(* Title: HOL/Probability/Projective_Limit.thy Author: Fabian Immler, TU München *) section ‹Projective Limit› theory Projective_Limit imports Fin_Map Infinite_Product_Measure "HOL-Library.Diagonal_Subsequence" begin subsection ‹Sequences of Finite Maps in Compact Sets› locale finmap_seqs_into_compact = fixes K::"nat ⇒ (nat ⇒⇩_{F}'a::metric_space) set" and f::"nat ⇒ (nat ⇒⇩_{F}'a)" and M assumes compact: "⋀n. compact (K n)" assumes f_in_K: "⋀n. K n ≠ {}" assumes domain_K: "⋀n. k ∈ K n ⟹ domain k = domain (f n)" assumes proj_in_K: "⋀t n m. m ≥ n ⟹ t ∈ domain (f n) ⟹ (f m)⇩_{F}t ∈ (λk. (k)⇩_{F}t) ` K n" begin lemma proj_in_K': "(∃n. ∀m ≥ n. (f m)⇩_{F}t ∈ (λk. (k)⇩_{F}t) ` K n)" using proj_in_K f_in_K proof cases obtain k where "k ∈ K (Suc 0)" using f_in_K by auto assume "∀n. t ∉ domain (f n)" thus ?thesis by (auto intro!: exI[where x=1] image_eqI[OF _ ‹k ∈ K (Suc 0)›] simp: domain_K[OF ‹k ∈ K (Suc 0)›]) qed blast lemma proj_in_KE: obtains n where "⋀m. m ≥ n ⟹ (f m)⇩_{F}t ∈ (λk. (k)⇩_{F}t) ` K n" using proj_in_K' by blast lemma compact_projset: shows "compact ((λk. (k)⇩_{F}i) ` K n)" using continuous_proj compact by (rule compact_continuous_image) end lemma compactE': fixes S :: "'a :: metric_space set" assumes "compact S" "∀n≥m. f n ∈ S" obtains l r where "l ∈ S" "strict_mono (r::nat⇒nat)" "((f ∘ r) ⤏ l) sequentially" proof atomize_elim have "strict_mono ((+) m)" by (simp add: strict_mono_def) have "∀n. (f o (λi. m + i)) n ∈ S" using assms by auto from seq_compactE[OF ‹compact S›[unfolded compact_eq_seq_compact_metric] this] obtain l r where "l ∈ S" "strict_mono r" "(f ∘ (+) m ∘ r) ⇢ l" by blast hence "l ∈ S" "strict_mono ((λi. m + i) o r) ∧ (f ∘ ((λi. m + i) o r)) ⇢ l" using strict_mono_o[OF ‹strict_mono ((+) m)› ‹strict_mono r›] by (auto simp: o_def) thus "∃l r. l ∈ S ∧ strict_mono r ∧ (f ∘ r) ⇢ l" by blast qed sublocale finmap_seqs_into_compact ⊆ subseqs "λn s. (∃l. (λi. ((f o s) i)⇩_{F}n) ⇢ l)" proof fix n and s :: "nat ⇒ nat" assume "strict_mono s" from proj_in_KE[of n] obtain n0 where n0: "⋀m. n0 ≤ m ⟹ (f m)⇩_{F}n ∈ (λk. (k)⇩_{F}n) ` K n0" by blast have "∀i ≥ n0. ((f ∘ s) i)⇩_{F}n ∈ (λk. (k)⇩_{F}n) ` K n0" proof safe fix i assume "n0 ≤ i" also have "… ≤ s i" by (rule seq_suble) fact finally have "n0 ≤ s i" . with n0 show "((f ∘ s) i)⇩_{F}n ∈ (λk. (k)⇩_{F}n) ` K n0 " by auto qed then obtain ls rs where "ls ∈ (λk. (k)⇩_{F}n) ` K n0" "strict_mono rs" "((λi. ((f ∘ s) i)⇩_{F}n) ∘ rs) ⇢ ls" by (rule compactE'[OF compact_projset]) thus "∃r'. strict_mono r' ∧ (∃l. (λi. ((f ∘ (s ∘ r')) i)⇩_{F}n) ⇢ l)" by (auto simp: o_def) qed lemma (in finmap_seqs_into_compact) diagonal_tendsto: "∃l. (λi. (f (diagseq i))⇩_{F}n) ⇢ l" proof - obtain l where "(λi. ((f o (diagseq o (+) (Suc n))) i)⇩_{F}n) ⇢ l" proof (atomize_elim, rule diagseq_holds) fix r s n assume "strict_mono (r :: nat ⇒ nat)" assume "∃l. (λi. ((f ∘ s) i)⇩_{F}n) ⇢ l" then obtain l where "((λi. (f i)⇩_{F}n) o s) ⇢ l" by (auto simp: o_def) hence "((λi. (f i)⇩_{F}n) o s o r) ⇢ l" using ‹strict_mono r› by (rule LIMSEQ_subseq_LIMSEQ) thus "∃l. (λi. ((f ∘ (s ∘ r)) i)⇩_{F}n) ⇢ l" by (auto simp add: o_def) qed hence "(λi. ((f (diagseq (i + Suc n))))⇩_{F}n) ⇢ l" by (simp add: ac_simps) hence "(λi. (f (diagseq i))⇩_{F}n) ⇢ l" by (rule LIMSEQ_offset) thus ?thesis .. qed subsection ‹Daniell-Kolmogorov Theorem› text ‹Existence of Projective Limit› locale polish_projective = projective_family I P "λ_. borel::'a::polish_space measure" for I::"'i set" and P begin lemma emeasure_lim_emb: assumes X: "J ⊆ I" "finite J" "X ∈ sets (Π⇩_{M}i∈J. borel)" shows "lim (emb I J X) = P J X" proof (rule emeasure_lim) write mu_G ("μG") interpret generator: algebra "space (PiM I (λi. borel))" generator by (rule algebra_generator) fix J and B :: "nat ⇒ ('i ⇒ 'a) set" assume J: "⋀n. finite (J n)" "⋀n. J n ⊆ I" "⋀n. B n ∈ sets (Π⇩_{M}i∈J n. borel)" "incseq J" and B: "decseq (λn. emb I (J n) (B n))" and "0 < (INF i. P (J i) (B i))" (is "0 < ?a") moreover have "?a ≤ 1" using J by (auto intro!: INF_lower2[of 0] prob_space_P[THEN prob_space.measure_le_1]) ultimately obtain r where r: "?a = ennreal r" "0 < r" "r ≤ 1" by (cases "?a") (auto simp: top_unique) define Z where "Z n = emb I (J n) (B n)" for n have Z_mono: "n ≤ m ⟹ Z m ⊆ Z n" for n m unfolding Z_def using B[THEN antimonoD, of n m] . have J_mono: "⋀n m. n ≤ m ⟹ J n ⊆ J m" using ‹incseq J› by (force simp: incseq_def) note [simp] = ‹⋀n. finite (J n)› interpret prob_space "P (J i)" for i using J prob_space_P by simp have P_eq[simp]: "sets (P (J i)) = sets (Π⇩_{M}i∈J i. borel)" "space (P (J i)) = space (Π⇩_{M}i∈J i. borel)" for i using J by (auto simp: sets_P space_P) have "Z i ∈ generator" for i unfolding Z_def by (auto intro!: generator.intros J) have countable_UN_J: "countable (⋃n. J n)" by (simp add: countable_finite) define Utn where "Utn = to_nat_on (⋃n. J n)" interpret function_to_finmap "J n" Utn "from_nat_into (⋃n. J n)" for n by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J]) have inj_on_Utn: "inj_on Utn (⋃n. J n)" unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on) hence inj_on_Utn_J: "⋀n. inj_on Utn (J n)" by (rule subset_inj_on) auto define P' where "P' n = mapmeasure n (P (J n)) (λ_. borel)" for n interpret P': prob_space "P' n" for n unfolding P'_def mapmeasure_def using J by (auto intro!: prob_space_distr fm_measurable simp: measurable_cong_sets[OF sets_P]) let ?SUP = "λn. SUP K ∈ {K. K ⊆ fm n ` (B n) ∧ compact K}. emeasure (P' n) K" { fix n have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))" using J by (auto simp: P'_def mapmeasure_PiM space_P sets_P) also have "… = ?SUP n" proof (rule inner_regular) show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def) next show "fm n ` B n ∈ sets borel" unfolding borel_eq_PiF_borel by (auto simp: P'_def fm_image_measurable_finite sets_P J(3)) qed simp finally have *: "emeasure (P (J n)) (B n) = ?SUP n" . have "?SUP n ≠ ∞" unfolding *[symmetric] by simp note * this } note R = this have "∀n. ∃K. emeasure (P (J n)) (B n) - emeasure (P' n) K ≤ 2 powr (-n) * ?a ∧ compact K ∧ K ⊆ fm n ` B n" proof fix n show "∃K. emeasure (P (J n)) (B n) - emeasure (P' n) K ≤ ennreal (2 powr - real n) * ?a ∧ compact K ∧ K ⊆ fm n ` B n" unfolding R[of n] proof (rule ccontr) assume H: "∄K'. ?SUP n - emeasure (P' n) K' ≤ ennreal (2 powr - real n) * ?a ∧ compact K' ∧ K' ⊆ fm n ` B n" have "?SUP n + 0 < ?SUP n + 2 powr (-n) * ?a" using R[of n] unfolding ennreal_add_left_cancel_less ennreal_zero_less_mult_iff by (auto intro: ‹0 < ?a›) also have "… = (SUP K∈{K. K ⊆ fm n ` B n ∧ compact K}. emeasure (P' n) K + 2 powr (-n) * ?a)" by (rule ennreal_SUP_add_left[symmetric]) auto also have "… ≤ ?SUP n" proof (intro SUP_least) fix K assume "K ∈ {K. K ⊆ fm n ` B n ∧ compact K}" with H have "2 powr (-n) * ?a < ?SUP n - emeasure (P' n) K" by auto then show "emeasure (P' n) K + (2 powr (-n)) * ?a ≤ ?SUP n" by (subst (asm) less_diff_eq_ennreal) (auto simp: less_top[symmetric] R(1)[symmetric] ac_simps) qed finally show False by simp qed qed then obtain K' where K': "⋀n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) ≤ ennreal (2 powr - real n) * ?a" "⋀n. compact (K' n)" "⋀n. K' n ⊆ fm n ` B n" unfolding choice_iff by blast define K where "K n = fm n -` K' n ∩ space (Pi⇩_{M}(J n) (λ_. borel))" for n have K_sets: "⋀n. K n ∈ sets (Pi⇩_{M}(J n) (λ_. borel))" unfolding K_def using compact_imp_closed[OF ‹compact (K' _)›] by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"]) (auto simp: borel_eq_PiF_borel[symmetric]) have K_B: "⋀n. K n ⊆ B n" proof fix x n assume "x ∈ K n" then have fm_in: "fm n x ∈ fm n ` B n" using K' by (force simp: K_def) show "x ∈ B n" using ‹x ∈ K n› K_sets sets.sets_into_space J(1,2,3)[of n] inj_on_image_mem_iff[OF inj_on_fm] by (metis (no_types) Int_iff K_def fm_in space_borel) qed define Z' where "Z' n = emb I (J n) (K n)" for n have Z': "⋀n. Z' n ⊆ Z n" unfolding Z'_def Z_def proof (rule prod_emb_mono, safe) fix n x assume "x ∈ K n" hence "fm n x ∈ K' n" "x ∈ space (Pi⇩_{M}(J n) (λ_. borel))" by (simp_all add: K_def space_P) note this(1) also have "K' n ⊆ fm n ` B n" by (simp add: K') finally have "fm n x ∈ fm n ` B n" . thus "x ∈ B n" proof safe fix y assume y: "y ∈ B n" hence "y ∈ space (Pi⇩_{M}(J n) (λ_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"] by (auto simp add: space_P sets_P) assume "fm n x = fm n y" note inj_onD[OF inj_on_fm[OF space_borel], OF ‹fm n x = fm n y› ‹x ∈ space _› ‹y ∈ space _›] with y show "x ∈ B n" by simp qed qed have "⋀n. Z' n ∈ generator" using J K'(2) unfolding Z'_def by (auto intro!: generator.intros measurable_sets[OF fm_measurable[of _ "Collect finite"]] simp: K_def borel_eq_PiF_borel[symmetric] compact_imp_closed) define Y where "Y n = (⋂i∈{1..n}. Z' i)" for n hence "⋀n k. Y (n + k) ⊆ Y n" by (induct_tac k) (auto simp: Y_def) hence Y_mono: "⋀n m. n ≤ m ⟹ Y m ⊆ Y n" by (auto simp: le_iff_add) have Y_Z': "⋀n. n ≥ 1 ⟹ Y n ⊆ Z' n" by (auto simp: Y_def) hence Y_Z: "⋀n. n ≥ 1 ⟹ Y n ⊆ Z n" using Z' by auto have Y_notempty: "⋀n. n ≥ 1 ⟹ (Y n) ≠ {}" proof - fix n::nat assume "n ≥ 1" hence "Y n ⊆ Z n" by fact have "Y n = (⋂i∈{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono by (auto simp: Y_def Z'_def) also have "… = prod_emb I (λ_. borel) (J n) (⋂i∈{1..n}. emb (J n) (J i) (K i))" using ‹n ≥ 1› by (subst prod_emb_INT) auto finally have Y_emb: "Y n = prod_emb I (λ_. borel) (J n) (⋂i∈{1..n}. prod_emb (J n) (λ_. borel) (J i) (K i))" . hence "Y n ∈ generator" using J J_mono K_sets ‹n ≥ 1› by (auto simp del: prod_emb_INT intro!: generator.intros) have *: "μG (Z n) = P (J n) (B n)" unfolding Z_def using J by (intro mu_G_spec) auto then have "μG (Z n) ≠ ∞" by auto note * moreover have *: "μG (Y n) = P (J n) (⋂i∈{Suc 0..n}. prod_emb (J n) (λ_. borel) (J i) (K i))" unfolding Y_emb using J J_mono K_sets ‹n ≥ 1› by (subst mu_G_spec) auto then have "μG (Y n) ≠ ∞" by auto note * moreover have "μG (Z n - Y n) = P (J n) (B n - (⋂i∈{Suc 0..n}. prod_emb (J n) (λ_. borel) (J i) (K i)))" unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets ‹n ≥ 1› by (subst mu_G_spec) (auto intro!: sets.Diff) ultimately have "μG (Z n) - μG (Y n) = μG (Z n - Y n)" using J J_mono K_sets ‹n ≥ 1› by (simp only: emeasure_eq_measure Z_def) (auto dest!: bspec[where x=n] intro!: measure_Diff[symmetric] subsetD[OF K_B] intro!: arg_cong[where f=ennreal] simp: extensional_restrict emeasure_eq_measure prod_emb_iff sets_P space_P ennreal_minus measure_nonneg) also have subs: "Z n - Y n ⊆ (⋃i∈{1..n}. (Z i - Z' i))" using ‹n ≥ 1› unfolding Y_def UN_extend_simps(7) by (intro UN_mono Diff_mono Z_mono order_refl) auto have "Z n - Y n ∈ generator" "(⋃i∈{1..n}. (Z i - Z' i)) ∈ generator" using ‹Z' _ ∈ generator› ‹Z _ ∈ generator› ‹Y _ ∈ generator› by auto hence "μG (Z n - Y n) ≤ μG (⋃i∈{1..n}. (Z i - Z' i))" using subs generator.additive_increasing[OF positive_mu_G additive_mu_G] unfolding increasing_def by auto also have "… ≤ (∑ i∈{1..n}. μG (Z i - Z' i))" using ‹Z _ ∈ generator› ‹Z' _ ∈ generator› by (intro generator.subadditive[OF positive_mu_G additive_mu_G]) auto also have "… ≤ (∑ i∈{1..n}. 2 powr -real i * ?a)" proof (rule sum_mono) fix i assume "i ∈ {1..n}" hence "i ≤ n" by simp have "μG (Z i - Z' i) = μG (prod_emb I (λ_. borel) (J i) (B i - K i))" unfolding Z'_def Z_def by simp also have "… = P (J i) (B i - K i)" using J K_sets by (subst mu_G_spec) auto also have "… = P (J i) (B i) - P (J i) (K i)" using K_sets J ‹K _ ⊆ B _› by (simp add: emeasure_Diff) also have "… = P (J i) (B i) - P' i (K' i)" unfolding K_def P'_def by (auto simp: mapmeasure_PiF borel_eq_PiF_borel[symmetric] compact_imp_closed[OF ‹compact (K' _)›] space_PiM PiE_def) also have "… ≤ ennreal (2 powr - real i) * ?a" using K'(1)[of i] . finally show "μG (Z i - Z' i) ≤ (2 powr - real i) * ?a" . qed also have "… = ennreal ((∑ i∈{1..n}. (2 powr -enn2real i)) * enn2real ?a)" using r by (simp add: sum_distrib_right ennreal_mult[symmetric]) also have "… < ennreal (1 * enn2real ?a)" proof (intro ennreal_lessI mult_strict_right_mono) have "(∑i = 1..n. 2 powr - real i) = (∑i = 1..<Suc n. (1/2) ^ i)" by (rule sum.cong) (auto simp: powr_realpow powr_divide power_divide powr_minus_divide) also have "{1..<Suc n} = {..<Suc n} - {0}" by auto also have "sum ((^) (1 / 2::real)) ({..<Suc n} - {0}) = sum ((^) (1 / 2)) ({..<Suc n}) - 1" by (auto simp: sum_diff1) also have "… < 1" by (subst geometric_sum) auto finally show "(∑i = 1..n. 2 powr - enn2real i) < 1" by simp qed (auto simp: r enn2real_positive_iff) also have "… = ?a" by (auto simp: r) also have "… ≤ μG (Z n)" using J by (auto intro: INF_lower simp: Z_def mu_G_spec) finally have "μG (Z n) - μG (Y n) < μG (Z n)" . hence R: "μG (Z n) < μG (Z n) + μG (Y n)" using ‹μG (Y n) ≠ ∞› by (auto simp: zero_less_iff_neq_zero) then have "μG (Y n) > 0" by simp thus "Y n ≠ {}" using positive_mu_G by (auto simp add: positive_def) qed hence "∀n∈{1..}. ∃y. y ∈ Y n" by auto then obtain y where y: "⋀n. n ≥ 1 ⟹ y n ∈ Y n" unfolding bchoice_iff by force { fix t and n m::nat assume "1 ≤ n" "n ≤ m" hence "1 ≤ m" by simp from Y_mono[OF ‹m ≥ n›] y[OF ‹1 ≤ m›] have "y m ∈ Y n" by auto also have "… ⊆ Z' n" using Y_Z'[OF ‹1 ≤ n›] . finally have "fm n (restrict (y m) (J n)) ∈ K' n" unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)" using J by (simp add: fm_def) ultimately have "fm n (y m) ∈ K' n" by simp } note fm_in_K' = this interpret finmap_seqs_into_compact "λn. K' (Suc n)" "λk. fm (Suc k) (y (Suc k))" borel proof fix n show "compact (K' n)" by fact next fix n from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) ∈ Y (Suc n)" by auto also have "… ⊆ Z' (Suc n)" using Y_Z' by auto finally have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) ∈ K' (Suc n)" unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) thus "K' (Suc n) ≠ {}" by auto fix k assume "k ∈ K' (Suc n)" with K'[of "Suc n"] sets.sets_into_space have "k ∈ fm (Suc n) ` B (Suc n)" by auto then obtain b where "k = fm (Suc n) b" by auto thus "domain k = domain (fm (Suc n) (y (Suc n)))" by (simp_all add: fm_def) next fix t and n m::nat assume "n ≤ m" hence "Suc n ≤ Suc m" by simp assume "t ∈ domain (fm (Suc n) (y (Suc n)))" then obtain j where j: "t = Utn j" "j ∈ J (Suc n)" by auto hence "j ∈ J (Suc m)" using J_mono[OF ‹Suc n ≤ Suc m›] by auto have img: "fm (Suc n) (y (Suc m)) ∈ K' (Suc n)" using ‹n ≤ m› by (intro fm_in_K') simp_all show "(fm (Suc m) (y (Suc m)))⇩_{F}t ∈ (λk. (k)⇩_{F}t) ` K' (Suc n)" apply (rule image_eqI[OF _ img]) using ‹j ∈ J (Suc n)› ‹j ∈ J (Suc m)› unfolding j by (subst proj_fm, auto)+ qed have "∀t. ∃z. (λi. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))⇩_{F}t) ⇢ z" using diagonal_tendsto .. then obtain z where z: "⋀t. (λi. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))⇩_{F}t) ⇢ z t" unfolding choice_iff by blast { fix n :: nat assume "n ≥ 1" have "⋀i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)" by simp moreover { fix t assume t: "t ∈ domain (finmap_of (Utn ` J n) z)" hence "t ∈ Utn ` J n" by simp then obtain j where j: "t = Utn j" "j ∈ J n" by auto have "(λi. (fm n (y (Suc (diagseq i))))⇩_{F}t) ⇢ z t" apply (subst (2) tendsto_iff, subst eventually_sequentially) proof safe fix e :: real assume "0 < e" { fix i and x :: "'i ⇒ 'a" assume i: "i ≥ n" assume "t ∈ domain (fm n x)" hence "t ∈ domain (fm i x)" using J_mono[OF ‹i ≥ n›] by auto with i have "(fm i x)⇩_{F}t = (fm n x)⇩_{F}t" using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn]) } note index_shift = this have I: "⋀i. i ≥ n ⟹ Suc (diagseq i) ≥ n" apply (rule le_SucI) apply (rule order_trans) apply simp apply (rule seq_suble[OF subseq_diagseq]) done from z have "∃N. ∀i≥N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))⇩_{F}t) (z t) < e" unfolding tendsto_iff eventually_sequentially using ‹0 < e› by auto then obtain N where N: "⋀i. i ≥ N ⟹ dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))⇩_{F}t) (z t) < e" by auto show "∃N. ∀na≥N. dist ((fm n (y (Suc (diagseq na))))⇩_{F}t) (z t) < e " proof (rule exI[where x="max N n"], safe) fix na assume "max N n ≤ na" hence "dist ((fm n (y (Suc (diagseq na))))⇩_{F}t) (z t) = dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))⇩_{F}t) (z t)" using t by (subst index_shift[OF I]) auto also have "… < e" using ‹max N n ≤ na› by (intro N) simp finally show "dist ((fm n (y (Suc (diagseq na))))⇩_{F}t) (z t) < e" . qed qed hence "(λi. (fm n (y (Suc (diagseq i))))⇩_{F}t) ⇢ (finmap_of (Utn ` J n) z)⇩_{F}t" by (simp add: tendsto_intros) } ultimately have "(λi. fm n (y (Suc (diagseq i)))) ⇢ finmap_of (Utn ` J n) z" by (rule tendsto_finmap) hence "((λi. fm n (y (Suc (diagseq i)))) o (λi. i + n)) ⇢ finmap_of (Utn ` J n) z" by (rule LIMSEQ_subseq_LIMSEQ) (simp add: strict_mono_def) moreover have "(∀i. ((λi. fm n (y (Suc (diagseq i)))) o (λi. i + n)) i ∈ K' n)" apply (auto simp add: o_def intro!: fm_in_K' ‹1 ≤ n› le_SucI) apply (rule le_trans) apply (rule le_add2) using seq_suble[OF subseq_diagseq] apply auto done moreover from ‹compact (K' n)› have "closed (K' n)" by (rule compact_imp_closed) ultimately have "finmap_of (Utn ` J n) z ∈ K' n" unfolding closed_sequential_limits by blast also have "finmap_of (Utn ` J n) z = fm n (λi. z (Utn i))" unfolding finmap_eq_iff proof clarsimp fix i assume i: "i ∈ J n" hence "from_nat_into (⋃n. J n) (Utn i) = i" unfolding Utn_def by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto with i show "z (Utn i) = (fm n (λi. z (Utn i)))⇩_{F}(Utn i)" by (simp add: finmap_eq_iff fm_def compose_def) qed finally have "fm n (λi. z (Utn i)) ∈ K' n" . moreover let ?J = "⋃n. J n" have "(?J ∩ J n) = J n" by auto ultimately have "restrict (λi. z (Utn i)) (?J ∩ J n) ∈ K n" unfolding K_def by (auto simp: space_P space_PiM) hence "restrict (λi. z (Utn i)) ?J ∈ Z' n" unfolding Z'_def using J by (auto simp: prod_emb_def PiE_def extensional_def) also have "… ⊆ Z n" using Z' by simp finally have "restrict (λi. z (Utn i)) ?J ∈ Z n" . } note in_Z = this hence "(⋂i∈{1..}. Z i) ≠ {}" by auto thus "(⋂i. Z i) ≠ {}" using INT_decseq_offset[OF antimonoI[OF Z_mono]] by simp qed fact+ lemma measure_lim_emb: "J ⊆ I ⟹ finite J ⟹ X ∈ sets (Π⇩_{M}i∈J. borel) ⟹ measure lim (emb I J X) = measure (P J) X" unfolding measure_def by (subst emeasure_lim_emb) auto end hide_const (open) PiF hide_const (open) Pi⇩_{F}hide_const (open) Pi' hide_const (open) finmap_of hide_const (open) proj hide_const (open) domain hide_const (open) basis_finmap sublocale polish_projective ⊆ P: prob_space lim proof have *: "emb I {} {λx. undefined} = space (Π⇩_{M}i∈I. borel)" by (auto simp: prod_emb_def space_PiM) interpret prob_space "P {}" using prob_space_P by simp show "emeasure lim (space lim) = 1" using emeasure_lim_emb[of "{}" "{λx. undefined}"] emeasure_space_1 by (simp add: * PiM_empty space_P) qed locale polish_product_prob_space = product_prob_space "λ_. borel::('a::polish_space) measure" I for I::"'i set" sublocale polish_product_prob_space ⊆ P: polish_projective I "λJ. PiM J (λ_. borel::('a) measure)" .. lemma (in polish_product_prob_space) limP_eq_PiM: "lim = PiM I (λ_. borel)" by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_lim_emb) end