src/HOL/Probability/Probability_Measure.thy
author hoelzl
Wed Apr 10 18:51:21 2013 +0200 (2013-04-10)
changeset 51683 baefa3b461c2
parent 51475 ebf9d4fd00ba
child 53015 a1119cf551e8
permissions -rw-r--r--
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
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(*  Title:      HOL/Probability/Probability_Measure.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
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header {*Probability measure*}
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theory Probability_Measure
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  imports Lebesgue_Measure Radon_Nikodym
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begin
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locale prob_space = finite_measure +
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  assumes emeasure_space_1: "emeasure M (space M) = 1"
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lemma prob_spaceI[Pure.intro!]:
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  assumes *: "emeasure M (space M) = 1"
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  shows "prob_space M"
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proof -
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  interpret finite_measure M
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  proof
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    show "emeasure M (space M) \<noteq> \<infinity>" using * by simp 
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  qed
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  show "prob_space M" by default fact
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qed
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abbreviation (in prob_space) "events \<equiv> sets M"
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abbreviation (in prob_space) "prob \<equiv> measure M"
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abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
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abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
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lemma (in prob_space) prob_space_distr:
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  assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
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proof (rule prob_spaceI)
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  have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
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  with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
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    by (auto simp: emeasure_distr emeasure_space_1)
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qed
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lemma (in prob_space) prob_space: "prob (space M) = 1"
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  using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
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lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
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  using bounded_measure[of A] by (simp add: prob_space)
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lemma (in prob_space) not_empty: "space M \<noteq> {}"
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  using prob_space by auto
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lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
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  using emeasure_space[of M X] by (simp add: emeasure_space_1)
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lemma (in prob_space) AE_I_eq_1:
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  assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
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  shows "AE x in M. P x"
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proof (rule AE_I)
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  show "emeasure M (space M - {x \<in> space M. P x}) = 0"
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    using assms emeasure_space_1 by (simp add: emeasure_compl)
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qed (insert assms, auto)
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lemma (in prob_space) prob_compl:
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  assumes A: "A \<in> events"
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  shows "prob (space M - A) = 1 - prob A"
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  using finite_measure_compl[OF A] by (simp add: prob_space)
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lemma (in prob_space) AE_in_set_eq_1:
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  assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
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proof
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  assume ae: "AE x in M. x \<in> A"
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  have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
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    using `A \<in> events`[THEN sets.sets_into_space] by auto
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  with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
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    by (simp add: emeasure_compl emeasure_space_1)
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  then show "prob A = 1"
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    using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
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next
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  assume prob: "prob A = 1"
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  show "AE x in M. x \<in> A"
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  proof (rule AE_I)
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    show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
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    show "emeasure M (space M - A) = 0"
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      using `A \<in> events` prob
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      by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
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    show "space M - A \<in> events"
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      using `A \<in> events` by auto
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  qed
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qed
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lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
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proof
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  assume "AE x in M. False"
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  then have "AE x in M. x \<in> {}" by simp
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  then show False
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    by (subst (asm) AE_in_set_eq_1) auto
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qed simp
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lemma (in prob_space) AE_prob_1:
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  assumes "prob A = 1" shows "AE x in M. x \<in> A"
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proof -
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  from `prob A = 1` have "A \<in> events"
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    by (metis measure_notin_sets zero_neq_one)
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  with AE_in_set_eq_1 assms show ?thesis by simp
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qed
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lemma (in prob_space) AE_const[simp]: "(AE x in M. P) \<longleftrightarrow> P"
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  by (cases P) (auto simp: AE_False)
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lemma (in prob_space) AE_contr:
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  assumes ae: "AE \<omega> in M. P \<omega>" "AE \<omega> in M. \<not> P \<omega>"
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  shows False
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proof -
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  from ae have "AE \<omega> in M. False" by eventually_elim auto
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  then show False by auto
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qed
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lemma (in prob_space) expectation_less:
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  assumes [simp]: "integrable M X"
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  assumes gt: "AE x in M. X x < b"
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  shows "expectation X < b"
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proof -
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  have "expectation X < expectation (\<lambda>x. b)"
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    using gt emeasure_space_1
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    by (intro integral_less_AE_space) auto
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  then show ?thesis using prob_space by simp
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qed
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lemma (in prob_space) expectation_greater:
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  assumes [simp]: "integrable M X"
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  assumes gt: "AE x in M. a < X x"
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  shows "a < expectation X"
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proof -
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  have "expectation (\<lambda>x. a) < expectation X"
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    using gt emeasure_space_1
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    by (intro integral_less_AE_space) auto
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  then show ?thesis using prob_space by simp
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qed
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lemma (in prob_space) jensens_inequality:
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  fixes a b :: real
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  assumes X: "integrable M X" "AE x in M. X x \<in> I"
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  assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
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  assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
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  shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
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proof -
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  let ?F = "\<lambda>x. Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
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  from X(2) AE_False have "I \<noteq> {}" by auto
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  from I have "open I" by auto
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  note I
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  moreover
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  { assume "I \<subseteq> {a <..}"
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    with X have "a < expectation X"
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      by (intro expectation_greater) auto }
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  moreover
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  { assume "I \<subseteq> {..< b}"
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    with X have "expectation X < b"
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      by (intro expectation_less) auto }
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  ultimately have "expectation X \<in> I"
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    by (elim disjE)  (auto simp: subset_eq)
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  moreover
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  { fix y assume y: "y \<in> I"
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    with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
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      by (auto intro!: cSup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) }
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  ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
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    by simp
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  also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
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  proof (rule cSup_least)
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    show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
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      using `I \<noteq> {}` by auto
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  next
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    fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
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    then guess x .. note x = this
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    have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
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      using prob_space by (simp add: X)
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    also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
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      using `x \<in> I` `open I` X(2)
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      apply (intro integral_mono_AE integral_add integral_cmult integral_diff
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                lebesgue_integral_const X q)
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      apply (elim eventually_elim1)
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      apply (intro convex_le_Inf_differential)
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      apply (auto simp: interior_open q)
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      done
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    finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
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  qed
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  finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
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qed
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subsection  {* Introduce binder for probability *}
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syntax
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  "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _'))")
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translations
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  "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
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definition
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  "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
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syntax
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  "_conditional_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _ \<bar>/ _'))")
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translations
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  "\<P>(x in M. P \<bar> Q)" => "CONST cond_prob M (\<lambda>x. P) (\<lambda>x. Q)"
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lemma (in prob_space) AE_E_prob:
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  assumes ae: "AE x in M. P x"
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  obtains S where "S \<subseteq> {x \<in> space M. P x}" "S \<in> events" "prob S = 1"
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proof -
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  from ae[THEN AE_E] guess N .
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  then show thesis
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    by (intro that[of "space M - N"])
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       (auto simp: prob_compl prob_space emeasure_eq_measure)
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qed
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lemma (in prob_space) prob_neg: "{x\<in>space M. P x} \<in> events \<Longrightarrow> \<P>(x in M. \<not> P x) = 1 - \<P>(x in M. P x)"
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  by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])
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lemma (in prob_space) prob_eq_AE:
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  "(AE x in M. P x \<longleftrightarrow> Q x) \<Longrightarrow> {x\<in>space M. P x} \<in> events \<Longrightarrow> {x\<in>space M. Q x} \<in> events \<Longrightarrow> \<P>(x in M. P x) = \<P>(x in M. Q x)"
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  by (rule finite_measure_eq_AE) auto
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lemma (in prob_space) prob_eq_0_AE:
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  assumes not: "AE x in M. \<not> P x" shows "\<P>(x in M. P x) = 0"
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proof cases
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  assume "{x\<in>space M. P x} \<in> events"
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  with not have "\<P>(x in M. P x) = \<P>(x in M. False)"
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    by (intro prob_eq_AE) auto
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  then show ?thesis by simp
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qed (simp add: measure_notin_sets)
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lemma (in prob_space) prob_Collect_eq_0:
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  "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 0 \<longleftrightarrow> (AE x in M. \<not> P x)"
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  using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"] by (simp add: emeasure_eq_measure)
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lemma (in prob_space) prob_Collect_eq_1:
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  "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 1 \<longleftrightarrow> (AE x in M. P x)"
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  using AE_in_set_eq_1[of "{x\<in>space M. P x}"] by simp
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lemma (in prob_space) prob_eq_0:
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  "A \<in> sets M \<Longrightarrow> prob A = 0 \<longleftrightarrow> (AE x in M. x \<notin> A)"
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  using AE_iff_measurable[OF _ refl, of M "\<lambda>x. x \<notin> A"]
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  by (auto simp add: emeasure_eq_measure Int_def[symmetric])
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lemma (in prob_space) prob_eq_1:
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  "A \<in> sets M \<Longrightarrow> prob A = 1 \<longleftrightarrow> (AE x in M. x \<in> A)"
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  using AE_in_set_eq_1[of A] by simp
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lemma (in prob_space) prob_sums:
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  assumes P: "\<And>n. {x\<in>space M. P n x} \<in> events"
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  assumes Q: "{x\<in>space M. Q x} \<in> events"
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  assumes ae: "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
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  shows "(\<lambda>n. \<P>(x in M. P n x)) sums \<P>(x in M. Q x)"
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proof -
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  from ae[THEN AE_E_prob] guess S . note S = this
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  then have disj: "disjoint_family (\<lambda>n. {x\<in>space M. P n x} \<inter> S)"
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    by (auto simp: disjoint_family_on_def)
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  from S have ae_S:
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    "AE x in M. x \<in> {x\<in>space M. Q x} \<longleftrightarrow> x \<in> (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
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    "\<And>n. AE x in M. x \<in> {x\<in>space M. P n x} \<longleftrightarrow> x \<in> {x\<in>space M. P n x} \<inter> S"
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    using ae by (auto dest!: AE_prob_1)
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  from ae_S have *:
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    "\<P>(x in M. Q x) = prob (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
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    using P Q S by (intro finite_measure_eq_AE) auto
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  from ae_S have **:
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    "\<And>n. \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
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    using P Q S by (intro finite_measure_eq_AE) auto
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  show ?thesis
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    unfolding * ** using S P disj
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    by (intro finite_measure_UNION) auto
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qed
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lemma (in prob_space) cond_prob_eq_AE:
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  assumes P: "AE x in M. Q x \<longrightarrow> P x \<longleftrightarrow> P' x" "{x\<in>space M. P x} \<in> events" "{x\<in>space M. P' x} \<in> events"
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  assumes Q: "AE x in M. Q x \<longleftrightarrow> Q' x" "{x\<in>space M. Q x} \<in> events" "{x\<in>space M. Q' x} \<in> events"
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   274
  shows "cond_prob M P Q = cond_prob M P' Q'"
hoelzl@50001
   275
  using P Q
immler@50244
   276
  by (auto simp: cond_prob_def intro!: arg_cong2[where f="op /"] prob_eq_AE sets.sets_Collect_conj)
hoelzl@50001
   277
hoelzl@50001
   278
hoelzl@40859
   279
lemma (in prob_space) joint_distribution_Times_le_fst:
hoelzl@47694
   280
  "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
hoelzl@47694
   281
    \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MX X) A"
hoelzl@47694
   282
  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
hoelzl@40859
   283
hoelzl@40859
   284
lemma (in prob_space) joint_distribution_Times_le_snd:
hoelzl@47694
   285
  "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
hoelzl@47694
   286
    \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MY Y) B"
hoelzl@47694
   287
  by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
hoelzl@40859
   288
hoelzl@45777
   289
locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
hoelzl@41689
   290
hoelzl@47694
   291
sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
hoelzl@45777
   292
proof
hoelzl@47694
   293
  show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
hoelzl@49776
   294
    by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
hoelzl@45777
   295
qed
hoelzl@40859
   296
hoelzl@47694
   297
locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@45777
   298
  fixes I :: "'i set"
hoelzl@45777
   299
  assumes prob_space: "\<And>i. prob_space (M i)"
hoelzl@42988
   300
hoelzl@45777
   301
sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
hoelzl@42988
   302
  by (rule prob_space)
hoelzl@42988
   303
hoelzl@45777
   304
locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I
hoelzl@42988
   305
hoelzl@42988
   306
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
hoelzl@45777
   307
proof
hoelzl@47694
   308
  show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
hoelzl@47694
   309
    by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
hoelzl@45777
   310
qed
hoelzl@42988
   311
hoelzl@42988
   312
lemma (in finite_product_prob_space) prob_times:
hoelzl@42988
   313
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
hoelzl@42988
   314
  shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
hoelzl@42988
   315
proof -
hoelzl@47694
   316
  have "ereal (measure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)) = emeasure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)"
hoelzl@47694
   317
    using X by (simp add: emeasure_eq_measure)
hoelzl@47694
   318
  also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
hoelzl@42988
   319
    using measure_times X by simp
hoelzl@47694
   320
  also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
hoelzl@47694
   321
    using X by (simp add: M.emeasure_eq_measure setprod_ereal)
hoelzl@42859
   322
  finally show ?thesis by simp
hoelzl@42859
   323
qed
hoelzl@42859
   324
hoelzl@47694
   325
section {* Distributions *}
hoelzl@42892
   326
hoelzl@47694
   327
definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and> 
hoelzl@47694
   328
  f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
hoelzl@36624
   329
hoelzl@47694
   330
lemma
hoelzl@50003
   331
  assumes "distributed M N X f"
hoelzl@50003
   332
  shows distributed_distr_eq_density: "distr M N X = density N f"
hoelzl@50003
   333
    and distributed_measurable: "X \<in> measurable M N"
hoelzl@50003
   334
    and distributed_borel_measurable: "f \<in> borel_measurable N"
hoelzl@50003
   335
    and distributed_AE: "(AE x in N. 0 \<le> f x)"
hoelzl@50003
   336
  using assms by (simp_all add: distributed_def)
hoelzl@50003
   337
hoelzl@50003
   338
lemma
hoelzl@50003
   339
  assumes D: "distributed M N X f"
hoelzl@50003
   340
  shows distributed_measurable'[measurable_dest]:
hoelzl@50003
   341
      "g \<in> measurable L M \<Longrightarrow> (\<lambda>x. X (g x)) \<in> measurable L N"
hoelzl@50003
   342
    and distributed_borel_measurable'[measurable_dest]:
hoelzl@50003
   343
      "h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
hoelzl@50003
   344
  using distributed_measurable[OF D] distributed_borel_measurable[OF D]
hoelzl@50003
   345
  by simp_all
hoelzl@39097
   346
hoelzl@47694
   347
lemma
hoelzl@47694
   348
  shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
hoelzl@47694
   349
    and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
hoelzl@47694
   350
  by (simp_all add: distributed_def borel_measurable_ereal_iff)
hoelzl@35977
   351
hoelzl@50003
   352
lemma
hoelzl@50003
   353
  assumes D: "distributed M N X (\<lambda>x. ereal (f x))"
hoelzl@50003
   354
  shows distributed_real_measurable'[measurable_dest]:
hoelzl@50003
   355
      "h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
hoelzl@50003
   356
  using distributed_real_measurable[OF D]
hoelzl@50003
   357
  by simp_all
hoelzl@50003
   358
hoelzl@50003
   359
lemma
hoelzl@50003
   360
  assumes D: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) f"
hoelzl@50003
   361
  shows joint_distributed_measurable1[measurable_dest]:
hoelzl@50003
   362
      "h1 \<in> measurable N M \<Longrightarrow> (\<lambda>x. X (h1 x)) \<in> measurable N S"
hoelzl@50003
   363
    and joint_distributed_measurable2[measurable_dest]:
hoelzl@50003
   364
      "h2 \<in> measurable N M \<Longrightarrow> (\<lambda>x. Y (h2 x)) \<in> measurable N T"
hoelzl@50003
   365
  using measurable_compose[OF distributed_measurable[OF D] measurable_fst]
hoelzl@50003
   366
  using measurable_compose[OF distributed_measurable[OF D] measurable_snd]
hoelzl@50003
   367
  by auto
hoelzl@50003
   368
hoelzl@47694
   369
lemma distributed_count_space:
hoelzl@47694
   370
  assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
hoelzl@47694
   371
  shows "P a = emeasure M (X -` {a} \<inter> space M)"
hoelzl@39097
   372
proof -
hoelzl@47694
   373
  have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
hoelzl@50003
   374
    using X a A by (simp add: emeasure_distr)
hoelzl@47694
   375
  also have "\<dots> = emeasure (density (count_space A) P) {a}"
hoelzl@47694
   376
    using X by (simp add: distributed_distr_eq_density)
hoelzl@47694
   377
  also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
hoelzl@47694
   378
    using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
hoelzl@47694
   379
  also have "\<dots> = P a"
hoelzl@47694
   380
    using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
hoelzl@47694
   381
  finally show ?thesis ..
hoelzl@39092
   382
qed
hoelzl@35977
   383
hoelzl@47694
   384
lemma distributed_cong_density:
hoelzl@47694
   385
  "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
hoelzl@47694
   386
    distributed M N X f \<longleftrightarrow> distributed M N X g"
hoelzl@47694
   387
  by (auto simp: distributed_def intro!: density_cong)
hoelzl@47694
   388
hoelzl@47694
   389
lemma subdensity:
hoelzl@47694
   390
  assumes T: "T \<in> measurable P Q"
hoelzl@47694
   391
  assumes f: "distributed M P X f"
hoelzl@47694
   392
  assumes g: "distributed M Q Y g"
hoelzl@47694
   393
  assumes Y: "Y = T \<circ> X"
hoelzl@47694
   394
  shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
hoelzl@47694
   395
proof -
hoelzl@47694
   396
  have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
hoelzl@47694
   397
    using g Y by (auto simp: null_sets_density_iff distributed_def)
hoelzl@47694
   398
  also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
hoelzl@47694
   399
    using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
hoelzl@47694
   400
  finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
hoelzl@47694
   401
    using T by (subst (asm) null_sets_distr_iff) auto
hoelzl@47694
   402
  also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
hoelzl@47694
   403
    using T by (auto dest: measurable_space)
hoelzl@47694
   404
  finally show ?thesis
hoelzl@47694
   405
    using f g by (auto simp add: null_sets_density_iff distributed_def)
hoelzl@35977
   406
qed
hoelzl@35977
   407
hoelzl@47694
   408
lemma subdensity_real:
hoelzl@47694
   409
  fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
hoelzl@47694
   410
  assumes T: "T \<in> measurable P Q"
hoelzl@47694
   411
  assumes f: "distributed M P X f"
hoelzl@47694
   412
  assumes g: "distributed M Q Y g"
hoelzl@47694
   413
  assumes Y: "Y = T \<circ> X"
hoelzl@47694
   414
  shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
hoelzl@47694
   415
  using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
hoelzl@47694
   416
hoelzl@49788
   417
lemma distributed_emeasure:
hoelzl@49788
   418
  "distributed M N X f \<Longrightarrow> A \<in> sets N \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>N)"
hoelzl@50003
   419
  by (auto simp: distributed_AE
hoelzl@49788
   420
                 distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)
hoelzl@49788
   421
hoelzl@49788
   422
lemma distributed_positive_integral:
hoelzl@49788
   423
  "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>\<^isup>+x. f x * g x \<partial>N) = (\<integral>\<^isup>+x. g (X x) \<partial>M)"
hoelzl@50003
   424
  by (auto simp: distributed_AE
hoelzl@49788
   425
                 distributed_distr_eq_density[symmetric] positive_integral_density[symmetric] positive_integral_distr)
hoelzl@49788
   426
hoelzl@47694
   427
lemma distributed_integral:
hoelzl@47694
   428
  "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
hoelzl@50003
   429
  by (auto simp: distributed_real_AE
hoelzl@47694
   430
                 distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
hoelzl@47694
   431
  
hoelzl@47694
   432
lemma distributed_transform_integral:
hoelzl@47694
   433
  assumes Px: "distributed M N X Px"
hoelzl@47694
   434
  assumes "distributed M P Y Py"
hoelzl@47694
   435
  assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
hoelzl@47694
   436
  shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
hoelzl@41689
   437
proof -
hoelzl@47694
   438
  have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
hoelzl@47694
   439
    by (rule distributed_integral) fact+
hoelzl@47694
   440
  also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
hoelzl@47694
   441
    using Y by simp
hoelzl@47694
   442
  also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
hoelzl@47694
   443
    using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
hoelzl@45777
   444
  finally show ?thesis .
hoelzl@39092
   445
qed
hoelzl@36624
   446
hoelzl@49788
   447
lemma (in prob_space) distributed_unique:
hoelzl@47694
   448
  assumes Px: "distributed M S X Px"
hoelzl@49788
   449
  assumes Py: "distributed M S X Py"
hoelzl@49788
   450
  shows "AE x in S. Px x = Py x"
hoelzl@49788
   451
proof -
hoelzl@49788
   452
  interpret X: prob_space "distr M S X"
hoelzl@50003
   453
    using Px by (intro prob_space_distr) simp
hoelzl@49788
   454
  have "sigma_finite_measure (distr M S X)" ..
hoelzl@49788
   455
  with sigma_finite_density_unique[of Px S Py ] Px Py
hoelzl@49788
   456
  show ?thesis
hoelzl@49788
   457
    by (auto simp: distributed_def)
hoelzl@49788
   458
qed
hoelzl@49788
   459
hoelzl@49788
   460
lemma (in prob_space) distributed_jointI:
hoelzl@49788
   461
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@50003
   462
  assumes X[measurable]: "X \<in> measurable M S" and Y[measurable]: "Y \<in> measurable M T"
hoelzl@50003
   463
  assumes [measurable]: "f \<in> borel_measurable (S \<Otimes>\<^isub>M T)" and f: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> f x"
hoelzl@49788
   464
  assumes eq: "\<And>A B. A \<in> sets S \<Longrightarrow> B \<in> sets T \<Longrightarrow> 
hoelzl@49788
   465
    emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. f (x, y) * indicator B y \<partial>T) * indicator A x \<partial>S)"
hoelzl@49788
   466
  shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) f"
hoelzl@49788
   467
  unfolding distributed_def
hoelzl@49788
   468
proof safe
hoelzl@49788
   469
  interpret S: sigma_finite_measure S by fact
hoelzl@49788
   470
  interpret T: sigma_finite_measure T by fact
hoelzl@49788
   471
  interpret ST: pair_sigma_finite S T by default
hoelzl@47694
   472
hoelzl@49788
   473
  from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('b \<times> 'c) set" .. note F = this
hoelzl@49788
   474
  let ?E = "{a \<times> b |a b. a \<in> sets S \<and> b \<in> sets T}"
hoelzl@49788
   475
  let ?P = "S \<Otimes>\<^isub>M T"
hoelzl@49788
   476
  show "distr M ?P (\<lambda>x. (X x, Y x)) = density ?P f" (is "?L = ?R")
hoelzl@49788
   477
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
hoelzl@49788
   478
    show "?E \<subseteq> Pow (space ?P)"
immler@50244
   479
      using sets.space_closed[of S] sets.space_closed[of T] by (auto simp: space_pair_measure)
hoelzl@49788
   480
    show "sets ?L = sigma_sets (space ?P) ?E"
hoelzl@49788
   481
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@49788
   482
    then show "sets ?R = sigma_sets (space ?P) ?E"
hoelzl@49788
   483
      by simp
hoelzl@49788
   484
  next
hoelzl@49788
   485
    interpret L: prob_space ?L
hoelzl@49788
   486
      by (rule prob_space_distr) (auto intro!: measurable_Pair)
hoelzl@49788
   487
    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?L (F i) \<noteq> \<infinity>"
hoelzl@49788
   488
      using F by (auto simp: space_pair_measure)
hoelzl@49788
   489
  next
hoelzl@49788
   490
    fix E assume "E \<in> ?E"
hoelzl@50003
   491
    then obtain A B where E[simp]: "E = A \<times> B"
hoelzl@50003
   492
      and A[measurable]: "A \<in> sets S" and B[measurable]: "B \<in> sets T" by auto
hoelzl@49788
   493
    have "emeasure ?L E = emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B}"
hoelzl@49788
   494
      by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
hoelzl@49788
   495
    also have "\<dots> = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
hoelzl@50003
   496
      using f by (auto simp add: eq positive_integral_multc intro!: positive_integral_cong)
hoelzl@49788
   497
    also have "\<dots> = emeasure ?R E"
hoelzl@50001
   498
      by (auto simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric]
hoelzl@49788
   499
               intro!: positive_integral_cong split: split_indicator)
hoelzl@49788
   500
    finally show "emeasure ?L E = emeasure ?R E" .
hoelzl@49788
   501
  qed
hoelzl@50003
   502
qed (auto simp: f)
hoelzl@49788
   503
hoelzl@49788
   504
lemma (in prob_space) distributed_swap:
hoelzl@49788
   505
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@49788
   506
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49788
   507
  shows "distributed M (T \<Otimes>\<^isub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
hoelzl@49788
   508
proof -
hoelzl@49788
   509
  interpret S: sigma_finite_measure S by fact
hoelzl@49788
   510
  interpret T: sigma_finite_measure T by fact
hoelzl@49788
   511
  interpret ST: pair_sigma_finite S T by default
hoelzl@49788
   512
  interpret TS: pair_sigma_finite T S by default
hoelzl@49788
   513
hoelzl@50003
   514
  note Pxy[measurable]
hoelzl@49788
   515
  show ?thesis 
hoelzl@49788
   516
    apply (subst TS.distr_pair_swap)
hoelzl@49788
   517
    unfolding distributed_def
hoelzl@49788
   518
  proof safe
hoelzl@49788
   519
    let ?D = "distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))"
hoelzl@49788
   520
    show 1: "(\<lambda>(x, y). Pxy (y, x)) \<in> borel_measurable ?D"
hoelzl@50003
   521
      by auto
hoelzl@49788
   522
    with Pxy
hoelzl@49788
   523
    show "AE x in distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x)). 0 \<le> (case x of (x, y) \<Rightarrow> Pxy (y, x))"
hoelzl@49788
   524
      by (subst AE_distr_iff)
hoelzl@49788
   525
         (auto dest!: distributed_AE
hoelzl@49788
   526
               simp: measurable_split_conv split_beta
hoelzl@51683
   527
               intro!: measurable_Pair)
hoelzl@49788
   528
    show 2: "random_variable (distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))) (\<lambda>x. (Y x, X x))"
hoelzl@50003
   529
      using Pxy by auto
hoelzl@49788
   530
    { fix A assume A: "A \<in> sets (T \<Otimes>\<^isub>M S)"
hoelzl@49788
   531
      let ?B = "(\<lambda>(x, y). (y, x)) -` A \<inter> space (S \<Otimes>\<^isub>M T)"
immler@50244
   532
      from sets.sets_into_space[OF A]
hoelzl@49788
   533
      have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
hoelzl@49788
   534
        emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
hoelzl@49788
   535
        by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49788
   536
      also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator ?B x \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@50003
   537
        using Pxy A by (intro distributed_emeasure) auto
hoelzl@49788
   538
      finally have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
hoelzl@49788
   539
        (\<integral>\<^isup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@49788
   540
        by (auto intro!: positive_integral_cong split: split_indicator) }
hoelzl@49788
   541
    note * = this
hoelzl@49788
   542
    show "distr M ?D (\<lambda>x. (Y x, X x)) = density ?D (\<lambda>(x, y). Pxy (y, x))"
hoelzl@49788
   543
      apply (intro measure_eqI)
hoelzl@49788
   544
      apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
hoelzl@49788
   545
      apply (subst positive_integral_distr)
hoelzl@50003
   546
      apply (auto intro!: * simp: comp_def split_beta)
hoelzl@49788
   547
      done
hoelzl@49788
   548
  qed
hoelzl@36624
   549
qed
hoelzl@36624
   550
hoelzl@47694
   551
lemma (in prob_space) distr_marginal1:
hoelzl@47694
   552
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@47694
   553
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49788
   554
  defines "Px \<equiv> \<lambda>x. (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
hoelzl@47694
   555
  shows "distributed M S X Px"
hoelzl@47694
   556
  unfolding distributed_def
hoelzl@47694
   557
proof safe
hoelzl@47694
   558
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
   559
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
   560
  interpret ST: pair_sigma_finite S T by default
hoelzl@47694
   561
hoelzl@50003
   562
  note Pxy[measurable]
hoelzl@50003
   563
  show X: "X \<in> measurable M S" by simp
hoelzl@47694
   564
hoelzl@50003
   565
  show borel: "Px \<in> borel_measurable S"
hoelzl@50003
   566
    by (auto intro!: T.positive_integral_fst_measurable simp: Px_def)
hoelzl@39097
   567
hoelzl@47694
   568
  interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@50003
   569
    by (intro prob_space_distr) simp
hoelzl@49788
   570
  have "(\<integral>\<^isup>+ x. max 0 (- Pxy x) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@47694
   571
    using Pxy
hoelzl@50003
   572
    by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_AE)
hoelzl@49788
   573
hoelzl@47694
   574
  show "distr M S X = density S Px"
hoelzl@47694
   575
  proof (rule measure_eqI)
hoelzl@47694
   576
    fix A assume A: "A \<in> sets (distr M S X)"
hoelzl@50003
   577
    with X measurable_space[of Y M T]
hoelzl@50003
   578
    have "emeasure (distr M S X) A = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (A \<times> space T)"
hoelzl@50003
   579
      by (auto simp add: emeasure_distr intro!: arg_cong[where f="emeasure M"])
hoelzl@47694
   580
    also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
hoelzl@47694
   581
      using Pxy by (simp add: distributed_def)
hoelzl@49788
   582
    also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
hoelzl@47694
   583
      using A borel Pxy
hoelzl@50003
   584
      by (simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric])
hoelzl@49788
   585
    also have "\<dots> = \<integral>\<^isup>+ x. Px x * indicator A x \<partial>S"
hoelzl@47694
   586
      apply (rule positive_integral_cong_AE)
hoelzl@49788
   587
      using Pxy[THEN distributed_AE, THEN ST.AE_pair] AE_space
hoelzl@47694
   588
    proof eventually_elim
hoelzl@49788
   589
      fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)"
hoelzl@47694
   590
      moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
hoelzl@47694
   591
        by (auto simp: indicator_def)
hoelzl@49788
   592
      ultimately have "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = (\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) * indicator A x"
hoelzl@50003
   593
        by (simp add: eq positive_integral_multc cong: positive_integral_cong)
hoelzl@49788
   594
      also have "(\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) = Px x"
hoelzl@49788
   595
        by (simp add: Px_def ereal_real positive_integral_positive)
hoelzl@49788
   596
      finally show "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = Px x * indicator A x" .
hoelzl@47694
   597
    qed
hoelzl@47694
   598
    finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
hoelzl@47694
   599
      using A borel Pxy by (simp add: emeasure_density)
hoelzl@47694
   600
  qed simp
hoelzl@47694
   601
  
hoelzl@49788
   602
  show "AE x in S. 0 \<le> Px x"
hoelzl@47694
   603
    by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
hoelzl@40859
   604
qed
hoelzl@40859
   605
hoelzl@49788
   606
lemma (in prob_space) distr_marginal2:
hoelzl@49788
   607
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49788
   608
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49788
   609
  shows "distributed M T Y (\<lambda>y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S))"
hoelzl@49788
   610
  using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp
hoelzl@49788
   611
hoelzl@49788
   612
lemma (in prob_space) distributed_marginal_eq_joint1:
hoelzl@49788
   613
  assumes T: "sigma_finite_measure T"
hoelzl@49788
   614
  assumes S: "sigma_finite_measure S"
hoelzl@49788
   615
  assumes Px: "distributed M S X Px"
hoelzl@49788
   616
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49788
   617
  shows "AE x in S. Px x = (\<integral>\<^isup>+y. Pxy (x, y) \<partial>T)"
hoelzl@49788
   618
  using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)
hoelzl@49788
   619
hoelzl@49788
   620
lemma (in prob_space) distributed_marginal_eq_joint2:
hoelzl@49788
   621
  assumes T: "sigma_finite_measure T"
hoelzl@49788
   622
  assumes S: "sigma_finite_measure S"
hoelzl@49788
   623
  assumes Py: "distributed M T Y Py"
hoelzl@49788
   624
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49788
   625
  shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
hoelzl@49788
   626
  using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)
hoelzl@49788
   627
hoelzl@49795
   628
lemma (in prob_space) distributed_joint_indep':
hoelzl@49795
   629
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@50003
   630
  assumes X[measurable]: "distributed M S X Px" and Y[measurable]: "distributed M T Y Py"
hoelzl@49795
   631
  assumes indep: "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@49795
   632
  shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
hoelzl@49795
   633
  unfolding distributed_def
hoelzl@49795
   634
proof safe
hoelzl@49795
   635
  interpret S: sigma_finite_measure S by fact
hoelzl@49795
   636
  interpret T: sigma_finite_measure T by fact
hoelzl@49795
   637
  interpret ST: pair_sigma_finite S T by default
hoelzl@49795
   638
hoelzl@49795
   639
  interpret X: prob_space "density S Px"
hoelzl@49795
   640
    unfolding distributed_distr_eq_density[OF X, symmetric]
hoelzl@50003
   641
    by (rule prob_space_distr) simp
hoelzl@49795
   642
  have sf_X: "sigma_finite_measure (density S Px)" ..
hoelzl@49795
   643
hoelzl@49795
   644
  interpret Y: prob_space "density T Py"
hoelzl@49795
   645
    unfolding distributed_distr_eq_density[OF Y, symmetric]
hoelzl@50003
   646
    by (rule prob_space_distr) simp
hoelzl@49795
   647
  have sf_Y: "sigma_finite_measure (density T Py)" ..
hoelzl@49795
   648
hoelzl@49795
   649
  show "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = density (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). Px x * Py y)"
hoelzl@49795
   650
    unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
hoelzl@49795
   651
    using distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49795
   652
    using distributed_borel_measurable[OF Y] distributed_AE[OF Y]
hoelzl@50003
   653
    by (rule pair_measure_density[OF _ _ _ _ T sf_Y])
hoelzl@49795
   654
hoelzl@50003
   655
  show "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" by auto
hoelzl@49795
   656
hoelzl@50003
   657
  show Pxy: "(\<lambda>(x, y). Px x * Py y) \<in> borel_measurable (S \<Otimes>\<^isub>M T)" by auto
hoelzl@49795
   658
hoelzl@49795
   659
  show "AE x in S \<Otimes>\<^isub>M T. 0 \<le> (case x of (x, y) \<Rightarrow> Px x * Py y)"
hoelzl@51683
   660
    apply (intro ST.AE_pair_measure borel_measurable_le Pxy borel_measurable_const)
hoelzl@49795
   661
    using distributed_AE[OF X]
hoelzl@49795
   662
    apply eventually_elim
hoelzl@49795
   663
    using distributed_AE[OF Y]
hoelzl@49795
   664
    apply eventually_elim
hoelzl@49795
   665
    apply auto
hoelzl@49795
   666
    done
hoelzl@49795
   667
qed
hoelzl@49795
   668
hoelzl@47694
   669
definition
hoelzl@47694
   670
  "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
hoelzl@47694
   671
    finite (X`space M)"
hoelzl@42902
   672
hoelzl@47694
   673
lemma simple_distributed:
hoelzl@47694
   674
  "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
hoelzl@47694
   675
  unfolding simple_distributed_def by auto
hoelzl@42902
   676
hoelzl@47694
   677
lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
hoelzl@47694
   678
  by (simp add: simple_distributed_def)
hoelzl@42902
   679
hoelzl@47694
   680
lemma (in prob_space) distributed_simple_function_superset:
hoelzl@47694
   681
  assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
hoelzl@47694
   682
  assumes A: "X`space M \<subseteq> A" "finite A"
hoelzl@47694
   683
  defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
hoelzl@47694
   684
  shows "distributed M S X P'"
hoelzl@47694
   685
  unfolding distributed_def
hoelzl@47694
   686
proof safe
hoelzl@47694
   687
  show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
hoelzl@47694
   688
  show "AE x in S. 0 \<le> ereal (P' x)"
hoelzl@47694
   689
    using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
hoelzl@47694
   690
  show "distr M S X = density S P'"
hoelzl@47694
   691
  proof (rule measure_eqI_finite)
hoelzl@47694
   692
    show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
hoelzl@47694
   693
      using A unfolding S_def by auto
hoelzl@47694
   694
    show "finite A" by fact
hoelzl@47694
   695
    fix a assume a: "a \<in> A"
hoelzl@47694
   696
    then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
hoelzl@47694
   697
    with A a X have "emeasure (distr M S X) {a} = P' a"
hoelzl@47694
   698
      by (subst emeasure_distr)
hoelzl@50002
   699
         (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure measurable_count_space_eq2
hoelzl@47694
   700
               intro!: arg_cong[where f=prob])
hoelzl@47694
   701
    also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
hoelzl@47694
   702
      using A X a
hoelzl@47694
   703
      by (subst positive_integral_cmult_indicator)
hoelzl@47694
   704
         (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
hoelzl@47694
   705
    also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
hoelzl@47694
   706
      by (auto simp: indicator_def intro!: positive_integral_cong)
hoelzl@47694
   707
    also have "\<dots> = emeasure (density S P') {a}"
hoelzl@47694
   708
      using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
hoelzl@47694
   709
    finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
hoelzl@47694
   710
  qed
hoelzl@47694
   711
  show "random_variable S X"
hoelzl@47694
   712
    using X(1) A by (auto simp: measurable_def simple_functionD S_def)
hoelzl@47694
   713
qed
hoelzl@42902
   714
hoelzl@47694
   715
lemma (in prob_space) simple_distributedI:
hoelzl@47694
   716
  assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
hoelzl@47694
   717
  shows "simple_distributed M X P"
hoelzl@47694
   718
  unfolding simple_distributed_def
hoelzl@47694
   719
proof
hoelzl@47694
   720
  have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
hoelzl@47694
   721
    (is "?A")
hoelzl@47694
   722
    using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
hoelzl@47694
   723
  also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
hoelzl@47694
   724
    by (rule distributed_cong_density) auto
hoelzl@47694
   725
  finally show "\<dots>" .
hoelzl@47694
   726
qed (rule simple_functionD[OF X(1)])
hoelzl@47694
   727
hoelzl@47694
   728
lemma simple_distributed_joint_finite:
hoelzl@47694
   729
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
hoelzl@47694
   730
  shows "finite (X ` space M)" "finite (Y ` space M)"
hoelzl@42902
   731
proof -
hoelzl@47694
   732
  have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
hoelzl@47694
   733
    using X by (auto simp: simple_distributed_def simple_functionD)
hoelzl@47694
   734
  then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
hoelzl@47694
   735
    by auto
hoelzl@47694
   736
  then show fin: "finite (X ` space M)" "finite (Y ` space M)"
hoelzl@47694
   737
    by (auto simp: image_image)
hoelzl@47694
   738
qed
hoelzl@47694
   739
hoelzl@47694
   740
lemma simple_distributed_joint2_finite:
hoelzl@47694
   741
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
hoelzl@47694
   742
  shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
hoelzl@47694
   743
proof -
hoelzl@47694
   744
  have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   745
    using X by (auto simp: simple_distributed_def simple_functionD)
hoelzl@47694
   746
  then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   747
    "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   748
    "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
hoelzl@47694
   749
    by auto
hoelzl@47694
   750
  then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
hoelzl@47694
   751
    by (auto simp: image_image)
hoelzl@42902
   752
qed
hoelzl@42902
   753
hoelzl@47694
   754
lemma simple_distributed_simple_function:
hoelzl@47694
   755
  "simple_distributed M X Px \<Longrightarrow> simple_function M X"
hoelzl@47694
   756
  unfolding simple_distributed_def distributed_def
hoelzl@50002
   757
  by (auto simp: simple_function_def measurable_count_space_eq2)
hoelzl@47694
   758
hoelzl@47694
   759
lemma simple_distributed_measure:
hoelzl@47694
   760
  "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
hoelzl@47694
   761
  using distributed_count_space[of M "X`space M" X P a, symmetric]
hoelzl@47694
   762
  by (auto simp: simple_distributed_def measure_def)
hoelzl@47694
   763
hoelzl@47694
   764
lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
hoelzl@47694
   765
  by (auto simp: simple_distributed_measure measure_nonneg)
hoelzl@42860
   766
hoelzl@47694
   767
lemma (in prob_space) simple_distributed_joint:
hoelzl@47694
   768
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
hoelzl@47694
   769
  defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
hoelzl@47694
   770
  defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
hoelzl@47694
   771
  shows "distributed M S (\<lambda>x. (X x, Y x)) P"
hoelzl@47694
   772
proof -
hoelzl@47694
   773
  from simple_distributed_joint_finite[OF X, simp]
hoelzl@47694
   774
  have S_eq: "S = count_space (X`space M \<times> Y`space M)"
hoelzl@47694
   775
    by (simp add: S_def pair_measure_count_space)
hoelzl@47694
   776
  show ?thesis
hoelzl@47694
   777
    unfolding S_eq P_def
hoelzl@47694
   778
  proof (rule distributed_simple_function_superset)
hoelzl@47694
   779
    show "simple_function M (\<lambda>x. (X x, Y x))"
hoelzl@47694
   780
      using X by (rule simple_distributed_simple_function)
hoelzl@47694
   781
    fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
hoelzl@47694
   782
    from simple_distributed_measure[OF X this]
hoelzl@47694
   783
    show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
hoelzl@47694
   784
  qed auto
hoelzl@47694
   785
qed
hoelzl@42860
   786
hoelzl@47694
   787
lemma (in prob_space) simple_distributed_joint2:
hoelzl@47694
   788
  assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
hoelzl@47694
   789
  defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
hoelzl@47694
   790
  defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
hoelzl@47694
   791
  shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
hoelzl@47694
   792
proof -
hoelzl@47694
   793
  from simple_distributed_joint2_finite[OF X, simp]
hoelzl@47694
   794
  have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
hoelzl@47694
   795
    by (simp add: S_def pair_measure_count_space)
hoelzl@47694
   796
  show ?thesis
hoelzl@47694
   797
    unfolding S_eq P_def
hoelzl@47694
   798
  proof (rule distributed_simple_function_superset)
hoelzl@47694
   799
    show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
hoelzl@47694
   800
      using X by (rule simple_distributed_simple_function)
hoelzl@47694
   801
    fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
hoelzl@47694
   802
    from simple_distributed_measure[OF X this]
hoelzl@47694
   803
    show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
hoelzl@47694
   804
  qed auto
hoelzl@47694
   805
qed
hoelzl@47694
   806
hoelzl@47694
   807
lemma (in prob_space) simple_distributed_setsum_space:
hoelzl@47694
   808
  assumes X: "simple_distributed M X f"
hoelzl@47694
   809
  shows "setsum f (X`space M) = 1"
hoelzl@47694
   810
proof -
hoelzl@47694
   811
  from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
hoelzl@47694
   812
    by (subst finite_measure_finite_Union)
hoelzl@47694
   813
       (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
hoelzl@47694
   814
             intro!: setsum_cong arg_cong[where f="prob"])
hoelzl@47694
   815
  also have "\<dots> = prob (space M)"
hoelzl@47694
   816
    by (auto intro!: arg_cong[where f=prob])
hoelzl@47694
   817
  finally show ?thesis
hoelzl@47694
   818
    using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
hoelzl@47694
   819
qed
hoelzl@42860
   820
hoelzl@47694
   821
lemma (in prob_space) distributed_marginal_eq_joint_simple:
hoelzl@47694
   822
  assumes Px: "simple_function M X"
hoelzl@47694
   823
  assumes Py: "simple_distributed M Y Py"
hoelzl@47694
   824
  assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
   825
  assumes y: "y \<in> Y`space M"
hoelzl@47694
   826
  shows "Py y = (\<Sum>x\<in>X`space M. if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
hoelzl@47694
   827
proof -
hoelzl@47694
   828
  note Px = simple_distributedI[OF Px refl]
hoelzl@47694
   829
  have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
hoelzl@47694
   830
    by (simp add: setsum_ereal[symmetric] zero_ereal_def)
hoelzl@49788
   831
  from distributed_marginal_eq_joint2[OF
hoelzl@49788
   832
    sigma_finite_measure_count_space_finite
hoelzl@49788
   833
    sigma_finite_measure_count_space_finite
hoelzl@49788
   834
    simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
hoelzl@47694
   835
    OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
hoelzl@49788
   836
    y
hoelzl@49788
   837
    Px[THEN simple_distributed_finite]
hoelzl@49788
   838
    Py[THEN simple_distributed_finite]
hoelzl@47694
   839
    Pxy[THEN simple_distributed, THEN distributed_real_AE]
hoelzl@47694
   840
  show ?thesis
hoelzl@47694
   841
    unfolding AE_count_space
hoelzl@47694
   842
    apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
hoelzl@47694
   843
    done
hoelzl@47694
   844
qed
hoelzl@42860
   845
hoelzl@50419
   846
lemma distributedI_real:
hoelzl@50419
   847
  fixes f :: "'a \<Rightarrow> real"
hoelzl@50419
   848
  assumes gen: "sets M1 = sigma_sets (space M1) E" and "Int_stable E"
hoelzl@50419
   849
    and A: "range A \<subseteq> E" "(\<Union>i::nat. A i) = space M1" "\<And>i. emeasure (distr M M1 X) (A i) \<noteq> \<infinity>"
hoelzl@50419
   850
    and X: "X \<in> measurable M M1"
hoelzl@50419
   851
    and f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
hoelzl@50419
   852
    and eq: "\<And>A. A \<in> E \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M1)"
hoelzl@50419
   853
  shows "distributed M M1 X f"
hoelzl@50419
   854
  unfolding distributed_def
hoelzl@50419
   855
proof (intro conjI)
hoelzl@50419
   856
  show "distr M M1 X = density M1 f"
hoelzl@50419
   857
  proof (rule measure_eqI_generator_eq[where A=A])
hoelzl@50419
   858
    { fix A assume A: "A \<in> E"
hoelzl@50419
   859
      then have "A \<in> sigma_sets (space M1) E" by auto
hoelzl@50419
   860
      then have "A \<in> sets M1"
hoelzl@50419
   861
        using gen by simp
hoelzl@50419
   862
      with f A eq[of A] X show "emeasure (distr M M1 X) A = emeasure (density M1 f) A"
hoelzl@50419
   863
        by (simp add: emeasure_distr emeasure_density borel_measurable_ereal
hoelzl@50419
   864
                      times_ereal.simps[symmetric] ereal_indicator
hoelzl@50419
   865
                 del: times_ereal.simps) }
hoelzl@50419
   866
    note eq_E = this
hoelzl@50419
   867
    show "Int_stable E" by fact
hoelzl@50419
   868
    { fix e assume "e \<in> E"
hoelzl@50419
   869
      then have "e \<in> sigma_sets (space M1) E" by auto
hoelzl@50419
   870
      then have "e \<in> sets M1" unfolding gen .
hoelzl@50419
   871
      then have "e \<subseteq> space M1" by (rule sets.sets_into_space) }
hoelzl@50419
   872
    then show "E \<subseteq> Pow (space M1)" by auto
hoelzl@50419
   873
    show "sets (distr M M1 X) = sigma_sets (space M1) E"
hoelzl@50419
   874
      "sets (density M1 (\<lambda>x. ereal (f x))) = sigma_sets (space M1) E"
hoelzl@50419
   875
      unfolding gen[symmetric] by auto
hoelzl@50419
   876
  qed fact+
hoelzl@50419
   877
qed (insert X f, auto)
hoelzl@50419
   878
hoelzl@50419
   879
lemma distributedI_borel_atMost:
hoelzl@50419
   880
  fixes f :: "real \<Rightarrow> real"
hoelzl@50419
   881
  assumes [measurable]: "X \<in> borel_measurable M"
hoelzl@50419
   882
    and [measurable]: "f \<in> borel_measurable borel" and f[simp]: "AE x in lborel. 0 \<le> f x"
hoelzl@50419
   883
    and g_eq: "\<And>a. (\<integral>\<^isup>+x. f x * indicator {..a} x \<partial>lborel)  = ereal (g a)"
hoelzl@50419
   884
    and M_eq: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = ereal (g a)"
hoelzl@50419
   885
  shows "distributed M lborel X f"
hoelzl@50419
   886
proof (rule distributedI_real)
hoelzl@50419
   887
  show "sets lborel = sigma_sets (space lborel) (range atMost)"
hoelzl@50419
   888
    by (simp add: borel_eq_atMost)
hoelzl@50419
   889
  show "Int_stable (range atMost :: real set set)"
hoelzl@50419
   890
    by (auto simp: Int_stable_def)
hoelzl@50419
   891
  have vimage_eq: "\<And>a. (X -` {..a} \<inter> space M) = {x\<in>space M. X x \<le> a}" by auto
hoelzl@50419
   892
  def A \<equiv> "\<lambda>i::nat. {.. real i}"
hoelzl@50419
   893
  then show "range A \<subseteq> range atMost" "(\<Union>i. A i) = space lborel"
hoelzl@50419
   894
    "\<And>i. emeasure (distr M lborel X) (A i) \<noteq> \<infinity>"
hoelzl@50419
   895
    by (auto simp: real_arch_simple emeasure_distr vimage_eq M_eq)
hoelzl@50419
   896
hoelzl@50419
   897
  fix A :: "real set" assume "A \<in> range atMost"
hoelzl@50419
   898
  then obtain a where A: "A = {..a}" by auto
hoelzl@50419
   899
  show "emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>lborel)"
hoelzl@50419
   900
    unfolding vimage_eq A M_eq g_eq ..
hoelzl@50419
   901
qed auto
hoelzl@50419
   902
hoelzl@50419
   903
lemma (in prob_space) uniform_distributed_params:
hoelzl@50419
   904
  assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)"
hoelzl@50419
   905
  shows "A \<in> sets MX" "measure MX A \<noteq> 0"
hoelzl@50419
   906
proof -
hoelzl@50419
   907
  interpret X: prob_space "distr M MX X"
hoelzl@50419
   908
    using distributed_measurable[OF X] by (rule prob_space_distr)
hoelzl@50419
   909
hoelzl@50419
   910
  show "measure MX A \<noteq> 0"
hoelzl@50419
   911
  proof
hoelzl@50419
   912
    assume "measure MX A = 0"
hoelzl@50419
   913
    with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
hoelzl@50419
   914
    show False
hoelzl@50419
   915
      by (simp add: emeasure_density zero_ereal_def[symmetric])
hoelzl@50419
   916
  qed
hoelzl@50419
   917
  with measure_notin_sets[of A MX] show "A \<in> sets MX"
hoelzl@50419
   918
    by blast
hoelzl@50419
   919
qed
hoelzl@50419
   920
hoelzl@47694
   921
lemma prob_space_uniform_measure:
hoelzl@47694
   922
  assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
hoelzl@47694
   923
  shows "prob_space (uniform_measure M A)"
hoelzl@47694
   924
proof
hoelzl@47694
   925
  show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
hoelzl@47694
   926
    using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
immler@50244
   927
    using sets.sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
hoelzl@47694
   928
    by (simp add: Int_absorb2 emeasure_nonneg)
hoelzl@47694
   929
qed
hoelzl@47694
   930
hoelzl@47694
   931
lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
hoelzl@47694
   932
  by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
hoelzl@42860
   933
hoelzl@35582
   934
end