imports Nonnegative_Lebesgue_Integration

(* Title: HOL/Analysis/Binary_Product_Measure.thy Author: Johannes Hölzl, TU München *) section ‹Binary Product Measure› theory Binary_Product_Measure imports Nonnegative_Lebesgue_Integration begin lemma Pair_vimage_times[simp]: "Pair x -` (A × B) = (if x ∈ A then B else {})" by auto lemma rev_Pair_vimage_times[simp]: "(λx. (x, y)) -` (A × B) = (if y ∈ B then A else {})" by auto subsection "Binary products" definition✐‹tag important› pair_measure (infixr "⨂⇩_{M}" 80) where "A ⨂⇩_{M}B = measure_of (space A × space B) {a × b | a b. a ∈ sets A ∧ b ∈ sets B} (λX. ∫⇧^{+}x. (∫⇧^{+}y. indicator X (x,y) ∂B) ∂A)" lemma pair_measure_closed: "{a × b | a b. a ∈ sets A ∧ b ∈ sets B} ⊆ Pow (space A × space B)" using sets.space_closed[of A] sets.space_closed[of B] by auto lemma space_pair_measure: "space (A ⨂⇩_{M}B) = space A × 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∈space N. P x y}) = {x∈space (M ⨂⇩_{M}N). P (fst x) (snd x)}" by (auto simp: space_pair_measure) lemma sets_pair_measure: "sets (A ⨂⇩_{M}B) = sigma_sets (space A × space B) {a × b | a b. a ∈ sets A ∧ b ∈ 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' ⟹ sets M2 = sets M2' ⟹ sets (M1 ⨂⇩_{M}M2) = sets (M1' ⨂⇩_{M}M2')" unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) lemma pair_measureI[intro, simp, measurable]: "x ∈ sets A ⟹ y ∈ sets B ⟹ x × y ∈ sets (A ⨂⇩_{M}B)" by (auto simp: sets_pair_measure) lemma sets_Pair: "{x} ∈ sets M1 ⟹ {y} ∈ sets M2 ⟹ {(x, y)} ∈ sets (M1 ⨂⇩_{M}M2)" using pair_measureI[of "{x}" M1 "{y}" M2] by simp lemma measurable_pair_measureI: assumes 1: "f ∈ space M → space M1 × space M2" assumes 2: "⋀A B. A ∈ sets M1 ⟹ B ∈ sets M2 ⟹ f -` (A × B) ∩ space M ∈ sets M" shows "f ∈ measurable M (M1 ⨂⇩_{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)]: "(λx. f x (fst (g x)) (snd (g x))) ∈ measurable M N ⟹ (λx. case_prod (f x) (g x)) ∈ measurable M N" unfolding split_beta' . lemma measurable_Pair[measurable (raw)]: assumes f: "f ∈ measurable M M1" and g: "g ∈ measurable M M2" shows "(λx. (f x, g x)) ∈ measurable M (M1 ⨂⇩_{M}M2)" proof (rule measurable_pair_measureI) show "(λx. (f x, g x)) ∈ space M → space M1 × space M2" using f g by (auto simp: measurable_def) fix A B assume *: "A ∈ sets M1" "B ∈ sets M2" have "(λx. (f x, g x)) -` (A × B) ∩ space M = (f -` A ∩ space M) ∩ (g -` B ∩ space M)" by auto also have "… ∈ sets M" by (rule sets.Int) (auto intro!: measurable_sets * f g) finally show "(λx. (f x, g x)) -` (A × B) ∩ space M ∈ sets M" . qed lemma measurable_fst[intro!, simp, measurable]: "fst ∈ measurable (M1 ⨂⇩_{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 ∈ measurable (M1 ⨂⇩_{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 ∈ measurable (M1 ⨂⇩_{M}M2) N" assumes g: "g ∈ measurable M M1" and h: "h ∈ measurable M M2" shows "(λx. f (g x) (h x)) ∈ measurable M N" using measurable_compose[OF measurable_Pair f, OF g h] by simp lemma measurable_Pair1_compose[measurable_dest]: assumes f: "(λx. (f x, g x)) ∈ measurable M (M1 ⨂⇩_{M}M2)" assumes [measurable]: "h ∈ measurable N M" shows "(λx. f (h x)) ∈ measurable N M1" using measurable_compose[OF f measurable_fst] by simp lemma measurable_Pair2_compose[measurable_dest]: assumes f: "(λx. (f x, g x)) ∈ measurable M (M1 ⨂⇩_{M}M2)" assumes [measurable]: "h ∈ measurable N M" shows "(λx. g (h x)) ∈ measurable N M2" using measurable_compose[OF f measurable_snd] by simp lemma measurable_pair: assumes "(fst ∘ f) ∈ measurable M M1" "(snd ∘ f) ∈ measurable M M2" shows "f ∈ measurable M (M1 ⨂⇩_{M}M2)" using measurable_Pair[OF assms] by simp lemma assumes f[measurable]: "f ∈ measurable M (N ⨂⇩_{M}P)" shows measurable_fst': "(λx. fst (f x)) ∈ measurable M N" and measurable_snd': "(λx. snd (f x)) ∈ measurable M P" by simp_all lemma assumes f[measurable]: "f ∈ measurable M N" shows measurable_fst'': "(λx. f (fst x)) ∈ measurable (M ⨂⇩_{M}P) N" and measurable_snd'': "(λx. f (snd x)) ∈ measurable (P ⨂⇩_{M}M) N" by simp_all lemma sets_pair_in_sets: assumes "⋀a b. a ∈ sets A ⟹ b ∈ sets B ⟹ a × b ∈ sets N" shows "sets (A ⨂⇩_{M}B) ⊆ sets N" unfolding sets_pair_measure by (intro sets.sigma_sets_subset') (auto intro!: assms) lemma sets_pair_eq_sets_fst_snd: "sets (A ⨂⇩_{M}B) = sets (Sup {vimage_algebra (space A × space B) fst A, vimage_algebra (space A × space B) snd B})" (is "?P = sets (Sup {?fst, ?snd})") proof - { fix a b assume ab: "a ∈ sets A" "b ∈ sets B" then have "a × b = (fst -` a ∩ (space A × space B)) ∩ (snd -` b ∩ (space A × space B))" by (auto dest: sets.sets_into_space) also have "… ∈ 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 × b ∈ sets (Sup {?fst, ?snd})" . } moreover have "sets ?fst ⊆ sets (A ⨂⇩_{M}B)" by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric]) moreover have "sets ?snd ⊆ sets (A ⨂⇩_{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 ∈ measurable M (M1 ⨂⇩_{M}M2) ⟷ (fst ∘ f) ∈ measurable M M1 ∧ (snd ∘ f) ∈ measurable M M2" by (auto intro: measurable_pair[of f M M1 M2]) lemma measurable_split_conv: "(λ(x, y). f x y) ∈ measurable A B ⟷ (λx. f (fst x) (snd x)) ∈ measurable A B" by (intro arg_cong2[where f="(∈)"]) auto lemma measurable_pair_swap': "(λ(x,y). (y, x)) ∈ measurable (M1 ⨂⇩_{M}M2) (M2 ⨂⇩_{M}M1)" by (auto intro!: measurable_Pair simp: measurable_split_conv) lemma measurable_pair_swap: assumes f: "f ∈ measurable (M1 ⨂⇩_{M}M2) M" shows "(λ(x,y). f (y, x)) ∈ measurable (M2 ⨂⇩_{M}M1) M" using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def) lemma measurable_pair_swap_iff: "f ∈ measurable (M2 ⨂⇩_{M}M1) M ⟷ (λ(x,y). f (y,x)) ∈ measurable (M1 ⨂⇩_{M}M2) M" by (auto dest: measurable_pair_swap) lemma measurable_Pair1': "x ∈ space M1 ⟹ Pair x ∈ measurable M2 (M1 ⨂⇩_{M}M2)" by simp lemma sets_Pair1[measurable (raw)]: assumes A: "A ∈ sets (M1 ⨂⇩_{M}M2)" shows "Pair x -` A ∈ sets M2" proof - have "Pair x -` A = (if x ∈ space M1 then Pair x -` A ∩ space M2 else {})" using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) also have "… ∈ 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 ∈ space M2 ⟹ (λx. (x, y)) ∈ measurable M1 (M1 ⨂⇩_{M}M2)" by (auto intro!: measurable_Pair) lemma sets_Pair2: assumes A: "A ∈ sets (M1 ⨂⇩_{M}M2)" shows "(λx. (x, y)) -` A ∈ sets M1" proof - have "(λx. (x, y)) -` A = (if y ∈ space M2 then (λx. (x, y)) -` A ∩ space M1 else {})" using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) also have "… ∈ 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 ∈ measurable (M1 ⨂⇩_{M}M2) M" and x: "x ∈ space M1" shows "(λy. f (x, y)) ∈ measurable M2 M" using measurable_comp[OF measurable_Pair1' f, OF x] by (simp add: comp_def) lemma measurable_Pair1: assumes f: "f ∈ measurable (M1 ⨂⇩_{M}M2) M" and y: "y ∈ space M2" shows "(λx. f (x, y)) ∈ 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 × b | a b. a ∈ sets A ∧ b ∈ 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 ∈ sets (N ⨂⇩_{M}M)" shows "(λx. emeasure M (Pair x -` Q)) ∈ borel_measurable N" (is "?s Q ∈ _") 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 "⋀x. emeasure M (Pair x -` ((space N × space M) - A)) = (if x ∈ 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 "⋀x. emeasure M (Pair x -` (⋃i. F i)) = (∑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 ∈ sets (N ⨂⇩_{M}M)" shows "(λx. emeasure M (Pair x -` Q)) ∈ borel_measurable N" (is "?s Q ∈ _") proof - from sigma_finite_disjoint guess F . note F = this then have F_sets: "⋀i. F i ∈ sets M" by auto let ?C = "λx i. F i ∩ Pair x -` Q" { fix i have [simp]: "space N × F i ∩ space N × space M = space N × 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 "(λx. emeasure ?R (Pair x -` (space N × space ?R ∩ Q))) ∈ borel_measurable N" by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q) moreover have "⋀x. emeasure ?R (Pair x -` (space N × space ?R ∩ Q)) = emeasure M (F i ∩ Pair x -` (space N × space ?R ∩ Q))" using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1) moreover have "⋀x. F i ∩ Pair x -` (space N × space ?R ∩ Q) = ?C x i" using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure) ultimately have "(λx. emeasure M (?C x i)) ∈ borel_measurable N" by simp } moreover { fix x have "(∑i. emeasure M (?C x i)) = emeasure M (⋃i. ?C x i)" proof (intro suminf_emeasure) show "range (?C x) ⊆ sets M" using F ‹Q ∈ sets (N ⨂⇩_{M}M)› 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 ‹disjoint_family F›]) auto qed also have "(⋃i. ?C x i) = Pair x -` Q" using F sets.sets_into_space[OF ‹Q ∈ sets (N ⨂⇩_{M}M)›] by (auto simp: space_pair_measure) finally have "emeasure M (Pair x -` Q) = (∑i. emeasure M (?C x i))" by simp } ultimately show ?thesis using ‹Q ∈ sets (N ⨂⇩_{M}M)› F_sets by auto qed lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]: assumes space: "⋀x. x ∈ space N ⟹ A x ⊆ space M" assumes A: "{x∈space (N ⨂⇩_{M}M). snd x ∈ A (fst x)} ∈ sets (N ⨂⇩_{M}M)" shows "(λx. emeasure M (A x)) ∈ borel_measurable N" proof - from space have "⋀x. x ∈ space N ⟹ Pair x -` {x ∈ space (N ⨂⇩_{M}M). snd x ∈ 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 ∈ sets (N ⨂⇩_{M}M)" shows "emeasure (N ⨂⇩_{M}M) X = (∫⇧^{+}x. ∫⇧^{+}y. indicator X (x, y) ∂M ∂N)" (is "_ = ?μ X") proof (rule emeasure_measure_of[OF pair_measure_def]) show "positive (sets (N ⨂⇩_{M}M)) ?μ" by (auto simp: positive_def) have eq[simp]: "⋀A x y. indicator A (x, y) = indicator (Pair x -` A) y" by (auto simp: indicator_def) show "countably_additive (sets (N ⨂⇩_{M}M)) ?μ" proof (rule countably_additiveI) fix F :: "nat ⇒ ('b × 'a) set" assume F: "range F ⊆ sets (N ⨂⇩_{M}M)" "disjoint_family F" from F have *: "⋀i. F i ∈ sets (N ⨂⇩_{M}M)" by auto moreover have "⋀x. disjoint_family (λi. Pair x -` F i)" by (intro disjoint_family_on_bisimulation[OF F(2)]) auto moreover have "⋀x. range (λi. Pair x -` F i) ⊆ sets M" using F by (auto simp: sets_Pair1) ultimately show "(∑n. ?μ (F n)) = ?μ (⋃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 × b |a b. a ∈ sets N ∧ b ∈ sets M} ⊆ Pow (space N × 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 ∈ sets (N ⨂⇩_{M}M)" shows "emeasure (N ⨂⇩_{M}M) X = (∫⇧^{+}x. emeasure M (Pair x -` X) ∂N)" proof - have [simp]: "⋀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 proposition (in sigma_finite_measure) emeasure_pair_measure_Times: assumes A: "A ∈ sets N" and B: "B ∈ sets M" shows "emeasure (N ⨂⇩_{M}M) (A × B) = emeasure N A * emeasure M B" proof - have "emeasure (N ⨂⇩_{M}M) (A × B) = (∫⇧^{+}x. emeasure M B * indicator A x ∂N)" using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt) also have "… = 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 ‹Binary products of ‹σ›-finite emeasure spaces› locale✐‹tag unimportant› 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 ∈ sets (M1 ⨂⇩_{M}M2) ⟹ (λx. emeasure M2 (Pair x -` Q)) ∈ borel_measurable M1" using M2.measurable_emeasure_Pair . lemma (in pair_sigma_finite) measurable_emeasure_Pair2: assumes Q: "Q ∈ sets (M1 ⨂⇩_{M}M2)" shows "(λy. emeasure M1 ((λx. (x, y)) -` Q)) ∈ borel_measurable M2" proof - have "(λ(x, y). (y, x)) -` Q ∩ space (M2 ⨂⇩_{M}M1) ∈ sets (M2 ⨂⇩_{M}M1)" using Q measurable_pair_swap' by (auto intro: measurable_sets) note M1.measurable_emeasure_Pair[OF this] moreover have "⋀y. Pair y -` ((λ(x, y). (y, x)) -` Q ∩ space (M2 ⨂⇩_{M}M1)) = (λx. (x, y)) -` Q" using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure) ultimately show ?thesis by simp qed proposition (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator: defines "E ≡ {A × B | A B. A ∈ sets M1 ∧ B ∈ sets M2}" shows "∃F::nat ⇒ ('a × 'b) set. range F ⊆ E ∧ incseq F ∧ (⋃i. F i) = space M1 × space M2 ∧ (∀i. emeasure (M1 ⨂⇩_{M}M2) (F i) ≠ ∞)" 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 = (⋃i. F1 i)" "space M2 = (⋃i. F2 i)" by auto let ?F = "λi. F1 i × F2 i" show ?thesis proof (intro exI[of _ ?F] conjI allI) show "range ?F ⊆ E" using F1 F2 by (auto simp: E_def) (metis range_subsetD) next have "space M1 × space M2 ⊆ (⋃i. ?F i)" proof (intro subsetI) fix x assume "x ∈ space M1 × space M2" then obtain i j where "fst x ∈ F1 i" "snd x ∈ F2 j" by (auto simp: space) then have "fst x ∈ F1 (max i j)" "snd x ∈ F2 (max j i)" using ‹incseq F1› ‹incseq F2› unfolding incseq_def by (force split: split_max)+ then have "(fst x, snd x) ∈ F1 (max i j) × F2 (max i j)" by (intro SigmaI) (auto simp add: max.commute) then show "x ∈ (⋃i. ?F i)" by auto qed then show "(⋃i. ?F i) = space M1 × space M2" using space by (auto simp: space) next fix i show "incseq (λi. F1 i × F2 i)" using ‹incseq F1› ‹incseq F2› unfolding incseq_Suc_iff by auto next fix i from F1 F2 have "F1 i ∈ sets M1" "F2 i ∈ sets M2" by auto with F1 F2 show "emeasure (M1 ⨂⇩_{M}M2) (F1 i × F2 i) ≠ ∞" by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff) qed qed sublocale✐‹tag unimportant› pair_sigma_finite ⊆ P?: sigma_finite_measure "M1 ⨂⇩_{M}M2" proof from M1.sigma_finite_countable guess F1 .. moreover from M2.sigma_finite_countable guess F2 .. ultimately show "∃A. countable A ∧ A ⊆ sets (M1 ⨂⇩_{M}M2) ∧ ⋃A = space (M1 ⨂⇩_{M}M2) ∧ (∀a∈A. emeasure (M1 ⨂⇩_{M}M2) a ≠ ∞)" by (intro exI[of _ "(λ(a, b). a × b) ` (F1 × 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 ⨂⇩_{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 ∈ sets (M1 ⨂⇩_{M}M2)" shows "(λ(x, y). (y, x)) -` A ∩ space (M2 ⨂⇩_{M}M1) ∈ sets (M2 ⨂⇩_{M}M1)" using measurable_pair_swap' assms by (rule measurable_sets) lemma (in pair_sigma_finite) distr_pair_swap: "M1 ⨂⇩_{M}M2 = distr (M2 ⨂⇩_{M}M1) (M1 ⨂⇩_{M}M2) (λ(x, y). (y, x))" (is "?P = ?D") proof - from sigma_finite_up_in_pair_measure_generator guess F :: "nat ⇒ ('a × 'b) set" .. note F = this let ?E = "{a × b |a b. a ∈ sets M1 ∧ b ∈ sets M2}" show ?thesis proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) show "?E ⊆ 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 ⊆ ?E" "(⋃i. F i) = space ?P" "⋀i. emeasure ?P (F i) ≠ ∞" using F by (auto simp: space_pair_measure) next fix X assume "X ∈ ?E" then obtain A B where X[simp]: "X = A × B" and A: "A ∈ sets M1" and B: "B ∈ sets M2" by auto have "(λ(y, x). (x, y)) -` X ∩ space (M2 ⨂⇩_{M}M1) = B × 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 ⨂⇩_{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 ∈ sets (M1 ⨂⇩_{M}M2)" shows "emeasure (M1 ⨂⇩_{M}M2) A = (∫⇧^{+}y. emeasure M1 ((λx. (x, y)) -` A) ∂M2)" (is "_ = ?ν A") proof - have [simp]: "⋀y. (Pair y -` ((λ(x, y). (y, x)) -` A ∩ space (M2 ⨂⇩_{M}M1))) = (λx. (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 ⨂⇩_{M}M2). Q x" shows "AE x in M1. (AE y in M2. Q (x, y))" proof - obtain N where N: "N ∈ sets (M1 ⨂⇩_{M}M2)" "emeasure (M1 ⨂⇩_{M}M2) N = 0" "{x∈space (M1 ⨂⇩_{M}M2). ¬ Q x} ⊆ N" using assms unfolding eventually_ae_filter by auto show ?thesis proof (rule AE_I) from N measurable_emeasure_Pair1[OF ‹N ∈ sets (M1 ⨂⇩_{M}M2)›] show "emeasure M1 {x∈space M1. emeasure M2 (Pair x -` N) ≠ 0} = 0" by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff) show "{x ∈ space M1. emeasure M2 (Pair x -` N) ≠ 0} ∈ sets M1" by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp { fix x assume "x ∈ 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 ∈ sets M2" using N(1) by (rule sets_Pair1) show "{y ∈ space M2. ¬ Q (x, y)} ⊆ Pair x -` N" using N ‹x ∈ space M1› unfolding space_pair_measure by auto qed } then show "{x ∈ space M1. ¬ (AE y in M2. Q (x, y))} ⊆ {x ∈ space M1. emeasure M2 (Pair x -` N) ≠ 0}" by auto qed qed lemma (in pair_sigma_finite) AE_pair_measure: assumes "{x∈space (M1 ⨂⇩_{M}M2). P x} ∈ sets (M1 ⨂⇩_{M}M2)" assumes ae: "AE x in M1. AE y in M2. P (x, y)" shows "AE x in M1 ⨂⇩_{M}M2. P x" proof (subst AE_iff_measurable[OF _ refl]) show "{x∈space (M1 ⨂⇩_{M}M2). ¬ P x} ∈ sets (M1 ⨂⇩_{M}M2)" by (rule sets.sets_Collect) fact then have "emeasure (M1 ⨂⇩_{M}M2) {x ∈ space (M1 ⨂⇩_{M}M2). ¬ P x} = (∫⇧^{+}x. ∫⇧^{+}y. indicator {x ∈ space (M1 ⨂⇩_{M}M2). ¬ P x} (x, y) ∂M2 ∂M1)" by (simp add: M2.emeasure_pair_measure) also have "… = (∫⇧^{+}x. ∫⇧^{+}y. 0 ∂M2 ∂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 ⨂⇩_{M}M2) {x ∈ space (M1 ⨂⇩_{M}M2). ¬ P x} = 0" by simp qed lemma (in pair_sigma_finite) AE_pair_iff: "{x∈space (M1 ⨂⇩_{M}M2). P (fst x) (snd x)} ∈ sets (M1 ⨂⇩_{M}M2) ⟹ (AE x in M1. AE y in M2. P x y) ⟷ (AE x in (M1 ⨂⇩_{M}M2). P (fst x) (snd x))" using AE_pair[of "λx. P (fst x) (snd x)"] AE_pair_measure[of "λx. P (fst x) (snd x)"] by auto lemma (in pair_sigma_finite) AE_commute: assumes P: "{x∈space (M1 ⨂⇩_{M}M2). P (fst x) (snd x)} ∈ sets (M1 ⨂⇩_{M}M2)" shows "(AE x in M1. AE y in M2. P x y) ⟷ (AE y in M2. AE x in M1. P x y)" proof - interpret Q: pair_sigma_finite M2 M1 .. have [simp]: "⋀x. (fst (case x of (x, y) ⇒ (y, x))) = snd x" "⋀x. (snd (case x of (x, y) ⇒ (y, x))) = fst x" by auto have "{x ∈ space (M2 ⨂⇩_{M}M1). P (snd x) (fst x)} = (λ(x, y). (y, x)) -` {x ∈ space (M1 ⨂⇩_{M}M2). P (fst x) (snd x)} ∩ space (M2 ⨂⇩_{M}M1)" by (auto simp: space_pair_measure) also have "… ∈ sets (M2 ⨂⇩_{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 ∈ space M1 ⟹ g ∈ measurable (M1 ⨂⇩_{M}M2) L ⟹ (λy. g (x, y)) ∈ measurable M2 L" by simp lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst: assumes f: "f ∈ borel_measurable (M1 ⨂⇩_{M}M)" shows "(λx. ∫⇧^{+}y. f (x, y) ∂M) ∈ borel_measurable M1" using f proof induct case (cong u v) then have "⋀w x. w ∈ space M1 ⟹ x ∈ space M ⟹ 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]: "⋀x y. indicator Q (x, y) = indicator (Pair x -` Q) y" by (auto simp: indicator_def) have "⋀x. x ∈ space M1 ⟹ emeasure M (Pair x -` Q) = ∫⇧^{+}y. indicator Q (x, y) ∂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 image_comp cong: measurable_cong) lemma (in sigma_finite_measure) nn_integral_fst: assumes f: "f ∈ borel_measurable (M1 ⨂⇩_{M}M)" shows "(∫⇧^{+}x. ∫⇧^{+}y. f (x, y) ∂M ∂M1) = integral⇧^{N}(M1 ⨂⇩_{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 image_comp cong: nn_integral_cong) lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]: "case_prod f ∈ borel_measurable (N ⨂⇩_{M}M) ⟹ (λx. ∫⇧^{+}y. f x y ∂M) ∈ borel_measurable N" using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp proposition (in pair_sigma_finite) nn_integral_snd: assumes f[measurable]: "f ∈ borel_measurable (M1 ⨂⇩_{M}M2)" shows "(∫⇧^{+}y. (∫⇧^{+}x. f (x, y) ∂M1) ∂M2) = integral⇧^{N}(M1 ⨂⇩_{M}M2) f" proof - note measurable_pair_swap[OF f] from M1.nn_integral_fst[OF this] have "(∫⇧^{+}y. (∫⇧^{+}x. f (x, y) ∂M1) ∂M2) = (∫⇧^{+}(x, y). f (y, x) ∂(M2 ⨂⇩_{M}M1))" by simp also have "(∫⇧^{+}(x, y). f (y, x) ∂(M2 ⨂⇩_{M}M1)) = integral⇧^{N}(M1 ⨂⇩_{M}M2) f" by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong) finally show ?thesis . qed theorem (in pair_sigma_finite) Fubini: assumes f: "f ∈ borel_measurable (M1 ⨂⇩_{M}M2)" shows "(∫⇧^{+}y. (∫⇧^{+}x. f (x, y) ∂M1) ∂M2) = (∫⇧^{+}x. (∫⇧^{+}y. f (x, y) ∂M2) ∂M1)" unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] .. theorem (in pair_sigma_finite) Fubini': assumes f: "case_prod f ∈ borel_measurable (M1 ⨂⇩_{M}M2)" shows "(∫⇧^{+}y. (∫⇧^{+}x. f x y ∂M1) ∂M2) = (∫⇧^{+}x. (∫⇧^{+}y. f x y ∂M2) ∂M1)" using Fubini[OF f] by simp subsection ‹Products on counting spaces, densities and distributions› proposition sigma_prod: assumes X_cover: "∃E⊆A. countable E ∧ X = ⋃E" and A: "A ⊆ Pow X" assumes Y_cover: "∃E⊆B. countable E ∧ Y = ⋃E" and B: "B ⊆ Pow Y" shows "sigma X A ⨂⇩_{M}sigma Y B = sigma (X × Y) {a × b | a b. a ∈ A ∧ b ∈ B}" (is "?P = ?S") proof (rule measure_eqI) have [simp]: "snd ∈ X × Y → Y" "fst ∈ X × Y → X" by auto let ?XY = "{{fst -` a ∩ X × Y | a. a ∈ A}, {snd -` b ∩ X × Y | b. b ∈ B}}" have "sets ?P = sets (SUP xy∈?XY. sigma (X × Y) xy)" by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B) also have "… = sets (sigma (X × Y) (⋃?XY))" by (intro Sup_sigma arg_cong[where f=sets]) auto also have "… = sets ?S" proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI) show "⋃?XY ⊆ Pow (X × Y)" "{a × b |a b. a ∈ A ∧ b ∈ B} ⊆ Pow (X × Y)" using A B by auto next interpret XY: sigma_algebra "X × Y" "sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}" using A B by (intro sigma_algebra_sigma_sets) auto fix Z assume "Z ∈ ⋃?XY" then show "Z ∈ sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}" proof safe fix a assume "a ∈ A" from Y_cover obtain E where E: "E ⊆ B" "countable E" and "Y = ⋃E" by auto with ‹a ∈ A› A have eq: "fst -` a ∩ X × Y = (⋃e∈E. a × e)" by auto show "fst -` a ∩ X × Y ∈ sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}" using ‹a ∈ A› E unfolding eq by (auto intro!: XY.countable_UN') next fix b assume "b ∈ B" from X_cover obtain E where E: "E ⊆ A" "countable E" and "X = ⋃E" by auto with ‹b ∈ B› B have eq: "snd -` b ∩ X × Y = (⋃e∈E. e × b)" by auto show "snd -` b ∩ X × Y ∈ sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}" using ‹b ∈ B› E unfolding eq by (auto intro!: XY.countable_UN') qed next fix Z assume "Z ∈ {a × b |a b. a ∈ A ∧ b ∈ B}" then obtain a b where "Z = a × b" and ab: "a ∈ A" "b ∈ B" by auto then have Z: "Z = (fst -` a ∩ X × Y) ∩ (snd -` b ∩ X × Y)" using A B by auto interpret XY: sigma_algebra "X × Y" "sigma_sets (X × Y) (⋃?XY)" by (intro sigma_algebra_sigma_sets) auto show "Z ∈ sigma_sets (X × Y) (⋃?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 ∈ 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 × B) { a × b | a b. a ⊆ A ∧ b ⊆ B} = Pow (A × B)" (is "sigma_sets ?prod ?sets = _") proof safe have fin: "finite (A × B)" using assms by (rule finite_cartesian_product) fix x assume subset: "x ⊆ A × B" hence "finite x" using fin by (rule finite_subset) from this subset show "x ∈ sigma_sets ?prod ?sets" proof (induct x) case empty show ?case by (rule sigma_sets.Empty) next case (insert a x) hence "{a} ∈ sigma_sets ?prod ?sets" by auto moreover have "x ∈ 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 ∈ sigma_sets ?prod ?sets" and "(a, b) ∈ x" from sigma_sets_into_sp[OF _ this(1)] this(2) show "a ∈ A" and "b ∈ B" by auto qed proposition sets_pair_eq: assumes Ea: "Ea ⊆ Pow (space A)" "sets A = sigma_sets (space A) Ea" and Ca: "countable Ca" "Ca ⊆ Ea" "⋃Ca = space A" and Eb: "Eb ⊆ Pow (space B)" "sets B = sigma_sets (space B) Eb" and Cb: "countable Cb" "Cb ⊆ Eb" "⋃Cb = space B" shows "sets (A ⨂⇩_{M}B) = sets (sigma (space A × space B) { a × b | a b. a ∈ Ea ∧ b ∈ Eb })" (is "_ = sets (sigma ?Ω ?E)") proof show "sets (sigma ?Ω ?E) ⊆ sets (A ⨂⇩_{M}B)" using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2)) have "?E ⊆ Pow ?Ω" using Ea(1) Eb(1) by auto then have E: "a ∈ Ea ⟹ b ∈ Eb ⟹ a × b ∈ sets (sigma ?Ω ?E)" for a b by auto have "sets (A ⨂⇩_{M}B) ⊆ sets (Sup {vimage_algebra ?Ω fst A, vimage_algebra ?Ω snd B})" unfolding sets_pair_eq_sets_fst_snd .. also have "vimage_algebra ?Ω fst A = vimage_algebra ?Ω fst (sigma (space A) Ea)" by (intro vimage_algebra_cong[OF refl refl]) (simp add: Ea) also have "… = sigma ?Ω {fst -` A ∩ ?Ω |A. A ∈ Ea}" by (intro Ea vimage_algebra_sigma) auto also have "vimage_algebra ?Ω snd B = vimage_algebra ?Ω snd (sigma (space B) Eb)" by (intro vimage_algebra_cong[OF refl refl]) (simp add: Eb) also have "… = sigma ?Ω {snd -` A ∩ ?Ω |A. A ∈ Eb}" by (intro Eb vimage_algebra_sigma) auto also have "{sigma ?Ω {fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, sigma ?Ω {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}} = sigma ?Ω ` {{fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}}" by auto also have "sets (SUP S∈{{fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}}. sigma ?Ω S) = sets (sigma ?Ω (⋃{{fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}}))" using Ea(1) Eb(1) by (intro sets_Sup_sigma) auto also have "… ⊆ sets (sigma ?Ω ?E)" proof (subst sigma_le_sets, safe intro!: space_in_measure_of) fix a assume "a ∈ Ea" then have "fst -` a ∩ ?Ω = (⋃b∈Cb. a × b)" using Cb(3)[symmetric] Ea(1) by auto then show "fst -` a ∩ ?Ω ∈ sets (sigma ?Ω ?E)" using Cb ‹a ∈ Ea› by (auto intro!: sets.countable_UN' E) next fix b assume "b ∈ Eb" then have "snd -` b ∩ ?Ω = (⋃a∈Ca. a × b)" using Ca(3)[symmetric] Eb(1) by auto then show "snd -` b ∩ ?Ω ∈ sets (sigma ?Ω ?E)" using Ca ‹b ∈ Eb› by (auto intro!: sets.countable_UN' E) qed finally show "sets (A ⨂⇩_{M}B) ⊆ sets (sigma ?Ω ?E)" . qed proposition borel_prod: "(borel ⨂⇩_{M}borel) = (borel :: ('a::second_countable_topology × 'b::second_countable_topology) measure)" (is "?P = ?B") proof - have "?B = sigma UNIV {A × B | A B. open A ∧ open B}" by (rule second_countable_borel_measurable[OF open_prod_generated]) also have "… = ?P" unfolding borel_def by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"]) finally show ?thesis .. qed proposition pair_measure_count_space: assumes A: "finite A" and B: "finite B" shows "count_space A ⨂⇩_{M}count_space B = count_space (A × 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 ∈ sets ?P" with eq have X_subset: "X ⊆ A × B" by simp with A B have fin_Pair: "⋀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 card: "0 < card (Pair a -` X)" if "(a, b) ∈ X" for a b using card_gt_0_iff fin_Pair that by auto then have "emeasure ?P X = ∫⇧^{+}x. emeasure (count_space B) (Pair x -` X) ∂count_space A" by (simp add: B.emeasure_pair_measure_alt X) also have "... = emeasure ?C X" apply (subst emeasure_count_space) using card X_subset A fin_Pair fin_X apply (auto simp add: nn_integral_count_space of_nat_sum[symmetric] card_SigmaI[symmetric] simp del: card_SigmaI intro!: arg_cong[where f=card]) done finally show "emeasure ?P X = emeasure ?C X" . qed lemma emeasure_prod_count_space: assumes A: "A ∈ sets (count_space UNIV ⨂⇩_{M}M)" (is "A ∈ sets (?A ⨂⇩_{M}?B)") shows "emeasure (?A ⨂⇩_{M}?B) A = (∫⇧^{+}x. ∫⇧^{+}y. indicator A (x, y) ∂?B ∂?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 ⨂⇩_{M}count_space UNIV) {x} = 1" proof - have [simp]: "⋀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 × 'b) set" assumes A: "A ∈ sets (count_space UNIV ⨂⇩_{M}count_space UNIV)" (is "A ∈ sets (?A ⨂⇩_{M}?B)") shows "emeasure (?A ⨂⇩_{M}?B) A = emeasure (count_space UNIV) A" proof - { fix A :: "('a × 'b) set" assume "countable A" then have "emeasure (?A ⨂⇩_{M}?B) (⋃a∈A. {a}) = (∫⇧^{+}a. emeasure (?A ⨂⇩_{M}?B) {a} ∂count_space A)" by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def) also have "… = (∫⇧^{+}a. indicator A a ∂count_space UNIV)" by (subst nn_integral_count_space_indicator) auto finally have "emeasure (?A ⨂⇩_{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 ⊆ A" by (auto dest: infinite_countable_subset') with A have "emeasure (?A ⨂⇩_{M}?B) C ≤ emeasure (?A ⨂⇩_{M}?B) A" by (intro emeasure_mono) auto also have "emeasure (?A ⨂⇩_{M}?B) C = emeasure (count_space UNIV) C" using ‹countable C› by (rule *) finally show ?thesis using ‹infinite C› ‹infinite A› by (simp add: top_unique) qed qed lemma nn_integral_count_space_prod_eq: "nn_integral (count_space UNIV ⨂⇩_{M}count_space UNIV) f = nn_integral (count_space UNIV) f" (is "nn_integral ?P f = _") proof cases assume cntbl: "countable {x. f x ≠ 0}" have [simp]: "⋀x. card ({x} ∩ {x. f x ≠ 0}) = (indicator {x. f x ≠ 0} x::ennreal)" by (auto split: split_indicator) have [measurable]: "⋀y. (λx. indicator {y} x) ∈ borel_measurable ?P" by (rule measurable_discrete_difference[of "λx. 0" _ borel "{y}" "λx. indicator {y} x" for y]) (auto intro: sets_Pair) have "(∫⇧^{+}x. f x ∂?P) = (∫⇧^{+}x. ∫⇧^{+}x'. f x * indicator {x} x' ∂count_space {x. f x ≠ 0} ∂?P)" by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator) also have "… = (∫⇧^{+}x. ∫⇧^{+}x'. f x' * indicator {x'} x ∂count_space {x. f x ≠ 0} ∂?P)" by (auto intro!: nn_integral_cong split: split_indicator) also have "… = (∫⇧^{+}x'. ∫⇧^{+}x. f x' * indicator {x'} x ∂?P ∂count_space {x. f x ≠ 0})" by (intro nn_integral_count_space_nn_integral cntbl) auto also have "… = (∫⇧^{+}x'. f x' ∂count_space {x. f x ≠ 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 ≠ 0" then have "(∃r≥0. 0 < r ∧ f x = ennreal r) ∨ f x = ∞" by (cases "f x" rule: ennreal_cases) (auto simp: less_le) then have "∃n. ennreal (1 / real (Suc n)) ≤ f x" by (auto elim!: nat_approx_posE intro!: less_imp_le) } note * = this assume cntbl: "uncountable {x. f x ≠ 0}" also have "{x. f x ≠ 0} = (⋃n. {x. 1/Suc n ≤ f x})" using * by auto finally obtain n where "infinite {x. 1/Suc n ≤ f x}" by (meson countableI_type countable_UN uncountable_infinite) then obtain C where C: "C ⊆ {x. 1/Suc n ≤ f x}" and "countable C" "infinite C" by (metis infinite_countable_subset') have [measurable]: "C ∈ sets ?P" using sets.countable[OF _ ‹countable C›, of ?P] by (auto simp: sets_Pair) have "(∫⇧^{+}x. ennreal (1/Suc n) * indicator C x ∂?P) ≤ nn_integral ?P f" using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) moreover have "(∫⇧^{+}x. ennreal (1/Suc n) * indicator C x ∂?P) = ∞" using ‹infinite C› by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top) moreover have "(∫⇧^{+}x. ennreal (1/Suc n) * indicator C x ∂count_space UNIV) ≤ nn_integral (count_space UNIV) f" using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric]) moreover have "(∫⇧^{+}x. ennreal (1/Suc n) * indicator C x ∂count_space UNIV) = ∞" using ‹infinite C› by (simp add: nn_integral_cmult ennreal_mult_top) ultimately show ?thesis by (simp add: top_unique) qed theorem pair_measure_density: assumes f: "f ∈ borel_measurable M1" assumes g: "g ∈ borel_measurable M2" assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)" shows "density M1 f ⨂⇩_{M}density M2 g = density (M1 ⨂⇩_{M}M2) (λ(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 ∈ sets ?L" with f g have "(∫⇧^{+}x. f x * ∫⇧^{+}y. g y * indicator A (x, y) ∂M2 ∂M1) = (∫⇧^{+}x. ∫⇧^{+}y. f x * g y * indicator A (x, y) ∂M2 ∂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 ∈ 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 "∃A. countable A ∧ A ⊆ sets M ∧ ⋃A = space M ∧ (∀a∈A. emeasure M a ≠ ∞)" using A f by (intro exI[of _ "(λa. f -` a ∩ space M) ` A"]) (auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space) qed qed lemma pair_measure_distr: assumes f: "f ∈ measurable M S" and g: "g ∈ measurable N T" assumes "sigma_finite_measure (distr N T g)" shows "distr M S f ⨂⇩_{M}distr N T g = distr (M ⨂⇩_{M}N) (S ⨂⇩_{M}T) (λ(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 ∈ 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 ⨂⇩_{M}M2) = sets M" assumes emeasure: "⋀A B. A ∈ sets M1 ⟹ B ∈ sets M2 ⟹ emeasure M1 A * emeasure M2 B = emeasure M (A × B)" shows "M1 ⨂⇩_{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 ⇒ ('a × 'b) set" .. note F = this let ?E = "{a × b |a b. a ∈ sets M1 ∧ b ∈ sets M2}" let ?P = "M1 ⨂⇩_{M}M2" show ?thesis proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) show "?E ⊆ 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 ⊆ ?E" "(⋃i. F i) = space ?P" "⋀i. emeasure ?P (F i) ≠ ∞" using F by (auto simp: space_pair_measure) next fix X assume "X ∈ ?E" then obtain A B where X[simp]: "X = A × B" and A: "A ∈ sets M1" and B: "B ∈ 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 "… = emeasure M (A × 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 ⨂⇩_{M}N) = Pow (S1 × S2)" proof auto fix x a b assume x: "x ∈ sets (M ⨂⇩_{M}N)" "(a, b) ∈ 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 ∈ S1" "b ∈ S2" by (auto simp: space_pair_measure) next fix X assume X: "X ⊆ S1 × S2" then have "countable X" by (metis countable_subset ‹countable S1› ‹countable S2› countable_SIGMA) have "X = (⋃(a, b)∈X. {a} × {b})" by auto also have "… ∈ sets (M ⨂⇩_{M}N)" using X by (safe intro!: sets.countable_UN' ‹countable X› subsetI pair_measureI) (auto simp: M N) finally show "X ∈ sets (M ⨂⇩_{M}N)" . qed lemma pair_measure_countable: assumes "countable S1" "countable S2" shows "count_space S1 ⨂⇩_{M}count_space S2 = count_space (S1 × 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 ⨂⇩_{M}count_space S2) = sets (count_space (S1 × S2))" by (subst sets_pair_countable[OF assms]) auto next fix A B assume "A ∈ sets (count_space S1)" "B ∈ sets (count_space S2)" then show "emeasure (count_space S1) A * emeasure (count_space S2) B = emeasure (count_space (S1 × S2)) (A × B)" by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult) qed proposition nn_integral_fst_count_space: "(∫⇧^{+}x. ∫⇧^{+}y. f (x, y) ∂count_space UNIV ∂count_space UNIV) = integral⇧^{N}(count_space UNIV) f" (is "?lhs = ?rhs") proof(cases) assume *: "countable {xy. f xy ≠ 0}" let ?A = "fst ` {xy. f xy ≠ 0}" let ?B = "snd ` {xy. f xy ≠ 0}" from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+ have "?lhs = (∫⇧^{+}x. ∫⇧^{+}y. f (x, y) ∂count_space UNIV ∂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 "… = (∫⇧^{+}x. ∫⇧^{+}y. f (x, y) ∂count_space ?B ∂count_space ?A)" by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI) also have "… = (∫⇧^{+}xy. f xy ∂count_space (?A × ?B))" by(subst sigma_finite_measure.nn_integral_fst) (simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable) also have "… = ?rhs" by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI) finally show ?thesis . next { fix xy assume "f xy ≠ 0" then have "(∃r≥0. 0 < r ∧ f xy = ennreal r) ∨ f xy = ∞" by (cases "f xy" rule: ennreal_cases) (auto simp: less_le) then have "∃n. ennreal (1 / real (Suc n)) ≤ f xy" by (auto elim!: nat_approx_posE intro!: less_imp_le) } note * = this assume cntbl: "uncountable {xy. f xy ≠ 0}" also have "{xy. f xy ≠ 0} = (⋃n. {xy. 1/Suc n ≤ f xy})" using * by auto finally obtain n where "infinite {xy. 1/Suc n ≤ f xy}" by (meson countableI_type countable_UN uncountable_infinite) then obtain C where C: "C ⊆ {xy. 1/Suc n ≤ f xy}" and "countable C" "infinite C" by (metis infinite_countable_subset') have "∞ = (∫⇧^{+}xy. ennreal (1 / Suc n) * indicator C xy ∂count_space UNIV)" using ‹infinite C› by(simp add: nn_integral_cmult ennreal_mult_top) also have "… ≤ ?rhs" using C by(intro nn_integral_mono)(auto split: split_indicator) finally have "?rhs = ∞" by (simp add: top_unique) moreover have "?lhs = ∞" proof(cases "finite (fst ` C)") case True then obtain x C' where x: "x ∈ fst ` C" and C': "C' = fst -` {x} ∩ C" and "infinite C'" using ‹infinite C› by(auto elim!: inf_img_fin_domE') from x C C' have **: "C' ⊆ {xy. 1 / Suc n ≤ f xy}" by auto from C' ‹infinite C'› have "infinite (snd ` C')" by(auto dest!: finite_imageD simp add: inj_on_def) then have "∞ = (∫⇧^{+}y. ennreal (1 / Suc n) * indicator (snd ` C') y ∂count_space UNIV)" by(simp add: nn_integral_cmult ennreal_mult_top) also have "… = (∫⇧^{+}y. ennreal (1 / Suc n) * indicator C' (x, y) ∂count_space UNIV)" by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C') also have "… = (∫⇧^{+}x'. (∫⇧^{+}y. ennreal (1 / Suc n) * indicator C' (x, y) ∂count_space UNIV) * indicator {x} x' ∂count_space UNIV)" by(simp add: one_ereal_def[symmetric]) also have "… ≤ (∫⇧^{+}x. ∫⇧^{+}y. ennreal (1 / Suc n) * indicator C' (x, y) ∂count_space UNIV ∂count_space UNIV)" by(rule nn_integral_mono)(simp split: split_indicator) also have "… ≤ ?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 "∞ = ∫⇧^{+}x. ennreal (1 / Suc n) * indicator C' x ∂count_space UNIV" using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top) also have "… = ∫⇧^{+}x. ∫⇧^{+}y. ennreal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) ∈ C} y ∂count_space UNIV ∂count_space UNIV" by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong) also have "… ≤ ∫⇧^{+}x. ∫⇧^{+}y. ennreal (1 / Suc n) * indicator C (x, y) ∂count_space UNIV ∂count_space UNIV" by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI) also have "… ≤ ?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 proposition nn_integral_snd_count_space: "(∫⇧^{+}y. ∫⇧^{+}x. f (x, y) ∂count_space UNIV ∂count_space UNIV) = integral⇧^{N}(count_space UNIV) f" (is "?lhs = ?rhs") proof - have "?lhs = (∫⇧^{+}y. ∫⇧^{+}x. (λ(y, x). f (x, y)) (y, x) ∂count_space UNIV ∂count_space UNIV)" by(simp) also have "… = ∫⇧^{+}yx. (λ(y, x). f (x, y)) yx ∂count_space UNIV" by(rule nn_integral_fst_count_space) also have "… = ∫⇧^{+}xy. f xy ∂count_space ((λ(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 "… = ?rhs" by(rule nn_integral_count_space_eq) auto finally show ?thesis . qed lemma measurable_pair_measure_countable1: assumes "countable A" and [measurable]: "⋀x. x ∈ A ⟹ (λy. f (x, y)) ∈ measurable N K" shows "f ∈ measurable (count_space A ⨂⇩_{M}N) K" using _ _ assms(1) by(rule measurable_compose_countable'[where f="λa b. f (a, snd b)" and g=fst and I=A, simplified])simp_all subsection ‹Product of Borel spaces› theorem borel_Times: fixes A :: "'a::topological_space set" and B :: "'b::topological_space set" assumes A: "A ∈ sets borel" and B: "B ∈ sets borel" shows "A × B ∈ sets borel" proof - have "A × B = (A×UNIV) ∩ (UNIV × B)" by auto moreover { have "A ∈ sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel) then have "A×UNIV ∈ 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) × UNIV = UNIV - (A × UNIV)" by auto ultimately show ?case unfolding * by auto next case (Union A) moreover have *: "(⋃(A ` UNIV)) × UNIV = ⋃((λi. A i × UNIV) ` UNIV)" by auto ultimately show ?case unfolding * by auto qed simp } moreover { have "B ∈ sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel) then have "UNIV×B ∈ 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 × (UNIV - B) = UNIV - (UNIV × B)" by auto ultimately show ?case unfolding * by auto next case (Union B) moreover have *: "UNIV × (⋃(B ` UNIV)) = ⋃((λi. UNIV × B i) ` UNIV)" 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 ⨂⇩_{M}M)" proof (rule finite_measureI) interpret M: finite_measure M by fact interpret N: finite_measure N by fact show "emeasure (N ⨂⇩_{M}M) (space (N ⨂⇩_{M}M)) ≠ ∞" by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff) qed end