src/HOL/Probability/Finite_Product_Measure.thy
author immler@in.tum.de
Tue Nov 06 11:03:28 2012 +0100 (2012-11-06)
changeset 50038 8e32c9254535
parent 50021 d96a3f468203
child 50041 afe886a04198
permissions -rw-r--r--
moved lemmas further up
hoelzl@42146
     1
(*  Title:      HOL/Probability/Finite_Product_Measure.thy
hoelzl@42067
     2
    Author:     Johannes Hölzl, TU München
hoelzl@42067
     3
*)
hoelzl@42067
     4
hoelzl@42146
     5
header {*Finite product measures*}
hoelzl@42067
     6
hoelzl@42146
     7
theory Finite_Product_Measure
hoelzl@42146
     8
imports Binary_Product_Measure
hoelzl@35833
     9
begin
hoelzl@35833
    10
hoelzl@47694
    11
lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
hoelzl@47694
    12
  by auto
hoelzl@47694
    13
hoelzl@40859
    14
abbreviation
hoelzl@40859
    15
  "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
hellerar@39094
    16
hoelzl@41689
    17
syntax
hoelzl@41689
    18
  "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
hoelzl@41689
    19
hoelzl@41689
    20
syntax (xsymbols)
hoelzl@41689
    21
  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
hoelzl@41689
    22
hoelzl@41689
    23
syntax (HTML output)
hoelzl@41689
    24
  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
hoelzl@41689
    25
hoelzl@41689
    26
translations
hoelzl@41689
    27
  "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
hoelzl@41689
    28
hoelzl@40859
    29
abbreviation
hoelzl@40859
    30
  funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
hoelzl@40859
    31
    (infixr "->\<^isub>E" 60) where
hoelzl@40859
    32
  "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
hoelzl@40859
    33
hoelzl@40859
    34
notation (xsymbols)
hoelzl@40859
    35
  funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
hoelzl@40859
    36
hoelzl@40859
    37
lemma extensional_insert[intro, simp]:
hoelzl@40859
    38
  assumes "a \<in> extensional (insert i I)"
hoelzl@40859
    39
  shows "a(i := b) \<in> extensional (insert i I)"
hoelzl@40859
    40
  using assms unfolding extensional_def by auto
hoelzl@40859
    41
hoelzl@40859
    42
lemma extensional_Int[simp]:
hoelzl@40859
    43
  "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
hoelzl@40859
    44
  unfolding extensional_def by auto
hoelzl@38656
    45
immler@50038
    46
lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
immler@50038
    47
  by (auto simp: extensional_def)
immler@50038
    48
immler@50038
    49
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
immler@50038
    50
  unfolding restrict_def extensional_def by auto
immler@50038
    51
immler@50038
    52
lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
immler@50038
    53
  unfolding restrict_def by (simp add: fun_eq_iff)
immler@50038
    54
hoelzl@35833
    55
definition
hoelzl@49780
    56
  "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
hoelzl@40859
    57
hoelzl@40859
    58
lemma merge_apply[simp]:
hoelzl@49780
    59
  "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
hoelzl@49780
    60
  "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
hoelzl@49780
    61
  "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i"
hoelzl@49780
    62
  "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i"
hoelzl@49780
    63
  "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined"
hoelzl@40859
    64
  unfolding merge_def by auto
hoelzl@40859
    65
hoelzl@40859
    66
lemma merge_commute:
hoelzl@49780
    67
  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)"
hoelzl@50003
    68
  by (force simp: merge_def)
hoelzl@40859
    69
hoelzl@40859
    70
lemma Pi_cancel_merge_range[simp]:
hoelzl@49780
    71
  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@49780
    72
  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@49780
    73
  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@49780
    74
  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    75
  by (auto simp: Pi_def)
hoelzl@40859
    76
hoelzl@40859
    77
lemma Pi_cancel_merge[simp]:
hoelzl@49780
    78
  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
hoelzl@49780
    79
  "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
hoelzl@49780
    80
  "I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
hoelzl@49780
    81
  "J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
hoelzl@40859
    82
  by (auto simp: Pi_def)
hoelzl@40859
    83
hoelzl@49780
    84
lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)"
hoelzl@40859
    85
  by (auto simp: extensional_def)
hoelzl@40859
    86
hoelzl@40859
    87
lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    88
  by (auto simp: restrict_def Pi_def)
hoelzl@40859
    89
hoelzl@40859
    90
lemma restrict_merge[simp]:
hoelzl@49780
    91
  "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
hoelzl@49780
    92
  "I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
hoelzl@49780
    93
  "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I"
hoelzl@49780
    94
  "J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J"
hoelzl@47694
    95
  by (auto simp: restrict_def)
hoelzl@40859
    96
hoelzl@40859
    97
lemma extensional_insert_undefined[intro, simp]:
hoelzl@40859
    98
  assumes "a \<in> extensional (insert i I)"
hoelzl@40859
    99
  shows "a(i := undefined) \<in> extensional I"
hoelzl@40859
   100
  using assms unfolding extensional_def by auto
hoelzl@40859
   101
hoelzl@40859
   102
lemma extensional_insert_cancel[intro, simp]:
hoelzl@40859
   103
  assumes "a \<in> extensional I"
hoelzl@40859
   104
  shows "a \<in> extensional (insert i I)"
hoelzl@40859
   105
  using assms unfolding extensional_def by auto
hoelzl@40859
   106
hoelzl@49780
   107
lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)"
hoelzl@41095
   108
  unfolding merge_def by (auto simp: fun_eq_iff)
hoelzl@41095
   109
hoelzl@41095
   110
lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
hoelzl@41095
   111
  by auto
hoelzl@41095
   112
hoelzl@40859
   113
lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)"
hoelzl@40859
   114
  by auto
hoelzl@40859
   115
hoelzl@40859
   116
lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
hoelzl@40859
   117
  by (auto simp: Pi_def)
hoelzl@40859
   118
hoelzl@40859
   119
lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
hoelzl@40859
   120
  by (auto simp: Pi_def)
hoelzl@39088
   121
hoelzl@40859
   122
lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
hoelzl@40859
   123
  by (auto simp: Pi_def)
hoelzl@40859
   124
hoelzl@40859
   125
lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
hoelzl@40859
   126
  by (auto simp: Pi_def)
hoelzl@40859
   127
hoelzl@41095
   128
lemma restrict_vimage:
hoelzl@41095
   129
  assumes "I \<inter> J = {}"
hoelzl@49780
   130
  shows "(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I J (E, F))"
hoelzl@41095
   131
  using assms by (auto simp: restrict_Pi_cancel)
hoelzl@41095
   132
hoelzl@41095
   133
lemma merge_vimage:
hoelzl@41095
   134
  assumes "I \<inter> J = {}"
hoelzl@49780
   135
  shows "merge I J -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
hoelzl@41095
   136
  using assms by (auto simp: restrict_Pi_cancel)
hoelzl@41095
   137
hoelzl@41095
   138
lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
hoelzl@47694
   139
  by (auto simp: restrict_def)
hoelzl@41095
   140
hoelzl@41095
   141
lemma merge_restrict[simp]:
hoelzl@49780
   142
  "merge I J (restrict x I, y) = merge I J (x, y)"
hoelzl@49780
   143
  "merge I J (x, restrict y J) = merge I J (x, y)"
hoelzl@47694
   144
  unfolding merge_def by auto
hoelzl@41095
   145
hoelzl@41095
   146
lemma merge_x_x_eq_restrict[simp]:
hoelzl@49780
   147
  "merge I J (x, x) = restrict x (I \<union> J)"
hoelzl@47694
   148
  unfolding merge_def by auto
hoelzl@41095
   149
hoelzl@41095
   150
lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
hoelzl@41095
   151
  apply auto
hoelzl@41095
   152
  apply (drule_tac x=x in Pi_mem)
hoelzl@41095
   153
  apply (simp_all split: split_if_asm)
hoelzl@41095
   154
  apply (drule_tac x=i in Pi_mem)
hoelzl@41095
   155
  apply (auto dest!: Pi_mem)
hoelzl@41095
   156
  done
hoelzl@41095
   157
hoelzl@41095
   158
lemma Pi_UN:
hoelzl@41095
   159
  fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
hoelzl@41095
   160
  assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
hoelzl@41095
   161
  shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
hoelzl@41095
   162
proof (intro set_eqI iffI)
hoelzl@41095
   163
  fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
hoelzl@41095
   164
  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
hoelzl@41095
   165
  from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
hoelzl@41095
   166
  obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
hoelzl@41095
   167
    using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
hoelzl@41095
   168
  have "f \<in> Pi I (A k)"
hoelzl@41095
   169
  proof (intro Pi_I)
hoelzl@41095
   170
    fix i assume "i \<in> I"
hoelzl@41095
   171
    from mono[OF this, of "n i" k] k[OF this] n[OF this]
hoelzl@41095
   172
    show "f i \<in> A k i" by auto
hoelzl@41095
   173
  qed
hoelzl@41095
   174
  then show "f \<in> (\<Union>n. Pi I (A n))" by auto
hoelzl@41095
   175
qed auto
hoelzl@41095
   176
hoelzl@41095
   177
lemma PiE_cong:
hoelzl@41095
   178
  assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
hoelzl@41095
   179
  shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
hoelzl@41095
   180
  using assms by (auto intro!: Pi_cong)
hoelzl@41095
   181
hoelzl@41095
   182
lemma restrict_upd[simp]:
hoelzl@41095
   183
  "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
hoelzl@41095
   184
  by (auto simp: fun_eq_iff)
hoelzl@41095
   185
hoelzl@41689
   186
lemma Pi_eq_subset:
hoelzl@41689
   187
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
hoelzl@41689
   188
  assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"
hoelzl@41689
   189
  shows "F i \<subseteq> F' i"
hoelzl@41689
   190
proof
hoelzl@41689
   191
  fix x assume "x \<in> F i"
hoelzl@41689
   192
  with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto
hoelzl@41689
   193
  from choice[OF this] guess f .. note f = this
hoelzl@41689
   194
  then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
hoelzl@41689
   195
  then have "f \<in> Pi\<^isub>E I F'" using assms by simp
hoelzl@41689
   196
  then show "x \<in> F' i" using f `i \<in> I` by auto
hoelzl@41689
   197
qed
hoelzl@41689
   198
hoelzl@41689
   199
lemma Pi_eq_iff_not_empty:
hoelzl@41689
   200
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
hoelzl@41689
   201
  shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
hoelzl@41689
   202
proof (intro iffI ballI)
hoelzl@41689
   203
  fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"
hoelzl@41689
   204
  show "F i = F' i"
hoelzl@41689
   205
    using Pi_eq_subset[of I F F', OF ne eq i]
hoelzl@41689
   206
    using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
hoelzl@41689
   207
    by auto
hoelzl@41689
   208
qed auto
hoelzl@41689
   209
hoelzl@41689
   210
lemma Pi_eq_empty_iff:
hoelzl@41689
   211
  "Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
hoelzl@41689
   212
proof
hoelzl@41689
   213
  assume "Pi\<^isub>E I F = {}"
hoelzl@41689
   214
  show "\<exists>i\<in>I. F i = {}"
hoelzl@41689
   215
  proof (rule ccontr)
hoelzl@41689
   216
    assume "\<not> ?thesis"
hoelzl@41689
   217
    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
hoelzl@41689
   218
    from choice[OF this] guess f ..
hoelzl@41689
   219
    then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
hoelzl@41689
   220
    with `Pi\<^isub>E I F = {}` show False by auto
hoelzl@41689
   221
  qed
hoelzl@41689
   222
qed auto
hoelzl@41689
   223
hoelzl@41689
   224
lemma Pi_eq_iff:
hoelzl@41689
   225
  "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
hoelzl@41689
   226
proof (intro iffI disjCI)
hoelzl@41689
   227
  assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"
hoelzl@41689
   228
  assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
hoelzl@41689
   229
  then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
hoelzl@41689
   230
    using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
hoelzl@41689
   231
  with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
hoelzl@41689
   232
next
hoelzl@41689
   233
  assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
hoelzl@41689
   234
  then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
hoelzl@41689
   235
    using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
hoelzl@41689
   236
qed
hoelzl@41689
   237
hoelzl@40859
   238
section "Finite product spaces"
hoelzl@40859
   239
hoelzl@40859
   240
section "Products"
hoelzl@40859
   241
hoelzl@47694
   242
definition prod_emb where
hoelzl@47694
   243
  "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
hoelzl@47694
   244
hoelzl@47694
   245
lemma prod_emb_iff: 
hoelzl@47694
   246
  "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))"
hoelzl@47694
   247
  unfolding prod_emb_def by auto
hoelzl@40859
   248
hoelzl@47694
   249
lemma
hoelzl@47694
   250
  shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
hoelzl@47694
   251
    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"
hoelzl@47694
   252
    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"
hoelzl@47694
   253
    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))"
hoelzl@47694
   254
    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))"
hoelzl@47694
   255
    and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
hoelzl@47694
   256
  by (auto simp: prod_emb_def)
hoelzl@40859
   257
hoelzl@47694
   258
lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
hoelzl@47694
   259
    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))"
hoelzl@47694
   260
  by (force simp: prod_emb_def Pi_iff split_if_mem2)
hoelzl@47694
   261
hoelzl@47694
   262
lemma prod_emb_PiE_same_index[simp]: "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E"
hoelzl@47694
   263
  by (auto simp: prod_emb_def Pi_iff)
hoelzl@41689
   264
immler@50038
   265
lemma prod_emb_trans[simp]:
immler@50038
   266
  "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"
immler@50038
   267
  by (auto simp add: Int_absorb1 prod_emb_def)
immler@50038
   268
immler@50038
   269
lemma prod_emb_Pi:
immler@50038
   270
  assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
immler@50038
   271
  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))"
immler@50038
   272
  using assms space_closed
immler@50038
   273
  by (auto simp: prod_emb_def Pi_iff split: split_if_asm) blast+
immler@50038
   274
immler@50038
   275
lemma prod_emb_id:
immler@50038
   276
  "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
immler@50038
   277
  by (auto simp: prod_emb_def Pi_iff subset_eq extensional_restrict)
immler@50038
   278
hoelzl@47694
   279
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
hoelzl@47694
   280
  "PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
hoelzl@47694
   281
    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
hoelzl@47694
   282
    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
hoelzl@47694
   283
    (\<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)))"
hoelzl@47694
   284
hoelzl@47694
   285
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
hoelzl@47694
   286
  "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) `
hoelzl@47694
   287
    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
hoelzl@47694
   288
hoelzl@47694
   289
abbreviation
hoelzl@47694
   290
  "Pi\<^isub>M I M \<equiv> PiM I M"
hoelzl@41689
   291
hoelzl@40859
   292
syntax
hoelzl@47694
   293
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIM _:_./ _)" 10)
hoelzl@40859
   294
hoelzl@40859
   295
syntax (xsymbols)
hoelzl@47694
   296
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
hoelzl@40859
   297
hoelzl@40859
   298
syntax (HTML output)
hoelzl@47694
   299
  "_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"  10)
hoelzl@40859
   300
hoelzl@40859
   301
translations
hoelzl@47694
   302
  "PIM x:I. M" == "CONST PiM I (%x. M)"
hoelzl@41689
   303
hoelzl@47694
   304
lemma prod_algebra_sets_into_space:
hoelzl@47694
   305
  "prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   306
  using assms by (auto simp: prod_emb_def prod_algebra_def)
hoelzl@40859
   307
hoelzl@47694
   308
lemma prod_algebra_eq_finite:
hoelzl@47694
   309
  assumes I: "finite I"
hoelzl@47694
   310
  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")
hoelzl@47694
   311
proof (intro iffI set_eqI)
hoelzl@47694
   312
  fix A assume "A \<in> ?L"
hoelzl@47694
   313
  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)"
hoelzl@47694
   314
    and A: "A = prod_emb I M J (PIE j:J. E j)"
hoelzl@47694
   315
    by (auto simp: prod_algebra_def)
hoelzl@47694
   316
  let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)"
hoelzl@47694
   317
  have A: "A = ?A"
hoelzl@47694
   318
    unfolding A using J by (intro prod_emb_PiE sets_into_space) auto
hoelzl@47694
   319
  show "A \<in> ?R" unfolding A using J top
hoelzl@47694
   320
    by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp
hoelzl@47694
   321
next
hoelzl@47694
   322
  fix A assume "A \<in> ?R"
hoelzl@47694
   323
  then obtain X where "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
hoelzl@47694
   324
  then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)"
hoelzl@47694
   325
    using sets_into_space by (force simp: prod_emb_def Pi_iff)
hoelzl@47694
   326
  from X I show "A \<in> ?L" unfolding A
hoelzl@47694
   327
    by (auto simp: prod_algebra_def)
hoelzl@47694
   328
qed
hoelzl@41095
   329
hoelzl@47694
   330
lemma prod_algebraI:
hoelzl@47694
   331
  "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
hoelzl@47694
   332
    \<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M"
hoelzl@47694
   333
  by (auto simp: prod_algebra_def Pi_iff)
hoelzl@41689
   334
immler@50038
   335
lemma prod_algebraI_finite:
immler@50038
   336
  "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M"
immler@50038
   337
  using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp
immler@50038
   338
immler@50038
   339
lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
immler@50038
   340
proof (safe intro!: Int_stableI)
immler@50038
   341
  fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
immler@50038
   342
  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))"
immler@50038
   343
    by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"])
immler@50038
   344
qed
immler@50038
   345
hoelzl@47694
   346
lemma prod_algebraE:
hoelzl@47694
   347
  assumes A: "A \<in> prod_algebra I M"
hoelzl@47694
   348
  obtains J E where "A = prod_emb I M J (PIE j:J. E j)"
hoelzl@47694
   349
    "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" 
hoelzl@47694
   350
  using A by (auto simp: prod_algebra_def)
hoelzl@42988
   351
hoelzl@47694
   352
lemma prod_algebraE_all:
hoelzl@47694
   353
  assumes A: "A \<in> prod_algebra I M"
hoelzl@47694
   354
  obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
hoelzl@47694
   355
proof -
hoelzl@47694
   356
  from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)"
hoelzl@47694
   357
    and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
hoelzl@47694
   358
    by (auto simp: prod_algebra_def)
hoelzl@47694
   359
  from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)"
hoelzl@47694
   360
    using sets_into_space by auto
hoelzl@47694
   361
  then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))"
hoelzl@47694
   362
    using A J by (auto simp: prod_emb_PiE)
hoelzl@47694
   363
  moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))"
hoelzl@47694
   364
    using top E by auto
hoelzl@47694
   365
  ultimately show ?thesis using that by auto
hoelzl@47694
   366
qed
hoelzl@40859
   367
hoelzl@47694
   368
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
hoelzl@47694
   369
proof (unfold Int_stable_def, safe)
hoelzl@47694
   370
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   371
  from prod_algebraE[OF this] guess J E . note A = this
hoelzl@47694
   372
  fix B assume "B \<in> prod_algebra I M"
hoelzl@47694
   373
  from prod_algebraE[OF this] guess K F . note B = this
hoelzl@47694
   374
  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> 
hoelzl@47694
   375
      (if i \<in> K then F i else space (M i)))"
hoelzl@47694
   376
    unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space]
hoelzl@47694
   377
    apply (subst (1 2 3) prod_emb_PiE)
hoelzl@47694
   378
    apply (simp_all add: subset_eq PiE_Int)
hoelzl@47694
   379
    apply blast
hoelzl@47694
   380
    apply (intro PiE_cong)
hoelzl@47694
   381
    apply auto
hoelzl@47694
   382
    done
hoelzl@47694
   383
  also have "\<dots> \<in> prod_algebra I M"
hoelzl@47694
   384
    using A B by (auto intro!: prod_algebraI)
hoelzl@47694
   385
  finally show "A \<inter> B \<in> prod_algebra I M" .
hoelzl@47694
   386
qed
hoelzl@47694
   387
hoelzl@47694
   388
lemma prod_algebra_mono:
hoelzl@47694
   389
  assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
hoelzl@47694
   390
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
hoelzl@47694
   391
  shows "prod_algebra I E \<subseteq> prod_algebra I F"
hoelzl@47694
   392
proof
hoelzl@47694
   393
  fix A assume "A \<in> prod_algebra I E"
hoelzl@47694
   394
  then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
hoelzl@47694
   395
    and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   396
    and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)"
hoelzl@47694
   397
    by (auto simp: prod_algebra_def)
hoelzl@47694
   398
  moreover
hoelzl@47694
   399
  from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))"
hoelzl@47694
   400
    by (rule PiE_cong)
hoelzl@47694
   401
  with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)"
hoelzl@47694
   402
    by (simp add: prod_emb_def)
hoelzl@47694
   403
  moreover
hoelzl@47694
   404
  from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)"
hoelzl@47694
   405
    by auto
hoelzl@47694
   406
  ultimately show "A \<in> prod_algebra I F"
hoelzl@47694
   407
    apply (simp add: prod_algebra_def image_iff)
hoelzl@47694
   408
    apply (intro exI[of _ J] exI[of _ G] conjI)
hoelzl@47694
   409
    apply auto
hoelzl@47694
   410
    done
hoelzl@41689
   411
qed
hoelzl@41689
   412
hoelzl@47694
   413
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   414
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
hoelzl@47694
   415
hoelzl@47694
   416
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)"
hoelzl@47694
   417
  using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
hoelzl@41689
   418
hoelzl@47694
   419
lemma sets_PiM_single: "sets (PiM I M) =
hoelzl@47694
   420
    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)}"
hoelzl@47694
   421
    (is "_ = sigma_sets ?\<Omega> ?R")
hoelzl@47694
   422
  unfolding sets_PiM
hoelzl@47694
   423
proof (rule sigma_sets_eqI)
hoelzl@47694
   424
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
hoelzl@47694
   425
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   426
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   427
  show "A \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   428
  proof cases
hoelzl@47694
   429
    assume "I = {}"
hoelzl@47694
   430
    with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
hoelzl@47694
   431
    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
hoelzl@47694
   432
  next
hoelzl@47694
   433
    assume "I \<noteq> {}"
hoelzl@47694
   434
    with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})"
hoelzl@47694
   435
      using sets_into_space[OF X(5)]
hoelzl@47694
   436
      by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast
hoelzl@47694
   437
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
hoelzl@47694
   438
      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
hoelzl@47694
   439
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
hoelzl@47694
   440
  qed
hoelzl@47694
   441
next
hoelzl@47694
   442
  fix A assume "A \<in> ?R"
hoelzl@47694
   443
  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)" 
hoelzl@47694
   444
    by auto
hoelzl@47694
   445
  then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)"
hoelzl@47694
   446
    using sets_into_space[OF A(3)]
hoelzl@47694
   447
    apply (subst prod_emb_PiE)
hoelzl@47694
   448
    apply (auto simp: Pi_iff split: split_if_asm)
hoelzl@47694
   449
    apply blast
hoelzl@47694
   450
    done
hoelzl@47694
   451
  also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)"
hoelzl@47694
   452
    using A by (intro sigma_sets.Basic prod_algebraI) auto
hoelzl@47694
   453
  finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
hoelzl@47694
   454
qed
hoelzl@47694
   455
hoelzl@47694
   456
lemma sets_PiM_I:
hoelzl@47694
   457
  assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
hoelzl@47694
   458
  shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   459
proof cases
hoelzl@47694
   460
  assume "J = {}"
hoelzl@47694
   461
  then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))"
hoelzl@47694
   462
    by (auto simp: prod_emb_def)
hoelzl@47694
   463
  then show ?thesis
hoelzl@47694
   464
    by (auto simp add: sets_PiM intro!: sigma_sets_top)
hoelzl@47694
   465
next
hoelzl@47694
   466
  assume "J \<noteq> {}" with assms show ?thesis
hoelzl@50003
   467
    by (force simp add: sets_PiM prod_algebra_def)
hoelzl@40859
   468
qed
hoelzl@40859
   469
hoelzl@47694
   470
lemma measurable_PiM:
hoelzl@47694
   471
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   472
  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>
hoelzl@47694
   473
    f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" 
hoelzl@47694
   474
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   475
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   476
proof (rule measurable_sigma_sets)
hoelzl@47694
   477
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   478
  from prod_algebraE[OF this] guess J X .
hoelzl@47694
   479
  with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
hoelzl@47694
   480
qed
hoelzl@47694
   481
hoelzl@47694
   482
lemma measurable_PiM_Collect:
hoelzl@47694
   483
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   484
  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>
hoelzl@47694
   485
    {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" 
hoelzl@47694
   486
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   487
  using sets_PiM prod_algebra_sets_into_space space
hoelzl@47694
   488
proof (rule measurable_sigma_sets)
hoelzl@47694
   489
  fix A assume "A \<in> prod_algebra I M"
hoelzl@47694
   490
  from prod_algebraE[OF this] guess J X . note X = this
hoelzl@47694
   491
  have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
hoelzl@47694
   492
    using sets_into_space[OF X(5)] X(2-) space unfolding X(1)
hoelzl@47694
   493
    by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm)
hoelzl@47694
   494
  also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets)
hoelzl@47694
   495
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@41689
   496
qed
hoelzl@41095
   497
hoelzl@47694
   498
lemma measurable_PiM_single:
hoelzl@47694
   499
  assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
   500
  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" 
hoelzl@47694
   501
  shows "f \<in> measurable N (PiM I M)"
hoelzl@47694
   502
  using sets_PiM_single
hoelzl@47694
   503
proof (rule measurable_sigma_sets)
hoelzl@47694
   504
  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)}"
hoelzl@47694
   505
  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)"
hoelzl@47694
   506
    by auto
hoelzl@47694
   507
  with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto
hoelzl@47694
   508
  also have "\<dots> \<in> sets N" using B by (rule sets)
hoelzl@47694
   509
  finally show "f -` A \<inter> space N \<in> sets N" .
hoelzl@47694
   510
qed (auto simp: space)
hoelzl@40859
   511
hoelzl@50003
   512
lemma sets_PiM_I_finite[measurable]:
hoelzl@47694
   513
  assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
hoelzl@47694
   514
  shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
hoelzl@47694
   515
  using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto
hoelzl@47694
   516
hoelzl@50021
   517
lemma measurable_component_singleton:
hoelzl@41689
   518
  assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
hoelzl@41689
   519
proof (unfold measurable_def, intro CollectI conjI ballI)
hoelzl@41689
   520
  fix A assume "A \<in> sets (M i)"
hoelzl@47694
   521
  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)"
hoelzl@47694
   522
    using sets_into_space `i \<in> I`
hoelzl@47694
   523
    by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
hoelzl@41689
   524
  then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
hoelzl@47694
   525
    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
hoelzl@47694
   526
qed (insert `i \<in> I`, auto simp: space_PiM)
hoelzl@47694
   527
hoelzl@50021
   528
lemma measurable_component_singleton'[measurable_app]:
hoelzl@50021
   529
  assumes f: "f \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@50021
   530
  assumes i: "i \<in> I"
hoelzl@50021
   531
  shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)"
hoelzl@50021
   532
  using measurable_compose[OF f measurable_component_singleton, OF i] .
hoelzl@50021
   533
hoelzl@50021
   534
lemma measurable_nat_case[measurable (raw)]:
hoelzl@50021
   535
  assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
hoelzl@50021
   536
    "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
hoelzl@50021
   537
  shows "(\<lambda>x. nat_case (f x) (g x) i) \<in> measurable M N"
hoelzl@50021
   538
  by (cases i) simp_all
hoelzl@50021
   539
hoelzl@50003
   540
lemma measurable_add_dim[measurable]:
hoelzl@49776
   541
  "(\<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)"
hoelzl@47694
   542
    (is "?f \<in> measurable ?P ?I")
hoelzl@47694
   543
proof (rule measurable_PiM_single)
hoelzl@47694
   544
  fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)"
hoelzl@47694
   545
  have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} =
hoelzl@47694
   546
    (if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)"
hoelzl@47694
   547
    using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM)
hoelzl@47694
   548
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   549
    using A j
hoelzl@47694
   550
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@47694
   551
  finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
hoelzl@47694
   552
qed (auto simp: space_pair_measure space_PiM)
hoelzl@41661
   553
hoelzl@50003
   554
lemma measurable_component_update:
hoelzl@50003
   555
  "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)"
hoelzl@50003
   556
  by simp
hoelzl@50003
   557
hoelzl@50003
   558
lemma measurable_merge[measurable]:
hoelzl@49780
   559
  "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)"
hoelzl@47694
   560
    (is "?f \<in> measurable ?P ?U")
hoelzl@47694
   561
proof (rule measurable_PiM_single)
hoelzl@47694
   562
  fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
hoelzl@49780
   563
  then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
hoelzl@47694
   564
    (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)"
hoelzl@49776
   565
    by (auto simp: merge_def)
hoelzl@47694
   566
  also have "\<dots> \<in> sets ?P"
hoelzl@47694
   567
    using A
hoelzl@47694
   568
    by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton])
hoelzl@49780
   569
  finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
hoelzl@49776
   570
qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def)
hoelzl@42988
   571
hoelzl@50003
   572
lemma measurable_restrict[measurable (raw)]:
hoelzl@47694
   573
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
hoelzl@47694
   574
  shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)"
hoelzl@47694
   575
proof (rule measurable_PiM_single)
hoelzl@47694
   576
  fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
hoelzl@47694
   577
  then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N"
hoelzl@47694
   578
    by auto
hoelzl@47694
   579
  then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N"
hoelzl@47694
   580
    using A X by (auto intro!: measurable_sets)
hoelzl@47694
   581
qed (insert X, auto dest: measurable_space)
hoelzl@47694
   582
immler@50038
   583
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)"
immler@50038
   584
  by (intro measurable_restrict measurable_component_singleton) auto
immler@50038
   585
immler@50038
   586
lemma measurable_prod_emb[intro, simp]:
immler@50038
   587
  "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)"
immler@50038
   588
  unfolding prod_emb_def space_PiM[symmetric]
immler@50038
   589
  by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
immler@50038
   590
hoelzl@50003
   591
lemma sets_in_Pi_aux:
hoelzl@50003
   592
  "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50003
   593
  {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
hoelzl@50003
   594
  by (simp add: subset_eq Pi_iff)
hoelzl@50003
   595
hoelzl@50003
   596
lemma sets_in_Pi[measurable (raw)]:
hoelzl@50003
   597
  "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
hoelzl@50003
   598
  (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
hoelzl@50003
   599
  Sigma_Algebra.pred N (\<lambda>x. f x \<in> Pi I F)"
hoelzl@50003
   600
  unfolding pred_def
hoelzl@50003
   601
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
hoelzl@50003
   602
hoelzl@50003
   603
lemma sets_in_extensional_aux:
hoelzl@50003
   604
  "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
hoelzl@50003
   605
proof -
hoelzl@50003
   606
  have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
hoelzl@50003
   607
    by (auto simp add: extensional_def space_PiM)
hoelzl@50003
   608
  then show ?thesis by simp
hoelzl@50003
   609
qed
hoelzl@50003
   610
hoelzl@50003
   611
lemma sets_in_extensional[measurable (raw)]:
hoelzl@50003
   612
  "f \<in> measurable N (PiM I M) \<Longrightarrow> Sigma_Algebra.pred N (\<lambda>x. f x \<in> extensional I)"
hoelzl@50003
   613
  unfolding pred_def
hoelzl@50003
   614
  by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
hoelzl@50003
   615
hoelzl@47694
   616
locale product_sigma_finite =
hoelzl@47694
   617
  fixes M :: "'i \<Rightarrow> 'a measure"
hoelzl@41689
   618
  assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
hoelzl@40859
   619
hoelzl@41689
   620
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
hoelzl@40859
   621
  by (rule sigma_finite_measures)
hoelzl@40859
   622
hoelzl@47694
   623
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
hoelzl@47694
   624
  fixes I :: "'i set"
hoelzl@47694
   625
  assumes finite_index: "finite I"
hoelzl@41689
   626
hoelzl@40859
   627
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
hoelzl@40859
   628
  "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
hoelzl@40859
   629
    (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
hoelzl@47694
   630
    (\<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>
hoelzl@47694
   631
    (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)"
hoelzl@40859
   632
proof -
hoelzl@47694
   633
  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>)"
hoelzl@47694
   634
    using M.sigma_finite_incseq by metis
hoelzl@40859
   635
  from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@47694
   636
  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>"
hoelzl@40859
   637
    by auto
hoelzl@40859
   638
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
hoelzl@47694
   639
  note space_PiM[simp]
hoelzl@40859
   640
  show ?thesis
hoelzl@41981
   641
  proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
hoelzl@40859
   642
    fix i show "range (F i) \<subseteq> sets (M i)" by fact
hoelzl@40859
   643
  next
hoelzl@47694
   644
    fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact
hoelzl@40859
   645
  next
hoelzl@47694
   646
    fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)"
hoelzl@47694
   647
      using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space
hoelzl@47694
   648
      by auto blast
hoelzl@40859
   649
  next
hoelzl@47694
   650
    fix f assume "f \<in> space (PiM I M)"
hoelzl@41981
   651
    with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
hoelzl@41981
   652
    show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
hoelzl@40859
   653
  next
hoelzl@40859
   654
    fix i show "?F i \<subseteq> ?F (Suc i)"
hoelzl@41981
   655
      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
hoelzl@40859
   656
  qed
hoelzl@40859
   657
qed
hoelzl@40859
   658
hoelzl@49780
   659
lemma
hoelzl@49780
   660
  shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}"
hoelzl@49780
   661
    and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }"
hoelzl@49780
   662
  by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
hoelzl@49780
   663
hoelzl@49780
   664
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
hoelzl@49780
   665
proof -
hoelzl@49780
   666
  let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)"
hoelzl@49780
   667
  have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1"
hoelzl@49780
   668
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49780
   669
    show "positive (PiM {} M) ?\<mu>"
hoelzl@49780
   670
      by (auto simp: positive_def)
hoelzl@49780
   671
    show "countably_additive (PiM {} M) ?\<mu>"
hoelzl@49780
   672
      by (rule countably_additiveI_finite)
hoelzl@49780
   673
         (auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: )
hoelzl@49780
   674
  qed (auto simp: prod_emb_def)
hoelzl@49780
   675
  also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}"
hoelzl@49780
   676
    by (auto simp: prod_emb_def)
hoelzl@49780
   677
  finally show ?thesis
hoelzl@49780
   678
    by simp
hoelzl@49780
   679
qed
hoelzl@49780
   680
hoelzl@49780
   681
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
hoelzl@49780
   682
  by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def)
hoelzl@49780
   683
hoelzl@49776
   684
lemma (in product_sigma_finite) emeasure_PiM:
hoelzl@49776
   685
  "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))"
hoelzl@49776
   686
proof (induct I arbitrary: A rule: finite_induct)
hoelzl@40859
   687
  case (insert i I)
hoelzl@41689
   688
  interpret finite_product_sigma_finite M I by default fact
hoelzl@40859
   689
  have "finite (insert i I)" using `finite I` by auto
hoelzl@41689
   690
  interpret I': finite_product_sigma_finite M "insert i I" by default fact
hoelzl@41661
   691
  let ?h = "(\<lambda>(f, y). f(i := y))"
hoelzl@47694
   692
hoelzl@47694
   693
  let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h"
hoelzl@47694
   694
  let ?\<mu> = "emeasure ?P"
hoelzl@47694
   695
  let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}"
hoelzl@47694
   696
  let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))"
hoelzl@47694
   697
hoelzl@49776
   698
  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)) =
hoelzl@49776
   699
    (\<Prod>i\<in>insert i I. emeasure (M i) (A i))"
hoelzl@49776
   700
  proof (subst emeasure_extend_measure_Pair[OF PiM_def])
hoelzl@49776
   701
    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))"
hoelzl@49776
   702
    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
hoelzl@49776
   703
    let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)"
hoelzl@49776
   704
    let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)"
hoelzl@49776
   705
    have "?\<mu> ?p =
hoelzl@49776
   706
      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))"
hoelzl@49776
   707
      by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+
hoelzl@49776
   708
    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))"
hoelzl@49776
   709
      using J E[rule_format, THEN sets_into_space]
hoelzl@49776
   710
      by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm)
hoelzl@49776
   711
    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))) =
hoelzl@49776
   712
      emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))"
hoelzl@49776
   713
      using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto
hoelzl@49776
   714
    also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))"
hoelzl@49776
   715
      using J E[rule_format, THEN sets_into_space]
hoelzl@49776
   716
      by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+
hoelzl@49776
   717
    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)) =
hoelzl@49776
   718
      (\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))"
hoelzl@49776
   719
      using E by (subst insert) (auto intro!: setprod_cong)
hoelzl@49776
   720
    also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
hoelzl@49776
   721
       emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
hoelzl@49776
   722
      using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong)
hoelzl@49776
   723
    also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
hoelzl@49776
   724
      using insert(1,2) J E by (intro setprod_mono_one_right) auto
hoelzl@49776
   725
    finally show "?\<mu> ?p = \<dots>" .
hoelzl@47694
   726
hoelzl@49776
   727
    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))"
hoelzl@49776
   728
      using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff)
hoelzl@49776
   729
  next
hoelzl@49776
   730
    show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>"
hoelzl@49776
   731
      using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all
hoelzl@49776
   732
  next
hoelzl@49776
   733
    show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and>
hoelzl@49776
   734
      insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))"
hoelzl@49776
   735
      using insert by auto
hoelzl@49776
   736
  qed (auto intro!: setprod_cong)
hoelzl@49776
   737
  with insert show ?case
hoelzl@49776
   738
    by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space)
hoelzl@50003
   739
qed simp
hoelzl@47694
   740
hoelzl@49776
   741
lemma (in product_sigma_finite) sigma_finite: 
hoelzl@49776
   742
  assumes "finite I"
hoelzl@49776
   743
  shows "sigma_finite_measure (PiM I M)"
hoelzl@49776
   744
proof -
hoelzl@49776
   745
  interpret finite_product_sigma_finite M I by default fact
hoelzl@49776
   746
hoelzl@49776
   747
  from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
hoelzl@49776
   748
  then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
hoelzl@49776
   749
    "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)"
hoelzl@49776
   750
    "(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)"
hoelzl@49776
   751
    "\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>"
hoelzl@47694
   752
    by blast+
hoelzl@49776
   753
  let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k"
hoelzl@47694
   754
hoelzl@49776
   755
  show ?thesis
hoelzl@47694
   756
  proof (unfold_locales, intro exI[of _ ?F] conjI allI)
hoelzl@49776
   757
    show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto
hoelzl@47694
   758
  next
hoelzl@49776
   759
    from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp
hoelzl@47694
   760
  next
hoelzl@47694
   761
    fix j
hoelzl@49776
   762
    from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M]
hoelzl@49776
   763
    show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>"
hoelzl@49776
   764
      by (subst emeasure_PiM) auto
hoelzl@40859
   765
  qed
hoelzl@40859
   766
qed
hoelzl@40859
   767
hoelzl@47694
   768
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M"
hoelzl@47694
   769
  using sigma_finite[OF finite_index] .
hoelzl@40859
   770
hoelzl@40859
   771
lemma (in finite_product_sigma_finite) measure_times:
hoelzl@47694
   772
  "(\<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))"
hoelzl@47694
   773
  using emeasure_PiM[OF finite_index] by auto
hoelzl@41096
   774
hoelzl@40859
   775
lemma (in product_sigma_finite) positive_integral_empty:
hoelzl@41981
   776
  assumes pos: "0 \<le> f (\<lambda>k. undefined)"
hoelzl@41981
   777
  shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
hoelzl@40859
   778
proof -
hoelzl@41689
   779
  interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
hoelzl@47694
   780
  have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
hoelzl@40859
   781
    using assms by (subst measure_times) auto
hoelzl@40859
   782
  then show ?thesis
hoelzl@47694
   783
    unfolding positive_integral_def simple_function_def simple_integral_def[abs_def]
hoelzl@47694
   784
  proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym)
hoelzl@41981
   785
    show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
hoelzl@44928
   786
      by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
hoelzl@41981
   787
    show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
hoelzl@44928
   788
      by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
hoelzl@40859
   789
  qed
hoelzl@40859
   790
qed
hoelzl@40859
   791
hoelzl@47694
   792
lemma (in product_sigma_finite) distr_merge:
hoelzl@40859
   793
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@49780
   794
  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"
hoelzl@47694
   795
   (is "?D = ?P")
hoelzl@40859
   796
proof -
hoelzl@41689
   797
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   798
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@40859
   799
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   800
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   801
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   802
  let ?g = "merge I J"
hoelzl@47694
   803
hoelzl@41661
   804
  from IJ.sigma_finite_pairs obtain F where
hoelzl@41661
   805
    F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
hoelzl@41981
   806
       "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
hoelzl@47694
   807
       "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P"
hoelzl@47694
   808
       "\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>"
hoelzl@41661
   809
    by auto
hoelzl@41661
   810
  let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
hoelzl@47694
   811
  
hoelzl@47694
   812
  show ?thesis
hoelzl@47694
   813
  proof (rule measure_eqI_generator_eq[symmetric])
hoelzl@47694
   814
    show "Int_stable (prod_algebra (I \<union> J) M)"
hoelzl@47694
   815
      by (rule Int_stable_prod_algebra)
hoelzl@47694
   816
    show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))"
hoelzl@47694
   817
      by (rule prod_algebra_sets_into_space)
hoelzl@47694
   818
    show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   819
      by (rule sets_PiM)
hoelzl@47694
   820
    then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)"
hoelzl@47694
   821
      by simp
hoelzl@47694
   822
hoelzl@47694
   823
    show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F
hoelzl@47694
   824
      using fin by (auto simp: prod_algebra_eq_finite)
hoelzl@47694
   825
    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))"
hoelzl@47694
   826
      using F(3) by (simp add: space_PiM)
hoelzl@41981
   827
  next
hoelzl@41981
   828
    fix k
hoelzl@47694
   829
    from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M]
hoelzl@47694
   830
    show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
hoelzl@41661
   831
  next
hoelzl@47694
   832
    fix A assume A: "A \<in> prod_algebra (I \<union> J) M"
hoelzl@50003
   833
    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)"
hoelzl@47694
   834
      by (auto simp add: prod_algebra_eq_finite)
hoelzl@47694
   835
    let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M"
hoelzl@47694
   836
    let ?X = "?g -` A \<inter> space ?B"
hoelzl@47694
   837
    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)"
hoelzl@50003
   838
      using F[rule_format, THEN sets_into_space] by (force simp: space_PiM)+
hoelzl@47694
   839
    then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
hoelzl@47694
   840
      unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM)
hoelzl@47694
   841
    have "emeasure ?D A = emeasure ?B ?X"
hoelzl@47694
   842
      using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM)
hoelzl@47694
   843
    also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))"
hoelzl@50003
   844
      using `finite J` `finite I` F unfolding X
hoelzl@49776
   845
      by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff)
hoelzl@47694
   846
    also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))"
hoelzl@41661
   847
      using `finite J` `finite I` `I \<inter> J = {}`  by (simp add: setprod_Un_one)
hoelzl@47694
   848
    also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)"
hoelzl@41661
   849
      using `finite J` `finite I` F unfolding A
hoelzl@41661
   850
      by (intro IJ.measure_times[symmetric]) auto
hoelzl@47694
   851
    finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp
hoelzl@47694
   852
  qed
hoelzl@41661
   853
qed
hoelzl@41026
   854
hoelzl@41026
   855
lemma (in product_sigma_finite) product_positive_integral_fold:
hoelzl@47694
   856
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
hoelzl@41689
   857
  and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@41689
   858
  shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
hoelzl@49780
   859
    (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
hoelzl@41026
   860
proof -
hoelzl@41689
   861
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   862
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41831
   863
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   864
  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)"
hoelzl@49776
   865
    using measurable_comp[OF measurable_merge f] by (simp add: comp_def)
hoelzl@41661
   866
  show ?thesis
hoelzl@47694
   867
    apply (subst distr_merge[OF IJ, symmetric])
hoelzl@49776
   868
    apply (subst positive_integral_distr[OF measurable_merge f])
hoelzl@49999
   869
    apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
hoelzl@47694
   870
    apply simp
hoelzl@47694
   871
    done
hoelzl@40859
   872
qed
hoelzl@40859
   873
hoelzl@47694
   874
lemma (in product_sigma_finite) distr_singleton:
hoelzl@47694
   875
  "distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
hoelzl@47694
   876
proof (intro measure_eqI[symmetric])
hoelzl@41831
   877
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   878
  fix A assume A: "A \<in> sets (M i)"
hoelzl@47694
   879
  moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)"
hoelzl@47694
   880
    using sets_into_space by (auto simp: space_PiM)
hoelzl@47694
   881
  ultimately show "emeasure (M i) A = emeasure ?D A"
hoelzl@47694
   882
    using A I.measure_times[of "\<lambda>_. A"]
hoelzl@47694
   883
    by (simp add: emeasure_distr measurable_component_singleton)
hoelzl@47694
   884
qed simp
hoelzl@41831
   885
hoelzl@41026
   886
lemma (in product_sigma_finite) product_positive_integral_singleton:
hoelzl@40859
   887
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   888
  shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
hoelzl@40859
   889
proof -
hoelzl@41689
   890
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@47694
   891
  from f show ?thesis
hoelzl@47694
   892
    apply (subst distr_singleton[symmetric])
hoelzl@47694
   893
    apply (subst positive_integral_distr[OF measurable_component_singleton])
hoelzl@47694
   894
    apply simp_all
hoelzl@47694
   895
    done
hoelzl@40859
   896
qed
hoelzl@40859
   897
hoelzl@41096
   898
lemma (in product_sigma_finite) product_positive_integral_insert:
hoelzl@49780
   899
  assumes I[simp]: "finite I" "i \<notin> I"
hoelzl@41689
   900
    and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@41689
   901
  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))"
hoelzl@41096
   902
proof -
hoelzl@41689
   903
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41689
   904
  interpret i: finite_product_sigma_finite M "{i}" by default auto
hoelzl@41689
   905
  have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
hoelzl@41689
   906
    using f by auto
hoelzl@41096
   907
  show ?thesis
hoelzl@49780
   908
    unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f]
hoelzl@49780
   909
  proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric])
hoelzl@47694
   910
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@49780
   911
    let ?f = "\<lambda>y. f (x(i := y))"
hoelzl@49780
   912
    show "?f \<in> borel_measurable (M i)"
hoelzl@47694
   913
      using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
hoelzl@47694
   914
      unfolding comp_def .
hoelzl@49780
   915
    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)"
hoelzl@49780
   916
      using x
hoelzl@49780
   917
      by (auto intro!: positive_integral_cong arg_cong[where f=f]
hoelzl@49780
   918
               simp add: space_PiM extensional_def)
hoelzl@41096
   919
  qed
hoelzl@41096
   920
qed
hoelzl@41096
   921
hoelzl@41096
   922
lemma (in product_sigma_finite) product_positive_integral_setprod:
hoelzl@43920
   923
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
hoelzl@41096
   924
  assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41981
   925
  and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
hoelzl@41689
   926
  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))"
hoelzl@41096
   927
using assms proof induct
hoelzl@41096
   928
  case (insert i I)
hoelzl@41096
   929
  note `finite I`[intro, simp]
hoelzl@41689
   930
  interpret I: finite_product_sigma_finite M I by default auto
hoelzl@41096
   931
  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))"
hoelzl@41096
   932
    using insert by (auto intro!: setprod_cong)
hoelzl@41689
   933
  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)"
hoelzl@41096
   934
    using sets_into_space insert
hoelzl@47694
   935
    by (intro borel_measurable_ereal_setprod
hoelzl@41689
   936
              measurable_comp[OF measurable_component_singleton, unfolded comp_def])
hoelzl@41096
   937
       auto
hoelzl@41981
   938
  then show ?case
hoelzl@41981
   939
    apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
hoelzl@47694
   940
    apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc)
hoelzl@47694
   941
    apply (subst positive_integral_cmult)
hoelzl@47694
   942
    apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive)
hoelzl@41981
   943
    done
hoelzl@47694
   944
qed (simp add: space_PiM)
hoelzl@41096
   945
hoelzl@41026
   946
lemma (in product_sigma_finite) product_integral_singleton:
hoelzl@41026
   947
  assumes f: "f \<in> borel_measurable (M i)"
hoelzl@41689
   948
  shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
hoelzl@41026
   949
proof -
hoelzl@41689
   950
  interpret I: finite_product_sigma_finite M "{i}" by default simp
hoelzl@43920
   951
  have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
hoelzl@43920
   952
    "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
hoelzl@41026
   953
    using assms by auto
hoelzl@41026
   954
  show ?thesis
hoelzl@41689
   955
    unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
hoelzl@41026
   956
qed
hoelzl@41026
   957
hoelzl@41026
   958
lemma (in product_sigma_finite) product_integral_fold:
hoelzl@41026
   959
  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
hoelzl@41689
   960
  and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
hoelzl@49780
   961
  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)"
hoelzl@41026
   962
proof -
hoelzl@41689
   963
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41689
   964
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@41026
   965
  have "finite (I \<union> J)" using fin by auto
hoelzl@41689
   966
  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
hoelzl@47694
   967
  interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
hoelzl@49780
   968
  let ?M = "merge I J"
hoelzl@41689
   969
  let ?f = "\<lambda>x. f (?M x)"
hoelzl@47694
   970
  from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
hoelzl@47694
   971
    by auto
hoelzl@49780
   972
  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)"
hoelzl@49776
   973
    using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
hoelzl@47694
   974
  have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f"
hoelzl@49776
   975
    by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
hoelzl@41026
   976
  show ?thesis
hoelzl@47694
   977
    apply (subst distr_merge[symmetric, OF IJ fin])
hoelzl@49776
   978
    apply (subst integral_distr[OF measurable_merge f_borel])
hoelzl@47694
   979
    apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
hoelzl@47694
   980
    apply simp
hoelzl@47694
   981
    done
hoelzl@41026
   982
qed
hoelzl@41026
   983
hoelzl@49776
   984
lemma (in product_sigma_finite)
hoelzl@49776
   985
  assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
hoelzl@49776
   986
  shows emeasure_fold_integral:
hoelzl@49780
   987
    "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)
hoelzl@49776
   988
    and emeasure_fold_measurable:
hoelzl@49780
   989
    "(\<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)
hoelzl@49776
   990
proof -
hoelzl@49776
   991
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@49776
   992
  interpret J: finite_product_sigma_finite M J by default fact
hoelzl@49776
   993
  interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
hoelzl@49780
   994
  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)"
hoelzl@49776
   995
    by (intro measurable_sets[OF _ A] measurable_merge assms)
hoelzl@49776
   996
hoelzl@49776
   997
  show ?I
hoelzl@49776
   998
    apply (subst distr_merge[symmetric, OF IJ])
hoelzl@49776
   999
    apply (subst emeasure_distr[OF measurable_merge A])
hoelzl@49776
  1000
    apply (subst J.emeasure_pair_measure_alt[OF merge])
hoelzl@49776
  1001
    apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
hoelzl@49776
  1002
    done
hoelzl@49776
  1003
hoelzl@49776
  1004
  show ?B
hoelzl@49776
  1005
    using IJ.measurable_emeasure_Pair1[OF merge]
hoelzl@49776
  1006
    by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
hoelzl@49776
  1007
qed
hoelzl@49776
  1008
hoelzl@41096
  1009
lemma (in product_sigma_finite) product_integral_insert:
hoelzl@47694
  1010
  assumes I: "finite I" "i \<notin> I"
hoelzl@41689
  1011
    and f: "integrable (Pi\<^isub>M (insert i I) M) f"
hoelzl@41689
  1012
  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)"
hoelzl@41096
  1013
proof -
hoelzl@47694
  1014
  have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f"
hoelzl@47694
  1015
    by simp
hoelzl@49780
  1016
  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)"
hoelzl@47694
  1017
    using f I by (intro product_integral_fold) auto
hoelzl@47694
  1018
  also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
hoelzl@47694
  1019
  proof (rule integral_cong, subst product_integral_singleton[symmetric])
hoelzl@47694
  1020
    fix x assume x: "x \<in> space (Pi\<^isub>M I M)"
hoelzl@47694
  1021
    have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
hoelzl@47694
  1022
      using f by auto
hoelzl@47694
  1023
    show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
hoelzl@47694
  1024
      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
hoelzl@47694
  1025
      unfolding comp_def .
hoelzl@49780
  1026
    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)"
hoelzl@47694
  1027
      by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def)
hoelzl@41096
  1028
  qed
hoelzl@47694
  1029
  finally show ?thesis .
hoelzl@41096
  1030
qed
hoelzl@41096
  1031
hoelzl@41096
  1032
lemma (in product_sigma_finite) product_integrable_setprod:
hoelzl@41096
  1033
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@41689
  1034
  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
  1035
  shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
hoelzl@41096
  1036
proof -
hoelzl@41689
  1037
  interpret finite_product_sigma_finite M I by default fact
hoelzl@41096
  1038
  have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
hoelzl@41689
  1039
    using integrable unfolding integrable_def by auto
hoelzl@47694
  1040
  have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@47694
  1041
    using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
hoelzl@41689
  1042
  moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
hoelzl@41096
  1043
  proof (unfold integrable_def, intro conjI)
hoelzl@47694
  1044
    show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)"
hoelzl@41096
  1045
      using borel by auto
hoelzl@47694
  1046
    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)"
hoelzl@43920
  1047
      by (simp add: setprod_ereal abs_setprod)
hoelzl@43920
  1048
    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
hoelzl@41096
  1049
      using f by (subst product_positive_integral_setprod) auto
hoelzl@41981
  1050
    also have "\<dots> < \<infinity>"
hoelzl@47694
  1051
      using integrable[THEN integrable_abs]
hoelzl@47694
  1052
      by (simp add: setprod_PInf integrable_def positive_integral_positive)
hoelzl@47694
  1053
    finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto
hoelzl@47694
  1054
    have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))"
hoelzl@41981
  1055
      by (intro positive_integral_cong_pos) auto
hoelzl@47694
  1056
    then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp
hoelzl@41096
  1057
  qed
hoelzl@41096
  1058
  ultimately show ?thesis
hoelzl@41096
  1059
    by (rule integrable_abs_iff[THEN iffD1])
hoelzl@41096
  1060
qed
hoelzl@41096
  1061
hoelzl@41096
  1062
lemma (in product_sigma_finite) product_integral_setprod:
hoelzl@41096
  1063
  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@49780
  1064
  assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
hoelzl@41689
  1065
  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))"
hoelzl@49780
  1066
using assms proof induct
hoelzl@49780
  1067
  case empty
hoelzl@49780
  1068
  interpret finite_measure "Pi\<^isub>M {} M"
hoelzl@49780
  1069
    by rule (simp add: space_PiM)
hoelzl@49780
  1070
  show ?case by (simp add: space_PiM measure_def)
hoelzl@41096
  1071
next
hoelzl@41096
  1072
  case (insert i I)
hoelzl@41096
  1073
  then have iI: "finite (insert i I)" by auto
hoelzl@41096
  1074
  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
hoelzl@41689
  1075
    integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
hoelzl@49780
  1076
    by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
hoelzl@41689
  1077
  interpret I: finite_product_sigma_finite M I by default fact
hoelzl@41096
  1078
  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))"
hoelzl@41096
  1079
    using `i \<notin> I` by (auto intro!: setprod_cong)
hoelzl@41096
  1080
  show ?case
hoelzl@49780
  1081
    unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
hoelzl@47694
  1082
    by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
hoelzl@41096
  1083
qed
hoelzl@41096
  1084
hoelzl@49776
  1085
lemma sets_Collect_single:
hoelzl@49776
  1086
  "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)"
hoelzl@50003
  1087
  by simp
hoelzl@49776
  1088
hoelzl@49776
  1089
lemma sigma_prod_algebra_sigma_eq_infinite:
hoelzl@49776
  1090
  fixes E :: "'i \<Rightarrow> 'a set set"
hoelzl@49779
  1091
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@49776
  1092
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@49776
  1093
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@49776
  1094
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@49776
  1095
  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}"
hoelzl@49776
  1096
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1097
proof
hoelzl@49776
  1098
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49776
  1099
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@49776
  1100
    using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
hoelzl@49776
  1101
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@49776
  1102
    by (simp add: space_PiM)
hoelzl@49776
  1103
  have "sets (PiM I M) =
hoelzl@49776
  1104
      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)}"
hoelzl@49776
  1105
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@49776
  1106
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@49776
  1107
  proof (safe intro!: sigma_sets_subset)
hoelzl@49776
  1108
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@49776
  1109
    then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@49776
  1110
      apply (subst measurable_iff_measure_of)
hoelzl@49776
  1111
      apply (simp_all add: P_closed)
hoelzl@49776
  1112
      using E_closed
hoelzl@49776
  1113
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1114
      apply (force simp: subset_eq space_PiM)
hoelzl@49776
  1115
      apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i])
hoelzl@49776
  1116
      apply (rule_tac x=Aa in exI)
hoelzl@49776
  1117
      apply (auto simp: space_PiM)
hoelzl@49776
  1118
      done
hoelzl@49776
  1119
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@49776
  1120
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@49776
  1121
      by (simp add: E_generates)
hoelzl@49776
  1122
    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}"
hoelzl@49776
  1123
      using P_closed by (auto simp: space_PiM)
hoelzl@49776
  1124
    finally show "\<dots> \<in> sets ?P" .
hoelzl@49776
  1125
  qed
hoelzl@49776
  1126
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@49776
  1127
    by (simp add: P_closed)
hoelzl@49776
  1128
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@49776
  1129
    unfolding P_def space_PiM[symmetric]
hoelzl@49776
  1130
    by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single)
hoelzl@49776
  1131
qed
hoelzl@49776
  1132
hoelzl@49779
  1133
lemma bchoice_iff: "(\<forall>a\<in>A. \<exists>b. P a b) \<longleftrightarrow> (\<exists>f. \<forall>a\<in>A. P a (f a))"
hoelzl@49779
  1134
  by metis
hoelzl@49779
  1135
hoelzl@47694
  1136
lemma sigma_prod_algebra_sigma_eq:
hoelzl@49779
  1137
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
hoelzl@47694
  1138
  assumes "finite I"
hoelzl@49779
  1139
  assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
hoelzl@47694
  1140
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
hoelzl@47694
  1141
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
hoelzl@47694
  1142
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
hoelzl@47694
  1143
  defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }"
hoelzl@47694
  1144
  shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1145
proof
hoelzl@47694
  1146
  let ?P = "sigma (space (Pi\<^isub>M I M)) P"
hoelzl@49779
  1147
  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
hoelzl@49779
  1148
  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"
hoelzl@49779
  1149
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f)
hoelzl@47694
  1150
  have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))"
hoelzl@47694
  1151
    using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq)
hoelzl@47694
  1152
  then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))"
hoelzl@47694
  1153
    by (simp add: space_PiM)
hoelzl@47694
  1154
  have "sets (PiM I M) =
hoelzl@47694
  1155
      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)}"
hoelzl@47694
  1156
    using sets_PiM_single[of I M] by (simp add: space_P)
hoelzl@47694
  1157
  also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)"
hoelzl@47694
  1158
  proof (safe intro!: sigma_sets_subset)
hoelzl@47694
  1159
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
hoelzl@47694
  1160
    have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))"
hoelzl@47694
  1161
    proof (subst measurable_iff_measure_of)
hoelzl@47694
  1162
      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
hoelzl@47694
  1163
      from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)"
hoelzl@47694
  1164
        by (auto simp: Pi_iff)
hoelzl@47694
  1165
      show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1166
      proof
hoelzl@47694
  1167
        fix A assume A: "A \<in> E i"
hoelzl@47694
  1168
        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))"
hoelzl@47694
  1169
          using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
hoelzl@47694
  1170
        also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)"
hoelzl@47694
  1171
          by (intro PiE_cong) (simp add: S_union)
hoelzl@49779
  1172
        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))"
hoelzl@49779
  1173
          using T
hoelzl@49779
  1174
          apply (auto simp: Pi_iff bchoice_iff)
hoelzl@49779
  1175
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
hoelzl@49779
  1176
          apply (auto simp: bij_betw_def)
hoelzl@49779
  1177
          done
hoelzl@47694
  1178
        also have "\<dots> \<in> sets ?P"
hoelzl@47694
  1179
        proof (safe intro!: countable_UN)
hoelzl@49779
  1180
          fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
hoelzl@47694
  1181
            using A S_in_E
hoelzl@47694
  1182
            by (simp add: P_closed)
hoelzl@49779
  1183
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
hoelzl@47694
  1184
        qed
hoelzl@47694
  1185
        finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1186
          using P_closed by simp
hoelzl@47694
  1187
      qed
hoelzl@47694
  1188
    qed
hoelzl@47694
  1189
    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
hoelzl@47694
  1190
    have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P"
hoelzl@47694
  1191
      by (simp add: E_generates)
hoelzl@47694
  1192
    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}"
hoelzl@47694
  1193
      using P_closed by (auto simp: space_PiM)
hoelzl@47694
  1194
    finally show "\<dots> \<in> sets ?P" .
hoelzl@47694
  1195
  qed
hoelzl@47694
  1196
  finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P"
hoelzl@47694
  1197
    by (simp add: P_closed)
hoelzl@47694
  1198
  show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)"
hoelzl@47694
  1199
    using `finite I`
hoelzl@50003
  1200
    by (auto intro!: sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def)
hoelzl@47694
  1201
qed
hoelzl@47694
  1202
hoelzl@47694
  1203
end