src/HOL/Probability/Finite_Product_Measure.thy
 author hoelzl Mon May 19 12:04:45 2014 +0200 (2014-05-19) changeset 56993 e5366291d6aa parent 56154 f0a927235162 child 56994 8d5e5ec1cac3 permissions -rw-r--r--
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
```     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>\<^sub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" and ne: "(\<Pi>\<^sub>E i\<in>I. S i) \<noteq> {}"
```
```    81   and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
```
```    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>\<^sub>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>\<^sub>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\<^sub>E I E \<times> Pi\<^sub>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\<^sub>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>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>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\<^sub>E I E) = Pi\<^sub>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\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
```
```   143   using assms sets.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>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>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\<^sub>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>\<^sub>M _\<in>_./ _)"  10)
```
```   172
```
```   173 syntax (HTML output)
```
```   174   "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^sub>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>\<^sub>E i\<in>I. space (M i))"
```
```   181   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>\<^sub>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>\<^sub>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.sets_into_space) auto
```
```   194   show "A \<in> ?R" unfolding A using J sets.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>\<^sub>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>\<^sub>E i\<in>I. X i)"
```
```   200     by (simp add: prod_emb_PiE_same_index[OF sets.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\<^sub>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.sets_into_space] by simp
```
```   213
```
```   214 lemma Int_stable_PiE: "Int_stable {Pi\<^sub>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\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
```
```   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\<^sub>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\<^sub>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.sets_into_space by auto
```
```   236   then have "A = (\<Pi>\<^sub>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 have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
```
```   239     using sets.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>\<^sub>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.sets_into_space] B(2,3,4)
```
```   252       B(5)[THEN sets.sets_into_space]
```
```   253     apply (subst (1 2 3) prod_emb_PiE)
```
```   254     apply (simp_all add: subset_eq PiE_Int)
```
```   255     apply blast
```
```   256     apply (intro PiE_cong)
```
```   257     apply auto
```
```   258     done
```
```   259   also have "\<dots> \<in> prod_algebra I M"
```
```   260     using A B by (auto intro!: prod_algebraI)
```
```   261   finally show "A \<inter> B \<in> prod_algebra I M" .
```
```   262 qed
```
```   263
```
```   264 lemma prod_algebra_mono:
```
```   265   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
```
```   266   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
```
```   267   shows "prod_algebra I E \<subseteq> prod_algebra I F"
```
```   268 proof
```
```   269   fix A assume "A \<in> prod_algebra I E"
```
```   270   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
```
```   271     and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   272     and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
```
```   273     by (auto simp: prod_algebra_def)
```
```   274   moreover
```
```   275   from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))"
```
```   276     by (rule PiE_cong)
```
```   277   with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)"
```
```   278     by (simp add: prod_emb_def)
```
```   279   moreover
```
```   280   from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
```
```   281     by auto
```
```   282   ultimately show "A \<in> prod_algebra I F"
```
```   283     apply (simp add: prod_algebra_def image_iff)
```
```   284     apply (intro exI[of _ J] exI[of _ G] conjI)
```
```   285     apply auto
```
```   286     done
```
```   287 qed
```
```   288
```
```   289 lemma prod_algebra_cong:
```
```   290   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
```
```   291   shows "prod_algebra I M = prod_algebra J N"
```
```   292 proof -
```
```   293   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
```
```   294     using sets_eq_imp_space_eq[OF sets] by auto
```
```   295   with sets show ?thesis unfolding `I = J`
```
```   296     by (intro antisym prod_algebra_mono) auto
```
```   297 qed
```
```   298
```
```   299 lemma space_in_prod_algebra:
```
```   300   "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
```
```   301 proof cases
```
```   302   assume "I = {}" then show ?thesis
```
```   303     by (auto simp add: prod_algebra_def image_iff prod_emb_def)
```
```   304 next
```
```   305   assume "I \<noteq> {}"
```
```   306   then obtain i where "i \<in> I" by auto
```
```   307   then have "(\<Pi>\<^sub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i))"
```
```   308     by (auto simp: prod_emb_def)
```
```   309   also have "\<dots> \<in> prod_algebra I M"
```
```   310     using `i \<in> I` by (intro prod_algebraI) auto
```
```   311   finally show ?thesis .
```
```   312 qed
```
```   313
```
```   314 lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   315   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
```
```   316
```
```   317 lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
```
```   318   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
```
```   319
```
```   320 lemma sets_PiM_single: "sets (PiM I M) =
```
```   321     sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   322     (is "_ = sigma_sets ?\<Omega> ?R")
```
```   323   unfolding sets_PiM
```
```   324 proof (rule sigma_sets_eqI)
```
```   325   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
```
```   326   fix A assume "A \<in> prod_algebra I M"
```
```   327   from prod_algebraE[OF this] guess J X . note X = this
```
```   328   show "A \<in> sigma_sets ?\<Omega> ?R"
```
```   329   proof cases
```
```   330     assume "I = {}"
```
```   331     with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
```
```   332     with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
```
```   333   next
```
```   334     assume "I \<noteq> {}"
```
```   335     with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
```
```   336       by (auto simp: prod_emb_def)
```
```   337     also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
```
```   338       using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
```
```   339     finally show "A \<in> sigma_sets ?\<Omega> ?R" .
```
```   340   qed
```
```   341 next
```
```   342   fix A assume "A \<in> ?R"
```
```   343   then obtain i B where A: "A = {f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
```
```   344     by auto
```
```   345   then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)"
```
```   346      by (auto simp: prod_emb_def)
```
```   347   also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
```
```   348     using A by (intro sigma_sets.Basic prod_algebraI) auto
```
```   349   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
```
```   350 qed
```
```   351
```
```   352 lemma sets_PiM_I:
```
```   353   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
```
```   354   shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
```
```   355 proof cases
```
```   356   assume "J = {}"
```
```   357   then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
```
```   358     by (auto simp: prod_emb_def)
```
```   359   then show ?thesis
```
```   360     by (auto simp add: sets_PiM intro!: sigma_sets_top)
```
```   361 next
```
```   362   assume "J \<noteq> {}" with assms show ?thesis
```
```   363     by (force simp add: sets_PiM prod_algebra_def)
```
```   364 qed
```
```   365
```
```   366 lemma measurable_PiM:
```
```   367   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   368   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>
```
```   369     f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
```
```   370   shows "f \<in> measurable N (PiM I M)"
```
```   371   using sets_PiM prod_algebra_sets_into_space space
```
```   372 proof (rule measurable_sigma_sets)
```
```   373   fix A assume "A \<in> prod_algebra I M"
```
```   374   from prod_algebraE[OF this] guess J X .
```
```   375   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
```
```   376 qed
```
```   377
```
```   378 lemma measurable_PiM_Collect:
```
```   379   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   380   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>
```
```   381     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
```
```   382   shows "f \<in> measurable N (PiM I M)"
```
```   383   using sets_PiM prod_algebra_sets_into_space space
```
```   384 proof (rule measurable_sigma_sets)
```
```   385   fix A assume "A \<in> prod_algebra I M"
```
```   386   from prod_algebraE[OF this] guess J X . note X = this
```
```   387   then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
```
```   388     using space by (auto simp: prod_emb_def del: PiE_I)
```
```   389   also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
```
```   390   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   391 qed
```
```   392
```
```   393 lemma measurable_PiM_single:
```
```   394   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   395   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"
```
```   396   shows "f \<in> measurable N (PiM I M)"
```
```   397   using sets_PiM_single
```
```   398 proof (rule measurable_sigma_sets)
```
```   399   fix A assume "A \<in> {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   400   then obtain B i where "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)"
```
```   401     by auto
```
```   402   with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
```
```   403   also have "\<dots> \<in> sets N" using B by (rule sets)
```
```   404   finally show "f -` A \<inter> space N \<in> sets N" .
```
```   405 qed (auto simp: space)
```
```   406
```
```   407 lemma measurable_PiM_single':
```
```   408   assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
```
```   409     and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   410   shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
```
```   411 proof (rule measurable_PiM_single)
```
```   412   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   413   then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
```
```   414     by auto
```
```   415   then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N"
```
```   416     using A f by (auto intro!: measurable_sets)
```
```   417 qed fact
```
```   418
```
```   419 lemma sets_PiM_I_finite[measurable]:
```
```   420   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
```
```   421   shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
```
```   422   using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto
```
```   423
```
```   424 lemma measurable_component_singleton:
```
```   425   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
```
```   426 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   427   fix A assume "A \<in> sets (M i)"
```
```   428   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. A)"
```
```   429     using sets.sets_into_space `i \<in> I`
```
```   430     by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
```
```   431   then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
```
```   432     using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
```
```   433 qed (insert `i \<in> I`, auto simp: space_PiM)
```
```   434
```
```   435 lemma measurable_component_singleton'[measurable_app]:
```
```   436   assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
```
```   437   assumes i: "i \<in> I"
```
```   438   shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
```
```   439   using measurable_compose[OF f measurable_component_singleton, OF i] .
```
```   440
```
```   441 lemma measurable_PiM_component_rev[measurable (raw)]:
```
```   442   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
```
```   443   by simp
```
```   444
```
```   445 lemma measurable_case_nat[measurable (raw)]:
```
```   446   assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
```
```   447     "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
```
```   448   shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
```
```   449   by (cases i) simp_all
```
```   450
```
```   451 lemma measurable_case_nat'[measurable (raw)]:
```
```   452   assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   453   shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
```
```   454   using fg[THEN measurable_space]
```
```   455   by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
```
```   456
```
```   457 lemma measurable_add_dim[measurable]:
```
```   458   "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
```
```   459     (is "?f \<in> measurable ?P ?I")
```
```   460 proof (rule measurable_PiM_single)
```
```   461   fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
```
```   462   have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
```
```   463     (if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
```
```   464     using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
```
```   465   also have "\<dots> \<in> sets ?P"
```
```   466     using A j
```
```   467     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   468   finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
```
```   469 qed (auto simp: space_pair_measure space_PiM PiE_def)
```
```   470
```
```   471 lemma measurable_component_update:
```
```   472   "x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)"
```
```   473   by simp
```
```   474
```
```   475 lemma measurable_merge[measurable]:
```
```   476   "merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)"
```
```   477     (is "?f \<in> measurable ?P ?U")
```
```   478 proof (rule measurable_PiM_single)
```
```   479   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
```
```   480   then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
```
```   481     (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)"
```
```   482     by (auto simp: merge_def)
```
```   483   also have "\<dots> \<in> sets ?P"
```
```   484     using A
```
```   485     by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
```
```   486   finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
```
```   487 qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
```
```   488
```
```   489 lemma measurable_restrict[measurable (raw)]:
```
```   490   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
```
```   491   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
```
```   492 proof (rule measurable_PiM_single)
```
```   493   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
```
```   494   then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
```
```   495     by auto
```
```   496   then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
```
```   497     using A X by (auto intro!: measurable_sets)
```
```   498 qed (insert X, auto simp add: PiE_def dest: measurable_space)
```
```   499
```
```   500 lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
```
```   501   by (intro measurable_restrict measurable_component_singleton) auto
```
```   502
```
```   503 lemma measurable_prod_emb[intro, simp]:
```
```   504   "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
```
```   505   unfolding prod_emb_def space_PiM[symmetric]
```
```   506   by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
```
```   507
```
```   508 lemma sets_in_Pi_aux:
```
```   509   "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   510   {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
```
```   511   by (simp add: subset_eq Pi_iff)
```
```   512
```
```   513 lemma sets_in_Pi[measurable (raw)]:
```
```   514   "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
```
```   515   (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
```
```   516   Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
```
```   517   unfolding pred_def
```
```   518   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
```
```   519
```
```   520 lemma sets_in_extensional_aux:
```
```   521   "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
```
```   522 proof -
```
```   523   have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
```
```   524     by (auto simp add: extensional_def space_PiM)
```
```   525   then show ?thesis by simp
```
```   526 qed
```
```   527
```
```   528 lemma sets_in_extensional[measurable (raw)]:
```
```   529   "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
```
```   530   unfolding pred_def
```
```   531   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
```
```   532
```
```   533 locale product_sigma_finite =
```
```   534   fixes M :: "'i \<Rightarrow> 'a measure"
```
```   535   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
```
```   536
```
```   537 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
```
```   538   by (rule sigma_finite_measures)
```
```   539
```
```   540 locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
```
```   541   fixes I :: "'i set"
```
```   542   assumes finite_index: "finite I"
```
```   543
```
```   544 lemma (in finite_product_sigma_finite) sigma_finite_pairs:
```
```   545   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
```
```   546     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
```
```   547     (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and>
```
```   548     (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
```
```   549 proof -
```
```   550   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>)"
```
```   551     using M.sigma_finite_incseq by metis
```
```   552   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   553   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>"
```
```   554     by auto
```
```   555   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k"
```
```   556   note space_PiM[simp]
```
```   557   show ?thesis
```
```   558   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
```
```   559     fix i show "range (F i) \<subseteq> sets (M i)" by fact
```
```   560   next
```
```   561     fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
```
```   562   next
```
```   563     fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)"
```
```   564       by (auto simp: PiE_def dest!: sets.sets_into_space)
```
```   565   next
```
```   566     fix f assume "f \<in> space (PiM I M)"
```
```   567     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
```
```   568     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
```
```   569   next
```
```   570     fix i show "?F i \<subseteq> ?F (Suc i)"
```
```   571       using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
```
```   572   qed
```
```   573 qed
```
```   574
```
```   575 lemma
```
```   576   shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
```
```   577     and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
```
```   578   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
```
```   579
```
```   580 lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
```
```   581 proof -
```
```   582   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
```
```   583   have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
```
```   584   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   585     show "positive (PiM {} M) ?\<mu>"
```
```   586       by (auto simp: positive_def)
```
```   587     show "countably_additive (PiM {} M) ?\<mu>"
```
```   588       by (rule sets.countably_additiveI_finite)
```
```   589          (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
```
```   590   qed (auto simp: prod_emb_def)
```
```   591   also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
```
```   592     by (auto simp: prod_emb_def)
```
```   593   finally show ?thesis
```
```   594     by simp
```
```   595 qed
```
```   596
```
```   597 lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
```
```   598   by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
```
```   599
```
```   600 lemma (in product_sigma_finite) emeasure_PiM:
```
```   601   "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   602 proof (induct I arbitrary: A rule: finite_induct)
```
```   603   case (insert i I)
```
```   604   interpret finite_product_sigma_finite M I by default fact
```
```   605   have "finite (insert i I)" using `finite I` by auto
```
```   606   interpret I': finite_product_sigma_finite M "insert i I" by default fact
```
```   607   let ?h = "(\<lambda>(f, y). f(i := y))"
```
```   608
```
```   609   let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h"
```
```   610   let ?\<mu> = "emeasure ?P"
```
```   611   let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
```
```   612   let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
```
```   613
```
```   614   have "emeasure (Pi\<^sub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^sub>E (insert i I) A)) =
```
```   615     (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
```
```   616   proof (subst emeasure_extend_measure_Pair[OF PiM_def])
```
```   617     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))"
```
```   618     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
```
```   619     let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)"
```
```   620     let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)"
```
```   621     have "?\<mu> ?p =
```
```   622       emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i))"
```
```   623       by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
```
```   624     also have "?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))"
```
```   625       using J E[rule_format, THEN sets.sets_into_space]
```
```   626       by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm)
```
```   627     also have "emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) =
```
```   628       emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
```
```   629       using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
```
```   630     also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
```
```   631       using J E[rule_format, THEN sets.sets_into_space]
```
```   632       by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+
```
```   633     also have "emeasure (Pi\<^sub>M I M) (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) =
```
```   634       (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
```
```   635       using E by (subst insert) (auto intro!: setprod_cong)
```
```   636     also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
```
```   637        emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
```
```   638       using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
```
```   639     also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
```
```   640       using insert(1,2) J E by (intro setprod_mono_one_right) auto
```
```   641     finally show "?\<mu> ?p = \<dots>" .
```
```   642
```
```   643     show "prod_emb (insert i I) M J (Pi\<^sub>E J E) \<in> Pow (\<Pi>\<^sub>E i\<in>insert i I. space (M i))"
```
```   644       using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def)
```
```   645   next
```
```   646     show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>"
```
```   647       using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
```
```   648   next
```
```   649     show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
```
```   650       insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
```
```   651       using insert by auto
```
```   652   qed (auto intro!: setprod_cong)
```
```   653   with insert show ?case
```
```   654     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
```
```   655 qed simp
```
```   656
```
```   657 lemma (in product_sigma_finite) sigma_finite:
```
```   658   assumes "finite I"
```
```   659   shows "sigma_finite_measure (PiM I M)"
```
```   660 proof -
```
```   661   interpret finite_product_sigma_finite M I by default fact
```
```   662
```
```   663   from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
```
```   664   then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
```
```   665     "incseq (\<lambda>k. \<Pi>\<^sub>E j \<in> I. F j k)"
```
```   666     "(\<Union>k. \<Pi>\<^sub>E j \<in> I. F j k) = space (Pi\<^sub>M I M)"
```
```   667     "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
```
```   668     by blast+
```
```   669   let ?F = "\<lambda>k. \<Pi>\<^sub>E j \<in> I. F j k"
```
```   670
```
```   671   show ?thesis
```
```   672   proof (unfold_locales, intro exI[of _ ?F] conjI allI)
```
```   673     show "range ?F \<subseteq> sets (Pi\<^sub>M I M)" using F(1) `finite I` by auto
```
```   674   next
```
```   675     from F(3) show "(\<Union>i. ?F i) = space (Pi\<^sub>M I M)" by simp
```
```   676   next
```
```   677     fix j
```
```   678     from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
```
```   679     show "emeasure (\<Pi>\<^sub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
```
```   680       by (subst emeasure_PiM) auto
```
```   681   qed
```
```   682 qed
```
```   683
```
```   684 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
```
```   685   using sigma_finite[OF finite_index] .
```
```   686
```
```   687 lemma (in finite_product_sigma_finite) measure_times:
```
```   688   "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
```
```   689   using emeasure_PiM[OF finite_index] by auto
```
```   690
```
```   691 lemma (in product_sigma_finite) positive_integral_empty:
```
```   692   assumes pos: "0 \<le> f (\<lambda>k. undefined)"
```
```   693   shows "integral\<^sup>P (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
```
```   694 proof -
```
```   695   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
```
```   696   have "\<And>A. emeasure (Pi\<^sub>M {} M) (Pi\<^sub>E {} A) = 1"
```
```   697     using assms by (subst measure_times) auto
```
```   698   then show ?thesis
```
```   699     unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
```
```   700   proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
```
```   701     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
```
```   702       by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
```
```   703     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
```
```   704       by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
```
```   705   qed
```
```   706 qed
```
```   707
```
```   708 lemma (in product_sigma_finite) distr_merge:
```
```   709   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
```
```   710   shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
```
```   711    (is "?D = ?P")
```
```   712 proof -
```
```   713   interpret I: finite_product_sigma_finite M I by default fact
```
```   714   interpret J: finite_product_sigma_finite M J by default fact
```
```   715   have "finite (I \<union> J)" using fin by auto
```
```   716   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
```
```   717   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
```
```   718   let ?g = "merge I J"
```
```   719
```
```   720   from IJ.sigma_finite_pairs obtain F where
```
```   721     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
```
```   722        "incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k)"
```
```   723        "(\<Union>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k) = space ?P"
```
```   724        "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
```
```   725     by auto
```
```   726   let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k"
```
```   727
```
```   728   show ?thesis
```
```   729   proof (rule measure_eqI_generator_eq[symmetric])
```
```   730     show "Int_stable (prod_algebra (I \<union> J) M)"
```
```   731       by (rule Int_stable_prod_algebra)
```
```   732     show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^sub>E i \<in> I \<union> J. space (M i))"
```
```   733       by (rule prod_algebra_sets_into_space)
```
```   734     show "sets ?P = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   735       by (rule sets_PiM)
```
```   736     then show "sets ?D = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
```
```   737       by simp
```
```   738
```
```   739     show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
```
```   740       using fin by (auto simp: prod_algebra_eq_finite)
```
```   741     show "(\<Union>i. \<Pi>\<^sub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i))"
```
```   742       using F(3) by (simp add: space_PiM)
```
```   743   next
```
```   744     fix k
```
```   745     from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
```
```   746     show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
```
```   747   next
```
```   748     fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
```
```   749     with fin obtain F where A_eq: "A = (Pi\<^sub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
```
```   750       by (auto simp add: prod_algebra_eq_finite)
```
```   751     let ?B = "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M"
```
```   752     let ?X = "?g -` A \<inter> space ?B"
```
```   753     have "Pi\<^sub>E I F \<subseteq> space (Pi\<^sub>M I M)" "Pi\<^sub>E J F \<subseteq> space (Pi\<^sub>M J M)"
```
```   754       using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+
```
```   755     then have X: "?X = (Pi\<^sub>E I F \<times> Pi\<^sub>E J F)"
```
```   756       unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
```
```   757     have "emeasure ?D A = emeasure ?B ?X"
```
```   758       using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
```
```   759     also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
```
```   760       using `finite J` `finite I` F unfolding X
```
```   761       by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times)
```
```   762     also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
```
```   763       using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
```
```   764     also have "\<dots> = emeasure ?P (Pi\<^sub>E (I \<union> J) F)"
```
```   765       using `finite J` `finite I` F unfolding A
```
```   766       by (intro IJ.measure_times[symmetric]) auto
```
```   767     finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
```
```   768   qed
```
```   769 qed
```
```   770
```
```   771 lemma (in product_sigma_finite) product_positive_integral_fold:
```
```   772   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
```
```   773   and f: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
```
```   774   shows "integral\<^sup>P (Pi\<^sub>M (I \<union> J) M) f =
```
```   775     (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
```
```   776 proof -
```
```   777   interpret I: finite_product_sigma_finite M I by default fact
```
```   778   interpret J: finite_product_sigma_finite M J by default fact
```
```   779   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
```
```   780   have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
```
```   781     using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
```
```   782   show ?thesis
```
```   783     apply (subst distr_merge[OF IJ, symmetric])
```
```   784     apply (subst positive_integral_distr[OF measurable_merge f])
```
```   785     apply (subst J.positive_integral_fst[symmetric, OF P_borel])
```
```   786     apply simp
```
```   787     done
```
```   788 qed
```
```   789
```
```   790 lemma (in product_sigma_finite) distr_singleton:
```
```   791   "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
```
```   792 proof (intro measure_eqI[symmetric])
```
```   793   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   794   fix A assume A: "A \<in> sets (M i)"
```
```   795   then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)"
```
```   796     using sets.sets_into_space by (auto simp: space_PiM)
```
```   797   then show "emeasure (M i) A = emeasure ?D A"
```
```   798     using A I.measure_times[of "\<lambda>_. A"]
```
```   799     by (simp add: emeasure_distr measurable_component_singleton)
```
```   800 qed simp
```
```   801
```
```   802 lemma (in product_sigma_finite) product_positive_integral_singleton:
```
```   803   assumes f: "f \<in> borel_measurable (M i)"
```
```   804   shows "integral\<^sup>P (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>P (M i) f"
```
```   805 proof -
```
```   806   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   807   from f show ?thesis
```
```   808     apply (subst distr_singleton[symmetric])
```
```   809     apply (subst positive_integral_distr[OF measurable_component_singleton])
```
```   810     apply simp_all
```
```   811     done
```
```   812 qed
```
```   813
```
```   814 lemma (in product_sigma_finite) product_positive_integral_insert:
```
```   815   assumes I[simp]: "finite I" "i \<notin> I"
```
```   816     and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
```
```   817   shows "integral\<^sup>P (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
```
```   818 proof -
```
```   819   interpret I: finite_product_sigma_finite M I by default auto
```
```   820   interpret i: finite_product_sigma_finite M "{i}" by default auto
```
```   821   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
```
```   822     using f by auto
```
```   823   show ?thesis
```
```   824     unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
```
```   825   proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
```
```   826     fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
```
```   827     let ?f = "\<lambda>y. f (x(i := y))"
```
```   828     show "?f \<in> borel_measurable (M i)"
```
```   829       using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
```
```   830       unfolding comp_def .
```
```   831     show "(\<integral>\<^sup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^sub>M {i} M) = (\<integral>\<^sup>+ y. f (x(i := y i)) \<partial>Pi\<^sub>M {i} M)"
```
```   832       using x
```
```   833       by (auto intro!: positive_integral_cong arg_cong[where f=f]
```
```   834                simp add: space_PiM extensional_def PiE_def)
```
```   835   qed
```
```   836 qed
```
```   837
```
```   838 lemma (in product_sigma_finite) product_positive_integral_setprod:
```
```   839   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
```
```   840   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
```
```   841   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
```
```   842   shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>P (M i) (f i))"
```
```   843 using assms proof induct
```
```   844   case (insert i I)
```
```   845   note `finite I`[intro, simp]
```
```   846   interpret I: finite_product_sigma_finite M I by default auto
```
```   847   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))"
```
```   848     using insert by (auto intro!: setprod_cong)
```
```   849   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^sub>M J M)"
```
```   850     using sets.sets_into_space insert
```
```   851     by (intro borel_measurable_ereal_setprod
```
```   852               measurable_comp[OF measurable_component_singleton, unfolded comp_def])
```
```   853        auto
```
```   854   then show ?case
```
```   855     apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
```
```   856     apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
```
```   857     apply (subst positive_integral_cmult)
```
```   858     apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
```
```   859     done
```
```   860 qed (simp add: space_PiM)
```
```   861
```
```   862 lemma (in product_sigma_finite) distr_component:
```
```   863   "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
```
```   864 proof (intro measure_eqI[symmetric])
```
```   865   interpret I: finite_product_sigma_finite M "{i}" by default simp
```
```   866
```
```   867   have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x"
```
```   868     by (auto simp: extensional_def restrict_def)
```
```   869
```
```   870   fix A assume A: "A \<in> sets ?P"
```
```   871   then have "emeasure ?P A = (\<integral>\<^sup>+x. indicator A x \<partial>?P)"
```
```   872     by simp
```
```   873   also have "\<dots> = (\<integral>\<^sup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)"
```
```   874     by (intro positive_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq)
```
```   875   also have "\<dots> = emeasure ?D A"
```
```   876     using A by (simp add: product_positive_integral_singleton emeasure_distr)
```
```   877   finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" .
```
```   878 qed simp
```
```   879
```
```   880 lemma (in product_sigma_finite)
```
```   881   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
```
```   882   shows emeasure_fold_integral:
```
```   883     "emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I)
```
```   884     and emeasure_fold_measurable:
```
```   885     "(\<lambda>x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M))) \<in> borel_measurable (Pi\<^sub>M I M)" (is ?B)
```
```   886 proof -
```
```   887   interpret I: finite_product_sigma_finite M I by default fact
```
```   888   interpret J: finite_product_sigma_finite M J by default fact
```
```   889   interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" ..
```
```   890   have merge: "merge I J -` A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
```
```   891     by (intro measurable_sets[OF _ A] measurable_merge assms)
```
```   892
```
```   893   show ?I
```
```   894     apply (subst distr_merge[symmetric, OF IJ])
```
```   895     apply (subst emeasure_distr[OF measurable_merge A])
```
```   896     apply (subst J.emeasure_pair_measure_alt[OF merge])
```
```   897     apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
```
```   898     done
```
```   899
```
```   900   show ?B
```
```   901     using IJ.measurable_emeasure_Pair1[OF merge]
```
```   902     by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
```
```   903 qed
```
```   904
```
```   905 lemma sets_Collect_single:
```
```   906   "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
```
```   907   by simp
```
```   908
```
```   909 lemma sigma_prod_algebra_sigma_eq_infinite:
```
```   910   fixes E :: "'i \<Rightarrow> 'a set set"
```
```   911   assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```   912     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```   913   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```   914     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```   915   defines "P == {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}"
```
```   916   shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
```
```   917 proof
```
```   918   let ?P = "sigma (space (Pi\<^sub>M I M)) P"
```
```   919   have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))"
```
```   920     using E_closed by (auto simp: space_PiM P_def subset_eq)
```
```   921   then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   922     by (simp add: space_PiM)
```
```   923   have "sets (PiM I M) =
```
```   924       sigma_sets (space ?P) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   925     using sets_PiM_single[of I M] by (simp add: space_P)
```
```   926   also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
```
```   927   proof (safe intro!: sets.sigma_sets_subset)
```
```   928     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```   929     then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```   930       apply (subst measurable_iff_measure_of)
```
```   931       apply (simp_all add: P_closed)
```
```   932       using E_closed
```
```   933       apply (force simp: subset_eq space_PiM)
```
```   934       apply (force simp: subset_eq space_PiM)
```
```   935       apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
```
```   936       apply (rule_tac x=Aa in exI)
```
```   937       apply (auto simp: space_PiM)
```
```   938       done
```
```   939     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```   940     have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```   941       by (simp add: E_generates)
```
```   942     also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A}"
```
```   943       using P_closed by (auto simp: space_PiM)
```
```   944     finally show "\<dots> \<in> sets ?P" .
```
```   945   qed
```
```   946   finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
```
```   947     by (simp add: P_closed)
```
```   948   show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
```
```   949     unfolding P_def space_PiM[symmetric]
```
```   950     by (intro sets.sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
```
```   951 qed
```
```   952
```
```   953 lemma sigma_prod_algebra_sigma_eq:
```
```   954   fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
```
```   955   assumes "finite I"
```
```   956   assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
```
```   957     and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
```
```   958   assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
```
```   959     and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
```
```   960   defines "P == { Pi\<^sub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
```
```   961   shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
```
```   962 proof
```
```   963   let ?P = "sigma (space (Pi\<^sub>M I M)) P"
```
```   964   from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
```
```   965   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"
```
```   966     by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
```
```   967   have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))"
```
```   968     using E_closed by (auto simp: space_PiM P_def subset_eq)
```
```   969   then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))"
```
```   970     by (simp add: space_PiM)
```
```   971   have "sets (PiM I M) =
```
```   972       sigma_sets (space ?P) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
```
```   973     using sets_PiM_single[of I M] by (simp add: space_P)
```
```   974   also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
```
```   975   proof (safe intro!: sets.sigma_sets_subset)
```
```   976     fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
```
```   977     have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
```
```   978     proof (subst measurable_iff_measure_of)
```
```   979       show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
```
```   980       from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto
```
```   981       show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```   982       proof
```
```   983         fix A assume A: "A \<in> E i"
```
```   984         then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^sub>E j\<in>I. if i = j then A else space (M j))"
```
```   985           using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm)
```
```   986         also have "\<dots> = (\<Pi>\<^sub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
```
```   987           by (intro PiE_cong) (simp add: S_union)
```
```   988         also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^sub>E j\<in>I. if i = j then A else S j (xs ! T j))"
```
```   989           using T
```
```   990           apply (auto simp: PiE_iff bchoice_iff)
```
```   991           apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
```
```   992           apply (auto simp: bij_betw_def)
```
```   993           done
```
```   994         also have "\<dots> \<in> sets ?P"
```
```   995         proof (safe intro!: sets.countable_UN)
```
```   996           fix xs show "(\<Pi>\<^sub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
```
```   997             using A S_in_E
```
```   998             by (simp add: P_closed)
```
```   999                (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
```
```  1000         qed
```
```  1001         finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1002           using P_closed by simp
```
```  1003       qed
```
```  1004     qed
```
```  1005     from measurable_sets[OF this, of A] A `i \<in> I` E_closed
```
```  1006     have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
```
```  1007       by (simp add: E_generates)
```
```  1008     also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A}"
```
```  1009       using P_closed by (auto simp: space_PiM)
```
```  1010     finally show "\<dots> \<in> sets ?P" .
```
```  1011   qed
```
```  1012   finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
```
```  1013     by (simp add: P_closed)
```
```  1014   show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
```
```  1015     using `finite I`
```
```  1016     by (auto intro!: sets.sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
```
```  1017 qed
```
```  1018
```
```  1019 lemma pair_measure_eq_distr_PiM:
```
```  1020   fixes M1 :: "'a measure" and M2 :: "'a measure"
```
```  1021   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
```
```  1022   shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))"
```
```  1023     (is "?P = ?D")
```
```  1024 proof (rule pair_measure_eqI[OF assms])
```
```  1025   interpret B: product_sigma_finite "case_bool M1 M2"
```
```  1026     unfolding product_sigma_finite_def using assms by (auto split: bool.split)
```
```  1027   let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)"
```
```  1028
```
```  1029   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)"
```
```  1030     by auto
```
```  1031   fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2"
```
```  1032   have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))"
```
```  1033     by (simp add: UNIV_bool ac_simps)
```
```  1034   also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))"
```
```  1035     using A B by (subst B.emeasure_PiM) (auto split: bool.split)
```
```  1036   also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B"
```
```  1037     using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space]
```
```  1038     by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split)
```
```  1039   finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)"
```
```  1040     using A B
```
```  1041       measurable_component_singleton[of True UNIV "case_bool M1 M2"]
```
```  1042       measurable_component_singleton[of False UNIV "case_bool M1 M2"]
```
```  1043     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
```
```  1044 qed simp
```
```  1045
```
```  1046 end
```