src/HOL/Probability/Product_Measure.thy
 author hoelzl Tue Jan 18 21:37:23 2011 +0100 (2011-01-18) changeset 41654 32fe42892983 parent 41544 c3b977fee8a3 child 41659 a5d1b2df5e97 permissions -rw-r--r--
Gauge measure removed
     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 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap:

   527   "sigma (pair_algebra M2 M1) = (vimage_algebra (space M2 \<times> space M1) (\<lambda>(x, y). (y, x)))"

   528   unfolding vimage_algebra_def

   529   apply (simp add: sets_sigma)

   530   apply (subst sigma_sets_vimage[symmetric])

   531   apply (fastsimp simp: pair_algebra_def)

   532   using M1.sets_into_space M2.sets_into_space apply (fastsimp simp: pair_algebra_def)

   533 proof -

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

   535     = sets (pair_algebra M2 M1)" (is "?S = _")

   536     by (rule pair_algebra_swap)

   537   then show "sigma (pair_algebra M2 M1) = \<lparr>space = space M2 \<times> space M1,

   538        sets = sigma_sets (space M2 \<times> space M1) ?S\<rparr>"

   539     by (simp add: pair_algebra_def sigma_def)

   540 qed

   541

   542 definition (in pair_sigma_finite)

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

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

   545

   546 lemma (in pair_sigma_finite) pair_measure_alt:

   547   assumes "A \<in> sets P"

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

   549   unfolding pair_measure_def

   550 proof (rule M1.positive_integral_cong)

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

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

   553     unfolding indicator_def by auto

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

   555     unfolding *

   556     apply (subst M2.positive_integral_indicator)

   557     apply (rule measurable_cut_fst[OF assms])

   558     by simp

   559 qed

   560

   561 lemma (in pair_sigma_finite) pair_measure_times:

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

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

   564 proof -

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

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

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

   568   with assms show ?thesis

   569     by (simp add: M1.positive_integral_cmult_indicator ac_simps)

   570 qed

   571

   572 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_algebra:

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

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

   575 proof -

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

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

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

   579     using M1.sigma_finite_up M2.sigma_finite_up by auto

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

   581     unfolding isoton_def by auto

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

   583   show ?thesis unfolding isoton_def space_pair_algebra

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

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

   586       by (fastsimp intro!: pair_algebraI)

   587   next

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

   589     proof (intro subsetI)

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

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

   592         by (auto simp: space)

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

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

   595         by (auto simp: max_def dest: isoton_mono_le)

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

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

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

   599     qed

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

   601       using space by (auto simp: space)

   602   next

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

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

   605       by auto

   606   next

   607     fix i

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

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

   610       by (simp add: pair_measure_times)

   611   qed

   612 qed

   613

   614 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P pair_measure

   615 proof

   616   show "pair_measure {} = 0"

   617     unfolding pair_measure_def by auto

   618

   619   show "countably_additive P pair_measure"

   620     unfolding countably_additive_def

   621   proof (intro allI impI)

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

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

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

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

   626       by (intro measure_cut_measurable_fst) auto

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

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

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

   630       using F by (auto intro!: measurable_cut_fst)

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

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

   633                     M2.measure_countably_additive

   634                cong: M1.positive_integral_cong)

   635   qed

   636

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

   638   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>)"

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

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

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

   642       using F by (auto simp: space_pair_algebra isoton_def)

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

   644   qed

   645 qed

   646

   647 lemma (in pair_sigma_finite) pair_measure_alt2:

   648   assumes "A \<in> sets P"

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

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

   651 proof -

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

   653   show ?thesis

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

   655          simp_all add: pair_sigma_algebra_def[symmetric])

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

   657       using F by auto

   658     show "measure_space P pair_measure" by default

   659   next

   660     show "measure_space P ?\<nu>"

   661     proof

   662       show "?\<nu> {} = 0" by auto

   663       show "countably_additive P ?\<nu>"

   664         unfolding countably_additive_def

   665       proof (intro allI impI)

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

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

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

   669         moreover from F have "\<And>i. (\<lambda>y. \<mu>1 ((\<lambda>x. (x, y)) - F i)) \<in> borel_measurable M2"

   670           by (intro measure_cut_measurable_snd) auto

   671         moreover have "\<And>y. disjoint_family (\<lambda>i. (\<lambda>x. (x, y)) - F i)"

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

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

   674           using F by (auto intro!: measurable_cut_snd)

   675         ultimately show "(\<Sum>\<^isub>\<infinity>n. ?\<nu> (F n)) = ?\<nu> (\<Union>i. F i)"

   676           by (simp add: vimage_UN M2.positive_integral_psuminf[symmetric]

   677                         M1.measure_countably_additive

   678                    cong: M2.positive_integral_cong)

   679       qed

   680     qed

   681   next

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

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

   684       unfolding pair_algebra_def by auto

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

   686     proof -

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

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

   689         by (auto intro!: M2.positive_integral_cong)

   690       with AB show ?thesis

   691         unfolding pair_measure_times[OF AB] X

   692         by (simp add: M2.positive_integral_cmult_indicator ac_simps)

   693     qed

   694   qed fact

   695 qed

   696

   697 section "Fubinis theorem"

   698

   699 lemma (in pair_sigma_finite) simple_function_cut:

   700   assumes f: "simple_function f"

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

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

   703       = positive_integral f"

   704 proof -

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

   706     using f by (rule borel_measurable_simple_function)

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

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

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

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

   711       by (auto simp: indicator_def)

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

   713       by (simp add: space_pair_algebra)

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

   715       by (intro borel_measurable_vimage measurable_cut_fst)

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

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

   718       apply (rule simple_function_indicator_representation[OF f])

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

   720   note M2_sf = this

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

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

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

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

   725       unfolding M2.simple_integral_def

   726     proof (safe intro!: setsum_mono_zero_cong_left)

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

   728     next

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

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

   731     next

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

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

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

   735         by (force simp: space_pair_algebra)

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

   737         unfolding * by simp

   738     qed (simp add: vimage_compose[symmetric] comp_def

   739                    space_pair_algebra) }

   740   note eq = this

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

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

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

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

   745   ultimately

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

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

   748     = positive_integral f"

   749     by (auto simp del: vimage_Int cong: measurable_cong

   750              intro!: M1.borel_measurable_pextreal_setsum

   751              simp add: M1.positive_integral_setsum simple_integral_def

   752                        M1.positive_integral_cmult

   753                        M1.positive_integral_cong[OF eq]

   754                        positive_integral_eq_simple_integral[OF f]

   755                        pair_measure_alt[symmetric])

   756 qed

   757

   758 lemma (in pair_sigma_finite) positive_integral_fst_measurable:

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

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

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

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

   763       positive_integral f"

   764 proof -

   765   from borel_measurable_implies_simple_function_sequence[OF f]

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

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

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

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

   770     unfolding isoton_fun_expand unfolding isoton_def le_fun_def

   771     by (auto intro: borel_measurable_simple_function)

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

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

   774     using F(1) by auto

   775   moreover

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

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

   778       using F \<up> f unfolding isoton_fun_expand

   779       by (auto simp: isoton_def)

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

   781     from M2.positive_integral_isoton[OF isotone this]

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

   783       by (simp add: isoton_def) }

   784   note SUPR_C = this

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

   786     by (simp cong: measurable_cong)

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

   788     using F_borel F_mono

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

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

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

   792     unfolding sf(2) by simp

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

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

   795                      M2.positive_integral_mono F_mono)

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

   797     using F_borel F_mono

   798     by (auto intro!: M2.positive_integral_monotone_convergence_SUP

   799                      M1.positive_integral_cong measurable_pair_image_snd)

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

   801       positive_integral f"

   802     unfolding F_SUPR by simp

   803 qed

   804

   805 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

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

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

   808       positive_integral f"

   809 proof -

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

   811   have s: "\<And>x y. (case (x, y) of (x, y) \<Rightarrow> f (y, x)) = f (y, x)" by simp

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

   813   have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"

   814     by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)

   815   note pair_sigma_algebra_measurable[OF f]

   816   from Q.positive_integral_fst_measurable[OF this]

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

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

   819     by simp

   820   also have "\<dots> = positive_integral f"

   821     unfolding positive_integral_vimage[OF bij, of f] t

   822     unfolding pair_sigma_algebra_swap[symmetric]

   823   proof (rule Q.positive_integral_cong_measure[symmetric])

   824     fix A assume "A \<in> sets Q.P"

   825     from this Q.sets_pair_sigma_algebra_swap[OF this]

   826     show "pair_measure ((\<lambda>(x, y). (y, x))  A) = Q.pair_measure A"

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

   828                simp: pair_measure_alt Q.pair_measure_alt2)

   829   qed

   830   finally show ?thesis .

   831 qed

   832

   833 lemma (in pair_sigma_finite) Fubini:

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

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

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

   837   unfolding positive_integral_snd_measurable[OF assms]

   838   unfolding positive_integral_fst_measurable[OF assms] ..

   839

   840 lemma (in pair_sigma_finite) AE_pair:

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

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

   843 proof -

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

   845     using assms unfolding almost_everywhere_def by auto

   846   show ?thesis

   847   proof (rule M1.AE_I)

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

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

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

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

   852       by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)

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

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

   855       proof (rule M2.AE_I)

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

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

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

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

   860       qed }

   861     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}"

   862       by auto

   863   qed

   864 qed

   865

   866 lemma (in pair_sigma_finite) positive_integral_product_swap:

   867   "measure_space.positive_integral

   868     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f =

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

   870 proof -

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

   872   have t: "(\<lambda>y. case case y of (y, x) \<Rightarrow> (x, y) of (x, y) \<Rightarrow> f (y, x)) = f"

   873     by (auto simp: fun_eq_iff)

   874   have bij: "bij_betw (\<lambda>(x, y). (y, x)) (space M2 \<times> space M1) (space P)"

   875     by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)

   876   show ?thesis

   877     unfolding positive_integral_vimage[OF bij, of "\<lambda>(y,x). f (x,y)"]

   878     unfolding pair_sigma_algebra_swap[symmetric] t

   879   proof (rule Q.positive_integral_cong_measure[symmetric])

   880     fix A assume "A \<in> sets Q.P"

   881     from this Q.sets_pair_sigma_algebra_swap[OF this]

   882     show "pair_measure ((\<lambda>(x, y). (y, x))  A) = Q.pair_measure A"

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

   884                simp: pair_measure_alt Q.pair_measure_alt2)

   885   qed

   886 qed

   887

   888 lemma (in pair_sigma_algebra) measurable_product_swap:

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

   890 proof -

   891   interpret Q: pair_sigma_algebra M2 M1 by default

   892   show ?thesis

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

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

   895 qed

   896

   897 lemma (in pair_sigma_finite) integrable_product_swap:

   898   "measure_space.integrable

   899     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f \<longleftrightarrow>

   900   integrable (\<lambda>(x,y). f (y,x))"

   901 proof -

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

   903   show ?thesis

   904     unfolding Q.integrable_def integrable_def

   905     unfolding positive_integral_product_swap

   906     unfolding measurable_product_swap

   907     by (simp add: case_prod_distrib)

   908 qed

   909

   910 lemma (in pair_sigma_finite) integral_product_swap:

   911   "measure_space.integral

   912     (sigma (pair_algebra M2 M1)) (pair_sigma_finite.pair_measure M2 \<mu>2 M1 \<mu>1) f =

   913   integral (\<lambda>(x,y). f (y,x))"

   914 proof -

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

   916   show ?thesis

   917     unfolding integral_def Q.integral_def positive_integral_product_swap

   918     by (simp add: case_prod_distrib)

   919 qed

   920

   921 lemma (in pair_sigma_finite) integrable_fst_measurable:

   922   assumes f: "integrable f"

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

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

   925 proof -

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

   927   have

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

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

   930     using assms by auto

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

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

   933     using borel[THEN positive_integral_fst_measurable(1)] int

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

   935   with borel[THEN positive_integral_fst_measurable(1)]

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

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

   938     by (auto intro!: M1.positive_integral_omega_AE)

   939   then show ?AE

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

   941     apply (rule M1.AE_cong)

   942     using assms unfolding M2.integrable_def

   943     by (auto intro!: measurable_pair_image_snd)

   944   have "M1.integrable

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

   946   proof (unfold M1.integrable_def, intro conjI)

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

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

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

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

   951       apply (rule M1.positive_integral_cong_AE)

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

   953       apply (rule M1.AE_cong)

   954       by (auto simp: Real_real)

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

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

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

   958       by (intro M1.positive_integral_cong) simp

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

   960   qed

   961   moreover have "M1.integrable

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

   963   proof (unfold M1.integrable_def, intro conjI)

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

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

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

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

   968       apply (rule M1.positive_integral_cong_AE)

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

   970       apply (rule M1.AE_cong)

   971       by (auto simp: Real_real)

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

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

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

   975       by (intro M1.positive_integral_cong) simp

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

   977   qed

   978   ultimately show ?INT

   979     unfolding M2.integral_def integral_def

   980       borel[THEN positive_integral_fst_measurable(2), symmetric]

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

   982 qed

   983

   984 lemma (in pair_sigma_finite) integrable_snd_measurable:

   985   assumes f: "integrable f"

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

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

   988 proof -

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

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

   991     using f unfolding integrable_product_swap by simp

   992   show ?INT

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

   994     unfolding integral_product_swap by simp

   995   show ?AE

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

   997     by simp

   998 qed

   999

  1000 lemma (in pair_sigma_finite) Fubini_integral:

  1001   assumes f: "integrable f"

  1002   shows "M2.integral (\<lambda>y. M1.integral (\<lambda>x. f (x, y))) =

  1003       M1.integral (\<lambda>x. M2.integral (\<lambda>y. f (x, y)))"

  1004   unfolding integrable_snd_measurable[OF assms]

  1005   unfolding integrable_fst_measurable[OF assms] ..

  1006

  1007 section "Finite product spaces"

  1008

  1009 section "Products"

  1010

  1011 locale product_sigma_algebra =

  1012   fixes M :: "'i \<Rightarrow> 'a algebra"

  1013   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"

  1014

  1015 locale finite_product_sigma_algebra = product_sigma_algebra M for M :: "'i \<Rightarrow> 'a algebra" +

  1016   fixes I :: "'i set"

  1017   assumes finite_index: "finite I"

  1018

  1019 syntax

  1020   "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)

  1021

  1022 syntax (xsymbols)

  1023   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)

  1024

  1025 syntax (HTML output)

  1026   "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^isub>E _\<in>_./ _)"   10)

  1027

  1028 translations

  1029   "PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"

  1030

  1031 definition

  1032   "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>"

  1033

  1034 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra M I"

  1035 abbreviation (in finite_product_sigma_algebra) "P \<equiv> sigma G"

  1036

  1037 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)

  1038

  1039 lemma (in finite_product_sigma_algebra) product_algebra_into_space:

  1040   "sets G \<subseteq> Pow (space G)"

  1041   using M.sets_into_space unfolding product_algebra_def

  1042   by auto blast

  1043

  1044 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P

  1045   using product_algebra_into_space by (rule sigma_algebra_sigma)

  1046

  1047 lemma product_algebraE:

  1048   assumes "A \<in> sets (product_algebra M I)"

  1049   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1050   using assms unfolding product_algebra_def by auto

  1051

  1052 lemma product_algebraI[intro]:

  1053   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1054   shows "Pi\<^isub>E I E \<in> sets (product_algebra M I)"

  1055   using assms unfolding product_algebra_def by auto

  1056

  1057 lemma space_product_algebra[simp]:

  1058   "space (product_algebra M I) = Pi\<^isub>E I (\<lambda>i. space (M i))"

  1059   unfolding product_algebra_def by simp

  1060

  1061 lemma product_algebra_sets_into_space:

  1062   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"

  1063   shows "sets (product_algebra M I) \<subseteq> Pow (space (product_algebra M I))"

  1064   using assms by (auto simp: product_algebra_def) blast

  1065

  1066 lemma (in finite_product_sigma_algebra) P_empty:

  1067   "I = {} \<Longrightarrow> P = \<lparr> space = {\<lambda>k. undefined}, sets = { {}, {\<lambda>k. undefined} }\<rparr>"

  1068   unfolding product_algebra_def by (simp add: sigma_def image_constant)

  1069

  1070 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:

  1071   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"

  1072   by (auto simp: product_algebra_def sets_sigma intro!: sigma_sets.Basic)

  1073

  1074 lemma (in product_sigma_algebra) bij_inv_restrict_merge:

  1075   assumes [simp]: "I \<inter> J = {}"

  1076   shows "bij_inv

  1077     (space (sigma (product_algebra M (I \<union> J))))

  1078     (space (sigma (pair_algebra (product_algebra M I) (product_algebra M J))))

  1079     (\<lambda>x. (restrict x I, restrict x J)) (\<lambda>(x, y). merge I x J y)"

  1080   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)

  1081

  1082 lemma (in product_sigma_algebra) bij_inv_singleton:

  1083   "bij_inv (space (sigma (product_algebra M {i}))) (space (M i))

  1084     (\<lambda>x. x i) (\<lambda>x. (\<lambda>j\<in>{i}. x))"

  1085   by (rule bij_invI) (auto simp: restrict_def extensional_def fun_eq_iff)

  1086

  1087 lemma (in product_sigma_algebra) bij_inv_restrict_insert:

  1088   assumes [simp]: "i \<notin> I"

  1089   shows "bij_inv

  1090     (space (sigma (product_algebra M (insert i I))))

  1091     (space (sigma (pair_algebra (product_algebra M I) (M i))))

  1092     (\<lambda>x. (restrict x I, x i)) (\<lambda>(x, y). x(i := y))"

  1093   by (rule bij_invI) (auto simp: space_pair_algebra extensional_restrict)

  1094

  1095 lemma (in product_sigma_algebra) measurable_restrict_on_generating:

  1096   assumes [simp]: "I \<inter> J = {}"

  1097   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable

  1098     (product_algebra M (I \<union> J))

  1099     (pair_algebra (product_algebra M I) (product_algebra M J))"

  1100     (is "?R \<in> measurable ?IJ ?P")

  1101 proof (unfold measurable_def, intro CollectI conjI ballI)

  1102   show "?R \<in> space ?IJ \<rightarrow> space ?P" by (auto simp: space_pair_algebra)

  1103   { fix F E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> (\<Pi> i\<in>J. sets (M i))"

  1104     then have "Pi (I \<union> J) (merge I E J F) \<inter> (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) =

  1105         Pi\<^isub>E (I \<union> J) (merge I E J F)"

  1106       using M.sets_into_space by auto blast+ }

  1107   note this[simp]

  1108   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?IJ \<in> sets ?IJ"

  1109     by (force elim!: pair_algebraE product_algebraE

  1110               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)

  1111   qed

  1112

  1113 lemma (in product_sigma_algebra) measurable_merge_on_generating:

  1114   assumes [simp]: "I \<inter> J = {}"

  1115   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable

  1116     (pair_algebra (product_algebra M I) (product_algebra M J))

  1117     (product_algebra M (I \<union> J))"

  1118     (is "?M \<in> measurable ?P ?IJ")

  1119 proof (unfold measurable_def, intro CollectI conjI ballI)

  1120   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)

  1121   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E \<in> (\<Pi> i\<in>J. sets (M i))"

  1122     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)) =

  1123         Pi\<^isub>E I E \<times> Pi\<^isub>E J E"

  1124       using M.sets_into_space by auto blast+ }

  1125   note this[simp]

  1126   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"

  1127     by (force elim!: pair_algebraE product_algebraE

  1128               simp del: vimage_Int simp: restrict_vimage merge_vimage space_pair_algebra)

  1129   qed

  1130

  1131 lemma (in product_sigma_algebra) measurable_singleton_on_generator:

  1132   "(\<lambda>x. \<lambda>j\<in>{i}. x) \<in> measurable (M i) (product_algebra M {i})"

  1133   (is "?f \<in> measurable _ ?P")

  1134 proof (unfold measurable_def, intro CollectI conjI)

  1135   show "?f \<in> space (M i) \<rightarrow> space ?P" by auto

  1136   have "\<And>E. E i \<in> sets (M i) \<Longrightarrow> ?f - Pi\<^isub>E {i} E \<inter> space (M i) = E i"

  1137     using M.sets_into_space by auto

  1138   then show "\<forall>A \<in> sets ?P. ?f - A \<inter> space (M i) \<in> sets (M i)"

  1139     by (auto elim!: product_algebraE)

  1140 qed

  1141

  1142 lemma (in product_sigma_algebra) measurable_component_on_generator:

  1143   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (product_algebra M I) (M i)"

  1144   (is "?f \<in> measurable ?P _")

  1145 proof (unfold measurable_def, intro CollectI conjI ballI)

  1146   show "?f \<in> space ?P \<rightarrow> space (M i)" using i \<in> I by auto

  1147   fix A assume "A \<in> sets (M i)"

  1148   moreover then have "(\<lambda>x. x i) - A \<inter> (\<Pi>\<^isub>E i\<in>I. space (M i)) =

  1149       (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"

  1150     using M.sets_into_space i \<in> I

  1151     by (fastsimp dest: Pi_mem split: split_if_asm)

  1152   ultimately show "?f - A \<inter> space ?P \<in> sets ?P"

  1153     by (auto intro!: product_algebraI)

  1154 qed

  1155

  1156 lemma (in product_sigma_algebra) measurable_restrict_singleton_on_generating:

  1157   assumes [simp]: "i \<notin> I"

  1158   shows "(\<lambda>x. (restrict x I, x i)) \<in> measurable

  1159     (product_algebra M (insert i I))

  1160     (pair_algebra (product_algebra M I) (M i))"

  1161     (is "?R \<in> measurable ?I ?P")

  1162 proof (unfold measurable_def, intro CollectI conjI ballI)

  1163   show "?R \<in> space ?I \<rightarrow> space ?P" by (auto simp: space_pair_algebra)

  1164   { fix E F assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "F \<in> sets (M i)"

  1165     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)) =

  1166         Pi\<^isub>E (insert i I) (E(i := F))"

  1167       using M.sets_into_space using i\<notin>I by (auto simp: restrict_Pi_cancel) blast+ }

  1168   note this[simp]

  1169   show "\<And>A. A \<in> sets ?P \<Longrightarrow> ?R - A \<inter> space ?I \<in> sets ?I"

  1170     by (force elim!: pair_algebraE product_algebraE

  1171               simp del: vimage_Int simp: space_pair_algebra)

  1172 qed

  1173

  1174 lemma (in product_sigma_algebra) measurable_merge_singleton_on_generating:

  1175   assumes [simp]: "i \<notin> I"

  1176   shows "(\<lambda>(x, y). x(i := y)) \<in> measurable

  1177     (pair_algebra (product_algebra M I) (M i))

  1178     (product_algebra M (insert i I))"

  1179     (is "?M \<in> measurable ?P ?IJ")

  1180 proof (unfold measurable_def, intro CollectI conjI ballI)

  1181   show "?M \<in> space ?P \<rightarrow> space ?IJ" by (auto simp: space_pair_algebra)

  1182   { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))" "E i \<in> sets (M i)"

  1183     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) =

  1184         Pi\<^isub>E I E \<times> E i"

  1185       using M.sets_into_space by auto blast+ }

  1186   note this[simp]

  1187   show "\<And>A. A \<in> sets ?IJ \<Longrightarrow> ?M - A \<inter> space ?P \<in> sets ?P"

  1188     by (force elim!: pair_algebraE product_algebraE

  1189               simp del: vimage_Int simp: space_pair_algebra)

  1190 qed

  1191

  1192 section "Generating set generates also product algebra"

  1193

  1194 lemma pair_sigma_algebra_sigma:

  1195   assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"

  1196   assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"

  1197   shows "sigma (pair_algebra (sigma E1) (sigma E2)) = sigma (pair_algebra E1 E2)"

  1198     (is "?S = ?E")

  1199 proof -

  1200   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)

  1201   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)

  1202   have P: "sets (pair_algebra E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"

  1203     using E1 E2 by (auto simp add: pair_algebra_def)

  1204   interpret E: sigma_algebra ?E unfolding pair_algebra_def

  1205     using E1 E2 by (intro sigma_algebra_sigma) auto

  1206   { fix A assume "A \<in> sets E1"

  1207     then have "fst - A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"

  1208       using E1 2 unfolding isoton_def pair_algebra_def by auto

  1209     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto

  1210     also have "\<dots> \<in> sets ?E" unfolding pair_algebra_def sets_sigma

  1211       using 2 A \<in> sets E1

  1212       by (intro sigma_sets.Union)

  1213          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1214     finally have "fst - A \<inter> space ?E \<in> sets ?E" . }

  1215   moreover

  1216   { fix B assume "B \<in> sets E2"

  1217     then have "snd - B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"

  1218       using E2 1 unfolding isoton_def pair_algebra_def by auto

  1219     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto

  1220     also have "\<dots> \<in> sets ?E"

  1221       using 1 B \<in> sets E2 unfolding pair_algebra_def sets_sigma

  1222       by (intro sigma_sets.Union)

  1223          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1224     finally have "snd - B \<inter> space ?E \<in> sets ?E" . }

  1225   ultimately have proj:

  1226     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"

  1227     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)

  1228                    (auto simp: pair_algebra_def sets_sigma)

  1229   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"

  1230     with proj have "fst - A \<inter> space ?E \<in> sets ?E" "snd - B \<inter> space ?E \<in> sets ?E"

  1231       unfolding measurable_def by simp_all

  1232     moreover have "A \<times> B = (fst - A \<inter> space ?E) \<inter> (snd - B \<inter> space ?E)"

  1233       using A B M1.sets_into_space M2.sets_into_space

  1234       by (auto simp: pair_algebra_def)

  1235     ultimately have "A \<times> B \<in> sets ?E" by auto }

  1236   then have "sigma_sets (space ?E) (sets (pair_algebra (sigma E1) (sigma E2))) \<subseteq> sets ?E"

  1237     by (intro E.sigma_sets_subset) (auto simp add: pair_algebra_def sets_sigma)

  1238   then have subset: "sets ?S \<subseteq> sets ?E"

  1239     by (simp add: sets_sigma pair_algebra_def)

  1240   have "sets ?S = sets ?E"

  1241   proof (intro set_eqI iffI)

  1242     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"

  1243       unfolding sets_sigma

  1244     proof induct

  1245       case (Basic A) then show ?case

  1246         by (auto simp: pair_algebra_def sets_sigma intro: sigma_sets.Basic)

  1247     qed (auto intro: sigma_sets.intros simp: pair_algebra_def)

  1248   next

  1249     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto

  1250   qed

  1251   then show ?thesis

  1252     by (simp add: pair_algebra_def sigma_def)

  1253 qed

  1254

  1255 lemma sigma_product_algebra_sigma_eq:

  1256   assumes "finite I"

  1257   assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))"

  1258   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"

  1259   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"

  1260   shows "sigma (product_algebra (\<lambda>i. sigma (E i)) I) = sigma (product_algebra E I)"

  1261     (is "?S = ?E")

  1262 proof cases

  1263   assume "I = {}" then show ?thesis by (simp add: product_algebra_def)

  1264 next

  1265   assume "I \<noteq> {}"

  1266   interpret E: sigma_algebra "sigma (E i)" for i

  1267     using E by (rule sigma_algebra_sigma)

  1268

  1269   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)"

  1270     using E by auto

  1271

  1272   interpret G: sigma_algebra ?E

  1273     unfolding product_algebra_def using E

  1274     by (intro sigma_algebra_sigma) (auto dest: Pi_mem)

  1275

  1276   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"

  1277     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"

  1278       using isotone unfolding isoton_def product_algebra_def by (auto dest: Pi_mem)

  1279     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"

  1280       unfolding product_algebra_def

  1281       apply simp

  1282       apply (subst Pi_UN[OF finite I])

  1283       using isotone[THEN isoton_mono_le] apply simp

  1284       apply (simp add: PiE_Int)

  1285       apply (intro PiE_cong)

  1286       using A sets_into by (auto intro!: into_space)

  1287     also have "\<dots> \<in> sets ?E" unfolding product_algebra_def sets_sigma

  1288       using sets_into A \<in> sets (E i)

  1289       by (intro sigma_sets.Union)

  1290          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1291     finally have "(\<lambda>x. x i) - A \<inter> space ?E \<in> sets ?E" . }

  1292   then have proj:

  1293     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"

  1294     using E by (subst G.measurable_iff_sigma)

  1295                (auto simp: product_algebra_def sets_sigma)

  1296

  1297   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"

  1298     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) - (A i) \<inter> space ?E \<in> sets ?E"

  1299       unfolding measurable_def by simp

  1300     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) - (A i) \<inter> space ?E)"

  1301       using A E.sets_into_space I \<noteq> {} unfolding product_algebra_def by auto blast

  1302     then have "Pi\<^isub>E I A \<in> sets ?E"

  1303       using G.finite_INT[OF finite I I \<noteq> {} basic, of "\<lambda>i. i"] by simp }

  1304   then have "sigma_sets (space ?E) (sets (product_algebra (\<lambda>i. sigma (E i)) I)) \<subseteq> sets ?E"

  1305     by (intro G.sigma_sets_subset) (auto simp add: sets_sigma product_algebra_def)

  1306   then have subset: "sets ?S \<subseteq> sets ?E"

  1307     by (simp add: sets_sigma product_algebra_def)

  1308

  1309   have "sets ?S = sets ?E"

  1310   proof (intro set_eqI iffI)

  1311     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"

  1312       unfolding sets_sigma

  1313     proof induct

  1314       case (Basic A) then show ?case

  1315         by (auto simp: sets_sigma product_algebra_def intro: sigma_sets.Basic)

  1316     qed (auto intro: sigma_sets.intros simp: product_algebra_def)

  1317   next

  1318     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto

  1319   qed

  1320   then show ?thesis

  1321     by (simp add: product_algebra_def sigma_def)

  1322 qed

  1323

  1324 lemma (in product_sigma_algebra) sigma_pair_algebra_sigma_eq:

  1325   "sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))) =

  1326    sigma (pair_algebra (product_algebra M I) (product_algebra M J))"

  1327   using M.sets_into_space

  1328   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)"])

  1329      (auto simp: isoton_const product_algebra_def, blast+)

  1330

  1331 lemma (in product_sigma_algebra) sigma_pair_algebra_product_singleton:

  1332   "sigma (pair_algebra (sigma (product_algebra M I)) (M i)) =

  1333    sigma (pair_algebra (product_algebra M I) (M i))"

  1334   using M.sets_into_space apply (subst M.sigma_eq[symmetric])

  1335   by (intro pair_sigma_algebra_sigma[of "\<lambda>_. \<Pi>\<^isub>E i\<in>I. space (M i)" _ "\<lambda>_. space (M i)"])

  1336      (auto simp: isoton_const product_algebra_def, blast+)

  1337

  1338 lemma (in product_sigma_algebra) measurable_restrict:

  1339   assumes [simp]: "I \<inter> J = {}"

  1340   shows "(\<lambda>x. (restrict x I, restrict x J)) \<in> measurable

  1341     (sigma (product_algebra M (I \<union> J)))

  1342     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))"

  1343   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space

  1344   by (intro measurable_sigma_sigma measurable_restrict_on_generating

  1345             pair_algebra_sets_into_space product_algebra_sets_into_space)

  1346      auto

  1347

  1348 lemma (in product_sigma_algebra) measurable_merge:

  1349   assumes [simp]: "I \<inter> J = {}"

  1350   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable

  1351     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))

  1352     (sigma (product_algebra M (I \<union> J)))"

  1353   unfolding sigma_pair_algebra_sigma_eq using M.sets_into_space

  1354   by (intro measurable_sigma_sigma measurable_merge_on_generating

  1355             pair_algebra_sets_into_space product_algebra_sets_into_space)

  1356      auto

  1357

  1358 lemma (in product_sigma_algebra) product_product_vimage_algebra:

  1359   assumes [simp]: "I \<inter> J = {}"

  1360   shows "sigma_algebra.vimage_algebra

  1361     (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))

  1362     (space (sigma (product_algebra M (I \<union> J)))) (\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))) =

  1363     sigma (product_algebra M (I \<union> J))"

  1364   unfolding sigma_pair_algebra_sigma_eq using sets_into_space

  1365   by (intro vimage_algebra_sigma[OF bij_inv_restrict_merge]

  1366             pair_algebra_sets_into_space product_algebra_sets_into_space

  1367             measurable_merge_on_generating measurable_restrict_on_generating)

  1368      auto

  1369

  1370 lemma (in product_sigma_algebra) pair_product_product_vimage_algebra:

  1371   assumes [simp]: "I \<inter> J = {}"

  1372   shows "sigma_algebra.vimage_algebra (sigma (product_algebra M (I \<union> J)))

  1373     (space (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J))))) (\<lambda>(x,y). merge I x J y) =

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

  1375   unfolding sigma_pair_algebra_sigma_eq using sets_into_space

  1376   by (intro vimage_algebra_sigma[OF bij_inv_restrict_merge[symmetric]]

  1377             pair_algebra_sets_into_space product_algebra_sets_into_space

  1378             measurable_merge_on_generating measurable_restrict_on_generating)

  1379      auto

  1380

  1381 lemma (in product_sigma_algebra) pair_product_singleton_vimage_algebra:

  1382   assumes [simp]: "i \<notin> I"

  1383   shows "sigma_algebra.vimage_algebra (sigma (pair_algebra (sigma (product_algebra M I)) (M i)))

  1384     (space (sigma (product_algebra M (insert i I)))) (\<lambda>x. (restrict x I, x i)) =

  1385     (sigma (product_algebra M (insert i I)))"

  1386   unfolding sigma_pair_algebra_product_singleton using sets_into_space

  1387   by (intro vimage_algebra_sigma[OF bij_inv_restrict_insert]

  1388             pair_algebra_sets_into_space product_algebra_sets_into_space

  1389             measurable_merge_singleton_on_generating measurable_restrict_singleton_on_generating)

  1390       auto

  1391

  1392 lemma (in product_sigma_algebra) singleton_vimage_algebra:

  1393   "sigma_algebra.vimage_algebra (sigma (product_algebra M {i})) (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i"

  1394   using sets_into_space

  1395   by (intro vimage_algebra_sigma[of "M i", unfolded M.sigma_eq, OF bij_inv_singleton[symmetric]]

  1396              product_algebra_sets_into_space measurable_singleton_on_generator measurable_component_on_generator)

  1397      auto

  1398

  1399 lemma (in product_sigma_algebra) measurable_restrict_iff:

  1400   assumes IJ[simp]: "I \<inter> J = {}"

  1401   shows "f \<in> measurable (sigma (pair_algebra

  1402       (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M' \<longleftrightarrow>

  1403     (\<lambda>x. f (restrict x I, restrict x J)) \<in> measurable (sigma (product_algebra M (I \<union> J))) M'"

  1404   using M.sets_into_space

  1405   apply (subst pair_product_product_vimage_algebra[OF IJ, symmetric])

  1406   apply (subst sigma_pair_algebra_sigma_eq)

  1407   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _

  1408       bij_inv_restrict_merge[symmetric]])

  1409   apply (intro sigma_algebra_sigma product_algebra_sets_into_space)

  1410   by auto

  1411

  1412 lemma (in product_sigma_algebra) measurable_merge_iff:

  1413   assumes IJ: "I \<inter> J = {}"

  1414   shows "f \<in> measurable (sigma (product_algebra M (I \<union> J))) M' \<longleftrightarrow>

  1415     (\<lambda>(x, y). f (merge I x J y)) \<in>

  1416       measurable (sigma (pair_algebra (sigma (product_algebra M I)) (sigma (product_algebra M J)))) M'"

  1417   unfolding measurable_restrict_iff[OF IJ]

  1418   by (rule measurable_cong) (auto intro!: arg_cong[where f=f] simp: extensional_restrict)

  1419

  1420 lemma (in product_sigma_algebra) measurable_component:

  1421   assumes "i \<in> I" and f: "f \<in> measurable (M i) M'"

  1422   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M'"

  1423     (is "?f \<in> measurable ?P M'")

  1424 proof -

  1425   have "f \<circ> (\<lambda>x. x i) \<in> measurable ?P M'"

  1426     apply (rule measurable_comp[OF _ f])

  1427     using measurable_up_sigma[of "product_algebra M I" "M i"]

  1428     using measurable_component_on_generator[OF i \<in> I]

  1429     by auto

  1430   then show "?f \<in> measurable ?P M'" by (simp add: comp_def)

  1431 qed

  1432

  1433 lemma (in product_sigma_algebra) measurable_component_singleton:

  1434   "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>

  1435     f \<in> measurable (M i) M'"

  1436   using sets_into_space

  1437   apply (subst singleton_vimage_algebra[symmetric])

  1438   apply (subst sigma_algebra.measurable_vimage_iff_inv[OF _ bij_inv_singleton[symmetric]])

  1439   by (auto intro!: sigma_algebra_sigma product_algebra_sets_into_space)

  1440

  1441 lemma (in product_sigma_algebra) measurable_component_iff:

  1442   assumes "i \<in> I" and not_empty: "\<forall>i\<in>I. space (M i) \<noteq> {}"

  1443   shows "(\<lambda>x. f (x i)) \<in> measurable (sigma (product_algebra M I)) M' \<longleftrightarrow>

  1444     f \<in> measurable (M i) M'"

  1445     (is "?f \<in> measurable ?P M' \<longleftrightarrow> _")

  1446 proof

  1447   assume "f \<in> measurable (M i) M'" then show "?f \<in> measurable ?P M'"

  1448     by (rule measurable_component[OF i \<in> I])

  1449 next

  1450   assume f: "?f \<in> measurable ?P M'"

  1451   def t \<equiv> "\<lambda>i. SOME x. x \<in> space (M i)"

  1452   have t: "\<And>i. i\<in>I \<Longrightarrow> t i \<in> space (M i)"

  1453      unfolding t_def using not_empty by (rule_tac someI_ex) auto

  1454   have "?f \<circ> (\<lambda>x. (\<lambda>j\<in>I. if j = i then x else t j)) \<in> measurable (M i) M'"

  1455     (is "?f \<circ> ?t \<in> measurable _ _")

  1456   proof (rule measurable_comp[OF _ f])

  1457     have "?t \<in> measurable (M i) (product_algebra M I)"

  1458     proof (unfold measurable_def, intro CollectI conjI ballI)

  1459       from t show "?t \<in> space (M i) \<rightarrow> (space (product_algebra M I))" by auto

  1460     next

  1461       fix A assume A: "A \<in> sets (product_algebra M I)"

  1462       { fix E assume "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1463         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 {})"

  1464           using i \<in> I sets_into_space by (auto dest: Pi_mem[where B=E]) }

  1465       note * = this

  1466       with A i \<in> I show "?t - A \<inter> space (M i) \<in> sets (M i)"

  1467         by (auto elim!: product_algebraE simp del: vimage_Int)

  1468     qed

  1469     also have "\<dots> \<subseteq> measurable (M i) (sigma (product_algebra M I))"

  1470       using M.sets_into_space by (intro measurable_subset) (auto simp: product_algebra_def, blast)

  1471     finally show "?t \<in> measurable (M i) (sigma (product_algebra M I))" .

  1472   qed

  1473   then show "f \<in> measurable (M i) M'" unfolding comp_def using i \<in> I by simp

  1474 qed

  1475

  1476 lemma (in product_sigma_algebra) measurable_singleton:

  1477   shows "f \<in> measurable (sigma (product_algebra M {i})) M' \<longleftrightarrow>

  1478     (\<lambda>x. f (\<lambda>j\<in>{i}. x)) \<in> measurable (M i) M'"

  1479   using sets_into_space unfolding measurable_component_singleton[symmetric]

  1480   by (auto intro!: measurable_cong arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1481

  1482 locale product_sigma_finite =

  1483   fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pextreal"

  1484   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i) (\<mu> i)"

  1485

  1486 locale finite_product_sigma_finite = product_sigma_finite M \<mu> for M :: "'i \<Rightarrow> 'a algebra" and \<mu> +

  1487   fixes I :: "'i set" assumes finite_index': "finite I"

  1488

  1489 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" "\<mu> i" for i

  1490   by (rule sigma_finite_measures)

  1491

  1492 sublocale product_sigma_finite \<subseteq> product_sigma_algebra

  1493   by default

  1494

  1495 sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra

  1496   by default (fact finite_index')

  1497

  1498 lemma (in finite_product_sigma_finite) sigma_finite_pairs:

  1499   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.

  1500     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>

  1501     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<omega>) \<and>

  1502     (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<up> space G"

  1503 proof -

  1504   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>)"

  1505     using M.sigma_finite_up by simp

  1506   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..

  1507   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>"

  1508     by auto

  1509   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"

  1510   note space_product_algebra[simp]

  1511   show ?thesis

  1512   proof (intro exI[of _ F] conjI allI isotoneI set_eqI iffI ballI)

  1513     fix i show "range (F i) \<subseteq> sets (M i)" by fact

  1514   next

  1515     fix i k show "\<mu> i (F i k) \<noteq> \<omega>" by fact

  1516   next

  1517     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"

  1518       using \<And>i. range (F i) \<subseteq> sets (M i) M.sets_into_space by auto blast

  1519   next

  1520     fix f assume "f \<in> space G"

  1521     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"]

  1522       \<And>i. F i \<up> space (M i)[THEN isotonD(2)]

  1523       \<And>i. F i \<up> space (M i)[THEN isoton_mono_le]

  1524     show "f \<in> (\<Union>i. ?F i)" by auto

  1525   next

  1526     fix i show "?F i \<subseteq> ?F (Suc i)"

  1527       using \<And>i. F i \<up> space (M i)[THEN isotonD(1)] by auto

  1528   qed

  1529 qed

  1530

  1531 lemma (in product_sigma_finite) product_measure_exists:

  1532   assumes "finite I"

  1533   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>

  1534      sigma_finite_measure (sigma (product_algebra M I)) \<nu>"

  1535 using finite I proof induct

  1536   case empty then show ?case unfolding product_algebra_def

  1537     by (auto intro!: exI[of _ "\<lambda>A. if A = {} then 0 else 1"] sigma_algebra_sigma

  1538                      sigma_algebra.finite_additivity_sufficient

  1539              simp add: positive_def additive_def sets_sigma sigma_finite_measure_def

  1540                        sigma_finite_measure_axioms_def image_constant)

  1541 next

  1542   case (insert i I)

  1543   interpret finite_product_sigma_finite M \<mu> I by default fact

  1544   have "finite (insert i I)" using finite I by auto

  1545   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default fact

  1546   from insert obtain \<nu> where

  1547     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

  1548     "sigma_finite_measure P \<nu>" by auto

  1549   interpret I: sigma_finite_measure P \<nu> by fact

  1550   interpret P: pair_sigma_finite P \<nu> "M i" "\<mu> i" ..

  1551

  1552   let ?h = "\<lambda>x. (restrict x I, x i)"

  1553   let ?\<nu> = "\<lambda>A. P.pair_measure (?h  A)"

  1554   interpret I': measure_space "sigma (product_algebra M (insert i I))" ?\<nu>

  1555     apply (subst pair_product_singleton_vimage_algebra[OF i \<notin> I, symmetric])

  1556     apply (intro P.measure_space_isomorphic bij_inv_bij_betw)

  1557     unfolding sigma_pair_algebra_product_singleton

  1558     by (rule bij_inv_restrict_insert[OF i \<notin> I])

  1559   show ?case

  1560   proof (intro exI[of _ ?\<nu>] conjI ballI)

  1561     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"

  1562       moreover then have "A \<in> (\<Pi> i\<in>I. sets (M i))" by auto

  1563       moreover have "(\<lambda>x. (restrict x I, x i))  Pi\<^isub>E (insert i I) A = Pi\<^isub>E I A \<times> A i"

  1564         using i \<notin> I

  1565         apply auto

  1566         apply (rule_tac x="a(i:=b)" in image_eqI)

  1567         apply (auto simp: extensional_def fun_eq_iff)

  1568         done

  1569       ultimately show "?\<nu> (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. \<mu> i (A i))"

  1570         apply simp

  1571         apply (subst P.pair_measure_times)

  1572         apply fastsimp

  1573         apply fastsimp

  1574         using i \<notin> I finite I prod[of A] by (auto simp: ac_simps) }

  1575     note product = this

  1576     show "sigma_finite_measure I'.P ?\<nu>"

  1577     proof

  1578       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..

  1579       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"

  1580         "(\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) \<up> space I'.G"

  1581         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<omega>"

  1582         by blast+

  1583       let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"

  1584       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>

  1585           (\<Union>i. F i) = space I'.P \<and> (\<forall>i. ?\<nu> (F i) \<noteq> \<omega>)"

  1586       proof (intro exI[of _ ?F] conjI allI)

  1587         show "range ?F \<subseteq> sets I'.P" using F(1) by auto

  1588       next

  1589         from F(2)[THEN isotonD(2)]

  1590         show "(\<Union>i. ?F i) = space I'.P" by simp

  1591       next

  1592         fix j

  1593         show "?\<nu> (?F j) \<noteq> \<omega>"

  1594           using F finite I

  1595           by (subst product) (auto simp: setprod_\<omega>)

  1596       qed

  1597     qed

  1598   qed

  1599 qed

  1600

  1601 definition (in finite_product_sigma_finite)

  1602   measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pextreal" where

  1603   "measure = (SOME \<nu>.

  1604      (\<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>

  1605      sigma_finite_measure P \<nu>)"

  1606

  1607 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P measure

  1608 proof -

  1609   show "sigma_finite_measure P measure"

  1610     unfolding measure_def

  1611     by (rule someI2_ex[OF product_measure_exists[OF finite_index]]) auto

  1612 qed

  1613

  1614 lemma (in finite_product_sigma_finite) measure_times:

  1615   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"

  1616   shows "measure (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))"

  1617 proof -

  1618   note ex = product_measure_exists[OF finite_index]

  1619   show ?thesis

  1620     unfolding measure_def

  1621   proof (rule someI2_ex[OF ex], elim conjE)

  1622     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))"

  1623     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)

  1624     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp

  1625     also have "\<dots> = (\<Prod>i\<in>I. \<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto

  1626     finally show "\<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))" by simp

  1627   qed

  1628 qed

  1629

  1630 abbreviation (in product_sigma_finite)

  1631   "product_measure I \<equiv> finite_product_sigma_finite.measure M \<mu> I"

  1632

  1633 abbreviation (in product_sigma_finite)

  1634   "product_positive_integral I \<equiv>

  1635     measure_space.positive_integral (sigma (product_algebra M I)) (product_measure I)"

  1636

  1637 abbreviation (in product_sigma_finite)

  1638   "product_integral I \<equiv>

  1639     measure_space.integral (sigma (product_algebra M I)) (product_measure I)"

  1640

  1641 abbreviation (in product_sigma_finite)

  1642   "product_integrable I \<equiv>

  1643     measure_space.integrable (sigma (product_algebra M I)) (product_measure I)"

  1644

  1645 lemma (in product_sigma_finite) product_measure_empty[simp]:

  1646   "product_measure {} {\<lambda>x. undefined} = 1"

  1647 proof -

  1648   interpret finite_product_sigma_finite M \<mu> "{}" by default auto

  1649   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp

  1650 qed

  1651

  1652 lemma (in product_sigma_finite) positive_integral_empty:

  1653   "product_positive_integral {} f = f (\<lambda>k. undefined)"

  1654 proof -

  1655   interpret finite_product_sigma_finite M \<mu> "{}" by default (fact finite.emptyI)

  1656   have "\<And>A. measure (Pi\<^isub>E {} A) = 1"

  1657     using assms by (subst measure_times) auto

  1658   then show ?thesis

  1659     unfolding positive_integral_def simple_function_def simple_integral_def_raw

  1660   proof (simp add: P_empty, intro antisym)

  1661     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))"

  1662       by (intro le_SUPI) auto

  1663     show "(SUP f:{g. g \<le> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)"

  1664       by (intro SUP_leI) (auto simp: le_fun_def)

  1665   qed

  1666 qed

  1667

  1668 lemma (in product_sigma_finite) measure_fold:

  1669   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1670   assumes A: "A \<in> sets (sigma (product_algebra M (I \<union> J)))"

  1671   shows "pair_sigma_finite.pair_measure

  1672      (sigma (product_algebra M I)) (product_measure I)

  1673      (sigma (product_algebra M J)) (product_measure J)

  1674      ((\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i)))  A) =

  1675    product_measure (I \<union> J) A"

  1676 proof -

  1677   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1678   interpret J: finite_product_sigma_finite M \<mu> J by default fact

  1679   have "finite (I \<union> J)" using fin by auto

  1680   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact

  1681   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default

  1682   let ?f = "\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))"

  1683     from IJ.sigma_finite_pairs obtain F where

  1684       F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"

  1685          "(\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) \<up> space IJ.G"

  1686          "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<omega>"

  1687       by auto

  1688     let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"

  1689   have split_f_image[simp]: "\<And>F. ?f  (Pi\<^isub>E (I \<union> J) F) = (Pi\<^isub>E I F) \<times> (Pi\<^isub>E J F)"

  1690     apply auto apply (rule_tac x="merge I a J b" in image_eqI)

  1691     by (auto dest: extensional_restrict)

  1692     show "P.pair_measure (?f  A) = IJ.measure A"

  1693   proof (rule measure_unique_Int_stable[OF _ _ _ _ _ _ _ _ A])

  1694       show "Int_stable IJ.G" by (simp add: PiE_Int Int_stable_def product_algebra_def) auto

  1695       show "range ?F \<subseteq> sets IJ.G" using F by (simp add: image_subset_iff product_algebra_def)

  1696       show "?F \<up> space IJ.G " using F(2) by simp

  1697       show "measure_space IJ.P (\<lambda>A. P.pair_measure (?f  A))"

  1698       apply (subst product_product_vimage_algebra[OF IJ, symmetric])

  1699       apply (intro P.measure_space_isomorphic bij_inv_bij_betw)

  1700       unfolding sigma_pair_algebra_sigma_eq

  1701       by (rule bij_inv_restrict_merge[OF I \<inter> J = {}])

  1702       show "measure_space IJ.P IJ.measure" by fact

  1703     next

  1704       fix A assume "A \<in> sets IJ.G"

  1705       then obtain F where A[simp]: "A = Pi\<^isub>E (I \<union> J) F" "F \<in> (\<Pi> i\<in>I \<union> J. sets (M i))"

  1706         by (auto simp: product_algebra_def)

  1707       then have F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (M i)" "\<And>i. i \<in> J \<Longrightarrow> F i \<in> sets (M i)"

  1708         by auto

  1709       have "P.pair_measure (?f  A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"

  1710         using finite J finite I F

  1711         by (simp add: P.pair_measure_times I.measure_times J.measure_times)

  1712       also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"

  1713         using finite J finite I I \<inter> J = {}  by (simp add: setprod_Un_one)

  1714       also have "\<dots> = IJ.measure A"

  1715         using finite J finite I F unfolding A

  1716         by (intro IJ.measure_times[symmetric]) auto

  1717       finally show "P.pair_measure (?f  A) = IJ.measure A" .

  1718     next

  1719       fix k

  1720       have "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i k \<in> sets (M i)" using F by auto

  1721       then have "P.pair_measure (?f  ?F k) = (\<Prod>i\<in>I. \<mu> i (F i k)) * (\<Prod>i\<in>J. \<mu> i (F i k))"

  1722         by (simp add: P.pair_measure_times I.measure_times J.measure_times)

  1723       then show "P.pair_measure (?f  ?F k) \<noteq> \<omega>"

  1724         using finite I F by (simp add: setprod_\<omega>)

  1725     qed simp

  1726   qed

  1727

  1728 lemma (in product_sigma_finite) product_positive_integral_fold:

  1729   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1730   and f: "f \<in> borel_measurable (sigma (product_algebra M (I \<union> J)))"

  1731   shows "product_positive_integral (I \<union> J) f =

  1732     product_positive_integral I (\<lambda>x. product_positive_integral J (\<lambda>y. f (merge I x J y)))"

  1733 proof -

  1734   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1735   interpret J: finite_product_sigma_finite M \<mu> J by default fact

  1736   have "finite (I \<union> J)" using fin by auto

  1737   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact

  1738   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default

  1739   let ?f = "\<lambda>x. ((\<lambda>i\<in>I. x i), (\<lambda>i\<in>J. x i))"

  1740   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"

  1741     unfolding case_prod_distrib measurable_merge_iff[OF IJ, symmetric] using f .

  1742   have bij: "bij_betw ?f (space IJ.P) (space P.P)"

  1743     unfolding sigma_pair_algebra_sigma_eq

  1744     by (intro bij_inv_bij_betw) (rule bij_inv_restrict_merge[OF IJ])

  1745   have "IJ.positive_integral f =  IJ.positive_integral (\<lambda>x. f (restrict x (I \<union> J)))"

  1746     by (auto intro!: IJ.positive_integral_cong arg_cong[where f=f] dest!: extensional_restrict)

  1747   also have "\<dots> = I.positive_integral (\<lambda>x. J.positive_integral (\<lambda>y. f (merge I x J y)))"

  1748     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]

  1749     unfolding P.positive_integral_vimage[OF bij]

  1750     unfolding product_product_vimage_algebra[OF IJ]

  1751     apply simp

  1752     apply (rule IJ.positive_integral_cong_measure[symmetric])

  1753     apply (rule measure_fold)

  1754     using assms by auto

  1755   finally show ?thesis .

  1756 qed

  1757

  1758 lemma (in product_sigma_finite) product_positive_integral_singleton:

  1759   assumes f: "f \<in> borel_measurable (M i)"

  1760   shows "product_positive_integral {i} (\<lambda>x. f (x i)) = M.positive_integral i f"

  1761 proof -

  1762   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp

  1763   have bij: "bij_betw (\<lambda>x. \<lambda>j\<in>{i}. x) (space (M i)) (space I.P)"

  1764     by (auto intro!: bij_betwI ext simp: extensional_def)

  1765   have *: "(\<lambda>x. (\<lambda>x. \<lambda>j\<in>{i}. x) - Pi\<^isub>E {i} x \<inter> space (M i))  (\<Pi> i\<in>{i}. sets (M i)) = sets (M i)"

  1766   proof (subst image_cong, rule refl)

  1767     fix x assume "x \<in> (\<Pi> i\<in>{i}. sets (M i))"

  1768     then show "(\<lambda>x. \<lambda>j\<in>{i}. x) - Pi\<^isub>E {i} x \<inter> space (M i) = x i"

  1769       using sets_into_space by auto

  1770   qed auto

  1771   have vimage: "I.vimage_algebra (space (M i)) (\<lambda>x. \<lambda>j\<in>{i}. x) = M i"

  1772     unfolding I.vimage_algebra_def

  1773     unfolding product_sigma_algebra_def sets_sigma

  1774     apply (subst sigma_sets_vimage[symmetric])

  1775     apply (simp_all add: image_image sigma_sets_eq product_algebra_def * del: vimage_Int)

  1776     using sets_into_space by blast

  1777   show "I.positive_integral (\<lambda>x. f (x i)) = M.positive_integral i f"

  1778     unfolding I.positive_integral_vimage[OF bij]

  1779     unfolding vimage

  1780     apply (subst positive_integral_cong_measure)

  1781   proof -

  1782     fix A assume A: "A \<in> sets (M i)"

  1783     have "(\<lambda>x. \<lambda>j\<in>{i}. x)  A = (\<Pi>\<^isub>E i\<in>{i}. A)"

  1784       by (auto intro!: image_eqI ext[where 'b='a] simp: extensional_def)

  1785     with A show "product_measure {i} ((\<lambda>x. \<lambda>j\<in>{i}. x)  A) = \<mu> i A"

  1786       using I.measure_times[of "\<lambda>i. A"] by simp

  1787   qed simp

  1788 qed

  1789

  1790 lemma (in product_sigma_finite) product_positive_integral_insert:

  1791   assumes [simp]: "finite I" "i \<notin> I"

  1792     and f: "f \<in> borel_measurable (sigma (product_algebra M (insert i I)))"

  1793   shows "product_positive_integral (insert i I) f

  1794     = product_positive_integral I (\<lambda>x. M.positive_integral i (\<lambda>y. f (x(i:=y))))"

  1795 proof -

  1796   interpret I: finite_product_sigma_finite M \<mu> I by default auto

  1797   interpret i: finite_product_sigma_finite M \<mu> "{i}" by default auto

  1798   interpret P: pair_sigma_algebra I.P i.P ..

  1799   have IJ: "I \<inter> {i} = {}" by auto

  1800   show ?thesis

  1801     unfolding product_positive_integral_fold[OF IJ, simplified, OF f]

  1802   proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)

  1803     fix x assume x: "x \<in> space I.P"

  1804     let "?f y" = "f (restrict (x(i := y)) (insert i I))"

  1805     have f'_eq: "\<And>y. ?f y = f (x(i := y))"

  1806       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1807     note fP = f[unfolded measurable_merge_iff[OF IJ, simplified]]

  1808     show "?f \<in> borel_measurable (M i)"

  1809       using P.measurable_pair_image_snd[OF fP x]

  1810       unfolding measurable_singleton f'_eq by (simp add: f'_eq)

  1811     show "M.positive_integral i ?f = M.positive_integral i (\<lambda>y. f (x(i := y)))"

  1812       unfolding f'_eq by simp

  1813   qed

  1814 qed

  1815

  1816 lemma (in product_sigma_finite) product_positive_integral_setprod:

  1817   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> pextreal"

  1818   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"

  1819   shows "product_positive_integral I (\<lambda>x. (\<Prod>i\<in>I. f i (x i))) =

  1820     (\<Prod>i\<in>I. M.positive_integral i (f i))"

  1821 using assms proof induct

  1822   case empty

  1823   interpret finite_product_sigma_finite M \<mu> "{}" by default auto

  1824   then show ?case by simp

  1825 next

  1826   case (insert i I)

  1827   note finite I[intro, simp]

  1828   interpret I: finite_product_sigma_finite M \<mu> I by default auto

  1829   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))"

  1830     using insert by (auto intro!: setprod_cong)

  1831   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>

  1832     (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (sigma (product_algebra M J))"

  1833     using sets_into_space insert

  1834     by (intro sigma_algebra.borel_measurable_pextreal_setprod

  1835               sigma_algebra_sigma product_algebra_sets_into_space

  1836               measurable_component)

  1837        auto

  1838   show ?case

  1839     by (simp add: product_positive_integral_insert[OF insert(1,2) prod])

  1840        (simp add: insert I.positive_integral_cmult M.positive_integral_multc * prod subset_insertI)

  1841 qed

  1842

  1843 lemma (in product_sigma_finite) product_integral_singleton:

  1844   assumes f: "f \<in> borel_measurable (M i)"

  1845   shows "product_integral {i} (\<lambda>x. f (x i)) = M.integral i f"

  1846 proof -

  1847   interpret I: finite_product_sigma_finite M \<mu> "{i}" by default simp

  1848   have *: "(\<lambda>x. Real (f x)) \<in> borel_measurable (M i)"

  1849     "(\<lambda>x. Real (- f x)) \<in> borel_measurable (M i)"

  1850     using assms by auto

  1851   show ?thesis

  1852     unfolding I.integral_def integral_def

  1853     unfolding *[THEN product_positive_integral_singleton] ..

  1854 qed

  1855

  1856 lemma (in product_sigma_finite) product_integral_fold:

  1857   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1858   and f: "measure_space.integrable (sigma (product_algebra M (I \<union> J))) (product_measure (I \<union> J)) f"

  1859   shows "product_integral (I \<union> J) f =

  1860     product_integral I (\<lambda>x. product_integral J (\<lambda>y. f (merge I x J y)))"

  1861 proof -

  1862   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1863   interpret J: finite_product_sigma_finite M \<mu> J by default fact

  1864   have "finite (I \<union> J)" using fin by auto

  1865   interpret IJ: finite_product_sigma_finite M \<mu> "I \<union> J" by default fact

  1866   interpret P: pair_sigma_finite I.P I.measure J.P J.measure by default

  1867   let ?f = "\<lambda>(x,y). f (merge I x J y)"

  1868   have f_borel: "f \<in> borel_measurable IJ.P"

  1869      "(\<lambda>x. Real (f x)) \<in> borel_measurable IJ.P"

  1870      "(\<lambda>x. Real (- f x)) \<in> borel_measurable IJ.P"

  1871     using f unfolding integrable_def by auto

  1872   have f_restrict: "(\<lambda>x. f (restrict x (I \<union> J))) \<in> borel_measurable IJ.P"

  1873     by (rule measurable_cong[THEN iffD2, OF _ f_borel(1)])

  1874        (auto intro!: arg_cong[where f=f] simp: extensional_restrict)

  1875   then have f'_borel:

  1876     "(\<lambda>x. Real (?f x)) \<in> borel_measurable P.P"

  1877     "(\<lambda>x. Real (- ?f x)) \<in> borel_measurable P.P"

  1878     unfolding measurable_restrict_iff[OF IJ]

  1879     by simp_all

  1880   have PI:

  1881     "P.positive_integral (\<lambda>x. Real (?f x)) = IJ.positive_integral (\<lambda>x. Real (f x))"

  1882     "P.positive_integral (\<lambda>x. Real (- ?f x)) = IJ.positive_integral (\<lambda>x. Real (- f x))"

  1883     using f'_borel[THEN P.positive_integral_fst_measurable(2)]

  1884     using f_borel(2,3)[THEN product_positive_integral_fold[OF assms(1-3)]]

  1885     by simp_all

  1886   have "P.integrable ?f" using IJ.integrable f

  1887     unfolding P.integrable_def IJ.integrable_def

  1888     unfolding measurable_restrict_iff[OF IJ]

  1889     using f_restrict PI by simp_all

  1890   show ?thesis

  1891     unfolding P.integrable_fst_measurable(2)[OF P.integrable ?f, simplified]

  1892     unfolding IJ.integral_def P.integral_def

  1893     unfolding PI by simp

  1894 qed

  1895

  1896 lemma (in product_sigma_finite) product_integral_insert:

  1897   assumes [simp]: "finite I" "i \<notin> I"

  1898     and f: "measure_space.integrable (sigma (product_algebra M (insert i I))) (product_measure (insert i I)) f"

  1899   shows "product_integral (insert i I) f

  1900     = product_integral I (\<lambda>x. M.integral i (\<lambda>y. f (x(i:=y))))"

  1901 proof -

  1902   interpret I: finite_product_sigma_finite M \<mu> I by default auto

  1903   interpret I': finite_product_sigma_finite M \<mu> "insert i I" by default auto

  1904   interpret i: finite_product_sigma_finite M \<mu> "{i}" by default auto

  1905   interpret P: pair_sigma_algebra I.P i.P ..

  1906   have IJ: "I \<inter> {i} = {}" by auto

  1907   show ?thesis

  1908     unfolding product_integral_fold[OF IJ, simplified, OF f]

  1909   proof (rule I.integral_cong, subst product_integral_singleton)

  1910     fix x assume x: "x \<in> space I.P"

  1911     let "?f y" = "f (restrict (x(i := y)) (insert i I))"

  1912     have f'_eq: "\<And>y. ?f y = f (x(i := y))"

  1913       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1914     have "f \<in> borel_measurable I'.P" using f unfolding I'.integrable_def by auto

  1915     note fP = this[unfolded measurable_merge_iff[OF IJ, simplified]]

  1916     show "?f \<in> borel_measurable (M i)"

  1917       using P.measurable_pair_image_snd[OF fP x]

  1918       unfolding measurable_singleton f'_eq by (simp add: f'_eq)

  1919     show "M.integral i ?f = M.integral i (\<lambda>y. f (x(i := y)))"

  1920       unfolding f'_eq by simp

  1921   qed

  1922 qed

  1923

  1924 lemma (in product_sigma_finite) product_integrable_setprod:

  1925   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"

  1926   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> M.integrable i (f i)"

  1927   shows "product_integrable I (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "product_integrable I ?f")

  1928 proof -

  1929   interpret finite_product_sigma_finite M \<mu> I by default fact

  1930   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"

  1931     using integrable unfolding M.integrable_def by auto

  1932   then have borel: "?f \<in> borel_measurable P"

  1933     by (intro borel_measurable_setprod measurable_component) auto

  1934   moreover have "integrable (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"

  1935   proof (unfold integrable_def, intro conjI)

  1936     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"

  1937       using borel by auto

  1938     have "positive_integral (\<lambda>x. Real (abs (?f x))) = positive_integral (\<lambda>x. \<Prod>i\<in>I. Real (abs (f i (x i))))"

  1939       by (simp add: Real_setprod abs_setprod)

  1940     also have "\<dots> = (\<Prod>i\<in>I. M.positive_integral i (\<lambda>x. Real (abs (f i x))))"

  1941       using f by (subst product_positive_integral_setprod) auto

  1942     also have "\<dots> < \<omega>"

  1943       using integrable[THEN M.integrable_abs]

  1944       unfolding pextreal_less_\<omega> setprod_\<omega> M.integrable_def by simp

  1945     finally show "positive_integral (\<lambda>x. Real (abs (?f x))) \<noteq> \<omega>" by auto

  1946     show "positive_integral (\<lambda>x. Real (- abs (?f x))) \<noteq> \<omega>" by simp

  1947   qed

  1948   ultimately show ?thesis

  1949     by (rule integrable_abs_iff[THEN iffD1])

  1950 qed

  1951

  1952 lemma (in product_sigma_finite) product_integral_setprod:

  1953   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"

  1954   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> M.integrable i (f i)"

  1955   shows "product_integral I (\<lambda>x. (\<Prod>i\<in>I. f i (x i))) = (\<Prod>i\<in>I. M.integral i (f i))"

  1956 using assms proof (induct rule: finite_ne_induct)

  1957   case (singleton i)

  1958   then show ?case by (simp add: product_integral_singleton integrable_def)

  1959 next

  1960   case (insert i I)

  1961   then have iI: "finite (insert i I)" by auto

  1962   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>

  1963     product_integrable J (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"

  1964     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)

  1965   interpret I: finite_product_sigma_finite M \<mu> I by default fact

  1966   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))"

  1967     using i \<notin> I by (auto intro!: setprod_cong)

  1968   show ?case

  1969     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]

  1970     by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)

  1971 qed

  1972

  1973 section "Products on finite spaces"

  1974

  1975 lemma sigma_sets_pair_algebra_finite:

  1976   assumes "finite A" and "finite B"

  1977   shows "sigma_sets (A \<times> B) ((\<lambda>(x,y). x \<times> y)  (Pow A \<times> Pow B)) = Pow (A \<times> B)"

  1978   (is "sigma_sets ?prod ?sets = _")

  1979 proof safe

  1980   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)

  1981   fix x assume subset: "x \<subseteq> A \<times> B"

  1982   hence "finite x" using fin by (rule finite_subset)

  1983   from this subset show "x \<in> sigma_sets ?prod ?sets"

  1984   proof (induct x)

  1985     case empty show ?case by (rule sigma_sets.Empty)

  1986   next

  1987     case (insert a x)

  1988     hence "{a} \<in> sigma_sets ?prod ?sets"

  1989       by (auto simp: pair_algebra_def intro!: sigma_sets.Basic)

  1990     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto

  1991     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)

  1992   qed

  1993 next

  1994   fix x a b

  1995   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"

  1996   from sigma_sets_into_sp[OF _ this(1)] this(2)

  1997   show "a \<in> A" and "b \<in> B" by auto

  1998 qed

  1999

  2000 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2

  2001

  2002 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default

  2003

  2004 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra[simp]:

  2005   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"

  2006 proof -

  2007   show ?thesis using M1.finite_space M2.finite_space

  2008     by (simp add: sigma_def space_pair_algebra sets_pair_algebra

  2009                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)

  2010 qed

  2011

  2012 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P

  2013 proof

  2014   show "finite (space P)" "sets P = Pow (space P)"

  2015     using M1.finite_space M2.finite_space by auto

  2016 qed

  2017

  2018 locale pair_finite_space = M1: finite_measure_space M1 \<mu>1 + M2: finite_measure_space M2 \<mu>2

  2019   for M1 \<mu>1 M2 \<mu>2

  2020

  2021 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra

  2022   by default

  2023

  2024 sublocale pair_finite_space \<subseteq> pair_sigma_finite

  2025   by default

  2026

  2027 lemma (in pair_finite_space) finite_pair_sigma_algebra[simp]:

  2028   shows "P = (| space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2) |)"

  2029 proof -

  2030   show ?thesis using M1.finite_space M2.finite_space

  2031     by (simp add: sigma_def space_pair_algebra sets_pair_algebra

  2032                   sigma_sets_pair_algebra_finite M1.sets_eq_Pow M2.sets_eq_Pow)

  2033 qed

  2034

  2035 lemma (in pair_finite_space) pair_measure_Pair[simp]:

  2036   assumes "a \<in> space M1" "b \<in> space M2"

  2037   shows "pair_measure {(a, b)} = \<mu>1 {a} * \<mu>2 {b}"

  2038 proof -

  2039   have "pair_measure ({a}\<times>{b}) = \<mu>1 {a} * \<mu>2 {b}"

  2040     using M1.sets_eq_Pow M2.sets_eq_Pow assms

  2041     by (subst pair_measure_times) auto

  2042   then show ?thesis by simp

  2043 qed

  2044

  2045 lemma (in pair_finite_space) pair_measure_singleton[simp]:

  2046   assumes "x \<in> space M1 \<times> space M2"

  2047   shows "pair_measure {x} = \<mu>1 {fst x} * \<mu>2 {snd x}"

  2048   using pair_measure_Pair assms by (cases x) auto

  2049

  2050 sublocale pair_finite_space \<subseteq> finite_measure_space P pair_measure

  2051   by default auto

  2052

  2053 lemma (in pair_finite_space) finite_measure_space_finite_prod_measure_alterantive:

  2054   "finite_measure_space \<lparr>space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2)\<rparr> pair_measure"

  2055   unfolding finite_pair_sigma_algebra[symmetric]

  2056   by default

  2057

  2058 end`