src/HOL/Probability/Binary_Product_Measure.thy
 author haftmann Sat Jul 05 11:01:53 2014 +0200 (2014-07-05) changeset 57514 bdc2c6b40bf2 parent 57447 87429bdecad5 child 58606 9c66f7c541fb permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
     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 Nonnegative_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 subsection "Binary products"

    18

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

    20   "A \<Otimes>\<^sub>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>\<^sup>+x. (\<integral>\<^sup>+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>\<^sub>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>\<^sub>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>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>M M2)"

    79   using measurable_Pair[OF assms] by simp

    80

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

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

    99   by simp_all

   100

   101 lemma measurable_pair_iff:

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

   121   by simp

   122

   123 lemma sets_Pair1[measurable (raw)]:

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

   134   by (auto intro!: measurable_Pair)

   135

   136 lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>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   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   with union 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>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>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>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>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>\<^sub>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>\<^sub>M M)"

   239   shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ 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>\<^sub>M M)) ?\<mu>"

   242     by (auto simp: positive_def nn_integral_nonneg)

   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>\<^sub>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>\<^sub>M M)" "disjoint_family F"

   248     from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^sub>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 nn_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1

   257                intro!: nn_integral_cong nn_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>\<^sub>M M)"

   265   shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+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!: nn_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>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"

   276 proof -

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

   278     using A B by (auto intro!: nn_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 nn_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>\<^sub>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>\<^sub>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>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>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>\<^sub>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>\<^sub>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: max.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>\<^sub>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>\<^sub>M M2"

   345 proof

   346   from M1.sigma_finite_countable guess F1 ..

   347   moreover from M2.sigma_finite_countable guess F2 ..

   348   ultimately show

   349     "\<exists>A. countable A \<and> A \<subseteq> sets (M1 \<Otimes>\<^sub>M M2) \<and> \<Union>A = space (M1 \<Otimes>\<^sub>M M2) \<and> (\<forall>a\<in>A. emeasure (M1 \<Otimes>\<^sub>M M2) a \<noteq> \<infinity>)"

   350     by (intro exI[of _ "(\<lambda>(a, b). a \<times> b)  (F1 \<times> F2)"] conjI)

   351        (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq

   352              dest: sets.sets_into_space)

   353 qed

   354

   355 lemma sigma_finite_pair_measure:

   356   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"

   357   shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"

   358 proof -

   359   interpret A: sigma_finite_measure A by fact

   360   interpret B: sigma_finite_measure B by fact

   361   interpret AB: pair_sigma_finite A  B ..

   362   show ?thesis ..

   363 qed

   364

   365 lemma sets_pair_swap:

   366   assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"

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

   368   using measurable_pair_swap' assms by (rule measurable_sets)

   369

   370 lemma (in pair_sigma_finite) distr_pair_swap:

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

   372 proof -

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

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

   375   show ?thesis

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

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

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

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

   380       by (simp add: sets_pair_measure space_pair_measure)

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

   382       by simp

   383   next

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

   385       using F by (auto simp: space_pair_measure)

   386   next

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

   388     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

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

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

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

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

   393                     measurable_pair_swap' ac_simps)

   394   qed

   395 qed

   396

   397 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:

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

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

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

   401 proof -

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

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

   404   show ?thesis using A

   405     by (subst distr_pair_swap)

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

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

   408 qed

   409

   410 lemma (in pair_sigma_finite) AE_pair:

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

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

   413 proof -

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

   415     using assms unfolding eventually_ae_filter by auto

   416   show ?thesis

   417   proof (rule AE_I)

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

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

   420       by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff emeasure_nonneg)

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

   422       by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)

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

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

   425       proof (rule AE_I)

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

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

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

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

   430       qed }

   431     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}"

   432       by auto

   433   qed

   434 qed

   435

   436 lemma (in pair_sigma_finite) AE_pair_measure:

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

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

   439   shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"

   440 proof (subst AE_iff_measurable[OF _ refl])

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

   442     by (rule sets.sets_Collect) fact

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

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

   445     by (simp add: M2.emeasure_pair_measure)

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

   447     using ae

   448     apply (safe intro!: nn_integral_cong_AE)

   449     apply (intro AE_I2)

   450     apply (safe intro!: nn_integral_cong_AE)

   451     apply auto

   452     done

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

   454 qed

   455

   456 lemma (in pair_sigma_finite) AE_pair_iff:

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

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

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

   460

   461 lemma (in pair_sigma_finite) AE_commute:

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

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

   464 proof -

   465   interpret Q: pair_sigma_finite M2 M1 ..

   466   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"

   467     by auto

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

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

   470     by (auto simp: space_pair_measure)

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

   472     by (intro sets_pair_swap P)

   473   finally show ?thesis

   474     apply (subst AE_pair_iff[OF P])

   475     apply (subst distr_pair_swap)

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

   477     apply (subst Q.AE_pair_iff)

   478     apply simp_all

   479     done

   480 qed

   481

   482 subsection "Fubinis theorem"

   483

   484 lemma measurable_compose_Pair1:

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

   486   by simp

   487

   488 lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst':

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

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

   491 using f proof induct

   492   case (cong u v)

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

   494     by (auto simp: space_pair_measure)

   495   show ?case

   496     apply (subst measurable_cong)

   497     apply (rule nn_integral_cong)

   498     apply fact+

   499     done

   500 next

   501   case (set Q)

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

   503     by (auto simp: indicator_def)

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

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

   506   from this measurable_emeasure_Pair[OF set] show ?case

   507     by (rule measurable_cong[THEN iffD1])

   508 qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1

   509                    nn_integral_monotone_convergence_SUP incseq_def le_fun_def

   510               cong: measurable_cong)

   511

   512 lemma (in sigma_finite_measure) nn_integral_fst':

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

   514   shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f" (is "?I f = _")

   515 using f proof induct

   516   case (cong u v)

   517   then have "?I u = ?I v"

   518     by (intro nn_integral_cong) (auto simp: space_pair_measure)

   519   with cong show ?case

   520     by (simp cong: nn_integral_cong)

   521 qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add

   522                    nn_integral_monotone_convergence_SUP

   523                    measurable_compose_Pair1 nn_integral_nonneg

   524                    borel_measurable_nn_integral_fst' nn_integral_mono incseq_def le_fun_def

   525               cong: nn_integral_cong)

   526

   527 lemma (in sigma_finite_measure) nn_integral_fst:

   528   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"

   529   shows "(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f"

   530   using f

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

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

   533   unfolding nn_integral_max_0 by auto

   534

   535 lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:

   536   "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N"

   537   using borel_measurable_nn_integral_fst'[of "\<lambda>x. max 0 (split f x)" N]

   538   by (simp add: nn_integral_max_0)

   539

   540 lemma (in pair_sigma_finite) nn_integral_snd:

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

   542   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"

   543 proof -

   544   note measurable_pair_swap[OF f]

   545   from M1.nn_integral_fst[OF this]

   546   have "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1))"

   547     by simp

   548   also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"

   549     by (subst distr_pair_swap)

   550        (auto simp: nn_integral_distr[OF measurable_pair_swap' f] intro!: nn_integral_cong)

   551   finally show ?thesis .

   552 qed

   553

   554 lemma (in pair_sigma_finite) Fubini:

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

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

   557   unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..

   558

   559 lemma (in pair_sigma_finite) Fubini':

   560   assumes f: "split f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"

   561   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f x y \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f x y \<partial>M2) \<partial>M1)"

   562   using Fubini[OF f] by simp

   563

   564 subsection {* Products on counting spaces, densities and distributions *}

   565

   566 lemma sigma_sets_pair_measure_generator_finite:

   567   assumes "finite A" and "finite B"

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

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

   570 proof safe

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

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

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

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

   575   proof (induct x)

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

   577   next

   578     case (insert a x)

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

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

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

   582   qed

   583 next

   584   fix x a b

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

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

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

   588 qed

   589

   590 lemma pair_measure_count_space:

   591   assumes A: "finite A" and B: "finite B"

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

   593 proof (rule measure_eqI)

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

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

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

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

   598     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)

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

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

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

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

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

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

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

   606     apply (subst emeasure_count_space)

   607     using X_subset apply auto []

   608     apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)

   609     apply (subst nn_integral_count_space)

   610     using A apply simp

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

   612     apply (subst card_gt_0_iff)

   613     apply (simp add: fin_Pair)

   614     apply (subst card_SigmaI[symmetric])

   615     using A apply simp

   616     using fin_Pair apply simp

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

   618     done

   619 qed

   620

   621 lemma pair_measure_density:

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

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

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

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

   626 proof (rule measure_eqI)

   627   interpret M2: sigma_finite_measure M2 by fact

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

   629

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

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

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

   633     by (intro nn_integral_cong_AE)

   634        (auto simp add: nn_integral_cmult[symmetric] ac_simps)

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

   636     by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density

   637                   M2.nn_integral_fst[symmetric]

   638              cong: nn_integral_cong)

   639 qed simp

   640

   641 lemma sigma_finite_measure_distr:

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

   643   shows "sigma_finite_measure M"

   644 proof -

   645   interpret sigma_finite_measure "distr M N f" by fact

   646   from sigma_finite_countable guess A .. note A = this

   647   show ?thesis

   648   proof

   649     show "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union>A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"

   650       using A f

   651       by (intro exI[of _ "(\<lambda>a. f - a \<inter> space M)  A"])

   652          (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)

   653   qed

   654 qed

   655

   656 lemma pair_measure_distr:

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

   658   assumes "sigma_finite_measure (distr N T g)"

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

   660 proof (rule measure_eqI)

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

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

   663

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

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

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

   667                        T.emeasure_pair_measure_alt nn_integral_distr

   668              intro!: nn_integral_cong arg_cong[where f="emeasure N"])

   669 qed simp

   670

   671 lemma pair_measure_eqI:

   672   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"

   673   assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"

   674   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)"

   675   shows "M1 \<Otimes>\<^sub>M M2 = M"

   676 proof -

   677   interpret M1: sigma_finite_measure M1 by fact

   678   interpret M2: sigma_finite_measure M2 by fact

   679   interpret pair_sigma_finite M1 M2 by default

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

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

   682   let ?P = "M1 \<Otimes>\<^sub>M M2"

   683   show ?thesis

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

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

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

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

   688       by (simp add: sets_pair_measure space_pair_measure)

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

   690       using sets[symmetric] by simp

   691   next

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

   693       using F by (auto simp: space_pair_measure)

   694   next

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

   696     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

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

   698        by (simp add: M2.emeasure_pair_measure_Times)

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

   700       using A B emeasure by auto

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

   702       by simp

   703   qed

   704 qed

   705

   706 lemma sets_pair_countable:

   707   assumes "countable S1" "countable S2"

   708   assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"

   709   shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"

   710 proof auto

   711   fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"

   712   from sets.sets_into_space[OF x(1)] x(2)

   713     sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N

   714   show "a \<in> S1" "b \<in> S2"

   715     by (auto simp: space_pair_measure)

   716 next

   717   fix X assume X: "X \<subseteq> S1 \<times> S2"

   718   then have "countable X"

   719     by (metis countable_subset countable S1 countable S2 countable_SIGMA)

   720   have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto

   721   also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)"

   722     using X

   723     by (safe intro!: sets.countable_UN' countable X subsetI pair_measureI) (auto simp: M N)

   724   finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .

   725 qed

   726

   727 lemma pair_measure_countable:

   728   assumes "countable S1" "countable S2"

   729   shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"

   730 proof (rule pair_measure_eqI)

   731   show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"

   732     using assms by (auto intro!: sigma_finite_measure_count_space_countable)

   733   show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"

   734     by (subst sets_pair_countable[OF assms]) auto

   735 next

   736   fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)"

   737   then show "emeasure (count_space S1) A * emeasure (count_space S2) B =

   738     emeasure (count_space (S1 \<times> S2)) (A \<times> B)"

   739     by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff)

   740 qed

   741

   742 subsection {* Product of Borel spaces *}

   743

   744 lemma borel_Times:

   745   fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"

   746   assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"

   747   shows "A \<times> B \<in> sets borel"

   748 proof -

   749   have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"

   750     by auto

   751   moreover

   752   { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)

   753     then have "A\<times>UNIV \<in> sets borel"

   754     proof (induct A)

   755       case (Basic S) then show ?case

   756         by (auto intro!: borel_open open_Times)

   757     next

   758       case (Compl A)

   759       moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)"

   760         by auto

   761       ultimately show ?case

   762         unfolding * by auto

   763     next

   764       case (Union A)

   765       moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)"

   766         by auto

   767       ultimately show ?case

   768         unfolding * by auto

   769     qed simp }

   770   moreover

   771   { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)

   772     then have "UNIV\<times>B \<in> sets borel"

   773     proof (induct B)

   774       case (Basic S) then show ?case

   775         by (auto intro!: borel_open open_Times)

   776     next

   777       case (Compl B)

   778       moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)"

   779         by auto

   780       ultimately show ?case

   781         unfolding * by auto

   782     next

   783       case (Union B)

   784       moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)"

   785         by auto

   786       ultimately show ?case

   787         unfolding * by auto

   788     qed simp }

   789   ultimately show ?thesis

   790     by auto

   791 qed

   792

   793 lemma borel_prod: "sets (borel \<Otimes>\<^sub>M borel) =

   794     (sets borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) set set)"

   795   (is "?l = ?r")

   796 proof -

   797   obtain A :: "'a set set" where A: "countable A" "topological_basis A"

   798     by (metis ex_countable_basis)

   799   moreover obtain B :: "'b set set" where B: "countable B" "topological_basis B"

   800     by (metis ex_countable_basis)

   801   ultimately have AB: "countable ((\<lambda>(a, b). a \<times> b)  (A \<times> B))" "topological_basis ((\<lambda>(a, b). a \<times> b)  (A \<times> B))"

   802     by (auto intro!: topological_basis_prod)

   803   have "sets (borel \<Otimes>\<^sub>M borel) = sigma_sets UNIV {a \<times> b |a b. a \<in> sigma_sets UNIV A \<and> b \<in> sigma_sets UNIV B}"

   804     by (simp add: sets_pair_measure

   805        borel_eq_countable_basis[OF A] borel_eq_countable_basis[OF B])

   806   also have "\<dots> \<supseteq> sigma_sets UNIV ((\<lambda>(a, b). a \<times> b)  (A \<times> B))" (is "... \<supseteq> ?A")

   807     by (auto intro!: sigma_sets_mono)

   808   also (xtrans) have "?A = sets borel"

   809     by (simp add: borel_eq_countable_basis[OF AB])

   810   finally have "?r \<subseteq> ?l" .

   811   moreover have "?l \<subseteq> ?r"

   812   proof (simp add: sets_pair_measure, safe intro!: sigma_sets_mono)

   813     fix A::"('a \<times> 'b) set" assume "A \<in> sigma_sets UNIV {a \<times> b |a b. a \<in> sets borel \<and> b \<in> sets borel}"

   814     then show "A \<in> sets borel"

   815       by (induct A) (auto intro!: borel_Times)

   816   qed

   817   ultimately show ?thesis by auto

   818 qed

   819

   820 lemma borel_prod':

   821   "borel \<Otimes>\<^sub>M borel = (borel ::

   822       ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"

   823 proof (rule measure_eqI[OF borel_prod])

   824   interpret sigma_finite_measure "borel :: 'b measure"

   825     proof qed (intro exI[of _ "{UNIV}"], auto simp: borel_def emeasure_sigma)

   826   fix X :: "('a \<times> 'b) set" assume asm: "X \<in> sets (borel \<Otimes>\<^sub>M borel)"

   827   have "UNIV \<times> UNIV \<in> sets (borel \<Otimes>\<^sub>M borel :: ('a \<times> 'b) measure)"

   828       by (simp add: borel_prod)

   829   moreover have "emeasure (borel \<Otimes>\<^sub>M borel) (UNIV \<times> UNIV :: ('a \<times> 'b) set) = 0"

   830       by (subst emeasure_pair_measure_Times, simp_all add: borel_def emeasure_sigma)

   831   moreover have "X \<subseteq> UNIV \<times> UNIV" by auto

   832   ultimately have "emeasure (borel \<Otimes>\<^sub>M borel) X = 0" by (rule emeasure_eq_0)

   833   thus "emeasure (borel \<Otimes>\<^sub>M borel) X = emeasure borel X"

   834       by (simp add: borel_def emeasure_sigma)

   835 qed

   836

   837 lemma finite_measure_pair_measure:

   838   assumes "finite_measure M" "finite_measure N"

   839   shows "finite_measure (N  \<Otimes>\<^sub>M M)"

   840 proof (rule finite_measureI)

   841   interpret M: finite_measure M by fact

   842   interpret N: finite_measure N by fact

   843   show "emeasure (N  \<Otimes>\<^sub>M M) (space (N  \<Otimes>\<^sub>M M)) \<noteq> \<infinity>"

   844     by (auto simp: space_pair_measure M.emeasure_pair_measure_Times)

   845 qed

   846

   847 end`