src/HOL/Probability/Finite_Product_Measure.thy
 author wenzelm Thu Jun 25 23:33:47 2015 +0200 (2015-06-25) changeset 60580 7e741e22d7fc parent 59425 c5e79df8cc21 child 61166 5976fe402824 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Probability/Finite_Product_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 section {*Finite product measures*}
```
```     6
```
```     7 theory Finite_Product_Measure
```
```     8 imports Binary_Product_Measure
```
```     9 begin
```
```    10
```
```    11 lemma PiE_choice: "(\<exists>f\<in>PiE 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 split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
```
```    16   by auto
```
```    17
```
```    18 subsubsection {* More about Function restricted by @{const extensional}  *}
```
```    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> PiE (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 `J \<subseteq> I` 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 `J \<subseteq> I` 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 {* Finite product spaces *}
```
```   113
```
```   114 subsubsection {* Products *}
```
```   115
```
```   116 definition prod_emb where
```
```   117   "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
```
```   118
```
```   119 lemma prod_emb_iff:
```
```   120   "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))"
```
```   121   unfolding prod_emb_def PiE_def by auto
```
```   122
```
```   123 lemma
```
```   124   shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
```
```   125     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"
```
```   126     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"
```
```   127     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))"
```
```   128     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))"
```
```   129     and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
```
```   130   by (auto simp: prod_emb_def)
```
```   131
```
```   132 lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
```
```   133     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))"
```
```   134   by (force simp: prod_emb_def PiE_iff split_if_mem2)
```
```   135
```
```   136 lemma prod_emb_PiE_same_index[simp]:
```
```   137     "(\<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"
```
```   138   by (auto simp: prod_emb_def PiE_iff)
```
```   139
```
```   140 lemma prod_emb_trans[simp]:
```
```   141   "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"
```
```   142   by (auto simp add: Int_absorb1 prod_emb_def PiE_def)
```
```   143
```
```   144 lemma prod_emb_Pi:
```
```   145   assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
```
```   146   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))"
```
```   147   using assms sets.space_closed
```
```   148   by (auto simp: prod_emb_def PiE_iff split: split_if_asm) blast+
```
```   149
```
```   150 lemma prod_emb_id:
```
```   151   "B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
```
```   152   by (auto simp: prod_emb_def subset_eq extensional_restrict)
```
```   153
```
```   154 lemma prod_emb_mono:
```
```   155   "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
```
```   156   by (auto simp: prod_emb_def)
```
```   157
```
```   158 definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
```
```   159   "PiM I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i))
```
```   160     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
```
```   161     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j))
```
```   162     (\<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)))"
```
```   163
```
```   164 definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
```
```   165   "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) `
```
```   166     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   167
```
```   168 abbreviation
```
```   169   "Pi\<^sub>M I M \<equiv> PiM I M"
```
```   170
```
```   171 syntax
```
```   172   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
```
```   173
```
```   174 syntax (xsymbols)
```
```   175   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^sub>M _\<in>_./ _)"  10)
```
```   176
```
```   177 syntax (HTML output)
```
```   178   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^sub>M _\<in>_./ _)"  10)
```
```   179
```
```   180 translations
```
```   181   "PIM x:I. M" == "CONST PiM I (%x. M)"
```
```   182
```
```   183 lemma extend_measure_cong:
```
```   184   assumes "\<Omega> = \<Omega>'" "I = I'" "G = G'" "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i"
```
```   185   shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'"
```
```   186   unfolding extend_measure_def by (auto simp add: assms)
```
```   187
```
```   188 lemma Pi_cong_sets:
```
```   189     "\<lbrakk>I = J; \<And>x. x \<in> I \<Longrightarrow> M x = N x\<rbrakk> \<Longrightarrow> Pi I M = Pi J N"
```
```   190   unfolding Pi_def by auto
```
```   191
```
```   192 lemma PiM_cong:
```
```   193   assumes "I = J" "\<And>x. x \<in> I \<Longrightarrow> M x = N x"
```
```   194   shows "PiM I M = PiM J N"
```
```   195   unfolding PiM_def
```
```   196 proof (rule extend_measure_cong, goals)
```
```   197   case 1
```
```   198   show ?case using assms
```
```   199     by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all
```
```   200 next
```
```   201   case 2
```
```   202   have "\<And>K. K \<subseteq> J \<Longrightarrow> (\<Pi> j\<in>K. sets (M j)) = (\<Pi> j\<in>K. sets (N j))"
```
```   203     using assms by (intro Pi_cong_sets) auto
```
```   204   thus ?case by (auto simp: assms)
```
```   205 next
```
```   206   case 3
```
```   207   show ?case using assms
```
```   208     by (intro ext) (auto simp: prod_emb_def dest: PiE_mem)
```
```   209 next
```
```   210   case (4 x)
```
```   211   thus ?case using assms
```
```   212     by (auto intro!: setprod.cong split: split_if_asm)
```
```   213 qed
```
```   214
```
```   215
```
```   216 lemma prod_algebra_sets_into_space:
```
```   217   "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   218   by (auto simp: prod_emb_def prod_algebra_def)
```
```   219
```
```   220 lemma prod_algebra_eq_finite:
```
```   221   assumes I: "finite I"
```
```   222   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")
```
```   223 proof (intro iffI set_eqI)
```
```   224   fix A assume "A \<in> ?L"
```
```   225   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)"
```
```   226     and A: "A = prod_emb I M J (PIE j:J. E j)"
```
```   227     by (auto simp: prod_algebra_def)
```
```   228   let ?A = "\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i)"
```
```   229   have A: "A = ?A"
```
```   230     unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto
```
```   231   show "A \<in> ?R" unfolding A using J sets.top
```
```   232     by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
```
```   233 next
```
```   234   fix A assume "A \<in> ?R"
```
```   235   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
```
```   236   then have A: "A = prod_emb I M I (\<Pi>\<^sub>E i\<in>I. X i)"
```
```   237     by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff)
```
```   238   from X I show "A \<in> ?L" unfolding A
```
```   239     by (auto simp: prod_algebra_def)
```
```   240 qed
```
```   241
```
```   242 lemma prod_algebraI:
```
```   243   "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
```
```   244     \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
```
```   245   by (auto simp: prod_algebra_def)
```
```   246
```
```   247 lemma prod_algebraI_finite:
```
```   248   "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M"
```
```   249   using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp
```
```   250
```
```   251 lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
```
```   252 proof (safe intro!: Int_stableI)
```
```   253   fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
```
```   254   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))"
```
```   255     by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
```
```   256 qed
```
```   257
```
```   258 lemma prod_algebraE:
```
```   259   assumes A: "A \<in> prod_algebra I M"
```
```   260   obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
```
```   261     "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
```
```   262   using A by (auto simp: prod_algebra_def)
```
```   263
```
```   264 lemma prod_algebraE_all:
```
```   265   assumes A: "A \<in> prod_algebra I M"
```
```   266   obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   267 proof -
```
```   268   from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)"
```
```   269     and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```   270     by (auto simp: prod_algebra_def)
```
```   271   from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
```
```   272     using sets.sets_into_space by auto
```
```   273   then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))"
```
```   274     using A J by (auto simp: prod_emb_PiE)
```
```   275   moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   276     using sets.top E by auto
```
```   277   ultimately show ?thesis using that by auto
```
```   278 qed
```
```   279
```
```   280 lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
```
```   281 proof (unfold Int_stable_def, safe)
```
```   282   fix A assume "A \<in> prod_algebra I M"
```
```   283   from prod_algebraE[OF this] guess J E . note A = this
```
```   284   fix B assume "B \<in> prod_algebra I M"
```
```   285   from prod_algebraE[OF this] guess K F . note B = this
```
```   286   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>
```
```   287       (if i \<in> K then F i else space (M i)))"
```
```   288     unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4)
```
```   289       B(5)[THEN sets.sets_into_space]
```
```   290     apply (subst (1 2 3) prod_emb_PiE)
```
```   291     apply (simp_all add: subset_eq PiE_Int)
```
```   292     apply blast
```
```   293     apply (intro PiE_cong)
```
```   294     apply auto
```
```   295     done
```
```   296   also have "\<dots> \<in> prod_algebra I M"
```
```   297     using A B by (auto intro!: prod_algebraI)
```
```   298   finally show "A \<inter> B \<in> prod_algebra I M" .
```
```   299 qed
```
```   300
```
```   301 lemma prod_algebra_mono:
```
```   302   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
```
```   303   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
```
```   304   shows "prod_algebra I E \<subseteq> prod_algebra I F"
```
```   305 proof
```
```   306   fix A assume "A \<in> prod_algebra I E"
```
```   307   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
```
```   308     and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   309     and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
```
```   310     by (auto simp: prod_algebra_def)
```
```   311   moreover
```
```   312   from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
```
```   313     by (rule PiE_cong)
```
```   314   with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   315     by (simp add: prod_emb_def)
```
```   316   moreover
```
```   317   from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
```
```   318     by auto
```
```   319   ultimately show "A \<in> prod_algebra I F"
```
```   320     apply (simp add: prod_algebra_def image_iff)
```
```   321     apply (intro exI[of _ J] exI[of _ G] conjI)
```
```   322     apply auto
```
```   323     done
```
```   324 qed
```
```   325
```
```   326 lemma prod_algebra_cong:
```
```   327   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
```
```   328   shows "prod_algebra I M = prod_algebra J N"
```
```   329 proof -
```
```   330   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
```
```   331     using sets_eq_imp_space_eq[OF sets] by auto
```
```   332   with sets show ?thesis unfolding `I = J`
```
```   333     by (intro antisym prod_algebra_mono) auto
```
```   334 qed
```
```   335
```
```   336 lemma space_in_prod_algebra:
```
```   337   "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
```
```   338 proof cases
```
```   339   assume "I = {}" then show ?thesis
```
```   340     by (auto simp add: prod_algebra_def image_iff prod_emb_def)
```
```   341 next
```
```   342   assume "I \<noteq> {}"
```
```   343   then obtain i where "i \<in> I" by auto
```
```   344   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))"
```
```   345     by (auto simp: prod_emb_def)
```
```   346   also have "\<dots> \<in> prod_algebra I M"
```
```   347     using `i \<in> I` by (intro prod_algebraI) auto
```
```   348   finally show ?thesis .
```
```   349 qed
```
```   350
```
```   351 lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   352   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
```
```   353
```
```   354 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)"
```
```   355   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
```
```   356
```
```   357 lemma sets_PiM_single: "sets (PiM I M) =
```
```   358     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)}"
```
```   359     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   360   unfolding sets_PiM
```
```   361 proof (rule sigma_sets_eqI)
```
```   362   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   363   fix A assume "A \<in> prod_algebra I M"
```
```   364   from prod_algebraE[OF this] guess J X . note X = this
```
```   365   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   366   proof cases
```
```   367     assume "I = {}"
```
```   368     with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
```
```   369     with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
```
```   370   next
```
```   371     assume "I \<noteq> {}"
```
```   372     with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
```
```   373       by (auto simp: prod_emb_def)
```
```   374     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```   375       using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
```
```   376     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```   377   qed
```
```   378 next
```
```   379   fix A assume "A \<in> ?R"
```
```   380   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)"
```
```   381     by auto
```
```   382   then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
```
```   383      by (auto simp: prod_emb_def)
```
```   384   also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
```
```   385     using A by (intro sigma_sets.Basic prod_algebraI) auto
```
```   386   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
```
```   387 qed
```
```   388
```
```   389 lemma sets_PiM_eq_proj:
```
```   390   "I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
```
```   391   apply (simp add: sets_PiM_single sets_Sup_sigma)
```
```   392   apply (subst SUP_cong[OF refl])
```
```   393   apply (rule sets_vimage_algebra2)
```
```   394   apply auto []
```
```   395   apply (auto intro!: arg_cong2[where f=sigma_sets])
```
```   396   done
```
```   397
```
```   398 lemma
```
```   399   shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
```
```   400     and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
```
```   401   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
```
```   402
```
```   403 lemma sets_PiM_sigma:
```
```   404   assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S"
```
```   405   assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)"
```
```   406   assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I"
```
```   407   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}"
```
```   408   shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
```
```   409 proof cases
```
```   410   assume "I = {}"
```
```   411   with `\<Union>J = I` have "P = {{\<lambda>_. undefined}} \<or> P = {}"
```
```   412     by (auto simp: P_def)
```
```   413   with `I = {}` show ?thesis
```
```   414     by (auto simp add: sets_PiM_empty sigma_sets_empty_eq)
```
```   415 next
```
```   416   let ?F = "\<lambda>i. {(\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega> |A. A \<in> E i}"
```
```   417   assume "I \<noteq> {}"
```
```   418   then have "sets (Pi\<^sub>M I (\<lambda>i. sigma (\<Omega> i) (E i))) =
```
```   419       sets (\<Squnion>\<^sub>\<sigma> i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<lambda>x. x i) (sigma (\<Omega> i) (E i)))"
```
```   420     by (subst sets_PiM_eq_proj) (auto simp: space_measure_of_conv)
```
```   421   also have "\<dots> = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. sigma (Pi\<^sub>E I \<Omega>) (?F i))"
```
```   422     using E by (intro SUP_sigma_cong arg_cong[where f=sets] vimage_algebra_sigma) auto
```
```   423   also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i))"
```
```   424     using `I \<noteq> {}` by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto
```
```   425   also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) P)"
```
```   426   proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
```
```   427     show "(\<Union>i\<in>I. ?F i) \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" "P \<subseteq> Pow (Pi\<^sub>E I \<Omega>)"
```
```   428       by (auto simp: P_def)
```
```   429   next
```
```   430     interpret P: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
```
```   431       by (auto intro!: sigma_algebra_sigma_sets simp: P_def)
```
```   432
```
```   433     fix Z assume "Z \<in> (\<Union>i\<in>I. ?F i)"
```
```   434     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>"
```
```   435       by auto
```
```   436     from `i \<in> I` J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j"
```
```   437       by auto
```
```   438     obtain S where S: "\<And>i. i \<in> j \<Longrightarrow> S i \<subseteq> E i" "\<And>i. i \<in> j \<Longrightarrow> countable (S i)"
```
```   439       "\<And>i. i \<in> j \<Longrightarrow> \<Omega> i = \<Union>(S i)"
```
```   440       by (metis subset_eq \<Omega>_cover `j \<subseteq> I`)
```
```   441     def A' \<equiv> "\<lambda>n. n(i := A)"
```
```   442     then have A'_i: "\<And>n. A' n i = A"
```
```   443       by simp
```
```   444     { fix n assume "n \<in> Pi\<^sub>E (j - {i}) S"
```
```   445       then have "A' n \<in> Pi j E"
```
```   446         unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def `A \<in> E i` )
```
```   447       with `j \<in> J` have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P"
```
```   448         by (auto simp: P_def) }
```
```   449     note A'_in_P = this
```
```   450
```
```   451     { fix x assume "x i \<in> A" "x \<in> Pi\<^sub>E I \<Omega>"
```
```   452       with S(3) `j \<subseteq> I` have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s"
```
```   453         by (auto simp: PiE_def Pi_def)
```
```   454       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"
```
```   455         by metis
```
```   456       with `x i \<in> A` have "\<exists>n\<in>PiE (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i"
```
```   457         by (intro bexI[of _ "restrict (s(i := A)) (j-{i})"]) (auto simp: A'_def split: if_splits) }
```
```   458     then have "Z = (\<Union>n\<in>PiE (j-{i}) S. {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A' n i})"
```
```   459       unfolding Z_def
```
```   460       by (auto simp add: set_eq_iff ball_conj_distrib `i\<in>j` A'_i dest: bspec[OF _ `i\<in>j`]
```
```   461                cong: conj_cong)
```
```   462     also have "\<dots> \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
```
```   463       using `finite j` S(2)
```
```   464       by (intro P.countable_UN' countable_PiE) (simp_all add: image_subset_iff A'_in_P)
```
```   465     finally show "Z \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" .
```
```   466   next
```
```   467     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)"
```
```   468       by (auto intro!: sigma_algebra_sigma_sets)
```
```   469
```
```   470     fix b assume "b \<in> P"
```
```   471     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"
```
```   472       by (auto simp: P_def)
```
```   473     show "b \<in> sigma_sets (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i)"
```
```   474     proof cases
```
```   475       assume "j = {}"
```
```   476       with b have "b = (\<Pi>\<^sub>E i\<in>I. \<Omega> i)"
```
```   477         by auto
```
```   478       then show ?thesis
```
```   479         by blast
```
```   480     next
```
```   481       assume "j \<noteq> {}"
```
```   482       with J b(2,3) have eq: "b = (\<Inter>i\<in>j. ((\<lambda>x. x i) -` A i \<inter> Pi\<^sub>E I \<Omega>))"
```
```   483         unfolding b(1)
```
```   484         by (auto simp: PiE_def Pi_def)
```
```   485       show ?thesis
```
```   486         unfolding eq using `A \<in> Pi j E` `j \<in> J` J(2)
```
```   487         by (intro F.finite_INT J `j \<in> J` `j \<noteq> {}` sigma_sets.Basic) blast
```
```   488     qed
```
```   489   qed
```
```   490   finally show "?thesis" .
```
```   491 qed
```
```   492
```
```   493 lemma sets_PiM_in_sets:
```
```   494   assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   495   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"
```
```   496   shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N"
```
```   497   unfolding sets_PiM_single space[symmetric]
```
```   498   by (intro sets.sigma_sets_subset subsetI) (auto intro: sets)
```
```   499
```
```   500 lemma sets_PiM_cong[measurable_cong]:
```
```   501   assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)"
```
```   502   using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong)
```
```   503
```
```   504 lemma sets_PiM_I:
```
```   505   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   506   shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
```
```   507 proof cases
```
```   508   assume "J = {}"
```
```   509   then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
```
```   510     by (auto simp: prod_emb_def)
```
```   511   then show ?thesis
```
```   512     by (auto simp add: sets_PiM intro!: sigma_sets_top)
```
```   513 next
```
```   514   assume "J \<noteq> {}" with assms show ?thesis
```
```   515     by (force simp add: sets_PiM prod_algebra_def)
```
```   516 qed
```
```   517
```
```   518 lemma measurable_PiM:
```
```   519   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   520   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>
```
```   521     f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
```
```   522   shows "f \<in> measurable N (PiM I M)"
```
```   523   using sets_PiM prod_algebra_sets_into_space space
```
```   524 proof (rule measurable_sigma_sets)
```
```   525   fix A assume "A \<in> prod_algebra I M"
```
```   526   from prod_algebraE[OF this] guess J X .
```
```   527   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
```
```   528 qed
```
```   529
```
```   530 lemma measurable_PiM_Collect:
```
```   531   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   532   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>
```
```   533     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
```
```   534   shows "f \<in> measurable N (PiM I M)"
```
```   535   using sets_PiM prod_algebra_sets_into_space space
```
```   536 proof (rule measurable_sigma_sets)
```
```   537   fix A assume "A \<in> prod_algebra I M"
```
```   538   from prod_algebraE[OF this] guess J X . note X = this
```
```   539   then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
```
```   540     using space by (auto simp: prod_emb_def del: PiE_I)
```
```   541   also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
```
```   542   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   543 qed
```
```   544
```
```   545 lemma measurable_PiM_single:
```
```   546   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   547   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"
```
```   548   shows "f \<in> measurable N (PiM I M)"
```
```   549   using sets_PiM_single
```
```   550 proof (rule measurable_sigma_sets)
```
```   551   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)}"
```
```   552   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)"
```
```   553     by auto
```
```   554   with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
```
```   555   also have "\<dots> \<in> sets N" using B by (rule sets)
```
```   556   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   557 qed (auto simp: space)
```
```   558
```
```   559 lemma measurable_PiM_single':
```
```   560   assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
```
```   561     and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   562   shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
```
```   563 proof (rule measurable_PiM_single)
```
```   564   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   565   then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
```
```   566     by auto
```
```   567   then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
```
```   568     using A f by (auto intro!: measurable_sets)
```
```   569 qed fact
```
```   570
```
```   571 lemma sets_PiM_I_finite[measurable]:
```
```   572   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
```
```   573   shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
```
```   574   using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto
```
```   575
```
```   576 lemma measurable_component_singleton[measurable (raw)]:
```
```   577   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
```
```   578 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   579   fix A assume "A \<in> sets (M i)"
```
```   580   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)"
```
```   581     using sets.sets_into_space `i \<in> I`
```
```   582     by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
```
```   583   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
```
```   584     using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
```
```   585 qed (insert `i \<in> I`, auto simp: space_PiM)
```
```   586
```
```   587 lemma measurable_component_singleton'[measurable_dest]:
```
```   588   assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
```
```   589   assumes g: "g \<in> measurable L N"
```
```   590   assumes i: "i \<in> I"
```
```   591   shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)"
```
```   592   using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] .
```
```   593
```
```   594 lemma measurable_PiM_component_rev:
```
```   595   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
```
```   596   by simp
```
```   597
```
```   598 lemma measurable_case_nat[measurable (raw)]:
```
```   599   assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
```
```   600     "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
```
```   601   shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
```
```   602   by (cases i) simp_all
```
```   603
```
```   604 lemma measurable_case_nat'[measurable (raw)]:
```
```   605   assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   606   shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   607   using fg[THEN measurable_space]
```
```   608   by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
```
```   609
```
```   610 lemma measurable_add_dim[measurable]:
```
```   611   "(\<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)"
```
```   612     (is "?f \<in> measurable ?P ?I")
```
```   613 proof (rule measurable_PiM_single)
```
```   614   fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
```
```   615   have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
```
```   616     (if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
```
```   617     using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
```
```   618   also have "\<dots> \<in> sets ?P"
```
```   619     using A j
```
```   620     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   621   finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
```
```   622 qed (auto simp: space_pair_measure space_PiM PiE_def)
```
```   623
```
```   624 lemma measurable_component_update:
```
```   625   "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)"
```
```   626   by simp
```
```   627
```
```   628 lemma measurable_merge[measurable]:
```
```   629   "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)"
```
```   630     (is "?f \<in> measurable ?P ?U")
```
```   631 proof (rule measurable_PiM_single)
```
```   632   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
```
```   633   then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
```
```   634     (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)"
```
```   635     by (auto simp: merge_def)
```
```   636   also have "\<dots> \<in> sets ?P"
```
```   637     using A
```
```   638     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   639   finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
```
```   640 qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
```
```   641
```
```   642 lemma measurable_restrict[measurable (raw)]:
```
```   643   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
```
```   644   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
```
```   645 proof (rule measurable_PiM_single)
```
```   646   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   647   then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
```
```   648     by auto
```
```   649   then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
```
```   650     using A X by (auto intro!: measurable_sets)
```
```   651 qed (insert X, auto simp add: PiE_def dest: measurable_space)
```
```   652
```
```   653 lemma measurable_abs_UNIV:
```
```   654   "(\<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)"
```
```   655   by (intro measurable_PiM_single) (auto dest: measurable_space)
```
```   656
```
```   657 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)"
```
```   658   by (intro measurable_restrict measurable_component_singleton) auto
```
```   659
```
```   660 lemma measurable_restrict_subset':
```
```   661   assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)"
```
```   662   shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
```
```   663 proof-
```
```   664   from assms(1) have "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
```
```   665     by (rule measurable_restrict_subset)
```
```   666   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)"
```
```   667     by (intro sets_PiM_cong measurable_cong_sets) simp_all
```
```   668   finally show ?thesis .
```
```   669 qed
```
```   670
```
```   671 lemma measurable_prod_emb[intro, simp]:
```
```   672   "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)"
```
```   673   unfolding prod_emb_def space_PiM[symmetric]
```
```   674   by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
```
```   675
```
```   676 lemma sets_in_Pi_aux:
```
```   677   "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   678   {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
```
```   679   by (simp add: subset_eq Pi_iff)
```
```   680
```
```   681 lemma sets_in_Pi[measurable (raw)]:
```
```   682   "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
```
```   683   (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   684   Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
```
```   685   unfolding pred_def
```
```   686   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
```
```   687
```
```   688 lemma sets_in_extensional_aux:
```
```   689   "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
```
```   690 proof -
```
```   691   have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
```
```   692     by (auto simp add: extensional_def space_PiM)
```
```   693   then show ?thesis by simp
```
```   694 qed
```
```   695
```
```   696 lemma sets_in_extensional[measurable (raw)]:
```
```   697   "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
```
```   698   unfolding pred_def
```
```   699   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
```
```   700
```
```   701 locale product_sigma_finite =
```
```   702   fixes M :: "'i \<Rightarrow> 'a measure"
```
```   703   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
```
```   704
```
```   705 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
```
```   706   by (rule sigma_finite_measures)
```
```   707
```
```   708 locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   709   fixes I :: "'i set"
```
```   710   assumes finite_index: "finite I"
```
```   711
```
```   712 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
```
```   713   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
```
```   714     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
```
```   715     (\<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>
```
```   716     (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
```
```   717 proof -
```
```   718   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>)"
```
```   719     using M.sigma_finite_incseq by metis
```
```   720   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   721   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>"
```
```   722     by auto
```
```   723   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
```
```   724   note space_PiM[simp]
```
```   725   show ?thesis
```
```   726   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
```
```   727     fix i show "range (F i) \<subseteq> sets (M i)" by fact
```
```   728   next
```
```   729     fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
```
```   730   next
```
```   731     fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
```
```   732       by (auto simp: PiE_def dest!: sets.sets_into_space)
```
```   733   next
```
```   734     fix f assume "f \<in> space (PiM I M)"
```
```   735     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
```
```   736     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
```
```   737   next
```
```   738     fix i show "?F i \<subseteq> ?F (Suc i)"
```
```   739       using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
```
```   740   qed
```
```   741 qed
```
```   742
```
```   743 lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
```
```   744 proof -
```
```   745   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
```
```   746   have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
```
```   747   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   748     show "positive (PiM {} M) ?\<mu>"
```
```   749       by (auto simp: positive_def)
```
```   750     show "countably_additive (PiM {} M) ?\<mu>"
```
```   751       by (rule sets.countably_additiveI_finite)
```
```   752          (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
```
```   753   qed (auto simp: prod_emb_def)
```
```   754   also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
```
```   755     by (auto simp: prod_emb_def)
```
```   756   finally show ?thesis
```
```   757     by simp
```
```   758 qed
```
```   759
```
```   760 lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
```
```   761   by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
```
```   762
```
```   763 lemma (in product_sigma_finite) emeasure_PiM:
```
```   764   "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))"
```
```   765 proof (induct I arbitrary: A rule: finite_induct)
```
```   766   case (insert i I)
```
```   767   interpret finite_product_sigma_finite M I by default fact
```
```   768   have "finite (insert i I)" using `finite I` by auto
```
```   769   interpret I': finite_product_sigma_finite M "insert i I" by default fact
```
```   770   let ?h = "(\<lambda>(f, y). f(i := y))"
```
```   771
```
```   772   let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
```
```   773   let ?\<mu> = "emeasure ?P"
```
```   774   let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
```
```   775   let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
```
```   776
```
```   777   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)) =
```
```   778     (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
```
```   779   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   780     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))"
```
```   781     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
```
```   782     let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
```
```   783     let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
```
```   784     have "?\<mu> ?p =
```
```   785       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))"
```
```   786       by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
```
```   787     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))"
```
```   788       using J E[rule_format, THEN sets.sets_into_space]
```
```   789       by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
```
```   790     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))) =
```
```   791       emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
```
```   792       using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
```
```   793     also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
```
```   794       using J E[rule_format, THEN sets.sets_into_space]
```
```   795       by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
```
```   796     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)) =
```
```   797       (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
```
```   798       using E by (subst insert) (auto intro!: setprod.cong)
```
```   799     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
```
```   800        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
```
```   801       using insert by (auto simp: mult.commute intro!: arg_cong2[where f="op *"] setprod.cong)
```
```   802     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
```
```   803       using insert(1,2) J E by (intro setprod.mono_neutral_right) auto
```
```   804     finally show "?\<mu> ?p = \<dots>" .
```
```   805
```
```   806     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))"
```
```   807       using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
```
```   808   next
```
```   809     show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
```
```   810       using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
```
```   811   next
```
```   812     show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
```
```   813       insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
```
```   814       using insert by auto
```
```   815   qed (auto intro!: setprod.cong)
```
```   816   with insert show ?case
```
```   817     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
```
```   818 qed simp
```
```   819
```
```   820 lemma (in product_sigma_finite) sigma_finite:
```
```   821   assumes "finite I"
```
```   822   shows "sigma_finite_measure (PiM I M)"
```
```   823 proof
```
```   824   interpret finite_product_sigma_finite M I by default fact
```
```   825
```
```   826   obtain F where F: "\<And>j. countable (F j)" "\<And>j f. f \<in> F j \<Longrightarrow> f \<in> sets (M j)"
```
```   827     "\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and
```
```   828     in_space: "\<And>j. space (M j) = (\<Union>F j)"
```
```   829     using sigma_finite_countable by (metis subset_eq)
```
```   830   moreover have "(\<Union>(PiE I ` PiE I F)) = space (Pi\<^sub>M I M)"
```
```   831     using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD2])
```
```   832   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>)"
```
```   833     by (intro exI[of _ "PiE I ` PiE I F"])
```
```   834        (auto intro!: countable_PiE sets_PiM_I_finite
```
```   835              simp: PiE_iff emeasure_PiM finite_index setprod_PInf emeasure_nonneg)
```
```   836 qed
```
```   837
```
```   838 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
```
```   839   using sigma_finite[OF finite_index] .
```
```   840
```
```   841 lemma (in finite_product_sigma_finite) measure_times:
```
```   842   "(\<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))"
```
```   843   using emeasure_PiM[OF finite_index] by auto
```
```   844
```
```   845 lemma (in product_sigma_finite) nn_integral_empty:
```
```   846   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
```
```   847   shows "integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
```
```   848 proof -
```
```   849   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
```
```   850   have "\<And>A. emeasure (Pi\<^sub>M {} M) (Pi\<^sub>E {} A) = 1"
```
```   851     using assms by (subst measure_times) auto
```
```   852   then show ?thesis
```
```   853     unfolding nn_integral_def simple_function_def simple_integral_def[abs_def]
```
```   854   proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
```
```   855     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
```
```   856       by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
```
```   857     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
```
```   858       by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
```
```   859   qed
```
```   860 qed
```
```   861
```
```   862 lemma (in product_sigma_finite) distr_merge:
```
```   863   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   864   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"
```
```   865    (is "?D = ?P")
```
```   866 proof -
```
```   867   interpret I: finite_product_sigma_finite M I by default fact
```
```   868   interpret J: finite_product_sigma_finite M J by default fact
```
```   869   have "finite (I \<union> J)" using fin by auto
```
```   870   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   871   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
```
```   872   let ?g = "merge I J"
```
```   873
```
```   874   from IJ.sigma_finite_pairs obtain F where
```
```   875     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
```
```   876        "incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k)"
```
```   877        "(\<Union>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k) = space ?P"
```
```   878        "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
```
```   879     by auto
```
```   880   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k"
```
```   881
```
```   882   show ?thesis
```
```   883   proof (rule measure_eqI_generator_eq[symmetric])
```
```   884     show "Int_stable (prod_algebra (I \<union> J) M)"
```
```   885       by (rule Int_stable_prod_algebra)
```
```   886     show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^sub>E i \<in> I \<union> J. space (M i))"
```
```   887       by (rule prod_algebra_sets_into_space)
```
```   888     show "sets ?P = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   889       by (rule sets_PiM)
```
```   890     then show "sets ?D = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   891       by simp
```
```   892
```
```   893     show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
```
```   894       using fin by (auto simp: prod_algebra_eq_finite)
```
```   895     show "(\<Union>i. \<Pi>\<^sub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i))"
```
```   896       using F(3) by (simp add: space_PiM)
```
```   897   next
```
```   898     fix k
```
```   899     from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
```
```   900     show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
```
```   901   next
```
```   902     fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
```
```   903     with fin obtain F where A_eq: "A = (Pi\<^sub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
```
```   904       by (auto simp add: prod_algebra_eq_finite)
```
```   905     let ?B = "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M"
```
```   906     let ?X = "?g -` A \<inter> space ?B"
```
```   907     have "Pi\<^sub>E I F \<subseteq> space (Pi\<^sub>M I M)" "Pi\<^sub>E J F \<subseteq> space (Pi\<^sub>M J M)"
```
```   908       using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+
```
```   909     then have X: "?X = (Pi\<^sub>E I F \<times> Pi\<^sub>E J F)"
```
```   910       unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
```
```   911     have "emeasure ?D A = emeasure ?B ?X"
```
```   912       using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
```
```   913     also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
```
```   914       using `finite J` `finite I` F unfolding X
```
```   915       by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
```
```   916     also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
```
```   917       using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod.union_inter_neutral)
```
```   918     also have "\<dots> = emeasure ?P (Pi\<^sub>E (I \<union> J) F)"
```
```   919       using `finite J` `finite I` F unfolding A
```
```   920       by (intro IJ.measure_times[symmetric]) auto
```
```   921     finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
```
```   922   qed
```
```   923 qed
```
```   924
```
```   925 lemma (in product_sigma_finite) product_nn_integral_fold:
```
```   926   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
```
```   927   and f: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
```
```   928   shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
```
```   929     (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
```
```   930 proof -
```
```   931   interpret I: finite_product_sigma_finite M I by default fact
```
```   932   interpret J: finite_product_sigma_finite M J by default fact
```
```   933   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
```
```   934   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)"
```
```   935     using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
```
```   936   show ?thesis
```
```   937     apply (subst distr_merge[OF IJ, symmetric])
```
```   938     apply (subst nn_integral_distr[OF measurable_merge f])
```
```   939     apply (subst J.nn_integral_fst[symmetric, OF P_borel])
```
```   940     apply simp
```
```   941     done
```
```   942 qed
```
```   943
```
```   944 lemma (in product_sigma_finite) distr_singleton:
```
```   945   "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
```
```   946 proof (intro measure_eqI[symmetric])
```
```   947   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   948   fix A assume A: "A \<in> sets (M i)"
```
```   949   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
```
```   950     using sets.sets_into_space by (auto simp: space_PiM)
```
```   951   then show "emeasure (M i) A = emeasure ?D A"
```
```   952     using A I.measure_times[of "\<lambda>_. A"]
```
```   953     by (simp add: emeasure_distr measurable_component_singleton)
```
```   954 qed simp
```
```   955
```
```   956 lemma (in product_sigma_finite) product_nn_integral_singleton:
```
```   957   assumes f: "f \<in> borel_measurable (M i)"
```
```   958   shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f"
```
```   959 proof -
```
```   960   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   961   from f show ?thesis
```
```   962     apply (subst distr_singleton[symmetric])
```
```   963     apply (subst nn_integral_distr[OF measurable_component_singleton])
```
```   964     apply simp_all
```
```   965     done
```
```   966 qed
```
```   967
```
```   968 lemma (in product_sigma_finite) product_nn_integral_insert:
```
```   969   assumes I[simp]: "finite I" "i \<notin> I"
```
```   970     and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```   971   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))"
```
```   972 proof -
```
```   973   interpret I: finite_product_sigma_finite M I by default auto
```
```   974   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   975   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
```
```   976     using f by auto
```
```   977   show ?thesis
```
```   978     unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
```
```   979   proof (rule nn_integral_cong, subst product_nn_integral_singleton[symmetric])
```
```   980     fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
```
```   981     let ?f = "\<lambda>y. f (x(i := y))"
```
```   982     show "?f \<in> borel_measurable (M i)"
```
```   983       using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
```
```   984       unfolding comp_def .
```
```   985     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)"
```
```   986       using x
```
```   987       by (auto intro!: nn_integral_cong arg_cong[where f=f]
```
```   988                simp add: space_PiM extensional_def PiE_def)
```
```   989   qed
```
```   990 qed
```
```   991
```
```   992 lemma (in product_sigma_finite) product_nn_integral_insert_rev:
```
```   993   assumes I[simp]: "finite I" "i \<notin> I"
```
```   994     and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```   995   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))"
```
```   996   apply (subst product_nn_integral_insert[OF assms])
```
```   997   apply (rule pair_sigma_finite.Fubini')
```
```   998   apply intro_locales []
```
```   999   apply (rule sigma_finite[OF I(1)])
```
```  1000   apply measurable
```
```  1001   done
```
```  1002
```
```  1003 lemma (in product_sigma_finite) product_nn_integral_setprod:
```
```  1004   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1005   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```  1006   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
```
```  1007   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))"
```
```  1008 using assms proof induct
```
```  1009   case (insert i I)
```
```  1010   note `finite I`[intro, simp]
```
```  1011   interpret I: finite_product_sigma_finite M I by default auto
```
```  1012   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))"
```
```  1013     using insert by (auto intro!: setprod.cong)
```
```  1014   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)"
```
```  1015     using sets.sets_into_space insert
```
```  1016     by (intro borel_measurable_ereal_setprod
```
```  1017               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
```
```  1018        auto
```
```  1019   then show ?case
```
```  1020     apply (simp add: product_nn_integral_insert[OF insert(1,2) prod])
```
```  1021     apply (simp add: insert(2-) * pos borel setprod_ereal_pos nn_integral_multc)
```
```  1022     apply (subst nn_integral_cmult)
```
```  1023     apply (auto simp add: pos borel insert(2-) setprod_ereal_pos nn_integral_nonneg)
```
```  1024     done
```
```  1025 qed (simp add: space_PiM)
```
```  1026
```
```  1027 lemma (in product_sigma_finite) product_nn_integral_pair:
```
```  1028   assumes [measurable]: "split f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
```
```  1029   assumes xy: "x \<noteq> y"
```
```  1030   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))"
```
```  1031 proof-
```
```  1032   interpret psm: pair_sigma_finite "M x" "M y"
```
```  1033     unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all
```
```  1034   have "{x, y} = {y, x}" by auto
```
```  1035   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)"
```
```  1036     using xy by (subst product_nn_integral_insert_rev) simp_all
```
```  1037   also have "... = (\<integral>\<^sup>+y. \<integral>\<^sup>+x. f x y \<partial>M x \<partial>M y)"
```
```  1038     by (intro nn_integral_cong, subst product_nn_integral_singleton) simp_all
```
```  1039   also have "... = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
```
```  1040     by (subst psm.nn_integral_snd[symmetric]) simp_all
```
```  1041   finally show ?thesis .
```
```  1042 qed
```
```  1043
```
```  1044 lemma (in product_sigma_finite) distr_component:
```
```  1045   "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
```
```  1046 proof (intro measure_eqI[symmetric])
```
```  1047   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```  1048
```
```  1049   have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
```
```  1050     by (auto simp: extensional_def restrict_def)
```
```  1051
```
```  1052   have [measurable]: "\<And>j. j \<in> {i} \<Longrightarrow> (\<lambda>x. x) \<in> measurable (M i) (M j)" by simp
```
```  1053
```
```  1054   fix A assume A: "A \<in> sets ?P"
```
```  1055   then have "emeasure ?P A = (\<integral>\<^sup>+x. indicator A x \<partial>?P)"
```
```  1056     by simp
```
```  1057   also have "\<dots> = (\<integral>\<^sup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)"
```
```  1058     by (intro nn_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
```
```  1059   also have "\<dots> = emeasure ?D A"
```
```  1060     using A by (simp add: product_nn_integral_singleton emeasure_distr)
```
```  1061   finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" .
```
```  1062 qed simp
```
```  1063
```
```  1064 lemma (in product_sigma_finite)
```
```  1065   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
```
```  1066   shows emeasure_fold_integral:
```
```  1067     "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)
```
```  1068     and emeasure_fold_measurable:
```
```  1069     "(\<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)
```
```  1070 proof -
```
```  1071   interpret I: finite_product_sigma_finite M I by default fact
```
```  1072   interpret J: finite_product_sigma_finite M J by default fact
```
```  1073   interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
```
```  1074   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)"
```
```  1075     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```  1076
```
```  1077   show ?I
```
```  1078     apply (subst distr_merge[symmetric, OF IJ])
```
```  1079     apply (subst emeasure_distr[OF measurable_merge A])
```
```  1080     apply (subst J.emeasure_pair_measure_alt[OF merge])
```
```  1081     apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```  1082     done
```
```  1083
```
```  1084   show ?B
```
```  1085     using IJ.measurable_emeasure_Pair1[OF merge]
```
```  1086     by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
```
```  1087 qed
```
```  1088
```
```  1089 lemma sets_Collect_single:
```
```  1090   "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)"
```
```  1091   by simp
```
```  1092
```
```  1093 lemma pair_measure_eq_distr_PiM:
```
```  1094   fixes M1 :: "'a measure" and M2 :: "'a measure"
```
```  1095   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
```
```  1096   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))"
```
```  1097     (is "?P = ?D")
```
```  1098 proof (rule pair_measure_eqI[OF assms])
```
```  1099   interpret B: product_sigma_finite "case_bool M1 M2"
```
```  1100     unfolding product_sigma_finite_def using assms by (auto split: bool.split)
```
```  1101   let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"
```
```  1102
```
```  1103   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)"
```
```  1104     by auto
```
```  1105   fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
```
```  1106   have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
```
```  1107     by (simp add: UNIV_bool ac_simps)
```
```  1108   also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
```
```  1109     using A B by (subst B.emeasure_PiM) (auto split: bool.split)
```
```  1110   also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
```
```  1111     using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
```
```  1112     by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
```
```  1113   finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
```
```  1114     using A B
```
```  1115       measurable_component_singleton[of True UNIV "case_bool M1 M2"]
```
```  1116       measurable_component_singleton[of False UNIV "case_bool M1 M2"]
```
```  1117     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
```
```  1118 qed simp
```
```  1119
```
```  1120 end
```