(* Title: HOL/Probability/Projective_Limit.thy
Author: Fabian Immler, TU München
*)
section \<open>Projective Limit\<close>
theory Projective_Limit
imports
Fin_Map
Infinite_Product_Measure
"HOL-Library.Diagonal_Subsequence"
begin
subsection \<open>Sequences of Finite Maps in Compact Sets\<close>
locale finmap_seqs_into_compact =
fixes K::"nat \ (nat \\<^sub>F 'a::metric_space) set" and f::"nat \ (nat \\<^sub>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)\<^sub>F t \ (\k. (k)\<^sub>F t) ` K n"
begin
lemma proj_in_K': "(\n. \m \ n. (f m)\<^sub>F t \ (\k. (k)\<^sub>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 _ \<open>k \<in> K (Suc 0)\<close>]
simp: domain_K[OF \<open>k \<in> K (Suc 0)\<close>])
qed blast
lemma proj_in_KE:
obtains n where "\m. m \ n \ (f m)\<^sub>F t \ (\k. (k)\<^sub>F t) ` K n"
using proj_in_K' by blast
lemma compact_projset:
shows "compact ((\k. (k)\<^sub>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 (\i. m + i)) n \ S" using assms by auto
from seq_compactE[OF \<open>compact S\<close>[unfolded compact_eq_seq_compact_metric] this] guess l r .
hence "l \ S" "strict_mono ((\i. m + i) o r) \ (f \ ((\i. m + i) o r)) \ l"
using strict_mono_o[OF \<open>strict_mono ((+) m)\<close> \<open>strict_mono r\<close>] by (auto simp: o_def)
thus "\l r. l \ S \ strict_mono r \ (f \ r) \ l" by blast
qed
sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^sub>F n) \<longlonglongrightarrow> l)"
proof
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)\<^sub>F n \ (\k. (k)\<^sub>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)\<^sub>F n \ (\k. (k)\<^sub>F n) ` K n0 "
by auto
qed
from compactE'[OF compact_projset this] guess ls rs .
thus "\r'. strict_mono r' \ (\l. (\i. ((f \ (s \ r')) i)\<^sub>F n) \ l)" by (auto simp: o_def)
qed
lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\l. (\i. (f (diagseq i))\<^sub>F n) \ l"
proof -
obtain l where "(\i. ((f o (diagseq o (+) (Suc n))) i)\<^sub>F n) \ l"
proof (atomize_elim, rule diagseq_holds)
fix r s n
assume "strict_mono (r :: nat \ nat)"
assume "\l. (\i. ((f \ s) i)\<^sub>F n) \ l"
then obtain l where "((\i. (f i)\<^sub>F n) o s) \ l"
by (auto simp: o_def)
hence "((\i. (f i)\<^sub>F n) o s o r) \ l" using \strict_mono r\
by (rule LIMSEQ_subseq_LIMSEQ)
thus "\l. (\i. ((f \ (s \ r)) i)\<^sub>F n) \ l" by (auto simp add: o_def)
qed
hence "(\i. ((f (diagseq (i + Suc n))))\<^sub>F n) \ l" by (simp add: ac_simps)
hence "(\i. (f (diagseq i))\<^sub>F n) \ l" by (rule LIMSEQ_offset)
thus ?thesis ..
qed
subsection \<open>Daniell-Kolmogorov Theorem\<close>
text \<open>Existence of Projective Limit\<close>
locale polish_projective = projective_family I P "\_. borel::'a::polish_space measure"
for I::"'i set" and P
begin
lemma emeasure_lim_emb:
assumes X: "J \ I" "finite J" "X \ sets (\\<^sub>M i\J. borel)"
shows "lim (emb I J X) = P J X"
proof (rule emeasure_lim)
write mu_G ("\G")
interpret generator: algebra "space (PiM I (\i. 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 (\\<^sub>M i\J n. borel)" "incseq J"
and B: "decseq (\n. 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 \<open>incseq J\<close> by (force simp: incseq_def)
note [simp] = \<open>\<And>n. finite (J n)\<close>
interpret prob_space "P (J i)" for i using J prob_space_P by simp
have P_eq[simp]:
"sets (P (J i)) = sets (\\<^sub>M i\J i. borel)" "space (P (J i)) = space (\\<^sub>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)) (\<lambda>_. 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 = "\n. 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 \<and> K \<subseteq> 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: \<open>0 < ?a\<close>)
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\<^sub>M (J n) (\_. borel))" for n
have K_sets: "\n. K n \ sets (Pi\<^sub>M (J n) (\_. borel))"
unfolding K_def
using compact_imp_closed[OF \<open>compact (K' _)\<close>]
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 \<open>x \<in> K n\<close> 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\<^sub>M (J n) (\_. 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\<^sub>M (J n) (\_. 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 \<open>fm n x = fm n y\<close> \<open>x \<in> space _\<close> \<open>y \<in> space _\<close>]
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 (\_. borel) (J n) (\i\{1..n}. emb (J n) (J i) (K i))"
using \<open>n \<ge> 1\<close>
by (subst prod_emb_INT) auto
finally
have Y_emb:
"Y n = prod_emb I (\_. borel) (J n) (\i\{1..n}. prod_emb (J n) (\_. 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 *: "\G (Z n) = P (J n) (B n)"
unfolding Z_def using J by (intro mu_G_spec) auto
then have "\G (Z n) \ \" by auto
note *
moreover have *: "\G (Y n) = P (J n) (\i\{Suc 0..n}. prod_emb (J n) (\_. borel) (J i) (K i))"
unfolding Y_emb using J J_mono K_sets \<open>n \<ge> 1\<close> by (subst mu_G_spec) auto
then have "\G (Y n) \ \" by auto
note *
moreover have "\G (Z n - Y n) =
P (J n) (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets \<open>n \<ge> 1\<close>
by (subst mu_G_spec) (auto intro!: sets.Diff)
ultimately
have "\G (Z n) - \G (Y n) = \G (Z n - Y n)"
using J J_mono K_sets \<open>n \<ge> 1\<close>
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 \<open>n \<ge> 1\<close> 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 \<open>Z' _ \<in> generator\<close> \<open>Z _ \<in> generator\<close> \<open>Y _ \<in> generator\<close> by auto
hence "\G (Z n - Y n) \ \G (\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}. \G (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 "\G (Z i - Z' i) = \G (prod_emb I (\_. 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 \<open>K _ \<subseteq> B _\<close> 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 \<open>compact (K' _)\<close>] space_PiM PiE_def)
also have "\ \ ennreal (2 powr - real i) * ?a" using K'(1)[of i] .
finally show "\G (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..
by (rule sum.cong) (auto simp: powr_realpow powr_divide power_divide powr_minus_divide)
also have "{1.. by auto
also have "sum ((^) (1 / 2::real)) ({..
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 "\ \ \G (Z n)"
using J by (auto intro: INF_lower simp: Z_def mu_G_spec)
finally have "\G (Z n) - \G (Y n) < \G (Z n)" .
hence R: "\G (Z n) < \G (Z n) + \G (Y n)"
using \<open>\<mu>G (Y n) \<noteq> \<infinity>\<close> by (auto simp: zero_less_iff_neq_zero)
then have "\G (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 \<open>m \<ge> n\<close>] y[OF \<open>1 \<le> m\<close>] have "y m \<in> 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 "\n. K' (Suc n)" "\k. 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)))\<^sub>F t \ (\k. (k)\<^sub>F t) ` K' (Suc n)"
apply (rule image_eqI[OF _ img])
using \<open>j \<in> J (Suc n)\<close> \<open>j \<in> J (Suc m)\<close>
unfolding j by (subst proj_fm, auto)+
qed
have "\t. \z. (\i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \ z"
using diagonal_tendsto ..
then obtain z where z:
"\t. (\i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>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 "(\i. (fm n (y (Suc (diagseq i))))\<^sub>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)\<^sub>F t = (fm n x)\<^sub>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))))\<^sub>F t) (z t) < e"
unfolding tendsto_iff eventually_sequentially using \<open>0 < e\<close> by auto
then obtain N where N: "\i. i \ N \
dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
show "\N. \na\N. dist ((fm n (y (Suc (diagseq na))))\<^sub>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))))\<^sub>F t) (z t) =
dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>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))))\<^sub>F t) (z t) < e" .
qed
qed
hence "(\i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \ (finmap_of (Utn ` J n) z)\<^sub>F t"
by (simp add: tendsto_intros)
} ultimately
have "(\i. fm n (y (Suc (diagseq i)))) \ finmap_of (Utn ` J n) z"
by (rule tendsto_finmap)
hence "((\i. fm n (y (Suc (diagseq i)))) o (\i. i + n)) \ finmap_of (Utn ` J n) z"
by (rule LIMSEQ_subseq_LIMSEQ) (simp add: strict_mono_def)
moreover
have "(\i. ((\i. fm n (y (Suc (diagseq i)))) o (\i. 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 \<open>compact (K' n)\<close> 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 (\i. 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 (\i. z (Utn i)))\<^sub>F (Utn i)"
by (simp add: finmap_eq_iff fm_def compose_def)
qed
finally have "fm n (\i. z (Utn i)) \ K' n" .
moreover
let ?J = "\n. J n"
have "(?J \ J n) = J n" by auto
ultimately have "restrict (\i. z (Utn i)) (?J \ J n) \ K n"
unfolding K_def by (auto simp: space_P space_PiM)
hence "restrict (\i. 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 (\i. 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 (\\<^sub>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\<^sub>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 \<subseteq> P: prob_space lim
proof
have *: "emb I {} {\x. undefined} = space (\\<^sub>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 "{}" "{\x. undefined}"] emeasure_space_1
by (simp add: * PiM_empty space_P)
qed
locale polish_product_prob_space =
product_prob_space "\_. borel::('a::polish_space) measure" I for I::"'i set"
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
..
lemma (in polish_product_prob_space) limP_eq_PiM: "lim = PiM I (\_. borel)"
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_lim_emb)
end
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