src/HOL/Probability/Finite_Product_Measure.thy
 author hoelzl Wed Oct 10 12:12:20 2012 +0200 (2012-10-10) changeset 49779 1484b4b82855 parent 49776 199d1d5bb17e child 49780 92a58f80b20c permissions -rw-r--r--
remove incseq assumption from sigma_prod_algebra_sigma_eq
```     1 (*  Title:      HOL/Probability/Finite_Product_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 header {*Finite product measures*}
```
```     6
```
```     7 theory Finite_Product_Measure
```
```     8 imports Binary_Product_Measure
```
```     9 begin
```
```    10
```
```    11 lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
```
```    12   by auto
```
```    13
```
```    14 abbreviation
```
```    15   "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
```
```    16
```
```    17 syntax
```
```    18   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
```
```    19
```
```    20 syntax (xsymbols)
```
```    21   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
```
```    22
```
```    23 syntax (HTML output)
```
```    24   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
```
```    25
```
```    26 translations
```
```    27   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
```
```    28
```
```    29 abbreviation
```
```    30   funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
```
```    31     (infixr "->\<^isub>E" 60) where
```
```    32   "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
```
```    33
```
```    34 notation (xsymbols)
```
```    35   funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
```
```    36
```
```    37 lemma extensional_insert[intro, simp]:
```
```    38   assumes "a \<in> extensional (insert i I)"
```
```    39   shows "a(i := b) \<in> extensional (insert i I)"
```
```    40   using assms unfolding extensional_def by auto
```
```    41
```
```    42 lemma extensional_Int[simp]:
```
```    43   "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
```
```    44   unfolding extensional_def by auto
```
```    45
```
```    46 definition
```
```    47   "merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
```
```    48
```
```    49 lemma merge_apply[simp]:
```
```    50   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
```
```    51   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
```
```    52   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
```
```    53   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
```
```    54   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"
```
```    55   unfolding merge_def by auto
```
```    56
```
```    57 lemma merge_commute:
```
```    58   "I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"
```
```    59   by (auto simp: merge_def intro!: ext)
```
```    60
```
```    61 lemma Pi_cancel_merge_range[simp]:
```
```    62   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
```
```    63   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
```
```    64   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
```
```    65   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
```
```    66   by (auto simp: Pi_def)
```
```    67
```
```    68 lemma Pi_cancel_merge[simp]:
```
```    69   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```    70   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```    71   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
```
```    72   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
```
```    73   by (auto simp: Pi_def)
```
```    74
```
```    75 lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"
```
```    76   by (auto simp: extensional_def)
```
```    77
```
```    78 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
```
```    79   by (auto simp: restrict_def Pi_def)
```
```    80
```
```    81 lemma restrict_merge[simp]:
```
```    82   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
```
```    83   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
```
```    84   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
```
```    85   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
```
```    86   by (auto simp: restrict_def)
```
```    87
```
```    88 lemma extensional_insert_undefined[intro, simp]:
```
```    89   assumes "a \<in> extensional (insert i I)"
```
```    90   shows "a(i := undefined) \<in> extensional I"
```
```    91   using assms unfolding extensional_def by auto
```
```    92
```
```    93 lemma extensional_insert_cancel[intro, simp]:
```
```    94   assumes "a \<in> extensional I"
```
```    95   shows "a \<in> extensional (insert i I)"
```
```    96   using assms unfolding extensional_def by auto
```
```    97
```
```    98 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)"
```
```    99   unfolding merge_def by (auto simp: fun_eq_iff)
```
```   100
```
```   101 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
```
```   102   by auto
```
```   103
```
```   104 lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)"
```
```   105   by auto
```
```   106
```
```   107 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
```
```   108   by (auto simp: Pi_def)
```
```   109
```
```   110 lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
```
```   111   by (auto simp: Pi_def)
```
```   112
```
```   113 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
```
```   114   by (auto simp: Pi_def)
```
```   115
```
```   116 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```   117   by (auto simp: Pi_def)
```
```   118
```
```   119 lemma restrict_vimage:
```
```   120   assumes "I \<inter> J = {}"
```
```   121   shows "(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I E J F)"
```
```   122   using assms by (auto simp: restrict_Pi_cancel)
```
```   123
```
```   124 lemma merge_vimage:
```
```   125   assumes "I \<inter> J = {}"
```
```   126   shows "(\<lambda>(x,y). merge I x J y) -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
```
```   127   using assms by (auto simp: restrict_Pi_cancel)
```
```   128
```
```   129 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
```
```   130   by (auto simp: restrict_def)
```
```   131
```
```   132 lemma merge_restrict[simp]:
```
```   133   "merge I (restrict x I) J y = merge I x J y"
```
```   134   "merge I x J (restrict y J) = merge I x J y"
```
```   135   unfolding merge_def by auto
```
```   136
```
```   137 lemma merge_x_x_eq_restrict[simp]:
```
```   138   "merge I x J x = restrict x (I \<union> J)"
```
```   139   unfolding merge_def by auto
```
```   140
```
```   141 lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
```
```   142   apply auto
```
```   143   apply (drule_tac x=x in Pi_mem)
```
```   144   apply (simp_all split: split_if_asm)
```
```   145   apply (drule_tac x=i in Pi_mem)
```
```   146   apply (auto dest!: Pi_mem)
```
```   147   done
```
```   148
```
```   149 lemma Pi_UN:
```
```   150   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
```
```   151   assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
```
```   152   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```   153 proof (intro set_eqI iffI)
```
```   154   fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```   155   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
```
```   156   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
```
```   157   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
```
```   158     using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
```
```   159   have "f \<in> Pi I (A k)"
```
```   160   proof (intro Pi_I)
```
```   161     fix i assume "i \<in> I"
```
```   162     from mono[OF this, of "n i" k] k[OF this] n[OF this]
```
```   163     show "f i \<in> A k i" by auto
```
```   164   qed
```
```   165   then show "f \<in> (\<Union>n. Pi I (A n))" by auto
```
```   166 qed auto
```
```   167
```
```   168 lemma PiE_cong:
```
```   169   assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
```
```   170   shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
```
```   171   using assms by (auto intro!: Pi_cong)
```
```   172
```
```   173 lemma restrict_upd[simp]:
```
```   174   "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
```
```   175   by (auto simp: fun_eq_iff)
```
```   176
```
```   177 lemma Pi_eq_subset:
```
```   178   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   179   assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"
```
```   180   shows "F i \<subseteq> F' i"
```
```   181 proof
```
```   182   fix x assume "x \<in> F i"
```
```   183   with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto
```
```   184   from choice[OF this] guess f .. note f = this
```
```   185   then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
```
```   186   then have "f \<in> Pi\<^isub>E I F'" using assms by simp
```
```   187   then show "x \<in> F' i" using f `i \<in> I` by auto
```
```   188 qed
```
```   189
```
```   190 lemma Pi_eq_iff_not_empty:
```
```   191   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   192   shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
```
```   193 proof (intro iffI ballI)
```
```   194   fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"
```
```   195   show "F i = F' i"
```
```   196     using Pi_eq_subset[of I F F', OF ne eq i]
```
```   197     using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
```
```   198     by auto
```
```   199 qed auto
```
```   200
```
```   201 lemma Pi_eq_empty_iff:
```
```   202   "Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
```
```   203 proof
```
```   204   assume "Pi\<^isub>E I F = {}"
```
```   205   show "\<exists>i\<in>I. F i = {}"
```
```   206   proof (rule ccontr)
```
```   207     assume "\<not> ?thesis"
```
```   208     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
```
```   209     from choice[OF this] guess f ..
```
```   210     then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
```
```   211     with `Pi\<^isub>E I F = {}` show False by auto
```
```   212   qed
```
```   213 qed auto
```
```   214
```
```   215 lemma Pi_eq_iff:
```
```   216   "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
```
```   217 proof (intro iffI disjCI)
```
```   218   assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"
```
```   219   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
```
```   220   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
```
```   221     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
```
```   222   with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
```
```   223 next
```
```   224   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
```
```   225   then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
```
```   226     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
```
```   227 qed
```
```   228
```
```   229 section "Finite product spaces"
```
```   230
```
```   231 section "Products"
```
```   232
```
```   233 definition prod_emb where
```
```   234   "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
```
```   235
```
```   236 lemma prod_emb_iff:
```
```   237   "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))"
```
```   238   unfolding prod_emb_def by auto
```
```   239
```
```   240 lemma
```
```   241   shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
```
```   242     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"
```
```   243     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"
```
```   244     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))"
```
```   245     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))"
```
```   246     and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
```
```   247   by (auto simp: prod_emb_def)
```
```   248
```
```   249 lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
```
```   250     prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))"
```
```   251   by (force simp: prod_emb_def Pi_iff split_if_mem2)
```
```   252
```
```   253 lemma prod_emb_PiE_same_index[simp]: "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E"
```
```   254   by (auto simp: prod_emb_def Pi_iff)
```
```   255
```
```   256 definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
```
```   257   "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
```
```   258     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
```
```   259     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
```
```   260     (\<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)))"
```
```   261
```
```   262 definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
```
```   263   "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
```
```   264     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   265
```
```   266 abbreviation
```
```   267   "Pi\<^isub>M I M \<equiv> PiM I M"
```
```   268
```
```   269 syntax
```
```   270   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
```
```   271
```
```   272 syntax (xsymbols)
```
```   273   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
```
```   274
```
```   275 syntax (HTML output)
```
```   276   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
```
```   277
```
```   278 translations
```
```   279   "PIM x:I. M" == "CONST PiM I (%x. M)"
```
```   280
```
```   281 lemma prod_algebra_sets_into_space:
```
```   282   "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   283   using assms by (auto simp: prod_emb_def prod_algebra_def)
```
```   284
```
```   285 lemma prod_algebra_eq_finite:
```
```   286   assumes I: "finite I"
```
```   287   shows "prod_algebra I M = {(\<Pi>\<^isub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
```
```   288 proof (intro iffI set_eqI)
```
```   289   fix A assume "A \<in> ?L"
```
```   290   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)"
```
```   291     and A: "A = prod_emb I M J (PIE j:J. E j)"
```
```   292     by (auto simp: prod_algebra_def)
```
```   293   let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
```
```   294   have A: "A = ?A"
```
```   295     unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
```
```   296   show "A \<in> ?R" unfolding A using J top
```
```   297     by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
```
```   298 next
```
```   299   fix A assume "A \<in> ?R"
```
```   300   then obtain X where "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
```
```   301   then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
```
```   302     using sets_into_space by (force simp: prod_emb_def Pi_iff)
```
```   303   from X I show "A \<in> ?L" unfolding A
```
```   304     by (auto simp: prod_algebra_def)
```
```   305 qed
```
```   306
```
```   307 lemma prod_algebraI:
```
```   308   "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
```
```   309     \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
```
```   310   by (auto simp: prod_algebra_def Pi_iff)
```
```   311
```
```   312 lemma prod_algebraE:
```
```   313   assumes A: "A \<in> prod_algebra I M"
```
```   314   obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
```
```   315     "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
```
```   316   using A by (auto simp: prod_algebra_def)
```
```   317
```
```   318 lemma prod_algebraE_all:
```
```   319   assumes A: "A \<in> prod_algebra I M"
```
```   320   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   321 proof -
```
```   322   from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
```
```   323     and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```   324     by (auto simp: prod_algebra_def)
```
```   325   from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
```
```   326     using sets_into_space by auto
```
```   327   then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
```
```   328     using A J by (auto simp: prod_emb_PiE)
```
```   329   moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   330     using top E by auto
```
```   331   ultimately show ?thesis using that by auto
```
```   332 qed
```
```   333
```
```   334 lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
```
```   335 proof (unfold Int_stable_def, safe)
```
```   336   fix A assume "A \<in> prod_algebra I M"
```
```   337   from prod_algebraE[OF this] guess J E . note A = this
```
```   338   fix B assume "B \<in> prod_algebra I M"
```
```   339   from prod_algebraE[OF this] guess K F . note B = this
```
```   340   have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^isub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter>
```
```   341       (if i \<in> K then F i else space (M i)))"
```
```   342     unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space]
```
```   343     apply (subst (1 2 3) prod_emb_PiE)
```
```   344     apply (simp_all add: subset_eq PiE_Int)
```
```   345     apply blast
```
```   346     apply (intro PiE_cong)
```
```   347     apply auto
```
```   348     done
```
```   349   also have "\<dots> \<in> prod_algebra I M"
```
```   350     using A B by (auto intro!: prod_algebraI)
```
```   351   finally show "A \<inter> B \<in> prod_algebra I M" .
```
```   352 qed
```
```   353
```
```   354 lemma prod_algebra_mono:
```
```   355   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
```
```   356   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
```
```   357   shows "prod_algebra I E \<subseteq> prod_algebra I F"
```
```   358 proof
```
```   359   fix A assume "A \<in> prod_algebra I E"
```
```   360   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
```
```   361     and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
```
```   362     and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
```
```   363     by (auto simp: prod_algebra_def)
```
```   364   moreover
```
```   365   from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
```
```   366     by (rule PiE_cong)
```
```   367   with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
```
```   368     by (simp add: prod_emb_def)
```
```   369   moreover
```
```   370   from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
```
```   371     by auto
```
```   372   ultimately show "A \<in> prod_algebra I F"
```
```   373     apply (simp add: prod_algebra_def image_iff)
```
```   374     apply (intro exI[of _ J] exI[of _ G] conjI)
```
```   375     apply auto
```
```   376     done
```
```   377 qed
```
```   378
```
```   379 lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   380   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
```
```   381
```
```   382 lemma sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)"
```
```   383   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
```
```   384
```
```   385 lemma sets_PiM_single: "sets (PiM I M) =
```
```   386     sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   387     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   388   unfolding sets_PiM
```
```   389 proof (rule sigma_sets_eqI)
```
```   390   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   391   fix A assume "A \<in> prod_algebra I M"
```
```   392   from prod_algebraE[OF this] guess J X . note X = this
```
```   393   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   394   proof cases
```
```   395     assume "I = {}"
```
```   396     with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
```
```   397     with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
```
```   398   next
```
```   399     assume "I \<noteq> {}"
```
```   400     with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})"
```
```   401       using sets_into_space[OF X(5)]
```
```   402       by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast
```
```   403     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```   404       using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
```
```   405     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```   406   qed
```
```   407 next
```
```   408   fix A assume "A \<in> ?R"
```
```   409   then obtain i B where A: "A = {f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
```
```   410     by auto
```
```   411   then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
```
```   412     using sets_into_space[OF A(3)]
```
```   413     apply (subst prod_emb_PiE)
```
```   414     apply (auto simp: Pi_iff split: split_if_asm)
```
```   415     apply blast
```
```   416     done
```
```   417   also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
```
```   418     using A by (intro sigma_sets.Basic prod_algebraI) auto
```
```   419   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
```
```   420 qed
```
```   421
```
```   422 lemma sets_PiM_I:
```
```   423   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   424   shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
```
```   425 proof cases
```
```   426   assume "J = {}"
```
```   427   then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
```
```   428     by (auto simp: prod_emb_def)
```
```   429   then show ?thesis
```
```   430     by (auto simp add: sets_PiM intro!: sigma_sets_top)
```
```   431 next
```
```   432   assume "J \<noteq> {}" with assms show ?thesis
```
```   433     by (auto simp add: sets_PiM prod_algebra_def intro!: sigma_sets.Basic)
```
```   434 qed
```
```   435
```
```   436 lemma measurable_PiM:
```
```   437   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   438   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>
```
```   439     f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N"
```
```   440   shows "f \<in> measurable N (PiM I M)"
```
```   441   using sets_PiM prod_algebra_sets_into_space space
```
```   442 proof (rule measurable_sigma_sets)
```
```   443   fix A assume "A \<in> prod_algebra I M"
```
```   444   from prod_algebraE[OF this] guess J X .
```
```   445   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
```
```   446 qed
```
```   447
```
```   448 lemma measurable_PiM_Collect:
```
```   449   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   450   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>
```
```   451     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
```
```   452   shows "f \<in> measurable N (PiM I M)"
```
```   453   using sets_PiM prod_algebra_sets_into_space space
```
```   454 proof (rule measurable_sigma_sets)
```
```   455   fix A assume "A \<in> prod_algebra I M"
```
```   456   from prod_algebraE[OF this] guess J X . note X = this
```
```   457   have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
```
```   458     using sets_into_space[OF X(5)] X(2-) space unfolding X(1)
```
```   459     by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm)
```
```   460   also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
```
```   461   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   462 qed
```
```   463
```
```   464 lemma measurable_PiM_single:
```
```   465   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   466   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"
```
```   467   shows "f \<in> measurable N (PiM I M)"
```
```   468   using sets_PiM_single
```
```   469 proof (rule measurable_sigma_sets)
```
```   470   fix A assume "A \<in> {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   471   then obtain B i where "A = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
```
```   472     by auto
```
```   473   with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
```
```   474   also have "\<dots> \<in> sets N" using B by (rule sets)
```
```   475   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   476 qed (auto simp: space)
```
```   477
```
```   478 lemma sets_PiM_I_finite[simp, intro]:
```
```   479   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
```
```   480   shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
```
```   481   using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
```
```   482
```
```   483 lemma measurable_component_update:
```
```   484   assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
```
```   485   shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
```
```   486 proof (intro measurable_PiM_single)
```
```   487   fix j A assume "j \<in> insert i I" "A \<in> sets (M j)"
```
```   488   moreover have "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} =
```
```   489     (if i = j then space (M i) \<inter> A else if x j \<in> A then space (M i) else {})"
```
```   490     by auto
```
```   491   ultimately show "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} \<in> sets (M i)"
```
```   492     by auto
```
```   493 qed (insert sets_into_space assms, auto simp: space_PiM)
```
```   494
```
```   495 lemma measurable_component_singleton:
```
```   496   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
```
```   497 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   498   fix A assume "A \<in> sets (M i)"
```
```   499   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)"
```
```   500     using sets_into_space `i \<in> I`
```
```   501     by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
```
```   502   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
```
```   503     using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
```
```   504 qed (insert `i \<in> I`, auto simp: space_PiM)
```
```   505
```
```   506 lemma measurable_add_dim:
```
```   507   "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
```
```   508     (is "?f \<in> measurable ?P ?I")
```
```   509 proof (rule measurable_PiM_single)
```
```   510   fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
```
```   511   have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
```
```   512     (if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
```
```   513     using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
```
```   514   also have "\<dots> \<in> sets ?P"
```
```   515     using A j
```
```   516     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   517   finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
```
```   518 qed (auto simp: space_pair_measure space_PiM)
```
```   519
```
```   520 lemma measurable_merge:
```
```   521   "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
```
```   522     (is "?f \<in> measurable ?P ?U")
```
```   523 proof (rule measurable_PiM_single)
```
```   524   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
```
```   525   then have "{\<omega> \<in> space ?P. prod_case (\<lambda>x y. merge I x J y) \<omega> i \<in> A} =
```
```   526     (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)"
```
```   527     by (auto simp: merge_def)
```
```   528   also have "\<dots> \<in> sets ?P"
```
```   529     using A
```
```   530     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   531   finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>x y. merge I x J y) \<omega> i \<in> A} \<in> sets ?P" .
```
```   532 qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def)
```
```   533
```
```   534 lemma measurable_restrict:
```
```   535   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
```
```   536   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
```
```   537 proof (rule measurable_PiM_single)
```
```   538   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   539   then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
```
```   540     by auto
```
```   541   then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
```
```   542     using A X by (auto intro!: measurable_sets)
```
```   543 qed (insert X, auto dest: measurable_space)
```
```   544
```
```   545 locale product_sigma_finite =
```
```   546   fixes M :: "'i \<Rightarrow> 'a measure"
```
```   547   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
```
```   548
```
```   549 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
```
```   550   by (rule sigma_finite_measures)
```
```   551
```
```   552 locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   553   fixes I :: "'i set"
```
```   554   assumes finite_index: "finite I"
```
```   555
```
```   556 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
```
```   557   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
```
```   558     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
```
```   559     (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
```
```   560     (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
```
```   561 proof -
```
```   562   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>)"
```
```   563     using M.sigma_finite_incseq by metis
```
```   564   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   565   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>"
```
```   566     by auto
```
```   567   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
```
```   568   note space_PiM[simp]
```
```   569   show ?thesis
```
```   570   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
```
```   571     fix i show "range (F i) \<subseteq> sets (M i)" by fact
```
```   572   next
```
```   573     fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
```
```   574   next
```
```   575     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)"
```
```   576       using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space
```
```   577       by auto blast
```
```   578   next
```
```   579     fix f assume "f \<in> space (PiM I M)"
```
```   580     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
```
```   581     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
```
```   582   next
```
```   583     fix i show "?F i \<subseteq> ?F (Suc i)"
```
```   584       using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
```
```   585   qed
```
```   586 qed
```
```   587
```
```   588 lemma (in product_sigma_finite) emeasure_PiM:
```
```   589   "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   590 proof (induct I arbitrary: A rule: finite_induct)
```
```   591   case empty
```
```   592   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
```
```   593   have "prod_algebra {} M = {{\<lambda>_. undefined}}"
```
```   594     by (auto simp: prod_algebra_def prod_emb_def intro!: image_eqI)
```
```   595   then have sets_empty: "sets (PiM {} M) = {{}, {\<lambda>_. undefined}}"
```
```   596     by (simp add: sets_PiM)
```
```   597   have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
```
```   598   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   599     have "finite (space (PiM {} M))"
```
```   600       by (simp add: space_PiM)
```
```   601     moreover show "positive (PiM {} M) ?\<mu>"
```
```   602       by (auto simp: positive_def)
```
```   603     ultimately show "countably_additive (PiM {} M) ?\<mu>"
```
```   604       by (rule countably_additiveI_finite) (auto simp: additive_def space_PiM sets_empty)
```
```   605   qed (auto simp: prod_emb_def)
```
```   606   also have *: "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
```
```   607     by (auto simp: prod_emb_def)
```
```   608   finally show ?case
```
```   609     using * by simp
```
```   610 next
```
```   611   case (insert i I)
```
```   612   interpret finite_product_sigma_finite M I by default fact
```
```   613   have "finite (insert i I)" using `finite I` by auto
```
```   614   interpret I': finite_product_sigma_finite M "insert i I" by default fact
```
```   615   let ?h = "(\<lambda>(f, y). f(i := y))"
```
```   616
```
```   617   let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
```
```   618   let ?\<mu> = "emeasure ?P"
```
```   619   let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
```
```   620   let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
```
```   621
```
```   622   have "emeasure (Pi\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>E (insert i I) A)) =
```
```   623     (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
```
```   624   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   625     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))"
```
```   626     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
```
```   627     let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
```
```   628     let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
```
```   629     have "?\<mu> ?p =
```
```   630       emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))"
```
```   631       by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
```
```   632     also have "?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
```
```   633       using J E[rule_format, THEN sets_into_space]
```
```   634       by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm)
```
```   635     also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
```
```   636       emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
```
```   637       using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
```
```   638     also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
```
```   639       using J E[rule_format, THEN sets_into_space]
```
```   640       by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+
```
```   641     also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
```
```   642       (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
```
```   643       using E by (subst insert) (auto intro!: setprod_cong)
```
```   644     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
```
```   645        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
```
```   646       using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
```
```   647     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
```
```   648       using insert(1,2) J E by (intro setprod_mono_one_right) auto
```
```   649     finally show "?\<mu> ?p = \<dots>" .
```
```   650
```
```   651     show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>E i\<in>insert i I. space (M i))"
```
```   652       using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff)
```
```   653   next
```
```   654     show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
```
```   655       using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
```
```   656   next
```
```   657     show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
```
```   658       insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
```
```   659       using insert by auto
```
```   660   qed (auto intro!: setprod_cong)
```
```   661   with insert show ?case
```
```   662     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space)
```
```   663 qed
```
```   664
```
```   665 lemma (in product_sigma_finite) sigma_finite:
```
```   666   assumes "finite I"
```
```   667   shows "sigma_finite_measure (PiM I M)"
```
```   668 proof -
```
```   669   interpret finite_product_sigma_finite M I by default fact
```
```   670
```
```   671   from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   672   then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
```
```   673     "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
```
```   674     "(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
```
```   675     "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
```
```   676     by blast+
```
```   677   let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"
```
```   678
```
```   679   show ?thesis
```
```   680   proof (unfold_locales, intro exI[of _ ?F] conjI allI)
```
```   681     show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
```
```   682   next
```
```   683     from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
```
```   684   next
```
```   685     fix j
```
```   686     from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
```
```   687     show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
```
```   688       by (subst emeasure_PiM) auto
```
```   689   qed
```
```   690 qed
```
```   691
```
```   692 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
```
```   693   using sigma_finite[OF finite_index] .
```
```   694
```
```   695 lemma (in finite_product_sigma_finite) measure_times:
```
```   696   "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^isub>M I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   697   using emeasure_PiM[OF finite_index] by auto
```
```   698
```
```   699 lemma (in product_sigma_finite) product_measure_empty[simp]:
```
```   700   "emeasure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
```
```   701 proof -
```
```   702   interpret finite_product_sigma_finite M "{}" by default auto
```
```   703   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
```
```   704 qed
```
```   705
```
```   706 lemma
```
```   707   shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
```
```   708     and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
```
```   709   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
```
```   710
```
```   711 lemma (in product_sigma_finite) positive_integral_empty:
```
```   712   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
```
```   713   shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
```
```   714 proof -
```
```   715   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
```
```   716   have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
```
```   717     using assms by (subst measure_times) auto
```
```   718   then show ?thesis
```
```   719     unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
```
```   720   proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
```
```   721     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
```
```   722       by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
```
```   723     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
```
```   724       by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
```
```   725   qed
```
```   726 qed
```
```   727
```
```   728 lemma (in product_sigma_finite) distr_merge:
```
```   729   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   730   shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (\<lambda>(x,y). merge I x J y) = Pi\<^isub>M (I \<union> J) M"
```
```   731    (is "?D = ?P")
```
```   732 proof -
```
```   733   interpret I: finite_product_sigma_finite M I by default fact
```
```   734   interpret J: finite_product_sigma_finite M J by default fact
```
```   735   have "finite (I \<union> J)" using fin by auto
```
```   736   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   737   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
```
```   738   let ?g = "\<lambda>(x,y). merge I x J y"
```
```   739
```
```   740   from IJ.sigma_finite_pairs obtain F where
```
```   741     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
```
```   742        "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
```
```   743        "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
```
```   744        "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
```
```   745     by auto
```
```   746   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
```
```   747
```
```   748   show ?thesis
```
```   749   proof (rule measure_eqI_generator_eq[symmetric])
```
```   750     show "Int_stable (prod_algebra (I \<union> J) M)"
```
```   751       by (rule Int_stable_prod_algebra)
```
```   752     show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
```
```   753       by (rule prod_algebra_sets_into_space)
```
```   754     show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   755       by (rule sets_PiM)
```
```   756     then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   757       by simp
```
```   758
```
```   759     show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
```
```   760       using fin by (auto simp: prod_algebra_eq_finite)
```
```   761     show "incseq ?F" by fact
```
```   762     show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))"
```
```   763       using F(3) by (simp add: space_PiM)
```
```   764   next
```
```   765     fix k
```
```   766     from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
```
```   767     show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
```
```   768   next
```
```   769     fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
```
```   770     with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>I \<union> J. F i \<in> sets (M i)"
```
```   771       by (auto simp add: prod_algebra_eq_finite)
```
```   772     let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
```
```   773     let ?X = "?g -` A \<inter> space ?B"
```
```   774     have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)"
```
```   775       using F[rule_format, THEN sets_into_space] by (auto simp: space_PiM)
```
```   776     then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
```
```   777       unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
```
```   778     have "emeasure ?D A = emeasure ?B ?X"
```
```   779       using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
```
```   780     also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
```
```   781       using `finite J` `finite I` F X
```
```   782       by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff)
```
```   783     also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
```
```   784       using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
```
```   785     also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
```
```   786       using `finite J` `finite I` F unfolding A
```
```   787       by (intro IJ.measure_times[symmetric]) auto
```
```   788     finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
```
```   789   qed
```
```   790 qed
```
```   791
```
```   792 lemma (in product_sigma_finite) product_positive_integral_fold:
```
```   793   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
```
```   794   and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
```
```   795   shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
```
```   796     (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
```
```   797 proof -
```
```   798   interpret I: finite_product_sigma_finite M I by default fact
```
```   799   interpret J: finite_product_sigma_finite M J by default fact
```
```   800   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
```
```   801   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
```
```   802     using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
```
```   803   show ?thesis
```
```   804     apply (subst distr_merge[OF IJ, symmetric])
```
```   805     apply (subst positive_integral_distr[OF measurable_merge f])
```
```   806     apply (subst P.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
```
```   807     apply simp
```
```   808     done
```
```   809 qed
```
```   810
```
```   811 lemma (in product_sigma_finite) distr_singleton:
```
```   812   "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
```
```   813 proof (intro measure_eqI[symmetric])
```
```   814   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   815   fix A assume A: "A \<in> sets (M i)"
```
```   816   moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
```
```   817     using sets_into_space by (auto simp: space_PiM)
```
```   818   ultimately show "emeasure (M i) A = emeasure ?D A"
```
```   819     using A I.measure_times[of "\<lambda>_. A"]
```
```   820     by (simp add: emeasure_distr measurable_component_singleton)
```
```   821 qed simp
```
```   822
```
```   823 lemma (in product_sigma_finite) product_positive_integral_singleton:
```
```   824   assumes f: "f \<in> borel_measurable (M i)"
```
```   825   shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
```
```   826 proof -
```
```   827   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   828   from f show ?thesis
```
```   829     apply (subst distr_singleton[symmetric])
```
```   830     apply (subst positive_integral_distr[OF measurable_component_singleton])
```
```   831     apply simp_all
```
```   832     done
```
```   833 qed
```
```   834
```
```   835 lemma (in product_sigma_finite) product_positive_integral_insert:
```
```   836   assumes [simp]: "finite I" "i \<notin> I"
```
```   837     and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
```
```   838   shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"
```
```   839 proof -
```
```   840   interpret I: finite_product_sigma_finite M I by default auto
```
```   841   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   842   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
```
```   843     using f by auto
```
```   844   show ?thesis
```
```   845     unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]
```
```   846   proof (rule positive_integral_cong, subst product_positive_integral_singleton)
```
```   847     fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
```
```   848     let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
```
```   849     have f'_eq: "\<And>y. ?f y = f (x(i := y))"
```
```   850       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def space_PiM)
```
```   851     show "?f \<in> borel_measurable (M i)" unfolding f'_eq
```
```   852       using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
```
```   853       unfolding comp_def .
```
```   854     show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"
```
```   855       unfolding f'_eq by simp
```
```   856   qed
```
```   857 qed
```
```   858
```
```   859 lemma (in product_sigma_finite) product_positive_integral_setprod:
```
```   860   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
```
```   861   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```   862   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
```
```   863   shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
```
```   864 using assms proof induct
```
```   865   case (insert i I)
```
```   866   note `finite I`[intro, simp]
```
```   867   interpret I: finite_product_sigma_finite M I by default auto
```
```   868   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))"
```
```   869     using insert by (auto intro!: setprod_cong)
```
```   870   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
```
```   871     using sets_into_space insert
```
```   872     by (intro borel_measurable_ereal_setprod
```
```   873               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
```
```   874        auto
```
```   875   then show ?case
```
```   876     apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
```
```   877     apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
```
```   878     apply (subst positive_integral_cmult)
```
```   879     apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
```
```   880     done
```
```   881 qed (simp add: space_PiM)
```
```   882
```
```   883 lemma (in product_sigma_finite) product_integral_singleton:
```
```   884   assumes f: "f \<in> borel_measurable (M i)"
```
```   885   shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
```
```   886 proof -
```
```   887   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   888   have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
```
```   889     "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
```
```   890     using assms by auto
```
```   891   show ?thesis
```
```   892     unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
```
```   893 qed
```
```   894
```
```   895 lemma (in product_sigma_finite) product_integral_fold:
```
```   896   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   897   and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
```
```   898   shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I x J y) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
```
```   899 proof -
```
```   900   interpret I: finite_product_sigma_finite M I by default fact
```
```   901   interpret J: finite_product_sigma_finite M J by default fact
```
```   902   have "finite (I \<union> J)" using fin by auto
```
```   903   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   904   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
```
```   905   let ?M = "\<lambda>(x, y). merge I x J y"
```
```   906   let ?f = "\<lambda>x. f (?M x)"
```
```   907   from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
```
```   908     by auto
```
```   909   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
```
```   910     using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
```
```   911   have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
```
```   912     by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
```
```   913   show ?thesis
```
```   914     apply (subst distr_merge[symmetric, OF IJ fin])
```
```   915     apply (subst integral_distr[OF measurable_merge f_borel])
```
```   916     apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
```
```   917     apply simp
```
```   918     done
```
```   919 qed
```
```   920
```
```   921 lemma (in product_sigma_finite)
```
```   922   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
```
```   923   shows emeasure_fold_integral:
```
```   924     "emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
```
```   925     and emeasure_fold_measurable:
```
```   926     "(\<lambda>x. emeasure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
```
```   927 proof -
```
```   928   interpret I: finite_product_sigma_finite M I by default fact
```
```   929   interpret J: finite_product_sigma_finite M J by default fact
```
```   930   interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
```
```   931   have merge: "(\<lambda>(x, y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
```
```   932     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   933
```
```   934   show ?I
```
```   935     apply (subst distr_merge[symmetric, OF IJ])
```
```   936     apply (subst emeasure_distr[OF measurable_merge A])
```
```   937     apply (subst J.emeasure_pair_measure_alt[OF merge])
```
```   938     apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```   939     done
```
```   940
```
```   941   show ?B
```
```   942     using IJ.measurable_emeasure_Pair1[OF merge]
```
```   943     by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
```
```   944 qed
```
```   945
```
```   946 lemma (in product_sigma_finite) product_integral_insert:
```
```   947   assumes I: "finite I" "i \<notin> I"
```
```   948     and f: "integrable (Pi\<^isub>M (insert i I) M) f"
```
```   949   shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
```
```   950 proof -
```
```   951   have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
```
```   952     by simp
```
```   953   also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I x {i} y) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)"
```
```   954     using f I by (intro product_integral_fold) auto
```
```   955   also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
```
```   956   proof (rule integral_cong, subst product_integral_singleton[symmetric])
```
```   957     fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
```
```   958     have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
```
```   959       using f by auto
```
```   960     show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
```
```   961       using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
```
```   962       unfolding comp_def .
```
```   963     from x I show "(\<integral> y. f (merge I x {i} y) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)"
```
```   964       by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def)
```
```   965   qed
```
```   966   finally show ?thesis .
```
```   967 qed
```
```   968
```
```   969 lemma (in product_sigma_finite) product_integrable_setprod:
```
```   970   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
```
```   971   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
```
```   972   shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
```
```   973 proof -
```
```   974   interpret finite_product_sigma_finite M I by default fact
```
```   975   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```   976     using integrable unfolding integrable_def by auto
```
```   977   have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
```
```   978     using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
```
```   979   moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
```
```   980   proof (unfold integrable_def, intro conjI)
```
```   981     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
```
```   982       using borel by auto
```
```   983     have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)"
```
```   984       by (simp add: setprod_ereal abs_setprod)
```
```   985     also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
```
```   986       using f by (subst product_positive_integral_setprod) auto
```
```   987     also have "\<dots> < \<infinity>"
```
```   988       using integrable[THEN integrable_abs]
```
```   989       by (simp add: setprod_PInf integrable_def positive_integral_positive)
```
```   990     finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
```
```   991     have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
```
```   992       by (intro positive_integral_cong_pos) auto
```
```   993     then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
```
```   994   qed
```
```   995   ultimately show ?thesis
```
```   996     by (rule integrable_abs_iff[THEN iffD1])
```
```   997 qed
```
```   998
```
```   999 lemma (in product_sigma_finite) product_integral_setprod:
```
```  1000   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
```
```  1001   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
```
```  1002   shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"
```
```  1003 using assms proof (induct rule: finite_ne_induct)
```
```  1004   case (singleton i)
```
```  1005   then show ?case by (simp add: product_integral_singleton integrable_def)
```
```  1006 next
```
```  1007   case (insert i I)
```
```  1008   then have iI: "finite (insert i I)" by auto
```
```  1009   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
```
```  1010     integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
```
```  1011     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)
```
```  1012   interpret I: finite_product_sigma_finite M I by default fact
```
```  1013   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))"
```
```  1014     using `i \<notin> I` by (auto intro!: setprod_cong)
```
```  1015   show ?case
```
```  1016     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
```
```  1017     by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
```
```  1018 qed
```
```  1019
```
```  1020 lemma sets_Collect_single:
```
```  1021   "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^isub>M I M). x i \<in> A } \<in> sets (Pi\<^isub>M I M)"
```
```  1022   unfolding sets_PiM_single
```
```  1023   by (auto intro!: sigma_sets.Basic exI[of _ i] exI[of _ A]) (auto simp: space_PiM)
```
```  1024
```
```  1025 lemma sigma_prod_algebra_sigma_eq_infinite:
```
```  1026   fixes E :: "'i \<Rightarrow> 'a set set"
```
```  1027   assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```  1028     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```  1029   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```  1030     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```  1031   defines "P == {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}"
```
```  1032   shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
```
```  1033 proof
```
```  1034   let ?P = "sigma (space (Pi\<^isub>M I M)) P"
```
```  1035   have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
```
```  1036     using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
```
```  1037   then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```  1038     by (simp add: space_PiM)
```
```  1039   have "sets (PiM I M) =
```
```  1040       sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```  1041     using sets_PiM_single[of I M] by (simp add: space_P)
```
```  1042   also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
```
```  1043   proof (safe intro!: sigma_sets_subset)
```
```  1044     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```  1045     then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```  1046       apply (subst measurable_iff_measure_of)
```
```  1047       apply (simp_all add: P_closed)
```
```  1048       using E_closed
```
```  1049       apply (force simp: subset_eq space_PiM)
```
```  1050       apply (force simp: subset_eq space_PiM)
```
```  1051       apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
```
```  1052       apply (rule_tac x=Aa in exI)
```
```  1053       apply (auto simp: space_PiM)
```
```  1054       done
```
```  1055     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```  1056     have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1057       by (simp add: E_generates)
```
```  1058     also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
```
```  1059       using P_closed by (auto simp: space_PiM)
```
```  1060     finally show "\<dots> \<in> sets ?P" .
```
```  1061   qed
```
```  1062   finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
```
```  1063     by (simp add: P_closed)
```
```  1064   show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
```
```  1065     unfolding P_def space_PiM[symmetric]
```
```  1066     by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
```
```  1067 qed
```
```  1068
```
```  1069 lemma bchoice_iff: "(\<forall>a\<in>A. \<exists>b. P a b) \<longleftrightarrow> (\<exists>f. \<forall>a\<in>A. P a (f a))"
```
```  1070   by metis
```
```  1071
```
```  1072 lemma sigma_prod_algebra_sigma_eq:
```
```  1073   fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
```
```  1074   assumes "finite I"
```
```  1075   assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```  1076     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```  1077   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```  1078     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```  1079   defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
```
```  1080   shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
```
```  1081 proof
```
```  1082   let ?P = "sigma (space (Pi\<^isub>M I M)) P"
```
```  1083   from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
```
```  1084   then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i"
```
```  1085     by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
```
```  1086   have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
```
```  1087     using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
```
```  1088   then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```  1089     by (simp add: space_PiM)
```
```  1090   have "sets (PiM I M) =
```
```  1091       sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```  1092     using sets_PiM_single[of I M] by (simp add: space_P)
```
```  1093   also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
```
```  1094   proof (safe intro!: sigma_sets_subset)
```
```  1095     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```  1096     have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```  1097     proof (subst measurable_iff_measure_of)
```
```  1098       show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
```
```  1099       from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)"
```
```  1100         by (auto simp: Pi_iff)
```
```  1101       show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1102       proof
```
```  1103         fix A assume A: "A \<in> E i"
```
```  1104         then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
```
```  1105           using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
```
```  1106         also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
```
```  1107           by (intro PiE_cong) (simp add: S_union)
```
```  1108         also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j))"
```
```  1109           using T
```
```  1110           apply (auto simp: Pi_iff bchoice_iff)
```
```  1111           apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
```
```  1112           apply (auto simp: bij_betw_def)
```
```  1113           done
```
```  1114         also have "\<dots> \<in> sets ?P"
```
```  1115         proof (safe intro!: countable_UN)
```
```  1116           fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
```
```  1117             using A S_in_E
```
```  1118             by (simp add: P_closed)
```
```  1119                (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
```
```  1120         qed
```
```  1121         finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1122           using P_closed by simp
```
```  1123       qed
```
```  1124     qed
```
```  1125     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```  1126     have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1127       by (simp add: E_generates)
```
```  1128     also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}"
```
```  1129       using P_closed by (auto simp: space_PiM)
```
```  1130     finally show "\<dots> \<in> sets ?P" .
```
```  1131   qed
```
```  1132   finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
```
```  1133     by (simp add: P_closed)
```
```  1134   show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
```
```  1135     using `finite I`
```
```  1136     by (auto intro!: sigma_sets_subset simp: E_generates P_def)
```
```  1137 qed
```
```  1138
```
```  1139 end
```