src/HOL/Probability/Product_Measure.thy
 author hoelzl Tue Mar 22 18:53:05 2011 +0100 (2011-03-22) changeset 42066 6db76c88907a parent 41981 cdf7693bbe08 child 42067 66c8281349ec permissions -rw-r--r--
generalized Caratheodory from algebra to ring_of_sets
     1 theory Product_Measure

     2 imports Lebesgue_Integration

     3 begin

     4

     5 lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"

     6 proof

     7   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"

     8     by induct (insert A \<subseteq> B, auto intro: sigma_sets.intros)

     9 qed

    10

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

    12   by auto

    13

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

    15   by auto

    16

    17 lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) - (A \<times> B) = (if y \<in> B then A else {})"

    18   by auto

    19

    20 lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"

    21   by (cases x) simp

    22

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

    24   by (auto simp: fun_eq_iff)

    25

    26 abbreviation

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

    28

    29 syntax

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

    31

    32 syntax (xsymbols)

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

    34

    35 syntax (HTML output)

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

    37

    38 translations

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

    40

    41 abbreviation

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

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

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

    45

    46 notation (xsymbols)

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

    48

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

    50   by safe (auto simp add: extensional_def fun_eq_iff)

    51

    52 lemma extensional_insert[intro, simp]:

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

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

    55   using assms unfolding extensional_def by auto

    56

    57 lemma extensional_Int[simp]:

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

    59   unfolding extensional_def by auto

    60

    61 definition

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

    63

    64 lemma merge_apply[simp]:

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

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

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

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

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

    70   unfolding merge_def by auto

    71

    72 lemma merge_commute:

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

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

    75

    76 lemma Pi_cancel_merge_range[simp]:

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

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

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

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

    81   by (auto simp: Pi_def)

    82

    83 lemma Pi_cancel_merge[simp]:

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

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

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

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

    88   by (auto simp: Pi_def)

    89

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

    91   by (auto simp: extensional_def)

    92

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

    94   by (auto simp: restrict_def Pi_def)

    95

    96 lemma restrict_merge[simp]:

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

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

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

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

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

   102

   103 lemma extensional_insert_undefined[intro, simp]:

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

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

   106   using assms unfolding extensional_def by auto

   107

   108 lemma extensional_insert_cancel[intro, simp]:

   109   assumes "a \<in> extensional I"

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

   111   using assms unfolding extensional_def by auto

   112

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

   114   unfolding merge_def by (auto simp: fun_eq_iff)

   115

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

   117   by auto

   118

   119 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)"

   120   by auto

   121

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

   123   by (auto simp: Pi_def)

   124

   125 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"

   126   by (auto simp: Pi_def)

   127

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

   129   by (auto simp: Pi_def)

   130

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

   132   by (auto simp: Pi_def)

   133

   134 lemma restrict_vimage:

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

   136   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)"

   137   using assms by (auto simp: restrict_Pi_cancel)

   138

   139 lemma merge_vimage:

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

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

   142   using assms by (auto simp: restrict_Pi_cancel)

   143

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

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

   146

   147 lemma merge_restrict[simp]:

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

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

   150   unfolding merge_def by (auto intro!: ext)

   151

   152 lemma merge_x_x_eq_restrict[simp]:

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

   154   unfolding merge_def by (auto intro!: ext)

   155

   156 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"

   157   apply auto

   158   apply (drule_tac x=x in Pi_mem)

   159   apply (simp_all split: split_if_asm)

   160   apply (drule_tac x=i in Pi_mem)

   161   apply (auto dest!: Pi_mem)

   162   done

   163

   164 lemma Pi_UN:

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

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

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

   168 proof (intro set_eqI iffI)

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

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

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

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

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

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

   175   proof (intro Pi_I)

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

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

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

   179   qed

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

   181 qed auto

   182

   183 lemma PiE_cong:

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

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

   186   using assms by (auto intro!: Pi_cong)

   187

   188 lemma restrict_upd[simp]:

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

   190   by (auto simp: fun_eq_iff)

   191

   192 lemma Pi_eq_subset:

   193   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"

   194   assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"

   195   shows "F i \<subseteq> F' i"

   196 proof

   197   fix x assume "x \<in> F i"

   198   with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto

   199   from choice[OF this] guess f .. note f = this

   200   then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)

   201   then have "f \<in> Pi\<^isub>E I F'" using assms by simp

   202   then show "x \<in> F' i" using f i \<in> I by auto

   203 qed

   204

   205 lemma Pi_eq_iff_not_empty:

   206   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"

   207   shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"

   208 proof (intro iffI ballI)

   209   fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"

   210   show "F i = F' i"

   211     using Pi_eq_subset[of I F F', OF ne eq i]

   212     using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]

   213     by auto

   214 qed auto

   215

   216 lemma Pi_eq_empty_iff:

   217   "Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"

   218 proof

   219   assume "Pi\<^isub>E I F = {}"

   220   show "\<exists>i\<in>I. F i = {}"

   221   proof (rule ccontr)

   222     assume "\<not> ?thesis"

   223     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto

   224     from choice[OF this] guess f ..

   225     then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)

   226     with Pi\<^isub>E I F = {} show False by auto

   227   qed

   228 qed auto

   229

   230 lemma Pi_eq_iff:

   231   "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"

   232 proof (intro iffI disjCI)

   233   assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"

   234   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"

   235   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"

   236     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto

   237   with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto

   238 next

   239   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"

   240   then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"

   241     using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto

   242 qed

   243

   244 section "Binary products"

   245

   246 definition

   247   "pair_measure_generator A B =

   248     \<lparr> space = space A \<times> space B,

   249       sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},

   250       measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"

   251

   252 definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where

   253   "A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"

   254

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

   256   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

   257

   258 abbreviation (in pair_sigma_algebra)

   259   "E \<equiv> pair_measure_generator M1 M2"

   260

   261 abbreviation (in pair_sigma_algebra)

   262   "P \<equiv> M1 \<Otimes>\<^isub>M M2"

   263

   264 lemma sigma_algebra_pair_measure:

   265   "sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"

   266   by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)

   267

   268 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P

   269   using M1.space_closed M2.space_closed

   270   by (rule sigma_algebra_pair_measure)

   271

   272 lemma pair_measure_generatorI[intro, simp]:

   273   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"

   274   by (auto simp add: pair_measure_generator_def)

   275

   276 lemma pair_measureI[intro, simp]:

   277   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"

   278   by (auto simp add: pair_measure_def)

   279

   280 lemma space_pair_measure:

   281   "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"

   282   by (simp add: pair_measure_def pair_measure_generator_def)

   283

   284 lemma sets_pair_measure_generator:

   285   "sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y)  (sets N \<times> sets M)"

   286   unfolding pair_measure_generator_def by auto

   287

   288 lemma pair_measure_generator_sets_into_space:

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

   290   shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"

   291   using assms by (auto simp: pair_measure_generator_def)

   292

   293 lemma pair_measure_generator_Int_snd:

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

   295   shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =

   296          sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"

   297   (is "?L = ?R")

   298   apply (auto simp: pair_measure_generator_def image_iff)

   299   using assms

   300   apply (rule_tac x="a \<times> xa" in exI)

   301   apply force

   302   using assms

   303   apply (rule_tac x="a" in exI)

   304   apply (rule_tac x="b \<inter> A" in exI)

   305   apply auto

   306   done

   307

   308 lemma (in pair_sigma_algebra)

   309   shows measurable_fst[intro!, simp]:

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

   311   and measurable_snd[intro!, simp]:

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

   313 proof -

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

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

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

   317       using M1.sets_into_space by force+ }

   318   moreover

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

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

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

   322       using M2.sets_into_space by force+ }

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

   324     by (fastsimp simp: measurable_def sets_sigma space_pair_measure

   325                  intro!: sigma_sets.Basic)

   326   then show ?fst ?snd by auto

   327 qed

   328

   329 lemma (in pair_sigma_algebra) measurable_pair_iff:

   330   assumes "sigma_algebra M"

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

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

   333 proof -

   334   interpret M: sigma_algebra M by fact

   335   from assms show ?thesis

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

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

   338     show "f \<in> measurable M P" unfolding pair_measure_def

   339     proof (rule M.measurable_sigma)

   340       show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"

   341         unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto

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

   343         using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)

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

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

   346         unfolding pair_measure_generator_def by auto

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

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

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

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

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

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

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

   354     qed

   355   qed

   356 qed

   357

   358 lemma (in pair_sigma_algebra) measurable_pair:

   359   assumes "sigma_algebra M"

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

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

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

   363

   364 lemma pair_measure_generatorE:

   365   assumes "X \<in> sets (pair_measure_generator M1 M2)"

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

   367   using assms unfolding pair_measure_generator_def by auto

   368

   369 lemma (in pair_sigma_algebra) pair_measure_generator_swap:

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

   371 proof (safe elim!: pair_measure_generatorE)

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

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

   374     using M1.sets_into_space M2.sets_into_space by auto

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

   376     by (auto intro: pair_measure_generatorI)

   377 next

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

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

   380     using M1.sets_into_space M2.sets_into_space

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

   382 qed

   383

   384 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:

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

   386   shows "(\<lambda>(x,y). (y, x)) - Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")

   387 proof -

   388   let "?f Q" = "(\<lambda>(x,y). (y, x)) - Q \<inter> space M2 \<times> space M1"

   389   have *: "(\<lambda>(x,y). (y, x)) - Q = ?f Q"

   390     using sets_into_space[OF Q] by (auto simp: space_pair_measure)

   391   have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"

   392     unfolding pair_measure_def ..

   393   also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f  sets E)"

   394     unfolding sigma_def pair_measure_generator_swap[symmetric]

   395     by (simp add: pair_measure_generator_def)

   396   also have "\<dots> = ?f  sigma_sets (space M1 \<times> space M2) (sets E)"

   397     using M1.sets_into_space M2.sets_into_space

   398     by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)

   399   also have "\<dots> = ?f  sets P"

   400     unfolding pair_measure_def pair_measure_generator_def sigma_def by simp

   401   finally show ?thesis

   402     using Q by (subst *) auto

   403 qed

   404

   405 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:

   406   shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)"

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

   408   unfolding measurable_def

   409 proof (intro CollectI conjI Pi_I ballI)

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

   411     unfolding pair_measure_generator_def pair_measure_def by auto

   412 next

   413   fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"

   414   interpret Q: pair_sigma_algebra M2 M1 by default

   415   with Q.sets_pair_sigma_algebra_swap[OF A \<in> sets (M2 \<Otimes>\<^isub>M M1)]

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

   417 qed

   418

   419 lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:

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

   421 proof -

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

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

   424   interpret Q: sigma_algebra ?Q

   425     proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)

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

   427     using M1.sets_into_space M2.sets_into_space

   428     by (auto simp: pair_measure_generator_def space_pair_measure)

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

   430     apply (subst pair_measure_def, intro Q.sets_sigma_subset)

   431     by (simp add: pair_measure_def)

   432   with assms show ?thesis by auto

   433 qed

   434

   435 lemma (in pair_sigma_algebra) measurable_cut_snd:

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

   437 proof -

   438   interpret Q: pair_sigma_algebra M2 M1 by default

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

   440   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)

   441 qed

   442

   443 lemma (in pair_sigma_algebra) measurable_pair_image_snd:

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

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

   446   unfolding measurable_def

   447 proof (intro CollectI conjI Pi_I ballI)

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

   449   show "f (x, y) \<in> space M"

   450     unfolding measurable_def pair_measure_generator_def pair_measure_def by auto

   451 next

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

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

   454     using f \<in> measurable P M

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

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

   457     using x \<in> space M1 by (auto simp: pair_measure_generator_def pair_measure_def)

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

   459 qed

   460

   461 lemma (in pair_sigma_algebra) measurable_pair_image_fst:

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

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

   464 proof -

   465   interpret Q: pair_sigma_algebra M2 M1 by default

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

   467                                       OF Q.pair_sigma_algebra_swap_measurable m]

   468   show ?thesis by simp

   469 qed

   470

   471 lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"

   472   unfolding Int_stable_def

   473 proof (intro ballI)

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

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

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

   477     unfolding pair_measure_generator_def by auto

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

   479     by (auto simp add: times_Int_times pair_measure_generator_def)

   480 qed

   481

   482 lemma finite_measure_cut_measurable:

   483   fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

   484   assumes "sigma_finite_measure M1" "finite_measure M2"

   485   assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"

   486   shows "(\<lambda>x. measure M2 (Pair x - Q)) \<in> borel_measurable M1"

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

   488 proof -

   489   interpret M1: sigma_finite_measure M1 by fact

   490   interpret M2: finite_measure M2 by fact

   491   interpret pair_sigma_algebra M1 M2 by default

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

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

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

   495   note space_pair_measure[simp]

   496   interpret dynkin_system ?D

   497   proof (intro dynkin_systemI)

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

   499       using sets_into_space by simp

   500   next

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

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

   503   next

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

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

   506         (if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"

   507       by (auto intro!: M2.measure_compl simp: vimage_Diff)

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

   509       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_extreal_diff)

   510   next

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

   512     moreover then have "\<And>x. measure M2 (\<Union>i. Pair x - F i) = (\<Sum>i. ?s (F i) x)"

   513       by (intro M2.measure_countably_additive[symmetric])

   514          (auto simp: disjoint_family_on_def)

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

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

   517   qed

   518   have "sets P = sets ?D" apply (subst pair_measure_def)

   519   proof (intro dynkin_lemma)

   520     show "Int_stable E" by (rule Int_stable_pair_measure_generator)

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

   522       by auto

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

   524       by (auto simp: pair_measure_generator_def sets_sigma if_distrib

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

   526   qed (auto simp: pair_measure_def)

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

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

   529 qed

   530

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

   532

   533 locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2

   534   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

   535

   536 sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2

   537   by default

   538

   539 lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"

   540   by auto

   541

   542 lemma (in pair_sigma_finite) measure_cut_measurable_fst:

   543   assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x - Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")

   544 proof -

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

   546   have M1: "sigma_finite_measure M1" by default

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

   548   then have F_sets: "\<And>i. F i \<in> sets M2" by auto

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

   550   { fix i

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

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

   553       using F M2.sets_into_space by auto

   554     let ?R2 = "M2.restricted_space (F i)"

   555     have "(\<lambda>x. measure ?R2 (Pair x - (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"

   556     proof (intro finite_measure_cut_measurable[OF M1])

   557       show "finite_measure ?R2"

   558         using F by (intro M2.restricted_to_finite_measure) auto

   559       have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i))  sets P"

   560         using Q \<in> sets P by (auto simp: image_iff)

   561       also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i))  sets E)"

   562         unfolding pair_measure_def pair_measure_generator_def sigma_def

   563         using F i \<in> sets M2 M2.sets_into_space

   564         by (auto intro!: sigma_sets_Int sigma_sets.Basic)

   565       also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"

   566         using M1.sets_into_space

   567         apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def

   568                     intro!: sigma_sets_subseteq)

   569         apply (rule_tac x="a" in exI)

   570         apply (rule_tac x="b \<inter> F i" in exI)

   571         by auto

   572       finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .

   573     qed

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

   575       using Q \<in> sets P sets_into_space by (auto simp: space_pair_measure)

   576     ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"

   577       by simp }

   578   moreover

   579   { fix x

   580     have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"

   581     proof (intro M2.measure_countably_additive)

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

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

   584       have "disjoint_family F" using F by auto

   585       show "disjoint_family (?C x)"

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

   587     qed

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

   589       using F sets_into_space Q \<in> sets P

   590       by (auto simp: space_pair_measure)

   591     finally have "measure M2 (Pair x - Q) = (\<Sum>i. measure M2 (?C x i))"

   592       by simp }

   593   ultimately show ?thesis using Q \<in> sets P F_sets

   594     by (auto intro!: M1.borel_measurable_psuminf M2.Int)

   595 qed

   596

   597 lemma (in pair_sigma_finite) measure_cut_measurable_snd:

   598   assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"

   599 proof -

   600   interpret Q: pair_sigma_finite M2 M1 by default

   601   note sets_pair_sigma_algebra_swap[OF assms]

   602   from Q.measure_cut_measurable_fst[OF this]

   603   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)

   604 qed

   605

   606 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:

   607   assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"

   608 proof -

   609   interpret Q: pair_sigma_algebra M2 M1 by default

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

   611   show ?thesis

   612     using Q.pair_sigma_algebra_swap_measurable assms

   613     unfolding * by (rule measurable_comp)

   614 qed

   615

   616 lemma (in pair_sigma_finite) pair_measure_alt:

   617   assumes "A \<in> sets P"

   618   shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x - A) \<partial>M1)"

   619   apply (simp add: pair_measure_def pair_measure_generator_def)

   620 proof (rule M1.positive_integral_cong)

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

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

   623     unfolding indicator_def by auto

   624   show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x - A)"

   625     unfolding *

   626     apply (subst M2.positive_integral_indicator)

   627     apply (rule measurable_cut_fst[OF assms])

   628     by simp

   629 qed

   630

   631 lemma (in pair_sigma_finite) pair_measure_times:

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

   633   shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"

   634 proof -

   635   have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"

   636     using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)

   637   with assms show ?thesis

   638     by (simp add: M1.positive_integral_cmult_indicator ac_simps)

   639 qed

   640

   641 lemma (in measure_space) measure_not_negative[simp,intro]:

   642   assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"

   643   using positive_measure[OF A] by auto

   644

   645 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:

   646   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>

   647     (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"

   648 proof -

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

   650     F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and

   651     F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"

   652     using M1.sigma_finite_up M2.sigma_finite_up by auto

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

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

   655   show ?thesis unfolding space_pair_measure

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

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

   658       by (fastsimp intro!: pair_measure_generatorI)

   659   next

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

   661     proof (intro subsetI)

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

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

   664         by (auto simp: space)

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

   666         using incseq F1 incseq F2 unfolding incseq_def

   667         by (force split: split_max)+

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

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

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

   671     qed

   672     then show "(\<Union>i. ?F i) = space E"

   673       using space by (auto simp: space pair_measure_generator_def)

   674   next

   675     fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"

   676       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto

   677   next

   678     fix i

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

   680     with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]

   681     show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"

   682       by (simp add: pair_measure_times)

   683   qed

   684 qed

   685

   686 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P

   687 proof

   688   show "positive P (measure P)"

   689     unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def

   690     by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)

   691

   692   show "countably_additive P (measure P)"

   693     unfolding countably_additive_def

   694   proof (intro allI impI)

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

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

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

   698     moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x - F i)) \<in> borel_measurable M1"

   699       by (intro measure_cut_measurable_fst) auto

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

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

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

   703       using F by auto

   704     ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"

   705       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]

   706                     M2.measure_countably_additive

   707                cong: M1.positive_integral_cong)

   708   qed

   709

   710   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this

   711   show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"

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

   713     show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)

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

   715       using F by (auto simp: pair_measure_def pair_measure_generator_def)

   716     show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto

   717   qed

   718 qed

   719

   720 lemma (in pair_sigma_algebra) sets_swap:

   721   assumes "A \<in> sets P"

   722   shows "(\<lambda>(x, y). (y, x)) - A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"

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

   724 proof -

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

   726     using A \<in> sets P sets_into_space by (auto simp: space_pair_measure)

   727   show ?thesis

   728     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)

   729 qed

   730

   731 lemma (in pair_sigma_finite) pair_measure_alt2:

   732   assumes A: "A \<in> sets P"

   733   shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) - A) \<partial>M2)"

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

   735 proof -

   736   interpret Q: pair_sigma_finite M2 M1 by default

   737   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this

   738   have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"

   739     unfolding pair_measure_def by simp

   740

   741   have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) - A \<inter> space Q.P)"

   742   proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])

   743     show "measure_space P" "measure_space Q.P" by default

   744     show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)

   745     show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"

   746       using assms unfolding pair_measure_def by auto

   747     show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"

   748       using F A \<in> sets P by (auto simp: pair_measure_def)

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

   750     then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"

   751       unfolding pair_measure_def pair_measure_generator_def by auto

   752     then have "(\<lambda>(y, x). (x, y)) - X \<inter> space Q.P = B \<times> A"

   753       using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)

   754     then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) - X \<inter> space Q.P)"

   755       using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)

   756   qed

   757   then show ?thesis

   758     using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]

   759     by (auto simp add: Q.pair_measure_alt space_pair_measure

   760              intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])

   761 qed

   762

   763 lemma pair_sigma_algebra_sigma:

   764   assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"

   765   assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"

   766   shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"

   767     (is "sets ?S = sets ?E")

   768 proof -

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

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

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

   772     using E1 E2 by (auto simp add: pair_measure_generator_def)

   773   interpret E: sigma_algebra ?E unfolding pair_measure_generator_def

   774     using E1 E2 by (intro sigma_algebra_sigma) auto

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

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

   777       using E1 2 unfolding pair_measure_generator_def by auto

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

   779     also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma

   780       using 2 A \<in> sets E1

   781       by (intro sigma_sets.Union)

   782          (force simp: image_subset_iff intro!: sigma_sets.Basic)

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

   784   moreover

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

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

   787       using E2 1 unfolding pair_measure_generator_def by auto

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

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

   790       using 1 B \<in> sets E2 unfolding pair_measure_generator_def sets_sigma

   791       by (intro sigma_sets.Union)

   792          (force simp: image_subset_iff intro!: sigma_sets.Basic)

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

   794   ultimately have proj:

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

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

   797                    (auto simp: pair_measure_generator_def sets_sigma)

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

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

   800       unfolding measurable_def by simp_all

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

   802       using A B M1.sets_into_space M2.sets_into_space

   803       by (auto simp: pair_measure_generator_def)

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

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

   806     by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)

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

   808     by (simp add: sets_sigma pair_measure_generator_def)

   809   show "sets ?S = sets ?E"

   810   proof (intro set_eqI iffI)

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

   812       unfolding sets_sigma

   813     proof induct

   814       case (Basic A) then show ?case

   815         by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)

   816     qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)

   817   next

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

   819   qed

   820 qed

   821

   822 section "Fubinis theorem"

   823

   824 lemma (in pair_sigma_finite) simple_function_cut:

   825   assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"

   826   shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"

   827     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"

   828 proof -

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

   830     using f(1) by (rule borel_measurable_simple_function)

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

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

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

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

   835       by (auto simp: indicator_def)

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

   837       by (simp add: space_pair_measure)

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

   839       by (intro borel_measurable_vimage measurable_cut_fst)

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

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

   842       apply (rule simple_function_indicator_representation[OF f(1)])

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

   844   note M2_sf = this

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

   846     then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f  space P. z * M2.\<mu> (?F' x z))"

   847       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]

   848       unfolding simple_integral_def

   849     proof (safe intro!: setsum_mono_zero_cong_left)

   850       from f(1) show "finite (f  space P)" by (rule simple_functionD)

   851     next

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

   853         using x \<in> space M1 by (auto simp: space_pair_measure)

   854     next

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

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

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

   858         by (force simp: space_pair_measure)

   859       show  "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"

   860         unfolding * by simp

   861     qed (simp add: vimage_compose[symmetric] comp_def

   862                    space_pair_measure) }

   863   note eq = this

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

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

   866   moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"

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

   868   moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x - (f - {i} \<inter> space P))"

   869     using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)

   870   moreover { fix i assume "i \<in> fspace P"

   871     with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x - (f - {i} \<inter> space P))"

   872       using f(2) by auto }

   873   ultimately

   874   show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"

   875     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)

   876     by (auto simp del: vimage_Int cong: measurable_cong

   877              intro!: M1.borel_measurable_extreal_setsum setsum_cong

   878              simp add: M1.positive_integral_setsum simple_integral_def

   879                        M1.positive_integral_cmult

   880                        M1.positive_integral_cong[OF eq]

   881                        positive_integral_eq_simple_integral[OF f]

   882                        pair_measure_alt[symmetric])

   883 qed

   884

   885 lemma (in pair_sigma_finite) positive_integral_fst_measurable:

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

   887   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"

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

   889     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"

   890 proof -

   891   from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this

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

   893     by (auto intro: borel_measurable_simple_function)

   894   note sf = simple_function_cut[OF F(1,5)]

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

   896     using F(1) by auto

   897   moreover

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

   899     from F measurable_pair_image_snd[OF F_borelx \<in> space M1]

   900     have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"

   901       by (intro M2.positive_integral_monotone_convergence_SUP)

   902          (auto simp: incseq_Suc_iff le_fun_def)

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

   904       unfolding F(4) positive_integral_max_0 by simp }

   905   note SUPR_C = this

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

   907     by (simp cong: measurable_cong)

   908   have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"

   909     using F_borel F

   910     by (intro positive_integral_monotone_convergence_SUP) auto

   911   also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"

   912     unfolding sf(2) by simp

   913   also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)

   914     by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])

   915        (auto intro!: M2.positive_integral_mono M2.positive_integral_positive

   916                 simp: incseq_Suc_iff le_fun_def)

   917   also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"

   918     using F_borel F(2,5)

   919     by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]

   920              simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)

   921   finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"

   922     using F by (simp add: positive_integral_max_0)

   923 qed

   924

   925 lemma (in pair_sigma_finite) measure_preserving_swap:

   926   "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"

   927 proof

   928   interpret Q: pair_sigma_finite M2 M1 by default

   929   show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"

   930     using pair_sigma_algebra_swap_measurable .

   931   fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"

   932   from measurable_sets[OF * this] this Q.sets_into_space[OF this]

   933   show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) - X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"

   934     by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]

   935       simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)

   936 qed

   937

   938 lemma (in pair_sigma_finite) positive_integral_product_swap:

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

   940   shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"

   941 proof -

   942   interpret Q: pair_sigma_finite M2 M1 by default

   943   have "sigma_algebra P" by default

   944   with f show ?thesis

   945     by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto

   946 qed

   947

   948 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

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

   950   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"

   951 proof -

   952   interpret Q: pair_sigma_finite M2 M1 by default

   953   note pair_sigma_algebra_measurable[OF f]

   954   from Q.positive_integral_fst_measurable[OF this]

   955   have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"

   956     by simp

   957   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"

   958     unfolding positive_integral_product_swap[OF f, symmetric]

   959     by (auto intro!: Q.positive_integral_cong)

   960   finally show ?thesis .

   961 qed

   962

   963 lemma (in pair_sigma_finite) Fubini:

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

   965   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"

   966   unfolding positive_integral_snd_measurable[OF assms]

   967   unfolding positive_integral_fst_measurable[OF assms] ..

   968

   969 lemma (in pair_sigma_finite) AE_pair:

   970   assumes "AE x in P. Q x"

   971   shows "AE x in M1. (AE y in M2. Q (x, y))"

   972 proof -

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

   974     using assms unfolding almost_everywhere_def by auto

   975   show ?thesis

   976   proof (rule M1.AE_I)

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

   978     show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x - N) \<noteq> 0} = 0"

   979       by (auto simp: pair_measure_alt M1.positive_integral_0_iff)

   980     show "{x \<in> space M1. M2.\<mu> (Pair x - N) \<noteq> 0} \<in> sets M1"

   981       by (intro M1.borel_measurable_extreal_neq_const measure_cut_measurable_fst N)

   982     { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x - N) = 0"

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

   984       proof (rule M2.AE_I)

   985         show "M2.\<mu> (Pair x - N) = 0" by fact

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

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

   988           using N x \<in> space M1 unfolding space_sigma space_pair_measure by auto

   989       qed }

   990     then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x - N) \<noteq> 0}"

   991       by auto

   992   qed

   993 qed

   994

   995 lemma (in pair_sigma_algebra) measurable_product_swap:

   996   "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"

   997 proof -

   998   interpret Q: pair_sigma_algebra M2 M1 by default

   999   show ?thesis

  1000     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]

  1001     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)

  1002 qed

  1003

  1004 lemma (in pair_sigma_finite) integrable_product_swap:

  1005   assumes "integrable P f"

  1006   shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"

  1007 proof -

  1008   interpret Q: pair_sigma_finite M2 M1 by default

  1009   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)

  1010   show ?thesis unfolding *

  1011     using assms unfolding integrable_def

  1012     apply (subst (1 2) positive_integral_product_swap)

  1013     using integrable P f unfolding integrable_def

  1014     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])

  1015 qed

  1016

  1017 lemma (in pair_sigma_finite) integrable_product_swap_iff:

  1018   "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"

  1019 proof -

  1020   interpret Q: pair_sigma_finite M2 M1 by default

  1021   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]

  1022   show ?thesis by auto

  1023 qed

  1024

  1025 lemma (in pair_sigma_finite) integral_product_swap:

  1026   assumes "integrable P f"

  1027   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"

  1028 proof -

  1029   interpret Q: pair_sigma_finite M2 M1 by default

  1030   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)

  1031   show ?thesis

  1032     unfolding lebesgue_integral_def *

  1033     apply (subst (1 2) positive_integral_product_swap)

  1034     using integrable P f unfolding integrable_def

  1035     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])

  1036 qed

  1037

  1038 lemma (in pair_sigma_finite) integrable_fst_measurable:

  1039   assumes f: "integrable P f"

  1040   shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")

  1041     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")

  1042 proof -

  1043   let "?pf x" = "extreal (f x)" and "?nf x" = "extreal (- f x)"

  1044   have

  1045     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and

  1046     int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"

  1047     using assms by auto

  1048   have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"

  1049      "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"

  1050     using borel[THEN positive_integral_fst_measurable(1)] int

  1051     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all

  1052   with borel[THEN positive_integral_fst_measurable(1)]

  1053   have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"

  1054     "AE x in M1. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"

  1055     by (auto intro!: M1.positive_integral_PInf_AE )

  1056   then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"

  1057     "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"

  1058     by (auto simp: M2.positive_integral_positive)

  1059   from AE_pos show ?AE using assms

  1060     by (simp add: measurable_pair_image_snd integrable_def)

  1061   { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"

  1062       using M2.positive_integral_positive

  1063       by (intro M1.positive_integral_cong_pos) (auto simp: extreal_uminus_le_reorder)

  1064     then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }

  1065   note this[simp]

  1066   { fix f assume borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable P"

  1067       and int: "integral\<^isup>P P (\<lambda>x. extreal (f x)) \<noteq> \<infinity>"

  1068       and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"

  1069     have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")

  1070     proof (intro integrable_def[THEN iffD2] conjI)

  1071       show "?f \<in> borel_measurable M1"

  1072         using borel by (auto intro!: M1.borel_measurable_real_of_extreal positive_integral_fst_measurable)

  1073       have "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y))  \<partial>M2) \<partial>M1)"

  1074         using AE M2.positive_integral_positive

  1075         by (auto intro!: M1.positive_integral_cong_AE simp: extreal_real)

  1076       then show "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) \<noteq> \<infinity>"

  1077         using positive_integral_fst_measurable[OF borel] int by simp

  1078       have "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"

  1079         by (intro M1.positive_integral_cong_pos)

  1080            (simp add: M2.positive_integral_positive real_of_extreal_pos)

  1081       then show "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp

  1082     qed }

  1083   with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]

  1084   show ?INT

  1085     unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]

  1086       borel[THEN positive_integral_fst_measurable(2), symmetric]

  1087     using AE[THEN M1.integral_real]

  1088     by simp

  1089 qed

  1090

  1091 lemma (in pair_sigma_finite) integrable_snd_measurable:

  1092   assumes f: "integrable P f"

  1093   shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")

  1094     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")

  1095 proof -

  1096   interpret Q: pair_sigma_finite M2 M1 by default

  1097   have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"

  1098     using f unfolding integrable_product_swap_iff .

  1099   show ?INT

  1100     using Q.integrable_fst_measurable(2)[OF Q_int]

  1101     using integral_product_swap[OF f] by simp

  1102   show ?AE

  1103     using Q.integrable_fst_measurable(1)[OF Q_int]

  1104     by simp

  1105 qed

  1106

  1107 lemma (in pair_sigma_finite) Fubini_integral:

  1108   assumes f: "integrable P f"

  1109   shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"

  1110   unfolding integrable_snd_measurable[OF assms]

  1111   unfolding integrable_fst_measurable[OF assms] ..

  1112

  1113 section "Finite product spaces"

  1114

  1115 section "Products"

  1116

  1117 locale product_sigma_algebra =

  1118   fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme"

  1119   assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"

  1120

  1121 locale finite_product_sigma_algebra = product_sigma_algebra M

  1122   for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +

  1123   fixes I :: "'i set"

  1124   assumes finite_index: "finite I"

  1125

  1126 definition

  1127   "product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)),

  1128     sets = Pi\<^isub>E I  (\<Pi> i \<in> I. sets (M i)),

  1129     measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>"

  1130

  1131 definition product_algebra_def:

  1132   "Pi\<^isub>M I M = sigma (product_algebra_generator I M)

  1133     \<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and>

  1134       (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>"

  1135

  1136 syntax

  1137   "_PiM"  :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>

  1138               ('i => 'a, 'b) measure_space_scheme"  ("(3PIM _:_./ _)" 10)

  1139

  1140 syntax (xsymbols)

  1141   "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>

  1142              ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)

  1143

  1144 syntax (HTML output)

  1145   "_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>

  1146              ('i => 'a, 'b) measure_space_scheme"  ("(3\<Pi>\<^isub>M _\<in>_./ _)"   10)

  1147

  1148 translations

  1149   "PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)"

  1150

  1151 abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M"

  1152 abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M"

  1153

  1154 sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)

  1155

  1156 lemma sigma_into_space:

  1157   assumes "sets M \<subseteq> Pow (space M)"

  1158   shows "sets (sigma M) \<subseteq> Pow (space M)"

  1159   using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto

  1160

  1161 lemma (in product_sigma_algebra) product_algebra_generator_into_space:

  1162   "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"

  1163   using M.sets_into_space unfolding product_algebra_generator_def

  1164   by auto blast

  1165

  1166 lemma (in product_sigma_algebra) product_algebra_into_space:

  1167   "sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))"

  1168   using product_algebra_generator_into_space

  1169   by (auto intro!: sigma_into_space simp add: product_algebra_def)

  1170

  1171 lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)"

  1172   using product_algebra_generator_into_space unfolding product_algebra_def

  1173   by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all

  1174

  1175 sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P

  1176   using sigma_algebra_product_algebra .

  1177

  1178 lemma product_algebraE:

  1179   assumes "A \<in> sets (product_algebra_generator I M)"

  1180   obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1181   using assms unfolding product_algebra_generator_def by auto

  1182

  1183 lemma product_algebra_generatorI[intro]:

  1184   assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"

  1185   shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)"

  1186   using assms unfolding product_algebra_generator_def by auto

  1187

  1188 lemma space_product_algebra_generator[simp]:

  1189   "space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))"

  1190   unfolding product_algebra_generator_def by simp

  1191

  1192 lemma space_product_algebra[simp]:

  1193   "space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))"

  1194   unfolding product_algebra_def product_algebra_generator_def by simp

  1195

  1196 lemma sets_product_algebra:

  1197   "sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))"

  1198   unfolding product_algebra_def sigma_def by simp

  1199

  1200 lemma product_algebra_generator_sets_into_space:

  1201   assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"

  1202   shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"

  1203   using assms by (auto simp: product_algebra_generator_def) blast

  1204

  1205 lemma (in finite_product_sigma_algebra) in_P[simp, intro]:

  1206   "\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"

  1207   by (auto simp: sets_product_algebra)

  1208

  1209 section "Generating set generates also product algebra"

  1210

  1211 lemma sigma_product_algebra_sigma_eq:

  1212   assumes "finite I"

  1213   assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"

  1214   assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"

  1215   assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"

  1216   and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"

  1217   shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"

  1218     (is "sets ?S = sets ?E")

  1219 proof cases

  1220   assume "I = {}" then show ?thesis

  1221     by (simp add: product_algebra_def product_algebra_generator_def)

  1222 next

  1223   assume "I \<noteq> {}"

  1224   interpret E: sigma_algebra "sigma (E i)" for i

  1225     using E by (rule sigma_algebra_sigma)

  1226   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)"

  1227     using E by auto

  1228   interpret G: sigma_algebra ?E

  1229     unfolding product_algebra_def product_algebra_generator_def using E

  1230     by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)

  1231   { fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"

  1232     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"

  1233       using mono union unfolding incseq_Suc_iff space_product_algebra

  1234       by (auto dest: Pi_mem)

  1235     also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"

  1236       unfolding space_product_algebra

  1237       apply simp

  1238       apply (subst Pi_UN[OF finite I])

  1239       using mono[THEN incseqD] apply simp

  1240       apply (simp add: PiE_Int)

  1241       apply (intro PiE_cong)

  1242       using A sets_into by (auto intro!: into_space)

  1243     also have "\<dots> \<in> sets ?E"

  1244       using sets_into A \<in> sets (E i)

  1245       unfolding sets_product_algebra sets_sigma

  1246       by (intro sigma_sets.Union)

  1247          (auto simp: image_subset_iff intro!: sigma_sets.Basic)

  1248     finally have "(\<lambda>x. x i) - A \<inter> space ?E \<in> sets ?E" . }

  1249   then have proj:

  1250     "\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"

  1251     using E by (subst G.measurable_iff_sigma)

  1252                (auto simp: sets_product_algebra sets_sigma)

  1253   { fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"

  1254     with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) - (A i) \<inter> space ?E \<in> sets ?E"

  1255       unfolding measurable_def by simp

  1256     have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) - (A i) \<inter> space ?E)"

  1257       using A E.sets_into_space I \<noteq> {} unfolding product_algebra_def by auto blast

  1258     then have "Pi\<^isub>E I A \<in> sets ?E"

  1259       using G.finite_INT[OF finite I I \<noteq> {} basic, of "\<lambda>i. i"] by simp }

  1260   then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E"

  1261     by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def)

  1262   then have subset: "sets ?S \<subseteq> sets ?E"

  1263     by (simp add: sets_sigma sets_product_algebra)

  1264   show "sets ?S = sets ?E"

  1265   proof (intro set_eqI iffI)

  1266     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"

  1267       unfolding sets_sigma sets_product_algebra

  1268     proof induct

  1269       case (Basic A) then show ?case

  1270         by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic)

  1271     qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def)

  1272   next

  1273     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto

  1274   qed

  1275 qed

  1276

  1277 lemma product_algebraI[intro]:

  1278     "E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M I M)"

  1279   using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI)

  1280

  1281 lemma (in product_sigma_algebra) measurable_component_update:

  1282   assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"

  1283   shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")

  1284   unfolding product_algebra_def apply simp

  1285 proof (intro measurable_sigma)

  1286   let ?G = "product_algebra_generator (insert i I) M"

  1287   show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space .

  1288   show "?f \<in> space (M i) \<rightarrow> space ?G"

  1289     using M.sets_into_space assms by auto

  1290   fix A assume "A \<in> sets ?G"

  1291   from product_algebraE[OF this] guess E . note E = this

  1292   then have "?f - A \<inter> space (M i) = E i \<or> ?f - A \<inter> space (M i) = {}"

  1293     using M.sets_into_space assms by auto

  1294   then show "?f - A \<inter> space (M i) \<in> sets (M i)"

  1295     using E by (auto intro!: product_algebraI)

  1296 qed

  1297

  1298 lemma (in product_sigma_algebra) measurable_add_dim:

  1299   assumes "i \<notin> I"

  1300   shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"

  1301 proof -

  1302   let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (insert i I) M"

  1303   interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M i"

  1304     unfolding pair_sigma_algebra_def

  1305     by (intro sigma_algebra_product_algebra sigma_algebras conjI)

  1306   have "?f \<in> measurable Ii.P (sigma ?G)"

  1307   proof (rule Ii.measurable_sigma)

  1308     show "sets ?G \<subseteq> Pow (space ?G)"

  1309       using product_algebra_generator_into_space .

  1310     show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G"

  1311       by (auto simp: space_pair_measure)

  1312   next

  1313     fix A assume "A \<in> sets ?G"

  1314     then obtain F where "A = Pi\<^isub>E (insert i I) F"

  1315       and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M i)"

  1316       by (auto elim!: product_algebraE)

  1317     then have "?f - A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)"

  1318       using sets_into_space i \<notin> I

  1319       by (auto simp add: space_pair_measure) blast+

  1320     then show "?f - A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)"

  1321       using F by (auto intro!: pair_measureI)

  1322   qed

  1323   then show ?thesis

  1324     by (simp add: product_algebra_def)

  1325 qed

  1326

  1327 lemma (in product_sigma_algebra) measurable_merge:

  1328   assumes [simp]: "I \<inter> J = {}"

  1329   shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"

  1330 proof -

  1331   let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M J M"

  1332   interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J"

  1333     by (intro sigma_algebra_pair_measure product_algebra_into_space)

  1334   let ?f = "\<lambda>(x, y). merge I x J y"

  1335   let ?G = "product_algebra_generator (I \<union> J) M"

  1336   have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)"

  1337   proof (rule P.measurable_sigma)

  1338     fix A assume "A \<in> sets ?G"

  1339     from product_algebraE[OF this]

  1340     obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .

  1341     then have *: "?f - A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"

  1342       using sets_into_space I \<inter> J = {}

  1343       by (auto simp add: space_pair_measure) fast+

  1344     then show "?f - A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"

  1345       using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI

  1346         simp: product_algebra_def)

  1347   qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure)

  1348   then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (Pi\<^isub>M (I \<union> J) M)"

  1349     unfolding product_algebra_def[of "I \<union> J"] by simp

  1350 qed

  1351

  1352 lemma (in product_sigma_algebra) measurable_component_singleton:

  1353   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"

  1354 proof (unfold measurable_def, intro CollectI conjI ballI)

  1355   fix A assume "A \<in> sets (M i)"

  1356   then have "(\<lambda>x. x i) - A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"

  1357     using M.sets_into_space i \<in> I by (fastsimp dest: Pi_mem split: split_if_asm)

  1358   then show "(\<lambda>x. x i) - A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"

  1359     using A \<in> sets (M i) by (auto intro!: product_algebraI)

  1360 qed (insert i \<in> I, auto)

  1361

  1362 locale product_sigma_finite =

  1363   fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme"

  1364   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"

  1365

  1366 locale finite_product_sigma_finite = product_sigma_finite M

  1367   for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" +

  1368   fixes I :: "'i set" assumes finite_index'[intro]: "finite I"

  1369

  1370 sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i

  1371   by (rule sigma_finite_measures)

  1372

  1373 sublocale product_sigma_finite \<subseteq> product_sigma_algebra

  1374   by default

  1375

  1376 sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra

  1377   by default (fact finite_index')

  1378

  1379 lemma setprod_extreal_0:

  1380   fixes f :: "'a \<Rightarrow> extreal"

  1381   shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"

  1382 proof cases

  1383   assume "finite A"

  1384   then show ?thesis by (induct A) auto

  1385 qed auto

  1386

  1387 lemma setprod_extreal_pos:

  1388   fixes f :: "'a \<Rightarrow> extreal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)"

  1389 proof cases

  1390   assume "finite I" from this pos show ?thesis by induct auto

  1391 qed simp

  1392

  1393 lemma setprod_PInf:

  1394   assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"

  1395   shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"

  1396 proof cases

  1397   assume "finite I" from this assms show ?thesis

  1398   proof (induct I)

  1399     case (insert i I)

  1400     then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_extreal_pos)

  1401     from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto

  1402     also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"

  1403       using setprod_extreal_pos[of I f] pos

  1404       by (cases rule: extreal2_cases[of "f i" "setprod f I"]) auto

  1405     also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)"

  1406       using insert by (auto simp: setprod_extreal_0)

  1407     finally show ?case .

  1408   qed simp

  1409 qed simp

  1410

  1411 lemma setprod_extreal: "(\<Prod>i\<in>A. extreal (f i)) = extreal (setprod f A)"

  1412 proof cases

  1413   assume "finite A" then show ?thesis

  1414     by induct (auto simp: one_extreal_def)

  1415 qed (simp add: one_extreal_def)

  1416

  1417 lemma (in finite_product_sigma_finite) product_algebra_generator_measure:

  1418   assumes "Pi\<^isub>E I F \<in> sets G"

  1419   shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"

  1420 proof cases

  1421   assume ne: "\<forall>i\<in>I. F i \<noteq> {}"

  1422   have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') i = F i"

  1423     by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])

  1424        (insert ne, auto simp: Pi_eq_iff)

  1425   then show ?thesis

  1426     unfolding product_algebra_generator_def by simp

  1427 next

  1428   assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"

  1429   then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"

  1430     by (auto simp: setprod_extreal_0 intro!: bexI)

  1431   moreover

  1432   have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"

  1433     by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])

  1434        (insert empty, auto simp: Pi_eq_empty_iff[symmetric])

  1435   then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"

  1436     by (auto simp: setprod_extreal_0 intro!: bexI)

  1437   ultimately show ?thesis

  1438     unfolding product_algebra_generator_def by simp

  1439 qed

  1440

  1441 lemma (in finite_product_sigma_finite) sigma_finite_pairs:

  1442   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.

  1443     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>

  1444     (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>

  1445     (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"

  1446 proof -

  1447   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)"

  1448     using M.sigma_finite_up by simp

  1449   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..

  1450   then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>"

  1451     by auto

  1452   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"

  1453   note space_product_algebra[simp]

  1454   show ?thesis

  1455   proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)

  1456     fix i show "range (F i) \<subseteq> sets (M i)" by fact

  1457   next

  1458     fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact

  1459   next

  1460     fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"

  1461       using \<And>i. range (F i) \<subseteq> sets (M i) M.sets_into_space

  1462       by (force simp: image_subset_iff)

  1463   next

  1464     fix f assume "f \<in> space G"

  1465     with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F

  1466     show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)

  1467   next

  1468     fix i show "?F i \<subseteq> ?F (Suc i)"

  1469       using \<And>i. incseq (F i)[THEN incseq_SucD] by auto

  1470   qed

  1471 qed

  1472

  1473 lemma sets_pair_cancel_measure[simp]:

  1474   "sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)"

  1475   "sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)"

  1476   unfolding pair_measure_def pair_measure_generator_def sets_sigma

  1477   by simp_all

  1478

  1479 lemma measurable_pair_cancel_measure[simp]:

  1480   "measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M"

  1481   "measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M"

  1482   "measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)"

  1483   "measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)"

  1484   unfolding measurable_def by (auto simp add: space_pair_measure)

  1485

  1486 lemma (in product_sigma_finite) product_measure_exists:

  1487   assumes "finite I"

  1488   shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and>

  1489     (\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))"

  1490 using finite I proof induct

  1491   case empty

  1492   interpret finite_product_sigma_finite M "{}" by default simp

  1493   let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> extreal"

  1494   show ?case

  1495   proof (intro exI conjI ballI)

  1496     have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"

  1497       by (rule sigma_algebra_cong) (simp_all add: product_algebra_def)

  1498     then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)"

  1499       by (rule finite_additivity_sufficient)

  1500          (simp_all add: positive_def additive_def sets_sigma

  1501                         product_algebra_generator_def image_constant)

  1502     then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)"

  1503       by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"]

  1504                simp: sigma_finite_measure_def sigma_finite_measure_axioms_def

  1505                      product_algebra_generator_def)

  1506   qed auto

  1507 next

  1508   case (insert i I)

  1509   interpret finite_product_sigma_finite M I by default fact

  1510   have "finite (insert i I)" using finite I by auto

  1511   interpret I': finite_product_sigma_finite M "insert i I" by default fact

  1512   from insert obtain \<nu> where

  1513     prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and

  1514     "sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by auto

  1515   then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp

  1516   interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" ..

  1517   let ?h = "(\<lambda>(f, y). f(i := y))"

  1518   let ?\<nu> = "\<lambda>A. P.\<mu> (?h - A \<inter> space P.P)"

  1519   have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)"

  1520     by (rule I'.sigma_algebra_cong) simp_all

  1521   interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>"

  1522     using measurable_add_dim[OF i \<notin> I]

  1523     by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def)

  1524   show ?case

  1525   proof (intro exI[of _ ?\<nu>] conjI ballI)

  1526     let "?m A" = "measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h - A \<inter> space P.P)"

  1527     { fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"

  1528       then have *: "?h - Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"

  1529         using i \<notin> I M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast

  1530       show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))"

  1531         unfolding * using A

  1532         apply (subst P.pair_measure_times)

  1533         using A apply fastsimp

  1534         using A apply fastsimp

  1535         using i \<notin> I finite I prod[of A] A by (auto simp: ac_simps) }

  1536     note product = this

  1537     have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>"

  1538       by (simp add: product_algebra_def)

  1539     show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)"

  1540     proof (unfold *, default, simp)

  1541       from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..

  1542       then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"

  1543         "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"

  1544         "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"

  1545         "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"

  1546         by blast+

  1547       let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"

  1548       show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>

  1549           (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"

  1550       proof (intro exI[of _ ?F] conjI allI)

  1551         show "range ?F \<subseteq> sets I'.P" using F(1) by auto

  1552       next

  1553         from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp

  1554       next

  1555         fix j

  1556         have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"

  1557           using F(1) by auto

  1558         with F finite I setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"

  1559           by (subst product) auto

  1560       qed

  1561     qed

  1562   qed

  1563 qed

  1564

  1565 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P

  1566   unfolding product_algebra_def

  1567   using product_measure_exists[OF finite_index]

  1568   by (rule someI2_ex) auto

  1569

  1570 lemma (in finite_product_sigma_finite) measure_times:

  1571   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"

  1572   shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"

  1573   unfolding product_algebra_def

  1574   using product_measure_exists[OF finite_index]

  1575   proof (rule someI2_ex, elim conjE)

  1576     fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"

  1577     have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)

  1578     then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp

  1579     also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto

  1580     finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"

  1581       by (simp add: sigma_def)

  1582   qed

  1583

  1584 lemma (in product_sigma_finite) product_measure_empty[simp]:

  1585   "measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"

  1586 proof -

  1587   interpret finite_product_sigma_finite M "{}" by default auto

  1588   from measure_times[of "\<lambda>x. {}"] show ?thesis by simp

  1589 qed

  1590

  1591 lemma (in finite_product_sigma_algebra) P_empty:

  1592   assumes "I = {}"

  1593   shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }"

  1594   unfolding product_algebra_def product_algebra_generator_def I = {}

  1595   by (simp_all add: sigma_def image_constant)

  1596

  1597 lemma (in product_sigma_finite) positive_integral_empty:

  1598   assumes pos: "0 \<le> f (\<lambda>k. undefined)"

  1599   shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"

  1600 proof -

  1601   interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)

  1602   have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"

  1603     using assms by (subst measure_times) auto

  1604   then show ?thesis

  1605     unfolding positive_integral_def simple_function_def simple_integral_def_raw

  1606   proof (simp add: P_empty, intro antisym)

  1607     show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"

  1608       by (intro le_SUPI) (auto simp: le_fun_def split: split_max)

  1609     show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos

  1610       by (intro SUP_leI) (auto simp: le_fun_def simp: max_def split: split_if_asm)

  1611   qed

  1612 qed

  1613

  1614 lemma (in product_sigma_finite) measure_fold:

  1615   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1616   assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"

  1617   shows "measure (Pi\<^isub>M (I \<union> J) M) A =

  1618     measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) - A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))"

  1619 proof -

  1620   interpret I: finite_product_sigma_finite M I by default fact

  1621   interpret J: finite_product_sigma_finite M J by default fact

  1622   have "finite (I \<union> J)" using fin by auto

  1623   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact

  1624   interpret P: pair_sigma_finite I.P J.P by default

  1625   let ?g = "\<lambda>(x,y). merge I x J y"

  1626   let "?X S" = "?g - S \<inter> space P.P"

  1627   from IJ.sigma_finite_pairs obtain F where

  1628     F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"

  1629        "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"

  1630        "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"

  1631        "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"

  1632     by auto

  1633   let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"

  1634   show "IJ.\<mu> A = P.\<mu> (?X A)"

  1635   proof (rule measure_unique_Int_stable_vimage)

  1636     show "measure_space IJ.P" "measure_space P.P" by default

  1637     show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)"

  1638       using A unfolding product_algebra_def by auto

  1639   next

  1640     show "Int_stable IJ.G"

  1641       by (simp add: PiE_Int Int_stable_def product_algebra_def

  1642                     product_algebra_generator_def)

  1643           auto

  1644     show "range ?F \<subseteq> sets IJ.G" using F

  1645       by (simp add: image_subset_iff product_algebra_def

  1646                     product_algebra_generator_def)

  1647     show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+

  1648   next

  1649     fix k

  1650     have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"

  1651       using F(1) by auto

  1652     with F finite I setprod_PInf[of "I \<union> J", OF this]

  1653     show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto

  1654   next

  1655     fix A assume "A \<in> sets IJ.G"

  1656     then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"

  1657       and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"

  1658       by (auto simp: product_algebra_generator_def)

  1659     then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"

  1660       using sets_into_space by (auto simp: space_pair_measure) blast+

  1661     then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"

  1662       using finite J finite I F

  1663       by (simp add: P.pair_measure_times I.measure_times J.measure_times)

  1664     also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"

  1665       using finite J finite I I \<inter> J = {}  by (simp add: setprod_Un_one)

  1666     also have "\<dots> = IJ.\<mu> A"

  1667       using finite J finite I F unfolding A

  1668       by (intro IJ.measure_times[symmetric]) auto

  1669     finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)

  1670   qed (rule measurable_merge[OF IJ])

  1671 qed

  1672

  1673 lemma (in product_sigma_finite) measure_preserving_merge:

  1674   assumes IJ: "I \<inter> J = {}" and "finite I" "finite J"

  1675   shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"

  1676   by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms)

  1677

  1678 lemma (in product_sigma_finite) product_positive_integral_fold:

  1679   assumes IJ[simp]: "I \<inter> J = {}" "finite I" "finite J"

  1680   and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"

  1681   shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =

  1682     (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"

  1683 proof -

  1684   interpret I: finite_product_sigma_finite M I by default fact

  1685   interpret J: finite_product_sigma_finite M J by default fact

  1686   interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default

  1687   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp

  1688   have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"

  1689     using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def)

  1690   show ?thesis

  1691     unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]

  1692   proof (rule P.positive_integral_vimage)

  1693     show "sigma_algebra IJ.P" by default

  1694     show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P"

  1695       using IJ by (rule measure_preserving_merge)

  1696     show "f \<in> borel_measurable IJ.P" using f by simp

  1697   qed

  1698 qed

  1699

  1700 lemma (in product_sigma_finite) measure_preserving_component_singelton:

  1701   "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"

  1702 proof (intro measure_preservingI measurable_component_singleton)

  1703   interpret I: finite_product_sigma_finite M "{i}" by default simp

  1704   fix A let ?P = "(\<lambda>x. x i) - A \<inter> space I.P"

  1705   assume A: "A \<in> sets (M i)"

  1706   then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto

  1707   show "I.\<mu> ?P = M.\<mu> i A" unfolding *

  1708     using A I.measure_times[of "\<lambda>_. A"] by auto

  1709 qed auto

  1710

  1711 lemma (in product_sigma_finite) product_positive_integral_singleton:

  1712   assumes f: "f \<in> borel_measurable (M i)"

  1713   shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"

  1714 proof -

  1715   interpret I: finite_product_sigma_finite M "{i}" by default simp

  1716   show ?thesis

  1717   proof (rule I.positive_integral_vimage[symmetric])

  1718     show "sigma_algebra (M i)" by (rule sigma_algebras)

  1719     show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"

  1720       by (rule measure_preserving_component_singelton)

  1721     show "f \<in> borel_measurable (M i)" by fact

  1722   qed

  1723 qed

  1724

  1725 lemma (in product_sigma_finite) product_positive_integral_insert:

  1726   assumes [simp]: "finite I" "i \<notin> I"

  1727     and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"

  1728   shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"

  1729 proof -

  1730   interpret I: finite_product_sigma_finite M I by default auto

  1731   interpret i: finite_product_sigma_finite M "{i}" by default auto

  1732   interpret P: pair_sigma_algebra I.P i.P ..

  1733   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"

  1734     using f by auto

  1735   show ?thesis

  1736     unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]

  1737   proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)

  1738     fix x assume x: "x \<in> space I.P"

  1739     let "?f y" = "f (restrict (x(i := y)) (insert i I))"

  1740     have f'_eq: "\<And>y. ?f y = f (x(i := y))"

  1741       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1742     show "?f \<in> borel_measurable (M i)" unfolding f'_eq

  1743       using measurable_comp[OF measurable_component_update f] x i \<notin> I

  1744       by (simp add: comp_def)

  1745     show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"

  1746       unfolding f'_eq by simp

  1747   qed

  1748 qed

  1749

  1750 lemma (in product_sigma_finite) product_positive_integral_setprod:

  1751   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> extreal"

  1752   assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"

  1753   and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"

  1754   shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"

  1755 using assms proof induct

  1756   case empty

  1757   interpret finite_product_sigma_finite M "{}" by default auto

  1758   then show ?case by simp

  1759 next

  1760   case (insert i I)

  1761   note finite I[intro, simp]

  1762   interpret I: finite_product_sigma_finite M I by default auto

  1763   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))"

  1764     using insert by (auto intro!: setprod_cong)

  1765   have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"

  1766     using sets_into_space insert

  1767     by (intro sigma_algebra.borel_measurable_extreal_setprod sigma_algebra_product_algebra

  1768               measurable_comp[OF measurable_component_singleton, unfolded comp_def])

  1769        auto

  1770   then show ?case

  1771     apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])

  1772     apply (simp add: insert * pos borel setprod_extreal_pos M.positive_integral_multc)

  1773     apply (subst I.positive_integral_cmult)

  1774     apply (auto simp add: pos borel insert setprod_extreal_pos positive_integral_positive)

  1775     done

  1776 qed

  1777

  1778 lemma (in product_sigma_finite) product_integral_singleton:

  1779   assumes f: "f \<in> borel_measurable (M i)"

  1780   shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"

  1781 proof -

  1782   interpret I: finite_product_sigma_finite M "{i}" by default simp

  1783   have *: "(\<lambda>x. extreal (f x)) \<in> borel_measurable (M i)"

  1784     "(\<lambda>x. extreal (- f x)) \<in> borel_measurable (M i)"

  1785     using assms by auto

  1786   show ?thesis

  1787     unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..

  1788 qed

  1789

  1790 lemma (in product_sigma_finite) product_integral_fold:

  1791   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"

  1792   and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"

  1793   shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I x J y) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"

  1794 proof -

  1795   interpret I: finite_product_sigma_finite M I by default fact

  1796   interpret J: finite_product_sigma_finite M J by default fact

  1797   have "finite (I \<union> J)" using fin by auto

  1798   interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact

  1799   interpret P: pair_sigma_finite I.P J.P by default

  1800   let ?M = "\<lambda>(x, y). merge I x J y"

  1801   let ?f = "\<lambda>x. f (?M x)"

  1802   show ?thesis

  1803   proof (subst P.integrable_fst_measurable(2)[of ?f, simplified])

  1804     have 1: "sigma_algebra IJ.P" by default

  1805     have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] .

  1806     have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact

  1807     then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"

  1808       by (simp add: integrable_def)

  1809     show "integrable P.P ?f"

  1810       by (rule P.integrable_vimage[where f=f, OF 1 2 3])

  1811     show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f"

  1812       by (rule P.integral_vimage[where f=f, OF 1 2 4])

  1813   qed

  1814 qed

  1815

  1816 lemma (in product_sigma_finite) product_integral_insert:

  1817   assumes [simp]: "finite I" "i \<notin> I"

  1818     and f: "integrable (Pi\<^isub>M (insert i I) M) f"

  1819   shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"

  1820 proof -

  1821   interpret I: finite_product_sigma_finite M I by default auto

  1822   interpret I': finite_product_sigma_finite M "insert i I" by default auto

  1823   interpret i: finite_product_sigma_finite M "{i}" by default auto

  1824   interpret P: pair_sigma_finite I.P i.P ..

  1825   have IJ: "I \<inter> {i} = {}" by auto

  1826   show ?thesis

  1827     unfolding product_integral_fold[OF IJ, simplified, OF f]

  1828   proof (rule I.integral_cong, subst product_integral_singleton)

  1829     fix x assume x: "x \<in> space I.P"

  1830     let "?f y" = "f (restrict (x(i := y)) (insert i I))"

  1831     have f'_eq: "\<And>y. ?f y = f (x(i := y))"

  1832       using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)

  1833     have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto

  1834     show "?f \<in> borel_measurable (M i)"

  1835       unfolding measurable_cong[OF f'_eq]

  1836       using measurable_comp[OF measurable_component_update f] x i \<notin> I

  1837       by (simp add: comp_def)

  1838     show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))"

  1839       unfolding f'_eq by simp

  1840   qed

  1841 qed

  1842

  1843 lemma (in product_sigma_finite) product_integrable_setprod:

  1844   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"

  1845   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"

  1846   shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")

  1847 proof -

  1848   interpret finite_product_sigma_finite M I by default fact

  1849   have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"

  1850     using integrable unfolding integrable_def by auto

  1851   then have borel: "?f \<in> borel_measurable P"

  1852     using measurable_comp[OF measurable_component_singleton f]

  1853     by (auto intro!: borel_measurable_setprod simp: comp_def)

  1854   moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"

  1855   proof (unfold integrable_def, intro conjI)

  1856     show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"

  1857       using borel by auto

  1858     have "(\<integral>\<^isup>+x. extreal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. extreal (abs (f i (x i)))) \<partial>P)"

  1859       by (simp add: setprod_extreal abs_setprod)

  1860     also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. extreal (abs (f i x)) \<partial>M i))"

  1861       using f by (subst product_positive_integral_setprod) auto

  1862     also have "\<dots> < \<infinity>"

  1863       using integrable[THEN M.integrable_abs]

  1864       by (simp add: setprod_PInf integrable_def M.positive_integral_positive)

  1865     finally show "(\<integral>\<^isup>+x. extreal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto

  1866     have "(\<integral>\<^isup>+x. extreal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"

  1867       by (intro positive_integral_cong_pos) auto

  1868     then show "(\<integral>\<^isup>+x. extreal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp

  1869   qed

  1870   ultimately show ?thesis

  1871     by (rule integrable_abs_iff[THEN iffD1])

  1872 qed

  1873

  1874 lemma (in product_sigma_finite) product_integral_setprod:

  1875   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"

  1876   assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"

  1877   shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"

  1878 using assms proof (induct rule: finite_ne_induct)

  1879   case (singleton i)

  1880   then show ?case by (simp add: product_integral_singleton integrable_def)

  1881 next

  1882   case (insert i I)

  1883   then have iI: "finite (insert i I)" by auto

  1884   then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>

  1885     integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"

  1886     by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)

  1887   interpret I: finite_product_sigma_finite M I by default fact

  1888   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))"

  1889     using i \<notin> I by (auto intro!: setprod_cong)

  1890   show ?case

  1891     unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]

  1892     by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)

  1893 qed

  1894

  1895 section "Products on finite spaces"

  1896

  1897 lemma sigma_sets_pair_measure_generator_finite:

  1898   assumes "finite A" and "finite B"

  1899   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"

  1900   (is "sigma_sets ?prod ?sets = _")

  1901 proof safe

  1902   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)

  1903   fix x assume subset: "x \<subseteq> A \<times> B"

  1904   hence "finite x" using fin by (rule finite_subset)

  1905   from this subset show "x \<in> sigma_sets ?prod ?sets"

  1906   proof (induct x)

  1907     case empty show ?case by (rule sigma_sets.Empty)

  1908   next

  1909     case (insert a x)

  1910     hence "{a} \<in> sigma_sets ?prod ?sets"

  1911       by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)

  1912     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto

  1913     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)

  1914   qed

  1915 next

  1916   fix x a b

  1917   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"

  1918   from sigma_sets_into_sp[OF _ this(1)] this(2)

  1919   show "a \<in> A" and "b \<in> B" by auto

  1920 qed

  1921

  1922 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2

  1923   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"

  1924

  1925 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default

  1926

  1927 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:

  1928   shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"

  1929 proof -

  1930   show ?thesis

  1931     using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]

  1932     by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)

  1933 qed

  1934

  1935 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P

  1936   apply default

  1937   using M1.finite_space M2.finite_space

  1938   apply (subst finite_pair_sigma_algebra) apply simp

  1939   apply (subst (1 2) finite_pair_sigma_algebra) apply simp

  1940   done

  1941

  1942 locale pair_finite_space = M1: finite_measure_space M1 + M2: finite_measure_space M2

  1943   for M1 M2

  1944

  1945 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra

  1946   by default

  1947

  1948 sublocale pair_finite_space \<subseteq> pair_sigma_finite

  1949   by default

  1950

  1951 lemma (in pair_finite_space) pair_measure_Pair[simp]:

  1952   assumes "a \<in> space M1" "b \<in> space M2"

  1953   shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"

  1954 proof -

  1955   have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"

  1956     using M1.sets_eq_Pow M2.sets_eq_Pow assms

  1957     by (subst pair_measure_times) auto

  1958   then show ?thesis by simp

  1959 qed

  1960

  1961 lemma (in pair_finite_space) pair_measure_singleton[simp]:

  1962   assumes "x \<in> space M1 \<times> space M2"

  1963   shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"

  1964   using pair_measure_Pair assms by (cases x) auto

  1965

  1966 sublocale pair_finite_space \<subseteq> finite_measure_space P

  1967   by default (auto simp: space_pair_measure)

  1968

  1969 end`