src/HOL/Probability/Binary_Product_Measure.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46898 1570b30ee040 child 47694 05663f75964c permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
     1 (*  Title:      HOL/Probability/Binary_Product_Measure.thy

     2     Author:     Johannes Hölzl, TU München

     3 *)

     4

     5 header {*Binary product measures*}

     6

     7 theory Binary_Product_Measure

     8 imports Lebesgue_Integration

     9 begin

    10

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

    12   by auto

    13

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

    15   by auto

    16

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

    18   by auto

    19

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

    21   by auto

    22

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

    24   by (cases x) simp

    25

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

    27   by (auto simp: fun_eq_iff)

    28

    29 section "Binary products"

    30

    31 definition

    32   "pair_measure_generator A B =

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

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

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

    36

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

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

    39

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

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

    42

    43 abbreviation (in pair_sigma_algebra)

    44   "E \<equiv> pair_measure_generator M1 M2"

    45

    46 abbreviation (in pair_sigma_algebra)

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

    48

    49 lemma sigma_algebra_pair_measure:

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

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

    52

    53 sublocale pair_sigma_algebra \<subseteq> sigma_algebra P

    54   using M1.space_closed M2.space_closed

    55   by (rule sigma_algebra_pair_measure)

    56

    57 lemma pair_measure_generatorI[intro, simp]:

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

    59   by (auto simp add: pair_measure_generator_def)

    60

    61 lemma pair_measureI[intro, simp]:

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

    63   by (auto simp add: pair_measure_def)

    64

    65 lemma space_pair_measure:

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

    67   by (simp add: pair_measure_def pair_measure_generator_def)

    68

    69 lemma sets_pair_measure_generator:

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

    71   unfolding pair_measure_generator_def by auto

    72

    73 lemma pair_measure_generator_sets_into_space:

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

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

    76   using assms by (auto simp: pair_measure_generator_def)

    77

    78 lemma pair_measure_generator_Int_snd:

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

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

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

    82   (is "?L = ?R")

    83   apply (auto simp: pair_measure_generator_def image_iff)

    84   using assms

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

    86   apply force

    87   using assms

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

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

    90   apply auto

    91   done

    92

    93 lemma (in pair_sigma_algebra)

    94   shows measurable_fst[intro!, simp]:

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

    96   and measurable_snd[intro!, simp]:

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

    98 proof -

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

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

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

   102       using M1.sets_into_space by force+ }

   103   moreover

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

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

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

   107       using M2.sets_into_space by force+ }

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

   109     by (fastforce simp: measurable_def sets_sigma space_pair_measure

   110                  intro!: sigma_sets.Basic)

   111   then show ?fst ?snd by auto

   112 qed

   113

   114 lemma (in pair_sigma_algebra) measurable_pair_iff:

   115   assumes "sigma_algebra M"

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

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

   118 proof -

   119   interpret M: sigma_algebra M by fact

   120   from assms show ?thesis

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

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

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

   124     proof (rule M.measurable_sigma)

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

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

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

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

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

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

   131         unfolding pair_measure_generator_def by auto

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

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

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

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

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

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

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

   139     qed

   140   qed

   141 qed

   142

   143 lemma (in pair_sigma_algebra) measurable_pair:

   144   assumes "sigma_algebra M"

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

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

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

   148

   149 lemma pair_measure_generatorE:

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

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

   152   using assms unfolding pair_measure_generator_def by auto

   153

   154 lemma (in pair_sigma_algebra) pair_measure_generator_swap:

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

   156 proof (safe elim!: pair_measure_generatorE)

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

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

   159     using M1.sets_into_space M2.sets_into_space by auto

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

   161     by (auto intro: pair_measure_generatorI)

   162 next

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

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

   165     using M1.sets_into_space M2.sets_into_space

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

   167 qed

   168

   169 lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:

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

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

   172 proof -

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

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

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

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

   177     unfolding pair_measure_def ..

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

   179     unfolding sigma_def pair_measure_generator_swap[symmetric]

   180     by (simp add: pair_measure_generator_def)

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

   182     using M1.sets_into_space M2.sets_into_space

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

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

   185     unfolding pair_measure_def pair_measure_generator_def sigma_def by simp

   186   finally show ?thesis

   187     using Q by (subst *) auto

   188 qed

   189

   190 lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:

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

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

   193   unfolding measurable_def

   194 proof (intro CollectI conjI Pi_I ballI)

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

   196     unfolding pair_measure_generator_def pair_measure_def by auto

   197 next

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

   199   interpret Q: pair_sigma_algebra M2 M1 by default

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

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

   202 qed

   203

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

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

   206 proof -

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

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

   209   interpret Q: sigma_algebra ?Q

   210     proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)

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

   212     using M1.sets_into_space M2.sets_into_space

   213     by (auto simp: pair_measure_generator_def space_pair_measure)

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

   215     apply (subst pair_measure_def, intro Q.sets_sigma_subset)

   216     by (simp add: pair_measure_def)

   217   with assms show ?thesis by auto

   218 qed

   219

   220 lemma (in pair_sigma_algebra) measurable_cut_snd:

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

   222 proof -

   223   interpret Q: pair_sigma_algebra M2 M1 by default

   224   from Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]

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

   226 qed

   227

   228 lemma (in pair_sigma_algebra) measurable_pair_image_snd:

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

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

   231   unfolding measurable_def

   232 proof (intro CollectI conjI Pi_I ballI)

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

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

   235     unfolding measurable_def pair_measure_generator_def pair_measure_def by auto

   236 next

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

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

   239     using f \<in> measurable P M

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

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

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

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

   244 qed

   245

   246 lemma (in pair_sigma_algebra) measurable_pair_image_fst:

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

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

   249 proof -

   250   interpret Q: pair_sigma_algebra M2 M1 by default

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

   252                                       OF Q.pair_sigma_algebra_swap_measurable m]

   253   show ?thesis by simp

   254 qed

   255

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

   257   unfolding Int_stable_def

   258 proof (intro ballI)

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

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

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

   262     unfolding pair_measure_generator_def by auto

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

   264     by (auto simp add: times_Int_times pair_measure_generator_def)

   265 qed

   266

   267 lemma finite_measure_cut_measurable:

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

   269   assumes "sigma_finite_measure M1" "finite_measure M2"

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

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

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

   273 proof -

   274   interpret M1: sigma_finite_measure M1 by fact

   275   interpret M2: finite_measure M2 by fact

   276   interpret pair_sigma_algebra M1 M2 by default

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

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

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

   280   note space_pair_measure[simp]

   281   interpret dynkin_system ?D

   282   proof (intro dynkin_systemI)

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

   284       using sets_into_space by simp

   285   next

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

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

   288   next

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

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

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

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

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

   294       by (auto intro!: Diff M1.measurable_If M1.borel_measurable_ereal_diff)

   295   next

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

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

   298       by (intro M2.measure_countably_additive[symmetric])

   299          (auto simp: disjoint_family_on_def)

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

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

   302   qed

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

   304   proof (intro dynkin_lemma)

   305     show "Int_stable E" by (rule Int_stable_pair_measure_generator)

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

   307       by auto

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

   309       by (auto simp: pair_measure_generator_def sets_sigma if_distrib

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

   311   qed (auto simp: pair_measure_def)

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

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

   314 qed

   315

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

   317

   318 locale pair_sigma_finite = pair_sigma_algebra M1 M2 + M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2

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

   320

   321 lemma (in pair_sigma_finite) measure_cut_measurable_fst:

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

   323 proof -

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

   325   have M1: "sigma_finite_measure M1" by default

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

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

   328   let ?C = "\<lambda>x i. F i \<inter> Pair x - Q"

   329   { fix i

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

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

   332       using F M2.sets_into_space by auto

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

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

   335     proof (intro finite_measure_cut_measurable[OF M1])

   336       show "finite_measure ?R2"

   337         using F by (intro M2.restricted_to_finite_measure) auto

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

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

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

   341         unfolding pair_measure_def pair_measure_generator_def sigma_def

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

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

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

   345         using M1.sets_into_space

   346         apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def

   347                     intro!: sigma_sets_subseteq)

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

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

   350         by auto

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

   352     qed

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

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

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

   356       by simp }

   357   moreover

   358   { fix x

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

   360     proof (intro M2.measure_countably_additive)

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

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

   363       have "disjoint_family F" using F by auto

   364       show "disjoint_family (?C x)"

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

   366     qed

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

   368       using F sets_into_space Q \<in> sets P

   369       by (auto simp: space_pair_measure)

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

   371       by simp }

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

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

   374 qed

   375

   376 lemma (in pair_sigma_finite) measure_cut_measurable_snd:

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

   378 proof -

   379   interpret Q: pair_sigma_finite M2 M1 by default

   380   note sets_pair_sigma_algebra_swap[OF assms]

   381   from Q.measure_cut_measurable_fst[OF this]

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

   383 qed

   384

   385 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:

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

   387 proof -

   388   interpret Q: pair_sigma_algebra M2 M1 by default

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

   390   show ?thesis

   391     using Q.pair_sigma_algebra_swap_measurable assms

   392     unfolding * by (rule measurable_comp)

   393 qed

   394

   395 lemma (in pair_sigma_finite) pair_measure_alt:

   396   assumes "A \<in> sets P"

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

   398   apply (simp add: pair_measure_def pair_measure_generator_def)

   399 proof (rule M1.positive_integral_cong)

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

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

   402     unfolding indicator_def by auto

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

   404     unfolding *

   405     apply (subst M2.positive_integral_indicator)

   406     apply (rule measurable_cut_fst[OF assms])

   407     by simp

   408 qed

   409

   410 lemma (in pair_sigma_finite) pair_measure_times:

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

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

   413 proof -

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

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

   416   with assms show ?thesis

   417     by (simp add: M1.positive_integral_cmult_indicator ac_simps)

   418 qed

   419

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

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

   422   using positive_measure[OF A] by auto

   423

   424 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:

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

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

   427 proof -

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

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

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

   431     using M1.sigma_finite_up M2.sigma_finite_up by auto

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

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

   434   show ?thesis unfolding space_pair_measure

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

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

   437       by (fastforce intro!: pair_measure_generatorI)

   438   next

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

   440     proof (intro subsetI)

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

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

   443         by (auto simp: space)

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

   445         using incseq F1 incseq F2 unfolding incseq_def

   446         by (force split: split_max)+

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

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

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

   450     qed

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

   452       using space by (auto simp: space pair_measure_generator_def)

   453   next

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

   455       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto

   456   next

   457     fix i

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

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

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

   461       by (simp add: pair_measure_times)

   462   qed

   463 qed

   464

   465 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P

   466 proof

   467   show "positive P (measure P)"

   468     unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def

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

   470

   471   show "countably_additive P (measure P)"

   472     unfolding countably_additive_def

   473   proof (intro allI impI)

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

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

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

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

   478       by (intro measure_cut_measurable_fst) auto

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

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

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

   482       using F by auto

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

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

   485                     M2.measure_countably_additive

   486                cong: M1.positive_integral_cong)

   487   qed

   488

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

   490   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>)"

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

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

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

   494       using F by (auto simp: pair_measure_def pair_measure_generator_def)

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

   496   qed

   497 qed

   498

   499 lemma (in pair_sigma_algebra) sets_swap:

   500   assumes "A \<in> sets P"

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

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

   503 proof -

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

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

   506   show ?thesis

   507     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)

   508 qed

   509

   510 lemma (in pair_sigma_finite) pair_measure_alt2:

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

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

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

   514 proof -

   515   interpret Q: pair_sigma_finite M2 M1 by default

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

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

   518     unfolding pair_measure_def by simp

   519

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

   521   proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])

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

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

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

   525       using assms unfolding pair_measure_def by auto

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

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

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

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

   530       unfolding pair_measure_def pair_measure_generator_def by auto

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

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

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

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

   535   qed

   536   then show ?thesis

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

   538     by (auto simp add: Q.pair_measure_alt space_pair_measure

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

   540 qed

   541

   542 lemma pair_sigma_algebra_sigma:

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

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

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

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

   547 proof -

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

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

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

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

   552   interpret E: sigma_algebra ?E unfolding pair_measure_generator_def

   553     using E1 E2 by (intro sigma_algebra_sigma) auto

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

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

   556       using E1 2 unfolding pair_measure_generator_def by auto

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

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

   559       using 2 A \<in> sets E1

   560       by (intro sigma_sets.Union)

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

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

   563   moreover

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

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

   566       using E2 1 unfolding pair_measure_generator_def by auto

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

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

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

   570       by (intro sigma_sets.Union)

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

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

   573   ultimately have proj:

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

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

   576                    (auto simp: pair_measure_generator_def sets_sigma)

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

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

   579       unfolding measurable_def by simp_all

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

   581       using A B M1.sets_into_space M2.sets_into_space

   582       by (auto simp: pair_measure_generator_def)

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

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

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

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

   587     by (simp add: sets_sigma pair_measure_generator_def)

   588   show "sets ?S = sets ?E"

   589   proof (intro set_eqI iffI)

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

   591       unfolding sets_sigma

   592     proof induct

   593       case (Basic A) then show ?case

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

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

   596   next

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

   598   qed

   599 qed

   600

   601 section "Fubinis theorem"

   602

   603 lemma (in pair_sigma_finite) simple_function_cut:

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

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

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

   607 proof -

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

   609     using f(1) by (rule borel_measurable_simple_function)

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

   611   let ?F' = "\<lambda>x z. Pair x - ?F z"

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

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

   614       by (auto simp: indicator_def)

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

   616       by (simp add: space_pair_measure)

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

   618       by (intro borel_measurable_vimage measurable_cut_fst)

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

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

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

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

   623   note M2_sf = this

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

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

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

   627       unfolding simple_integral_def

   628     proof (safe intro!: setsum_mono_zero_cong_left)

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

   630     next

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

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

   633     next

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

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

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

   637         by (force simp: space_pair_measure)

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

   639         unfolding * by simp

   640     qed (simp add: vimage_compose[symmetric] comp_def

   641                    space_pair_measure) }

   642   note eq = this

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

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

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

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

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

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

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

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

   651       using f(2) by auto }

   652   ultimately

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

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

   655     by (auto simp del: vimage_Int cong: measurable_cong

   656              intro!: M1.borel_measurable_ereal_setsum setsum_cong

   657              simp add: M1.positive_integral_setsum simple_integral_def

   658                        M1.positive_integral_cmult

   659                        M1.positive_integral_cong[OF eq]

   660                        positive_integral_eq_simple_integral[OF f]

   661                        pair_measure_alt[symmetric])

   662 qed

   663

   664 lemma (in pair_sigma_finite) positive_integral_fst_measurable:

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

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

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

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

   669 proof -

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

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

   672     by (auto intro: borel_measurable_simple_function)

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

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

   675     using F(1) by auto

   676   moreover

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

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

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

   680       by (intro M2.positive_integral_monotone_convergence_SUP)

   681          (auto simp: incseq_Suc_iff le_fun_def)

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

   683       unfolding F(4) positive_integral_max_0 by simp }

   684   note SUPR_C = this

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

   686     by (simp cong: measurable_cong)

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

   688     using F_borel F

   689     by (intro positive_integral_monotone_convergence_SUP) auto

   690   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)"

   691     unfolding sf(2) by simp

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

   693     by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])

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

   695                 simp: incseq_Suc_iff le_fun_def)

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

   697     using F_borel F(2,5)

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

   699              simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)

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

   701     using F by (simp add: positive_integral_max_0)

   702 qed

   703

   704 lemma (in pair_sigma_finite) measure_preserving_swap:

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

   706 proof

   707   interpret Q: pair_sigma_finite M2 M1 by default

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

   709     using pair_sigma_algebra_swap_measurable .

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

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

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

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

   714       simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)

   715 qed

   716

   717 lemma (in pair_sigma_finite) positive_integral_product_swap:

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

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

   720 proof -

   721   interpret Q: pair_sigma_finite M2 M1 by default

   722   have "sigma_algebra P" by default

   723   with f show ?thesis

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

   725 qed

   726

   727 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

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

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

   730 proof -

   731   interpret Q: pair_sigma_finite M2 M1 by default

   732   note pair_sigma_algebra_measurable[OF f]

   733   from Q.positive_integral_fst_measurable[OF this]

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

   735     by simp

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

   737     unfolding positive_integral_product_swap[OF f, symmetric]

   738     by (auto intro!: Q.positive_integral_cong)

   739   finally show ?thesis .

   740 qed

   741

   742 lemma (in pair_sigma_finite) Fubini:

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

   744   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)"

   745   unfolding positive_integral_snd_measurable[OF assms]

   746   unfolding positive_integral_fst_measurable[OF assms] ..

   747

   748 lemma (in pair_sigma_finite) AE_pair:

   749   assumes "AE x in P. Q x"

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

   751 proof -

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

   753     using assms unfolding almost_everywhere_def by auto

   754   show ?thesis

   755   proof (rule M1.AE_I)

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

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

   758       by (auto simp: pair_measure_alt M1.positive_integral_0_iff)

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

   760       by (intro M1.borel_measurable_ereal_neq_const measure_cut_measurable_fst N)

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

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

   763       proof (rule M2.AE_I)

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

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

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

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

   768       qed }

   769     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}"

   770       by auto

   771   qed

   772 qed

   773

   774 lemma (in pair_sigma_algebra) measurable_product_swap:

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

   776 proof -

   777   interpret Q: pair_sigma_algebra M2 M1 by default

   778   show ?thesis

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

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

   781 qed

   782

   783 lemma (in pair_sigma_finite) integrable_product_swap:

   784   assumes "integrable P f"

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

   786 proof -

   787   interpret Q: pair_sigma_finite M2 M1 by default

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

   789   show ?thesis unfolding *

   790     using assms unfolding integrable_def

   791     apply (subst (1 2) positive_integral_product_swap)

   792     using integrable P f unfolding integrable_def

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

   794 qed

   795

   796 lemma (in pair_sigma_finite) integrable_product_swap_iff:

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

   798 proof -

   799   interpret Q: pair_sigma_finite M2 M1 by default

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

   801   show ?thesis by auto

   802 qed

   803

   804 lemma (in pair_sigma_finite) integral_product_swap:

   805   assumes "integrable P f"

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

   807 proof -

   808   interpret Q: pair_sigma_finite M2 M1 by default

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

   810   show ?thesis

   811     unfolding lebesgue_integral_def *

   812     apply (subst (1 2) positive_integral_product_swap)

   813     using integrable P f unfolding integrable_def

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

   815 qed

   816

   817 lemma (in pair_sigma_finite) integrable_fst_measurable:

   818   assumes f: "integrable P f"

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

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

   821 proof -

   822   let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"

   823   have

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

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

   826     using assms by auto

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

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

   829     using borel[THEN positive_integral_fst_measurable(1)] int

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

   831   with borel[THEN positive_integral_fst_measurable(1)]

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

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

   834     by (auto intro!: M1.positive_integral_PInf_AE )

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

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

   837     by (auto simp: M2.positive_integral_positive)

   838   from AE_pos show ?AE using assms

   839     by (simp add: measurable_pair_image_snd integrable_def)

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

   841       using M2.positive_integral_positive

   842       by (intro M1.positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)

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

   844   note this[simp]

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

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

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

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

   849     proof (intro integrable_def[THEN iffD2] conjI)

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

   851         using borel by (auto intro!: M1.borel_measurable_real_of_ereal positive_integral_fst_measurable)

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

   853         using AE M2.positive_integral_positive

   854         by (auto intro!: M1.positive_integral_cong_AE simp: ereal_real)

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

   856         using positive_integral_fst_measurable[OF borel] int by simp

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

   858         by (intro M1.positive_integral_cong_pos)

   859            (simp add: M2.positive_integral_positive real_of_ereal_pos)

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

   861     qed }

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

   863   show ?INT

   864     unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]

   865       borel[THEN positive_integral_fst_measurable(2), symmetric]

   866     using AE[THEN M1.integral_real]

   867     by simp

   868 qed

   869

   870 lemma (in pair_sigma_finite) integrable_snd_measurable:

   871   assumes f: "integrable P f"

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

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

   874 proof -

   875   interpret Q: pair_sigma_finite M2 M1 by default

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

   877     using f unfolding integrable_product_swap_iff .

   878   show ?INT

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

   880     using integral_product_swap[OF f] by simp

   881   show ?AE

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

   883     by simp

   884 qed

   885

   886 lemma (in pair_sigma_finite) Fubini_integral:

   887   assumes f: "integrable P f"

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

   889   unfolding integrable_snd_measurable[OF assms]

   890   unfolding integrable_fst_measurable[OF assms] ..

   891

   892 section "Products on finite spaces"

   893

   894 lemma sigma_sets_pair_measure_generator_finite:

   895   assumes "finite A" and "finite B"

   896   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"

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

   898 proof safe

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

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

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

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

   903   proof (induct x)

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

   905   next

   906     case (insert a x)

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

   908       by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)

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

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

   911   qed

   912 next

   913   fix x a b

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

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

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

   917 qed

   918

   919 locale pair_finite_sigma_algebra = pair_sigma_algebra M1 M2 + M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 for M1 M2

   920

   921 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:

   922   shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"

   923 proof -

   924   show ?thesis

   925     using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]

   926     by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)

   927 qed

   928

   929 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P

   930 proof

   931   show "finite (space P)"

   932     using M1.finite_space M2.finite_space

   933     by (subst finite_pair_sigma_algebra) simp

   934   show "sets P = Pow (space P)"

   935     by (subst (1 2) finite_pair_sigma_algebra) simp

   936 qed

   937

   938 locale pair_finite_space = pair_sigma_finite M1 M2 + pair_finite_sigma_algebra M1 M2 +

   939   M1: finite_measure_space M1 + M2: finite_measure_space M2 for M1 M2

   940

   941 lemma (in pair_finite_space) pair_measure_Pair[simp]:

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

   943   shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"

   944 proof -

   945   have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"

   946     using M1.sets_eq_Pow M2.sets_eq_Pow assms

   947     by (subst pair_measure_times) auto

   948   then show ?thesis by simp

   949 qed

   950

   951 lemma (in pair_finite_space) pair_measure_singleton[simp]:

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

   953   shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"

   954   using pair_measure_Pair assms by (cases x) auto

   955

   956 sublocale pair_finite_space \<subseteq> finite_measure_space P

   957 proof unfold_locales

   958   show "measure P (space P) \<noteq> \<infinity>"

   959     by (subst (2) finite_pair_sigma_algebra)

   960        (simp add: pair_measure_times)

   961 qed

   962

   963 end