src/HOL/Probability/Binary_Product_Measure.thy
author huffman
Fri Aug 19 14:17:28 2011 -0700 (2011-08-19)
changeset 44311 42c5cbf68052
parent 43920 cedb5cb948fd
child 44890 22f665a2e91c
permissions -rw-r--r--
Transcendental.thy: add tendsto_intros lemmas;
new isCont theorems;
simplify some proofs.
     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 (fastsimp 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 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   with 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   with 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 = 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 sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
   322   by default
   323 
   324 lemma (in pair_sigma_finite) measure_cut_measurable_fst:
   325   assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
   326 proof -
   327   have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
   328   have M1: "sigma_finite_measure M1" by default
   329   from M2.disjoint_sigma_finite guess F .. note F = this
   330   then have F_sets: "\<And>i. F i \<in> sets M2" by auto
   331   let "?C x i" = "F i \<inter> Pair x -` Q"
   332   { fix i
   333     let ?R = "M2.restricted_space (F i)"
   334     have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
   335       using F M2.sets_into_space by auto
   336     let ?R2 = "M2.restricted_space (F i)"
   337     have "(\<lambda>x. measure ?R2 (Pair x -` (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"
   338     proof (intro finite_measure_cut_measurable[OF M1])
   339       show "finite_measure ?R2"
   340         using F by (intro M2.restricted_to_finite_measure) auto
   341       have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i)) ` sets P"
   342         using `Q \<in> sets P` by (auto simp: image_iff)
   343       also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i)) ` sets E)"
   344         unfolding pair_measure_def pair_measure_generator_def sigma_def
   345         using `F i \<in> sets M2` M2.sets_into_space
   346         by (auto intro!: sigma_sets_Int sigma_sets.Basic)
   347       also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"
   348         using M1.sets_into_space
   349         apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def
   350                     intro!: sigma_sets_subseteq)
   351         apply (rule_tac x="a" in exI)
   352         apply (rule_tac x="b \<inter> F i" in exI)
   353         by auto
   354       finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .
   355     qed
   356     moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"
   357       using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
   358     ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"
   359       by simp }
   360   moreover
   361   { fix x
   362     have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
   363     proof (intro M2.measure_countably_additive)
   364       show "range (?C x) \<subseteq> sets M2"
   365         using F `Q \<in> sets P` by (auto intro!: M2.Int)
   366       have "disjoint_family F" using F by auto
   367       show "disjoint_family (?C x)"
   368         by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
   369     qed
   370     also have "(\<Union>i. ?C x i) = Pair x -` Q"
   371       using F sets_into_space `Q \<in> sets P`
   372       by (auto simp: space_pair_measure)
   373     finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))"
   374       by simp }
   375   ultimately show ?thesis using `Q \<in> sets P` F_sets
   376     by (auto intro!: M1.borel_measurable_psuminf M2.Int)
   377 qed
   378 
   379 lemma (in pair_sigma_finite) measure_cut_measurable_snd:
   380   assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
   381 proof -
   382   interpret Q: pair_sigma_finite M2 M1 by default
   383   note sets_pair_sigma_algebra_swap[OF assms]
   384   from Q.measure_cut_measurable_fst[OF this]
   385   show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
   386 qed
   387 
   388 lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
   389   assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
   390 proof -
   391   interpret Q: pair_sigma_algebra M2 M1 by default
   392   have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
   393   show ?thesis
   394     using Q.pair_sigma_algebra_swap_measurable assms
   395     unfolding * by (rule measurable_comp)
   396 qed
   397 
   398 lemma (in pair_sigma_finite) pair_measure_alt:
   399   assumes "A \<in> sets P"
   400   shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x -` A) \<partial>M1)"
   401   apply (simp add: pair_measure_def pair_measure_generator_def)
   402 proof (rule M1.positive_integral_cong)
   403   fix x assume "x \<in> space M1"
   404   have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: ereal)"
   405     unfolding indicator_def by auto
   406   show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)"
   407     unfolding *
   408     apply (subst M2.positive_integral_indicator)
   409     apply (rule measurable_cut_fst[OF assms])
   410     by simp
   411 qed
   412 
   413 lemma (in pair_sigma_finite) pair_measure_times:
   414   assumes A: "A \<in> sets M1" and "B \<in> sets M2"
   415   shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"
   416 proof -
   417   have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"
   418     using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
   419   with assms show ?thesis
   420     by (simp add: M1.positive_integral_cmult_indicator ac_simps)
   421 qed
   422 
   423 lemma (in measure_space) measure_not_negative[simp,intro]:
   424   assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
   425   using positive_measure[OF A] by auto
   426 
   427 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
   428   "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
   429     (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
   430 proof -
   431   obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
   432     F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
   433     F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
   434     using M1.sigma_finite_up M2.sigma_finite_up by auto
   435   then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
   436   let ?F = "\<lambda>i. F1 i \<times> F2 i"
   437   show ?thesis unfolding space_pair_measure
   438   proof (intro exI[of _ ?F] conjI allI)
   439     show "range ?F \<subseteq> sets E" using F1 F2
   440       by (fastsimp intro!: pair_measure_generatorI)
   441   next
   442     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
   443     proof (intro subsetI)
   444       fix x assume "x \<in> space M1 \<times> space M2"
   445       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
   446         by (auto simp: space)
   447       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
   448         using `incseq F1` `incseq F2` unfolding incseq_def
   449         by (force split: split_max)+
   450       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
   451         by (intro SigmaI) (auto simp add: min_max.sup_commute)
   452       then show "x \<in> (\<Union>i. ?F i)" by auto
   453     qed
   454     then show "(\<Union>i. ?F i) = space E"
   455       using space by (auto simp: space pair_measure_generator_def)
   456   next
   457     fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
   458       using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
   459   next
   460     fix i
   461     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
   462     with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
   463     show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
   464       by (simp add: pair_measure_times)
   465   qed
   466 qed
   467 
   468 sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
   469 proof
   470   show "positive P (measure P)"
   471     unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
   472     by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
   473 
   474   show "countably_additive P (measure P)"
   475     unfolding countably_additive_def
   476   proof (intro allI impI)
   477     fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
   478     assume F: "range F \<subseteq> sets P" "disjoint_family F"
   479     from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
   480     moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x -` F i)) \<in> borel_measurable M1"
   481       by (intro measure_cut_measurable_fst) auto
   482     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
   483       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
   484     moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
   485       using F by auto
   486     ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
   487       by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
   488                     M2.measure_countably_additive
   489                cong: M1.positive_integral_cong)
   490   qed
   491 
   492   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   493   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>)"
   494   proof (rule exI[of _ F], intro conjI)
   495     show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
   496     show "(\<Union>i. F i) = space P"
   497       using F by (auto simp: pair_measure_def pair_measure_generator_def)
   498     show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
   499   qed
   500 qed
   501 
   502 lemma (in pair_sigma_algebra) sets_swap:
   503   assumes "A \<in> sets P"
   504   shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
   505     (is "_ -` A \<inter> space ?Q \<in> sets ?Q")
   506 proof -
   507   have *: "(\<lambda>(x, y). (y, x)) -` A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) -` A"
   508     using `A \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
   509   show ?thesis
   510     unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
   511 qed
   512 
   513 lemma (in pair_sigma_finite) pair_measure_alt2:
   514   assumes A: "A \<in> sets P"
   515   shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
   516     (is "_ = ?\<nu> A")
   517 proof -
   518   interpret Q: pair_sigma_finite M2 M1 by default
   519   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   520   have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"
   521     unfolding pair_measure_def by simp
   522 
   523   have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` A \<inter> space Q.P)"
   524   proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])
   525     show "measure_space P" "measure_space Q.P" by default
   526     show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
   527     show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
   528       using assms unfolding pair_measure_def by auto
   529     show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
   530       using F `A \<in> sets P` by (auto simp: pair_measure_def)
   531     fix X assume "X \<in> sets E"
   532     then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
   533       unfolding pair_measure_def pair_measure_generator_def by auto
   534     then have "(\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P = B \<times> A"
   535       using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)
   536     then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P)"
   537       using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)
   538   qed
   539   then show ?thesis
   540     using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]
   541     by (auto simp add: Q.pair_measure_alt space_pair_measure
   542              intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])
   543 qed
   544 
   545 lemma pair_sigma_algebra_sigma:
   546   assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
   547   assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
   548   shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
   549     (is "sets ?S = sets ?E")
   550 proof -
   551   interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
   552   interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
   553   have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
   554     using E1 E2 by (auto simp add: pair_measure_generator_def)
   555   interpret E: sigma_algebra ?E unfolding pair_measure_generator_def
   556     using E1 E2 by (intro sigma_algebra_sigma) auto
   557   { fix A assume "A \<in> sets E1"
   558     then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
   559       using E1 2 unfolding pair_measure_generator_def by auto
   560     also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
   561     also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
   562       using 2 `A \<in> sets E1`
   563       by (intro sigma_sets.Union)
   564          (force simp: image_subset_iff intro!: sigma_sets.Basic)
   565     finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
   566   moreover
   567   { fix B assume "B \<in> sets E2"
   568     then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
   569       using E2 1 unfolding pair_measure_generator_def by auto
   570     also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
   571     also have "\<dots> \<in> sets ?E"
   572       using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma
   573       by (intro sigma_sets.Union)
   574          (force simp: image_subset_iff intro!: sigma_sets.Basic)
   575     finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
   576   ultimately have proj:
   577     "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
   578     using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
   579                    (auto simp: pair_measure_generator_def sets_sigma)
   580   { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
   581     with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
   582       unfolding measurable_def by simp_all
   583     moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"
   584       using A B M1.sets_into_space M2.sets_into_space
   585       by (auto simp: pair_measure_generator_def)
   586     ultimately have "A \<times> B \<in> sets ?E" by auto }
   587   then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"
   588     by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)
   589   then have subset: "sets ?S \<subseteq> sets ?E"
   590     by (simp add: sets_sigma pair_measure_generator_def)
   591   show "sets ?S = sets ?E"
   592   proof (intro set_eqI iffI)
   593     fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
   594       unfolding sets_sigma
   595     proof induct
   596       case (Basic A) then show ?case
   597         by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)
   598     qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)
   599   next
   600     fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
   601   qed
   602 qed
   603 
   604 section "Fubinis theorem"
   605 
   606 lemma (in pair_sigma_finite) simple_function_cut:
   607   assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
   608   shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
   609     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
   610 proof -
   611   have f_borel: "f \<in> borel_measurable P"
   612     using f(1) by (rule borel_measurable_simple_function)
   613   let "?F z" = "f -` {z} \<inter> space P"
   614   let "?F' x z" = "Pair x -` ?F z"
   615   { fix x assume "x \<in> space M1"
   616     have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
   617       by (auto simp: indicator_def)
   618     have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1`
   619       by (simp add: space_pair_measure)
   620     moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
   621       by (intro borel_measurable_vimage measurable_cut_fst)
   622     ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
   623       apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
   624       apply (rule simple_function_indicator_representation[OF f(1)])
   625       using `x \<in> space M1` by (auto simp del: space_sigma) }
   626   note M2_sf = this
   627   { fix x assume x: "x \<in> space M1"
   628     then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space P. z * M2.\<mu> (?F' x z))"
   629       unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
   630       unfolding simple_integral_def
   631     proof (safe intro!: setsum_mono_zero_cong_left)
   632       from f(1) show "finite (f ` space P)" by (rule simple_functionD)
   633     next
   634       fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
   635         using `x \<in> space M1` by (auto simp: space_pair_measure)
   636     next
   637       fix x' y assume "(x', y) \<in> space P"
   638         "f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
   639       then have *: "?F' x (f (x', y)) = {}"
   640         by (force simp: space_pair_measure)
   641       show  "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"
   642         unfolding * by simp
   643     qed (simp add: vimage_compose[symmetric] comp_def
   644                    space_pair_measure) }
   645   note eq = this
   646   moreover have "\<And>z. ?F z \<in> sets P"
   647     by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
   648   moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
   649     by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
   650   moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
   651     using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
   652   moreover { fix i assume "i \<in> f`space P"
   653     with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
   654       using f(2) by auto }
   655   ultimately
   656   show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
   657     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
   658     by (auto simp del: vimage_Int cong: measurable_cong
   659              intro!: M1.borel_measurable_ereal_setsum setsum_cong
   660              simp add: M1.positive_integral_setsum simple_integral_def
   661                        M1.positive_integral_cmult
   662                        M1.positive_integral_cong[OF eq]
   663                        positive_integral_eq_simple_integral[OF f]
   664                        pair_measure_alt[symmetric])
   665 qed
   666 
   667 lemma (in pair_sigma_finite) positive_integral_fst_measurable:
   668   assumes f: "f \<in> borel_measurable P"
   669   shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
   670       (is "?C f \<in> borel_measurable M1")
   671     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
   672 proof -
   673   from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
   674   then have F_borel: "\<And>i. F i \<in> borel_measurable P"
   675     by (auto intro: borel_measurable_simple_function)
   676   note sf = simple_function_cut[OF F(1,5)]
   677   then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
   678     using F(1) by auto
   679   moreover
   680   { fix x assume "x \<in> space M1"
   681     from F measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
   682     have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
   683       by (intro M2.positive_integral_monotone_convergence_SUP)
   684          (auto simp: incseq_Suc_iff le_fun_def)
   685     then have "(SUP i. ?C (F i) x) = ?C f x"
   686       unfolding F(4) positive_integral_max_0 by simp }
   687   note SUPR_C = this
   688   ultimately show "?C f \<in> borel_measurable M1"
   689     by (simp cong: measurable_cong)
   690   have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
   691     using F_borel F
   692     by (intro positive_integral_monotone_convergence_SUP) auto
   693   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)"
   694     unfolding sf(2) by simp
   695   also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
   696     by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
   697        (auto intro!: M2.positive_integral_mono M2.positive_integral_positive
   698                 simp: incseq_Suc_iff le_fun_def)
   699   also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
   700     using F_borel F(2,5)
   701     by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
   702              simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
   703   finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
   704     using F by (simp add: positive_integral_max_0)
   705 qed
   706 
   707 lemma (in pair_sigma_finite) measure_preserving_swap:
   708   "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
   709 proof
   710   interpret Q: pair_sigma_finite M2 M1 by default
   711   show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
   712     using pair_sigma_algebra_swap_measurable .
   713   fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"
   714   from measurable_sets[OF * this] this Q.sets_into_space[OF this]
   715   show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) -` X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"
   716     by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]
   717       simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)
   718 qed
   719 
   720 lemma (in pair_sigma_finite) positive_integral_product_swap:
   721   assumes f: "f \<in> borel_measurable P"
   722   shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"
   723 proof -
   724   interpret Q: pair_sigma_finite M2 M1 by default
   725   have "sigma_algebra P" by default
   726   with f show ?thesis
   727     by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto
   728 qed
   729 
   730 lemma (in pair_sigma_finite) positive_integral_snd_measurable:
   731   assumes f: "f \<in> borel_measurable P"
   732   shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"
   733 proof -
   734   interpret Q: pair_sigma_finite M2 M1 by default
   735   note pair_sigma_algebra_measurable[OF f]
   736   from Q.positive_integral_fst_measurable[OF this]
   737   have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"
   738     by simp
   739   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"
   740     unfolding positive_integral_product_swap[OF f, symmetric]
   741     by (auto intro!: Q.positive_integral_cong)
   742   finally show ?thesis .
   743 qed
   744 
   745 lemma (in pair_sigma_finite) Fubini:
   746   assumes f: "f \<in> borel_measurable P"
   747   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)"
   748   unfolding positive_integral_snd_measurable[OF assms]
   749   unfolding positive_integral_fst_measurable[OF assms] ..
   750 
   751 lemma (in pair_sigma_finite) AE_pair:
   752   assumes "AE x in P. Q x"
   753   shows "AE x in M1. (AE y in M2. Q (x, y))"
   754 proof -
   755   obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
   756     using assms unfolding almost_everywhere_def by auto
   757   show ?thesis
   758   proof (rule M1.AE_I)
   759     from N measure_cut_measurable_fst[OF `N \<in> sets P`]
   760     show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0"
   761       by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
   762     show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1"
   763       by (intro M1.borel_measurable_ereal_neq_const measure_cut_measurable_fst N)
   764     { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0"
   765       have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
   766       proof (rule M2.AE_I)
   767         show "M2.\<mu> (Pair x -` N) = 0" by fact
   768         show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)
   769         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
   770           using N `x \<in> space M1` unfolding space_sigma space_pair_measure by auto
   771       qed }
   772     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}"
   773       by auto
   774   qed
   775 qed
   776 
   777 lemma (in pair_sigma_algebra) measurable_product_swap:
   778   "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
   779 proof -
   780   interpret Q: pair_sigma_algebra M2 M1 by default
   781   show ?thesis
   782     using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
   783     by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
   784 qed
   785 
   786 lemma (in pair_sigma_finite) integrable_product_swap:
   787   assumes "integrable P f"
   788   shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
   789 proof -
   790   interpret Q: pair_sigma_finite M2 M1 by default
   791   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
   792   show ?thesis unfolding *
   793     using assms unfolding integrable_def
   794     apply (subst (1 2) positive_integral_product_swap)
   795     using `integrable P f` unfolding integrable_def
   796     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
   797 qed
   798 
   799 lemma (in pair_sigma_finite) integrable_product_swap_iff:
   800   "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"
   801 proof -
   802   interpret Q: pair_sigma_finite M2 M1 by default
   803   from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
   804   show ?thesis by auto
   805 qed
   806 
   807 lemma (in pair_sigma_finite) integral_product_swap:
   808   assumes "integrable P f"
   809   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"
   810 proof -
   811   interpret Q: pair_sigma_finite M2 M1 by default
   812   have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
   813   show ?thesis
   814     unfolding lebesgue_integral_def *
   815     apply (subst (1 2) positive_integral_product_swap)
   816     using `integrable P f` unfolding integrable_def
   817     by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
   818 qed
   819 
   820 lemma (in pair_sigma_finite) integrable_fst_measurable:
   821   assumes f: "integrable P f"
   822   shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
   823     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
   824 proof -
   825   let "?pf x" = "ereal (f x)" and "?nf x" = "ereal (- f x)"
   826   have
   827     borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
   828     int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
   829     using assms by auto
   830   have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
   831      "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
   832     using borel[THEN positive_integral_fst_measurable(1)] int
   833     unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
   834   with borel[THEN positive_integral_fst_measurable(1)]
   835   have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
   836     "AE x in M1. (\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
   837     by (auto intro!: M1.positive_integral_PInf_AE )
   838   then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
   839     "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
   840     by (auto simp: M2.positive_integral_positive)
   841   from AE_pos show ?AE using assms
   842     by (simp add: measurable_pair_image_snd integrable_def)
   843   { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
   844       using M2.positive_integral_positive
   845       by (intro M1.positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
   846     then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
   847   note this[simp]
   848   { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable P"
   849       and int: "integral\<^isup>P P (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
   850       and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
   851     have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
   852     proof (intro integrable_def[THEN iffD2] conjI)
   853       show "?f \<in> borel_measurable M1"
   854         using borel by (auto intro!: M1.borel_measurable_real_of_ereal positive_integral_fst_measurable)
   855       have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y))  \<partial>M2) \<partial>M1)"
   856         using AE M2.positive_integral_positive
   857         by (auto intro!: M1.positive_integral_cong_AE simp: ereal_real)
   858       then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
   859         using positive_integral_fst_measurable[OF borel] int by simp
   860       have "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
   861         by (intro M1.positive_integral_cong_pos)
   862            (simp add: M2.positive_integral_positive real_of_ereal_pos)
   863       then show "(\<integral>\<^isup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
   864     qed }
   865   with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
   866   show ?INT
   867     unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
   868       borel[THEN positive_integral_fst_measurable(2), symmetric]
   869     using AE[THEN M1.integral_real]
   870     by simp
   871 qed
   872 
   873 lemma (in pair_sigma_finite) integrable_snd_measurable:
   874   assumes f: "integrable P f"
   875   shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")
   876     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")
   877 proof -
   878   interpret Q: pair_sigma_finite M2 M1 by default
   879   have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"
   880     using f unfolding integrable_product_swap_iff .
   881   show ?INT
   882     using Q.integrable_fst_measurable(2)[OF Q_int]
   883     using integral_product_swap[OF f] by simp
   884   show ?AE
   885     using Q.integrable_fst_measurable(1)[OF Q_int]
   886     by simp
   887 qed
   888 
   889 lemma (in pair_sigma_finite) Fubini_integral:
   890   assumes f: "integrable P f"
   891   shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
   892   unfolding integrable_snd_measurable[OF assms]
   893   unfolding integrable_fst_measurable[OF assms] ..
   894 
   895 section "Products on finite spaces"
   896 
   897 lemma sigma_sets_pair_measure_generator_finite:
   898   assumes "finite A" and "finite B"
   899   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"
   900   (is "sigma_sets ?prod ?sets = _")
   901 proof safe
   902   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
   903   fix x assume subset: "x \<subseteq> A \<times> B"
   904   hence "finite x" using fin by (rule finite_subset)
   905   from this subset show "x \<in> sigma_sets ?prod ?sets"
   906   proof (induct x)
   907     case empty show ?case by (rule sigma_sets.Empty)
   908   next
   909     case (insert a x)
   910     hence "{a} \<in> sigma_sets ?prod ?sets"
   911       by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)
   912     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
   913     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
   914   qed
   915 next
   916   fix x a b
   917   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
   918   from sigma_sets_into_sp[OF _ this(1)] this(2)
   919   show "a \<in> A" and "b \<in> B" by auto
   920 qed
   921 
   922 locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2
   923   for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
   924 
   925 sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default
   926 
   927 lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:
   928   shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"
   929 proof -
   930   show ?thesis
   931     using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]
   932     by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)
   933 qed
   934 
   935 sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
   936   apply default
   937   using M1.finite_space M2.finite_space
   938   apply (subst finite_pair_sigma_algebra) apply simp
   939   apply (subst (1 2) finite_pair_sigma_algebra) apply simp
   940   done
   941 
   942 locale pair_finite_space = M1: finite_measure_space M1 + M2: finite_measure_space M2
   943   for M1 M2
   944 
   945 sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra
   946   by default
   947 
   948 sublocale pair_finite_space \<subseteq> pair_sigma_finite
   949   by default
   950 
   951 lemma (in pair_finite_space) pair_measure_Pair[simp]:
   952   assumes "a \<in> space M1" "b \<in> space M2"
   953   shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"
   954 proof -
   955   have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"
   956     using M1.sets_eq_Pow M2.sets_eq_Pow assms
   957     by (subst pair_measure_times) auto
   958   then show ?thesis by simp
   959 qed
   960 
   961 lemma (in pair_finite_space) pair_measure_singleton[simp]:
   962   assumes "x \<in> space M1 \<times> space M2"
   963   shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"
   964   using pair_measure_Pair assms by (cases x) auto
   965 
   966 sublocale pair_finite_space \<subseteq> finite_measure_space P
   967   by default (auto simp: space_pair_measure)
   968 
   969 end