src/HOL/Probability/Binary_Product_Measure.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60066 14efa7f4ee7b child 60727 53697011b03a permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     1 (*  Title:      HOL/Probability/Binary_Product_Measure.thy

     2     Author:     Johannes Hölzl, TU München

     3 *)

     4

     5 section {*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 SIGMA_Collect_eq: "(SIGMA x:space M. {y\<in>space N. P x y}) = {x\<in>space (M \<Otimes>\<^sub>M N). P (fst x) (snd x)}"

    33   by (auto simp: space_pair_measure)

    34

    35 lemma sets_pair_measure:

    36   "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}"

    37   unfolding pair_measure_def using pair_measure_closed[of A B]

    38   by (rule sets_measure_of)

    39

    40 lemma sets_pair_in_sets:

    41   assumes N: "space A \<times> space B = space N"

    42   assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"

    43   shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"

    44   using assms by (auto intro!: sets.sigma_sets_subset simp: sets_pair_measure N)

    45

    46 lemma sets_pair_measure_cong[measurable_cong, cong]:

    47   "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"

    48   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)

    49

    50 lemma pair_measureI[intro, simp, measurable]:

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

    52   by (auto simp: sets_pair_measure)

    53

    54 lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"

    55   using pair_measureI[of "{x}" M1 "{y}" M2] by simp

    56

    57 lemma measurable_pair_measureI:

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

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

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

    61   unfolding pair_measure_def using 1 2

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

    63

    64 lemma measurable_split_replace[measurable (raw)]:

    65   "(\<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"

    66   unfolding split_beta' .

    67

    68 lemma measurable_Pair[measurable (raw)]:

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

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

    71 proof (rule measurable_pair_measureI)

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

    73     using f g by (auto simp: measurable_def)

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

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

    76     by auto

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

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

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

    80 qed

    81

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

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

    84     measurable_def)

    85

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

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

    88     measurable_def)

    89

    90 lemma measurable_Pair_compose_split[measurable_dest]:

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

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

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

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

    95

    96 lemma measurable_Pair1_compose[measurable_dest]:

    97   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"

    98   assumes [measurable]: "h \<in> measurable N M"

    99   shows "(\<lambda>x. f (h x)) \<in> measurable N M1"

   100   using measurable_compose[OF f measurable_fst] by simp

   101

   102 lemma measurable_Pair2_compose[measurable_dest]:

   103   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"

   104   assumes [measurable]: "h \<in> measurable N M"

   105   shows "(\<lambda>x. g (h x)) \<in> measurable N M2"

   106   using measurable_compose[OF f measurable_snd] by simp

   107

   108 lemma measurable_pair:

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

   110   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"

   111   using measurable_Pair[OF assms] by simp

   112

   113 lemma

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

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

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

   117   by simp_all

   118

   119 lemma

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

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

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

   123   by simp_all

   124

   125 lemma sets_pair_eq_sets_fst_snd:

   126   "sets (A \<Otimes>\<^sub>M B) = sets (Sup_sigma {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})"

   127     (is "?P = sets (Sup_sigma {?fst, ?snd})")

   128 proof -

   129   { fix a b assume ab: "a \<in> sets A" "b \<in> sets B"

   130     then have "a \<times> b = (fst - a \<inter> (space A \<times> space B)) \<inter> (snd - b \<inter> (space A \<times> space B))"

   131       by (auto dest: sets.sets_into_space)

   132     also have "\<dots> \<in> sets (Sup_sigma {?fst, ?snd})"

   133       using ab by (auto intro: in_Sup_sigma in_vimage_algebra)

   134     finally have "a \<times> b \<in> sets (Sup_sigma {?fst, ?snd})" . }

   135   moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)"

   136     by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric])

   137   moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)"

   138     by (rule sets_image_in_sets) (auto simp: space_pair_measure)

   139   ultimately show ?thesis

   140     by (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets )

   141        (auto simp add: space_Sup_sigma space_pair_measure)

   142 qed

   143

   144 lemma measurable_pair_iff:

   145   "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"

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

   147

   148 lemma measurable_split_conv:

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

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

   151

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

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

   154

   155 lemma measurable_pair_swap:

   156   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"

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

   158

   159 lemma measurable_pair_swap_iff:

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

   161   by (auto dest: measurable_pair_swap)

   162

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

   164   by simp

   165

   166 lemma sets_Pair1[measurable (raw)]:

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

   168 proof -

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

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

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

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

   173   finally show ?thesis .

   174 qed

   175

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

   177   by (auto intro!: measurable_Pair)

   178

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

   180 proof -

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

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

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

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

   185   finally show ?thesis .

   186 qed

   187

   188 lemma measurable_Pair2:

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

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

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

   192   by (simp add: comp_def)

   193

   194 lemma measurable_Pair1:

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

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

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

   198   by (simp add: comp_def)

   199

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

   201   unfolding Int_stable_def

   202   by safe (auto simp add: times_Int_times)

   203

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

   205   by (auto simp: disjoint_family_on_def)

   206

   207 lemma (in finite_measure) finite_measure_cut_measurable:

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

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

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

   211   using Int_stable_pair_measure_generator pair_measure_closed assms

   212   unfolding sets_pair_measure

   213 proof (induct rule: sigma_sets_induct_disjoint)

   214   case (compl A)

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

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

   217     unfolding sets_pair_measure[symmetric]

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

   219   with compl sets.top show ?case

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

   221 next

   222   case (union F)

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

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

   225   with union show ?case

   226     unfolding sets_pair_measure[symmetric] by simp

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

   228

   229 lemma (in sigma_finite_measure) measurable_emeasure_Pair:

   230   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> _")

   231 proof -

   232   from sigma_finite_disjoint guess F . note F = this

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

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

   235   { fix i

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

   237       using F sets.sets_into_space by auto

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

   239     have "finite_measure ?R"

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

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

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

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

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

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

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

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

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

   249       by simp }

   250   moreover

   251   { fix x

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

   253     proof (intro suminf_emeasure)

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

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

   256       have "disjoint_family F" using F by auto

   257       show "disjoint_family (?C x)"

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

   259     qed

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

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

   262       by (auto simp: space_pair_measure)

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

   264       by simp }

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

   266     by auto

   267 qed

   268

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

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

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

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

   273 proof -

   274   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"

   275     by (auto simp: space_pair_measure)

   276   with measurable_emeasure_Pair[OF A] show ?thesis

   277     by (auto cong: measurable_cong)

   278 qed

   279

   280 lemma (in sigma_finite_measure) emeasure_pair_measure:

   281   assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"

   282   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")

   283 proof (rule emeasure_measure_of[OF pair_measure_def])

   284   show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"

   285     by (auto simp: positive_def nn_integral_nonneg)

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

   287     by (auto simp: indicator_def)

   288   show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"

   289   proof (rule countably_additiveI)

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

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

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

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

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

   295       using F by (auto simp: sets_Pair1)

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

   297       by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure emeasure_nonneg

   298                intro!: nn_integral_cong nn_integral_indicator[symmetric])

   299   qed

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

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

   302 qed fact

   303

   304 lemma (in sigma_finite_measure) emeasure_pair_measure_alt:

   305   assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"

   306   shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x - X) \<partial>N)"

   307 proof -

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

   309     by (auto simp: indicator_def)

   310   show ?thesis

   311     using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)

   312 qed

   313

   314 lemma (in sigma_finite_measure) emeasure_pair_measure_Times:

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

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

   317 proof -

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

   319     using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)

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

   321     using A by (simp add: emeasure_nonneg nn_integral_cmult_indicator)

   322   finally show ?thesis

   323     by (simp add: ac_simps)

   324 qed

   325

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

   327

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

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

   330

   331 lemma (in pair_sigma_finite) measurable_emeasure_Pair1:

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

   333   using M2.measurable_emeasure_Pair .

   334

   335 lemma (in pair_sigma_finite) measurable_emeasure_Pair2:

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

   337 proof -

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

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

   340   note M1.measurable_emeasure_Pair[OF this]

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

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

   343   ultimately show ?thesis by simp

   344 qed

   345

   346 lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:

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

   348   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>

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

   350 proof -

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

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

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

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

   355   show ?thesis

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

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

   358   next

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

   360     proof (intro subsetI)

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

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

   363         by (auto simp: space)

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

   365         using incseq F1 incseq F2 unfolding incseq_def

   366         by (force split: split_max)+

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

   368         by (intro SigmaI) (auto simp add: max.commute)

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

   370     qed

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

   372       using space by (auto simp: space)

   373   next

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

   375       using incseq F1 incseq F2 unfolding incseq_Suc_iff by auto

   376   next

   377     fix i

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

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

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

   381       by (auto simp add: emeasure_pair_measure_Times)

   382   qed

   383 qed

   384

   385 sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"

   386 proof

   387   from M1.sigma_finite_countable guess F1 ..

   388   moreover from M2.sigma_finite_countable guess F2 ..

   389   ultimately show

   390     "\<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>)"

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

   392        (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq

   393              dest: sets.sets_into_space)

   394 qed

   395

   396 lemma sigma_finite_pair_measure:

   397   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"

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

   399 proof -

   400   interpret A: sigma_finite_measure A by fact

   401   interpret B: sigma_finite_measure B by fact

   402   interpret AB: pair_sigma_finite A  B ..

   403   show ?thesis ..

   404 qed

   405

   406 lemma sets_pair_swap:

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

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

   409   using measurable_pair_swap' assms by (rule measurable_sets)

   410

   411 lemma (in pair_sigma_finite) distr_pair_swap:

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

   413 proof -

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

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

   416   show ?thesis

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

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

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

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

   421       by (simp add: sets_pair_measure space_pair_measure)

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

   423       by simp

   424   next

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

   426       using F by (auto simp: space_pair_measure)

   427   next

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

   429     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

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

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

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

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

   434                     measurable_pair_swap' ac_simps)

   435   qed

   436 qed

   437

   438 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:

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

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

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

   442 proof -

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

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

   445   show ?thesis using A

   446     by (subst distr_pair_swap)

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

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

   449 qed

   450

   451 lemma (in pair_sigma_finite) AE_pair:

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

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

   454 proof -

   455   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"

   456     using assms unfolding eventually_ae_filter by auto

   457   show ?thesis

   458   proof (rule AE_I)

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

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

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

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

   463       by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N)

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

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

   466       proof (rule AE_I)

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

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

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

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

   471       qed }

   472     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}"

   473       by auto

   474   qed

   475 qed

   476

   477 lemma (in pair_sigma_finite) AE_pair_measure:

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

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

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

   481 proof (subst AE_iff_measurable[OF _ refl])

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

   483     by (rule sets.sets_Collect) fact

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

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

   486     by (simp add: M2.emeasure_pair_measure)

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

   488     using ae

   489     apply (safe intro!: nn_integral_cong_AE)

   490     apply (intro AE_I2)

   491     apply (safe intro!: nn_integral_cong_AE)

   492     apply auto

   493     done

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

   495 qed

   496

   497 lemma (in pair_sigma_finite) AE_pair_iff:

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

   499     (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))"

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

   501

   502 lemma (in pair_sigma_finite) AE_commute:

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

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

   505 proof -

   506   interpret Q: pair_sigma_finite M2 M1 ..

   507   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"

   508     by auto

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

   510     (\<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)"

   511     by (auto simp: space_pair_measure)

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

   513     by (intro sets_pair_swap P)

   514   finally show ?thesis

   515     apply (subst AE_pair_iff[OF P])

   516     apply (subst distr_pair_swap)

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

   518     apply (subst Q.AE_pair_iff)

   519     apply simp_all

   520     done

   521 qed

   522

   523 subsection "Fubinis theorem"

   524

   525 lemma measurable_compose_Pair1:

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

   527   by simp

   528

   529 lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst':

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

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

   532 using f proof induct

   533   case (cong u v)

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

   535     by (auto simp: space_pair_measure)

   536   show ?case

   537     apply (subst measurable_cong)

   538     apply (rule nn_integral_cong)

   539     apply fact+

   540     done

   541 next

   542   case (set Q)

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

   544     by (auto simp: indicator_def)

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

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

   547   from this measurable_emeasure_Pair[OF set] show ?case

   548     by (rule measurable_cong[THEN iffD1])

   549 qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1

   550                    nn_integral_monotone_convergence_SUP incseq_def le_fun_def

   551               cong: measurable_cong)

   552

   553 lemma (in sigma_finite_measure) nn_integral_fst':

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

   555   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 = _")

   556 using f proof induct

   557   case (cong u v)

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

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

   560   with cong show ?case

   561     by (simp cong: nn_integral_cong)

   562 qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add

   563                    nn_integral_monotone_convergence_SUP

   564                    measurable_compose_Pair1 nn_integral_nonneg

   565                    borel_measurable_nn_integral_fst' nn_integral_mono incseq_def le_fun_def

   566               cong: nn_integral_cong)

   567

   568 lemma (in sigma_finite_measure) nn_integral_fst:

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

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

   571   using f

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

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

   574   unfolding nn_integral_max_0 by auto

   575

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

   577   "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"

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

   579   by (simp add: nn_integral_max_0)

   580

   581 lemma (in pair_sigma_finite) nn_integral_snd:

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

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

   584 proof -

   585   note measurable_pair_swap[OF f]

   586   from M1.nn_integral_fst[OF this]

   587   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))"

   588     by simp

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

   590     by (subst distr_pair_swap)

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

   592   finally show ?thesis .

   593 qed

   594

   595 lemma (in pair_sigma_finite) Fubini:

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

   597   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)"

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

   599

   600 lemma (in pair_sigma_finite) Fubini':

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

   602   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)"

   603   using Fubini[OF f] by simp

   604

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

   606

   607 lemma sigma_prod:

   608   assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"

   609   assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y"

   610   shows "sigma X A \<Otimes>\<^sub>M sigma Y B = sigma (X \<times> Y) {a \<times> b | a b. a \<in> A \<and> b \<in> B}"

   611     (is "?P = ?S")

   612 proof (rule measure_eqI)

   613   have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X"

   614     by auto

   615   let ?XY = "{{fst - a \<inter> X \<times> Y | a. a \<in> A}, {snd - b \<inter> X \<times> Y | b. b \<in> B}}"

   616   have "sets ?P =

   617     sets (\<Squnion>\<^sub>\<sigma> xy\<in>?XY. sigma (X \<times> Y) xy)"

   618     by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)

   619   also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))"

   620     by (intro Sup_sigma_sigma arg_cong[where f=sets]) auto

   621   also have "\<dots> = sets ?S"

   622   proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)

   623     show "\<Union>?XY \<subseteq> Pow (X \<times> Y)" "{a \<times> b |a b. a \<in> A \<and> b \<in> B} \<subseteq> Pow (X \<times> Y)"

   624       using A B by auto

   625   next

   626     interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"

   627       using A B by (intro sigma_algebra_sigma_sets) auto

   628     fix Z assume "Z \<in> \<Union>?XY"

   629     then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"

   630     proof safe

   631       fix a assume "a \<in> A"

   632       from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E"

   633         by auto

   634       with a \<in> A A have eq: "fst - a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"

   635         by auto

   636       show "fst - a \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"

   637         using a \<in> A E unfolding eq by (auto intro!: XY.countable_UN')

   638     next

   639       fix b assume "b \<in> B"

   640       from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E"

   641         by auto

   642       with b \<in> B B have eq: "snd - b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"

   643         by auto

   644       show "snd - b \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"

   645         using b \<in> B E unfolding eq by (auto intro!: XY.countable_UN')

   646     qed

   647   next

   648     fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}"

   649     then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B"

   650       by auto

   651     then have Z: "Z = (fst - a \<inter> X \<times> Y) \<inter> (snd - b \<inter> X \<times> Y)"

   652       using A B by auto

   653     interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)"

   654       by (intro sigma_algebra_sigma_sets) auto

   655     show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)"

   656       unfolding Z by (rule XY.Int) (blast intro: ab)+

   657   qed

   658   finally show "sets ?P = sets ?S" .

   659 next

   660   interpret finite_measure "sigma X A" for X A

   661     proof qed (simp add: emeasure_sigma)

   662   fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A"

   663     by (simp add: emeasure_pair_measure_alt emeasure_sigma)

   664 qed

   665

   666 lemma sigma_sets_pair_measure_generator_finite:

   667   assumes "finite A" and "finite B"

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

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

   670 proof safe

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

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

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

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

   675   proof (induct x)

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

   677   next

   678     case (insert a x)

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

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

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

   682   qed

   683 next

   684   fix x a b

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

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

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

   688 qed

   689

   690 lemma borel_prod:

   691   "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"

   692   (is "?P = ?B")

   693 proof -

   694   have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"

   695     by (rule second_countable_borel_measurable[OF open_prod_generated])

   696   also have "\<dots> = ?P"

   697     unfolding borel_def

   698     by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])

   699   finally show ?thesis ..

   700 qed

   701

   702 lemma pair_measure_count_space:

   703   assumes A: "finite A" and B: "finite B"

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

   705 proof (rule measure_eqI)

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

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

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

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

   710     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)

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

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

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

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

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

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

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

   718     apply (subst emeasure_count_space)

   719     using X_subset apply auto []

   720     apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)

   721     apply (subst nn_integral_count_space)

   722     using A apply simp

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

   724     apply (subst card_gt_0_iff)

   725     apply (simp add: fin_Pair)

   726     apply (subst card_SigmaI[symmetric])

   727     using A apply simp

   728     using fin_Pair apply simp

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

   730     done

   731 qed

   732

   733

   734 lemma emeasure_prod_count_space:

   735   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")

   736   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"

   737   by (rule emeasure_measure_of[OF pair_measure_def])

   738      (auto simp: countably_additive_def positive_def suminf_indicator nn_integral_nonneg A

   739                  nn_integral_suminf[symmetric] dest: sets.sets_into_space)

   740

   741 lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"

   742 proof -

   743   have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ereal)"

   744     by (auto split: split_indicator)

   745   show ?thesis

   746     by (cases x)

   747        (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair nn_integral_max_0 one_ereal_def[symmetric])

   748 qed

   749

   750 lemma emeasure_count_space_prod_eq:

   751   fixes A :: "('a \<times> 'b) set"

   752   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")

   753   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"

   754 proof -

   755   { fix A :: "('a \<times> 'b) set" assume "countable A"

   756     then have "emeasure (?A \<Otimes>\<^sub>M ?B) (\<Union>a\<in>A. {a}) = (\<integral>\<^sup>+a. emeasure (?A \<Otimes>\<^sub>M ?B) {a} \<partial>count_space A)"

   757       by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)

   758     also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)"

   759       by (subst nn_integral_count_space_indicator) auto

   760     finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"

   761       by simp }

   762   note * = this

   763

   764   show ?thesis

   765   proof cases

   766     assume "finite A" then show ?thesis

   767       by (intro * countable_finite)

   768   next

   769     assume "infinite A"

   770     then obtain C where "countable C" and "infinite C" and "C \<subseteq> A"

   771       by (auto dest: infinite_countable_subset')

   772     with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A"

   773       by (intro emeasure_mono) auto

   774     also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C"

   775       using countable C by (rule *)

   776     finally show ?thesis

   777       using infinite C infinite A by simp

   778   qed

   779 qed

   780

   781 lemma nn_intergal_count_space_prod_eq':

   782   assumes [simp]: "\<And>x. 0 \<le> f x"

   783   shows "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"

   784     (is "nn_integral ?P f = _")

   785 proof cases

   786   assume cntbl: "countable {x. f x \<noteq> 0}"

   787   have [simp]: "\<And>x. ereal (real (card ({x} \<inter> {x. f x \<noteq> 0}))) = indicator {x. f x \<noteq> 0} x"

   788     by (auto split: split_indicator)

   789   have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P"

   790     by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y])

   791        (auto intro: sets_Pair)

   792

   793   have "(\<integral>\<^sup>+x. f x \<partial>?P) = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x * indicator {x} x' \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)"

   794     by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator)

   795   also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)"

   796     by (auto intro!: nn_integral_cong split: split_indicator)

   797   also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})"

   798     by (intro nn_integral_count_space_nn_integral cntbl) auto

   799   also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})"

   800     by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair)

   801   finally show ?thesis

   802     by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)

   803 next

   804   { fix x assume "f x \<noteq> 0"

   805     with 0 \<le> f x have "(\<exists>r. 0 < r \<and> f x = ereal r) \<or> f x = \<infinity>"

   806       by (cases "f x") (auto simp: less_le)

   807     then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f x"

   808       by (auto elim!: nat_approx_posE intro!: less_imp_le) }

   809   note * = this

   810

   811   assume cntbl: "uncountable {x. f x \<noteq> 0}"

   812   also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})"

   813     using * by auto

   814   finally obtain n where "infinite {x. 1/Suc n \<le> f x}"

   815     by (meson countableI_type countable_UN uncountable_infinite)

   816   then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C"

   817     by (metis infinite_countable_subset')

   818

   819   have [measurable]: "C \<in> sets ?P"

   820     using sets.countable[OF _ countable C, of ?P] by (auto simp: sets_Pair)

   821

   822   have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f"

   823     using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])

   824   moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>"

   825     using infinite C by (simp add: nn_integral_cmult emeasure_count_space_prod_eq)

   826   moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f"

   827     using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])

   828   moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>"

   829     using infinite C by (simp add: nn_integral_cmult)

   830   ultimately show ?thesis

   831     by simp

   832 qed

   833

   834 lemma nn_intergal_count_space_prod_eq:

   835   "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"

   836   by (subst (1 2) nn_integral_max_0[symmetric]) (auto intro!: nn_intergal_count_space_prod_eq')

   837

   838 lemma pair_measure_density:

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

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

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

   842   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")

   843 proof (rule measure_eqI)

   844   interpret M2: sigma_finite_measure M2 by fact

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

   846

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

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

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

   850     by (intro nn_integral_cong_AE)

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

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

   853     by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density

   854                   M2.nn_integral_fst[symmetric]

   855              cong: nn_integral_cong)

   856 qed simp

   857

   858 lemma sigma_finite_measure_distr:

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

   860   shows "sigma_finite_measure M"

   861 proof -

   862   interpret sigma_finite_measure "distr M N f" by fact

   863   from sigma_finite_countable guess A .. note A = this

   864   show ?thesis

   865   proof

   866     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>)"

   867       using A f

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

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

   870   qed

   871 qed

   872

   873 lemma pair_measure_distr:

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

   875   assumes "sigma_finite_measure (distr N T g)"

   876   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")

   877 proof (rule measure_eqI)

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

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

   880

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

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

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

   884                        T.emeasure_pair_measure_alt nn_integral_distr

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

   886 qed simp

   887

   888 lemma pair_measure_eqI:

   889   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"

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

   891   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)"

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

   893 proof -

   894   interpret M1: sigma_finite_measure M1 by fact

   895   interpret M2: sigma_finite_measure M2 by fact

   896   interpret pair_sigma_finite M1 M2 by default

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

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

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

   900   show ?thesis

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

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

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

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

   905       by (simp add: sets_pair_measure space_pair_measure)

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

   907       using sets[symmetric] by simp

   908   next

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

   910       using F by (auto simp: space_pair_measure)

   911   next

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

   913     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

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

   915        by (simp add: M2.emeasure_pair_measure_Times)

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

   917       using A B emeasure by auto

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

   919       by simp

   920   qed

   921 qed

   922

   923 lemma sets_pair_countable:

   924   assumes "countable S1" "countable S2"

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

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

   927 proof auto

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

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

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

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

   932     by (auto simp: space_pair_measure)

   933 next

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

   935   then have "countable X"

   936     by (metis countable_subset countable S1 countable S2 countable_SIGMA)

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

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

   939     using X

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

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

   942 qed

   943

   944 lemma pair_measure_countable:

   945   assumes "countable S1" "countable S2"

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

   947 proof (rule pair_measure_eqI)

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

   949     using assms by (auto intro!: sigma_finite_measure_count_space_countable)

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

   951     by (subst sets_pair_countable[OF assms]) auto

   952 next

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

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

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

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

   957 qed

   958

   959 lemma nn_integral_fst_count_space':

   960   assumes nonneg: "\<And>xy. 0 \<le> f xy"

   961   shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"

   962   (is "?lhs = ?rhs")

   963 proof(cases)

   964   assume *: "countable {xy. f xy \<noteq> 0}"

   965   let ?A = "fst  {xy. f xy \<noteq> 0}"

   966   let ?B = "snd  {xy. f xy \<noteq> 0}"

   967   from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+

   968   from nonneg have f_neq_0: "\<And>xy. f xy \<noteq> 0 \<longleftrightarrow> f xy > 0"

   969     by(auto simp add: order.order_iff_strict)

   970   have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)"

   971     by(rule nn_integral_count_space_eq)

   972       (auto simp add: f_neq_0 nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI)

   973   also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)"

   974     by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI)

   975   also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))"

   976     by(subst sigma_finite_measure.nn_integral_fst)

   977       (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable)

   978   also have "\<dots> = ?rhs"

   979     by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI)

   980   finally show ?thesis .

   981 next

   982   { fix xy assume "f xy \<noteq> 0"

   983     with 0 \<le> f xy have "(\<exists>r. 0 < r \<and> f xy = ereal r) \<or> f xy = \<infinity>"

   984       by (cases "f xy") (auto simp: less_le)

   985     then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f xy"

   986       by (auto elim!: nat_approx_posE intro!: less_imp_le) }

   987   note * = this

   988

   989   assume cntbl: "uncountable {xy. f xy \<noteq> 0}"

   990   also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})"

   991     using * by auto

   992   finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}"

   993     by (meson countableI_type countable_UN uncountable_infinite)

   994   then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C"

   995     by (metis infinite_countable_subset')

   996

   997   have "\<infinity> = (\<integral>\<^sup>+ xy. ereal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)"

   998     using \<open>infinite C\<close> by(simp add: nn_integral_cmult)

   999   also have "\<dots> \<le> ?rhs" using C

  1000     by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg)

  1001   finally have "?rhs = \<infinity>" by simp

  1002   moreover have "?lhs = \<infinity>"

  1003   proof(cases "finite (fst  C)")

  1004     case True

  1005     then obtain x C' where x: "x \<in> fst  C"

  1006       and C': "C' = fst - {x} \<inter> C"

  1007       and "infinite C'"

  1008       using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE')

  1009     from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto

  1010

  1011     from C' \<open>infinite C'\<close> have "infinite (snd  C')"

  1012       by(auto dest!: finite_imageD simp add: inj_on_def)

  1013     then have "\<infinity> = (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator (snd  C') y \<partial>count_space UNIV)"

  1014       by(simp add: nn_integral_cmult)

  1015     also have "\<dots> = (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)"

  1016       by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C')

  1017     also have "\<dots> = (\<integral>\<^sup>+ x'. (\<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV) * indicator {x} x' \<partial>count_space UNIV)"

  1018       by(simp add: one_ereal_def[symmetric] nn_integral_nonneg nn_integral_cmult_indicator)

  1019     also have "\<dots> \<le> (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV \<partial>count_space UNIV)"

  1020       by(rule nn_integral_mono)(simp split: split_indicator add: nn_integral_nonneg)

  1021     also have "\<dots> \<le> ?lhs" using **

  1022       by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg)

  1023     finally show ?thesis by simp

  1024   next

  1025     case False

  1026     def C' \<equiv> "fst  C"

  1027     have "\<infinity> = \<integral>\<^sup>+ x. ereal (1 / Suc n) * indicator C' x \<partial>count_space UNIV"

  1028       using C'_def False by(simp add: nn_integral_cmult)

  1029     also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) \<in> C} y \<partial>count_space UNIV \<partial>count_space UNIV"

  1030       by(auto simp add: one_ereal_def[symmetric] nn_integral_cmult_indicator intro: nn_integral_cong)

  1031     also have "\<dots> \<le> \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ereal (1 / Suc n) * indicator C (x, y) \<partial>count_space UNIV \<partial>count_space UNIV"

  1032       by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI)

  1033     also have "\<dots> \<le> ?lhs" using C

  1034       by(intro nn_integral_mono)(auto split: split_indicator simp add: nonneg)

  1035     finally show ?thesis by simp

  1036   qed

  1037   ultimately show ?thesis by simp

  1038 qed

  1039

  1040 lemma nn_integral_fst_count_space:

  1041   "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"

  1042 by(subst (2 3) nn_integral_max_0[symmetric])(rule nn_integral_fst_count_space', simp)

  1043

  1044 lemma nn_integral_snd_count_space:

  1045   "(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"

  1046   (is "?lhs = ?rhs")

  1047 proof -

  1048   have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)"

  1049     by(simp)

  1050   also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV"

  1051     by(rule nn_integral_fst_count_space)

  1052   also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x))  UNIV)"

  1053     by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])

  1054       (simp_all add: inj_on_def split_def)

  1055   also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto

  1056   finally show ?thesis .

  1057 qed

  1058

  1059 lemma measurable_pair_measure_countable1:

  1060   assumes "countable A"

  1061   and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"

  1062   shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"

  1063 using _ _ assms(1)

  1064 by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all

  1065

  1066 subsection {* Product of Borel spaces *}

  1067

  1068 lemma borel_Times:

  1069   fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"

  1070   assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"

  1071   shows "A \<times> B \<in> sets borel"

  1072 proof -

  1073   have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"

  1074     by auto

  1075   moreover

  1076   { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)

  1077     then have "A\<times>UNIV \<in> sets borel"

  1078     proof (induct A)

  1079       case (Basic S) then show ?case

  1080         by (auto intro!: borel_open open_Times)

  1081     next

  1082       case (Compl A)

  1083       moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)"

  1084         by auto

  1085       ultimately show ?case

  1086         unfolding * by auto

  1087     next

  1088       case (Union A)

  1089       moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)"

  1090         by auto

  1091       ultimately show ?case

  1092         unfolding * by auto

  1093     qed simp }

  1094   moreover

  1095   { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)

  1096     then have "UNIV\<times>B \<in> sets borel"

  1097     proof (induct B)

  1098       case (Basic S) then show ?case

  1099         by (auto intro!: borel_open open_Times)

  1100     next

  1101       case (Compl B)

  1102       moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)"

  1103         by auto

  1104       ultimately show ?case

  1105         unfolding * by auto

  1106     next

  1107       case (Union B)

  1108       moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)"

  1109         by auto

  1110       ultimately show ?case

  1111         unfolding * by auto

  1112     qed simp }

  1113   ultimately show ?thesis

  1114     by auto

  1115 qed

  1116

  1117 lemma finite_measure_pair_measure:

  1118   assumes "finite_measure M" "finite_measure N"

  1119   shows "finite_measure (N  \<Otimes>\<^sub>M M)"

  1120 proof (rule finite_measureI)

  1121   interpret M: finite_measure M by fact

  1122   interpret N: finite_measure N by fact

  1123   show "emeasure (N  \<Otimes>\<^sub>M M) (space (N  \<Otimes>\<^sub>M M)) \<noteq> \<infinity>"

  1124     by (auto simp: space_pair_measure M.emeasure_pair_measure_Times)

  1125 qed

  1126

  1127 end