src/HOL/Probability/Product_Measure.thy
author hoelzl
Fri Dec 03 15:25:14 2010 +0100 (2010-12-03)
changeset 41023 9118eb4eb8dc
parent 40873 1ef85f4e7097
child 41026 bea75746dc9d
permissions -rw-r--r--
it is known as the extended reals, not the infinite reals
hoelzl@35833
     1
theory Product_Measure
hoelzl@38656
     2
imports Lebesgue_Integration
hoelzl@35833
     3
begin
hoelzl@35833
     4
hoelzl@40859
     5
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@40859
     6
  by auto
hoelzl@40859
     7
hoelzl@40859
     8
lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
hoelzl@40859
     9
  by auto
hoelzl@40859
    10
hoelzl@40859
    11
lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
hoelzl@40859
    12
  by auto
hoelzl@40859
    13
hoelzl@40859
    14
lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
hoelzl@40859
    15
  by (cases x) simp
hoelzl@40859
    16
hoelzl@40859
    17
abbreviation
hoelzl@40859
    18
  "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
hellerar@39094
    19
hoelzl@40859
    20
abbreviation
hoelzl@40859
    21
  funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
hoelzl@40859
    22
    (infixr "->\<^isub>E" 60) where
hoelzl@40859
    23
  "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
hoelzl@40859
    24
hoelzl@40859
    25
notation (xsymbols)
hoelzl@40859
    26
  funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)
hoelzl@40859
    27
hoelzl@40859
    28
lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
hoelzl@40859
    29
  by safe (auto simp add: extensional_def fun_eq_iff)
hoelzl@40859
    30
hoelzl@40859
    31
lemma extensional_insert[intro, simp]:
hoelzl@40859
    32
  assumes "a \<in> extensional (insert i I)"
hoelzl@40859
    33
  shows "a(i := b) \<in> extensional (insert i I)"
hoelzl@40859
    34
  using assms unfolding extensional_def by auto
hoelzl@40859
    35
hoelzl@40859
    36
lemma extensional_Int[simp]:
hoelzl@40859
    37
  "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
hoelzl@40859
    38
  unfolding extensional_def by auto
hoelzl@38656
    39
hoelzl@35833
    40
definition
hoelzl@40859
    41
  "merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
hoelzl@40859
    42
hoelzl@40859
    43
lemma merge_apply[simp]:
hoelzl@40859
    44
  "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
hoelzl@40859
    45
  "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
hoelzl@40859
    46
  "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
hoelzl@40859
    47
  "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
hoelzl@40859
    48
  "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"
hoelzl@40859
    49
  unfolding merge_def by auto
hoelzl@40859
    50
hoelzl@40859
    51
lemma merge_commute:
hoelzl@40859
    52
  "I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"
hoelzl@40859
    53
  by (auto simp: merge_def intro!: ext)
hoelzl@40859
    54
hoelzl@40859
    55
lemma Pi_cancel_merge_range[simp]:
hoelzl@40859
    56
  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    57
  "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    58
  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    59
  "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    60
  by (auto simp: Pi_def)
hoelzl@40859
    61
hoelzl@40859
    62
lemma Pi_cancel_merge[simp]:
hoelzl@40859
    63
  "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
hoelzl@40859
    64
  "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
hoelzl@40859
    65
  "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
hoelzl@40859
    66
  "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
hoelzl@40859
    67
  by (auto simp: Pi_def)
hoelzl@40859
    68
hoelzl@40859
    69
lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"
hoelzl@40859
    70
  by (auto simp: extensional_def)
hoelzl@40859
    71
hoelzl@40859
    72
lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
hoelzl@40859
    73
  by (auto simp: restrict_def Pi_def)
hoelzl@40859
    74
hoelzl@40859
    75
lemma restrict_merge[simp]:
hoelzl@40859
    76
  "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
hoelzl@40859
    77
  "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
hoelzl@40859
    78
  "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
hoelzl@40859
    79
  "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
hoelzl@40859
    80
  by (auto simp: restrict_def intro!: ext)
hoelzl@40859
    81
hoelzl@40859
    82
lemma extensional_insert_undefined[intro, simp]:
hoelzl@40859
    83
  assumes "a \<in> extensional (insert i I)"
hoelzl@40859
    84
  shows "a(i := undefined) \<in> extensional I"
hoelzl@40859
    85
  using assms unfolding extensional_def by auto
hoelzl@40859
    86
hoelzl@40859
    87
lemma extensional_insert_cancel[intro, simp]:
hoelzl@40859
    88
  assumes "a \<in> extensional I"
hoelzl@40859
    89
  shows "a \<in> extensional (insert i I)"
hoelzl@40859
    90
  using assms unfolding extensional_def by auto
hoelzl@40859
    91
hoelzl@40859
    92
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
    93
  by auto
hoelzl@40859
    94
hoelzl@40859
    95
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
    96
  by (auto simp: Pi_def)
hoelzl@40859
    97
hoelzl@40859
    98
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
    99
  by (auto simp: Pi_def)
hoelzl@39088
   100
hoelzl@40859
   101
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
   102
  by (auto simp: Pi_def)
hoelzl@40859
   103
hoelzl@40859
   104
lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
hoelzl@40859
   105
  by (auto simp: Pi_def)
hoelzl@40859
   106
hoelzl@40859
   107
section "Binary products"
hoelzl@40859
   108
hoelzl@40859
   109
definition
hoelzl@40859
   110
  "pair_algebra A B = \<lparr> space = space A \<times> space B,
hoelzl@40859
   111
                           sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<rparr>"
hoelzl@40859
   112
hoelzl@40859
   113
locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
hoelzl@40859
   114
  for M1 M2
hoelzl@40859
   115
hoelzl@40859
   116
abbreviation (in pair_sigma_algebra)
hoelzl@40859
   117
  "E \<equiv> pair_algebra M1 M2"
hoelzl@40859
   118
hoelzl@40859
   119
abbreviation (in pair_sigma_algebra)
hoelzl@40859
   120
  "P \<equiv> sigma E"
hoelzl@40859
   121
hoelzl@40859
   122
sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
hoelzl@40859
   123
  using M1.sets_into_space M2.sets_into_space
hoelzl@40859
   124
  by (force simp: pair_algebra_def intro!: sigma_algebra_sigma)
hoelzl@40859
   125
hoelzl@40859
   126
lemma pair_algebraI[intro, simp]:
hoelzl@40859
   127
  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_algebra A B)"
hoelzl@40859
   128
  by (auto simp add: pair_algebra_def)
hoelzl@40859
   129
hoelzl@40859
   130
lemma space_pair_algebra:
hoelzl@40859
   131
  "space (pair_algebra A B) = space A \<times> space B"
hoelzl@40859
   132
  by (simp add: pair_algebra_def)
hoelzl@40859
   133
hoelzl@40859
   134
lemma pair_algebra_Int_snd:
hoelzl@40859
   135
  assumes "sets S1 \<subseteq> Pow (space S1)"
hoelzl@40859
   136
  shows "pair_algebra S1 (algebra.restricted_space S2 A) =
hoelzl@40859
   137
         algebra.restricted_space (pair_algebra S1 S2) (space S1 \<times> A)"
hoelzl@40859
   138
  (is "?L = ?R")
hoelzl@40859
   139
proof (intro algebra.equality set_eqI iffI)
hoelzl@40859
   140
  fix X assume "X \<in> sets ?L"
hoelzl@40859
   141
  then obtain A1 A2 where X: "X = A1 \<times> (A \<inter> A2)" and "A1 \<in> sets S1" "A2 \<in> sets S2"
hoelzl@40859
   142
    by (auto simp: pair_algebra_def)
hoelzl@40859
   143
  then show "X \<in> sets ?R" unfolding pair_algebra_def
hoelzl@40859
   144
    using assms apply simp by (intro image_eqI[of _ _ "A1 \<times> A2"]) auto
hoelzl@40859
   145
next
hoelzl@40859
   146
  fix X assume "X \<in> sets ?R"
hoelzl@40859
   147
  then obtain A1 A2 where "X = space S1 \<times> A \<inter> A1 \<times> A2" "A1 \<in> sets S1" "A2 \<in> sets S2"
hoelzl@40859
   148
    by (auto simp: pair_algebra_def)
hoelzl@40859
   149
  moreover then have "X = A1 \<times> (A \<inter> A2)"
hoelzl@40859
   150
    using assms by auto
hoelzl@40859
   151
  ultimately show "X \<in> sets ?L"
hoelzl@40859
   152
    unfolding pair_algebra_def by auto
hoelzl@40859
   153
qed (auto simp add: pair_algebra_def)
hoelzl@40859
   154
hoelzl@40859
   155
lemma (in pair_sigma_algebra)
hoelzl@40859
   156
  shows measurable_fst[intro!, simp]:
hoelzl@40859
   157
    "fst \<in> measurable P M1" (is ?fst)
hoelzl@40859
   158
  and measurable_snd[intro!, simp]:
hoelzl@40859
   159
    "snd \<in> measurable P M2" (is ?snd)
hoelzl@39088
   160
proof -
hoelzl@39088
   161
  { fix X assume "X \<in> sets M1"
hoelzl@39088
   162
    then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
hoelzl@39088
   163
      apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
hoelzl@39088
   164
      using M1.sets_into_space by force+ }
hoelzl@39088
   165
  moreover
hoelzl@39088
   166
  { fix X assume "X \<in> sets M2"
hoelzl@39088
   167
    then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
hoelzl@39088
   168
      apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
hoelzl@39088
   169
      using M2.sets_into_space by force+ }
hoelzl@40859
   170
  ultimately have "?fst \<and> ?snd"
hoelzl@40859
   171
    by (fastsimp simp: measurable_def sets_sigma space_pair_algebra
hoelzl@40859
   172
                 intro!: sigma_sets.Basic)
hoelzl@40859
   173
  then show ?fst ?snd by auto
hoelzl@40859
   174
qed
hoelzl@40859
   175
hoelzl@40859
   176
lemma (in pair_sigma_algebra) measurable_pair:
hoelzl@40859
   177
  assumes "sigma_algebra M"
hoelzl@40859
   178
  shows "f \<in> measurable M P \<longleftrightarrow>
hoelzl@40859
   179
    (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
hoelzl@40859
   180
proof -
hoelzl@40859
   181
  interpret M: sigma_algebra M by fact
hoelzl@40859
   182
  from assms show ?thesis
hoelzl@40859
   183
  proof (safe intro!: measurable_comp[where b=P])
hoelzl@40859
   184
    assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
hoelzl@40859
   185
    show "f \<in> measurable M P"
hoelzl@40859
   186
    proof (rule M.measurable_sigma)
hoelzl@40859
   187
      show "sets (pair_algebra M1 M2) \<subseteq> Pow (space E)"
hoelzl@40859
   188
        unfolding pair_algebra_def using M1.sets_into_space M2.sets_into_space by auto
hoelzl@40859
   189
      show "f \<in> space M \<rightarrow> space E"
hoelzl@40859
   190
        using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma space_pair_algebra)
hoelzl@40859
   191
      fix A assume "A \<in> sets E"
hoelzl@40859
   192
      then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
hoelzl@40859
   193
        unfolding pair_algebra_def by auto
hoelzl@40859
   194
      moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M"
hoelzl@40859
   195
        using f `B \<in> sets M1` unfolding measurable_def by auto
hoelzl@40859
   196
      moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M"
hoelzl@40859
   197
        using s `C \<in> sets M2` unfolding measurable_def by auto
hoelzl@40859
   198
      moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)"
hoelzl@40859
   199
        unfolding `A = B \<times> C` by (auto simp: vimage_Times)
hoelzl@40859
   200
      ultimately show "f -` A \<inter> space M \<in> sets M" by auto
hoelzl@40859
   201
    qed
hoelzl@40859
   202
  qed
hoelzl@40859
   203
qed
hoelzl@40859
   204
hoelzl@40859
   205
lemma (in pair_sigma_algebra) measurable_prod_sigma:
hoelzl@40859
   206
  assumes "sigma_algebra M"
hoelzl@40859
   207
  assumes 1: "(fst \<circ> f) \<in> measurable M M1" and 2: "(snd \<circ> f) \<in> measurable M M2"
hoelzl@40859
   208
  shows "f \<in> measurable M P"
hoelzl@40859
   209
proof -
hoelzl@40859
   210
  interpret M: sigma_algebra M by fact
hoelzl@40859
   211
  from 1 have fn1: "fst \<circ> f \<in> space M \<rightarrow> space M1"
hoelzl@40859
   212
     and q1: "\<forall>y\<in>sets M1. (fst \<circ> f) -` y \<inter> space M \<in> sets M"
hoelzl@40859
   213
    by (auto simp add: measurable_def)
hoelzl@40859
   214
  from 2 have fn2: "snd \<circ> f \<in> space M \<rightarrow> space M2"
hoelzl@40859
   215
     and q2: "\<forall>y\<in>sets M2. (snd \<circ> f) -` y \<inter> space M \<in> sets M"
hoelzl@40859
   216
    by (auto simp add: measurable_def)
hoelzl@40859
   217
  show ?thesis
hoelzl@40859
   218
  proof (rule M.measurable_sigma)
hoelzl@40859
   219
    show "sets E \<subseteq> Pow (space E)"
hoelzl@40859
   220
      using M1.space_closed M2.space_closed
hoelzl@40859
   221
      by (auto simp add: sigma_algebra_iff pair_algebra_def)
hoelzl@40859
   222
  next
hoelzl@40859
   223
    show "f \<in> space M \<rightarrow> space E"
hoelzl@40859
   224
      by (simp add: space_pair_algebra) (rule prod_final [OF fn1 fn2])
hoelzl@40859
   225
  next
hoelzl@40859
   226
    fix z
hoelzl@40859
   227
    assume z: "z \<in> sets E"
hoelzl@40859
   228
    thus "f -` z \<inter> space M \<in> sets M"
hoelzl@40859
   229
    proof (auto simp add: pair_algebra_def vimage_Times)
hoelzl@40859
   230
      fix x y
hoelzl@40859
   231
      assume x: "x \<in> sets M1" and y: "y \<in> sets M2"
hoelzl@40859
   232
      have "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M =
hoelzl@40859
   233
            ((fst \<circ> f) -` x \<inter> space M) \<inter> ((snd \<circ> f) -` y \<inter> space M)"
hoelzl@40859
   234
        by blast
hoelzl@40859
   235
      also have "...  \<in> sets M" using x y q1 q2
hoelzl@40859
   236
        by blast
hoelzl@40859
   237
      finally show "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M \<in> sets M" .
hoelzl@40859
   238
    qed
hoelzl@40859
   239
  qed
hoelzl@40859
   240
qed
hoelzl@40859
   241
hoelzl@40859
   242
lemma pair_algebraE:
hoelzl@40859
   243
  assumes "X \<in> sets (pair_algebra M1 M2)"
hoelzl@40859
   244
  obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
hoelzl@40859
   245
  using assms unfolding pair_algebra_def by auto
hoelzl@40859
   246
hoelzl@40859
   247
lemma (in pair_sigma_algebra) pair_algebra_swap:
hoelzl@40859
   248
  "(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_algebra M2 M1)"
hoelzl@40859
   249
proof (safe elim!: pair_algebraE)
hoelzl@40859
   250
  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
hoelzl@40859
   251
  moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
hoelzl@40859
   252
    using M1.sets_into_space M2.sets_into_space by auto
hoelzl@40859
   253
  ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_algebra M2 M1)"
hoelzl@40859
   254
    by (auto intro: pair_algebraI)
hoelzl@40859
   255
next
hoelzl@40859
   256
  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
hoelzl@40859
   257
  then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E"
hoelzl@40859
   258
    using M1.sets_into_space M2.sets_into_space
hoelzl@40859
   259
    by (auto intro!: image_eqI[where x="A \<times> B"] pair_algebraI)
hoelzl@40859
   260
qed
hoelzl@40859
   261
hoelzl@40859
   262
lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
hoelzl@40859
   263
  assumes Q: "Q \<in> sets P"
hoelzl@40859
   264
  shows "(\<lambda>(x,y). (y, x)) ` Q \<in> sets (sigma (pair_algebra M2 M1))" (is "_ \<in> sets ?Q")
hoelzl@40859
   265
proof -
hoelzl@40859
   266
  have *: "(\<lambda>(x,y). (y, x)) \<in> space M2 \<times> space M1 \<rightarrow> (space M1 \<times> space M2)"
hoelzl@40859
   267
       "sets (pair_algebra M1 M2) \<subseteq> Pow (space M1 \<times> space M2)"
hoelzl@40859
   268
    using M1.sets_into_space M2.sets_into_space by (auto elim!: pair_algebraE)
hoelzl@40859
   269
  from Q sets_into_space show ?thesis
hoelzl@40859
   270
    by (auto intro!: image_eqI[where x=Q]
hoelzl@40859
   271
             simp: pair_algebra_swap[symmetric] sets_sigma
hoelzl@40859
   272
                   sigma_sets_vimage[OF *] space_pair_algebra)
hoelzl@40859
   273
qed
hoelzl@40859
   274
hoelzl@40859
   275
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
hoelzl@40859
   276
  shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (sigma (pair_algebra M2 M1))"
hoelzl@40859
   277
    (is "?f \<in> measurable ?P ?Q")
hoelzl@40859
   278
  unfolding measurable_def
hoelzl@40859
   279
proof (intro CollectI conjI Pi_I ballI)
hoelzl@40859
   280
  fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
hoelzl@40859
   281
    unfolding pair_algebra_def by auto
hoelzl@40859
   282
next
hoelzl@40859
   283
  fix A assume "A \<in> sets ?Q"
hoelzl@40859
   284
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   285
  have "?f -` A \<inter> space ?P = (\<lambda>(x,y). (y, x)) ` A"
hoelzl@40859
   286
    using Q.sets_into_space `A \<in> sets ?Q` by (auto simp: pair_algebra_def)
hoelzl@40859
   287
  with Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets ?Q`]
hoelzl@40859
   288
  show "?f -` A \<inter> space ?P \<in> sets ?P" by simp
hoelzl@40859
   289
qed
hoelzl@40859
   290
hoelzl@40859
   291
lemma (in pair_sigma_algebra) measurable_cut_fst:
hoelzl@40859
   292
  assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"
hoelzl@40859
   293
proof -
hoelzl@40859
   294
  let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"
hoelzl@40859
   295
  let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
hoelzl@40859
   296
  interpret Q: sigma_algebra ?Q
hoelzl@40859
   297
    proof qed (auto simp: vimage_UN vimage_Diff space_pair_algebra)
hoelzl@40859
   298
  have "sets E \<subseteq> sets ?Q"
hoelzl@40859
   299
    using M1.sets_into_space M2.sets_into_space
hoelzl@40859
   300
    by (auto simp: pair_algebra_def space_pair_algebra)
hoelzl@40859
   301
  then have "sets P \<subseteq> sets ?Q"
hoelzl@40859
   302
    by (subst pair_algebra_def, intro Q.sets_sigma_subset)
hoelzl@40859
   303
       (simp_all add: pair_algebra_def)
hoelzl@40859
   304
  with assms show ?thesis by auto
hoelzl@40859
   305
qed
hoelzl@40859
   306
hoelzl@40859
   307
lemma (in pair_sigma_algebra) measurable_cut_snd:
hoelzl@40859
   308
  assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1")
hoelzl@40859
   309
proof -
hoelzl@40859
   310
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   311
  have "Pair y -` (\<lambda>(x, y). (y, x)) ` Q = (\<lambda>x. (x, y)) -` Q" by auto
hoelzl@40859
   312
  with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
hoelzl@40859
   313
  show ?thesis by simp
hoelzl@40859
   314
qed
hoelzl@40859
   315
hoelzl@40859
   316
lemma (in pair_sigma_algebra) measurable_pair_image_snd:
hoelzl@40859
   317
  assumes m: "f \<in> measurable P M" and "x \<in> space M1"
hoelzl@40859
   318
  shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
hoelzl@40859
   319
  unfolding measurable_def
hoelzl@40859
   320
proof (intro CollectI conjI Pi_I ballI)
hoelzl@40859
   321
  fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1`
hoelzl@40859
   322
  show "f (x, y) \<in> space M" unfolding measurable_def pair_algebra_def by auto
hoelzl@40859
   323
next
hoelzl@40859
   324
  fix A assume "A \<in> sets M"
hoelzl@40859
   325
  then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
hoelzl@40859
   326
    using `f \<in> measurable P M`
hoelzl@40859
   327
    by (intro measurable_cut_fst) (auto simp: measurable_def)
hoelzl@40859
   328
  also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2"
hoelzl@40859
   329
    using `x \<in> space M1` by (auto simp: pair_algebra_def)
hoelzl@40859
   330
  finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" .
hoelzl@40859
   331
qed
hoelzl@40859
   332
hoelzl@40859
   333
lemma (in pair_sigma_algebra) measurable_pair_image_fst:
hoelzl@40859
   334
  assumes m: "f \<in> measurable P M" and "y \<in> space M2"
hoelzl@40859
   335
  shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
hoelzl@40859
   336
proof -
hoelzl@40859
   337
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   338
  from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`,
hoelzl@40859
   339
                                      OF Q.pair_sigma_algebra_swap_measurable m]
hoelzl@40859
   340
  show ?thesis by simp
hoelzl@40859
   341
qed
hoelzl@40859
   342
hoelzl@40859
   343
lemma (in pair_sigma_algebra) Int_stable_pair_algebra: "Int_stable E"
hoelzl@40859
   344
  unfolding Int_stable_def
hoelzl@40859
   345
proof (intro ballI)
hoelzl@40859
   346
  fix A B assume "A \<in> sets E" "B \<in> sets E"
hoelzl@40859
   347
  then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
hoelzl@40859
   348
    "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
hoelzl@40859
   349
    unfolding pair_algebra_def by auto
hoelzl@40859
   350
  then show "A \<inter> B \<in> sets E"
hoelzl@40859
   351
    by (auto simp add: times_Int_times pair_algebra_def)
hoelzl@40859
   352
qed
hoelzl@40859
   353
hoelzl@40859
   354
lemma finite_measure_cut_measurable:
hoelzl@40859
   355
  fixes M1 :: "'a algebra" and M2 :: "'b algebra"
hoelzl@40859
   356
  assumes "sigma_finite_measure M1 \<mu>1" "finite_measure M2 \<mu>2"
hoelzl@40859
   357
  assumes "Q \<in> sets (sigma (pair_algebra M1 M2))"
hoelzl@40859
   358
  shows "(\<lambda>x. \<mu>2 (Pair x -` Q)) \<in> borel_measurable M1"
hoelzl@40859
   359
    (is "?s Q \<in> _")
hoelzl@40859
   360
proof -
hoelzl@40859
   361
  interpret M1: sigma_finite_measure M1 \<mu>1 by fact
hoelzl@40859
   362
  interpret M2: finite_measure M2 \<mu>2 by fact
hoelzl@40859
   363
  interpret pair_sigma_algebra M1 M2 by default
hoelzl@40859
   364
  have [intro]: "sigma_algebra M1" by fact
hoelzl@40859
   365
  have [intro]: "sigma_algebra M2" by fact
hoelzl@40859
   366
  let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"
hoelzl@40859
   367
  note space_pair_algebra[simp]
hoelzl@40859
   368
  interpret dynkin_system ?D
hoelzl@40859
   369
  proof (intro dynkin_systemI)
hoelzl@40859
   370
    fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
hoelzl@40859
   371
      using sets_into_space by simp
hoelzl@40859
   372
  next
hoelzl@40859
   373
    from top show "space ?D \<in> sets ?D"
hoelzl@40859
   374
      by (auto simp add: if_distrib intro!: M1.measurable_If)
hoelzl@40859
   375
  next
hoelzl@40859
   376
    fix A assume "A \<in> sets ?D"
hoelzl@40859
   377
    with sets_into_space have "\<And>x. \<mu>2 (Pair x -` (space M1 \<times> space M2 - A)) =
hoelzl@40859
   378
        (if x \<in> space M1 then \<mu>2 (space M2) - ?s A x else 0)"
hoelzl@40859
   379
      by (auto intro!: M2.finite_measure_compl measurable_cut_fst
hoelzl@40859
   380
               simp: vimage_Diff)
hoelzl@40859
   381
    with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"
hoelzl@41023
   382
      by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pextreal_diff)
hoelzl@40859
   383
  next
hoelzl@40859
   384
    fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
hoelzl@40859
   385
    moreover then have "\<And>x. \<mu>2 (\<Union>i. Pair x -` F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)"
hoelzl@40859
   386
      by (intro M2.measure_countably_additive[symmetric])
hoelzl@40859
   387
         (auto intro!: measurable_cut_fst simp: disjoint_family_on_def)
hoelzl@40859
   388
    ultimately show "(\<Union>i. F i) \<in> sets ?D"
hoelzl@40859
   389
      by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
hoelzl@40859
   390
  qed
hoelzl@40859
   391
  have "P = ?D"
hoelzl@40859
   392
  proof (intro dynkin_lemma)
hoelzl@40859
   393
    show "Int_stable E" by (rule Int_stable_pair_algebra)
hoelzl@40859
   394
    from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
hoelzl@40859
   395
      by auto
hoelzl@40859
   396
    then show "sets E \<subseteq> sets ?D"
hoelzl@40859
   397
      by (auto simp: pair_algebra_def sets_sigma if_distrib
hoelzl@40859
   398
               intro: sigma_sets.Basic intro!: M1.measurable_If)
hoelzl@40859
   399
  qed auto
hoelzl@40859
   400
  with `Q \<in> sets P` have "Q \<in> sets ?D" by simp
hoelzl@40859
   401
  then show "?s Q \<in> borel_measurable M1" by simp
hoelzl@40859
   402
qed
hoelzl@40859
   403
hoelzl@40859
   404
subsection {* Binary products of $\sigma$-finite measure spaces *}
hoelzl@40859
   405
hoelzl@40859
   406
locale pair_sigma_finite = M1: sigma_finite_measure M1 \<mu>1 + M2: sigma_finite_measure M2 \<mu>2
hoelzl@40859
   407
  for M1 \<mu>1 M2 \<mu>2
hoelzl@40859
   408
hoelzl@40859
   409
sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
hoelzl@40859
   410
  by default
hoelzl@40859
   411
hoelzl@40859
   412
lemma (in pair_sigma_finite) measure_cut_measurable_fst:
hoelzl@40859
   413
  assumes "Q \<in> sets P" shows "(\<lambda>x. \<mu>2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
hoelzl@40859
   414
proof -
hoelzl@40859
   415
  have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
hoelzl@40859
   416
  have M1: "sigma_finite_measure M1 \<mu>1" by default
hoelzl@40859
   417
hoelzl@40859
   418
  from M2.disjoint_sigma_finite guess F .. note F = this
hoelzl@40859
   419
  let "?C x i" = "F i \<inter> Pair x -` Q"
hoelzl@40859
   420
  { fix i
hoelzl@40859
   421
    let ?R = "M2.restricted_space (F i)"
hoelzl@40859
   422
    have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
hoelzl@40859
   423
      using F M2.sets_into_space by auto
hoelzl@40859
   424
    have "(\<lambda>x. \<mu>2 (Pair x -` (space M1 \<times> F i \<inter> Q))) \<in> borel_measurable M1"
hoelzl@40859
   425
    proof (intro finite_measure_cut_measurable[OF M1])
hoelzl@40859
   426
      show "finite_measure (M2.restricted_space (F i)) \<mu>2"
hoelzl@40859
   427
        using F by (intro M2.restricted_to_finite_measure) auto
hoelzl@40859
   428
      have "space M1 \<times> F i \<in> sets P"
hoelzl@40859
   429
        using M1.top F by blast
hoelzl@40859
   430
      from sigma_sets_Int[symmetric,
hoelzl@40859
   431
        OF this[unfolded pair_sigma_algebra_def sets_sigma]]
hoelzl@40859
   432
      show "(space M1 \<times> F i) \<inter> Q \<in> sets (sigma (pair_algebra M1 ?R))"
hoelzl@40859
   433
        using `Q \<in> sets P`
hoelzl@40859
   434
        using pair_algebra_Int_snd[OF M1.space_closed, of "F i" M2]
hoelzl@40859
   435
        by (auto simp: pair_algebra_def sets_sigma)
hoelzl@40859
   436
    qed
hoelzl@40859
   437
    moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"
hoelzl@40859
   438
      using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_algebra)
hoelzl@40859
   439
    ultimately have "(\<lambda>x. \<mu>2 (?C x i)) \<in> borel_measurable M1"
hoelzl@40859
   440
      by simp }
hoelzl@40859
   441
  moreover
hoelzl@40859
   442
  { fix x
hoelzl@40859
   443
    have "(\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i)) = \<mu>2 (\<Union>i. ?C x i)"
hoelzl@40859
   444
    proof (intro M2.measure_countably_additive)
hoelzl@40859
   445
      show "range (?C x) \<subseteq> sets M2"
hoelzl@40859
   446
        using F `Q \<in> sets P` by (auto intro!: M2.Int measurable_cut_fst)
hoelzl@40859
   447
      have "disjoint_family F" using F by auto
hoelzl@40859
   448
      show "disjoint_family (?C x)"
hoelzl@40859
   449
        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
hoelzl@40859
   450
    qed
hoelzl@40859
   451
    also have "(\<Union>i. ?C x i) = Pair x -` Q"
hoelzl@40859
   452
      using F sets_into_space `Q \<in> sets P`
hoelzl@40859
   453
      by (auto simp: space_pair_algebra)
hoelzl@40859
   454
    finally have "\<mu>2 (Pair x -` Q) = (\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i))"
hoelzl@40859
   455
      by simp }
hoelzl@40859
   456
  ultimately show ?thesis
hoelzl@40859
   457
    by (auto intro!: M1.borel_measurable_psuminf)
hoelzl@40859
   458
qed
hoelzl@40859
   459
hoelzl@40859
   460
lemma (in pair_sigma_finite) measure_cut_measurable_snd:
hoelzl@40859
   461
  assumes "Q \<in> sets P" shows "(\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
hoelzl@40859
   462
proof -
hoelzl@40859
   463
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@40859
   464
  have [simp]: "\<And>y. (Pair y -` (\<lambda>(x, y). (y, x)) ` Q) = (\<lambda>x. (x, y)) -` Q"
hoelzl@40859
   465
    by auto
hoelzl@40859
   466
  note sets_pair_sigma_algebra_swap[OF assms]
hoelzl@40859
   467
  from Q.measure_cut_measurable_fst[OF this]
hoelzl@40859
   468
  show ?thesis by simp
hoelzl@40859
   469
qed
hoelzl@40859
   470
hoelzl@40859
   471
lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
hoelzl@40859
   472
  assumes "f \<in> measurable P M"
hoelzl@40859
   473
  shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (sigma (pair_algebra M2 M1)) M"
hoelzl@40859
   474
proof -
hoelzl@40859
   475
  interpret Q: pair_sigma_algebra M2 M1 by default
hoelzl@40859
   476
  have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
hoelzl@40859
   477
  show ?thesis
hoelzl@40859
   478
    using Q.pair_sigma_algebra_swap_measurable assms
hoelzl@40859
   479
    unfolding * by (rule measurable_comp)
hoelzl@39088
   480
qed
hoelzl@39088
   481
hoelzl@40859
   482
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap:
hoelzl@40859
   483
  "sigma (pair_algebra M2 M1) = (vimage_algebra (space M2 \<times> space M1) (\<lambda>(x, y). (y, x)))"
hoelzl@40859
   484
  unfolding vimage_algebra_def
hoelzl@40859
   485
  apply (simp add: sets_sigma)
hoelzl@40859
   486
  apply (subst sigma_sets_vimage[symmetric])
hoelzl@40859
   487
  apply (fastsimp simp: pair_algebra_def)
hoelzl@40859
   488
  using M1.sets_into_space M2.sets_into_space apply (fastsimp simp: pair_algebra_def)
hoelzl@40859
   489
proof -
hoelzl@40859
   490
  have "(\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E
hoelzl@40859
   491
    = sets (pair_algebra M2 M1)" (is "?S = _")
hoelzl@40859
   492
    by (rule pair_algebra_swap)
hoelzl@40859
   493
  then show "sigma (pair_algebra M2 M1) = \<lparr>space = space M2 \<times> space M1,
hoelzl@40859
   494
       sets = sigma_sets (space M2 \<times> space M1) ?S\<rparr>"
hoelzl@40859
   495
    by (simp add: pair_algebra_def sigma_def)
hoelzl@40859
   496
qed
hoelzl@40859
   497
hoelzl@40859
   498
definition (in pair_sigma_finite)
hoelzl@40859
   499
  "pair_measure A = M1.positive_integral (\<lambda>x.
hoelzl@40859
   500
    M2.positive_integral (\<lambda>y. indicator A (x, y)))"
hoelzl@40859
   501
hoelzl@40859
   502
lemma (in pair_sigma_finite) pair_measure_alt:
hoelzl@40859
   503
  assumes "A \<in> sets P"
hoelzl@40859
   504
  shows "pair_measure A = M1.positive_integral (\<lambda>x. \<mu>2 (Pair x -` A))"
hoelzl@40859
   505
  unfolding pair_measure_def
hoelzl@40859
   506
proof (rule M1.positive_integral_cong)
hoelzl@40859
   507
  fix x assume "x \<in> space M1"
hoelzl@41023
   508
  have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: pextreal)"
hoelzl@40859
   509
    unfolding indicator_def by auto
hoelzl@40859
   510
  show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x -` A)"
hoelzl@40859
   511
    unfolding *
hoelzl@40859
   512
    apply (subst M2.positive_integral_indicator)
hoelzl@40859
   513
    apply (rule measurable_cut_fst[OF assms])
hoelzl@40859
   514
    by simp
hoelzl@40859
   515
qed
hoelzl@40859
   516
hoelzl@40859
   517
lemma (in pair_sigma_finite) pair_measure_times:
hoelzl@40859
   518
  assumes A: "A \<in> sets M1" and "B \<in> sets M2"
hoelzl@40859
   519
  shows "pair_measure (A \<times> B) = \<mu>1 A * \<mu>2 B"
hoelzl@40859
   520
proof -
hoelzl@40859
   521
  from assms have "pair_measure (A \<times> B) =
hoelzl@40859
   522
      M1.positive_integral (\<lambda>x. \<mu>2 B * indicator A x)"
hoelzl@40859
   523
    by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
hoelzl@40859
   524
  with assms show ?thesis
hoelzl@40859
   525
    by (simp add: M1.positive_integral_cmult_indicator ac_simps)
hoelzl@40859
   526
qed
hoelzl@40859
   527
hoelzl@40859
   528
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra:
hoelzl@40859
   529
  "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and>
hoelzl@40859
   530
    (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"
hoelzl@40859
   531
proof -
hoelzl@40859
   532
  obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
hoelzl@40859
   533
    F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. \<mu>1 (F1 i) \<noteq> \<omega>" and
hoelzl@40859
   534
    F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. \<mu>2 (F2 i) \<noteq> \<omega>"
hoelzl@40859
   535
    using M1.sigma_finite_up M2.sigma_finite_up by auto
hoelzl@40859
   536
  then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)"
hoelzl@40859
   537
    unfolding isoton_def by auto
hoelzl@40859
   538
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
hoelzl@40859
   539
  show ?thesis unfolding isoton_def space_pair_algebra
hoelzl@40859
   540
  proof (intro exI[of _ ?F] conjI allI)
hoelzl@40859
   541
    show "range ?F \<subseteq> sets E" using F1 F2
hoelzl@40859
   542
      by (fastsimp intro!: pair_algebraI)
hoelzl@40859
   543
  next
hoelzl@40859
   544
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
hoelzl@40859
   545
    proof (intro subsetI)
hoelzl@40859
   546
      fix x assume "x \<in> space M1 \<times> space M2"
hoelzl@40859
   547
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
hoelzl@40859
   548
        by (auto simp: space)
hoelzl@40859
   549
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
hoelzl@40859
   550
        using `F1 \<up> space M1` `F2 \<up> space M2`
hoelzl@40859
   551
        by (auto simp: max_def dest: isoton_mono_le)
hoelzl@40859
   552
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
hoelzl@40859
   553
        by (intro SigmaI) (auto simp add: min_max.sup_commute)
hoelzl@40859
   554
      then show "x \<in> (\<Union>i. ?F i)" by auto
hoelzl@40859
   555
    qed
hoelzl@40859
   556
    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
hoelzl@40859
   557
      using space by (auto simp: space)
hoelzl@40859
   558
  next
hoelzl@40859
   559
    fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)"
hoelzl@40859
   560
      using `F1 \<up> space M1` `F2 \<up> space M2` unfolding isoton_def
hoelzl@40859
   561
      by auto
hoelzl@40859
   562
  next
hoelzl@40859
   563
    fix i
hoelzl@40859
   564
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
hoelzl@40859
   565
    with F1 F2 show "pair_measure (F1 i \<times> F2 i) \<noteq> \<omega>"
hoelzl@40859
   566
      by (simp add: pair_measure_times)
hoelzl@40859
   567
  qed
hoelzl@40859
   568
qed
hoelzl@40859
   569
hoelzl@40859
   570
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure
hoelzl@40859
   571
proof
hoelzl@40859
   572
  show "pair_measure {} = 0"
hoelzl@40859
   573
    unfolding pair_measure_def by auto
hoelzl@40859
   574
hoelzl@40859
   575
  show "countably_additive P pair_measure"
hoelzl@40859
   576
    unfolding countably_additive_def
hoelzl@40859
   577
  proof (intro allI impI)
hoelzl@40859
   578
    fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
hoelzl@40859
   579
    assume F: "range F \<subseteq> sets P" "disjoint_family F"
hoelzl@40859
   580
    from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
hoelzl@40859
   581
    moreover from F have "\<And>i. (\<lambda>x. \<mu>2 (Pair x -` F i)) \<in> borel_measurable M1"
hoelzl@40859
   582
      by (intro measure_cut_measurable_fst) auto
hoelzl@40859
   583
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
hoelzl@40859
   584
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@40859
   585
    moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
hoelzl@40859
   586
      using F by (auto intro!: measurable_cut_fst)
hoelzl@40859
   587
    ultimately show "(\<Sum>\<^isub>\<infinity>n. pair_measure (F n)) = pair_measure (\<Union>i. F i)"
hoelzl@40859
   588
      by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric]
hoelzl@40859
   589
                    M2.measure_countably_additive
hoelzl@40859
   590
               cong: M1.positive_integral_cong)
hoelzl@40859
   591
  qed
hoelzl@40859
   592
hoelzl@40859
   593
  from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
hoelzl@40859
   594
  show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"
hoelzl@40859
   595
  proof (rule exI[of _ F], intro conjI)
hoelzl@40859
   596
    show "range F \<subseteq> sets P" using F by auto
hoelzl@40859
   597
    show "(\<Union>i. F i) = space P"
hoelzl@40859
   598
      using F by (auto simp: space_pair_algebra isoton_def)
hoelzl@40859
   599
    show "\<forall>i. pair_measure (F i) \<noteq> \<omega>" using F by auto
hoelzl@40859
   600
  qed
hoelzl@40859
   601
qed
hoelzl@39088
   602
hoelzl@40859
   603
lemma (in pair_sigma_finite) pair_measure_alt2:
hoelzl@40859
   604
  assumes "A \<in> sets P"
hoelzl@40859
   605
  shows "pair_measure A = M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` A))"
hoelzl@40859
   606
    (is "_ = ?\<nu> A")
hoelzl@40859
   607
proof -
hoelzl@40859
   608
  from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this
hoelzl@40859
   609
  show ?thesis
hoelzl@40859
   610
  proof (rule measure_unique_Int_stable[where \<nu>="?\<nu>", OF Int_stable_pair_algebra],
hoelzl@40859
   611
         simp_all add: pair_sigma_algebra_def[symmetric])
hoelzl@40859
   612
    show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. pair_measure (F i) \<noteq> \<omega>"
hoelzl@40859
   613
      using F by auto
hoelzl@40859
   614
    show "measure_space P pair_measure" by default
hoelzl@40859
   615
  next
hoelzl@40859
   616
    show "measure_space P ?\<nu>"
hoelzl@40859
   617
    proof
hoelzl@40859
   618
      show "?\<nu> {} = 0" by auto
hoelzl@40859
   619
      show "countably_additive P ?\<nu>"
hoelzl@40859
   620
        unfolding countably_additive_def
hoelzl@40859
   621
      proof (intro allI impI)
hoelzl@40859
   622
        fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
hoelzl@40859
   623
        assume F: "range F \<subseteq> sets P" "disjoint_family F"
hoelzl@40859
   624
        from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
hoelzl@40859
   625
        moreover from F have "\<And>i. (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) -` F i)) \<in> borel_measurable M2"
hoelzl@40859
   626
          by (intro measure_cut_measurable_snd) auto
hoelzl@40859
   627
        moreover have "\<And>y. disjoint_family (\<lambda>i. (\<lambda>x. (x, y)) -` F i)"
hoelzl@40859
   628
          by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
hoelzl@40859
   629
        moreover have "\<And>y. y \<in> space M2 \<Longrightarrow> range (\<lambda>i. (\<lambda>x. (x, y)) -` F i) \<subseteq> sets M1"
hoelzl@40859
   630
          using F by (auto intro!: measurable_cut_snd)
hoelzl@40859
   631
        ultimately show "(\<Sum>\<^isub>\<infinity>n. ?\<nu> (F n)) = ?\<nu> (\<Union>i. F i)"
hoelzl@40859
   632
          by (simp add: vimage_UN M2.positive_integral_psuminf[symmetric]
hoelzl@40859
   633
                        M1.measure_countably_additive
hoelzl@40859
   634
                   cong: M2.positive_integral_cong)
hoelzl@40859
   635
      qed
hoelzl@40859
   636
    qed
hoelzl@40859
   637
  next
hoelzl@40859
   638
    fix X assume "X \<in> sets E"
hoelzl@40859
   639
    then obtain A B where X: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
hoelzl@40859
   640
      unfolding pair_algebra_def by auto
hoelzl@40859
   641
    show "pair_measure X = ?\<nu> X"
hoelzl@40859
   642
    proof -
hoelzl@40859
   643
      from AB have "?\<nu> (A \<times> B) =
hoelzl@40859
   644
          M2.positive_integral (\<lambda>y. \<mu>1 A * indicator B y)"
hoelzl@40859
   645
        by (auto intro!: M2.positive_integral_cong)
hoelzl@40859
   646
      with AB show ?thesis
hoelzl@40859
   647
        unfolding pair_measure_times[OF AB] X
hoelzl@40859
   648
        by (simp add: M2.positive_integral_cmult_indicator ac_simps)
hoelzl@40859
   649
    qed
hoelzl@40859
   650
  qed fact
hoelzl@40859
   651
qed
hoelzl@40859
   652
hoelzl@40859
   653
section "Fubinis theorem"
hoelzl@40859
   654
hoelzl@40859
   655
lemma (in pair_sigma_finite) simple_function_cut:
hoelzl@40859
   656
  assumes f: "simple_function f"
hoelzl@40859
   657
  shows "(\<lambda>x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
hoelzl@40859
   658
    and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
hoelzl@40859
   659
      = positive_integral f"
hoelzl@40859
   660
proof -
hoelzl@40859
   661
  have f_borel: "f \<in> borel_measurable P"
hoelzl@40859
   662
    using f by (rule borel_measurable_simple_function)
hoelzl@40859
   663
  let "?F z" = "f -` {z} \<inter> space P"
hoelzl@40859
   664
  let "?F' x z" = "Pair x -` ?F z"
hoelzl@40859
   665
  { fix x assume "x \<in> space M1"
hoelzl@40859
   666
    have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
hoelzl@40859
   667
      by (auto simp: indicator_def)
hoelzl@40859
   668
    have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1`
hoelzl@40859
   669
      by (simp add: space_pair_algebra)
hoelzl@40859
   670
    moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
hoelzl@40859
   671
      by (intro borel_measurable_vimage measurable_cut_fst)
hoelzl@40859
   672
    ultimately have "M2.simple_function (\<lambda> y. f (x, y))"
hoelzl@40859
   673
      apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
hoelzl@40859
   674
      apply (rule simple_function_indicator_representation[OF f])
hoelzl@40859
   675
      using `x \<in> space M1` by (auto simp del: space_sigma) }
hoelzl@40859
   676
  note M2_sf = this
hoelzl@40859
   677
  { fix x assume x: "x \<in> space M1"
hoelzl@40859
   678
    then have "M2.positive_integral (\<lambda> y. f (x, y)) =
hoelzl@40859
   679
        (\<Sum>z\<in>f ` space P. z * \<mu>2 (?F' x z))"
hoelzl@40859
   680
      unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]]
hoelzl@40859
   681
      unfolding M2.simple_integral_def
hoelzl@40859
   682
    proof (safe intro!: setsum_mono_zero_cong_left)
hoelzl@40859
   683
      from f show "finite (f ` space P)" by (rule simple_functionD)
hoelzl@40859
   684
    next
hoelzl@40859
   685
      fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
hoelzl@40859
   686
        using `x \<in> space M1` by (auto simp: space_pair_algebra)
hoelzl@40859
   687
    next
hoelzl@40859
   688
      fix x' y assume "(x', y) \<in> space P"
hoelzl@40859
   689
        "f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
hoelzl@40859
   690
      then have *: "?F' x (f (x', y)) = {}"
hoelzl@40859
   691
        by (force simp: space_pair_algebra)
hoelzl@40859
   692
      show  "f (x', y) * \<mu>2 (?F' x (f (x', y))) = 0"
hoelzl@40859
   693
        unfolding * by simp
hoelzl@40859
   694
    qed (simp add: vimage_compose[symmetric] comp_def
hoelzl@40859
   695
                   space_pair_algebra) }
hoelzl@40859
   696
  note eq = this
hoelzl@40859
   697
  moreover have "\<And>z. ?F z \<in> sets P"
hoelzl@40859
   698
    by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
hoelzl@40859
   699
  moreover then have "\<And>z. (\<lambda>x. \<mu>2 (?F' x z)) \<in> borel_measurable M1"
hoelzl@40859
   700
    by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
hoelzl@40859
   701
  ultimately
hoelzl@40859
   702
  show "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
hoelzl@40859
   703
    and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
hoelzl@40859
   704
    = positive_integral f"
hoelzl@40859
   705
    by (auto simp del: vimage_Int cong: measurable_cong
hoelzl@41023
   706
             intro!: M1.borel_measurable_pextreal_setsum
hoelzl@40859
   707
             simp add: M1.positive_integral_setsum simple_integral_def
hoelzl@40859
   708
                       M1.positive_integral_cmult
hoelzl@40859
   709
                       M1.positive_integral_cong[OF eq]
hoelzl@40859
   710
                       positive_integral_eq_simple_integral[OF f]
hoelzl@40859
   711
                       pair_measure_alt[symmetric])
hoelzl@40859
   712
qed
hoelzl@40859
   713
hoelzl@40859
   714
lemma (in pair_sigma_finite) positive_integral_fst_measurable:
hoelzl@40859
   715
  assumes f: "f \<in> borel_measurable P"
hoelzl@40859
   716
  shows "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"
hoelzl@40859
   717
      (is "?C f \<in> borel_measurable M1")
hoelzl@40859
   718
    and "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
hoelzl@40859
   719
      positive_integral f"
hoelzl@40859
   720
proof -
hoelzl@40859
   721
  from borel_measurable_implies_simple_function_sequence[OF f]
hoelzl@40859
   722
  obtain F where F: "\<And>i. simple_function (F i)" "F \<up> f" by auto
hoelzl@40859
   723
  then have F_borel: "\<And>i. F i \<in> borel_measurable P"
hoelzl@40859
   724
    and F_mono: "\<And>i x. F i x \<le> F (Suc i) x"
hoelzl@40859
   725
    and F_SUPR: "\<And>x. (SUP i. F i x) = f x"
hoelzl@40859
   726
    unfolding isoton_def le_fun_def SUPR_fun_expand
hoelzl@40859
   727
    by (auto intro: borel_measurable_simple_function)
hoelzl@40859
   728
  note sf = simple_function_cut[OF F(1)]
hoelzl@40859
   729
  then have "(SUP i. ?C (F i)) \<in> borel_measurable M1"
hoelzl@40859
   730
    using F(1) by (auto intro!: M1.borel_measurable_SUP)
hoelzl@40859
   731
  moreover
hoelzl@40859
   732
  { fix x assume "x \<in> space M1"
hoelzl@40859
   733
    have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))"
hoelzl@40859
   734
      using `F \<up> f` unfolding isoton_fun_expand
hoelzl@40859
   735
      by (auto simp: isoton_def)
hoelzl@40859
   736
    note measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
hoelzl@40859
   737
    from M2.positive_integral_isoton[OF isotone this]
hoelzl@40859
   738
    have "(SUP i. ?C (F i) x) = ?C f x"
hoelzl@40859
   739
      by (simp add: isoton_def) }
hoelzl@40859
   740
  note SUPR_C = this
hoelzl@40859
   741
  ultimately show "?C f \<in> borel_measurable M1"
hoelzl@40859
   742
    unfolding SUPR_fun_expand by (simp cong: measurable_cong)
hoelzl@40859
   743
  have "positive_integral (\<lambda>x. SUP i. F i x) = (SUP i. positive_integral (F i))"
hoelzl@40859
   744
    using F_borel F_mono
hoelzl@40859
   745
    by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])
hoelzl@40859
   746
  also have "(SUP i. positive_integral (F i)) =
hoelzl@40859
   747
    (SUP i. M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. F i (x, y))))"
hoelzl@40859
   748
    unfolding sf(2) by simp
hoelzl@40859
   749
  also have "\<dots> = M1.positive_integral (\<lambda>x. SUP i. M2.positive_integral (\<lambda>y. F i (x, y)))"
hoelzl@40859
   750
    by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)]
hoelzl@40859
   751
                     M2.positive_integral_mono F_mono)
hoelzl@40859
   752
  also have "\<dots> = M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. SUP i. F i (x, y)))"
hoelzl@40859
   753
    using F_borel F_mono
hoelzl@40859
   754
    by (auto intro!: M2.positive_integral_monotone_convergence_SUP
hoelzl@40859
   755
                     M1.positive_integral_cong measurable_pair_image_snd)
hoelzl@40859
   756
  finally show "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =
hoelzl@40859
   757
      positive_integral f"
hoelzl@40859
   758
    unfolding F_SUPR by simp
hoelzl@40859
   759
qed
hoelzl@40859
   760
hoelzl@40859
   761
lemma (in pair_sigma_finite) positive_integral_snd_measurable:
hoelzl@40859
   762
  assumes f: "f \<in> borel_measurable P"
hoelzl@40859
   763
  shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
hoelzl@40859
   764
      positive_integral f"
hoelzl@40859
   765
proof -
hoelzl@40859
   766
  interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default
hoelzl@40859
   767
  have s: "\<And>x y. (case (x, y) of (x, y) \<Rightarrow> f (y, x)) = f (y, x)" by simp
hoelzl@40859
   768
  have t: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> (y, x))) = (\<lambda>(x, y). f (y, x))" by (auto simp: fun_eq_iff)
hoelzl@40859
   769
  have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"
hoelzl@40859
   770
    by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
hoelzl@40859
   771
  note pair_sigma_algebra_measurable[OF f]
hoelzl@40859
   772
  from Q.positive_integral_fst_measurable[OF this]
hoelzl@40859
   773
  have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
hoelzl@40859
   774
    Q.positive_integral (\<lambda>(x, y). f (y, x))"
hoelzl@40859
   775
    by simp
hoelzl@40859
   776
  also have "\<dots> = positive_integral f"
hoelzl@40859
   777
    unfolding positive_integral_vimage[OF bij, of f] t
hoelzl@40859
   778
    unfolding pair_sigma_algebra_swap[symmetric]
hoelzl@40859
   779
  proof (rule Q.positive_integral_cong_measure[symmetric])
hoelzl@40859
   780
    fix A assume "A \<in> sets Q.P"
hoelzl@40859
   781
    from this Q.sets_pair_sigma_algebra_swap[OF this]
hoelzl@40859
   782
    show "pair_measure ((\<lambda>(x, y). (y, x)) ` A) = Q.pair_measure A"
hoelzl@40859
   783
      by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]
hoelzl@40859
   784
               simp: pair_measure_alt Q.pair_measure_alt2)
hoelzl@40859
   785
  qed
hoelzl@40859
   786
  finally show ?thesis .
hoelzl@40859
   787
qed
hoelzl@40859
   788
hoelzl@40859
   789
lemma (in pair_sigma_finite) Fubini:
hoelzl@40859
   790
  assumes f: "f \<in> borel_measurable P"
hoelzl@40859
   791
  shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =
hoelzl@40859
   792
      M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))"
hoelzl@40859
   793
  unfolding positive_integral_snd_measurable[OF assms]
hoelzl@40859
   794
  unfolding positive_integral_fst_measurable[OF assms] ..
hoelzl@40859
   795
hoelzl@40859
   796
lemma (in pair_sigma_finite) AE_pair:
hoelzl@40859
   797
  assumes "almost_everywhere (\<lambda>x. Q x)"
hoelzl@40859
   798
  shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))"
hoelzl@40859
   799
proof -
hoelzl@40859
   800
  obtain N where N: "N \<in> sets P" "pair_measure N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
hoelzl@40859
   801
    using assms unfolding almost_everywhere_def by auto
hoelzl@40859
   802
  show ?thesis
hoelzl@40859
   803
  proof (rule M1.AE_I)
hoelzl@40859
   804
    from N measure_cut_measurable_fst[OF `N \<in> sets P`]
hoelzl@40859
   805
    show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x -` N) \<noteq> 0} = 0"
hoelzl@40859
   806
      by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)
hoelzl@40859
   807
    show "{x \<in> space M1. \<mu>2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
hoelzl@41023
   808
      by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)
hoelzl@40859
   809
    { fix x assume "x \<in> space M1" "\<mu>2 (Pair x -` N) = 0"
hoelzl@40859
   810
      have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
hoelzl@40859
   811
      proof (rule M2.AE_I)
hoelzl@40859
   812
        show "\<mu>2 (Pair x -` N) = 0" by fact
hoelzl@40859
   813
        show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)
hoelzl@40859
   814
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
hoelzl@40859
   815
          using N `x \<in> space M1` unfolding space_sigma space_pair_algebra by auto
hoelzl@40859
   816
      qed }
hoelzl@40859
   817
    then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. \<mu>2 (Pair x -` N) \<noteq> 0}"
hoelzl@40859
   818
      by auto
hoelzl@39088
   819
  qed
hoelzl@39088
   820
qed
hoelzl@35833
   821
hoelzl@40859
   822
section "Finite product spaces"
hoelzl@40859
   823
hoelzl@40859
   824
section "Products"
hoelzl@40859
   825
hoelzl@40859
   826
locale product_sigma_algebra =
hoelzl@40859
   827
  fixes M :: "'i \<Rightarrow> 'a algebra"
hoelzl@40859
   828
  assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
hoelzl@40859
   829
hoelzl@40859
   830
locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" +
hoelzl@40859
   831
  fixes I :: "'i set"
hoelzl@40859
   832
  assumes finite_index: "finite I"
hoelzl@40859
   833
hoelzl@40859
   834
syntax
hoelzl@40859
   835
  "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
hoelzl@40859
   836
hoelzl@40859
   837
syntax (xsymbols)
hoelzl@40859
   838
  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
hoelzl@40859
   839
hoelzl@40859
   840
syntax (HTML output)
hoelzl@40859
   841
  "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)
hoelzl@40859
   842
hoelzl@40859
   843
translations
hoelzl@40859
   844
  "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
hoelzl@40859
   845
hoelzl@35833
   846
definition
hoelzl@40859
   847
  "product_algebra M I = \<lparr> space = (\<Pi>\<^isub>E i\<in>I. space (M i)), sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)) \<rparr>"
hoelzl@40859
   848
hoelzl@40859
   849
abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I"
hoelzl@40859
   850
abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G"
hoelzl@40859
   851
hoelzl@40859
   852
sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
hoelzl@40859
   853
hoelzl@40859
   854
lemma (in finite_product_sigma_algebra) product_algebra_into_space:
hoelzl@40859
   855
  "sets G \<subseteq> Pow (space G)"
hoelzl@40859
   856
  using M.sets_into_space unfolding product_algebra_def
hoelzl@40859
   857
  by auto blast
hoelzl@40859
   858
hoelzl@40859
   859
sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
hoelzl@40859
   860
  using product_algebra_into_space by (rule sigma_algebra_sigma)
hoelzl@40859
   861
hoelzl@40859
   862
lemma space_product_algebra[simp]:
hoelzl@40859
   863
  "space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))"
hoelzl@40859
   864
  unfolding product_algebra_def by simp
hoelzl@40859
   865
hoelzl@40859
   866
lemma (in finite_product_sigma_algebra) P_empty:
hoelzl@40859
   867
  "I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"
hoelzl@40872
   868
  unfolding product_algebra_def by (simp add: sigma_def image_constant)
hoelzl@40859
   869
hoelzl@40859
   870
lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
hoelzl@40859
   871
  "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
hoelzl@40859
   872
  by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic)
hoelzl@40859
   873
hoelzl@40859
   874
lemma bij_betw_prod_fold:
hoelzl@40859
   875
  assumes "i \<notin> I"
hoelzl@40859
   876
  shows "bij_betw (\<lambda>x. (x(i:=undefined), x i)) (\<Pi>\<^isub>E j\<in>insert i I. space (M j)) ((\<Pi>\<^isub>E j\<in>I. space (M j)) \<times> space (M i))"
hoelzl@40859
   877
    (is "bij_betw ?f ?P ?F")
hoelzl@40859
   878
    using `i \<notin> I`
hoelzl@40859
   879
proof (unfold bij_betw_def, intro conjI set_eqI iffI)
hoelzl@40859
   880
  show "inj_on ?f ?P"
hoelzl@40859
   881
  proof (safe intro!: inj_onI ext)
hoelzl@40859
   882
    fix a b x assume "a(i:=undefined) = b(i:=undefined)" "a i = b i"
hoelzl@40859
   883
    then show "a x = b x"
hoelzl@40859
   884
      by (cases "x = i") (auto simp: fun_eq_iff split: split_if_asm)
hoelzl@40859
   885
  qed
hoelzl@40859
   886
next
hoelzl@40859
   887
  fix X assume *: "X \<in> ?F" show "X \<in> ?f ` ?P"
hoelzl@40859
   888
  proof (cases X)
hoelzl@40859
   889
    case (Pair a b) with * `i \<notin> I` show ?thesis
hoelzl@40859
   890
      by (auto intro!: image_eqI[where x="a (i := b)"] ext simp: extensional_def)
hoelzl@40859
   891
  qed
hoelzl@40859
   892
qed auto
hoelzl@40859
   893
hoelzl@40859
   894
section "Generating set generates also product algebra"
hoelzl@40859
   895
hoelzl@40859
   896
lemma pair_sigma_algebra_sigma:
hoelzl@40859
   897
  assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
hoelzl@40859
   898
  assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
hoelzl@40859
   899
  shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)"
hoelzl@40859
   900
    (is "?S = ?E")
hoelzl@40859
   901
proof -
hoelzl@40859
   902
  interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
hoelzl@40859
   903
  interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
hoelzl@40859
   904
  have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
hoelzl@40859
   905
    using E1 E2 by (auto simp add: pair_algebra_def)
hoelzl@40859
   906
  interpret E: sigma_algebra ?E unfolding pair_algebra_def
hoelzl@40859
   907
    using E1 E2 by (intro sigma_algebra_sigma) auto
hoelzl@40859
   908
  { fix A assume "A \<in> sets E1"
hoelzl@40859
   909
    then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
hoelzl@40859
   910
      using E1 2 unfolding isoton_def pair_algebra_def by auto
hoelzl@40859
   911
    also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
hoelzl@40859
   912
    also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma
hoelzl@40859
   913
      using 2 `A \<in> sets E1`
hoelzl@40859
   914
      by (intro sigma_sets.Union)
hoelzl@40859
   915
         (auto simp: image_subset_iff intro!: sigma_sets.Basic)
hoelzl@40859
   916
    finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
hoelzl@40859
   917
  moreover
hoelzl@40859
   918
  { fix B assume "B \<in> sets E2"
hoelzl@40859
   919
    then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
hoelzl@40859
   920
      using E2 1 unfolding isoton_def pair_algebra_def by auto
hoelzl@40859
   921
    also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
hoelzl@40859
   922
    also have "\<dots> \<in> sets ?E"
hoelzl@40859
   923
      using 1 `B \<in> sets E2` unfolding pair_algebra_def sets_sigma
hoelzl@40859
   924
      by (intro sigma_sets.Union)
hoelzl@40859
   925
         (auto simp: image_subset_iff intro!: sigma_sets.Basic)
hoelzl@40859
   926
    finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
hoelzl@40859
   927
  ultimately have proj:
hoelzl@40859
   928
    "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
hoelzl@40859
   929
    using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
hoelzl@40859
   930
                   (auto simp: pair_algebra_def sets_sigma)
hoelzl@40859
   931
  { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
hoelzl@40859
   932
    with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
hoelzl@40859
   933
      unfolding measurable_def by simp_all
hoelzl@40859
   934
    moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"
hoelzl@40859
   935
      using A B M1.sets_into_space M2.sets_into_space
hoelzl@40859
   936
      by (auto simp: pair_algebra_def)
hoelzl@40859
   937
    ultimately have "A \<times> B \<in> sets ?E" by auto }
hoelzl@40859
   938
  then have "sigma_sets (space ?E) (sets (pair_algebra (sigma E1) (sigma E2))) \<subseteq> sets ?E"
hoelzl@40859
   939
    by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma)
hoelzl@40859
   940
  then have subset: "sets ?S \<subseteq> sets ?E"
hoelzl@40859
   941
    by (simp add: sets_sigma pair_algebra_def)
hoelzl@40859
   942
  have "sets ?S = sets ?E"
hoelzl@40859
   943
  proof (intro set_eqI iffI)
hoelzl@40859
   944
    fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
hoelzl@40859
   945
      unfolding sets_sigma
hoelzl@40859
   946
    proof induct
hoelzl@40859
   947
      case (Basic A) then show ?case
hoelzl@40859
   948
        by (auto simp: pair_algebra_def sets_sigma intro: sigma_sets.Basic)
hoelzl@40859
   949
    qed (auto intro: sigma_sets.intros simp: pair_algebra_def)
hoelzl@40859
   950
  next
hoelzl@40859
   951
    fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
hoelzl@40859
   952
  qed
hoelzl@40859
   953
  then show ?thesis
hoelzl@40859
   954
    by (simp add: pair_algebra_def sigma_def)
hoelzl@40859
   955
qed
hoelzl@40859
   956
hoelzl@40859
   957
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@40859
   958
  apply auto
hoelzl@40859
   959
  apply (drule_tac x=x in Pi_mem)
hoelzl@40859
   960
  apply (simp_all split: split_if_asm)
hoelzl@40859
   961
  apply (drule_tac x=i in Pi_mem)
hoelzl@40859
   962
  apply (auto dest!: Pi_mem)
hoelzl@40859
   963
  done
hoelzl@40859
   964
hoelzl@40859
   965
lemma Pi_UN:
hoelzl@40859
   966
  fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
hoelzl@40859
   967
  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@40859
   968
  shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
hoelzl@40859
   969
proof (intro set_eqI iffI)
hoelzl@40859
   970
  fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
hoelzl@40859
   971
  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
hoelzl@40859
   972
  from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
hoelzl@40859
   973
  obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
hoelzl@40859
   974
    using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
hoelzl@40859
   975
  have "f \<in> Pi I (A k)"
hoelzl@40859
   976
  proof (intro Pi_I)
hoelzl@40859
   977
    fix i assume "i \<in> I"
hoelzl@40859
   978
    from mono[OF this, of "n i" k] k[OF this] n[OF this]
hoelzl@40859
   979
    show "f i \<in> A k i" by auto
hoelzl@40859
   980
  qed
hoelzl@40859
   981
  then show "f \<in> (\<Union>n. Pi I (A n))" by auto
hoelzl@40859
   982
qed auto
hoelzl@40859
   983
hoelzl@40859
   984
lemma PiE_cong:
hoelzl@40859
   985
  assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
hoelzl@40859
   986
  shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
hoelzl@40859
   987
  using assms by (auto intro!: Pi_cong)
hoelzl@40859
   988
hoelzl@40859
   989
lemma sigma_product_algebra_sigma_eq:
hoelzl@40859
   990
  assumes "finite I"
hoelzl@40859
   991
  assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))"
hoelzl@40859
   992
  assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
hoelzl@40859
   993
  and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
hoelzl@40859
   994
  shows "sigma (product_algebra (\<lambda>i. sigma (E i)) I) = sigma (product_algebra E I)"
hoelzl@40859
   995
    (is "?S = ?E")
hoelzl@40859
   996
proof cases
hoelzl@40859
   997
  assume "I = {}" then show ?thesis by (simp add: product_algebra_def)
hoelzl@40859
   998
next
hoelzl@40859
   999
  assume "I \<noteq> {}"
hoelzl@40859
  1000
  interpret E: sigma_algebra "sigma (E i)" for i
hoelzl@40859
  1001
    using E by (rule sigma_algebra_sigma)
hoelzl@40859
  1002