src/HOL/Probability/Finite_Product_Measure.thy
 author hoelzl Thu Jan 22 14:51:08 2015 +0100 (2015-01-22) changeset 59425 c5e79df8cc21 parent 59353 f0707dc3d9aa child 60580 7e741e22d7fc permissions -rw-r--r--
import general thms from Density_Compiler
```     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)
```
```   197   case goal1 show ?case using assms
```
```   198     by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all
```
```   199 next
```
```   200   case goal2
```
```   201   have "\<And>K. K \<subseteq> J \<Longrightarrow> (\<Pi> j\<in>K. sets (M j)) = (\<Pi> j\<in>K. sets (N j))"
```
```   202     using assms by (intro Pi_cong_sets) auto
```
```   203   thus ?case by (auto simp: assms)
```
```   204 next
```
```   205   case goal3 show ?case using assms
```
```   206     by (intro ext) (auto simp: prod_emb_def dest: PiE_mem)
```
```   207 next
```
```   208   case (goal4 x)
```
```   209   thus ?case using assms
```
```   210     by (auto intro!: setprod.cong split: split_if_asm)
```
```   211 qed
```
```   212
```
```   213
```
```   214 lemma prod_algebra_sets_into_space:
```
```   215   "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   216   by (auto simp: prod_emb_def prod_algebra_def)
```
```   217
```
```   218 lemma prod_algebra_eq_finite:
```
```   219   assumes I: "finite I"
```
```   220   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")
```
```   221 proof (intro iffI set_eqI)
```
```   222   fix A assume "A \<in> ?L"
```
```   223   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)"
```
```   224     and A: "A = prod_emb I M J (PIE j:J. E j)"
```
```   225     by (auto simp: prod_algebra_def)
```
```   226   let ?A = "\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i)"
```
```   227   have A: "A = ?A"
```
```   228     unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto
```
```   229   show "A \<in> ?R" unfolding A using J sets.top
```
```   230     by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
```
```   231 next
```
```   232   fix A assume "A \<in> ?R"
```
```   233   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
```
```   234   then have A: "A = prod_emb I M I (\<Pi>\<^sub>E i\<in>I. X i)"
```
```   235     by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff)
```
```   236   from X I show "A \<in> ?L" unfolding A
```
```   237     by (auto simp: prod_algebra_def)
```
```   238 qed
```
```   239
```
```   240 lemma prod_algebraI:
```
```   241   "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
```
```   242     \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
```
```   243   by (auto simp: prod_algebra_def)
```
```   244
```
```   245 lemma prod_algebraI_finite:
```
```   246   "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M"
```
```   247   using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp
```
```   248
```
```   249 lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
```
```   250 proof (safe intro!: Int_stableI)
```
```   251   fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
```
```   252   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))"
```
```   253     by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
```
```   254 qed
```
```   255
```
```   256 lemma prod_algebraE:
```
```   257   assumes A: "A \<in> prod_algebra I M"
```
```   258   obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
```
```   259     "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
```
```   260   using A by (auto simp: prod_algebra_def)
```
```   261
```
```   262 lemma prod_algebraE_all:
```
```   263   assumes A: "A \<in> prod_algebra I M"
```
```   264   obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   265 proof -
```
```   266   from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)"
```
```   267     and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```   268     by (auto simp: prod_algebra_def)
```
```   269   from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
```
```   270     using sets.sets_into_space by auto
```
```   271   then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))"
```
```   272     using A J by (auto simp: prod_emb_PiE)
```
```   273   moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   274     using sets.top E by auto
```
```   275   ultimately show ?thesis using that by auto
```
```   276 qed
```
```   277
```
```   278 lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
```
```   279 proof (unfold Int_stable_def, safe)
```
```   280   fix A assume "A \<in> prod_algebra I M"
```
```   281   from prod_algebraE[OF this] guess J E . note A = this
```
```   282   fix B assume "B \<in> prod_algebra I M"
```
```   283   from prod_algebraE[OF this] guess K F . note B = this
```
```   284   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>
```
```   285       (if i \<in> K then F i else space (M i)))"
```
```   286     unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4)
```
```   287       B(5)[THEN sets.sets_into_space]
```
```   288     apply (subst (1 2 3) prod_emb_PiE)
```
```   289     apply (simp_all add: subset_eq PiE_Int)
```
```   290     apply blast
```
```   291     apply (intro PiE_cong)
```
```   292     apply auto
```
```   293     done
```
```   294   also have "\<dots> \<in> prod_algebra I M"
```
```   295     using A B by (auto intro!: prod_algebraI)
```
```   296   finally show "A \<inter> B \<in> prod_algebra I M" .
```
```   297 qed
```
```   298
```
```   299 lemma prod_algebra_mono:
```
```   300   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
```
```   301   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
```
```   302   shows "prod_algebra I E \<subseteq> prod_algebra I F"
```
```   303 proof
```
```   304   fix A assume "A \<in> prod_algebra I E"
```
```   305   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
```
```   306     and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   307     and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
```
```   308     by (auto simp: prod_algebra_def)
```
```   309   moreover
```
```   310   from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
```
```   311     by (rule PiE_cong)
```
```   312   with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   313     by (simp add: prod_emb_def)
```
```   314   moreover
```
```   315   from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
```
```   316     by auto
```
```   317   ultimately show "A \<in> prod_algebra I F"
```
```   318     apply (simp add: prod_algebra_def image_iff)
```
```   319     apply (intro exI[of _ J] exI[of _ G] conjI)
```
```   320     apply auto
```
```   321     done
```
```   322 qed
```
```   323
```
```   324 lemma prod_algebra_cong:
```
```   325   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
```
```   326   shows "prod_algebra I M = prod_algebra J N"
```
```   327 proof -
```
```   328   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
```
```   329     using sets_eq_imp_space_eq[OF sets] by auto
```
```   330   with sets show ?thesis unfolding `I = J`
```
```   331     by (intro antisym prod_algebra_mono) auto
```
```   332 qed
```
```   333
```
```   334 lemma space_in_prod_algebra:
```
```   335   "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
```
```   336 proof cases
```
```   337   assume "I = {}" then show ?thesis
```
```   338     by (auto simp add: prod_algebra_def image_iff prod_emb_def)
```
```   339 next
```
```   340   assume "I \<noteq> {}"
```
```   341   then obtain i where "i \<in> I" by auto
```
```   342   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))"
```
```   343     by (auto simp: prod_emb_def)
```
```   344   also have "\<dots> \<in> prod_algebra I M"
```
```   345     using `i \<in> I` by (intro prod_algebraI) auto
```
```   346   finally show ?thesis .
```
```   347 qed
```
```   348
```
```   349 lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   350   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
```
```   351
```
```   352 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)"
```
```   353   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
```
```   354
```
```   355 lemma sets_PiM_single: "sets (PiM I M) =
```
```   356     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)}"
```
```   357     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   358   unfolding sets_PiM
```
```   359 proof (rule sigma_sets_eqI)
```
```   360   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   361   fix A assume "A \<in> prod_algebra I M"
```
```   362   from prod_algebraE[OF this] guess J X . note X = this
```
```   363   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   364   proof cases
```
```   365     assume "I = {}"
```
```   366     with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
```
```   367     with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
```
```   368   next
```
```   369     assume "I \<noteq> {}"
```
```   370     with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
```
```   371       by (auto simp: prod_emb_def)
```
```   372     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```   373       using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
```
```   374     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```   375   qed
```
```   376 next
```
```   377   fix A assume "A \<in> ?R"
```
```   378   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)"
```
```   379     by auto
```
```   380   then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
```
```   381      by (auto simp: prod_emb_def)
```
```   382   also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
```
```   383     using A by (intro sigma_sets.Basic prod_algebraI) auto
```
```   384   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
```
```   385 qed
```
```   386
```
```   387 lemma sets_PiM_eq_proj:
```
```   388   "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))"
```
```   389   apply (simp add: sets_PiM_single sets_Sup_sigma)
```
```   390   apply (subst SUP_cong[OF refl])
```
```   391   apply (rule sets_vimage_algebra2)
```
```   392   apply auto []
```
```   393   apply (auto intro!: arg_cong2[where f=sigma_sets])
```
```   394   done
```
```   395
```
```   396 lemma
```
```   397   shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
```
```   398     and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
```
```   399   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
```
```   400
```
```   401 lemma sets_PiM_sigma:
```
```   402   assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S"
```
```   403   assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)"
```
```   404   assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I"
```
```   405   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}"
```
```   406   shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
```
```   407 proof cases
```
```   408   assume "I = {}"
```
```   409   with `\<Union>J = I` have "P = {{\<lambda>_. undefined}} \<or> P = {}"
```
```   410     by (auto simp: P_def)
```
```   411   with `I = {}` show ?thesis
```
```   412     by (auto simp add: sets_PiM_empty sigma_sets_empty_eq)
```
```   413 next
```
```   414   let ?F = "\<lambda>i. {(\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega> |A. A \<in> E i}"
```
```   415   assume "I \<noteq> {}"
```
```   416   then have "sets (Pi\<^sub>M I (\<lambda>i. sigma (\<Omega> i) (E i))) =
```
```   417       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)))"
```
```   418     by (subst sets_PiM_eq_proj) (auto simp: space_measure_of_conv)
```
```   419   also have "\<dots> = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. sigma (Pi\<^sub>E I \<Omega>) (?F i))"
```
```   420     using E by (intro SUP_sigma_cong arg_cong[where f=sets] vimage_algebra_sigma) auto
```
```   421   also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i))"
```
```   422     using `I \<noteq> {}` by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto
```
```   423   also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) P)"
```
```   424   proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
```
```   425     show "(\<Union>i\<in>I. ?F i) \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" "P \<subseteq> Pow (Pi\<^sub>E I \<Omega>)"
```
```   426       by (auto simp: P_def)
```
```   427   next
```
```   428     interpret P: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
```
```   429       by (auto intro!: sigma_algebra_sigma_sets simp: P_def)
```
```   430
```
```   431     fix Z assume "Z \<in> (\<Union>i\<in>I. ?F i)"
```
```   432     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>"
```
```   433       by auto
```
```   434     from `i \<in> I` J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j"
```
```   435       by auto
```
```   436     obtain S where S: "\<And>i. i \<in> j \<Longrightarrow> S i \<subseteq> E i" "\<And>i. i \<in> j \<Longrightarrow> countable (S i)"
```
```   437       "\<And>i. i \<in> j \<Longrightarrow> \<Omega> i = \<Union>(S i)"
```
```   438       by (metis subset_eq \<Omega>_cover `j \<subseteq> I`)
```
```   439     def A' \<equiv> "\<lambda>n. n(i := A)"
```
```   440     then have A'_i: "\<And>n. A' n i = A"
```
```   441       by simp
```
```   442     { fix n assume "n \<in> Pi\<^sub>E (j - {i}) S"
```
```   443       then have "A' n \<in> Pi j E"
```
```   444         unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def `A \<in> E i` )
```
```   445       with `j \<in> J` have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P"
```
```   446         by (auto simp: P_def) }
```
```   447     note A'_in_P = this
```
```   448
```
```   449     { fix x assume "x i \<in> A" "x \<in> Pi\<^sub>E I \<Omega>"
```
```   450       with S(3) `j \<subseteq> I` have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s"
```
```   451         by (auto simp: PiE_def Pi_def)
```
```   452       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"
```
```   453         by metis
```
```   454       with `x i \<in> A` have "\<exists>n\<in>PiE (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i"
```
```   455         by (intro bexI[of _ "restrict (s(i := A)) (j-{i})"]) (auto simp: A'_def split: if_splits) }
```
```   456     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})"
```
```   457       unfolding Z_def
```
```   458       by (auto simp add: set_eq_iff ball_conj_distrib `i\<in>j` A'_i dest: bspec[OF _ `i\<in>j`]
```
```   459                cong: conj_cong)
```
```   460     also have "\<dots> \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
```
```   461       using `finite j` S(2)
```
```   462       by (intro P.countable_UN' countable_PiE) (simp_all add: image_subset_iff A'_in_P)
```
```   463     finally show "Z \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" .
```
```   464   next
```
```   465     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)"
```
```   466       by (auto intro!: sigma_algebra_sigma_sets)
```
```   467
```
```   468     fix b assume "b \<in> P"
```
```   469     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"
```
```   470       by (auto simp: P_def)
```
```   471     show "b \<in> sigma_sets (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i)"
```
```   472     proof cases
```
```   473       assume "j = {}"
```
```   474       with b have "b = (\<Pi>\<^sub>E i\<in>I. \<Omega> i)"
```
```   475         by auto
```
```   476       then show ?thesis
```
```   477         by blast
```
```   478     next
```
```   479       assume "j \<noteq> {}"
```
```   480       with J b(2,3) have eq: "b = (\<Inter>i\<in>j. ((\<lambda>x. x i) -` A i \<inter> Pi\<^sub>E I \<Omega>))"
```
```   481         unfolding b(1)
```
```   482         by (auto simp: PiE_def Pi_def)
```
```   483       show ?thesis
```
```   484         unfolding eq using `A \<in> Pi j E` `j \<in> J` J(2)
```
```   485         by (intro F.finite_INT J `j \<in> J` `j \<noteq> {}` sigma_sets.Basic) blast
```
```   486     qed
```
```   487   qed
```
```   488   finally show "?thesis" .
```
```   489 qed
```
```   490
```
```   491 lemma sets_PiM_in_sets:
```
```   492   assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   493   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"
```
```   494   shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N"
```
```   495   unfolding sets_PiM_single space[symmetric]
```
```   496   by (intro sets.sigma_sets_subset subsetI) (auto intro: sets)
```
```   497
```
```   498 lemma sets_PiM_cong[measurable_cong]:
```
```   499   assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)"
```
```   500   using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong)
```
```   501
```
```   502 lemma sets_PiM_I:
```
```   503   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   504   shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
```
```   505 proof cases
```
```   506   assume "J = {}"
```
```   507   then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
```
```   508     by (auto simp: prod_emb_def)
```
```   509   then show ?thesis
```
```   510     by (auto simp add: sets_PiM intro!: sigma_sets_top)
```
```   511 next
```
```   512   assume "J \<noteq> {}" with assms show ?thesis
```
```   513     by (force simp add: sets_PiM prod_algebra_def)
```
```   514 qed
```
```   515
```
```   516 lemma measurable_PiM:
```
```   517   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   518   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>
```
```   519     f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
```
```   520   shows "f \<in> measurable N (PiM I M)"
```
```   521   using sets_PiM prod_algebra_sets_into_space space
```
```   522 proof (rule measurable_sigma_sets)
```
```   523   fix A assume "A \<in> prod_algebra I M"
```
```   524   from prod_algebraE[OF this] guess J X .
```
```   525   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
```
```   526 qed
```
```   527
```
```   528 lemma measurable_PiM_Collect:
```
```   529   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   530   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>
```
```   531     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
```
```   532   shows "f \<in> measurable N (PiM I M)"
```
```   533   using sets_PiM prod_algebra_sets_into_space space
```
```   534 proof (rule measurable_sigma_sets)
```
```   535   fix A assume "A \<in> prod_algebra I M"
```
```   536   from prod_algebraE[OF this] guess J X . note X = this
```
```   537   then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
```
```   538     using space by (auto simp: prod_emb_def del: PiE_I)
```
```   539   also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
```
```   540   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   541 qed
```
```   542
```
```   543 lemma measurable_PiM_single:
```
```   544   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   545   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"
```
```   546   shows "f \<in> measurable N (PiM I M)"
```
```   547   using sets_PiM_single
```
```   548 proof (rule measurable_sigma_sets)
```
```   549   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)}"
```
```   550   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)"
```
```   551     by auto
```
```   552   with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
```
```   553   also have "\<dots> \<in> sets N" using B by (rule sets)
```
```   554   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   555 qed (auto simp: space)
```
```   556
```
```   557 lemma measurable_PiM_single':
```
```   558   assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
```
```   559     and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   560   shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
```
```   561 proof (rule measurable_PiM_single)
```
```   562   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   563   then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
```
```   564     by auto
```
```   565   then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
```
```   566     using A f by (auto intro!: measurable_sets)
```
```   567 qed fact
```
```   568
```
```   569 lemma sets_PiM_I_finite[measurable]:
```
```   570   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
```
```   571   shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
```
```   572   using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto
```
```   573
```
```   574 lemma measurable_component_singleton[measurable (raw)]:
```
```   575   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
```
```   576 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   577   fix A assume "A \<in> sets (M i)"
```
```   578   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)"
```
```   579     using sets.sets_into_space `i \<in> I`
```
```   580     by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
```
```   581   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
```
```   582     using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
```
```   583 qed (insert `i \<in> I`, auto simp: space_PiM)
```
```   584
```
```   585 lemma measurable_component_singleton'[measurable_dest]:
```
```   586   assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
```
```   587   assumes g: "g \<in> measurable L N"
```
```   588   assumes i: "i \<in> I"
```
```   589   shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)"
```
```   590   using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] .
```
```   591
```
```   592 lemma measurable_PiM_component_rev:
```
```   593   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
```
```   594   by simp
```
```   595
```
```   596 lemma measurable_case_nat[measurable (raw)]:
```
```   597   assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
```
```   598     "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
```
```   599   shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
```
```   600   by (cases i) simp_all
```
```   601
```
```   602 lemma measurable_case_nat'[measurable (raw)]:
```
```   603   assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   604   shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   605   using fg[THEN measurable_space]
```
```   606   by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
```
```   607
```
```   608 lemma measurable_add_dim[measurable]:
```
```   609   "(\<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)"
```
```   610     (is "?f \<in> measurable ?P ?I")
```
```   611 proof (rule measurable_PiM_single)
```
```   612   fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
```
```   613   have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
```
```   614     (if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
```
```   615     using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
```
```   616   also have "\<dots> \<in> sets ?P"
```
```   617     using A j
```
```   618     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   619   finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
```
```   620 qed (auto simp: space_pair_measure space_PiM PiE_def)
```
```   621
```
```   622 lemma measurable_component_update:
```
```   623   "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)"
```
```   624   by simp
```
```   625
```
```   626 lemma measurable_merge[measurable]:
```
```   627   "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)"
```
```   628     (is "?f \<in> measurable ?P ?U")
```
```   629 proof (rule measurable_PiM_single)
```
```   630   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
```
```   631   then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
```
```   632     (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)"
```
```   633     by (auto simp: merge_def)
```
```   634   also have "\<dots> \<in> sets ?P"
```
```   635     using A
```
```   636     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   637   finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
```
```   638 qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
```
```   639
```
```   640 lemma measurable_restrict[measurable (raw)]:
```
```   641   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
```
```   642   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
```
```   643 proof (rule measurable_PiM_single)
```
```   644   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   645   then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
```
```   646     by auto
```
```   647   then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
```
```   648     using A X by (auto intro!: measurable_sets)
```
```   649 qed (insert X, auto simp add: PiE_def dest: measurable_space)
```
```   650
```
```   651 lemma measurable_abs_UNIV:
```
```   652   "(\<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)"
```
```   653   by (intro measurable_PiM_single) (auto dest: measurable_space)
```
```   654
```
```   655 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)"
```
```   656   by (intro measurable_restrict measurable_component_singleton) auto
```
```   657
```
```   658 lemma measurable_restrict_subset':
```
```   659   assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)"
```
```   660   shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
```
```   661 proof-
```
```   662   from assms(1) have "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
```
```   663     by (rule measurable_restrict_subset)
```
```   664   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)"
```
```   665     by (intro sets_PiM_cong measurable_cong_sets) simp_all
```
```   666   finally show ?thesis .
```
```   667 qed
```
```   668
```
```   669 lemma measurable_prod_emb[intro, simp]:
```
```   670   "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)"
```
```   671   unfolding prod_emb_def space_PiM[symmetric]
```
```   672   by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
```
```   673
```
```   674 lemma sets_in_Pi_aux:
```
```   675   "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   676   {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
```
```   677   by (simp add: subset_eq Pi_iff)
```
```   678
```
```   679 lemma sets_in_Pi[measurable (raw)]:
```
```   680   "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
```
```   681   (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   682   Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
```
```   683   unfolding pred_def
```
```   684   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
```
```   685
```
```   686 lemma sets_in_extensional_aux:
```
```   687   "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
```
```   688 proof -
```
```   689   have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
```
```   690     by (auto simp add: extensional_def space_PiM)
```
```   691   then show ?thesis by simp
```
```   692 qed
```
```   693
```
```   694 lemma sets_in_extensional[measurable (raw)]:
```
```   695   "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
```
```   696   unfolding pred_def
```
```   697   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
```
```   698
```
```   699 locale product_sigma_finite =
```
```   700   fixes M :: "'i \<Rightarrow> 'a measure"
```
```   701   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
```
```   702
```
```   703 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
```
```   704   by (rule sigma_finite_measures)
```
```   705
```
```   706 locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   707   fixes I :: "'i set"
```
```   708   assumes finite_index: "finite I"
```
```   709
```
```   710 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
```
```   711   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
```
```   712     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
```
```   713     (\<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>
```
```   714     (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
```
```   715 proof -
```
```   716   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>)"
```
```   717     using M.sigma_finite_incseq by metis
```
```   718   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   719   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>"
```
```   720     by auto
```
```   721   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
```
```   722   note space_PiM[simp]
```
```   723   show ?thesis
```
```   724   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
```
```   725     fix i show "range (F i) \<subseteq> sets (M i)" by fact
```
```   726   next
```
```   727     fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
```
```   728   next
```
```   729     fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
```
```   730       by (auto simp: PiE_def dest!: sets.sets_into_space)
```
```   731   next
```
```   732     fix f assume "f \<in> space (PiM I M)"
```
```   733     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
```
```   734     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
```
```   735   next
```
```   736     fix i show "?F i \<subseteq> ?F (Suc i)"
```
```   737       using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
```
```   738   qed
```
```   739 qed
```
```   740
```
```   741 lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
```
```   742 proof -
```
```   743   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
```
```   744   have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
```
```   745   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   746     show "positive (PiM {} M) ?\<mu>"
```
```   747       by (auto simp: positive_def)
```
```   748     show "countably_additive (PiM {} M) ?\<mu>"
```
```   749       by (rule sets.countably_additiveI_finite)
```
```   750          (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
```
```   751   qed (auto simp: prod_emb_def)
```
```   752   also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
```
```   753     by (auto simp: prod_emb_def)
```
```   754   finally show ?thesis
```
```   755     by simp
```
```   756 qed
```
```   757
```
```   758 lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
```
```   759   by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
```
```   760
```
```   761 lemma (in product_sigma_finite) emeasure_PiM:
```
```   762   "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))"
```
```   763 proof (induct I arbitrary: A rule: finite_induct)
```
```   764   case (insert i I)
```
```   765   interpret finite_product_sigma_finite M I by default fact
```
```   766   have "finite (insert i I)" using `finite I` by auto
```
```   767   interpret I': finite_product_sigma_finite M "insert i I" by default fact
```
```   768   let ?h = "(\<lambda>(f, y). f(i := y))"
```
```   769
```
```   770   let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
```
```   771   let ?\<mu> = "emeasure ?P"
```
```   772   let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
```
```   773   let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
```
```   774
```
```   775   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)) =
```
```   776     (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
```
```   777   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   778     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))"
```
```   779     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
```
```   780     let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
```
```   781     let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
```
```   782     have "?\<mu> ?p =
```
```   783       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))"
```
```   784       by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
```
```   785     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))"
```
```   786       using J E[rule_format, THEN sets.sets_into_space]
```
```   787       by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
```
```   788     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))) =
```
```   789       emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
```
```   790       using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
```
```   791     also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
```
```   792       using J E[rule_format, THEN sets.sets_into_space]
```
```   793       by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
```
```   794     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)) =
```
```   795       (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
```
```   796       using E by (subst insert) (auto intro!: setprod.cong)
```
```   797     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
```
```   798        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
```
```   799       using insert by (auto simp: mult.commute intro!: arg_cong2[where f="op *"] setprod.cong)
```
```   800     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
```
```   801       using insert(1,2) J E by (intro setprod.mono_neutral_right) auto
```
```   802     finally show "?\<mu> ?p = \<dots>" .
```
```   803
```
```   804     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))"
```
```   805       using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
```
```   806   next
```
```   807     show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
```
```   808       using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
```
```   809   next
```
```   810     show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
```
```   811       insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
```
```   812       using insert by auto
```
```   813   qed (auto intro!: setprod.cong)
```
```   814   with insert show ?case
```
```   815     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
```
```   816 qed simp
```
```   817
```
```   818 lemma (in product_sigma_finite) sigma_finite:
```
```   819   assumes "finite I"
```
```   820   shows "sigma_finite_measure (PiM I M)"
```
```   821 proof
```
```   822   interpret finite_product_sigma_finite M I by default fact
```
```   823
```
```   824   obtain F where F: "\<And>j. countable (F j)" "\<And>j f. f \<in> F j \<Longrightarrow> f \<in> sets (M j)"
```
```   825     "\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and
```
```   826     in_space: "\<And>j. space (M j) = (\<Union>F j)"
```
```   827     using sigma_finite_countable by (metis subset_eq)
```
```   828   moreover have "(\<Union>(PiE I ` PiE I F)) = space (Pi\<^sub>M I M)"
```
```   829     using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD2])
```
```   830   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>)"
```
```   831     by (intro exI[of _ "PiE I ` PiE I F"])
```
```   832        (auto intro!: countable_PiE sets_PiM_I_finite
```
```   833              simp: PiE_iff emeasure_PiM finite_index setprod_PInf emeasure_nonneg)
```
```   834 qed
```
```   835
```
```   836 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
```
```   837   using sigma_finite[OF finite_index] .
```
```   838
```
```   839 lemma (in finite_product_sigma_finite) measure_times:
```
```   840   "(\<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))"
```
```   841   using emeasure_PiM[OF finite_index] by auto
```
```   842
```
```   843 lemma (in product_sigma_finite) nn_integral_empty:
```
```   844   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
```
```   845   shows "integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
```
```   846 proof -
```
```   847   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
```
```   848   have "\<And>A. emeasure (Pi\<^sub>M {} M) (Pi\<^sub>E {} A) = 1"
```
```   849     using assms by (subst measure_times) auto
```
```   850   then show ?thesis
```
```   851     unfolding nn_integral_def simple_function_def simple_integral_def[abs_def]
```
```   852   proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
```
```   853     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
```
```   854       by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
```
```   855     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
```
```   856       by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
```
```   857   qed
```
```   858 qed
```
```   859
```
```   860 lemma (in product_sigma_finite) distr_merge:
```
```   861   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   862   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"
```
```   863    (is "?D = ?P")
```
```   864 proof -
```
```   865   interpret I: finite_product_sigma_finite M I by default fact
```
```   866   interpret J: finite_product_sigma_finite M J by default fact
```
```   867   have "finite (I \<union> J)" using fin by auto
```
```   868   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   869   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
```
```   870   let ?g = "merge I J"
```
```   871
```
```   872   from IJ.sigma_finite_pairs obtain F where
```
```   873     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
```
```   874        "incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k)"
```
```   875        "(\<Union>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k) = space ?P"
```
```   876        "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
```
```   877     by auto
```
```   878   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k"
```
```   879
```
```   880   show ?thesis
```
```   881   proof (rule measure_eqI_generator_eq[symmetric])
```
```   882     show "Int_stable (prod_algebra (I \<union> J) M)"
```
```   883       by (rule Int_stable_prod_algebra)
```
```   884     show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^sub>E i \<in> I \<union> J. space (M i))"
```
```   885       by (rule prod_algebra_sets_into_space)
```
```   886     show "sets ?P = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   887       by (rule sets_PiM)
```
```   888     then show "sets ?D = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   889       by simp
```
```   890
```
```   891     show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
```
```   892       using fin by (auto simp: prod_algebra_eq_finite)
```
```   893     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))"
```
```   894       using F(3) by (simp add: space_PiM)
```
```   895   next
```
```   896     fix k
```
```   897     from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
```
```   898     show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
```
```   899   next
```
```   900     fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
```
```   901     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)"
```
```   902       by (auto simp add: prod_algebra_eq_finite)
```
```   903     let ?B = "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M"
```
```   904     let ?X = "?g -` A \<inter> space ?B"
```
```   905     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)"
```
```   906       using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+
```
```   907     then have X: "?X = (Pi\<^sub>E I F \<times> Pi\<^sub>E J F)"
```
```   908       unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
```
```   909     have "emeasure ?D A = emeasure ?B ?X"
```
```   910       using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
```
```   911     also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
```
```   912       using `finite J` `finite I` F unfolding X
```
```   913       by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
```
```   914     also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
```
```   915       using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod.union_inter_neutral)
```
```   916     also have "\<dots> = emeasure ?P (Pi\<^sub>E (I \<union> J) F)"
```
```   917       using `finite J` `finite I` F unfolding A
```
```   918       by (intro IJ.measure_times[symmetric]) auto
```
```   919     finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
```
```   920   qed
```
```   921 qed
```
```   922
```
```   923 lemma (in product_sigma_finite) product_nn_integral_fold:
```
```   924   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
```
```   925   and f: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
```
```   926   shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
```
```   927     (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
```
```   928 proof -
```
```   929   interpret I: finite_product_sigma_finite M I by default fact
```
```   930   interpret J: finite_product_sigma_finite M J by default fact
```
```   931   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
```
```   932   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)"
```
```   933     using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
```
```   934   show ?thesis
```
```   935     apply (subst distr_merge[OF IJ, symmetric])
```
```   936     apply (subst nn_integral_distr[OF measurable_merge f])
```
```   937     apply (subst J.nn_integral_fst[symmetric, OF P_borel])
```
```   938     apply simp
```
```   939     done
```
```   940 qed
```
```   941
```
```   942 lemma (in product_sigma_finite) distr_singleton:
```
```   943   "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
```
```   944 proof (intro measure_eqI[symmetric])
```
```   945   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   946   fix A assume A: "A \<in> sets (M i)"
```
```   947   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
```
```   948     using sets.sets_into_space by (auto simp: space_PiM)
```
```   949   then show "emeasure (M i) A = emeasure ?D A"
```
```   950     using A I.measure_times[of "\<lambda>_. A"]
```
```   951     by (simp add: emeasure_distr measurable_component_singleton)
```
```   952 qed simp
```
```   953
```
```   954 lemma (in product_sigma_finite) product_nn_integral_singleton:
```
```   955   assumes f: "f \<in> borel_measurable (M i)"
```
```   956   shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f"
```
```   957 proof -
```
```   958   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   959   from f show ?thesis
```
```   960     apply (subst distr_singleton[symmetric])
```
```   961     apply (subst nn_integral_distr[OF measurable_component_singleton])
```
```   962     apply simp_all
```
```   963     done
```
```   964 qed
```
```   965
```
```   966 lemma (in product_sigma_finite) product_nn_integral_insert:
```
```   967   assumes I[simp]: "finite I" "i \<notin> I"
```
```   968     and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```   969   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))"
```
```   970 proof -
```
```   971   interpret I: finite_product_sigma_finite M I by default auto
```
```   972   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   973   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
```
```   974     using f by auto
```
```   975   show ?thesis
```
```   976     unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
```
```   977   proof (rule nn_integral_cong, subst product_nn_integral_singleton[symmetric])
```
```   978     fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
```
```   979     let ?f = "\<lambda>y. f (x(i := y))"
```
```   980     show "?f \<in> borel_measurable (M i)"
```
```   981       using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
```
```   982       unfolding comp_def .
```
```   983     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)"
```
```   984       using x
```
```   985       by (auto intro!: nn_integral_cong arg_cong[where f=f]
```
```   986                simp add: space_PiM extensional_def PiE_def)
```
```   987   qed
```
```   988 qed
```
```   989
```
```   990 lemma (in product_sigma_finite) product_nn_integral_insert_rev:
```
```   991   assumes I[simp]: "finite I" "i \<notin> I"
```
```   992     and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```   993   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))"
```
```   994   apply (subst product_nn_integral_insert[OF assms])
```
```   995   apply (rule pair_sigma_finite.Fubini')
```
```   996   apply intro_locales []
```
```   997   apply (rule sigma_finite[OF I(1)])
```
```   998   apply measurable
```
```   999   done
```
```  1000
```
```  1001 lemma (in product_sigma_finite) product_nn_integral_setprod:
```
```  1002   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1003   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```  1004   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
```
```  1005   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))"
```
```  1006 using assms proof induct
```
```  1007   case (insert i I)
```
```  1008   note `finite I`[intro, simp]
```
```  1009   interpret I: finite_product_sigma_finite M I by default auto
```
```  1010   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))"
```
```  1011     using insert by (auto intro!: setprod.cong)
```
```  1012   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)"
```
```  1013     using sets.sets_into_space insert
```
```  1014     by (intro borel_measurable_ereal_setprod
```
```  1015               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
```
```  1016        auto
```
```  1017   then show ?case
```
```  1018     apply (simp add: product_nn_integral_insert[OF insert(1,2) prod])
```
```  1019     apply (simp add: insert(2-) * pos borel setprod_ereal_pos nn_integral_multc)
```
```  1020     apply (subst nn_integral_cmult)
```
```  1021     apply (auto simp add: pos borel insert(2-) setprod_ereal_pos nn_integral_nonneg)
```
```  1022     done
```
```  1023 qed (simp add: space_PiM)
```
```  1024
```
```  1025 lemma (in product_sigma_finite) product_nn_integral_pair:
```
```  1026   assumes [measurable]: "split f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
```
```  1027   assumes xy: "x \<noteq> y"
```
```  1028   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))"
```
```  1029 proof-
```
```  1030   interpret psm: pair_sigma_finite "M x" "M y"
```
```  1031     unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all
```
```  1032   have "{x, y} = {y, x}" by auto
```
```  1033   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)"
```
```  1034     using xy by (subst product_nn_integral_insert_rev) simp_all
```
```  1035   also have "... = (\<integral>\<^sup>+y. \<integral>\<^sup>+x. f x y \<partial>M x \<partial>M y)"
```
```  1036     by (intro nn_integral_cong, subst product_nn_integral_singleton) simp_all
```
```  1037   also have "... = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
```
```  1038     by (subst psm.nn_integral_snd[symmetric]) simp_all
```
```  1039   finally show ?thesis .
```
```  1040 qed
```
```  1041
```
```  1042 lemma (in product_sigma_finite) distr_component:
```
```  1043   "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
```
```  1044 proof (intro measure_eqI[symmetric])
```
```  1045   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```  1046
```
```  1047   have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
```
```  1048     by (auto simp: extensional_def restrict_def)
```
```  1049
```
```  1050   have [measurable]: "\<And>j. j \<in> {i} \<Longrightarrow> (\<lambda>x. x) \<in> measurable (M i) (M j)" by simp
```
```  1051
```
```  1052   fix A assume A: "A \<in> sets ?P"
```
```  1053   then have "emeasure ?P A = (\<integral>\<^sup>+x. indicator A x \<partial>?P)"
```
```  1054     by simp
```
```  1055   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)"
```
```  1056     by (intro nn_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
```
```  1057   also have "\<dots> = emeasure ?D A"
```
```  1058     using A by (simp add: product_nn_integral_singleton emeasure_distr)
```
```  1059   finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" .
```
```  1060 qed simp
```
```  1061
```
```  1062 lemma (in product_sigma_finite)
```
```  1063   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
```
```  1064   shows emeasure_fold_integral:
```
```  1065     "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)
```
```  1066     and emeasure_fold_measurable:
```
```  1067     "(\<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)
```
```  1068 proof -
```
```  1069   interpret I: finite_product_sigma_finite M I by default fact
```
```  1070   interpret J: finite_product_sigma_finite M J by default fact
```
```  1071   interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
```
```  1072   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)"
```
```  1073     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```  1074
```
```  1075   show ?I
```
```  1076     apply (subst distr_merge[symmetric, OF IJ])
```
```  1077     apply (subst emeasure_distr[OF measurable_merge A])
```
```  1078     apply (subst J.emeasure_pair_measure_alt[OF merge])
```
```  1079     apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```  1080     done
```
```  1081
```
```  1082   show ?B
```
```  1083     using IJ.measurable_emeasure_Pair1[OF merge]
```
```  1084     by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
```
```  1085 qed
```
```  1086
```
```  1087 lemma sets_Collect_single:
```
```  1088   "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)"
```
```  1089   by simp
```
```  1090
```
```  1091 lemma pair_measure_eq_distr_PiM:
```
```  1092   fixes M1 :: "'a measure" and M2 :: "'a measure"
```
```  1093   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
```
```  1094   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))"
```
```  1095     (is "?P = ?D")
```
```  1096 proof (rule pair_measure_eqI[OF assms])
```
```  1097   interpret B: product_sigma_finite "case_bool M1 M2"
```
```  1098     unfolding product_sigma_finite_def using assms by (auto split: bool.split)
```
```  1099   let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"
```
```  1100
```
```  1101   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)"
```
```  1102     by auto
```
```  1103   fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
```
```  1104   have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
```
```  1105     by (simp add: UNIV_bool ac_simps)
```
```  1106   also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
```
```  1107     using A B by (subst B.emeasure_PiM) (auto split: bool.split)
```
```  1108   also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
```
```  1109     using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
```
```  1110     by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
```
```  1111   finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
```
```  1112     using A B
```
```  1113       measurable_component_singleton[of True UNIV "case_bool M1 M2"]
```
```  1114       measurable_component_singleton[of False UNIV "case_bool M1 M2"]
```
```  1115     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
```
```  1116 qed simp
```
```  1117
```
```  1118 end
```