src/HOL/Analysis/Finite_Product_Measure.thy
 author wenzelm Mon Mar 25 17:21:26 2019 +0100 (3 months ago) changeset 69981 3dced198b9ec parent 69918 eddcc7c726f3 child 70136 f03a01a18c6e permissions -rw-r--r--
more strict AFP properties;
```     1 (*  Title:      HOL/Analysis/Finite_Product_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Finite Product Measure\<close>
```
```     6
```
```     7 theory Finite_Product_Measure
```
```     8 imports Binary_Product_Measure Function_Topology
```
```     9 begin
```
```    10
```
```    11 lemma PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
```
```    12   by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1])
```
```    13      (force intro: exI[of _ "restrict f I" for f])
```
```    14
```
```    15 lemma case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
```
```    16   by auto
```
```    17
```
```    18 subsection%unimportant \<open>More about Function restricted by \<^const>\<open>extensional\<close>\<close>
```
```    19
```
```    20 definition
```
```    21   "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
```
```    22
```
```    23 lemma merge_apply[simp]:
```
```    24   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
```
```    25   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
```
```    26   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
```
```    27   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
```
```    28   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined"
```
```    29   unfolding merge_def by auto
```
```    30
```
```    31 lemma merge_commute:
```
```    32   "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)"
```
```    33   by (force simp: merge_def)
```
```    34
```
```    35 lemma Pi_cancel_merge_range[simp]:
```
```    36   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
```
```    37   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
```
```    38   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
```
```    39   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
```
```    40   by (auto simp: Pi_def)
```
```    41
```
```    42 lemma Pi_cancel_merge[simp]:
```
```    43   "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```    44   "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```    45   "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
```
```    46   "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
```
```    47   by (auto simp: Pi_def)
```
```    48
```
```    49 lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)"
```
```    50   by (auto simp: extensional_def)
```
```    51
```
```    52 lemma restrict_merge[simp]:
```
```    53   "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
```
```    54   "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
```
```    55   "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
```
```    56   "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
```
```    57   by (auto simp: restrict_def)
```
```    58
```
```    59 lemma split_merge: "P (merge I J (x,y) i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J - I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)"
```
```    60   unfolding merge_def by auto
```
```    61
```
```    62 lemma PiE_cancel_merge[simp]:
```
```    63   "I \<inter> J = {} \<Longrightarrow>
```
```    64     merge I J (x, y) \<in> Pi\<^sub>E (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B"
```
```    65   by (auto simp: PiE_def restrict_Pi_cancel)
```
```    66
```
```    67 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)"
```
```    68   unfolding merge_def by (auto simp: fun_eq_iff)
```
```    69
```
```    70 lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K"
```
```    71   unfolding merge_def extensional_def by auto
```
```    72
```
```    73 lemma merge_restrict[simp]:
```
```    74   "merge I J (restrict x I, y) = merge I J (x, y)"
```
```    75   "merge I J (x, restrict y J) = merge I J (x, y)"
```
```    76   unfolding merge_def by auto
```
```    77
```
```    78 lemma merge_x_x_eq_restrict[simp]:
```
```    79   "merge I J (x, x) = restrict x (I \<union> J)"
```
```    80   unfolding merge_def by auto
```
```    81
```
```    82 lemma injective_vimage_restrict:
```
```    83   assumes J: "J \<subseteq> I"
```
```    84   and sets: "A \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" and ne: "(\<Pi>\<^sub>E i\<in>I. S i) \<noteq> {}"
```
```    85   and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
```
```    86   shows "A = B"
```
```    87 proof  (intro set_eqI)
```
```    88   fix x
```
```    89   from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
```
```    90   have "J \<inter> (I - J) = {}" by auto
```
```    91   show "x \<in> A \<longleftrightarrow> x \<in> B"
```
```    92   proof cases
```
```    93     assume x: "x \<in> (\<Pi>\<^sub>E i\<in>J. S i)"
```
```    94     have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
```
```    95       using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S]
```
```    96       by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
```
```    97     then show "x \<in> A \<longleftrightarrow> x \<in> B"
```
```    98       using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S]
```
```    99       by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq)
```
```   100   qed (insert sets, auto)
```
```   101 qed
```
```   102
```
```   103 lemma restrict_vimage:
```
```   104   "I \<inter> J = {} \<Longrightarrow>
```
```   105     (\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^sub>E I E \<times> Pi\<^sub>E J F) = Pi (I \<union> J) (merge I J (E, F))"
```
```   106   by (auto simp: restrict_Pi_cancel PiE_def)
```
```   107
```
```   108 lemma merge_vimage:
```
```   109   "I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^sub>E (I \<union> J) E = Pi I E \<times> Pi J E"
```
```   110   by (auto simp: restrict_Pi_cancel PiE_def)
```
```   111
```
```   112 subsection \<open>Finite product spaces\<close>
```
```   113
```
```   114 definition%important prod_emb where
```
```   115   "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   116
```
```   117 lemma prod_emb_iff:
```
```   118   "f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))"
```
```   119   unfolding%unimportant prod_emb_def PiE_def by auto
```
```   120
```
```   121 lemma
```
```   122   shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
```
```   123     and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
```
```   124     and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
```
```   125     and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))"
```
```   126     and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
```
```   127     and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
```
```   128   by (auto simp: prod_emb_def)
```
```   129
```
```   130 lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
```
```   131     prod_emb I M J (\<Pi>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
```
```   132   by (force simp: prod_emb_def PiE_iff if_split_mem2)
```
```   133
```
```   134 lemma prod_emb_PiE_same_index[simp]:
```
```   135     "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E"
```
```   136   by (auto simp: prod_emb_def PiE_iff)
```
```   137
```
```   138 lemma prod_emb_trans[simp]:
```
```   139   "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
```
```   140   by (auto simp add: Int_absorb1 prod_emb_def PiE_def)
```
```   141
```
```   142 lemma prod_emb_Pi:
```
```   143   assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
```
```   144   shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
```
```   145   using assms sets.space_closed
```
```   146   by (auto simp: prod_emb_def PiE_iff split: if_split_asm) blast+
```
```   147
```
```   148 lemma prod_emb_id:
```
```   149   "B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
```
```   150   by (auto simp: prod_emb_def subset_eq extensional_restrict)
```
```   151
```
```   152 lemma prod_emb_mono:
```
```   153   "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
```
```   154   by (auto simp: prod_emb_def)
```
```   155
```
```   156 definition%important PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
```
```   157   "PiM I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i))
```
```   158     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
```
```   159     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j))
```
```   160     (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
```
```   161
```
```   162 definition%important prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
```
```   163   "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) `
```
```   164     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   165
```
```   166 abbreviation
```
```   167   "Pi\<^sub>M I M \<equiv> PiM I M"
```
```   168
```
```   169 syntax
```
```   170   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^sub>M _\<in>_./ _)"  10)
```
```   171 translations
```
```   172   "\<Pi>\<^sub>M x\<in>I. M" == "CONST PiM I (%x. M)"
```
```   173
```
```   174 lemma extend_measure_cong:
```
```   175   assumes "\<Omega> = \<Omega>'" "I = I'" "G = G'" "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i"
```
```   176   shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'"
```
```   177   unfolding extend_measure_def by (auto simp add: assms)
```
```   178
```
```   179 lemma Pi_cong_sets:
```
```   180     "\<lbrakk>I = J; \<And>x. x \<in> I \<Longrightarrow> M x = N x\<rbrakk> \<Longrightarrow> Pi I M = Pi J N"
```
```   181   unfolding Pi_def by auto
```
```   182
```
```   183 lemma PiM_cong:
```
```   184   assumes "I = J" "\<And>x. x \<in> I \<Longrightarrow> M x = N x"
```
```   185   shows "PiM I M = PiM J N"
```
```   186   unfolding PiM_def
```
```   187 proof (rule extend_measure_cong, goal_cases)
```
```   188   case 1
```
```   189   show ?case using assms
```
```   190     by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all
```
```   191 next
```
```   192   case 2
```
```   193   have "\<And>K. K \<subseteq> J \<Longrightarrow> (\<Pi> j\<in>K. sets (M j)) = (\<Pi> j\<in>K. sets (N j))"
```
```   194     using assms by (intro Pi_cong_sets) auto
```
```   195   thus ?case by (auto simp: assms)
```
```   196 next
```
```   197   case 3
```
```   198   show ?case using assms
```
```   199     by (intro ext) (auto simp: prod_emb_def dest: PiE_mem)
```
```   200 next
```
```   201   case (4 x)
```
```   202   thus ?case using assms
```
```   203     by (auto intro!: prod.cong split: if_split_asm)
```
```   204 qed
```
```   205
```
```   206
```
```   207 lemma prod_algebra_sets_into_space:
```
```   208   "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   209   by (auto simp: prod_emb_def prod_algebra_def)
```
```   210
```
```   211 lemma prod_algebra_eq_finite:
```
```   212   assumes I: "finite I"
```
```   213   shows "prod_algebra I M = {(\<Pi>\<^sub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
```
```   214 proof (intro iffI set_eqI)
```
```   215   fix A assume "A \<in> ?L"
```
```   216   then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   217     and A: "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
```
```   218     by (auto simp: prod_algebra_def)
```
```   219   let ?A = "\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i)"
```
```   220   have A: "A = ?A"
```
```   221     unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto
```
```   222   show "A \<in> ?R" unfolding A using J sets.top
```
```   223     by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
```
```   224 next
```
```   225   fix A assume "A \<in> ?R"
```
```   226   then obtain X where A: "A = (\<Pi>\<^sub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
```
```   227   then have A: "A = prod_emb I M I (\<Pi>\<^sub>E i\<in>I. X i)"
```
```   228     by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff)
```
```   229   from X I show "A \<in> ?L" unfolding A
```
```   230     by (auto simp: prod_algebra_def)
```
```   231 qed
```
```   232
```
```   233 lemma prod_algebraI:
```
```   234   "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
```
```   235     \<Longrightarrow> prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> prod_algebra I M"
```
```   236   by (auto simp: prod_algebra_def)
```
```   237
```
```   238 lemma prod_algebraI_finite:
```
```   239   "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M"
```
```   240   using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp
```
```   241
```
```   242 lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
```
```   243 proof (safe intro!: Int_stableI)
```
```   244   fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
```
```   245   then show "\<exists>G. Pi\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
```
```   246     by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
```
```   247 qed
```
```   248
```
```   249 lemma prod_algebraE:
```
```   250   assumes A: "A \<in> prod_algebra I M"
```
```   251   obtains J E where "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
```
```   252     "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
```
```   253   using A by (auto simp: prod_algebra_def)
```
```   254
```
```   255 lemma prod_algebraE_all:
```
```   256   assumes A: "A \<in> prod_algebra I M"
```
```   257   obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   258 proof -
```
```   259   from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)"
```
```   260     and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```   261     by (auto simp: prod_algebra_def)
```
```   262   from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
```
```   263     using sets.sets_into_space by auto
```
```   264   then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))"
```
```   265     using A J by (auto simp: prod_emb_PiE)
```
```   266   moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   267     using sets.top E by auto
```
```   268   ultimately show ?thesis using that by auto
```
```   269 qed
```
```   270
```
```   271 lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
```
```   272 proof (unfold Int_stable_def, safe)
```
```   273   fix A assume "A \<in> prod_algebra I M"
```
```   274   from prod_algebraE[OF this] guess J E . note A = this
```
```   275   fix B assume "B \<in> prod_algebra I M"
```
```   276   from prod_algebraE[OF this] guess K F . note B = this
```
```   277   have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^sub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter>
```
```   278       (if i \<in> K then F i else space (M i)))"
```
```   279     unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4)
```
```   280       B(5)[THEN sets.sets_into_space]
```
```   281     apply (subst (1 2 3) prod_emb_PiE)
```
```   282     apply (simp_all add: subset_eq PiE_Int)
```
```   283     apply blast
```
```   284     apply (intro PiE_cong)
```
```   285     apply auto
```
```   286     done
```
```   287   also have "\<dots> \<in> prod_algebra I M"
```
```   288     using A B by (auto intro!: prod_algebraI)
```
```   289   finally show "A \<inter> B \<in> prod_algebra I M" .
```
```   290 qed
```
```   291
```
```   292 proposition prod_algebra_mono:
```
```   293   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
```
```   294   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
```
```   295   shows "prod_algebra I E \<subseteq> prod_algebra I F"
```
```   296 proof
```
```   297   fix A assume "A \<in> prod_algebra I E"
```
```   298   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
```
```   299     and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   300     and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
```
```   301     by (auto simp: prod_algebra_def)
```
```   302   moreover
```
```   303   from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
```
```   304     by (rule PiE_cong)
```
```   305   with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   306     by (simp add: prod_emb_def)
```
```   307   moreover
```
```   308   from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
```
```   309     by auto
```
```   310   ultimately show "A \<in> prod_algebra I F"
```
```   311     apply (simp add: prod_algebra_def image_iff)
```
```   312     apply (intro exI[of _ J] exI[of _ G] conjI)
```
```   313     apply auto
```
```   314     done
```
```   315 qed
```
```   316 proposition prod_algebra_cong:
```
```   317   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
```
```   318   shows "prod_algebra I M = prod_algebra J N"
```
```   319 proof -
```
```   320   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
```
```   321     using sets_eq_imp_space_eq[OF sets] by auto
```
```   322   with sets show ?thesis unfolding \<open>I = J\<close>
```
```   323     by (intro antisym prod_algebra_mono) auto
```
```   324 qed
```
```   325
```
```   326 lemma space_in_prod_algebra:
```
```   327   "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
```
```   328 proof cases
```
```   329   assume "I = {}" then show ?thesis
```
```   330     by (auto simp add: prod_algebra_def image_iff prod_emb_def)
```
```   331 next
```
```   332   assume "I \<noteq> {}"
```
```   333   then obtain i where "i \<in> I" by auto
```
```   334   then have "(\<Pi>\<^sub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i))"
```
```   335     by (auto simp: prod_emb_def)
```
```   336   also have "\<dots> \<in> prod_algebra I M"
```
```   337     using \<open>i \<in> I\<close> by (intro prod_algebraI) auto
```
```   338   finally show ?thesis .
```
```   339 qed
```
```   340
```
```   341 lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   342   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
```
```   343
```
```   344 lemma prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)"
```
```   345   by (auto simp: prod_emb_def space_PiM)
```
```   346
```
```   347 lemma space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow>  (\<exists>i\<in>I. space (M i) = {})"
```
```   348   by (auto simp: space_PiM PiE_eq_empty_iff)
```
```   349
```
```   350 lemma undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)"
```
```   351   by (auto simp: space_PiM)
```
```   352
```
```   353 lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
```
```   354   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
```
```   355
```
```   356 proposition sets_PiM_single: "sets (PiM I M) =
```
```   357     sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   358     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   359   unfolding sets_PiM
```
```   360 proof (rule sigma_sets_eqI)
```
```   361   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   362   fix A assume "A \<in> prod_algebra I M"
```
```   363   from prod_algebraE[OF this] guess J X . note X = this
```
```   364   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   365   proof cases
```
```   366     assume "I = {}"
```
```   367     with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
```
```   368     with \<open>I = {}\<close> show ?thesis by (auto intro!: sigma_sets_top)
```
```   369   next
```
```   370     assume "I \<noteq> {}"
```
```   371     with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
```
```   372       by (auto simp: prod_emb_def)
```
```   373     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```   374       using X \<open>I \<noteq> {}\<close> by (intro R.finite_INT sigma_sets.Basic) auto
```
```   375     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```   376   qed
```
```   377 next
```
```   378   fix A assume "A \<in> ?R"
```
```   379   then obtain i B where A: "A = {f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
```
```   380     by auto
```
```   381   then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
```
```   382      by (auto simp: prod_emb_def)
```
```   383   also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
```
```   384     using A by (intro sigma_sets.Basic prod_algebraI) auto
```
```   385   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
```
```   386 qed
```
```   387
```
```   388 lemma sets_PiM_eq_proj:
```
```   389   "I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
```
```   390   apply (simp add: sets_PiM_single)
```
```   391   apply (subst sets_Sup_eq[where X="\<Pi>\<^sub>E i\<in>I. space (M i)"])
```
```   392   apply auto []
```
```   393   apply auto []
```
```   394   apply simp
```
```   395   apply (subst arg_cong [of _ _ Sup, OF image_cong, OF refl])
```
```   396   apply (rule sets_vimage_algebra2)
```
```   397   apply (auto intro!: arg_cong2[where f=sigma_sets])
```
```   398   done
```
```   399
```
```   400 lemma
```
```   401   shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
```
```   402     and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
```
```   403   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
```
```   404
```
```   405 proposition sets_PiM_sigma:
```
```   406   assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S"
```
```   407   assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)"
```
```   408   assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I"
```
```   409   defines "P \<equiv> {{f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i} | A j. j \<in> J \<and> A \<in> Pi j E}"
```
```   410   shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
```
```   411 proof cases
```
```   412   assume "I = {}"
```
```   413   with \<open>\<Union>J = I\<close> have "P = {{\<lambda>_. undefined}} \<or> P = {}"
```
```   414     by (auto simp: P_def)
```
```   415   with \<open>I = {}\<close> show ?thesis
```
```   416     by (auto simp add: sets_PiM_empty sigma_sets_empty_eq)
```
```   417 next
```
```   418   let ?F = "\<lambda>i. {(\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega> |A. A \<in> E i}"
```
```   419   assume "I \<noteq> {}"
```
```   420   then have "sets (Pi\<^sub>M I (\<lambda>i. sigma (\<Omega> i) (E i))) =
```
```   421       sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<lambda>x. x i) (sigma (\<Omega> i) (E i)))"
```
```   422     by (subst sets_PiM_eq_proj) (auto simp: space_measure_of_conv)
```
```   423   also have "\<dots> = sets (SUP i\<in>I. sigma (Pi\<^sub>E I \<Omega>) (?F i))"
```
```   424     using E by (intro sets_SUP_cong arg_cong[where f=sets] vimage_algebra_sigma) auto
```
```   425   also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i))"
```
```   426     using \<open>I \<noteq> {}\<close> by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto
```
```   427   also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) P)"
```
```   428   proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
```
```   429     show "(\<Union>i\<in>I. ?F i) \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" "P \<subseteq> Pow (Pi\<^sub>E I \<Omega>)"
```
```   430       by (auto simp: P_def)
```
```   431   next
```
```   432     interpret P: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
```
```   433       by (auto intro!: sigma_algebra_sigma_sets simp: P_def)
```
```   434
```
```   435     fix Z assume "Z \<in> (\<Union>i\<in>I. ?F i)"
```
```   436     then obtain i A where i: "i \<in> I" "A \<in> E i" and Z_def: "Z = (\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega>"
```
```   437       by auto
```
```   438     from \<open>i \<in> I\<close> J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j"
```
```   439       by auto
```
```   440     obtain S where S: "\<And>i. i \<in> j \<Longrightarrow> S i \<subseteq> E i" "\<And>i. i \<in> j \<Longrightarrow> countable (S i)"
```
```   441       "\<And>i. i \<in> j \<Longrightarrow> \<Omega> i = \<Union>(S i)"
```
```   442       by (metis subset_eq \<Omega>_cover \<open>j \<subseteq> I\<close>)
```
```   443     define A' where "A' n = n(i := A)" for n
```
```   444     then have A'_i: "\<And>n. A' n i = A"
```
```   445       by simp
```
```   446     { fix n assume "n \<in> Pi\<^sub>E (j - {i}) S"
```
```   447       then have "A' n \<in> Pi j E"
```
```   448         unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def \<open>A \<in> E i\<close> )
```
```   449       with \<open>j \<in> J\<close> have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P"
```
```   450         by (auto simp: P_def) }
```
```   451     note A'_in_P = this
```
```   452
```
```   453     { fix x assume "x i \<in> A" "x \<in> Pi\<^sub>E I \<Omega>"
```
```   454       with S(3) \<open>j \<subseteq> I\<close> have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s"
```
```   455         by (auto simp: PiE_def Pi_def)
```
```   456       then obtain s where s: "\<And>i. i \<in> j \<Longrightarrow> s i \<in> S i" "\<And>i. i \<in> j \<Longrightarrow> x i \<in> s i"
```
```   457         by metis
```
```   458       with \<open>x i \<in> A\<close> have "\<exists>n\<in>Pi\<^sub>E (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i"
```
```   459         by (intro bexI[of _ "restrict (s(i := A)) (j-{i})"]) (auto simp: A'_def split: if_splits) }
```
```   460     then have "Z = (\<Union>n\<in>Pi\<^sub>E (j-{i}) S. {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A' n i})"
```
```   461       unfolding Z_def
```
```   462       by (auto simp add: set_eq_iff ball_conj_distrib \<open>i\<in>j\<close> A'_i dest: bspec[OF _ \<open>i\<in>j\<close>]
```
```   463                cong: conj_cong)
```
```   464     also have "\<dots> \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
```
```   465       using \<open>finite j\<close> S(2)
```
```   466       by (intro P.countable_UN' countable_PiE) (simp_all add: image_subset_iff A'_in_P)
```
```   467     finally show "Z \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" .
```
```   468   next
```
```   469     interpret F: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<Union>i\<in>I. ?F i)"
```
```   470       by (auto intro!: sigma_algebra_sigma_sets)
```
```   471
```
```   472     fix b assume "b \<in> P"
```
```   473     then obtain A j where b: "b = {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i}" "j \<in> J" "A \<in> Pi j E"
```
```   474       by (auto simp: P_def)
```
```   475     show "b \<in> sigma_sets (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i)"
```
```   476     proof cases
```
```   477       assume "j = {}"
```
```   478       with b have "b = (\<Pi>\<^sub>E i\<in>I. \<Omega> i)"
```
```   479         by auto
```
```   480       then show ?thesis
```
```   481         by blast
```
```   482     next
```
```   483       assume "j \<noteq> {}"
```
```   484       with J b(2,3) have eq: "b = (\<Inter>i\<in>j. ((\<lambda>x. x i) -` A i \<inter> Pi\<^sub>E I \<Omega>))"
```
```   485         unfolding b(1)
```
```   486         by (auto simp: PiE_def Pi_def)
```
```   487       show ?thesis
```
```   488         unfolding eq using \<open>A \<in> Pi j E\<close> \<open>j \<in> J\<close> J(2)
```
```   489         by (intro F.finite_INT J \<open>j \<in> J\<close> \<open>j \<noteq> {}\<close> sigma_sets.Basic) blast
```
```   490     qed
```
```   491   qed
```
```   492   finally show "?thesis" .
```
```   493 qed
```
```   494
```
```   495 lemma sets_PiM_in_sets:
```
```   496   assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   497   assumes sets: "\<And>i A. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {x\<in>space N. x i \<in> A} \<in> sets N"
```
```   498   shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N"
```
```   499   unfolding sets_PiM_single space[symmetric]
```
```   500   by (intro sets.sigma_sets_subset subsetI) (auto intro: sets)
```
```   501
```
```   502 lemma sets_PiM_cong[measurable_cong]:
```
```   503   assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)"
```
```   504   using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong)
```
```   505
```
```   506 lemma sets_PiM_I:
```
```   507   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   508   shows "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
```
```   509 proof cases
```
```   510   assume "J = {}"
```
```   511   then have "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) = (\<Pi>\<^sub>E j\<in>I. space (M j))"
```
```   512     by (auto simp: prod_emb_def)
```
```   513   then show ?thesis
```
```   514     by (auto simp add: sets_PiM intro!: sigma_sets_top)
```
```   515 next
```
```   516   assume "J \<noteq> {}" with assms show ?thesis
```
```   517     by (force simp add: sets_PiM prod_algebra_def)
```
```   518 qed
```
```   519
```
```   520 proposition measurable_PiM:
```
```   521   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   522   assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
```
```   523     f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
```
```   524   shows "f \<in> measurable N (PiM I M)"
```
```   525   using sets_PiM prod_algebra_sets_into_space space
```
```   526 proof (rule measurable_sigma_sets)
```
```   527   fix A assume "A \<in> prod_algebra I M"
```
```   528   from prod_algebraE[OF this] guess J X .
```
```   529   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
```
```   530 qed
```
```   531
```
```   532 lemma measurable_PiM_Collect:
```
```   533   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   534   assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
```
```   535     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
```
```   536   shows "f \<in> measurable N (PiM I M)"
```
```   537   using sets_PiM prod_algebra_sets_into_space space
```
```   538 proof (rule measurable_sigma_sets)
```
```   539   fix A assume "A \<in> prod_algebra I M"
```
```   540   from prod_algebraE[OF this] guess J X . note X = this
```
```   541   then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
```
```   542     using space by (auto simp: prod_emb_def del: PiE_I)
```
```   543   also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
```
```   544   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   545 qed
```
```   546
```
```   547 lemma measurable_PiM_single:
```
```   548   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   549   assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N"
```
```   550   shows "f \<in> measurable N (PiM I M)"
```
```   551   using sets_PiM_single
```
```   552 proof (rule measurable_sigma_sets)
```
```   553   fix A assume "A \<in> {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   554   then obtain B i where "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
```
```   555     by auto
```
```   556   with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
```
```   557   also have "\<dots> \<in> sets N" using B by (rule sets)
```
```   558   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   559 qed (auto simp: space)
```
```   560
```
```   561 lemma measurable_PiM_single':
```
```   562   assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
```
```   563     and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   564   shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
```
```   565 proof (rule measurable_PiM_single)
```
```   566   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   567   then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
```
```   568     by auto
```
```   569   then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
```
```   570     using A f by (auto intro!: measurable_sets)
```
```   571 qed fact
```
```   572
```
```   573 lemma sets_PiM_I_finite[measurable]:
```
```   574   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
```
```   575   shows "(\<Pi>\<^sub>E j\<in>I. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
```
```   576   using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] \<open>finite I\<close> sets by auto
```
```   577
```
```   578 lemma measurable_component_singleton[measurable (raw)]:
```
```   579   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
```
```   580 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   581   fix A assume "A \<in> sets (M i)"
```
```   582   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. A)"
```
```   583     using sets.sets_into_space \<open>i \<in> I\<close>
```
```   584     by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: if_split_asm)
```
```   585   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
```
```   586     using \<open>A \<in> sets (M i)\<close> \<open>i \<in> I\<close> by (auto intro!: sets_PiM_I)
```
```   587 qed (insert \<open>i \<in> I\<close>, auto simp: space_PiM)
```
```   588
```
```   589 lemma measurable_component_singleton'[measurable_dest]:
```
```   590   assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
```
```   591   assumes g: "g \<in> measurable L N"
```
```   592   assumes i: "i \<in> I"
```
```   593   shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)"
```
```   594   using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] .
```
```   595
```
```   596 lemma measurable_PiM_component_rev:
```
```   597   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
```
```   598   by simp
```
```   599
```
```   600 lemma measurable_case_nat[measurable (raw)]:
```
```   601   assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
```
```   602     "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
```
```   603   shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
```
```   604   by (cases i) simp_all
```
```   605
```
```   606 lemma measurable_case_nat'[measurable (raw)]:
```
```   607   assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   608   shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   609   using fg[THEN measurable_space]
```
```   610   by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
```
```   611
```
```   612 lemma measurable_add_dim[measurable]:
```
```   613   "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
```
```   614     (is "?f \<in> measurable ?P ?I")
```
```   615 proof (rule measurable_PiM_single)
```
```   616   fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
```
```   617   have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
```
```   618     (if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
```
```   619     using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
```
```   620   also have "\<dots> \<in> sets ?P"
```
```   621     using A j
```
```   622     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   623   finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
```
```   624 qed (auto simp: space_pair_measure space_PiM PiE_def)
```
```   625
```
```   626 proposition measurable_fun_upd:
```
```   627   assumes I: "I = J \<union> {i}"
```
```   628   assumes f[measurable]: "f \<in> measurable N (PiM J M)"
```
```   629   assumes h[measurable]: "h \<in> measurable N (M i)"
```
```   630   shows "(\<lambda>x. (f x) (i := h x)) \<in> measurable N (PiM I M)"
```
```   631 proof (intro measurable_PiM_single')
```
```   632   fix j assume "j \<in> I" then show "(\<lambda>\<omega>. ((f \<omega>)(i := h \<omega>)) j) \<in> measurable N (M j)"
```
```   633     unfolding I by (cases "j = i") auto
```
```   634 next
```
```   635   show "(\<lambda>x. (f x)(i := h x)) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   636     using I f[THEN measurable_space] h[THEN measurable_space]
```
```   637     by (auto simp: space_PiM PiE_iff extensional_def)
```
```   638 qed
```
```   639
```
```   640 lemma measurable_component_update:
```
```   641   "x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)"
```
```   642   by simp
```
```   643
```
```   644 lemma measurable_merge[measurable]:
```
```   645   "merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)"
```
```   646     (is "?f \<in> measurable ?P ?U")
```
```   647 proof (rule measurable_PiM_single)
```
```   648   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
```
```   649   then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
```
```   650     (if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)"
```
```   651     by (auto simp: merge_def)
```
```   652   also have "\<dots> \<in> sets ?P"
```
```   653     using A
```
```   654     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   655   finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
```
```   656 qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
```
```   657
```
```   658 lemma measurable_restrict[measurable (raw)]:
```
```   659   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
```
```   660   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
```
```   661 proof (rule measurable_PiM_single)
```
```   662   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   663   then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
```
```   664     by auto
```
```   665   then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
```
```   666     using A X by (auto intro!: measurable_sets)
```
```   667 qed (insert X, auto simp add: PiE_def dest: measurable_space)
```
```   668
```
```   669 lemma measurable_abs_UNIV:
```
```   670   "(\<And>n. (\<lambda>\<omega>. f n \<omega>) \<in> measurable M (N n)) \<Longrightarrow> (\<lambda>\<omega> n. f n \<omega>) \<in> measurable M (PiM UNIV N)"
```
```   671   by (intro measurable_PiM_single) (auto dest: measurable_space)
```
```   672
```
```   673 lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
```
```   674   by (intro measurable_restrict measurable_component_singleton) auto
```
```   675
```
```   676 lemma measurable_restrict_subset':
```
```   677   assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)"
```
```   678   shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
```
```   679 proof-
```
```   680   from assms(1) have "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
```
```   681     by (rule measurable_restrict_subset)
```
```   682   also from assms(2) have "measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M) = measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
```
```   683     by (intro sets_PiM_cong measurable_cong_sets) simp_all
```
```   684   finally show ?thesis .
```
```   685 qed
```
```   686
```
```   687 lemma measurable_prod_emb[intro, simp]:
```
```   688   "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
```
```   689   unfolding prod_emb_def space_PiM[symmetric]
```
```   690   by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
```
```   691
```
```   692 lemma merge_in_prod_emb:
```
```   693   assumes "y \<in> space (PiM I M)" "x \<in> X" and X: "X \<in> sets (Pi\<^sub>M J M)" and "J \<subseteq> I"
```
```   694   shows "merge J I (x, y) \<in> prod_emb I M J X"
```
```   695   using assms sets.sets_into_space[OF X]
```
```   696   by (simp add: merge_def prod_emb_def subset_eq space_PiM PiE_def extensional_restrict Pi_iff
```
```   697            cong: if_cong restrict_cong)
```
```   698      (simp add: extensional_def)
```
```   699
```
```   700 lemma prod_emb_eq_emptyD:
```
```   701   assumes J: "J \<subseteq> I" and ne: "space (PiM I M) \<noteq> {}" and X: "X \<in> sets (Pi\<^sub>M J M)"
```
```   702     and *: "prod_emb I M J X = {}"
```
```   703   shows "X = {}"
```
```   704 proof safe
```
```   705   fix x assume "x \<in> X"
```
```   706   obtain \<omega> where "\<omega> \<in> space (PiM I M)"
```
```   707     using ne by blast
```
```   708   from merge_in_prod_emb[OF this \<open>x\<in>X\<close> X J] * show "x \<in> {}" by auto
```
```   709 qed
```
```   710
```
```   711 lemma sets_in_Pi_aux:
```
```   712   "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   713   {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
```
```   714   by (simp add: subset_eq Pi_iff)
```
```   715
```
```   716 lemma sets_in_Pi[measurable (raw)]:
```
```   717   "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
```
```   718   (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   719   Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
```
```   720   unfolding pred_def
```
```   721   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
```
```   722
```
```   723 lemma sets_in_extensional_aux:
```
```   724   "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
```
```   725 proof -
```
```   726   have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
```
```   727     by (auto simp add: extensional_def space_PiM)
```
```   728   then show ?thesis by simp
```
```   729 qed
```
```   730
```
```   731 lemma sets_in_extensional[measurable (raw)]:
```
```   732   "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
```
```   733   unfolding pred_def
```
```   734   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
```
```   735
```
```   736 lemma sets_PiM_I_countable:
```
```   737   assumes I: "countable I" and E: "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
```
```   738 proof cases
```
```   739   assume "I \<noteq> {}"
```
```   740   then have "Pi\<^sub>E I E = (\<Inter>i\<in>I. prod_emb I M {i} (Pi\<^sub>E {i} E))"
```
```   741     using E[THEN sets.sets_into_space] by (auto simp: PiE_iff prod_emb_def fun_eq_iff)
```
```   742   also have "\<dots> \<in> sets (PiM I M)"
```
```   743     using I \<open>I \<noteq> {}\<close> by (safe intro!: sets.countable_INT' measurable_prod_emb sets_PiM_I_finite E)
```
```   744   finally show ?thesis .
```
```   745 qed (simp add: sets_PiM_empty)
```
```   746
```
```   747 lemma sets_PiM_D_countable:
```
```   748   assumes A: "A \<in> PiM I M"
```
```   749   shows "\<exists>J\<subseteq>I. \<exists>X\<in>PiM J M. countable J \<and> A = prod_emb I M J X"
```
```   750   using A[unfolded sets_PiM_single]
```
```   751 proof induction
```
```   752   case (Basic A)
```
```   753   then obtain i X where *: "i \<in> I" "X \<in> sets (M i)" and "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> X}"
```
```   754     by auto
```
```   755   then have A: "A = prod_emb I M {i} (\<Pi>\<^sub>E _\<in>{i}. X)"
```
```   756     by (auto simp: prod_emb_def)
```
```   757   then show ?case
```
```   758     by (intro exI[of _ "{i}"] conjI bexI[of _ "\<Pi>\<^sub>E _\<in>{i}. X"])
```
```   759        (auto intro: countable_finite * sets_PiM_I_finite)
```
```   760 next
```
```   761   case Empty then show ?case
```
```   762     by (intro exI[of _ "{}"] conjI bexI[of _ "{}"]) auto
```
```   763 next
```
```   764   case (Compl A)
```
```   765   then obtain J X where "J \<subseteq> I" "X \<in> sets (Pi\<^sub>M J M)" "countable J" "A = prod_emb I M J X"
```
```   766     by auto
```
```   767   then show ?case
```
```   768     by (intro exI[of _ J] bexI[of _ "space (PiM J M) - X"] conjI)
```
```   769        (auto simp add: space_PiM prod_emb_PiE intro!: sets_PiM_I_countable)
```
```   770 next
```
```   771   case (Union K)
```
```   772   obtain J X where J: "\<And>i. J i \<subseteq> I" "\<And>i. countable (J i)" and X: "\<And>i. X i \<in> sets (Pi\<^sub>M (J i) M)"
```
```   773     and K: "\<And>i. K i = prod_emb I M (J i) (X i)"
```
```   774     by (metis Union.IH)
```
```   775   show ?case
```
```   776   proof (intro exI[of _ "\<Union>i. J i"] bexI[of _ "\<Union>i. prod_emb (\<Union>i. J i) M (J i) (X i)"] conjI)
```
```   777     show "(\<Union>i. J i) \<subseteq> I" "countable (\<Union>i. J i)" using J by auto
```
```   778     with J show "\<Union>(K ` UNIV) = prod_emb I M (\<Union>i. J i) (\<Union>i. prod_emb (\<Union>i. J i) M (J i) (X i))"
```
```   779       by (simp add: K[abs_def] SUP_upper)
```
```   780   qed(auto intro: X)
```
```   781 qed
```
```   782
```
```   783 proposition measure_eqI_PiM_finite:
```
```   784   assumes [simp]: "finite I" "sets P = PiM I M" "sets Q = PiM I M"
```
```   785   assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = Q (Pi\<^sub>E I A)"
```
```   786   assumes A: "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = space (PiM I M)" "\<And>i::nat. P (A i) \<noteq> \<infinity>"
```
```   787   shows "P = Q"
```
```   788 proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
```
```   789   show "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. P (A i) \<noteq> \<infinity>"
```
```   790     unfolding space_PiM[symmetric] by fact+
```
```   791   fix X assume "X \<in> prod_algebra I M"
```
```   792   then obtain J E where X: "X = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
```
```   793     and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)"
```
```   794     by (force elim!: prod_algebraE)
```
```   795   then show "emeasure P X = emeasure Q X"
```
```   796     unfolding X by (subst (1 2) prod_emb_Pi) (auto simp: eq)
```
```   797 qed (simp_all add: sets_PiM)
```
```   798
```
```   799 proposition measure_eqI_PiM_infinite:
```
```   800   assumes [simp]: "sets P = PiM I M" "sets Q = PiM I M"
```
```   801   assumes eq: "\<And>A J. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow>
```
```   802     P (prod_emb I M J (Pi\<^sub>E J A)) = Q (prod_emb I M J (Pi\<^sub>E J A))"
```
```   803   assumes A: "finite_measure P"
```
```   804   shows "P = Q"
```
```   805 proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
```
```   806   interpret finite_measure P by fact
```
```   807   define i where "i = (SOME i. i \<in> I)"
```
```   808   have i: "I \<noteq> {} \<Longrightarrow> i \<in> I"
```
```   809     unfolding i_def by (rule someI_ex) auto
```
```   810   define A where "A n =
```
```   811     (if I = {} then prod_emb I M {} (\<Pi>\<^sub>E i\<in>{}. {}) else prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i)))"
```
```   812     for n :: nat
```
```   813   then show "range A \<subseteq> prod_algebra I M"
```
```   814     using prod_algebraI[of "{}" I "\<lambda>i. space (M i)" M] by (auto intro!: prod_algebraI i)
```
```   815   have "\<And>i. A i = space (PiM I M)"
```
```   816     by (auto simp: prod_emb_def space_PiM PiE_iff A_def i ex_in_conv[symmetric] exI)
```
```   817   then show "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. emeasure P (A i) \<noteq> \<infinity>"
```
```   818     by (auto simp: space_PiM)
```
```   819 next
```
```   820   fix X assume X: "X \<in> prod_algebra I M"
```
```   821   then obtain J E where X: "X = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
```
```   822     and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)"
```
```   823     by (force elim!: prod_algebraE)
```
```   824   then show "emeasure P X = emeasure Q X"
```
```   825     by (auto intro!: eq)
```
```   826 qed (auto simp: sets_PiM)
```
```   827
```
```   828 locale%unimportant product_sigma_finite =
```
```   829   fixes M :: "'i \<Rightarrow> 'a measure"
```
```   830   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
```
```   831
```
```   832 sublocale%unimportant product_sigma_finite \<subseteq> M?: sigma_finite_measure "M i" for i
```
```   833   by (rule sigma_finite_measures)
```
```   834
```
```   835 locale%unimportant finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   836   fixes I :: "'i set"
```
```   837   assumes finite_index: "finite I"
```
```   838
```
```   839 proposition (in finite_product_sigma_finite) sigma_finite_pairs:
```
```   840   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
```
```   841     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
```
```   842     (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and>
```
```   843     (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
```
```   844 proof -
```
```   845   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)"
```
```   846     using M.sigma_finite_incseq by metis
```
```   847   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   848   then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. emeasure (M i) (F i k) \<noteq> \<infinity>"
```
```   849     by auto
```
```   850   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
```
```   851   note space_PiM[simp]
```
```   852   show ?thesis
```
```   853   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
```
```   854     fix i show "range (F i) \<subseteq> sets (M i)" by fact
```
```   855   next
```
```   856     fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
```
```   857   next
```
```   858     fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
```
```   859       by (auto simp: PiE_def dest!: sets.sets_into_space)
```
```   860   next
```
```   861     fix f assume "f \<in> space (PiM I M)"
```
```   862     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
```
```   863     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
```
```   864   next
```
```   865     fix i show "?F i \<subseteq> ?F (Suc i)"
```
```   866       using \<open>\<And>i. incseq (F i)\<close>[THEN incseq_SucD] by auto
```
```   867   qed
```
```   868 qed
```
```   869
```
```   870 lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
```
```   871 proof -
```
```   872   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ennreal)"
```
```   873   have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
```
```   874   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   875     show "positive (PiM {} M) ?\<mu>"
```
```   876       by (auto simp: positive_def)
```
```   877     show "countably_additive (PiM {} M) ?\<mu>"
```
```   878       by (rule sets.countably_additiveI_finite)
```
```   879          (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
```
```   880   qed (auto simp: prod_emb_def)
```
```   881   also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
```
```   882     by (auto simp: prod_emb_def)
```
```   883   finally show ?thesis
```
```   884     by simp
```
```   885 qed
```
```   886
```
```   887 lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
```
```   888   by (rule measure_eqI) (auto simp add: sets_PiM_empty)
```
```   889
```
```   890 lemma (in product_sigma_finite) emeasure_PiM:
```
```   891   "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   892 proof (induct I arbitrary: A rule: finite_induct)
```
```   893   case (insert i I)
```
```   894   interpret finite_product_sigma_finite M I by standard fact
```
```   895   have "finite (insert i I)" using \<open>finite I\<close> by auto
```
```   896   interpret I': finite_product_sigma_finite M "insert i I" by standard fact
```
```   897   let ?h = "(\<lambda>(f, y). f(i := y))"
```
```   898
```
```   899   let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
```
```   900   let ?\<mu> = "emeasure ?P"
```
```   901   let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
```
```   902   let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
```
```   903
```
```   904   have "emeasure (Pi\<^sub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^sub>E (insert i I) A)) =
```
```   905     (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
```
```   906   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   907     fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))"
```
```   908     then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto
```
```   909     let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
```
```   910     let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
```
```   911     have "?\<mu> ?p =
```
```   912       emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i))"
```
```   913       by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
```
```   914     also have "?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
```
```   915       using J E[rule_format, THEN sets.sets_into_space]
```
```   916       by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: if_split_asm)
```
```   917     also have "emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
```
```   918       emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
```
```   919       using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
```
```   920     also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
```
```   921       using J E[rule_format, THEN sets.sets_into_space]
```
```   922       by (auto simp: prod_emb_iff PiE_def Pi_iff split: if_split_asm) blast+
```
```   923     also have "emeasure (Pi\<^sub>M I M) (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
```
```   924       (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
```
```   925       using E by (subst insert) (auto intro!: prod.cong)
```
```   926     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
```
```   927        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
```
```   928       using insert by (auto simp: mult.commute intro!: arg_cong2[where f="(*)"] prod.cong)
```
```   929     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
```
```   930       using insert(1,2) J E by (intro prod.mono_neutral_right) auto
```
```   931     finally show "?\<mu> ?p = \<dots>" .
```
```   932
```
```   933     show "prod_emb (insert i I) M J (Pi\<^sub>E J E) \<in> Pow (\<Pi>\<^sub>E i\<in>insert i I. space (M i))"
```
```   934       using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
```
```   935   next
```
```   936     show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
```
```   937       using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
```
```   938   next
```
```   939     show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
```
```   940       insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
```
```   941       using insert by auto
```
```   942   qed (auto intro!: prod.cong)
```
```   943   with insert show ?case
```
```   944     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
```
```   945 qed simp
```
```   946
```
```   947 lemma (in product_sigma_finite) PiM_eqI:
```
```   948   assumes I[simp]: "finite I" and P: "sets P = PiM I M"
```
```   949   assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   950   shows "P = PiM I M"
```
```   951 proof -
```
```   952   interpret finite_product_sigma_finite M I
```
```   953     proof qed fact
```
```   954   from sigma_finite_pairs guess C .. note C = this
```
```   955   show ?thesis
```
```   956   proof (rule measure_eqI_PiM_finite[OF I refl P, symmetric])
```
```   957     show "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>M I M) (Pi\<^sub>E I A) = P (Pi\<^sub>E I A)" for A
```
```   958       by (simp add: eq emeasure_PiM)
```
```   959     define A where "A n = (\<Pi>\<^sub>E i\<in>I. C i n)" for n
```
```   960     with C show "range A \<subseteq> prod_algebra I M" "\<And>i. emeasure (Pi\<^sub>M I M) (A i) \<noteq> \<infinity>" "(\<Union>i. A i) = space (PiM I M)"
```
```   961       by (auto intro!: prod_algebraI_finite simp: emeasure_PiM subset_eq ennreal_prod_eq_top)
```
```   962   qed
```
```   963 qed
```
```   964
```
```   965 lemma (in product_sigma_finite) sigma_finite:
```
```   966   assumes "finite I"
```
```   967   shows "sigma_finite_measure (PiM I M)"
```
```   968 proof
```
```   969   interpret finite_product_sigma_finite M I by standard fact
```
```   970
```
```   971   obtain F where F: "\<And>j. countable (F j)" "\<And>j f. f \<in> F j \<Longrightarrow> f \<in> sets (M j)"
```
```   972     "\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and
```
```   973     in_space: "\<And>j. space (M j) = \<Union>(F j)"
```
```   974     using sigma_finite_countable by (metis subset_eq)
```
```   975   moreover have "(\<Union>(Pi\<^sub>E I ` Pi\<^sub>E I F)) = space (Pi\<^sub>M I M)"
```
```   976     using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD2])
```
```   977   ultimately show "\<exists>A. countable A \<and> A \<subseteq> sets (Pi\<^sub>M I M) \<and> \<Union>A = space (Pi\<^sub>M I M) \<and> (\<forall>a\<in>A. emeasure (Pi\<^sub>M I M) a \<noteq> \<infinity>)"
```
```   978     by (intro exI[of _ "Pi\<^sub>E I ` Pi\<^sub>E I F"])
```
```   979        (auto intro!: countable_PiE sets_PiM_I_finite
```
```   980              simp: PiE_iff emeasure_PiM finite_index ennreal_prod_eq_top)
```
```   981 qed
```
```   982
```
```   983 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
```
```   984   using sigma_finite[OF finite_index] .
```
```   985
```
```   986 lemma (in finite_product_sigma_finite) measure_times:
```
```   987   "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   988   using emeasure_PiM[OF finite_index] by auto
```
```   989
```
```   990 lemma (in product_sigma_finite) nn_integral_empty:
```
```   991   "0 \<le> f (\<lambda>k. undefined) \<Longrightarrow> integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
```
```   992   by (simp add: PiM_empty nn_integral_count_space_finite max.absorb2)
```
```   993
```
```   994 lemma%important (in product_sigma_finite) distr_merge:
```
```   995   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   996   shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
```
```   997    (is "?D = ?P")
```
```   998 proof (rule PiM_eqI)
```
```   999   interpret I: finite_product_sigma_finite M I by standard fact
```
```  1000   interpret J: finite_product_sigma_finite M J by standard fact
```
```  1001   fix A assume A: "\<And>i. i \<in> I \<union> J \<Longrightarrow> A i \<in> sets (M i)"
```
```  1002   have *: "(merge I J -` Pi\<^sub>E (I \<union> J) A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)) = Pi\<^sub>E I A \<times> Pi\<^sub>E J A"
```
```  1003     using A[THEN sets.sets_into_space] by (auto simp: space_PiM space_pair_measure)
```
```  1004   from A fin show "emeasure (distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J)) (Pi\<^sub>E (I \<union> J) A) =
```
```  1005       (\<Prod>i\<in>I \<union> J. emeasure (M i) (A i))"
```
```  1006     by (subst emeasure_distr)
```
```  1007        (auto simp: * J.emeasure_pair_measure_Times I.measure_times J.measure_times prod.union_disjoint)
```
```  1008 qed (insert fin, simp_all)
```
```  1009
```
```  1010 proposition (in product_sigma_finite) product_nn_integral_fold:
```
```  1011   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
```
```  1012   and f[measurable]: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
```
```  1013   shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
```
```  1014     (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
```
```  1015 proof -
```
```  1016   interpret I: finite_product_sigma_finite M I by standard fact
```
```  1017   interpret J: finite_product_sigma_finite M J by standard fact
```
```  1018   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by standard
```
```  1019   have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
```
```  1020     using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
```
```  1021   show ?thesis
```
```  1022     apply (subst distr_merge[OF IJ, symmetric])
```
```  1023     apply (subst nn_integral_distr[OF measurable_merge])
```
```  1024     apply measurable []
```
```  1025     apply (subst J.nn_integral_fst[symmetric, OF P_borel])
```
```  1026     apply simp
```
```  1027     done
```
```  1028 qed
```
```  1029
```
```  1030 lemma (in product_sigma_finite) distr_singleton:
```
```  1031   "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
```
```  1032 proof (intro measure_eqI[symmetric])
```
```  1033   interpret I: finite_product_sigma_finite M "{i}" by standard simp
```
```  1034   fix A assume A: "A \<in> sets (M i)"
```
```  1035   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
```
```  1036     using sets.sets_into_space by (auto simp: space_PiM)
```
```  1037   then show "emeasure (M i) A = emeasure ?D A"
```
```  1038     using A I.measure_times[of "\<lambda>_. A"]
```
```  1039     by (simp add: emeasure_distr measurable_component_singleton)
```
```  1040 qed simp
```
```  1041
```
```  1042 lemma (in product_sigma_finite) product_nn_integral_singleton:
```
```  1043   assumes f: "f \<in> borel_measurable (M i)"
```
```  1044   shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f"
```
```  1045 proof -
```
```  1046   interpret I: finite_product_sigma_finite M "{i}" by standard simp
```
```  1047   from f show ?thesis
```
```  1048     apply (subst distr_singleton[symmetric])
```
```  1049     apply (subst nn_integral_distr[OF measurable_component_singleton])
```
```  1050     apply simp_all
```
```  1051     done
```
```  1052 qed
```
```  1053
```
```  1054 proposition (in product_sigma_finite) product_nn_integral_insert:
```
```  1055   assumes I[simp]: "finite I" "i \<notin> I"
```
```  1056     and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```  1057   shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
```
```  1058 proof -
```
```  1059   interpret I: finite_product_sigma_finite M I by standard auto
```
```  1060   interpret i: finite_product_sigma_finite M "{i}" by standard auto
```
```  1061   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
```
```  1062     using f by auto
```
```  1063   show ?thesis
```
```  1064     unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
```
```  1065   proof (rule nn_integral_cong, subst product_nn_integral_singleton[symmetric])
```
```  1066     fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
```
```  1067     let ?f = "\<lambda>y. f (x(i := y))"
```
```  1068     show "?f \<in> borel_measurable (M i)"
```
```  1069       using measurable_comp[OF measurable_component_update f, OF x \<open>i \<notin> I\<close>]
```
```  1070       unfolding comp_def .
```
```  1071     show "(\<integral>\<^sup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^sub>M {i} M) = (\<integral>\<^sup>+ y. f (x(i := y i)) \<partial>Pi\<^sub>M {i} M)"
```
```  1072       using x
```
```  1073       by (auto intro!: nn_integral_cong arg_cong[where f=f]
```
```  1074                simp add: space_PiM extensional_def PiE_def)
```
```  1075   qed
```
```  1076 qed
```
```  1077
```
```  1078 lemma (in product_sigma_finite) product_nn_integral_insert_rev:
```
```  1079   assumes I[simp]: "finite I" "i \<notin> I"
```
```  1080     and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```  1081   shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x(i := y)) \<partial>(Pi\<^sub>M I M)) \<partial>(M i))"
```
```  1082   apply (subst product_nn_integral_insert[OF assms])
```
```  1083   apply (rule pair_sigma_finite.Fubini')
```
```  1084   apply intro_locales []
```
```  1085   apply (rule sigma_finite[OF I(1)])
```
```  1086   apply measurable
```
```  1087   done
```
```  1088
```
```  1089 lemma (in product_sigma_finite) product_nn_integral_prod:
```
```  1090   assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```  1091   shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
```
```  1092 using assms proof (induction I)
```
```  1093   case (insert i I)
```
```  1094   note insert.prems[measurable]
```
```  1095   note \<open>finite I\<close>[intro, simp]
```
```  1096   interpret I: finite_product_sigma_finite M I by standard auto
```
```  1097   have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
```
```  1098     using insert by (auto intro!: prod.cong)
```
```  1099   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^sub>M J M)"
```
```  1100     using sets.sets_into_space insert
```
```  1101     by (intro borel_measurable_prod_ennreal
```
```  1102               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
```
```  1103        auto
```
```  1104   then show ?case
```
```  1105     apply (simp add: product_nn_integral_insert[OF insert(1,2)])
```
```  1106     apply (simp add: insert(2-) * nn_integral_multc)
```
```  1107     apply (subst nn_integral_cmult)
```
```  1108     apply (auto simp add: insert(2-))
```
```  1109     done
```
```  1110 qed (simp add: space_PiM)
```
```  1111
```
```  1112 proposition (in product_sigma_finite) product_nn_integral_pair:
```
```  1113   assumes [measurable]: "case_prod f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
```
```  1114   assumes xy: "x \<noteq> y"
```
```  1115   shows "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {x, y} M) = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
```
```  1116 proof -
```
```  1117   interpret psm: pair_sigma_finite "M x" "M y"
```
```  1118     unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all
```
```  1119   have "{x, y} = {y, x}" by auto
```
```  1120   also have "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {y, x} M) = (\<integral>\<^sup>+y. \<integral>\<^sup>+\<sigma>. f (\<sigma> x) y \<partial>PiM {x} M \<partial>M y)"
```
```  1121     using xy by (subst product_nn_integral_insert_rev) simp_all
```
```  1122   also have "... = (\<integral>\<^sup>+y. \<integral>\<^sup>+x. f x y \<partial>M x \<partial>M y)"
```
```  1123     by (intro nn_integral_cong, subst product_nn_integral_singleton) simp_all
```
```  1124   also have "... = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
```
```  1125     by (subst psm.nn_integral_snd[symmetric]) simp_all
```
```  1126   finally show ?thesis .
```
```  1127 qed
```
```  1128
```
```  1129 lemma (in product_sigma_finite) distr_component:
```
```  1130   "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
```
```  1131 proof (intro PiM_eqI)
```
```  1132   fix A assume A: "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)"
```
```  1133   then have "(\<lambda>x. \<lambda>i\<in>{i}. x) -` Pi\<^sub>E {i} A \<inter> space (M i) = A i"
```
```  1134     by (fastforce dest: sets.sets_into_space)
```
```  1135   with A show "emeasure (distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x)) (Pi\<^sub>E {i} A) = (\<Prod>i\<in>{i}. emeasure (M i) (A i))"
```
```  1136     by (subst emeasure_distr) (auto intro!: sets_PiM_I_finite measurable_restrict)
```
```  1137 qed simp_all
```
```  1138
```
```  1139 lemma (in product_sigma_finite)
```
```  1140   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
```
```  1141   shows emeasure_fold_integral:
```
```  1142     "emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I)
```
```  1143     and emeasure_fold_measurable:
```
```  1144     "(\<lambda>x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M))) \<in> borel_measurable (Pi\<^sub>M I M)" (is ?B)
```
```  1145 proof -
```
```  1146   interpret I: finite_product_sigma_finite M I by standard fact
```
```  1147   interpret J: finite_product_sigma_finite M J by standard fact
```
```  1148   interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
```
```  1149   have merge: "merge I J -` A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
```
```  1150     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```  1151
```
```  1152   show ?I
```
```  1153     apply (subst distr_merge[symmetric, OF IJ])
```
```  1154     apply (subst emeasure_distr[OF measurable_merge A])
```
```  1155     apply (subst J.emeasure_pair_measure_alt[OF merge])
```
```  1156     apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```  1157     done
```
```  1158
```
```  1159   show ?B
```
```  1160     using IJ.measurable_emeasure_Pair1[OF merge]
```
```  1161     by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
```
```  1162 qed
```
```  1163
```
```  1164 lemma sets_Collect_single:
```
```  1165   "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
```
```  1166   by simp
```
```  1167
```
```  1168 lemma pair_measure_eq_distr_PiM:
```
```  1169   fixes M1 :: "'a measure" and M2 :: "'a measure"
```
```  1170   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
```
```  1171   shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))"
```
```  1172     (is "?P = ?D")
```
```  1173 proof (rule pair_measure_eqI[OF assms])
```
```  1174   interpret B: product_sigma_finite "case_bool M1 M2"
```
```  1175     unfolding product_sigma_finite_def using assms by (auto split: bool.split)
```
```  1176   let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"
```
```  1177
```
```  1178   have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)"
```
```  1179     by auto
```
```  1180   fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
```
```  1181   have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
```
```  1182     by (simp add: UNIV_bool ac_simps)
```
```  1183   also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
```
```  1184     using A B by (subst B.emeasure_PiM) (auto split: bool.split)
```
```  1185   also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
```
```  1186     using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
```
```  1187     by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
```
```  1188   finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
```
```  1189     using A B
```
```  1190       measurable_component_singleton[of True UNIV "case_bool M1 M2"]
```
```  1191       measurable_component_singleton[of False UNIV "case_bool M1 M2"]
```
```  1192     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
```
```  1193 qed simp
```
```  1194
```
```  1195 lemma infprod_in_sets[intro]:
```
```  1196   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
```
```  1197   shows "Pi UNIV E \<in> sets (\<Pi>\<^sub>M i\<in>UNIV::nat set. M i)"
```
```  1198 proof -
```
```  1199   have "Pi UNIV E = (\<Inter>i. prod_emb UNIV M {..i} (\<Pi>\<^sub>E j\<in>{..i}. E j))"
```
```  1200     using E E[THEN sets.sets_into_space]
```
```  1201     by (auto simp: prod_emb_def Pi_iff extensional_def)
```
```  1202   with E show ?thesis by auto
```
```  1203 qed
```
```  1204
```
```  1205
```
```  1206
```
```  1207 subsection \<open>Measurability\<close>
```
```  1208
```
```  1209 text \<open>There are two natural sigma-algebras on a product space: the borel sigma algebra,
```
```  1210 generated by open sets in the product, and the product sigma algebra, countably generated by
```
```  1211 products of measurable sets along finitely many coordinates. The second one is defined and studied
```
```  1212 in \<^file>\<open>Finite_Product_Measure.thy\<close>.
```
```  1213
```
```  1214 These sigma-algebra share a lot of natural properties (measurability of coordinates, for instance),
```
```  1215 but there is a fundamental difference: open sets are generated by arbitrary unions, not only
```
```  1216 countable ones, so typically many open sets will not be measurable with respect to the product sigma
```
```  1217 algebra (while all sets in the product sigma algebra are borel). The two sigma algebras coincide
```
```  1218 only when everything is countable (i.e., the product is countable, and the borel sigma algebra in
```
```  1219 the factor is countably generated).
```
```  1220
```
```  1221 In this paragraph, we develop basic measurability properties for the borel sigma algebra, and
```
```  1222 compare it with the product sigma algebra as explained above.
```
```  1223 \<close>
```
```  1224
```
```  1225 lemma measurable_product_coordinates [measurable (raw)]:
```
```  1226   "(\<lambda>x. x i) \<in> measurable borel borel"
```
```  1227 by (rule borel_measurable_continuous_on1[OF continuous_on_product_coordinates])
```
```  1228
```
```  1229 lemma measurable_product_then_coordinatewise:
```
```  1230   fixes f::"'a \<Rightarrow> 'b \<Rightarrow> ('c::topological_space)"
```
```  1231   assumes [measurable]: "f \<in> borel_measurable M"
```
```  1232   shows "(\<lambda>x. f x i) \<in> borel_measurable M"
```
```  1233 proof -
```
```  1234   have "(\<lambda>x. f x i) = (\<lambda>y. y i) o f"
```
```  1235     unfolding comp_def by auto
```
```  1236   then show ?thesis by simp
```
```  1237 qed
```
```  1238
```
```  1239 text \<open>To compare the Borel sigma algebra with the product sigma algebra, we give a presentation
```
```  1240 of the product sigma algebra that is more similar to the one we used above for the product
```
```  1241 topology.\<close>
```
```  1242
```
```  1243 lemma sets_PiM_finite:
```
```  1244   "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i))
```
```  1245         {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
```
```  1246 proof
```
```  1247   have "{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}} \<subseteq> sets (Pi\<^sub>M I M)"
```
```  1248   proof (auto)
```
```  1249     fix X assume H: "\<forall>i. X i \<in> sets (M i)" "finite {i. X i \<noteq> space (M i)}"
```
```  1250     then have *: "X i \<in> sets (M i)" for i by simp
```
```  1251     define J where "J = {i \<in> I. X i \<noteq> space (M i)}"
```
```  1252     have "finite J" "J \<subseteq> I" unfolding J_def using H by auto
```
```  1253     define Y where "Y = (\<Pi>\<^sub>E j\<in>J. X j)"
```
```  1254     have "prod_emb I M J Y \<in> sets (Pi\<^sub>M I M)"
```
```  1255       unfolding Y_def apply (rule sets_PiM_I) using \<open>finite J\<close> \<open>J \<subseteq> I\<close> * by auto
```
```  1256     moreover have "prod_emb I M J Y = (\<Pi>\<^sub>E i\<in>I. X i)"
```
```  1257       unfolding prod_emb_def Y_def J_def using H sets.sets_into_space[OF *]
```
```  1258       by (auto simp add: PiE_iff, blast)
```
```  1259     ultimately show "Pi\<^sub>E I X \<in> sets (Pi\<^sub>M I M)" by simp
```
```  1260   qed
```
```  1261   then show "sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}
```
```  1262               \<subseteq> sets (Pi\<^sub>M I M)"
```
```  1263     by (metis (mono_tags, lifting) sets.sigma_sets_subset' sets.top space_PiM)
```
```  1264
```
```  1265   have *: "\<exists>X. {f. (\<forall>i\<in>I. f i \<in> space (M i)) \<and> f \<in> extensional I \<and> f i \<in> A} = Pi\<^sub>E I X \<and>
```
```  1266                 (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}"
```
```  1267     if "i \<in> I" "A \<in> sets (M i)" for i A
```
```  1268   proof -
```
```  1269     define X where "X = (\<lambda>j. if j = i then A else space (M j))"
```
```  1270     have "{f. (\<forall>i\<in>I. f i \<in> space (M i)) \<and> f \<in> extensional I \<and> f i \<in> A} = Pi\<^sub>E I X"
```
```  1271       unfolding X_def using sets.sets_into_space[OF \<open>A \<in> sets (M i)\<close>] \<open>i \<in> I\<close>
```
```  1272       by (auto simp add: PiE_iff extensional_def, metis subsetCE, metis)
```
```  1273     moreover have "X j \<in> sets (M j)" for j
```
```  1274       unfolding X_def using \<open>A \<in> sets (M i)\<close> by auto
```
```  1275     moreover have "finite {j. X j \<noteq> space (M j)}"
```
```  1276       unfolding X_def by simp
```
```  1277     ultimately show ?thesis by auto
```
```  1278   qed
```
```  1279   show "sets (Pi\<^sub>M I M) \<subseteq> sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
```
```  1280     unfolding sets_PiM_single
```
```  1281     apply (rule sigma_sets_mono')
```
```  1282     apply (auto simp add: PiE_iff *)
```
```  1283     done
```
```  1284 qed
```
```  1285
```
```  1286 lemma sets_PiM_subset_borel:
```
```  1287   "sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) \<subseteq> sets borel"
```
```  1288 proof -
```
```  1289   have *: "Pi\<^sub>E UNIV X \<in> sets borel" if [measurable]: "\<And>i. X i \<in> sets borel" "finite {i. X i \<noteq> UNIV}" for X::"'a \<Rightarrow> 'b set"
```
```  1290   proof -
```
```  1291     define I where "I = {i. X i \<noteq> UNIV}"
```
```  1292     have "finite I" unfolding I_def using that by simp
```
```  1293     have "Pi\<^sub>E UNIV X = (\<Inter>i\<in>I. (\<lambda>x. x i)-`(X i) \<inter> space borel) \<inter> space borel"
```
```  1294       unfolding I_def by auto
```
```  1295     also have "... \<in> sets borel"
```
```  1296       using that \<open>finite I\<close> by measurable
```
```  1297     finally show ?thesis by simp
```
```  1298   qed
```
```  1299   then have "{(\<Pi>\<^sub>E i\<in>UNIV. X i) |X::('a \<Rightarrow> 'b set). (\<forall>i. X i \<in> sets borel) \<and> finite {i. X i \<noteq> space borel}} \<subseteq> sets borel"
```
```  1300     by auto
```
```  1301   then show ?thesis unfolding sets_PiM_finite space_borel
```
```  1302     by (simp add: * sets.sigma_sets_subset')
```
```  1303 qed
```
```  1304
```
```  1305 proposition sets_PiM_equal_borel:
```
```  1306   "sets (Pi\<^sub>M UNIV (\<lambda>i::('a::countable). borel::('b::second_countable_topology measure))) = sets borel"
```
```  1307 proof
```
```  1308   obtain K::"('a \<Rightarrow> 'b) set set" where K: "topological_basis K" "countable K"
```
```  1309             "\<And>k. k \<in> K \<Longrightarrow> \<exists>X. (k = Pi\<^sub>E UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
```
```  1310     using product_topology_countable_basis by fast
```
```  1311   have *: "k \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "k \<in> K" for k
```
```  1312   proof -
```
```  1313     obtain X where H: "k = Pi\<^sub>E UNIV X" "\<And>i. open (X i)" "finite {i. X i \<noteq> UNIV}"
```
```  1314       using K(3)[OF \<open>k \<in> K\<close>] by blast
```
```  1315     show ?thesis unfolding H(1) sets_PiM_finite space_borel using borel_open[OF H(2)] H(3) by auto
```
```  1316   qed
```
```  1317   have **: "U \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "open U" for U::"('a \<Rightarrow> 'b) set"
```
```  1318   proof -
```
```  1319     obtain B where "B \<subseteq> K" "U = (\<Union>B)"
```
```  1320       using \<open>open U\<close> \<open>topological_basis K\<close> by (metis topological_basis_def)
```
```  1321     have "countable B" using \<open>B \<subseteq> K\<close> \<open>countable K\<close> countable_subset by blast
```
```  1322     moreover have "k \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "k \<in> B" for k
```
```  1323       using \<open>B \<subseteq> K\<close> * that by auto
```
```  1324     ultimately show ?thesis unfolding \<open>U = (\<Union>B)\<close> by auto
```
```  1325   qed
```
```  1326   have "sigma_sets UNIV (Collect open) \<subseteq> sets (Pi\<^sub>M UNIV (\<lambda>i::'a. (borel::('b measure))))"
```
```  1327     apply (rule sets.sigma_sets_subset') using ** by auto
```
```  1328   then show "sets (borel::('a \<Rightarrow> 'b) measure) \<subseteq> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))"
```
```  1329     unfolding borel_def by auto
```
```  1330 qed (simp add: sets_PiM_subset_borel)
```
```  1331
```
```  1332 lemma measurable_coordinatewise_then_product:
```
```  1333   fixes f::"'a \<Rightarrow> ('b::countable) \<Rightarrow> ('c::second_countable_topology)"
```
```  1334   assumes [measurable]: "\<And>i. (\<lambda>x. f x i) \<in> borel_measurable M"
```
```  1335   shows "f \<in> borel_measurable M"
```
```  1336 proof -
```
```  1337   have "f \<in> measurable M (Pi\<^sub>M UNIV (\<lambda>_. borel))"
```
```  1338     by (rule measurable_PiM_single', auto simp add: assms)
```
```  1339   then show ?thesis using sets_PiM_equal_borel measurable_cong_sets by blast
```
```  1340 qed
```
```  1341
```
```  1342
```
```  1343 end
```