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src/HOL/Multivariate_Analysis/Binary_Product_Measure.thy

changeset 63627 | 6ddb43c6b711 |

parent 63626 | 44ce6b524ff3 |

child 63631 | 2edc8da89edc |

child 63633 | 2accfb71e33b |

--- a/src/HOL/Multivariate_Analysis/Binary_Product_Measure.thy Fri Aug 05 18:34:57 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1110 +0,0 @@ -(* Title: HOL/Probability/Binary_Product_Measure.thy - Author: Johannes Hölzl, TU München -*) - -section \<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 {})" - by auto - -lemma 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 - "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: - "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 simp - -lemma 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" . -qed - -lemma 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 simp - -lemma 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]: - 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: - 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 - 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 - 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: - 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) - done -qed - -lemma 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>"]) auto - -lemma 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 simp - -lemma 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)" - 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 . -qed - -lemma 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_measure -proof (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 simp -qed (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 auto -qed - -lemma (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: - 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 auto -qed fact - -lemma (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: - 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) -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 - 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 simp -qed - -lemma (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) - qed -qed - -sublocale 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: - 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: - 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) - qed -qed - -lemma (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]) -qed - -lemma (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 - qed -qed - -lemma (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 simp -qed - -lemma (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: - 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 - done -qed - -subsection "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 simp - -lemma (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+ - done -next - 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 simp - -lemma (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 . -qed - -lemma (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 simp - -subsection \<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) -qed - -lemma 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) - qed -next - 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 auto -qed - -lemma 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 .. -qed - -lemma 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]) - done -qed - - -lemma 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) -qed - -lemma 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) - qed -qed - -lemma 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) -qed - -lemma 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 simp - -lemma 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) - qed -qed - -lemma 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 simp - -lemma 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 - qed -qed - -lemma 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)" . -qed - -lemma 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]) auto -next - 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) -qed - -lemma 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 simp -qed - -lemma 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 . -qed - -lemma 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: - 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 auto -qed - -lemma 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) -qed - -end