1(* Title: HOL/Probability/Independent_Family.thy 2 Author: Johannes H��lzl, TU M��nchen 3 Author: Sudeep Kanav, TU M��nchen 4*) 5 6section \<open>Independent families of events, event sets, and random variables\<close> 7 8theory Independent_Family 9 imports Infinite_Product_Measure 10begin 11 12definition (in prob_space) 13 "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and> 14 (\<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))))" 15 16definition (in prob_space) 17 "indep_set A B \<longleftrightarrow> indep_sets (case_bool A B) UNIV" 18 19definition (in prob_space) 20 indep_events_def_alt: "indep_events A I \<longleftrightarrow> indep_sets (\<lambda>i. {A i}) I" 21 22lemma (in prob_space) indep_events_def: 23 "indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and> 24 (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))" 25 unfolding indep_events_def_alt indep_sets_def 26 apply (simp add: Ball_def Pi_iff image_subset_iff_funcset) 27 apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong) 28 apply auto 29 done 30 31lemma (in prob_space) indep_eventsI: 32 "(\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M) \<Longrightarrow> (\<And>J. J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> J \<noteq> {} \<Longrightarrow> prob (\<Inter>i\<in>J. F i) = (\<Prod>i\<in>J. prob (F i))) \<Longrightarrow> indep_events F I" 33 by (auto simp: indep_events_def) 34 35definition (in prob_space) 36 "indep_event A B \<longleftrightarrow> indep_events (case_bool A B) UNIV" 37 38lemma (in prob_space) indep_sets_cong: 39 "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J" 40 by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+ 41 42lemma (in prob_space) indep_events_finite_index_events: 43 "indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)" 44 by (auto simp: indep_events_def) 45 46lemma (in prob_space) indep_sets_finite_index_sets: 47 "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)" 48proof (intro iffI allI impI) 49 assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J" 50 show "indep_sets F I" unfolding indep_sets_def 51 proof (intro conjI ballI allI impI) 52 fix i assume "i \<in> I" 53 with *[THEN spec, of "{i}"] show "F i \<subseteq> events" 54 by (auto simp: indep_sets_def) 55 qed (insert *, auto simp: indep_sets_def) 56qed (auto simp: indep_sets_def) 57 58lemma (in prob_space) indep_sets_mono_index: 59 "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J" 60 unfolding indep_sets_def by auto 61 62lemma (in prob_space) indep_sets_mono_sets: 63 assumes indep: "indep_sets F I" 64 assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i" 65 shows "indep_sets G I" 66proof - 67 have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)" 68 using mono by auto 69 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)" 70 using mono by (auto simp: Pi_iff) 71 ultimately show ?thesis 72 using indep by (auto simp: indep_sets_def) 73qed 74 75lemma (in prob_space) indep_sets_mono: 76 assumes indep: "indep_sets F I" 77 assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i" 78 shows "indep_sets G J" 79 apply (rule indep_sets_mono_sets) 80 apply (rule indep_sets_mono_index) 81 apply (fact +) 82 done 83 84lemma (in prob_space) indep_setsI: 85 assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" 86 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))" 87 shows "indep_sets F I" 88 using assms unfolding indep_sets_def by (auto simp: Pi_iff) 89 90lemma (in prob_space) indep_setsD: 91 assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j" 92 shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" 93 using assms unfolding indep_sets_def by auto 94 95lemma (in prob_space) indep_setI: 96 assumes ev: "A \<subseteq> events" "B \<subseteq> events" 97 and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b" 98 shows "indep_set A B" 99 unfolding indep_set_def 100proof (rule indep_setsI) 101 fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV" 102 and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)" 103 have "J \<in> Pow UNIV" by auto 104 with F \<open>J \<noteq> {}\<close> indep[of "F True" "F False"] 105 show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))" 106 unfolding UNIV_bool Pow_insert by (auto simp: ac_simps) 107qed (auto split: bool.split simp: ev) 108 109lemma (in prob_space) indep_setD: 110 assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B" 111 shows "prob (a \<inter> b) = prob a * prob b" 112 using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "case_bool a b"] ev 113 by (simp add: ac_simps UNIV_bool) 114 115lemma (in prob_space) 116 assumes indep: "indep_set A B" 117 shows indep_setD_ev1: "A \<subseteq> events" 118 and indep_setD_ev2: "B \<subseteq> events" 119 using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto 120 121lemma (in prob_space) indep_sets_Dynkin: 122 assumes indep: "indep_sets F I" 123 shows "indep_sets (\<lambda>i. Dynkin (space M) (F i)) I" 124 (is "indep_sets ?F I") 125proof (subst indep_sets_finite_index_sets, intro allI impI ballI) 126 fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}" 127 with indep have "indep_sets F J" 128 by (subst (asm) indep_sets_finite_index_sets) auto 129 { fix J K assume "indep_sets F K" 130 let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i" 131 assume "finite J" "J \<subseteq> K" 132 then have "indep_sets (?G J) K" 133 proof induct 134 case (insert j J) 135 moreover define G where "G = ?G J" 136 ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K" 137 by (auto simp: indep_sets_def) 138 let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }" 139 { fix X assume X: "X \<in> events" 140 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) 141 \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))" 142 have "indep_sets (G(j := {X})) K" 143 proof (rule indep_setsI) 144 fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events" 145 using G X by auto 146 next 147 fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i" 148 show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" 149 proof cases 150 assume "j \<in> J" 151 with J have "A j = X" by auto 152 show ?thesis 153 proof cases 154 assume "J = {j}" then show ?thesis by simp 155 next 156 assume "J \<noteq> {j}" 157 have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)" 158 using \<open>j \<in> J\<close> \<open>A j = X\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_asm) 159 also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))" 160 proof (rule indep) 161 show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}" 162 using J \<open>J \<noteq> {j}\<close> \<open>j \<in> J\<close> by auto 163 show "\<forall>i\<in>J - {j}. A i \<in> G i" 164 using J by auto 165 qed 166 also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))" 167 using \<open>A j = X\<close> by simp 168 also have "\<dots> = (\<Prod>i\<in>J. prob (A i))" 169 unfolding prod.insert_remove[OF \<open>finite J\<close>, symmetric, of "\<lambda>i. prob (A i)"] 170 using \<open>j \<in> J\<close> by (simp add: insert_absorb) 171 finally show ?thesis . 172 qed 173 next 174 assume "j \<notin> J" 175 with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: if_split_asm) 176 with J show ?thesis 177 by (intro indep_setsD[OF G(1)]) auto 178 qed 179 qed } 180 note indep_sets_insert = this 181 have "Dynkin_system (space M) ?D" 182 proof (rule Dynkin_systemI', simp_all cong del: indep_sets_cong, safe) 183 show "indep_sets (G(j := {{}})) K" 184 by (rule indep_sets_insert) auto 185 next 186 fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K" 187 show "indep_sets (G(j := {space M - X})) K" 188 proof (rule indep_sets_insert) 189 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" 190 then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events" 191 using G by auto 192 have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = 193 prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))" 194 using A_sets sets.sets_into_space[of _ M] X \<open>J \<noteq> {}\<close> 195 by (auto intro!: arg_cong[where f=prob] split: if_split_asm) 196 also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" 197 using J \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> A_sets X sets.sets_into_space 198 by (auto intro!: finite_measure_Diff sets.finite_INT split: if_split_asm) 199 finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = 200 prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" . 201 moreover { 202 have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" 203 using J A \<open>finite J\<close> by (intro indep_setsD[OF G(1)]) auto 204 then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))" 205 using prob_space by simp } 206 moreover { 207 have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))" 208 using J A \<open>j \<in> K\<close> by (intro indep_setsD[OF G']) auto 209 then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))" 210 using \<open>finite J\<close> \<open>j \<notin> J\<close> by (auto intro!: prod.cong) } 211 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))" 212 by (simp add: field_simps) 213 also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" 214 using X A by (simp add: finite_measure_compl) 215 finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" . 216 qed (insert X, auto) 217 next 218 fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D" 219 then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto 220 show "indep_sets (G(j := {\<Union>k. F k})) K" 221 proof (rule indep_sets_insert) 222 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" 223 then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events" 224 using G by auto 225 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))" 226 using \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> \<open>j \<in> K\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_asm) 227 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))" 228 proof (rule finite_measure_UNION) 229 show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)" 230 using disj by (rule disjoint_family_on_bisimulation) auto 231 show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events" 232 using A_sets F \<open>finite J\<close> \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> by (auto intro!: sets.Int) 233 qed 234 moreover { fix k 235 from J A \<open>j \<in> K\<close> have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))" 236 by (subst indep_setsD[OF F(2)]) (auto intro!: prod.cong split: if_split_asm) 237 also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)" 238 using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1)]) auto 239 finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . } 240 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)))" 241 by simp 242 moreover 243 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))" 244 using disj F(1) by (intro finite_measure_UNION sums_mult2) auto 245 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)))" 246 using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1), symmetric]) auto 247 ultimately 248 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))" 249 by (auto dest!: sums_unique) 250 qed (insert F, auto) 251 qed (insert sets.sets_into_space, auto) 252 then have mono: "Dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}" 253 proof (rule Dynkin_system.Dynkin_subset, safe) 254 fix X assume "X \<in> G j" 255 then show "X \<in> events" using G \<open>j \<in> K\<close> by auto 256 from \<open>indep_sets G K\<close> 257 show "indep_sets (G(j := {X})) K" 258 by (rule indep_sets_mono_sets) (insert \<open>X \<in> G j\<close>, auto) 259 qed 260 have "indep_sets (G(j:=?D)) K" 261 proof (rule indep_setsI) 262 fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events" 263 using G(2) by auto 264 next 265 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" 266 show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" 267 proof cases 268 assume "j \<in> J" 269 with A have indep: "indep_sets (G(j := {A j})) K" by auto 270 from J A show ?thesis 271 by (intro indep_setsD[OF indep]) auto 272 next 273 assume "j \<notin> J" 274 with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: if_split_asm) 275 with J show ?thesis 276 by (intro indep_setsD[OF G(1)]) auto 277 qed 278 qed 279 then have "indep_sets (G(j := Dynkin (space M) (G j))) K" 280 by (rule indep_sets_mono_sets) (insert mono, auto) 281 then show ?case 282 by (rule indep_sets_mono_sets) (insert \<open>j \<in> K\<close> \<open>j \<notin> J\<close>, auto simp: G_def) 283 qed (insert \<open>indep_sets F K\<close>, simp) } 284 from this[OF \<open>indep_sets F J\<close> \<open>finite J\<close> subset_refl] 285 show "indep_sets ?F J" 286 by (rule indep_sets_mono_sets) auto 287qed 288 289lemma (in prob_space) indep_sets_sigma: 290 assumes indep: "indep_sets F I" 291 assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" 292 shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" 293proof - 294 from indep_sets_Dynkin[OF indep] 295 show ?thesis 296 proof (rule indep_sets_mono_sets, subst sigma_eq_Dynkin, simp_all add: stable) 297 fix i assume "i \<in> I" 298 with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def) 299 with sets.sets_into_space show "F i \<subseteq> Pow (space M)" by auto 300 qed 301qed 302 303lemma (in prob_space) indep_sets_sigma_sets_iff: 304 assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" 305 shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I" 306proof 307 assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" 308 by (rule indep_sets_sigma) fact 309next 310 assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I" 311 by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic) 312qed 313 314definition (in prob_space) 315 indep_vars_def2: "indep_vars M' X I \<longleftrightarrow> 316 (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> 317 indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I" 318 319definition (in prob_space) 320 "indep_var Ma A Mb B \<longleftrightarrow> indep_vars (case_bool Ma Mb) (case_bool A B) UNIV" 321 322lemma (in prob_space) indep_vars_def: 323 "indep_vars M' X I \<longleftrightarrow> 324 (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> 325 indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I" 326 unfolding indep_vars_def2 327 apply (rule conj_cong[OF refl]) 328 apply (rule indep_sets_sigma_sets_iff[symmetric]) 329 apply (auto simp: Int_stable_def) 330 apply (rule_tac x="A \<inter> Aa" in exI) 331 apply auto 332 done 333 334lemma (in prob_space) indep_var_eq: 335 "indep_var S X T Y \<longleftrightarrow> 336 (random_variable S X \<and> random_variable T Y) \<and> 337 indep_set 338 (sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S}) 339 (sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})" 340 unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool 341 by (intro arg_cong2[where f="(\<and>)"] arg_cong2[where f=indep_sets] ext) 342 (auto split: bool.split) 343 344lemma (in prob_space) indep_sets2_eq: 345 "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)" 346 unfolding indep_set_def 347proof (intro iffI ballI conjI) 348 assume indep: "indep_sets (case_bool A B) UNIV" 349 { fix a b assume "a \<in> A" "b \<in> B" 350 with indep_setsD[OF indep, of UNIV "case_bool a b"] 351 show "prob (a \<inter> b) = prob a * prob b" 352 unfolding UNIV_bool by (simp add: ac_simps) } 353 from indep show "A \<subseteq> events" "B \<subseteq> events" 354 unfolding indep_sets_def UNIV_bool by auto 355next 356 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)" 357 show "indep_sets (case_bool A B) UNIV" 358 proof (rule indep_setsI) 359 fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events" 360 using * by (auto split: bool.split) 361 next 362 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)" 363 then have "J = {True} \<or> J = {False} \<or> J = {True,False}" 364 by (auto simp: UNIV_bool) 365 then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))" 366 using X * by auto 367 qed 368qed 369 370lemma (in prob_space) indep_set_sigma_sets: 371 assumes "indep_set A B" 372 assumes A: "Int_stable A" and B: "Int_stable B" 373 shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)" 374proof - 375 have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV" 376 proof (rule indep_sets_sigma) 377 show "indep_sets (case_bool A B) UNIV" 378 by (rule \<open>indep_set A B\<close>[unfolded indep_set_def]) 379 fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)" 380 using A B by (cases i) auto 381 qed 382 then show ?thesis 383 unfolding indep_set_def 384 by (rule indep_sets_mono_sets) (auto split: bool.split) 385qed 386 387lemma (in prob_space) indep_eventsI_indep_vars: 388 assumes indep: "indep_vars N X I" 389 assumes P: "\<And>i. i \<in> I \<Longrightarrow> {x\<in>space (N i). P i x} \<in> sets (N i)" 390 shows "indep_events (\<lambda>i. {x\<in>space M. P i (X i x)}) I" 391proof - 392 have "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (N i)}) I" 393 using indep unfolding indep_vars_def2 by auto 394 then show ?thesis 395 unfolding indep_events_def_alt 396 proof (rule indep_sets_mono_sets) 397 fix i assume "i \<in> I" 398 then have "{{x \<in> space M. P i (X i x)}} = {X i -` {x\<in>space (N i). P i x} \<inter> space M}" 399 using indep by (auto simp: indep_vars_def dest: measurable_space) 400 also have "\<dots> \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}" 401 using P[OF \<open>i \<in> I\<close>] by blast 402 finally show "{{x \<in> space M. P i (X i x)}} \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}" . 403 qed 404qed 405 406lemma (in prob_space) indep_sets_collect_sigma: 407 fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set" 408 assumes indep: "indep_sets E (\<Union>j\<in>J. I j)" 409 assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)" 410 assumes disjoint: "disjoint_family_on I J" 411 shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J" 412proof - 413 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) }" 414 415 from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events" 416 unfolding indep_sets_def by auto 417 { fix j 418 let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)" 419 assume "j \<in> J" 420 from E[OF this] interpret S: sigma_algebra "space M" ?S 421 using sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto 422 423 have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)" 424 proof (rule sigma_sets_eqI) 425 fix A assume "A \<in> (\<Union>i\<in>I j. E i)" 426 then guess i .. 427 then show "A \<in> sigma_sets (space M) (?E j)" 428 by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"]) 429 next 430 fix A assume "A \<in> ?E j" 431 then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k" 432 and A: "A = (\<Inter>k\<in>K. E' k)" 433 by auto 434 then have "A \<in> ?S" unfolding A 435 by (safe intro!: S.finite_INT) auto 436 then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)" 437 by simp 438 qed } 439 moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J" 440 proof (rule indep_sets_sigma) 441 show "indep_sets ?E J" 442 proof (intro indep_setsI) 443 fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force intro!: sets.finite_INT) 444 next 445 fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K" 446 and "\<forall>j\<in>K. A j \<in> ?E j" 447 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)" 448 by simp 449 from bchoice[OF this] guess E' .. 450 from bchoice[OF this] obtain L 451 where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)" 452 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" 453 and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l" 454 by auto 455 456 { fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k" 457 have "k = j" 458 proof (rule ccontr) 459 assume "k \<noteq> j" 460 with disjoint \<open>K \<subseteq> J\<close> \<open>k \<in> K\<close> \<open>j \<in> K\<close> have "I k \<inter> I j = {}" 461 unfolding disjoint_family_on_def by auto 462 with L(2,3)[OF \<open>j \<in> K\<close>] L(2,3)[OF \<open>k \<in> K\<close>] 463 show False using \<open>l \<in> L k\<close> \<open>l \<in> L j\<close> by auto 464 qed } 465 note L_inj = this 466 467 define k where "k l = (SOME k. k \<in> K \<and> l \<in> L k)" for l 468 { fix x j l assume *: "j \<in> K" "l \<in> L j" 469 have "k l = j" unfolding k_def 470 proof (rule some_equality) 471 fix k assume "k \<in> K \<and> l \<in> L k" 472 with * L_inj show "k = j" by auto 473 qed (insert *, simp) } 474 note k_simp[simp] = this 475 let ?E' = "\<lambda>l. E' (k l) l" 476 have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)" 477 by (auto simp: A intro!: arg_cong[where f=prob]) 478 also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))" 479 using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono) 480 also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))" 481 using K L L_inj by (subst prod.UNION_disjoint) auto 482 also have "\<dots> = (\<Prod>j\<in>K. prob (A j))" 483 using K L E' by (auto simp add: A intro!: prod.cong indep_setsD[OF indep, symmetric]) blast 484 finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" . 485 qed 486 next 487 fix j assume "j \<in> J" 488 show "Int_stable (?E j)" 489 proof (rule Int_stableI) 490 fix a assume "a \<in> ?E j" then obtain Ka Ea 491 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 492 fix b assume "b \<in> ?E j" then obtain Kb Eb 493 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 494 let ?f = "\<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 {})" 495 have "Ka \<union> Kb = (Ka \<inter> Kb) \<union> (Kb - Ka) \<union> (Ka - Kb)" 496 by blast 497 moreover have "(\<Inter>x\<in>Ka \<inter> Kb. Ea x \<inter> Eb x) \<inter> 498 (\<Inter>x\<in>Kb - Ka. Eb x) \<inter> (\<Inter>x\<in>Ka - Kb. Ea x) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)" 499 by auto 500 ultimately have "(\<Inter>k\<in>Ka \<union> Kb. ?f k) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)" (is "?lhs = ?rhs") 501 by (simp only: image_Un Inter_Un_distrib) simp 502 then have "a \<inter> b = (\<Inter>k\<in>Ka \<union> Kb. ?f k)" 503 by (simp only: a(1) b(1)) 504 with a b \<open>j \<in> J\<close> Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j" 505 by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?f]) auto 506 qed 507 qed 508 ultimately show ?thesis 509 by (simp cong: indep_sets_cong) 510qed 511 512lemma (in prob_space) indep_vars_restrict: 513 assumes ind: "indep_vars M' X I" and K: "\<And>j. j \<in> L \<Longrightarrow> K j \<subseteq> I" and J: "disjoint_family_on K L" 514 shows "indep_vars (\<lambda>j. PiM (K j) M') (\<lambda>j \<omega>. restrict (\<lambda>i. X i \<omega>) (K j)) L" 515 unfolding indep_vars_def 516proof safe 517 fix j assume "j \<in> L" then show "random_variable (Pi\<^sub>M (K j) M') (\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>)" 518 using K ind by (auto simp: indep_vars_def intro!: measurable_restrict) 519next 520 have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (M' i)" 521 using ind by (auto simp: indep_vars_def) 522 let ?proj = "\<lambda>j S. {(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` A \<inter> space M |A. A \<in> S}" 523 let ?UN = "\<lambda>j. sigma_sets (space M) (\<Union>i\<in>K j. { X i -` A \<inter> space M| A. A \<in> sets (M' i) })" 524 show "indep_sets (\<lambda>i. sigma_sets (space M) (?proj i (sets (Pi\<^sub>M (K i) M')))) L" 525 proof (rule indep_sets_mono_sets) 526 fix j assume j: "j \<in> L" 527 have "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) = 528 sigma_sets (space M) (sigma_sets (space M) (?proj j (prod_algebra (K j) M')))" 529 using j K X[THEN measurable_space] unfolding sets_PiM 530 by (subst sigma_sets_vimage_commute) (auto simp add: Pi_iff) 531 also have "\<dots> = sigma_sets (space M) (?proj j (prod_algebra (K j) M'))" 532 by (rule sigma_sets_sigma_sets_eq) auto 533 also have "\<dots> \<subseteq> ?UN j" 534 proof (rule sigma_sets_mono, safe del: disjE elim!: prod_algebraE) 535 fix J E assume J: "finite J" "J \<noteq> {} \<or> K j = {}" "J \<subseteq> K j" and E: "\<forall>i. i \<in> J \<longrightarrow> E i \<in> sets (M' i)" 536 show "(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \<inter> space M \<in> ?UN j" 537 proof cases 538 assume "K j = {}" with J show ?thesis 539 by (auto simp add: sigma_sets_empty_eq prod_emb_def) 540 next 541 assume "K j \<noteq> {}" with J have "J \<noteq> {}" 542 by auto 543 { interpret sigma_algebra "space M" "?UN j" 544 by (rule sigma_algebra_sigma_sets) auto 545 have "\<And>A. (\<And>i. i \<in> J \<Longrightarrow> A i \<in> ?UN j) \<Longrightarrow> \<Inter>(A ` J) \<in> ?UN j" 546 using \<open>finite J\<close> \<open>J \<noteq> {}\<close> by (rule finite_INT) blast } 547 note INT = this 548 549 from \<open>J \<noteq> {}\<close> J K E[rule_format, THEN sets.sets_into_space] j 550 have "(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \<inter> space M 551 = (\<Inter>i\<in>J. X i -` E i \<inter> space M)" 552 apply (subst prod_emb_PiE[OF _ ]) 553 apply auto [] 554 apply auto [] 555 apply (auto simp add: Pi_iff intro!: X[THEN measurable_space]) 556 apply (erule_tac x=i in ballE) 557 apply auto 558 done 559 also have "\<dots> \<in> ?UN j" 560 apply (rule INT) 561 apply (rule sigma_sets.Basic) 562 using \<open>J \<subseteq> K j\<close> E 563 apply auto 564 done 565 finally show ?thesis . 566 qed 567 qed 568 finally show "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) \<subseteq> ?UN j" . 569 next 570 show "indep_sets ?UN L" 571 proof (rule indep_sets_collect_sigma) 572 show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) (\<Union>j\<in>L. K j)" 573 proof (rule indep_sets_mono_index) 574 show "indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" 575 using ind unfolding indep_vars_def2 by auto 576 show "(\<Union>l\<in>L. K l) \<subseteq> I" 577 using K by auto 578 qed 579 next 580 fix l i assume "l \<in> L" "i \<in> K l" 581 show "Int_stable {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" 582 apply (auto simp: Int_stable_def) 583 apply (rule_tac x="A \<inter> Aa" in exI) 584 apply auto 585 done 586 qed fact 587 qed 588qed 589 590lemma (in prob_space) indep_var_restrict: 591 assumes ind: "indep_vars M' X I" and AB: "A \<inter> B = {}" "A \<subseteq> I" "B \<subseteq> I" 592 shows "indep_var (PiM A M') (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) A) (PiM B M') (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) B)" 593proof - 594 have *: 595 "case_bool (Pi\<^sub>M A M') (Pi\<^sub>M B M') = (\<lambda>b. PiM (case_bool A B b) M')" 596 "case_bool (\<lambda>\<omega>. \<lambda>i\<in>A. X i \<omega>) (\<lambda>\<omega>. \<lambda>i\<in>B. X i \<omega>) = (\<lambda>b \<omega>. \<lambda>i\<in>case_bool A B b. X i \<omega>)" 597 by (simp_all add: fun_eq_iff split: bool.split) 598 show ?thesis 599 unfolding indep_var_def * using AB 600 by (intro indep_vars_restrict[OF ind]) (auto simp: disjoint_family_on_def split: bool.split) 601qed 602 603lemma (in prob_space) indep_vars_subset: 604 assumes "indep_vars M' X I" "J \<subseteq> I" 605 shows "indep_vars M' X J" 606 using assms unfolding indep_vars_def indep_sets_def 607 by auto 608 609lemma (in prob_space) indep_vars_cong: 610 "I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> X i = Y i) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> M' i = N' i) \<Longrightarrow> indep_vars M' X I \<longleftrightarrow> indep_vars N' Y J" 611 unfolding indep_vars_def2 by (intro conj_cong indep_sets_cong) auto 612 613definition (in prob_space) tail_events where 614 "tail_events A = (\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))" 615 616lemma (in prob_space) tail_events_sets: 617 assumes A: "\<And>i::nat. A i \<subseteq> events" 618 shows "tail_events A \<subseteq> events" 619proof 620 fix X assume X: "X \<in> tail_events A" 621 let ?A = "(\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))" 622 from X have "\<And>n::nat. X \<in> sigma_sets (space M) (\<Union> (A ` {n..}))" by (auto simp: tail_events_def) 623 from this[of 0] have "X \<in> sigma_sets (space M) (\<Union>(A ` UNIV))" by simp 624 then show "X \<in> events" 625 by induct (insert A, auto) 626qed 627 628lemma (in prob_space) sigma_algebra_tail_events: 629 assumes "\<And>i::nat. sigma_algebra (space M) (A i)" 630 shows "sigma_algebra (space M) (tail_events A)" 631 unfolding tail_events_def 632proof (simp add: sigma_algebra_iff2, safe) 633 let ?A = "(\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))" 634 interpret A: sigma_algebra "space M" "A i" for i by fact 635 { fix X x assume "X \<in> ?A" "x \<in> X" 636 then have "\<And>n. X \<in> sigma_sets (space M) (\<Union> (A ` {n..}))" by auto 637 from this[of 0] have "X \<in> sigma_sets (space M) (\<Union>(A ` UNIV))" by simp 638 then have "X \<subseteq> space M" 639 by induct (insert A.sets_into_space, auto) 640 with \<open>x \<in> X\<close> show "x \<in> space M" by auto } 641 { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A" 642 then show "(\<Union>(F ` UNIV)) \<in> sigma_sets (space M) (\<Union> (A ` {n..}))" 643 by (intro sigma_sets.Union) auto } 644qed (auto intro!: sigma_sets.Compl sigma_sets.Empty) 645 646lemma (in prob_space) kolmogorov_0_1_law: 647 fixes A :: "nat \<Rightarrow> 'a set set" 648 assumes "\<And>i::nat. sigma_algebra (space M) (A i)" 649 assumes indep: "indep_sets A UNIV" 650 and X: "X \<in> tail_events A" 651 shows "prob X = 0 \<or> prob X = 1" 652proof - 653 have A: "\<And>i. A i \<subseteq> events" 654 using indep unfolding indep_sets_def by simp 655 656 let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}" 657 interpret A: sigma_algebra "space M" "A i" for i by fact 658 interpret T: sigma_algebra "space M" "tail_events A" 659 by (rule sigma_algebra_tail_events) fact 660 have "X \<subseteq> space M" using T.space_closed X by auto 661 662 have X_in: "X \<in> events" 663 using tail_events_sets A X by auto 664 665 interpret D: Dynkin_system "space M" ?D 666 proof (rule Dynkin_systemI) 667 fix D assume "D \<in> ?D" then show "D \<subseteq> space M" 668 using sets.sets_into_space by auto 669 next 670 show "space M \<in> ?D" 671 using prob_space \<open>X \<subseteq> space M\<close> by (simp add: Int_absorb2) 672 next 673 fix A assume A: "A \<in> ?D" 674 have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))" 675 using \<open>X \<subseteq> space M\<close> by (auto intro!: arg_cong[where f=prob]) 676 also have "\<dots> = prob X - prob (X \<inter> A)" 677 using X_in A by (intro finite_measure_Diff) auto 678 also have "\<dots> = prob X * prob (space M) - prob X * prob A" 679 using A prob_space by auto 680 also have "\<dots> = prob X * prob (space M - A)" 681 using X_in A sets.sets_into_space 682 by (subst finite_measure_Diff) (auto simp: field_simps) 683 finally show "space M - A \<in> ?D" 684 using A \<open>X \<subseteq> space M\<close> by auto 685 next 686 fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D" 687 then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)" 688 by auto 689 have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)" 690 proof (rule finite_measure_UNION) 691 show "range (\<lambda>i. X \<inter> F i) \<subseteq> events" 692 using F X_in by auto 693 show "disjoint_family (\<lambda>i. X \<inter> F i)" 694 using dis by (rule disjoint_family_on_bisimulation) auto 695 qed 696 with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))" 697 by simp 698 moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))" 699 by (intro sums_mult finite_measure_UNION F dis) 700 ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)" 701 by (auto dest!: sums_unique) 702 with F show "(\<Union>i. F i) \<in> ?D" 703 by auto 704 qed 705 706 { fix n 707 have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>case_bool {..n} {Suc n..} b. A m)) UNIV" 708 proof (rule indep_sets_collect_sigma) 709 have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _") 710 by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq) 711 with indep show "indep_sets A ?U" by simp 712 show "disjoint_family (case_bool {..n} {Suc n..})" 713 unfolding disjoint_family_on_def by (auto split: bool.split) 714 fix m 715 show "Int_stable (A m)" 716 unfolding Int_stable_def using A.Int by auto 717 qed 718 also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>case_bool {..n} {Suc n..} b. A m)) = 719 case_bool (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))" 720 by (auto intro!: ext split: bool.split) 721 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))" 722 unfolding indep_set_def by simp 723 724 have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D" 725 proof (simp add: subset_eq, rule) 726 fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" 727 have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)" 728 using X unfolding tail_events_def by simp 729 from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D 730 show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D" 731 by (auto simp add: ac_simps) 732 qed } 733 then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _") 734 by auto 735 736 note \<open>X \<in> tail_events A\<close> 737 also { 738 have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A" 739 by (intro sigma_sets_subseteq UN_mono) auto 740 then have "tail_events A \<subseteq> sigma_sets (space M) ?A" 741 unfolding tail_events_def by auto } 742 also have "sigma_sets (space M) ?A = Dynkin (space M) ?A" 743 proof (rule sigma_eq_Dynkin) 744 { fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" 745 then have "B \<subseteq> space M" 746 by induct (insert A sets.sets_into_space[of _ M], auto) } 747 then show "?A \<subseteq> Pow (space M)" by auto 748 show "Int_stable ?A" 749 proof (rule Int_stableI) 750 fix a assume "a \<in> ?A" then guess n .. note a = this 751 fix b assume "b \<in> ?A" then guess m .. note b = this 752 interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" 753 using A sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto 754 have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" 755 by (intro sigma_sets_subseteq UN_mono) auto 756 with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto 757 moreover 758 have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" 759 by (intro sigma_sets_subseteq UN_mono) auto 760 with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto 761 ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" 762 using Amn.Int[of a b] by simp 763 then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto 764 qed 765 qed 766 also have "Dynkin (space M) ?A \<subseteq> ?D" 767 using \<open>?A \<subseteq> ?D\<close> by (auto intro!: D.Dynkin_subset) 768 finally show ?thesis by auto 769qed 770 771lemma (in prob_space) borel_0_1_law: 772 fixes F :: "nat \<Rightarrow> 'a set" 773 assumes F2: "indep_events F UNIV" 774 shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1" 775proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"]) 776 have F1: "range F \<subseteq> events" 777 using F2 by (simp add: indep_events_def subset_eq) 778 { fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})" 779 using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets.sets_into_space 780 by auto } 781 show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV" 782 proof (rule indep_sets_sigma) 783 show "indep_sets (\<lambda>i. {F i}) UNIV" 784 unfolding indep_events_def_alt[symmetric] by fact 785 fix i show "Int_stable {F i}" 786 unfolding Int_stable_def by simp 787 qed 788 let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i" 789 show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})" 790 unfolding tail_events_def 791 proof 792 fix j 793 interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" 794 using order_trans[OF F1 sets.space_closed] 795 by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq) 796 have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)" 797 by (intro decseq_SucI INT_decseq_offset UN_mono) auto 798 also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" 799 using order_trans[OF F1 sets.space_closed] 800 by (safe intro!: S.countable_INT S.countable_UN) 801 (auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI) 802 finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" 803 by simp 804 qed 805qed 806 807lemma (in prob_space) borel_0_1_law_AE: 808 fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool" 809 assumes "indep_events (\<lambda>m. {x\<in>space M. P m x}) UNIV" (is "indep_events ?P _") 810 shows "(AE x in M. infinite {m. P m x}) \<or> (AE x in M. finite {m. P m x})" 811proof - 812 have [measurable]: "\<And>m. {x\<in>space M. P m x} \<in> sets M" 813 using assms by (auto simp: indep_events_def) 814 have *: "(\<Inter>n. \<Union>m\<in>{n..}. {x \<in> space M. P m x}) \<in> events" 815 by simp 816 from assms have "prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 1" 817 by (rule borel_0_1_law) 818 also have "prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 1 \<longleftrightarrow> (AE x in M. infinite {m. P m x})" 819 using * by (simp add: prob_eq_1) 820 (simp add: Bex_def infinite_nat_iff_unbounded_le) 821 also have "prob (\<Inter>n. \<Union>m\<in>{n..}. ?P m) = 0 \<longleftrightarrow> (AE x in M. finite {m. P m x})" 822 using * by (simp add: prob_eq_0) 823 (auto simp add: Ball_def finite_nat_iff_bounded not_less [symmetric]) 824 finally show ?thesis 825 by blast 826qed 827 828lemma (in prob_space) indep_sets_finite: 829 assumes I: "I \<noteq> {}" "finite I" 830 and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" 831 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)))" 832proof 833 assume *: "indep_sets F I" 834 from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" 835 by (intro indep_setsD[OF *] ballI) auto 836next 837 assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" 838 show "indep_sets F I" 839 proof (rule indep_setsI[OF F(1)]) 840 fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" 841 assume A: "\<forall>j\<in>J. A j \<in> F j" 842 let ?A = "\<lambda>j. if j \<in> J then A j else space M" 843 have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)" 844 using subset_trans[OF F(1) sets.space_closed] J A 845 by (auto intro!: arg_cong[where f=prob] split: if_split_asm) blast 846 also 847 from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _") 848 by (auto split: if_split_asm) 849 with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))" 850 by auto 851 also have "\<dots> = (\<Prod>j\<in>J. prob (A j))" 852 unfolding if_distrib prod.If_cases[OF \<open>finite I\<close>] 853 using prob_space \<open>J \<subseteq> I\<close> by (simp add: Int_absorb1 prod.neutral_const) 854 finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" .. 855 qed 856qed 857 858lemma (in prob_space) indep_vars_finite: 859 fixes I :: "'i set" 860 assumes I: "I \<noteq> {}" "finite I" 861 and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)" 862 and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)" 863 and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)" 864 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))" 865 shows "indep_vars M' X I \<longleftrightarrow> 866 (\<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)))" 867proof - 868 from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)" 869 unfolding measurable_def by simp 870 871 { fix i assume "i\<in>I" 872 from closed[OF \<open>i \<in> I\<close>] 873 have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} 874 = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}" 875 unfolding sigma_sets_vimage_commute[OF X, OF \<open>i \<in> I\<close>, symmetric] M'[OF \<open>i \<in> I\<close>] 876 by (subst sigma_sets_sigma_sets_eq) auto } 877 note sigma_sets_X = this 878 879 { fix i assume "i\<in>I" 880 have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}" 881 proof (rule Int_stableI) 882 fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}" 883 then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto 884 moreover 885 fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}" 886 then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto 887 moreover 888 have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto 889 moreover note Int_stable[OF \<open>i \<in> I\<close>] 890 ultimately 891 show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}" 892 by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD) 893 qed } 894 note indep_sets_X = indep_sets_sigma_sets_iff[OF this] 895 896 { fix i assume "i \<in> I" 897 { fix A assume "A \<in> E i" 898 with M'[OF \<open>i \<in> I\<close>] have "A \<in> sets (M' i)" by auto 899 moreover 900 from rv[OF \<open>i\<in>I\<close>] have "X i \<in> measurable M (M' i)" by auto 901 ultimately 902 have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) } 903 with X[OF \<open>i\<in>I\<close>] space[OF \<open>i\<in>I\<close>] 904 have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events" 905 "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}" 906 by (auto intro!: exI[of _ "space (M' i)"]) } 907 note indep_sets_finite_X = indep_sets_finite[OF I this] 908 909 have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (\<Inter>(A ` I)) = (\<Prod>j\<in>I. prob (A j))) = 910 (\<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)))" 911 (is "?L = ?R") 912 proof safe 913 fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)" 914 from \<open>?L\<close>[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A \<open>I \<noteq> {}\<close> 915 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))" 916 by (auto simp add: Pi_iff) 917 next 918 fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})" 919 from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto 920 from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M" 921 "B \<in> (\<Pi> i\<in>I. E i)" by auto 922 from \<open>?R\<close>[THEN bspec, OF B(2)] B(1) \<open>I \<noteq> {}\<close> 923 show "prob (\<Inter>(A ` I)) = (\<Prod>j\<in>I. prob (A j))" 924 by simp 925 qed 926 then show ?thesis using \<open>I \<noteq> {}\<close> 927 by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong) 928qed 929 930lemma (in prob_space) indep_vars_compose: 931 assumes "indep_vars M' X I" 932 assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)" 933 shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I" 934 unfolding indep_vars_def 935proof 936 from rv \<open>indep_vars M' X I\<close> 937 show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)" 938 by (auto simp: indep_vars_def) 939 940 have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" 941 using \<open>indep_vars M' X I\<close> by (simp add: indep_vars_def) 942 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" 943 proof (rule indep_sets_mono_sets) 944 fix i assume "i \<in> I" 945 with \<open>indep_vars M' X I\<close> have X: "X i \<in> space M \<rightarrow> space (M' i)" 946 unfolding indep_vars_def measurable_def by auto 947 { fix A assume "A \<in> sets (N i)" 948 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)" 949 by (intro exI[of _ "Y i -` A \<inter> space (M' i)"]) 950 (auto simp: vimage_comp intro!: measurable_sets rv \<open>i \<in> I\<close> funcset_mem[OF X]) } 951 then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq> 952 sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" 953 by (intro sigma_sets_subseteq) (auto simp: vimage_comp) 954 qed 955qed 956 957lemma (in prob_space) indep_vars_compose2: 958 assumes "indep_vars M' X I" 959 assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)" 960 shows "indep_vars N (\<lambda>i x. Y i (X i x)) I" 961 using indep_vars_compose [OF assms] by (simp add: comp_def) 962 963lemma (in prob_space) indep_var_compose: 964 assumes "indep_var M1 X1 M2 X2" "Y1 \<in> measurable M1 N1" "Y2 \<in> measurable M2 N2" 965 shows "indep_var N1 (Y1 \<circ> X1) N2 (Y2 \<circ> X2)" 966proof - 967 have "indep_vars (case_bool N1 N2) (\<lambda>b. case_bool Y1 Y2 b \<circ> case_bool X1 X2 b) UNIV" 968 using assms 969 by (intro indep_vars_compose[where M'="case_bool M1 M2"]) 970 (auto simp: indep_var_def split: bool.split) 971 also have "(\<lambda>b. case_bool Y1 Y2 b \<circ> case_bool X1 X2 b) = case_bool (Y1 \<circ> X1) (Y2 \<circ> X2)" 972 by (simp add: fun_eq_iff split: bool.split) 973 finally show ?thesis 974 unfolding indep_var_def . 975qed 976 977lemma (in prob_space) indep_vars_Min: 978 fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real" 979 assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)" 980 shows "indep_var borel (X i) borel (\<lambda>\<omega>. Min ((\<lambda>i. X i \<omega>)`I))" 981proof - 982 have "indep_var 983 borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) 984 borel ((\<lambda>f. Min (f`I)) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))" 985 using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] borel_measurable_Min) auto 986 also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i" 987 by auto 988 also have "((\<lambda>f. Min (f`I)) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I)) = (\<lambda>\<omega>. Min ((\<lambda>i. X i \<omega>)`I))" 989 by (auto cong: rev_conj_cong) 990 finally show ?thesis 991 unfolding indep_var_def . 992qed 993 994lemma (in prob_space) indep_vars_sum: 995 fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real" 996 assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)" 997 shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Sum>i\<in>I. X i \<omega>)" 998proof - 999 have "indep_var 1000 borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) 1001 borel ((\<lambda>f. \<Sum>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))" 1002 using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto 1003 also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i" 1004 by auto 1005 also have "((\<lambda>f. \<Sum>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I)) = (\<lambda>\<omega>. \<Sum>i\<in>I. X i \<omega>)" 1006 by (auto cong: rev_conj_cong) 1007 finally show ?thesis . 1008qed 1009 1010lemma (in prob_space) indep_vars_prod: 1011 fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> real" 1012 assumes I: "finite I" "i \<notin> I" and indep: "indep_vars (\<lambda>_. borel) X (insert i I)" 1013 shows "indep_var borel (X i) borel (\<lambda>\<omega>. \<Prod>i\<in>I. X i \<omega>)" 1014proof - 1015 have "indep_var 1016 borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) 1017 borel ((\<lambda>f. \<Prod>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))" 1018 using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto 1019 also have "((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i})) = X i" 1020 by auto 1021 also have "((\<lambda>f. \<Prod>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I)) = (\<lambda>\<omega>. \<Prod>i\<in>I. X i \<omega>)" 1022 by (auto cong: rev_conj_cong) 1023 finally show ?thesis . 1024qed 1025 1026lemma (in prob_space) indep_varsD_finite: 1027 assumes X: "indep_vars M' X I" 1028 assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)" 1029 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))" 1030proof (rule indep_setsD) 1031 show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" 1032 using X by (auto simp: indep_vars_def) 1033 show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto 1034 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)}" 1035 using I by auto 1036qed 1037 1038lemma (in prob_space) indep_varsD: 1039 assumes X: "indep_vars M' X I" 1040 assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)" 1041 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))" 1042proof (rule indep_setsD) 1043 show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" 1044 using X by (auto simp: indep_vars_def) 1045 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)}" 1046 using I by auto 1047qed fact+ 1048 1049lemma (in prob_space) indep_vars_iff_distr_eq_PiM: 1050 fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b" 1051 assumes "I \<noteq> {}" 1052 assumes rv: "\<And>i. random_variable (M' i) (X i)" 1053 shows "indep_vars M' X I \<longleftrightarrow> 1054 distr M (\<Pi>\<^sub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i))" 1055proof - 1056 let ?P = "\<Pi>\<^sub>M i\<in>I. M' i" 1057 let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x" 1058 let ?D = "distr M ?P ?X" 1059 have X: "random_variable ?P ?X" by (intro measurable_restrict rv) 1060 interpret D: prob_space ?D by (intro prob_space_distr X) 1061 1062 let ?D' = "\<lambda>i. distr M (M' i) (X i)" 1063 let ?P' = "\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i)" 1064 interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv) 1065 interpret P: product_prob_space ?D' I .. 1066 1067 show ?thesis 1068 proof 1069 assume "indep_vars M' X I" 1070 show "?D = ?P'" 1071 proof (rule measure_eqI_generator_eq) 1072 show "Int_stable (prod_algebra I M')" 1073 by (rule Int_stable_prod_algebra) 1074 show "prod_algebra I M' \<subseteq> Pow (space ?P)" 1075 using prod_algebra_sets_into_space by (simp add: space_PiM) 1076 show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')" 1077 by (simp add: sets_PiM space_PiM) 1078 show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')" 1079 by (simp add: sets_PiM space_PiM cong: prod_algebra_cong) 1080 let ?A = "\<lambda>i. \<Pi>\<^sub>E i\<in>I. space (M' i)" 1081 show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^sub>M I M')" 1082 by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong) 1083 { fix i show "emeasure ?D (\<Pi>\<^sub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto } 1084 next 1085 fix E assume E: "E \<in> prod_algebra I M'" 1086 from prod_algebraE[OF E] guess J Y . note J = this 1087 1088 from E have "E \<in> sets ?P" by (auto simp: sets_PiM) 1089 then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)" 1090 by (simp add: emeasure_distr X) 1091 also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)" 1092 using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm) 1093 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))" 1094 using \<open>indep_vars M' X I\<close> J \<open>I \<noteq> {}\<close> using indep_varsD[of M' X I J] 1095 by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg) 1096 also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))" 1097 using rv J by (simp add: emeasure_distr) 1098 also have "\<dots> = emeasure ?P' E" 1099 using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def) 1100 finally show "emeasure ?D E = emeasure ?P' E" . 1101 qed 1102 next 1103 assume "?D = ?P'" 1104 show "indep_vars M' X I" unfolding indep_vars_def 1105 proof (intro conjI indep_setsI ballI rv) 1106 fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events" 1107 by (auto intro!: sets.sigma_sets_subset measurable_sets rv) 1108 next 1109 fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" 1110 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)}" 1111 have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)" 1112 proof 1113 fix j assume "j \<in> J" 1114 from Y'[rule_format, OF this] rv[of j] 1115 show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)" 1116 by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"]) 1117 (auto dest: measurable_space simp: sets.sigma_sets_eq) 1118 qed 1119 from bchoice[OF this] obtain Y where 1120 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 1121 let ?E = "prod_emb I M' J (Pi\<^sub>E J Y)" 1122 from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M" 1123 using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm) 1124 then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)" 1125 by simp 1126 also have "\<dots> = emeasure ?D ?E" 1127 using Y J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto 1128 also have "\<dots> = emeasure ?P' ?E" 1129 using \<open>?D = ?P'\<close> by simp 1130 also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))" 1131 using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def) 1132 also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))" 1133 using rv J Y by (simp add: emeasure_distr) 1134 finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" . 1135 then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))" 1136 by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg) 1137 qed 1138 qed 1139qed 1140 1141lemma (in prob_space) indep_varD: 1142 assumes indep: "indep_var Ma A Mb B" 1143 assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb" 1144 shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = 1145 prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)" 1146proof - 1147 have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = 1148 prob (\<Inter>i\<in>UNIV. (case_bool A B i -` case_bool Xa Xb i \<inter> space M))" 1149 by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool) 1150 also have "\<dots> = (\<Prod>i\<in>UNIV. prob (case_bool A B i -` case_bool Xa Xb i \<inter> space M))" 1151 using indep unfolding indep_var_def 1152 by (rule indep_varsD) (auto split: bool.split intro: sets) 1153 also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)" 1154 unfolding UNIV_bool by simp 1155 finally show ?thesis . 1156qed 1157 1158lemma (in prob_space) prob_indep_random_variable: 1159 assumes ind[simp]: "indep_var N X N Y" 1160 assumes [simp]: "A \<in> sets N" "B \<in> sets N" 1161 shows "\<P>(x in M. X x \<in> A \<and> Y x \<in> B) = \<P>(x in M. X x \<in> A) * \<P>(x in M. Y x \<in> B)" 1162proof- 1163 have " \<P>(x in M. (X x)\<in>A \<and> (Y x)\<in> B ) = prob ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)" 1164 by (auto intro!: arg_cong[where f= prob]) 1165 also have "...= prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)" 1166 by (auto intro!: indep_varD[where Ma=N and Mb=N]) 1167 also have "... = \<P>(x in M. X x \<in> A) * \<P>(x in M. Y x \<in> B)" 1168 by (auto intro!: arg_cong2[where f= "(*)"] arg_cong[where f= prob]) 1169 finally show ?thesis . 1170qed 1171 1172lemma (in prob_space) 1173 assumes "indep_var S X T Y" 1174 shows indep_var_rv1: "random_variable S X" 1175 and indep_var_rv2: "random_variable T Y" 1176proof - 1177 have "\<forall>i\<in>UNIV. random_variable (case_bool S T i) (case_bool X Y i)" 1178 using assms unfolding indep_var_def indep_vars_def by auto 1179 then show "random_variable S X" "random_variable T Y" 1180 unfolding UNIV_bool by auto 1181qed 1182 1183lemma (in prob_space) indep_var_distribution_eq: 1184 "indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and> 1185 distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^sub>M ?T = ?J") 1186proof safe 1187 assume "indep_var S X T Y" 1188 then show rvs: "random_variable S X" "random_variable T Y" 1189 by (blast dest: indep_var_rv1 indep_var_rv2)+ 1190 then have XY: "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" 1191 by (rule measurable_Pair) 1192 1193 interpret X: prob_space ?S by (rule prob_space_distr) fact 1194 interpret Y: prob_space ?T by (rule prob_space_distr) fact 1195 interpret XY: pair_prob_space ?S ?T .. 1196 show "?S \<Otimes>\<^sub>M ?T = ?J" 1197 proof (rule pair_measure_eqI) 1198 show "sigma_finite_measure ?S" .. 1199 show "sigma_finite_measure ?T" .. 1200 1201 fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T" 1202 have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)" 1203 using A B by (intro emeasure_distr[OF XY]) auto 1204 also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)" 1205 using indep_varD[OF \<open>indep_var S X T Y\<close>, of A B] A B 1206 by (simp add: emeasure_eq_measure measure_nonneg ennreal_mult) 1207 also have "\<dots> = emeasure ?S A * emeasure ?T B" 1208 using rvs A B by (simp add: emeasure_distr) 1209 finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp 1210 qed simp 1211next 1212 assume rvs: "random_variable S X" "random_variable T Y" 1213 then have XY: "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" 1214 by (rule measurable_Pair) 1215 1216 let ?S = "distr M S X" and ?T = "distr M T Y" 1217 interpret X: prob_space ?S by (rule prob_space_distr) fact 1218 interpret Y: prob_space ?T by (rule prob_space_distr) fact 1219 interpret XY: pair_prob_space ?S ?T .. 1220 1221 assume "?S \<Otimes>\<^sub>M ?T = ?J" 1222 1223 { fix S and X 1224 have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}" 1225 proof (safe intro!: Int_stableI) 1226 fix A B assume "A \<in> sets S" "B \<in> sets S" 1227 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" 1228 by (intro exI[of _ "A \<inter> B"]) auto 1229 qed } 1230 note Int_stable = this 1231 1232 show "indep_var S X T Y" unfolding indep_var_eq 1233 proof (intro conjI indep_set_sigma_sets Int_stable rvs) 1234 show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}" 1235 proof (safe intro!: indep_setI) 1236 { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M" 1237 using \<open>X \<in> measurable M S\<close> by (auto intro: measurable_sets) } 1238 { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M" 1239 using \<open>Y \<in> measurable M T\<close> by (auto intro: measurable_sets) } 1240 next 1241 fix A B assume ab: "A \<in> sets S" "B \<in> sets T" 1242 then have "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) = emeasure ?J (A \<times> B)" 1243 using XY by (auto simp add: emeasure_distr emeasure_eq_measure measure_nonneg intro!: arg_cong[where f="prob"]) 1244 also have "\<dots> = emeasure (?S \<Otimes>\<^sub>M ?T) (A \<times> B)" 1245 unfolding \<open>?S \<Otimes>\<^sub>M ?T = ?J\<close> .. 1246 also have "\<dots> = emeasure ?S A * emeasure ?T B" 1247 using ab by (simp add: Y.emeasure_pair_measure_Times) 1248 finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) = 1249 prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)" 1250 using rvs ab by (simp add: emeasure_eq_measure emeasure_distr measure_nonneg ennreal_mult[symmetric]) 1251 qed 1252 qed 1253qed 1254 1255lemma (in prob_space) distributed_joint_indep: 1256 assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" 1257 assumes X: "distributed M S X Px" and Y: "distributed M T Y Py" 1258 assumes indep: "indep_var S X T Y" 1259 shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)" 1260 using indep_var_distribution_eq[of S X T Y] indep 1261 by (intro distributed_joint_indep'[OF S T X Y]) auto 1262 1263lemma (in prob_space) indep_vars_nn_integral: 1264 assumes I: "finite I" "indep_vars (\<lambda>_. borel) X I" "\<And>i \<omega>. i \<in> I \<Longrightarrow> 0 \<le> X i \<omega>" 1265 shows "(\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<Prod>i\<in>I. \<integral>\<^sup>+\<omega>. X i \<omega> \<partial>M)" 1266proof cases 1267 assume "I \<noteq> {}" 1268 define Y where [abs_def]: "Y i \<omega> = (if i \<in> I then X i \<omega> else 0)" for i \<omega> 1269 { fix i have "i \<in> I \<Longrightarrow> random_variable borel (X i)" 1270 using I(2) by (cases "i\<in>I") (auto simp: indep_vars_def) } 1271 note rv_X = this 1272 1273 { fix i have "random_variable borel (Y i)" 1274 using I(2) by (cases "i\<in>I") (auto simp: Y_def rv_X) } 1275 note rv_Y = this[measurable] 1276 1277 interpret Y: prob_space "distr M borel (Y i)" for i 1278 using I(2) by (cases "i \<in> I") (auto intro!: prob_space_distr simp: indep_vars_def prob_space_return) 1279 interpret product_sigma_finite "\<lambda>i. distr M borel (Y i)" 1280 .. 1281 1282 have indep_Y: "indep_vars (\<lambda>i. borel) Y I" 1283 by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def) 1284 1285 have "(\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. Y i \<omega>) \<partial>M)" 1286 using I(3) by (auto intro!: nn_integral_cong prod.cong simp add: Y_def max_def) 1287 also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x))" 1288 by (subst nn_integral_distr) auto 1289 also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))" 1290 unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] .. 1291 also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^sup>+\<omega>. \<omega> \<partial>distr M borel (Y i)))" 1292 by (rule product_nn_integral_prod) (auto intro: \<open>finite I\<close>) 1293 also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<^sup>+\<omega>. X i \<omega> \<partial>M)" 1294 by (intro prod.cong nn_integral_cong) (auto simp: nn_integral_distr Y_def rv_X) 1295 finally show ?thesis . 1296qed (simp add: emeasure_space_1) 1297 1298lemma (in prob_space) 1299 fixes X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{real_normed_field, banach, second_countable_topology}" 1300 assumes I: "finite I" "indep_vars (\<lambda>_. borel) X I" "\<And>i. i \<in> I \<Longrightarrow> integrable M (X i)" 1301 shows indep_vars_lebesgue_integral: "(\<integral>\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<Prod>i\<in>I. \<integral>\<omega>. X i \<omega> \<partial>M)" (is ?eq) 1302 and indep_vars_integrable: "integrable M (\<lambda>\<omega>. (\<Prod>i\<in>I. X i \<omega>))" (is ?int) 1303proof (induct rule: case_split) 1304 assume "I \<noteq> {}" 1305 define Y where [abs_def]: "Y i \<omega> = (if i \<in> I then X i \<omega> else 0)" for i \<omega> 1306 { fix i have "i \<in> I \<Longrightarrow> random_variable borel (X i)" 1307 using I(2) by (cases "i\<in>I") (auto simp: indep_vars_def) } 1308 note rv_X = this[measurable] 1309 1310 { fix i have "random_variable borel (Y i)" 1311 using I(2) by (cases "i\<in>I") (auto simp: Y_def rv_X) } 1312 note rv_Y = this[measurable] 1313 1314 { fix i have "integrable M (Y i)" 1315 using I(3) by (cases "i\<in>I") (auto simp: Y_def) } 1316 note int_Y = this 1317 1318 interpret Y: prob_space "distr M borel (Y i)" for i 1319 using I(2) by (cases "i \<in> I") (auto intro!: prob_space_distr simp: indep_vars_def prob_space_return) 1320 interpret product_sigma_finite "\<lambda>i. distr M borel (Y i)" 1321 .. 1322 1323 have indep_Y: "indep_vars (\<lambda>i. borel) Y I" 1324 by (rule indep_vars_cong[THEN iffD1, OF _ _ _ I(2)]) (auto simp: Y_def) 1325 1326 have "(\<integral>\<omega>. (\<Prod>i\<in>I. X i \<omega>) \<partial>M) = (\<integral>\<omega>. (\<Prod>i\<in>I. Y i \<omega>) \<partial>M)" 1327 using I(3) by (simp add: Y_def) 1328 also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x))" 1329 by (subst integral_distr) auto 1330 also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))" 1331 unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] .. 1332 also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<omega>. \<omega> \<partial>distr M borel (Y i)))" 1333 by (rule product_integral_prod) (auto intro: \<open>finite I\<close> simp: integrable_distr_eq int_Y) 1334 also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<omega>. X i \<omega> \<partial>M)" 1335 by (intro prod.cong integral_cong) 1336 (auto simp: integral_distr Y_def rv_X) 1337 finally show ?eq . 1338 1339 have "integrable (distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x)) (\<lambda>\<omega>. (\<Prod>i\<in>I. \<omega> i))" 1340 unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] 1341 by (intro product_integrable_prod[OF \<open>finite I\<close>]) 1342 (simp add: integrable_distr_eq int_Y) 1343 then show ?int 1344 by (simp add: integrable_distr_eq Y_def) 1345qed (simp_all add: prob_space) 1346 1347lemma (in prob_space) 1348 fixes X1 X2 :: "'a \<Rightarrow> 'b::{real_normed_field, banach, second_countable_topology}" 1349 assumes "indep_var borel X1 borel X2" "integrable M X1" "integrable M X2" 1350 shows indep_var_lebesgue_integral: "(\<integral>\<omega>. X1 \<omega> * X2 \<omega> \<partial>M) = (\<integral>\<omega>. X1 \<omega> \<partial>M) * (\<integral>\<omega>. X2 \<omega> \<partial>M)" (is ?eq) 1351 and indep_var_integrable: "integrable M (\<lambda>\<omega>. X1 \<omega> * X2 \<omega>)" (is ?int) 1352unfolding indep_var_def 1353proof - 1354 have *: "(\<lambda>\<omega>. X1 \<omega> * X2 \<omega>) = (\<lambda>\<omega>. \<Prod>i\<in>UNIV. (case_bool X1 X2 i \<omega>))" 1355 by (simp add: UNIV_bool mult.commute) 1356 have **: "(\<lambda> _. borel) = case_bool borel borel" 1357 by (rule ext, metis (full_types) bool.simps(3) bool.simps(4)) 1358 show ?eq 1359 apply (subst *) 1360 apply (subst indep_vars_lebesgue_integral) 1361 apply (auto) 1362 apply (subst **, subst indep_var_def [symmetric], rule assms) 1363 apply (simp split: bool.split add: assms) 1364 by (simp add: UNIV_bool mult.commute) 1365 show ?int 1366 apply (subst *) 1367 apply (rule indep_vars_integrable) 1368 apply auto 1369 apply (subst **, subst indep_var_def [symmetric], rule assms) 1370 by (simp split: bool.split add: assms) 1371qed 1372 1373end 1374