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src/HOL/Probability/Probability_Space.thy

author | hoelzl |

Fri, 27 Aug 2010 16:23:51 +0200 | |

changeset 39091 | 11314c196e11 |

parent 39090 | a2d38b8b693e |

child 39092 | 98de40859858 |

permissions | -rw-r--r-- |

factorizable measurable functions

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