1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Probability/Fin_Map.thy Thu Nov 15 11:16:58 2012 +0100
1.3 @@ -0,0 +1,1503 @@
1.4 +(* Title: HOL/Probability/Projective_Family.thy
1.5 + Author: Fabian Immler, TU München
1.6 +*)
1.7 +
1.8 +theory Fin_Map
1.9 +imports Finite_Product_Measure
1.10 +begin
1.11 +
1.12 +section {* Finite Maps *}
1.13 +
1.14 +text {* Auxiliary type that is instantiated to @{class polish_space}, needed for the proof of
1.15 + projective limit. @{const extensional} functions are used for the representation in order to
1.16 + stay close to the developments of (finite) products @{const Pi\<^isub>E} and their sigma-algebra
1.17 + @{const Pi\<^isub>M}. *}
1.18 +
1.19 +typedef ('i, 'a) finmap ("(_ \<Rightarrow>\<^isub>F /_)" [22, 21] 21) =
1.20 + "{(I::'i set, f::'i \<Rightarrow> 'a). finite I \<and> f \<in> extensional I}" by auto
1.21 +
1.22 +subsection {* Domain and Application *}
1.23 +
1.24 +definition domain where "domain P = fst (Rep_finmap P)"
1.25 +
1.26 +lemma finite_domain[simp, intro]: "finite (domain P)"
1.27 + by (cases P) (auto simp: domain_def Abs_finmap_inverse)
1.28 +
1.29 +definition proj ("_\<^isub>F" [1000] 1000) where "proj P i = snd (Rep_finmap P) i"
1.30 +
1.31 +declare [[coercion proj]]
1.32 +
1.33 +lemma extensional_proj[simp, intro]: "(P)\<^isub>F \<in> extensional (domain P)"
1.34 + by (cases P) (auto simp: domain_def Abs_finmap_inverse proj_def[abs_def])
1.35 +
1.36 +lemma proj_undefined[simp, intro]: "i \<notin> domain P \<Longrightarrow> P i = undefined"
1.37 + using extensional_proj[of P] unfolding extensional_def by auto
1.38 +
1.39 +lemma finmap_eq_iff: "P = Q \<longleftrightarrow> (domain P = domain Q \<and> (\<forall>i\<in>domain P. P i = Q i))"
1.40 + by (cases P, cases Q)
1.41 + (auto simp add: Abs_finmap_inject extensional_def domain_def proj_def Abs_finmap_inverse
1.42 + intro: extensionalityI)
1.43 +
1.44 +subsection {* Countable Finite Maps *}
1.45 +
1.46 +instance finmap :: (countable, countable) countable
1.47 +proof
1.48 + obtain mapper where mapper: "\<And>fm :: 'a \<Rightarrow>\<^isub>F 'b. set (mapper fm) = domain fm"
1.49 + by (metis finite_list[OF finite_domain])
1.50 + have "inj (\<lambda>fm. map (\<lambda>i. (i, (fm)\<^isub>F i)) (mapper fm))" (is "inj ?F")
1.51 + proof (rule inj_onI)
1.52 + fix f1 f2 assume "?F f1 = ?F f2"
1.53 + then have "map fst (?F f1) = map fst (?F f2)" by simp
1.54 + then have "mapper f1 = mapper f2" by (simp add: comp_def)
1.55 + then have "domain f1 = domain f2" by (simp add: mapper[symmetric])
1.56 + with `?F f1 = ?F f2` show "f1 = f2"
1.57 + unfolding `mapper f1 = mapper f2` map_eq_conv mapper
1.58 + by (simp add: finmap_eq_iff)
1.59 + qed
1.60 + then show "\<exists>to_nat :: 'a \<Rightarrow>\<^isub>F 'b \<Rightarrow> nat. inj to_nat"
1.61 + by (intro exI[of _ "to_nat \<circ> ?F"] inj_comp) auto
1.62 +qed
1.63 +
1.64 +subsection {* Constructor of Finite Maps *}
1.65 +
1.66 +definition "finmap_of inds f = Abs_finmap (inds, restrict f inds)"
1.67 +
1.68 +lemma proj_finmap_of[simp]:
1.69 + assumes "finite inds"
1.70 + shows "(finmap_of inds f)\<^isub>F = restrict f inds"
1.71 + using assms
1.72 + by (auto simp: Abs_finmap_inverse finmap_of_def proj_def)
1.73 +
1.74 +lemma domain_finmap_of[simp]:
1.75 + assumes "finite inds"
1.76 + shows "domain (finmap_of inds f) = inds"
1.77 + using assms
1.78 + by (auto simp: Abs_finmap_inverse finmap_of_def domain_def)
1.79 +
1.80 +lemma finmap_of_eq_iff[simp]:
1.81 + assumes "finite i" "finite j"
1.82 + shows "finmap_of i m = finmap_of j n \<longleftrightarrow> i = j \<and> restrict m i = restrict n i"
1.83 + using assms
1.84 + apply (auto simp: finmap_eq_iff restrict_def) by metis
1.85 +
1.86 +lemma
1.87 + finmap_of_inj_on_extensional_finite:
1.88 + assumes "finite K"
1.89 + assumes "S \<subseteq> extensional K"
1.90 + shows "inj_on (finmap_of K) S"
1.91 +proof (rule inj_onI)
1.92 + fix x y::"'a \<Rightarrow> 'b"
1.93 + assume "finmap_of K x = finmap_of K y"
1.94 + hence "(finmap_of K x)\<^isub>F = (finmap_of K y)\<^isub>F" by simp
1.95 + moreover
1.96 + assume "x \<in> S" "y \<in> S" hence "x \<in> extensional K" "y \<in> extensional K" using assms by auto
1.97 + ultimately
1.98 + show "x = y" using assms by (simp add: extensional_restrict)
1.99 +qed
1.100 +
1.101 +lemma finmap_choice:
1.102 + assumes *: "\<And>i. i \<in> I \<Longrightarrow> \<exists>x. P i x" and I: "finite I"
1.103 + shows "\<exists>fm. domain fm = I \<and> (\<forall>i\<in>I. P i (fm i))"
1.104 +proof -
1.105 + have "\<exists>f. \<forall>i\<in>I. P i (f i)"
1.106 + unfolding bchoice_iff[symmetric] using * by auto
1.107 + then guess f ..
1.108 + with I show ?thesis
1.109 + by (intro exI[of _ "finmap_of I f"]) auto
1.110 +qed
1.111 +
1.112 +subsection {* Product set of Finite Maps *}
1.113 +
1.114 +text {* This is @{term Pi} for Finite Maps, most of this is copied *}
1.115 +
1.116 +definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^isub>F 'a) set" where
1.117 + "Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^isub>F i \<in> A i) } "
1.118 +
1.119 +syntax
1.120 + "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PI' _:_./ _)" 10)
1.121 +
1.122 +syntax (xsymbols)
1.123 + "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>' _\<in>_./ _)" 10)
1.124 +
1.125 +syntax (HTML output)
1.126 + "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>' _\<in>_./ _)" 10)
1.127 +
1.128 +translations
1.129 + "PI' x:A. B" == "CONST Pi' A (%x. B)"
1.130 +
1.131 +abbreviation
1.132 + finmapset :: "['a set, 'b set] => ('a \<Rightarrow>\<^isub>F 'b) set"
1.133 + (infixr "~>" 60) where
1.134 + "A ~> B \<equiv> Pi' A (%_. B)"
1.135 +
1.136 +notation (xsymbols)
1.137 + finmapset (infixr "\<leadsto>" 60)
1.138 +
1.139 +subsubsection{*Basic Properties of @{term Pi'}*}
1.140 +
1.141 +lemma Pi'_I[intro!]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"
1.142 + by (simp add: Pi'_def)
1.143 +
1.144 +lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"
1.145 + by (simp add:Pi'_def)
1.146 +
1.147 +lemma finmapsetI: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<leadsto> B"
1.148 + by (simp add: Pi_def)
1.149 +
1.150 +lemma Pi'_mem: "f\<in> Pi' A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
1.151 + by (simp add: Pi'_def)
1.152 +
1.153 +lemma Pi'_iff: "f \<in> Pi' I X \<longleftrightarrow> domain f = I \<and> (\<forall>i\<in>I. f i \<in> X i)"
1.154 + unfolding Pi'_def by auto
1.155 +
1.156 +lemma Pi'E [elim]:
1.157 + "f \<in> Pi' A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> domain f = A \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
1.158 + by(auto simp: Pi'_def)
1.159 +
1.160 +lemma in_Pi'_cong:
1.161 + "domain f = domain g \<Longrightarrow> (\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi' A B \<longleftrightarrow> g \<in> Pi' A B"
1.162 + by (auto simp: Pi'_def)
1.163 +
1.164 +lemma funcset_mem: "[|f \<in> A \<leadsto> B; x \<in> A|] ==> f x \<in> B"
1.165 + by (simp add: Pi'_def)
1.166 +
1.167 +lemma funcset_image: "f \<in> A \<leadsto> B ==> f ` A \<subseteq> B"
1.168 +by auto
1.169 +
1.170 +lemma Pi'_eq_empty[simp]:
1.171 + assumes "finite A" shows "(Pi' A B) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
1.172 + using assms
1.173 + apply (simp add: Pi'_def, auto)
1.174 + apply (drule_tac x = "finmap_of A (\<lambda>u. SOME y. y \<in> B u)" in spec, auto)
1.175 + apply (cut_tac P= "%y. y \<in> B i" in some_eq_ex, auto)
1.176 + done
1.177 +
1.178 +lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C"
1.179 + by (auto simp: Pi'_def)
1.180 +
1.181 +lemma Pi_Pi': "finite A \<Longrightarrow> (Pi\<^isub>E A B) = proj ` Pi' A B"
1.182 + apply (auto simp: Pi'_def Pi_def extensional_def)
1.183 + apply (rule_tac x = "finmap_of A (restrict x A)" in image_eqI)
1.184 + apply auto
1.185 + done
1.186 +
1.187 +subsection {* Metric Space of Finite Maps *}
1.188 +
1.189 +instantiation finmap :: (type, metric_space) metric_space
1.190 +begin
1.191 +
1.192 +definition dist_finmap where
1.193 + "dist P Q = (\<Sum>i\<in>domain P \<union> domain Q. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)) +
1.194 + card ((domain P - domain Q) \<union> (domain Q - domain P))"
1.195 +
1.196 +lemma dist_finmap_extend:
1.197 + assumes "finite X"
1.198 + shows "dist P Q = (\<Sum>i\<in>domain P \<union> domain Q \<union> X. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)) +
1.199 + card ((domain P - domain Q) \<union> (domain Q - domain P))"
1.200 + unfolding dist_finmap_def add_right_cancel
1.201 + using assms extensional_arb[of "(P)\<^isub>F"] extensional_arb[of "(Q)\<^isub>F" "domain Q"]
1.202 + by (intro setsum_mono_zero_cong_left) auto
1.203 +
1.204 +definition open_finmap :: "('a \<Rightarrow>\<^isub>F 'b) set \<Rightarrow> bool" where
1.205 + "open_finmap S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
1.206 +
1.207 +lemma add_eq_zero_iff[simp]:
1.208 + fixes a b::real
1.209 + assumes "a \<ge> 0" "b \<ge> 0"
1.210 + shows "a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
1.211 +using assms by auto
1.212 +
1.213 +lemma dist_le_1_imp_domain_eq:
1.214 + assumes "dist P Q < 1"
1.215 + shows "domain P = domain Q"
1.216 +proof -
1.217 + have "0 \<le> (\<Sum>i\<in>domain P \<union> domain Q. dist (P i) (Q i))"
1.218 + by (simp add: setsum_nonneg)
1.219 + with assms have "card (domain P - domain Q \<union> (domain Q - domain P)) = 0"
1.220 + unfolding dist_finmap_def by arith
1.221 + thus "domain P = domain Q" by auto
1.222 +qed
1.223 +
1.224 +lemma dist_proj:
1.225 + shows "dist ((x)\<^isub>F i) ((y)\<^isub>F i) \<le> dist x y"
1.226 +proof -
1.227 + have "dist (x i) (y i) = (\<Sum>i\<in>{i}. dist (x i) (y i))" by simp
1.228 + also have "\<dots> \<le> (\<Sum>i\<in>domain x \<union> domain y \<union> {i}. dist (x i) (y i))"
1.229 + by (intro setsum_mono2) auto
1.230 + also have "\<dots> \<le> dist x y" by (simp add: dist_finmap_extend[of "{i}"])
1.231 + finally show ?thesis by simp
1.232 +qed
1.233 +
1.234 +lemma open_Pi'I:
1.235 + assumes open_component: "\<And>i. i \<in> I \<Longrightarrow> open (A i)"
1.236 + shows "open (Pi' I A)"
1.237 +proof (subst open_finmap_def, safe)
1.238 + fix x assume x: "x \<in> Pi' I A"
1.239 + hence dim_x: "domain x = I" by (simp add: Pi'_def)
1.240 + hence [simp]: "finite I" unfolding dim_x[symmetric] by simp
1.241 + have "\<exists>ei. \<forall>i\<in>I. 0 < ei i \<and> (\<forall>y. dist y (x i) < ei i \<longrightarrow> y \<in> A i)"
1.242 + proof (safe intro!: bchoice)
1.243 + fix i assume i: "i \<in> I"
1.244 + moreover with open_component have "open (A i)" by simp
1.245 + moreover have "x i \<in> A i" using x i
1.246 + by (auto simp: proj_def)
1.247 + ultimately show "\<exists>e>0. \<forall>y. dist y (x i) < e \<longrightarrow> y \<in> A i"
1.248 + using x by (auto simp: open_dist Ball_def)
1.249 + qed
1.250 + then guess ei .. note ei = this
1.251 + def es \<equiv> "ei ` I"
1.252 + def e \<equiv> "if es = {} then 0.5 else min 0.5 (Min es)"
1.253 + from ei have "e > 0" using x
1.254 + by (auto simp add: e_def es_def Pi'_def Ball_def)
1.255 + moreover have "\<forall>y. dist y x < e \<longrightarrow> y \<in> Pi' I A"
1.256 + proof (intro allI impI)
1.257 + fix y
1.258 + assume "dist y x < e"
1.259 + also have "\<dots> < 1" by (auto simp: e_def)
1.260 + finally have "domain y = domain x" by (rule dist_le_1_imp_domain_eq)
1.261 + with dim_x have dims: "domain y = domain x" "domain x = I" by auto
1.262 + show "y \<in> Pi' I A"
1.263 + proof
1.264 + show "domain y = I" using dims by simp
1.265 + next
1.266 + fix i
1.267 + assume "i \<in> I"
1.268 + have "dist (y i) (x i) \<le> dist y x" using dims `i \<in> I`
1.269 + by (auto intro: dist_proj)
1.270 + also have "\<dots> < e" using `dist y x < e` dims
1.271 + by (simp add: dist_finmap_def)
1.272 + also have "e \<le> Min (ei ` I)" using dims `i \<in> I`
1.273 + by (auto simp: e_def es_def)
1.274 + also have "\<dots> \<le> ei i" using `i \<in> I` by (simp add: e_def)
1.275 + finally have "dist (y i) (x i) < ei i" .
1.276 + with ei `i \<in> I` show "y i \<in> A i" by simp
1.277 + qed
1.278 + qed
1.279 + ultimately
1.280 + show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> Pi' I A" by blast
1.281 +qed
1.282 +
1.283 +instance
1.284 +proof
1.285 + fix S::"('a \<Rightarrow>\<^isub>F 'b) set"
1.286 + show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
1.287 + unfolding open_finmap_def ..
1.288 +next
1.289 + fix P Q::"'a \<Rightarrow>\<^isub>F 'b"
1.290 + show "dist P Q = 0 \<longleftrightarrow> P = Q"
1.291 + by (auto simp: finmap_eq_iff dist_finmap_def setsum_nonneg setsum_nonneg_eq_0_iff)
1.292 +next
1.293 + fix P Q R::"'a \<Rightarrow>\<^isub>F 'b"
1.294 + let ?symdiff = "\<lambda>a b. domain a - domain b \<union> (domain b - domain a)"
1.295 + def E \<equiv> "domain P \<union> domain Q \<union> domain R"
1.296 + hence "finite E" by (simp add: E_def)
1.297 + have "card (?symdiff P Q) \<le> card (?symdiff P R \<union> ?symdiff Q R)"
1.298 + by (auto intro: card_mono)
1.299 + also have "\<dots> \<le> card (?symdiff P R) + card (?symdiff Q R)"
1.300 + by (subst card_Un_Int) auto
1.301 + finally have "dist P Q \<le> (\<Sum>i\<in>E. dist (P i) (R i) + dist (Q i) (R i)) +
1.302 + real (card (?symdiff P R) + card (?symdiff Q R))"
1.303 + unfolding dist_finmap_extend[OF `finite E`]
1.304 + by (intro add_mono) (auto simp: E_def intro: setsum_mono dist_triangle_le)
1.305 + also have "\<dots> \<le> dist P R + dist Q R"
1.306 + unfolding dist_finmap_extend[OF `finite E`] by (simp add: ac_simps E_def setsum_addf[symmetric])
1.307 + finally show "dist P Q \<le> dist P R + dist Q R" by simp
1.308 +qed
1.309 +
1.310 +end
1.311 +
1.312 +lemma open_restricted_space:
1.313 + shows "open {m. P (domain m)}"
1.314 +proof -
1.315 + have "{m. P (domain m)} = (\<Union>i \<in> Collect P. {m. domain m = i})" by auto
1.316 + also have "open \<dots>"
1.317 + proof (rule, safe, cases)
1.318 + fix i::"'a set"
1.319 + assume "finite i"
1.320 + hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def)
1.321 + also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`)
1.322 + finally show "open {m. domain m = i}" .
1.323 + next
1.324 + fix i::"'a set"
1.325 + assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto
1.326 + also have "open \<dots>" by simp
1.327 + finally show "open {m. domain m = i}" .
1.328 + qed
1.329 + finally show ?thesis .
1.330 +qed
1.331 +
1.332 +lemma closed_restricted_space:
1.333 + shows "closed {m. P (domain m)}"
1.334 +proof -
1.335 + have "{m. P (domain m)} = - (\<Union>i \<in> - Collect P. {m. domain m = i})" by auto
1.336 + also have "closed \<dots>"
1.337 + proof (rule, rule, rule, cases)
1.338 + fix i::"'a set"
1.339 + assume "finite i"
1.340 + hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def)
1.341 + also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`)
1.342 + finally show "open {m. domain m = i}" .
1.343 + next
1.344 + fix i::"'a set"
1.345 + assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto
1.346 + also have "open \<dots>" by simp
1.347 + finally show "open {m. domain m = i}" .
1.348 + qed
1.349 + finally show ?thesis .
1.350 +qed
1.351 +
1.352 +lemma continuous_proj:
1.353 + shows "continuous_on s (\<lambda>x. (x)\<^isub>F i)"
1.354 + unfolding continuous_on_topological
1.355 +proof safe
1.356 + fix x B assume "x \<in> s" "open B" "x i \<in> B"
1.357 + let ?A = "Pi' (domain x) (\<lambda>j. if i = j then B else UNIV)"
1.358 + have "open ?A" using `open B` by (auto intro: open_Pi'I)
1.359 + moreover have "x \<in> ?A" using `x i \<in> B` by auto
1.360 + moreover have "(\<forall>y\<in>s. y \<in> ?A \<longrightarrow> y i \<in> B)"
1.361 + proof (cases, safe)
1.362 + fix y assume "y \<in> s"
1.363 + assume "i \<notin> domain x" hence "undefined \<in> B" using `x i \<in> B`
1.364 + by simp
1.365 + moreover
1.366 + assume "y \<in> ?A" hence "domain y = domain x" by (simp add: Pi'_def)
1.367 + hence "y i = undefined" using `i \<notin> domain x` by simp
1.368 + ultimately
1.369 + show "y i \<in> B" by simp
1.370 + qed force
1.371 + ultimately
1.372 + show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> y i \<in> B)" by blast
1.373 +qed
1.374 +
1.375 +subsection {* Complete Space of Finite Maps *}
1.376 +
1.377 +lemma tendsto_dist_zero:
1.378 + assumes "(\<lambda>i. dist (f i) g) ----> 0"
1.379 + shows "f ----> g"
1.380 + using assms by (auto simp: tendsto_iff dist_real_def)
1.381 +
1.382 +lemma tendsto_dist_zero':
1.383 + assumes "(\<lambda>i. dist (f i) g) ----> x"
1.384 + assumes "0 = x"
1.385 + shows "f ----> g"
1.386 + using assms tendsto_dist_zero by simp
1.387 +
1.388 +lemma tendsto_finmap:
1.389 + fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^isub>F ('a::metric_space))"
1.390 + assumes ind_f: "\<And>n. domain (f n) = domain g"
1.391 + assumes proj_g: "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) ----> g i"
1.392 + shows "f ----> g"
1.393 + apply (rule tendsto_dist_zero')
1.394 + unfolding dist_finmap_def assms
1.395 + apply (rule tendsto_intros proj_g | simp)+
1.396 + done
1.397 +
1.398 +instance finmap :: (type, complete_space) complete_space
1.399 +proof
1.400 + fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^isub>F 'b"
1.401 + assume "Cauchy P"
1.402 + then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1"
1.403 + by (force simp: cauchy)
1.404 + def d \<equiv> "domain (P Nd)"
1.405 + with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto
1.406 + have [simp]: "finite d" unfolding d_def by simp
1.407 + def p \<equiv> "\<lambda>i n. (P n) i"
1.408 + def q \<equiv> "\<lambda>i. lim (p i)"
1.409 + def Q \<equiv> "finmap_of d q"
1.410 + have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_finmap_inverse)
1.411 + {
1.412 + fix i assume "i \<in> d"
1.413 + have "Cauchy (p i)" unfolding cauchy p_def
1.414 + proof safe
1.415 + fix e::real assume "0 < e"
1.416 + with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1"
1.417 + by (force simp: cauchy min_def)
1.418 + hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto
1.419 + with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear)
1.420 + show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e"
1.421 + proof (safe intro!: exI[where x="N"])
1.422 + fix n assume "N \<le> n" have "N \<le> N" by simp
1.423 + have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)"
1.424 + using dim[OF `N \<le> n`] dim[OF `N \<le> N`] `i \<in> d`
1.425 + by (auto intro!: dist_proj)
1.426 + also have "\<dots> < e" using N[OF `N \<le> n`] by simp
1.427 + finally show "dist ((P n) i) ((P N) i) < e" .
1.428 + qed
1.429 + qed
1.430 + hence "convergent (p i)" by (metis Cauchy_convergent_iff)
1.431 + hence "p i ----> q i" unfolding q_def convergent_def by (metis limI)
1.432 + } note p = this
1.433 + have "P ----> Q"
1.434 + proof (rule metric_LIMSEQ_I)
1.435 + fix e::real assume "0 < e"
1.436 + def e' \<equiv> "min 1 (e / (card d + 1))"
1.437 + hence "0 < e'" using `0 < e` by (auto simp: e'_def intro: divide_pos_pos)
1.438 + have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e'"
1.439 + proof (safe intro!: bchoice)
1.440 + fix i assume "i \<in> d"
1.441 + from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e'`]
1.442 + show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e'" .
1.443 + qed then guess ni .. note ni = this
1.444 + def N \<equiv> "max Nd (Max (ni ` d))"
1.445 + show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e"
1.446 + proof (safe intro!: exI[where x="N"])
1.447 + fix n assume "N \<le> n"
1.448 + hence "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q"
1.449 + using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse)
1.450 + hence "dist (P n) Q = (\<Sum>i\<in>d. dist ((P n) i) (Q i))" by (simp add: dist_finmap_def)
1.451 + also have "\<dots> \<le> (\<Sum>i\<in>d. e')"
1.452 + proof (intro setsum_mono less_imp_le)
1.453 + fix i assume "i \<in> d"
1.454 + hence "ni i \<le> Max (ni ` d)" by simp
1.455 + also have "\<dots> \<le> N" by (simp add: N_def)
1.456 + also have "\<dots> \<le> n" using `N \<le> n` .
1.457 + finally
1.458 + show "dist ((P n) i) (Q i) < e'"
1.459 + using ni `i \<in> d` by (auto simp: p_def q N_def)
1.460 + qed
1.461 + also have "\<dots> = card d * e'" by (simp add: real_eq_of_nat)
1.462 + also have "\<dots> < e" using `0 < e` by (simp add: e'_def field_simps min_def)
1.463 + finally show "dist (P n) Q < e" .
1.464 + qed
1.465 + qed
1.466 + thus "convergent P" by (auto simp: convergent_def)
1.467 +qed
1.468 +
1.469 +subsection {* Polish Space of Finite Maps *}
1.470 +
1.471 +instantiation finmap :: (countable, polish_space) polish_space
1.472 +begin
1.473 +
1.474 +definition enum_basis_finmap :: "nat \<Rightarrow> ('a \<Rightarrow>\<^isub>F 'b) set" where
1.475 + "enum_basis_finmap n =
1.476 + (let m = from_nat n::('a \<Rightarrow>\<^isub>F nat) in Pi' (domain m) (enum_basis o (m)\<^isub>F))"
1.477 +
1.478 +lemma range_enum_basis_eq:
1.479 + "range enum_basis_finmap = {Pi' I S|I S. finite I \<and> (\<forall>i \<in> I. S i \<in> range enum_basis)}"
1.480 +proof (auto simp: enum_basis_finmap_def[abs_def])
1.481 + fix S::"('a \<Rightarrow> 'b set)" and I
1.482 + assume "\<forall>i\<in>I. S i \<in> range enum_basis"
1.483 + hence "\<forall>i\<in>I. \<exists>n. S i = enum_basis n" by auto
1.484 + then obtain n where n: "\<forall>i\<in>I. S i = enum_basis (n i)"
1.485 + unfolding bchoice_iff by blast
1.486 + assume [simp]: "finite I"
1.487 + have "\<exists>fm. domain fm = I \<and> (\<forall>i\<in>I. n i = (fm i))"
1.488 + by (rule finmap_choice) auto
1.489 + then obtain m where "Pi' I S = Pi' (domain m) (enum_basis o m)"
1.490 + using n by (auto simp: Pi'_def)
1.491 + hence "Pi' I S = (let m = from_nat (to_nat m) in Pi' (domain m) (enum_basis \<circ> m))"
1.492 + by simp
1.493 + thus "Pi' I S \<in> range (\<lambda>n. let m = from_nat n in Pi' (domain m) (enum_basis \<circ> m))"
1.494 + by blast
1.495 +qed (metis finite_domain o_apply rangeI)
1.496 +
1.497 +lemma in_enum_basis_finmapI:
1.498 + assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> range enum_basis"
1.499 + shows "Pi' I S \<in> range enum_basis_finmap"
1.500 + using assms unfolding range_enum_basis_eq by auto
1.501 +
1.502 +lemma finmap_topological_basis:
1.503 + "topological_basis (range (enum_basis_finmap))"
1.504 +proof (subst topological_basis_iff, safe)
1.505 + fix n::nat
1.506 + show "open (enum_basis_finmap n::('a \<Rightarrow>\<^isub>F 'b) set)" using enumerable_basis
1.507 + by (auto intro!: open_Pi'I simp: topological_basis_def enum_basis_finmap_def Let_def)
1.508 +next
1.509 + fix O'::"('a \<Rightarrow>\<^isub>F 'b) set" and x
1.510 + assume "open O'" "x \<in> O'"
1.511 + then obtain e where e: "e > 0" "\<And>y. dist y x < e \<Longrightarrow> y \<in> O'" unfolding open_dist by blast
1.512 + def e' \<equiv> "e / (card (domain x) + 1)"
1.513 +
1.514 + have "\<exists>B.
1.515 + (\<forall>i\<in>domain x. x i \<in> enum_basis (B i) \<and> enum_basis (B i) \<subseteq> ball (x i) e')"
1.516 + proof (rule bchoice, safe)
1.517 + fix i assume "i \<in> domain x"
1.518 + have "open (ball (x i) e')" "x i \<in> ball (x i) e'" using e
1.519 + by (auto simp add: e'_def intro!: divide_pos_pos)
1.520 + from enumerable_basisE[OF this] guess b' .
1.521 + thus "\<exists>y. x i \<in> enum_basis y \<and>
1.522 + enum_basis y \<subseteq> ball (x i) e'" by auto
1.523 + qed
1.524 + then guess B .. note B = this
1.525 + def B' \<equiv> "Pi' (domain x) (\<lambda>i. enum_basis (B i)::'b set)"
1.526 + hence "B' \<in> range enum_basis_finmap" unfolding B'_def
1.527 + by (intro in_enum_basis_finmapI) auto
1.528 + moreover have "x \<in> B'" unfolding B'_def using B by auto
1.529 + moreover have "B' \<subseteq> O'"
1.530 + proof
1.531 + fix y assume "y \<in> B'" with B have "domain y = domain x" unfolding B'_def
1.532 + by (simp add: Pi'_def)
1.533 + show "y \<in> O'"
1.534 + proof (rule e)
1.535 + have "dist y x = (\<Sum>i \<in> domain x. dist (y i) (x i))"
1.536 + using `domain y = domain x` by (simp add: dist_finmap_def)
1.537 + also have "\<dots> \<le> (\<Sum>i \<in> domain x. e')"
1.538 + proof (rule setsum_mono)
1.539 + fix i assume "i \<in> domain x"
1.540 + with `y \<in> B'` B have "y i \<in> enum_basis (B i)"
1.541 + by (simp add: Pi'_def B'_def)
1.542 + hence "y i \<in> ball (x i) e'" using B `domain y = domain x` `i \<in> domain x`
1.543 + by force
1.544 + thus "dist (y i) (x i) \<le> e'" by (simp add: dist_commute)
1.545 + qed
1.546 + also have "\<dots> = card (domain x) * e'" by (simp add: real_eq_of_nat)
1.547 + also have "\<dots> < e" using e by (simp add: e'_def field_simps)
1.548 + finally show "dist y x < e" .
1.549 + qed
1.550 + qed
1.551 + ultimately
1.552 + show "\<exists>B'\<in>range enum_basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" by blast
1.553 +qed
1.554 +
1.555 +lemma range_enum_basis_finmap_imp_open:
1.556 + assumes "x \<in> range enum_basis_finmap"
1.557 + shows "open x"
1.558 + using finmap_topological_basis assms by (auto simp: topological_basis_def)
1.559 +
1.560 +lemma
1.561 + open_imp_ex_UNION_of_enum:
1.562 + fixes X::"('a \<Rightarrow>\<^isub>F 'b) set"
1.563 + assumes "open X" assumes "X \<noteq> {}"
1.564 + shows "\<exists>A::nat\<Rightarrow>'a set. \<exists>B::nat\<Rightarrow>('a \<Rightarrow> 'b set) . X = UNION UNIV (\<lambda>i. Pi' (A i) (B i)) \<and>
1.565 + (\<forall>n. \<forall>i\<in>A n. (B n) i \<in> range enum_basis) \<and> (\<forall>n. finite (A n))"
1.566 +proof -
1.567 + from `open X` obtain B' where B': "B'\<subseteq>range enum_basis_finmap" "\<Union>B' = X"
1.568 + using finmap_topological_basis by (force simp add: topological_basis_def)
1.569 + then obtain B where B: "B' = enum_basis_finmap ` B" by (auto simp: subset_image_iff)
1.570 + show ?thesis
1.571 + proof cases
1.572 + assume "B = {}" with B have "B' = {}" by simp hence False using B' assms by simp
1.573 + thus ?thesis by simp
1.574 + next
1.575 + assume "B \<noteq> {}" then obtain b where b: "b \<in> B" by auto
1.576 + def NA \<equiv> "\<lambda>n::nat. if n \<in> B
1.577 + then domain ((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) n)
1.578 + else domain ((from_nat::_\<Rightarrow>'a\<Rightarrow>\<^isub>F nat) b)"
1.579 + def NB \<equiv> "\<lambda>n::nat. if n \<in> B
1.580 + then (\<lambda>i. (enum_basis::nat\<Rightarrow>'b set) (((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) n) i))
1.581 + else (\<lambda>i. (enum_basis::nat\<Rightarrow>'b set) (((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) b) i))"
1.582 + have "X = UNION UNIV (\<lambda>i. Pi' (NA i) (NB i))" unfolding B'(2)[symmetric] using b
1.583 + unfolding B
1.584 + by safe
1.585 + (auto simp add: NA_def NB_def enum_basis_finmap_def Let_def o_def split: split_if_asm)
1.586 + moreover
1.587 + have "(\<forall>n. \<forall>i\<in>NA n. (NB n) i \<in> range enum_basis)"
1.588 + using enumerable_basis by (auto simp: topological_basis_def NA_def NB_def)
1.589 + moreover have "(\<forall>n. finite (NA n))" by (simp add: NA_def)
1.590 + ultimately show ?thesis by auto
1.591 + qed
1.592 +qed
1.593 +
1.594 +lemma
1.595 + open_imp_ex_UNION:
1.596 + fixes X::"('a \<Rightarrow>\<^isub>F 'b) set"
1.597 + assumes "open X" assumes "X \<noteq> {}"
1.598 + shows "\<exists>A::nat\<Rightarrow>'a set. \<exists>B::nat\<Rightarrow>('a \<Rightarrow> 'b set) . X = UNION UNIV (\<lambda>i. Pi' (A i) (B i)) \<and>
1.599 + (\<forall>n. \<forall>i\<in>A n. open ((B n) i)) \<and> (\<forall>n. finite (A n))"
1.600 + using open_imp_ex_UNION_of_enum[OF assms]
1.601 + apply auto
1.602 + apply (rule_tac x = A in exI)
1.603 + apply (rule_tac x = B in exI)
1.604 + apply (auto simp: open_enum_basis)
1.605 + done
1.606 +
1.607 +lemma
1.608 + open_basisE:
1.609 + assumes "open X" assumes "X \<noteq> {}"
1.610 + obtains A::"nat\<Rightarrow>'a set" and B::"nat\<Rightarrow>('a \<Rightarrow> 'b set)" where
1.611 + "X = UNION UNIV (\<lambda>i. Pi' (A i) (B i))" "\<And>n i. i\<in>A n \<Longrightarrow> open ((B n) i)" "\<And>n. finite (A n)"
1.612 +using open_imp_ex_UNION[OF assms] by auto
1.613 +
1.614 +lemma
1.615 + open_basis_of_enumE:
1.616 + assumes "open X" assumes "X \<noteq> {}"
1.617 + obtains A::"nat\<Rightarrow>'a set" and B::"nat\<Rightarrow>('a \<Rightarrow> 'b set)" where
1.618 + "X = UNION UNIV (\<lambda>i. Pi' (A i) (B i))" "\<And>n i. i\<in>A n \<Longrightarrow> (B n) i \<in> range enum_basis"
1.619 + "\<And>n. finite (A n)"
1.620 +using open_imp_ex_UNION_of_enum[OF assms] by auto
1.621 +
1.622 +instance proof qed (blast intro: finmap_topological_basis)
1.623 +
1.624 +end
1.625 +
1.626 +subsection {* Product Measurable Space of Finite Maps *}
1.627 +
1.628 +definition "PiF I M \<equiv>
1.629 + sigma
1.630 + (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))
1.631 + {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
1.632 +
1.633 +abbreviation
1.634 + "Pi\<^isub>F I M \<equiv> PiF I M"
1.635 +
1.636 +syntax
1.637 + "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIF _:_./ _)" 10)
1.638 +
1.639 +syntax (xsymbols)
1.640 + "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10)
1.641 +
1.642 +syntax (HTML output)
1.643 + "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10)
1.644 +
1.645 +translations
1.646 + "PIF x:I. M" == "CONST PiF I (%x. M)"
1.647 +
1.648 +lemma PiF_gen_subset: "{(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} \<subseteq>
1.649 + Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
1.650 + by (auto simp: Pi'_def) (blast dest: sets_into_space)
1.651 +
1.652 +lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
1.653 + unfolding PiF_def using PiF_gen_subset by (rule space_measure_of)
1.654 +
1.655 +lemma sets_PiF:
1.656 + "sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))
1.657 + {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
1.658 + unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of)
1.659 +
1.660 +lemma sets_PiF_singleton:
1.661 + "sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j))
1.662 + {(\<Pi>' j\<in>I. X j) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
1.663 + unfolding sets_PiF by simp
1.664 +
1.665 +lemma in_sets_PiFI:
1.666 + assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
1.667 + shows "X \<in> sets (PiF I M)"
1.668 + unfolding sets_PiF
1.669 + using assms by blast
1.670 +
1.671 +lemma product_in_sets_PiFI:
1.672 + assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
1.673 + shows "(Pi' J S) \<in> sets (PiF I M)"
1.674 + unfolding sets_PiF
1.675 + using assms by blast
1.676 +
1.677 +lemma singleton_space_subset_in_sets:
1.678 + fixes J
1.679 + assumes "J \<in> I"
1.680 + assumes "finite J"
1.681 + shows "space (PiF {J} M) \<in> sets (PiF I M)"
1.682 + using assms
1.683 + by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"])
1.684 + (auto simp: product_def space_PiF)
1.685 +
1.686 +lemma singleton_subspace_set_in_sets:
1.687 + assumes A: "A \<in> sets (PiF {J} M)"
1.688 + assumes "finite J"
1.689 + assumes "J \<in> I"
1.690 + shows "A \<in> sets (PiF I M)"
1.691 + using A[unfolded sets_PiF]
1.692 + apply (induct A)
1.693 + unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
1.694 + using assms
1.695 + by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets)
1.696 +
1.697 +lemma
1.698 + finite_measurable_singletonI:
1.699 + assumes "finite I"
1.700 + assumes "\<And>J. J \<in> I \<Longrightarrow> finite J"
1.701 + assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
1.702 + shows "A \<in> measurable (PiF I M) N"
1.703 + unfolding measurable_def
1.704 +proof safe
1.705 + fix y assume "y \<in> sets N"
1.706 + have "A -` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A -` y \<inter> space (PiF {J} M))"
1.707 + by (auto simp: space_PiF)
1.708 + also have "\<dots> \<in> sets (PiF I M)"
1.709 + proof
1.710 + show "finite I" by fact
1.711 + fix J assume "J \<in> I"
1.712 + with assms have "finite J" by simp
1.713 + show "A -` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)"
1.714 + by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+
1.715 + qed
1.716 + finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
1.717 +next
1.718 + fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
1.719 + using MN[of "domain x"]
1.720 + by (auto simp: space_PiF measurable_space Pi'_def)
1.721 +qed
1.722 +
1.723 +lemma
1.724 + countable_finite_comprehension:
1.725 + fixes f :: "'a::countable set \<Rightarrow> _"
1.726 + assumes "\<And>s. P s \<Longrightarrow> finite s"
1.727 + assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M"
1.728 + shows "\<Union>{f s|s. P s} \<in> sets M"
1.729 +proof -
1.730 + have "\<Union>{f s|s. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})"
1.731 + proof safe
1.732 + fix x X s assume "x \<in> f s" "P s"
1.733 + moreover with assms obtain l where "s = set l" using finite_list by blast
1.734 + ultimately show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s`
1.735 + by (auto intro!: exI[where x="to_nat l"])
1.736 + next
1.737 + fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})"
1.738 + thus "x \<in> \<Union>{f s|s. P s}" using assms by (auto simp: Let_def split: split_if_asm)
1.739 + qed
1.740 + hence "\<Union>{f s|s. P s} = (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" by simp
1.741 + also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def)
1.742 + finally show ?thesis .
1.743 +qed
1.744 +
1.745 +lemma space_subset_in_sets:
1.746 + fixes J::"'a::countable set set"
1.747 + assumes "J \<subseteq> I"
1.748 + assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
1.749 + shows "space (PiF J M) \<in> sets (PiF I M)"
1.750 +proof -
1.751 + have "space (PiF J M) = \<Union>{space (PiF {j} M)|j. j \<in> J}"
1.752 + unfolding space_PiF by blast
1.753 + also have "\<dots> \<in> sets (PiF I M)" using assms
1.754 + by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets)
1.755 + finally show ?thesis .
1.756 +qed
1.757 +
1.758 +lemma subspace_set_in_sets:
1.759 + fixes J::"'a::countable set set"
1.760 + assumes A: "A \<in> sets (PiF J M)"
1.761 + assumes "J \<subseteq> I"
1.762 + assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
1.763 + shows "A \<in> sets (PiF I M)"
1.764 + using A[unfolded sets_PiF]
1.765 + apply (induct A)
1.766 + unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
1.767 + using assms
1.768 + by (auto intro: in_sets_PiFI intro!: space_subset_in_sets)
1.769 +
1.770 +lemma
1.771 + countable_measurable_PiFI:
1.772 + fixes I::"'a::countable set set"
1.773 + assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
1.774 + shows "A \<in> measurable (PiF I M) N"
1.775 + unfolding measurable_def
1.776 +proof safe
1.777 + fix y assume "y \<in> sets N"
1.778 + have "A -` y = (\<Union>{A -` y \<inter> {x. domain x = J}|J. finite J})" by auto
1.779 + hence "A -` y \<inter> space (PiF I M) = (\<Union>n. A -` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))"
1.780 + apply (auto simp: space_PiF Pi'_def)
1.781 + proof -
1.782 + case goal1
1.783 + from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto
1.784 + thus ?case
1.785 + apply (intro exI[where x="to_nat xs"])
1.786 + apply auto
1.787 + done
1.788 + qed
1.789 + also have "\<dots> \<in> sets (PiF I M)"
1.790 + apply (intro Int countable_nat_UN subsetI, safe)
1.791 + apply (case_tac "set (from_nat i) \<in> I")
1.792 + apply simp_all
1.793 + apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]])
1.794 + using assms `y \<in> sets N`
1.795 + apply (auto simp: space_PiF)
1.796 + done
1.797 + finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
1.798 +next
1.799 + fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
1.800 + using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def)
1.801 +qed
1.802 +
1.803 +lemma measurable_PiF:
1.804 + assumes f: "\<And>x. x \<in> space N \<Longrightarrow> domain (f x) \<in> I \<and> (\<forall>i\<in>domain (f x). (f x) i \<in> space (M i))"
1.805 + assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow>
1.806 + f -` (Pi' J S) \<inter> space N \<in> sets N"
1.807 + shows "f \<in> measurable N (PiF I M)"
1.808 + unfolding PiF_def
1.809 + using PiF_gen_subset
1.810 + apply (rule measurable_measure_of)
1.811 + using f apply force
1.812 + apply (insert S, auto)
1.813 + done
1.814 +
1.815 +lemma
1.816 + restrict_sets_measurable:
1.817 + assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I"
1.818 + shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
1.819 + using A[unfolded sets_PiF]
1.820 + apply (induct A)
1.821 + unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
1.822 +proof -
1.823 + fix a assume "a \<in> {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
1.824 + then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))"
1.825 + by auto
1.826 + show "a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
1.827 + proof cases
1.828 + assume "K \<in> J"
1.829 + hence "a \<inter> {m. domain m \<in> J} \<in> {Pi' K X |X K. K \<in> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))}" using S
1.830 + by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def)
1.831 + also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto
1.832 + finally show ?thesis .
1.833 + next
1.834 + assume "K \<notin> J"
1.835 + hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def)
1.836 + also have "\<dots> \<in> sets (PiF J M)" by simp
1.837 + finally show ?thesis .
1.838 + qed
1.839 +next
1.840 + show "{} \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" by simp
1.841 +next
1.842 + fix a :: "nat \<Rightarrow> _"
1.843 + assume a: "(\<And>i. a i \<inter> {m. domain m \<in> J} \<in> sets (PiF J M))"
1.844 + have "UNION UNIV a \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))"
1.845 + by simp
1.846 + also have "\<dots> \<in> sets (PiF J M)" using a by (intro countable_nat_UN) auto
1.847 + finally show "UNION UNIV a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" .
1.848 +next
1.849 + fix a assume a: "a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
1.850 + have "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} = (space (PiF J M) - (a \<inter> {m. domain m \<in> J}))"
1.851 + using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def)
1.852 + also have "\<dots> \<in> sets (PiF J M)" using a by auto
1.853 + finally show "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" .
1.854 +qed
1.855 +
1.856 +lemma measurable_finmap_of:
1.857 + assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
1.858 + assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)"
1.859 + assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N"
1.860 + shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)"
1.861 +proof (rule measurable_PiF)
1.862 + fix x assume "x \<in> space N"
1.863 + with J[of x] measurable_space[OF f]
1.864 + show "domain (finmap_of (J x) (f x)) \<in> I \<and>
1.865 + (\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))"
1.866 + by auto
1.867 +next
1.868 + fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)"
1.869 + with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N =
1.870 + (if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K}
1.871 + else (\<Inter>i\<in>K. (\<lambda>x. f x i) -` S i \<inter> {x \<in> space N. J x = K}) else {})"
1.872 + by (auto simp: Pi'_def)
1.873 + have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto
1.874 + show "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N \<in> sets N"
1.875 + unfolding eq r
1.876 + apply (simp del: INT_simps add: )
1.877 + apply (intro conjI impI finite_INT JN Int[OF top])
1.878 + apply simp apply assumption
1.879 + apply (subst Int_assoc[symmetric])
1.880 + apply (rule Int)
1.881 + apply (intro measurable_sets[OF f] *) apply force apply assumption
1.882 + apply (intro JN)
1.883 + done
1.884 +qed
1.885 +
1.886 +lemma measurable_PiM_finmap_of:
1.887 + assumes "finite J"
1.888 + shows "finmap_of J \<in> measurable (Pi\<^isub>M J M) (PiF {J} M)"
1.889 + apply (rule measurable_finmap_of)
1.890 + apply (rule measurable_component_singleton)
1.891 + apply simp
1.892 + apply rule
1.893 + apply (rule `finite J`)
1.894 + apply simp
1.895 + done
1.896 +
1.897 +lemma proj_measurable_singleton:
1.898 + assumes "A \<in> sets (M i)" "finite I"
1.899 + shows "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)"
1.900 +proof cases
1.901 + assume "i \<in> I"
1.902 + hence "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) =
1.903 + Pi' I (\<lambda>x. if x = i then A else space (M x))"
1.904 + using sets_into_space[OF ] `A \<in> sets (M i)` assms
1.905 + by (auto simp: space_PiF Pi'_def)
1.906 + thus ?thesis using assms `A \<in> sets (M i)`
1.907 + by (intro in_sets_PiFI) auto
1.908 +next
1.909 + assume "i \<notin> I"
1.910 + hence "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) =
1.911 + (if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def)
1.912 + thus ?thesis by simp
1.913 +qed
1.914 +
1.915 +lemma measurable_proj_singleton:
1.916 + fixes I
1.917 + assumes "finite I" "i \<in> I"
1.918 + shows "(\<lambda>x. (x)\<^isub>F i) \<in> measurable (PiF {I} M) (M i)"
1.919 +proof (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms)
1.920 +qed (insert `i \<in> I`, auto simp: space_PiF)
1.921 +
1.922 +lemma measurable_proj_countable:
1.923 + fixes I::"'a::countable set set"
1.924 + assumes "y \<in> space (M i)"
1.925 + shows "(\<lambda>x. if i \<in> domain x then (x)\<^isub>F i else y) \<in> measurable (PiF I M) (M i)"
1.926 +proof (rule countable_measurable_PiFI)
1.927 + fix J assume "J \<in> I" "finite J"
1.928 + show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)"
1.929 + unfolding measurable_def
1.930 + proof safe
1.931 + fix z assume "z \<in> sets (M i)"
1.932 + have "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) =
1.933 + (\<lambda>x. if i \<in> J then (x)\<^isub>F i else y) -` z \<inter> space (PiF {J} M)"
1.934 + by (auto simp: space_PiF Pi'_def)
1.935 + also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J`
1.936 + by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton])
1.937 + finally show "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) \<in>
1.938 + sets (PiF {J} M)" .
1.939 + qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def)
1.940 +qed
1.941 +
1.942 +lemma measurable_restrict_proj:
1.943 + assumes "J \<in> II" "finite J"
1.944 + shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)"
1.945 + using assms
1.946 + by (intro measurable_finmap_of measurable_component_singleton) auto
1.947 +
1.948 +lemma
1.949 + measurable_proj_PiM:
1.950 + fixes J K ::"'a::countable set" and I::"'a set set"
1.951 + assumes "finite J" "J \<in> I"
1.952 + assumes "x \<in> space (PiM J M)"
1.953 + shows "proj \<in>
1.954 + measurable (PiF {J} M) (PiM J M)"
1.955 +proof (rule measurable_PiM_single)
1.956 + show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^isub>E i \<in> J. space (M i))"
1.957 + using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def)
1.958 +next
1.959 + fix A i assume A: "i \<in> J" "A \<in> sets (M i)"
1.960 + show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} \<in> sets (PiF {J} M)"
1.961 + proof
1.962 + have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} =
1.963 + (\<lambda>\<omega>. (\<omega>)\<^isub>F i) -` A \<inter> space (PiF {J} M)" by auto
1.964 + also have "\<dots> \<in> sets (PiF {J} M)"
1.965 + using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM)
1.966 + finally show ?thesis .
1.967 + qed simp
1.968 +qed
1.969 +
1.970 +lemma sets_subspaceI:
1.971 + assumes "A \<inter> space M \<in> sets M"
1.972 + assumes "B \<in> sets M"
1.973 + shows "A \<inter> B \<in> sets M" using assms
1.974 +proof -
1.975 + have "A \<inter> B = (A \<inter> space M) \<inter> B"
1.976 + using assms sets_into_space by auto
1.977 + thus ?thesis using assms by auto
1.978 +qed
1.979 +
1.980 +lemma space_PiF_singleton_eq_product:
1.981 + assumes "finite I"
1.982 + shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))"
1.983 + by (auto simp: product_def space_PiF assms)
1.984 +
1.985 +text {* adapted from @{thm sets_PiM_single} *}
1.986 +
1.987 +lemma sets_PiF_single:
1.988 + assumes "finite I" "I \<noteq> {}"
1.989 + shows "sets (PiF {I} M) =
1.990 + sigma_sets (\<Pi>' i\<in>I. space (M i))
1.991 + {{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
1.992 + (is "_ = sigma_sets ?\<Omega> ?R")
1.993 + unfolding sets_PiF_singleton
1.994 +proof (rule sigma_sets_eqI)
1.995 + interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
1.996 + fix A assume "A \<in> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
1.997 + then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
1.998 + show "A \<in> sigma_sets ?\<Omega> ?R"
1.999 + proof -
1.1000 + from `I \<noteq> {}` X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})"
1.1001 + using sets_into_space
1.1002 + by (auto simp: space_PiF product_def) blast
1.1003 + also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
1.1004 + using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF)
1.1005 + finally show "A \<in> sigma_sets ?\<Omega> ?R" .
1.1006 + qed
1.1007 +next
1.1008 + fix A assume "A \<in> ?R"
1.1009 + then obtain i B where A: "A = {f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
1.1010 + by auto
1.1011 + then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))"
1.1012 + using sets_into_space[OF A(3)]
1.1013 + apply (auto simp: Pi'_iff split: split_if_asm)
1.1014 + apply blast
1.1015 + done
1.1016 + also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
1.1017 + using A
1.1018 + by (intro sigma_sets.Basic )
1.1019 + (auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"])
1.1020 + finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" .
1.1021 +qed
1.1022 +
1.1023 +text {* adapted from @{thm PiE_cong} *}
1.1024 +
1.1025 +lemma Pi'_cong:
1.1026 + assumes "finite I"
1.1027 + assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i"
1.1028 + shows "Pi' I f = Pi' I g"
1.1029 +using assms by (auto simp: Pi'_def)
1.1030 +
1.1031 +text {* adapted from @{thm Pi_UN} *}
1.1032 +
1.1033 +lemma Pi'_UN:
1.1034 + fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
1.1035 + assumes "finite I"
1.1036 + assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
1.1037 + shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)"
1.1038 +proof (intro set_eqI iffI)
1.1039 + fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)"
1.1040 + then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def)
1.1041 + from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
1.1042 + obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
1.1043 + using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
1.1044 + have "f \<in> Pi' I (\<lambda>i. A k i)"
1.1045 + proof
1.1046 + fix i assume "i \<in> I"
1.1047 + from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I`
1.1048 + show "f i \<in> A k i " by (auto simp: `finite I`)
1.1049 + qed (simp add: `domain f = I` `finite I`)
1.1050 + then show "f \<in> (\<Union>n. Pi' I (A n))" by auto
1.1051 +qed (auto simp: Pi'_def `finite I`)
1.1052 +
1.1053 +text {* adapted from @{thm sigma_prod_algebra_sigma_eq} *}
1.1054 +
1.1055 +lemma sigma_fprod_algebra_sigma_eq:
1.1056 + fixes E :: "'i \<Rightarrow> 'a set set"
1.1057 + assumes [simp]: "finite I" "I \<noteq> {}"
1.1058 + assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
1.1059 + and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
1.1060 + and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
1.1061 + assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
1.1062 + and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
1.1063 + defines "P == { Pi' I F | F. \<forall>i\<in>I. F i \<in> E i }"
1.1064 + shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
1.1065 +proof
1.1066 + let ?P = "sigma (space (Pi\<^isub>F {I} M)) P"
1.1067 + have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>F {I} M))"
1.1068 + using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
1.1069 + then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))"
1.1070 + by (simp add: space_PiF)
1.1071 + have "sets (PiF {I} M) =
1.1072 + sigma_sets (space ?P) {{f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
1.1073 + using sets_PiF_single[of I M] by (simp add: space_P)
1.1074 + also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)"
1.1075 + proof (safe intro!: sigma_sets_subset)
1.1076 + fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
1.1077 + have "(\<lambda>x. (x)\<^isub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))"
1.1078 + proof (subst measurable_iff_measure_of)
1.1079 + show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
1.1080 + from space_P `i \<in> I` show "(\<lambda>x. (x)\<^isub>F i) \<in> space ?P \<rightarrow> space (M i)"
1.1081 + by auto
1.1082 + show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
1.1083 + proof
1.1084 + fix A assume A: "A \<in> E i"
1.1085 + then have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P = (\<Pi>' j\<in>I. if i = j then A else space (M j))"
1.1086 + using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
1.1087 + also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)"
1.1088 + by (intro Pi'_cong) (simp_all add: S_union)
1.1089 + also have "\<dots> = (\<Union>n. \<Pi>' j\<in>I. if i = j then A else S j n)"
1.1090 + using S_mono
1.1091 + by (subst Pi'_UN[symmetric, OF `finite I`]) (auto simp: incseq_def)
1.1092 + also have "\<dots> \<in> sets ?P"
1.1093 + proof (safe intro!: countable_UN)
1.1094 + fix n show "(\<Pi>' j\<in>I. if i = j then A else S j n) \<in> sets ?P"
1.1095 + using A S_in_E
1.1096 + by (simp add: P_closed)
1.1097 + (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j n"])
1.1098 + qed
1.1099 + finally show "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
1.1100 + using P_closed by simp
1.1101 + qed
1.1102 + qed
1.1103 + from measurable_sets[OF this, of A] A `i \<in> I` E_closed
1.1104 + have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
1.1105 + by (simp add: E_generates)
1.1106 + also have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P = {f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A}"
1.1107 + using P_closed by (auto simp: space_PiF)
1.1108 + finally show "\<dots> \<in> sets ?P" .
1.1109 + qed
1.1110 + finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P"
1.1111 + by (simp add: P_closed)
1.1112 + show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)"
1.1113 + using `finite I` `I \<noteq> {}`
1.1114 + by (auto intro!: sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def)
1.1115 +qed
1.1116 +
1.1117 +lemma enumerable_sigma_fprod_algebra_sigma_eq:
1.1118 + assumes "I \<noteq> {}"
1.1119 + assumes [simp]: "finite I"
1.1120 + shows "sets (PiF {I} (\<lambda>_. borel)) = sigma_sets (space (PiF {I} (\<lambda>_. borel)))
1.1121 + {Pi' I F |F. (\<forall>i\<in>I. F i \<in> range enum_basis)}"
1.1122 +proof -
1.1123 + from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
1.1124 + show ?thesis
1.1125 + proof (rule sigma_fprod_algebra_sigma_eq)
1.1126 + show "finite I" by simp
1.1127 + show "I \<noteq> {}" by fact
1.1128 + show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> range enum_basis"
1.1129 + using S by simp_all
1.1130 + show "range enum_basis \<subseteq> Pow (space borel)" by simp
1.1131 + show "sets borel = sigma_sets (space borel) (range enum_basis)"
1.1132 + by (simp add: borel_eq_enum_basis)
1.1133 + qed
1.1134 +qed
1.1135 +
1.1136 +text {* adapted from @{thm enumerable_sigma_fprod_algebra_sigma_eq} *}
1.1137 +
1.1138 +lemma enumerable_sigma_prod_algebra_sigma_eq:
1.1139 + assumes "I \<noteq> {}"
1.1140 + assumes [simp]: "finite I"
1.1141 + shows "sets (PiM I (\<lambda>_. borel)) = sigma_sets (space (PiM I (\<lambda>_. borel)))
1.1142 + {Pi\<^isub>E I F |F. \<forall>i\<in>I. F i \<in> range enum_basis}"
1.1143 +proof -
1.1144 + from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
1.1145 + show ?thesis
1.1146 + proof (rule sigma_prod_algebra_sigma_eq)
1.1147 + show "finite I" by simp note[[show_types]]
1.1148 + fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> range enum_basis"
1.1149 + using S by simp_all
1.1150 + show "range enum_basis \<subseteq> Pow (space borel)" by simp
1.1151 + show "sets borel = sigma_sets (space borel) (range enum_basis)"
1.1152 + by (simp add: borel_eq_enum_basis)
1.1153 + qed
1.1154 +qed
1.1155 +
1.1156 +lemma product_open_generates_sets_PiF_single:
1.1157 + assumes "I \<noteq> {}"
1.1158 + assumes [simp]: "finite I"
1.1159 + shows "sets (PiF {I} (\<lambda>_. borel::'b::enumerable_basis measure)) =
1.1160 + sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F |F. (\<forall>i\<in>I. F i \<in> Collect open)}"
1.1161 +proof -
1.1162 + from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
1.1163 + show ?thesis
1.1164 + proof (rule sigma_fprod_algebra_sigma_eq)
1.1165 + show "finite I" by simp
1.1166 + show "I \<noteq> {}" by fact
1.1167 + show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open"
1.1168 + using S by (auto simp: open_enum_basis)
1.1169 + show "Collect open \<subseteq> Pow (space borel)" by simp
1.1170 + show "sets borel = sigma_sets (space borel) (Collect open)"
1.1171 + by (simp add: borel_def)
1.1172 + qed
1.1173 +qed
1.1174 +
1.1175 +lemma product_open_generates_sets_PiM:
1.1176 + assumes "I \<noteq> {}"
1.1177 + assumes [simp]: "finite I"
1.1178 + shows "sets (PiM I (\<lambda>_. borel::'b::enumerable_basis measure)) =
1.1179 + sigma_sets (space (PiM I (\<lambda>_. borel))) {Pi\<^isub>E I F |F. \<forall>i\<in>I. F i \<in> Collect open}"
1.1180 +proof -
1.1181 + from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
1.1182 + show ?thesis
1.1183 + proof (rule sigma_prod_algebra_sigma_eq)
1.1184 + show "finite I" by simp note[[show_types]]
1.1185 + fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open"
1.1186 + using S by (auto simp: open_enum_basis)
1.1187 + show "Collect open \<subseteq> Pow (space borel)" by simp
1.1188 + show "sets borel = sigma_sets (space borel) (Collect open)"
1.1189 + by (simp add: borel_def)
1.1190 + qed
1.1191 +qed
1.1192 +
1.1193 +lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. J \<leadsto> UNIV) = UNIV" by auto
1.1194 +
1.1195 +lemma borel_eq_PiF_borel:
1.1196 + shows "(borel :: ('i::countable \<Rightarrow>\<^isub>F 'a::polish_space) measure) =
1.1197 + PiF (Collect finite) (\<lambda>_. borel :: 'a measure)"
1.1198 +proof (rule measure_eqI)
1.1199 + have C: "Collect finite \<noteq> {}" by auto
1.1200 + show "sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure) = sets (PiF (Collect finite) (\<lambda>_. borel))"
1.1201 + proof
1.1202 + show "sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure) \<subseteq> sets (PiF (Collect finite) (\<lambda>_. borel))"
1.1203 + apply (simp add: borel_def sets_PiF)
1.1204 + proof (rule sigma_sets_mono, safe, cases)
1.1205 + fix X::"('i \<Rightarrow>\<^isub>F 'a) set" assume "open X" "X \<noteq> {}"
1.1206 + from open_basisE[OF this] guess NA NB . note N = this
1.1207 + hence "X = (\<Union>i. Pi' (NA i) (NB i))" by simp
1.1208 + also have "\<dots> \<in>
1.1209 + sigma_sets UNIV {Pi' J S |S J. finite J \<and> S \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
1.1210 + using N by (intro Union sigma_sets.Basic) blast
1.1211 + finally show "X \<in> sigma_sets UNIV
1.1212 + {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" .
1.1213 + qed (auto simp: Empty)
1.1214 + next
1.1215 + show "sets (PiF (Collect finite) (\<lambda>_. borel)) \<subseteq> sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure)"
1.1216 + proof
1.1217 + fix x assume x: "x \<in> sets (PiF (Collect finite::'i set set) (\<lambda>_. borel::'a measure))"
1.1218 + hence x_sp: "x \<subseteq> space (PiF (Collect finite) (\<lambda>_. borel))" by (rule sets_into_space)
1.1219 + let ?x = "\<lambda>J. x \<inter> {x. domain x = J}"
1.1220 + have "x = \<Union>{?x J |J. finite J}" by auto
1.1221 + also have "\<dots> \<in> sets borel"
1.1222 + proof (rule countable_finite_comprehension, assumption)
1.1223 + fix J::"'i set" assume "finite J"
1.1224 + { assume ef: "J = {}"
1.1225 + { assume e: "?x J = {}"
1.1226 + hence "?x J \<in> sets borel" by simp
1.1227 + } moreover {
1.1228 + assume "?x J \<noteq> {}"
1.1229 + then obtain f where "f \<in> x" "domain f = {}" using ef by auto
1.1230 + hence "?x J = {f}" using `J = {}`
1.1231 + by (auto simp: finmap_eq_iff)
1.1232 + also have "{f} \<in> sets borel" by simp
1.1233 + finally have "?x J \<in> sets borel" .
1.1234 + } ultimately have "?x J \<in> sets borel" by blast
1.1235 + } moreover {
1.1236 + assume "J \<noteq> ({}::'i set)"
1.1237 + from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'a set" . note S = this
1.1238 + have "(?x J) = x \<inter> {m. domain m \<in> {J}}" by auto
1.1239 + also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))"
1.1240 + using x by (rule restrict_sets_measurable) (auto simp: `finite J`)
1.1241 + also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel)))
1.1242 + {Pi' (J) F |F. (\<forall>j\<in>J. F j \<in> range enum_basis)}"
1.1243 + (is "_ = sigma_sets _ ?P")
1.1244 + by (rule enumerable_sigma_fprod_algebra_sigma_eq[OF `J \<noteq> {}` `finite J`])
1.1245 + also have "\<dots> \<subseteq> sets borel"
1.1246 + proof
1.1247 + fix x
1.1248 + assume "x \<in> sigma_sets (space (PiF {J} (\<lambda>_. borel))) ?P"
1.1249 + thus "x \<in> sets borel"
1.1250 + proof (rule sigma_sets.induct, safe)
1.1251 + fix F::"'i \<Rightarrow> 'a set"
1.1252 + assume "\<forall>j\<in>J. F j \<in> range enum_basis"
1.1253 + hence "Pi' J F \<in> range enum_basis_finmap"
1.1254 + unfolding range_enum_basis_eq
1.1255 + by (auto simp: `finite J` intro!: exI[where x=J] exI[where x=F])
1.1256 + hence "open (Pi' (J) F)" by (rule range_enum_basis_finmap_imp_open)
1.1257 + thus "Pi' (J) F \<in> sets borel" by simp
1.1258 + next
1.1259 + fix a::"('i \<Rightarrow>\<^isub>F 'a) set"
1.1260 + have "space (PiF {J::'i set} (\<lambda>_. borel::'a measure)) =
1.1261 + Pi' (J) (\<lambda>_. UNIV)"
1.1262 + by (auto simp: space_PiF product_def)
1.1263 + moreover have "open (Pi' (J::'i set) (\<lambda>_. UNIV::'a set))"
1.1264 + by (intro open_Pi'I) auto
1.1265 + ultimately
1.1266 + have "space (PiF {J::'i set} (\<lambda>_. borel::'a measure)) \<in> sets borel"
1.1267 + by simp
1.1268 + moreover
1.1269 + assume "a \<in> sets borel"
1.1270 + ultimately show "space (PiF {J} (\<lambda>_. borel)) - a \<in> sets borel" ..
1.1271 + qed auto
1.1272 + qed
1.1273 + finally have "(?x J) \<in> sets borel" .
1.1274 + } ultimately show "(?x J) \<in> sets borel" by blast
1.1275 + qed
1.1276 + finally show "x \<in> sets (borel)" .
1.1277 + qed
1.1278 + qed
1.1279 +qed (simp add: emeasure_sigma borel_def PiF_def)
1.1280 +
1.1281 +subsection {* Isomorphism between Functions and Finite Maps *}
1.1282 +
1.1283 +lemma
1.1284 + measurable_compose:
1.1285 + fixes f::"'a \<Rightarrow> 'b"
1.1286 + assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j"
1.1287 + assumes "finite J"
1.1288 + shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
1.1289 +proof (rule measurable_PiM)
1.1290 + show "(\<lambda>m. compose J m f)
1.1291 + \<in> space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) \<rightarrow>
1.1292 + (J \<rightarrow> space M) \<inter> extensional J"
1.1293 + proof safe
1.1294 + fix x and i
1.1295 + assume x: "x \<in> space (PiM (f ` J) (\<lambda>_. M))" "i \<in> J"
1.1296 + with inj show "compose J x f i \<in> space M"
1.1297 + by (auto simp: space_PiM compose_def)
1.1298 + next
1.1299 + fix x assume "x \<in> space (PiM (f ` J) (\<lambda>_. M))"
1.1300 + show "(compose J x f) \<in> extensional J" by (rule compose_extensional)
1.1301 + qed
1.1302 +next
1.1303 + fix S X
1.1304 + have inv: "\<And>j. j \<in> f ` J \<Longrightarrow> f (f' j) = j" using assms by auto
1.1305 + assume S: "S \<noteq> {} \<or> J = {}" "finite S" "S \<subseteq> J" and P: "\<And>i. i \<in> S \<Longrightarrow> X i \<in> sets M"
1.1306 + have "(\<lambda>m. compose J m f) -` prod_emb J (\<lambda>_. M) S (Pi\<^isub>E S X) \<inter>
1.1307 + space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) = prod_emb (f ` J) (\<lambda>_. M) (f ` S) (Pi\<^isub>E (f ` S) (\<lambda>b. X (f' b)))"
1.1308 + using assms inv S sets_into_space[OF P]
1.1309 + by (force simp: prod_emb_iff compose_def space_PiM extensional_def Pi_def intro: imageI)
1.1310 + also have "\<dots> \<in> sets (Pi\<^isub>M (f ` J) (\<lambda>_. M))"
1.1311 + proof
1.1312 + from S show "f ` S \<subseteq> f ` J" by auto
1.1313 + show "(\<Pi>\<^isub>E b\<in>f ` S. X (f' b)) \<in> sets (Pi\<^isub>M (f ` S) (\<lambda>_. M))"
1.1314 + proof (rule sets_PiM_I_finite)
1.1315 + show "finite (f ` S)" using S by simp
1.1316 + fix i assume "i \<in> f ` S" hence "f' i \<in> S" using S assms by auto
1.1317 + thus "X (f' i) \<in> sets M" by (rule P)
1.1318 + qed
1.1319 + qed
1.1320 + finally show "(\<lambda>m. compose J m f) -` prod_emb J (\<lambda>_. M) S (Pi\<^isub>E S X) \<inter>
1.1321 + space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) \<in> sets (Pi\<^isub>M (f ` J) (\<lambda>_. M))" .
1.1322 +qed
1.1323 +
1.1324 +lemma
1.1325 + measurable_compose_inv:
1.1326 + fixes f::"'a \<Rightarrow> 'b"
1.1327 + assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j"
1.1328 + assumes "finite J"
1.1329 + shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))"
1.1330 +proof -
1.1331 + have "(\<lambda>m. compose (f ` J) m f') \<in> measurable (Pi\<^isub>M (f' ` f ` J) (\<lambda>_. M)) (Pi\<^isub>M (f ` J) (\<lambda>_. M))"
1.1332 + using assms by (auto intro: measurable_compose)
1.1333 + moreover
1.1334 + from inj have "f' ` f ` J = J" by (metis (hide_lams, mono_tags) image_iff set_eqI)
1.1335 + ultimately show ?thesis by simp
1.1336 +qed
1.1337 +
1.1338 +locale function_to_finmap =
1.1339 + fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f'
1.1340 + assumes [simp]: "finite J"
1.1341 + assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i"
1.1342 +begin
1.1343 +
1.1344 +text {* to measure finmaps *}
1.1345 +
1.1346 +definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')"
1.1347 +
1.1348 +lemma domain_fm[simp]: "domain (fm x) = f ` J"
1.1349 + unfolding fm_def by simp
1.1350 +
1.1351 +lemma fm_restrict[simp]: "fm (restrict y J) = fm y"
1.1352 + unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext)
1.1353 +
1.1354 +lemma fm_product:
1.1355 + assumes "\<And>i. space (M i) = UNIV"
1.1356 + shows "fm -` Pi' (f ` J) S \<inter> space (Pi\<^isub>M J M) = (\<Pi>\<^isub>E j \<in> J. S (f j))"
1.1357 + using assms
1.1358 + by (auto simp: inv fm_def compose_def space_PiM Pi'_def)
1.1359 +
1.1360 +lemma fm_measurable:
1.1361 + assumes "f ` J \<in> N"
1.1362 + shows "fm \<in> measurable (Pi\<^isub>M J (\<lambda>_. M)) (Pi\<^isub>F N (\<lambda>_. M))"
1.1363 + unfolding fm_def
1.1364 +proof (rule measurable_comp, rule measurable_compose_inv)
1.1365 + show "finmap_of (f ` J) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) "
1.1366 + using assms by (intro measurable_finmap_of measurable_component_singleton) auto
1.1367 +qed (simp_all add: inv)
1.1368 +
1.1369 +lemma proj_fm:
1.1370 + assumes "x \<in> J"
1.1371 + shows "fm m (f x) = m x"
1.1372 + using assms by (auto simp: fm_def compose_def o_def inv)
1.1373 +
1.1374 +lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)"
1.1375 +proof (rule inj_on_inverseI)
1.1376 + fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J"
1.1377 + thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x"
1.1378 + by (auto simp: compose_def inv extensional_def)
1.1379 +qed
1.1380 +
1.1381 +lemma inj_on_fm:
1.1382 + assumes "\<And>i. space (M i) = UNIV"
1.1383 + shows "inj_on fm (space (Pi\<^isub>M J M))"
1.1384 + using assms
1.1385 + apply (auto simp: fm_def space_PiM)
1.1386 + apply (rule comp_inj_on)
1.1387 + apply (rule inj_on_compose_f')
1.1388 + apply (rule finmap_of_inj_on_extensional_finite)
1.1389 + apply simp
1.1390 + apply (auto)
1.1391 + done
1.1392 +
1.1393 +text {* to measure functions *}
1.1394 +
1.1395 +definition "mf = (\<lambda>g. compose J g f) o proj"
1.1396 +
1.1397 +lemma
1.1398 + assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))" "finite J"
1.1399 + shows "proj (finmap_of J x) = x"
1.1400 + using assms by (auto simp: space_PiM extensional_def)
1.1401 +
1.1402 +lemma
1.1403 + assumes "x \<in> space (Pi\<^isub>F {J} (\<lambda>_. M))"
1.1404 + shows "finmap_of J (proj x) = x"
1.1405 + using assms by (auto simp: space_PiF Pi'_def finmap_eq_iff)
1.1406 +
1.1407 +lemma mf_fm:
1.1408 + assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))"
1.1409 + shows "mf (fm x) = x"
1.1410 +proof -
1.1411 + have "mf (fm x) \<in> extensional J"
1.1412 + by (auto simp: mf_def extensional_def compose_def)
1.1413 + moreover
1.1414 + have "x \<in> extensional J" using assms sets_into_space
1.1415 + by (force simp: space_PiM)
1.1416 + moreover
1.1417 + { fix i assume "i \<in> J"
1.1418 + hence "mf (fm x) i = x i"
1.1419 + by (auto simp: inv mf_def compose_def fm_def)
1.1420 + }
1.1421 + ultimately
1.1422 + show ?thesis by (rule extensionalityI)
1.1423 +qed
1.1424 +
1.1425 +lemma mf_measurable:
1.1426 + assumes "space M = UNIV"
1.1427 + shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
1.1428 + unfolding mf_def
1.1429 +proof (rule measurable_comp, rule measurable_proj_PiM)
1.1430 + show "(\<lambda>g. compose J g f) \<in>
1.1431 + measurable (Pi\<^isub>M (f ` J) (\<lambda>x. M)) (Pi\<^isub>M J (\<lambda>_. M))"
1.1432 + by (rule measurable_compose, rule inv) auto
1.1433 +qed (auto simp add: space_PiM extensional_def assms)
1.1434 +
1.1435 +lemma fm_image_measurable:
1.1436 + assumes "space M = UNIV"
1.1437 + assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M))"
1.1438 + shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))"
1.1439 +proof -
1.1440 + have "fm ` X = (mf) -` X \<inter> space (PiF {f ` J} (\<lambda>_. M))"
1.1441 + proof safe
1.1442 + fix x assume "x \<in> X"
1.1443 + with mf_fm[of x] sets_into_space[OF assms(2)] show "fm x \<in> mf -` X" by auto
1.1444 + show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms)
1.1445 + next
1.1446 + fix y x
1.1447 + assume x: "mf y \<in> X"
1.1448 + assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))"
1.1449 + thus "y \<in> fm ` X"
1.1450 + by (intro image_eqI[OF _ x], unfold finmap_eq_iff)
1.1451 + (auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def)
1.1452 + qed
1.1453 + also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))"
1.1454 + using assms
1.1455 + by (intro measurable_sets[OF mf_measurable]) auto
1.1456 + finally show ?thesis .
1.1457 +qed
1.1458 +
1.1459 +lemma fm_image_measurable_finite:
1.1460 + assumes "space M = UNIV"
1.1461 + assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M::'c measure))"
1.1462 + shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))"
1.1463 + using fm_image_measurable[OF assms]
1.1464 + by (rule subspace_set_in_sets) (auto simp: finite_subset)
1.1465 +
1.1466 +text {* measure on finmaps *}
1.1467 +
1.1468 +definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)"
1.1469 +
1.1470 +lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)"
1.1471 + unfolding mapmeasure_def by simp
1.1472 +
1.1473 +lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"
1.1474 + unfolding mapmeasure_def by simp
1.1475 +
1.1476 +lemma mapmeasure_PiF:
1.1477 + assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))"
1.1478 + assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))"
1.1479 + assumes "space N = UNIV"
1.1480 + assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
1.1481 + shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm -` X \<inter> extensional J))"
1.1482 + using assms
1.1483 + by (auto simp: measurable_eqI[OF s1 refl s2 refl] mapmeasure_def emeasure_distr
1.1484 + fm_measurable space_PiM)
1.1485 +
1.1486 +lemma mapmeasure_PiM:
1.1487 + fixes N::"'c measure"
1.1488 + assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))"
1.1489 + assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))"
1.1490 + assumes N: "space N = UNIV"
1.1491 + assumes X: "X \<in> sets M"
1.1492 + shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)"
1.1493 + unfolding mapmeasure_def
1.1494 +proof (subst emeasure_distr, subst measurable_eqI[OF s1 refl s2 refl], rule fm_measurable)
1.1495 + have "X \<subseteq> space (Pi\<^isub>M J (\<lambda>_. N))" using assms by (simp add: sets_into_space)
1.1496 + from assms inj_on_fm[of "\<lambda>_. N"] set_mp[OF this] have "fm -` fm ` X \<inter> space (Pi\<^isub>M J (\<lambda>_. N)) = X"
1.1497 + by (auto simp: vimage_image_eq inj_on_def)
1.1498 + thus "emeasure M X = emeasure M (fm -` fm ` X \<inter> space M)" using s1
1.1499 + by simp
1.1500 + show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
1.1501 + by (rule fm_image_measurable_finite[OF N X[simplified s2]])
1.1502 +qed simp
1.1503 +
1.1504 +end
1.1505 +
1.1506 +end
2.1 --- a/src/HOL/Probability/Probability.thy Thu Nov 15 10:49:58 2012 +0100
2.2 +++ b/src/HOL/Probability/Probability.thy Thu Nov 15 11:16:58 2012 +0100
2.3 @@ -3,7 +3,7 @@
2.4 Complete_Measure
2.5 Probability_Measure
2.6 Infinite_Product_Measure
2.7 - Regularity
2.8 + Projective_Limit
2.9 Independent_Family
2.10 Information
2.11 begin
3.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
3.2 +++ b/src/HOL/Probability/Projective_Limit.thy Thu Nov 15 11:16:58 2012 +0100
3.3 @@ -0,0 +1,690 @@
3.4 +(* Title: HOL/Probability/Projective_Family.thy
3.5 + Author: Fabian Immler, TU München
3.6 +*)
3.7 +
3.8 +header {* Projective Limit *}
3.9 +
3.10 +theory Projective_Limit
3.11 + imports
3.12 + Caratheodory
3.13 + Fin_Map
3.14 + Regularity
3.15 + Projective_Family
3.16 + Infinite_Product_Measure
3.17 +begin
3.18 +
3.19 +subsection {* Enumeration of Countable Union of Finite Sets *}
3.20 +
3.21 +locale finite_set_sequence =
3.22 + fixes Js::"nat \<Rightarrow> 'a set"
3.23 + assumes finite_seq[simp]: "finite (Js n)"
3.24 +begin
3.25 +
3.26 +text {* Enumerate finite set *}
3.27 +
3.28 +definition "enum_finite_max J = (SOME n. \<exists> f. J = f ` {i. i < n} \<and> inj_on f {i. i < n})"
3.29 +
3.30 +definition enum_finite where
3.31 + "enum_finite J =
3.32 + (SOME f. J = f ` {i::nat. i < enum_finite_max J} \<and> inj_on f {i. i < enum_finite_max J})"
3.33 +
3.34 +lemma enum_finite_max:
3.35 + assumes "finite J"
3.36 + shows "\<exists>f::nat\<Rightarrow>_. J = f ` {i. i < enum_finite_max J} \<and> inj_on f {i. i < enum_finite_max J}"
3.37 + unfolding enum_finite_max_def
3.38 + by (rule someI_ex) (rule finite_imp_nat_seg_image_inj_on[OF `finite J`])
3.39 +
3.40 +lemma enum_finite:
3.41 + assumes "finite J"
3.42 + shows "J = enum_finite J ` {i::nat. i < enum_finite_max J} \<and>
3.43 + inj_on (enum_finite J) {i::nat. i < enum_finite_max J}"
3.44 + unfolding enum_finite_def
3.45 + by (rule someI_ex[of "\<lambda>f. J = f ` {i::nat. i < enum_finite_max J} \<and>
3.46 + inj_on f {i. i < enum_finite_max J}"]) (rule enum_finite_max[OF `finite J`])
3.47 +
3.48 +lemma in_set_enum_exist:
3.49 + assumes "finite A"
3.50 + assumes "y \<in> A"
3.51 + shows "\<exists>i. y = enum_finite A i"
3.52 + using assms enum_finite by auto
3.53 +
3.54 +definition set_of_Un where "set_of_Un j = (LEAST n. j \<in> Js n)"
3.55 +
3.56 +definition index_in_set where "index_in_set J j = (SOME n. j = enum_finite J n)"
3.57 +
3.58 +definition Un_to_nat where
3.59 + "Un_to_nat j = to_nat (set_of_Un j, index_in_set (Js (set_of_Un j)) j)"
3.60 +
3.61 +lemma inj_on_Un_to_nat:
3.62 + shows "inj_on Un_to_nat (\<Union>n::nat. Js n)"
3.63 +proof (rule inj_onI)
3.64 + fix x y
3.65 + assume "x \<in> (\<Union>n. Js n)" "y \<in> (\<Union>n. Js n)"
3.66 + then obtain ix iy where ix: "x \<in> Js ix" and iy: "y \<in> Js iy" by blast
3.67 + assume "Un_to_nat x = Un_to_nat y"
3.68 + hence "set_of_Un x = set_of_Un y"
3.69 + "index_in_set (Js (set_of_Un y)) y = index_in_set (Js (set_of_Un x)) x"
3.70 + by (auto simp: Un_to_nat_def)
3.71 + moreover
3.72 + {
3.73 + fix x assume "x \<in> Js (set_of_Un x)"
3.74 + have "x = enum_finite (Js (set_of_Un x)) (index_in_set (Js (set_of_Un x)) x)"
3.75 + unfolding index_in_set_def
3.76 + apply (rule someI_ex)
3.77 + using `x \<in> Js (set_of_Un x)` finite_seq
3.78 + apply (auto intro!: in_set_enum_exist)
3.79 + done
3.80 + } note H = this
3.81 + moreover
3.82 + have "y \<in> Js (set_of_Un y)" unfolding set_of_Un_def using iy by (rule LeastI)
3.83 + note H[OF this]
3.84 + moreover
3.85 + have "x \<in> Js (set_of_Un x)" unfolding set_of_Un_def using ix by (rule LeastI)
3.86 + note H[OF this]
3.87 + ultimately show "x = y" by simp
3.88 +qed
3.89 +
3.90 +lemma inj_Un[simp]:
3.91 + shows "inj_on (Un_to_nat) (Js n)"
3.92 + by (intro subset_inj_on[OF inj_on_Un_to_nat]) (auto simp: assms)
3.93 +
3.94 +lemma Un_to_nat_injectiveD:
3.95 + assumes "Un_to_nat x = Un_to_nat y"
3.96 + assumes "x \<in> Js i" "y \<in> Js j"
3.97 + shows "x = y"
3.98 + using assms
3.99 + by (intro inj_onD[OF inj_on_Un_to_nat]) auto
3.100 +
3.101 +end
3.102 +
3.103 +subsection {* Sequences of Finite Maps in Compact Sets *}
3.104 +
3.105 +locale finmap_seqs_into_compact =
3.106 + fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^isub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^isub>F 'a)" and M
3.107 + assumes compact: "\<And>n. compact (K n)"
3.108 + assumes f_in_K: "\<And>n. K n \<noteq> {}"
3.109 + assumes domain_K: "\<And>n. k \<in> K n \<Longrightarrow> domain k = domain (f n)"
3.110 + assumes proj_in_K:
3.111 + "\<And>t n m. m \<ge> n \<Longrightarrow> t \<in> domain (f n) \<Longrightarrow> (f m)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K n"
3.112 +begin
3.113 +
3.114 +lemma proj_in_K': "(\<exists>n. \<forall>m \<ge> n. (f m)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K n)"
3.115 + using proj_in_K f_in_K
3.116 +proof cases
3.117 + obtain k where "k \<in> K (Suc 0)" using f_in_K by auto
3.118 + assume "\<forall>n. t \<notin> domain (f n)"
3.119 + thus ?thesis
3.120 + by (auto intro!: exI[where x=1] image_eqI[OF _ `k \<in> K (Suc 0)`]
3.121 + simp: domain_K[OF `k \<in> K (Suc 0)`])
3.122 +qed blast
3.123 +
3.124 +lemma proj_in_KE:
3.125 + obtains n where "\<And>m. m \<ge> n \<Longrightarrow> (f m)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K n"
3.126 + using proj_in_K' by blast
3.127 +
3.128 +lemma compact_projset:
3.129 + shows "compact ((\<lambda>k. (k)\<^isub>F i) ` K n)"
3.130 + using continuous_proj compact by (rule compact_continuous_image)
3.131 +
3.132 +end
3.133 +
3.134 +lemma compactE':
3.135 + assumes "compact S" "\<forall>n\<ge>m. f n \<in> S"
3.136 + obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
3.137 +proof atomize_elim
3.138 + have "subseq (op + m)" by (simp add: subseq_def)
3.139 + have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
3.140 + from compactE[OF `compact S` this] guess l r .
3.141 + hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l"
3.142 + using subseq_o[OF `subseq (op + m)` `subseq r`] by (auto simp: o_def)
3.143 + thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) ----> l" by blast
3.144 +qed
3.145 +
3.146 +sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^isub>F n) ----> l)"
3.147 +proof
3.148 + fix n s
3.149 + assume "subseq s"
3.150 + from proj_in_KE[of n] guess n0 . note n0 = this
3.151 + have "\<forall>i \<ge> n0. ((f \<circ> s) i)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>F n) ` K n0"
3.152 + proof safe
3.153 + fix i assume "n0 \<le> i"
3.154 + also have "\<dots> \<le> s i" by (rule seq_suble) fact
3.155 + finally have "n0 \<le> s i" .
3.156 + with n0 show "((f \<circ> s) i)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>F n) ` K n0 "
3.157 + by auto
3.158 + qed
3.159 + from compactE'[OF compact_projset this] guess ls rs .
3.160 + thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^isub>F n) ----> l)" by (auto simp: o_def)
3.161 +qed
3.162 +
3.163 +lemma (in finmap_seqs_into_compact)
3.164 + diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^isub>F n) ----> l"
3.165 +proof -
3.166 + have "\<And>i n0. (f o seqseq i) i = f (diagseq i)" unfolding diagseq_def by simp
3.167 + from reducer_reduces obtain l where l: "(\<lambda>i. ((f \<circ> seqseq (Suc n)) i)\<^isub>F n) ----> l"
3.168 + unfolding seqseq_reducer
3.169 + by auto
3.170 + have "(\<lambda>i. (f (diagseq (i + Suc n)))\<^isub>F n) =
3.171 + (\<lambda>i. ((f o (diagseq o (op + (Suc n)))) i)\<^isub>F n)" by (simp add: add_commute)
3.172 + also have "\<dots> =
3.173 + (\<lambda>i. ((f o ((seqseq (Suc n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))))) i)\<^isub>F n)"
3.174 + unfolding diagseq_seqseq by simp
3.175 + also have "\<dots> = (\<lambda>i. ((f o ((seqseq (Suc n)))) i)\<^isub>F n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))"
3.176 + by (simp add: o_def)
3.177 + also have "\<dots> ----> l"
3.178 + proof (rule LIMSEQ_subseq_LIMSEQ[OF _ subseq_diagonal_rest], rule tendstoI)
3.179 + fix e::real assume "0 < e"
3.180 + from tendstoD[OF l `0 < e`]
3.181 + show "eventually (\<lambda>x. dist (((f \<circ> seqseq (Suc n)) x)\<^isub>F n) l < e)
3.182 + sequentially" .
3.183 + qed
3.184 + finally show ?thesis by (intro exI) (rule LIMSEQ_offset)
3.185 +qed
3.186 +
3.187 +subsection {* Daniell-Kolmogorov Theorem *}
3.188 +
3.189 +text {* Existence of Projective Limit *}
3.190 +
3.191 +locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure"
3.192 + for I::"'i set" and P
3.193 +begin
3.194 +
3.195 +abbreviation "PiB \<equiv> (\<lambda>J P. PiP J (\<lambda>_. borel) P)"
3.196 +
3.197 +lemma
3.198 + emeasure_PiB_emb_not_empty:
3.199 + assumes "I \<noteq> {}"
3.200 + assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
3.201 + shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)"
3.202 +proof -
3.203 + let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel"
3.204 + let ?G = generator
3.205 + interpret G!: algebra ?\<Omega> generator by (intro algebra_generator) fact
3.206 + note \<mu>G_mono =
3.207 + G.additive_increasing[OF positive_\<mu>G[OF `I \<noteq> {}`] additive_\<mu>G[OF `I \<noteq> {}`], THEN increasingD]
3.208 + have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
3.209 + proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G,
3.210 + OF `I \<noteq> {}`, OF `I \<noteq> {}`])
3.211 + fix A assume "A \<in> ?G"
3.212 + with generatorE guess J X . note JX = this
3.213 + interpret prob_space "P J" using prob_space[OF `finite J`] .
3.214 + show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: PiP_finite)
3.215 + next
3.216 + fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}"
3.217 + then have "decseq (\<lambda>i. \<mu>G (Z i))"
3.218 + by (auto intro!: \<mu>G_mono simp: decseq_def)
3.219 + moreover
3.220 + have "(INF i. \<mu>G (Z i)) = 0"
3.221 + proof (rule ccontr)
3.222 + assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0")
3.223 + moreover have "0 \<le> ?a"
3.224 + using Z positive_\<mu>G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def)
3.225 + ultimately have "0 < ?a" by auto
3.226 + hence "?a \<noteq> -\<infinity>" by auto
3.227 + have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^isub>M J (\<lambda>_. borel)) \<and>
3.228 + Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (PiB J P) B"
3.229 + using Z by (intro allI generator_Ex) auto
3.230 + then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I"
3.231 + "\<And>n. B' n \<in> sets (\<Pi>\<^isub>M i\<in>J' n. borel)"
3.232 + and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)"
3.233 + unfolding choice_iff by blast
3.234 + moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
3.235 + moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)"
3.236 + ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I"
3.237 + "\<And>n. B n \<in> sets (\<Pi>\<^isub>M i\<in>J n. borel)"
3.238 + by auto
3.239 + have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
3.240 + unfolding J_def by force
3.241 + have "\<forall>n. \<exists>j. j \<in> J n" using J by blast
3.242 + then obtain j where j: "\<And>n. j n \<in> J n"
3.243 + unfolding choice_iff by blast
3.244 + note [simp] = `\<And>n. finite (J n)`
3.245 + from J Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G"
3.246 + unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
3.247 + interpret prob_space "P (J i)" for i using prob_space by simp
3.248 + have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower)
3.249 + also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq \<mu>G_eq PiP_finite proj_sets)
3.250 + finally have "?a \<noteq> \<infinity>" by simp
3.251 + have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono
3.252 + by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq)
3.253 +
3.254 + interpret finite_set_sequence J by unfold_locales simp
3.255 + def Utn \<equiv> Un_to_nat
3.256 + interpret function_to_finmap "J n" Utn "inv_into (J n) Utn" for n
3.257 + by unfold_locales (auto simp: Utn_def)
3.258 + def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
3.259 + let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
3.260 + {
3.261 + fix n
3.262 + interpret finite_measure "P (J n)" by unfold_locales
3.263 + have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
3.264 + using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets)
3.265 + also
3.266 + have "\<dots> = ?SUP n"
3.267 + proof (rule inner_regular)
3.268 + show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
3.269 + unfolding P'_def
3.270 + by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets)
3.271 + show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
3.272 + next
3.273 + show "fm n ` B n \<in> sets borel"
3.274 + unfolding borel_eq_PiF_borel
3.275 + by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J)
3.276 + qed
3.277 + finally
3.278 + have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
3.279 + } note R = this
3.280 + have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a
3.281 + \<and> compact K \<and> K \<subseteq> fm n ` B n"
3.282 + proof
3.283 + fix n
3.284 + have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
3.285 + by (simp add: mapmeasure_PiF P'_def proj_space proj_sets)
3.286 + then interpret finite_measure "P' n" ..
3.287 + show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
3.288 + compact K \<and> K \<subseteq> fm n ` B n"
3.289 + unfolding R
3.290 + proof (rule ccontr)
3.291 + assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and>
3.292 + compact K' \<and> K' \<subseteq> fm n ` B n)"
3.293 + have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
3.294 + proof (intro SUP_least)
3.295 + fix K
3.296 + assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
3.297 + with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
3.298 + by auto
3.299 + hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
3.300 + unfolding not_less[symmetric] by simp
3.301 + hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
3.302 + using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
3.303 + thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
3.304 + qed
3.305 + hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
3.306 + hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
3.307 + hence "0 \<le> - (2 powr (-n) * ?a)"
3.308 + using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>`
3.309 + by (subst (asm) ereal_add_le_add_iff) (auto simp:)
3.310 + moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
3.311 + by (auto simp: ereal_zero_less_0_iff)
3.312 + ultimately show False by simp
3.313 + qed
3.314 + qed
3.315 + then obtain K' where K':
3.316 + "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
3.317 + "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
3.318 + unfolding choice_iff by blast
3.319 + def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^isub>M (J n) (\<lambda>_. borel))"
3.320 + have K_sets: "\<And>n. K n \<in> sets (Pi\<^isub>M (J n) (\<lambda>_. borel))"
3.321 + unfolding K_def
3.322 + using compact_imp_closed[OF `compact (K' _)`]
3.323 + by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
3.324 + (auto simp: borel_eq_PiF_borel[symmetric])
3.325 + have K_B: "\<And>n. K n \<subseteq> B n"
3.326 + proof
3.327 + fix x n
3.328 + assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n"
3.329 + using K' by (force simp: K_def)
3.330 + show "x \<in> B n"
3.331 + apply (rule inj_on_image_mem_iff[OF inj_on_fm _ fm_in])
3.332 + using `x \<in> K n` K_sets J[of n] sets_into_space
3.333 + apply (auto simp: proj_space)
3.334 + using J[of n] sets_into_space apply auto
3.335 + done
3.336 + qed
3.337 + def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
3.338 + have Z': "\<And>n. Z' n \<subseteq> Z n"
3.339 + unfolding Z_eq unfolding Z'_def
3.340 + proof (rule prod_emb_mono, safe)
3.341 + fix n x assume "x \<in> K n"
3.342 + hence "fm n x \<in> K' n" "x \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))"
3.343 + by (simp_all add: K_def proj_space)
3.344 + note this(1)
3.345 + also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
3.346 + finally have "fm n x \<in> fm n ` B n" .
3.347 + thus "x \<in> B n"
3.348 + proof safe
3.349 + fix y assume "y \<in> B n"
3.350 + moreover
3.351 + hence "y \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" using J sets_into_space[of "B n" "P (J n)"]
3.352 + by (auto simp add: proj_space proj_sets)
3.353 + assume "fm n x = fm n y"
3.354 + note inj_onD[OF inj_on_fm[OF space_borel],
3.355 + OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
3.356 + ultimately show "x \<in> B n" by simp
3.357 + qed
3.358 + qed
3.359 + { fix n
3.360 + have "Z' n \<in> ?G" using K' unfolding Z'_def
3.361 + apply (intro generatorI'[OF J(1-3)])
3.362 + unfolding K_def proj_space
3.363 + apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]])
3.364 + apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
3.365 + done
3.366 + }
3.367 + def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
3.368 + hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
3.369 + hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
3.370 + have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
3.371 + hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
3.372 + have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
3.373 + proof -
3.374 + fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
3.375 + have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
3.376 + by (auto simp: Y_def Z'_def)
3.377 + also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))"
3.378 + using `n \<ge> 1`
3.379 + by (subst prod_emb_INT) auto
3.380 + finally
3.381 + have Y_emb:
3.382 + "Y n = prod_emb I (\<lambda>_. borel) (J n)
3.383 + (\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
3.384 + hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto
3.385 + hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1`
3.386 + by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq)
3.387 + interpret finite_measure "(PiP (J n) (\<lambda>_. borel) P)"
3.388 + proof
3.389 + have "emeasure (PiP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>"
3.390 + using J by (subst emeasure_PiP) auto
3.391 + thus "emeasure (PiP (J n) (\<lambda>_. borel) P) (space (PiP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
3.392 + by (simp add: space_PiM)
3.393 + qed
3.394 + have "\<mu>G (Z n) = PiP (J n) (\<lambda>_. borel) P (B n)"
3.395 + unfolding Z_eq using J by (auto simp: \<mu>G_eq)
3.396 + moreover have "\<mu>G (Y n) =
3.397 + PiP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
3.398 + unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst \<mu>G_eq) auto
3.399 + moreover have "\<mu>G (Z n - Y n) = PiP (J n) (\<lambda>_. borel) P
3.400 + (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
3.401 + unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
3.402 + by (subst \<mu>G_eq) (auto intro!: Diff)
3.403 + ultimately
3.404 + have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
3.405 + using J J_mono K_sets `n \<ge> 1`
3.406 + by (simp only: emeasure_eq_measure)
3.407 + (auto dest!: bspec[where x=n]
3.408 + simp: extensional_restrict emeasure_eq_measure prod_emb_iff
3.409 + intro!: measure_Diff[symmetric] set_mp[OF K_B])
3.410 + also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1`
3.411 + unfolding Y_def by (force simp: decseq_def)
3.412 + have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G"
3.413 + using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto
3.414 + hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))"
3.415 + using subs G.additive_increasing[OF positive_\<mu>G[OF `I \<noteq> {}`] additive_\<mu>G[OF `I \<noteq> {}`]]
3.416 + unfolding increasing_def by auto
3.417 + also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G`
3.418 + by (intro G.subadditive[OF positive_\<mu>G additive_\<mu>G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto
3.419 + also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
3.420 + proof (rule setsum_mono)
3.421 + fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
3.422 + have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
3.423 + unfolding Z'_def Z_eq by simp
3.424 + also have "\<dots> = P (J i) (B i - K i)"
3.425 + apply (subst \<mu>G_eq) using J K_sets apply auto
3.426 + apply (subst PiP_finite) apply auto
3.427 + done
3.428 + also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
3.429 + apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets)
3.430 + done
3.431 + also have "\<dots> = P (J i) (B i) - P' i (K' i)"
3.432 + unfolding K_def P'_def
3.433 + by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric]
3.434 + compact_imp_closed[OF `compact (K' _)`] space_PiM)
3.435 + also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
3.436 + finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
3.437 + qed
3.438 + also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))"
3.439 + using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto
3.440 + also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp
3.441 + also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
3.442 + by (simp add: setsum_left_distrib)
3.443 + also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
3.444 + proof (rule mult_strict_right_mono)
3.445 + have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)"
3.446 + by (rule setsum_cong)
3.447 + (auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide)
3.448 + also have "{1..<Suc n} = {0..<Suc n} - {0}" by auto
3.449 + also have "setsum (op ^ (1 / 2::real)) ({0..<Suc n} - {0}) =
3.450 + setsum (op ^ (1 / 2)) ({0..<Suc n}) - 1" by (auto simp: setsum_diff1)
3.451 + also have "\<dots> < 1" by (subst sumr_geometric) auto
3.452 + finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
3.453 + qed (auto simp:
3.454 + `0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric])
3.455 + also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real')
3.456 + also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower)
3.457 + finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
3.458 + hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
3.459 + using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
3.460 + have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
3.461 + also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
3.462 + apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
3.463 + finally have "\<mu>G (Y n) > 0"
3.464 + using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
3.465 + thus "Y n \<noteq> {}" using positive_\<mu>G `I \<noteq> {}` by (auto simp add: positive_def)
3.466 + qed
3.467 + hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
3.468 + then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
3.469 + {
3.470 + fix t and n m::nat
3.471 + assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
3.472 + from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
3.473 + also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
3.474 + finally
3.475 + have "fm n (restrict (y m) (J n)) \<in> K' n"
3.476 + unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
3.477 + moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
3.478 + using J by (simp add: fm_def)
3.479 + ultimately have "fm n (y m) \<in> K' n" by simp
3.480 + } note fm_in_K' = this
3.481 + interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
3.482 + proof
3.483 + fix n show "compact (K' n)" by fact
3.484 + next
3.485 + fix n
3.486 + from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
3.487 + also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
3.488 + finally
3.489 + have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
3.490 + unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
3.491 + thus "K' (Suc n) \<noteq> {}" by auto
3.492 + fix k
3.493 + assume "k \<in> K' (Suc n)"
3.494 + with K'[of "Suc n"] sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
3.495 + then obtain b where "k = fm (Suc n) b" by auto
3.496 + thus "domain k = domain (fm (Suc n) (y (Suc n)))"
3.497 + by (simp_all add: fm_def)
3.498 + next
3.499 + fix t and n m::nat
3.500 + assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
3.501 + assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
3.502 + then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
3.503 + hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
3.504 + have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
3.505 + by (intro fm_in_K') simp_all
3.506 + show "(fm (Suc m) (y (Suc m)))\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K' (Suc n)"
3.507 + apply (rule image_eqI[OF _ img])
3.508 + using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
3.509 + unfolding j by (subst proj_fm, auto)+
3.510 + qed
3.511 + have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z"
3.512 + using diagonal_tendsto ..
3.513 + then obtain z where z:
3.514 + "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z t"
3.515 + unfolding choice_iff by blast
3.516 + {
3.517 + fix n :: nat assume "n \<ge> 1"
3.518 + have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
3.519 + by simp
3.520 + moreover
3.521 + {
3.522 + fix t
3.523 + assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
3.524 + hence "t \<in> Utn ` J n" by simp
3.525 + then obtain j where j: "t = Utn j" "j \<in> J n" by auto
3.526 + have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> z t"
3.527 + apply (subst (2) tendsto_iff, subst eventually_sequentially)
3.528 + proof safe
3.529 + fix e :: real assume "0 < e"
3.530 + { fix i x assume "i \<ge> n" "t \<in> domain (fm n x)"
3.531 + moreover
3.532 + hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
3.533 + ultimately have "(fm i x)\<^isub>F t = (fm n x)\<^isub>F t"
3.534 + using j by (auto simp: proj_fm dest!:
3.535 + Un_to_nat_injectiveD[simplified Utn_def[symmetric]])
3.536 + } note index_shift = this
3.537 + have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
3.538 + apply (rule le_SucI)
3.539 + apply (rule order_trans) apply simp
3.540 + apply (rule seq_suble[OF subseq_diagseq])
3.541 + done
3.542 + from z
3.543 + have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e"
3.544 + unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
3.545 + then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
3.546 + dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" by auto
3.547 + show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e "
3.548 + proof (rule exI[where x="max N n"], safe)
3.549 + fix na assume "max N n \<le> na"
3.550 + hence "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) =
3.551 + dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^isub>F t) (z t)" using t
3.552 + by (subst index_shift[OF I]) auto
3.553 + also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
3.554 + finally show "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e" .
3.555 + qed
3.556 + qed
3.557 + hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> (finmap_of (Utn ` J n) z)\<^isub>F t"
3.558 + by (simp add: tendsto_intros)
3.559 + } ultimately
3.560 + have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
3.561 + by (rule tendsto_finmap)
3.562 + hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
3.563 + by (intro lim_subseq) (simp add: subseq_def)
3.564 + moreover
3.565 + have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
3.566 + apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
3.567 + apply (rule le_trans)
3.568 + apply (rule le_add2)
3.569 + using seq_suble[OF subseq_diagseq]
3.570 + apply auto
3.571 + done
3.572 + moreover
3.573 + from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
3.574 + ultimately
3.575 + have "finmap_of (Utn ` J n) z \<in> K' n"
3.576 + unfolding closed_sequential_limits by blast
3.577 + also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))"
3.578 + by (auto simp: finmap_eq_iff fm_def compose_def f_inv_into_f)
3.579 + finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
3.580 + moreover
3.581 + let ?J = "\<Union>n. J n"
3.582 + have "(?J \<inter> J n) = J n" by auto
3.583 + ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
3.584 + unfolding K_def by (auto simp: proj_space space_PiM)
3.585 + hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
3.586 + using J by (auto simp: prod_emb_def extensional_def)
3.587 + also have "\<dots> \<subseteq> Z n" using Z' by simp
3.588 + finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
3.589 + } note in_Z = this
3.590 + hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
3.591 + hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp
3.592 + thus False using Z by simp
3.593 + qed
3.594 + ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0"
3.595 + using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (Z i)"] by simp
3.596 + qed
3.597 + then guess \<mu> .. note \<mu> = this
3.598 + def f \<equiv> "finmap_of J B"
3.599 + show "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)"
3.600 + proof (subst emeasure_extend_measure_Pair[OF PiP_def, of I "\<lambda>_. borel" \<mu>])
3.601 + show "positive (sets (PiB I P)) \<mu>" "countably_additive (sets (PiB I P)) \<mu>"
3.602 + using \<mu> unfolding sets_PiP sets_PiM_generator by (auto simp: measure_space_def)
3.603 + next
3.604 + show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel"
3.605 + using assms by (auto simp: f_def)
3.606 + next
3.607 + fix J and X::"'i \<Rightarrow> 'a set"
3.608 + show "prod_emb I (\<lambda>_. borel) J (Pi\<^isub>E J X) \<in> Pow ((I \<rightarrow> space borel) \<inter> extensional I)"
3.609 + by (auto simp: prod_emb_def)
3.610 + assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel"
3.611 + hence "emb I J (Pi\<^isub>E J X) \<in> generator" using assms
3.612 + by (intro generatorI[where J=J and X="Pi\<^isub>E J X"]) (auto intro: sets_PiM_I_finite)
3.613 + hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp
3.614 + also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)"
3.615 + using JX assms proj_sets
3.616 + by (subst \<mu>G_eq) (auto simp: \<mu>G_eq PiP_finite intro: sets_PiM_I_finite)
3.617 + finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" .
3.618 + next
3.619 + show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (PiP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)"
3.620 + using assms by (simp add: f_def PiP_finite Pi_def)
3.621 + qed
3.622 +qed
3.623 +
3.624 +end
3.625 +
3.626 +sublocale polish_projective \<subseteq> P!: prob_space "(PiB I P)"
3.627 +proof
3.628 + show "emeasure (PiB I P) (space (PiB I P)) = 1"
3.629 + proof cases
3.630 + assume "I = {}"
3.631 + interpret prob_space "P {}" using prob_space by simp
3.632 + show ?thesis
3.633 + by (simp add: space_PiM_empty PiP_finite emeasure_space_1 `I = {}`)
3.634 + next
3.635 + assume "I \<noteq> {}"
3.636 + then obtain i where "i \<in> I" by auto
3.637 + interpret prob_space "P {i}" using prob_space by simp
3.638 + have R: "(space (PiB I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))"
3.639 + by (auto simp: prod_emb_def space_PiM)
3.640 + moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM)
3.641 + ultimately show ?thesis using `i \<in> I`
3.642 + apply (subst R)
3.643 + apply (subst emeasure_PiB_emb_not_empty)
3.644 + apply (auto simp: PiP_finite emeasure_space_1)
3.645 + done
3.646 + qed
3.647 +qed
3.648 +
3.649 +context polish_projective begin
3.650 +
3.651 +lemma emeasure_PiB_emb:
3.652 + assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
3.653 + shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)"
3.654 +proof cases
3.655 + interpret prob_space "P {}" using prob_space by simp
3.656 + assume "J = {}"
3.657 + moreover have "emb I {} {\<lambda>x. undefined} = space (PiB I P)"
3.658 + by (auto simp: space_PiM prod_emb_def)
3.659 + moreover have "{\<lambda>x. undefined} = space (PiB {} P)"
3.660 + by (auto simp: space_PiM prod_emb_def)
3.661 + ultimately show ?thesis
3.662 + by (simp add: P.emeasure_space_1 PiP_finite emeasure_space_1 del: space_PiP)
3.663 +next
3.664 + assume "J \<noteq> {}" with X show ?thesis
3.665 + by (subst emeasure_PiB_emb_not_empty) (auto simp: PiP_finite)
3.666 +qed
3.667 +
3.668 +lemma measure_PiB_emb:
3.669 + assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
3.670 + shows "measure (PiB I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)"
3.671 +proof -
3.672 + interpret prob_space "P J" using prob_space assms by simp
3.673 + show ?thesis
3.674 + using emeasure_PiB_emb[OF assms]
3.675 + unfolding emeasure_eq_measure PiP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
3.676 + by simp
3.677 +qed
3.678 +
3.679 +end
3.680 +
3.681 +locale polish_product_prob_space =
3.682 + product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
3.683 +
3.684 +sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
3.685 +proof qed
3.686 +
3.687 +lemma (in polish_product_prob_space)
3.688 + PiP_eq_PiM:
3.689 + "I \<noteq> {} \<Longrightarrow> PiP I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
3.690 + PiM I (\<lambda>_. borel)"
3.691 + by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_PiB_emb)
3.692 +
3.693 +end