src/HOL/Probability/Binary_Product_Measure.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 49999 dfb63b9b8908 child 50003 8c213922ed49 permissions -rw-r--r--
     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 pair_measure (infixr "\<Otimes>\<^isub>M" 80) where

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

    33       {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}

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

    35

    36 lemma pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"

    37   using space_closed[of A] space_closed[of B] by auto

    38

    39 lemma space_pair_measure:

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

    41   unfolding pair_measure_def using pair_measure_closed[of A B]

    42   by (rule space_measure_of)

    43

    44 lemma sets_pair_measure:

    45   "sets (A \<Otimes>\<^isub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"

    46   unfolding pair_measure_def using pair_measure_closed[of A B]

    47   by (rule sets_measure_of)

    48

    49 lemma sets_pair_measure_cong[cong]:

    50   "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^isub>M M2) = sets (M1' \<Otimes>\<^isub>M M2')"

    51   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)

    52

    53 lemma pair_measureI[intro, simp]:

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

    55   by (auto simp: sets_pair_measure)

    56

    57 lemma measurable_pair_measureI:

    58   assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"

    59   assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f - (A \<times> B) \<inter> space M \<in> sets M"

    60   shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"

    61   unfolding pair_measure_def using 1 2

    62   by (intro measurable_measure_of) (auto dest: sets_into_space)

    63

    64 lemma measurable_Pair:

    65   assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"

    66   shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"

    67 proof (rule measurable_pair_measureI)

    68   show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"

    69     using f g by (auto simp: measurable_def)

    70   fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"

    71   have "(\<lambda>x. (f x, g x)) - (A \<times> B) \<inter> space M = (f - A \<inter> space M) \<inter> (g - B \<inter> space M)"

    72     by auto

    73   also have "\<dots> \<in> sets M"

    74     by (rule Int) (auto intro!: measurable_sets * f g)

    75   finally show "(\<lambda>x. (f x, g x)) - (A \<times> B) \<inter> space M \<in> sets M" .

    76 qed

    77

    78 lemma measurable_pair:

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

    80   shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)"

    81   using measurable_Pair[OF assms] by simp

    82

    83 lemma measurable_fst[intro!, simp]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1"

    84   by (auto simp: fst_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)

    85

    86 lemma measurable_snd[intro!, simp]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2"

    87   by (auto simp: snd_vimage_eq_Times space_pair_measure sets_into_space times_Int_times measurable_def)

    88

    89 lemma measurable_fst': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. fst (f x)) \<in> measurable M N"

    90   using measurable_comp[OF _ measurable_fst] by (auto simp: comp_def)

    91

    92 lemma measurable_snd': "f \<in> measurable M (N \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. snd (f x)) \<in> measurable M P"

    93     using measurable_comp[OF _ measurable_snd] by (auto simp: comp_def)

    94

    95 lemma measurable_fst'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^isub>M P) N"

    96   using measurable_comp[OF measurable_fst _] by (auto simp: comp_def)

    97

    98 lemma measurable_snd'': "f \<in> measurable M N \<Longrightarrow> (\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^isub>M M) N"

    99   using measurable_comp[OF measurable_snd _] by (auto simp: comp_def)

   100

   101 lemma measurable_pair_iff:

   102   "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"

   103   using measurable_pair[of f M M1 M2]

   104   using [[simproc del: measurable]] (* the measurable method is nonterminating when using measurable_pair *)

   105   by auto

   106

   107 lemma measurable_split_conv:

   108   "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"

   109   by (intro arg_cong2[where f="op \<in>"]) auto

   110

   111 lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"

   112   by (auto intro!: measurable_Pair simp: measurable_split_conv)

   113

   114 lemma measurable_pair_swap:

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

   116   using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)

   117

   118 lemma measurable_pair_swap_iff:

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

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

   121   by (auto intro!: measurable_pair_swap)

   122

   123 lemma measurable_ident[intro, simp]: "(\<lambda>x. x) \<in> measurable M M"

   124   unfolding measurable_def by auto

   125

   126 lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)"

   127   by (auto intro!: measurable_Pair)

   128

   129 lemma sets_Pair1: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "Pair x - A \<in> sets M2"

   130 proof -

   131   have "Pair x - A = (if x \<in> space M1 then Pair x - A \<inter> space M2 else {})"

   132     using A[THEN sets_into_space] by (auto simp: space_pair_measure)

   133   also have "\<dots> \<in> sets M2"

   134     using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm)

   135   finally show ?thesis .

   136 qed

   137

   138 lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^isub>M M2)"

   139   by (auto intro!: measurable_Pair)

   140

   141 lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. (x, y)) - A \<in> sets M1"

   142 proof -

   143   have "(\<lambda>x. (x, y)) - A = (if y \<in> space M2 then (\<lambda>x. (x, y)) - A \<inter> space M1 else {})"

   144     using A[THEN sets_into_space] by (auto simp: space_pair_measure)

   145   also have "\<dots> \<in> sets M1"

   146     using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm)

   147   finally show ?thesis .

   148 qed

   149

   150 lemma measurable_Pair2:

   151   assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and x: "x \<in> space M1"

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

   153   using measurable_comp[OF measurable_Pair1' f, OF x]

   154   by (simp add: comp_def)

   155

   156 lemma measurable_Pair1:

   157   assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and y: "y \<in> space M2"

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

   159   using measurable_comp[OF measurable_Pair2' f, OF y]

   160   by (simp add: comp_def)

   161

   162 lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"

   163   unfolding Int_stable_def

   164   by safe (auto simp add: times_Int_times)

   165

   166 lemma (in finite_measure) finite_measure_cut_measurable:

   167   assumes "Q \<in> sets (N \<Otimes>\<^isub>M M)"

   168   shows "(\<lambda>x. emeasure M (Pair x - Q)) \<in> borel_measurable N"

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

   170   using Int_stable_pair_measure_generator pair_measure_closed assms

   171   unfolding sets_pair_measure

   172 proof (induct rule: sigma_sets_induct_disjoint)

   173   case (compl A)

   174   with sets_into_space have "\<And>x. emeasure M (Pair x - ((space N \<times> space M) - A)) =

   175       (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"

   176     unfolding sets_pair_measure[symmetric]

   177     by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)

   178   with compl top show ?case

   179     by (auto intro!: measurable_If simp: space_pair_measure)

   180 next

   181   case (union F)

   182   moreover then have "\<And>x. emeasure M (\<Union>i. Pair x - F i) = (\<Sum>i. ?s (F i) x)"

   183     unfolding sets_pair_measure[symmetric]

   184     by (intro suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def sets_Pair1)

   185   ultimately show ?case

   186     by (auto simp: vimage_UN)

   187 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)

   188

   189 lemma (in sigma_finite_measure) measurable_emeasure_Pair:

   190   assumes Q: "Q \<in> sets (N \<Otimes>\<^isub>M M)" shows "(\<lambda>x. emeasure M (Pair x - Q)) \<in> borel_measurable N" (is "?s Q \<in> _")

   191 proof -

   192   from sigma_finite_disjoint guess F . note F = this

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

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

   195   { fix i

   196     have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"

   197       using F sets_into_space by auto

   198     let ?R = "density M (indicator (F i))"

   199     have "finite_measure ?R"

   200       using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)

   201     then have "(\<lambda>x. emeasure ?R (Pair x - (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"

   202      by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)

   203     moreover have "\<And>x. emeasure ?R (Pair x - (space N \<times> space ?R \<inter> Q))

   204         = emeasure M (F i \<inter> Pair x - (space N \<times> space ?R \<inter> Q))"

   205       using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)

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

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

   208     ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"

   209       by simp }

   210   moreover

   211   { fix x

   212     have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"

   213     proof (intro suminf_emeasure)

   214       show "range (?C x) \<subseteq> sets M"

   215         using F Q \<in> sets (N \<Otimes>\<^isub>M M) by (auto intro!: sets_Pair1)

   216       have "disjoint_family F" using F by auto

   217       show "disjoint_family (?C x)"

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

   219     qed

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

   221       using F sets_into_space[OF Q \<in> sets (N \<Otimes>\<^isub>M M)]

   222       by (auto simp: space_pair_measure)

   223     finally have "emeasure M (Pair x - Q) = (\<Sum>i. emeasure M (?C x i))"

   224       by simp }

   225   ultimately show ?thesis using Q \<in> sets (N \<Otimes>\<^isub>M M) F_sets

   226     by auto

   227 qed

   228

   229 lemma (in sigma_finite_measure) emeasure_pair_measure:

   230   assumes "X \<in> sets (N \<Otimes>\<^isub>M M)"

   231   shows "emeasure (N \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X")

   232 proof (rule emeasure_measure_of[OF pair_measure_def])

   233   show "positive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"

   234     by (auto simp: positive_def positive_integral_positive)

   235   have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x - A) y"

   236     by (auto simp: indicator_def)

   237   show "countably_additive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>"

   238   proof (rule countably_additiveI)

   239     fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^isub>M M)" "disjoint_family F"

   240     from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^isub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^isub>M M)" by auto

   241     moreover from F have "\<And>i. (\<lambda>x. emeasure M (Pair x - F i)) \<in> borel_measurable N"

   242       by (intro measurable_emeasure_Pair) auto

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

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

   245     moreover have "\<And>x. range (\<lambda>i. Pair x - F i) \<subseteq> sets M"

   246       using F by (auto simp: sets_Pair1)

   247     ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"

   248       by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1

   249                intro!: positive_integral_cong positive_integral_indicator[symmetric])

   250   qed

   251   show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"

   252     using space_closed[of N] space_closed[of M] by auto

   253 qed fact

   254

   255 lemma (in sigma_finite_measure) emeasure_pair_measure_alt:

   256   assumes X: "X \<in> sets (N \<Otimes>\<^isub>M M)"

   257   shows "emeasure (N  \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+x. emeasure M (Pair x - X) \<partial>N)"

   258 proof -

   259   have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x - X) y"

   260     by (auto simp: indicator_def)

   261   show ?thesis

   262     using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1)

   263 qed

   264

   265 lemma (in sigma_finite_measure) emeasure_pair_measure_Times:

   266   assumes A: "A \<in> sets N" and B: "B \<in> sets M"

   267   shows "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = emeasure N A * emeasure M B"

   268 proof -

   269   have "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M B * indicator A x \<partial>N)"

   270     using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt)

   271   also have "\<dots> = emeasure M B * emeasure N A"

   272     using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)

   273   finally show ?thesis

   274     by (simp add: ac_simps)

   275 qed

   276

   277 subsection {* Binary products of $\sigma$-finite emeasure spaces *}

   278

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

   280   for M1 :: "'a measure" and M2 :: "'b measure"

   281

   282 lemma (in pair_sigma_finite) measurable_emeasure_Pair1:

   283   "Q \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x - Q)) \<in> borel_measurable M1"

   284   using M2.measurable_emeasure_Pair .

   285

   286 lemma (in pair_sigma_finite) measurable_emeasure_Pair2:

   287   assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) - Q)) \<in> borel_measurable M2"

   288 proof -

   289   have "(\<lambda>(x, y). (y, x)) - Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"

   290     using Q measurable_pair_swap' by (auto intro: measurable_sets)

   291   note M1.measurable_emeasure_Pair[OF this]

   292   moreover have "\<And>y. Pair y - ((\<lambda>(x, y). (y, x)) - Q \<inter> space (M2 \<Otimes>\<^isub>M M1)) = (\<lambda>x. (x, y)) - Q"

   293     using Q[THEN sets_into_space] by (auto simp: space_pair_measure)

   294   ultimately show ?thesis by simp

   295 qed

   296

   297 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:

   298   defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"

   299   shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and>

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

   301 proof -

   302   from M1.sigma_finite_incseq guess F1 . note F1 = this

   303   from M2.sigma_finite_incseq guess F2 . note F2 = this

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

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

   306   show ?thesis

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

   308     show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)

   309   next

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

   311     proof (intro subsetI)

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

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

   314         by (auto simp: space)

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

   316         using incseq F1 incseq F2 unfolding incseq_def

   317         by (force split: split_max)+

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

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

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

   321     qed

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

   323       using space by (auto simp: space)

   324   next

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

   326       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto

   327   next

   328     fix i

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

   330     with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"]

   331     show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"

   332       by (auto simp add: emeasure_pair_measure_Times)

   333   qed

   334 qed

   335

   336 sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^isub>M M2"

   337 proof

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

   339   show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"

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

   341     show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def)

   342     show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)"

   343       using F by (auto simp: space_pair_measure)

   344     show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto

   345   qed

   346 qed

   347

   348 lemma sigma_finite_pair_measure:

   349   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"

   350   shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)"

   351 proof -

   352   interpret A: sigma_finite_measure A by fact

   353   interpret B: sigma_finite_measure B by fact

   354   interpret AB: pair_sigma_finite A  B ..

   355   show ?thesis ..

   356 qed

   357

   358 lemma sets_pair_swap:

   359   assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"

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

   361   using measurable_pair_swap' assms by (rule measurable_sets)

   362

   363 lemma (in pair_sigma_finite) distr_pair_swap:

   364   "M1 \<Otimes>\<^isub>M M2 = distr (M2 \<Otimes>\<^isub>M M1) (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")

   365 proof -

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

   367   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"

   368   show ?thesis

   369   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])

   370     show "?E \<subseteq> Pow (space ?P)"

   371       using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure)

   372     show "sets ?P = sigma_sets (space ?P) ?E"

   373       by (simp add: sets_pair_measure space_pair_measure)

   374     then show "sets ?D = sigma_sets (space ?P) ?E"

   375       by simp

   376   next

   377     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"

   378       using F by (auto simp: space_pair_measure)

   379   next

   380     fix X assume "X \<in> ?E"

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

   382     have "(\<lambda>(y, x). (x, y)) - X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A"

   383       using sets_into_space[OF A] sets_into_space[OF B] by (auto simp: space_pair_measure)

   384     with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X"

   385       by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr

   386                     measurable_pair_swap' ac_simps)

   387   qed

   388 qed

   389

   390 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:

   391   assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)"

   392   shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) - A) \<partial>M2)"

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

   394 proof -

   395   have [simp]: "\<And>y. (Pair y - ((\<lambda>(x, y). (y, x)) - A \<inter> space (M2 \<Otimes>\<^isub>M M1))) = (\<lambda>x. (x, y)) - A"

   396     using sets_into_space[OF A] by (auto simp: space_pair_measure)

   397   show ?thesis using A

   398     by (subst distr_pair_swap)

   399        (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']

   400                  M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])

   401 qed

   402

   403 lemma (in pair_sigma_finite) AE_pair:

   404   assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x"

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

   406 proof -

   407   obtain N where N: "N \<in> sets (M1 \<Otimes>\<^isub>M M2)" "emeasure (M1 \<Otimes>\<^isub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> Q x} \<subseteq> N"

   408     using assms unfolding eventually_ae_filter by auto

   409   show ?thesis

   410   proof (rule AE_I)

   411     from N measurable_emeasure_Pair1[OF N \<in> sets (M1 \<Otimes>\<^isub>M M2)]

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

   413       by (auto simp: M2.emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg)

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

   415       by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)

   416     { fix x assume "x \<in> space M1" "emeasure M2 (Pair x - N) = 0"

   417       have "AE y in M2. Q (x, y)"

   418       proof (rule AE_I)

   419         show "emeasure M2 (Pair x - N) = 0" by fact

   420         show "Pair x - N \<in> sets M2" using N(1) by (rule sets_Pair1)

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

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

   423       qed }

   424     then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x - N) \<noteq> 0}"

   425       by auto

   426   qed

   427 qed

   428

   429 lemma (in pair_sigma_finite) AE_pair_measure:

   430   assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"

   431   assumes ae: "AE x in M1. AE y in M2. P (x, y)"

   432   shows "AE x in M1 \<Otimes>\<^isub>M M2. P x"

   433 proof (subst AE_iff_measurable[OF _ refl])

   434   show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)"

   435     by (rule sets_Collect) fact

   436   then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} =

   437       (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"

   438     by (simp add: M2.emeasure_pair_measure)

   439   also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)"

   440     using ae

   441     apply (safe intro!: positive_integral_cong_AE)

   442     apply (intro AE_I2)

   443     apply (safe intro!: positive_integral_cong_AE)

   444     apply auto

   445     done

   446   finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp

   447 qed

   448

   449 lemma (in pair_sigma_finite) AE_pair_iff:

   450   "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow>

   451     (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x))"

   452   using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto

   453

   454 lemma (in pair_sigma_finite) AE_commute:

   455   assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)"

   456   shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"

   457 proof -

   458   interpret Q: pair_sigma_finite M2 M1 ..

   459   have [simp]: "\<And>x. (fst (case x of (x, y) \<Rightarrow> (y, x))) = snd x" "\<And>x. (snd (case x of (x, y) \<Rightarrow> (y, x))) = fst x"

   460     by auto

   461   have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} =

   462     (\<lambda>(x, y). (y, x)) - {x \<in> space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^isub>M M1)"

   463     by (auto simp: space_pair_measure)

   464   also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)"

   465     by (intro sets_pair_swap P)

   466   finally show ?thesis

   467     apply (subst AE_pair_iff[OF P])

   468     apply (subst distr_pair_swap)

   469     apply (subst AE_distr_iff[OF measurable_pair_swap' P])

   470     apply (subst Q.AE_pair_iff)

   471     apply simp_all

   472     done

   473 qed

   474

   475 section "Fubinis theorem"

   476

   477 lemma measurable_compose_Pair1:

   478   "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^isub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"

   479   by (rule measurable_compose[OF measurable_Pair]) auto

   480

   481 lemma (in sigma_finite_measure) borel_measurable_positive_integral_fst':

   482   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x"

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

   484 using f proof induct

   485   case (cong u v)

   486   then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"

   487     by (auto simp: space_pair_measure)

   488   show ?case

   489     apply (subst measurable_cong)

   490     apply (rule positive_integral_cong)

   491     apply fact+

   492     done

   493 next

   494   case (set Q)

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

   496     by (auto simp: indicator_def)

   497   have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x - Q) = \<integral>\<^isup>+ y. indicator Q (x, y) \<partial>M"

   498     by (simp add: sets_Pair1[OF set])

   499   from this measurable_emeasure_Pair[OF set] show ?case

   500     by (rule measurable_cong[THEN iffD1])

   501 qed (simp_all add: positive_integral_add positive_integral_cmult measurable_compose_Pair1

   502                    positive_integral_monotone_convergence_SUP incseq_def le_fun_def

   503               cong: measurable_cong)

   504

   505 lemma (in sigma_finite_measure) positive_integral_fst:

   506   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x"

   507   shows "(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f" (is "?I f = _")

   508 using f proof induct

   509   case (cong u v)

   510   moreover then have "?I u = ?I v"

   511     by (intro positive_integral_cong) (auto simp: space_pair_measure)

   512   ultimately show ?case

   513     by (simp cong: positive_integral_cong)

   514 qed (simp_all add: emeasure_pair_measure positive_integral_cmult positive_integral_add

   515                    positive_integral_monotone_convergence_SUP

   516                    measurable_compose_Pair1 positive_integral_positive

   517                    borel_measurable_positive_integral_fst' positive_integral_mono incseq_def le_fun_def

   518               cong: positive_integral_cong)

   519

   520 lemma (in sigma_finite_measure) positive_integral_fst_measurable:

   521   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)"

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

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

   524     and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f"

   525   using f

   526     borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (f x)"]

   527     positive_integral_fst[of "\<lambda>x. max 0 (f x)"]

   528   unfolding positive_integral_max_0 by auto

   529

   530 lemma (in sigma_finite_measure) borel_measurable_positive_integral:

   531   "(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M) \<in> borel_measurable M1"

   532   using positive_integral_fst_measurable(1)[of "split f" M1] by simp

   533

   534 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

   535   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

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

   537 proof -

   538   note measurable_pair_swap[OF f]

   539   from M1.positive_integral_fst_measurable[OF this]

   540   have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1))"

   541     by simp

   542   also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f"

   543     by (subst distr_pair_swap)

   544        (auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong)

   545   finally show ?thesis .

   546 qed

   547

   548 lemma (in pair_sigma_finite) Fubini:

   549   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

   550   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)"

   551   unfolding positive_integral_snd_measurable[OF assms]

   552   unfolding M2.positive_integral_fst_measurable[OF assms] ..

   553

   554 lemma (in pair_sigma_finite) integrable_product_swap:

   555   assumes "integrable (M1 \<Otimes>\<^isub>M M2) f"

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

   557 proof -

   558   interpret Q: pair_sigma_finite M2 M1 by default

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

   560   show ?thesis unfolding *

   561     by (rule integrable_distr[OF measurable_pair_swap'])

   562        (simp add: distr_pair_swap[symmetric] assms)

   563 qed

   564

   565 lemma (in pair_sigma_finite) integrable_product_swap_iff:

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

   567 proof -

   568   interpret Q: pair_sigma_finite M2 M1 by default

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

   570   show ?thesis by auto

   571 qed

   572

   573 lemma (in pair_sigma_finite) integral_product_swap:

   574   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

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

   576 proof -

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

   578   show ?thesis unfolding *

   579     by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])

   580 qed

   581

   582 lemma (in pair_sigma_finite) integrable_fst_measurable:

   583   assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"

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

   585     and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")

   586 proof -

   587   have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

   588     using f by auto

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

   590   have

   591     borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and

   592     int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?nf \<noteq> \<infinity>" "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?pf \<noteq> \<infinity>"

   593     using assms by auto

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

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

   596     using borel[THEN M2.positive_integral_fst_measurable(1)] int

   597     unfolding borel[THEN M2.positive_integral_fst_measurable(2)] by simp_all

   598   with borel[THEN M2.positive_integral_fst_measurable(1)]

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

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

   601     by (auto intro!: positive_integral_PInf_AE )

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

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

   604     by (auto simp: positive_integral_positive)

   605   from AE_pos show ?AE using assms

   606     by (simp add: measurable_Pair2[OF f_borel] integrable_def)

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

   608       using positive_integral_positive

   609       by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)

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

   611   note this[simp]

   612   { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

   613       and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"

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

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

   616     proof (intro integrable_def[THEN iffD2] conjI)

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

   618         using borel by (auto intro!: M2.positive_integral_fst_measurable)

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

   620         using AE positive_integral_positive[of M2]

   621         by (auto intro!: positive_integral_cong_AE simp: ereal_real)

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

   623         using M2.positive_integral_fst_measurable[OF borel] int by simp

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

   625         by (intro positive_integral_cong_pos)

   626            (simp add: positive_integral_positive real_of_ereal_pos)

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

   628     qed }

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

   630   show ?INT

   631     unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2]

   632       borel[THEN M2.positive_integral_fst_measurable(2), symmetric]

   633     using AE[THEN integral_real]

   634     by simp

   635 qed

   636

   637 lemma (in pair_sigma_finite) integrable_snd_measurable:

   638   assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"

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

   640     and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT")

   641 proof -

   642   interpret Q: pair_sigma_finite M2 M1 by default

   643   have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))"

   644     using f unfolding integrable_product_swap_iff .

   645   show ?INT

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

   647     using integral_product_swap[of f] f by auto

   648   show ?AE

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

   650     by simp

   651 qed

   652

   653 lemma (in pair_sigma_finite) positive_integral_fst_measurable':

   654   assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

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

   656   using M2.positive_integral_fst_measurable(1)[OF f] by simp

   657

   658 lemma (in pair_sigma_finite) integral_fst_measurable:

   659   "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M2) \<in> borel_measurable M1"

   660   by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_fst_measurable')

   661

   662 lemma (in pair_sigma_finite) positive_integral_snd_measurable':

   663   assumes f: "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

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

   665 proof -

   666   interpret Q: pair_sigma_finite M2 M1 ..

   667   show ?thesis

   668     using measurable_pair_swap[OF f]

   669     by (intro Q.positive_integral_fst_measurable') (simp add: split_beta')

   670 qed

   671

   672 lemma (in pair_sigma_finite) integral_snd_measurable:

   673   "(\<lambda>x. f (fst x) (snd x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>y. \<integral> x. f x y \<partial>M1) \<in> borel_measurable M2"

   674   by (auto simp: lebesgue_integral_def intro!: borel_measurable_diff positive_integral_snd_measurable')

   675

   676 lemma (in pair_sigma_finite) Fubini_integral:

   677   assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f"

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

   679   unfolding integrable_snd_measurable[OF assms]

   680   unfolding integrable_fst_measurable[OF assms] ..

   681

   682 section {* Products on counting spaces, densities and distributions *}

   683

   684 lemma sigma_sets_pair_measure_generator_finite:

   685   assumes "finite A" and "finite B"

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

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

   688 proof safe

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

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

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

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

   693   proof (induct x)

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

   695   next

   696     case (insert a x)

   697     hence "{a} \<in> sigma_sets ?prod ?sets" by auto

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

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

   700   qed

   701 next

   702   fix x a b

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

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

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

   706 qed

   707

   708 lemma pair_measure_count_space:

   709   assumes A: "finite A" and B: "finite B"

   710   shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")

   711 proof (rule measure_eqI)

   712   interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact

   713   interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact

   714   interpret P: pair_sigma_finite "count_space A" "count_space B" by default

   715   show eq: "sets ?P = sets ?C"

   716     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)

   717   fix X assume X: "X \<in> sets ?P"

   718   with eq have X_subset: "X \<subseteq> A \<times> B" by simp

   719   with A B have fin_Pair: "\<And>x. finite (Pair x - X)"

   720     by (intro finite_subset[OF _ B]) auto

   721   have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)

   722   show "emeasure ?P X = emeasure ?C X"

   723     apply (subst B.emeasure_pair_measure_alt[OF X])

   724     apply (subst emeasure_count_space)

   725     using X_subset apply auto []

   726     apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)

   727     apply (subst positive_integral_count_space)

   728     using A apply simp

   729     apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])

   730     apply (subst card_gt_0_iff)

   731     apply (simp add: fin_Pair)

   732     apply (subst card_SigmaI[symmetric])

   733     using A apply simp

   734     using fin_Pair apply simp

   735     using X_subset apply (auto intro!: arg_cong[where f=card])

   736     done

   737 qed

   738

   739 lemma pair_measure_density:

   740   assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"

   741   assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x"

   742   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"

   743   assumes "sigma_finite_measure (density M1 f)" "sigma_finite_measure (density M2 g)"

   744   shows "density M1 f \<Otimes>\<^isub>M density M2 g = density (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")

   745 proof (rule measure_eqI)

   746   interpret M1: sigma_finite_measure M1 by fact

   747   interpret M2: sigma_finite_measure M2 by fact

   748   interpret D1: sigma_finite_measure "density M1 f" by fact

   749   interpret D2: sigma_finite_measure "density M2 g" by fact

   750   interpret L: pair_sigma_finite "density M1 f" "density M2 g" ..

   751   interpret R: pair_sigma_finite M1 M2 ..

   752

   753   fix A assume A: "A \<in> sets ?L"

   754   then have indicator_eq: "\<And>x y. indicator A (x, y) = indicator (Pair x - A) y"

   755    and Pair_A: "\<And>x. Pair x - A \<in> sets M2"

   756     by (auto simp: indicator_def sets_Pair1)

   757   have f_fst: "(\<lambda>p. f (fst p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

   758     using measurable_comp[OF measurable_fst f(1)] by (simp add: comp_def)

   759   have g_snd: "(\<lambda>p. g (snd p)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)"

   760     using measurable_comp[OF measurable_snd g(1)] by (simp add: comp_def)

   761   have "(\<lambda>x. \<integral>\<^isup>+ y. g (snd (x, y)) * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"

   762     using g_snd Pair_A A by (intro M2.positive_integral_fst_measurable) auto

   763   then have int_g: "(\<lambda>x. \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2) \<in> borel_measurable M1"

   764     by simp

   765

   766   show "emeasure ?L A = emeasure ?R A"

   767     apply (subst D2.emeasure_pair_measure[OF A])

   768     apply (subst emeasure_density)

   769         using f_fst g_snd apply (simp add: split_beta')

   770       using A apply simp

   771     apply (subst positive_integral_density[OF g])

   772       apply (simp add: indicator_eq Pair_A)

   773     apply (subst positive_integral_density[OF f])

   774       apply (rule int_g)

   775     apply (subst M2.positive_integral_fst_measurable(2)[symmetric])

   776       using f g A Pair_A f_fst g_snd

   777       apply (auto intro!: positive_integral_cong_AE R.measurable_emeasure_Pair1

   778                   simp: positive_integral_cmult indicator_eq split_beta')

   779     apply (intro AE_I2 impI)

   780     apply (subst mult_assoc)

   781     apply (subst positive_integral_cmult)

   782           apply auto

   783     done

   784 qed simp

   785

   786 lemma sigma_finite_measure_distr:

   787   assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"

   788   shows "sigma_finite_measure M"

   789 proof -

   790   interpret sigma_finite_measure "distr M N f" by fact

   791   from sigma_finite_disjoint guess A . note A = this

   792   show ?thesis

   793   proof (unfold_locales, intro conjI exI allI)

   794     show "range (\<lambda>i. f - A i \<inter> space M) \<subseteq> sets M"

   795       using A f by (auto intro!: measurable_sets)

   796     show "(\<Union>i. f - A i \<inter> space M) = space M"

   797       using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def)

   798     fix i show "emeasure M (f - A i \<inter> space M) \<noteq> \<infinity>"

   799       using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq)

   800   qed

   801 qed

   802

   803 lemma measurable_cong':

   804   assumes sets: "sets M = sets M'" "sets N = sets N'"

   805   shows "measurable M N = measurable M' N'"

   806   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

   807

   808 lemma pair_measure_distr:

   809   assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"

   810   assumes "sigma_finite_measure (distr M S f)" "sigma_finite_measure (distr N T g)"

   811   shows "distr M S f \<Otimes>\<^isub>M distr N T g = distr (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")

   812 proof (rule measure_eqI)

   813   show "sets ?P = sets ?D"

   814     by simp

   815   interpret S: sigma_finite_measure "distr M S f" by fact

   816   interpret T: sigma_finite_measure "distr N T g" by fact

   817   interpret ST: pair_sigma_finite "distr M S f"  "distr N T g" ..

   818   interpret M: sigma_finite_measure M by (rule sigma_finite_measure_distr) fact+

   819   interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+

   820   interpret MN: pair_sigma_finite M N ..

   821   interpret SN: pair_sigma_finite "distr M S f" N ..

   822   have [simp]:

   823     "\<And>f g. fst \<circ> (\<lambda>(x, y). (f x, g y)) = f \<circ> fst" "\<And>f g. snd \<circ> (\<lambda>(x, y). (f x, g y)) = g \<circ> snd"

   824     by auto

   825   then have fg: "(\<lambda>(x, y). (f x, g y)) \<in> measurable (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T)"

   826     using measurable_comp[OF measurable_fst f] measurable_comp[OF measurable_snd g]

   827     by (auto simp: measurable_pair_iff)

   828   fix A assume A: "A \<in> sets ?P"

   829   then have "emeasure ?P A = (\<integral>\<^isup>+x. emeasure (distr N T g) (Pair x - A) \<partial>distr M S f)"

   830     by (rule T.emeasure_pair_measure_alt)

   831   also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g - (Pair x - A) \<inter> space N) \<partial>distr M S f)"

   832     using g A by (simp add: sets_Pair1 emeasure_distr)

   833   also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (g - (Pair (f x) - A) \<inter> space N) \<partial>M)"

   834     using f g A ST.measurable_emeasure_Pair1[OF A]

   835     by (intro positive_integral_distr) (auto simp add: sets_Pair1 emeasure_distr)

   836   also have "\<dots> = (\<integral>\<^isup>+x. emeasure N (Pair x - ((\<lambda>(x, y). (f x, g y)) - A \<inter> space (M \<Otimes>\<^isub>M N))) \<partial>M)"

   837     by (intro positive_integral_cong arg_cong2[where f=emeasure]) (auto simp: space_pair_measure)

   838   also have "\<dots> = emeasure (M \<Otimes>\<^isub>M N) ((\<lambda>(x, y). (f x, g y)) - A \<inter> space (M \<Otimes>\<^isub>M N))"

   839     using fg by (intro N.emeasure_pair_measure_alt[symmetric] measurable_sets[OF _ A])

   840                 (auto cong: measurable_cong')

   841   also have "\<dots> = emeasure ?D A"

   842     using fg A by (subst emeasure_distr) auto

   843   finally show "emeasure ?P A = emeasure ?D A" .

   844 qed

   845

   846 end