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author | hoelzl |

Tue, 29 Mar 2011 14:27:42 +0200 | |

changeset 42148 | d596e7bb251f |

parent 42147 | 61d5d50ca74c |

child 42149 | 7e6f4ca198bb |

rename Probability_Space to Probability_Measure

--- 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