src/HOL/Multivariate_Analysis/Binary_Product_Measure.thy
changeset 63627 6ddb43c6b711
parent 63626 44ce6b524ff3
child 63631 2edc8da89edc
child 63633 2accfb71e33b
     1.1 --- a/src/HOL/Multivariate_Analysis/Binary_Product_Measure.thy	Fri Aug 05 18:34:57 2016 +0200
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,1110 +0,0 @@
     1.4 -(*  Title:      HOL/Probability/Binary_Product_Measure.thy
     1.5 -    Author:     Johannes Hölzl, TU München
     1.6 -*)
     1.7 -
     1.8 -section \<open>Binary product measures\<close>
     1.9 -
    1.10 -theory Binary_Product_Measure
    1.11 -imports Nonnegative_Lebesgue_Integration
    1.12 -begin
    1.13 -
    1.14 -lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
    1.15 -  by auto
    1.16 -
    1.17 -lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
    1.18 -  by auto
    1.19 -
    1.20 -subsection "Binary products"
    1.21 -
    1.22 -definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
    1.23 -  "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B)
    1.24 -      {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
    1.25 -      (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
    1.26 -
    1.27 -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)"
    1.28 -  using sets.space_closed[of A] sets.space_closed[of B] by auto
    1.29 -
    1.30 -lemma space_pair_measure:
    1.31 -  "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
    1.32 -  unfolding pair_measure_def using pair_measure_closed[of A B]
    1.33 -  by (rule space_measure_of)
    1.34 -
    1.35 -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)}"
    1.36 -  by (auto simp: space_pair_measure)
    1.37 -
    1.38 -lemma sets_pair_measure:
    1.39 -  "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}"
    1.40 -  unfolding pair_measure_def using pair_measure_closed[of A B]
    1.41 -  by (rule sets_measure_of)
    1.42 -
    1.43 -lemma sets_pair_measure_cong[measurable_cong, cong]:
    1.44 -  "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
    1.45 -  unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
    1.46 -
    1.47 -lemma pair_measureI[intro, simp, measurable]:
    1.48 -  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
    1.49 -  by (auto simp: sets_pair_measure)
    1.50 -
    1.51 -lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
    1.52 -  using pair_measureI[of "{x}" M1 "{y}" M2] by simp
    1.53 -
    1.54 -lemma measurable_pair_measureI:
    1.55 -  assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
    1.56 -  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"
    1.57 -  shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
    1.58 -  unfolding pair_measure_def using 1 2
    1.59 -  by (intro measurable_measure_of) (auto dest: sets.sets_into_space)
    1.60 -
    1.61 -lemma measurable_split_replace[measurable (raw)]:
    1.62 -  "(\<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"
    1.63 -  unfolding split_beta' .
    1.64 -
    1.65 -lemma measurable_Pair[measurable (raw)]:
    1.66 -  assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
    1.67 -  shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
    1.68 -proof (rule measurable_pair_measureI)
    1.69 -  show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
    1.70 -    using f g by (auto simp: measurable_def)
    1.71 -  fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
    1.72 -  have "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
    1.73 -    by auto
    1.74 -  also have "\<dots> \<in> sets M"
    1.75 -    by (rule sets.Int) (auto intro!: measurable_sets * f g)
    1.76 -  finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .
    1.77 -qed
    1.78 -
    1.79 -lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
    1.80 -  by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
    1.81 -    measurable_def)
    1.82 -
    1.83 -lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
    1.84 -  by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
    1.85 -    measurable_def)
    1.86 -
    1.87 -lemma measurable_Pair_compose_split[measurable_dest]:
    1.88 -  assumes f: "case_prod f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N"
    1.89 -  assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"
    1.90 -  shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"
    1.91 -  using measurable_compose[OF measurable_Pair f, OF g h] by simp
    1.92 -
    1.93 -lemma measurable_Pair1_compose[measurable_dest]:
    1.94 -  assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
    1.95 -  assumes [measurable]: "h \<in> measurable N M"
    1.96 -  shows "(\<lambda>x. f (h x)) \<in> measurable N M1"
    1.97 -  using measurable_compose[OF f measurable_fst] by simp
    1.98 -
    1.99 -lemma measurable_Pair2_compose[measurable_dest]:
   1.100 -  assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   1.101 -  assumes [measurable]: "h \<in> measurable N M"
   1.102 -  shows "(\<lambda>x. g (h x)) \<in> measurable N M2"
   1.103 -  using measurable_compose[OF f measurable_snd] by simp
   1.104 -
   1.105 -lemma measurable_pair:
   1.106 -  assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
   1.107 -  shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   1.108 -  using measurable_Pair[OF assms] by simp
   1.109 -
   1.110 -lemma
   1.111 -  assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)"
   1.112 -  shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"
   1.113 -    and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"
   1.114 -  by simp_all
   1.115 -
   1.116 -lemma
   1.117 -  assumes f[measurable]: "f \<in> measurable M N"
   1.118 -  shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N"
   1.119 -    and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N"
   1.120 -  by simp_all
   1.121 -
   1.122 -lemma sets_pair_in_sets:
   1.123 -  assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"
   1.124 -  shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
   1.125 -  unfolding sets_pair_measure
   1.126 -  by (intro sets.sigma_sets_subset') (auto intro!: assms)
   1.127 -
   1.128 -lemma sets_pair_eq_sets_fst_snd:
   1.129 -  "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})"
   1.130 -    (is "?P = sets (Sup {?fst, ?snd})")
   1.131 -proof -
   1.132 -  { fix a b assume ab: "a \<in> sets A" "b \<in> sets B"
   1.133 -    then have "a \<times> b = (fst -` a \<inter> (space A \<times> space B)) \<inter> (snd -` b \<inter> (space A \<times> space B))"
   1.134 -      by (auto dest: sets.sets_into_space)
   1.135 -    also have "\<dots> \<in> sets (Sup {?fst, ?snd})"
   1.136 -      apply (rule sets.Int)
   1.137 -      apply (rule in_sets_Sup)
   1.138 -      apply auto []
   1.139 -      apply (rule insertI1)
   1.140 -      apply (auto intro: ab in_vimage_algebra) []
   1.141 -      apply (rule in_sets_Sup)
   1.142 -      apply auto []
   1.143 -      apply (rule insertI2)
   1.144 -      apply (auto intro: ab in_vimage_algebra)
   1.145 -      done
   1.146 -    finally have "a \<times> b \<in> sets (Sup {?fst, ?snd})" . }
   1.147 -  moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)"
   1.148 -    by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric])
   1.149 -  moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)"
   1.150 -    by (rule sets_image_in_sets) (auto simp: space_pair_measure)
   1.151 -  ultimately show ?thesis
   1.152 -    apply (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets)
   1.153 -    apply simp
   1.154 -    apply simp
   1.155 -    apply simp
   1.156 -    apply (elim disjE)
   1.157 -    apply (simp add: space_pair_measure)
   1.158 -    apply (simp add: space_pair_measure)
   1.159 -    apply (auto simp add: space_pair_measure)
   1.160 -    done
   1.161 -qed
   1.162 -
   1.163 -lemma measurable_pair_iff:
   1.164 -  "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"
   1.165 -  by (auto intro: measurable_pair[of f M M1 M2])
   1.166 -
   1.167 -lemma measurable_split_conv:
   1.168 -  "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
   1.169 -  by (intro arg_cong2[where f="op \<in>"]) auto
   1.170 -
   1.171 -lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
   1.172 -  by (auto intro!: measurable_Pair simp: measurable_split_conv)
   1.173 -
   1.174 -lemma measurable_pair_swap:
   1.175 -  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"
   1.176 -  using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
   1.177 -
   1.178 -lemma measurable_pair_swap_iff:
   1.179 -  "f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M"
   1.180 -  by (auto dest: measurable_pair_swap)
   1.181 -
   1.182 -lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
   1.183 -  by simp
   1.184 -
   1.185 -lemma sets_Pair1[measurable (raw)]:
   1.186 -  assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2"
   1.187 -proof -
   1.188 -  have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
   1.189 -    using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   1.190 -  also have "\<dots> \<in> sets M2"
   1.191 -    using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: if_split_asm)
   1.192 -  finally show ?thesis .
   1.193 -qed
   1.194 -
   1.195 -lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
   1.196 -  by (auto intro!: measurable_Pair)
   1.197 -
   1.198 -lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
   1.199 -proof -
   1.200 -  have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
   1.201 -    using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   1.202 -  also have "\<dots> \<in> sets M1"
   1.203 -    using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: if_split_asm)
   1.204 -  finally show ?thesis .
   1.205 -qed
   1.206 -
   1.207 -lemma measurable_Pair2:
   1.208 -  assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1"
   1.209 -  shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
   1.210 -  using measurable_comp[OF measurable_Pair1' f, OF x]
   1.211 -  by (simp add: comp_def)
   1.212 -
   1.213 -lemma measurable_Pair1:
   1.214 -  assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2"
   1.215 -  shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
   1.216 -  using measurable_comp[OF measurable_Pair2' f, OF y]
   1.217 -  by (simp add: comp_def)
   1.218 -
   1.219 -lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
   1.220 -  unfolding Int_stable_def
   1.221 -  by safe (auto simp add: times_Int_times)
   1.222 -
   1.223 -lemma (in finite_measure) finite_measure_cut_measurable:
   1.224 -  assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)"
   1.225 -  shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
   1.226 -    (is "?s Q \<in> _")
   1.227 -  using Int_stable_pair_measure_generator pair_measure_closed assms
   1.228 -  unfolding sets_pair_measure
   1.229 -proof (induct rule: sigma_sets_induct_disjoint)
   1.230 -  case (compl A)
   1.231 -  with sets.sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) =
   1.232 -      (if x \<in> space N then emeasure M (space M) - ?s A x else 0)"
   1.233 -    unfolding sets_pair_measure[symmetric]
   1.234 -    by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
   1.235 -  with compl sets.top show ?case
   1.236 -    by (auto intro!: measurable_If simp: space_pair_measure)
   1.237 -next
   1.238 -  case (union F)
   1.239 -  then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)"
   1.240 -    by (simp add: suminf_emeasure disjoint_family_on_vimageI subset_eq vimage_UN sets_pair_measure[symmetric])
   1.241 -  with union show ?case
   1.242 -    unfolding sets_pair_measure[symmetric] by simp
   1.243 -qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
   1.244 -
   1.245 -lemma (in sigma_finite_measure) measurable_emeasure_Pair:
   1.246 -  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> _")
   1.247 -proof -
   1.248 -  from sigma_finite_disjoint guess F . note F = this
   1.249 -  then have F_sets: "\<And>i. F i \<in> sets M" by auto
   1.250 -  let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"
   1.251 -  { fix i
   1.252 -    have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i"
   1.253 -      using F sets.sets_into_space by auto
   1.254 -    let ?R = "density M (indicator (F i))"
   1.255 -    have "finite_measure ?R"
   1.256 -      using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
   1.257 -    then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N"
   1.258 -     by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)
   1.259 -    moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))
   1.260 -        = emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))"
   1.261 -      using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
   1.262 -    moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i"
   1.263 -      using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure)
   1.264 -    ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N"
   1.265 -      by simp }
   1.266 -  moreover
   1.267 -  { fix x
   1.268 -    have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
   1.269 -    proof (intro suminf_emeasure)
   1.270 -      show "range (?C x) \<subseteq> sets M"
   1.271 -        using F \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> by (auto intro!: sets_Pair1)
   1.272 -      have "disjoint_family F" using F by auto
   1.273 -      show "disjoint_family (?C x)"
   1.274 -        by (rule disjoint_family_on_bisimulation[OF \<open>disjoint_family F\<close>]) auto
   1.275 -    qed
   1.276 -    also have "(\<Union>i. ?C x i) = Pair x -` Q"
   1.277 -      using F sets.sets_into_space[OF \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close>]
   1.278 -      by (auto simp: space_pair_measure)
   1.279 -    finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"
   1.280 -      by simp }
   1.281 -  ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets
   1.282 -    by auto
   1.283 -qed
   1.284 -
   1.285 -lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
   1.286 -  assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
   1.287 -  assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)"
   1.288 -  shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
   1.289 -proof -
   1.290 -  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"
   1.291 -    by (auto simp: space_pair_measure)
   1.292 -  with measurable_emeasure_Pair[OF A] show ?thesis
   1.293 -    by (auto cong: measurable_cong)
   1.294 -qed
   1.295 -
   1.296 -lemma (in sigma_finite_measure) emeasure_pair_measure:
   1.297 -  assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
   1.298 -  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")
   1.299 -proof (rule emeasure_measure_of[OF pair_measure_def])
   1.300 -  show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
   1.301 -    by (auto simp: positive_def)
   1.302 -  have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y"
   1.303 -    by (auto simp: indicator_def)
   1.304 -  show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
   1.305 -  proof (rule countably_additiveI)
   1.306 -    fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F"
   1.307 -    from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto
   1.308 -    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
   1.309 -      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
   1.310 -    moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M"
   1.311 -      using F by (auto simp: sets_Pair1)
   1.312 -    ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)"
   1.313 -      by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure
   1.314 -               intro!: nn_integral_cong nn_integral_indicator[symmetric])
   1.315 -  qed
   1.316 -  show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
   1.317 -    using sets.space_closed[of N] sets.space_closed[of M] by auto
   1.318 -qed fact
   1.319 -
   1.320 -lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
   1.321 -  assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
   1.322 -  shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
   1.323 -proof -
   1.324 -  have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
   1.325 -    by (auto simp: indicator_def)
   1.326 -  show ?thesis
   1.327 -    using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
   1.328 -qed
   1.329 -
   1.330 -lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
   1.331 -  assumes A: "A \<in> sets N" and B: "B \<in> sets M"
   1.332 -  shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
   1.333 -proof -
   1.334 -  have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
   1.335 -    using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
   1.336 -  also have "\<dots> = emeasure M B * emeasure N A"
   1.337 -    using A by (simp add: nn_integral_cmult_indicator)
   1.338 -  finally show ?thesis
   1.339 -    by (simp add: ac_simps)
   1.340 -qed
   1.341 -
   1.342 -subsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
   1.343 -
   1.344 -locale pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
   1.345 -  for M1 :: "'a measure" and M2 :: "'b measure"
   1.346 -
   1.347 -lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
   1.348 -  "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
   1.349 -  using M2.measurable_emeasure_Pair .
   1.350 -
   1.351 -lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
   1.352 -  assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
   1.353 -proof -
   1.354 -  have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   1.355 -    using Q measurable_pair_swap' by (auto intro: measurable_sets)
   1.356 -  note M1.measurable_emeasure_Pair[OF this]
   1.357 -  moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1)) = (\<lambda>x. (x, y)) -` Q"
   1.358 -    using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   1.359 -  ultimately show ?thesis by simp
   1.360 -qed
   1.361 -
   1.362 -lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
   1.363 -  defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
   1.364 -  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>
   1.365 -    (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
   1.366 -proof -
   1.367 -  from M1.sigma_finite_incseq guess F1 . note F1 = this
   1.368 -  from M2.sigma_finite_incseq guess F2 . note F2 = this
   1.369 -  from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
   1.370 -  let ?F = "\<lambda>i. F1 i \<times> F2 i"
   1.371 -  show ?thesis
   1.372 -  proof (intro exI[of _ ?F] conjI allI)
   1.373 -    show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
   1.374 -  next
   1.375 -    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
   1.376 -    proof (intro subsetI)
   1.377 -      fix x assume "x \<in> space M1 \<times> space M2"
   1.378 -      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
   1.379 -        by (auto simp: space)
   1.380 -      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
   1.381 -        using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_def
   1.382 -        by (force split: split_max)+
   1.383 -      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
   1.384 -        by (intro SigmaI) (auto simp add: max.commute)
   1.385 -      then show "x \<in> (\<Union>i. ?F i)" by auto
   1.386 -    qed
   1.387 -    then show "(\<Union>i. ?F i) = space M1 \<times> space M2"
   1.388 -      using space by (auto simp: space)
   1.389 -  next
   1.390 -    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
   1.391 -      using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_Suc_iff by auto
   1.392 -  next
   1.393 -    fix i
   1.394 -    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
   1.395 -    with F1 F2 show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
   1.396 -      by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
   1.397 -  qed
   1.398 -qed
   1.399 -
   1.400 -sublocale pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
   1.401 -proof
   1.402 -  from M1.sigma_finite_countable guess F1 ..
   1.403 -  moreover from M2.sigma_finite_countable guess F2 ..
   1.404 -  ultimately show
   1.405 -    "\<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>)"
   1.406 -    by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI)
   1.407 -       (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
   1.408 -qed
   1.409 -
   1.410 -lemma sigma_finite_pair_measure:
   1.411 -  assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
   1.412 -  shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
   1.413 -proof -
   1.414 -  interpret A: sigma_finite_measure A by fact
   1.415 -  interpret B: sigma_finite_measure B by fact
   1.416 -  interpret AB: pair_sigma_finite A  B ..
   1.417 -  show ?thesis ..
   1.418 -qed
   1.419 -
   1.420 -lemma sets_pair_swap:
   1.421 -  assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.422 -  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   1.423 -  using measurable_pair_swap' assms by (rule measurable_sets)
   1.424 -
   1.425 -lemma (in pair_sigma_finite) distr_pair_swap:
   1.426 -  "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
   1.427 -proof -
   1.428 -  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   1.429 -  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
   1.430 -  show ?thesis
   1.431 -  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
   1.432 -    show "?E \<subseteq> Pow (space ?P)"
   1.433 -      using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
   1.434 -    show "sets ?P = sigma_sets (space ?P) ?E"
   1.435 -      by (simp add: sets_pair_measure space_pair_measure)
   1.436 -    then show "sets ?D = sigma_sets (space ?P) ?E"
   1.437 -      by simp
   1.438 -  next
   1.439 -    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
   1.440 -      using F by (auto simp: space_pair_measure)
   1.441 -  next
   1.442 -    fix X assume "X \<in> ?E"
   1.443 -    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
   1.444 -    have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A"
   1.445 -      using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure)
   1.446 -    with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X"
   1.447 -      by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
   1.448 -                    measurable_pair_swap' ac_simps)
   1.449 -  qed
   1.450 -qed
   1.451 -
   1.452 -lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
   1.453 -  assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.454 -  shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
   1.455 -    (is "_ = ?\<nu> A")
   1.456 -proof -
   1.457 -  have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A"
   1.458 -    using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
   1.459 -  show ?thesis using A
   1.460 -    by (subst distr_pair_swap)
   1.461 -       (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
   1.462 -                 M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
   1.463 -qed
   1.464 -
   1.465 -lemma (in pair_sigma_finite) AE_pair:
   1.466 -  assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
   1.467 -  shows "AE x in M1. (AE y in M2. Q (x, y))"
   1.468 -proof -
   1.469 -  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"
   1.470 -    using assms unfolding eventually_ae_filter by auto
   1.471 -  show ?thesis
   1.472 -  proof (rule AE_I)
   1.473 -    from N measurable_emeasure_Pair1[OF \<open>N \<in> sets (M1 \<Otimes>\<^sub>M M2)\<close>]
   1.474 -    show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
   1.475 -      by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff)
   1.476 -    show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
   1.477 -      by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp
   1.478 -    { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0"
   1.479 -      have "AE y in M2. Q (x, y)"
   1.480 -      proof (rule AE_I)
   1.481 -        show "emeasure M2 (Pair x -` N) = 0" by fact
   1.482 -        show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
   1.483 -        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
   1.484 -          using N \<open>x \<in> space M1\<close> unfolding space_pair_measure by auto
   1.485 -      qed }
   1.486 -    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}"
   1.487 -      by auto
   1.488 -  qed
   1.489 -qed
   1.490 -
   1.491 -lemma (in pair_sigma_finite) AE_pair_measure:
   1.492 -  assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.493 -  assumes ae: "AE x in M1. AE y in M2. P (x, y)"
   1.494 -  shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
   1.495 -proof (subst AE_iff_measurable[OF _ refl])
   1.496 -  show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.497 -    by (rule sets.sets_Collect) fact
   1.498 -  then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
   1.499 -      (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
   1.500 -    by (simp add: M2.emeasure_pair_measure)
   1.501 -  also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)"
   1.502 -    using ae
   1.503 -    apply (safe intro!: nn_integral_cong_AE)
   1.504 -    apply (intro AE_I2)
   1.505 -    apply (safe intro!: nn_integral_cong_AE)
   1.506 -    apply auto
   1.507 -    done
   1.508 -  finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
   1.509 -qed
   1.510 -
   1.511 -lemma (in pair_sigma_finite) AE_pair_iff:
   1.512 -  "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
   1.513 -    (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))"
   1.514 -  using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
   1.515 -
   1.516 -lemma (in pair_sigma_finite) AE_commute:
   1.517 -  assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.518 -  shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
   1.519 -proof -
   1.520 -  interpret Q: pair_sigma_finite M2 M1 ..
   1.521 -  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"
   1.522 -    by auto
   1.523 -  have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} =
   1.524 -    (\<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)"
   1.525 -    by (auto simp: space_pair_measure)
   1.526 -  also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   1.527 -    by (intro sets_pair_swap P)
   1.528 -  finally show ?thesis
   1.529 -    apply (subst AE_pair_iff[OF P])
   1.530 -    apply (subst distr_pair_swap)
   1.531 -    apply (subst AE_distr_iff[OF measurable_pair_swap' P])
   1.532 -    apply (subst Q.AE_pair_iff)
   1.533 -    apply simp_all
   1.534 -    done
   1.535 -qed
   1.536 -
   1.537 -subsection "Fubinis theorem"
   1.538 -
   1.539 -lemma measurable_compose_Pair1:
   1.540 -  "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
   1.541 -  by simp
   1.542 -
   1.543 -lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst:
   1.544 -  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   1.545 -  shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
   1.546 -using f proof induct
   1.547 -  case (cong u v)
   1.548 -  then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
   1.549 -    by (auto simp: space_pair_measure)
   1.550 -  show ?case
   1.551 -    apply (subst measurable_cong)
   1.552 -    apply (rule nn_integral_cong)
   1.553 -    apply fact+
   1.554 -    done
   1.555 -next
   1.556 -  case (set Q)
   1.557 -  have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y"
   1.558 -    by (auto simp: indicator_def)
   1.559 -  have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M"
   1.560 -    by (simp add: sets_Pair1[OF set])
   1.561 -  from this measurable_emeasure_Pair[OF set] show ?case
   1.562 -    by (rule measurable_cong[THEN iffD1])
   1.563 -qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1
   1.564 -                   nn_integral_monotone_convergence_SUP incseq_def le_fun_def
   1.565 -              cong: measurable_cong)
   1.566 -
   1.567 -lemma (in sigma_finite_measure) nn_integral_fst:
   1.568 -  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   1.569 -  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 = _")
   1.570 -using f proof induct
   1.571 -  case (cong u v)
   1.572 -  then have "?I u = ?I v"
   1.573 -    by (intro nn_integral_cong) (auto simp: space_pair_measure)
   1.574 -  with cong show ?case
   1.575 -    by (simp cong: nn_integral_cong)
   1.576 -qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add
   1.577 -                   nn_integral_monotone_convergence_SUP measurable_compose_Pair1
   1.578 -                   borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def
   1.579 -              cong: nn_integral_cong)
   1.580 -
   1.581 -lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
   1.582 -  "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"
   1.583 -  using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp
   1.584 -
   1.585 -lemma (in pair_sigma_finite) nn_integral_snd:
   1.586 -  assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   1.587 -  shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
   1.588 -proof -
   1.589 -  note measurable_pair_swap[OF f]
   1.590 -  from M1.nn_integral_fst[OF this]
   1.591 -  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))"
   1.592 -    by simp
   1.593 -  also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
   1.594 -    by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong)
   1.595 -  finally show ?thesis .
   1.596 -qed
   1.597 -
   1.598 -lemma (in pair_sigma_finite) Fubini:
   1.599 -  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   1.600 -  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)"
   1.601 -  unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..
   1.602 -
   1.603 -lemma (in pair_sigma_finite) Fubini':
   1.604 -  assumes f: "case_prod f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   1.605 -  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)"
   1.606 -  using Fubini[OF f] by simp
   1.607 -
   1.608 -subsection \<open>Products on counting spaces, densities and distributions\<close>
   1.609 -
   1.610 -lemma sigma_prod:
   1.611 -  assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"
   1.612 -  assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y"
   1.613 -  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}"
   1.614 -    (is "?P = ?S")
   1.615 -proof (rule measure_eqI)
   1.616 -  have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X"
   1.617 -    by auto
   1.618 -  let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}"
   1.619 -  have "sets ?P = sets (SUP xy:?XY. sigma (X \<times> Y) xy)"
   1.620 -    by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)
   1.621 -  also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))"
   1.622 -    by (intro Sup_sigma arg_cong[where f=sets]) auto
   1.623 -  also have "\<dots> = sets ?S"
   1.624 -  proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
   1.625 -    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)"
   1.626 -      using A B by auto
   1.627 -  next
   1.628 -    interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
   1.629 -      using A B by (intro sigma_algebra_sigma_sets) auto
   1.630 -    fix Z assume "Z \<in> \<Union>?XY"
   1.631 -    then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
   1.632 -    proof safe
   1.633 -      fix a assume "a \<in> A"
   1.634 -      from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E"
   1.635 -        by auto
   1.636 -      with \<open>a \<in> A\<close> A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"
   1.637 -        by auto
   1.638 -      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}"
   1.639 -        using \<open>a \<in> A\<close> E unfolding eq by (auto intro!: XY.countable_UN')
   1.640 -    next
   1.641 -      fix b assume "b \<in> B"
   1.642 -      from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E"
   1.643 -        by auto
   1.644 -      with \<open>b \<in> B\<close> B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"
   1.645 -        by auto
   1.646 -      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}"
   1.647 -        using \<open>b \<in> B\<close> E unfolding eq by (auto intro!: XY.countable_UN')
   1.648 -    qed
   1.649 -  next
   1.650 -    fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
   1.651 -    then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B"
   1.652 -      by auto
   1.653 -    then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)"
   1.654 -      using A B by auto
   1.655 -    interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)"
   1.656 -      by (intro sigma_algebra_sigma_sets) auto
   1.657 -    show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)"
   1.658 -      unfolding Z by (rule XY.Int) (blast intro: ab)+
   1.659 -  qed
   1.660 -  finally show "sets ?P = sets ?S" .
   1.661 -next
   1.662 -  interpret finite_measure "sigma X A" for X A
   1.663 -    proof qed (simp add: emeasure_sigma)
   1.664 -  fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A"
   1.665 -    by (simp add: emeasure_pair_measure_alt emeasure_sigma)
   1.666 -qed
   1.667 -
   1.668 -lemma sigma_sets_pair_measure_generator_finite:
   1.669 -  assumes "finite A" and "finite B"
   1.670 -  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
   1.671 -  (is "sigma_sets ?prod ?sets = _")
   1.672 -proof safe
   1.673 -  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
   1.674 -  fix x assume subset: "x \<subseteq> A \<times> B"
   1.675 -  hence "finite x" using fin by (rule finite_subset)
   1.676 -  from this subset show "x \<in> sigma_sets ?prod ?sets"
   1.677 -  proof (induct x)
   1.678 -    case empty show ?case by (rule sigma_sets.Empty)
   1.679 -  next
   1.680 -    case (insert a x)
   1.681 -    hence "{a} \<in> sigma_sets ?prod ?sets" by auto
   1.682 -    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
   1.683 -    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
   1.684 -  qed
   1.685 -next
   1.686 -  fix x a b
   1.687 -  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
   1.688 -  from sigma_sets_into_sp[OF _ this(1)] this(2)
   1.689 -  show "a \<in> A" and "b \<in> B" by auto
   1.690 -qed
   1.691 -
   1.692 -lemma borel_prod:
   1.693 -  "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
   1.694 -  (is "?P = ?B")
   1.695 -proof -
   1.696 -  have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
   1.697 -    by (rule second_countable_borel_measurable[OF open_prod_generated])
   1.698 -  also have "\<dots> = ?P"
   1.699 -    unfolding borel_def
   1.700 -    by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])
   1.701 -  finally show ?thesis ..
   1.702 -qed
   1.703 -
   1.704 -lemma pair_measure_count_space:
   1.705 -  assumes A: "finite A" and B: "finite B"
   1.706 -  shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
   1.707 -proof (rule measure_eqI)
   1.708 -  interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
   1.709 -  interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
   1.710 -  interpret P: pair_sigma_finite "count_space A" "count_space B" ..
   1.711 -  show eq: "sets ?P = sets ?C"
   1.712 -    by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
   1.713 -  fix X assume X: "X \<in> sets ?P"
   1.714 -  with eq have X_subset: "X \<subseteq> A \<times> B" by simp
   1.715 -  with A B have fin_Pair: "\<And>x. finite (Pair x -` X)"
   1.716 -    by (intro finite_subset[OF _ B]) auto
   1.717 -  have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
   1.718 -  have pos_card: "(0::ennreal) < of_nat (card (Pair x -` X)) \<longleftrightarrow> Pair x -` X \<noteq> {}" for x
   1.719 -    by (auto simp: card_eq_0_iff fin_Pair) blast
   1.720 -
   1.721 -  show "emeasure ?P X = emeasure ?C X"
   1.722 -    using X_subset A fin_Pair fin_X
   1.723 -    apply (subst B.emeasure_pair_measure_alt[OF X])
   1.724 -    apply (subst emeasure_count_space)
   1.725 -    apply (auto simp add: emeasure_count_space nn_integral_count_space
   1.726 -                          pos_card of_nat_setsum[symmetric] card_SigmaI[symmetric]
   1.727 -                simp del: of_nat_setsum card_SigmaI
   1.728 -                intro!: arg_cong[where f=card])
   1.729 -    done
   1.730 -qed
   1.731 -
   1.732 -
   1.733 -lemma emeasure_prod_count_space:
   1.734 -  assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   1.735 -  shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
   1.736 -  by (rule emeasure_measure_of[OF pair_measure_def])
   1.737 -     (auto simp: countably_additive_def positive_def suminf_indicator A
   1.738 -                 nn_integral_suminf[symmetric] dest: sets.sets_into_space)
   1.739 -
   1.740 -lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
   1.741 -proof -
   1.742 -  have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
   1.743 -    by (auto split: split_indicator)
   1.744 -  show ?thesis
   1.745 -    by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
   1.746 -qed
   1.747 -
   1.748 -lemma emeasure_count_space_prod_eq:
   1.749 -  fixes A :: "('a \<times> 'b) set"
   1.750 -  assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   1.751 -  shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
   1.752 -proof -
   1.753 -  { fix A :: "('a \<times> 'b) set" assume "countable A"
   1.754 -    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)"
   1.755 -      by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
   1.756 -    also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)"
   1.757 -      by (subst nn_integral_count_space_indicator) auto
   1.758 -    finally have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
   1.759 -      by simp }
   1.760 -  note * = this
   1.761 -
   1.762 -  show ?thesis
   1.763 -  proof cases
   1.764 -    assume "finite A" then show ?thesis
   1.765 -      by (intro * countable_finite)
   1.766 -  next
   1.767 -    assume "infinite A"
   1.768 -    then obtain C where "countable C" and "infinite C" and "C \<subseteq> A"
   1.769 -      by (auto dest: infinite_countable_subset')
   1.770 -    with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A"
   1.771 -      by (intro emeasure_mono) auto
   1.772 -    also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C"
   1.773 -      using \<open>countable C\<close> by (rule *)
   1.774 -    finally show ?thesis
   1.775 -      using \<open>infinite C\<close> \<open>infinite A\<close> by (simp add: top_unique)
   1.776 -  qed
   1.777 -qed
   1.778 -
   1.779 -lemma nn_integral_count_space_prod_eq:
   1.780 -  "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
   1.781 -    (is "nn_integral ?P f = _")
   1.782 -proof cases
   1.783 -  assume cntbl: "countable {x. f x \<noteq> 0}"
   1.784 -  have [simp]: "\<And>x. card ({x} \<inter> {x. f x \<noteq> 0}) = (indicator {x. f x \<noteq> 0} x::ennreal)"
   1.785 -    by (auto split: split_indicator)
   1.786 -  have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P"
   1.787 -    by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y])
   1.788 -       (auto intro: sets_Pair)
   1.789 -
   1.790 -  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)"
   1.791 -    by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator)
   1.792 -  also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)"
   1.793 -    by (auto intro!: nn_integral_cong split: split_indicator)
   1.794 -  also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})"
   1.795 -    by (intro nn_integral_count_space_nn_integral cntbl) auto
   1.796 -  also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})"
   1.797 -    by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair)
   1.798 -  finally show ?thesis
   1.799 -    by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
   1.800 -next
   1.801 -  { fix x assume "f x \<noteq> 0"
   1.802 -    then have "(\<exists>r\<ge>0. 0 < r \<and> f x = ennreal r) \<or> f x = \<infinity>"
   1.803 -      by (cases "f x" rule: ennreal_cases) (auto simp: less_le)
   1.804 -    then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f x"
   1.805 -      by (auto elim!: nat_approx_posE intro!: less_imp_le) }
   1.806 -  note * = this
   1.807 -
   1.808 -  assume cntbl: "uncountable {x. f x \<noteq> 0}"
   1.809 -  also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})"
   1.810 -    using * by auto
   1.811 -  finally obtain n where "infinite {x. 1/Suc n \<le> f x}"
   1.812 -    by (meson countableI_type countable_UN uncountable_infinite)
   1.813 -  then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C"
   1.814 -    by (metis infinite_countable_subset')
   1.815 -
   1.816 -  have [measurable]: "C \<in> sets ?P"
   1.817 -    using sets.countable[OF _ \<open>countable C\<close>, of ?P] by (auto simp: sets_Pair)
   1.818 -
   1.819 -  have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f"
   1.820 -    using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
   1.821 -  moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>"
   1.822 -    using \<open>infinite C\<close> by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top)
   1.823 -  moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f"
   1.824 -    using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
   1.825 -  moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>"
   1.826 -    using \<open>infinite C\<close> by (simp add: nn_integral_cmult ennreal_mult_top)
   1.827 -  ultimately show ?thesis
   1.828 -    by (simp add: top_unique)
   1.829 -qed
   1.830 -
   1.831 -lemma pair_measure_density:
   1.832 -  assumes f: "f \<in> borel_measurable M1"
   1.833 -  assumes g: "g \<in> borel_measurable M2"
   1.834 -  assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
   1.835 -  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")
   1.836 -proof (rule measure_eqI)
   1.837 -  interpret M2: sigma_finite_measure M2 by fact
   1.838 -  interpret D2: sigma_finite_measure "density M2 g" by fact
   1.839 -
   1.840 -  fix A assume A: "A \<in> sets ?L"
   1.841 -  with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) =
   1.842 -    (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)"
   1.843 -    by (intro nn_integral_cong_AE)
   1.844 -       (auto simp add: nn_integral_cmult[symmetric] ac_simps)
   1.845 -  with A f g show "emeasure ?L A = emeasure ?R A"
   1.846 -    by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density
   1.847 -                  M2.nn_integral_fst[symmetric]
   1.848 -             cong: nn_integral_cong)
   1.849 -qed simp
   1.850 -
   1.851 -lemma sigma_finite_measure_distr:
   1.852 -  assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
   1.853 -  shows "sigma_finite_measure M"
   1.854 -proof -
   1.855 -  interpret sigma_finite_measure "distr M N f" by fact
   1.856 -  from sigma_finite_countable guess A .. note A = this
   1.857 -  show ?thesis
   1.858 -  proof
   1.859 -    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>)"
   1.860 -      using A f
   1.861 -      by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"])
   1.862 -         (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)
   1.863 -  qed
   1.864 -qed
   1.865 -
   1.866 -lemma pair_measure_distr:
   1.867 -  assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
   1.868 -  assumes "sigma_finite_measure (distr N T g)"
   1.869 -  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")
   1.870 -proof (rule measure_eqI)
   1.871 -  interpret T: sigma_finite_measure "distr N T g" by fact
   1.872 -  interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+
   1.873 -
   1.874 -  fix A assume A: "A \<in> sets ?P"
   1.875 -  with f g show "emeasure ?P A = emeasure ?D A"
   1.876 -    by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr
   1.877 -                       T.emeasure_pair_measure_alt nn_integral_distr
   1.878 -             intro!: nn_integral_cong arg_cong[where f="emeasure N"])
   1.879 -qed simp
   1.880 -
   1.881 -lemma pair_measure_eqI:
   1.882 -  assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
   1.883 -  assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
   1.884 -  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)"
   1.885 -  shows "M1 \<Otimes>\<^sub>M M2 = M"
   1.886 -proof -
   1.887 -  interpret M1: sigma_finite_measure M1 by fact
   1.888 -  interpret M2: sigma_finite_measure M2 by fact
   1.889 -  interpret pair_sigma_finite M1 M2 ..
   1.890 -  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   1.891 -  let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
   1.892 -  let ?P = "M1 \<Otimes>\<^sub>M M2"
   1.893 -  show ?thesis
   1.894 -  proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
   1.895 -    show "?E \<subseteq> Pow (space ?P)"
   1.896 -      using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
   1.897 -    show "sets ?P = sigma_sets (space ?P) ?E"
   1.898 -      by (simp add: sets_pair_measure space_pair_measure)
   1.899 -    then show "sets M = sigma_sets (space ?P) ?E"
   1.900 -      using sets[symmetric] by simp
   1.901 -  next
   1.902 -    show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>"
   1.903 -      using F by (auto simp: space_pair_measure)
   1.904 -  next
   1.905 -    fix X assume "X \<in> ?E"
   1.906 -    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
   1.907 -    then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
   1.908 -       by (simp add: M2.emeasure_pair_measure_Times)
   1.909 -    also have "\<dots> = emeasure M (A \<times> B)"
   1.910 -      using A B emeasure by auto
   1.911 -    finally show "emeasure ?P X = emeasure M X"
   1.912 -      by simp
   1.913 -  qed
   1.914 -qed
   1.915 -
   1.916 -lemma sets_pair_countable:
   1.917 -  assumes "countable S1" "countable S2"
   1.918 -  assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
   1.919 -  shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
   1.920 -proof auto
   1.921 -  fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
   1.922 -  from sets.sets_into_space[OF x(1)] x(2)
   1.923 -    sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
   1.924 -  show "a \<in> S1" "b \<in> S2"
   1.925 -    by (auto simp: space_pair_measure)
   1.926 -next
   1.927 -  fix X assume X: "X \<subseteq> S1 \<times> S2"
   1.928 -  then have "countable X"
   1.929 -    by (metis countable_subset \<open>countable S1\<close> \<open>countable S2\<close> countable_SIGMA)
   1.930 -  have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto
   1.931 -  also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)"
   1.932 -    using X
   1.933 -    by (safe intro!: sets.countable_UN' \<open>countable X\<close> subsetI pair_measureI) (auto simp: M N)
   1.934 -  finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
   1.935 -qed
   1.936 -
   1.937 -lemma pair_measure_countable:
   1.938 -  assumes "countable S1" "countable S2"
   1.939 -  shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
   1.940 -proof (rule pair_measure_eqI)
   1.941 -  show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
   1.942 -    using assms by (auto intro!: sigma_finite_measure_count_space_countable)
   1.943 -  show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
   1.944 -    by (subst sets_pair_countable[OF assms]) auto
   1.945 -next
   1.946 -  fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)"
   1.947 -  then show "emeasure (count_space S1) A * emeasure (count_space S2) B =
   1.948 -    emeasure (count_space (S1 \<times> S2)) (A \<times> B)"
   1.949 -    by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
   1.950 -qed
   1.951 -
   1.952 -lemma nn_integral_fst_count_space:
   1.953 -  "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
   1.954 -  (is "?lhs = ?rhs")
   1.955 -proof(cases)
   1.956 -  assume *: "countable {xy. f xy \<noteq> 0}"
   1.957 -  let ?A = "fst ` {xy. f xy \<noteq> 0}"
   1.958 -  let ?B = "snd ` {xy. f xy \<noteq> 0}"
   1.959 -  from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+
   1.960 -  have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)"
   1.961 -    by(rule nn_integral_count_space_eq)
   1.962 -      (auto simp add: nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI)
   1.963 -  also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)"
   1.964 -    by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI)
   1.965 -  also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))"
   1.966 -    by(subst sigma_finite_measure.nn_integral_fst)
   1.967 -      (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable)
   1.968 -  also have "\<dots> = ?rhs"
   1.969 -    by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI)
   1.970 -  finally show ?thesis .
   1.971 -next
   1.972 -  { fix xy assume "f xy \<noteq> 0"
   1.973 -    then have "(\<exists>r\<ge>0. 0 < r \<and> f xy = ennreal r) \<or> f xy = \<infinity>"
   1.974 -      by (cases "f xy" rule: ennreal_cases) (auto simp: less_le)
   1.975 -    then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f xy"
   1.976 -      by (auto elim!: nat_approx_posE intro!: less_imp_le) }
   1.977 -  note * = this
   1.978 -
   1.979 -  assume cntbl: "uncountable {xy. f xy \<noteq> 0}"
   1.980 -  also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})"
   1.981 -    using * by auto
   1.982 -  finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}"
   1.983 -    by (meson countableI_type countable_UN uncountable_infinite)
   1.984 -  then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C"
   1.985 -    by (metis infinite_countable_subset')
   1.986 -
   1.987 -  have "\<infinity> = (\<integral>\<^sup>+ xy. ennreal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)"
   1.988 -    using \<open>infinite C\<close> by(simp add: nn_integral_cmult ennreal_mult_top)
   1.989 -  also have "\<dots> \<le> ?rhs" using C
   1.990 -    by(intro nn_integral_mono)(auto split: split_indicator)
   1.991 -  finally have "?rhs = \<infinity>" by (simp add: top_unique)
   1.992 -  moreover have "?lhs = \<infinity>"
   1.993 -  proof(cases "finite (fst ` C)")
   1.994 -    case True
   1.995 -    then obtain x C' where x: "x \<in> fst ` C"
   1.996 -      and C': "C' = fst -` {x} \<inter> C"
   1.997 -      and "infinite C'"
   1.998 -      using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE')
   1.999 -    from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto
  1.1000 -
  1.1001 -    from C' \<open>infinite C'\<close> have "infinite (snd ` C')"
  1.1002 -      by(auto dest!: finite_imageD simp add: inj_on_def)
  1.1003 -    then have "\<infinity> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)"
  1.1004 -      by(simp add: nn_integral_cmult ennreal_mult_top)
  1.1005 -    also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)"
  1.1006 -      by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C')
  1.1007 -    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)"
  1.1008 -      by(simp add: one_ereal_def[symmetric])
  1.1009 -    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)"
  1.1010 -      by(rule nn_integral_mono)(simp split: split_indicator)
  1.1011 -    also have "\<dots> \<le> ?lhs" using **
  1.1012 -      by(intro nn_integral_mono)(auto split: split_indicator)
  1.1013 -    finally show ?thesis by (simp add: top_unique)
  1.1014 -  next
  1.1015 -    case False
  1.1016 -    define C' where "C' = fst ` C"
  1.1017 -    have "\<infinity> = \<integral>\<^sup>+ x. ennreal (1 / Suc n) * indicator C' x \<partial>count_space UNIV"
  1.1018 -      using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top)
  1.1019 -    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"
  1.1020 -      by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong)
  1.1021 -    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"
  1.1022 -      by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI)
  1.1023 -    also have "\<dots> \<le> ?lhs" using C
  1.1024 -      by(intro nn_integral_mono)(auto split: split_indicator)
  1.1025 -    finally show ?thesis by (simp add: top_unique)
  1.1026 -  qed
  1.1027 -  ultimately show ?thesis by simp
  1.1028 -qed
  1.1029 -
  1.1030 -lemma nn_integral_snd_count_space:
  1.1031 -  "(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
  1.1032 -  (is "?lhs = ?rhs")
  1.1033 -proof -
  1.1034 -  have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)"
  1.1035 -    by(simp)
  1.1036 -  also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV"
  1.1037 -    by(rule nn_integral_fst_count_space)
  1.1038 -  also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)"
  1.1039 -    by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
  1.1040 -      (simp_all add: inj_on_def split_def)
  1.1041 -  also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto
  1.1042 -  finally show ?thesis .
  1.1043 -qed
  1.1044 -
  1.1045 -lemma measurable_pair_measure_countable1:
  1.1046 -  assumes "countable A"
  1.1047 -  and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
  1.1048 -  shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
  1.1049 -using _ _ assms(1)
  1.1050 -by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
  1.1051 -
  1.1052 -subsection \<open>Product of Borel spaces\<close>
  1.1053 -
  1.1054 -lemma borel_Times:
  1.1055 -  fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
  1.1056 -  assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
  1.1057 -  shows "A \<times> B \<in> sets borel"
  1.1058 -proof -
  1.1059 -  have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
  1.1060 -    by auto
  1.1061 -  moreover
  1.1062 -  { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)
  1.1063 -    then have "A\<times>UNIV \<in> sets borel"
  1.1064 -    proof (induct A)
  1.1065 -      case (Basic S) then show ?case
  1.1066 -        by (auto intro!: borel_open open_Times)
  1.1067 -    next
  1.1068 -      case (Compl A)
  1.1069 -      moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)"
  1.1070 -        by auto
  1.1071 -      ultimately show ?case
  1.1072 -        unfolding * by auto
  1.1073 -    next
  1.1074 -      case (Union A)
  1.1075 -      moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)"
  1.1076 -        by auto
  1.1077 -      ultimately show ?case
  1.1078 -        unfolding * by auto
  1.1079 -    qed simp }
  1.1080 -  moreover
  1.1081 -  { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)
  1.1082 -    then have "UNIV\<times>B \<in> sets borel"
  1.1083 -    proof (induct B)
  1.1084 -      case (Basic S) then show ?case
  1.1085 -        by (auto intro!: borel_open open_Times)
  1.1086 -    next
  1.1087 -      case (Compl B)
  1.1088 -      moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)"
  1.1089 -        by auto
  1.1090 -      ultimately show ?case
  1.1091 -        unfolding * by auto
  1.1092 -    next
  1.1093 -      case (Union B)
  1.1094 -      moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)"
  1.1095 -        by auto
  1.1096 -      ultimately show ?case
  1.1097 -        unfolding * by auto
  1.1098 -    qed simp }
  1.1099 -  ultimately show ?thesis
  1.1100 -    by auto
  1.1101 -qed
  1.1102 -
  1.1103 -lemma finite_measure_pair_measure:
  1.1104 -  assumes "finite_measure M" "finite_measure N"
  1.1105 -  shows "finite_measure (N  \<Otimes>\<^sub>M M)"
  1.1106 -proof (rule finite_measureI)
  1.1107 -  interpret M: finite_measure M by fact
  1.1108 -  interpret N: finite_measure N by fact
  1.1109 -  show "emeasure (N  \<Otimes>\<^sub>M M) (space (N  \<Otimes>\<^sub>M M)) \<noteq> \<infinity>"
  1.1110 -    by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
  1.1111 -qed
  1.1112 -
  1.1113 -end