src/HOL/Probability/Finite_Product_Measure.thy
 author hoelzl Mon Nov 19 12:29:02 2012 +0100 (2012-11-19) changeset 50123 69b35a75caf3 parent 50104 de19856feb54 child 50244 de72bbe42190 permissions -rw-r--r--
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
```     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 subsubsection {* Merge two extensional functions *}
```
```    15
```
```    16 definition
```
```    17   "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
```
```    18
```
```    19 lemma merge_apply[simp]:
```
```    20   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
```
```    21   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
```
```    22   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
```
```    23   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
```
```    24   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined"
```
```    25   unfolding merge_def by auto
```
```    26
```
```    27 lemma merge_commute:
```
```    28   "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)"
```
```    29   by (force simp: merge_def)
```
```    30
```
```    31 lemma Pi_cancel_merge_range[simp]:
```
```    32   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
```
```    33   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
```
```    34   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
```
```    35   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
```
```    36   by (auto simp: Pi_def)
```
```    37
```
```    38 lemma Pi_cancel_merge[simp]:
```
```    39   "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```    40   "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```    41   "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
```
```    42   "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
```
```    43   by (auto simp: Pi_def)
```
```    44
```
```    45 lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)"
```
```    46   by (auto simp: extensional_def)
```
```    47
```
```    48 lemma restrict_merge[simp]:
```
```    49   "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
```
```    50   "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
```
```    51   "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
```
```    52   "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
```
```    53   by (auto simp: restrict_def)
```
```    54
```
```    55 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)"
```
```    56   unfolding merge_def by auto
```
```    57
```
```    58 lemma PiE_cancel_merge[simp]:
```
```    59   "I \<inter> J = {} \<Longrightarrow>
```
```    60     merge I J (x, y) \<in> PiE (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B"
```
```    61   by (auto simp: PiE_def restrict_Pi_cancel)
```
```    62
```
```    63 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)"
```
```    64   unfolding merge_def by (auto simp: fun_eq_iff)
```
```    65
```
```    66 lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K"
```
```    67   unfolding merge_def extensional_def by auto
```
```    68
```
```    69 lemma merge_restrict[simp]:
```
```    70   "merge I J (restrict x I, y) = merge I J (x, y)"
```
```    71   "merge I J (x, restrict y J) = merge I J (x, y)"
```
```    72   unfolding merge_def by auto
```
```    73
```
```    74 lemma merge_x_x_eq_restrict[simp]:
```
```    75   "merge I J (x, x) = restrict x (I \<union> J)"
```
```    76   unfolding merge_def by auto
```
```    77
```
```    78 lemma injective_vimage_restrict:
```
```    79   assumes J: "J \<subseteq> I"
```
```    80   and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}"
```
```    81   and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
```
```    82   shows "A = B"
```
```    83 proof  (intro set_eqI)
```
```    84   fix x
```
```    85   from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
```
```    86   have "J \<inter> (I - J) = {}" by auto
```
```    87   show "x \<in> A \<longleftrightarrow> x \<in> B"
```
```    88   proof cases
```
```    89     assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)"
```
```    90     have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
```
```    91       using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S]
```
```    92       by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
```
```    93     then show "x \<in> A \<longleftrightarrow> x \<in> B"
```
```    94       using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S]
```
```    95       by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq)
```
```    96   qed (insert sets, auto)
```
```    97 qed
```
```    98
```
```    99 lemma restrict_vimage:
```
```   100   "I \<inter> J = {} \<Longrightarrow>
```
```   101     (\<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 J (E, F))"
```
```   102   by (auto simp: restrict_Pi_cancel PiE_def)
```
```   103
```
```   104 lemma merge_vimage:
```
```   105   "I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
```
```   106   by (auto simp: restrict_Pi_cancel PiE_def)
```
```   107
```
```   108 section "Finite product spaces"
```
```   109
```
```   110 section "Products"
```
```   111
```
```   112 definition prod_emb where
```
```   113   "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
```
```   114
```
```   115 lemma prod_emb_iff:
```
```   116   "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))"
```
```   117   unfolding prod_emb_def PiE_def by auto
```
```   118
```
```   119 lemma
```
```   120   shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
```
```   121     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"
```
```   122     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"
```
```   123     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))"
```
```   124     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))"
```
```   125     and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
```
```   126   by (auto simp: prod_emb_def)
```
```   127
```
```   128 lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
```
```   129     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))"
```
```   130   by (force simp: prod_emb_def PiE_iff split_if_mem2)
```
```   131
```
```   132 lemma prod_emb_PiE_same_index[simp]:
```
```   133     "(\<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"
```
```   134   by (auto simp: prod_emb_def PiE_iff)
```
```   135
```
```   136 lemma prod_emb_trans[simp]:
```
```   137   "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"
```
```   138   by (auto simp add: Int_absorb1 prod_emb_def PiE_def)
```
```   139
```
```   140 lemma prod_emb_Pi:
```
```   141   assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
```
```   142   shows "prod_emb K M J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))"
```
```   143   using assms space_closed
```
```   144   by (auto simp: prod_emb_def PiE_iff split: split_if_asm) blast+
```
```   145
```
```   146 lemma prod_emb_id:
```
```   147   "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
```
```   148   by (auto simp: prod_emb_def subset_eq extensional_restrict)
```
```   149
```
```   150 lemma prod_emb_mono:
```
```   151   "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
```
```   152   by (auto simp: prod_emb_def)
```
```   153
```
```   154 definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
```
```   155   "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
```
```   156     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
```
```   157     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
```
```   158     (\<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)))"
```
```   159
```
```   160 definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
```
```   161   "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
```
```   162     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
```
```   163
```
```   164 abbreviation
```
```   165   "Pi\<^isub>M I M \<equiv> PiM I M"
```
```   166
```
```   167 syntax
```
```   168   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
```
```   169
```
```   170 syntax (xsymbols)
```
```   171   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
```
```   172
```
```   173 syntax (HTML output)
```
```   174   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
```
```   175
```
```   176 translations
```
```   177   "PIM x:I. M" == "CONST PiM I (%x. M)"
```
```   178
```
```   179 lemma prod_algebra_sets_into_space:
```
```   180   "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   181   using assms by (auto simp: prod_emb_def prod_algebra_def)
```
```   182
```
```   183 lemma prod_algebra_eq_finite:
```
```   184   assumes I: "finite I"
```
```   185   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")
```
```   186 proof (intro iffI set_eqI)
```
```   187   fix A assume "A \<in> ?L"
```
```   188   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)"
```
```   189     and A: "A = prod_emb I M J (PIE j:J. E j)"
```
```   190     by (auto simp: prod_algebra_def)
```
```   191   let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
```
```   192   have A: "A = ?A"
```
```   193     unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
```
```   194   show "A \<in> ?R" unfolding A using J top
```
```   195     by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
```
```   196 next
```
```   197   fix A assume "A \<in> ?R"
```
```   198   then obtain X where A: "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
```
```   199   then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
```
```   200     by (simp add: prod_emb_PiE_same_index[OF sets_into_space] Pi_iff)
```
```   201   from X I show "A \<in> ?L" unfolding A
```
```   202     by (auto simp: prod_algebra_def)
```
```   203 qed
```
```   204
```
```   205 lemma prod_algebraI:
```
```   206   "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
```
```   207     \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
```
```   208   by (auto simp: prod_algebra_def)
```
```   209
```
```   210 lemma prod_algebraI_finite:
```
```   211   "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M"
```
```   212   using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp
```
```   213
```
```   214 lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
```
```   215 proof (safe intro!: Int_stableI)
```
```   216   fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
```
```   217   then show "\<exists>G. Pi\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
```
```   218     by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
```
```   219 qed
```
```   220
```
```   221 lemma prod_algebraE:
```
```   222   assumes A: "A \<in> prod_algebra I M"
```
```   223   obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
```
```   224     "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
```
```   225   using A by (auto simp: prod_algebra_def)
```
```   226
```
```   227 lemma prod_algebraE_all:
```
```   228   assumes A: "A \<in> prod_algebra I M"
```
```   229   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   230 proof -
```
```   231   from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
```
```   232     and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
```
```   233     by (auto simp: prod_algebra_def)
```
```   234   from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
```
```   235     using sets_into_space by auto
```
```   236   then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
```
```   237     using A J by (auto simp: prod_emb_PiE)
```
```   238   moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   239     using top E by auto
```
```   240   ultimately show ?thesis using that by auto
```
```   241 qed
```
```   242
```
```   243 lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
```
```   244 proof (unfold Int_stable_def, safe)
```
```   245   fix A assume "A \<in> prod_algebra I M"
```
```   246   from prod_algebraE[OF this] guess J E . note A = this
```
```   247   fix B assume "B \<in> prod_algebra I M"
```
```   248   from prod_algebraE[OF this] guess K F . note B = this
```
```   249   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>
```
```   250       (if i \<in> K then F i else space (M i)))"
```
```   251     unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space]
```
```   252     apply (subst (1 2 3) prod_emb_PiE)
```
```   253     apply (simp_all add: subset_eq PiE_Int)
```
```   254     apply blast
```
```   255     apply (intro PiE_cong)
```
```   256     apply auto
```
```   257     done
```
```   258   also have "\<dots> \<in> prod_algebra I M"
```
```   259     using A B by (auto intro!: prod_algebraI)
```
```   260   finally show "A \<inter> B \<in> prod_algebra I M" .
```
```   261 qed
```
```   262
```
```   263 lemma prod_algebra_mono:
```
```   264   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
```
```   265   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
```
```   266   shows "prod_algebra I E \<subseteq> prod_algebra I F"
```
```   267 proof
```
```   268   fix A assume "A \<in> prod_algebra I E"
```
```   269   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
```
```   270     and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
```
```   271     and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
```
```   272     by (auto simp: prod_algebra_def)
```
```   273   moreover
```
```   274   from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
```
```   275     by (rule PiE_cong)
```
```   276   with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
```
```   277     by (simp add: prod_emb_def)
```
```   278   moreover
```
```   279   from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
```
```   280     by auto
```
```   281   ultimately show "A \<in> prod_algebra I F"
```
```   282     apply (simp add: prod_algebra_def image_iff)
```
```   283     apply (intro exI[of _ J] exI[of _ G] conjI)
```
```   284     apply auto
```
```   285     done
```
```   286 qed
```
```   287
```
```   288 lemma prod_algebra_cong:
```
```   289   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
```
```   290   shows "prod_algebra I M = prod_algebra J N"
```
```   291 proof -
```
```   292   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
```
```   293     using sets_eq_imp_space_eq[OF sets] by auto
```
```   294   with sets show ?thesis unfolding `I = J`
```
```   295     by (intro antisym prod_algebra_mono) auto
```
```   296 qed
```
```   297
```
```   298 lemma space_in_prod_algebra:
```
```   299   "(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
```
```   300 proof cases
```
```   301   assume "I = {}" then show ?thesis
```
```   302     by (auto simp add: prod_algebra_def image_iff prod_emb_def)
```
```   303 next
```
```   304   assume "I \<noteq> {}"
```
```   305   then obtain i where "i \<in> I" by auto
```
```   306   then have "(\<Pi>\<^isub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i))"
```
```   307     by (auto simp: prod_emb_def)
```
```   308   also have "\<dots> \<in> prod_algebra I M"
```
```   309     using `i \<in> I` by (intro prod_algebraI) auto
```
```   310   finally show ?thesis .
```
```   311 qed
```
```   312
```
```   313 lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   314   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
```
```   315
```
```   316 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)"
```
```   317   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
```
```   318
```
```   319 lemma sets_PiM_single: "sets (PiM I M) =
```
```   320     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)}"
```
```   321     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   322   unfolding sets_PiM
```
```   323 proof (rule sigma_sets_eqI)
```
```   324   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   325   fix A assume "A \<in> prod_algebra I M"
```
```   326   from prod_algebraE[OF this] guess J X . note X = this
```
```   327   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   328   proof cases
```
```   329     assume "I = {}"
```
```   330     with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
```
```   331     with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
```
```   332   next
```
```   333     assume "I \<noteq> {}"
```
```   334     with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})"
```
```   335       by (auto simp: prod_emb_def)
```
```   336     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```   337       using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
```
```   338     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```   339   qed
```
```   340 next
```
```   341   fix A assume "A \<in> ?R"
```
```   342   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)"
```
```   343     by auto
```
```   344   then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
```
```   345      by (auto simp: prod_emb_def)
```
```   346   also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
```
```   347     using A by (intro sigma_sets.Basic prod_algebraI) auto
```
```   348   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
```
```   349 qed
```
```   350
```
```   351 lemma sets_PiM_I:
```
```   352   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   353   shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
```
```   354 proof cases
```
```   355   assume "J = {}"
```
```   356   then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
```
```   357     by (auto simp: prod_emb_def)
```
```   358   then show ?thesis
```
```   359     by (auto simp add: sets_PiM intro!: sigma_sets_top)
```
```   360 next
```
```   361   assume "J \<noteq> {}" with assms show ?thesis
```
```   362     by (force simp add: sets_PiM prod_algebra_def)
```
```   363 qed
```
```   364
```
```   365 lemma measurable_PiM:
```
```   366   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   367   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>
```
```   368     f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N"
```
```   369   shows "f \<in> measurable N (PiM I M)"
```
```   370   using sets_PiM prod_algebra_sets_into_space space
```
```   371 proof (rule measurable_sigma_sets)
```
```   372   fix A assume "A \<in> prod_algebra I M"
```
```   373   from prod_algebraE[OF this] guess J X .
```
```   374   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
```
```   375 qed
```
```   376
```
```   377 lemma measurable_PiM_Collect:
```
```   378   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   379   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>
```
```   380     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
```
```   381   shows "f \<in> measurable N (PiM I M)"
```
```   382   using sets_PiM prod_algebra_sets_into_space space
```
```   383 proof (rule measurable_sigma_sets)
```
```   384   fix A assume "A \<in> prod_algebra I M"
```
```   385   from prod_algebraE[OF this] guess J X . note X = this
```
```   386   then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
```
```   387     using space by (auto simp: prod_emb_def del: PiE_I)
```
```   388   also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
```
```   389   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   390 qed
```
```   391
```
```   392 lemma measurable_PiM_single:
```
```   393   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   394   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"
```
```   395   shows "f \<in> measurable N (PiM I M)"
```
```   396   using sets_PiM_single
```
```   397 proof (rule measurable_sigma_sets)
```
```   398   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)}"
```
```   399   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)"
```
```   400     by auto
```
```   401   with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
```
```   402   also have "\<dots> \<in> sets N" using B by (rule sets)
```
```   403   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   404 qed (auto simp: space)
```
```   405
```
```   406 lemma measurable_PiM_single':
```
```   407   assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
```
```   408     and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```   409   shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^isub>M I M)"
```
```   410 proof (rule measurable_PiM_single)
```
```   411   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   412   then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
```
```   413     by auto
```
```   414   then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
```
```   415     using A f by (auto intro!: measurable_sets)
```
```   416 qed fact
```
```   417
```
```   418 lemma sets_PiM_I_finite[measurable]:
```
```   419   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
```
```   420   shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
```
```   421   using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
```
```   422
```
```   423 lemma measurable_component_singleton:
```
```   424   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
```
```   425 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   426   fix A assume "A \<in> sets (M i)"
```
```   427   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)"
```
```   428     using sets_into_space `i \<in> I`
```
```   429     by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
```
```   430   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
```
```   431     using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
```
```   432 qed (insert `i \<in> I`, auto simp: space_PiM)
```
```   433
```
```   434 lemma measurable_component_singleton'[measurable_app]:
```
```   435   assumes f: "f \<in> measurable N (Pi\<^isub>M I M)"
```
```   436   assumes i: "i \<in> I"
```
```   437   shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
```
```   438   using measurable_compose[OF f measurable_component_singleton, OF i] .
```
```   439
```
```   440 lemma measurable_PiM_component_rev[measurable (raw)]:
```
```   441   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
```
```   442   by simp
```
```   443
```
```   444 lemma measurable_nat_case[measurable (raw)]:
```
```   445   assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
```
```   446     "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
```
```   447   shows "(\<lambda>x. nat_case (f x) (g x) i) \<in> measurable M N"
```
```   448   by (cases i) simp_all
```
```   449
```
```   450 lemma measurable_nat_case'[measurable (raw)]:
```
```   451   assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
```
```   452   shows "(\<lambda>x. nat_case (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)"
```
```   453   using fg[THEN measurable_space]
```
```   454   by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
```
```   455
```
```   456 lemma measurable_add_dim[measurable]:
```
```   457   "(\<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)"
```
```   458     (is "?f \<in> measurable ?P ?I")
```
```   459 proof (rule measurable_PiM_single)
```
```   460   fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
```
```   461   have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
```
```   462     (if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
```
```   463     using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
```
```   464   also have "\<dots> \<in> sets ?P"
```
```   465     using A j
```
```   466     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   467   finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
```
```   468 qed (auto simp: space_pair_measure space_PiM PiE_def)
```
```   469
```
```   470 lemma measurable_component_update:
```
```   471   "x \<in> space (Pi\<^isub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)"
```
```   472   by simp
```
```   473
```
```   474 lemma measurable_merge[measurable]:
```
```   475   "merge I J \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
```
```   476     (is "?f \<in> measurable ?P ?U")
```
```   477 proof (rule measurable_PiM_single)
```
```   478   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
```
```   479   then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
```
```   480     (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)"
```
```   481     by (auto simp: merge_def)
```
```   482   also have "\<dots> \<in> sets ?P"
```
```   483     using A
```
```   484     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   485   finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
```
```   486 qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
```
```   487
```
```   488 lemma measurable_restrict[measurable (raw)]:
```
```   489   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
```
```   490   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
```
```   491 proof (rule measurable_PiM_single)
```
```   492   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   493   then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
```
```   494     by auto
```
```   495   then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
```
```   496     using A X by (auto intro!: measurable_sets)
```
```   497 qed (insert X, auto simp add: PiE_def dest: measurable_space)
```
```   498
```
```   499 lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)"
```
```   500   by (intro measurable_restrict measurable_component_singleton) auto
```
```   501
```
```   502 lemma measurable_prod_emb[intro, simp]:
```
```   503   "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^isub>M L M)"
```
```   504   unfolding prod_emb_def space_PiM[symmetric]
```
```   505   by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
```
```   506
```
```   507 lemma sets_in_Pi_aux:
```
```   508   "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   509   {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
```
```   510   by (simp add: subset_eq Pi_iff)
```
```   511
```
```   512 lemma sets_in_Pi[measurable (raw)]:
```
```   513   "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
```
```   514   (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   515   Sigma_Algebra.pred N (\<lambda>x. f x \<in> Pi I F)"
```
```   516   unfolding pred_def
```
```   517   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
```
```   518
```
```   519 lemma sets_in_extensional_aux:
```
```   520   "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
```
```   521 proof -
```
```   522   have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
```
```   523     by (auto simp add: extensional_def space_PiM)
```
```   524   then show ?thesis by simp
```
```   525 qed
```
```   526
```
```   527 lemma sets_in_extensional[measurable (raw)]:
```
```   528   "f \<in> measurable N (PiM I M) \<Longrightarrow> Sigma_Algebra.pred N (\<lambda>x. f x \<in> extensional I)"
```
```   529   unfolding pred_def
```
```   530   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
```
```   531
```
```   532 locale product_sigma_finite =
```
```   533   fixes M :: "'i \<Rightarrow> 'a measure"
```
```   534   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
```
```   535
```
```   536 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
```
```   537   by (rule sigma_finite_measures)
```
```   538
```
```   539 locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   540   fixes I :: "'i set"
```
```   541   assumes finite_index: "finite I"
```
```   542
```
```   543 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
```
```   544   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
```
```   545     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
```
```   546     (\<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>
```
```   547     (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
```
```   548 proof -
```
```   549   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>)"
```
```   550     using M.sigma_finite_incseq by metis
```
```   551   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   552   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>"
```
```   553     by auto
```
```   554   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
```
```   555   note space_PiM[simp]
```
```   556   show ?thesis
```
```   557   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
```
```   558     fix i show "range (F i) \<subseteq> sets (M i)" by fact
```
```   559   next
```
```   560     fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
```
```   561   next
```
```   562     fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
```
```   563       by (auto simp: PiE_def dest!: sets_into_space)
```
```   564   next
```
```   565     fix f assume "f \<in> space (PiM I M)"
```
```   566     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
```
```   567     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
```
```   568   next
```
```   569     fix i show "?F i \<subseteq> ?F (Suc i)"
```
```   570       using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
```
```   571   qed
```
```   572 qed
```
```   573
```
```   574 lemma
```
```   575   shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
```
```   576     and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
```
```   577   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
```
```   578
```
```   579 lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
```
```   580 proof -
```
```   581   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
```
```   582   have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
```
```   583   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   584     show "positive (PiM {} M) ?\<mu>"
```
```   585       by (auto simp: positive_def)
```
```   586     show "countably_additive (PiM {} M) ?\<mu>"
```
```   587       by (rule countably_additiveI_finite)
```
```   588          (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
```
```   589   qed (auto simp: prod_emb_def)
```
```   590   also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
```
```   591     by (auto simp: prod_emb_def)
```
```   592   finally show ?thesis
```
```   593     by simp
```
```   594 qed
```
```   595
```
```   596 lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
```
```   597   by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
```
```   598
```
```   599 lemma (in product_sigma_finite) emeasure_PiM:
```
```   600   "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))"
```
```   601 proof (induct I arbitrary: A rule: finite_induct)
```
```   602   case (insert i I)
```
```   603   interpret finite_product_sigma_finite M I by default fact
```
```   604   have "finite (insert i I)" using `finite I` by auto
```
```   605   interpret I': finite_product_sigma_finite M "insert i I" by default fact
```
```   606   let ?h = "(\<lambda>(f, y). f(i := y))"
```
```   607
```
```   608   let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
```
```   609   let ?\<mu> = "emeasure ?P"
```
```   610   let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
```
```   611   let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
```
```   612
```
```   613   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)) =
```
```   614     (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
```
```   615   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   616     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))"
```
```   617     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
```
```   618     let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
```
```   619     let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
```
```   620     have "?\<mu> ?p =
```
```   621       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))"
```
```   622       by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
```
```   623     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))"
```
```   624       using J E[rule_format, THEN sets_into_space]
```
```   625       by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
```
```   626     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))) =
```
```   627       emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
```
```   628       using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
```
```   629     also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
```
```   630       using J E[rule_format, THEN sets_into_space]
```
```   631       by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
```
```   632     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)) =
```
```   633       (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
```
```   634       using E by (subst insert) (auto intro!: setprod_cong)
```
```   635     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
```
```   636        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
```
```   637       using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
```
```   638     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
```
```   639       using insert(1,2) J E by (intro setprod_mono_one_right) auto
```
```   640     finally show "?\<mu> ?p = \<dots>" .
```
```   641
```
```   642     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))"
```
```   643       using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff PiE_def)
```
```   644   next
```
```   645     show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
```
```   646       using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
```
```   647   next
```
```   648     show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
```
```   649       insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
```
```   650       using insert by auto
```
```   651   qed (auto intro!: setprod_cong)
```
```   652   with insert show ?case
```
```   653     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space)
```
```   654 qed simp
```
```   655
```
```   656 lemma (in product_sigma_finite) sigma_finite:
```
```   657   assumes "finite I"
```
```   658   shows "sigma_finite_measure (PiM I M)"
```
```   659 proof -
```
```   660   interpret finite_product_sigma_finite M I by default fact
```
```   661
```
```   662   from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   663   then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
```
```   664     "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
```
```   665     "(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
```
```   666     "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
```
```   667     by blast+
```
```   668   let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"
```
```   669
```
```   670   show ?thesis
```
```   671   proof (unfold_locales, intro exI[of _ ?F] conjI allI)
```
```   672     show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
```
```   673   next
```
```   674     from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
```
```   675   next
```
```   676     fix j
```
```   677     from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
```
```   678     show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
```
```   679       by (subst emeasure_PiM) auto
```
```   680   qed
```
```   681 qed
```
```   682
```
```   683 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
```
```   684   using sigma_finite[OF finite_index] .
```
```   685
```
```   686 lemma (in finite_product_sigma_finite) measure_times:
```
```   687   "(\<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))"
```
```   688   using emeasure_PiM[OF finite_index] by auto
```
```   689
```
```   690 lemma (in product_sigma_finite) positive_integral_empty:
```
```   691   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
```
```   692   shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
```
```   693 proof -
```
```   694   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
```
```   695   have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
```
```   696     using assms by (subst measure_times) auto
```
```   697   then show ?thesis
```
```   698     unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
```
```   699   proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
```
```   700     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
```
```   701       by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
```
```   702     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
```
```   703       by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
```
```   704   qed
```
```   705 qed
```
```   706
```
```   707 lemma (in product_sigma_finite) distr_merge:
```
```   708   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   709   shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (merge I J) = Pi\<^isub>M (I \<union> J) M"
```
```   710    (is "?D = ?P")
```
```   711 proof -
```
```   712   interpret I: finite_product_sigma_finite M I by default fact
```
```   713   interpret J: finite_product_sigma_finite M J by default fact
```
```   714   have "finite (I \<union> J)" using fin by auto
```
```   715   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   716   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
```
```   717   let ?g = "merge I J"
```
```   718
```
```   719   from IJ.sigma_finite_pairs obtain F where
```
```   720     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
```
```   721        "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
```
```   722        "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
```
```   723        "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
```
```   724     by auto
```
```   725   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
```
```   726
```
```   727   show ?thesis
```
```   728   proof (rule measure_eqI_generator_eq[symmetric])
```
```   729     show "Int_stable (prod_algebra (I \<union> J) M)"
```
```   730       by (rule Int_stable_prod_algebra)
```
```   731     show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
```
```   732       by (rule prod_algebra_sets_into_space)
```
```   733     show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   734       by (rule sets_PiM)
```
```   735     then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   736       by simp
```
```   737
```
```   738     show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
```
```   739       using fin by (auto simp: prod_algebra_eq_finite)
```
```   740     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))"
```
```   741       using F(3) by (simp add: space_PiM)
```
```   742   next
```
```   743     fix k
```
```   744     from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
```
```   745     show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
```
```   746   next
```
```   747     fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
```
```   748     with fin obtain F where A_eq: "A = (Pi\<^isub>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)"
```
```   749       by (auto simp add: prod_algebra_eq_finite)
```
```   750     let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
```
```   751     let ?X = "?g -` A \<inter> space ?B"
```
```   752     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)"
```
```   753       using F[rule_format, THEN sets_into_space] by (force simp: space_PiM)+
```
```   754     then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
```
```   755       unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
```
```   756     have "emeasure ?D A = emeasure ?B ?X"
```
```   757       using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
```
```   758     also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
```
```   759       using `finite J` `finite I` F unfolding X
```
```   760       by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
```
```   761     also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
```
```   762       using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
```
```   763     also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
```
```   764       using `finite J` `finite I` F unfolding A
```
```   765       by (intro IJ.measure_times[symmetric]) auto
```
```   766     finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
```
```   767   qed
```
```   768 qed
```
```   769
```
```   770 lemma (in product_sigma_finite) product_positive_integral_fold:
```
```   771   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
```
```   772   and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
```
```   773   shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
```
```   774     (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
```
```   775 proof -
```
```   776   interpret I: finite_product_sigma_finite M I by default fact
```
```   777   interpret J: finite_product_sigma_finite M J by default fact
```
```   778   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
```
```   779   have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
```
```   780     using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
```
```   781   show ?thesis
```
```   782     apply (subst distr_merge[OF IJ, symmetric])
```
```   783     apply (subst positive_integral_distr[OF measurable_merge f])
```
```   784     apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
```
```   785     apply simp
```
```   786     done
```
```   787 qed
```
```   788
```
```   789 lemma (in product_sigma_finite) distr_singleton:
```
```   790   "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
```
```   791 proof (intro measure_eqI[symmetric])
```
```   792   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   793   fix A assume A: "A \<in> sets (M i)"
```
```   794   moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
```
```   795     using sets_into_space by (auto simp: space_PiM)
```
```   796   ultimately show "emeasure (M i) A = emeasure ?D A"
```
```   797     using A I.measure_times[of "\<lambda>_. A"]
```
```   798     by (simp add: emeasure_distr measurable_component_singleton)
```
```   799 qed simp
```
```   800
```
```   801 lemma (in product_sigma_finite) product_positive_integral_singleton:
```
```   802   assumes f: "f \<in> borel_measurable (M i)"
```
```   803   shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
```
```   804 proof -
```
```   805   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   806   from f show ?thesis
```
```   807     apply (subst distr_singleton[symmetric])
```
```   808     apply (subst positive_integral_distr[OF measurable_component_singleton])
```
```   809     apply simp_all
```
```   810     done
```
```   811 qed
```
```   812
```
```   813 lemma (in product_sigma_finite) product_positive_integral_insert:
```
```   814   assumes I[simp]: "finite I" "i \<notin> I"
```
```   815     and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
```
```   816   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))"
```
```   817 proof -
```
```   818   interpret I: finite_product_sigma_finite M I by default auto
```
```   819   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   820   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
```
```   821     using f by auto
```
```   822   show ?thesis
```
```   823     unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
```
```   824   proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
```
```   825     fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
```
```   826     let ?f = "\<lambda>y. f (x(i := y))"
```
```   827     show "?f \<in> borel_measurable (M i)"
```
```   828       using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
```
```   829       unfolding comp_def .
```
```   830     show "(\<integral>\<^isup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^isub>M {i} M) = (\<integral>\<^isup>+ y. f (x(i := y i)) \<partial>Pi\<^isub>M {i} M)"
```
```   831       using x
```
```   832       by (auto intro!: positive_integral_cong arg_cong[where f=f]
```
```   833                simp add: space_PiM extensional_def PiE_def)
```
```   834   qed
```
```   835 qed
```
```   836
```
```   837 lemma (in product_sigma_finite) product_positive_integral_setprod:
```
```   838   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
```
```   839   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```   840   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
```
```   841   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))"
```
```   842 using assms proof induct
```
```   843   case (insert i I)
```
```   844   note `finite I`[intro, simp]
```
```   845   interpret I: finite_product_sigma_finite M I by default auto
```
```   846   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))"
```
```   847     using insert by (auto intro!: setprod_cong)
```
```   848   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)"
```
```   849     using sets_into_space insert
```
```   850     by (intro borel_measurable_ereal_setprod
```
```   851               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
```
```   852        auto
```
```   853   then show ?case
```
```   854     apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
```
```   855     apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
```
```   856     apply (subst positive_integral_cmult)
```
```   857     apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
```
```   858     done
```
```   859 qed (simp add: space_PiM)
```
```   860
```
```   861 lemma (in product_sigma_finite) product_integral_singleton:
```
```   862   assumes f: "f \<in> borel_measurable (M i)"
```
```   863   shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
```
```   864 proof -
```
```   865   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   866   have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
```
```   867     "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
```
```   868     using assms by auto
```
```   869   show ?thesis
```
```   870     unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
```
```   871 qed
```
```   872 lemma (in product_sigma_finite) distr_component:
```
```   873   "distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P")
```
```   874 proof (intro measure_eqI[symmetric])
```
```   875   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   876
```
```   877   have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
```
```   878     by (auto simp: extensional_def restrict_def)
```
```   879
```
```   880   fix A assume A: "A \<in> sets ?P"
```
```   881   then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)"
```
```   882     by simp
```
```   883   also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)"
```
```   884     by (intro positive_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
```
```   885   also have "\<dots> = emeasure ?D A"
```
```   886     using A by (simp add: product_positive_integral_singleton emeasure_distr)
```
```   887   finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" .
```
```   888 qed simp
```
```   889
```
```   890 lemma (in product_sigma_finite) product_integral_fold:
```
```   891   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   892   and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
```
```   893   shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
```
```   894 proof -
```
```   895   interpret I: finite_product_sigma_finite M I by default fact
```
```   896   interpret J: finite_product_sigma_finite M J by default fact
```
```   897   have "finite (I \<union> J)" using fin by auto
```
```   898   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   899   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
```
```   900   let ?M = "merge I J"
```
```   901   let ?f = "\<lambda>x. f (?M x)"
```
```   902   from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
```
```   903     by auto
```
```   904   have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
```
```   905     using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
```
```   906   have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
```
```   907     by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
```
```   908   show ?thesis
```
```   909     apply (subst distr_merge[symmetric, OF IJ fin])
```
```   910     apply (subst integral_distr[OF measurable_merge f_borel])
```
```   911     apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
```
```   912     apply simp
```
```   913     done
```
```   914 qed
```
```   915
```
```   916 lemma (in product_sigma_finite)
```
```   917   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
```
```   918   shows emeasure_fold_integral:
```
```   919     "emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
```
```   920     and emeasure_fold_measurable:
```
```   921     "(\<lambda>x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
```
```   922 proof -
```
```   923   interpret I: finite_product_sigma_finite M I by default fact
```
```   924   interpret J: finite_product_sigma_finite M J by default fact
```
```   925   interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
```
```   926   have merge: "merge I J -` 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)"
```
```   927     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   928
```
```   929   show ?I
```
```   930     apply (subst distr_merge[symmetric, OF IJ])
```
```   931     apply (subst emeasure_distr[OF measurable_merge A])
```
```   932     apply (subst J.emeasure_pair_measure_alt[OF merge])
```
```   933     apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```   934     done
```
```   935
```
```   936   show ?B
```
```   937     using IJ.measurable_emeasure_Pair1[OF merge]
```
```   938     by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
```
```   939 qed
```
```   940
```
```   941 lemma (in product_sigma_finite) product_integral_insert:
```
```   942   assumes I: "finite I" "i \<notin> I"
```
```   943     and f: "integrable (Pi\<^isub>M (insert i I) M) f"
```
```   944   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)"
```
```   945 proof -
```
```   946   have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
```
```   947     by simp
```
```   948   also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)"
```
```   949     using f I by (intro product_integral_fold) auto
```
```   950   also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
```
```   951   proof (rule integral_cong, subst product_integral_singleton[symmetric])
```
```   952     fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
```
```   953     have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
```
```   954       using f by auto
```
```   955     show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
```
```   956       using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
```
```   957       unfolding comp_def .
```
```   958     from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)"
```
```   959       by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
```
```   960   qed
```
```   961   finally show ?thesis .
```
```   962 qed
```
```   963
```
```   964 lemma (in product_sigma_finite) product_integrable_setprod:
```
```   965   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
```
```   966   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
```
```   967   shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
```
```   968 proof -
```
```   969   interpret finite_product_sigma_finite M I by default fact
```
```   970   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```   971     using integrable unfolding integrable_def by auto
```
```   972   have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
```
```   973     using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
```
```   974   moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
```
```   975   proof (unfold integrable_def, intro conjI)
```
```   976     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
```
```   977       using borel by auto
```
```   978     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)"
```
```   979       by (simp add: setprod_ereal abs_setprod)
```
```   980     also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
```
```   981       using f by (subst product_positive_integral_setprod) auto
```
```   982     also have "\<dots> < \<infinity>"
```
```   983       using integrable[THEN integrable_abs]
```
```   984       by (simp add: setprod_PInf integrable_def positive_integral_positive)
```
```   985     finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
```
```   986     have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
```
```   987       by (intro positive_integral_cong_pos) auto
```
```   988     then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
```
```   989   qed
```
```   990   ultimately show ?thesis
```
```   991     by (rule integrable_abs_iff[THEN iffD1])
```
```   992 qed
```
```   993
```
```   994 lemma (in product_sigma_finite) product_integral_setprod:
```
```   995   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
```
```   996   assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
```
```   997   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))"
```
```   998 using assms proof induct
```
```   999   case empty
```
```  1000   interpret finite_measure "Pi\<^isub>M {} M"
```
```  1001     by rule (simp add: space_PiM)
```
```  1002   show ?case by (simp add: space_PiM measure_def)
```
```  1003 next
```
```  1004   case (insert i I)
```
```  1005   then have iI: "finite (insert i I)" by auto
```
```  1006   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
```
```  1007     integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
```
```  1008     by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
```
```  1009   interpret I: finite_product_sigma_finite M I by default fact
```
```  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 `i \<notin> I` by (auto intro!: setprod_cong)
```
```  1012   show ?case
```
```  1013     unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
```
```  1014     by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
```
```  1015 qed
```
```  1016
```
```  1017 lemma sets_Collect_single:
```
```  1018   "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)"
```
```  1019   by simp
```
```  1020
```
```  1021 lemma sigma_prod_algebra_sigma_eq_infinite:
```
```  1022   fixes E :: "'i \<Rightarrow> 'a set set"
```
```  1023   assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```  1024     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```  1025   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```  1026     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```  1027   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}"
```
```  1028   shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
```
```  1029 proof
```
```  1030   let ?P = "sigma (space (Pi\<^isub>M I M)) P"
```
```  1031   have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
```
```  1032     using E_closed by (auto simp: space_PiM P_def subset_eq)
```
```  1033   then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```  1034     by (simp add: space_PiM)
```
```  1035   have "sets (PiM I M) =
```
```  1036       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)}"
```
```  1037     using sets_PiM_single[of I M] by (simp add: space_P)
```
```  1038   also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
```
```  1039   proof (safe intro!: sigma_sets_subset)
```
```  1040     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```  1041     then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```  1042       apply (subst measurable_iff_measure_of)
```
```  1043       apply (simp_all add: P_closed)
```
```  1044       using E_closed
```
```  1045       apply (force simp: subset_eq space_PiM)
```
```  1046       apply (force simp: subset_eq space_PiM)
```
```  1047       apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
```
```  1048       apply (rule_tac x=Aa in exI)
```
```  1049       apply (auto simp: space_PiM)
```
```  1050       done
```
```  1051     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```  1052     have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1053       by (simp add: E_generates)
```
```  1054     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}"
```
```  1055       using P_closed by (auto simp: space_PiM)
```
```  1056     finally show "\<dots> \<in> sets ?P" .
```
```  1057   qed
```
```  1058   finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
```
```  1059     by (simp add: P_closed)
```
```  1060   show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
```
```  1061     unfolding P_def space_PiM[symmetric]
```
```  1062     by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
```
```  1063 qed
```
```  1064
```
```  1065 lemma sigma_prod_algebra_sigma_eq:
```
```  1066   fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
```
```  1067   assumes "finite I"
```
```  1068   assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```  1069     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```  1070   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```  1071     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```  1072   defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
```
```  1073   shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
```
```  1074 proof
```
```  1075   let ?P = "sigma (space (Pi\<^isub>M I M)) P"
```
```  1076   from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
```
```  1077   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"
```
```  1078     by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
```
```  1079   have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
```
```  1080     using E_closed by (auto simp: space_PiM P_def subset_eq)
```
```  1081   then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
```
```  1082     by (simp add: space_PiM)
```
```  1083   have "sets (PiM I M) =
```
```  1084       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)}"
```
```  1085     using sets_PiM_single[of I M] by (simp add: space_P)
```
```  1086   also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
```
```  1087   proof (safe intro!: sigma_sets_subset)
```
```  1088     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```  1089     have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```  1090     proof (subst measurable_iff_measure_of)
```
```  1091       show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
```
```  1092       from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto
```
```  1093       show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1094       proof
```
```  1095         fix A assume A: "A \<in> E i"
```
```  1096         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))"
```
```  1097           using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm)
```
```  1098         also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
```
```  1099           by (intro PiE_cong) (simp add: S_union)
```
```  1100         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))"
```
```  1101           using T
```
```  1102           apply (auto simp: PiE_iff bchoice_iff)
```
```  1103           apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
```
```  1104           apply (auto simp: bij_betw_def)
```
```  1105           done
```
```  1106         also have "\<dots> \<in> sets ?P"
```
```  1107         proof (safe intro!: countable_UN)
```
```  1108           fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
```
```  1109             using A S_in_E
```
```  1110             by (simp add: P_closed)
```
```  1111                (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
```
```  1112         qed
```
```  1113         finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1114           using P_closed by simp
```
```  1115       qed
```
```  1116     qed
```
```  1117     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```  1118     have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1119       by (simp add: E_generates)
```
```  1120     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}"
```
```  1121       using P_closed by (auto simp: space_PiM)
```
```  1122     finally show "\<dots> \<in> sets ?P" .
```
```  1123   qed
```
```  1124   finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
```
```  1125     by (simp add: P_closed)
```
```  1126   show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
```
```  1127     using `finite I`
```
```  1128     by (auto intro!: sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
```
```  1129 qed
```
```  1130
```
```  1131 lemma pair_measure_eq_distr_PiM:
```
```  1132   fixes M1 :: "'a measure" and M2 :: "'a measure"
```
```  1133   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
```
```  1134   shows "(M1 \<Otimes>\<^isub>M M2) = distr (Pi\<^isub>M UNIV (bool_case M1 M2)) (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. (x True, x False))"
```
```  1135     (is "?P = ?D")
```
```  1136 proof (rule pair_measure_eqI[OF assms])
```
```  1137   interpret B: product_sigma_finite "bool_case M1 M2"
```
```  1138     unfolding product_sigma_finite_def using assms by (auto split: bool.split)
```
```  1139   let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)"
```
```  1140
```
```  1141   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)"
```
```  1142     by auto
```
```  1143   fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
```
```  1144   have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))"
```
```  1145     by (simp add: UNIV_bool ac_simps)
```
```  1146   also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))"
```
```  1147     using A B by (subst B.emeasure_PiM) (auto split: bool.split)
```
```  1148   also have "Pi\<^isub>E UNIV (bool_case A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
```
```  1149     using A[THEN sets_into_space] B[THEN sets_into_space]
```
```  1150     by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
```
```  1151   finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
```
```  1152     using A B
```
```  1153       measurable_component_singleton[of True UNIV "bool_case M1 M2"]
```
```  1154       measurable_component_singleton[of False UNIV "bool_case M1 M2"]
```
```  1155     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
```
```  1156 qed simp
```
```  1157
```
```  1158 end
```