src/HOL/Analysis/Binary_Product_Measure.thy
 author wenzelm Mon Mar 25 17:21:26 2019 +0100 (2 months ago) changeset 69981 3dced198b9ec parent 69939 812ce526da33 child 70136 f03a01a18c6e permissions -rw-r--r--
more strict AFP properties;
```     1 (*  Title:      HOL/Analysis/Binary_Product_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Binary Product Measure\<close>
```
```     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%important 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_measure_cong[measurable_cong, cong]:
```
```    41   "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
```
```    42   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
```
```    43
```
```    44 lemma pair_measureI[intro, simp, measurable]:
```
```    45   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
```
```    46   by (auto simp: sets_pair_measure)
```
```    47
```
```    48 lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```    49   using pair_measureI[of "{x}" M1 "{y}" M2] by simp
```
```    50
```
```    51 lemma measurable_pair_measureI:
```
```    52   assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
```
```    53   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"
```
```    54   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
```
```    55   unfolding pair_measure_def using 1 2
```
```    56   by (intro measurable_measure_of) (auto dest: sets.sets_into_space)
```
```    57
```
```    58 lemma measurable_split_replace[measurable (raw)]:
```
```    59   "(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. case_prod (f x) (g x)) \<in> measurable M N"
```
```    60   unfolding split_beta' .
```
```    61
```
```    62 lemma measurable_Pair[measurable (raw)]:
```
```    63   assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
```
```    64   shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
```
```    65 proof (rule measurable_pair_measureI)
```
```    66   show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
```
```    67     using f g by (auto simp: measurable_def)
```
```    68   fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
```
```    69   have "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
```
```    70     by auto
```
```    71   also have "\<dots> \<in> sets M"
```
```    72     by (rule sets.Int) (auto intro!: measurable_sets * f g)
```
```    73   finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .
```
```    74 qed
```
```    75
```
```    76 lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
```
```    77   by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space Times_Int_Times
```
```    78     measurable_def)
```
```    79
```
```    80 lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
```
```    81   by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space Times_Int_Times
```
```    82     measurable_def)
```
```    83
```
```    84 lemma measurable_Pair_compose_split[measurable_dest]:
```
```    85   assumes f: "case_prod f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N"
```
```    86   assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"
```
```    87   shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"
```
```    88   using measurable_compose[OF measurable_Pair f, OF g h] by simp
```
```    89
```
```    90 lemma measurable_Pair1_compose[measurable_dest]:
```
```    91   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
```
```    92   assumes [measurable]: "h \<in> measurable N M"
```
```    93   shows "(\<lambda>x. f (h x)) \<in> measurable N M1"
```
```    94   using measurable_compose[OF f measurable_fst] by simp
```
```    95
```
```    96 lemma measurable_Pair2_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. g (h x)) \<in> measurable N M2"
```
```   100   using measurable_compose[OF f measurable_snd] by simp
```
```   101
```
```   102 lemma measurable_pair:
```
```   103   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
```
```   104   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
```
```   105   using measurable_Pair[OF assms] by simp
```
```   106
```
```   107 lemma
```
```   108   assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)"
```
```   109   shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"
```
```   110     and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"
```
```   111   by simp_all
```
```   112
```
```   113 lemma
```
```   114   assumes f[measurable]: "f \<in> measurable M N"
```
```   115   shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N"
```
```   116     and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N"
```
```   117   by simp_all
```
```   118
```
```   119 lemma sets_pair_in_sets:
```
```   120   assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"
```
```   121   shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
```
```   122   unfolding sets_pair_measure
```
```   123   by (intro sets.sigma_sets_subset') (auto intro!: assms)
```
```   124
```
```   125 lemma  sets_pair_eq_sets_fst_snd:
```
```   126   "sets (A \<Otimes>\<^sub>M B) = sets (Sup {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})"
```
```   127     (is "?P = sets (Sup {?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 {?fst, ?snd})"
```
```   133       apply (rule sets.Int)
```
```   134       apply (rule in_sets_Sup)
```
```   135       apply auto []
```
```   136       apply (rule insertI1)
```
```   137       apply (auto intro: ab in_vimage_algebra) []
```
```   138       apply (rule in_sets_Sup)
```
```   139       apply auto []
```
```   140       apply (rule insertI2)
```
```   141       apply (auto intro: ab in_vimage_algebra)
```
```   142       done
```
```   143     finally have "a \<times> b \<in> sets (Sup {?fst, ?snd})" . }
```
```   144   moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)"
```
```   145     by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric])
```
```   146   moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)"
```
```   147     by (rule sets_image_in_sets) (auto simp: space_pair_measure)
```
```   148   ultimately show ?thesis
```
```   149     apply (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets)
```
```   150     apply simp
```
```   151     apply simp
```
```   152     apply simp
```
```   153     apply (elim disjE)
```
```   154     apply (simp add: space_pair_measure)
```
```   155     apply (simp add: space_pair_measure)
```
```   156     apply (auto simp add: space_pair_measure)
```
```   157     done
```
```   158 qed
```
```   159
```
```   160 lemma measurable_pair_iff:
```
```   161   "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"
```
```   162   by (auto intro: measurable_pair[of f M M1 M2])
```
```   163
```
```   164 lemma  measurable_split_conv:
```
```   165   "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
```
```   166   by (intro arg_cong2[where f="(\<in>)"]) auto
```
```   167
```
```   168 lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
```
```   169   by (auto intro!: measurable_Pair simp: measurable_split_conv)
```
```   170
```
```   171 lemma  measurable_pair_swap:
```
```   172   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"
```
```   173   using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
```
```   174
```
```   175 lemma measurable_pair_swap_iff:
```
```   176   "f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M"
```
```   177   by (auto dest: measurable_pair_swap)
```
```   178
```
```   179 lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
```
```   180   by simp
```
```   181
```
```   182 lemma sets_Pair1[measurable (raw)]:
```
```   183   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2"
```
```   184 proof -
```
```   185   have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
```
```   186     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
```
```   187   also have "\<dots> \<in> sets M2"
```
```   188     using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: if_split_asm)
```
```   189   finally show ?thesis .
```
```   190 qed
```
```   191
```
```   192 lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
```
```   193   by (auto intro!: measurable_Pair)
```
```   194
```
```   195 lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
```
```   196 proof -
```
```   197   have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
```
```   198     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
```
```   199   also have "\<dots> \<in> sets M1"
```
```   200     using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: if_split_asm)
```
```   201   finally show ?thesis .
```
```   202 qed
```
```   203
```
```   204 lemma measurable_Pair2:
```
```   205   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1"
```
```   206   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
```
```   207   using measurable_comp[OF measurable_Pair1' f, OF x]
```
```   208   by (simp add: comp_def)
```
```   209
```
```   210 lemma measurable_Pair1:
```
```   211   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2"
```
```   212   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
```
```   213   using measurable_comp[OF measurable_Pair2' f, OF y]
```
```   214   by (simp add: comp_def)
```
```   215
```
```   216 lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
```
```   217   unfolding Int_stable_def
```
```   218   by safe (auto simp add: Times_Int_Times)
```
```   219
```
```   220 lemma (in finite_measure) finite_measure_cut_measurable:
```
```   221   assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)"
```
```   222   shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
```
```   223     (is "?s Q \<in> _")
```
```   224   using Int_stable_pair_measure_generator pair_measure_closed assms
```
```   225   unfolding sets_pair_measure
```
```   226 proof (induct rule: sigma_sets_induct_disjoint)
```
```   227   case (compl A)
```
```   228   with sets.sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) =
```
```   229       (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"
```
```   230     unfolding sets_pair_measure[symmetric]
```
```   231     by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
```
```   232   with compl sets.top show ?case
```
```   233     by (auto intro!: measurable_If simp: space_pair_measure)
```
```   234 next
```
```   235   case (union F)
```
```   236   then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)"
```
```   237     by (simp add: suminf_emeasure disjoint_family_on_vimageI subset_eq vimage_UN sets_pair_measure[symmetric])
```
```   238   with union show ?case
```
```   239     unfolding sets_pair_measure[symmetric] by simp
```
```   240 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
```
```   241
```
```   242 lemma (in sigma_finite_measure) measurable_emeasure_Pair:
```
```   243   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> _")
```
```   244 proof -
```
```   245   from sigma_finite_disjoint guess F . note F = this
```
```   246   then have F_sets: "\<And>i. F i \<in> sets M" by auto
```
```   247   let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"
```
```   248   { fix i
```
```   249     have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"
```
```   250       using F sets.sets_into_space by auto
```
```   251     let ?R = "density M (indicator (F i))"
```
```   252     have "finite_measure ?R"
```
```   253       using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
```
```   254     then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"
```
```   255      by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)
```
```   256     moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))
```
```   257         = emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))"
```
```   258       using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
```
```   259     moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i"
```
```   260       using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure)
```
```   261     ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"
```
```   262       by simp }
```
```   263   moreover
```
```   264   { fix x
```
```   265     have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
```
```   266     proof (intro suminf_emeasure)
```
```   267       show "range (?C x) \<subseteq> sets M"
```
```   268         using F \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> by (auto intro!: sets_Pair1)
```
```   269       have "disjoint_family F" using F by auto
```
```   270       show "disjoint_family (?C x)"
```
```   271         by (rule disjoint_family_on_bisimulation[OF \<open>disjoint_family F\<close>]) auto
```
```   272     qed
```
```   273     also have "(\<Union>i. ?C x i) = Pair x -` Q"
```
```   274       using F sets.sets_into_space[OF \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close>]
```
```   275       by (auto simp: space_pair_measure)
```
```   276     finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"
```
```   277       by simp }
```
```   278   ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets
```
```   279     by auto
```
```   280 qed
```
```   281
```
```   282 lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
```
```   283   assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
```
```   284   assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)"
```
```   285   shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
```
```   286 proof -
```
```   287   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"
```
```   288     by (auto simp: space_pair_measure)
```
```   289   with measurable_emeasure_Pair[OF A] show ?thesis
```
```   290     by (auto cong: measurable_cong)
```
```   291 qed
```
```   292
```
```   293 lemma (in sigma_finite_measure) emeasure_pair_measure:
```
```   294   assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
```
```   295   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")
```
```   296 proof (rule emeasure_measure_of[OF pair_measure_def])
```
```   297   show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
```
```   298     by (auto simp: positive_def)
```
```   299   have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
```
```   300     by (auto simp: indicator_def)
```
```   301   show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
```
```   302   proof (rule countably_additiveI)
```
```   303     fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F"
```
```   304     from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto
```
```   305     moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
```
```   306       by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
```
```   307     moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M"
```
```   308       using F by (auto simp: sets_Pair1)
```
```   309     ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
```
```   310       by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure
```
```   311                intro!: nn_integral_cong nn_integral_indicator[symmetric])
```
```   312   qed
```
```   313   show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
```
```   314     using sets.space_closed[of N] sets.space_closed[of M] by auto
```
```   315 qed fact
```
```   316
```
```   317 lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
```
```   318   assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
```
```   319   shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
```
```   320 proof -
```
```   321   have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
```
```   322     by (auto simp: indicator_def)
```
```   323   show ?thesis
```
```   324     using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
```
```   325 qed
```
```   326
```
```   327 proposition (in sigma_finite_measure) emeasure_pair_measure_Times:
```
```   328   assumes A: "A \<in> sets N" and B: "B \<in> sets M"
```
```   329   shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
```
```   330 proof -
```
```   331   have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
```
```   332     using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
```
```   333   also have "\<dots> = emeasure M B * emeasure N A"
```
```   334     using A by (simp add: nn_integral_cmult_indicator)
```
```   335   finally show ?thesis
```
```   336     by (simp add: ac_simps)
```
```   337 qed
```
```   338
```
```   339 subsection \<open>Binary products of \<open>\<sigma>\<close>-finite emeasure spaces\<close>
```
```   340
```
```   341 locale%unimportant pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
```
```   342   for M1 :: "'a measure" and M2 :: "'b measure"
```
```   343
```
```   344 lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
```
```   345   "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
```
```   346   using M2.measurable_emeasure_Pair .
```
```   347
```
```   348 lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
```
```   349   assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
```
```   350 proof -
```
```   351   have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
```
```   352     using Q measurable_pair_swap' by (auto intro: measurable_sets)
```
```   353   note M1.measurable_emeasure_Pair[OF this]
```
```   354   moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1)) = (\<lambda>x. (x, y)) -` Q"
```
```   355     using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
```
```   356   ultimately show ?thesis by simp
```
```   357 qed
```
```   358
```
```   359 proposition (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
```
```   360   defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
```
```   361   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>
```
```   362     (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
```
```   363 proof -
```
```   364   from M1.sigma_finite_incseq guess F1 . note F1 = this
```
```   365   from M2.sigma_finite_incseq guess F2 . note F2 = this
```
```   366   from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
```
```   367   let ?F = "\<lambda>i. F1 i \<times> F2 i"
```
```   368   show ?thesis
```
```   369   proof (intro exI[of _ ?F] conjI allI)
```
```   370     show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
```
```   371   next
```
```   372     have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
```
```   373     proof (intro subsetI)
```
```   374       fix x assume "x \<in> space M1 \<times> space M2"
```
```   375       then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
```
```   376         by (auto simp: space)
```
```   377       then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
```
```   378         using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_def
```
```   379         by (force split: split_max)+
```
```   380       then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
```
```   381         by (intro SigmaI) (auto simp add: max.commute)
```
```   382       then show "x \<in> (\<Union>i. ?F i)" by auto
```
```   383     qed
```
```   384     then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
```
```   385       using space by (auto simp: space)
```
```   386   next
```
```   387     fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
```
```   388       using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_Suc_iff by auto
```
```   389   next
```
```   390     fix i
```
```   391     from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
```
```   392     with F1 F2 show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
```
```   393       by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
```
```   394   qed
```
```   395 qed
```
```   396
```
```   397 sublocale%unimportant pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
```
```   398 proof
```
```   399   from M1.sigma_finite_countable guess F1 ..
```
```   400   moreover from M2.sigma_finite_countable guess F2 ..
```
```   401   ultimately show
```
```   402     "\<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>)"
```
```   403     by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI)
```
```   404        (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
```
```   405 qed
```
```   406
```
```   407 lemma sigma_finite_pair_measure:
```
```   408   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
```
```   409   shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
```
```   410 proof -
```
```   411   interpret A: sigma_finite_measure A by fact
```
```   412   interpret B: sigma_finite_measure B by fact
```
```   413   interpret AB: pair_sigma_finite A  B ..
```
```   414   show ?thesis ..
```
```   415 qed
```
```   416
```
```   417 lemma sets_pair_swap:
```
```   418   assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```   419   shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
```
```   420   using measurable_pair_swap' assms by (rule measurable_sets)
```
```   421
```
```   422 lemma (in pair_sigma_finite) distr_pair_swap:
```
```   423   "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
```
```   424 proof -
```
```   425   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
```
```   426   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
```
```   427   show ?thesis
```
```   428   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
```
```   429     show "?E \<subseteq> Pow (space ?P)"
```
```   430       using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
```
```   431     show "sets ?P = sigma_sets (space ?P) ?E"
```
```   432       by (simp add: sets_pair_measure space_pair_measure)
```
```   433     then show "sets ?D = sigma_sets (space ?P) ?E"
```
```   434       by simp
```
```   435   next
```
```   436     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
```
```   437       using F by (auto simp: space_pair_measure)
```
```   438   next
```
```   439     fix X assume "X \<in> ?E"
```
```   440     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
```
```   441     have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A"
```
```   442       using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure)
```
```   443     with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X"
```
```   444       by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
```
```   445                     measurable_pair_swap' ac_simps)
```
```   446   qed
```
```   447 qed
```
```   448
```
```   449 lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
```
```   450   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```   451   shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
```
```   452     (is "_ = ?\<nu> A")
```
```   453 proof -
```
```   454   have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A"
```
```   455     using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
```
```   456   show ?thesis using A
```
```   457     by (subst distr_pair_swap)
```
```   458        (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
```
```   459                  M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
```
```   460 qed
```
```   461
```
```   462 lemma (in pair_sigma_finite) AE_pair:
```
```   463   assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
```
```   464   shows "AE x in M1. (AE y in M2. Q (x, y))"
```
```   465 proof -
```
```   466   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"
```
```   467     using assms unfolding eventually_ae_filter by auto
```
```   468   show ?thesis
```
```   469   proof (rule AE_I)
```
```   470     from N measurable_emeasure_Pair1[OF \<open>N \<in> sets (M1 \<Otimes>\<^sub>M M2)\<close>]
```
```   471     show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
```
```   472       by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff)
```
```   473     show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
```
```   474       by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp
```
```   475     { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
```
```   476       have "AE y in M2. Q (x, y)"
```
```   477       proof (rule AE_I)
```
```   478         show "emeasure M2 (Pair x -` N) = 0" by fact
```
```   479         show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
```
```   480         show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
```
```   481           using N \<open>x \<in> space M1\<close> unfolding space_pair_measure by auto
```
```   482       qed }
```
```   483     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}"
```
```   484       by auto
```
```   485   qed
```
```   486 qed
```
```   487
```
```   488 lemma (in pair_sigma_finite) AE_pair_measure:
```
```   489   assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```   490   assumes ae: "AE x in M1. AE y in M2. P (x, y)"
```
```   491   shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
```
```   492 proof (subst AE_iff_measurable[OF _ refl])
```
```   493   show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```   494     by (rule sets.sets_Collect) fact
```
```   495   then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
```
```   496       (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
```
```   497     by (simp add: M2.emeasure_pair_measure)
```
```   498   also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)"
```
```   499     using ae
```
```   500     apply (safe intro!: nn_integral_cong_AE)
```
```   501     apply (intro AE_I2)
```
```   502     apply (safe intro!: nn_integral_cong_AE)
```
```   503     apply auto
```
```   504     done
```
```   505   finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
```
```   506 qed
```
```   507
```
```   508 lemma (in pair_sigma_finite) AE_pair_iff:
```
```   509   "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
```
```   510     (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))"
```
```   511   using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
```
```   512
```
```   513 lemma (in pair_sigma_finite) AE_commute:
```
```   514   assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```   515   shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
```
```   516 proof -
```
```   517   interpret Q: pair_sigma_finite M2 M1 ..
```
```   518   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"
```
```   519     by auto
```
```   520   have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} =
```
```   521     (\<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)"
```
```   522     by (auto simp: space_pair_measure)
```
```   523   also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)"
```
```   524     by (intro sets_pair_swap P)
```
```   525   finally show ?thesis
```
```   526     apply (subst AE_pair_iff[OF P])
```
```   527     apply (subst distr_pair_swap)
```
```   528     apply (subst AE_distr_iff[OF measurable_pair_swap' P])
```
```   529     apply (subst Q.AE_pair_iff)
```
```   530     apply simp_all
```
```   531     done
```
```   532 qed
```
```   533
```
```   534 subsection "Fubinis theorem"
```
```   535
```
```   536 lemma measurable_compose_Pair1:
```
```   537   "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
```
```   538   by simp
```
```   539
```
```   540 lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst:
```
```   541   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
```
```   542   shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
```
```   543 using f proof induct
```
```   544   case (cong u v)
```
```   545   then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
```
```   546     by (auto simp: space_pair_measure)
```
```   547   show ?case
```
```   548     apply (subst measurable_cong)
```
```   549     apply (rule nn_integral_cong)
```
```   550     apply fact+
```
```   551     done
```
```   552 next
```
```   553   case (set Q)
```
```   554   have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y"
```
```   555     by (auto simp: indicator_def)
```
```   556   have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M"
```
```   557     by (simp add: sets_Pair1[OF set])
```
```   558   from this measurable_emeasure_Pair[OF set] show ?case
```
```   559     by (rule measurable_cong[THEN iffD1])
```
```   560 qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1
```
```   561                    nn_integral_monotone_convergence_SUP incseq_def le_fun_def image_comp
```
```   562               cong: measurable_cong)
```
```   563
```
```   564 lemma (in sigma_finite_measure) nn_integral_fst:
```
```   565   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
```
```   566   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 = _")
```
```   567   using f proof induct
```
```   568   case (cong u v)
```
```   569   then have "?I u = ?I v"
```
```   570     by (intro nn_integral_cong) (auto simp: space_pair_measure)
```
```   571   with cong show ?case
```
```   572     by (simp cong: nn_integral_cong)
```
```   573 qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add
```
```   574                    nn_integral_monotone_convergence_SUP measurable_compose_Pair1
```
```   575                    borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def image_comp
```
```   576               cong: nn_integral_cong)
```
```   577
```
```   578 lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
```
```   579   "case_prod f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N"
```
```   580   using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp
```
```   581
```
```   582 proposition (in pair_sigma_finite) nn_integral_snd:
```
```   583   assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
```
```   584   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
```
```   585 proof -
```
```   586   note measurable_pair_swap[OF f]
```
```   587   from M1.nn_integral_fst[OF this]
```
```   588   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))"
```
```   589     by simp
```
```   590   also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
```
```   591     by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong)
```
```   592   finally show ?thesis .
```
```   593 qed
```
```   594
```
```   595 theorem (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 theorem (in pair_sigma_finite) Fubini':
```
```   601   assumes f: "case_prod 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 \<open>Products on counting spaces, densities and distributions\<close>
```
```   606
```
```   607 proposition 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 = sets (SUP xy\<in>?XY. sigma (X \<times> Y) xy)"
```
```   617     by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)
```
```   618   also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))"
```
```   619     by (intro Sup_sigma arg_cong[where f=sets]) auto
```
```   620   also have "\<dots> = sets ?S"
```
```   621   proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
```
```   622     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)"
```
```   623       using A B by auto
```
```   624   next
```
```   625     interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
```
```   626       using A B by (intro sigma_algebra_sigma_sets) auto
```
```   627     fix Z assume "Z \<in> \<Union>?XY"
```
```   628     then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
```
```   629     proof safe
```
```   630       fix a assume "a \<in> A"
```
```   631       from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E"
```
```   632         by auto
```
```   633       with \<open>a \<in> A\<close> A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"
```
```   634         by auto
```
```   635       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}"
```
```   636         using \<open>a \<in> A\<close> E unfolding eq by (auto intro!: XY.countable_UN')
```
```   637     next
```
```   638       fix b assume "b \<in> B"
```
```   639       from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E"
```
```   640         by auto
```
```   641       with \<open>b \<in> B\<close> B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"
```
```   642         by auto
```
```   643       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}"
```
```   644         using \<open>b \<in> B\<close> E unfolding eq by (auto intro!: XY.countable_UN')
```
```   645     qed
```
```   646   next
```
```   647     fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
```
```   648     then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B"
```
```   649       by auto
```
```   650     then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)"
```
```   651       using A B by auto
```
```   652     interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)"
```
```   653       by (intro sigma_algebra_sigma_sets) auto
```
```   654     show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)"
```
```   655       unfolding Z by (rule XY.Int) (blast intro: ab)+
```
```   656   qed
```
```   657   finally show "sets ?P = sets ?S" .
```
```   658 next
```
```   659   interpret finite_measure "sigma X A" for X A
```
```   660     proof qed (simp add: emeasure_sigma)
```
```   661   fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A"
```
```   662     by (simp add: emeasure_pair_measure_alt emeasure_sigma)
```
```   663 qed
```
```   664
```
```   665 lemma sigma_sets_pair_measure_generator_finite:
```
```   666   assumes "finite A" and "finite B"
```
```   667   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
```
```   668   (is "sigma_sets ?prod ?sets = _")
```
```   669 proof safe
```
```   670   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
```
```   671   fix x assume subset: "x \<subseteq> A \<times> B"
```
```   672   hence "finite x" using fin by (rule finite_subset)
```
```   673   from this subset show "x \<in> sigma_sets ?prod ?sets"
```
```   674   proof (induct x)
```
```   675     case empty show ?case by (rule sigma_sets.Empty)
```
```   676   next
```
```   677     case (insert a x)
```
```   678     hence "{a} \<in> sigma_sets ?prod ?sets" by auto
```
```   679     moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
```
```   680     ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
```
```   681   qed
```
```   682 next
```
```   683   fix x a b
```
```   684   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
```
```   685   from sigma_sets_into_sp[OF _ this(1)] this(2)
```
```   686   show "a \<in> A" and "b \<in> B" by auto
```
```   687 qed
```
```   688
```
```   689 proposition  sets_pair_eq:
```
```   690   assumes Ea: "Ea \<subseteq> Pow (space A)" "sets A = sigma_sets (space A) Ea"
```
```   691     and Ca: "countable Ca" "Ca \<subseteq> Ea" "\<Union>Ca = space A"
```
```   692     and Eb: "Eb \<subseteq> Pow (space B)" "sets B = sigma_sets (space B) Eb"
```
```   693     and Cb: "countable Cb" "Cb \<subseteq> Eb" "\<Union>Cb = space B"
```
```   694   shows "sets (A \<Otimes>\<^sub>M B) = sets (sigma (space A \<times> space B) { a \<times> b | a b. a \<in> Ea \<and> b \<in> Eb })"
```
```   695     (is "_ = sets (sigma ?\<Omega> ?E)")
```
```   696 proof
```
```   697   show "sets (sigma ?\<Omega> ?E) \<subseteq> sets (A \<Otimes>\<^sub>M B)"
```
```   698     using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
```
```   699   have "?E \<subseteq> Pow ?\<Omega>"
```
```   700     using Ea(1) Eb(1) by auto
```
```   701   then have E: "a \<in> Ea \<Longrightarrow> b \<in> Eb \<Longrightarrow> a \<times> b \<in> sets (sigma ?\<Omega> ?E)" for a b
```
```   702     by auto
```
```   703   have "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (Sup {vimage_algebra ?\<Omega> fst A, vimage_algebra ?\<Omega> snd B})"
```
```   704     unfolding sets_pair_eq_sets_fst_snd ..
```
```   705   also have "vimage_algebra ?\<Omega> fst A = vimage_algebra ?\<Omega> fst (sigma (space A) Ea)"
```
```   706     by (intro vimage_algebra_cong[OF refl refl]) (simp add: Ea)
```
```   707   also have "\<dots> = sigma ?\<Omega> {fst -` A \<inter> ?\<Omega> |A. A \<in> Ea}"
```
```   708     by (intro Ea vimage_algebra_sigma) auto
```
```   709   also have "vimage_algebra ?\<Omega> snd B = vimage_algebra ?\<Omega> snd (sigma (space B) Eb)"
```
```   710     by (intro vimage_algebra_cong[OF refl refl]) (simp add: Eb)
```
```   711   also have "\<dots> = sigma ?\<Omega> {snd -` A \<inter> ?\<Omega> |A. A \<in> Eb}"
```
```   712     by (intro Eb vimage_algebra_sigma) auto
```
```   713   also have "{sigma ?\<Omega> {fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, sigma ?\<Omega> {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}} =
```
```   714     sigma ?\<Omega> ` {{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}"
```
```   715     by auto
```
```   716   also have "sets (SUP S\<in>{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}. sigma ?\<Omega> S) =
```
```   717     sets (sigma ?\<Omega> (\<Union>{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}))"
```
```   718     using Ea(1) Eb(1) by (intro sets_Sup_sigma) auto
```
```   719   also have "\<dots> \<subseteq> sets (sigma ?\<Omega> ?E)"
```
```   720   proof (subst sigma_le_sets, safe intro!: space_in_measure_of)
```
```   721     fix a assume "a \<in> Ea"
```
```   722     then have "fst -` a \<inter> ?\<Omega> = (\<Union>b\<in>Cb. a \<times> b)"
```
```   723       using Cb(3)[symmetric] Ea(1) by auto
```
```   724     then show "fst -` a \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)"
```
```   725       using Cb \<open>a \<in> Ea\<close> by (auto intro!: sets.countable_UN' E)
```
```   726   next
```
```   727     fix b assume "b \<in> Eb"
```
```   728     then have "snd -` b \<inter> ?\<Omega> = (\<Union>a\<in>Ca. a \<times> b)"
```
```   729       using Ca(3)[symmetric] Eb(1) by auto
```
```   730     then show "snd -` b \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)"
```
```   731       using Ca \<open>b \<in> Eb\<close> by (auto intro!: sets.countable_UN' E)
```
```   732   qed
```
```   733   finally show "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (sigma ?\<Omega> ?E)" .
```
```   734 qed
```
```   735
```
```   736 proposition  borel_prod:
```
```   737   "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
```
```   738   (is "?P = ?B")
```
```   739 proof -
```
```   740   have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
```
```   741     by (rule second_countable_borel_measurable[OF open_prod_generated])
```
```   742   also have "\<dots> = ?P"
```
```   743     unfolding borel_def
```
```   744     by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])
```
```   745   finally show ?thesis ..
```
```   746 qed
```
```   747
```
```   748 proposition pair_measure_count_space:
```
```   749   assumes A: "finite A" and B: "finite B"
```
```   750   shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
```
```   751 proof (rule measure_eqI)
```
```   752   interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
```
```   753   interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
```
```   754   interpret P: pair_sigma_finite "count_space A" "count_space B" ..
```
```   755   show eq: "sets ?P = sets ?C"
```
```   756     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
```
```   757   fix X assume X: "X \<in> sets ?P"
```
```   758   with eq have X_subset: "X \<subseteq> A \<times> B" by simp
```
```   759   with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
```
```   760     by (intro finite_subset[OF _ B]) auto
```
```   761   have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
```
```   762   have card: "0 < card (Pair a -` X)" if "(a, b) \<in> X" for a b
```
```   763     using card_gt_0_iff fin_Pair that by auto
```
```   764   then have "emeasure ?P X = \<integral>\<^sup>+ x. emeasure (count_space B) (Pair x -` X)
```
```   765             \<partial>count_space A"
```
```   766     by (simp add: B.emeasure_pair_measure_alt X)
```
```   767   also have "... = emeasure ?C X"
```
```   768     apply (subst emeasure_count_space)
```
```   769     using card X_subset A fin_Pair fin_X
```
```   770     apply (auto simp add: nn_integral_count_space
```
```   771                            of_nat_sum[symmetric] card_SigmaI[symmetric]
```
```   772                 simp del:  card_SigmaI
```
```   773                 intro!: arg_cong[where f=card])
```
```   774     done
```
```   775   finally show "emeasure ?P X = emeasure ?C X" .
```
```   776 qed
```
```   777
```
```   778
```
```   779 lemma emeasure_prod_count_space:
```
```   780   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
```
```   781   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
```
```   782   by (rule emeasure_measure_of[OF pair_measure_def])
```
```   783      (auto simp: countably_additive_def positive_def suminf_indicator A
```
```   784                  nn_integral_suminf[symmetric] dest: sets.sets_into_space)
```
```   785
```
```   786 lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
```
```   787 proof -
```
```   788   have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
```
```   789     by (auto split: split_indicator)
```
```   790   show ?thesis
```
```   791     by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
```
```   792 qed
```
```   793
```
```   794 lemma emeasure_count_space_prod_eq:
```
```   795   fixes A :: "('a \<times> 'b) set"
```
```   796   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
```
```   797   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
```
```   798 proof -
```
```   799   { fix A :: "('a \<times> 'b) set" assume "countable A"
```
```   800     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)"
```
```   801       by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
```
```   802     also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)"
```
```   803       by (subst nn_integral_count_space_indicator) auto
```
```   804     finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
```
```   805       by simp }
```
```   806   note * = this
```
```   807
```
```   808   show ?thesis
```
```   809   proof cases
```
```   810     assume "finite A" then show ?thesis
```
```   811       by (intro * countable_finite)
```
```   812   next
```
```   813     assume "infinite A"
```
```   814     then obtain C where "countable C" and "infinite C" and "C \<subseteq> A"
```
```   815       by (auto dest: infinite_countable_subset')
```
```   816     with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A"
```
```   817       by (intro emeasure_mono) auto
```
```   818     also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C"
```
```   819       using \<open>countable C\<close> by (rule *)
```
```   820     finally show ?thesis
```
```   821       using \<open>infinite C\<close> \<open>infinite A\<close> by (simp add: top_unique)
```
```   822   qed
```
```   823 qed
```
```   824
```
```   825 lemma nn_integral_count_space_prod_eq:
```
```   826   "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
```
```   827     (is "nn_integral ?P f = _")
```
```   828 proof cases
```
```   829   assume cntbl: "countable {x. f x \<noteq> 0}"
```
```   830   have [simp]: "\<And>x. card ({x} \<inter> {x. f x \<noteq> 0}) = (indicator {x. f x \<noteq> 0} x::ennreal)"
```
```   831     by (auto split: split_indicator)
```
```   832   have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P"
```
```   833     by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y])
```
```   834        (auto intro: sets_Pair)
```
```   835
```
```   836   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)"
```
```   837     by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator)
```
```   838   also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)"
```
```   839     by (auto intro!: nn_integral_cong split: split_indicator)
```
```   840   also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})"
```
```   841     by (intro nn_integral_count_space_nn_integral cntbl) auto
```
```   842   also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})"
```
```   843     by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair)
```
```   844   finally show ?thesis
```
```   845     by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
```
```   846 next
```
```   847   { fix x assume "f x \<noteq> 0"
```
```   848     then have "(\<exists>r\<ge>0. 0 < r \<and> f x = ennreal r) \<or> f x = \<infinity>"
```
```   849       by (cases "f x" rule: ennreal_cases) (auto simp: less_le)
```
```   850     then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f x"
```
```   851       by (auto elim!: nat_approx_posE intro!: less_imp_le) }
```
```   852   note * = this
```
```   853
```
```   854   assume cntbl: "uncountable {x. f x \<noteq> 0}"
```
```   855   also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})"
```
```   856     using * by auto
```
```   857   finally obtain n where "infinite {x. 1/Suc n \<le> f x}"
```
```   858     by (meson countableI_type countable_UN uncountable_infinite)
```
```   859   then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C"
```
```   860     by (metis infinite_countable_subset')
```
```   861
```
```   862   have [measurable]: "C \<in> sets ?P"
```
```   863     using sets.countable[OF _ \<open>countable C\<close>, of ?P] by (auto simp: sets_Pair)
```
```   864
```
```   865   have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f"
```
```   866     using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
```
```   867   moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>"
```
```   868     using \<open>infinite C\<close> by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top)
```
```   869   moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f"
```
```   870     using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
```
```   871   moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>"
```
```   872     using \<open>infinite C\<close> by (simp add: nn_integral_cmult ennreal_mult_top)
```
```   873   ultimately show ?thesis
```
```   874     by (simp add: top_unique)
```
```   875 qed
```
```   876
```
```   877 theorem pair_measure_density:
```
```   878   assumes f: "f \<in> borel_measurable M1"
```
```   879   assumes g: "g \<in> borel_measurable M2"
```
```   880   assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
```
```   881   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")
```
```   882 proof (rule measure_eqI)
```
```   883   interpret M2: sigma_finite_measure M2 by fact
```
```   884   interpret D2: sigma_finite_measure "density M2 g" by fact
```
```   885
```
```   886   fix A assume A: "A \<in> sets ?L"
```
```   887   with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) =
```
```   888     (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)"
```
```   889     by (intro nn_integral_cong_AE)
```
```   890        (auto simp add: nn_integral_cmult[symmetric] ac_simps)
```
```   891   with A f g show "emeasure ?L A = emeasure ?R A"
```
```   892     by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density
```
```   893                   M2.nn_integral_fst[symmetric]
```
```   894              cong: nn_integral_cong)
```
```   895 qed simp
```
```   896
```
```   897 lemma sigma_finite_measure_distr:
```
```   898   assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
```
```   899   shows "sigma_finite_measure M"
```
```   900 proof -
```
```   901   interpret sigma_finite_measure "distr M N f" by fact
```
```   902   from sigma_finite_countable guess A .. note A = this
```
```   903   show ?thesis
```
```   904   proof
```
```   905     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>)"
```
```   906       using A f
```
```   907       by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"])
```
```   908          (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)
```
```   909   qed
```
```   910 qed
```
```   911
```
```   912 lemma pair_measure_distr:
```
```   913   assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
```
```   914   assumes "sigma_finite_measure (distr N T g)"
```
```   915   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")
```
```   916 proof (rule measure_eqI)
```
```   917   interpret T: sigma_finite_measure "distr N T g" by fact
```
```   918   interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
```
```   919
```
```   920   fix A assume A: "A \<in> sets ?P"
```
```   921   with f g show "emeasure ?P A = emeasure ?D A"
```
```   922     by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr
```
```   923                        T.emeasure_pair_measure_alt nn_integral_distr
```
```   924              intro!: nn_integral_cong arg_cong[where f="emeasure N"])
```
```   925 qed simp
```
```   926
```
```   927 lemma pair_measure_eqI:
```
```   928   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
```
```   929   assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
```
```   930   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)"
```
```   931   shows "M1 \<Otimes>\<^sub>M M2 = M"
```
```   932 proof -
```
```   933   interpret M1: sigma_finite_measure M1 by fact
```
```   934   interpret M2: sigma_finite_measure M2 by fact
```
```   935   interpret pair_sigma_finite M1 M2 ..
```
```   936   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
```
```   937   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
```
```   938   let ?P = "M1 \<Otimes>\<^sub>M M2"
```
```   939   show ?thesis
```
```   940   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
```
```   941     show "?E \<subseteq> Pow (space ?P)"
```
```   942       using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
```
```   943     show "sets ?P = sigma_sets (space ?P) ?E"
```
```   944       by (simp add: sets_pair_measure space_pair_measure)
```
```   945     then show "sets M = sigma_sets (space ?P) ?E"
```
```   946       using sets[symmetric] by simp
```
```   947   next
```
```   948     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
```
```   949       using F by (auto simp: space_pair_measure)
```
```   950   next
```
```   951     fix X assume "X \<in> ?E"
```
```   952     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
```
```   953     then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
```
```   954        by (simp add: M2.emeasure_pair_measure_Times)
```
```   955     also have "\<dots> = emeasure M (A \<times> B)"
```
```   956       using A B emeasure by auto
```
```   957     finally show "emeasure ?P X = emeasure M X"
```
```   958       by simp
```
```   959   qed
```
```   960 qed
```
```   961
```
```   962 lemma sets_pair_countable:
```
```   963   assumes "countable S1" "countable S2"
```
```   964   assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
```
```   965   shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
```
```   966 proof auto
```
```   967   fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
```
```   968   from sets.sets_into_space[OF x(1)] x(2)
```
```   969     sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
```
```   970   show "a \<in> S1" "b \<in> S2"
```
```   971     by (auto simp: space_pair_measure)
```
```   972 next
```
```   973   fix X assume X: "X \<subseteq> S1 \<times> S2"
```
```   974   then have "countable X"
```
```   975     by (metis countable_subset \<open>countable S1\<close> \<open>countable S2\<close> countable_SIGMA)
```
```   976   have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto
```
```   977   also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)"
```
```   978     using X
```
```   979     by (safe intro!: sets.countable_UN' \<open>countable X\<close> subsetI pair_measureI) (auto simp: M N)
```
```   980   finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
```
```   981 qed
```
```   982
```
```   983 lemma pair_measure_countable:
```
```   984   assumes "countable S1" "countable S2"
```
```   985   shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
```
```   986 proof (rule pair_measure_eqI)
```
```   987   show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
```
```   988     using assms by (auto intro!: sigma_finite_measure_count_space_countable)
```
```   989   show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
```
```   990     by (subst sets_pair_countable[OF assms]) auto
```
```   991 next
```
```   992   fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)"
```
```   993   then show "emeasure (count_space S1) A * emeasure (count_space S2) B =
```
```   994     emeasure (count_space (S1 \<times> S2)) (A \<times> B)"
```
```   995     by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
```
```   996 qed
```
```   997
```
```   998 proposition nn_integral_fst_count_space:
```
```   999   "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
```
```  1000   (is "?lhs = ?rhs")
```
```  1001 proof(cases)
```
```  1002   assume *: "countable {xy. f xy \<noteq> 0}"
```
```  1003   let ?A = "fst ` {xy. f xy \<noteq> 0}"
```
```  1004   let ?B = "snd ` {xy. f xy \<noteq> 0}"
```
```  1005   from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+
```
```  1006   have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)"
```
```  1007     by(rule nn_integral_count_space_eq)
```
```  1008       (auto simp add: nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI)
```
```  1009   also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)"
```
```  1010     by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI)
```
```  1011   also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))"
```
```  1012     by(subst sigma_finite_measure.nn_integral_fst)
```
```  1013       (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable)
```
```  1014   also have "\<dots> = ?rhs"
```
```  1015     by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI)
```
```  1016   finally show ?thesis .
```
```  1017 next
```
```  1018   { fix xy assume "f xy \<noteq> 0"
```
```  1019     then have "(\<exists>r\<ge>0. 0 < r \<and> f xy = ennreal r) \<or> f xy = \<infinity>"
```
```  1020       by (cases "f xy" rule: ennreal_cases) (auto simp: less_le)
```
```  1021     then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f xy"
```
```  1022       by (auto elim!: nat_approx_posE intro!: less_imp_le) }
```
```  1023   note * = this
```
```  1024
```
```  1025   assume cntbl: "uncountable {xy. f xy \<noteq> 0}"
```
```  1026   also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})"
```
```  1027     using * by auto
```
```  1028   finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}"
```
```  1029     by (meson countableI_type countable_UN uncountable_infinite)
```
```  1030   then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C"
```
```  1031     by (metis infinite_countable_subset')
```
```  1032
```
```  1033   have "\<infinity> = (\<integral>\<^sup>+ xy. ennreal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)"
```
```  1034     using \<open>infinite C\<close> by(simp add: nn_integral_cmult ennreal_mult_top)
```
```  1035   also have "\<dots> \<le> ?rhs" using C
```
```  1036     by(intro nn_integral_mono)(auto split: split_indicator)
```
```  1037   finally have "?rhs = \<infinity>" by (simp add: top_unique)
```
```  1038   moreover have "?lhs = \<infinity>"
```
```  1039   proof(cases "finite (fst ` C)")
```
```  1040     case True
```
```  1041     then obtain x C' where x: "x \<in> fst ` C"
```
```  1042       and C': "C' = fst -` {x} \<inter> C"
```
```  1043       and "infinite C'"
```
```  1044       using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE')
```
```  1045     from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto
```
```  1046
```
```  1047     from C' \<open>infinite C'\<close> have "infinite (snd ` C')"
```
```  1048       by(auto dest!: finite_imageD simp add: inj_on_def)
```
```  1049     then have "\<infinity> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)"
```
```  1050       by(simp add: nn_integral_cmult ennreal_mult_top)
```
```  1051     also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)"
```
```  1052       by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C')
```
```  1053     also have "\<dots> = (\<integral>\<^sup>+ x'. (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV) * indicator {x} x' \<partial>count_space UNIV)"
```
```  1054       by(simp add: one_ereal_def[symmetric])
```
```  1055     also have "\<dots> \<le> (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV \<partial>count_space UNIV)"
```
```  1056       by(rule nn_integral_mono)(simp split: split_indicator)
```
```  1057     also have "\<dots> \<le> ?lhs" using **
```
```  1058       by(intro nn_integral_mono)(auto split: split_indicator)
```
```  1059     finally show ?thesis by (simp add: top_unique)
```
```  1060   next
```
```  1061     case False
```
```  1062     define C' where "C' = fst ` C"
```
```  1063     have "\<infinity> = \<integral>\<^sup>+ x. ennreal (1 / Suc n) * indicator C' x \<partial>count_space UNIV"
```
```  1064       using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top)
```
```  1065     also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) \<in> C} y \<partial>count_space UNIV \<partial>count_space UNIV"
```
```  1066       by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong)
```
```  1067     also have "\<dots> \<le> \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C (x, y) \<partial>count_space UNIV \<partial>count_space UNIV"
```
```  1068       by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI)
```
```  1069     also have "\<dots> \<le> ?lhs" using C
```
```  1070       by(intro nn_integral_mono)(auto split: split_indicator)
```
```  1071     finally show ?thesis by (simp add: top_unique)
```
```  1072   qed
```
```  1073   ultimately show ?thesis by simp
```
```  1074 qed
```
```  1075
```
```  1076 proposition nn_integral_snd_count_space:
```
```  1077   "(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
```
```  1078   (is "?lhs = ?rhs")
```
```  1079 proof -
```
```  1080   have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)"
```
```  1081     by(simp)
```
```  1082   also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV"
```
```  1083     by(rule nn_integral_fst_count_space)
```
```  1084   also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)"
```
```  1085     by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
```
```  1086       (simp_all add: inj_on_def split_def)
```
```  1087   also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto
```
```  1088   finally show ?thesis .
```
```  1089 qed
```
```  1090
```
```  1091 lemma measurable_pair_measure_countable1:
```
```  1092   assumes "countable A"
```
```  1093   and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
```
```  1094   shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
```
```  1095 using _ _ assms(1)
```
```  1096 by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
```
```  1097
```
```  1098 subsection \<open>Product of Borel spaces\<close>
```
```  1099
```
```  1100 theorem borel_Times:
```
```  1101   fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
```
```  1102   assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
```
```  1103   shows "A \<times> B \<in> sets borel"
```
```  1104 proof -
```
```  1105   have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
```
```  1106     by auto
```
```  1107   moreover
```
```  1108   { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)
```
```  1109     then have "A\<times>UNIV \<in> sets borel"
```
```  1110     proof (induct A)
```
```  1111       case (Basic S) then show ?case
```
```  1112         by (auto intro!: borel_open open_Times)
```
```  1113     next
```
```  1114       case (Compl A)
```
```  1115       moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)"
```
```  1116         by auto
```
```  1117       ultimately show ?case
```
```  1118         unfolding * by auto
```
```  1119     next
```
```  1120       case (Union A)
```
```  1121       moreover have *: "(\<Union>(A ` UNIV)) \<times> UNIV = \<Union>((\<lambda>i. A i \<times> UNIV) ` UNIV)"
```
```  1122         by auto
```
```  1123       ultimately show ?case
```
```  1124         unfolding * by auto
```
```  1125     qed simp }
```
```  1126   moreover
```
```  1127   { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)
```
```  1128     then have "UNIV\<times>B \<in> sets borel"
```
```  1129     proof (induct B)
```
```  1130       case (Basic S) then show ?case
```
```  1131         by (auto intro!: borel_open open_Times)
```
```  1132     next
```
```  1133       case (Compl B)
```
```  1134       moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)"
```
```  1135         by auto
```
```  1136       ultimately show ?case
```
```  1137         unfolding * by auto
```
```  1138     next
```
```  1139       case (Union B)
```
```  1140       moreover have *: "UNIV \<times> (\<Union>(B ` UNIV)) = \<Union>((\<lambda>i. UNIV \<times> B i) ` UNIV)"
```
```  1141         by auto
```
```  1142       ultimately show ?case
```
```  1143         unfolding * by auto
```
```  1144     qed simp }
```
```  1145   ultimately show ?thesis
```
```  1146     by auto
```
```  1147 qed
```
```  1148
```
```  1149 lemma finite_measure_pair_measure:
```
```  1150   assumes "finite_measure M" "finite_measure N"
```
```  1151   shows "finite_measure (N  \<Otimes>\<^sub>M M)"
```
```  1152 proof (rule finite_measureI)
```
```  1153   interpret M: finite_measure M by fact
```
```  1154   interpret N: finite_measure N by fact
```
```  1155   show "emeasure (N  \<Otimes>\<^sub>M M) (space (N  \<Otimes>\<^sub>M M)) \<noteq> \<infinity>"
```
```  1156     by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
```
```  1157 qed
```
```  1158
```
```  1159 end
```