src/HOL/Analysis/Binary_Product_Measure.thy

 (*  Title:      HOL/Analysis/Binary_Product_Measure.thy
     Author:     Johannes Hölzl, TU München
 *)
 
-section \<open>Binary product measures\<close>
+section%important \<open>Binary product measures\<close>
 
 theory Binary_Product_Measure
 imports Nonnegative_Lebesgue_Integration
 begin
 
-lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
+lemma%unimportant Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
   by auto
 
-lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
+lemma%unimportant rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
   by auto
 
-subsection "Binary products"
-
-definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
+subsection%important "Binary products"
+
+definition%important pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
   "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B)
       {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
       (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
 
-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)"
-  using sets.space_closed[of A] sets.space_closed[of B] by auto
-
-lemma space_pair_measure:
+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)"
+  using%unimportant sets.space_closed[of A] sets.space_closed[of B] by auto
+
+lemma%important space_pair_measure:
   "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
   unfolding pair_measure_def using pair_measure_closed[of A B]
-  by (rule space_measure_of)
-
-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)}"
+  by%unimportant (rule space_measure_of)
+
+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)}"
   by (auto simp: space_pair_measure)
 
-lemma sets_pair_measure:
+lemma%unimportant sets_pair_measure:
   "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}"
   unfolding pair_measure_def using pair_measure_closed[of A B]
   by (rule sets_measure_of)
 
-lemma sets_pair_measure_cong[measurable_cong, cong]:
+lemma%unimportant sets_pair_measure_cong[measurable_cong, cong]:
   "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
 
-lemma pair_measureI[intro, simp, measurable]:
+lemma%unimportant pair_measureI[intro, simp, measurable]:
   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
   by (auto simp: sets_pair_measure)
 
-lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
+lemma%unimportant sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   using pair_measureI[of "{x}" M1 "{y}" M2] by simp
 
-lemma measurable_pair_measureI:
+lemma%unimportant measurable_pair_measureI:
   assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
   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"
   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   unfolding pair_measure_def using 1 2
   by (intro measurable_measure_of) (auto dest: sets.sets_into_space)
 
-lemma measurable_split_replace[measurable (raw)]:
+lemma%unimportant measurable_split_replace[measurable (raw)]:
   "(\<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"
   unfolding split_beta' .
 
-lemma measurable_Pair[measurable (raw)]:
+lemma%important measurable_Pair[measurable (raw)]:
   assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
   shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
-proof (rule measurable_pair_measureI)
+proof%unimportant (rule measurable_pair_measureI)
   show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
     using f g by (auto simp: measurable_def)
   fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
   have "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
     by auto
   also have "\<dots> \<in> sets M"
     by (rule sets.Int) (auto intro!: measurable_sets * f g)
   finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .
 qed
 
-lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
+lemma%unimportant measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
   by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
     measurable_def)
 
-lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
+lemma%unimportant measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
   by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
     measurable_def)
 
-lemma measurable_Pair_compose_split[measurable_dest]:
+lemma%unimportant measurable_Pair_compose_split[measurable_dest]:
   assumes f: "case_prod f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N"
   assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"
   shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"
   using measurable_compose[OF measurable_Pair f, OF g h] by simp
 
-lemma measurable_Pair1_compose[measurable_dest]:
+lemma%unimportant measurable_Pair1_compose[measurable_dest]:
   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   assumes [measurable]: "h \<in> measurable N M"
   shows "(\<lambda>x. f (h x)) \<in> measurable N M1"
   using measurable_compose[OF f measurable_fst] by simp
 
-lemma measurable_Pair2_compose[measurable_dest]:
+lemma%unimportant measurable_Pair2_compose[measurable_dest]:
   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   assumes [measurable]: "h \<in> measurable N M"
   shows "(\<lambda>x. g (h x)) \<in> measurable N M2"
   using measurable_compose[OF f measurable_snd] by simp
 
-lemma measurable_pair:
+lemma%unimportant measurable_pair:
   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   using measurable_Pair[OF assms] by simp
 
-lemma
+lemma%unimportant (*FIX ME needs a name *)
   assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)"
   shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"
     and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"
   by simp_all
 
-lemma
+lemma%unimportant (*FIX ME needs a name *)
   assumes f[measurable]: "f \<in> measurable M N"
   shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N"
     and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N"
   by simp_all
 
-lemma sets_pair_in_sets:
+lemma%unimportant sets_pair_in_sets:
   assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"
   shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
   unfolding sets_pair_measure
   by (intro sets.sigma_sets_subset') (auto intro!: assms)
 
-lemma sets_pair_eq_sets_fst_snd:
+lemma%important  sets_pair_eq_sets_fst_snd:
   "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})"
     (is "?P = sets (Sup {?fst, ?snd})")
-proof -
+proof%unimportant -
   { fix a b assume ab: "a \<in> sets A" "b \<in> sets B"
     then have "a \<times> b = (fst -` a \<inter> (space A \<times> space B)) \<inter> (snd -` b \<inter> (space A \<times> space B))"
       by (auto dest: sets.sets_into_space)
     also have "\<dots> \<in> sets (Sup {?fst, ?snd})"
       apply (rule sets.Int)

     apply (simp add: space_pair_measure)
     apply (auto simp add: space_pair_measure)
     done
 qed
 
-lemma measurable_pair_iff:
+lemma%unimportant measurable_pair_iff:
   "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"
   by (auto intro: measurable_pair[of f M M1 M2])
 
-lemma measurable_split_conv:
+lemma%unimportant  measurable_split_conv:
   "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
   by (intro arg_cong2[where f="(\<in>)"]) auto
 
-lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
+lemma%unimportant measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
   by (auto intro!: measurable_Pair simp: measurable_split_conv)
 
-lemma measurable_pair_swap:
+lemma%unimportant  measurable_pair_swap:
   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"
   using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
 
-lemma measurable_pair_swap_iff:
+lemma%unimportant measurable_pair_swap_iff:
   "f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M"
   by (auto dest: measurable_pair_swap)
 
-lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
+lemma%unimportant measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
   by simp
 
-lemma sets_Pair1[measurable (raw)]:
+lemma%unimportant sets_Pair1[measurable (raw)]:
   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2"
 proof -
   have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   also have "\<dots> \<in> sets M2"
     using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: if_split_asm)
   finally show ?thesis .
 qed
 
-lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
+lemma%unimportant measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
   by (auto intro!: measurable_Pair)
 
-lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
+lemma%unimportant sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
 proof -
   have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   also have "\<dots> \<in> sets M1"
     using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: if_split_asm)
   finally show ?thesis .
 qed
 
-lemma measurable_Pair2:
+lemma%unimportant measurable_Pair2:
   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1"
   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
   using measurable_comp[OF measurable_Pair1' f, OF x]
   by (simp add: comp_def)
 
-lemma measurable_Pair1:
+lemma%unimportant measurable_Pair1:
   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2"
   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
   using measurable_comp[OF measurable_Pair2' f, OF y]
   by (simp add: comp_def)
 
-lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
+lemma%unimportant Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
   unfolding Int_stable_def
   by safe (auto simp add: times_Int_times)
 
-lemma (in finite_measure) finite_measure_cut_measurable:
+lemma%unimportant (in finite_measure) finite_measure_cut_measurable:
   assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
     (is "?s Q \<in> _")
   using Int_stable_pair_measure_generator pair_measure_closed assms
   unfolding sets_pair_measure

     by (simp add: suminf_emeasure disjoint_family_on_vimageI subset_eq vimage_UN sets_pair_measure[symmetric])
   with union show ?case
     unfolding sets_pair_measure[symmetric] by simp
 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
 
-lemma (in sigma_finite_measure) measurable_emeasure_Pair:
+lemma%unimportant (in sigma_finite_measure) measurable_emeasure_Pair:
   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> _")
 proof -
   from sigma_finite_disjoint guess F . note F = this
   then have F_sets: "\<And>i. F i \<in> sets M" by auto
   let ?C = "\<lambda>x i. F i \<inter> Pair x -` Q"

       by simp }
   ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets
     by auto
 qed
 
-lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
+lemma%unimportant (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
   assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
   assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
 proof -
   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"
     by (auto simp: space_pair_measure)
   with measurable_emeasure_Pair[OF A] show ?thesis
     by (auto cong: measurable_cong)
 qed
 
-lemma (in sigma_finite_measure) emeasure_pair_measure:
+lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure:
   assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
   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")
 proof (rule emeasure_measure_of[OF pair_measure_def])
   show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>"
     by (auto simp: positive_def)

   qed
   show "{a \<times> b |a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)"
     using sets.space_closed[of N] sets.space_closed[of M] by auto
 qed fact
 
-lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
+lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure_alt:
   assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
 proof -
   have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x -` X) y"
     by (auto simp: indicator_def)
   show ?thesis
     using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
 qed
 
-lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
+lemma%important (in sigma_finite_measure) emeasure_pair_measure_Times:
   assumes A: "A \<in> sets N" and B: "B \<in> sets M"
   shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
-proof -
+proof%unimportant -
   have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
     using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
   also have "\<dots> = emeasure M B * emeasure N A"
     using A by (simp add: nn_integral_cmult_indicator)
   finally show ?thesis
     by (simp add: ac_simps)
 qed
 
-subsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
-
-locale pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
+subsection%important \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
+
+locale%important pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
   for M1 :: "'a measure" and M2 :: "'b measure"
 
-lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
+lemma%unimportant (in pair_sigma_finite) measurable_emeasure_Pair1:
   "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
   using M2.measurable_emeasure_Pair .
 
-lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
+lemma%important (in pair_sigma_finite) measurable_emeasure_Pair2:
   assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
-proof -
+proof%unimportant -
   have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
     using Q measurable_pair_swap' by (auto intro: measurable_sets)
   note M1.measurable_emeasure_Pair[OF this]
   moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1)) = (\<lambda>x. (x, y)) -` Q"
     using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
   ultimately show ?thesis by simp
 qed
 
-lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
+lemma%important (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
   defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
   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>
     (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
-proof -
+proof%unimportant -
   from M1.sigma_finite_incseq guess F1 . note F1 = this
   from M2.sigma_finite_incseq guess F2 . note F2 = this
   from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
   let ?F = "\<lambda>i. F1 i \<times> F2 i"
   show ?thesis

     with F1 F2 show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>"
       by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
   qed
 qed
 
-sublocale pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
+sublocale%important pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
 proof
   from M1.sigma_finite_countable guess F1 ..
   moreover from M2.sigma_finite_countable guess F2 ..
   ultimately show
     "\<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>)"
     by (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (F1 \<times> F2)"] conjI)
        (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
 qed
 
-lemma sigma_finite_pair_measure:
+lemma%unimportant sigma_finite_pair_measure:
   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
   shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
 proof -
   interpret A: sigma_finite_measure A by fact
   interpret B: sigma_finite_measure B by fact
   interpret AB: pair_sigma_finite A  B ..
   show ?thesis ..
 qed
 
-lemma sets_pair_swap:
+lemma%unimportant sets_pair_swap:
   assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   using measurable_pair_swap' assms by (rule measurable_sets)
 
-lemma (in pair_sigma_finite) distr_pair_swap:
+lemma%important (in pair_sigma_finite) distr_pair_swap:
   "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
-proof -
+proof%unimportant -
   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
   show ?thesis
   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
     show "?E \<subseteq> Pow (space ?P)"

       by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
                     measurable_pair_swap' ac_simps)
   qed
 qed
 
-lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
+lemma%unimportant (in pair_sigma_finite) emeasure_pair_measure_alt2:
   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
     (is "_ = ?\<nu> A")
-proof -
+proof%unimportant -
   have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A"
     using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
   show ?thesis using A
     by (subst distr_pair_swap)
        (simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
                  M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
 qed
 
-lemma (in pair_sigma_finite) AE_pair:
+lemma%unimportant (in pair_sigma_finite) AE_pair:
   assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
   shows "AE x in M1. (AE y in M2. Q (x, y))"
 proof -
   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"
     using assms unfolding eventually_ae_filter by auto

     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}"
       by auto
   qed
 qed
 
-lemma (in pair_sigma_finite) AE_pair_measure:
+lemma%important (in pair_sigma_finite) AE_pair_measure:
   assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   assumes ae: "AE x in M1. AE y in M2. P (x, y)"
   shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
-proof (subst AE_iff_measurable[OF _ refl])
+proof%unimportant (subst AE_iff_measurable[OF _ refl])
   show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
     by (rule sets.sets_Collect) fact
   then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
       (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)"
     by (simp add: M2.emeasure_pair_measure)

     apply auto
     done
   finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
 qed
 
-lemma (in pair_sigma_finite) AE_pair_iff:
+lemma%unimportant (in pair_sigma_finite) AE_pair_iff:
   "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
     (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))"
   using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
 
-lemma (in pair_sigma_finite) AE_commute:
+lemma%unimportant (in pair_sigma_finite) AE_commute:
   assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
 proof -
   interpret Q: pair_sigma_finite M2 M1 ..
   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"

     apply (subst Q.AE_pair_iff)
     apply simp_all
     done
 qed
 
-subsection "Fubinis theorem"
-
-lemma measurable_compose_Pair1:
+subsection%important "Fubinis theorem"
+
+lemma%unimportant measurable_compose_Pair1:
   "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
   by simp
 
-lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst:
+lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral_fst:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
-using f proof induct
+using f proof%unimportant induct
   case (cong u v)
   then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
     by (auto simp: space_pair_measure)
   show ?case
     apply (subst measurable_cong)

     by (rule measurable_cong[THEN iffD1])
 qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1
                    nn_integral_monotone_convergence_SUP incseq_def le_fun_def
               cong: measurable_cong)
 
-lemma (in sigma_finite_measure) nn_integral_fst:
+lemma%unimportant (in sigma_finite_measure) nn_integral_fst:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   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 = _")
-using f proof induct
+  using f proof induct
   case (cong u v)
   then have "?I u = ?I v"
     by (intro nn_integral_cong) (auto simp: space_pair_measure)
   with cong show ?case
     by (simp cong: nn_integral_cong)
 qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add
                    nn_integral_monotone_convergence_SUP measurable_compose_Pair1
                    borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def
               cong: nn_integral_cong)
 
-lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
+lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
   "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"
   using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp
 
-lemma (in pair_sigma_finite) nn_integral_snd:
+lemma%important (in pair_sigma_finite) nn_integral_snd:
   assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
-proof -
+proof%unimportant -
   note measurable_pair_swap[OF f]
   from M1.nn_integral_fst[OF this]
   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))"
     by simp
   also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
     by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong)
   finally show ?thesis .
 qed
 
-lemma (in pair_sigma_finite) Fubini:
+lemma%important (in pair_sigma_finite) Fubini:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   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)"
   unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..
 
-lemma (in pair_sigma_finite) Fubini':
+lemma%important (in pair_sigma_finite) Fubini':
   assumes f: "case_prod f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   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)"
   using Fubini[OF f] by simp
 
-subsection \<open>Products on counting spaces, densities and distributions\<close>
-
-lemma sigma_prod:
+subsection%important \<open>Products on counting spaces, densities and distributions\<close>
+
+lemma%important sigma_prod:
   assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"
   assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y"
   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}"
     (is "?P = ?S")
-proof (rule measure_eqI)
+proof%unimportant (rule measure_eqI)
   have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X"
     by auto
   let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}"
   have "sets ?P = sets (SUP xy:?XY. sigma (X \<times> Y) xy)"
     by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)

     proof qed (simp add: emeasure_sigma)
   fix A assume "A \<in> sets ?P" then show "emeasure ?P A = emeasure ?S A"
     by (simp add: emeasure_pair_measure_alt emeasure_sigma)
 qed
 
-lemma sigma_sets_pair_measure_generator_finite:
+lemma%unimportant sigma_sets_pair_measure_generator_finite:
   assumes "finite A" and "finite B"
   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
   (is "sigma_sets ?prod ?sets = _")
 proof safe
   have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)

   assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
   from sigma_sets_into_sp[OF _ this(1)] this(2)
   show "a \<in> A" and "b \<in> B" by auto
 qed
 
-lemma sets_pair_eq:
+lemma%important sets_pair_eq:
   assumes Ea: "Ea \<subseteq> Pow (space A)" "sets A = sigma_sets (space A) Ea"
     and Ca: "countable Ca" "Ca \<subseteq> Ea" "\<Union>Ca = space A"
     and Eb: "Eb \<subseteq> Pow (space B)" "sets B = sigma_sets (space B) Eb"
     and Cb: "countable Cb" "Cb \<subseteq> Eb" "\<Union>Cb = space B"
   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 })"
     (is "_ = sets (sigma ?\<Omega> ?E)")
-proof
+proof%unimportant
   show "sets (sigma ?\<Omega> ?E) \<subseteq> sets (A \<Otimes>\<^sub>M B)"
     using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
   have "?E \<subseteq> Pow ?\<Omega>"
     using Ea(1) Eb(1) by auto
   then have E: "a \<in> Ea \<Longrightarrow> b \<in> Eb \<Longrightarrow> a \<times> b \<in> sets (sigma ?\<Omega> ?E)" for a b

       using Ca \<open>b \<in> Eb\<close> by (auto intro!: sets.countable_UN' E)
   qed
   finally show "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (sigma ?\<Omega> ?E)" .
 qed
 
-lemma borel_prod:
+lemma%important borel_prod:
   "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
   (is "?P = ?B")
-proof -
+proof%unimportant -
   have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
     by (rule second_countable_borel_measurable[OF open_prod_generated])
   also have "\<dots> = ?P"
     unfolding borel_def
     by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])
   finally show ?thesis ..
 qed
 
-lemma pair_measure_count_space:
+lemma%important pair_measure_count_space:
   assumes A: "finite A" and B: "finite B"
   shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
-proof (rule measure_eqI)
+proof%unimportant (rule measure_eqI)
   interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
   interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
   interpret P: pair_sigma_finite "count_space A" "count_space B" ..
   show eq: "sets ?P = sets ?C"
     by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)

     done
   finally show "emeasure ?P X = emeasure ?C X" .
 qed
 
 
-lemma emeasure_prod_count_space:
+lemma%unimportant emeasure_prod_count_space:
   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
   by (rule emeasure_measure_of[OF pair_measure_def])
      (auto simp: countably_additive_def positive_def suminf_indicator A
                  nn_integral_suminf[symmetric] dest: sets.sets_into_space)
 
-lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
+lemma%unimportant emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
 proof -
   have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
     by (auto split: split_indicator)
   show ?thesis
     by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
 qed
 
-lemma emeasure_count_space_prod_eq:
+lemma%important emeasure_count_space_prod_eq:
   fixes A :: "('a \<times> 'b) set"
   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
-proof -
+proof%unimportant -
   { fix A :: "('a \<times> 'b) set" assume "countable A"
     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)"
       by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
     also have "\<dots> = (\<integral>\<^sup>+a. indicator A a \<partial>count_space UNIV)"
       by (subst nn_integral_count_space_indicator) auto

     finally show ?thesis
       using \<open>infinite C\<close> \<open>infinite A\<close> by (simp add: top_unique)
   qed
 qed
 
-lemma nn_integral_count_space_prod_eq:
+lemma%unimportant nn_integral_count_space_prod_eq:
   "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
     (is "nn_integral ?P f = _")
 proof cases
   assume cntbl: "countable {x. f x \<noteq> 0}"
   have [simp]: "\<And>x. card ({x} \<inter> {x. f x \<noteq> 0}) = (indicator {x. f x \<noteq> 0} x::ennreal)"

     using \<open>infinite C\<close> by (simp add: nn_integral_cmult ennreal_mult_top)
   ultimately show ?thesis
     by (simp add: top_unique)
 qed
 
-lemma pair_measure_density:
+lemma%important pair_measure_density:
   assumes f: "f \<in> borel_measurable M1"
   assumes g: "g \<in> borel_measurable M2"
   assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
   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")
-proof (rule measure_eqI)
+proof%unimportant (rule measure_eqI)
   interpret M2: sigma_finite_measure M2 by fact
   interpret D2: sigma_finite_measure "density M2 g" by fact
 
   fix A assume A: "A \<in> sets ?L"
   with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) =

     by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density
                   M2.nn_integral_fst[symmetric]
              cong: nn_integral_cong)
 qed simp
 
-lemma sigma_finite_measure_distr:
+lemma%unimportant sigma_finite_measure_distr:
   assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
   shows "sigma_finite_measure M"
 proof -
   interpret sigma_finite_measure "distr M N f" by fact
   from sigma_finite_countable guess A .. note A = this

       by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"])
          (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)
   qed
 qed
 
-lemma pair_measure_distr:
+lemma%unimportant pair_measure_distr:
   assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
   assumes "sigma_finite_measure (distr N T g)"
   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")
 proof (rule measure_eqI)
   interpret T: sigma_finite_measure "distr N T g" by fact

     by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr
                        T.emeasure_pair_measure_alt nn_integral_distr
              intro!: nn_integral_cong arg_cong[where f="emeasure N"])
 qed simp
 
-lemma pair_measure_eqI:
+lemma%important pair_measure_eqI:
   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
   assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
   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)"
   shows "M1 \<Otimes>\<^sub>M M2 = M"
-proof -
+proof%unimportant -
   interpret M1: sigma_finite_measure M1 by fact
   interpret M2: sigma_finite_measure M2 by fact
   interpret pair_sigma_finite M1 M2 ..
   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"

     finally show "emeasure ?P X = emeasure M X"
       by simp
   qed
 qed
 
-lemma sets_pair_countable:
+lemma%important sets_pair_countable:
   assumes "countable S1" "countable S2"
   assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
   shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
-proof auto
+proof%unimportant auto
   fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
   from sets.sets_into_space[OF x(1)] x(2)
     sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
   show "a \<in> S1" "b \<in> S2"
     by (auto simp: space_pair_measure)

     using X
     by (safe intro!: sets.countable_UN' \<open>countable X\<close> subsetI pair_measureI) (auto simp: M N)
   finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
 qed
 
-lemma pair_measure_countable:
+lemma%important pair_measure_countable:
   assumes "countable S1" "countable S2"
   shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
-proof (rule pair_measure_eqI)
+proof%unimportant (rule pair_measure_eqI)
   show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
     using assms by (auto intro!: sigma_finite_measure_count_space_countable)
   show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
     by (subst sets_pair_countable[OF assms]) auto
 next

   then show "emeasure (count_space S1) A * emeasure (count_space S2) B =
     emeasure (count_space (S1 \<times> S2)) (A \<times> B)"
     by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
 qed
 
-lemma nn_integral_fst_count_space:
+lemma%important nn_integral_fst_count_space:
   "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
   (is "?lhs = ?rhs")
-proof(cases)
+proof%unimportant(cases)
   assume *: "countable {xy. f xy \<noteq> 0}"
   let ?A = "fst ` {xy. f xy \<noteq> 0}"
   let ?B = "snd ` {xy. f xy \<noteq> 0}"
   from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+
   have "?lhs = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space ?A)"

       (simp_all add: inj_on_def split_def)
   also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto
   finally show ?thesis .
 qed
 
-lemma measurable_pair_measure_countable1:
+lemma%unimportant measurable_pair_measure_countable1:
   assumes "countable A"
   and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
   shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
 using _ _ assms(1)
 by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
 
-subsection \<open>Product of Borel spaces\<close>
-
-lemma borel_Times:
+subsection%important \<open>Product of Borel spaces\<close>
+
+lemma%important borel_Times:
   fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
   assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
   shows "A \<times> B \<in> sets borel"
-proof -
+proof%unimportant -
   have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
     by auto
   moreover
   { have "A \<in> sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)
     then have "A\<times>UNIV \<in> sets borel"

     qed simp }
   ultimately show ?thesis
     by auto
 qed
 
-lemma finite_measure_pair_measure:
+lemma%unimportant finite_measure_pair_measure:
   assumes "finite_measure M" "finite_measure N"
   shows "finite_measure (N  \<Otimes>\<^sub>M M)"
 proof (rule finite_measureI)
   interpret M: finite_measure M by fact
   interpret N: finite_measure N by fact