src/HOL/Probability/Independent_Family.thy
 author hoelzl Mon Nov 19 12:29:02 2012 +0100 (2012-11-19) changeset 50123 69b35a75caf3 parent 50104 de19856feb54 child 50244 de72bbe42190 permissions -rw-r--r--
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
```     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 definition (in prob_space)
```
```    12   "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and>
```
```    13     (\<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))))"
```
```    14
```
```    15 definition (in prob_space)
```
```    16   "indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV"
```
```    17
```
```    18 definition (in prob_space)
```
```    19   indep_events_def_alt: "indep_events A I \<longleftrightarrow> indep_sets (\<lambda>i. {A i}) I"
```
```    20
```
```    21 lemma (in prob_space) indep_events_def:
```
```    22   "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and>
```
```    23     (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
```
```    24   unfolding indep_events_def_alt indep_sets_def
```
```    25   apply (simp add: Ball_def Pi_iff image_subset_iff_funcset)
```
```    26   apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong)
```
```    27   apply auto
```
```    28   done
```
```    29
```
```    30 definition (in prob_space)
```
```    31   "indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV"
```
```    32
```
```    33 lemma (in prob_space) indep_sets_cong:
```
```    34   "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J"
```
```    35   by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
```
```    36
```
```    37 lemma (in prob_space) indep_events_finite_index_events:
```
```    38   "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
```
```    39   by (auto simp: indep_events_def)
```
```    40
```
```    41 lemma (in prob_space) indep_sets_finite_index_sets:
```
```    42   "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
```
```    43 proof (intro iffI allI impI)
```
```    44   assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
```
```    45   show "indep_sets F I" unfolding indep_sets_def
```
```    46   proof (intro conjI ballI allI impI)
```
```    47     fix i assume "i \<in> I"
```
```    48     with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
```
```    49       by (auto simp: indep_sets_def)
```
```    50   qed (insert *, auto simp: indep_sets_def)
```
```    51 qed (auto simp: indep_sets_def)
```
```    52
```
```    53 lemma (in prob_space) indep_sets_mono_index:
```
```    54   "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
```
```    55   unfolding indep_sets_def by auto
```
```    56
```
```    57 lemma (in prob_space) indep_sets_mono_sets:
```
```    58   assumes indep: "indep_sets F I"
```
```    59   assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
```
```    60   shows "indep_sets G I"
```
```    61 proof -
```
```    62   have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
```
```    63     using mono by auto
```
```    64   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)"
```
```    65     using mono by (auto simp: Pi_iff)
```
```    66   ultimately show ?thesis
```
```    67     using indep by (auto simp: indep_sets_def)
```
```    68 qed
```
```    69
```
```    70 lemma (in prob_space) indep_sets_mono:
```
```    71   assumes indep: "indep_sets F I"
```
```    72   assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i"
```
```    73   shows "indep_sets G J"
```
```    74   apply (rule indep_sets_mono_sets)
```
```    75   apply (rule indep_sets_mono_index)
```
```    76   apply (fact +)
```
```    77   done
```
```    78
```
```    79 lemma (in prob_space) indep_setsI:
```
```    80   assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
```
```    81     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))"
```
```    82   shows "indep_sets F I"
```
```    83   using assms unfolding indep_sets_def by (auto simp: Pi_iff)
```
```    84
```
```    85 lemma (in prob_space) indep_setsD:
```
```    86   assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
```
```    87   shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
```
```    88   using assms unfolding indep_sets_def by auto
```
```    89
```
```    90 lemma (in prob_space) indep_setI:
```
```    91   assumes ev: "A \<subseteq> events" "B \<subseteq> events"
```
```    92     and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
```
```    93   shows "indep_set A B"
```
```    94   unfolding indep_set_def
```
```    95 proof (rule indep_setsI)
```
```    96   fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
```
```    97     and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
```
```    98   have "J \<in> Pow UNIV" by auto
```
```    99   with F `J \<noteq> {}` indep[of "F True" "F False"]
```
```   100   show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
```
```   101     unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
```
```   102 qed (auto split: bool.split simp: ev)
```
```   103
```
```   104 lemma (in prob_space) indep_setD:
```
```   105   assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B"
```
```   106   shows "prob (a \<inter> b) = prob a * prob b"
```
```   107   using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev
```
```   108   by (simp add: ac_simps UNIV_bool)
```
```   109
```
```   110 lemma (in prob_space)
```
```   111   assumes indep: "indep_set A B"
```
```   112   shows indep_setD_ev1: "A \<subseteq> events"
```
```   113     and indep_setD_ev2: "B \<subseteq> events"
```
```   114   using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
```
```   115
```
```   116 lemma (in prob_space) indep_sets_dynkin:
```
```   117   assumes indep: "indep_sets F I"
```
```   118   shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I"
```
```   119     (is "indep_sets ?F I")
```
```   120 proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
```
```   121   fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
```
```   122   with indep have "indep_sets F J"
```
```   123     by (subst (asm) indep_sets_finite_index_sets) auto
```
```   124   { fix J K assume "indep_sets F K"
```
```   125     let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i"
```
```   126     assume "finite J" "J \<subseteq> K"
```
```   127     then have "indep_sets (?G J) K"
```
```   128     proof induct
```
```   129       case (insert j J)
```
```   130       moreover def G \<equiv> "?G J"
```
```   131       ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
```
```   132         by (auto simp: indep_sets_def)
```
```   133       let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
```
```   134       { fix X assume X: "X \<in> events"
```
```   135         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)
```
```   136           \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
```
```   137         have "indep_sets (G(j := {X})) K"
```
```   138         proof (rule indep_setsI)
```
```   139           fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
```
```   140             using G X by auto
```
```   141         next
```
```   142           fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
```
```   143           show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
```
```   144           proof cases
```
```   145             assume "j \<in> J"
```
```   146             with J have "A j = X" by auto
```
```   147             show ?thesis
```
```   148             proof cases
```
```   149               assume "J = {j}" then show ?thesis by simp
```
```   150             next
```
```   151               assume "J \<noteq> {j}"
```
```   152               have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
```
```   153                 using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
```
```   154               also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
```
```   155               proof (rule indep)
```
```   156                 show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
```
```   157                   using J `J \<noteq> {j}` `j \<in> J` by auto
```
```   158                 show "\<forall>i\<in>J - {j}. A i \<in> G i"
```
```   159                   using J by auto
```
```   160               qed
```
```   161               also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
```
```   162                 using `A j = X` by simp
```
```   163               also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
```
```   164                 unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
```
```   165                 using `j \<in> J` by (simp add: insert_absorb)
```
```   166               finally show ?thesis .
```
```   167             qed
```
```   168           next
```
```   169             assume "j \<notin> J"
```
```   170             with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
```
```   171             with J show ?thesis
```
```   172               by (intro indep_setsD[OF G(1)]) auto
```
```   173           qed
```
```   174         qed }
```
```   175       note indep_sets_insert = this
```
```   176       have "dynkin_system (space M) ?D"
```
```   177       proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
```
```   178         show "indep_sets (G(j := {{}})) K"
```
```   179           by (rule indep_sets_insert) auto
```
```   180       next
```
```   181         fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
```
```   182         show "indep_sets (G(j := {space M - X})) K"
```
```   183         proof (rule indep_sets_insert)
```
```   184           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"
```
```   185           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
```
```   186             using G by auto
```
```   187           have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
```
```   188               prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
```
```   189             using A_sets sets_into_space[of _ M] X `J \<noteq> {}`
```
```   190             by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
```
```   191           also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
```
```   192             using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
```
```   193             by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)
```
```   194           finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
```
```   195               prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
```
```   196           moreover {
```
```   197             have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
```
```   198               using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
```
```   199             then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
```
```   200               using prob_space by simp }
```
```   201           moreover {
```
```   202             have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
```
```   203               using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
```
```   204             then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
```
```   205               using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }
```
```   206           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))"
```
```   207             by (simp add: field_simps)
```
```   208           also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
```
```   209             using X A by (simp add: finite_measure_compl)
```
```   210           finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
```
```   211         qed (insert X, auto)
```
```   212       next
```
```   213         fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
```
```   214         then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
```
```   215         show "indep_sets (G(j := {\<Union>k. F k})) K"
```
```   216         proof (rule indep_sets_insert)
```
```   217           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"
```
```   218           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
```
```   219             using G by auto
```
```   220           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))"
```
```   221             using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
```
```   222           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))"
```
```   223           proof (rule finite_measure_UNION)
```
```   224             show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
```
```   225               using disj by (rule disjoint_family_on_bisimulation) auto
```
```   226             show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
```
```   227               using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
```
```   228           qed
```
```   229           moreover { fix k
```
```   230             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))"
```
```   231               by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)
```
```   232             also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
```
```   233               using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
```
```   234             finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
```
```   235           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)))"
```
```   236             by simp
```
```   237           moreover
```
```   238           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))"
```
```   239             using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
```
```   240           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)))"
```
```   241             using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
```
```   242           ultimately
```
```   243           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))"
```
```   244             by (auto dest!: sums_unique)
```
```   245         qed (insert F, auto)
```
```   246       qed (insert sets_into_space, auto)
```
```   247       then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
```
```   248       proof (rule dynkin_system.dynkin_subset, safe)
```
```   249         fix X assume "X \<in> G j"
```
```   250         then show "X \<in> events" using G `j \<in> K` by auto
```
```   251         from `indep_sets G K`
```
```   252         show "indep_sets (G(j := {X})) K"
```
```   253           by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
```
```   254       qed
```
```   255       have "indep_sets (G(j:=?D)) K"
```
```   256       proof (rule indep_setsI)
```
```   257         fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
```
```   258           using G(2) by auto
```
```   259       next
```
```   260         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"
```
```   261         show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
```
```   262         proof cases
```
```   263           assume "j \<in> J"
```
```   264           with A have indep: "indep_sets (G(j := {A j})) K" by auto
```
```   265           from J A show ?thesis
```
```   266             by (intro indep_setsD[OF indep]) auto
```
```   267         next
```
```   268           assume "j \<notin> J"
```
```   269           with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
```
```   270           with J show ?thesis
```
```   271             by (intro indep_setsD[OF G(1)]) auto
```
```   272         qed
```
```   273       qed
```
```   274       then have "indep_sets (G(j := dynkin (space M) (G j))) K"
```
```   275         by (rule indep_sets_mono_sets) (insert mono, auto)
```
```   276       then show ?case
```
```   277         by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
```
```   278     qed (insert `indep_sets F K`, simp) }
```
```   279   from this[OF `indep_sets F J` `finite J` subset_refl]
```
```   280   show "indep_sets ?F J"
```
```   281     by (rule indep_sets_mono_sets) auto
```
```   282 qed
```
```   283
```
```   284 lemma (in prob_space) indep_sets_sigma:
```
```   285   assumes indep: "indep_sets F I"
```
```   286   assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
```
```   287   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
```
```   288 proof -
```
```   289   from indep_sets_dynkin[OF indep]
```
```   290   show ?thesis
```
```   291   proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
```
```   292     fix i assume "i \<in> I"
```
```   293     with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
```
```   294     with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
```
```   295   qed
```
```   296 qed
```
```   297
```
```   298 lemma (in prob_space) indep_sets_sigma_sets_iff:
```
```   299   assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)"
```
```   300   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I"
```
```   301 proof
```
```   302   assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
```
```   303     by (rule indep_sets_sigma) fact
```
```   304 next
```
```   305   assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
```
```   306     by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
```
```   307 qed
```
```   308
```
```   309 definition (in prob_space)
```
```   310   indep_vars_def2: "indep_vars M' X I \<longleftrightarrow>
```
```   311     (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
```
```   312     indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
```
```   313
```
```   314 definition (in prob_space)
```
```   315   "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV"
```
```   316
```
```   317 lemma (in prob_space) indep_vars_def:
```
```   318   "indep_vars M' X I \<longleftrightarrow>
```
```   319     (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
```
```   320     indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
```
```   321   unfolding indep_vars_def2
```
```   322   apply (rule conj_cong[OF refl])
```
```   323   apply (rule indep_sets_sigma_sets_iff[symmetric])
```
```   324   apply (auto simp: Int_stable_def)
```
```   325   apply (rule_tac x="A \<inter> Aa" in exI)
```
```   326   apply auto
```
```   327   done
```
```   328
```
```   329 lemma (in prob_space) indep_var_eq:
```
```   330   "indep_var S X T Y \<longleftrightarrow>
```
```   331     (random_variable S X \<and> random_variable T Y) \<and>
```
```   332     indep_set
```
```   333       (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})
```
```   334       (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"
```
```   335   unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
```
```   336   by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext)
```
```   337      (auto split: bool.split)
```
```   338
```
```   339 lemma (in prob_space) indep_sets2_eq:
```
```   340   "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)"
```
```   341   unfolding indep_set_def
```
```   342 proof (intro iffI ballI conjI)
```
```   343   assume indep: "indep_sets (bool_case A B) UNIV"
```
```   344   { fix a b assume "a \<in> A" "b \<in> B"
```
```   345     with indep_setsD[OF indep, of UNIV "bool_case a b"]
```
```   346     show "prob (a \<inter> b) = prob a * prob b"
```
```   347       unfolding UNIV_bool by (simp add: ac_simps) }
```
```   348   from indep show "A \<subseteq> events" "B \<subseteq> events"
```
```   349     unfolding indep_sets_def UNIV_bool by auto
```
```   350 next
```
```   351   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)"
```
```   352   show "indep_sets (bool_case A B) UNIV"
```
```   353   proof (rule indep_setsI)
```
```   354     fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
```
```   355       using * by (auto split: bool.split)
```
```   356   next
```
```   357     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)"
```
```   358     then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
```
```   359       by (auto simp: UNIV_bool)
```
```   360     then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
```
```   361       using X * by auto
```
```   362   qed
```
```   363 qed
```
```   364
```
```   365 lemma (in prob_space) indep_set_sigma_sets:
```
```   366   assumes "indep_set A B"
```
```   367   assumes A: "Int_stable A" and B: "Int_stable B"
```
```   368   shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
```
```   369 proof -
```
```   370   have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
```
```   371   proof (rule indep_sets_sigma)
```
```   372     show "indep_sets (bool_case A B) UNIV"
```
```   373       by (rule `indep_set A B`[unfolded indep_set_def])
```
```   374     fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
```
```   375       using A B by (cases i) auto
```
```   376   qed
```
```   377   then show ?thesis
```
```   378     unfolding indep_set_def
```
```   379     by (rule indep_sets_mono_sets) (auto split: bool.split)
```
```   380 qed
```
```   381
```
```   382 lemma (in prob_space) indep_sets_collect_sigma:
```
```   383   fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
```
```   384   assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
```
```   385   assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)"
```
```   386   assumes disjoint: "disjoint_family_on I J"
```
```   387   shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
```
```   388 proof -
```
```   389   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) }"
```
```   390
```
```   391   from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events"
```
```   392     unfolding indep_sets_def by auto
```
```   393   { fix j
```
```   394     let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)"
```
```   395     assume "j \<in> J"
```
```   396     from E[OF this] interpret S: sigma_algebra "space M" ?S
```
```   397       using sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
```
```   398
```
```   399     have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
```
```   400     proof (rule sigma_sets_eqI)
```
```   401       fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
```
```   402       then guess i ..
```
```   403       then show "A \<in> sigma_sets (space M) (?E j)"
```
```   404         by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
```
```   405     next
```
```   406       fix A assume "A \<in> ?E j"
```
```   407       then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
```
```   408         and A: "A = (\<Inter>k\<in>K. E' k)"
```
```   409         by auto
```
```   410       then have "A \<in> ?S" unfolding A
```
```   411         by (safe intro!: S.finite_INT) auto
```
```   412       then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
```
```   413         by simp
```
```   414     qed }
```
```   415   moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
```
```   416   proof (rule indep_sets_sigma)
```
```   417     show "indep_sets ?E J"
```
```   418     proof (intro indep_setsI)
```
```   419       fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force  intro!: finite_INT)
```
```   420     next
```
```   421       fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
```
```   422         and "\<forall>j\<in>K. A j \<in> ?E j"
```
```   423       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)"
```
```   424         by simp
```
```   425       from bchoice[OF this] guess E' ..
```
```   426       from bchoice[OF this] obtain L
```
```   427         where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
```
```   428         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"
```
```   429         and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
```
```   430         by auto
```
```   431
```
```   432       { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
```
```   433         have "k = j"
```
```   434         proof (rule ccontr)
```
```   435           assume "k \<noteq> j"
```
```   436           with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
```
```   437             unfolding disjoint_family_on_def by auto
```
```   438           with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
```
```   439           show False using `l \<in> L k` `l \<in> L j` by auto
```
```   440         qed }
```
```   441       note L_inj = this
```
```   442
```
```   443       def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
```
```   444       { fix x j l assume *: "j \<in> K" "l \<in> L j"
```
```   445         have "k l = j" unfolding k_def
```
```   446         proof (rule some_equality)
```
```   447           fix k assume "k \<in> K \<and> l \<in> L k"
```
```   448           with * L_inj show "k = j" by auto
```
```   449         qed (insert *, simp) }
```
```   450       note k_simp[simp] = this
```
```   451       let ?E' = "\<lambda>l. E' (k l) l"
```
```   452       have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
```
```   453         by (auto simp: A intro!: arg_cong[where f=prob])
```
```   454       also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
```
```   455         using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
```
```   456       also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
```
```   457         using K L L_inj by (subst setprod_UN_disjoint) auto
```
```   458       also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
```
```   459         using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast
```
```   460       finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
```
```   461     qed
```
```   462   next
```
```   463     fix j assume "j \<in> J"
```
```   464     show "Int_stable (?E j)"
```
```   465     proof (rule Int_stableI)
```
```   466       fix a assume "a \<in> ?E j" then obtain Ka Ea
```
```   467         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
```
```   468       fix b assume "b \<in> ?E j" then obtain Kb Eb
```
```   469         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
```
```   470       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 {})"
```
```   471       have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
```
```   472         by (simp add: a b set_eq_iff) auto
```
```   473       with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
```
```   474         by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
```
```   475     qed
```
```   476   qed
```
```   477   ultimately show ?thesis
```
```   478     by (simp cong: indep_sets_cong)
```
```   479 qed
```
```   480
```
```   481 definition (in prob_space) tail_events where
```
```   482   "tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
```
```   483
```
```   484 lemma (in prob_space) tail_events_sets:
```
```   485   assumes A: "\<And>i::nat. A i \<subseteq> events"
```
```   486   shows "tail_events A \<subseteq> events"
```
```   487 proof
```
```   488   fix X assume X: "X \<in> tail_events A"
```
```   489   let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
```
```   490   from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def)
```
```   491   from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
```
```   492   then show "X \<in> events"
```
```   493     by induct (insert A, auto)
```
```   494 qed
```
```   495
```
```   496 lemma (in prob_space) sigma_algebra_tail_events:
```
```   497   assumes "\<And>i::nat. sigma_algebra (space M) (A i)"
```
```   498   shows "sigma_algebra (space M) (tail_events A)"
```
```   499   unfolding tail_events_def
```
```   500 proof (simp add: sigma_algebra_iff2, safe)
```
```   501   let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
```
```   502   interpret A: sigma_algebra "space M" "A i" for i by fact
```
```   503   { fix X x assume "X \<in> ?A" "x \<in> X"
```
```   504     then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
```
```   505     from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
```
```   506     then have "X \<subseteq> space M"
```
```   507       by induct (insert A.sets_into_space, auto)
```
```   508     with `x \<in> X` show "x \<in> space M" by auto }
```
```   509   { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
```
```   510     then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
```
```   511       by (intro sigma_sets.Union) auto }
```
```   512 qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
```
```   513
```
```   514 lemma (in prob_space) kolmogorov_0_1_law:
```
```   515   fixes A :: "nat \<Rightarrow> 'a set set"
```
```   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   have A: "\<And>i. A i \<subseteq> events"
```
```   522     using indep unfolding indep_sets_def by simp
```
```   523
```
```   524   let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}"
```
```   525   interpret A: sigma_algebra "space M" "A i" for i by fact
```
```   526   interpret T: sigma_algebra "space M" "tail_events A"
```
```   527     by (rule sigma_algebra_tail_events) fact
```
```   528   have "X \<subseteq> space M" using T.space_closed X by auto
```
```   529
```
```   530   have X_in: "X \<in> events"
```
```   531     using tail_events_sets A X by auto
```
```   532
```
```   533   interpret D: dynkin_system "space M" ?D
```
```   534   proof (rule dynkin_systemI)
```
```   535     fix D assume "D \<in> ?D" then show "D \<subseteq> space M"
```
```   536       using sets_into_space by auto
```
```   537   next
```
```   538     show "space M \<in> ?D"
```
```   539       using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
```
```   540   next
```
```   541     fix A assume A: "A \<in> ?D"
```
```   542     have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
```
```   543       using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
```
```   544     also have "\<dots> = prob X - prob (X \<inter> A)"
```
```   545       using X_in A by (intro finite_measure_Diff) auto
```
```   546     also have "\<dots> = prob X * prob (space M) - prob X * prob A"
```
```   547       using A prob_space by auto
```
```   548     also have "\<dots> = prob X * prob (space M - A)"
```
```   549       using X_in A sets_into_space
```
```   550       by (subst finite_measure_Diff) (auto simp: field_simps)
```
```   551     finally show "space M - A \<in> ?D"
```
```   552       using A `X \<subseteq> space M` by auto
```
```   553   next
```
```   554     fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
```
```   555     then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
```
```   556       by auto
```
```   557     have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
```
```   558     proof (rule finite_measure_UNION)
```
```   559       show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
```
```   560         using F X_in by auto
```
```   561       show "disjoint_family (\<lambda>i. X \<inter> F i)"
```
```   562         using dis by (rule disjoint_family_on_bisimulation) auto
```
```   563     qed
```
```   564     with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
```
```   565       by simp
```
```   566     moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
```
```   567       by (intro sums_mult finite_measure_UNION F dis)
```
```   568     ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
```
```   569       by (auto dest!: sums_unique)
```
```   570     with F show "(\<Union>i. F i) \<in> ?D"
```
```   571       by auto
```
```   572   qed
```
```   573
```
```   574   { fix n
```
```   575     have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV"
```
```   576     proof (rule indep_sets_collect_sigma)
```
```   577       have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _")
```
```   578         by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
```
```   579       with indep show "indep_sets A ?U" by simp
```
```   580       show "disjoint_family (bool_case {..n} {Suc n..})"
```
```   581         unfolding disjoint_family_on_def by (auto split: bool.split)
```
```   582       fix m
```
```   583       show "Int_stable (A m)"
```
```   584         unfolding Int_stable_def using A.Int by auto
```
```   585     qed
```
```   586     also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) =
```
```   587       bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
```
```   588       by (auto intro!: ext split: bool.split)
```
```   589     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))"
```
```   590       unfolding indep_set_def by simp
```
```   591
```
```   592     have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D"
```
```   593     proof (simp add: subset_eq, rule)
```
```   594       fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
```
```   595       have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
```
```   596         using X unfolding tail_events_def by simp
```
```   597       from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
```
```   598       show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
```
```   599         by (auto simp add: ac_simps)
```
```   600     qed }
```
```   601   then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
```
```   602     by auto
```
```   603
```
```   604   note `X \<in> tail_events A`
```
```   605   also {
```
```   606     have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
```
```   607       by (intro sigma_sets_subseteq UN_mono) auto
```
```   608    then have "tail_events A \<subseteq> sigma_sets (space M) ?A"
```
```   609       unfolding tail_events_def by auto }
```
```   610   also have "sigma_sets (space M) ?A = dynkin (space M) ?A"
```
```   611   proof (rule sigma_eq_dynkin)
```
```   612     { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
```
```   613       then have "B \<subseteq> space M"
```
```   614         by induct (insert A sets_into_space[of _ M], auto) }
```
```   615     then show "?A \<subseteq> Pow (space M)" by auto
```
```   616     show "Int_stable ?A"
```
```   617     proof (rule Int_stableI)
```
```   618       fix a assume "a \<in> ?A" then guess n .. note a = this
```
```   619       fix b assume "b \<in> ?A" then guess m .. note b = this
```
```   620       interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
```
```   621         using A sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
```
```   622       have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
```
```   623         by (intro sigma_sets_subseteq UN_mono) auto
```
```   624       with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
```
```   625       moreover
```
```   626       have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
```
```   627         by (intro sigma_sets_subseteq UN_mono) auto
```
```   628       with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
```
```   629       ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
```
```   630         using Amn.Int[of a b] by simp
```
```   631       then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
```
```   632     qed
```
```   633   qed
```
```   634   also have "dynkin (space M) ?A \<subseteq> ?D"
```
```   635     using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)
```
```   636   finally show ?thesis by auto
```
```   637 qed
```
```   638
```
```   639 lemma (in prob_space) borel_0_1_law:
```
```   640   fixes F :: "nat \<Rightarrow> 'a set"
```
```   641   assumes F2: "indep_events F UNIV"
```
```   642   shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1"
```
```   643 proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"])
```
```   644   have F1: "range F \<subseteq> events"
```
```   645     using F2 by (simp add: indep_events_def subset_eq)
```
```   646   { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
```
```   647       using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets_into_space
```
```   648       by auto }
```
```   649   show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV"
```
```   650   proof (rule indep_sets_sigma)
```
```   651     show "indep_sets (\<lambda>i. {F i}) UNIV"
```
```   652       unfolding indep_events_def_alt[symmetric] by fact
```
```   653     fix i show "Int_stable {F i}"
```
```   654       unfolding Int_stable_def by simp
```
```   655   qed
```
```   656   let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i"
```
```   657   show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})"
```
```   658     unfolding tail_events_def
```
```   659   proof
```
```   660     fix j
```
```   661     interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
```
```   662       using order_trans[OF F1 space_closed]
```
```   663       by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
```
```   664     have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)"
```
```   665       by (intro decseq_SucI INT_decseq_offset UN_mono) auto
```
```   666     also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
```
```   667       using order_trans[OF F1 space_closed]
```
```   668       by (safe intro!: S.countable_INT S.countable_UN)
```
```   669          (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
```
```   670     finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})"
```
```   671       by simp
```
```   672   qed
```
```   673 qed
```
```   674
```
```   675 lemma (in prob_space) indep_sets_finite:
```
```   676   assumes I: "I \<noteq> {}" "finite I"
```
```   677     and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i"
```
```   678   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)))"
```
```   679 proof
```
```   680   assume *: "indep_sets F I"
```
```   681   from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
```
```   682     by (intro indep_setsD[OF *] ballI) auto
```
```   683 next
```
```   684   assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))"
```
```   685   show "indep_sets F I"
```
```   686   proof (rule indep_setsI[OF F(1)])
```
```   687     fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
```
```   688     assume A: "\<forall>j\<in>J. A j \<in> F j"
```
```   689     let ?A = "\<lambda>j. if j \<in> J then A j else space M"
```
```   690     have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)"
```
```   691       using subset_trans[OF F(1) space_closed] J A
```
```   692       by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast
```
```   693     also
```
```   694     from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _")
```
```   695       by (auto split: split_if_asm)
```
```   696     with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
```
```   697       by auto
```
```   698     also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
```
```   699       unfolding if_distrib setprod.If_cases[OF `finite I`]
```
```   700       using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1)
```
```   701     finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
```
```   702   qed
```
```   703 qed
```
```   704
```
```   705 lemma (in prob_space) indep_vars_finite:
```
```   706   fixes I :: "'i set"
```
```   707   assumes I: "I \<noteq> {}" "finite I"
```
```   708     and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)"
```
```   709     and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
```
```   710     and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)"
```
```   711     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))"
```
```   712   shows "indep_vars M' X I \<longleftrightarrow>
```
```   713     (\<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)))"
```
```   714 proof -
```
```   715   from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)"
```
```   716     unfolding measurable_def by simp
```
```   717
```
```   718   { fix i assume "i\<in>I"
```
```   719     from closed[OF `i \<in> I`]
```
```   720     have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
```
```   721       = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
```
```   722       unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]
```
```   723       by (subst sigma_sets_sigma_sets_eq) auto }
```
```   724   note sigma_sets_X = this
```
```   725
```
```   726   { fix i assume "i\<in>I"
```
```   727     have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}"
```
```   728     proof (rule Int_stableI)
```
```   729       fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
```
```   730       then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto
```
```   731       moreover
```
```   732       fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
```
```   733       then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
```
```   734       moreover
```
```   735       have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
```
```   736       moreover note Int_stable[OF `i \<in> I`]
```
```   737       ultimately
```
```   738       show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
```
```   739         by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
```
```   740     qed }
```
```   741   note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
```
```   742
```
```   743   { fix i assume "i \<in> I"
```
```   744     { fix A assume "A \<in> E i"
```
```   745       with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto
```
```   746       moreover
```
```   747       from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto
```
```   748       ultimately
```
```   749       have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
```
```   750     with X[OF `i\<in>I`] space[OF `i\<in>I`]
```
```   751     have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
```
```   752       "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
```
```   753       by (auto intro!: exI[of _ "space (M' i)"]) }
```
```   754   note indep_sets_finite_X = indep_sets_finite[OF I this]
```
```   755
```
```   756   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))) =
```
```   757     (\<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)))"
```
```   758     (is "?L = ?R")
```
```   759   proof safe
```
```   760     fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
```
```   761     from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
```
```   762     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))"
```
```   763       by (auto simp add: Pi_iff)
```
```   764   next
```
```   765     fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})"
```
```   766     from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
```
```   767     from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
```
```   768       "B \<in> (\<Pi> i\<in>I. E i)" by auto
```
```   769     from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
```
```   770     show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
```
```   771       by simp
```
```   772   qed
```
```   773   then show ?thesis using `I \<noteq> {}`
```
```   774     by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
```
```   775 qed
```
```   776
```
```   777 lemma (in prob_space) indep_vars_compose:
```
```   778   assumes "indep_vars M' X I"
```
```   779   assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)"
```
```   780   shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
```
```   781   unfolding indep_vars_def
```
```   782 proof
```
```   783   from rv `indep_vars M' X I`
```
```   784   show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
```
```   785     by (auto simp: indep_vars_def)
```
```   786
```
```   787   have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
```
```   788     using `indep_vars M' X I` by (simp add: indep_vars_def)
```
```   789   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"
```
```   790   proof (rule indep_sets_mono_sets)
```
```   791     fix i assume "i \<in> I"
```
```   792     with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)"
```
```   793       unfolding indep_vars_def measurable_def by auto
```
```   794     { fix A assume "A \<in> sets (N i)"
```
```   795       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)"
```
```   796         by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])
```
```   797            (auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) }
```
```   798     then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>
```
```   799       sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
```
```   800       by (intro sigma_sets_subseteq) (auto simp: vimage_compose)
```
```   801   qed
```
```   802 qed
```
```   803
```
```   804 lemma (in prob_space) indep_varsD_finite:
```
```   805   assumes X: "indep_vars M' X I"
```
```   806   assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)"
```
```   807   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))"
```
```   808 proof (rule indep_setsD)
```
```   809   show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
```
```   810     using X by (auto simp: indep_vars_def)
```
```   811   show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto
```
```   812   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)}"
```
```   813     using I by auto
```
```   814 qed
```
```   815
```
```   816 lemma (in prob_space) indep_varsD:
```
```   817   assumes X: "indep_vars M' X I"
```
```   818   assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)"
```
```   819   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))"
```
```   820 proof (rule indep_setsD)
```
```   821   show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
```
```   822     using X by (auto simp: indep_vars_def)
```
```   823   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)}"
```
```   824     using I by auto
```
```   825 qed fact+
```
```   826
```
```   827 lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
```
```   828   fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   829   assumes "I \<noteq> {}"
```
```   830   assumes rv: "\<And>i. random_variable (M' i) (X i)"
```
```   831   shows "indep_vars M' X I \<longleftrightarrow>
```
```   832     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))"
```
```   833 proof -
```
```   834   let ?P = "\<Pi>\<^isub>M i\<in>I. M' i"
```
```   835   let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x"
```
```   836   let ?D = "distr M ?P ?X"
```
```   837   have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
```
```   838   interpret D: prob_space ?D by (intro prob_space_distr X)
```
```   839
```
```   840   let ?D' = "\<lambda>i. distr M (M' i) (X i)"
```
```   841   let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)"
```
```   842   interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
```
```   843   interpret P: product_prob_space ?D' I ..
```
```   844
```
```   845   show ?thesis
```
```   846   proof
```
```   847     assume "indep_vars M' X I"
```
```   848     show "?D = ?P'"
```
```   849     proof (rule measure_eqI_generator_eq)
```
```   850       show "Int_stable (prod_algebra I M')"
```
```   851         by (rule Int_stable_prod_algebra)
```
```   852       show "prod_algebra I M' \<subseteq> Pow (space ?P)"
```
```   853         using prod_algebra_sets_into_space by (simp add: space_PiM)
```
```   854       show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
```
```   855         by (simp add: sets_PiM space_PiM)
```
```   856       show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
```
```   857         by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
```
```   858       let ?A = "\<lambda>i. \<Pi>\<^isub>E i\<in>I. space (M' i)"
```
```   859       show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')"
```
```   860         by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
```
```   861       { fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto }
```
```   862     next
```
```   863       fix E assume E: "E \<in> prod_algebra I M'"
```
```   864       from prod_algebraE[OF E] guess J Y . note J = this
```
```   865
```
```   866       from E have "E \<in> sets ?P" by (auto simp: sets_PiM)
```
```   867       then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
```
```   868         by (simp add: emeasure_distr X)
```
```   869       also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
```
```   870         using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
```
```   871       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))"
```
```   872         using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]
```
```   873         by (auto simp: emeasure_eq_measure setprod_ereal)
```
```   874       also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
```
```   875         using rv J by (simp add: emeasure_distr)
```
```   876       also have "\<dots> = emeasure ?P' E"
```
```   877         using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
```
```   878       finally show "emeasure ?D E = emeasure ?P' E" .
```
```   879     qed
```
```   880   next
```
```   881     assume "?D = ?P'"
```
```   882     show "indep_vars M' X I" unfolding indep_vars_def
```
```   883     proof (intro conjI indep_setsI ballI rv)
```
```   884       fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events"
```
```   885         by (auto intro!: sigma_sets_subset measurable_sets rv)
```
```   886     next
```
```   887       fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J"
```
```   888       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)}"
```
```   889       have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
```
```   890       proof
```
```   891         fix j assume "j \<in> J"
```
```   892         from Y'[rule_format, OF this] rv[of j]
```
```   893         show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)"
```
```   894           by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
```
```   895              (auto dest: measurable_space simp: sigma_sets_eq)
```
```   896       qed
```
```   897       from bchoice[OF this] obtain Y where
```
```   898         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
```
```   899       let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)"
```
```   900       from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
```
```   901         using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
```
```   902       then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
```
```   903         by simp
```
```   904       also have "\<dots> = emeasure ?D ?E"
```
```   905         using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
```
```   906       also have "\<dots> = emeasure ?P' ?E"
```
```   907         using `?D = ?P'` by simp
```
```   908       also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
```
```   909         using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
```
```   910       also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
```
```   911         using rv J Y by (simp add: emeasure_distr)
```
```   912       finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" .
```
```   913       then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))"
```
```   914         by (auto simp: emeasure_eq_measure setprod_ereal)
```
```   915     qed
```
```   916   qed
```
```   917 qed
```
```   918
```
```   919 lemma (in prob_space) indep_varD:
```
```   920   assumes indep: "indep_var Ma A Mb B"
```
```   921   assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb"
```
```   922   shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
```
```   923     prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
```
```   924 proof -
```
```   925   have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) =
```
```   926     prob (\<Inter>i\<in>UNIV. (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"
```
```   927     by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
```
```   928   also have "\<dots> = (\<Prod>i\<in>UNIV. prob (bool_case A B i -` bool_case Xa Xb i \<inter> space M))"
```
```   929     using indep unfolding indep_var_def
```
```   930     by (rule indep_varsD) (auto split: bool.split intro: sets)
```
```   931   also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
```
```   932     unfolding UNIV_bool by simp
```
```   933   finally show ?thesis .
```
```   934 qed
```
```   935
```
```   936 lemma (in prob_space)
```
```   937   assumes "indep_var S X T Y"
```
```   938   shows indep_var_rv1: "random_variable S X"
```
```   939     and indep_var_rv2: "random_variable T Y"
```
```   940 proof -
```
```   941   have "\<forall>i\<in>UNIV. random_variable (bool_case S T i) (bool_case X Y i)"
```
```   942     using assms unfolding indep_var_def indep_vars_def by auto
```
```   943   then show "random_variable S X" "random_variable T Y"
```
```   944     unfolding UNIV_bool by auto
```
```   945 qed
```
```   946
```
```   947 lemma (in prob_space) indep_var_distribution_eq:
```
```   948   "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and>
```
```   949     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")
```
```   950 proof safe
```
```   951   assume "indep_var S X T Y"
```
```   952   then show rvs: "random_variable S X" "random_variable T Y"
```
```   953     by (blast dest: indep_var_rv1 indep_var_rv2)+
```
```   954   then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
```
```   955     by (rule measurable_Pair)
```
```   956
```
```   957   interpret X: prob_space ?S by (rule prob_space_distr) fact
```
```   958   interpret Y: prob_space ?T by (rule prob_space_distr) fact
```
```   959   interpret XY: pair_prob_space ?S ?T ..
```
```   960   show "?S \<Otimes>\<^isub>M ?T = ?J"
```
```   961   proof (rule pair_measure_eqI)
```
```   962     show "sigma_finite_measure ?S" ..
```
```   963     show "sigma_finite_measure ?T" ..
```
```   964
```
```   965     fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T"
```
```   966     have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
```
```   967       using A B by (intro emeasure_distr[OF XY]) auto
```
```   968     also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
```
```   969       using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)
```
```   970     also have "\<dots> = emeasure ?S A * emeasure ?T B"
```
```   971       using rvs A B by (simp add: emeasure_distr)
```
```   972     finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
```
```   973   qed simp
```
```   974 next
```
```   975   assume rvs: "random_variable S X" "random_variable T Y"
```
```   976   then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
```
```   977     by (rule measurable_Pair)
```
```   978
```
```   979   let ?S = "distr M S X" and ?T = "distr M T Y"
```
```   980   interpret X: prob_space ?S by (rule prob_space_distr) fact
```
```   981   interpret Y: prob_space ?T by (rule prob_space_distr) fact
```
```   982   interpret XY: pair_prob_space ?S ?T ..
```
```   983
```
```   984   assume "?S \<Otimes>\<^isub>M ?T = ?J"
```
```   985
```
```   986   { fix S and X
```
```   987     have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}"
```
```   988     proof (safe intro!: Int_stableI)
```
```   989       fix A B assume "A \<in> sets S" "B \<in> sets S"
```
```   990       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"
```
```   991         by (intro exI[of _ "A \<inter> B"]) auto
```
```   992     qed }
```
```   993   note Int_stable = this
```
```   994
```
```   995   show "indep_var S X T Y" unfolding indep_var_eq
```
```   996   proof (intro conjI indep_set_sigma_sets Int_stable rvs)
```
```   997     show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
```
```   998     proof (safe intro!: indep_setI)
```
```   999       { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
```
```  1000         using `X \<in> measurable M S` by (auto intro: measurable_sets) }
```
```  1001       { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
```
```  1002         using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
```
```  1003     next
```
```  1004       fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
```
```  1005       then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)"
```
```  1006         using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"])
```
```  1007       also have "\<dots> = emeasure (?S \<Otimes>\<^isub>M ?T) (A \<times> B)"
```
```  1008         unfolding `?S \<Otimes>\<^isub>M ?T = ?J` ..
```
```  1009       also have "\<dots> = emeasure ?S A * emeasure ?T B"
```
```  1010         using ab by (simp add: Y.emeasure_pair_measure_Times)
```
```  1011       finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
```
```  1012         prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
```
```  1013         using rvs ab by (simp add: emeasure_eq_measure emeasure_distr)
```
```  1014     qed
```
```  1015   qed
```
```  1016 qed
```
```  1017
```
```  1018 lemma (in prob_space) distributed_joint_indep:
```
```  1019   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
```
```  1020   assumes X: "distributed M S X Px" and Y: "distributed M T Y Py"
```
```  1021   assumes indep: "indep_var S X T Y"
```
```  1022   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
```
```  1023   using indep_var_distribution_eq[of S X T Y] indep
```
```  1024   by (intro distributed_joint_indep'[OF S T X Y]) auto
```
```  1025
```
```  1026 end
```