HOL-Probability: more about probability, prepare for Markov processes in the AFP
(* Title: HOL/Analysis/Binary_Product_Measure.thy Author: Johannes Hölzl, TU München*)section \<open>Binary product measures\<close>theory Binary_Product_Measureimports Nonnegative_Lebesgue_Integrationbeginlemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})" by autolemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" by autosubsection "Binary products"definition 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 autolemma 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 (auto simp: space_pair_measure)lemma 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]: "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]: "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)" using pair_measureI[of "{x}" M1 "{y}" M2] by simplemma 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)]: "(\<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)]: 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) 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" .qedlemma 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" 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]: 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 simplemma 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 simplemma 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 simplemma 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 simplemma 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_alllemma 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_alllemma 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: "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 - { 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 (rule in_sets_Sup) apply auto [] apply (rule insertI1) apply (auto intro: ab in_vimage_algebra) [] apply (rule in_sets_Sup) apply auto [] apply (rule insertI2) apply (auto intro: ab in_vimage_algebra) done finally have "a \<times> b \<in> sets (Sup {?fst, ?snd})" . } moreover have "sets ?fst \<subseteq> sets (A \<Otimes>\<^sub>M B)" by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric]) moreover have "sets ?snd \<subseteq> sets (A \<Otimes>\<^sub>M B)" by (rule sets_image_in_sets) (auto simp: space_pair_measure) ultimately show ?thesis apply (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets) apply simp apply simp apply simp apply (elim disjE) apply (simp add: space_pair_measure) apply (simp add: space_pair_measure) apply (auto simp add: space_pair_measure) doneqedlemma 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: "(\<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="op \<in>"]) autolemma 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: 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: "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)" by simplemma 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 .qedlemma 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"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 .qedlemma 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: 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}" unfolding Int_stable_def by safe (auto simp add: times_Int_times)lemma (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_measureproof (induct rule: sigma_sets_induct_disjoint) case (compl A) with sets.sets_into_space have "\<And>x. emeasure M (Pair x -` ((space N \<times> space M) - A)) = (if x \<in> space N then emeasure M (space M) - ?s A x else 0)" unfolding sets_pair_measure[symmetric] by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1) with compl sets.top show ?case by (auto intro!: measurable_If simp: space_pair_measure)next case (union F) then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)" 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 simpqed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)lemma (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" { fix i have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i" using F sets.sets_into_space by auto let ?R = "density M (indicator (F i))" have "finite_measure ?R" using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq) then have "(\<lambda>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N" by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q) moreover have "\<And>x. emeasure ?R (Pair x -` (space N \<times> space ?R \<inter> Q)) = emeasure M (F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q))" using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1) moreover have "\<And>x. F i \<inter> Pair x -` (space N \<times> space ?R \<inter> Q) = ?C x i" using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure) ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N" by simp } moreover { fix x have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)" proof (intro suminf_emeasure) show "range (?C x) \<subseteq> sets M" using F \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> by (auto intro!: sets_Pair1) have "disjoint_family F" using F by auto show "disjoint_family (?C x)" by (rule disjoint_family_on_bisimulation[OF \<open>disjoint_family F\<close>]) auto qed also have "(\<Union>i. ?C x i) = Pair x -` Q" using F sets.sets_into_space[OF \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close>] by (auto simp: space_pair_measure) finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))" by simp } ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets by autoqedlemma (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)qedlemma (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) have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x -` A) y" by (auto simp: indicator_def) show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" proof (rule countably_additiveI) fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F" from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" by auto moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)" by (intro disjoint_family_on_bisimulation[OF F(2)]) auto moreover have "\<And>x. range (\<lambda>i. Pair x -` F i) \<subseteq> sets M" using F by (auto simp: sets_Pair1) ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)" by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure intro!: nn_integral_cong nn_integral_indicator[symmetric]) 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 autoqed factlemma (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)qedlemma (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 - 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)qedsubsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close>locale 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: "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: 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 - 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 simpqedlemma (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 - 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 proof (intro exI[of _ ?F] conjI allI) show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD) next have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)" proof (intro subsetI) fix x assume "x \<in> space M1 \<times> space M2" then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j" by (auto simp: space) then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)" using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_def by (force split: split_max)+ then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" by (intro SigmaI) (auto simp add: max.commute) then show "x \<in> (\<Union>i. ?F i)" by auto qed then show "(\<Union>i. ?F i) = space M1 \<times> space M2" using space by (auto simp: space) next fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)" using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_Suc_iff by auto next fix i from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto 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) qedqedsublocale 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)qedlemma 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 ..qedlemma 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: "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 - 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)" using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) show "sets ?P = sigma_sets (space ?P) ?E" by (simp add: sets_pair_measure space_pair_measure) then show "sets ?D = sigma_sets (space ?P) ?E" by simp next show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" using F by (auto simp: space_pair_measure) next fix X assume "X \<in> ?E" 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 have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A" using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure) with A B show "emeasure (M1 \<Otimes>\<^sub>M M2) X = emeasure ?D X" by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr measurable_pair_swap' ac_simps) qedqedlemma (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 - 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])qedlemma (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 show ?thesis proof (rule AE_I) from N measurable_emeasure_Pair1[OF \<open>N \<in> sets (M1 \<Otimes>\<^sub>M M2)\<close>] show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0" by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff) show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1" by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp { fix x assume "x \<in> space M1" "emeasure M2 (Pair x -` N) = 0" have "AE y in M2. Q (x, y)" proof (rule AE_I) show "emeasure M2 (Pair x -` N) = 0" by fact show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1) show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N" using N \<open>x \<in> space M1\<close> unfolding space_pair_measure by auto qed } 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 qedqedlemma (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]) 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) also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)" using ae apply (safe intro!: nn_integral_cong_AE) apply (intro AE_I2) apply (safe intro!: nn_integral_cong_AE) apply auto done finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simpqedlemma (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 autolemma (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" by auto have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} = (\<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)" by (auto simp: space_pair_measure) also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)" by (intro sets_pair_swap P) finally show ?thesis apply (subst AE_pair_iff[OF P]) apply (subst distr_pair_swap) apply (subst AE_distr_iff[OF measurable_pair_swap' P]) apply (subst Q.AE_pair_iff) apply simp_all doneqedsubsection "Fubinis theorem"lemma 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 simplemma (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 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) apply (rule nn_integral_cong) apply fact+ donenext case (set Q) have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x -` Q) y" by (auto simp: indicator_def) have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ y. indicator Q (x, y) \<partial>M" by (simp add: sets_Pair1[OF set]) from this measurable_emeasure_Pair[OF set] show ?case 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: 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 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)]: "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 simplemma (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 - 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 .qedlemma (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': 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 simpsubsection \<open>Products on counting spaces, densities and distributions\<close>lemma 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) 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) also have "\<dots> = sets (sigma (X \<times> Y) (\<Union>?XY))" by (intro Sup_sigma arg_cong[where f=sets]) auto also have "\<dots> = sets ?S" proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI) 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)" using A B by auto next interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" using A B by (intro sigma_algebra_sigma_sets) auto fix Z assume "Z \<in> \<Union>?XY" then show "Z \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}" proof safe fix a assume "a \<in> A" from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E" by auto with \<open>a \<in> A\<close> A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)" by auto 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}" using \<open>a \<in> A\<close> E unfolding eq by (auto intro!: XY.countable_UN') next fix b assume "b \<in> B" from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E" by auto with \<open>b \<in> B\<close> B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)" by auto 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}" using \<open>b \<in> B\<close> E unfolding eq by (auto intro!: XY.countable_UN') qed next fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}" then obtain a b where "Z = a \<times> b" and ab: "a \<in> A" "b \<in> B" by auto then have Z: "Z = (fst -` a \<inter> X \<times> Y) \<inter> (snd -` b \<inter> X \<times> Y)" using A B by auto interpret XY: sigma_algebra "X \<times> Y" "sigma_sets (X \<times> Y) (\<Union>?XY)" by (intro sigma_algebra_sigma_sets) auto show "Z \<in> sigma_sets (X \<times> Y) (\<Union>?XY)" unfolding Z by (rule XY.Int) (blast intro: ab)+ qed finally show "sets ?P = sets ?S" .next interpret finite_measure "sigma X A" for X A 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)qedlemma 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) fix x assume subset: "x \<subseteq> A \<times> B" hence "finite x" using fin by (rule finite_subset) from this subset show "x \<in> sigma_sets ?prod ?sets" proof (induct x) case empty show ?case by (rule sigma_sets.Empty) next case (insert a x) hence "{a} \<in> sigma_sets ?prod ?sets" by auto moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) qednext fix x a b 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 autoqedlemma 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 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 by auto have "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (Sup {vimage_algebra ?\<Omega> fst A, vimage_algebra ?\<Omega> snd B})" unfolding sets_pair_eq_sets_fst_snd .. also have "vimage_algebra ?\<Omega> fst A = vimage_algebra ?\<Omega> fst (sigma (space A) Ea)" by (intro vimage_algebra_cong[OF refl refl]) (simp add: Ea) also have "\<dots> = sigma ?\<Omega> {fst -` A \<inter> ?\<Omega> |A. A \<in> Ea}" by (intro Ea vimage_algebra_sigma) auto also have "vimage_algebra ?\<Omega> snd B = vimage_algebra ?\<Omega> snd (sigma (space B) Eb)" by (intro vimage_algebra_cong[OF refl refl]) (simp add: Eb) also have "\<dots> = sigma ?\<Omega> {snd -` A \<inter> ?\<Omega> |A. A \<in> Eb}" by (intro Eb vimage_algebra_sigma) auto also have "{sigma ?\<Omega> {fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, sigma ?\<Omega> {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}} = sigma ?\<Omega> ` {{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}" by auto also have "sets (SUP S:{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}. sigma ?\<Omega> S) = sets (sigma ?\<Omega> (\<Union>{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}))" using Ea(1) Eb(1) by (intro sets_Sup_sigma) auto also have "\<dots> \<subseteq> sets (sigma ?\<Omega> ?E)" proof (subst sigma_le_sets, safe intro!: space_in_measure_of) fix a assume "a \<in> Ea" then have "fst -` a \<inter> ?\<Omega> = (\<Union>b\<in>Cb. a \<times> b)" using Cb(3)[symmetric] Ea(1) by auto then show "fst -` a \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)" using Cb \<open>a \<in> Ea\<close> by (auto intro!: sets.countable_UN' E) next fix b assume "b \<in> Eb" then have "snd -` b \<inter> ?\<Omega> = (\<Union>a\<in>Ca. a \<times> b)" using Ca(3)[symmetric] Eb(1) by auto then show "snd -` b \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)" 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)" .qedlemma borel_prod: "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)" (is "?P = ?B")proof - 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 ..qedlemma 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) 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) fix X assume X: "X \<in> sets ?P" with eq have X_subset: "X \<subseteq> A \<times> B" by simp with A B have fin_Pair: "\<And>x. finite (Pair x -` X)" by (intro finite_subset[OF _ B]) auto have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B) have pos_card: "(0::ennreal) < of_nat (card (Pair x -` X)) \<longleftrightarrow> Pair x -` X \<noteq> {}" for x by (auto simp: card_eq_0_iff fin_Pair) blast show "emeasure ?P X = emeasure ?C X" using X_subset A fin_Pair fin_X apply (subst B.emeasure_pair_measure_alt[OF X]) apply (subst emeasure_count_space) apply (auto simp add: emeasure_count_space nn_integral_count_space pos_card of_nat_setsum[symmetric] card_SigmaI[symmetric] simp del: of_nat_setsum card_SigmaI intro!: arg_cong[where f=card]) doneqedlemma 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"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)qedlemma 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 - { 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 have "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A" by simp } note * = this show ?thesis proof cases assume "finite A" then show ?thesis by (intro * countable_finite) next assume "infinite A" then obtain C where "countable C" and "infinite C" and "C \<subseteq> A" by (auto dest: infinite_countable_subset') with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A" by (intro emeasure_mono) auto also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C" using \<open>countable C\<close> by (rule *) finally show ?thesis using \<open>infinite C\<close> \<open>infinite A\<close> by (simp add: top_unique) qedqedlemma 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)" by (auto split: split_indicator) have [measurable]: "\<And>y. (\<lambda>x. indicator {y} x) \<in> borel_measurable ?P" by (rule measurable_discrete_difference[of "\<lambda>x. 0" _ borel "{y}" "\<lambda>x. indicator {y} x" for y]) (auto intro: sets_Pair) 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)" by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator) also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+x'. f x' * indicator {x'} x \<partial>count_space {x. f x \<noteq> 0} \<partial>?P)" by (auto intro!: nn_integral_cong split: split_indicator) also have "\<dots> = (\<integral>\<^sup>+x'. \<integral>\<^sup>+x. f x' * indicator {x'} x \<partial>?P \<partial>count_space {x. f x \<noteq> 0})" by (intro nn_integral_count_space_nn_integral cntbl) auto also have "\<dots> = (\<integral>\<^sup>+x'. f x' \<partial>count_space {x. f x \<noteq> 0})" by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair) finally show ?thesis by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)next { fix x assume "f x \<noteq> 0" then have "(\<exists>r\<ge>0. 0 < r \<and> f x = ennreal r) \<or> f x = \<infinity>" by (cases "f x" rule: ennreal_cases) (auto simp: less_le) then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f x" by (auto elim!: nat_approx_posE intro!: less_imp_le) } note * = this assume cntbl: "uncountable {x. f x \<noteq> 0}" also have "{x. f x \<noteq> 0} = (\<Union>n. {x. 1/Suc n \<le> f x})" using * by auto finally obtain n where "infinite {x. 1/Suc n \<le> f x}" by (meson countableI_type countable_UN uncountable_infinite) then obtain C where C: "C \<subseteq> {x. 1/Suc n \<le> f x}" and "countable C" "infinite C" by (metis infinite_countable_subset') have [measurable]: "C \<in> sets ?P" using sets.countable[OF _ \<open>countable C\<close>, of ?P] by (auto simp: sets_Pair) have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f" using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>" using \<open>infinite C\<close> by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top) moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f" using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) moreover have "(\<integral>\<^sup>+x. ennreal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>" using \<open>infinite C\<close> by (simp add: nn_integral_cmult ennreal_mult_top) ultimately show ?thesis by (simp add: top_unique)qedlemma 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) 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) = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)" by (intro nn_integral_cong_AE) (auto simp add: nn_integral_cmult[symmetric] ac_simps) with A f g show "emeasure ?L A = emeasure ?R A" by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density M2.nn_integral_fst[symmetric] cong: nn_integral_cong)qed simplemma 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 show ?thesis proof 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>)" using A f by (intro exI[of _ "(\<lambda>a. f -` a \<inter> space M) ` A"]) (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space) qedqedlemma 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 interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+ fix A assume A: "A \<in> sets ?P" with f g show "emeasure ?P A = emeasure ?D A" 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 simplemma 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 - 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}" let ?P = "M1 \<Otimes>\<^sub>M M2" show ?thesis proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) show "?E \<subseteq> Pow (space ?P)" using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) show "sets ?P = sigma_sets (space ?P) ?E" by (simp add: sets_pair_measure space_pair_measure) then show "sets M = sigma_sets (space ?P) ?E" using sets[symmetric] by simp next show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" using F by (auto simp: space_pair_measure) next fix X assume "X \<in> ?E" 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 then have "emeasure ?P X = emeasure M1 A * emeasure M2 B" by (simp add: M2.emeasure_pair_measure_Times) also have "\<dots> = emeasure M (A \<times> B)" using A B emeasure by auto finally show "emeasure ?P X = emeasure M X" by simp qedqedlemma 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 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)next fix X assume X: "X \<subseteq> S1 \<times> S2" then have "countable X" by (metis countable_subset \<open>countable S1\<close> \<open>countable S2\<close> countable_SIGMA) have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)" 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)" .qedlemma 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) 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]) autonext fix A B assume "A \<in> sets (count_space S1)" "B \<in> sets (count_space S2)" 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)qedlemma 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) 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)" by(rule nn_integral_count_space_eq) (auto simp add: nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI) also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space ?B \<partial>count_space ?A)" by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI) also have "\<dots> = (\<integral>\<^sup>+ xy. f xy \<partial>count_space (?A \<times> ?B))" by(subst sigma_finite_measure.nn_integral_fst) (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable) also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI) finally show ?thesis .next { fix xy assume "f xy \<noteq> 0" then have "(\<exists>r\<ge>0. 0 < r \<and> f xy = ennreal r) \<or> f xy = \<infinity>" by (cases "f xy" rule: ennreal_cases) (auto simp: less_le) then have "\<exists>n. ennreal (1 / real (Suc n)) \<le> f xy" by (auto elim!: nat_approx_posE intro!: less_imp_le) } note * = this assume cntbl: "uncountable {xy. f xy \<noteq> 0}" also have "{xy. f xy \<noteq> 0} = (\<Union>n. {xy. 1/Suc n \<le> f xy})" using * by auto finally obtain n where "infinite {xy. 1/Suc n \<le> f xy}" by (meson countableI_type countable_UN uncountable_infinite) then obtain C where C: "C \<subseteq> {xy. 1/Suc n \<le> f xy}" and "countable C" "infinite C" by (metis infinite_countable_subset') have "\<infinity> = (\<integral>\<^sup>+ xy. ennreal (1 / Suc n) * indicator C xy \<partial>count_space UNIV)" using \<open>infinite C\<close> by(simp add: nn_integral_cmult ennreal_mult_top) also have "\<dots> \<le> ?rhs" using C by(intro nn_integral_mono)(auto split: split_indicator) finally have "?rhs = \<infinity>" by (simp add: top_unique) moreover have "?lhs = \<infinity>" proof(cases "finite (fst ` C)") case True then obtain x C' where x: "x \<in> fst ` C" and C': "C' = fst -` {x} \<inter> C" and "infinite C'" using \<open>infinite C\<close> by(auto elim!: inf_img_fin_domE') from x C C' have **: "C' \<subseteq> {xy. 1 / Suc n \<le> f xy}" by auto from C' \<open>infinite C'\<close> have "infinite (snd ` C')" by(auto dest!: finite_imageD simp add: inj_on_def) then have "\<infinity> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator (snd ` C') y \<partial>count_space UNIV)" by(simp add: nn_integral_cmult ennreal_mult_top) also have "\<dots> = (\<integral>\<^sup>+ y. ennreal (1 / Suc n) * indicator C' (x, y) \<partial>count_space UNIV)" by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C') 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)" by(simp add: one_ereal_def[symmetric]) 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)" by(rule nn_integral_mono)(simp split: split_indicator) also have "\<dots> \<le> ?lhs" using ** by(intro nn_integral_mono)(auto split: split_indicator) finally show ?thesis by (simp add: top_unique) next case False define C' where "C' = fst ` C" have "\<infinity> = \<integral>\<^sup>+ x. ennreal (1 / Suc n) * indicator C' x \<partial>count_space UNIV" using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top) 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" by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong) 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" by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI) also have "\<dots> \<le> ?lhs" using C by(intro nn_integral_mono)(auto split: split_indicator) finally show ?thesis by (simp add: top_unique) qed ultimately show ?thesis by simpqedlemma nn_integral_snd_count_space: "(\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f" (is "?lhs = ?rhs")proof - have "?lhs = (\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. (\<lambda>(y, x). f (x, y)) (y, x) \<partial>count_space UNIV \<partial>count_space UNIV)" by(simp) also have "\<dots> = \<integral>\<^sup>+ yx. (\<lambda>(y, x). f (x, y)) yx \<partial>count_space UNIV" by(rule nn_integral_fst_count_space) also have "\<dots> = \<integral>\<^sup>+ xy. f xy \<partial>count_space ((\<lambda>(x, y). (y, x)) ` UNIV)" by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric]) (simp_all add: inj_on_def split_def) also have "\<dots> = ?rhs" by(rule nn_integral_count_space_eq) auto finally show ?thesis .qedlemma 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_allsubsection \<open>Product of Borel spaces\<close>lemma 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 - 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" proof (induct A) case (Basic S) then show ?case by (auto intro!: borel_open open_Times) next case (Compl A) moreover have *: "(UNIV - A) \<times> UNIV = UNIV - (A \<times> UNIV)" by auto ultimately show ?case unfolding * by auto next case (Union A) moreover have *: "(UNION UNIV A) \<times> UNIV = UNION UNIV (\<lambda>i. A i \<times> UNIV)" by auto ultimately show ?case unfolding * by auto qed simp } moreover { have "B \<in> sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel) then have "UNIV\<times>B \<in> sets borel" proof (induct B) case (Basic S) then show ?case by (auto intro!: borel_open open_Times) next case (Compl B) moreover have *: "UNIV \<times> (UNIV - B) = UNIV - (UNIV \<times> B)" by auto ultimately show ?case unfolding * by auto next case (Union B) moreover have *: "UNIV \<times> (UNION UNIV B) = UNION UNIV (\<lambda>i. UNIV \<times> B i)" by auto ultimately show ?case unfolding * by auto qed simp } ultimately show ?thesis by autoqedlemma 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 show "emeasure (N \<Otimes>\<^sub>M M) (space (N \<Otimes>\<^sub>M M)) \<noteq> \<infinity>" by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff)qedend