src/HOL/Probability/Product_Measure.thy
 author hoelzl Mon Jan 24 22:29:50 2011 +0100 (2011-01-24) changeset 41661 baf1964bc468 parent 41659 a5d1b2df5e97 child 41689 3e39b0e730d6 permissions -rw-r--r--
use pre-image measure, instead of image
     1 theory Product_Measure

     2 imports Lebesgue_Integration

     3 begin

     4

     5 lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"

     6   by auto

     7

     8 lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x - (A \<times> B) = (if x \<in> A then B else {})"

     9   by auto

    10

    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 {})"

    12   by auto

    13

    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))"

    15   by (cases x) simp

    16

    17 lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"

    18   by (auto simp: fun_eq_iff)

    19

    20 abbreviation

    21   "Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"

    22

    23 abbreviation

    24   funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"

    25     (infixr "->\<^isub>E" 60) where

    26   "A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"

    27

    28 notation (xsymbols)

    29   funcset_extensional  (infixr "\<rightarrow>\<^isub>E" 60)

    30

    31 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"

    32   by safe (auto simp add: extensional_def fun_eq_iff)

    33

    34 lemma extensional_insert[intro, simp]:

    35   assumes "a \<in> extensional (insert i I)"

    36   shows "a(i := b) \<in> extensional (insert i I)"

    37   using assms unfolding extensional_def by auto

    38

    39 lemma extensional_Int[simp]:

    40   "extensional I \<inter> extensional I' = extensional (I \<inter> I')"

    41   unfolding extensional_def by auto

    42

    43 definition

    44   "merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"

    45

    46 lemma merge_apply[simp]:

    47   "I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"

    48   "I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"

    49   "J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"

    50   "J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"

    51   "i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"

    52   unfolding merge_def by auto

    53

    54 lemma merge_commute:

    55   "I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"

    56   by (auto simp: merge_def intro!: ext)

    57

    58 lemma Pi_cancel_merge_range[simp]:

    59   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"

    60   "I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"

    61   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"

    62   "J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"

    63   by (auto simp: Pi_def)

    64

    65 lemma Pi_cancel_merge[simp]:

    66   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"

    67   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"

    68   "I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"

    69   "J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"

    70   by (auto simp: Pi_def)

    71

    72 lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"

    73   by (auto simp: extensional_def)

    74

    75 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"

    76   by (auto simp: restrict_def Pi_def)

    77

    78 lemma restrict_merge[simp]:

    79   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"

    80   "I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"

    81   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"

    82   "J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"

    83   by (auto simp: restrict_def intro!: ext)

    84

    85 lemma extensional_insert_undefined[intro, simp]:

    86   assumes "a \<in> extensional (insert i I)"

    87   shows "a(i := undefined) \<in> extensional I"

    88   using assms unfolding extensional_def by auto

    89

    90 lemma extensional_insert_cancel[intro, simp]:

    91   assumes "a \<in> extensional I"

    92   shows "a \<in> extensional (insert i I)"

    93   using assms unfolding extensional_def by auto

    94

    95 lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)"

    96   unfolding merge_def by (auto simp: fun_eq_iff)

    97

    98 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"

    99   by auto

   100

   101 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)"

   102   by auto

   103

   104 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"

   105   by (auto simp: Pi_def)

   106

   107 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"

   108   by (auto simp: Pi_def)

   109

   110 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"

   111   by (auto simp: Pi_def)

   112

   113 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"

   114   by (auto simp: Pi_def)

   115

   116 lemma restrict_vimage:

   117   assumes "I \<inter> J = {}"

   118   shows "(\<lambda>x. (restrict x I, restrict x J)) - (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I E J F)"

   119   using assms by (auto simp: restrict_Pi_cancel)

   120

   121 lemma merge_vimage:

   122   assumes "I \<inter> J = {}"

   123   shows "(\<lambda>(x,y). merge I x J y) - Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"

   124   using assms by (auto simp: restrict_Pi_cancel)

   125

   126 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"

   127   by (auto simp: restrict_def intro!: ext)

   128

   129 lemma merge_restrict[simp]:

   130   "merge I (restrict x I) J y = merge I x J y"

   131   "merge I x J (restrict y J) = merge I x J y"

   132   unfolding merge_def by (auto intro!: ext)

   133

   134 lemma merge_x_x_eq_restrict[simp]:

   135   "merge I x J x = restrict x (I \<union> J)"

   136   unfolding merge_def by (auto intro!: ext)

   137

   138 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"

   139   apply auto

   140   apply (drule_tac x=x in Pi_mem)

   141   apply (simp_all split: split_if_asm)

   142   apply (drule_tac x=i in Pi_mem)

   143   apply (auto dest!: Pi_mem)

   144   done

   145

   146 lemma Pi_UN:

   147   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"

   148   assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"

   149   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"

   150 proof (intro set_eqI iffI)

   151   fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"

   152   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto

   153   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto

   154   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"

   155     using finite I finite_nat_set_iff_bounded_le[of "nI"] by auto

   156   have "f \<in> Pi I (A k)"

   157   proof (intro Pi_I)

   158     fix i assume "i \<in> I"

   159     from mono[OF this, of "n i" k] k[OF this] n[OF this]

   160     show "f i \<in> A k i" by auto

   161   qed

   162   then show "f \<in> (\<Union>n. Pi I (A n))" by auto

   163 qed auto

   164

   165 lemma PiE_cong:

   166   assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"

   167   shows "Pi\<^isub>E I A = Pi\<^isub>E I B"

   168   using assms by (auto intro!: Pi_cong)

   169

   170 lemma restrict_upd[simp]:

   171   "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"

   172   by (auto simp: fun_eq_iff)

   173

   174 section "Binary products"

   175

   176 definition

   177   "pair_algebra A B = \<lparr> space = space A \<times> space B,

   178                            sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<rparr>"

   179

   180 locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2

   181   for M1 M2

   182

   183 abbreviation (in pair_sigma_algebra)

   184   "E \<equiv> pair_algebra M1 M2"

   185

   186 abbreviation (in pair_sigma_algebra)

   187   "P \<equiv> sigma E"

   188

   189 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P

   190   using M1.sets_into_space M2.sets_into_space

   191   by (force simp: pair_algebra_def intro!: sigma_algebra_sigma)

   192

   193 lemma pair_algebraI[intro, simp]:

   194   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_algebra A B)"

   195   by (auto simp add: pair_algebra_def)

   196

   197 lemma space_pair_algebra:

   198   "space (pair_algebra A B) = space A \<times> space B"

   199   by (simp add: pair_algebra_def)

   200

   201 lemma sets_pair_algebra: "sets (pair_algebra N M) = (\<lambda>(x, y). x \<times> y)  (sets N \<times> sets M)"

   202   unfolding pair_algebra_def by auto

   203

   204 lemma pair_algebra_sets_into_space:

   205   assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"

   206   shows "sets (pair_algebra M N) \<subseteq> Pow (space (pair_algebra M N))"

   207   using assms by (auto simp: pair_algebra_def)

   208

   209 lemma pair_algebra_Int_snd:

   210   assumes "sets S1 \<subseteq> Pow (space S1)"

   211   shows "pair_algebra S1 (algebra.restricted_space S2 A) =

   212          algebra.restricted_space (pair_algebra S1 S2) (space S1 \<times> A)"

   213   (is "?L = ?R")

   214 proof (intro algebra.equality set_eqI iffI)

   215   fix X assume "X \<in> sets ?L"

   216   then obtain A1 A2 where X: "X = A1 \<times> (A \<inter> A2)" and "A1 \<in> sets S1" "A2 \<in> sets S2"

   217     by (auto simp: pair_algebra_def)

   218   then show "X \<in> sets ?R" unfolding pair_algebra_def

   219     using assms apply simp by (intro image_eqI[of _ _ "A1 \<times> A2"]) auto

   220 next

   221   fix X assume "X \<in> sets ?R"

   222   then obtain A1 A2 where "X = space S1 \<times> A \<inter> A1 \<times> A2" "A1 \<in> sets S1" "A2 \<in> sets S2"

   223     by (auto simp: pair_algebra_def)

   224   moreover then have "X = A1 \<times> (A \<inter> A2)"

   225     using assms by auto

   226   ultimately show "X \<in> sets ?L"

   227     unfolding pair_algebra_def by auto

   228 qed (auto simp add: pair_algebra_def)

   229

   230 lemma (in pair_sigma_algebra)

   231   shows measurable_fst[intro!, simp]:

   232     "fst \<in> measurable P M1" (is ?fst)

   233   and measurable_snd[intro!, simp]:

   234     "snd \<in> measurable P M2" (is ?snd)

   235 proof -

   236   { fix X assume "X \<in> sets M1"

   237     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"

   238       apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])

   239       using M1.sets_into_space by force+ }

   240   moreover

   241   { fix X assume "X \<in> sets M2"

   242     then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd - X \<inter> space M1 \<times> space M2 = X1 \<times> X2"

   243       apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])

   244       using M2.sets_into_space by force+ }

   245   ultimately have "?fst \<and> ?snd"

   246     by (fastsimp simp: measurable_def sets_sigma space_pair_algebra

   247                  intro!: sigma_sets.Basic)

   248   then show ?fst ?snd by auto

   249 qed

   250

   251 lemma (in pair_sigma_algebra) measurable_pair_iff:

   252   assumes "sigma_algebra M"

   253   shows "f \<in> measurable M P \<longleftrightarrow>

   254     (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"

   255 proof -

   256   interpret M: sigma_algebra M by fact

   257   from assms show ?thesis

   258   proof (safe intro!: measurable_comp[where b=P])

   259     assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"

   260     show "f \<in> measurable M P"

   261     proof (rule M.measurable_sigma)

   262       show "sets (pair_algebra M1 M2) \<subseteq> Pow (space E)"

   263         unfolding pair_algebra_def using M1.sets_into_space M2.sets_into_space by auto

   264       show "f \<in> space M \<rightarrow> space E"

   265         using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma space_pair_algebra)

   266       fix A assume "A \<in> sets E"

   267       then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"

   268         unfolding pair_algebra_def by auto

   269       moreover have "(fst \<circ> f) - B \<inter> space M \<in> sets M"

   270         using f B \<in> sets M1 unfolding measurable_def by auto

   271       moreover have "(snd \<circ> f) - C \<inter> space M \<in> sets M"

   272         using s C \<in> sets M2 unfolding measurable_def by auto

   273       moreover have "f - A \<inter> space M = ((fst \<circ> f) - B \<inter> space M) \<inter> ((snd \<circ> f) - C \<inter> space M)"

   274         unfolding A = B \<times> C by (auto simp: vimage_Times)

   275       ultimately show "f - A \<inter> space M \<in> sets M" by auto

   276     qed

   277   qed

   278 qed

   279

   280 lemma (in pair_sigma_algebra) measurable_pair:

   281   assumes "sigma_algebra M"

   282   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"

   283   shows "f \<in> measurable M P"

   284   unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp

   285

   286 lemma pair_algebraE:

   287   assumes "X \<in> sets (pair_algebra M1 M2)"

   288   obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"

   289   using assms unfolding pair_algebra_def by auto

   290

   291 lemma (in pair_sigma_algebra) pair_algebra_swap:

   292   "(\<lambda>X. (\<lambda>(x,y). (y,x)) - X \<inter> space M2 \<times> space M1)  sets E = sets (pair_algebra M2 M1)"

   293 proof (safe elim!: pair_algebraE)

   294   fix A B assume "A \<in> sets M1" "B \<in> sets M2"

   295   moreover then have "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"

   296     using M1.sets_into_space M2.sets_into_space by auto

   297   ultimately show "(\<lambda>(x, y). (y, x)) - (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_algebra M2 M1)"

   298     by (auto intro: pair_algebraI)

   299 next

   300   fix A B assume "A \<in> sets M1" "B \<in> sets M2"

   301   then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) - X \<inter> space M2 \<times> space M1)  sets E"

   302     using M1.sets_into_space M2.sets_into_space

   303     by (auto intro!: image_eqI[where x="A \<times> B"] pair_algebraI)

   304 qed

   305

   306 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:

   307   assumes Q: "Q \<in> sets P"

   308   shows "(\<lambda>(x,y). (y, x))  Q \<in> sets (sigma (pair_algebra M2 M1))" (is "_ \<in> sets ?Q")

   309 proof -

   310   have *: "(\<lambda>(x,y). (y, x)) \<in> space M2 \<times> space M1 \<rightarrow> (space M1 \<times> space M2)"

   311        "sets (pair_algebra M1 M2) \<subseteq> Pow (space M1 \<times> space M2)"

   312     using M1.sets_into_space M2.sets_into_space by (auto elim!: pair_algebraE)

   313   from Q sets_into_space show ?thesis

   314     by (auto intro!: image_eqI[where x=Q]

   315              simp: pair_algebra_swap[symmetric] sets_sigma

   316                    sigma_sets_vimage[OF *] space_pair_algebra)

   317 qed

   318

   319 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:

   320   shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (sigma (pair_algebra M2 M1))"

   321     (is "?f \<in> measurable ?P ?Q")

   322   unfolding measurable_def

   323 proof (intro CollectI conjI Pi_I ballI)

   324   fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"

   325     unfolding pair_algebra_def by auto

   326 next

   327   fix A assume "A \<in> sets ?Q"

   328   interpret Q: pair_sigma_algebra M2 M1 by default

   329   have "?f - A \<inter> space ?P = (\<lambda>(x,y). (y, x))  A"

   330     using Q.sets_into_space A \<in> sets ?Q by (auto simp: pair_algebra_def)

   331   with Q.sets_pair_sigma_algebra_swap[OF A \<in> sets ?Q]

   332   show "?f - A \<inter> space ?P \<in> sets ?P" by simp

   333 qed

   334

   335 lemma (in pair_sigma_algebra) measurable_cut_fst:

   336   assumes "Q \<in> sets P" shows "Pair x - Q \<in> sets M2"

   337 proof -

   338   let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x - Q \<in> sets M2}"

   339   let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"

   340   interpret Q: sigma_algebra ?Q

   341     proof qed (auto simp: vimage_UN vimage_Diff space_pair_algebra)

   342   have "sets E \<subseteq> sets ?Q"

   343     using M1.sets_into_space M2.sets_into_space

   344     by (auto simp: pair_algebra_def space_pair_algebra)

   345   then have "sets P \<subseteq> sets ?Q"

   346     by (subst pair_algebra_def, intro Q.sets_sigma_subset)

   347        (simp_all add: pair_algebra_def)

   348   with assms show ?thesis by auto

   349 qed

   350

   351 lemma (in pair_sigma_algebra) measurable_cut_snd:

   352   assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) - Q \<in> sets M1" (is "?cut Q \<in> sets M1")

   353 proof -

   354   interpret Q: pair_sigma_algebra M2 M1 by default

   355   have "Pair y - (\<lambda>(x, y). (y, x))  Q = (\<lambda>x. (x, y)) - Q" by auto

   356   with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]

   357   show ?thesis by simp

   358 qed

   359

   360 lemma (in pair_sigma_algebra) measurable_pair_image_snd:

   361   assumes m: "f \<in> measurable P M" and "x \<in> space M1"

   362   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"

   363   unfolding measurable_def

   364 proof (intro CollectI conjI Pi_I ballI)

   365   fix y assume "y \<in> space M2" with f \<in> measurable P M x \<in> space M1

   366   show "f (x, y) \<in> space M" unfolding measurable_def pair_algebra_def by auto

   367 next

   368   fix A assume "A \<in> sets M"

   369   then have "Pair x - (f - A \<inter> space P) \<in> sets M2" (is "?C \<in> _")

   370     using f \<in> measurable P M

   371     by (intro measurable_cut_fst) (auto simp: measurable_def)

   372   also have "?C = (\<lambda>y. f (x, y)) - A \<inter> space M2"

   373     using x \<in> space M1 by (auto simp: pair_algebra_def)

   374   finally show "(\<lambda>y. f (x, y)) - A \<inter> space M2 \<in> sets M2" .

   375 qed

   376

   377 lemma (in pair_sigma_algebra) measurable_pair_image_fst:

   378   assumes m: "f \<in> measurable P M" and "y \<in> space M2"

   379   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"

   380 proof -

   381   interpret Q: pair_sigma_algebra M2 M1 by default

   382   from Q.measurable_pair_image_snd[OF measurable_comp y \<in> space M2,

   383                                       OF Q.pair_sigma_algebra_swap_measurable m]

   384   show ?thesis by simp

   385 qed

   386

   387 lemma (in pair_sigma_algebra) Int_stable_pair_algebra: "Int_stable E"

   388   unfolding Int_stable_def

   389 proof (intro ballI)

   390   fix A B assume "A \<in> sets E" "B \<in> sets E"

   391   then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"

   392     "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"

   393     unfolding pair_algebra_def by auto

   394   then show "A \<inter> B \<in> sets E"

   395     by (auto simp add: times_Int_times pair_algebra_def)

   396 qed

   397

   398 lemma finite_measure_cut_measurable:

   399   fixes M1 :: "'a algebra" and M2 :: "'b algebra"

   400   assumes "sigma_finite_measure M1 \<mu>1" "finite_measure M2 \<mu>2"

   401   assumes "Q \<in> sets (sigma (pair_algebra M1 M2))"

   402   shows "(\<lambda>x. \<mu>2 (Pair x - Q)) \<in> borel_measurable M1"

   403     (is "?s Q \<in> _")

   404 proof -

   405   interpret M1: sigma_finite_measure M1 \<mu>1 by fact

   406   interpret M2: finite_measure M2 \<mu>2 by fact

   407   interpret pair_sigma_algebra M1 M2 by default

   408   have [intro]: "sigma_algebra M1" by fact

   409   have [intro]: "sigma_algebra M2" by fact

   410   let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"

   411   note space_pair_algebra[simp]

   412   interpret dynkin_system ?D

   413   proof (intro dynkin_systemI)

   414     fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"

   415       using sets_into_space by simp

   416   next

   417     from top show "space ?D \<in> sets ?D"

   418       by (auto simp add: if_distrib intro!: M1.measurable_If)

   419   next

   420     fix A assume "A \<in> sets ?D"

   421     with sets_into_space have "\<And>x. \<mu>2 (Pair x - (space M1 \<times> space M2 - A)) =

   422         (if x \<in> space M1 then \<mu>2 (space M2) - ?s A x else 0)"

   423       by (auto intro!: M2.finite_measure_compl measurable_cut_fst

   424                simp: vimage_Diff)

   425     with A \<in> sets ?D top show "space ?D - A \<in> sets ?D"

   426       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pextreal_diff)

   427   next

   428     fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"

   429     moreover then have "\<And>x. \<mu>2 (\<Union>i. Pair x - F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)"

   430       by (intro M2.measure_countably_additive[symmetric])

   431          (auto intro!: measurable_cut_fst simp: disjoint_family_on_def)

   432     ultimately show "(\<Union>i. F i) \<in> sets ?D"

   433       by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)

   434   qed

   435   have "P = ?D"

   436   proof (intro dynkin_lemma)

   437     show "Int_stable E" by (rule Int_stable_pair_algebra)

   438     from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"

   439       by auto

   440     then show "sets E \<subseteq> sets ?D"

   441       by (auto simp: pair_algebra_def sets_sigma if_distrib

   442                intro: sigma_sets.Basic intro!: M1.measurable_If)

   443   qed auto

   444   with Q \<in> sets P have "Q \<in> sets ?D" by simp

   445   then show "?s Q \<in> borel_measurable M1" by simp

   446 qed

   447

   448 subsection {* Binary products of $\sigma$-finite measure spaces *}

   449

   450 locale pair_sigma_finite = M1: sigma_finite_measure M1 \<mu>1 + M2: sigma_finite_measure M2 \<mu>2

   451   for M1 \<mu>1 M2 \<mu>2

   452

   453 sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2

   454   by default

   455

   456 lemma (in pair_sigma_finite) measure_cut_measurable_fst:

   457   assumes "Q \<in> sets P" shows "(\<lambda>x. \<mu>2 (Pair x - Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")

   458 proof -

   459   have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+

   460   have M1: "sigma_finite_measure M1 \<mu>1" by default

   461

   462   from M2.disjoint_sigma_finite guess F .. note F = this

   463   let "?C x i" = "F i \<inter> Pair x - Q"

   464   { fix i

   465     let ?R = "M2.restricted_space (F i)"

   466     have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"

   467       using F M2.sets_into_space by auto

   468     have "(\<lambda>x. \<mu>2 (Pair x - (space M1 \<times> F i \<inter> Q))) \<in> borel_measurable M1"

   469     proof (intro finite_measure_cut_measurable[OF M1])

   470       show "finite_measure (M2.restricted_space (F i)) \<mu>2"

   471         using F by (intro M2.restricted_to_finite_measure) auto

   472       have "space M1 \<times> F i \<in> sets P"

   473         using M1.top F by blast

   474       from sigma_sets_Int[symmetric,

   475         OF this[unfolded pair_sigma_algebra_def sets_sigma]]

   476       show "(space M1 \<times> F i) \<inter> Q \<in> sets (sigma (pair_algebra M1 ?R))"

   477         using Q \<in> sets P

   478         using pair_algebra_Int_snd[OF M1.space_closed, of "F i" M2]

   479         by (auto simp: pair_algebra_def sets_sigma)

   480     qed

   481     moreover have "\<And>x. Pair x - (space M1 \<times> F i \<inter> Q) = ?C x i"

   482       using Q \<in> sets P sets_into_space by (auto simp: space_pair_algebra)

   483     ultimately have "(\<lambda>x. \<mu>2 (?C x i)) \<in> borel_measurable M1"

   484       by simp }

   485   moreover

   486   { fix x

   487     have "(\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i)) = \<mu>2 (\<Union>i. ?C x i)"

   488     proof (intro M2.measure_countably_additive)

   489       show "range (?C x) \<subseteq> sets M2"

   490         using F Q \<in> sets P by (auto intro!: M2.Int measurable_cut_fst)

   491       have "disjoint_family F" using F by auto

   492       show "disjoint_family (?C x)"

   493         by (rule disjoint_family_on_bisimulation[OF disjoint_family F]) auto

   494     qed

   495     also have "(\<Union>i. ?C x i) = Pair x - Q"

   496       using F sets_into_space Q \<in> sets P

   497       by (auto simp: space_pair_algebra)

   498     finally have "\<mu>2 (Pair x - Q) = (\<Sum>\<^isub>\<infinity>i. \<mu>2 (?C x i))"

   499       by simp }

   500   ultimately show ?thesis

   501     by (auto intro!: M1.borel_measurable_psuminf)

   502 qed

   503

   504 lemma (in pair_sigma_finite) measure_cut_measurable_snd:

   505   assumes "Q \<in> sets P" shows "(\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"

   506 proof -

   507   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   508   have [simp]: "\<And>y. (Pair y - (\<lambda>(x, y). (y, x))  Q) = (\<lambda>x. (x, y)) - Q"

   509     by auto

   510   note sets_pair_sigma_algebra_swap[OF assms]

   511   from Q.measure_cut_measurable_fst[OF this]

   512   show ?thesis by simp

   513 qed

   514

   515 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:

   516   assumes "f \<in> measurable P M"

   517   shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (sigma (pair_algebra M2 M1)) M"

   518 proof -

   519   interpret Q: pair_sigma_algebra M2 M1 by default

   520   have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)

   521   show ?thesis

   522     using Q.pair_sigma_algebra_swap_measurable assms

   523     unfolding * by (rule measurable_comp)

   524 qed

   525

   526 definition (in pair_sigma_finite)

   527   "pair_measure A = M1.positive_integral (\<lambda>x.

   528     M2.positive_integral (\<lambda>y. indicator A (x, y)))"

   529

   530 lemma (in pair_sigma_finite) pair_measure_alt:

   531   assumes "A \<in> sets P"

   532   shows "pair_measure A = M1.positive_integral (\<lambda>x. \<mu>2 (Pair x - A))"

   533   unfolding pair_measure_def

   534 proof (rule M1.positive_integral_cong)

   535   fix x assume "x \<in> space M1"

   536   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x - A) y :: pextreal)"

   537     unfolding indicator_def by auto

   538   show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x - A)"

   539     unfolding *

   540     apply (subst M2.positive_integral_indicator)

   541     apply (rule measurable_cut_fst[OF assms])

   542     by simp

   543 qed

   544

   545 lemma (in pair_sigma_finite) pair_measure_times:

   546   assumes A: "A \<in> sets M1" and "B \<in> sets M2"

   547   shows "pair_measure (A \<times> B) = \<mu>1 A * \<mu>2 B"

   548 proof -

   549   from assms have "pair_measure (A \<times> B) =

   550       M1.positive_integral (\<lambda>x. \<mu>2 B * indicator A x)"

   551     by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)

   552   with assms show ?thesis

   553     by (simp add: M1.positive_integral_cmult_indicator ac_simps)

   554 qed

   555

   556 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra:

   557   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and>

   558     (\<forall>i. pair_measure (F i) \<noteq> \<omega>)"

   559 proof -

   560   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where

   561     F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. \<mu>1 (F1 i) \<noteq> \<omega>" and

   562     F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. \<mu>2 (F2 i) \<noteq> \<omega>"

   563     using M1.sigma_finite_up M2.sigma_finite_up by auto

   564   then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)"

   565     unfolding isoton_def by auto

   566   let ?F = "\<lambda>i. F1 i \<times> F2 i"

   567   show ?thesis unfolding isoton_def space_pair_algebra

   568   proof (intro exI[of _ ?F] conjI allI)

   569     show "range ?F \<subseteq> sets E" using F1 F2

   570       by (fastsimp intro!: pair_algebraI)

   571   next

   572     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"

   573     proof (intro subsetI)

   574       fix x assume "x \<in> space M1 \<times> space M2"

   575       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"

   576         by (auto simp: space)

   577       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"

   578         using F1 \<up> space M1 F2 \<up> space M2

   579         by (auto simp: max_def dest: isoton_mono_le)

   580       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"

   581         by (intro SigmaI) (auto simp add: min_max.sup_commute)

   582       then show "x \<in> (\<Union>i. ?F i)" by auto

   583     qed

   584     then show "(\<Union>i. ?F i) = space M1 \<times> space M2"

   585       using space by (auto simp: space)

   586   next

   587     fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)"

   588       using F1 \<up> space M1 F2 \<up> space M2 unfolding isoton_def

   589       by auto

   590   next

   591     fix i

   592     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto

   593     with F1 F2 show "pair_measure (F1 i \<times> F2 i) \<noteq> \<omega>"

   594       by (simp add: pair_measure_times)

   595   qed

   596 qed

   597

   598 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure

   599 proof

   600   show "pair_measure {} = 0"

   601     unfolding pair_measure_def by auto

   602

   603   show "countably_additive P pair_measure"

   604     unfolding countably_additive_def

   605   proof (intro allI impI)

   606     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"

   607     assume F: "range F \<subseteq> sets P" "disjoint_family F"

   608     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto

   609     moreover from F have "\<And>i. (\<lambda>x. \<mu>2 (Pair x - F i)) \<in> borel_measurable M1"

   610       by (intro measure_cut_measurable_fst) auto

   611     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x - F i)"

   612       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto

   613     moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x - F i) \<subseteq> sets M2"

   614       using F by (auto intro!: measurable_cut_fst)

   615     ultimately show "(\<Sum>\<^isub>\<infinity>n. pair_measure (F n)) = pair_measure (\<Union>i. F i)"

   616       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric]

   617                     M2.measure_countably_additive

   618                cong: M1.positive_integral_cong)

   619   qed

   620

   621   from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this

   622   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>)"

   623   proof (rule exI[of _ F], intro conjI)

   624     show "range F \<subseteq> sets P" using F by auto

   625     show "(\<Union>i. F i) = space P"

   626       using F by (auto simp: space_pair_algebra isoton_def)

   627     show "\<forall>i. pair_measure (F i) \<noteq> \<omega>" using F by auto

   628   qed

   629 qed

   630

   631 lemma (in pair_sigma_algebra) sets_swap:

   632   assumes "A \<in> sets P"

   633   shows "(\<lambda>(x, y). (y, x)) - A \<inter> space (sigma (pair_algebra M2 M1)) \<in> sets (sigma (pair_algebra M2 M1))"

   634     (is "_ - A \<inter> space ?Q \<in> sets ?Q")

   635 proof -

   636   have *: "(\<lambda>(x, y). (y, x)) - A \<inter> space ?Q = (\<lambda>(x, y). (y, x))  A"

   637     using A \<in> sets P sets_into_space by (auto simp: space_pair_algebra)

   638   show ?thesis

   639     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)

   640 qed

   641

   642 lemma (in pair_sigma_finite) pair_measure_alt2:

   643   assumes "A \<in> sets P"

   644   shows "pair_measure A = M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - A))"

   645     (is "_ = ?\<nu> A")

   646 proof -

   647   from sigma_finite_up_in_pair_algebra guess F :: "nat \<Rightarrow> ('a \<times> 'c) set" .. note F = this

   648   show ?thesis

   649   proof (rule measure_unique_Int_stable[where \<nu>="?\<nu>", OF Int_stable_pair_algebra],

   650          simp_all add: pair_sigma_algebra_def[symmetric])

   651     show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. pair_measure (F i) \<noteq> \<omega>"

   652       using F by auto

   653     show "measure_space P pair_measure" by default

   654     interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   655     have P: "sigma_algebra P" by default

   656     show "measure_space P ?\<nu>"

   657       apply (rule Q.measure_space_vimage[OF P])

   658       apply (rule Q.pair_sigma_algebra_swap_measurable)

   659     proof -

   660       fix A assume "A \<in> sets P"

   661       from sets_swap[OF this]

   662       show "M2.positive_integral (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - A)) =

   663             Q.pair_measure ((\<lambda>(x, y). (y, x)) - A \<inter> space Q.P)"

   664         using sets_into_space[OF A \<in> sets P]

   665         by (auto simp add: Q.pair_measure_alt space_pair_algebra

   666                  intro!: M2.positive_integral_cong arg_cong[where f=\<mu>1])

   667     qed

   668     fix X assume "X \<in> sets E"

   669     then obtain A B where X: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"

   670       unfolding pair_algebra_def by auto

   671     show "pair_measure X = ?\<nu> X"

   672     proof -

   673       from AB have "?\<nu> (A \<times> B) =

   674           M2.positive_integral (\<lambda>y. \<mu>1 A * indicator B y)"

   675         by (auto intro!: M2.positive_integral_cong)

   676       with AB show ?thesis

   677         unfolding pair_measure_times[OF AB] X

   678         by (simp add: M2.positive_integral_cmult_indicator ac_simps)

   679     qed

   680   qed fact

   681 qed

   682

   683 section "Fubinis theorem"

   684

   685 lemma (in pair_sigma_finite) simple_function_cut:

   686   assumes f: "simple_function f"

   687   shows "(\<lambda>x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"

   688     and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))

   689       = positive_integral f"

   690 proof -

   691   have f_borel: "f \<in> borel_measurable P"

   692     using f by (rule borel_measurable_simple_function)

   693   let "?F z" = "f - {z} \<inter> space P"

   694   let "?F' x z" = "Pair x - ?F z"

   695   { fix x assume "x \<in> space M1"

   696     have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"

   697       by (auto simp: indicator_def)

   698     have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using x \<in> space M1

   699       by (simp add: space_pair_algebra)

   700     moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel

   701       by (intro borel_measurable_vimage measurable_cut_fst)

   702     ultimately have "M2.simple_function (\<lambda> y. f (x, y))"

   703       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])

   704       apply (rule simple_function_indicator_representation[OF f])

   705       using x \<in> space M1 by (auto simp del: space_sigma) }

   706   note M2_sf = this

   707   { fix x assume x: "x \<in> space M1"

   708     then have "M2.positive_integral (\<lambda> y. f (x, y)) =

   709         (\<Sum>z\<in>f  space P. z * \<mu>2 (?F' x z))"

   710       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]]

   711       unfolding M2.simple_integral_def

   712     proof (safe intro!: setsum_mono_zero_cong_left)

   713       from f show "finite (f  space P)" by (rule simple_functionD)

   714     next

   715       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f  space P"

   716         using x \<in> space M1 by (auto simp: space_pair_algebra)

   717     next

   718       fix x' y assume "(x', y) \<in> space P"

   719         "f (x', y) \<notin> (\<lambda>y. f (x, y))  space M2"

   720       then have *: "?F' x (f (x', y)) = {}"

   721         by (force simp: space_pair_algebra)

   722       show  "f (x', y) * \<mu>2 (?F' x (f (x', y))) = 0"

   723         unfolding * by simp

   724     qed (simp add: vimage_compose[symmetric] comp_def

   725                    space_pair_algebra) }

   726   note eq = this

   727   moreover have "\<And>z. ?F z \<in> sets P"

   728     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)

   729   moreover then have "\<And>z. (\<lambda>x. \<mu>2 (?F' x z)) \<in> borel_measurable M1"

   730     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)

   731   ultimately

   732   show "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"

   733     and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))

   734     = positive_integral f"

   735     by (auto simp del: vimage_Int cong: measurable_cong

   736              intro!: M1.borel_measurable_pextreal_setsum

   737              simp add: M1.positive_integral_setsum simple_integral_def

   738                        M1.positive_integral_cmult

   739                        M1.positive_integral_cong[OF eq]

   740                        positive_integral_eq_simple_integral[OF f]

   741                        pair_measure_alt[symmetric])

   742 qed

   743

   744 lemma (in pair_sigma_finite) positive_integral_fst_measurable:

   745   assumes f: "f \<in> borel_measurable P"

   746   shows "(\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) \<in> borel_measurable M1"

   747       (is "?C f \<in> borel_measurable M1")

   748     and "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =

   749       positive_integral f"

   750 proof -

   751   from borel_measurable_implies_simple_function_sequence[OF f]

   752   obtain F where F: "\<And>i. simple_function (F i)" "F \<up> f" by auto

   753   then have F_borel: "\<And>i. F i \<in> borel_measurable P"

   754     and F_mono: "\<And>i x. F i x \<le> F (Suc i) x"

   755     and F_SUPR: "\<And>x. (SUP i. F i x) = f x"

   756     unfolding isoton_fun_expand unfolding isoton_def le_fun_def

   757     by (auto intro: borel_measurable_simple_function)

   758   note sf = simple_function_cut[OF F(1)]

   759   then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"

   760     using F(1) by auto

   761   moreover

   762   { fix x assume "x \<in> space M1"

   763     have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))"

   764       using F \<up> f unfolding isoton_fun_expand

   765       by (auto simp: isoton_def)

   766     note measurable_pair_image_snd[OF F_borelx \<in> space M1]

   767     from M2.positive_integral_isoton[OF isotone this]

   768     have "(SUP i. ?C (F i) x) = ?C f x"

   769       by (simp add: isoton_def) }

   770   note SUPR_C = this

   771   ultimately show "?C f \<in> borel_measurable M1"

   772     by (simp cong: measurable_cong)

   773   have "positive_integral (\<lambda>x. (SUP i. F i x)) = (SUP i. positive_integral (F i))"

   774     using F_borel F_mono

   775     by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])

   776   also have "(SUP i. positive_integral (F i)) =

   777     (SUP i. M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. F i (x, y))))"

   778     unfolding sf(2) by simp

   779   also have "\<dots> = M1.positive_integral (\<lambda>x. SUP i. M2.positive_integral (\<lambda>y. F i (x, y)))"

   780     by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)]

   781                      M2.positive_integral_mono F_mono)

   782   also have "\<dots> = M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. SUP i. F i (x, y)))"

   783     using F_borel F_mono

   784     by (auto intro!: M2.positive_integral_monotone_convergence_SUP

   785                      M1.positive_integral_cong measurable_pair_image_snd)

   786   finally show "M1.positive_integral (\<lambda> x. M2.positive_integral (\<lambda> y. f (x, y))) =

   787       positive_integral f"

   788     unfolding F_SUPR by simp

   789 qed

   790

   791 lemma (in pair_sigma_finite) positive_integral_product_swap:

   792   assumes f: "f \<in> borel_measurable P"

   793   shows "measure_space.positive_integral

   794     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x))) =

   795   positive_integral f"

   796 proof -

   797   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   798   have P: "sigma_algebra P" by default

   799   show ?thesis

   800     unfolding Q.positive_integral_vimage[OF P Q.pair_sigma_algebra_swap_measurable f, symmetric]

   801   proof (rule positive_integral_cong_measure)

   802     fix A

   803     assume A: "A \<in> sets P"

   804     from Q.pair_sigma_algebra_swap_measurable[THEN measurable_sets, OF this] this sets_into_space[OF this]

   805     show "Q.pair_measure ((\<lambda>(x, y). (y, x)) - A \<inter> space Q.P) = pair_measure A"

   806       by (auto intro!: M1.positive_integral_cong arg_cong[where f=\<mu>2]

   807                simp: pair_measure_alt Q.pair_measure_alt2 space_pair_algebra)

   808   qed

   809 qed

   810

   811 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

   812   assumes f: "f \<in> borel_measurable P"

   813   shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =

   814       positive_integral f"

   815 proof -

   816   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   817   note pair_sigma_algebra_measurable[OF f]

   818   from Q.positive_integral_fst_measurable[OF this]

   819   have "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =

   820     Q.positive_integral (\<lambda>(x, y). f (y, x))"

   821     by simp

   822   also have "Q.positive_integral (\<lambda>(x, y). f (y, x)) = positive_integral f"

   823     unfolding positive_integral_product_swap[OF f, symmetric]

   824     by (auto intro!: Q.positive_integral_cong)

   825   finally show ?thesis .

   826 qed

   827

   828 lemma (in pair_sigma_finite) Fubini:

   829   assumes f: "f \<in> borel_measurable P"

   830   shows "M2.positive_integral (\<lambda>y. M1.positive_integral (\<lambda>x. f (x, y))) =

   831       M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))"

   832   unfolding positive_integral_snd_measurable[OF assms]

   833   unfolding positive_integral_fst_measurable[OF assms] ..

   834

   835 lemma (in pair_sigma_finite) AE_pair:

   836   assumes "almost_everywhere (\<lambda>x. Q x)"

   837   shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))"

   838 proof -

   839   obtain N where N: "N \<in> sets P" "pair_measure N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"

   840     using assms unfolding almost_everywhere_def by auto

   841   show ?thesis

   842   proof (rule M1.AE_I)

   843     from N measure_cut_measurable_fst[OF N \<in> sets P]

   844     show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x - N) \<noteq> 0} = 0"

   845       by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)

   846     show "{x \<in> space M1. \<mu>2 (Pair x - N) \<noteq> 0} \<in> sets M1"

   847       by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)

   848     { fix x assume "x \<in> space M1" "\<mu>2 (Pair x - N) = 0"

   849       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"

   850       proof (rule M2.AE_I)

   851         show "\<mu>2 (Pair x - N) = 0" by fact

   852         show "Pair x - N \<in> sets M2" by (intro measurable_cut_fst N)

   853         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x - N"

   854           using N x \<in> space M1 unfolding space_sigma space_pair_algebra by auto

   855       qed }

   856     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}"

   857       by auto

   858   qed

   859 qed

   860

   861 lemma (in pair_sigma_algebra) measurable_product_swap:

   862   "f \<in> measurable (sigma (pair_algebra M2 M1)) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"

   863 proof -

   864   interpret Q: pair_sigma_algebra M2 M1 by default

   865   show ?thesis

   866     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]

   867     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)

   868 qed

   869

   870 lemma (in pair_sigma_finite) integrable_product_swap:

   871   assumes "integrable f"

   872   shows "measure_space.integrable

   873     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>(x,y). f (y,x))"

   874 proof -

   875   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   876   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)

   877   show ?thesis unfolding *

   878     using assms unfolding Q.integrable_def integrable_def

   879     apply (subst (1 2) positive_integral_product_swap)

   880     using integrable f unfolding integrable_def

   881     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])

   882 qed

   883

   884 lemma (in pair_sigma_finite) integrable_product_swap_iff:

   885   "measure_space.integrable

   886     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow>

   887   integrable f"

   888 proof -

   889   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   890   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]

   891   show ?thesis by auto

   892 qed

   893

   894 lemma (in pair_sigma_finite) integral_product_swap:

   895   assumes "integrable f"

   896   shows "measure_space.integral

   897     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) (\<lambda>(x,y). f (y,x)) =

   898   integral f"

   899 proof -

   900   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   901   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)

   902   show ?thesis

   903     unfolding integral_def Q.integral_def *

   904     apply (subst (1 2) positive_integral_product_swap)

   905     using integrable f unfolding integrable_def

   906     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])

   907 qed

   908

   909 lemma (in pair_sigma_finite) integrable_fst_measurable:

   910   assumes f: "integrable f"

   911   shows "M1.almost_everywhere (\<lambda>x. M2.integrable (\<lambda> y. f (x, y)))" (is "?AE")

   912     and "M1.integral (\<lambda> x. M2.integral (\<lambda> y. f (x, y))) = integral f" (is "?INT")

   913 proof -

   914   let "?pf x" = "Real (f x)" and "?nf x" = "Real (- f x)"

   915   have

   916     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and

   917     int: "positive_integral ?nf \<noteq> \<omega>" "positive_integral ?pf \<noteq> \<omega>"

   918     using assms by auto

   919   have "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y)))) \<noteq> \<omega>"

   920      "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y)))) \<noteq> \<omega>"

   921     using borel[THEN positive_integral_fst_measurable(1)] int

   922     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all

   923   with borel[THEN positive_integral_fst_measurable(1)]

   924   have AE: "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (f (x, y))) \<noteq> \<omega>)"

   925     "M1.almost_everywhere (\<lambda>x. M2.positive_integral (\<lambda>y. Real (- f (x, y))) \<noteq> \<omega>)"

   926     by (auto intro!: M1.positive_integral_omega_AE)

   927   then show ?AE

   928     apply (rule M1.AE_mp[OF _ M1.AE_mp])

   929     apply (rule M1.AE_cong)

   930     using assms unfolding M2.integrable_def

   931     by (auto intro!: measurable_pair_image_snd)

   932   have "M1.integrable

   933      (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (f (x, xa)))))" (is "M1.integrable ?f")

   934   proof (unfold M1.integrable_def, intro conjI)

   935     show "?f \<in> borel_measurable M1"

   936       using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)

   937     have "M1.positive_integral (\<lambda>x. Real (?f x)) =

   938         M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (f (x, xa))))"

   939       apply (rule M1.positive_integral_cong_AE)

   940       apply (rule M1.AE_mp[OF AE(1)])

   941       apply (rule M1.AE_cong)

   942       by (auto simp: Real_real)

   943     then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"

   944       using positive_integral_fst_measurable[OF borel(2)] int by simp

   945     have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"

   946       by (intro M1.positive_integral_cong) simp

   947     then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp

   948   qed

   949   moreover have "M1.integrable

   950      (\<lambda>x. real (M2.positive_integral (\<lambda>xa. Real (- f (x, xa)))))" (is "M1.integrable ?f")

   951   proof (unfold M1.integrable_def, intro conjI)

   952     show "?f \<in> borel_measurable M1"

   953       using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)

   954     have "M1.positive_integral (\<lambda>x. Real (?f x)) =

   955         M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>xa. Real (- f (x, xa))))"

   956       apply (rule M1.positive_integral_cong_AE)

   957       apply (rule M1.AE_mp[OF AE(2)])

   958       apply (rule M1.AE_cong)

   959       by (auto simp: Real_real)

   960     then show "M1.positive_integral (\<lambda>x. Real (?f x)) \<noteq> \<omega>"

   961       using positive_integral_fst_measurable[OF borel(1)] int by simp

   962     have "M1.positive_integral (\<lambda>x. Real (- ?f x)) = M1.positive_integral (\<lambda>x. 0)"

   963       by (intro M1.positive_integral_cong) simp

   964     then show "M1.positive_integral (\<lambda>x. Real (- ?f x)) \<noteq> \<omega>" by simp

   965   qed

   966   ultimately show ?INT

   967     unfolding M2.integral_def integral_def

   968       borel[THEN positive_integral_fst_measurable(2), symmetric]

   969     by (simp add: M1.integral_real[OF AE(1)] M1.integral_real[OF AE(2)])

   970 qed

   971

   972 lemma (in pair_sigma_finite) integrable_snd_measurable:

   973   assumes f: "integrable f"

   974   shows "M2.almost_everywhere (\<lambda>y. M1.integrable (\<lambda>x. f (x, y)))" (is "?AE")

   975     and "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) = integral f" (is "?INT")

   976 proof -

   977   interpret Q: pair_sigma_finite M2 \<mu>2 M1 \<mu>1 by default

   978   have Q_int: "Q.integrable (\<lambda>(x, y). f (y, x))"

   979     using f unfolding integrable_product_swap_iff .

   980   show ?INT

   981     using Q.integrable_fst_measurable(2)[OF Q_int]

   982     using integral_product_swap[OF f] by simp

   983   show ?AE

   984     using Q.integrable_fst_measurable(1)[OF Q_int]

   985     by simp

   986 qed

   987

   988 lemma (in pair_sigma_finite) Fubini_integral:

   989   assumes f: "integrable f"

   990   shows "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) =

   991       M1.integral (\<lambda>x. M2.integral (\<lambda>y. f (x, y)))"

   992   unfolding integrable_snd_measurable[OF assms]

   993   unfolding integrable_fst_measurable[OF assms] ..

   994

   995 section "Finite product spaces"

   996

   997 section "Products"

   998

   999 locale product_sigma_algebra =

  1000   fixes M :: "'i \<Rightarrow> 'a algebra"

  1001   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"

  1002

  1003 locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" +

  1004   fixes I :: "'i set"

  1005   assumes finite_index: "finite I"

  1006

  1007 syntax

  1008   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)

  1009

  1010 syntax (xsymbols)

  1011   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)

  1012

  1013 syntax (HTML output)

  1014   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)

  1015

  1016 translations

  1017   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"

  1018

  1019 definition

  1020   "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>"

  1021

  1022 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I"

  1023 abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G"

  1024

  1025 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)

  1026

  1027 lemma (in finite_product_sigma_algebra) product_algebra_into_space:

  1028   "sets G \<subseteq> Pow (space G)"

  1029   using M.sets_into_space unfolding product_algebra_def

  1030   by auto blast

  1031

  1032 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P

  1033   using product_algebra_into_space by (rule sigma_algebra_sigma)

  1034

  1035 lemma product_algebraE:

  1036   assumes "A \<in> sets (product_algebra M I)"

  1037   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1038   using assms unfolding product_algebra_def by auto

  1039

  1040 lemma product_algebraI[intro]:

  1041   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1042   shows "Pi\<^isub>E I E \<in> sets (product_algebra M I)"

  1043   using assms unfolding product_algebra_def by auto

  1044

  1045 lemma space_product_algebra[simp]:

  1046   "space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))"

  1047   unfolding product_algebra_def by simp

  1048

  1049 lemma product_algebra_sets_into_space:

  1050   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"

  1051   shows "sets (product_algebra M I) \<subseteq> Pow (space (product_algebra M I))"

  1052   using assms by (auto simp: product_algebra_def) blast

  1053

  1054 lemma (in finite_product_sigma_algebra) P_empty:

  1055   "I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"

  1056   unfolding product_algebra_def by (simp add: sigma_def image_constant)

  1057

  1058 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:

  1059   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"

  1060   by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic)

  1061

  1062 lemma (in product_sigma_algebra) bij_inv_restrict_merge:

  1063   assumes [simp]: "I \<inter> J = {}"

  1064   shows "bij_inv

  1065     (space (sigma (product_algebra M (I \<union> J))))

  1066     (space (sigma (pair_algebra (product_algebra M I) (product_algebra M J))))

  1067     (\<lambda>x. (restrict x I, restrict x J)) (\<lambda>(x, y). merge I x J y)"

  1068   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)

  1069

  1070 lemma (in product_sigma_algebra) bij_inv_singleton:

  1071   "bij_inv (space (sigma (product_algebra M {i}))) (space (M i))

  1072     (\<lambda>x. x i) (\<lambda>x. (\<lambda>j\<in>{i}. x))"

  1073   by (rule bij_invI) (auto simp: restrict_def extensional_def fun_eq_iff)

  1074

  1075 lemma (in product_sigma_algebra) bij_inv_restrict_insert:

  1076   assumes [simp]: "i \<notin> I"

  1077   shows "bij_inv

  1078     (space (sigma (product_algebra M (insert i I))))

  1079     (space (sigma (pair_algebra (product_algebra M I) (M i))))

  1080     (\<lambda>x. (restrict x I, x i)) (\<lambda>(x, y). x(i := y))"

  1081   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)

  1082

  1083 lemma (in product_sigma_algebra) measurable_restrict_on_generating:

  1084   assumes [simp]: "I \<inter> J = {}"

  1085   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable

  1086     (product_algebra M (I \<union> J))

  1087     (pair_algebra (product_algebra M I) (product_algebra M J))"

  1088     (is "?R \<in> measurable ?IJ ?P")

  1089 proof (unfold measurable_def, intro CollectI conjI ballI)

  1090   show "?R \<in> space ?IJ \<rightarrow> space ?P" by (auto simp: space_pair_algebra)

  1091   { fix F E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> (\<Pi> i\<in>J. sets (M i))"

  1092     then have "Pi (I \<union> J) (merge I E J F) \<inter> (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) =

  1093         Pi\<^isub>E (I \<union> J) (merge I E J F)"

  1094       using M.sets_into_space by auto blast+ }

  1095   note this[simp]

  1096   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?IJ \<in> sets ?IJ"

  1097     by (force elim!: pair_algebraE product_algebraE

  1098               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)

  1099   qed

  1100

  1101 lemma (in product_sigma_algebra) measurable_merge_on_generating:

  1102   assumes [simp]: "I \<inter> J = {}"

  1103   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable

  1104     (pair_algebra (product_algebra M I) (product_algebra M J))

  1105     (product_algebra M (I \<union> J))"

  1106     (is "?M \<in> measurable ?P ?IJ")

  1107 proof (unfold measurable_def, intro CollectI conjI ballI)

  1108   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)

  1109   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E \<in> (\<Pi> i\<in>J. sets (M i))"

  1110     then have "Pi I E \<times> Pi J E \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) \<times> (\<Pi>\<^isub>E i\<in>J. space (M i)) =

  1111         Pi\<^isub>E I E \<times> Pi\<^isub>E J E"

  1112       using M.sets_into_space by auto blast+ }

  1113   note this[simp]

  1114   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"

  1115     by (force elim!: pair_algebraE product_algebraE

  1116               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)

  1117   qed

  1118

  1119 lemma (in product_sigma_algebra) measurable_singleton_on_generator:

  1120   "(\<lambda>x. \<lambda>j\<in>{i}. x) \<in> measurable (M i) (product_algebra M {i})"

  1121   (is "?f \<in> measurable _ ?P")

  1122 proof (unfold measurable_def, intro CollectI conjI)

  1123   show "?f \<in> space (M i) \<rightarrow> space ?P" by auto

  1124   have "\<And>E. E i \<in> sets (M i) \<Longrightarrow> ?f - Pi\<^isub>E {i} E \<inter> space (M i) = E i"

  1125     using M.sets_into_space by auto

  1126   then show "\<forall>A \<in> sets ?P. ?f - A \<inter> space (M i) \<in> sets (M i)"

  1127     by (auto elim!: product_algebraE)

  1128 qed

  1129

  1130 lemma (in product_sigma_algebra) measurable_component_on_generator:

  1131   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (product_algebra M I) (M i)"

  1132   (is "?f \<in> measurable ?P _")

  1133 proof (unfold measurable_def, intro CollectI conjI ballI)

  1134   show "?f \<in> space ?P \<rightarrow> space (M i)" using i \<in> I by auto

  1135   fix A assume "A \<in> sets (M i)"

  1136   moreover then have "(\<lambda>x. x i) - A \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) =

  1137       (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"

  1138     using M.sets_into_space i \<in> I

  1139     by (fastsimp dest: Pi_mem split: split_if_asm)

  1140   ultimately show "?f - A \<inter> space ?P \<in> sets ?P"

  1141     by (auto intro!: product_algebraI)

  1142 qed

  1143

  1144 lemma (in product_sigma_algebra) measurable_restrict_singleton_on_generating:

  1145   assumes [simp]: "i \<notin> I"

  1146   shows "(\<lambda>x. (restrict x I, x i)) \<in> measurable

  1147     (product_algebra M (insert i I))

  1148     (pair_algebra (product_algebra M I) (M i))"

  1149     (is "?R \<in> measurable ?I ?P")

  1150 proof (unfold measurable_def, intro CollectI conjI ballI)

  1151   show "?R \<in> space ?I \<rightarrow> space ?P" by (auto simp: space_pair_algebra)

  1152   { fix E F assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> sets (M i)"

  1153     then have "(\<lambda>x. (restrict x I, x i)) - (Pi\<^isub>E I E \<times> F) \<inter> (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) =

  1154         Pi\<^isub>E (insert i I) (E(i := F))"

  1155       using M.sets_into_space using i\<notin>I by (auto simp: restrict_Pi_cancel) blast+ }

  1156   note this[simp]

  1157   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?I \<in> sets ?I"

  1158     by (force elim!: pair_algebraE product_algebraE

  1159               simp del: vimage_Int simp: space_pair_algebra)

  1160 qed

  1161

  1162 lemma (in product_sigma_algebra) measurable_merge_singleton_on_generating:

  1163   assumes [simp]: "i \<notin> I"

  1164   shows "(\<lambda>(x, y). x(i := y)) \<in> measurable

  1165     (pair_algebra (product_algebra M I) (M i))

  1166     (product_algebra M (insert i I))"

  1167     (is "?M \<in> measurable ?P ?IJ")

  1168 proof (unfold measurable_def, intro CollectI conjI ballI)

  1169   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)

  1170   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E i \<in> sets (M i)"

  1171     then have "(\<lambda>(x, y). x(i := y)) - Pi\<^isub>E (insert i I) E \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) \<times> space (M i) =

  1172         Pi\<^isub>E I E \<times> E i"

  1173       using M.sets_into_space by auto blast+ }

  1174   note this[simp]

  1175   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"

  1176     by (force elim!: pair_algebraE product_algebraE

  1177               simp del: vimage_Int simp: space_pair_algebra)

  1178 qed

  1179

  1180 section "Generating set generates also product algebra"

  1181

  1182 lemma pair_sigma_algebra_sigma:

  1183   assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"

  1184   assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"

  1185   shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)"

  1186     (is "?S = ?E")

  1187 proof -

  1188   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)

  1189   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)

  1190   have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"

  1191     using E1 E2 by (auto simp add: pair_algebra_def)

  1192   interpret E: sigma_algebra ?E unfolding pair_algebra_def

  1193     using E1 E2 by (intro sigma_algebra_sigma) auto

  1194   { fix A assume "A \<in> sets E1"

  1195     then have "fst - A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"

  1196       using E1 2 unfolding isoton_def pair_algebra_def by auto

  1197     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto

  1198     also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma

  1199       using 2 A \<in> sets E1

  1200       by (intro sigma_sets.Union)

  1201          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1202     finally have "fst - A \<inter> space ?E \<in> sets ?E" . }

  1203   moreover

  1204   { fix B assume "B \<in> sets E2"

  1205     then have "snd - B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"

  1206       using E2 1 unfolding isoton_def pair_algebra_def by auto

  1207     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto

  1208     also have "\<dots> \<in> sets ?E"

  1209       using 1 B \<in> sets E2 unfolding pair_algebra_def sets_sigma

  1210       by (intro sigma_sets.Union)

  1211          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1212     finally have "snd - B \<inter> space ?E \<in> sets ?E" . }

  1213   ultimately have proj:

  1214     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"

  1215     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)

  1216                    (auto simp: pair_algebra_def sets_sigma)

  1217   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"

  1218     with proj have "fst - A \<inter> space ?E \<in> sets ?E" "snd - B \<inter> space ?E \<in> sets ?E"

  1219       unfolding measurable_def by simp_all

  1220     moreover have "A \<times> B = (fst - A \<inter> space ?E) \<inter> (snd - B \<inter> space ?E)"

  1221       using A B M1.sets_into_space M2.sets_into_space

  1222       by (auto simp: pair_algebra_def)

  1223     ultimately have "A \<times> B \<in> sets ?E" by auto }

  1224   then have "sigma_sets (space ?E) (sets (pair_algebra (sigma E1) (sigma E2))) \<subseteq> sets ?E"

  1225     by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma)

  1226   then have subset: "sets ?S \<subseteq> sets ?E"

  1227     by (simp add: sets_sigma pair_algebra_def)

  1228   have "sets ?S = sets ?E"

  1229   proof (intro set_eqI iffI)

  1230     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"

  1231       unfolding sets_sigma

  1232     proof induct

  1233       case (Basic A) then show ?case

  1234         by (auto simp: pair_algebra_def sets_sigma intro: sigma_sets.Basic)

  1235     qed (auto intro: sigma_sets.intros simp: pair_algebra_def)

  1236   next

  1237     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto

  1238   qed

  1239   then show ?thesis

  1240     by (simp add: pair_algebra_def sigma_def)

  1241 qed

  1242

  1243 lemma sigma_product_algebra_sigma_eq:

  1244   assumes "finite I"

  1245   assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))"

  1246   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"

  1247   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"

  1248   shows "sigma (product_algebra (\<lambda>i. sigma (E i)) I) = sigma (product_algebra E I)"

  1249     (is "?S = ?E")

  1250 proof cases

  1251   assume "I = {}" then show ?thesis by (simp add: product_algebra_def)

  1252 next

  1253   assume "I \<noteq> {}"

  1254   interpret E: sigma_algebra "sigma (E i)" for i

  1255     using E by (rule sigma_algebra_sigma)

  1256

  1257   have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)"

  1258     using E by auto

  1259

  1260   interpret G: sigma_algebra ?E

  1261     unfolding product_algebra_def using E

  1262     by (intro sigma_algebra_sigma) (auto dest: Pi_mem)

  1263

  1264   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"

  1265     then have "(\<lambda>x. x i) - A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E"

  1266       using isotone unfolding isoton_def product_algebra_def by (auto dest: Pi_mem)

  1267     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"

  1268       unfolding product_algebra_def

  1269       apply simp

  1270       apply (subst Pi_UN[OF finite I])

  1271       using isotone[THEN isoton_mono_le] apply simp

  1272       apply (simp add: PiE_Int)

  1273       apply (intro PiE_cong)

  1274       using A sets_into by (auto intro!: into_space)

  1275     also have "\<dots> \<in> sets ?E" unfolding product_algebra_def sets_sigma

  1276       using sets_into A \<in> sets (E i)

  1277       by (intro sigma_sets.Union)

  1278          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1279     finally have "(\<lambda>x. x i) - A \<inter> space ?E \<in> sets ?E" . }

  1280   then have proj:

  1281     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"

  1282     using E by (subst G.measurable_iff_sigma)

  1283                (auto simp: product_algebra_def sets_sigma)

  1284

  1285   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"

  1286     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) - (A i) \<inter> space ?E \<in> sets ?E"

  1287       unfolding measurable_def by simp

  1288     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) - (A i) \<inter> space ?E)"

  1289       using A E.sets_into_space I \<noteq> {} unfolding product_algebra_def by auto blast

  1290     then have "Pi\<^isub>E I A \<in> sets ?E"

  1291       using G.finite_INT[OF finite I I \<noteq> {} basic, of "\<lambda>i. i"] by simp }

  1292   then have "sigma_sets (space ?E) (sets (product_algebra (\<lambda>i. sigma (E i)) I)) \<subseteq> sets ?E"

  1293     by (intro G.sigma_sets_subset) (auto simp add: sets_sigma product_algebra_def)

  1294   then have subset: "sets ?S \<subseteq> sets ?E"

  1295     by (simp add: sets_sigma product_algebra_def)

  1296

  1297   have "sets ?S = sets ?E"

  1298   proof (intro set_eqI iffI)

  1299     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"

  1300       unfolding sets_sigma

  1301     proof induct

  1302       case (Basic A) then show ?case

  1303         by (auto simp: sets_sigma product_algebra_def intro: sigma_sets.Basic)

  1304     qed (auto intro: sigma_sets.intros simp: product_algebra_def)

  1305   next

  1306     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto

  1307   qed

  1308   then show ?thesis

  1309     by (simp add: product_algebra_def sigma_def)

  1310 qed

  1311

  1312 lemma (in product_sigma_algebra) sigma_pair_algebra_sigma_eq:

  1313   "sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))) =

  1314    sigma (pair_algebra (product_algebra M I) (product_algebra M J))"

  1315   using M.sets_into_space

  1316   by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)", of _ "\<lambda>_. \<Pi>\<^isub>E i\<in>J. space (M i)"])

  1317      (auto simp: isoton_const product_algebra_def, blast+)

  1318

  1319 lemma (in product_sigma_algebra) sigma_pair_algebra_product_singleton:

  1320   "sigma (pair_algebra (sigma (product_algebra M I)) (M i)) =

  1321    sigma (pair_algebra (product_algebra M I) (M i))"

  1322   using M.sets_into_space apply (subst M.sigma_eq[symmetric])

  1323   by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)" _ "\<lambda>_. space (M i)"])

  1324      (auto simp: isoton_const product_algebra_def, blast+)

  1325

  1326 lemma (in product_sigma_algebra) measurable_restrict:

  1327   assumes [simp]: "I \<inter> J = {}"

  1328   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable

  1329     (sigma (product_algebra M (I \<union> J)))

  1330     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"

  1331   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space

  1332   by (intro measurable_sigma_sigma measurable_restrict_on_generating

  1333             pair_algebra_sets_into_space product_algebra_sets_into_space)

  1334      auto

  1335

  1336 lemma (in product_sigma_algebra) measurable_merge:

  1337   assumes [simp]: "I \<inter> J = {}"

  1338   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable

  1339     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))

  1340     (sigma (product_algebra M (I \<union> J)))"

  1341   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space

  1342   by (intro measurable_sigma_sigma measurable_merge_on_generating

  1343             pair_algebra_sets_into_space product_algebra_sets_into_space)

  1344      auto

  1345

  1346 lemma (in product_sigma_algebra) pair_product_product_vimage_algebra:

  1347   assumes [simp]: "I \<inter> J = {}"

  1348   shows "sigma_algebra.vimage_algebra (sigma (product_algebra M (I \<union> J)))

  1349     (space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) (\<lambda>(x,y). merge I x J y) =

  1350     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"

  1351   unfolding sigma_pair_algebra_sigma_eq using sets_into_space

  1352   by (intro vimage_algebra_sigma[OF bij_inv_restrict_merge[symmetric]]

  1353             pair_algebra_sets_into_space product_algebra_sets_into_space

  1354             measurable_merge_on_generating measurable_restrict_on_generating)

  1355      auto

  1356

  1357 lemma (in product_sigma_algebra) measurable_restrict_iff:

  1358   assumes IJ[simp]: "I \<inter> J = {}"

  1359   shows "f \<in> measurable (sigma (pair_algebra

  1360       (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M' \<longleftrightarrow>

  1361     (\<lambda>x. f (restrict x I, restrict x J)) \<in> measurable (sigma (product_algebra M (I \<union> J))) M'"

  1362   using M.sets_into_space

  1363   apply (subst pair_product_product_vimage_algebra[OF IJ, symmetric])

  1364   apply (subst sigma_pair_algebra_sigma_eq)

  1365   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _

  1366       bij_inv_restrict_merge[symmetric]])

  1367   apply (intro sigma_algebra_sigma product_algebra_sets_into_space)

  1368   by auto

  1369

  1370 lemma (in product_sigma_algebra) measurable_merge_iff:

  1371   assumes IJ: "I \<inter> J = {}"

  1372   shows "f \<in> measurable (sigma (product_algebra M (I \<union> J))) M' \<longleftrightarrow>

  1373     (\<lambda>(x, y). f (merge I x J y)) \<in>

  1374       measurable (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M'"

  1375   unfolding measurable_restrict_iff[OF IJ]

  1376   by (rule measurable_cong) (auto intro!: arg_cong[where f=f] simp: extensional_restrict)

  1377

  1378 lemma (in product_sigma_algebra) measurable_component:

  1379   assumes "i \<in> I" and f: "f \<in> measurable (M i) M'"

  1380   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M'"

  1381     (is "?f \<in> measurable ?P M'")

  1382 proof -

  1383   have "f \<circ> (\<lambda>x. x i) \<in> measurable ?P M'"

  1384     apply (rule measurable_comp[OF _ f])

  1385     using measurable_up_sigma[of "product_algebra M I" "M i"]

  1386     using measurable_component_on_generator[OF i \<in> I]

  1387     by auto

  1388   then show "?f \<in> measurable ?P M'" by (simp add: comp_def)

  1389 qed

  1390

  1391 lemma (in product_sigma_algebra) singleton_vimage_algebra:

  1392   "sigma_algebra.vimage_algebra (sigma (product_algebra M {i})) (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i"

  1393   using sets_into_space

  1394   by (intro vimage_algebra_sigma[of "M i", unfolded M.sigma_eq, OF bij_inv_singleton[symmetric]]

  1395             product_algebra_sets_into_space measurable_singleton_on_generator measurable_component_on_generator)

  1396      auto

  1397

  1398 lemma (in product_sigma_algebra) measurable_component_singleton:

  1399   "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>

  1400     f \<in> measurable (M i) M'"

  1401   using sets_into_space

  1402   apply (subst singleton_vimage_algebra[symmetric])

  1403   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _ bij_inv_singleton[symmetric]])

  1404   by (auto intro!: sigma_algebra_sigma product_algebra_sets_into_space)

  1405

  1406 lemma (in product_sigma_algebra) measurable_component_iff:

  1407   assumes "i \<in> I" and not_empty: "\<forall>i\<in>I. space (M i) \<noteq> {}"

  1408   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M' \<longleftrightarrow>

  1409     f \<in> measurable (M i) M'"

  1410     (is "?f \<in> measurable ?P M' \<longleftrightarrow> _")

  1411 proof

  1412   assume "f \<in> measurable (M i) M'" then show "?f \<in> measurable ?P M'"

  1413     by (rule measurable_component[OF i \<in> I])

  1414 next

  1415   assume f: "?f \<in> measurable ?P M'"

  1416   def t \<equiv> "\<lambda>i. SOME x. x \<in> space (M i)"

  1417   have t: "\<And>i. i\<in>I \<Longrightarrow> t i \<in> space (M i)"

  1418      unfolding t_def using not_empty by (rule_tac someI_ex) auto

  1419   have "?f \<circ> (\<lambda>x. (\<lambda>j\<in>I. if j = i then x else t j)) \<in> measurable (M i) M'"

  1420     (is "?f \<circ> ?t \<in> measurable _ _")

  1421   proof (rule measurable_comp[OF _ f])

  1422     have "?t \<in> measurable (M i) (product_algebra M I)"

  1423     proof (unfold measurable_def, intro CollectI conjI ballI)

  1424       from t show "?t \<in> space (M i) \<rightarrow> (space (product_algebra M I))" by auto

  1425     next

  1426       fix A assume A: "A \<in> sets (product_algebra M I)"

  1427       { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1428         then have "?t - Pi\<^isub>E I E \<inter> space (M i) = (if (\<forall>j\<in>I-{i}. t j \<in> E j) then E i else {})"

  1429           using i \<in> I sets_into_space by (auto dest: Pi_mem[where B=E]) }

  1430       note * = this

  1431       with A i \<in> I show "?t - A \<inter> space (M i) \<in> sets (M i)"

  1432         by (auto elim!: product_algebraE simp del: vimage_Int)

  1433     qed

  1434     also have "\<dots> \<subseteq> measurable (M i) (sigma (product_algebra M I))"

  1435       using M.sets_into_space by (intro measurable_subset) (auto simp: product_algebra_def, blast)

  1436     finally show "?t \<in> measurable (M i) (sigma (product_algebra M I))" .

  1437   qed

  1438   then show "f \<in> measurable (M i) M'" unfolding comp_def using i \<in> I by simp

  1439 qed

  1440

  1441 lemma (in product_sigma_algebra) measurable_singleton:

  1442   shows "f \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>

  1443     (\<lambda>x. f (\<lambda>j\<in>{i}. x)) \<in> measurable (M i) M'"

  1444   using sets_into_space unfolding measurable_component_singleton[symmetric]

  1445   by (auto intro!: measurable_cong arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1446

  1447

  1448 lemma (in pair_sigma_algebra) measurable_pair_split:

  1449   assumes "sigma_algebra M1'" "sigma_algebra M2'"

  1450   assumes f: "f \<in> measurable M1 M1'" and g: "g \<in> measurable M2 M2'"

  1451   shows "(\<lambda>(x, y). (f x, g y)) \<in> measurable P (sigma (pair_algebra M1' M2'))"

  1452 proof (rule measurable_sigma)

  1453   interpret M1': sigma_algebra M1' by fact

  1454   interpret M2': sigma_algebra M2' by fact

  1455   interpret Q: pair_sigma_algebra M1' M2' by default

  1456   show "sets Q.E \<subseteq> Pow (space Q.E)"

  1457     using M1'.sets_into_space M2'.sets_into_space by (auto simp: pair_algebra_def)

  1458   show "(\<lambda>(x, y). (f x, g y)) \<in> space P \<rightarrow> space Q.E"

  1459     using f g unfolding measurable_def pair_algebra_def by auto

  1460   fix A assume "A \<in> sets Q.E"

  1461   then obtain X Y where A: "A = X \<times> Y" "X \<in> sets M1'" "Y \<in> sets M2'"

  1462     unfolding pair_algebra_def by auto

  1463   then have *: "(\<lambda>(x, y). (f x, g y)) - A \<inter> space P =

  1464       (f - X \<inter> space M1) \<times> (g - Y \<inter> space M2)"

  1465     by (auto simp: pair_algebra_def)

  1466   then show "(\<lambda>(x, y). (f x, g y)) - A \<inter> space P \<in> sets P"

  1467     using f g A unfolding measurable_def *

  1468     by (auto intro!: pair_algebraI in_sigma)

  1469 qed

  1470

  1471 lemma (in product_sigma_algebra) measurable_add_dim:

  1472   assumes "i \<notin> I" "finite I"

  1473   shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (sigma (pair_algebra (sigma (product_algebra M I)) (M i)))

  1474                          (sigma (product_algebra M (insert i I)))"

  1475 proof (subst measurable_cong)

  1476   interpret I: finite_product_sigma_algebra M I by default fact

  1477   interpret i: finite_product_sigma_algebra M "{i}" by default auto

  1478   interpret Ii: pair_sigma_algebra I.P "M i" by default

  1479   interpret Ii': pair_sigma_algebra I.P i.P by default

  1480   have disj: "I \<inter> {i} = {}" using i \<notin> I by auto

  1481   have "(\<lambda>(x, y). (id x, \<lambda>_\<in>{i}. y)) \<in> measurable Ii.P Ii'.P"

  1482   proof (intro Ii.measurable_pair_split I.measurable_ident)

  1483     show "(\<lambda>y. \<lambda>_\<in>{i}. y) \<in> measurable (M i) i.P"

  1484       apply (rule measurable_singleton[THEN iffD1])

  1485       using i.measurable_ident unfolding id_def .

  1486   qed default

  1487   from measurable_comp[OF this measurable_merge[OF disj]]

  1488   show "(\<lambda>(x, y). merge I x {i} y) \<circ> (\<lambda>(x, y). (id x, \<lambda>_\<in>{i}. y))

  1489     \<in> measurable (sigma (pair_algebra I.P (M i))) (sigma (product_algebra M (insert i I)))"

  1490     (is "?f \<in> _") by simp

  1491   fix x assume "x \<in> space Ii.P"

  1492   with assms show "(\<lambda>(f, y). f(i := y)) x = ?f x"

  1493     by (cases x) (simp add: merge_def fun_eq_iff pair_algebra_def extensional_def)

  1494 qed

  1495

  1496 locale product_sigma_finite =

  1497   fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pextreal"

  1498   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i) (\<mu> i)"

  1499

  1500 locale finite_product_sigma_finite = product_sigma_finite M \<mu> for M :: "'i \<Rightarrow> 'a algebra" and \<mu> +

  1501   fixes I :: "'i set" assumes finite_index': "finite I"

  1502

  1503 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" "\<mu> i" for i

  1504   by (rule sigma_finite_measures)

  1505

  1506 sublocale product_sigma_finite \<subseteq> product_sigma_algebra

  1507   by default

  1508

  1509 sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra

  1510   by default (fact finite_index')

  1511

  1512 lemma (in finite_product_sigma_finite) sigma_finite_pairs:

  1513   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.

  1514     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>

  1515     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<omega>) \<and>

  1516     (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<up> space G"

  1517 proof -

  1518   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> F \<up> space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<omega>)"

  1519     using M.sigma_finite_up by simp

  1520   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..

  1521   then have "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. F i \<up> space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<omega>"

  1522     by auto

  1523   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"

  1524   note space_product_algebra[simp]

  1525   show ?thesis

  1526   proof (intro exI[of _ F] conjI allI isotoneI set_eqI iffI ballI)

  1527     fix i show "range (F i) \<subseteq> sets (M i)" by fact

  1528   next

  1529     fix i k show "\<mu> i (F i k) \<noteq> \<omega>" by fact

  1530   next

  1531     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"

  1532       using \<And>i. range (F i) \<subseteq> sets (M i) M.sets_into_space by auto blast

  1533   next

  1534     fix f assume "f \<in> space G"

  1535     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"]

  1536       \<And>i. F i \<up> space (M i)[THEN isotonD(2)]

  1537       \<And>i. F i \<up> space (M i)[THEN isoton_mono_le]

  1538     show "f \<in> (\<Union>i. ?F i)" by auto

  1539   next

  1540     fix i show "?F i \<subseteq> ?F (Suc i)"

  1541       using \<And>i. F i \<up> space (M i)[THEN isotonD(1)] by auto

  1542   qed

  1543 qed

  1544

  1545 lemma (in product_sigma_finite) product_measure_exists:

  1546   assumes "finite I"

  1547   shows "\<exists>\<nu>. (\<forall>A\<in>(\<Pi> i\<in>I. sets (M i)). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and>

  1548      sigma_finite_measure (sigma (product_algebra M I)) \<nu>"

  1549 using finite I proof induct

  1550   case empty then show ?case unfolding product_algebra_def

  1551     by (auto intro!: exI[of _ "\<lambda>A. if A = {} then 0 else 1"] sigma_algebra_sigma

  1552                      sigma_algebra.finite_additivity_sufficient

  1553              simp add: positive_def additive_def sets_sigma sigma_finite_measure_def

  1554                        sigma_finite_measure_axioms_def image_constant)

  1555 next

  1556   case (insert i I)

  1557   interpret finite_product_sigma_finite M \<mu> I by default fact

  1558   have "finite (insert i I)" using finite I by auto

  1559   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default fact

  1560   from insert obtain \<nu> where

  1561     prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" and

  1562     "sigma_finite_measure P \<nu>" by auto

  1563   interpret I: sigma_finite_measure P \<nu> by fact

  1564   interpret P: pair_sigma_finite P \<nu> "M i" "\<mu> i" ..

  1565

  1566   let ?h = "(\<lambda>(f, y). f(i := y))"

  1567   let ?\<nu> = "\<lambda>A. P.pair_measure (?h - A \<inter> space P.P)"

  1568   have I': "sigma_algebra I'.P" by default

  1569   interpret I': measure_space "sigma (product_algebra M (insert i I))" ?\<nu>

  1570     apply (rule P.measure_space_vimage[OF I'])

  1571     apply (rule measurable_add_dim[OF i \<notin> I finite I])

  1572     by simp

  1573   show ?case

  1574   proof (intro exI[of _ ?\<nu>] conjI ballI)

  1575     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"

  1576       then have *: "?h - Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"

  1577         using i \<notin> I M.sets_into_space by (auto simp: pair_algebra_def) blast

  1578       show "?\<nu> (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. \<mu> i (A i))"

  1579         unfolding * using A

  1580         apply (subst P.pair_measure_times)

  1581         using A apply fastsimp

  1582         using A apply fastsimp

  1583         using i \<notin> I finite I prod[of A] A by (auto simp: ac_simps) }

  1584     note product = this

  1585     show "sigma_finite_measure I'.P ?\<nu>"

  1586     proof

  1587       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..

  1588       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"

  1589         "(\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) \<up> space I'.G"

  1590         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<omega>"

  1591         by blast+

  1592       let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"

  1593       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>

  1594           (\<Union>i. F i) = space I'.P \<and> (\<forall>i. ?\<nu> (F i) \<noteq> \<omega>)"

  1595       proof (intro exI[of _ ?F] conjI allI)

  1596         show "range ?F \<subseteq> sets I'.P" using F(1) by auto

  1597       next

  1598         from F(2)[THEN isotonD(2)]

  1599         show "(\<Union>i. ?F i) = space I'.P" by simp

  1600       next

  1601         fix j

  1602         show "?\<nu> (?F j) \<noteq> \<omega>"

  1603           using F finite I

  1604           by (subst product) (auto simp: setprod_\<omega>)

  1605       qed

  1606     qed

  1607   qed

  1608 qed

  1609

  1610 definition (in finite_product_sigma_finite)

  1611   measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pextreal" where

  1612   "measure = (SOME \<nu>.

  1613      (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and>

  1614      sigma_finite_measure P \<nu>)"

  1615

  1616 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P measure

  1617 proof -

  1618   show "sigma_finite_measure P measure"

  1619     unfolding measure_def

  1620     by (rule someI2_ex[OF product_measure_exists[OF finite_index]]) auto

  1621 qed

  1622

  1623 lemma (in finite_product_sigma_finite) measure_times:

  1624   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"

  1625   shows "measure (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))"

  1626 proof -

  1627   note ex = product_measure_exists[OF finite_index]

  1628   show ?thesis

  1629     unfolding measure_def

  1630   proof (rule someI2_ex[OF ex], elim conjE)

  1631     fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))"

  1632     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)

  1633     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp

  1634     also have "\<dots> = (\<Prod>i\<in>I. \<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto

  1635     finally show "\<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" by simp

  1636   qed

  1637 qed

  1638

  1639 abbreviation (in product_sigma_finite)

  1640   "product_measure I \<equiv> finite_product_sigma_finite.measure M \<mu> I"

  1641

  1642 abbreviation (in product_sigma_finite)

  1643   "product_positive_integral I \<equiv>

  1644     measure_space.positive_integral (sigma (product_algebra M I)) (product_measure I)"

  1645

  1646 abbreviation (in product_sigma_finite)

  1647   "product_integral I \<equiv>

  1648     measure_space.integral (sigma (product_algebra M I)) (product_measure I)"

  1649

  1650 abbreviation (in product_sigma_finite)

  1651   "product_integrable I \<equiv>

  1652     measure_space.integrable (sigma (product_algebra M I)) (product_measure I)"

  1653

  1654 lemma (in product_sigma_finite) product_measure_empty[simp]:

  1655   "product_measure {} {\<lambda>x. undefined} = 1"

  1656 proof -

  1657   interpret finite_product_sigma_finite M \<mu> "{}" by default auto

  1658   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp

  1659 qed

  1660

  1661 lemma (in product_sigma_finite) positive_integral_empty:

  1662   "product_positive_integral {} f = f (\<lambda>k. undefined)"

  1663 proof -

  1664   interpret finite_product_sigma_finite M \<mu> "{}" by default (fact finite.emptyI)

  1665   have "\<And>A. measure (Pi\<^isub>E {} A) = 1"

  1666     using assms by (subst measure_times) auto

  1667   then show ?thesis

  1668     unfolding positive_integral_def simple_function_def simple_integral_def_raw

  1669   proof (simp add: P_empty, intro antisym)

  1670     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))"

  1671       by (intro le_SUPI) auto

  1672     show "(SUP f:{g. g \<le> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)"

  1673       by (intro SUP_leI) (auto simp: le_fun_def)

  1674   qed

  1675 qed

  1676

  1677 lemma (in product_sigma_finite) measure_fold:

  1678   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1679   assumes A: "A \<in> sets (sigma (product_algebra M (I \<union> J)))"

  1680   shows "pair_sigma_finite.pair_measure

  1681      (sigma (product_algebra M I)) (product_measure I)

  1682      (sigma (product_algebra M J)) (product_measure J)

  1683      ((\<lambda>(x,y). merge I x J y) - A \<inter> space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) =

  1684    product_measure (I \<union> J) A"

  1685 proof -

  1686   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1687   interpret J: finite_product_sigma_finite M \<mu> J by default fact

  1688   have "finite (I \<union> J)" using fin by auto

  1689   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact

  1690   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default

  1691   let ?g = "\<lambda>(x,y). merge I x J y"

  1692   let "?X S" = "?g - S \<inter> space P.P"

  1693   from IJ.sigma_finite_pairs obtain F where

  1694     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"

  1695        "(\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) \<up> space IJ.G"

  1696        "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<omega>"

  1697     by auto

  1698   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"

  1699   show "P.pair_measure (?X A) = IJ.measure A"

  1700   proof (rule measure_unique_Int_stable[OF _ _ _ _ _ _ _ _ A])

  1701     show "Int_stable IJ.G" by (simp add: PiE_Int Int_stable_def product_algebra_def) auto

  1702     show "range ?F \<subseteq> sets IJ.G" using F by (simp add: image_subset_iff product_algebra_def)

  1703     show "?F \<up> space IJ.G " using F(2) by simp

  1704     have "sigma_algebra IJ.P" by default

  1705     then show "measure_space IJ.P (\<lambda>A. P.pair_measure (?X A))"

  1706       apply (rule P.measure_space_vimage)

  1707       apply (rule measurable_merge[OF I \<inter> J = {}])

  1708       by simp

  1709     show "measure_space IJ.P IJ.measure" by fact

  1710   next

  1711     fix A assume "A \<in> sets IJ.G"

  1712     then obtain F where A[simp]: "A = Pi\<^isub>E (I \<union> J) F"

  1713       and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"

  1714       by (auto simp: product_algebra_def)

  1715     then have "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"

  1716       using sets_into_space by (auto simp: space_pair_algebra) blast+

  1717     then have "P.pair_measure (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"

  1718       using finite J finite I F

  1719       by (simp add: P.pair_measure_times I.measure_times J.measure_times)

  1720     also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"

  1721       using finite J finite I I \<inter> J = {}  by (simp add: setprod_Un_one)

  1722     also have "\<dots> = IJ.measure A"

  1723       using finite J finite I F unfolding A

  1724       by (intro IJ.measure_times[symmetric]) auto

  1725     finally show "P.pair_measure (?X A) = IJ.measure A" .

  1726   next

  1727     fix k

  1728     have k: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i k \<in> sets (M i)" using F by auto

  1729     then have "?X (?F k) = (\<Pi>\<^isub>E i\<in>I. F i k) \<times> (\<Pi>\<^isub>E i\<in>J. F i k)"

  1730       using sets_into_space by (auto simp: space_pair_algebra) blast+

  1731     with k have "P.pair_measure (?X (?F k)) = (\<Prod>i\<in>I. \<mu> i (F i k)) * (\<Prod>i\<in>J. \<mu> i (F i k))"

  1732      by (simp add: P.pair_measure_times I.measure_times J.measure_times)

  1733     then show "P.pair_measure (?X (?F k)) \<noteq> \<omega>"

  1734       using finite I F by (simp add: setprod_\<omega>)

  1735   qed simp

  1736 qed

  1737

  1738 lemma (in product_sigma_finite) product_positive_integral_fold:

  1739   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1740   and f: "f \<in> borel_measurable (sigma (product_algebra M (I \<union> J)))"

  1741   shows "product_positive_integral (I \<union> J) f =

  1742     product_positive_integral I (\<lambda>x. product_positive_integral J (\<lambda>y. f (merge I x J y)))"

  1743 proof -

  1744   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1745   interpret J: finite_product_sigma_finite M \<mu> J by default fact

  1746   have "finite (I \<union> J)" using fin by auto

  1747   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact

  1748   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default

  1749   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"

  1750     unfolding case_prod_distrib measurable_merge_iff[OF IJ, symmetric] using f .

  1751   show ?thesis

  1752     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]

  1753     apply (subst IJ.positive_integral_cong_measure[symmetric])

  1754     apply (rule measure_fold[OF IJ fin])

  1755     apply assumption

  1756   proof (rule P.positive_integral_vimage)

  1757     show "sigma_algebra IJ.P" by default

  1758     show "(\<lambda>(x, y). merge I x J y) \<in> measurable P.P IJ.P" by (rule measurable_merge[OF IJ])

  1759     show "f \<in> borel_measurable IJ.P" using f .

  1760   qed

  1761 qed

  1762

  1763 lemma (in product_sigma_finite) product_positive_integral_singleton:

  1764   assumes f: "f \<in> borel_measurable (M i)"

  1765   shows "product_positive_integral {i} (\<lambda>x. f (x i)) = M.positive_integral i f"

  1766 proof -

  1767   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp

  1768   have T: "(\<lambda>x. x i) \<in> measurable (sigma (product_algebra M {i})) (M i)"

  1769     using measurable_component_singleton[of "\<lambda>x. x" i]

  1770           measurable_ident[unfolded id_def] by auto

  1771   show "I.positive_integral (\<lambda>x. f (x i)) = M.positive_integral i f"

  1772     unfolding I.positive_integral_vimage[OF sigma_algebras T f, symmetric]

  1773   proof (rule positive_integral_cong_measure)

  1774     fix A let ?P = "(\<lambda>x. x i) - A \<inter> space (sigma (product_algebra M {i}))"

  1775     assume A: "A \<in> sets (M i)"

  1776     then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto

  1777     show "product_measure {i} ?P = \<mu> i A" unfolding *

  1778       using A I.measure_times[of "\<lambda>_. A"] by auto

  1779   qed

  1780 qed

  1781

  1782 lemma (in product_sigma_finite) product_positive_integral_insert:

  1783   assumes [simp]: "finite I" "i \<notin> I"

  1784     and f: "f \<in> borel_measurable (sigma (product_algebra M (insert i I)))"

  1785   shows "product_positive_integral (insert i I) f

  1786     = product_positive_integral I (\<lambda>x. M.positive_integral i (\<lambda>y. f (x(i:=y))))"

  1787 proof -

  1788   interpret I: finite_product_sigma_finite M \<mu> I by default auto

  1789   interpret i: finite_product_sigma_finite M \<mu> "{i}" by default auto

  1790   interpret P: pair_sigma_algebra I.P i.P ..

  1791   have IJ: "I \<inter> {i} = {}" by auto

  1792   show ?thesis

  1793     unfolding product_positive_integral_fold[OF IJ, simplified, OF f]

  1794   proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)

  1795     fix x assume x: "x \<in> space I.P"

  1796     let "?f y" = "f (restrict (x(i := y)) (insert i I))"

  1797     have f'_eq: "\<And>y. ?f y = f (x(i := y))"

  1798       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1799     note fP = f[unfolded measurable_merge_iff[OF IJ, simplified]]

  1800     show "?f \<in> borel_measurable (M i)"

  1801       using P.measurable_pair_image_snd[OF fP x]

  1802       unfolding measurable_singleton f'_eq by (simp add: f'_eq)

  1803     show "M.positive_integral i ?f = M.positive_integral i (\<lambda>y. f (x(i := y)))"

  1804       unfolding f'_eq by simp

  1805   qed

  1806 qed

  1807

  1808 lemma (in product_sigma_finite) product_positive_integral_setprod:

  1809   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> pextreal"

  1810   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"

  1811   shows "product_positive_integral I (\<lambda>x. (\<Prod>i\<in>I. f i (x i))) =

  1812     (\<Prod>i\<in>I. M.positive_integral i (f i))"

  1813 using assms proof induct

  1814   case empty

  1815   interpret finite_product_sigma_finite M \<mu> "{}" by default auto

  1816   then show ?case by simp

  1817 next

  1818   case (insert i I)

  1819   note finite I[intro, simp]

  1820   interpret I: finite_product_sigma_finite M \<mu> I by default auto

  1821   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))"

  1822     using insert by (auto intro!: setprod_cong)

  1823   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>

  1824     (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (sigma (product_algebra M J))"

  1825     using sets_into_space insert

  1826     by (intro sigma_algebra.borel_measurable_pextreal_setprod

  1827               sigma_algebra_sigma product_algebra_sets_into_space

  1828               measurable_component)

  1829        auto

  1830   show ?case

  1831     by (simp add: product_positive_integral_insert[OF insert(1,2) prod])

  1832        (simp add: insert I.positive_integral_cmult M.positive_integral_multc * prod subset_insertI)

  1833 qed

  1834

  1835 lemma (in product_sigma_finite) product_integral_singleton:

  1836   assumes f: "f \<in> borel_measurable (M i)"

  1837   shows "product_integral {i} (\<lambda>x. f (x i)) = M.integral i f"

  1838 proof -

  1839   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp

  1840   have *: "(\<lambda>x. Real (f x)) \<in> borel_measurable (M i)"

  1841     "(\<lambda>x. Real (- f x)) \<in> borel_measurable (M i)"

  1842     using assms by auto

  1843   show ?thesis

  1844     unfolding I.integral_def integral_def

  1845     unfolding *[THEN product_positive_integral_singleton] ..

  1846 qed

  1847

  1848 lemma (in product_sigma_finite) product_integral_fold:

  1849   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1850   and f: "measure_space.integrable (sigma (product_algebra M (I \<union> J))) (product_measure (I \<union> J)) f"

  1851   shows "product_integral (I \<union> J) f =

  1852     product_integral I (\<lambda>x. product_integral J (\<lambda>y. f (merge I x J y)))"

  1853 proof -

  1854   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1855   interpret J: finite_product_sigma_finite M \<mu> J by default fact

  1856   have "finite (I \<union> J)" using fin by auto

  1857   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact

  1858   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default

  1859   let ?f = "\<lambda>(x,y). f (merge I x J y)"

  1860   have f_borel: "f \<in> borel_measurable IJ.P"

  1861      "(\<lambda>x. Real (f x)) \<in> borel_measurable IJ.P"

  1862      "(\<lambda>x. Real (- f x)) \<in> borel_measurable IJ.P"

  1863     using f unfolding integrable_def by auto

  1864   have f_restrict: "(\<lambda>x. f (restrict x (I \<union> J))) \<in> borel_measurable IJ.P"

  1865     by (rule measurable_cong[THEN iffD2, OF _ f_borel(1)])

  1866        (auto intro!: arg_cong[where f=f] simp: extensional_restrict)

  1867   then have f'_borel:

  1868     "(\<lambda>x. Real (?f x)) \<in> borel_measurable P.P"

  1869     "(\<lambda>x. Real (- ?f x)) \<in> borel_measurable P.P"

  1870     unfolding measurable_restrict_iff[OF IJ]

  1871     by simp_all

  1872   have PI:

  1873     "P.positive_integral (\<lambda>x. Real (?f x)) = IJ.positive_integral (\<lambda>x. Real (f x))"

  1874     "P.positive_integral (\<lambda>x. Real (- ?f x)) = IJ.positive_integral (\<lambda>x. Real (- f x))"

  1875     using f'_borel[THEN P.positive_integral_fst_measurable(2)]

  1876     using f_borel(2,3)[THEN product_positive_integral_fold[OF assms(1-3)]]

  1877     by simp_all

  1878   have "P.integrable ?f" using IJ.integrable f

  1879     unfolding P.integrable_def IJ.integrable_def

  1880     unfolding measurable_restrict_iff[OF IJ]

  1881     using f_restrict PI by simp_all

  1882   show ?thesis

  1883     unfolding P.integrable_fst_measurable(2)[OF P.integrable ?f, simplified]

  1884     unfolding IJ.integral_def P.integral_def

  1885     unfolding PI by simp

  1886 qed

  1887

  1888 lemma (in product_sigma_finite) product_integral_insert:

  1889   assumes [simp]: "finite I" "i \<notin> I"

  1890     and f: "measure_space.integrable (sigma (product_algebra M (insert i I))) (product_measure (insert i I)) f"

  1891   shows "product_integral (insert i I) f

  1892     = product_integral I (\<lambda>x. M.integral i (\<lambda>y. f (x(i:=y))))"

  1893 proof -

  1894   interpret I: finite_product_sigma_finite M \<mu> I by default auto

  1895   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default auto

  1896   interpret i: finite_product_sigma_finite M \<mu> "{i}" by default auto

  1897   interpret P: pair_sigma_algebra I.P i.P ..

  1898   have IJ: "I \<inter> {i} = {}" by auto

  1899   show ?thesis

  1900     unfolding product_integral_fold[OF IJ, simplified, OF f]

  1901   proof (rule I.integral_cong, subst product_integral_singleton)

  1902     fix x assume x: "x \<in> space I.P"

  1903     let "?f y" = "f (restrict (x(i := y)) (insert i I))"

  1904     have f'_eq: "\<And>y. ?f y = f (x(i := y))"

  1905       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1906     have "f \<in> borel_measurable I'.P" using f unfolding I'.integrable_def by auto

  1907     note fP = this[unfolded measurable_merge_iff[OF IJ, simplified]]

  1908     show "?f \<in> borel_measurable (M i)"

  1909       using P.measurable_pair_image_snd[OF fP x]

  1910       unfolding measurable_singleton f'_eq by (simp add: f'_eq)

  1911     show "M.integral i ?f = M.integral i (\<lambda>y. f (x(i := y)))"

  1912       unfolding f'_eq by simp

  1913   qed

  1914 qed

  1915

  1916 lemma (in product_sigma_finite) product_integrable_setprod:

  1917   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"

  1918   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> M.integrable i (f i)"

  1919   shows "product_integrable I (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "product_integrable I ?f")

  1920 proof -

  1921   interpret finite_product_sigma_finite M \<mu> I by default fact

  1922   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"

  1923     using integrable unfolding M.integrable_def by auto

  1924   then have borel: "?f \<in> borel_measurable P"

  1925     by (intro borel_measurable_setprod measurable_component) auto

  1926   moreover have "integrable (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"

  1927   proof (unfold integrable_def, intro conjI)

  1928     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"

  1929       using borel by auto

  1930     have "positive_integral (\<lambda>x. Real (abs (?f x))) = positive_integral (\<lambda>x. \<Prod>i\<in>I. Real (abs (f i (x i))))"

  1931       by (simp add: Real_setprod abs_setprod)

  1932     also have "\<dots> = (\<Prod>i\<in>I. M.positive_integral i (\<lambda>x. Real (abs (f i x))))"

  1933       using f by (subst product_positive_integral_setprod) auto

  1934     also have "\<dots> < \<omega>"

  1935       using integrable[THEN M.integrable_abs]

  1936       unfolding pextreal_less_\<omega> setprod_\<omega> M.integrable_def by simp

  1937     finally show "positive_integral (\<lambda>x. Real (abs (?f x))) \<noteq> \<omega>" by auto

  1938     show "positive_integral (\<lambda>x. Real (- abs (?f x))) \<noteq> \<omega>" by simp

  1939   qed

  1940   ultimately show ?thesis

  1941     by (rule integrable_abs_iff[THEN iffD1])

  1942 qed

  1943

  1944 lemma (in product_sigma_finite) product_integral_setprod:

  1945   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"

  1946   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> M.integrable i (f i)"

  1947   shows "product_integral I (\<lambda>x. (\<Prod>i\<in>I. f i (x i))) = (\<Prod>i\<in>I. M.integral i (f i))"

  1948 using assms proof (induct rule: finite_ne_induct)

  1949   case (singleton i)

  1950   then show ?case by (simp add: product_integral_singleton integrable_def)

  1951 next

  1952   case (insert i I)

  1953   then have iI: "finite (insert i I)" by auto

  1954   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>

  1955     product_integrable J (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"

  1956     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)

  1957   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1958   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))"

  1959     using i \<notin> I by (auto intro!: setprod_cong)

  1960   show ?case

  1961     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]

  1962     by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)

  1963 qed

  1964

  1965 section "Products on finite spaces"

  1966

  1967 lemma sigma_sets_pair_algebra_finite:

  1968   assumes "finite A" and "finite B"

  1969   shows "sigma_sets (A \<times> B) ((\<lambda>(x,y). x \<times> y)  (Pow A \<times> Pow B)) = Pow (A \<times> B)"

  1970   (is "sigma_sets ?prod ?sets = _")

  1971 proof safe

  1972   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)

  1973   fix x assume subset: "x \<subseteq> A \<times> B"

  1974   hence "finite x" using fin by (rule finite_subset)

  1975   from this subset show "x \<in> sigma_sets ?prod ?sets"

  1976   proof (induct x)

  1977     case empty show ?case by (rule sigma_sets.Empty)

  1978   next

  1979     case (insert a x)

  1980     hence "{a} \<in> sigma_sets ?prod ?sets"

  1981       by (auto simp: pair_algebra_def intro!: sigma_sets.Basic)

  1982     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto

  1983     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)

  1984   qed

  1985 next

  1986   fix x a b

  1987   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"

  1988   from sigma_sets_into_sp[OF _ this(1)] this(2)

  1989   show "a \<in> A" and "b \<in> B" by auto

  1990 qed

  1991

  1992 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2

  1993

  1994 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default

  1995

  1996 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra[simp]:

  1997   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"

  1998 proof -

  1999   show ?thesis using M1.finite_space M2.finite_space

  2000     by (simp add: sigma_def space_pair_algebra sets_pair_algebra

  2001                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)

  2002 qed

  2003

  2004 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P

  2005 proof

  2006   show "finite (space P)" "sets P = Pow (space P)"

  2007     using M1.finite_space M2.finite_space by auto

  2008 qed

  2009

  2010 locale pair_finite_space = M1: finite_measure_space M1 \<mu>1 + M2: finite_measure_space M2 \<mu>2

  2011   for M1 \<mu>1 M2 \<mu>2

  2012

  2013 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra

  2014   by default

  2015

  2016 sublocale pair_finite_space \<subseteq> pair_sigma_finite

  2017   by default

  2018

  2019 lemma (in pair_finite_space) finite_pair_sigma_algebra[simp]:

  2020   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"

  2021 proof -

  2022   show ?thesis using M1.finite_space M2.finite_space

  2023     by (simp add: sigma_def space_pair_algebra sets_pair_algebra

  2024                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)

  2025 qed

  2026

  2027 lemma (in pair_finite_space) pair_measure_Pair[simp]:

  2028   assumes "a \<in> space M1" "b \<in> space M2"

  2029   shows "pair_measure {(a, b)} = \<mu>1 {a} * \<mu>2 {b}"

  2030 proof -

  2031   have "pair_measure ({a}\<times>{b}) = \<mu>1 {a} * \<mu>2 {b}"

  2032     using M1.sets_eq_Pow M2.sets_eq_Pow assms

  2033     by (subst pair_measure_times) auto

  2034   then show ?thesis by simp

  2035 qed

  2036

  2037 lemma (in pair_finite_space) pair_measure_singleton[simp]:

  2038   assumes "x \<in> space M1 \<times> space M2"

  2039   shows "pair_measure {x} = \<mu>1 {fst x} * \<mu>2 {snd x}"

  2040   using pair_measure_Pair assms by (cases x) auto

  2041

  2042 sublocale pair_finite_space \<subseteq> finite_measure_space P pair_measure

  2043   by default auto

  2044

  2045 lemma (in pair_finite_space) finite_measure_space_finite_prod_measure_alterantive:

  2046   "finite_measure_space \<lparr>space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2)\<rparr> pair_measure"

  2047   unfolding finite_pair_sigma_algebra[symmetric]

  2048   by default

  2049

  2050 end`