src/HOL/Probability/Fin_Map.thy
 author immler Thu Nov 15 11:16:58 2012 +0100 (2012-11-15) changeset 50088 32d1795cc77a child 50091 b3b5dc2350b7 permissions -rw-r--r--
proof is based on auxiliary type finmap::polish_space
```     1 (*  Title:      HOL/Probability/Projective_Family.thy
```
```     2     Author:     Fabian Immler, TU München
```
```     3 *)
```
```     4
```
```     5 theory Fin_Map
```
```     6 imports Finite_Product_Measure
```
```     7 begin
```
```     8
```
```     9 section {* Finite Maps *}
```
```    10
```
```    11 text {* Auxiliary type that is instantiated to @{class polish_space}, needed for the proof of
```
```    12   projective limit. @{const extensional} functions are used for the representation in order to
```
```    13   stay close to the developments of (finite) products @{const Pi\<^isub>E} and their sigma-algebra
```
```    14   @{const Pi\<^isub>M}. *}
```
```    15
```
```    16 typedef ('i, 'a) finmap ("(_ \<Rightarrow>\<^isub>F /_)" [22, 21] 21) =
```
```    17   "{(I::'i set, f::'i \<Rightarrow> 'a). finite I \<and> f \<in> extensional I}" by auto
```
```    18
```
```    19 subsection {* Domain and Application *}
```
```    20
```
```    21 definition domain where "domain P = fst (Rep_finmap P)"
```
```    22
```
```    23 lemma finite_domain[simp, intro]: "finite (domain P)"
```
```    24   by (cases P) (auto simp: domain_def Abs_finmap_inverse)
```
```    25
```
```    26 definition proj ("_\<^isub>F"  1000) where "proj P i = snd (Rep_finmap P) i"
```
```    27
```
```    28 declare [[coercion proj]]
```
```    29
```
```    30 lemma extensional_proj[simp, intro]: "(P)\<^isub>F \<in> extensional (domain P)"
```
```    31   by (cases P) (auto simp: domain_def Abs_finmap_inverse proj_def[abs_def])
```
```    32
```
```    33 lemma proj_undefined[simp, intro]: "i \<notin> domain P \<Longrightarrow> P i = undefined"
```
```    34   using extensional_proj[of P] unfolding extensional_def by auto
```
```    35
```
```    36 lemma finmap_eq_iff: "P = Q \<longleftrightarrow> (domain P = domain Q \<and> (\<forall>i\<in>domain P. P i = Q i))"
```
```    37   by (cases P, cases Q)
```
```    38      (auto simp add: Abs_finmap_inject extensional_def domain_def proj_def Abs_finmap_inverse
```
```    39               intro: extensionalityI)
```
```    40
```
```    41 subsection {* Countable Finite Maps *}
```
```    42
```
```    43 instance finmap :: (countable, countable) countable
```
```    44 proof
```
```    45   obtain mapper where mapper: "\<And>fm :: 'a \<Rightarrow>\<^isub>F 'b. set (mapper fm) = domain fm"
```
```    46     by (metis finite_list[OF finite_domain])
```
```    47   have "inj (\<lambda>fm. map (\<lambda>i. (i, (fm)\<^isub>F i)) (mapper fm))" (is "inj ?F")
```
```    48   proof (rule inj_onI)
```
```    49     fix f1 f2 assume "?F f1 = ?F f2"
```
```    50     then have "map fst (?F f1) = map fst (?F f2)" by simp
```
```    51     then have "mapper f1 = mapper f2" by (simp add: comp_def)
```
```    52     then have "domain f1 = domain f2" by (simp add: mapper[symmetric])
```
```    53     with `?F f1 = ?F f2` show "f1 = f2"
```
```    54       unfolding `mapper f1 = mapper f2` map_eq_conv mapper
```
```    55       by (simp add: finmap_eq_iff)
```
```    56   qed
```
```    57   then show "\<exists>to_nat :: 'a \<Rightarrow>\<^isub>F 'b \<Rightarrow> nat. inj to_nat"
```
```    58     by (intro exI[of _ "to_nat \<circ> ?F"] inj_comp) auto
```
```    59 qed
```
```    60
```
```    61 subsection {* Constructor of Finite Maps *}
```
```    62
```
```    63 definition "finmap_of inds f = Abs_finmap (inds, restrict f inds)"
```
```    64
```
```    65 lemma proj_finmap_of[simp]:
```
```    66   assumes "finite inds"
```
```    67   shows "(finmap_of inds f)\<^isub>F = restrict f inds"
```
```    68   using assms
```
```    69   by (auto simp: Abs_finmap_inverse finmap_of_def proj_def)
```
```    70
```
```    71 lemma domain_finmap_of[simp]:
```
```    72   assumes "finite inds"
```
```    73   shows "domain (finmap_of inds f) = inds"
```
```    74   using assms
```
```    75   by (auto simp: Abs_finmap_inverse finmap_of_def domain_def)
```
```    76
```
```    77 lemma finmap_of_eq_iff[simp]:
```
```    78   assumes "finite i" "finite j"
```
```    79   shows "finmap_of i m = finmap_of j n \<longleftrightarrow> i = j \<and> restrict m i = restrict n i"
```
```    80   using assms
```
```    81   apply (auto simp: finmap_eq_iff restrict_def) by metis
```
```    82
```
```    83 lemma
```
```    84   finmap_of_inj_on_extensional_finite:
```
```    85   assumes "finite K"
```
```    86   assumes "S \<subseteq> extensional K"
```
```    87   shows "inj_on (finmap_of K) S"
```
```    88 proof (rule inj_onI)
```
```    89   fix x y::"'a \<Rightarrow> 'b"
```
```    90   assume "finmap_of K x = finmap_of K y"
```
```    91   hence "(finmap_of K x)\<^isub>F = (finmap_of K y)\<^isub>F" by simp
```
```    92   moreover
```
```    93   assume "x \<in> S" "y \<in> S" hence "x \<in> extensional K" "y \<in> extensional K" using assms by auto
```
```    94   ultimately
```
```    95   show "x = y" using assms by (simp add: extensional_restrict)
```
```    96 qed
```
```    97
```
```    98 lemma finmap_choice:
```
```    99   assumes *: "\<And>i. i \<in> I \<Longrightarrow> \<exists>x. P i x" and I: "finite I"
```
```   100   shows "\<exists>fm. domain fm = I \<and> (\<forall>i\<in>I. P i (fm i))"
```
```   101 proof -
```
```   102   have "\<exists>f. \<forall>i\<in>I. P i (f i)"
```
```   103     unfolding bchoice_iff[symmetric] using * by auto
```
```   104   then guess f ..
```
```   105   with I show ?thesis
```
```   106     by (intro exI[of _ "finmap_of I f"]) auto
```
```   107 qed
```
```   108
```
```   109 subsection {* Product set of Finite Maps *}
```
```   110
```
```   111 text {* This is @{term Pi} for Finite Maps, most of this is copied *}
```
```   112
```
```   113 definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^isub>F 'a) set" where
```
```   114   "Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^isub>F i \<in> A i) } "
```
```   115
```
```   116 syntax
```
```   117   "_Pi'"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI' _:_./ _)" 10)
```
```   118
```
```   119 syntax (xsymbols)
```
```   120   "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>' _\<in>_./ _)"   10)
```
```   121
```
```   122 syntax (HTML output)
```
```   123   "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>' _\<in>_./ _)"   10)
```
```   124
```
```   125 translations
```
```   126   "PI' x:A. B" == "CONST Pi' A (%x. B)"
```
```   127
```
```   128 abbreviation
```
```   129   finmapset :: "['a set, 'b set] => ('a \<Rightarrow>\<^isub>F 'b) set"
```
```   130     (infixr "~>" 60) where
```
```   131   "A ~> B \<equiv> Pi' A (%_. B)"
```
```   132
```
```   133 notation (xsymbols)
```
```   134   finmapset  (infixr "\<leadsto>" 60)
```
```   135
```
```   136 subsubsection{*Basic Properties of @{term Pi'}*}
```
```   137
```
```   138 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"
```
```   139   by (simp add: Pi'_def)
```
```   140
```
```   141 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"
```
```   142   by (simp add:Pi'_def)
```
```   143
```
```   144 lemma finmapsetI: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<leadsto> B"
```
```   145   by (simp add: Pi_def)
```
```   146
```
```   147 lemma Pi'_mem: "f\<in> Pi' A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
```
```   148   by (simp add: Pi'_def)
```
```   149
```
```   150 lemma Pi'_iff: "f \<in> Pi' I X \<longleftrightarrow> domain f = I \<and> (\<forall>i\<in>I. f i \<in> X i)"
```
```   151   unfolding Pi'_def by auto
```
```   152
```
```   153 lemma Pi'E [elim]:
```
```   154   "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"
```
```   155   by(auto simp: Pi'_def)
```
```   156
```
```   157 lemma in_Pi'_cong:
```
```   158   "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"
```
```   159   by (auto simp: Pi'_def)
```
```   160
```
```   161 lemma funcset_mem: "[|f \<in> A \<leadsto> B; x \<in> A|] ==> f x \<in> B"
```
```   162   by (simp add: Pi'_def)
```
```   163
```
```   164 lemma funcset_image: "f \<in> A \<leadsto> B ==> f ` A \<subseteq> B"
```
```   165 by auto
```
```   166
```
```   167 lemma Pi'_eq_empty[simp]:
```
```   168   assumes "finite A" shows "(Pi' A B) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
```
```   169   using assms
```
```   170   apply (simp add: Pi'_def, auto)
```
```   171   apply (drule_tac x = "finmap_of A (\<lambda>u. SOME y. y \<in> B u)" in spec, auto)
```
```   172   apply (cut_tac P= "%y. y \<in> B i" in some_eq_ex, auto)
```
```   173   done
```
```   174
```
```   175 lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C"
```
```   176   by (auto simp: Pi'_def)
```
```   177
```
```   178 lemma Pi_Pi': "finite A \<Longrightarrow> (Pi\<^isub>E A B) = proj ` Pi' A B"
```
```   179   apply (auto simp: Pi'_def Pi_def extensional_def)
```
```   180   apply (rule_tac x = "finmap_of A (restrict x A)" in image_eqI)
```
```   181   apply auto
```
```   182   done
```
```   183
```
```   184 subsection {* Metric Space of Finite Maps *}
```
```   185
```
```   186 instantiation finmap :: (type, metric_space) metric_space
```
```   187 begin
```
```   188
```
```   189 definition dist_finmap where
```
```   190   "dist P Q = (\<Sum>i\<in>domain P \<union> domain Q. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)) +
```
```   191     card ((domain P - domain Q) \<union> (domain Q - domain P))"
```
```   192
```
```   193 lemma dist_finmap_extend:
```
```   194   assumes "finite X"
```
```   195   shows "dist P Q = (\<Sum>i\<in>domain P \<union> domain Q \<union> X. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)) +
```
```   196     card ((domain P - domain Q) \<union> (domain Q - domain P))"
```
```   197     unfolding dist_finmap_def add_right_cancel
```
```   198     using assms extensional_arb[of "(P)\<^isub>F"] extensional_arb[of "(Q)\<^isub>F" "domain Q"]
```
```   199     by (intro setsum_mono_zero_cong_left) auto
```
```   200
```
```   201 definition open_finmap :: "('a \<Rightarrow>\<^isub>F 'b) set \<Rightarrow> bool" where
```
```   202   "open_finmap S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   203
```
```   204 lemma add_eq_zero_iff[simp]:
```
```   205   fixes a b::real
```
```   206   assumes "a \<ge> 0" "b \<ge> 0"
```
```   207   shows "a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
```
```   208 using assms by auto
```
```   209
```
```   210 lemma dist_le_1_imp_domain_eq:
```
```   211   assumes "dist P Q < 1"
```
```   212   shows "domain P = domain Q"
```
```   213 proof -
```
```   214   have "0 \<le> (\<Sum>i\<in>domain P \<union> domain Q. dist (P i) (Q i))"
```
```   215     by (simp add: setsum_nonneg)
```
```   216   with assms have "card (domain P - domain Q \<union> (domain Q - domain P)) = 0"
```
```   217     unfolding dist_finmap_def by arith
```
```   218   thus "domain P = domain Q" by auto
```
```   219 qed
```
```   220
```
```   221 lemma dist_proj:
```
```   222   shows "dist ((x)\<^isub>F i) ((y)\<^isub>F i) \<le> dist x y"
```
```   223 proof -
```
```   224   have "dist (x i) (y i) = (\<Sum>i\<in>{i}. dist (x i) (y i))" by simp
```
```   225   also have "\<dots> \<le> (\<Sum>i\<in>domain x \<union> domain y \<union> {i}. dist (x i) (y i))"
```
```   226     by (intro setsum_mono2) auto
```
```   227   also have "\<dots> \<le> dist x y" by (simp add: dist_finmap_extend[of "{i}"])
```
```   228   finally show ?thesis by simp
```
```   229 qed
```
```   230
```
```   231 lemma open_Pi'I:
```
```   232   assumes open_component: "\<And>i. i \<in> I \<Longrightarrow> open (A i)"
```
```   233   shows "open (Pi' I A)"
```
```   234 proof (subst open_finmap_def, safe)
```
```   235   fix x assume x: "x \<in> Pi' I A"
```
```   236   hence dim_x: "domain x = I" by (simp add: Pi'_def)
```
```   237   hence [simp]: "finite I" unfolding dim_x[symmetric] by simp
```
```   238   have "\<exists>ei. \<forall>i\<in>I. 0 < ei i \<and> (\<forall>y. dist y (x i) < ei i \<longrightarrow> y \<in> A i)"
```
```   239   proof (safe intro!: bchoice)
```
```   240     fix i assume i: "i \<in> I"
```
```   241     moreover with open_component have "open (A i)" by simp
```
```   242     moreover have "x i \<in> A i" using x i
```
```   243       by (auto simp: proj_def)
```
```   244     ultimately show "\<exists>e>0. \<forall>y. dist y (x i) < e \<longrightarrow> y \<in> A i"
```
```   245       using x by (auto simp: open_dist Ball_def)
```
```   246   qed
```
```   247   then guess ei .. note ei = this
```
```   248   def es \<equiv> "ei ` I"
```
```   249   def e \<equiv> "if es = {} then 0.5 else min 0.5 (Min es)"
```
```   250   from ei have "e > 0" using x
```
```   251     by (auto simp add: e_def es_def Pi'_def Ball_def)
```
```   252   moreover have "\<forall>y. dist y x < e \<longrightarrow> y \<in> Pi' I A"
```
```   253   proof (intro allI impI)
```
```   254     fix y
```
```   255     assume "dist y x < e"
```
```   256     also have "\<dots> < 1" by (auto simp: e_def)
```
```   257     finally have "domain y = domain x" by (rule dist_le_1_imp_domain_eq)
```
```   258     with dim_x have dims: "domain y = domain x" "domain x = I" by auto
```
```   259     show "y \<in> Pi' I A"
```
```   260     proof
```
```   261       show "domain y = I" using dims by simp
```
```   262     next
```
```   263       fix i
```
```   264       assume "i \<in> I"
```
```   265       have "dist (y i) (x i) \<le> dist y x" using dims `i \<in> I`
```
```   266         by (auto intro: dist_proj)
```
```   267       also have "\<dots> < e" using `dist y x < e` dims
```
```   268         by (simp add: dist_finmap_def)
```
```   269       also have "e \<le> Min (ei ` I)" using dims `i \<in> I`
```
```   270         by (auto simp: e_def es_def)
```
```   271       also have "\<dots> \<le> ei i" using `i \<in> I` by (simp add: e_def)
```
```   272       finally have "dist (y i) (x i) < ei i" .
```
```   273       with ei `i \<in> I` show "y i \<in> A  i" by simp
```
```   274     qed
```
```   275   qed
```
```   276   ultimately
```
```   277   show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> Pi' I A" by blast
```
```   278 qed
```
```   279
```
```   280 instance
```
```   281 proof
```
```   282   fix S::"('a \<Rightarrow>\<^isub>F 'b) set"
```
```   283   show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   284     unfolding open_finmap_def ..
```
```   285 next
```
```   286   fix P Q::"'a \<Rightarrow>\<^isub>F 'b"
```
```   287   show "dist P Q = 0 \<longleftrightarrow> P = Q"
```
```   288     by (auto simp: finmap_eq_iff dist_finmap_def setsum_nonneg setsum_nonneg_eq_0_iff)
```
```   289 next
```
```   290   fix P Q R::"'a \<Rightarrow>\<^isub>F 'b"
```
```   291   let ?symdiff = "\<lambda>a b. domain a - domain b \<union> (domain b - domain a)"
```
```   292   def E \<equiv> "domain P \<union> domain Q \<union> domain R"
```
```   293   hence "finite E" by (simp add: E_def)
```
```   294   have "card (?symdiff P Q) \<le> card (?symdiff P R \<union> ?symdiff Q R)"
```
```   295     by (auto intro: card_mono)
```
```   296   also have "\<dots> \<le> card (?symdiff P R) + card (?symdiff Q R)"
```
```   297     by (subst card_Un_Int) auto
```
```   298   finally have "dist P Q \<le> (\<Sum>i\<in>E. dist (P i) (R i) + dist (Q i) (R i)) +
```
```   299     real (card (?symdiff P R) + card (?symdiff Q R))"
```
```   300     unfolding dist_finmap_extend[OF `finite E`]
```
```   301     by (intro add_mono) (auto simp: E_def intro: setsum_mono dist_triangle_le)
```
```   302   also have "\<dots> \<le> dist P R + dist Q R"
```
```   303     unfolding dist_finmap_extend[OF `finite E`] by (simp add: ac_simps E_def setsum_addf[symmetric])
```
```   304   finally show "dist P Q \<le> dist P R + dist Q R" by simp
```
```   305 qed
```
```   306
```
```   307 end
```
```   308
```
```   309 lemma open_restricted_space:
```
```   310   shows "open {m. P (domain m)}"
```
```   311 proof -
```
```   312   have "{m. P (domain m)} = (\<Union>i \<in> Collect P. {m. domain m = i})" by auto
```
```   313   also have "open \<dots>"
```
```   314   proof (rule, safe, cases)
```
```   315     fix i::"'a set"
```
```   316     assume "finite i"
```
```   317     hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def)
```
```   318     also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`)
```
```   319     finally show "open {m. domain m = i}" .
```
```   320   next
```
```   321     fix i::"'a set"
```
```   322     assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto
```
```   323     also have "open \<dots>" by simp
```
```   324     finally show "open {m. domain m = i}" .
```
```   325   qed
```
```   326   finally show ?thesis .
```
```   327 qed
```
```   328
```
```   329 lemma closed_restricted_space:
```
```   330   shows "closed {m. P (domain m)}"
```
```   331 proof -
```
```   332   have "{m. P (domain m)} = - (\<Union>i \<in> - Collect P. {m. domain m = i})" by auto
```
```   333   also have "closed \<dots>"
```
```   334   proof (rule, rule, rule, cases)
```
```   335     fix i::"'a set"
```
```   336     assume "finite i"
```
```   337     hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def)
```
```   338     also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`)
```
```   339     finally show "open {m. domain m = i}" .
```
```   340   next
```
```   341     fix i::"'a set"
```
```   342     assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto
```
```   343     also have "open \<dots>" by simp
```
```   344     finally show "open {m. domain m = i}" .
```
```   345   qed
```
```   346   finally show ?thesis .
```
```   347 qed
```
```   348
```
```   349 lemma continuous_proj:
```
```   350   shows "continuous_on s (\<lambda>x. (x)\<^isub>F i)"
```
```   351   unfolding continuous_on_topological
```
```   352 proof safe
```
```   353   fix x B assume "x \<in> s" "open B" "x i \<in> B"
```
```   354   let ?A = "Pi' (domain x) (\<lambda>j. if i = j then B else UNIV)"
```
```   355   have "open ?A" using `open B` by (auto intro: open_Pi'I)
```
```   356   moreover have "x \<in> ?A" using `x i \<in> B` by auto
```
```   357   moreover have "(\<forall>y\<in>s. y \<in> ?A \<longrightarrow> y i \<in> B)"
```
```   358   proof (cases, safe)
```
```   359     fix y assume "y \<in> s"
```
```   360     assume "i \<notin> domain x" hence "undefined \<in> B" using `x i \<in> B`
```
```   361       by simp
```
```   362     moreover
```
```   363     assume "y \<in> ?A" hence "domain y = domain x" by (simp add: Pi'_def)
```
```   364     hence "y i = undefined" using `i \<notin> domain x` by simp
```
```   365     ultimately
```
```   366     show "y i \<in> B" by simp
```
```   367   qed force
```
```   368   ultimately
```
```   369   show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> y i \<in> B)" by blast
```
```   370 qed
```
```   371
```
```   372 subsection {* Complete Space of Finite Maps *}
```
```   373
```
```   374 lemma tendsto_dist_zero:
```
```   375   assumes "(\<lambda>i. dist (f i) g) ----> 0"
```
```   376   shows "f ----> g"
```
```   377   using assms by (auto simp: tendsto_iff dist_real_def)
```
```   378
```
```   379 lemma tendsto_dist_zero':
```
```   380   assumes "(\<lambda>i. dist (f i) g) ----> x"
```
```   381   assumes "0 = x"
```
```   382   shows "f ----> g"
```
```   383   using assms tendsto_dist_zero by simp
```
```   384
```
```   385 lemma tendsto_finmap:
```
```   386   fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^isub>F ('a::metric_space))"
```
```   387   assumes ind_f:  "\<And>n. domain (f n) = domain g"
```
```   388   assumes proj_g:  "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) ----> g i"
```
```   389   shows "f ----> g"
```
```   390   apply (rule tendsto_dist_zero')
```
```   391   unfolding dist_finmap_def assms
```
```   392   apply (rule tendsto_intros proj_g | simp)+
```
```   393   done
```
```   394
```
```   395 instance finmap :: (type, complete_space) complete_space
```
```   396 proof
```
```   397   fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^isub>F 'b"
```
```   398   assume "Cauchy P"
```
```   399   then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1"
```
```   400     by (force simp: cauchy)
```
```   401   def d \<equiv> "domain (P Nd)"
```
```   402   with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto
```
```   403   have [simp]: "finite d" unfolding d_def by simp
```
```   404   def p \<equiv> "\<lambda>i n. (P n) i"
```
```   405   def q \<equiv> "\<lambda>i. lim (p i)"
```
```   406   def Q \<equiv> "finmap_of d q"
```
```   407   have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_finmap_inverse)
```
```   408   {
```
```   409     fix i assume "i \<in> d"
```
```   410     have "Cauchy (p i)" unfolding cauchy p_def
```
```   411     proof safe
```
```   412       fix e::real assume "0 < e"
```
```   413       with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1"
```
```   414         by (force simp: cauchy min_def)
```
```   415       hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto
```
```   416       with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear)
```
```   417       show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e"
```
```   418       proof (safe intro!: exI[where x="N"])
```
```   419         fix n assume "N \<le> n" have "N \<le> N" by simp
```
```   420         have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)"
```
```   421           using dim[OF `N \<le> n`]  dim[OF `N \<le> N`] `i \<in> d`
```
```   422           by (auto intro!: dist_proj)
```
```   423         also have "\<dots> < e" using N[OF `N \<le> n`] by simp
```
```   424         finally show "dist ((P n) i) ((P N) i) < e" .
```
```   425       qed
```
```   426     qed
```
```   427     hence "convergent (p i)" by (metis Cauchy_convergent_iff)
```
```   428     hence "p i ----> q i" unfolding q_def convergent_def by (metis limI)
```
```   429   } note p = this
```
```   430   have "P ----> Q"
```
```   431   proof (rule metric_LIMSEQ_I)
```
```   432     fix e::real assume "0 < e"
```
```   433     def e' \<equiv> "min 1 (e / (card d + 1))"
```
```   434     hence "0 < e'" using `0 < e` by (auto simp: e'_def intro: divide_pos_pos)
```
```   435     have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e'"
```
```   436     proof (safe intro!: bchoice)
```
```   437       fix i assume "i \<in> d"
```
```   438       from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e'`]
```
```   439       show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e'" .
```
```   440     qed then guess ni .. note ni = this
```
```   441     def N \<equiv> "max Nd (Max (ni ` d))"
```
```   442     show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e"
```
```   443     proof (safe intro!: exI[where x="N"])
```
```   444       fix n assume "N \<le> n"
```
```   445       hence "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q"
```
```   446         using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse)
```
```   447       hence "dist (P n) Q = (\<Sum>i\<in>d. dist ((P n) i) (Q i))" by (simp add: dist_finmap_def)
```
```   448       also have "\<dots> \<le> (\<Sum>i\<in>d. e')"
```
```   449       proof (intro setsum_mono less_imp_le)
```
```   450         fix i assume "i \<in> d"
```
```   451         hence "ni i \<le> Max (ni ` d)" by simp
```
```   452         also have "\<dots> \<le> N" by (simp add: N_def)
```
```   453         also have "\<dots> \<le> n" using `N \<le> n` .
```
```   454         finally
```
```   455         show "dist ((P n) i) (Q i) < e'"
```
```   456           using ni `i \<in> d` by (auto simp: p_def q N_def)
```
```   457       qed
```
```   458       also have "\<dots> = card d * e'" by (simp add: real_eq_of_nat)
```
```   459       also have "\<dots> < e" using `0 < e` by (simp add: e'_def field_simps min_def)
```
```   460       finally show "dist (P n) Q < e" .
```
```   461     qed
```
```   462   qed
```
```   463   thus "convergent P" by (auto simp: convergent_def)
```
```   464 qed
```
```   465
```
```   466 subsection {* Polish Space of Finite Maps *}
```
```   467
```
```   468 instantiation finmap :: (countable, polish_space) polish_space
```
```   469 begin
```
```   470
```
```   471 definition enum_basis_finmap :: "nat \<Rightarrow> ('a \<Rightarrow>\<^isub>F 'b) set" where
```
```   472   "enum_basis_finmap n =
```
```   473   (let m = from_nat n::('a \<Rightarrow>\<^isub>F nat) in Pi' (domain m) (enum_basis o (m)\<^isub>F))"
```
```   474
```
```   475 lemma range_enum_basis_eq:
```
```   476   "range enum_basis_finmap = {Pi' I S|I S. finite I \<and> (\<forall>i \<in> I. S i \<in> range enum_basis)}"
```
```   477 proof (auto simp: enum_basis_finmap_def[abs_def])
```
```   478   fix S::"('a \<Rightarrow> 'b set)" and I
```
```   479   assume "\<forall>i\<in>I. S i \<in> range enum_basis"
```
```   480   hence "\<forall>i\<in>I. \<exists>n. S i = enum_basis n" by auto
```
```   481   then obtain n where n: "\<forall>i\<in>I. S i = enum_basis (n i)"
```
```   482     unfolding bchoice_iff by blast
```
```   483   assume [simp]: "finite I"
```
```   484   have "\<exists>fm. domain fm = I \<and> (\<forall>i\<in>I. n i = (fm i))"
```
```   485     by (rule finmap_choice) auto
```
```   486   then obtain m where "Pi' I S = Pi' (domain m) (enum_basis o m)"
```
```   487     using n by (auto simp: Pi'_def)
```
```   488   hence "Pi' I S = (let m = from_nat (to_nat m) in Pi' (domain m) (enum_basis \<circ> m))"
```
```   489     by simp
```
```   490   thus "Pi' I S \<in> range (\<lambda>n. let m = from_nat n in Pi' (domain m) (enum_basis \<circ> m))"
```
```   491     by blast
```
```   492 qed (metis finite_domain o_apply rangeI)
```
```   493
```
```   494 lemma in_enum_basis_finmapI:
```
```   495   assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> range enum_basis"
```
```   496   shows "Pi' I S \<in> range enum_basis_finmap"
```
```   497   using assms unfolding range_enum_basis_eq by auto
```
```   498
```
```   499 lemma finmap_topological_basis:
```
```   500   "topological_basis (range (enum_basis_finmap))"
```
```   501 proof (subst topological_basis_iff, safe)
```
```   502   fix n::nat
```
```   503   show "open (enum_basis_finmap n::('a \<Rightarrow>\<^isub>F 'b) set)" using enumerable_basis
```
```   504     by (auto intro!: open_Pi'I simp: topological_basis_def enum_basis_finmap_def Let_def)
```
```   505 next
```
```   506   fix O'::"('a \<Rightarrow>\<^isub>F 'b) set" and x
```
```   507   assume "open O'" "x \<in> O'"
```
```   508   then obtain e where e: "e > 0" "\<And>y. dist y x < e \<Longrightarrow> y \<in> O'"  unfolding open_dist by blast
```
```   509   def e' \<equiv> "e / (card (domain x) + 1)"
```
```   510
```
```   511   have "\<exists>B.
```
```   512     (\<forall>i\<in>domain x. x i \<in> enum_basis (B i) \<and> enum_basis (B i) \<subseteq> ball (x i) e')"
```
```   513   proof (rule bchoice, safe)
```
```   514     fix i assume "i \<in> domain x"
```
```   515     have "open (ball (x i) e')" "x i \<in> ball (x i) e'" using e
```
```   516       by (auto simp add: e'_def intro!: divide_pos_pos)
```
```   517     from enumerable_basisE[OF this] guess b' .
```
```   518     thus "\<exists>y. x i \<in> enum_basis y \<and>
```
```   519             enum_basis y \<subseteq> ball (x i) e'" by auto
```
```   520   qed
```
```   521   then guess B .. note B = this
```
```   522   def B' \<equiv> "Pi' (domain x) (\<lambda>i. enum_basis (B i)::'b set)"
```
```   523   hence "B' \<in> range enum_basis_finmap" unfolding B'_def
```
```   524     by (intro in_enum_basis_finmapI) auto
```
```   525   moreover have "x \<in> B'" unfolding B'_def using B by auto
```
```   526   moreover have "B' \<subseteq> O'"
```
```   527   proof
```
```   528     fix y assume "y \<in> B'" with B have "domain y = domain x" unfolding B'_def
```
```   529       by (simp add: Pi'_def)
```
```   530     show "y \<in> O'"
```
```   531     proof (rule e)
```
```   532       have "dist y x = (\<Sum>i \<in> domain x. dist (y i) (x i))"
```
```   533         using `domain y = domain x` by (simp add: dist_finmap_def)
```
```   534       also have "\<dots> \<le> (\<Sum>i \<in> domain x. e')"
```
```   535       proof (rule setsum_mono)
```
```   536         fix i assume "i \<in> domain x"
```
```   537         with `y \<in> B'` B have "y i \<in> enum_basis (B i)"
```
```   538           by (simp add: Pi'_def B'_def)
```
```   539         hence "y i \<in> ball (x i) e'" using B `domain y = domain x` `i \<in> domain x`
```
```   540           by force
```
```   541         thus "dist (y i) (x i) \<le> e'" by (simp add: dist_commute)
```
```   542       qed
```
```   543       also have "\<dots> = card (domain x) * e'" by (simp add: real_eq_of_nat)
```
```   544       also have "\<dots> < e" using e by (simp add: e'_def field_simps)
```
```   545       finally show "dist y x < e" .
```
```   546     qed
```
```   547   qed
```
```   548   ultimately
```
```   549   show "\<exists>B'\<in>range enum_basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" by blast
```
```   550 qed
```
```   551
```
```   552 lemma range_enum_basis_finmap_imp_open:
```
```   553   assumes "x \<in> range enum_basis_finmap"
```
```   554   shows "open x"
```
```   555   using finmap_topological_basis assms by (auto simp: topological_basis_def)
```
```   556
```
```   557 lemma
```
```   558   open_imp_ex_UNION_of_enum:
```
```   559   fixes X::"('a \<Rightarrow>\<^isub>F 'b) set"
```
```   560   assumes "open X" assumes "X \<noteq> {}"
```
```   561   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>
```
```   562     (\<forall>n. \<forall>i\<in>A n. (B n) i \<in> range enum_basis) \<and> (\<forall>n. finite (A n))"
```
```   563 proof -
```
```   564   from `open X` obtain B' where B': "B'\<subseteq>range enum_basis_finmap" "\<Union>B' = X"
```
```   565     using finmap_topological_basis by (force simp add: topological_basis_def)
```
```   566   then obtain B where B: "B' = enum_basis_finmap ` B" by (auto simp: subset_image_iff)
```
```   567   show ?thesis
```
```   568   proof cases
```
```   569     assume "B = {}" with B have "B' = {}" by simp hence False using B' assms by simp
```
```   570     thus ?thesis by simp
```
```   571   next
```
```   572     assume "B \<noteq> {}" then obtain b where b: "b \<in> B" by auto
```
```   573     def NA \<equiv> "\<lambda>n::nat. if n \<in> B
```
```   574       then domain ((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) n)
```
```   575       else domain ((from_nat::_\<Rightarrow>'a\<Rightarrow>\<^isub>F nat) b)"
```
```   576     def NB \<equiv> "\<lambda>n::nat. if n \<in> B
```
```   577       then (\<lambda>i. (enum_basis::nat\<Rightarrow>'b set) (((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) n) i))
```
```   578       else (\<lambda>i. (enum_basis::nat\<Rightarrow>'b set) (((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) b) i))"
```
```   579     have "X = UNION UNIV (\<lambda>i. Pi' (NA i) (NB i))" unfolding B'(2)[symmetric] using b
```
```   580       unfolding B
```
```   581       by safe
```
```   582          (auto simp add: NA_def NB_def enum_basis_finmap_def Let_def o_def split: split_if_asm)
```
```   583     moreover
```
```   584     have "(\<forall>n. \<forall>i\<in>NA n. (NB n) i \<in> range enum_basis)"
```
```   585       using enumerable_basis by (auto simp: topological_basis_def NA_def NB_def)
```
```   586     moreover have "(\<forall>n. finite (NA n))" by (simp add: NA_def)
```
```   587     ultimately show ?thesis by auto
```
```   588   qed
```
```   589 qed
```
```   590
```
```   591 lemma
```
```   592   open_imp_ex_UNION:
```
```   593   fixes X::"('a \<Rightarrow>\<^isub>F 'b) set"
```
```   594   assumes "open X" assumes "X \<noteq> {}"
```
```   595   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>
```
```   596     (\<forall>n. \<forall>i\<in>A n. open ((B n) i)) \<and> (\<forall>n. finite (A n))"
```
```   597   using open_imp_ex_UNION_of_enum[OF assms]
```
```   598   apply auto
```
```   599   apply (rule_tac x = A in exI)
```
```   600   apply (rule_tac x = B in exI)
```
```   601   apply (auto simp: open_enum_basis)
```
```   602   done
```
```   603
```
```   604 lemma
```
```   605   open_basisE:
```
```   606   assumes "open X" assumes "X \<noteq> {}"
```
```   607   obtains A::"nat\<Rightarrow>'a set" and B::"nat\<Rightarrow>('a \<Rightarrow> 'b set)" where
```
```   608   "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)"
```
```   609 using open_imp_ex_UNION[OF assms] by auto
```
```   610
```
```   611 lemma
```
```   612   open_basis_of_enumE:
```
```   613   assumes "open X" assumes "X \<noteq> {}"
```
```   614   obtains A::"nat\<Rightarrow>'a set" and B::"nat\<Rightarrow>('a \<Rightarrow> 'b set)" where
```
```   615   "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"
```
```   616   "\<And>n. finite (A n)"
```
```   617 using open_imp_ex_UNION_of_enum[OF assms] by auto
```
```   618
```
```   619 instance proof qed (blast intro: finmap_topological_basis)
```
```   620
```
```   621 end
```
```   622
```
```   623 subsection {* Product Measurable Space of Finite Maps *}
```
```   624
```
```   625 definition "PiF I M \<equiv>
```
```   626   sigma
```
```   627     (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))
```
```   628     {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   629
```
```   630 abbreviation
```
```   631   "Pi\<^isub>F I M \<equiv> PiF I M"
```
```   632
```
```   633 syntax
```
```   634   "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIF _:_./ _)" 10)
```
```   635
```
```   636 syntax (xsymbols)
```
```   637   "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>F _\<in>_./ _)"  10)
```
```   638
```
```   639 syntax (HTML output)
```
```   640   "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>F _\<in>_./ _)"  10)
```
```   641
```
```   642 translations
```
```   643   "PIF x:I. M" == "CONST PiF I (%x. M)"
```
```   644
```
```   645 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>
```
```   646     Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
```
```   647   by (auto simp: Pi'_def) (blast dest: sets_into_space)
```
```   648
```
```   649 lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
```
```   650   unfolding PiF_def using PiF_gen_subset by (rule space_measure_of)
```
```   651
```
```   652 lemma sets_PiF:
```
```   653   "sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))
```
```   654     {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   655   unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of)
```
```   656
```
```   657 lemma sets_PiF_singleton:
```
```   658   "sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j))
```
```   659     {(\<Pi>' j\<in>I. X j) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
```
```   660   unfolding sets_PiF by simp
```
```   661
```
```   662 lemma in_sets_PiFI:
```
```   663   assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
```
```   664   shows "X \<in> sets (PiF I M)"
```
```   665   unfolding sets_PiF
```
```   666   using assms by blast
```
```   667
```
```   668 lemma product_in_sets_PiFI:
```
```   669   assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
```
```   670   shows "(Pi' J S) \<in> sets (PiF I M)"
```
```   671   unfolding sets_PiF
```
```   672   using assms by blast
```
```   673
```
```   674 lemma singleton_space_subset_in_sets:
```
```   675   fixes J
```
```   676   assumes "J \<in> I"
```
```   677   assumes "finite J"
```
```   678   shows "space (PiF {J} M) \<in> sets (PiF I M)"
```
```   679   using assms
```
```   680   by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"])
```
```   681       (auto simp: product_def space_PiF)
```
```   682
```
```   683 lemma singleton_subspace_set_in_sets:
```
```   684   assumes A: "A \<in> sets (PiF {J} M)"
```
```   685   assumes "finite J"
```
```   686   assumes "J \<in> I"
```
```   687   shows "A \<in> sets (PiF I M)"
```
```   688   using A[unfolded sets_PiF]
```
```   689   apply (induct A)
```
```   690   unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
```
```   691   using assms
```
```   692   by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets)
```
```   693
```
```   694 lemma
```
```   695   finite_measurable_singletonI:
```
```   696   assumes "finite I"
```
```   697   assumes "\<And>J. J \<in> I \<Longrightarrow> finite J"
```
```   698   assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
```
```   699   shows "A \<in> measurable (PiF I M) N"
```
```   700   unfolding measurable_def
```
```   701 proof safe
```
```   702   fix y assume "y \<in> sets N"
```
```   703   have "A -` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A -` y \<inter> space (PiF {J} M))"
```
```   704     by (auto simp: space_PiF)
```
```   705   also have "\<dots> \<in> sets (PiF I M)"
```
```   706   proof
```
```   707     show "finite I" by fact
```
```   708     fix J assume "J \<in> I"
```
```   709     with assms have "finite J" by simp
```
```   710     show "A -` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)"
```
```   711       by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+
```
```   712   qed
```
```   713   finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
```
```   714 next
```
```   715   fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
```
```   716     using MN[of "domain x"]
```
```   717     by (auto simp: space_PiF measurable_space Pi'_def)
```
```   718 qed
```
```   719
```
```   720 lemma
```
```   721   countable_finite_comprehension:
```
```   722   fixes f :: "'a::countable set \<Rightarrow> _"
```
```   723   assumes "\<And>s. P s \<Longrightarrow> finite s"
```
```   724   assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M"
```
```   725   shows "\<Union>{f s|s. P s} \<in> sets M"
```
```   726 proof -
```
```   727   have "\<Union>{f s|s. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})"
```
```   728   proof safe
```
```   729     fix x X s assume "x \<in> f s" "P s"
```
```   730     moreover with assms obtain l where "s = set l" using finite_list by blast
```
```   731     ultimately show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s`
```
```   732       by (auto intro!: exI[where x="to_nat l"])
```
```   733   next
```
```   734     fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})"
```
```   735     thus "x \<in> \<Union>{f s|s. P s}" using assms by (auto simp: Let_def split: split_if_asm)
```
```   736   qed
```
```   737   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
```
```   738   also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def)
```
```   739   finally show ?thesis .
```
```   740 qed
```
```   741
```
```   742 lemma space_subset_in_sets:
```
```   743   fixes J::"'a::countable set set"
```
```   744   assumes "J \<subseteq> I"
```
```   745   assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
```
```   746   shows "space (PiF J M) \<in> sets (PiF I M)"
```
```   747 proof -
```
```   748   have "space (PiF J M) = \<Union>{space (PiF {j} M)|j. j \<in> J}"
```
```   749     unfolding space_PiF by blast
```
```   750   also have "\<dots> \<in> sets (PiF I M)" using assms
```
```   751     by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets)
```
```   752   finally show ?thesis .
```
```   753 qed
```
```   754
```
```   755 lemma subspace_set_in_sets:
```
```   756   fixes J::"'a::countable set set"
```
```   757   assumes A: "A \<in> sets (PiF J M)"
```
```   758   assumes "J \<subseteq> I"
```
```   759   assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
```
```   760   shows "A \<in> sets (PiF I M)"
```
```   761   using A[unfolded sets_PiF]
```
```   762   apply (induct A)
```
```   763   unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
```
```   764   using assms
```
```   765   by (auto intro: in_sets_PiFI intro!: space_subset_in_sets)
```
```   766
```
```   767 lemma
```
```   768   countable_measurable_PiFI:
```
```   769   fixes I::"'a::countable set set"
```
```   770   assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
```
```   771   shows "A \<in> measurable (PiF I M) N"
```
```   772   unfolding measurable_def
```
```   773 proof safe
```
```   774   fix y assume "y \<in> sets N"
```
```   775   have "A -` y = (\<Union>{A -` y \<inter> {x. domain x = J}|J. finite J})" by auto
```
```   776   hence "A -` y \<inter> space (PiF I M) = (\<Union>n. A -` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))"
```
```   777     apply (auto simp: space_PiF Pi'_def)
```
```   778   proof -
```
```   779     case goal1
```
```   780     from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto
```
```   781     thus ?case
```
```   782       apply (intro exI[where x="to_nat xs"])
```
```   783       apply auto
```
```   784       done
```
```   785   qed
```
```   786   also have "\<dots> \<in> sets (PiF I M)"
```
```   787     apply (intro Int countable_nat_UN subsetI, safe)
```
```   788     apply (case_tac "set (from_nat i) \<in> I")
```
```   789     apply simp_all
```
```   790     apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]])
```
```   791     using assms `y \<in> sets N`
```
```   792     apply (auto simp: space_PiF)
```
```   793     done
```
```   794   finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
```
```   795 next
```
```   796   fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
```
```   797     using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def)
```
```   798 qed
```
```   799
```
```   800 lemma measurable_PiF:
```
```   801   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))"
```
```   802   assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow>
```
```   803     f -` (Pi' J S) \<inter> space N \<in> sets N"
```
```   804   shows "f \<in> measurable N (PiF I M)"
```
```   805   unfolding PiF_def
```
```   806   using PiF_gen_subset
```
```   807   apply (rule measurable_measure_of)
```
```   808   using f apply force
```
```   809   apply (insert S, auto)
```
```   810   done
```
```   811
```
```   812 lemma
```
```   813   restrict_sets_measurable:
```
```   814   assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I"
```
```   815   shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
```
```   816   using A[unfolded sets_PiF]
```
```   817   apply (induct A)
```
```   818   unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
```
```   819 proof -
```
```   820   fix a assume "a \<in> {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   821   then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))"
```
```   822     by auto
```
```   823   show "a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
```
```   824   proof cases
```
```   825     assume "K \<in> J"
```
```   826     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
```
```   827       by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def)
```
```   828     also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto
```
```   829     finally show ?thesis .
```
```   830   next
```
```   831     assume "K \<notin> J"
```
```   832     hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def)
```
```   833     also have "\<dots> \<in> sets (PiF J M)" by simp
```
```   834     finally show ?thesis .
```
```   835   qed
```
```   836 next
```
```   837   show "{} \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" by simp
```
```   838 next
```
```   839   fix a :: "nat \<Rightarrow> _"
```
```   840   assume a: "(\<And>i. a i \<inter> {m. domain m \<in> J} \<in> sets (PiF J M))"
```
```   841   have "UNION UNIV a \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))"
```
```   842     by simp
```
```   843   also have "\<dots> \<in> sets (PiF J M)" using a by (intro countable_nat_UN) auto
```
```   844   finally show "UNION UNIV a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" .
```
```   845 next
```
```   846   fix a assume a: "a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
```
```   847   have "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} = (space (PiF J M) - (a \<inter> {m. domain m \<in> J}))"
```
```   848     using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def)
```
```   849   also have "\<dots> \<in> sets (PiF J M)" using a by auto
```
```   850   finally show "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" .
```
```   851 qed
```
```   852
```
```   853 lemma measurable_finmap_of:
```
```   854   assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
```
```   855   assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)"
```
```   856   assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N"
```
```   857   shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)"
```
```   858 proof (rule measurable_PiF)
```
```   859   fix x assume "x \<in> space N"
```
```   860   with J[of x] measurable_space[OF f]
```
```   861   show "domain (finmap_of (J x) (f x)) \<in> I \<and>
```
```   862         (\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))"
```
```   863     by auto
```
```   864 next
```
```   865   fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)"
```
```   866   with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N =
```
```   867     (if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K}
```
```   868       else (\<Inter>i\<in>K. (\<lambda>x. f x i) -` S i \<inter> {x \<in> space N. J x = K}) else {})"
```
```   869     by (auto simp: Pi'_def)
```
```   870   have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto
```
```   871   show "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N \<in> sets N"
```
```   872     unfolding eq r
```
```   873     apply (simp del: INT_simps add: )
```
```   874     apply (intro conjI impI finite_INT JN Int[OF top])
```
```   875     apply simp apply assumption
```
```   876     apply (subst Int_assoc[symmetric])
```
```   877     apply (rule Int)
```
```   878     apply (intro measurable_sets[OF f] *) apply force apply assumption
```
```   879     apply (intro JN)
```
```   880     done
```
```   881 qed
```
```   882
```
```   883 lemma measurable_PiM_finmap_of:
```
```   884   assumes "finite J"
```
```   885   shows "finmap_of J \<in> measurable (Pi\<^isub>M J M) (PiF {J} M)"
```
```   886   apply (rule measurable_finmap_of)
```
```   887   apply (rule measurable_component_singleton)
```
```   888   apply simp
```
```   889   apply rule
```
```   890   apply (rule `finite J`)
```
```   891   apply simp
```
```   892   done
```
```   893
```
```   894 lemma proj_measurable_singleton:
```
```   895   assumes "A \<in> sets (M i)" "finite I"
```
```   896   shows "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)"
```
```   897 proof cases
```
```   898   assume "i \<in> I"
```
```   899   hence "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) =
```
```   900     Pi' I (\<lambda>x. if x = i then A else space (M x))"
```
```   901     using sets_into_space[OF ] `A \<in> sets (M i)` assms
```
```   902     by (auto simp: space_PiF Pi'_def)
```
```   903   thus ?thesis  using assms `A \<in> sets (M i)`
```
```   904     by (intro in_sets_PiFI) auto
```
```   905 next
```
```   906   assume "i \<notin> I"
```
```   907   hence "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) =
```
```   908     (if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def)
```
```   909   thus ?thesis by simp
```
```   910 qed
```
```   911
```
```   912 lemma measurable_proj_singleton:
```
```   913   fixes I
```
```   914   assumes "finite I" "i \<in> I"
```
```   915   shows "(\<lambda>x. (x)\<^isub>F i) \<in> measurable (PiF {I} M) (M i)"
```
```   916 proof (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms)
```
```   917 qed (insert `i \<in> I`, auto simp: space_PiF)
```
```   918
```
```   919 lemma measurable_proj_countable:
```
```   920   fixes I::"'a::countable set set"
```
```   921   assumes "y \<in> space (M i)"
```
```   922   shows "(\<lambda>x. if i \<in> domain x then (x)\<^isub>F i else y) \<in> measurable (PiF I M) (M i)"
```
```   923 proof (rule countable_measurable_PiFI)
```
```   924   fix J assume "J \<in> I" "finite J"
```
```   925   show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)"
```
```   926     unfolding measurable_def
```
```   927   proof safe
```
```   928     fix z assume "z \<in> sets (M i)"
```
```   929     have "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) =
```
```   930       (\<lambda>x. if i \<in> J then (x)\<^isub>F i else y) -` z \<inter> space (PiF {J} M)"
```
```   931       by (auto simp: space_PiF Pi'_def)
```
```   932     also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J`
```
```   933       by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton])
```
```   934     finally show "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) \<in>
```
```   935       sets (PiF {J} M)" .
```
```   936   qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def)
```
```   937 qed
```
```   938
```
```   939 lemma measurable_restrict_proj:
```
```   940   assumes "J \<in> II" "finite J"
```
```   941   shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)"
```
```   942   using assms
```
```   943   by (intro measurable_finmap_of measurable_component_singleton) auto
```
```   944
```
```   945 lemma
```
```   946   measurable_proj_PiM:
```
```   947   fixes J K ::"'a::countable set" and I::"'a set set"
```
```   948   assumes "finite J" "J \<in> I"
```
```   949   assumes "x \<in> space (PiM J M)"
```
```   950   shows "proj \<in>
```
```   951     measurable (PiF {J} M) (PiM J M)"
```
```   952 proof (rule measurable_PiM_single)
```
```   953   show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^isub>E i \<in> J. space (M i))"
```
```   954     using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def)
```
```   955 next
```
```   956   fix A i assume A: "i \<in> J" "A \<in> sets (M i)"
```
```   957   show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} \<in> sets (PiF {J} M)"
```
```   958   proof
```
```   959     have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} =
```
```   960       (\<lambda>\<omega>. (\<omega>)\<^isub>F i) -` A \<inter> space (PiF {J} M)" by auto
```
```   961     also have "\<dots> \<in> sets (PiF {J} M)"
```
```   962       using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM)
```
```   963     finally show ?thesis .
```
```   964   qed simp
```
```   965 qed
```
```   966
```
```   967 lemma sets_subspaceI:
```
```   968   assumes "A \<inter> space M \<in> sets M"
```
```   969   assumes "B \<in> sets M"
```
```   970   shows "A \<inter> B \<in> sets M" using assms
```
```   971 proof -
```
```   972   have "A \<inter> B = (A \<inter> space M) \<inter> B"
```
```   973     using assms sets_into_space by auto
```
```   974   thus ?thesis using assms by auto
```
```   975 qed
```
```   976
```
```   977 lemma space_PiF_singleton_eq_product:
```
```   978   assumes "finite I"
```
```   979   shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))"
```
```   980   by (auto simp: product_def space_PiF assms)
```
```   981
```
```   982 text {* adapted from @{thm sets_PiM_single} *}
```
```   983
```
```   984 lemma sets_PiF_single:
```
```   985   assumes "finite I" "I \<noteq> {}"
```
```   986   shows "sets (PiF {I} M) =
```
```   987     sigma_sets (\<Pi>' i\<in>I. space (M i))
```
```   988       {{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   989     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   990   unfolding sets_PiF_singleton
```
```   991 proof (rule sigma_sets_eqI)
```
```   992   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   993   fix A assume "A \<in> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
```
```   994   then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
```
```   995   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   996   proof -
```
```   997     from `I \<noteq> {}` X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})"
```
```   998       using sets_into_space
```
```   999       by (auto simp: space_PiF product_def) blast
```
```  1000     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```  1001       using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF)
```
```  1002     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```  1003   qed
```
```  1004 next
```
```  1005   fix A assume "A \<in> ?R"
```
```  1006   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)"
```
```  1007     by auto
```
```  1008   then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))"
```
```  1009     using sets_into_space[OF A(3)]
```
```  1010     apply (auto simp: Pi'_iff split: split_if_asm)
```
```  1011     apply blast
```
```  1012     done
```
```  1013   also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
```
```  1014     using A
```
```  1015     by (intro sigma_sets.Basic )
```
```  1016        (auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"])
```
```  1017   finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" .
```
```  1018 qed
```
```  1019
```
```  1020 text {* adapted from @{thm PiE_cong} *}
```
```  1021
```
```  1022 lemma Pi'_cong:
```
```  1023   assumes "finite I"
```
```  1024   assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i"
```
```  1025   shows "Pi' I f = Pi' I g"
```
```  1026 using assms by (auto simp: Pi'_def)
```
```  1027
```
```  1028 text {* adapted from @{thm Pi_UN} *}
```
```  1029
```
```  1030 lemma Pi'_UN:
```
```  1031   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
```
```  1032   assumes "finite I"
```
```  1033   assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
```
```  1034   shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)"
```
```  1035 proof (intro set_eqI iffI)
```
```  1036   fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)"
```
```  1037   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def)
```
```  1038   from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
```
```  1039   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
```
```  1040     using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
```
```  1041   have "f \<in> Pi' I (\<lambda>i. A k i)"
```
```  1042   proof
```
```  1043     fix i assume "i \<in> I"
```
```  1044     from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I`
```
```  1045     show "f i \<in> A k i " by (auto simp: `finite I`)
```
```  1046   qed (simp add: `domain f = I` `finite I`)
```
```  1047   then show "f \<in> (\<Union>n. Pi' I (A n))" by auto
```
```  1048 qed (auto simp: Pi'_def `finite I`)
```
```  1049
```
```  1050 text {* adapted from @{thm sigma_prod_algebra_sigma_eq} *}
```
```  1051
```
```  1052 lemma sigma_fprod_algebra_sigma_eq:
```
```  1053   fixes E :: "'i \<Rightarrow> 'a set set"
```
```  1054   assumes [simp]: "finite I" "I \<noteq> {}"
```
```  1055   assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
```
```  1056     and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```  1057     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```  1058   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```  1059     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```  1060   defines "P == { Pi' I F | F. \<forall>i\<in>I. F i \<in> E i }"
```
```  1061   shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
```
```  1062 proof
```
```  1063   let ?P = "sigma (space (Pi\<^isub>F {I} M)) P"
```
```  1064   have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>F {I} M))"
```
```  1065     using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
```
```  1066   then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))"
```
```  1067     by (simp add: space_PiF)
```
```  1068   have "sets (PiF {I} M) =
```
```  1069       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)}"
```
```  1070     using sets_PiF_single[of I M] by (simp add: space_P)
```
```  1071   also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)"
```
```  1072   proof (safe intro!: sigma_sets_subset)
```
```  1073     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```  1074     have "(\<lambda>x. (x)\<^isub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```  1075     proof (subst measurable_iff_measure_of)
```
```  1076       show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
```
```  1077       from space_P `i \<in> I` show "(\<lambda>x. (x)\<^isub>F i) \<in> space ?P \<rightarrow> space (M i)"
```
```  1078         by auto
```
```  1079       show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1080       proof
```
```  1081         fix A assume A: "A \<in> E i"
```
```  1082         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))"
```
```  1083           using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
```
```  1084         also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)"
```
```  1085           by (intro Pi'_cong) (simp_all add: S_union)
```
```  1086         also have "\<dots> = (\<Union>n. \<Pi>' j\<in>I. if i = j then A else S j n)"
```
```  1087           using S_mono
```
```  1088           by (subst Pi'_UN[symmetric, OF `finite I`]) (auto simp: incseq_def)
```
```  1089         also have "\<dots> \<in> sets ?P"
```
```  1090         proof (safe intro!: countable_UN)
```
```  1091           fix n show "(\<Pi>' j\<in>I. if i = j then A else S j n) \<in> sets ?P"
```
```  1092             using A S_in_E
```
```  1093             by (simp add: P_closed)
```
```  1094                (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j n"])
```
```  1095         qed
```
```  1096         finally show "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1097           using P_closed by simp
```
```  1098       qed
```
```  1099     qed
```
```  1100     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```  1101     have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1102       by (simp add: E_generates)
```
```  1103     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}"
```
```  1104       using P_closed by (auto simp: space_PiF)
```
```  1105     finally show "\<dots> \<in> sets ?P" .
```
```  1106   qed
```
```  1107   finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P"
```
```  1108     by (simp add: P_closed)
```
```  1109   show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)"
```
```  1110     using `finite I` `I \<noteq> {}`
```
```  1111     by (auto intro!: sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def)
```
```  1112 qed
```
```  1113
```
```  1114 lemma enumerable_sigma_fprod_algebra_sigma_eq:
```
```  1115   assumes "I \<noteq> {}"
```
```  1116   assumes [simp]: "finite I"
```
```  1117   shows "sets (PiF {I} (\<lambda>_. borel)) = sigma_sets (space (PiF {I} (\<lambda>_. borel)))
```
```  1118     {Pi' I F |F. (\<forall>i\<in>I. F i \<in> range enum_basis)}"
```
```  1119 proof -
```
```  1120   from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
```
```  1121   show ?thesis
```
```  1122   proof (rule sigma_fprod_algebra_sigma_eq)
```
```  1123     show "finite I" by simp
```
```  1124     show "I \<noteq> {}" by fact
```
```  1125     show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> range enum_basis"
```
```  1126       using S by simp_all
```
```  1127     show "range enum_basis \<subseteq> Pow (space borel)" by simp
```
```  1128     show "sets borel = sigma_sets (space borel) (range enum_basis)"
```
```  1129       by (simp add: borel_eq_enum_basis)
```
```  1130   qed
```
```  1131 qed
```
```  1132
```
```  1133 text {* adapted from @{thm enumerable_sigma_fprod_algebra_sigma_eq} *}
```
```  1134
```
```  1135 lemma enumerable_sigma_prod_algebra_sigma_eq:
```
```  1136   assumes "I \<noteq> {}"
```
```  1137   assumes [simp]: "finite I"
```
```  1138   shows "sets (PiM I (\<lambda>_. borel)) = sigma_sets (space (PiM I (\<lambda>_. borel)))
```
```  1139     {Pi\<^isub>E I F |F. \<forall>i\<in>I. F i \<in> range enum_basis}"
```
```  1140 proof -
```
```  1141   from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
```
```  1142   show ?thesis
```
```  1143   proof (rule sigma_prod_algebra_sigma_eq)
```
```  1144     show "finite I" by simp note[[show_types]]
```
```  1145     fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> range enum_basis"
```
```  1146       using S by simp_all
```
```  1147     show "range enum_basis \<subseteq> Pow (space borel)" by simp
```
```  1148     show "sets borel = sigma_sets (space borel) (range enum_basis)"
```
```  1149       by (simp add: borel_eq_enum_basis)
```
```  1150   qed
```
```  1151 qed
```
```  1152
```
```  1153 lemma product_open_generates_sets_PiF_single:
```
```  1154   assumes "I \<noteq> {}"
```
```  1155   assumes [simp]: "finite I"
```
```  1156   shows "sets (PiF {I} (\<lambda>_. borel::'b::enumerable_basis measure)) =
```
```  1157     sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F |F. (\<forall>i\<in>I. F i \<in> Collect open)}"
```
```  1158 proof -
```
```  1159   from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
```
```  1160   show ?thesis
```
```  1161   proof (rule sigma_fprod_algebra_sigma_eq)
```
```  1162     show "finite I" by simp
```
```  1163     show "I \<noteq> {}" by fact
```
```  1164     show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open"
```
```  1165       using S by (auto simp: open_enum_basis)
```
```  1166     show "Collect open \<subseteq> Pow (space borel)" by simp
```
```  1167     show "sets borel = sigma_sets (space borel) (Collect open)"
```
```  1168       by (simp add: borel_def)
```
```  1169   qed
```
```  1170 qed
```
```  1171
```
```  1172 lemma product_open_generates_sets_PiM:
```
```  1173   assumes "I \<noteq> {}"
```
```  1174   assumes [simp]: "finite I"
```
```  1175   shows "sets (PiM I (\<lambda>_. borel::'b::enumerable_basis measure)) =
```
```  1176     sigma_sets (space (PiM I (\<lambda>_. borel))) {Pi\<^isub>E I F |F. \<forall>i\<in>I. F i \<in> Collect open}"
```
```  1177 proof -
```
```  1178   from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this
```
```  1179   show ?thesis
```
```  1180   proof (rule sigma_prod_algebra_sigma_eq)
```
```  1181     show "finite I" by simp note[[show_types]]
```
```  1182     fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open"
```
```  1183       using S by (auto simp: open_enum_basis)
```
```  1184     show "Collect open \<subseteq> Pow (space borel)" by simp
```
```  1185     show "sets borel = sigma_sets (space borel) (Collect open)"
```
```  1186       by (simp add: borel_def)
```
```  1187   qed
```
```  1188 qed
```
```  1189
```
```  1190 lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. J \<leadsto> UNIV) = UNIV" by auto
```
```  1191
```
```  1192 lemma borel_eq_PiF_borel:
```
```  1193   shows "(borel :: ('i::countable \<Rightarrow>\<^isub>F 'a::polish_space) measure) =
```
```  1194   PiF (Collect finite) (\<lambda>_. borel :: 'a measure)"
```
```  1195 proof (rule measure_eqI)
```
```  1196   have C: "Collect finite \<noteq> {}" by auto
```
```  1197   show "sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure) = sets (PiF (Collect finite) (\<lambda>_. borel))"
```
```  1198   proof
```
```  1199     show "sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure) \<subseteq> sets (PiF (Collect finite) (\<lambda>_. borel))"
```
```  1200       apply (simp add: borel_def sets_PiF)
```
```  1201     proof (rule sigma_sets_mono, safe, cases)
```
```  1202       fix X::"('i \<Rightarrow>\<^isub>F 'a) set" assume "open X" "X \<noteq> {}"
```
```  1203       from open_basisE[OF this] guess NA NB . note N = this
```
```  1204       hence "X = (\<Union>i. Pi' (NA i) (NB i))" by simp
```
```  1205       also have "\<dots> \<in>
```
```  1206         sigma_sets UNIV {Pi' J S |S J. finite J \<and> S \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
```
```  1207         using N by (intro Union sigma_sets.Basic) blast
```
```  1208       finally show "X \<in> sigma_sets UNIV
```
```  1209         {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" .
```
```  1210     qed (auto simp: Empty)
```
```  1211   next
```
```  1212     show "sets (PiF (Collect finite) (\<lambda>_. borel)) \<subseteq> sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure)"
```
```  1213     proof
```
```  1214       fix x assume x: "x \<in> sets (PiF (Collect finite::'i set set) (\<lambda>_. borel::'a measure))"
```
```  1215       hence x_sp: "x \<subseteq> space (PiF (Collect finite) (\<lambda>_. borel))" by (rule sets_into_space)
```
```  1216       let ?x = "\<lambda>J. x \<inter> {x. domain x = J}"
```
```  1217       have "x = \<Union>{?x J |J. finite J}" by auto
```
```  1218       also have "\<dots> \<in> sets borel"
```
```  1219       proof (rule countable_finite_comprehension, assumption)
```
```  1220         fix J::"'i set" assume "finite J"
```
```  1221         { assume ef: "J = {}"
```
```  1222           { assume e: "?x J = {}"
```
```  1223             hence "?x J \<in> sets borel" by simp
```
```  1224           } moreover {
```
```  1225             assume "?x J \<noteq> {}"
```
```  1226             then obtain f where "f \<in> x" "domain f = {}" using ef by auto
```
```  1227             hence "?x J = {f}" using `J = {}`
```
```  1228               by (auto simp: finmap_eq_iff)
```
```  1229             also have "{f} \<in> sets borel" by simp
```
```  1230             finally have "?x J \<in> sets borel" .
```
```  1231           } ultimately have "?x J \<in> sets borel" by blast
```
```  1232         } moreover {
```
```  1233           assume "J \<noteq> ({}::'i set)"
```
```  1234           from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'a set" . note S = this
```
```  1235           have "(?x J) = x \<inter> {m. domain m \<in> {J}}" by auto
```
```  1236           also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))"
```
```  1237             using x by (rule restrict_sets_measurable) (auto simp: `finite J`)
```
```  1238           also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel)))
```
```  1239             {Pi' (J) F |F. (\<forall>j\<in>J. F j \<in> range enum_basis)}"
```
```  1240             (is "_ = sigma_sets _ ?P")
```
```  1241             by (rule enumerable_sigma_fprod_algebra_sigma_eq[OF `J \<noteq> {}` `finite J`])
```
```  1242           also have "\<dots> \<subseteq> sets borel"
```
```  1243           proof
```
```  1244             fix x
```
```  1245             assume "x \<in> sigma_sets (space (PiF {J} (\<lambda>_. borel))) ?P"
```
```  1246             thus "x \<in> sets borel"
```
```  1247             proof (rule sigma_sets.induct, safe)
```
```  1248               fix F::"'i \<Rightarrow> 'a set"
```
```  1249               assume "\<forall>j\<in>J. F j \<in> range enum_basis"
```
```  1250               hence "Pi' J F \<in> range enum_basis_finmap"
```
```  1251                 unfolding range_enum_basis_eq
```
```  1252                 by (auto simp: `finite J` intro!: exI[where x=J] exI[where x=F])
```
```  1253               hence "open (Pi' (J) F)" by (rule range_enum_basis_finmap_imp_open)
```
```  1254               thus "Pi' (J) F \<in> sets borel" by simp
```
```  1255             next
```
```  1256               fix a::"('i \<Rightarrow>\<^isub>F 'a) set"
```
```  1257               have "space (PiF {J::'i set} (\<lambda>_. borel::'a measure)) =
```
```  1258                 Pi' (J) (\<lambda>_. UNIV)"
```
```  1259                 by (auto simp: space_PiF product_def)
```
```  1260               moreover have "open (Pi' (J::'i set) (\<lambda>_. UNIV::'a set))"
```
```  1261                 by (intro open_Pi'I) auto
```
```  1262               ultimately
```
```  1263               have "space (PiF {J::'i set} (\<lambda>_. borel::'a measure)) \<in> sets borel"
```
```  1264                 by simp
```
```  1265               moreover
```
```  1266               assume "a \<in> sets borel"
```
```  1267               ultimately show "space (PiF {J} (\<lambda>_. borel)) - a \<in> sets borel" ..
```
```  1268             qed auto
```
```  1269           qed
```
```  1270           finally have "(?x J) \<in> sets borel" .
```
```  1271         } ultimately show "(?x J) \<in> sets borel" by blast
```
```  1272       qed
```
```  1273       finally show "x \<in> sets (borel)" .
```
```  1274     qed
```
```  1275   qed
```
```  1276 qed (simp add: emeasure_sigma borel_def PiF_def)
```
```  1277
```
```  1278 subsection {* Isomorphism between Functions and Finite Maps *}
```
```  1279
```
```  1280 lemma
```
```  1281   measurable_compose:
```
```  1282   fixes f::"'a \<Rightarrow> 'b"
```
```  1283   assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j"
```
```  1284   assumes "finite J"
```
```  1285   shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
```
```  1286 proof (rule measurable_PiM)
```
```  1287   show "(\<lambda>m. compose J m f)
```
```  1288     \<in> space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) \<rightarrow>
```
```  1289       (J \<rightarrow> space M) \<inter> extensional J"
```
```  1290   proof safe
```
```  1291     fix x and i
```
```  1292     assume x: "x \<in> space (PiM (f ` J) (\<lambda>_. M))" "i \<in> J"
```
```  1293     with inj show  "compose J x f i \<in> space M"
```
```  1294       by (auto simp: space_PiM compose_def)
```
```  1295   next
```
```  1296     fix x assume "x \<in> space (PiM (f ` J) (\<lambda>_. M))"
```
```  1297     show "(compose J x f) \<in> extensional J" by (rule compose_extensional)
```
```  1298   qed
```
```  1299 next
```
```  1300   fix S X
```
```  1301   have inv: "\<And>j. j \<in> f ` J \<Longrightarrow> f (f' j) = j" using assms by auto
```
```  1302   assume S: "S \<noteq> {} \<or> J = {}" "finite S" "S \<subseteq> J" and P: "\<And>i. i \<in> S \<Longrightarrow> X i \<in> sets M"
```
```  1303   have "(\<lambda>m. compose J m f) -` prod_emb J (\<lambda>_. M) S (Pi\<^isub>E S X) \<inter>
```
```  1304     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)))"
```
```  1305     using assms inv S sets_into_space[OF P]
```
```  1306     by (force simp: prod_emb_iff compose_def space_PiM extensional_def Pi_def intro: imageI)
```
```  1307   also have "\<dots> \<in> sets (Pi\<^isub>M (f ` J) (\<lambda>_. M))"
```
```  1308   proof
```
```  1309     from S show "f ` S \<subseteq> f `  J" by auto
```
```  1310     show "(\<Pi>\<^isub>E b\<in>f ` S. X (f' b)) \<in> sets (Pi\<^isub>M (f ` S) (\<lambda>_. M))"
```
```  1311     proof (rule sets_PiM_I_finite)
```
```  1312       show "finite (f ` S)" using S by simp
```
```  1313       fix i assume "i \<in> f ` S" hence "f' i \<in> S" using S assms by auto
```
```  1314       thus "X (f' i) \<in> sets M" by (rule P)
```
```  1315     qed
```
```  1316   qed
```
```  1317   finally show "(\<lambda>m. compose J m f) -` prod_emb J (\<lambda>_. M) S (Pi\<^isub>E S X) \<inter>
```
```  1318     space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) \<in> sets (Pi\<^isub>M (f ` J) (\<lambda>_. M))" .
```
```  1319 qed
```
```  1320
```
```  1321 lemma
```
```  1322   measurable_compose_inv:
```
```  1323   fixes f::"'a \<Rightarrow> 'b"
```
```  1324   assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j"
```
```  1325   assumes "finite J"
```
```  1326   shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))"
```
```  1327 proof -
```
```  1328   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))"
```
```  1329     using assms by (auto intro: measurable_compose)
```
```  1330   moreover
```
```  1331   from inj have "f' ` f ` J = J" by (metis (hide_lams, mono_tags) image_iff set_eqI)
```
```  1332   ultimately show ?thesis by simp
```
```  1333 qed
```
```  1334
```
```  1335 locale function_to_finmap =
```
```  1336   fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f'
```
```  1337   assumes [simp]: "finite J"
```
```  1338   assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i"
```
```  1339 begin
```
```  1340
```
```  1341 text {* to measure finmaps *}
```
```  1342
```
```  1343 definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')"
```
```  1344
```
```  1345 lemma domain_fm[simp]: "domain (fm x) = f ` J"
```
```  1346   unfolding fm_def by simp
```
```  1347
```
```  1348 lemma fm_restrict[simp]: "fm (restrict y J) = fm y"
```
```  1349   unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext)
```
```  1350
```
```  1351 lemma fm_product:
```
```  1352   assumes "\<And>i. space (M i) = UNIV"
```
```  1353   shows "fm -` Pi' (f ` J) S \<inter> space (Pi\<^isub>M J M) = (\<Pi>\<^isub>E j \<in> J. S (f j))"
```
```  1354   using assms
```
```  1355   by (auto simp: inv fm_def compose_def space_PiM Pi'_def)
```
```  1356
```
```  1357 lemma fm_measurable:
```
```  1358   assumes "f ` J \<in> N"
```
```  1359   shows "fm \<in> measurable (Pi\<^isub>M J (\<lambda>_. M)) (Pi\<^isub>F N (\<lambda>_. M))"
```
```  1360   unfolding fm_def
```
```  1361 proof (rule measurable_comp, rule measurable_compose_inv)
```
```  1362   show "finmap_of (f ` J) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) "
```
```  1363     using assms by (intro measurable_finmap_of measurable_component_singleton) auto
```
```  1364 qed (simp_all add: inv)
```
```  1365
```
```  1366 lemma proj_fm:
```
```  1367   assumes "x \<in> J"
```
```  1368   shows "fm m (f x) = m x"
```
```  1369   using assms by (auto simp: fm_def compose_def o_def inv)
```
```  1370
```
```  1371 lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)"
```
```  1372 proof (rule inj_on_inverseI)
```
```  1373   fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J"
```
```  1374   thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x"
```
```  1375     by (auto simp: compose_def inv extensional_def)
```
```  1376 qed
```
```  1377
```
```  1378 lemma inj_on_fm:
```
```  1379   assumes "\<And>i. space (M i) = UNIV"
```
```  1380   shows "inj_on fm (space (Pi\<^isub>M J M))"
```
```  1381   using assms
```
```  1382   apply (auto simp: fm_def space_PiM)
```
```  1383   apply (rule comp_inj_on)
```
```  1384   apply (rule inj_on_compose_f')
```
```  1385   apply (rule finmap_of_inj_on_extensional_finite)
```
```  1386   apply simp
```
```  1387   apply (auto)
```
```  1388   done
```
```  1389
```
```  1390 text {* to measure functions *}
```
```  1391
```
```  1392 definition "mf = (\<lambda>g. compose J g f) o proj"
```
```  1393
```
```  1394 lemma
```
```  1395   assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))" "finite J"
```
```  1396   shows "proj (finmap_of J x) = x"
```
```  1397   using assms by (auto simp: space_PiM extensional_def)
```
```  1398
```
```  1399 lemma
```
```  1400   assumes "x \<in> space (Pi\<^isub>F {J} (\<lambda>_. M))"
```
```  1401   shows "finmap_of J (proj x) = x"
```
```  1402   using assms by (auto simp: space_PiF Pi'_def finmap_eq_iff)
```
```  1403
```
```  1404 lemma mf_fm:
```
```  1405   assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))"
```
```  1406   shows "mf (fm x) = x"
```
```  1407 proof -
```
```  1408   have "mf (fm x) \<in> extensional J"
```
```  1409     by (auto simp: mf_def extensional_def compose_def)
```
```  1410   moreover
```
```  1411   have "x \<in> extensional J" using assms sets_into_space
```
```  1412     by (force simp: space_PiM)
```
```  1413   moreover
```
```  1414   { fix i assume "i \<in> J"
```
```  1415     hence "mf (fm x) i = x i"
```
```  1416       by (auto simp: inv mf_def compose_def fm_def)
```
```  1417   }
```
```  1418   ultimately
```
```  1419   show ?thesis by (rule extensionalityI)
```
```  1420 qed
```
```  1421
```
```  1422 lemma mf_measurable:
```
```  1423   assumes "space M = UNIV"
```
```  1424   shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
```
```  1425   unfolding mf_def
```
```  1426 proof (rule measurable_comp, rule measurable_proj_PiM)
```
```  1427   show "(\<lambda>g. compose J g f) \<in>
```
```  1428     measurable (Pi\<^isub>M (f ` J) (\<lambda>x. M)) (Pi\<^isub>M J (\<lambda>_. M))"
```
```  1429     by (rule measurable_compose, rule inv) auto
```
```  1430 qed (auto simp add: space_PiM extensional_def assms)
```
```  1431
```
```  1432 lemma fm_image_measurable:
```
```  1433   assumes "space M = UNIV"
```
```  1434   assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M))"
```
```  1435   shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))"
```
```  1436 proof -
```
```  1437   have "fm ` X = (mf) -` X \<inter> space (PiF {f ` J} (\<lambda>_. M))"
```
```  1438   proof safe
```
```  1439     fix x assume "x \<in> X"
```
```  1440     with mf_fm[of x] sets_into_space[OF assms(2)] show "fm x \<in> mf -` X" by auto
```
```  1441     show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms)
```
```  1442   next
```
```  1443     fix y x
```
```  1444     assume x: "mf y \<in> X"
```
```  1445     assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))"
```
```  1446     thus "y \<in> fm ` X"
```
```  1447       by (intro image_eqI[OF _ x], unfold finmap_eq_iff)
```
```  1448          (auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def)
```
```  1449   qed
```
```  1450   also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))"
```
```  1451     using assms
```
```  1452     by (intro measurable_sets[OF mf_measurable]) auto
```
```  1453   finally show ?thesis .
```
```  1454 qed
```
```  1455
```
```  1456 lemma fm_image_measurable_finite:
```
```  1457   assumes "space M = UNIV"
```
```  1458   assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M::'c measure))"
```
```  1459   shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))"
```
```  1460   using fm_image_measurable[OF assms]
```
```  1461   by (rule subspace_set_in_sets) (auto simp: finite_subset)
```
```  1462
```
```  1463 text {* measure on finmaps *}
```
```  1464
```
```  1465 definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)"
```
```  1466
```
```  1467 lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)"
```
```  1468   unfolding mapmeasure_def by simp
```
```  1469
```
```  1470 lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"
```
```  1471   unfolding mapmeasure_def by simp
```
```  1472
```
```  1473 lemma mapmeasure_PiF:
```
```  1474   assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))"
```
```  1475   assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))"
```
```  1476   assumes "space N = UNIV"
```
```  1477   assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
```
```  1478   shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm -` X \<inter> extensional J))"
```
```  1479   using assms
```
```  1480   by (auto simp: measurable_eqI[OF s1 refl s2 refl] mapmeasure_def emeasure_distr
```
```  1481     fm_measurable space_PiM)
```
```  1482
```
```  1483 lemma mapmeasure_PiM:
```
```  1484   fixes N::"'c measure"
```
```  1485   assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))"
```
```  1486   assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))"
```
```  1487   assumes N: "space N = UNIV"
```
```  1488   assumes X: "X \<in> sets M"
```
```  1489   shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)"
```
```  1490   unfolding mapmeasure_def
```
```  1491 proof (subst emeasure_distr, subst measurable_eqI[OF s1 refl s2 refl], rule fm_measurable)
```
```  1492   have "X \<subseteq> space (Pi\<^isub>M J (\<lambda>_. N))" using assms by (simp add: sets_into_space)
```
```  1493   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"
```
```  1494     by (auto simp: vimage_image_eq inj_on_def)
```
```  1495   thus "emeasure M X = emeasure M (fm -` fm ` X \<inter> space M)" using s1
```
```  1496     by simp
```
```  1497   show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
```
```  1498     by (rule fm_image_measurable_finite[OF N X[simplified s2]])
```
```  1499 qed simp
```
```  1500
```
```  1501 end
```
```  1502
```
```  1503 end
```