Theory Projective_Limit

(*  Title:      HOL/Probability/Projective_Limit.thy
    Author:     Fabian Immler, TU München

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" "nm. f n  S"
  obtains l r where "l  S" "strict_mono (r::natnat)" "((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

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] 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
  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)

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)
  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 iJ. 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 iJ 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 iJ i. borel)" "space (P (J i)) = space (ΠM iJ 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] 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" .
    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)
  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. iN. 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. naN. 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 iJ. 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 iI. 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)