src/HOL/Probability/Binary_Product_Measure.thy
 author haftmann Sun Jun 23 21:16:07 2013 +0200 (2013-06-23) changeset 52435 6646bb548c6b parent 50244 de72bbe42190 child 53015 a1119cf551e8 permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
     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 Pair_vimage_times[simp]: "Pair x - (A \<times> B) = (if x \<in> A then B else {})"

    12   by auto

    13

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

    15   by auto

    16

    17 section "Binary products"

    18

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

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

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

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

    23

    24 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)"

    25   using sets.space_closed[of A] sets.space_closed[of B] by auto

    26

    27 lemma space_pair_measure:

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

    29   unfolding pair_measure_def using pair_measure_closed[of A B]

    30   by (rule space_measure_of)

    31

    32 lemma sets_pair_measure:

    33   "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}"

    34   unfolding pair_measure_def using pair_measure_closed[of A B]

    35   by (rule sets_measure_of)

    36

    37 lemma sets_pair_measure_cong[cong]:

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

    39   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)

    40

    41 lemma pair_measureI[intro, simp, measurable]:

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

    43   by (auto simp: sets_pair_measure)

    44

    45 lemma measurable_pair_measureI:

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

    47   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"

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

    49   unfolding pair_measure_def using 1 2

    50   by (intro measurable_measure_of) (auto dest: sets.sets_into_space)

    51

    52 lemma measurable_split_replace[measurable (raw)]:

    53   "(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. split (f x) (g x)) \<in> measurable M N"

    54   unfolding split_beta' .

    55

    56 lemma measurable_Pair[measurable (raw)]:

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

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

    59 proof (rule measurable_pair_measureI)

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

    61     using f g by (auto simp: measurable_def)

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

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

    64     by auto

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

    66     by (rule sets.Int) (auto intro!: measurable_sets * f g)

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

    68 qed

    69

    70 lemma measurable_Pair_compose_split[measurable_dest]:

    71   assumes f: "split f \<in> measurable (M1 \<Otimes>\<^isub>M M2) N"

    72   assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"

    73   shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"

    74   using measurable_compose[OF measurable_Pair f, OF g h] by simp

    75

    76 lemma measurable_pair:

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

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

    79   using measurable_Pair[OF assms] by simp

    80

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

    82   by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times

    83     measurable_def)

    84

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

    86   by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times

    87     measurable_def)

    88

    89 lemma

    90   assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^isub>M P)"

    91   shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"

    92     and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"

    93   by simp_all

    94

    95 lemma

    96   assumes f[measurable]: "f \<in> measurable M N"

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

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

    99   by simp_all

   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   by (auto intro: measurable_pair[of f M M1 M2])

   104

   105 lemma measurable_split_conv:

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

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

   108

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

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

   111

   112 lemma measurable_pair_swap:

   113   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"

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

   115

   116 lemma measurable_pair_swap_iff:

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

   118   by (auto dest: measurable_pair_swap)

   119

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

   121   by simp

   122

   123 lemma sets_Pair1[measurable (raw)]:

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

   125 proof -

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

   127     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)

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

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

   130   finally show ?thesis .

   131 qed

   132

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

   134   by (auto intro!: measurable_Pair)

   135

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

   137 proof -

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

   139     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)

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

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

   142   finally show ?thesis .

   143 qed

   144

   145 lemma measurable_Pair2:

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

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

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

   149   by (simp add: comp_def)

   150

   151 lemma measurable_Pair1:

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

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

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

   155   by (simp add: comp_def)

   156

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

   158   unfolding Int_stable_def

   159   by safe (auto simp add: times_Int_times)

   160

   161 lemma disjoint_family_vimageI: "disjoint_family F \<Longrightarrow> disjoint_family (\<lambda>i. f - F i)"

   162   by (auto simp: disjoint_family_on_def)

   163

   164 lemma (in finite_measure) finite_measure_cut_measurable:

   165   assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^isub>M M)"

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

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

   168   using Int_stable_pair_measure_generator pair_measure_closed assms

   169   unfolding sets_pair_measure

   170 proof (induct rule: sigma_sets_induct_disjoint)

   171   case (compl A)

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

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

   174     unfolding sets_pair_measure[symmetric]

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

   176   with compl sets.top show ?case

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

   178 next

   179   case (union F)

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

   181     by (simp add: suminf_emeasure disjoint_family_vimageI subset_eq vimage_UN sets_pair_measure[symmetric])

   182   ultimately show ?case

   183     unfolding sets_pair_measure[symmetric] by simp

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

   185

   186 lemma (in sigma_finite_measure) measurable_emeasure_Pair:

   187   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> _")

   188 proof -

   189   from sigma_finite_disjoint guess F . note F = this

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

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

   192   { fix i

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

   194       using F sets.sets_into_space by auto

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

   196     have "finite_measure ?R"

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

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

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

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

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

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

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

   204       using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure)

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

   206       by simp }

   207   moreover

   208   { fix x

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

   210     proof (intro suminf_emeasure)

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

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

   213       have "disjoint_family F" using F by auto

   214       show "disjoint_family (?C x)"

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

   216     qed

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

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

   219       by (auto simp: space_pair_measure)

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

   221       by simp }

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

   223     by auto

   224 qed

   225

   226 lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:

   227   assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"

   228   assumes A: "{x\<in>space (N \<Otimes>\<^isub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^isub>M M)"

   229   shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"

   230 proof -

   231   from space have "\<And>x. x \<in> space N \<Longrightarrow> Pair x - {x \<in> space (N \<Otimes>\<^isub>M M). snd x \<in> A (fst x)} = A x"

   232     by (auto simp: space_pair_measure)

   233   with measurable_emeasure_Pair[OF A] show ?thesis

   234     by (auto cong: measurable_cong)

   235 qed

   236

   237 lemma (in sigma_finite_measure) emeasure_pair_measure:

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

   239   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")

   240 proof (rule emeasure_measure_of[OF pair_measure_def])

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

   242     by (auto simp: positive_def positive_integral_positive)

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

   244     by (auto simp: indicator_def)

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

   246   proof (rule countably_additiveI)

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

   248     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

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

   250       by (intro measurable_emeasure_Pair) auto

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

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

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

   254       using F by (auto simp: sets_Pair1)

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

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

   257                intro!: positive_integral_cong positive_integral_indicator[symmetric])

   258   qed

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

   260     using sets.space_closed[of N] sets.space_closed[of M] by auto

   261 qed fact

   262

   263 lemma (in sigma_finite_measure) emeasure_pair_measure_alt:

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

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

   266 proof -

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

   268     by (auto simp: indicator_def)

   269   show ?thesis

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

   271 qed

   272

   273 lemma (in sigma_finite_measure) emeasure_pair_measure_Times:

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

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

   276 proof -

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

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

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

   280     using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator)

   281   finally show ?thesis

   282     by (simp add: ac_simps)

   283 qed

   284

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

   286

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

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

   289

   290 lemma (in pair_sigma_finite) measurable_emeasure_Pair1:

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

   292   using M2.measurable_emeasure_Pair .

   293

   294 lemma (in pair_sigma_finite) measurable_emeasure_Pair2:

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

   296 proof -

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

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

   299   note M1.measurable_emeasure_Pair[OF this]

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

   301     using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)

   302   ultimately show ?thesis by simp

   303 qed

   304

   305 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:

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

   307   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>

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

   309 proof -

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

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

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

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

   314   show ?thesis

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

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

   317   next

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

   319     proof (intro subsetI)

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

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

   322         by (auto simp: space)

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

   324         using incseq F1 incseq F2 unfolding incseq_def

   325         by (force split: split_max)+

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

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

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

   329     qed

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

   331       using space by (auto simp: space)

   332   next

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

   334       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto

   335   next

   336     fix i

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

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

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

   340       by (auto simp add: emeasure_pair_measure_Times)

   341   qed

   342 qed

   343

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

   345 proof

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

   347   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>)"

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

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

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

   351       using F by (auto simp: space_pair_measure)

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

   353   qed

   354 qed

   355

   356 lemma sigma_finite_pair_measure:

   357   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"

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

   359 proof -

   360   interpret A: sigma_finite_measure A by fact

   361   interpret B: sigma_finite_measure B by fact

   362   interpret AB: pair_sigma_finite A  B ..

   363   show ?thesis ..

   364 qed

   365

   366 lemma sets_pair_swap:

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

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

   369   using measurable_pair_swap' assms by (rule measurable_sets)

   370

   371 lemma (in pair_sigma_finite) distr_pair_swap:

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

   373 proof -

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

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

   376   show ?thesis

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

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

   379       using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)

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

   381       by (simp add: sets_pair_measure space_pair_measure)

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

   383       by simp

   384   next

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

   386       using F by (auto simp: space_pair_measure)

   387   next

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

   389     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

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

   391       using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure)

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

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

   394                     measurable_pair_swap' ac_simps)

   395   qed

   396 qed

   397

   398 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:

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

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

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

   402 proof -

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

   404     using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)

   405   show ?thesis using A

   406     by (subst distr_pair_swap)

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

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

   409 qed

   410

   411 lemma (in pair_sigma_finite) AE_pair:

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

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

   414 proof -

   415   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"

   416     using assms unfolding eventually_ae_filter by auto

   417   show ?thesis

   418   proof (rule AE_I)

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

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

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

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

   423       by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)

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

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

   426       proof (rule AE_I)

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

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

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

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

   431       qed }

   432     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}"

   433       by auto

   434   qed

   435 qed

   436

   437 lemma (in pair_sigma_finite) AE_pair_measure:

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

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

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

   441 proof (subst AE_iff_measurable[OF _ refl])

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

   443     by (rule sets.sets_Collect) fact

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

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

   446     by (simp add: M2.emeasure_pair_measure)

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

   448     using ae

   449     apply (safe intro!: positive_integral_cong_AE)

   450     apply (intro AE_I2)

   451     apply (safe intro!: positive_integral_cong_AE)

   452     apply auto

   453     done

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

   455 qed

   456

   457 lemma (in pair_sigma_finite) AE_pair_iff:

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

   459     (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))"

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

   461

   462 lemma (in pair_sigma_finite) AE_commute:

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

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

   465 proof -

   466   interpret Q: pair_sigma_finite M2 M1 ..

   467   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"

   468     by auto

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

   470     (\<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)"

   471     by (auto simp: space_pair_measure)

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

   473     by (intro sets_pair_swap P)

   474   finally show ?thesis

   475     apply (subst AE_pair_iff[OF P])

   476     apply (subst distr_pair_swap)

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

   478     apply (subst Q.AE_pair_iff)

   479     apply simp_all

   480     done

   481 qed

   482

   483 section "Fubinis theorem"

   484

   485 lemma measurable_compose_Pair1:

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

   487   by simp

   488

   489 lemma (in sigma_finite_measure) borel_measurable_positive_integral_fst':

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

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

   492 using f proof induct

   493   case (cong u v)

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

   495     by (auto simp: space_pair_measure)

   496   show ?case

   497     apply (subst measurable_cong)

   498     apply (rule positive_integral_cong)

   499     apply fact+

   500     done

   501 next

   502   case (set Q)

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

   504     by (auto simp: indicator_def)

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

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

   507   from this measurable_emeasure_Pair[OF set] show ?case

   508     by (rule measurable_cong[THEN iffD1])

   509 qed (simp_all add: positive_integral_add positive_integral_cmult measurable_compose_Pair1

   510                    positive_integral_monotone_convergence_SUP incseq_def le_fun_def

   511               cong: measurable_cong)

   512

   513 lemma (in sigma_finite_measure) positive_integral_fst:

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

   515   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 = _")

   516 using f proof induct

   517   case (cong u v)

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

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

   520   ultimately show ?case

   521     by (simp cong: positive_integral_cong)

   522 qed (simp_all add: emeasure_pair_measure positive_integral_cmult positive_integral_add

   523                    positive_integral_monotone_convergence_SUP

   524                    measurable_compose_Pair1 positive_integral_positive

   525                    borel_measurable_positive_integral_fst' positive_integral_mono incseq_def le_fun_def

   526               cong: positive_integral_cong)

   527

   528 lemma (in sigma_finite_measure) positive_integral_fst_measurable:

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

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

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

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

   533   using f

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

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

   536   unfolding positive_integral_max_0 by auto

   537

   538 lemma (in sigma_finite_measure) borel_measurable_positive_integral[measurable (raw)]:

   539   "split f \<in> borel_measurable (N \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M) \<in> borel_measurable N"

   540   using positive_integral_fst_measurable(1)[of "split f" N] by simp

   541

   542 lemma (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]:

   543   "split f \<in> borel_measurable (N \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M) \<in> borel_measurable N"

   544   by (simp add: lebesgue_integral_def)

   545

   546 lemma (in pair_sigma_finite) positive_integral_snd_measurable:

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

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

   549 proof -

   550   note measurable_pair_swap[OF f]

   551   from M1.positive_integral_fst_measurable[OF this]

   552   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))"

   553     by simp

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

   555     by (subst distr_pair_swap)

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

   557   finally show ?thesis .

   558 qed

   559

   560 lemma (in pair_sigma_finite) Fubini:

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

   562   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)"

   563   unfolding positive_integral_snd_measurable[OF assms]

   564   unfolding M2.positive_integral_fst_measurable[OF assms] ..

   565

   566 lemma (in pair_sigma_finite) integrable_product_swap:

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

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

   569 proof -

   570   interpret Q: pair_sigma_finite M2 M1 by default

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

   572   show ?thesis unfolding *

   573     by (rule integrable_distr[OF measurable_pair_swap'])

   574        (simp add: distr_pair_swap[symmetric] assms)

   575 qed

   576

   577 lemma (in pair_sigma_finite) integrable_product_swap_iff:

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

   579 proof -

   580   interpret Q: pair_sigma_finite M2 M1 by default

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

   582   show ?thesis by auto

   583 qed

   584

   585 lemma (in pair_sigma_finite) integral_product_swap:

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

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

   588 proof -

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

   590   show ?thesis unfolding *

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

   592 qed

   593

   594 lemma (in pair_sigma_finite) integrable_fst_measurable:

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

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

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

   598 proof -

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

   600     using f by auto

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

   602   have

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

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

   605     using assms by auto

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

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

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

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

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

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

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

   613     by (auto intro!: positive_integral_PInf_AE )

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

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

   616     by (auto simp: positive_integral_positive)

   617   from AE_pos show ?AE using assms

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

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

   620       using positive_integral_positive

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

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

   623   note this[simp]

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

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

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

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

   628     proof (intro integrable_def[THEN iffD2] conjI)

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

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

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

   632         using AE positive_integral_positive[of M2]

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

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

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

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

   637         by (intro positive_integral_cong_pos)

   638            (simp add: positive_integral_positive real_of_ereal_pos)

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

   640     qed }

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

   642   show ?INT

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

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

   645     using AE[THEN integral_real]

   646     by simp

   647 qed

   648

   649 lemma (in pair_sigma_finite) integrable_snd_measurable:

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

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

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

   653 proof -

   654   interpret Q: pair_sigma_finite M2 M1 by default

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

   656     using f unfolding integrable_product_swap_iff .

   657   show ?INT

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

   659     using integral_product_swap[of f] f by auto

   660   show ?AE

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

   662     by simp

   663 qed

   664

   665 lemma (in pair_sigma_finite) Fubini_integral:

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

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

   668   unfolding integrable_snd_measurable[OF assms]

   669   unfolding integrable_fst_measurable[OF assms] ..

   670

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

   672

   673 lemma sigma_sets_pair_measure_generator_finite:

   674   assumes "finite A" and "finite B"

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

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

   677 proof safe

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

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

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

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

   682   proof (induct x)

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

   684   next

   685     case (insert a x)

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

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

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

   689   qed

   690 next

   691   fix x a b

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

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

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

   695 qed

   696

   697 lemma pair_measure_count_space:

   698   assumes A: "finite A" and B: "finite B"

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

   700 proof (rule measure_eqI)

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

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

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

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

   705     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)

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

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

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

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

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

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

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

   713     apply (subst emeasure_count_space)

   714     using X_subset apply auto []

   715     apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)

   716     apply (subst positive_integral_count_space)

   717     using A apply simp

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

   719     apply (subst card_gt_0_iff)

   720     apply (simp add: fin_Pair)

   721     apply (subst card_SigmaI[symmetric])

   722     using A apply simp

   723     using fin_Pair apply simp

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

   725     done

   726 qed

   727

   728 lemma pair_measure_density:

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

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

   731   assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"

   732   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")

   733 proof (rule measure_eqI)

   734   interpret M2: sigma_finite_measure M2 by fact

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

   736

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

   738   with f g have "(\<integral>\<^isup>+ x. f x * \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) =

   739     (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)"

   740     by (intro positive_integral_cong_AE)

   741        (auto simp add: positive_integral_cmult[symmetric] ac_simps)

   742   with A f g show "emeasure ?L A = emeasure ?R A"

   743     by (simp add: D2.emeasure_pair_measure emeasure_density positive_integral_density

   744                   M2.positive_integral_fst_measurable(2)[symmetric]

   745              cong: positive_integral_cong)

   746 qed simp

   747

   748 lemma sigma_finite_measure_distr:

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

   750   shows "sigma_finite_measure M"

   751 proof -

   752   interpret sigma_finite_measure "distr M N f" by fact

   753   from sigma_finite_disjoint guess A . note A = this

   754   show ?thesis

   755   proof (unfold_locales, intro conjI exI allI)

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

   757       using A f by auto

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

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

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

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

   762   qed

   763 qed

   764

   765 lemma pair_measure_distr:

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

   767   assumes "sigma_finite_measure (distr N T g)"

   768   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")

   769 proof (rule measure_eqI)

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

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

   772

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

   774   with f g show "emeasure ?P A = emeasure ?D A"

   775     by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr

   776                        T.emeasure_pair_measure_alt positive_integral_distr

   777              intro!: positive_integral_cong arg_cong[where f="emeasure N"])

   778 qed simp

   779

   780 lemma pair_measure_eqI:

   781   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"

   782   assumes sets: "sets (M1 \<Otimes>\<^isub>M M2) = sets M"

   783   assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)"

   784   shows "M1 \<Otimes>\<^isub>M M2 = M"

   785 proof -

   786   interpret M1: sigma_finite_measure M1 by fact

   787   interpret M2: sigma_finite_measure M2 by fact

   788   interpret pair_sigma_finite M1 M2 by default

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

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

   791   let ?P = "M1 \<Otimes>\<^isub>M M2"

   792   show ?thesis

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

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

   795       using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)

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

   797       by (simp add: sets_pair_measure space_pair_measure)

   798     then show "sets M = sigma_sets (space ?P) ?E"

   799       using sets[symmetric] by simp

   800   next

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

   802       using F by (auto simp: space_pair_measure)

   803   next

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

   805     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

   806     then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"

   807        by (simp add: M2.emeasure_pair_measure_Times)

   808     also have "\<dots> = emeasure M (A \<times> B)"

   809       using A B emeasure by auto

   810     finally show "emeasure ?P X = emeasure M X"

   811       by simp

   812   qed

   813 qed

   814

   815 end