src/HOL/Probability/Independent_Family.thy
author hoelzl
Wed Oct 10 12:12:18 2012 +0200 (2012-10-10)
changeset 49776 199d1d5bb17e
parent 49772 75660d89c339
child 49781 59b219ca3513
permissions -rw-r--r--
tuned product measurability
     1 (*  Title:      HOL/Probability/Independent_Family.thy
     2     Author:     Johannes Hölzl, TU München
     3 *)
     4 
     5 header {* Independent families of events, event sets, and random variables *}
     6 
     7 theory Independent_Family
     8   imports Probability_Measure Infinite_Product_Measure
     9 begin
    10 
    11 lemma INT_decseq_offset:
    12   assumes "decseq F"
    13   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
    14 proof safe
    15   fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
    16   show "x \<in> F i"
    17   proof cases
    18     from x have "x \<in> F n" by auto
    19     also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
    20       unfolding decseq_def by simp
    21     finally show ?thesis .
    22   qed (insert x, simp)
    23 qed auto
    24 
    25 definition (in prob_space)
    26   "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and>
    27     (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
    28 
    29 definition (in prob_space)
    30   "indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV"
    31 
    32 definition (in prob_space)
    33   "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and>
    34     (\<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))))"
    35 
    36 definition (in prob_space)
    37   "indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV"
    38 
    39 definition (in prob_space)
    40   "indep_vars M' X I \<longleftrightarrow>
    41     (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
    42     indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
    43 
    44 definition (in prob_space)
    45   "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV"
    46 
    47 lemma (in prob_space) indep_sets_cong:
    48   "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
    49   by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
    50 
    51 lemma (in prob_space) indep_sets_singleton_iff_indep_events:
    52   "indep_sets (\<lambda>i. {F i}) I \<longleftrightarrow> indep_events F I"
    53   unfolding indep_sets_def indep_events_def
    54   by (simp, intro conj_cong ball_cong all_cong imp_cong) (auto simp: Pi_iff)
    55 
    56 lemma (in prob_space) indep_events_finite_index_events:
    57   "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
    58   by (auto simp: indep_events_def)
    59 
    60 lemma (in prob_space) indep_sets_finite_index_sets:
    61   "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
    62 proof (intro iffI allI impI)
    63   assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
    64   show "indep_sets F I" unfolding indep_sets_def
    65   proof (intro conjI ballI allI impI)
    66     fix i assume "i \<in> I"
    67     with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
    68       by (auto simp: indep_sets_def)
    69   qed (insert *, auto simp: indep_sets_def)
    70 qed (auto simp: indep_sets_def)
    71 
    72 lemma (in prob_space) indep_sets_mono_index:
    73   "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
    74   unfolding indep_sets_def by auto
    75 
    76 lemma (in prob_space) indep_sets_mono_sets:
    77   assumes indep: "indep_sets F I"
    78   assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
    79   shows "indep_sets G I"
    80 proof -
    81   have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
    82     using mono by auto
    83   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)"
    84     using mono by (auto simp: Pi_iff)
    85   ultimately show ?thesis
    86     using indep by (auto simp: indep_sets_def)
    87 qed
    88 
    89 lemma (in prob_space) indep_sets_mono:
    90   assumes indep: "indep_sets F I"
    91   assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i"
    92   shows "indep_sets G J"
    93   apply (rule indep_sets_mono_sets)
    94   apply (rule indep_sets_mono_index)
    95   apply (fact +)
    96   done
    97 
    98 lemma (in prob_space) indep_setsI:
    99   assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
   100     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))"
   101   shows "indep_sets F I"
   102   using assms unfolding indep_sets_def by (auto simp: Pi_iff)
   103 
   104 lemma (in prob_space) indep_setsD:
   105   assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
   106   shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   107   using assms unfolding indep_sets_def by auto
   108 
   109 lemma (in prob_space) indep_setI:
   110   assumes ev: "A \<subseteq> events" "B \<subseteq> events"
   111     and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
   112   shows "indep_set A B"
   113   unfolding indep_set_def
   114 proof (rule indep_setsI)
   115   fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
   116     and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
   117   have "J \<in> Pow UNIV" by auto
   118   with F `J \<noteq> {}` indep[of "F True" "F False"]
   119   show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
   120     unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
   121 qed (auto split: bool.split simp: ev)
   122 
   123 lemma (in prob_space) indep_setD:
   124   assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"
   125   shows "prob (a \<inter> b) = prob a * prob b"
   126   using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev
   127   by (simp add: ac_simps UNIV_bool)
   128 
   129 lemma (in prob_space) indep_var_eq:
   130   "indep_var S X T Y \<longleftrightarrow>
   131     (random_variable S X \<and> random_variable T Y) \<and>
   132     indep_set
   133       (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})
   134       (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"
   135   unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
   136   by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext)
   137      (auto split: bool.split)
   138 
   139 lemma (in prob_space)
   140   assumes indep: "indep_set A B"
   141   shows indep_setD_ev1: "A \<subseteq> events"
   142     and indep_setD_ev2: "B \<subseteq> events"
   143   using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
   144 
   145 lemma (in prob_space) indep_sets_dynkin:
   146   assumes indep: "indep_sets F I"
   147   shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I"
   148     (is "indep_sets ?F I")
   149 proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
   150   fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
   151   with indep have "indep_sets F J"
   152     by (subst (asm) indep_sets_finite_index_sets) auto
   153   { fix J K assume "indep_sets F K"
   154     let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i"
   155     assume "finite J" "J \<subseteq> K"
   156     then have "indep_sets (?G J) K"
   157     proof induct
   158       case (insert j J)
   159       moreover def G \<equiv> "?G J"
   160       ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
   161         by (auto simp: indep_sets_def)
   162       let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
   163       { fix X assume X: "X \<in> events"
   164         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)
   165           \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
   166         have "indep_sets (G(j := {X})) K"
   167         proof (rule indep_setsI)
   168           fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
   169             using G X by auto
   170         next
   171           fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
   172           show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   173           proof cases
   174             assume "j \<in> J"
   175             with J have "A j = X" by auto
   176             show ?thesis
   177             proof cases
   178               assume "J = {j}" then show ?thesis by simp
   179             next
   180               assume "J \<noteq> {j}"
   181               have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
   182                 using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   183               also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
   184               proof (rule indep)
   185                 show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
   186                   using J `J \<noteq> {j}` `j \<in> J` by auto
   187                 show "\<forall>i\<in>J - {j}. A i \<in> G i"
   188                   using J by auto
   189               qed
   190               also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
   191                 using `A j = X` by simp
   192               also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
   193                 unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
   194                 using `j \<in> J` by (simp add: insert_absorb)
   195               finally show ?thesis .
   196             qed
   197           next
   198             assume "j \<notin> J"
   199             with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
   200             with J show ?thesis
   201               by (intro indep_setsD[OF G(1)]) auto
   202           qed
   203         qed }
   204       note indep_sets_insert = this
   205       have "dynkin_system (space M) ?D"
   206       proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
   207         show "indep_sets (G(j := {{}})) K"
   208           by (rule indep_sets_insert) auto
   209       next
   210         fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
   211         show "indep_sets (G(j := {space M - X})) K"
   212         proof (rule indep_sets_insert)
   213           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"
   214           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   215             using G by auto
   216           have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   217               prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
   218             using A_sets sets_into_space[of _ M] X `J \<noteq> {}`
   219             by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   220           also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
   221             using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
   222             by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)
   223           finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   224               prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
   225           moreover {
   226             have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   227               using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
   228             then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
   229               using prob_space by simp }
   230           moreover {
   231             have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
   232               using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
   233             then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
   234               using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }
   235           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))"
   236             by (simp add: field_simps)
   237           also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
   238             using X A by (simp add: finite_measure_compl)
   239           finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
   240         qed (insert X, auto)
   241       next
   242         fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
   243         then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
   244         show "indep_sets (G(j := {\<Union>k. F k})) K"
   245         proof (rule indep_sets_insert)
   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"
   247           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   248             using G by auto
   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))"
   250             using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   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))"
   252           proof (rule finite_measure_UNION)
   253             show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
   254               using disj by (rule disjoint_family_on_bisimulation) auto
   255             show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
   256               using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
   257           qed
   258           moreover { fix k
   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))"
   260               by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)
   261             also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
   262               using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
   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)" . }
   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)))"
   265             by simp
   266           moreover
   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))"
   268             using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
   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)))"
   270             using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
   271           ultimately
   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))"
   273             by (auto dest!: sums_unique)
   274         qed (insert F, auto)
   275       qed (insert sets_into_space, auto)
   276       then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
   277       proof (rule dynkin_system.dynkin_subset, safe)
   278         fix X assume "X \<in> G j"
   279         then show "X \<in> events" using G `j \<in> K` by auto
   280         from `indep_sets G K`
   281         show "indep_sets (G(j := {X})) K"
   282           by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
   283       qed
   284       have "indep_sets (G(j:=?D)) K"
   285       proof (rule indep_setsI)
   286         fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
   287           using G(2) by auto
   288       next
   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"
   290         show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   291         proof cases
   292           assume "j \<in> J"
   293           with A have indep: "indep_sets (G(j := {A j})) K" by auto
   294           from J A show ?thesis
   295             by (intro indep_setsD[OF indep]) auto
   296         next
   297           assume "j \<notin> J"
   298           with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
   299           with J show ?thesis
   300             by (intro indep_setsD[OF G(1)]) auto
   301         qed
   302       qed
   303       then have "indep_sets (G(j := dynkin (space M) (G j))) K"
   304         by (rule indep_sets_mono_sets) (insert mono, auto)
   305       then show ?case
   306         by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
   307     qed (insert `indep_sets F K`, simp) }
   308   from this[OF `indep_sets F J` `finite J` subset_refl]
   309   show "indep_sets ?F J"
   310     by (rule indep_sets_mono_sets) auto
   311 qed
   312 
   313 lemma (in prob_space) indep_sets_sigma:
   314   assumes indep: "indep_sets F I"
   315   assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
   316   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
   317 proof -
   318   from indep_sets_dynkin[OF indep]
   319   show ?thesis
   320   proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
   321     fix i assume "i \<in> I"
   322     with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
   323     with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
   324   qed
   325 qed
   326 
   327 lemma (in prob_space) indep_sets_sigma_sets_iff:
   328   assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
   329   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
   330 proof
   331   assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
   332     by (rule indep_sets_sigma) fact
   333 next
   334   assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
   335     by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
   336 qed
   337 
   338 lemma (in prob_space) indep_sets2_eq:
   339   "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)"
   340   unfolding indep_set_def
   341 proof (intro iffI ballI conjI)
   342   assume indep: "indep_sets (bool_case A B) UNIV"
   343   { fix a b assume "a \<in> A" "b \<in> B"
   344     with indep_setsD[OF indep, of UNIV "bool_case a b"]
   345     show "prob (a \<inter> b) = prob a * prob b"
   346       unfolding UNIV_bool by (simp add: ac_simps) }
   347   from indep show "A \<subseteq> events" "B \<subseteq> events"
   348     unfolding indep_sets_def UNIV_bool by auto
   349 next
   350   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)"
   351   show "indep_sets (bool_case A B) UNIV"
   352   proof (rule indep_setsI)
   353     fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
   354       using * by (auto split: bool.split)
   355   next
   356     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)"
   357     then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
   358       by (auto simp: UNIV_bool)
   359     then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
   360       using X * by auto
   361   qed
   362 qed
   363 
   364 lemma (in prob_space) indep_set_sigma_sets:
   365   assumes "indep_set A B"
   366   assumes A: "Int_stable A" and B: "Int_stable B"
   367   shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
   368 proof -
   369   have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
   370   proof (rule indep_sets_sigma)
   371     show "indep_sets (bool_case A B) UNIV"
   372       by (rule `indep_set A B`[unfolded indep_set_def])
   373     fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
   374       using A B by (cases i) auto
   375   qed
   376   then show ?thesis
   377     unfolding indep_set_def
   378     by (rule indep_sets_mono_sets) (auto split: bool.split)
   379 qed
   380 
   381 lemma (in prob_space) indep_sets_collect_sigma:
   382   fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
   383   assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
   384   assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"
   385   assumes disjoint: "disjoint_family_on I J"
   386   shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
   387 proof -
   388   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) }"
   389 
   390   from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
   391     unfolding indep_sets_def by auto
   392   { fix j
   393     let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"
   394     assume "j \<in> J"
   395     from E[OF this] interpret S: sigma_algebra "space M" ?S
   396       using sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
   397 
   398     have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
   399     proof (rule sigma_sets_eqI)
   400       fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
   401       then guess i ..
   402       then show "A \<in> sigma_sets (space M) (?E j)"
   403         by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
   404     next
   405       fix A assume "A \<in> ?E j"
   406       then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
   407         and A: "A = (\<Inter>k\<in>K. E' k)"
   408         by auto
   409       then have "A \<in> ?S" unfolding A
   410         by (safe intro!: S.finite_INT) auto
   411       then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
   412         by simp
   413     qed }
   414   moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
   415   proof (rule indep_sets_sigma)
   416     show "indep_sets ?E J"
   417     proof (intro indep_setsI)
   418       fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: finite_INT)
   419     next
   420       fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
   421         and "\<forall>j\<in>K. A j \<in> ?E j"
   422       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)"
   423         by simp
   424       from bchoice[OF this] guess E' ..
   425       from bchoice[OF this] obtain L
   426         where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
   427         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"
   428         and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
   429         by auto
   430 
   431       { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
   432         have "k = j"
   433         proof (rule ccontr)
   434           assume "k \<noteq> j"
   435           with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
   436             unfolding disjoint_family_on_def by auto
   437           with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
   438           show False using `l \<in> L k` `l \<in> L j` by auto
   439         qed }
   440       note L_inj = this
   441 
   442       def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
   443       { fix x j l assume *: "j \<in> K" "l \<in> L j"
   444         have "k l = j" unfolding k_def
   445         proof (rule some_equality)
   446           fix k assume "k \<in> K \<and> l \<in> L k"
   447           with * L_inj show "k = j" by auto
   448         qed (insert *, simp) }
   449       note k_simp[simp] = this
   450       let ?E' = "\<lambda>l. E' (k l) l"
   451       have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
   452         by (auto simp: A intro!: arg_cong[where f=prob])
   453       also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
   454         using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
   455       also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
   456         using K L L_inj by (subst setprod_UN_disjoint) auto
   457       also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
   458         using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast
   459       finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
   460     qed
   461   next
   462     fix j assume "j \<in> J"
   463     show "Int_stable (?E j)"
   464     proof (rule Int_stableI)
   465       fix a assume "a \<in> ?E j" then obtain Ka Ea
   466         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
   467       fix b assume "b \<in> ?E j" then obtain Kb Eb
   468         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
   469       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 {})"
   470       have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
   471         by (simp add: a b set_eq_iff) auto
   472       with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
   473         by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
   474     qed
   475   qed
   476   ultimately show ?thesis
   477     by (simp cong: indep_sets_cong)
   478 qed
   479 
   480 definition (in prob_space) tail_events where
   481   "tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
   482 
   483 lemma (in prob_space) tail_events_sets:
   484   assumes A: "\<And>i::nat. A i \<subseteq> events"
   485   shows "tail_events A \<subseteq> events"
   486 proof
   487   fix X assume X: "X \<in> tail_events A"
   488   let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
   489   from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def)
   490   from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
   491   then show "X \<in> events"
   492     by induct (insert A, auto)
   493 qed
   494 
   495 lemma (in prob_space) sigma_algebra_tail_events:
   496   assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
   497   shows "sigma_algebra (space M) (tail_events A)"
   498   unfolding tail_events_def
   499 proof (simp add: sigma_algebra_iff2, safe)
   500   let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
   501   interpret A: sigma_algebra "space M" "A i" for i by fact
   502   { fix X x assume "X \<in> ?A" "x \<in> X"
   503     then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
   504     from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
   505     then have "X \<subseteq> space M"
   506       by induct (insert A.sets_into_space, auto)
   507     with `x \<in> X` show "x \<in> space M" by auto }
   508   { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
   509     then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
   510       by (intro sigma_sets.Union) auto }
   511 qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
   512 
   513 lemma (in prob_space) kolmogorov_0_1_law:
   514   fixes A :: "nat \<Rightarrow> 'a set set"
   515   assumes A: "\<And>i. A i \<subseteq> events"
   516   assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
   517   assumes indep: "indep_sets A UNIV"
   518   and X: "X \<in> tail_events A"
   519   shows "prob X = 0 \<or> prob X = 1"
   520 proof -
   521   let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"
   522   interpret A: sigma_algebra "space M" "A i" for i by fact
   523   interpret T: sigma_algebra "space M" "tail_events A"
   524     by (rule sigma_algebra_tail_events) fact
   525   have "X \<subseteq> space M" using T.space_closed X by auto
   526 
   527   have X_in: "X \<in> events"
   528     using tail_events_sets A X by auto
   529 
   530   interpret D: dynkin_system "space M" ?D
   531   proof (rule dynkin_systemI)
   532     fix D assume "D \<in> ?D" then show "D \<subseteq> space M"
   533       using sets_into_space by auto
   534   next
   535     show "space M \<in> ?D"
   536       using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
   537   next
   538     fix A assume A: "A \<in> ?D"
   539     have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
   540       using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
   541     also have "\<dots> = prob X - prob (X \<inter> A)"
   542       using X_in A by (intro finite_measure_Diff) auto
   543     also have "\<dots> = prob X * prob (space M) - prob X * prob A"
   544       using A prob_space by auto
   545     also have "\<dots> = prob X * prob (space M - A)"
   546       using X_in A sets_into_space
   547       by (subst finite_measure_Diff) (auto simp: field_simps)
   548     finally show "space M - A \<in> ?D"
   549       using A `X \<subseteq> space M` by auto
   550   next
   551     fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
   552     then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
   553       by auto
   554     have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
   555     proof (rule finite_measure_UNION)
   556       show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
   557         using F X_in by auto
   558       show "disjoint_family (\<lambda>i. X \<inter> F i)"
   559         using dis by (rule disjoint_family_on_bisimulation) auto
   560     qed
   561     with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
   562       by simp
   563     moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
   564       by (intro sums_mult finite_measure_UNION F dis)
   565     ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
   566       by (auto dest!: sums_unique)
   567     with F show "(\<Union>i. F i) \<in> ?D"
   568       by auto
   569   qed
   570 
   571   { fix n
   572     have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV"
   573     proof (rule indep_sets_collect_sigma)
   574       have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")
   575         by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
   576       with indep show "indep_sets A ?U" by simp
   577       show "disjoint_family (bool_case {..n} {Suc n..})"
   578         unfolding disjoint_family_on_def by (auto split: bool.split)
   579       fix m
   580       show "Int_stable (A m)"
   581         unfolding Int_stable_def using A.Int by auto
   582     qed
   583     also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) =
   584       bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
   585       by (auto intro!: ext split: bool.split)
   586     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))"
   587       unfolding indep_set_def by simp
   588 
   589     have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"
   590     proof (simp add: subset_eq, rule)
   591       fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
   592       have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
   593         using X unfolding tail_events_def by simp
   594       from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
   595       show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
   596         by (auto simp add: ac_simps)
   597     qed }
   598   then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
   599     by auto
   600 
   601   note `X \<in> tail_events A`
   602   also {
   603     have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
   604       by (intro sigma_sets_subseteq UN_mono) auto
   605    then have "tail_events A \<subseteq> sigma_sets (space M) ?A"
   606       unfolding tail_events_def by auto }
   607   also have "sigma_sets (space M) ?A = dynkin (space M) ?A"
   608   proof (rule sigma_eq_dynkin)
   609     { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
   610       then have "B \<subseteq> space M"
   611         by induct (insert A sets_into_space[of _ M], auto) }
   612     then show "?A \<subseteq> Pow (space M)" by auto
   613     show "Int_stable ?A"
   614     proof (rule Int_stableI)
   615       fix a assume "a \<in> ?A" then guess n .. note a = this
   616       fix b assume "b \<in> ?A" then guess m .. note b = this
   617       interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   618         using A sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
   619       have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   620         by (intro sigma_sets_subseteq UN_mono) auto
   621       with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
   622       moreover
   623       have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   624         by (intro sigma_sets_subseteq UN_mono) auto
   625       with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
   626       ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
   627         using Amn.Int[of a b] by simp
   628       then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
   629     qed
   630   qed
   631   also have "dynkin (space M) ?A \<subseteq> ?D"
   632     using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)
   633   finally show ?thesis by auto
   634 qed
   635 
   636 lemma (in prob_space) borel_0_1_law:
   637   fixes F :: "nat \<Rightarrow> 'a set"
   638   assumes F: "range F \<subseteq> events" "indep_events F UNIV"
   639   shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"
   640 proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])
   641   show "\<And>i. sigma_sets (space M) {F i} \<subseteq> events"
   642     using F(1) sets_into_space
   643     by (subst sigma_sets_singleton) auto
   644   { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
   645       using sigma_algebra_sigma_sets[of "{F i}" "space M"] F sets_into_space
   646       by auto }
   647   show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
   648   proof (rule indep_sets_sigma)
   649     show "indep_sets (\<lambda>i. {F i}) UNIV"
   650       unfolding indep_sets_singleton_iff_indep_events by fact
   651     fix i show "Int_stable {F i}"
   652       unfolding Int_stable_def by simp
   653   qed
   654   let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"
   655   show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})"
   656     unfolding tail_events_def
   657   proof
   658     fix j
   659     interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
   660       using order_trans[OF F(1) space_closed]
   661       by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
   662     have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
   663       by (intro decseq_SucI INT_decseq_offset UN_mono) auto
   664     also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
   665       using order_trans[OF F(1) space_closed]
   666       by (safe intro!: S.countable_INT S.countable_UN)
   667          (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
   668     finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
   669       by simp
   670   qed
   671 qed
   672 
   673 lemma (in prob_space) indep_sets_finite:
   674   assumes I: "I \<noteq> {}" "finite I"
   675     and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"
   676   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)))"
   677 proof
   678   assume *: "indep_sets F I"
   679   from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
   680     by (intro indep_setsD[OF *] ballI) auto
   681 next
   682   assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
   683   show "indep_sets F I"
   684   proof (rule indep_setsI[OF F(1)])
   685     fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
   686     assume A: "\<forall>j\<in>J. A j \<in> F j"
   687     let ?A = "\<lambda>j. if j \<in> J then A j else space M"
   688     have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"
   689       using subset_trans[OF F(1) space_closed] J A
   690       by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast
   691     also
   692     from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")
   693       by (auto split: split_if_asm)
   694     with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
   695       by auto
   696     also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
   697       unfolding if_distrib setprod.If_cases[OF `finite I`]
   698       using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1)
   699     finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
   700   qed
   701 qed
   702 
   703 lemma (in prob_space) indep_vars_finite:
   704   fixes I :: "'i set"
   705   assumes I: "I \<noteq> {}" "finite I"
   706     and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"
   707     and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
   708     and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"
   709     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))"
   710   shows "indep_vars M' X I \<longleftrightarrow>
   711     (\<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)))"
   712 proof -
   713   from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
   714     unfolding measurable_def by simp
   715 
   716   { fix i assume "i\<in>I"
   717     from closed[OF `i \<in> I`]
   718     have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
   719       = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
   720       unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]
   721       by (subst sigma_sets_sigma_sets_eq) auto }
   722   note sigma_sets_X = this
   723 
   724   { fix i assume "i\<in>I"
   725     have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"
   726     proof (rule Int_stableI)
   727       fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   728       then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto
   729       moreover
   730       fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   731       then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
   732       moreover
   733       have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
   734       moreover note Int_stable[OF `i \<in> I`]
   735       ultimately
   736       show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   737         by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
   738     qed }
   739   note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
   740 
   741   { fix i assume "i \<in> I"
   742     { fix A assume "A \<in> E i"
   743       with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto
   744       moreover
   745       from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto
   746       ultimately
   747       have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
   748     with X[OF `i\<in>I`] space[OF `i\<in>I`]
   749     have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
   750       "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
   751       by (auto intro!: exI[of _ "space (M' i)"]) }
   752   note indep_sets_finite_X = indep_sets_finite[OF I this]
   753 
   754   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))) =
   755     (\<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)))"
   756     (is "?L = ?R")
   757   proof safe
   758     fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
   759     from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
   760     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))"
   761       by (auto simp add: Pi_iff)
   762   next
   763     fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"
   764     from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
   765     from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
   766       "B \<in> (\<Pi> i\<in>I. E i)" by auto
   767     from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
   768     show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
   769       by simp
   770   qed
   771   then show ?thesis using `I \<noteq> {}`
   772     by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
   773 qed
   774 
   775 lemma (in prob_space) indep_vars_compose:
   776   assumes "indep_vars M' X I"
   777   assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
   778   shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
   779   unfolding indep_vars_def
   780 proof
   781   from rv `indep_vars M' X I`
   782   show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
   783     by (auto simp: indep_vars_def)
   784 
   785   have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   786     using `indep_vars M' X I` by (simp add: indep_vars_def)
   787   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"
   788   proof (rule indep_sets_mono_sets)
   789     fix i assume "i \<in> I"
   790     with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)"
   791       unfolding indep_vars_def measurable_def by auto
   792     { fix A assume "A \<in> sets (N i)"
   793       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)"
   794         by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])
   795            (auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) }
   796     then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>
   797       sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
   798       by (intro sigma_sets_subseteq) (auto simp: vimage_compose)
   799   qed
   800 qed
   801 
   802 lemma (in prob_space) indep_varsD_finite:
   803   assumes X: "indep_vars M' X I"
   804   assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"
   805   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))"
   806 proof (rule indep_setsD)
   807   show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   808     using X by (auto simp: indep_vars_def)
   809   show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto
   810   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)}"
   811     using I by auto
   812 qed
   813 
   814 lemma (in prob_space) indep_varsD:
   815   assumes X: "indep_vars M' X I"
   816   assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"
   817   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))"
   818 proof (rule indep_setsD)
   819   show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
   820     using X by (auto simp: indep_vars_def)
   821   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)}"
   822     using I by auto
   823 qed fact+
   824 
   825 lemma prod_algebra_cong:
   826   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
   827   shows "prod_algebra I M = prod_algebra J N"
   828 proof -
   829   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
   830     using sets_eq_imp_space_eq[OF sets] by auto
   831   with sets show ?thesis unfolding `I = J`
   832     by (intro antisym prod_algebra_mono) auto
   833 qed
   834 
   835 lemma space_in_prod_algebra:
   836   "(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
   837 proof cases
   838   assume "I = {}" then show ?thesis
   839     by (auto simp add: prod_algebra_def image_iff prod_emb_def)
   840 next
   841   assume "I \<noteq> {}"
   842   then obtain i where "i \<in> I" by auto
   843   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))"
   844     by (auto simp: prod_emb_def Pi_iff)
   845   also have "\<dots> \<in> prod_algebra I M"
   846     using `i \<in> I` by (intro prod_algebraI) auto
   847   finally show ?thesis .
   848 qed
   849 
   850 lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
   851   fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
   852   assumes "I \<noteq> {}"
   853   assumes rv: "\<And>i. random_variable (M' i) (X i)"
   854   shows "indep_vars M' X I \<longleftrightarrow>
   855     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))"
   856 proof -
   857   let ?P = "\<Pi>\<^isub>M i\<in>I. M' i"
   858   let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"
   859   let ?D = "distr M ?P ?X"
   860   have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
   861   interpret D: prob_space ?D by (intro prob_space_distr X)
   862 
   863   let ?D' = "\<lambda>i. distr M (M' i) (X i)"
   864   let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)"
   865   interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
   866   interpret P: product_prob_space ?D' I ..
   867     
   868   show ?thesis
   869   proof
   870     assume "indep_vars M' X I"
   871     show "?D = ?P'"
   872     proof (rule measure_eqI_generator_eq)
   873       show "Int_stable (prod_algebra I M')"
   874         by (rule Int_stable_prod_algebra)
   875       show "prod_algebra I M' \<subseteq> Pow (space ?P)"
   876         using prod_algebra_sets_into_space by (simp add: space_PiM)
   877       show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
   878         by (simp add: sets_PiM space_PiM)
   879       show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
   880         by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
   881       let ?A = "\<lambda>i. \<Pi>\<^isub>E i\<in>I. space (M' i)"
   882       show "range ?A \<subseteq> prod_algebra I M'" "incseq ?A" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')"
   883         by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
   884       { fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }
   885     next
   886       fix E assume E: "E \<in> prod_algebra I M'"
   887       from prod_algebraE[OF E] guess J Y . note J = this
   888 
   889       from E have "E \<in> sets ?P" by (auto simp: sets_PiM)
   890       then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
   891         by (simp add: emeasure_distr X)
   892       also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
   893         using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm)
   894       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))"
   895         using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]
   896         by (auto simp: emeasure_eq_measure setprod_ereal)
   897       also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
   898         using rv J by (simp add: emeasure_distr)
   899       also have "\<dots> = emeasure ?P' E"
   900         using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
   901       finally show "emeasure ?D E = emeasure ?P' E" .
   902     qed
   903   next
   904     assume "?D = ?P'"
   905     show "indep_vars M' X I" unfolding indep_vars_def
   906     proof (intro conjI indep_setsI ballI rv)
   907       fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
   908         by (auto intro!: sigma_sets_subset measurable_sets rv)
   909     next
   910       fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
   911       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)}"
   912       have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
   913       proof
   914         fix j assume "j \<in> J"
   915         from Y'[rule_format, OF this] rv[of j]
   916         show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
   917           by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
   918              (auto dest: measurable_space simp: sigma_sets_eq)
   919       qed
   920       from bchoice[OF this] obtain Y where
   921         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
   922       let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)"
   923       from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
   924         using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm)
   925       then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
   926         by simp
   927       also have "\<dots> = emeasure ?D ?E"
   928         using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
   929       also have "\<dots> = emeasure ?P' ?E"
   930         using `?D = ?P'` by simp
   931       also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
   932         using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
   933       also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
   934         using rv J Y by (simp add: emeasure_distr)
   935       finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .
   936       then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"
   937         by (auto simp: emeasure_eq_measure setprod_ereal)
   938     qed
   939   qed
   940 qed
   941 
   942 lemma (in prob_space) indep_varD:
   943   assumes indep: "indep_var Ma A Mb B"
   944   assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb"
   945   shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
   946     prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
   947 proof -
   948   have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
   949     prob (\<Inter>i\<in>UNIV. (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"
   950     by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
   951   also have "\<dots> = (\<Prod>i\<in>UNIV. prob (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"
   952     using indep unfolding indep_var_def
   953     by (rule indep_varsD) (auto split: bool.split intro: sets)
   954   also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
   955     unfolding UNIV_bool by simp
   956   finally show ?thesis .
   957 qed
   958 
   959 lemma (in prob_space)
   960   assumes "indep_var S X T Y"
   961   shows indep_var_rv1: "random_variable S X"
   962     and indep_var_rv2: "random_variable T Y"
   963 proof -
   964   have "\<forall>i\<in>UNIV. random_variable (bool_case S T i) (bool_case X Y i)"
   965     using assms unfolding indep_var_def indep_vars_def by auto
   966   then show "random_variable S X" "random_variable T Y"
   967     unfolding UNIV_bool by auto
   968 qed
   969 
   970 lemma measurable_bool_case[simp, intro]:
   971   "(\<lambda>(x, y). bool_case x y) \<in> measurable (M \<Otimes>\<^isub>M N) (Pi\<^isub>M UNIV (bool_case M N))"
   972     (is "?f \<in> measurable ?B ?P")
   973 proof (rule measurable_PiM_single)
   974   show "?f \<in> space ?B \<rightarrow> (\<Pi>\<^isub>E i\<in>UNIV. space (bool_case M N i))"
   975     by (auto simp: space_pair_measure extensional_def split: bool.split)
   976   fix i A assume "A \<in> sets (case i of True \<Rightarrow> M | False \<Rightarrow> N)"
   977   moreover then have "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A}
   978     = (case i of True \<Rightarrow> A \<times> space N | False \<Rightarrow> space M \<times> A)" 
   979     by (auto simp: space_pair_measure split: bool.split dest!: sets_into_space)
   980   ultimately show "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A} \<in> sets ?B"
   981     by (auto split: bool.split)
   982 qed
   983 
   984 lemma borel_measurable_indicator':
   985   "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M"
   986   using measurable_comp[OF _ borel_measurable_indicator, of f M N A] by (auto simp add: comp_def)
   987 
   988 lemma (in product_sigma_finite) distr_component:
   989   "distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P")
   990 proof (intro measure_eqI[symmetric])
   991   interpret I: finite_product_sigma_finite M "{i}" by default simp
   992 
   993   have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
   994     by (auto simp: extensional_def restrict_def)
   995 
   996   fix A assume A: "A \<in> sets ?P"
   997   then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)" 
   998     by simp
   999   also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) x \<partial>M i)" 
  1000     apply (subst product_positive_integral_singleton[symmetric])
  1001     apply (force intro!: measurable_restrict measurable_sets A)
  1002     apply (auto intro!: positive_integral_cong simp: space_PiM indicator_def simp: eq)
  1003     done
  1004   also have "\<dots> = emeasure (M i) ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i))"
  1005     by (force intro!: measurable_restrict measurable_sets A positive_integral_indicator)
  1006   also have "\<dots> = emeasure ?D A"
  1007     using A by (auto intro!: emeasure_distr[symmetric] measurable_restrict) 
  1008   finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" .
  1009 qed simp
  1010 
  1011 lemma pair_measure_eqI:
  1012   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
  1013   assumes sets: "sets (M1 \<Otimes>\<^isub>M M2) = sets M"
  1014   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)"
  1015   shows "M1 \<Otimes>\<^isub>M M2 = M"
  1016 proof -
  1017   interpret M1: sigma_finite_measure M1 by fact
  1018   interpret M2: sigma_finite_measure M2 by fact
  1019   interpret pair_sigma_finite M1 M2 by default
  1020   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
  1021   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
  1022   let ?P = "M1 \<Otimes>\<^isub>M M2"
  1023   show ?thesis
  1024   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
  1025     show "?E \<subseteq> Pow (space ?P)"
  1026       using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)
  1027     show "sets ?P = sigma_sets (space ?P) ?E"
  1028       by (simp add: sets_pair_measure space_pair_measure)
  1029     then show "sets M = sigma_sets (space ?P) ?E"
  1030       using sets[symmetric] by simp
  1031   next
  1032     show "range F \<subseteq> ?E" "incseq F" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
  1033       using F by (auto simp: space_pair_measure)
  1034   next
  1035     fix X assume "X \<in> ?E"
  1036     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
  1037     then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
  1038        by (simp add: M2.emeasure_pair_measure_Times)
  1039     also have "\<dots> = emeasure M (A \<times> B)"
  1040       using A B emeasure by auto
  1041     finally show "emeasure ?P X = emeasure M X"
  1042       by simp
  1043   qed
  1044 qed
  1045 
  1046 lemma pair_measure_eq_distr_PiM:
  1047   fixes M1 :: "'a measure" and M2 :: "'a measure"
  1048   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
  1049   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))"
  1050     (is "?P = ?D")
  1051 proof (rule pair_measure_eqI[OF assms])
  1052   interpret B: product_sigma_finite "bool_case M1 M2"
  1053     unfolding product_sigma_finite_def using assms by (auto split: bool.split)
  1054   let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)"
  1055 
  1056   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)"
  1057     by auto
  1058   fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
  1059   have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))"
  1060     by (simp add: UNIV_bool ac_simps)
  1061   also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))"
  1062     using A B by (subst B.emeasure_PiM) (auto split: bool.split)
  1063   also have "Pi\<^isub>E UNIV (bool_case A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
  1064     using A[THEN sets_into_space] B[THEN sets_into_space]
  1065     by (auto simp: Pi_iff all_bool_eq space_PiM split: bool.split)
  1066   finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
  1067     using A B
  1068       measurable_component_singleton[of True UNIV "bool_case M1 M2"]
  1069       measurable_component_singleton[of False UNIV "bool_case M1 M2"]
  1070     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
  1071 qed simp
  1072 
  1073 lemma measurable_Pair:
  1074   assumes rvs: "X \<in> measurable M S" "Y \<in> measurable M T"
  1075   shows "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
  1076 proof -
  1077   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)"
  1078     by auto
  1079   show " (\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
  1080     by (auto simp: measurable_pair_iff rvs)
  1081 qed
  1082 
  1083 lemma (in prob_space) indep_var_distribution_eq:
  1084   "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>
  1085     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")
  1086 proof safe
  1087   assume "indep_var S X T Y"
  1088   then show rvs: "random_variable S X" "random_variable T Y"
  1089     by (blast dest: indep_var_rv1 indep_var_rv2)+
  1090   then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
  1091     by (rule measurable_Pair)
  1092 
  1093   interpret X: prob_space ?S by (rule prob_space_distr) fact
  1094   interpret Y: prob_space ?T by (rule prob_space_distr) fact
  1095   interpret XY: pair_prob_space ?S ?T ..
  1096   show "?S \<Otimes>\<^isub>M ?T = ?J"
  1097   proof (rule pair_measure_eqI)
  1098     show "sigma_finite_measure ?S" ..
  1099     show "sigma_finite_measure ?T" ..
  1100 
  1101     fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"
  1102     have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
  1103       using A B by (intro emeasure_distr[OF XY]) auto
  1104     also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
  1105       using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)
  1106     also have "\<dots> = emeasure ?S A * emeasure ?T B"
  1107       using rvs A B by (simp add: emeasure_distr)
  1108     finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
  1109   qed simp
  1110 next
  1111   assume rvs: "random_variable S X" "random_variable T Y"
  1112   then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
  1113     by (rule measurable_Pair)
  1114 
  1115   let ?S = "distr M S X" and ?T = "distr M T Y"
  1116   interpret X: prob_space ?S by (rule prob_space_distr) fact
  1117   interpret Y: prob_space ?T by (rule prob_space_distr) fact
  1118   interpret XY: pair_prob_space ?S ?T ..
  1119 
  1120   assume "?S \<Otimes>\<^isub>M ?T = ?J"
  1121 
  1122   { fix S and X
  1123     have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"
  1124     proof (safe intro!: Int_stableI)
  1125       fix A B assume "A \<in> sets S" "B \<in> sets S"
  1126       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"
  1127         by (intro exI[of _ "A \<inter> B"]) auto
  1128     qed }
  1129   note Int_stable = this
  1130 
  1131   show "indep_var S X T Y" unfolding indep_var_eq
  1132   proof (intro conjI indep_set_sigma_sets Int_stable rvs)
  1133     show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
  1134     proof (safe intro!: indep_setI)
  1135       { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
  1136         using `X \<in> measurable M S` by (auto intro: measurable_sets) }
  1137       { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
  1138         using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
  1139     next
  1140       fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
  1141       then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)"
  1142         using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"])
  1143       also have "\<dots> = emeasure (?S \<Otimes>\<^isub>M ?T) (A \<times> B)"
  1144         unfolding `?S \<Otimes>\<^isub>M ?T = ?J` ..
  1145       also have "\<dots> = emeasure ?S A * emeasure ?T B"
  1146         using ab by (simp add: Y.emeasure_pair_measure_Times)
  1147       finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
  1148         prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
  1149         using rvs ab by (simp add: emeasure_eq_measure emeasure_distr)
  1150     qed
  1151   qed
  1152 qed
  1153 
  1154 end