src/HOL/Probability/Independent_Family.thy
author hoelzl
Wed Oct 10 12:12:21 2012 +0200 (2012-10-10)
changeset 49781 59b219ca3513
parent 49776 199d1d5bb17e
child 49784 5e5b2da42a69
permissions -rw-r--r--
simplified assumptions for kolmogorov_0_1_law
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(*  Title:      HOL/Probability/Independent_Family.thy
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    Author:     Johannes Hölzl, TU München
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*)
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header {* Independent families of events, event sets, and random variables *}
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theory Independent_Family
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  imports Probability_Measure Infinite_Product_Measure
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begin
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lemma INT_decseq_offset:
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  assumes "decseq F"
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  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
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proof safe
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  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
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  show "x \<in> F i"
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  proof cases
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    from x have "x \<in> F n" by auto
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    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
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      unfolding decseq_def by simp
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    finally show ?thesis .
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  qed (insert x, simp)
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qed auto
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definition (in prob_space)
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  "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and>
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    (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
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definition (in prob_space)
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  "indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV"
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definition (in prob_space)
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  "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and>
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    (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))))"
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definition (in prob_space)
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  "indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV"
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definition (in prob_space)
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  "indep_vars M' X I \<longleftrightarrow>
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    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
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    indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
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definition (in prob_space)
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  "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV"
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lemma (in prob_space) indep_sets_cong:
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  "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
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  by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
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lemma (in prob_space) indep_sets_singleton_iff_indep_events:
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  "indep_sets (\<lambda>i. {F i}) I \<longleftrightarrow> indep_events F I"
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  unfolding indep_sets_def indep_events_def
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  by (simp, intro conj_cong ball_cong all_cong imp_cong) (auto simp: Pi_iff)
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lemma (in prob_space) indep_events_finite_index_events:
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  "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
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  by (auto simp: indep_events_def)
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lemma (in prob_space) indep_sets_finite_index_sets:
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  "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
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proof (intro iffI allI impI)
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  assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
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  show "indep_sets F I" unfolding indep_sets_def
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  proof (intro conjI ballI allI impI)
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    fix i assume "i \<in> I"
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    with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
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      by (auto simp: indep_sets_def)
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  qed (insert *, auto simp: indep_sets_def)
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qed (auto simp: indep_sets_def)
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lemma (in prob_space) indep_sets_mono_index:
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  "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
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  unfolding indep_sets_def by auto
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lemma (in prob_space) indep_sets_mono_sets:
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  assumes indep: "indep_sets F I"
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  assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
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  shows "indep_sets G I"
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proof -
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  have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
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    using mono by auto
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  moreover have "\<And>A J. J \<subseteq> I \<Longrightarrow> A \<in> (\<Pi> j\<in>J. G j) \<Longrightarrow> A \<in> (\<Pi> j\<in>J. F j)"
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    using mono by (auto simp: Pi_iff)
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  ultimately show ?thesis
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    using indep by (auto simp: indep_sets_def)
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qed
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lemma (in prob_space) indep_sets_mono:
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  assumes indep: "indep_sets F I"
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  assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i"
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  shows "indep_sets G J"
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  apply (rule indep_sets_mono_sets)
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  apply (rule indep_sets_mono_index)
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  apply (fact +)
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  done
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lemma (in prob_space) indep_setsI:
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  assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
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    and "\<And>A J. J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> (\<forall>j\<in>J. A j \<in> F j) \<Longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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  shows "indep_sets F I"
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  using assms unfolding indep_sets_def by (auto simp: Pi_iff)
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lemma (in prob_space) indep_setsD:
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  assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
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  shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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  using assms unfolding indep_sets_def by auto
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lemma (in prob_space) indep_setI:
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  assumes ev: "A \<subseteq> events" "B \<subseteq> events"
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    and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
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  shows "indep_set A B"
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  unfolding indep_set_def
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proof (rule indep_setsI)
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  fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
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    and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
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  have "J \<in> Pow UNIV" by auto
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  with F `J \<noteq> {}` indep[of "F True" "F False"]
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  show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
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    unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
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qed (auto split: bool.split simp: ev)
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lemma (in prob_space) indep_setD:
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  assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"
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  shows "prob (a \<inter> b) = prob a * prob b"
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  using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev
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  by (simp add: ac_simps UNIV_bool)
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lemma (in prob_space) indep_var_eq:
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  "indep_var S X T Y \<longleftrightarrow>
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    (random_variable S X \<and> random_variable T Y) \<and>
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    indep_set
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      (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})
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      (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"
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  unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
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  by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext)
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     (auto split: bool.split)
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lemma (in prob_space)
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  assumes indep: "indep_set A B"
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  shows indep_setD_ev1: "A \<subseteq> events"
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    and indep_setD_ev2: "B \<subseteq> events"
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  using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
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lemma (in prob_space) indep_sets_dynkin:
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  assumes indep: "indep_sets F I"
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  shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I"
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    (is "indep_sets ?F I")
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proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
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  fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
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  with indep have "indep_sets F J"
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    by (subst (asm) indep_sets_finite_index_sets) auto
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  { fix J K assume "indep_sets F K"
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    let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i"
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    assume "finite J" "J \<subseteq> K"
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    then have "indep_sets (?G J) K"
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    proof induct
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      case (insert j J)
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      moreover def G \<equiv> "?G J"
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      ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
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        by (auto simp: indep_sets_def)
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      let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
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      { fix X assume X: "X \<in> events"
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        assume indep: "\<And>J A. J \<noteq> {} \<Longrightarrow> J \<subseteq> K \<Longrightarrow> finite J \<Longrightarrow> j \<notin> J \<Longrightarrow> (\<forall>i\<in>J. A i \<in> G i)
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          \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
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        have "indep_sets (G(j := {X})) K"
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        proof (rule indep_setsI)
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          fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
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            using G X by auto
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        next
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          fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
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          show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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          proof cases
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            assume "j \<in> J"
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            with J have "A j = X" by auto
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            show ?thesis
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            proof cases
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              assume "J = {j}" then show ?thesis by simp
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            next
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              assume "J \<noteq> {j}"
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              have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
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                using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
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              also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
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              proof (rule indep)
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                show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
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                  using J `J \<noteq> {j}` `j \<in> J` by auto
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                show "\<forall>i\<in>J - {j}. A i \<in> G i"
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                  using J by auto
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              qed
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              also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
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                using `A j = X` by simp
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              also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
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                unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
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                using `j \<in> J` by (simp add: insert_absorb)
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              finally show ?thesis .
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            qed
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          next
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            assume "j \<notin> J"
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            with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
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            with J show ?thesis
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              by (intro indep_setsD[OF G(1)]) auto
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          qed
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        qed }
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      note indep_sets_insert = this
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      have "dynkin_system (space M) ?D"
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      proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
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        show "indep_sets (G(j := {{}})) K"
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          by (rule indep_sets_insert) auto
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      next
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        fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
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        show "indep_sets (G(j := {space M - X})) K"
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        proof (rule indep_sets_insert)
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          fix J A assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "j \<notin> J" and A: "\<forall>i\<in>J. A i \<in> G i"
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          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
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            using G by auto
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          have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
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              prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
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            using A_sets sets_into_space[of _ M] X `J \<noteq> {}`
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            by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
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          also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
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            using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
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            by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)
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          finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
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              prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
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          moreover {
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            have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
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              using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
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            then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
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              using prob_space by simp }
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          moreover {
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            have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
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              using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
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            then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
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              using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }
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          ultimately have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = (prob (space M) - prob X) * (\<Prod>i\<in>J. prob (A i))"
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            by (simp add: field_simps)
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          also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
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            using X A by (simp add: finite_measure_compl)
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          finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
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        qed (insert X, auto)
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      next
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        fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
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        then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
hoelzl@42861
   244
        show "indep_sets (G(j := {\<Union>k. F k})) K"
hoelzl@42861
   245
        proof (rule indep_sets_insert)
hoelzl@42861
   246
          fix J A assume J: "j \<notin> J" "J \<noteq> {}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> G i"
hoelzl@42861
   247
          then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
hoelzl@42861
   248
            using G by auto
hoelzl@42861
   249
          have "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"
hoelzl@42861
   250
            using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
hoelzl@42861
   251
          moreover have "(\<lambda>k. prob (\<Inter>i\<in>insert j J. (A(j := F k)) i)) sums prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"
hoelzl@42861
   252
          proof (rule finite_measure_UNION)
hoelzl@42861
   253
            show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
hoelzl@42861
   254
              using disj by (rule disjoint_family_on_bisimulation) auto
hoelzl@42861
   255
            show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
hoelzl@42861
   256
              using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
hoelzl@42861
   257
          qed
hoelzl@42861
   258
          moreover { fix k
hoelzl@42861
   259
            from J A `j \<in> K` have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))"
hoelzl@42861
   260
              by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)
hoelzl@42861
   261
            also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
hoelzl@42861
   262
              using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
hoelzl@42861
   263
            finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
hoelzl@42861
   264
          ultimately have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)))"
hoelzl@42861
   265
            by simp
hoelzl@42861
   266
          moreover
hoelzl@42861
   267
          have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * prob (\<Inter>i\<in>J. A i))"
hoelzl@42861
   268
            using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
hoelzl@42861
   269
          then have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * (\<Prod>i\<in>J. prob (A i)))"
hoelzl@42861
   270
            using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
hoelzl@42861
   271
          ultimately
hoelzl@42861
   272
          show "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. F k) * (\<Prod>j\<in>J. prob (A j))"
hoelzl@42861
   273
            by (auto dest!: sums_unique)
hoelzl@42861
   274
        qed (insert F, auto)
hoelzl@42861
   275
      qed (insert sets_into_space, auto)
hoelzl@47694
   276
      then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
hoelzl@47694
   277
      proof (rule dynkin_system.dynkin_subset, safe)
hoelzl@42861
   278
        fix X assume "X \<in> G j"
hoelzl@42861
   279
        then show "X \<in> events" using G `j \<in> K` by auto
hoelzl@42861
   280
        from `indep_sets G K`
hoelzl@42861
   281
        show "indep_sets (G(j := {X})) K"
hoelzl@42861
   282
          by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
hoelzl@42861
   283
      qed
hoelzl@42861
   284
      have "indep_sets (G(j:=?D)) K"
hoelzl@42861
   285
      proof (rule indep_setsI)
hoelzl@42861
   286
        fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
hoelzl@42861
   287
          using G(2) by auto
hoelzl@42861
   288
      next
hoelzl@42861
   289
        fix A J assume J: "J\<noteq>{}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> (G(j := ?D)) i"
hoelzl@42861
   290
        show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
hoelzl@42861
   291
        proof cases
hoelzl@42861
   292
          assume "j \<in> J"
hoelzl@42861
   293
          with A have indep: "indep_sets (G(j := {A j})) K" by auto
hoelzl@42861
   294
          from J A show ?thesis
hoelzl@42861
   295
            by (intro indep_setsD[OF indep]) auto
hoelzl@42861
   296
        next
hoelzl@42861
   297
          assume "j \<notin> J"
hoelzl@42861
   298
          with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
hoelzl@42861
   299
          with J show ?thesis
hoelzl@42861
   300
            by (intro indep_setsD[OF G(1)]) auto
hoelzl@42861
   301
        qed
hoelzl@42861
   302
      qed
hoelzl@47694
   303
      then have "indep_sets (G(j := dynkin (space M) (G j))) K"
hoelzl@42861
   304
        by (rule indep_sets_mono_sets) (insert mono, auto)
hoelzl@42861
   305
      then show ?case
hoelzl@42861
   306
        by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
hoelzl@42861
   307
    qed (insert `indep_sets F K`, simp) }
hoelzl@42861
   308
  from this[OF `indep_sets F J` `finite J` subset_refl]
hoelzl@47694
   309
  show "indep_sets ?F J"
hoelzl@42861
   310
    by (rule indep_sets_mono_sets) auto
hoelzl@42861
   311
qed
hoelzl@42861
   312
hoelzl@42861
   313
lemma (in prob_space) indep_sets_sigma:
hoelzl@42861
   314
  assumes indep: "indep_sets F I"
hoelzl@47694
   315
  assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
hoelzl@47694
   316
  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
hoelzl@42861
   317
proof -
hoelzl@42861
   318
  from indep_sets_dynkin[OF indep]
hoelzl@42861
   319
  show ?thesis
hoelzl@42861
   320
  proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
hoelzl@42861
   321
    fix i assume "i \<in> I"
hoelzl@42861
   322
    with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
hoelzl@42861
   323
    with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
hoelzl@42861
   324
  qed
hoelzl@42861
   325
qed
hoelzl@42861
   326
hoelzl@42987
   327
lemma (in prob_space) indep_sets_sigma_sets_iff:
hoelzl@47694
   328
  assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
hoelzl@42987
   329
  shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
hoelzl@42987
   330
proof
hoelzl@42987
   331
  assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
hoelzl@47694
   332
    by (rule indep_sets_sigma) fact
hoelzl@42987
   333
next
hoelzl@42987
   334
  assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
hoelzl@42987
   335
    by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
hoelzl@42987
   336
qed
hoelzl@42987
   337
hoelzl@49781
   338
lemma (in prob_space)
hoelzl@49781
   339
  "indep_vars M' X I \<longleftrightarrow>
hoelzl@49781
   340
    (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
hoelzl@49781
   341
    indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
hoelzl@49781
   342
  unfolding indep_vars_def
hoelzl@49781
   343
  apply (rule conj_cong[OF refl])
hoelzl@49781
   344
  apply (rule indep_sets_sigma_sets_iff)
hoelzl@49781
   345
  apply (auto simp: Int_stable_def)
hoelzl@49781
   346
  apply (rule_tac x="A \<inter> Aa" in exI)
hoelzl@49781
   347
  apply auto
hoelzl@49781
   348
  done
hoelzl@49781
   349
hoelzl@42861
   350
lemma (in prob_space) indep_sets2_eq:
hoelzl@42981
   351
  "indep_set A B \<longleftrightarrow> A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)"
hoelzl@42981
   352
  unfolding indep_set_def
hoelzl@42861
   353
proof (intro iffI ballI conjI)
hoelzl@42861
   354
  assume indep: "indep_sets (bool_case A B) UNIV"
hoelzl@42861
   355
  { fix a b assume "a \<in> A" "b \<in> B"
hoelzl@42861
   356
    with indep_setsD[OF indep, of UNIV "bool_case a b"]
hoelzl@42861
   357
    show "prob (a \<inter> b) = prob a * prob b"
hoelzl@42861
   358
      unfolding UNIV_bool by (simp add: ac_simps) }
hoelzl@42861
   359
  from indep show "A \<subseteq> events" "B \<subseteq> events"
hoelzl@42861
   360
    unfolding indep_sets_def UNIV_bool by auto
hoelzl@42861
   361
next
hoelzl@42861
   362
  assume *: "A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)"
hoelzl@42861
   363
  show "indep_sets (bool_case A B) UNIV"
hoelzl@42861
   364
  proof (rule indep_setsI)
hoelzl@42861
   365
    fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
hoelzl@42861
   366
      using * by (auto split: bool.split)
hoelzl@42861
   367
  next
hoelzl@42861
   368
    fix J X assume "J \<noteq> {}" "J \<subseteq> UNIV" and X: "\<forall>j\<in>J. X j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
hoelzl@42861
   369
    then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
hoelzl@42861
   370
      by (auto simp: UNIV_bool)
hoelzl@42861
   371
    then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
hoelzl@42861
   372
      using X * by auto
hoelzl@42861
   373
  qed
hoelzl@42861
   374
qed
hoelzl@42861
   375
hoelzl@42981
   376
lemma (in prob_space) indep_set_sigma_sets:
hoelzl@42981
   377
  assumes "indep_set A B"
hoelzl@47694
   378
  assumes A: "Int_stable A" and B: "Int_stable B"
hoelzl@42981
   379
  shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
hoelzl@42861
   380
proof -
hoelzl@42861
   381
  have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
hoelzl@47694
   382
  proof (rule indep_sets_sigma)
hoelzl@42861
   383
    show "indep_sets (bool_case A B) UNIV"
hoelzl@42981
   384
      by (rule `indep_set A B`[unfolded indep_set_def])
hoelzl@47694
   385
    fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
hoelzl@42861
   386
      using A B by (cases i) auto
hoelzl@42861
   387
  qed
hoelzl@42861
   388
  then show ?thesis
hoelzl@42981
   389
    unfolding indep_set_def
hoelzl@42861
   390
    by (rule indep_sets_mono_sets) (auto split: bool.split)
hoelzl@42861
   391
qed
hoelzl@42861
   392
hoelzl@42981
   393
lemma (in prob_space) indep_sets_collect_sigma:
hoelzl@42981
   394
  fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
hoelzl@42981
   395
  assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
hoelzl@47694
   396
  assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"
hoelzl@42981
   397
  assumes disjoint: "disjoint_family_on I J"
hoelzl@42981
   398
  shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
hoelzl@42981
   399
proof -
wenzelm@46731
   400
  let ?E = "\<lambda>j. {\<Inter>k\<in>K. E' k| E' K. finite K \<and> K \<noteq> {} \<and> K \<subseteq> I j \<and> (\<forall>k\<in>K. E' k \<in> E k) }"
hoelzl@42981
   401
hoelzl@42983
   402
  from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
hoelzl@42981
   403
    unfolding indep_sets_def by auto
hoelzl@42981
   404
  { fix j
hoelzl@47694
   405
    let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"
hoelzl@42981
   406
    assume "j \<in> J"
hoelzl@47694
   407
    from E[OF this] interpret S: sigma_algebra "space M" ?S
hoelzl@47694
   408
      using sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
hoelzl@42981
   409
hoelzl@42981
   410
    have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
hoelzl@42981
   411
    proof (rule sigma_sets_eqI)
hoelzl@42981
   412
      fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
hoelzl@42981
   413
      then guess i ..
hoelzl@42981
   414
      then show "A \<in> sigma_sets (space M) (?E j)"
hoelzl@47694
   415
        by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
hoelzl@42981
   416
    next
hoelzl@42981
   417
      fix A assume "A \<in> ?E j"
hoelzl@42981
   418
      then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
hoelzl@42981
   419
        and A: "A = (\<Inter>k\<in>K. E' k)"
hoelzl@42981
   420
        by auto
hoelzl@47694
   421
      then have "A \<in> ?S" unfolding A
hoelzl@47694
   422
        by (safe intro!: S.finite_INT) auto
hoelzl@42981
   423
      then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
hoelzl@47694
   424
        by simp
hoelzl@42981
   425
    qed }
hoelzl@42981
   426
  moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
hoelzl@47694
   427
  proof (rule indep_sets_sigma)
hoelzl@42981
   428
    show "indep_sets ?E J"
hoelzl@42981
   429
    proof (intro indep_setsI)
hoelzl@42981
   430
      fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: finite_INT)
hoelzl@42981
   431
    next
hoelzl@42981
   432
      fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
hoelzl@42981
   433
        and "\<forall>j\<in>K. A j \<in> ?E j"
hoelzl@42981
   434
      then have "\<forall>j\<in>K. \<exists>E' L. A j = (\<Inter>l\<in>L. E' l) \<and> finite L \<and> L \<noteq> {} \<and> L \<subseteq> I j \<and> (\<forall>l\<in>L. E' l \<in> E l)"
hoelzl@42981
   435
        by simp
hoelzl@42981
   436
      from bchoice[OF this] guess E' ..
hoelzl@42981
   437
      from bchoice[OF this] obtain L
hoelzl@42981
   438
        where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
hoelzl@42981
   439
        and L: "\<And>j. j\<in>K \<Longrightarrow> finite (L j)" "\<And>j. j\<in>K \<Longrightarrow> L j \<noteq> {}" "\<And>j. j\<in>K \<Longrightarrow> L j \<subseteq> I j"
hoelzl@42981
   440
        and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
hoelzl@42981
   441
        by auto
hoelzl@42981
   442
hoelzl@42981
   443
      { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
hoelzl@42981
   444
        have "k = j"
hoelzl@42981
   445
        proof (rule ccontr)
hoelzl@42981
   446
          assume "k \<noteq> j"
hoelzl@42981
   447
          with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
hoelzl@42981
   448
            unfolding disjoint_family_on_def by auto
hoelzl@42981
   449
          with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
hoelzl@42981
   450
          show False using `l \<in> L k` `l \<in> L j` by auto
hoelzl@42981
   451
        qed }
hoelzl@42981
   452
      note L_inj = this
hoelzl@42981
   453
hoelzl@42981
   454
      def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
hoelzl@42981
   455
      { fix x j l assume *: "j \<in> K" "l \<in> L j"
hoelzl@42981
   456
        have "k l = j" unfolding k_def
hoelzl@42981
   457
        proof (rule some_equality)
hoelzl@42981
   458
          fix k assume "k \<in> K \<and> l \<in> L k"
hoelzl@42981
   459
          with * L_inj show "k = j" by auto
hoelzl@42981
   460
        qed (insert *, simp) }
hoelzl@42981
   461
      note k_simp[simp] = this
wenzelm@46731
   462
      let ?E' = "\<lambda>l. E' (k l) l"
hoelzl@42981
   463
      have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
hoelzl@42981
   464
        by (auto simp: A intro!: arg_cong[where f=prob])
hoelzl@42981
   465
      also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
hoelzl@42981
   466
        using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
hoelzl@42981
   467
      also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
hoelzl@42981
   468
        using K L L_inj by (subst setprod_UN_disjoint) auto
hoelzl@42981
   469
      also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
hoelzl@42981
   470
        using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast
hoelzl@42981
   471
      finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
hoelzl@42981
   472
    qed
hoelzl@42981
   473
  next
hoelzl@42981
   474
    fix j assume "j \<in> J"
hoelzl@47694
   475
    show "Int_stable (?E j)"
hoelzl@42981
   476
    proof (rule Int_stableI)
hoelzl@42981
   477
      fix a assume "a \<in> ?E j" then obtain Ka Ea
hoelzl@42981
   478
        where a: "a = (\<Inter>k\<in>Ka. Ea k)" "finite Ka" "Ka \<noteq> {}" "Ka \<subseteq> I j" "\<And>k. k\<in>Ka \<Longrightarrow> Ea k \<in> E k" by auto
hoelzl@42981
   479
      fix b assume "b \<in> ?E j" then obtain Kb Eb
hoelzl@42981
   480
        where b: "b = (\<Inter>k\<in>Kb. Eb k)" "finite Kb" "Kb \<noteq> {}" "Kb \<subseteq> I j" "\<And>k. k\<in>Kb \<Longrightarrow> Eb k \<in> E k" by auto
hoelzl@42981
   481
      let ?A = "\<lambda>k. (if k \<in> Ka \<inter> Kb then Ea k \<inter> Eb k else if k \<in> Kb then Eb k else if k \<in> Ka then Ea k else {})"
hoelzl@42981
   482
      have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
hoelzl@42981
   483
        by (simp add: a b set_eq_iff) auto
hoelzl@42981
   484
      with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
hoelzl@42981
   485
        by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
hoelzl@42981
   486
    qed
hoelzl@42981
   487
  qed
hoelzl@42981
   488
  ultimately show ?thesis
hoelzl@42981
   489
    by (simp cong: indep_sets_cong)
hoelzl@42981
   490
qed
hoelzl@42981
   491
hoelzl@49772
   492
definition (in prob_space) tail_events where
hoelzl@49772
   493
  "tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
hoelzl@42982
   494
hoelzl@49772
   495
lemma (in prob_space) tail_events_sets:
hoelzl@49772
   496
  assumes A: "\<And>i::nat. A i \<subseteq> events"
hoelzl@49772
   497
  shows "tail_events A \<subseteq> events"
hoelzl@49772
   498
proof
hoelzl@49772
   499
  fix X assume X: "X \<in> tail_events A"
hoelzl@42982
   500
  let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
hoelzl@49772
   501
  from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def)
hoelzl@42982
   502
  from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
hoelzl@42983
   503
  then show "X \<in> events"
hoelzl@42982
   504
    by induct (insert A, auto)
hoelzl@42982
   505
qed
hoelzl@42982
   506
hoelzl@49772
   507
lemma (in prob_space) sigma_algebra_tail_events:
hoelzl@47694
   508
  assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
hoelzl@49772
   509
  shows "sigma_algebra (space M) (tail_events A)"
hoelzl@49772
   510
  unfolding tail_events_def
hoelzl@42982
   511
proof (simp add: sigma_algebra_iff2, safe)
hoelzl@42982
   512
  let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
hoelzl@47694
   513
  interpret A: sigma_algebra "space M" "A i" for i by fact
hoelzl@43340
   514
  { fix X x assume "X \<in> ?A" "x \<in> X"
hoelzl@42982
   515
    then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
hoelzl@42982
   516
    from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
hoelzl@42982
   517
    then have "X \<subseteq> space M"
hoelzl@42982
   518
      by induct (insert A.sets_into_space, auto)
hoelzl@42982
   519
    with `x \<in> X` show "x \<in> space M" by auto }
hoelzl@42982
   520
  { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
hoelzl@42982
   521
    then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
hoelzl@42982
   522
      by (intro sigma_sets.Union) auto }
hoelzl@42982
   523
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
hoelzl@42982
   524
hoelzl@42982
   525
lemma (in prob_space) kolmogorov_0_1_law:
hoelzl@42982
   526
  fixes A :: "nat \<Rightarrow> 'a set set"
hoelzl@47694
   527
  assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
hoelzl@42982
   528
  assumes indep: "indep_sets A UNIV"
hoelzl@49772
   529
  and X: "X \<in> tail_events A"
hoelzl@42982
   530
  shows "prob X = 0 \<or> prob X = 1"
hoelzl@42982
   531
proof -
hoelzl@49781
   532
  have A: "\<And>i. A i \<subseteq> events"
hoelzl@49781
   533
    using indep unfolding indep_sets_def by simp
hoelzl@49781
   534
hoelzl@47694
   535
  let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"
hoelzl@47694
   536
  interpret A: sigma_algebra "space M" "A i" for i by fact
hoelzl@49772
   537
  interpret T: sigma_algebra "space M" "tail_events A"
hoelzl@49772
   538
    by (rule sigma_algebra_tail_events) fact
hoelzl@42982
   539
  have "X \<subseteq> space M" using T.space_closed X by auto
hoelzl@42982
   540
hoelzl@42983
   541
  have X_in: "X \<in> events"
hoelzl@49772
   542
    using tail_events_sets A X by auto
hoelzl@42982
   543
hoelzl@47694
   544
  interpret D: dynkin_system "space M" ?D
hoelzl@42982
   545
  proof (rule dynkin_systemI)
hoelzl@47694
   546
    fix D assume "D \<in> ?D" then show "D \<subseteq> space M"
hoelzl@42982
   547
      using sets_into_space by auto
hoelzl@42982
   548
  next
hoelzl@47694
   549
    show "space M \<in> ?D"
hoelzl@42982
   550
      using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
hoelzl@42982
   551
  next
hoelzl@47694
   552
    fix A assume A: "A \<in> ?D"
hoelzl@42982
   553
    have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
hoelzl@42982
   554
      using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
hoelzl@42982
   555
    also have "\<dots> = prob X - prob (X \<inter> A)"
hoelzl@42982
   556
      using X_in A by (intro finite_measure_Diff) auto
hoelzl@42982
   557
    also have "\<dots> = prob X * prob (space M) - prob X * prob A"
hoelzl@42982
   558
      using A prob_space by auto
hoelzl@42982
   559
    also have "\<dots> = prob X * prob (space M - A)"
hoelzl@42982
   560
      using X_in A sets_into_space
hoelzl@42982
   561
      by (subst finite_measure_Diff) (auto simp: field_simps)
hoelzl@47694
   562
    finally show "space M - A \<in> ?D"
hoelzl@42982
   563
      using A `X \<subseteq> space M` by auto
hoelzl@42982
   564
  next
hoelzl@47694
   565
    fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
hoelzl@42982
   566
    then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
hoelzl@42982
   567
      by auto
hoelzl@42982
   568
    have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
hoelzl@42982
   569
    proof (rule finite_measure_UNION)
hoelzl@42982
   570
      show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
hoelzl@42982
   571
        using F X_in by auto
hoelzl@42982
   572
      show "disjoint_family (\<lambda>i. X \<inter> F i)"
hoelzl@42982
   573
        using dis by (rule disjoint_family_on_bisimulation) auto
hoelzl@42982
   574
    qed
hoelzl@42982
   575
    with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
hoelzl@42982
   576
      by simp
hoelzl@42982
   577
    moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
huffman@44282
   578
      by (intro sums_mult finite_measure_UNION F dis)
hoelzl@42982
   579
    ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
hoelzl@42982
   580
      by (auto dest!: sums_unique)
hoelzl@47694
   581
    with F show "(\<Union>i. F i) \<in> ?D"
hoelzl@42982
   582
      by auto
hoelzl@42982
   583
  qed
hoelzl@42982
   584
hoelzl@42982
   585
  { fix n
hoelzl@42982
   586
    have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV"
hoelzl@42982
   587
    proof (rule indep_sets_collect_sigma)
hoelzl@42982
   588
      have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")
hoelzl@42982
   589
        by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
hoelzl@42982
   590
      with indep show "indep_sets A ?U" by simp
hoelzl@42982
   591
      show "disjoint_family (bool_case {..n} {Suc n..})"
hoelzl@42982
   592
        unfolding disjoint_family_on_def by (auto split: bool.split)
hoelzl@42982
   593
      fix m
hoelzl@47694
   594
      show "Int_stable (A m)"
hoelzl@42982
   595
        unfolding Int_stable_def using A.Int by auto
hoelzl@42982
   596
    qed
hoelzl@43340
   597
    also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) =
hoelzl@42982
   598
      bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
hoelzl@42982
   599
      by (auto intro!: ext split: bool.split)
hoelzl@42982
   600
    finally have indep: "indep_set (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
hoelzl@42982
   601
      unfolding indep_set_def by simp
hoelzl@42982
   602
hoelzl@47694
   603
    have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"
hoelzl@42982
   604
    proof (simp add: subset_eq, rule)
hoelzl@42982
   605
      fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
hoelzl@42982
   606
      have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
hoelzl@49772
   607
        using X unfolding tail_events_def by simp
hoelzl@42982
   608
      from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
hoelzl@42982
   609
      show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
hoelzl@42982
   610
        by (auto simp add: ac_simps)
hoelzl@42982
   611
    qed }
hoelzl@47694
   612
  then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
hoelzl@42982
   613
    by auto
hoelzl@42982
   614
hoelzl@49772
   615
  note `X \<in> tail_events A`
hoelzl@47694
   616
  also {
hoelzl@47694
   617
    have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
hoelzl@47694
   618
      by (intro sigma_sets_subseteq UN_mono) auto
hoelzl@49772
   619
   then have "tail_events A \<subseteq> sigma_sets (space M) ?A"
hoelzl@49772
   620
      unfolding tail_events_def by auto }
hoelzl@47694
   621
  also have "sigma_sets (space M) ?A = dynkin (space M) ?A"
hoelzl@42982
   622
  proof (rule sigma_eq_dynkin)
hoelzl@42982
   623
    { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
hoelzl@42982
   624
      then have "B \<subseteq> space M"
hoelzl@47694
   625
        by induct (insert A sets_into_space[of _ M], auto) }
hoelzl@47694
   626
    then show "?A \<subseteq> Pow (space M)" by auto
hoelzl@47694
   627
    show "Int_stable ?A"
hoelzl@42982
   628
    proof (rule Int_stableI)
hoelzl@42982
   629
      fix a assume "a \<in> ?A" then guess n .. note a = this
hoelzl@42982
   630
      fix b assume "b \<in> ?A" then guess m .. note b = this
hoelzl@47694
   631
      interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@47694
   632
        using A sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
hoelzl@42982
   633
      have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@42982
   634
        by (intro sigma_sets_subseteq UN_mono) auto
hoelzl@42982
   635
      with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
hoelzl@42982
   636
      moreover
hoelzl@42982
   637
      have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@42982
   638
        by (intro sigma_sets_subseteq UN_mono) auto
hoelzl@42982
   639
      with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
hoelzl@42982
   640
      ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
hoelzl@47694
   641
        using Amn.Int[of a b] by simp
hoelzl@42982
   642
      then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
hoelzl@42982
   643
    qed
hoelzl@42982
   644
  qed
hoelzl@47694
   645
  also have "dynkin (space M) ?A \<subseteq> ?D"
hoelzl@47694
   646
    using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)
hoelzl@47694
   647
  finally show ?thesis by auto
hoelzl@42982
   648
qed
hoelzl@42982
   649
hoelzl@42985
   650
lemma (in prob_space) borel_0_1_law:
hoelzl@42985
   651
  fixes F :: "nat \<Rightarrow> 'a set"
hoelzl@49781
   652
  assumes F2: "indep_events F UNIV"
hoelzl@42985
   653
  shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"
hoelzl@42985
   654
proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])
hoelzl@49781
   655
  have F1: "range F \<subseteq> events"
hoelzl@49781
   656
    using F2 by (simp add: indep_events_def subset_eq)
hoelzl@47694
   657
  { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
hoelzl@49781
   658
      using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets_into_space
hoelzl@47694
   659
      by auto }
hoelzl@42985
   660
  show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
hoelzl@47694
   661
  proof (rule indep_sets_sigma)
hoelzl@42985
   662
    show "indep_sets (\<lambda>i. {F i}) UNIV"
hoelzl@42985
   663
      unfolding indep_sets_singleton_iff_indep_events by fact
hoelzl@47694
   664
    fix i show "Int_stable {F i}"
hoelzl@42985
   665
      unfolding Int_stable_def by simp
hoelzl@42985
   666
  qed
wenzelm@46731
   667
  let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"
hoelzl@49772
   668
  show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})"
hoelzl@49772
   669
    unfolding tail_events_def
hoelzl@42985
   670
  proof
hoelzl@42985
   671
    fix j
hoelzl@47694
   672
    interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
hoelzl@49781
   673
      using order_trans[OF F1 space_closed]
hoelzl@47694
   674
      by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
hoelzl@42985
   675
    have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
hoelzl@42985
   676
      by (intro decseq_SucI INT_decseq_offset UN_mono) auto
hoelzl@47694
   677
    also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
hoelzl@49781
   678
      using order_trans[OF F1 space_closed]
hoelzl@42985
   679
      by (safe intro!: S.countable_INT S.countable_UN)
hoelzl@47694
   680
         (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
hoelzl@42985
   681
    finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
hoelzl@47694
   682
      by simp
hoelzl@42985
   683
  qed
hoelzl@42985
   684
qed
hoelzl@42985
   685
hoelzl@42987
   686
lemma (in prob_space) indep_sets_finite:
hoelzl@42987
   687
  assumes I: "I \<noteq> {}" "finite I"
hoelzl@42987
   688
    and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"
hoelzl@42987
   689
  shows "indep_sets F I \<longleftrightarrow> (\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j)))"
hoelzl@42987
   690
proof
hoelzl@42987
   691
  assume *: "indep_sets F I"
hoelzl@42987
   692
  from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
hoelzl@42987
   693
    by (intro indep_setsD[OF *] ballI) auto
hoelzl@42987
   694
next
hoelzl@42987
   695
  assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
hoelzl@42987
   696
  show "indep_sets F I"
hoelzl@42987
   697
  proof (rule indep_setsI[OF F(1)])
hoelzl@42987
   698
    fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
hoelzl@42987
   699
    assume A: "\<forall>j\<in>J. A j \<in> F j"
wenzelm@46731
   700
    let ?A = "\<lambda>j. if j \<in> J then A j else space M"
hoelzl@42987
   701
    have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"
hoelzl@42987
   702
      using subset_trans[OF F(1) space_closed] J A
hoelzl@42987
   703
      by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast
hoelzl@42987
   704
    also
hoelzl@42987
   705
    from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")
hoelzl@42987
   706
      by (auto split: split_if_asm)
hoelzl@42987
   707
    with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
hoelzl@42987
   708
      by auto
hoelzl@42987
   709
    also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
hoelzl@42987
   710
      unfolding if_distrib setprod.If_cases[OF `finite I`]
hoelzl@42987
   711
      using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1)
hoelzl@42987
   712
    finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
hoelzl@42987
   713
  qed
hoelzl@42987
   714
qed
hoelzl@42987
   715
hoelzl@42989
   716
lemma (in prob_space) indep_vars_finite:
hoelzl@42987
   717
  fixes I :: "'i set"
hoelzl@42987
   718
  assumes I: "I \<noteq> {}" "finite I"
hoelzl@47694
   719
    and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"
hoelzl@47694
   720
    and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
hoelzl@47694
   721
    and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"
hoelzl@47694
   722
    and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> E i" and closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M' i))"
hoelzl@47694
   723
  shows "indep_vars M' X I \<longleftrightarrow>
hoelzl@47694
   724
    (\<forall>A\<in>(\<Pi> i\<in>I. E i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))"
hoelzl@42987
   725
proof -
hoelzl@42987
   726
  from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
hoelzl@42987
   727
    unfolding measurable_def by simp
hoelzl@42987
   728
hoelzl@42987
   729
  { fix i assume "i\<in>I"
hoelzl@47694
   730
    from closed[OF `i \<in> I`]
hoelzl@47694
   731
    have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
hoelzl@47694
   732
      = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
hoelzl@47694
   733
      unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]
hoelzl@42987
   734
      by (subst sigma_sets_sigma_sets_eq) auto }
hoelzl@47694
   735
  note sigma_sets_X = this
hoelzl@42987
   736
hoelzl@42987
   737
  { fix i assume "i\<in>I"
hoelzl@47694
   738
    have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"
hoelzl@42987
   739
    proof (rule Int_stableI)
hoelzl@47694
   740
      fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
hoelzl@47694
   741
      then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto
hoelzl@42987
   742
      moreover
hoelzl@47694
   743
      fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
hoelzl@47694
   744
      then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
hoelzl@42987
   745
      moreover
hoelzl@42987
   746
      have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
hoelzl@42987
   747
      moreover note Int_stable[OF `i \<in> I`]
hoelzl@42987
   748
      ultimately
hoelzl@47694
   749
      show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
hoelzl@42987
   750
        by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
hoelzl@42987
   751
    qed }
hoelzl@47694
   752
  note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
hoelzl@43340
   753
hoelzl@42987
   754
  { fix i assume "i \<in> I"
hoelzl@47694
   755
    { fix A assume "A \<in> E i"
hoelzl@47694
   756
      with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto
hoelzl@42987
   757
      moreover
hoelzl@47694
   758
      from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto
hoelzl@42987
   759
      ultimately
hoelzl@42987
   760
      have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
hoelzl@42987
   761
    with X[OF `i\<in>I`] space[OF `i\<in>I`]
hoelzl@47694
   762
    have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
hoelzl@47694
   763
      "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
hoelzl@42987
   764
      by (auto intro!: exI[of _ "space (M' i)"]) }
hoelzl@47694
   765
  note indep_sets_finite_X = indep_sets_finite[OF I this]
hoelzl@43340
   766
hoelzl@47694
   767
  have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) =
hoelzl@47694
   768
    (\<forall>A\<in>\<Pi> i\<in>I. E i. prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"
hoelzl@42987
   769
    (is "?L = ?R")
hoelzl@42987
   770
  proof safe
hoelzl@47694
   771
    fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
hoelzl@42987
   772
    from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
hoelzl@42987
   773
    show "prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M))"
hoelzl@42987
   774
      by (auto simp add: Pi_iff)
hoelzl@42987
   775
  next
hoelzl@47694
   776
    fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"
hoelzl@47694
   777
    from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
hoelzl@42987
   778
    from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
hoelzl@47694
   779
      "B \<in> (\<Pi> i\<in>I. E i)" by auto
hoelzl@42987
   780
    from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
hoelzl@42987
   781
    show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
hoelzl@42987
   782
      by simp
hoelzl@42987
   783
  qed
hoelzl@42987
   784
  then show ?thesis using `I \<noteq> {}`
hoelzl@47694
   785
    by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
hoelzl@42988
   786
qed
hoelzl@42988
   787
hoelzl@42989
   788
lemma (in prob_space) indep_vars_compose:
hoelzl@42989
   789
  assumes "indep_vars M' X I"
hoelzl@47694
   790
  assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
hoelzl@42989
   791
  shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
hoelzl@42989
   792
  unfolding indep_vars_def
hoelzl@42988
   793
proof
hoelzl@42989
   794
  from rv `indep_vars M' X I`
hoelzl@42988
   795
  show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
hoelzl@47694
   796
    by (auto simp: indep_vars_def)
hoelzl@42988
   797
hoelzl@42988
   798
  have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
hoelzl@42989
   799
    using `indep_vars M' X I` by (simp add: indep_vars_def)
hoelzl@42988
   800
  then show "indep_sets (\<lambda>i. sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)}) I"
hoelzl@42988
   801
  proof (rule indep_sets_mono_sets)
hoelzl@42988
   802
    fix i assume "i \<in> I"
hoelzl@42989
   803
    with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)"
hoelzl@42989
   804
      unfolding indep_vars_def measurable_def by auto
hoelzl@42988
   805
    { fix A assume "A \<in> sets (N i)"
hoelzl@42988
   806
      then have "\<exists>B. (Y i \<circ> X i) -` A \<inter> space M = X i -` B \<inter> space M \<and> B \<in> sets (M' i)"
hoelzl@42988
   807
        by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])
hoelzl@42988
   808
           (auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) }
hoelzl@42988
   809
    then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>
hoelzl@42988
   810
      sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@42988
   811
      by (intro sigma_sets_subseteq) (auto simp: vimage_compose)
hoelzl@42988
   812
  qed
hoelzl@42988
   813
qed
hoelzl@42988
   814
hoelzl@47694
   815
lemma (in prob_space) indep_varsD_finite:
hoelzl@42989
   816
  assumes X: "indep_vars M' X I"
hoelzl@42988
   817
  assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"
hoelzl@42988
   818
  shows "prob (\<Inter>i\<in>I. X i -` A i \<inter> space M) = (\<Prod>i\<in>I. prob (X i -` A i \<inter> space M))"
hoelzl@42988
   819
proof (rule indep_setsD)
hoelzl@42988
   820
  show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
hoelzl@42989
   821
    using X by (auto simp: indep_vars_def)
hoelzl@42988
   822
  show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto
hoelzl@42988
   823
  show "\<forall>i\<in>I. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@47694
   824
    using I by auto
hoelzl@42988
   825
qed
hoelzl@42988
   826
hoelzl@47694
   827
lemma (in prob_space) indep_varsD:
hoelzl@47694
   828
  assumes X: "indep_vars M' X I"
hoelzl@47694
   829
  assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"
hoelzl@47694
   830
  shows "prob (\<Inter>i\<in>J. X i -` A i \<inter> space M) = (\<Prod>i\<in>J. prob (X i -` A i \<inter> space M))"
hoelzl@47694
   831
proof (rule indep_setsD)
hoelzl@47694
   832
  show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
hoelzl@47694
   833
    using X by (auto simp: indep_vars_def)
hoelzl@47694
   834
  show "\<forall>i\<in>J. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
hoelzl@47694
   835
    using I by auto
hoelzl@47694
   836
qed fact+
hoelzl@47694
   837
hoelzl@47694
   838
lemma prod_algebra_cong:
hoelzl@47694
   839
  assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
hoelzl@47694
   840
  shows "prod_algebra I M = prod_algebra J N"
hoelzl@47694
   841
proof -
hoelzl@47694
   842
  have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
hoelzl@47694
   843
    using sets_eq_imp_space_eq[OF sets] by auto
hoelzl@47694
   844
  with sets show ?thesis unfolding `I = J`
hoelzl@47694
   845
    by (intro antisym prod_algebra_mono) auto
hoelzl@47694
   846
qed
hoelzl@47694
   847
hoelzl@47694
   848
lemma space_in_prod_algebra:
hoelzl@47694
   849
  "(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
hoelzl@47694
   850
proof cases
hoelzl@47694
   851
  assume "I = {}" then show ?thesis
hoelzl@47694
   852
    by (auto simp add: prod_algebra_def image_iff prod_emb_def)
hoelzl@47694
   853
next
hoelzl@47694
   854
  assume "I \<noteq> {}"
hoelzl@47694
   855
  then obtain i where "i \<in> I" by auto
hoelzl@47694
   856
  then have "(\<Pi>\<^isub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i))"
hoelzl@47694
   857
    by (auto simp: prod_emb_def Pi_iff)
hoelzl@47694
   858
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   859
    using `i \<in> I` by (intro prod_algebraI) auto
hoelzl@47694
   860
  finally show ?thesis .
hoelzl@47694
   861
qed
hoelzl@47694
   862
hoelzl@47694
   863
lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
hoelzl@47694
   864
  fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
hoelzl@47694
   865
  assumes "I \<noteq> {}"
hoelzl@42988
   866
  assumes rv: "\<And>i. random_variable (M' i) (X i)"
hoelzl@42989
   867
  shows "indep_vars M' X I \<longleftrightarrow>
hoelzl@47694
   868
    distr M (\<Pi>\<^isub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i))"
hoelzl@42988
   869
proof -
hoelzl@47694
   870
  let ?P = "\<Pi>\<^isub>M i\<in>I. M' i"
hoelzl@47694
   871
  let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"
hoelzl@47694
   872
  let ?D = "distr M ?P ?X"
hoelzl@47694
   873
  have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
hoelzl@47694
   874
  interpret D: prob_space ?D by (intro prob_space_distr X)
hoelzl@42988
   875
hoelzl@47694
   876
  let ?D' = "\<lambda>i. distr M (M' i) (X i)"
hoelzl@47694
   877
  let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)"
hoelzl@47694
   878
  interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
hoelzl@47694
   879
  interpret P: product_prob_space ?D' I ..
hoelzl@47694
   880
    
hoelzl@42988
   881
  show ?thesis
hoelzl@47694
   882
  proof
hoelzl@42989
   883
    assume "indep_vars M' X I"
hoelzl@47694
   884
    show "?D = ?P'"
hoelzl@47694
   885
    proof (rule measure_eqI_generator_eq)
hoelzl@47694
   886
      show "Int_stable (prod_algebra I M')"
hoelzl@47694
   887
        by (rule Int_stable_prod_algebra)
hoelzl@47694
   888
      show "prod_algebra I M' \<subseteq> Pow (space ?P)"
hoelzl@47694
   889
        using prod_algebra_sets_into_space by (simp add: space_PiM)
hoelzl@47694
   890
      show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
hoelzl@47694
   891
        by (simp add: sets_PiM space_PiM)
hoelzl@47694
   892
      show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
hoelzl@47694
   893
        by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
hoelzl@47694
   894
      let ?A = "\<lambda>i. \<Pi>\<^isub>E i\<in>I. space (M' i)"
hoelzl@47694
   895
      show "range ?A \<subseteq> prod_algebra I M'" "incseq ?A" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')"
hoelzl@47694
   896
        by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
hoelzl@47694
   897
      { fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }
hoelzl@47694
   898
    next
hoelzl@47694
   899
      fix E assume E: "E \<in> prod_algebra I M'"
hoelzl@47694
   900
      from prod_algebraE[OF E] guess J Y . note J = this
hoelzl@43340
   901
hoelzl@47694
   902
      from E have "E \<in> sets ?P" by (auto simp: sets_PiM)
hoelzl@47694
   903
      then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
hoelzl@47694
   904
        by (simp add: emeasure_distr X)
hoelzl@47694
   905
      also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
hoelzl@47694
   906
        using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm)
hoelzl@47694
   907
      also have "emeasure M (\<Inter>i\<in>J. X i -` Y i \<inter> space M) = (\<Prod> i\<in>J. emeasure M (X i -` Y i \<inter> space M))"
hoelzl@47694
   908
        using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]
hoelzl@47694
   909
        by (auto simp: emeasure_eq_measure setprod_ereal)
hoelzl@47694
   910
      also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
hoelzl@47694
   911
        using rv J by (simp add: emeasure_distr)
hoelzl@47694
   912
      also have "\<dots> = emeasure ?P' E"
hoelzl@47694
   913
        using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
hoelzl@47694
   914
      finally show "emeasure ?D E = emeasure ?P' E" .
hoelzl@42988
   915
    qed
hoelzl@42988
   916
  next
hoelzl@47694
   917
    assume "?D = ?P'"
hoelzl@47694
   918
    show "indep_vars M' X I" unfolding indep_vars_def
hoelzl@47694
   919
    proof (intro conjI indep_setsI ballI rv)
hoelzl@47694
   920
      fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
hoelzl@47694
   921
        by (auto intro!: sigma_sets_subset measurable_sets rv)
hoelzl@42988
   922
    next
hoelzl@47694
   923
      fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
hoelzl@47694
   924
      assume Y': "\<forall>j\<in>J. Y' j \<in> sigma_sets (space M) {X j -` A \<inter> space M |A. A \<in> sets (M' j)}"
hoelzl@47694
   925
      have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
hoelzl@42988
   926
      proof
hoelzl@47694
   927
        fix j assume "j \<in> J"
hoelzl@47694
   928
        from Y'[rule_format, OF this] rv[of j]
hoelzl@47694
   929
        show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
hoelzl@47694
   930
          by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
hoelzl@47694
   931
             (auto dest: measurable_space simp: sigma_sets_eq)
hoelzl@42988
   932
      qed
hoelzl@47694
   933
      from bchoice[OF this] obtain Y where
hoelzl@47694
   934
        Y: "\<And>j. j \<in> J \<Longrightarrow> Y' j = X j -` Y j \<inter> space M" "\<And>j. j \<in> J \<Longrightarrow> Y j \<in> sets (M' j)" by auto
hoelzl@47694
   935
      let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)"
hoelzl@47694
   936
      from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
hoelzl@47694
   937
        using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm)
hoelzl@47694
   938
      then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
hoelzl@47694
   939
        by simp
hoelzl@47694
   940
      also have "\<dots> = emeasure ?D ?E"
hoelzl@47694
   941
        using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
hoelzl@47694
   942
      also have "\<dots> = emeasure ?P' ?E"
hoelzl@47694
   943
        using `?D = ?P'` by simp
hoelzl@47694
   944
      also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
hoelzl@47694
   945
        using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
hoelzl@47694
   946
      also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
hoelzl@47694
   947
        using rv J Y by (simp add: emeasure_distr)
hoelzl@47694
   948
      finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .
hoelzl@47694
   949
      then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"
hoelzl@47694
   950
        by (auto simp: emeasure_eq_measure setprod_ereal)
hoelzl@42988
   951
    qed
hoelzl@42988
   952
  qed
hoelzl@42987
   953
qed
hoelzl@42987
   954
hoelzl@42989
   955
lemma (in prob_space) indep_varD:
hoelzl@42989
   956
  assumes indep: "indep_var Ma A Mb B"
hoelzl@42989
   957
  assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb"
hoelzl@42989
   958
  shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
hoelzl@42989
   959
    prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
hoelzl@42989
   960
proof -
hoelzl@42989
   961
  have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
hoelzl@42989
   962
    prob (\<Inter>i\<in>UNIV. (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"
hoelzl@42989
   963
    by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
hoelzl@42989
   964
  also have "\<dots> = (\<Prod>i\<in>UNIV. prob (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"
hoelzl@42989
   965
    using indep unfolding indep_var_def
hoelzl@42989
   966
    by (rule indep_varsD) (auto split: bool.split intro: sets)
hoelzl@42989
   967
  also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
hoelzl@42989
   968
    unfolding UNIV_bool by simp
hoelzl@42989
   969
  finally show ?thesis .
hoelzl@42989
   970
qed
hoelzl@42989
   971
hoelzl@43340
   972
lemma (in prob_space)
hoelzl@43340
   973
  assumes "indep_var S X T Y"
hoelzl@43340
   974
  shows indep_var_rv1: "random_variable S X"
hoelzl@43340
   975
    and indep_var_rv2: "random_variable T Y"
hoelzl@43340
   976
proof -
hoelzl@43340
   977
  have "\<forall>i\<in>UNIV. random_variable (bool_case S T i) (bool_case X Y i)"
hoelzl@43340
   978
    using assms unfolding indep_var_def indep_vars_def by auto
hoelzl@43340
   979
  then show "random_variable S X" "random_variable T Y"
hoelzl@43340
   980
    unfolding UNIV_bool by auto
hoelzl@43340
   981
qed
hoelzl@43340
   982
hoelzl@47694
   983
lemma measurable_bool_case[simp, intro]:
hoelzl@47694
   984
  "(\<lambda>(x, y). bool_case x y) \<in> measurable (M \<Otimes>\<^isub>M N) (Pi\<^isub>M UNIV (bool_case M N))"
hoelzl@47694
   985
    (is "?f \<in> measurable ?B ?P")
hoelzl@47694
   986
proof (rule measurable_PiM_single)
hoelzl@47694
   987
  show "?f \<in> space ?B \<rightarrow> (\<Pi>\<^isub>E i\<in>UNIV. space (bool_case M N i))"
hoelzl@47694
   988
    by (auto simp: space_pair_measure extensional_def split: bool.split)
hoelzl@47694
   989
  fix i A assume "A \<in> sets (case i of True \<Rightarrow> M | False \<Rightarrow> N)"
hoelzl@47694
   990
  moreover then have "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A}
hoelzl@47694
   991
    = (case i of True \<Rightarrow> A \<times> space N | False \<Rightarrow> space M \<times> A)" 
hoelzl@47694
   992
    by (auto simp: space_pair_measure split: bool.split dest!: sets_into_space)
hoelzl@47694
   993
  ultimately show "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A} \<in> sets ?B"
hoelzl@47694
   994
    by (auto split: bool.split)
hoelzl@47694
   995
qed
hoelzl@47694
   996
hoelzl@47694
   997
lemma borel_measurable_indicator':
hoelzl@47694
   998
  "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M"
hoelzl@47694
   999
  using measurable_comp[OF _ borel_measurable_indicator, of f M N A] by (auto simp add: comp_def)
hoelzl@47694
  1000
hoelzl@47694
  1001
lemma (in product_sigma_finite) distr_component:
hoelzl@47694
  1002
  "distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P")
hoelzl@47694
  1003
proof (intro measure_eqI[symmetric])
hoelzl@47694
  1004
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
  1005
hoelzl@47694
  1006
  have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
hoelzl@47694
  1007
    by (auto simp: extensional_def restrict_def)
hoelzl@47694
  1008
hoelzl@47694
  1009
  fix A assume A: "A \<in> sets ?P"
hoelzl@47694
  1010
  then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)" 
hoelzl@47694
  1011
    by simp
hoelzl@47694
  1012
  also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) x \<partial>M i)" 
hoelzl@47694
  1013
    apply (subst product_positive_integral_singleton[symmetric])
hoelzl@47694
  1014
    apply (force intro!: measurable_restrict measurable_sets A)
hoelzl@47694
  1015
    apply (auto intro!: positive_integral_cong simp: space_PiM indicator_def simp: eq)
hoelzl@47694
  1016
    done
hoelzl@47694
  1017
  also have "\<dots> = emeasure (M i) ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i))"
hoelzl@47694
  1018
    by (force intro!: measurable_restrict measurable_sets A positive_integral_indicator)
hoelzl@47694
  1019
  also have "\<dots> = emeasure ?D A"
hoelzl@47694
  1020
    using A by (auto intro!: emeasure_distr[symmetric] measurable_restrict) 
hoelzl@47694
  1021
  finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" .
hoelzl@47694
  1022
qed simp
hoelzl@43340
  1023
hoelzl@47694
  1024
lemma pair_measure_eqI:
hoelzl@47694
  1025
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
hoelzl@47694
  1026
  assumes sets: "sets (M1 \<Otimes>\<^isub>M M2) = sets M"
hoelzl@47694
  1027
  assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)"
hoelzl@47694
  1028
  shows "M1 \<Otimes>\<^isub>M M2 = M"
hoelzl@47694
  1029
proof -
hoelzl@47694
  1030
  interpret M1: sigma_finite_measure M1 by fact
hoelzl@47694
  1031
  interpret M2: sigma_finite_measure M2 by fact
hoelzl@47694
  1032
  interpret pair_sigma_finite M1 M2 by default
hoelzl@47694
  1033
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
hoelzl@47694
  1034
  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
hoelzl@47694
  1035
  let ?P = "M1 \<Otimes>\<^isub>M M2"
hoelzl@47694
  1036
  show ?thesis
hoelzl@47694
  1037
  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
hoelzl@47694
  1038
    show "?E \<subseteq> Pow (space ?P)"
hoelzl@47694
  1039
      using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
hoelzl@47694
  1040
    show "sets ?P = sigma_sets (space ?P) ?E"
hoelzl@47694
  1041
      by (simp add: sets_pair_measure space_pair_measure)
hoelzl@47694
  1042
    then show "sets M = sigma_sets (space ?P) ?E"
hoelzl@47694
  1043
      using sets[symmetric] by simp
hoelzl@47694
  1044
  next
hoelzl@47694
  1045
    show "range F \<subseteq> ?E" "incseq F" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
hoelzl@47694
  1046
      using F by (auto simp: space_pair_measure)
hoelzl@47694
  1047
  next
hoelzl@47694
  1048
    fix X assume "X \<in> ?E"
hoelzl@47694
  1049
    then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto
hoelzl@47694
  1050
    then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
hoelzl@49776
  1051
       by (simp add: M2.emeasure_pair_measure_Times)
hoelzl@47694
  1052
    also have "\<dots> = emeasure M (A \<times> B)"
hoelzl@47694
  1053
      using A B emeasure by auto
hoelzl@47694
  1054
    finally show "emeasure ?P X = emeasure M X"
hoelzl@47694
  1055
      by simp
hoelzl@47694
  1056
  qed
hoelzl@47694
  1057
qed
hoelzl@43340
  1058
hoelzl@47694
  1059
lemma pair_measure_eq_distr_PiM:
hoelzl@47694
  1060
  fixes M1 :: "'a measure" and M2 :: "'a measure"
hoelzl@47694
  1061
  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
hoelzl@47694
  1062
  shows "(M1 \<Otimes>\<^isub>M M2) = distr (Pi\<^isub>M UNIV (bool_case M1 M2)) (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. (x True, x False))"
hoelzl@47694
  1063
    (is "?P = ?D")
hoelzl@47694
  1064
proof (rule pair_measure_eqI[OF assms])
hoelzl@47694
  1065
  interpret B: product_sigma_finite "bool_case M1 M2"
hoelzl@47694
  1066
    unfolding product_sigma_finite_def using assms by (auto split: bool.split)
hoelzl@47694
  1067
  let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)"
hoelzl@43340
  1068
hoelzl@47694
  1069
  have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)"
hoelzl@47694
  1070
    by auto
hoelzl@47694
  1071
  fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
hoelzl@47694
  1072
  have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))"
hoelzl@47694
  1073
    by (simp add: UNIV_bool ac_simps)
hoelzl@47694
  1074
  also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))"
hoelzl@47694
  1075
    using A B by (subst B.emeasure_PiM) (auto split: bool.split)
hoelzl@47694
  1076
  also have "Pi\<^isub>E UNIV (bool_case A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
hoelzl@47694
  1077
    using A[THEN sets_into_space] B[THEN sets_into_space]
hoelzl@47694
  1078
    by (auto simp: Pi_iff all_bool_eq space_PiM split: bool.split)
hoelzl@47694
  1079
  finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
hoelzl@47694
  1080
    using A B
hoelzl@47694
  1081
      measurable_component_singleton[of True UNIV "bool_case M1 M2"]
hoelzl@47694
  1082
      measurable_component_singleton[of False UNIV "bool_case M1 M2"]
hoelzl@47694
  1083
    by (subst emeasure_distr) (auto simp: measurable_pair_iff)
hoelzl@47694
  1084
qed simp
hoelzl@43340
  1085
hoelzl@47694
  1086
lemma measurable_Pair:
hoelzl@47694
  1087
  assumes rvs: "X \<in> measurable M S" "Y \<in> measurable M T"
hoelzl@47694
  1088
  shows "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
hoelzl@47694
  1089
proof -
hoelzl@47694
  1090
  have [simp]: "fst \<circ> (\<lambda>x. (X x, Y x)) = (\<lambda>x. X x)" "snd \<circ> (\<lambda>x. (X x, Y x)) = (\<lambda>x. Y x)"
hoelzl@47694
  1091
    by auto
hoelzl@47694
  1092
  show " (\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
hoelzl@47694
  1093
    by (auto simp: measurable_pair_iff rvs)
hoelzl@47694
  1094
qed
hoelzl@47694
  1095
hoelzl@47694
  1096
lemma (in prob_space) indep_var_distribution_eq:
hoelzl@47694
  1097
  "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>
hoelzl@47694
  1098
    distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^isub>M ?T = ?J")
hoelzl@47694
  1099
proof safe
hoelzl@47694
  1100
  assume "indep_var S X T Y"
hoelzl@47694
  1101
  then show rvs: "random_variable S X" "random_variable T Y"
hoelzl@47694
  1102
    by (blast dest: indep_var_rv1 indep_var_rv2)+
hoelzl@47694
  1103
  then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@47694
  1104
    by (rule measurable_Pair)
hoelzl@47694
  1105
hoelzl@47694
  1106
  interpret X: prob_space ?S by (rule prob_space_distr) fact
hoelzl@47694
  1107
  interpret Y: prob_space ?T by (rule prob_space_distr) fact
hoelzl@47694
  1108
  interpret XY: pair_prob_space ?S ?T ..
hoelzl@47694
  1109
  show "?S \<Otimes>\<^isub>M ?T = ?J"
hoelzl@47694
  1110
  proof (rule pair_measure_eqI)
hoelzl@47694
  1111
    show "sigma_finite_measure ?S" ..
hoelzl@47694
  1112
    show "sigma_finite_measure ?T" ..
hoelzl@43340
  1113
hoelzl@47694
  1114
    fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"
hoelzl@47694
  1115
    have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
hoelzl@47694
  1116
      using A B by (intro emeasure_distr[OF XY]) auto
hoelzl@47694
  1117
    also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
hoelzl@47694
  1118
      using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)
hoelzl@47694
  1119
    also have "\<dots> = emeasure ?S A * emeasure ?T B"
hoelzl@47694
  1120
      using rvs A B by (simp add: emeasure_distr)
hoelzl@47694
  1121
    finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
hoelzl@47694
  1122
  qed simp
hoelzl@47694
  1123
next
hoelzl@47694
  1124
  assume rvs: "random_variable S X" "random_variable T Y"
hoelzl@47694
  1125
  then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@47694
  1126
    by (rule measurable_Pair)
hoelzl@43340
  1127
hoelzl@47694
  1128
  let ?S = "distr M S X" and ?T = "distr M T Y"
hoelzl@47694
  1129
  interpret X: prob_space ?S by (rule prob_space_distr) fact
hoelzl@47694
  1130
  interpret Y: prob_space ?T by (rule prob_space_distr) fact
hoelzl@47694
  1131
  interpret XY: pair_prob_space ?S ?T ..
hoelzl@47694
  1132
hoelzl@47694
  1133
  assume "?S \<Otimes>\<^isub>M ?T = ?J"
hoelzl@43340
  1134
hoelzl@47694
  1135
  { fix S and X
hoelzl@47694
  1136
    have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"
hoelzl@47694
  1137
    proof (safe intro!: Int_stableI)
hoelzl@47694
  1138
      fix A B assume "A \<in> sets S" "B \<in> sets S"
hoelzl@47694
  1139
      then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
hoelzl@47694
  1140
        by (intro exI[of _ "A \<inter> B"]) auto
hoelzl@47694
  1141
    qed }
hoelzl@47694
  1142
  note Int_stable = this
hoelzl@47694
  1143
hoelzl@47694
  1144
  show "indep_var S X T Y" unfolding indep_var_eq
hoelzl@47694
  1145
  proof (intro conjI indep_set_sigma_sets Int_stable rvs)
hoelzl@47694
  1146
    show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
hoelzl@47694
  1147
    proof (safe intro!: indep_setI)
hoelzl@47694
  1148
      { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
hoelzl@47694
  1149
        using `X \<in> measurable M S` by (auto intro: measurable_sets) }
hoelzl@47694
  1150
      { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
hoelzl@47694
  1151
        using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
hoelzl@47694
  1152
    next
hoelzl@47694
  1153
      fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
hoelzl@47694
  1154
      then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)"
hoelzl@47694
  1155
        using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"])
hoelzl@47694
  1156
      also have "\<dots> = emeasure (?S \<Otimes>\<^isub>M ?T) (A \<times> B)"
hoelzl@47694
  1157
        unfolding `?S \<Otimes>\<^isub>M ?T = ?J` ..
hoelzl@47694
  1158
      also have "\<dots> = emeasure ?S A * emeasure ?T B"
hoelzl@49776
  1159
        using ab by (simp add: Y.emeasure_pair_measure_Times)
hoelzl@47694
  1160
      finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
hoelzl@47694
  1161
        prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
hoelzl@47694
  1162
        using rvs ab by (simp add: emeasure_eq_measure emeasure_distr)
hoelzl@47694
  1163
    qed
hoelzl@43340
  1164
  qed
hoelzl@43340
  1165
qed
hoelzl@42989
  1166
hoelzl@42861
  1167
end