# Theory Projective_Limit

```(*  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)"
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))"
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))"
unfolding increasing_def by auto
also have "… ≤ (∑ i∈{1..n}. μG (Z i - Z' i))" using ‹Z _ ∈ generator› ‹Z' _ ∈ generator›
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)))"
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"
} 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)
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
```