_finite_sigma_algebra MX MY by default
+ show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
+ have sa: "sigma_algebra M" by default
+ show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
+ unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
+qed
+
+lemma (in prob_space) finite_random_variable_imp_sets:
+ "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
+ unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
+
+lemma (in prob_space) finite_random_variable_measurable:
+ assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
+proof -
+ interpret X: finite_sigma_algebra MX using X by simp
+ from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
+ "X \<in> space M \<rightarrow> space MX"
+ by (auto simp: measurable_def)
+ then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
+ by auto
+ show "X -` A \<inter> space M \<in> events"
+ unfolding * by (intro vimage) auto
+qed
+
+lemma (in prob_space) joint_distribution_finite_Times_le_fst:
+ assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
+ shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
+ unfolding distribution_def
+proof (intro finite_measure_mono)
+ show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
+ show "X -` A \<inter> space M \<in> events"
+ using finite_random_variable_measurable[OF X] .
+ have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
+ (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
+qed
+
+lemma (in prob_space) joint_distribution_finite_Times_le_snd:
+ assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
+ shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
+ using assms
+ by (subst joint_distribution_commute)
+ (simp add: swap_product joint_distribution_finite_Times_le_fst)
+
+lemma (in prob_space) finite_distribution_order:
+ fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
+ assumes "finite_random_variable MX X" "finite_random_variable MY Y"
+ shows "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 joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
+ using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
+ by (auto intro: antisym)
+
+lemma (in prob_space) setsum_joint_distribution:
+ assumes X: "finite_random_variable MX X"
+ assumes Y: "random_variable MY Y" "B \<in> sets MY"
+ shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
+ unfolding distribution_def
+proof (subst finite_measure_finite_Union[symmetric])
+ interpret MX: finite_sigma_algebra MX using X by auto
+ show "finite (space MX)" using MX.finite_space .
+ let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
+ { fix i assume "i \<in> space MX"
+ moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
+ ultimately show "?d i \<in> events"
+ using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
+ using MX.sets_eq_Pow by auto }
+ show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
+ show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
+ using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
+qed
+
+lemma (in prob_space) setsum_joint_distribution_singleton:
+ assumes X: "finite_random_variable MX X"
+ assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
+ shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
+ using setsum_joint_distribution[OF X
+ finite_random_variableD[OF Y(1)]
+ finite_random_variable_imp_sets[OF Y]] by simp
+
+locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
+
+sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default
+sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2 by default
+sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
+
+lemma (in prob_space) joint_distribution_finite_prob_space:
+ assumes X: "finite_random_variable MX X"
+ assumes Y: "finite_random_variable MY Y"
+ shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
+ by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
+
+lemma finite_prob_space_eq:
+ "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (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) joint_distribution_restriction_fst:
+ "joint_distribution X Y A \<le> distribution X (fst ` A)"
+ unfolding distribution_def
+proof (safe intro!: finite_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!: finite_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
+ joint_distribution_restriction_fst[of X Y "{(x, y)}"]
+ joint_distribution_restriction_snd[of X Y "{(x, y)}"]
+ by (auto intro: antisym)
+
+lemma (in finite_prob_space) distribution_mono:
+ assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
+ shows "distribution X x \<le> distribution Y y"
+ unfolding distribution_def
+ using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
+
+lemma (in finite_prob_space) distribution_mono_gt_0:
+ assumes gt_0: "0 < distribution X x"
+ assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
+ shows "0 < distribution Y y"
+ by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
+
+lemma (in finite_prob_space) sum_over_space_distrib:
+ "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
+ unfolding distribution_def prob_space[symmetric] using finite_space
+ by (subst finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow
+ intro!: arg_cong[where f=\<mu>'])
+
+lemma (in finite_prob_space) sum_over_space_real_distribution:
+ "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
+ unfolding distribution_def prob_space[symmetric] using finite_space
+ by (subst finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+
+lemma (in finite_prob_space) finite_sum_over_space_eq_1:
+ "(\<Sum>x\<in>space M. prob {x}) = 1"
+ using prob_space finite_space
+ by (subst (asm) finite_measure_finite_singleton) auto
+
+lemma (in prob_space) distribution_remove_const:
+ shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
+ and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
+ and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
+ and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
+ and "distribution (\<lambda>x. ()) {()} = 1"
+ by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
+
+lemma (in finite_prob_space) setsum_distribution_gen:
+ assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
+ and "inj_on f (X`space M)"
+ shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
+ unfolding distribution_def assms
+ using finite_space assms
+ by (subst finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
+ intro!: arg_cong[where f=prob])
+
+lemma (in finite_prob_space) setsum_distribution:
+ "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
+ "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
+ "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
+ "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
+ "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
+ by (auto intro!: inj_onI setsum_distribution_gen)
+
+lemma (in finite_prob_space) uniform_prob:
+ assumes "x \<in> space M"
+ assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
+ shows "prob {x} = 1 / card (space M)"
+proof -
+ have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
+ using assms(2)[OF _ `x \<in> space M`] by blast
+ have "1 = prob (space M)"
+ using prob_space by auto
+ also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
+ using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
+ sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
+ finite_space unfolding disjoint_family_on_def prob_space[symmetric]
+ by (auto simp add:setsum_restrict_set)
+ also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
+ using prob_x by auto
+ also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
+ finally have one: "1 = real (card (space M)) * prob {x}"
+ using real_eq_of_nat by auto
+ hence two: "real (card (space M)) \<noteq> 0" by fastsimp
+ from one have three: "prob {x} \<noteq> 0" by fastsimp
+ thus ?thesis using one two three divide_cancel_right
+ by (auto simp:field_simps)
+qed
+
+lemma (in prob_space) prob_space_subalgebra:
+ assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M"
+ and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
+ shows "prob_space N"
+proof -
+ interpret N: measure_space N
+ by (rule measure_space_subalgebra[OF assms])
+ show ?thesis
+ proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1)
+qed
+
+lemma (in prob_space) prob_space_of_restricted_space:
+ assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
+ shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
+ (is "prob_space ?P")
+proof -
+ interpret A: measure_space "restricted_space A"
+ using `A \<in> sets M` by (rule restricted_measure_space)
+ interpret A': sigma_algebra ?P
+ by (rule A.sigma_algebra_cong) auto
+ show "prob_space ?P"
+ proof
+ show "measure ?P (space ?P) = 1"
+ using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
+ show "positive ?P (measure ?P)"
+ proof (simp add: positive_def, safe)
+ show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def)
+ fix B assume "B \<in> events"
+ with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
+ show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
+ qed
+ show "countably_additive ?P (measure ?P)"
+ proof (simp add: countably_additive_def, safe)
+ fix B and F :: "nat \<Rightarrow> 'a set"
+ assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
+ { fix i
+ from F have "F i \<in> op \<inter> A ` events" by auto
+ with `A \<in> events` have "F i \<in> events" by auto }
+ moreover then have "range F \<subseteq> events" by auto
+ moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
+ by (simp add: mult_commute divide_extreal_def)
+ moreover have "0 \<le> inverse (\<mu> A)"
+ using real_measure[OF `A \<in> events`] by auto
+ ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
+ using measure_countably_additive[of F] F
+ by (auto simp: suminf_cmult_extreal)
+ qed
+ qed
+qed
+
+lemma finite_prob_spaceI:
+ assumes "finite (space M)" "sets M = Pow(space M)"
+ and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
+ 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"
+ shows "finite_prob_space M"
+ unfolding finite_prob_space_eq
+proof
+ show "finite_measure_space M" using assms
+ by (auto intro!: finite_measure_spaceI)
+ show "measure M (space M) = 1" by fact
+qed
+
+lemma (in finite_prob_space) finite_measure_space:
+ fixes X :: "'a \<Rightarrow> 'x"
+ shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>"
+ (is "finite_measure_space ?S")
+proof (rule finite_measure_spaceI, simp_all)
+ show "finite (X ` space M)" using finite_space by simp
+next
+ fix A B :: "'x set" assume "A \<inter> B = {}"
+ then show "distribution X (A \<union> B) = distribution X A + distribution X B"
+ unfolding distribution_def
+ by (subst finite_measure_Union[symmetric])
+ (auto intro!: arg_cong[where f=\<mu>'] 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), measure = extreal \<circ> distribution X \<rparr>"
+ by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def)
+
+lemma (in finite_prob_space) finite_product_measure_space:
+ fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
+ assumes "finite s1" "finite s2"
+ shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>"
+ (is "finite_measure_space ?M")
+proof (rule finite_measure_spaceI, simp_all)
+ show "finite (s1 \<times> s2)"
+ using assms by auto
+next
+ fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
+ then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
+ unfolding distribution_def
+ by (subst finite_measure_Union[symmetric])
+ (auto intro!: arg_cong[where f=\<mu>'] 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),
+ measure = extreal \<circ> joint_distribution X Y \<rparr>"
+ using finite_space by (auto intro!: finite_product_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),
+ measure = extreal \<circ> joint_distribution X Y \<rparr>"
+ (is "finite_prob_space ?S")
+proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def)
+ 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
+
+section "Conditional Expectation and Probability"
+
+lemma (in prob_space) conditional_expectation_exists:
+ fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
+ assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
+ and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
+ shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<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))"
+proof -
+ note N(4)[simp]
+ interpret P: prob_space N
+ using prob_space_subalgebra[OF N] .
+
+ let "?f A" = "\<lambda>x. X x * indicator A x"
+ let "?Q A" = "integral\<^isup>P M (?f A)"
+
+ from measure_space_density[OF borel]
+ have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)"
+ apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"])
+ using N by (auto intro!: P.sigma_algebra_cong)
+ then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" .
+
+ have "P.absolutely_continuous ?Q"
+ unfolding P.absolutely_continuous_def
+ proof safe
+ fix A assume "A \<in> sets N" "P.\<mu> A = 0"
+ then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
+ using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
+ then show "?Q A = 0"
+ by (auto simp add: positive_integral_0_iff_AE)
+ qed
+ from P.Radon_Nikodym[OF Q this]
+ obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x"
+ "\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
+ by blast
+ with N(2) show ?thesis
+ by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
+qed
+
+definition (in prob_space)
+ "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x)
+ \<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)))"
+
+abbreviation (in prob_space)
+ "conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
+
+lemma (in prob_space)
+ fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
+ assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
+ and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
+ shows borel_measurable_conditional_expectation:
+ "conditional_expectation N X \<in> borel_measurable N"
+ and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow>
+ (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) =
+ (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
+ (is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C")
+proof -
+ note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
+ then show "conditional_expectation N X \<in> borel_measurable N"
+ unfolding conditional_expectation_def by (rule someI2_ex) blast
+
+ from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C"
+ unfolding conditional_expectation_def by (rule someI2_ex) blast
+qed
+
+lemma (in sigma_algebra) factorize_measurable_function_pos:
+ fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
+ assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
+ assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
+ shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
+proof -
+ interpret M': sigma_algebra M' by fact
+ have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
+ from M'.sigma_algebra_vimage[OF this]
+ interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
+
+ from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
+
+ have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+ proof
+ fix i
+ from f(1)[of i] have "finite (f i`space M)" and B_ex:
+ "\<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"
+ unfolding simple_function_def by auto
+ from B_ex[THEN bchoice] guess B .. note B = this
+
+ let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
+
+ show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+ proof (intro exI[of _ ?g] conjI ballI)
+ show "simple_function M' ?g" using B by auto
+
+ fix x assume "x \<in> space M"
+ 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)"
+ unfolding indicator_def using B by auto
+ then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
+ by (subst va.simple_function_indicator_representation) auto
+ qed
+ qed
+ from choice[OF this] guess g .. note g = this
+
+ show ?thesis
+ proof (intro ballI bexI)
+ show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
+ using g by (auto intro: M'.borel_measurable_simple_function)
+ fix x assume "x \<in> space M"
+ have "max 0 (Z x) = (SUP i. f i x)" using f by simp
+ also have "\<dots> = (SUP i. g i (Y x))"
+ using g `x \<in> space M` by simp
+ finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
+ qed
+qed
+
+lemma extreal_0_le_iff_le_0[simp]:
+ fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
+ by (cases rule: extreal2_cases[of a]) auto
+
+lemma (in sigma_algebra) factorize_measurable_function:
+ fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
+ assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
+ shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
+ \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
+proof safe
+ interpret M': sigma_algebra M' by fact
+ have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
+ from M'.sigma_algebra_vimage[OF this]
+ interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
+
+ { fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'"
+ with M'.measurable_vimage_algebra[OF Y]
+ have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by (rule measurable_comp)
+ moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
+ then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
+ g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by (auto intro!: measurable_cong)
+ ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by simp }
+
+ assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
+ "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by auto
+ from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
+ from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
+ let "?g x" = "p x - n x"
+ show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
+ proof (intro bexI ballI)
+ show "?g \<in> borel_measurable M'" using p n by auto
+ fix x assume "x \<in> space M"
+ then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
+ using p n by auto
+ then show "Z x = ?g (Y x)"
+ by (auto split: split_max)
+ qed
+qed
+
+end
--- a/src/HOL/Probability/Probability_Space.thy Tue Mar 29 14:27:41 2011 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,935 +0,0 @@
-(* Title: HOL/Probability/Probability_Space.thy
- Author: Johannes Hölzl, TU München
- Author: Armin Heller, TU München
-*)
-
-header {*Probability spaces*}
-
-theory Probability_Space
-imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure
-begin
-
-lemma real_of_extreal_inverse[simp]:
- fixes X :: extreal
- shows "real (inverse X) = 1 / real X"
- by (cases X) (auto simp: inverse_eq_divide)
-
-lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)"
- by (cases X) auto
-
-lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>"
- by (cases X) auto
-
-lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)"
- by (cases X) auto
-
-lemma real_of_extreal_le_1: fixes X :: extreal shows "X \<le> 1 \<Longrightarrow> real X \<le> 1"
- by (cases X) (auto simp: one_extreal_def)
-
-locale prob_space = measure_space +
- assumes measure_space_1: "measure M (space M) = 1"
-
-sublocale prob_space < finite_measure
-proof
- from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp
-qed
-
-abbreviation (in prob_space) "events \<equiv> sets M"
-abbreviation (in prob_space) "prob \<equiv> \<mu>'"
-abbreviation (in prob_space) "prob_preserving \<equiv> measure_preserving"
-abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
-abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
-
-definition (in prob_space)
- "indep A B \<longleftrightarrow> A \<in> events \<and> B \<in> events \<and> prob (A \<inter> B) = prob A * prob B"
-
-definition (in prob_space)
- "indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)"
-
-definition (in prob_space)
- "distribution X A = \<mu>' (X -` A \<inter> space M)"
-
-abbreviation (in prob_space)
- "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
-
-declare (in finite_measure) positive_measure'[intro, simp]
-
-lemma (in prob_space) distribution_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
- shows "distribution X = distribution Y"
- unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) joint_distribution_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
- assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
- shows "joint_distribution X Y = joint_distribution X' Y'"
- unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) distribution_id[simp]:
- "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
- by (auto simp: distribution_def intro!: arg_cong[where f=prob])
-
-lemma (in prob_space) prob_space: "prob (space M) = 1"
- using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def)
-
-lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
- using bounded_measure[of A] by (simp add: prob_space)
-
-lemma (in prob_space) distribution_positive[simp, intro]:
- "0 \<le> distribution X A" unfolding distribution_def by auto
-
-lemma (in prob_space) joint_distribution_remove[simp]:
- "joint_distribution X X {(x, x)} = distribution X {x}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) distribution_1:
- "distribution X A \<le> 1"
- unfolding distribution_def by simp
-
-lemma (in prob_space) prob_compl:
- assumes A: "A \<in> events"
- shows "prob (space M - A) = 1 - prob A"
- using finite_measure_compl[OF A] by (simp add: prob_space)
-
-lemma (in prob_space) indep_space: "s \<in> events \<Longrightarrow> indep (space M) s"
- by (simp add: indep_def prob_space)
-
-lemma (in prob_space) prob_space_increasing: "increasing M prob"
- by (auto intro!: finite_measure_mono simp: increasing_def)
-
-lemma (in prob_space) 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 finite_measure_subadditive[of s t] assms by auto
- moreover have "prob (s \<union> t) \<ge> prob s"
- using assms by (blast intro: finite_measure_mono)
- ultimately show ?thesis by simp
-qed
-
-lemma (in prob_space) 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 (in prob_space) 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 (in prob_space) 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 finite_measure_UNION[OF assms(1,3)])
- have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))"
- by (rule finite_measure_UNION[OF assms(2,4)])
- show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
-qed
-
-lemma (in prob_space) 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 finite_measure_countably_subadditive[OF assms(1)]
- by (simp add: assms(2) suminf_zero summable_zero)
-qed simp
-
-lemma (in prob_space) indep_sym:
- "indep a b \<Longrightarrow> indep b a"
-unfolding indep_def using Int_commute[of a b] by auto
-
-lemma (in prob_space) 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 (in prob_space) 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 finite_measure_finite_singleton[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 (in prob_space) 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 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 (in prob_space) distribution_prob_space:
- assumes "random_variable S X"
- shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
-proof -
- interpret S: measure_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
- proof (rule measure_space.measure_space_cong)
- show "measure_space (S\<lparr> measure := \<lambda>A. \<mu> (X -` A \<inter> space M) \<rparr>)"
- using assms by (auto intro!: measure_space_vimage simp: measure_preserving_def)
- qed (insert assms, auto simp add: finite_measure_eq distribution_def measurable_sets)
- show ?thesis
- proof (default, simp)
- have "X -` space S \<inter> space M = space M"
- using `random_variable S X` by (auto simp: measurable_def)
- then show "extreal (distribution X (space S)) = 1"
- by (simp add: distribution_def one_extreal_def prob_space)
- qed
-qed
-
-lemma (in prob_space) AE_distribution:
- assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x"
- shows "AE x. Q (X x)"
-proof -
- interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
- obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
- using assms unfolding X.almost_everywhere_def by auto
- from X[unfolded measurable_def] N show "AE x. Q (X x)"
- by (intro AE_I'[where N="X -` N \<inter> space M"])
- (auto simp: finite_measure_eq distribution_def measurable_sets)
-qed
-
-lemma (in prob_space) distribution_eq_integral:
- "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
- using finite_measure_eq[of "X -` A \<inter> space M"]
- by (auto simp: measurable_sets distribution_def)
-
-lemma (in prob_space) distribution_eq_translated_integral:
- assumes "random_variable S X" "A \<in> sets S"
- shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)"
-proof -
- interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
- using assms(1) by (rule distribution_prob_space)
- show ?thesis
- using S.positive_integral_indicator(1)[of A] assms by simp
-qed
-
-lemma (in prob_space) finite_expectation1:
- assumes f: "finite (X`space M)" and rv: "random_variable borel X"
- shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
-proof (subst integral_on_finite)
- show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
- show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
- "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
- using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
-qed
-
-lemma (in prob_space) finite_expectation:
- assumes "finite (X`space M)" "random_variable borel X"
- shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
- using assms unfolding distribution_def using finite_expectation1 by auto
-
-lemma (in prob_space) 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 (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
- unfolding distribution_def by simp
-
-lemma (in prob_space) 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: prob_space)
-qed
-
-lemma (in prob_space) distribution_one:
- assumes "random_variable M' X" and "A \<in> sets M'"
- 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!: finite_measure_mono)
- thus ?thesis by (simp add: prob_space)
-qed
-
-lemma (in prob_space) 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 cases
- { fix x have "X -` {x} \<inter> space M \<in> sets M"
- proof cases
- assume "x \<in> X`space M" with X show ?thesis
- by (auto simp: measurable_def image_iff)
- next
- assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
- then show ?thesis by auto
- qed } note single = this
- have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
- "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
- using `y \<noteq> x` by auto
- with finite_measure_inter_full_set[OF single single, of x y] assms(2)
- show ?thesis by (auto simp: distribution_def prob_space)
-next
- assume "{y} \<notin> sets ?S"
- then have "X -` {y} \<inter> space M = {}" by auto
- thus "distribution X {y} = 0" unfolding distribution_def by auto
-qed
-
-lemma (in prob_space) joint_distribution_Times_le_fst:
- assumes X: "random_variable MX X" and Y: "random_variable MY Y"
- and A: "A \<in> sets MX" and B: "B \<in> sets MY"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
- unfolding distribution_def
-proof (intro finite_measure_mono)
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
- show "X -` A \<inter> space M \<in> events"
- using X A unfolding measurable_def by simp
- have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
- (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_commute:
- "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) joint_distribution_Times_le_snd:
- assumes X: "random_variable MX X" and Y: "random_variable MY Y"
- and A: "A \<in> sets MX" and B: "B \<in> sets MY"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
- using assms
- by (subst joint_distribution_commute)
- (simp add: swap_product joint_distribution_Times_le_fst)
-
-lemma (in prob_space) random_variable_pairI:
- assumes "random_variable MX X"
- assumes "random_variable MY Y"
- shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
- interpret MX: sigma_algebra MX using assms by simp
- interpret MY: sigma_algebra MY using assms by simp
- interpret P: pair_sigma_algebra MX MY by default
- show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
- have sa: "sigma_algebra M" by default
- show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
- unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) joint_distribution_commute_singleton:
- "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) joint_distribution_assoc_singleton:
- "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
- joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2
-
-sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default
-
-sublocale pair_prob_space \<subseteq> P: prob_space P
-by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
-
-lemma countably_additiveI[case_names countably]:
- assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
- (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
- shows "countably_additive M \<mu>"
- using assms unfolding countably_additive_def by auto
-
-lemma (in prob_space) joint_distribution_prob_space:
- assumes "random_variable MX X" "random_variable MY Y"
- shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
- using random_variable_pairI[OF assms] by (rule distribution_prob_space)
-
-section "Probability spaces on finite sets"
-
-locale finite_prob_space = prob_space + finite_measure_space
-
-abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
-
-lemma (in prob_space) finite_random_variableD:
- assumes "finite_random_variable M' X" shows "random_variable M' X"
-proof -
- interpret M': finite_sigma_algebra M' using assms by simp
- then show "random_variable M' X" using assms by simp default
-qed
-
-lemma (in prob_space) distribution_finite_prob_space:
- assumes "finite_random_variable MX X"
- shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
-proof -
- interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>"
- using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
- interpret MX: finite_sigma_algebra MX
- using assms by auto
- show ?thesis by default (simp_all add: MX.finite_space)
-qed
-
-lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
- assumes "simple_function M X"
- shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
- (is "finite_random_variable ?X _")
-proof (intro conjI)
- have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp
- interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
- show "finite_sigma_algebra ?X"
- by default auto
- show "X \<in> measurable M ?X"
- proof (unfold measurable_def, clarsimp)
- fix A assume A: "A \<subseteq> X`space M"
- then have "finite A" by (rule finite_subset) simp
- then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
- unfolding vimage_UN UN_extend_simps
- apply (rule finite_UN)
- using A assms unfolding simple_function_def by auto
- then show "X -` A \<inter> space M \<in> events" by simp
- qed
-qed
-
-lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
- assumes "simple_function M X"
- shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
- using simple_function_imp_finite_random_variable[OF assms, of ext]
- by (auto dest!: finite_random_variableD)
-
-lemma (in prob_space) sum_over_space_real_distribution:
- "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric]
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def simple_function_def
- intro!: arg_cong[where f=prob])
-
-lemma (in prob_space) finite_random_variable_pairI:
- assumes "finite_random_variable MX X"
- assumes "finite_random_variable MY Y"
- shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
- interpret MX: finite_sigma_algebra MX using assms by simp
- interpret MY: finite_sigma_algebra MY using assms by simp
- interpret P: pair_finite_sigma_algebra MX MY by default
- show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
- have sa: "sigma_algebra M" by default
- show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
- unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) finite_random_variable_imp_sets:
- "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
- unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
-
-lemma (in prob_space) finite_random_variable_measurable:
- assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
-proof -
- interpret X: finite_sigma_algebra MX using X by simp
- from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
- "X \<in> space M \<rightarrow> space MX"
- by (auto simp: measurable_def)
- then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
- by auto
- show "X -` A \<inter> space M \<in> events"
- unfolding * by (intro vimage) auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_fst:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
- unfolding distribution_def
-proof (intro finite_measure_mono)
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
- show "X -` A \<inter> space M \<in> events"
- using finite_random_variable_measurable[OF X] .
- have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
- (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_snd:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
- using assms
- by (subst joint_distribution_commute)
- (simp add: swap_product joint_distribution_finite_Times_le_fst)
-
-lemma (in prob_space) finite_distribution_order:
- fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
- assumes "finite_random_variable MX X" "finite_random_variable MY Y"
- shows "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 joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
- using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
- by (auto intro: antisym)
-
-lemma (in prob_space) setsum_joint_distribution:
- assumes X: "finite_random_variable MX X"
- assumes Y: "random_variable MY Y" "B \<in> sets MY"
- shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
- unfolding distribution_def
-proof (subst finite_measure_finite_Union[symmetric])
- interpret MX: finite_sigma_algebra MX using X by auto
- show "finite (space MX)" using MX.finite_space .
- let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
- { fix i assume "i \<in> space MX"
- moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
- ultimately show "?d i \<in> events"
- using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
- using MX.sets_eq_Pow by auto }
- show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
- show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
- using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
-qed
-
-lemma (in prob_space) setsum_joint_distribution_singleton:
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
- shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
- using setsum_joint_distribution[OF X
- finite_random_variableD[OF Y(1)]
- finite_random_variable_imp_sets[OF Y]] by simp
-
-locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
-
-sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default
-sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2 by default
-sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
-
-lemma (in prob_space) joint_distribution_finite_prob_space:
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y"
- shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
- by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
-
-lemma finite_prob_space_eq:
- "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (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) joint_distribution_restriction_fst:
- "joint_distribution X Y A \<le> distribution X (fst ` A)"
- unfolding distribution_def
-proof (safe intro!: finite_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!: finite_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
- joint_distribution_restriction_fst[of X Y "{(x, y)}"]
- joint_distribution_restriction_snd[of X Y "{(x, y)}"]
- by (auto intro: antisym)
-
-lemma (in finite_prob_space) distribution_mono:
- assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
- shows "distribution X x \<le> distribution Y y"
- unfolding distribution_def
- using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
-
-lemma (in finite_prob_space) distribution_mono_gt_0:
- assumes gt_0: "0 < distribution X x"
- assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
- shows "0 < distribution Y y"
- by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
-
-lemma (in finite_prob_space) sum_over_space_distrib:
- "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow
- intro!: arg_cong[where f=\<mu>'])
-
-lemma (in finite_prob_space) sum_over_space_real_distribution:
- "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
-
-lemma (in finite_prob_space) finite_sum_over_space_eq_1:
- "(\<Sum>x\<in>space M. prob {x}) = 1"
- using prob_space finite_space
- by (subst (asm) finite_measure_finite_singleton) auto
-
-lemma (in prob_space) distribution_remove_const:
- shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
- and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
- and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
- and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
- and "distribution (\<lambda>x. ()) {()} = 1"
- by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
-
-lemma (in finite_prob_space) setsum_distribution_gen:
- assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
- and "inj_on f (X`space M)"
- shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
- unfolding distribution_def assms
- using finite_space assms
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
- intro!: arg_cong[where f=prob])
-
-lemma (in finite_prob_space) setsum_distribution:
- "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
- "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
- "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
- "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
- "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
- by (auto intro!: inj_onI setsum_distribution_gen)
-
-lemma (in finite_prob_space) uniform_prob:
- assumes "x \<in> space M"
- assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
- shows "prob {x} = 1 / card (space M)"
-proof -
- have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
- using assms(2)[OF _ `x \<in> space M`] by blast
- have "1 = prob (space M)"
- using prob_space by auto
- also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
- using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
- sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
- finite_space unfolding disjoint_family_on_def prob_space[symmetric]
- by (auto simp add:setsum_restrict_set)
- also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
- using prob_x by auto
- also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
- finally have one: "1 = real (card (space M)) * prob {x}"
- using real_eq_of_nat by auto
- hence two: "real (card (space M)) \<noteq> 0" by fastsimp
- from one have three: "prob {x} \<noteq> 0" by fastsimp
- thus ?thesis using one two three divide_cancel_right
- by (auto simp:field_simps)
-qed
-
-lemma (in prob_space) prob_space_subalgebra:
- assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M"
- and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
- shows "prob_space N"
-proof -
- interpret N: measure_space N
- by (rule measure_space_subalgebra[OF assms])
- show ?thesis
- proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1)
-qed
-
-lemma (in prob_space) prob_space_of_restricted_space:
- assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
- shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
- (is "prob_space ?P")
-proof -
- interpret A: measure_space "restricted_space A"
- using `A \<in> sets M` by (rule restricted_measure_space)
- interpret A': sigma_algebra ?P
- by (rule A.sigma_algebra_cong) auto
- show "prob_space ?P"
- proof
- show "measure ?P (space ?P) = 1"
- using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
- show "positive ?P (measure ?P)"
- proof (simp add: positive_def, safe)
- show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def)
- fix B assume "B \<in> events"
- with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
- show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
- qed
- show "countably_additive ?P (measure ?P)"
- proof (simp add: countably_additive_def, safe)
- fix B and F :: "nat \<Rightarrow> 'a set"
- assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
- { fix i
- from F have "F i \<in> op \<inter> A ` events" by auto
- with `A \<in> events` have "F i \<in> events" by auto }
- moreover then have "range F \<subseteq> events" by auto
- moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
- by (simp add: mult_commute divide_extreal_def)
- moreover have "0 \<le> inverse (\<mu> A)"
- using real_measure[OF `A \<in> events`] by auto
- ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
- using measure_countably_additive[of F] F
- by (auto simp: suminf_cmult_extreal)
- qed
- qed
-qed
-
-lemma finite_prob_spaceI:
- assumes "finite (space M)" "sets M = Pow(space M)"
- and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
- 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"
- shows "finite_prob_space M"
- unfolding finite_prob_space_eq
-proof
- show "finite_measure_space M" using assms
- by (auto intro!: finite_measure_spaceI)
- show "measure M (space M) = 1" by fact
-qed
-
-lemma (in finite_prob_space) finite_measure_space:
- fixes X :: "'a \<Rightarrow> 'x"
- shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>"
- (is "finite_measure_space ?S")
-proof (rule finite_measure_spaceI, simp_all)
- show "finite (X ` space M)" using finite_space by simp
-next
- fix A B :: "'x set" assume "A \<inter> B = {}"
- then show "distribution X (A \<union> B) = distribution X A + distribution X B"
- unfolding distribution_def
- by (subst finite_measure_Union[symmetric])
- (auto intro!: arg_cong[where f=\<mu>'] 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), measure = extreal \<circ> distribution X \<rparr>"
- by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def)
-
-lemma (in finite_prob_space) finite_product_measure_space:
- fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
- assumes "finite s1" "finite s2"
- shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>"
- (is "finite_measure_space ?M")
-proof (rule finite_measure_spaceI, simp_all)
- show "finite (s1 \<times> s2)"
- using assms by auto
-next
- fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
- then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
- unfolding distribution_def
- by (subst finite_measure_Union[symmetric])
- (auto intro!: arg_cong[where f=\<mu>'] 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),
- measure = extreal \<circ> joint_distribution X Y \<rparr>"
- using finite_space by (auto intro!: finite_product_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),
- measure = extreal \<circ> joint_distribution X Y \<rparr>"
- (is "finite_prob_space ?S")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def)
- 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
-
-section "Conditional Expectation and Probability"
-
-lemma (in prob_space) conditional_expectation_exists:
- fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
- assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
- and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
- shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<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))"
-proof -
- note N(4)[simp]
- interpret P: prob_space N
- using prob_space_subalgebra[OF N] .
-
- let "?f A" = "\<lambda>x. X x * indicator A x"
- let "?Q A" = "integral\<^isup>P M (?f A)"
-
- from measure_space_density[OF borel]
- have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)"
- apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"])
- using N by (auto intro!: P.sigma_algebra_cong)
- then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" .
-
- have "P.absolutely_continuous ?Q"
- unfolding P.absolutely_continuous_def
- proof safe
- fix A assume "A \<in> sets N" "P.\<mu> A = 0"
- then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
- using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
- then show "?Q A = 0"
- by (auto simp add: positive_integral_0_iff_AE)
- qed
- from P.Radon_Nikodym[OF Q this]
- obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x"
- "\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
- by blast
- with N(2) show ?thesis
- by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
-qed
-
-definition (in prob_space)
- "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x)
- \<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)))"
-
-abbreviation (in prob_space)
- "conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
-
-lemma (in prob_space)
- fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
- assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
- and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
- shows borel_measurable_conditional_expectation:
- "conditional_expectation N X \<in> borel_measurable N"
- and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow>
- (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) =
- (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
- (is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C")
-proof -
- note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
- then show "conditional_expectation N X \<in> borel_measurable N"
- unfolding conditional_expectation_def by (rule someI2_ex) blast
-
- from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C"
- unfolding conditional_expectation_def by (rule someI2_ex) blast
-qed
-
-lemma (in sigma_algebra) factorize_measurable_function_pos:
- fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
- assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
- assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
- shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
-proof -
- interpret M': sigma_algebra M' by fact
- have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
- from M'.sigma_algebra_vimage[OF this]
- interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
-
- from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
-
- have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
- proof
- fix i
- from f(1)[of i] have "finite (f i`space M)" and B_ex:
- "\<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"
- unfolding simple_function_def by auto
- from B_ex[THEN bchoice] guess B .. note B = this
-
- let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
-
- show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
- proof (intro exI[of _ ?g] conjI ballI)
- show "simple_function M' ?g" using B by auto
-
- fix x assume "x \<in> space M"
- 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)"
- unfolding indicator_def using B by auto
- then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
- by (subst va.simple_function_indicator_representation) auto
- qed
- qed
- from choice[OF this] guess g .. note g = this
-
- show ?thesis
- proof (intro ballI bexI)
- show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
- using g by (auto intro: M'.borel_measurable_simple_function)
- fix x assume "x \<in> space M"
- have "max 0 (Z x) = (SUP i. f i x)" using f by simp
- also have "\<dots> = (SUP i. g i (Y x))"
- using g `x \<in> space M` by simp
- finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
- qed
-qed
-
-lemma extreal_0_le_iff_le_0[simp]:
- fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
- by (cases rule: extreal2_cases[of a]) auto
-
-lemma (in sigma_algebra) factorize_measurable_function:
- fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
- assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
- shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
- \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
-proof safe
- interpret M': sigma_algebra M' by fact
- have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
- from M'.sigma_algebra_vimage[OF this]
- interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
-
- { fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'"
- with M'.measurable_vimage_algebra[OF Y]
- have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by (rule measurable_comp)
- moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
- then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
- g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by (auto intro!: measurable_cong)
- ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by simp }
-
- assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
- "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by auto
- from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
- from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
- let "?g x" = "p x - n x"
- show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
- proof (intro bexI ballI)
- show "?g \<in> borel_measurable M'" using p n by auto
- fix x assume "x \<in> space M"
- then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
- using p n by auto
- then show "Z x = ?g (Y x)"
- by (auto split: split_max)
- qed
-qed
-
-end