src/HOL/Analysis/Binary_Product_Measure.thy
changeset 68833 fde093888c16
parent 67693 4fa9d5ef95bc
child 69260 0a9688695a1b
     1.1 --- a/src/HOL/Analysis/Binary_Product_Measure.thy	Mon Aug 27 22:58:36 2018 +0200
     1.2 +++ b/src/HOL/Analysis/Binary_Product_Measure.thy	Tue Aug 28 13:28:39 2018 +0100
     1.3 @@ -2,67 +2,67 @@
     1.4      Author:     Johannes Hölzl, TU München
     1.5  *)
     1.6  
     1.7 -section \<open>Binary product measures\<close>
     1.8 +section%important \<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 +lemma%unimportant Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
    1.16    by auto
    1.17  
    1.18 -lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
    1.19 +lemma%unimportant rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
    1.20    by auto
    1.21  
    1.22 -subsection "Binary products"
    1.23 +subsection%important "Binary products"
    1.24  
    1.25 -definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
    1.26 +definition%important pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
    1.27    "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B)
    1.28        {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
    1.29        (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
    1.30  
    1.31 -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)"
    1.32 -  using sets.space_closed[of A] sets.space_closed[of B] by auto
    1.33 +lemma%important pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"
    1.34 +  using%unimportant sets.space_closed[of A] sets.space_closed[of B] by auto
    1.35  
    1.36 -lemma space_pair_measure:
    1.37 +lemma%important space_pair_measure:
    1.38    "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
    1.39    unfolding pair_measure_def using pair_measure_closed[of A B]
    1.40 -  by (rule space_measure_of)
    1.41 +  by%unimportant (rule space_measure_of)
    1.42  
    1.43 -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)}"
    1.44 +lemma%unimportant 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)}"
    1.45    by (auto simp: space_pair_measure)
    1.46  
    1.47 -lemma sets_pair_measure:
    1.48 +lemma%unimportant sets_pair_measure:
    1.49    "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}"
    1.50    unfolding pair_measure_def using pair_measure_closed[of A B]
    1.51    by (rule sets_measure_of)
    1.52  
    1.53 -lemma sets_pair_measure_cong[measurable_cong, cong]:
    1.54 +lemma%unimportant sets_pair_measure_cong[measurable_cong, cong]:
    1.55    "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
    1.56    unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
    1.57  
    1.58 -lemma pair_measureI[intro, simp, measurable]:
    1.59 +lemma%unimportant pair_measureI[intro, simp, measurable]:
    1.60    "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
    1.61    by (auto simp: sets_pair_measure)
    1.62  
    1.63 -lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
    1.64 +lemma%unimportant sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
    1.65    using pair_measureI[of "{x}" M1 "{y}" M2] by simp
    1.66  
    1.67 -lemma measurable_pair_measureI:
    1.68 +lemma%unimportant measurable_pair_measureI:
    1.69    assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
    1.70    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"
    1.71    shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
    1.72    unfolding pair_measure_def using 1 2
    1.73    by (intro measurable_measure_of) (auto dest: sets.sets_into_space)
    1.74  
    1.75 -lemma measurable_split_replace[measurable (raw)]:
    1.76 +lemma%unimportant measurable_split_replace[measurable (raw)]:
    1.77    "(\<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"
    1.78    unfolding split_beta' .
    1.79  
    1.80 -lemma measurable_Pair[measurable (raw)]:
    1.81 +lemma%important measurable_Pair[measurable (raw)]:
    1.82    assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
    1.83    shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
    1.84 -proof (rule measurable_pair_measureI)
    1.85 +proof%unimportant (rule measurable_pair_measureI)
    1.86    show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
    1.87      using f g by (auto simp: measurable_def)
    1.88    fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
    1.89 @@ -73,59 +73,59 @@
    1.90    finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .
    1.91  qed
    1.92  
    1.93 -lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
    1.94 +lemma%unimportant measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
    1.95    by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
    1.96      measurable_def)
    1.97  
    1.98 -lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
    1.99 +lemma%unimportant measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
   1.100    by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
   1.101      measurable_def)
   1.102  
   1.103 -lemma measurable_Pair_compose_split[measurable_dest]:
   1.104 +lemma%unimportant measurable_Pair_compose_split[measurable_dest]:
   1.105    assumes f: "case_prod f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N"
   1.106    assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"
   1.107    shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"
   1.108    using measurable_compose[OF measurable_Pair f, OF g h] by simp
   1.109  
   1.110 -lemma measurable_Pair1_compose[measurable_dest]:
   1.111 +lemma%unimportant measurable_Pair1_compose[measurable_dest]:
   1.112    assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   1.113    assumes [measurable]: "h \<in> measurable N M"
   1.114    shows "(\<lambda>x. f (h x)) \<in> measurable N M1"
   1.115    using measurable_compose[OF f measurable_fst] by simp
   1.116  
   1.117 -lemma measurable_Pair2_compose[measurable_dest]:
   1.118 +lemma%unimportant measurable_Pair2_compose[measurable_dest]:
   1.119    assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   1.120    assumes [measurable]: "h \<in> measurable N M"
   1.121    shows "(\<lambda>x. g (h x)) \<in> measurable N M2"
   1.122    using measurable_compose[OF f measurable_snd] by simp
   1.123  
   1.124 -lemma measurable_pair:
   1.125 +lemma%unimportant measurable_pair:
   1.126    assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
   1.127    shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   1.128    using measurable_Pair[OF assms] by simp
   1.129  
   1.130 -lemma
   1.131 +lemma%unimportant (*FIX ME needs a name *)
   1.132    assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)"
   1.133    shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"
   1.134      and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"
   1.135    by simp_all
   1.136  
   1.137 -lemma
   1.138 +lemma%unimportant (*FIX ME needs a name *)
   1.139    assumes f[measurable]: "f \<in> measurable M N"
   1.140    shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N"
   1.141      and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N"
   1.142    by simp_all
   1.143  
   1.144 -lemma sets_pair_in_sets:
   1.145 +lemma%unimportant sets_pair_in_sets:
   1.146    assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"
   1.147    shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
   1.148    unfolding sets_pair_measure
   1.149    by (intro sets.sigma_sets_subset') (auto intro!: assms)
   1.150  
   1.151 -lemma sets_pair_eq_sets_fst_snd:
   1.152 +lemma%important  sets_pair_eq_sets_fst_snd:
   1.153    "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})"
   1.154      (is "?P = sets (Sup {?fst, ?snd})")
   1.155 -proof -
   1.156 +proof%unimportant -
   1.157    { fix a b assume ab: "a \<in> sets A" "b \<in> sets B"
   1.158      then have "a \<times> b = (fst -` a \<inter> (space A \<times> space B)) \<inter> (snd -` b \<inter> (space A \<times> space B))"
   1.159        by (auto dest: sets.sets_into_space)
   1.160 @@ -157,29 +157,29 @@
   1.161      done
   1.162  qed
   1.163  
   1.164 -lemma measurable_pair_iff:
   1.165 +lemma%unimportant measurable_pair_iff:
   1.166    "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"
   1.167    by (auto intro: measurable_pair[of f M M1 M2])
   1.168  
   1.169 -lemma measurable_split_conv:
   1.170 +lemma%unimportant  measurable_split_conv:
   1.171    "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
   1.172    by (intro arg_cong2[where f="(\<in>)"]) auto
   1.173  
   1.174 -lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
   1.175 +lemma%unimportant measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
   1.176    by (auto intro!: measurable_Pair simp: measurable_split_conv)
   1.177  
   1.178 -lemma measurable_pair_swap:
   1.179 +lemma%unimportant  measurable_pair_swap:
   1.180    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"
   1.181    using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
   1.182  
   1.183 -lemma measurable_pair_swap_iff:
   1.184 +lemma%unimportant measurable_pair_swap_iff:
   1.185    "f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M"
   1.186    by (auto dest: measurable_pair_swap)
   1.187  
   1.188 -lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
   1.189 +lemma%unimportant measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
   1.190    by simp
   1.191  
   1.192 -lemma sets_Pair1[measurable (raw)]:
   1.193 +lemma%unimportant sets_Pair1[measurable (raw)]:
   1.194    assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2"
   1.195  proof -
   1.196    have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
   1.197 @@ -189,10 +189,10 @@
   1.198    finally show ?thesis .
   1.199  qed
   1.200  
   1.201 -lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
   1.202 +lemma%unimportant measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
   1.203    by (auto intro!: measurable_Pair)
   1.204  
   1.205 -lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
   1.206 +lemma%unimportant sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
   1.207  proof -
   1.208    have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
   1.209      using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   1.210 @@ -201,23 +201,23 @@
   1.211    finally show ?thesis .
   1.212  qed
   1.213  
   1.214 -lemma measurable_Pair2:
   1.215 +lemma%unimportant measurable_Pair2:
   1.216    assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1"
   1.217    shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
   1.218    using measurable_comp[OF measurable_Pair1' f, OF x]
   1.219    by (simp add: comp_def)
   1.220  
   1.221 -lemma measurable_Pair1:
   1.222 +lemma%unimportant measurable_Pair1:
   1.223    assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2"
   1.224    shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
   1.225    using measurable_comp[OF measurable_Pair2' f, OF y]
   1.226    by (simp add: comp_def)
   1.227  
   1.228 -lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
   1.229 +lemma%unimportant Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
   1.230    unfolding Int_stable_def
   1.231    by safe (auto simp add: times_Int_times)
   1.232  
   1.233 -lemma (in finite_measure) finite_measure_cut_measurable:
   1.234 +lemma%unimportant (in finite_measure) finite_measure_cut_measurable:
   1.235    assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)"
   1.236    shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
   1.237      (is "?s Q \<in> _")
   1.238 @@ -239,7 +239,7 @@
   1.239      unfolding sets_pair_measure[symmetric] by simp
   1.240  qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
   1.241  
   1.242 -lemma (in sigma_finite_measure) measurable_emeasure_Pair:
   1.243 +lemma%unimportant (in sigma_finite_measure) measurable_emeasure_Pair:
   1.244    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> _")
   1.245  proof -
   1.246    from sigma_finite_disjoint guess F . note F = this
   1.247 @@ -279,7 +279,7 @@
   1.248      by auto
   1.249  qed
   1.250  
   1.251 -lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
   1.252 +lemma%unimportant (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
   1.253    assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
   1.254    assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)"
   1.255    shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
   1.256 @@ -290,7 +290,7 @@
   1.257      by (auto cong: measurable_cong)
   1.258  qed
   1.259  
   1.260 -lemma (in sigma_finite_measure) emeasure_pair_measure:
   1.261 +lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure:
   1.262    assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
   1.263    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")
   1.264  proof (rule emeasure_measure_of[OF pair_measure_def])
   1.265 @@ -314,7 +314,7 @@
   1.266      using sets.space_closed[of N] sets.space_closed[of M] by auto
   1.267  qed fact
   1.268  
   1.269 -lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
   1.270 +lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure_alt:
   1.271    assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
   1.272    shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
   1.273  proof -
   1.274 @@ -324,10 +324,10 @@
   1.275      using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
   1.276  qed
   1.277  
   1.278 -lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
   1.279 +lemma%important (in sigma_finite_measure) emeasure_pair_measure_Times:
   1.280    assumes A: "A \<in> sets N" and B: "B \<in> sets M"
   1.281    shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
   1.282 -proof -
   1.283 +proof%unimportant -
   1.284    have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
   1.285      using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
   1.286    also have "\<dots> = emeasure M B * emeasure N A"
   1.287 @@ -336,18 +336,18 @@
   1.288      by (simp add: ac_simps)
   1.289  qed
   1.290  
   1.291 -subsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
   1.292 +subsection%important \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
   1.293  
   1.294 -locale pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
   1.295 +locale%important pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
   1.296    for M1 :: "'a measure" and M2 :: "'b measure"
   1.297  
   1.298 -lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
   1.299 +lemma%unimportant (in pair_sigma_finite) measurable_emeasure_Pair1:
   1.300    "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
   1.301    using M2.measurable_emeasure_Pair .
   1.302  
   1.303 -lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
   1.304 +lemma%important (in pair_sigma_finite) measurable_emeasure_Pair2:
   1.305    assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
   1.306 -proof -
   1.307 +proof%unimportant -
   1.308    have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   1.309      using Q measurable_pair_swap' by (auto intro: measurable_sets)
   1.310    note M1.measurable_emeasure_Pair[OF this]
   1.311 @@ -356,11 +356,11 @@
   1.312    ultimately show ?thesis by simp
   1.313  qed
   1.314  
   1.315 -lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
   1.316 +lemma%important (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
   1.317    defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
   1.318    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>
   1.319      (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
   1.320 -proof -
   1.321 +proof%unimportant -
   1.322    from M1.sigma_finite_incseq guess F1 . note F1 = this
   1.323    from M2.sigma_finite_incseq guess F2 . note F2 = this
   1.324    from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
   1.325 @@ -394,7 +394,7 @@
   1.326    qed
   1.327  qed
   1.328  
   1.329 -sublocale pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
   1.330 +sublocale%important pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
   1.331  proof
   1.332    from M1.sigma_finite_countable guess F1 ..
   1.333    moreover from M2.sigma_finite_countable guess F2 ..
   1.334 @@ -404,7 +404,7 @@
   1.335         (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
   1.336  qed
   1.337  
   1.338 -lemma sigma_finite_pair_measure:
   1.339 +lemma%unimportant sigma_finite_pair_measure:
   1.340    assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
   1.341    shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
   1.342  proof -
   1.343 @@ -414,14 +414,14 @@
   1.344    show ?thesis ..
   1.345  qed
   1.346  
   1.347 -lemma sets_pair_swap:
   1.348 +lemma%unimportant sets_pair_swap:
   1.349    assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.350    shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   1.351    using measurable_pair_swap' assms by (rule measurable_sets)
   1.352  
   1.353 -lemma (in pair_sigma_finite) distr_pair_swap:
   1.354 +lemma%important (in pair_sigma_finite) distr_pair_swap:
   1.355    "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
   1.356 -proof -
   1.357 +proof%unimportant -
   1.358    from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   1.359    let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
   1.360    show ?thesis
   1.361 @@ -446,11 +446,11 @@
   1.362    qed
   1.363  qed
   1.364  
   1.365 -lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
   1.366 +lemma%unimportant (in pair_sigma_finite) emeasure_pair_measure_alt2:
   1.367    assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.368    shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
   1.369      (is "_ = ?\<nu> A")
   1.370 -proof -
   1.371 +proof%unimportant -
   1.372    have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A"
   1.373      using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
   1.374    show ?thesis using A
   1.375 @@ -459,7 +459,7 @@
   1.376                   M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
   1.377  qed
   1.378  
   1.379 -lemma (in pair_sigma_finite) AE_pair:
   1.380 +lemma%unimportant (in pair_sigma_finite) AE_pair:
   1.381    assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
   1.382    shows "AE x in M1. (AE y in M2. Q (x, y))"
   1.383  proof -
   1.384 @@ -485,11 +485,11 @@
   1.385    qed
   1.386  qed
   1.387  
   1.388 -lemma (in pair_sigma_finite) AE_pair_measure:
   1.389 +lemma%important (in pair_sigma_finite) AE_pair_measure:
   1.390    assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.391    assumes ae: "AE x in M1. AE y in M2. P (x, y)"
   1.392    shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
   1.393 -proof (subst AE_iff_measurable[OF _ refl])
   1.394 +proof%unimportant (subst AE_iff_measurable[OF _ refl])
   1.395    show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.396      by (rule sets.sets_Collect) fact
   1.397    then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
   1.398 @@ -505,12 +505,12 @@
   1.399    finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
   1.400  qed
   1.401  
   1.402 -lemma (in pair_sigma_finite) AE_pair_iff:
   1.403 +lemma%unimportant (in pair_sigma_finite) AE_pair_iff:
   1.404    "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
   1.405      (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))"
   1.406    using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
   1.407  
   1.408 -lemma (in pair_sigma_finite) AE_commute:
   1.409 +lemma%unimportant (in pair_sigma_finite) AE_commute:
   1.410    assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   1.411    shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
   1.412  proof -
   1.413 @@ -531,16 +531,16 @@
   1.414      done
   1.415  qed
   1.416  
   1.417 -subsection "Fubinis theorem"
   1.418 +subsection%important "Fubinis theorem"
   1.419  
   1.420 -lemma measurable_compose_Pair1:
   1.421 +lemma%unimportant measurable_compose_Pair1:
   1.422    "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
   1.423    by simp
   1.424  
   1.425 -lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst:
   1.426 +lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral_fst:
   1.427    assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   1.428    shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
   1.429 -using f proof induct
   1.430 +using f proof%unimportant induct
   1.431    case (cong u v)
   1.432    then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
   1.433      by (auto simp: space_pair_measure)
   1.434 @@ -561,10 +561,10 @@
   1.435                     nn_integral_monotone_convergence_SUP incseq_def le_fun_def
   1.436                cong: measurable_cong)
   1.437  
   1.438 -lemma (in sigma_finite_measure) nn_integral_fst:
   1.439 +lemma%unimportant (in sigma_finite_measure) nn_integral_fst:
   1.440    assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   1.441    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 = _")
   1.442 -using f proof induct
   1.443 +  using f proof induct
   1.444    case (cong u v)
   1.445    then have "?I u = ?I v"
   1.446      by (intro nn_integral_cong) (auto simp: space_pair_measure)
   1.447 @@ -575,14 +575,14 @@
   1.448                     borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def
   1.449                cong: nn_integral_cong)
   1.450  
   1.451 -lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
   1.452 +lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
   1.453    "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"
   1.454    using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp
   1.455  
   1.456 -lemma (in pair_sigma_finite) nn_integral_snd:
   1.457 +lemma%important (in pair_sigma_finite) nn_integral_snd:
   1.458    assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   1.459    shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
   1.460 -proof -
   1.461 +proof%unimportant -
   1.462    note measurable_pair_swap[OF f]
   1.463    from M1.nn_integral_fst[OF this]
   1.464    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))"
   1.465 @@ -592,24 +592,24 @@
   1.466    finally show ?thesis .
   1.467  qed
   1.468  
   1.469 -lemma (in pair_sigma_finite) Fubini:
   1.470 +lemma%important (in pair_sigma_finite) Fubini:
   1.471    assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   1.472    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)"
   1.473    unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..
   1.474  
   1.475 -lemma (in pair_sigma_finite) Fubini':
   1.476 +lemma%important (in pair_sigma_finite) Fubini':
   1.477    assumes f: "case_prod f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   1.478    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)"
   1.479    using Fubini[OF f] by simp
   1.480  
   1.481 -subsection \<open>Products on counting spaces, densities and distributions\<close>
   1.482 +subsection%important \<open>Products on counting spaces, densities and distributions\<close>
   1.483  
   1.484 -lemma sigma_prod:
   1.485 +lemma%important sigma_prod:
   1.486    assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"
   1.487    assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y"
   1.488    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}"
   1.489      (is "?P = ?S")
   1.490 -proof (rule measure_eqI)
   1.491 +proof%unimportant (rule measure_eqI)
   1.492    have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X"
   1.493      by auto
   1.494    let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}"
   1.495 @@ -662,7 +662,7 @@
   1.496      by (simp add: emeasure_pair_measure_alt emeasure_sigma)
   1.497  qed
   1.498  
   1.499 -lemma sigma_sets_pair_measure_generator_finite:
   1.500 +lemma%unimportant sigma_sets_pair_measure_generator_finite:
   1.501    assumes "finite A" and "finite B"
   1.502    shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
   1.503    (is "sigma_sets ?prod ?sets = _")
   1.504 @@ -686,14 +686,14 @@
   1.505    show "a \<in> A" and "b \<in> B" by auto
   1.506  qed
   1.507  
   1.508 -lemma sets_pair_eq:
   1.509 +lemma%important sets_pair_eq:
   1.510    assumes Ea: "Ea \<subseteq> Pow (space A)" "sets A = sigma_sets (space A) Ea"
   1.511      and Ca: "countable Ca" "Ca \<subseteq> Ea" "\<Union>Ca = space A"
   1.512      and Eb: "Eb \<subseteq> Pow (space B)" "sets B = sigma_sets (space B) Eb"
   1.513      and Cb: "countable Cb" "Cb \<subseteq> Eb" "\<Union>Cb = space B"
   1.514    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 })"
   1.515      (is "_ = sets (sigma ?\<Omega> ?E)")
   1.516 -proof
   1.517 +proof%unimportant
   1.518    show "sets (sigma ?\<Omega> ?E) \<subseteq> sets (A \<Otimes>\<^sub>M B)"
   1.519      using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
   1.520    have "?E \<subseteq> Pow ?\<Omega>"
   1.521 @@ -733,10 +733,10 @@
   1.522    finally show "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (sigma ?\<Omega> ?E)" .
   1.523  qed
   1.524  
   1.525 -lemma borel_prod:
   1.526 +lemma%important borel_prod:
   1.527    "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
   1.528    (is "?P = ?B")
   1.529 -proof -
   1.530 +proof%unimportant -
   1.531    have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
   1.532      by (rule second_countable_borel_measurable[OF open_prod_generated])
   1.533    also have "\<dots> = ?P"
   1.534 @@ -745,10 +745,10 @@
   1.535    finally show ?thesis ..
   1.536  qed
   1.537  
   1.538 -lemma pair_measure_count_space:
   1.539 +lemma%important pair_measure_count_space:
   1.540    assumes A: "finite A" and B: "finite B"
   1.541    shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
   1.542 -proof (rule measure_eqI)
   1.543 +proof%unimportant (rule measure_eqI)
   1.544    interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
   1.545    interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
   1.546    interpret P: pair_sigma_finite "count_space A" "count_space B" ..
   1.547 @@ -776,14 +776,14 @@
   1.548  qed
   1.549  
   1.550  
   1.551 -lemma emeasure_prod_count_space:
   1.552 +lemma%unimportant emeasure_prod_count_space:
   1.553    assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   1.554    shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
   1.555    by (rule emeasure_measure_of[OF pair_measure_def])
   1.556       (auto simp: countably_additive_def positive_def suminf_indicator A
   1.557                   nn_integral_suminf[symmetric] dest: sets.sets_into_space)
   1.558  
   1.559 -lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
   1.560 +lemma%unimportant emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
   1.561  proof -
   1.562    have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
   1.563      by (auto split: split_indicator)
   1.564 @@ -791,11 +791,11 @@
   1.565      by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
   1.566  qed
   1.567  
   1.568 -lemma emeasure_count_space_prod_eq:
   1.569 +lemma%important emeasure_count_space_prod_eq:
   1.570    fixes A :: "('a \<times> 'b) set"
   1.571    assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   1.572    shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
   1.573 -proof -
   1.574 +proof%unimportant -
   1.575    { fix A :: "('a \<times> 'b) set" assume "countable A"
   1.576      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)"
   1.577        by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
   1.578 @@ -822,7 +822,7 @@
   1.579    qed
   1.580  qed
   1.581  
   1.582 -lemma nn_integral_count_space_prod_eq:
   1.583 +lemma%unimportant nn_integral_count_space_prod_eq:
   1.584    "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
   1.585      (is "nn_integral ?P f = _")
   1.586  proof cases
   1.587 @@ -874,12 +874,12 @@
   1.588      by (simp add: top_unique)
   1.589  qed
   1.590  
   1.591 -lemma pair_measure_density:
   1.592 +lemma%important pair_measure_density:
   1.593    assumes f: "f \<in> borel_measurable M1"
   1.594    assumes g: "g \<in> borel_measurable M2"
   1.595    assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
   1.596    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")
   1.597 -proof (rule measure_eqI)
   1.598 +proof%unimportant (rule measure_eqI)
   1.599    interpret M2: sigma_finite_measure M2 by fact
   1.600    interpret D2: sigma_finite_measure "density M2 g" by fact
   1.601  
   1.602 @@ -894,7 +894,7 @@
   1.603               cong: nn_integral_cong)
   1.604  qed simp
   1.605  
   1.606 -lemma sigma_finite_measure_distr:
   1.607 +lemma%unimportant sigma_finite_measure_distr:
   1.608    assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
   1.609    shows "sigma_finite_measure M"
   1.610  proof -
   1.611 @@ -909,7 +909,7 @@
   1.612    qed
   1.613  qed
   1.614  
   1.615 -lemma pair_measure_distr:
   1.616 +lemma%unimportant pair_measure_distr:
   1.617    assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
   1.618    assumes "sigma_finite_measure (distr N T g)"
   1.619    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")
   1.620 @@ -924,12 +924,12 @@
   1.621               intro!: nn_integral_cong arg_cong[where f="emeasure N"])
   1.622  qed simp
   1.623  
   1.624 -lemma pair_measure_eqI:
   1.625 +lemma%important pair_measure_eqI:
   1.626    assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
   1.627    assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
   1.628    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)"
   1.629    shows "M1 \<Otimes>\<^sub>M M2 = M"
   1.630 -proof -
   1.631 +proof%unimportant -
   1.632    interpret M1: sigma_finite_measure M1 by fact
   1.633    interpret M2: sigma_finite_measure M2 by fact
   1.634    interpret pair_sigma_finite M1 M2 ..
   1.635 @@ -959,11 +959,11 @@
   1.636    qed
   1.637  qed
   1.638  
   1.639 -lemma sets_pair_countable:
   1.640 +lemma%important sets_pair_countable:
   1.641    assumes "countable S1" "countable S2"
   1.642    assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
   1.643    shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
   1.644 -proof auto
   1.645 +proof%unimportant auto
   1.646    fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
   1.647    from sets.sets_into_space[OF x(1)] x(2)
   1.648      sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
   1.649 @@ -980,10 +980,10 @@
   1.650    finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
   1.651  qed
   1.652  
   1.653 -lemma pair_measure_countable:
   1.654 +lemma%important pair_measure_countable:
   1.655    assumes "countable S1" "countable S2"
   1.656    shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
   1.657 -proof (rule pair_measure_eqI)
   1.658 +proof%unimportant (rule pair_measure_eqI)
   1.659    show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
   1.660      using assms by (auto intro!: sigma_finite_measure_count_space_countable)
   1.661    show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
   1.662 @@ -995,10 +995,10 @@
   1.663      by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
   1.664  qed
   1.665  
   1.666 -lemma nn_integral_fst_count_space:
   1.667 +lemma%important nn_integral_fst_count_space:
   1.668    "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
   1.669    (is "?lhs = ?rhs")
   1.670 -proof(cases)
   1.671 +proof%unimportant(cases)
   1.672    assume *: "countable {xy. f xy \<noteq> 0}"
   1.673    let ?A = "fst ` {xy. f xy \<noteq> 0}"
   1.674    let ?B = "snd ` {xy. f xy \<noteq> 0}"
   1.675 @@ -1088,20 +1088,20 @@
   1.676    finally show ?thesis .
   1.677  qed
   1.678  
   1.679 -lemma measurable_pair_measure_countable1:
   1.680 +lemma%unimportant measurable_pair_measure_countable1:
   1.681    assumes "countable A"
   1.682    and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
   1.683    shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
   1.684  using _ _ assms(1)
   1.685  by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
   1.686  
   1.687 -subsection \<open>Product of Borel spaces\<close>
   1.688 +subsection%important \<open>Product of Borel spaces\<close>
   1.689  
   1.690 -lemma borel_Times:
   1.691 +lemma%important borel_Times:
   1.692    fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
   1.693    assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
   1.694    shows "A \<times> B \<in> sets borel"
   1.695 -proof -
   1.696 +proof%unimportant -
   1.697    have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
   1.698      by auto
   1.699    moreover
   1.700 @@ -1146,7 +1146,7 @@
   1.701      by auto
   1.702  qed
   1.703  
   1.704 -lemma finite_measure_pair_measure:
   1.705 +lemma%unimportant finite_measure_pair_measure:
   1.706    assumes "finite_measure M" "finite_measure N"
   1.707    shows "finite_measure (N  \<Otimes>\<^sub>M M)"
   1.708  proof (rule finite_measureI)