Theory Projective_Limit

theory Projective_Limit
imports Fin_Map Infinite_Product_Measure Diagonal_Subsequence
(*  Title:      HOL/Probability/Projective_Limit.thy
    Author:     Fabian Immler, TU M√ľnchen

section ‹Projective Limit›

theory Projective_Limit

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"

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)


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 (op + 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] guess l r .
  hence "l ∈ S" "strict_mono ((λi. m + i) o r) ∧ (f ∘ ((λi. m + i) o r)) ⇢ l"
    using strict_mono_o[OF ‹strict_mono (op + m)› ‹strict_mono r›] by (auto simp: o_def)
  thus "∃l r. l ∈ S ∧ strict_mono r ∧ (f ∘ r) ⇢ l" by blast

sublocale finmap_seqs_into_compact  subseqs "λn s. (∃l. (λi. ((f o s) i)F n) ⇢ l)"
  fix n and s :: "nat ⇒ nat"
  assume "strict_mono s"
  from proj_in_KE[of n] guess n0 . note n0 = this
  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
  from compactE'[OF compact_projset this] guess ls rs .
  thus "∃r'. strict_mono r' ∧ (∃l. (λi. ((f ∘ (s ∘ r')) i)F n) ⇢ l)" by (auto simp: o_def)

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 op + (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)
  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 ..

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

lemma emeasure_lim_emb:
  assumes X: "J ⊆ I" "finite J" "X ∈ sets (ΠM 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 (ΠM 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 (ΠM i∈J i. borel)" "space (P (J i)) = space (ΠM 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)
    have "… = ?SUP n"
    proof (rule inner_regular)
      show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
      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"
    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)
      finally show False by simp
  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 (PiM (J n) (λ_. borel))" for n
  have K_sets: "⋀n. K n ∈ sets (PiM (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"
    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)
  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 (PiM (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 (PiM (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
  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
    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)
    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] set_mp[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" .
    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 (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
        sum (op ^ (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)
  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›] .
    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
    fix n show "compact (K' n)" by fact
    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
    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)
    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)+
  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
      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])
        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" .
      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)
    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
    from ‹compact (K' n)› have "closed (K' n)" by (rule compact_imp_closed)
    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)
    finally have "fm n (λi. z (Utn i)) ∈ K' n" .
    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 (ΠM i∈J. borel) ⟹ measure lim (emb I J X) = measure (P J) X"
  unfolding measure_def by (subst emeasure_lim_emb) auto


hide_const (open) PiF
hide_const (open) PiF
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
  have *: "emb I {} {λx. undefined} = space (ΠM 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)

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)