standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
(* Title: HOL/Probability/Projective_Limit.thy
Author: Fabian Immler, TU München
*)
header {* Projective Limit *}
theory Projective_Limit
imports
Caratheodory
Fin_Map
Regularity
Projective_Family
Infinite_Product_Measure
"~~/src/HOL/Library/Diagonal_Subsequence"
begin
subsection {* Sequences of Finite Maps in Compact Sets *}
locale finmap_seqs_into_compact =
fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a)" and M
assumes compact: "\<And>n. compact (K n)"
assumes f_in_K: "\<And>n. K n \<noteq> {}"
assumes domain_K: "\<And>n. k \<in> K n \<Longrightarrow> domain k = domain (f n)"
assumes proj_in_K:
"\<And>t n m. m \<ge> n \<Longrightarrow> t \<in> domain (f n) \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
begin
lemma proj_in_K': "(\<exists>n. \<forall>m \<ge> n. (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n)"
using proj_in_K f_in_K
proof cases
obtain k where "k \<in> K (Suc 0)" using f_in_K by auto
assume "\<forall>n. t \<notin> domain (f n)"
thus ?thesis
by (auto intro!: exI[where x=1] image_eqI[OF _ `k \<in> K (Suc 0)`]
simp: domain_K[OF `k \<in> K (Suc 0)`])
qed blast
lemma proj_in_KE:
obtains n where "\<And>m. m \<ge> n \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
using proj_in_K' by blast
lemma compact_projset:
shows "compact ((\<lambda>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" "\<forall>n\<ge>m. f n \<in> S"
obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
proof atomize_elim
have "subseq (op + m)" by (simp add: subseq_def)
have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
from seq_compactE[OF `compact S`[unfolded compact_eq_seq_compact_metric] this] guess l r .
hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l"
using subseq_o[OF `subseq (op + m)` `subseq r`] by (auto simp: o_def)
thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> 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) ----> l)"
proof
fix n s
assume "subseq s"
from proj_in_KE[of n] guess n0 . note n0 = this
have "\<forall>i \<ge> n0. ((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0"
proof safe
fix i assume "n0 \<le> i"
also have "\<dots> \<le> s i" by (rule seq_suble) fact
finally have "n0 \<le> s i" .
with n0 show "((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0 "
by auto
qed
from compactE'[OF compact_projset this] guess ls rs .
thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^sub>F n) ----> l)" by (auto simp: o_def)
qed
lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l"
proof -
obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) ----> l"
proof (atomize_elim, rule diagseq_holds)
fix r s n
assume "subseq r"
assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) ----> l"
then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) ----> l"
by (auto simp: o_def)
hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) ----> l" using `subseq r`
by (rule LIMSEQ_subseq_LIMSEQ)
thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) ----> l" by (auto simp add: o_def)
qed
hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) ----> l" by (simp add: ac_simps)
hence "(\<lambda>i. (f (diagseq i))\<^sub>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 "\<lambda>_. borel::'a::polish_space measure"
for I::"'i set" and P
begin
abbreviation "lim\<^sub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)"
lemma emeasure_limB_emb_not_empty:
assumes "I \<noteq> {}"
assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
shows "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (lim\<^sub>B J P) (Pi\<^sub>E J B)"
proof -
let ?\<Omega> = "\<Pi>\<^sub>E i\<in>I. space borel"
let ?G = generator
interpret G!: algebra ?\<Omega> generator by (intro algebra_generator) fact
note mu_G_mono =
G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`],
THEN increasingD]
write mu_G ("\<mu>G")
have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G,
OF `I \<noteq> {}`, OF `I \<noteq> {}`])
fix A assume "A \<in> ?G"
with generatorE guess J X . note JX = this
interpret prob_space "P J" using proj_prob_space[OF `finite J`] .
show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite)
next
fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}"
then have "decseq (\<lambda>i. \<mu>G (Z i))"
by (auto intro!: mu_G_mono simp: decseq_def)
moreover
have "(INF i. \<mu>G (Z i)) = 0"
proof (rule ccontr)
assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0")
moreover have "0 \<le> ?a"
using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def)
ultimately have "0 < ?a" by auto
hence "?a \<noteq> -\<infinity>" by auto
have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^sub>M J (\<lambda>_. borel)) \<and>
Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^sub>B J P) B"
using Z by (intro allI generator_Ex) auto
then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I"
"\<And>n. B' n \<in> sets (\<Pi>\<^sub>M i\<in>J' n. borel)"
and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)"
unfolding choice_iff by blast
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)"
ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I"
"\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)"
by auto
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
unfolding J_def by force
have "\<forall>n. \<exists>j. j \<in> J n" using J by blast
then obtain j where j: "\<And>n. j n \<in> J n"
unfolding choice_iff by blast
note [simp] = `\<And>n. finite (J n)`
from J Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G"
unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
interpret prob_space "P (J i)" for i using proj_prob_space by simp
have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower)
also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq mu_G_eq limP_finite proj_sets)
finally have "?a \<noteq> \<infinity>" by simp
have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono
by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>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 (\<Union>n. J n)"
unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
{
fix n
interpret finite_measure "P (J n)" by unfold_locales
have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets)
also
have "\<dots> = ?SUP n"
proof (rule inner_regular)
show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
unfolding P'_def
by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets)
show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
next
show "fm n ` B n \<in> sets borel"
unfolding borel_eq_PiF_borel
by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J)
qed
finally
have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
} note R = this
have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a
\<and> compact K \<and> K \<subseteq> fm n ` B n"
proof
fix n
have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
by (simp add: mapmeasure_PiF P'_def proj_space proj_sets)
then interpret finite_measure "P' n" ..
show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
compact K \<and> K \<subseteq> fm n ` B n"
unfolding R
proof (rule ccontr)
assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and>
compact K' \<and> K' \<subseteq> fm n ` B n)"
have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
proof (intro SUP_least)
fix K
assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
by auto
hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
unfolding not_less[symmetric] by simp
hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
qed
hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
hence "0 \<le> - (2 powr (-n) * ?a)"
using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>`
by (subst (asm) ereal_add_le_add_iff) (auto simp:)
moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
by (auto simp: ereal_zero_less_0_iff)
ultimately show False by simp
qed
qed
then obtain K' where K':
"\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
"\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
unfolding choice_iff by blast
def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. 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: "\<And>n. K n \<subseteq> B n"
proof
fix x n
assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n"
using K' by (force simp: K_def)
show "x \<in> B n"
using `x \<in> K n` K_sets sets.sets_into_space J[of n]
by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto
qed
def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
have Z': "\<And>n. Z' n \<subseteq> Z n"
unfolding Z_eq unfolding Z'_def
proof (rule prod_emb_mono, safe)
fix n x assume "x \<in> K n"
hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
by (simp_all add: K_def proj_space)
note this(1)
also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
finally have "fm n x \<in> fm n ` B n" .
thus "x \<in> B n"
proof safe
fix y assume "y \<in> B n"
moreover
hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
by (auto simp add: proj_space proj_sets)
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 \<in> space _` `y \<in> space _`]
ultimately show "x \<in> B n" by simp
qed
qed
{ fix n
have "Z' n \<in> ?G" using K' unfolding Z'_def
apply (intro generatorI'[OF J(1-3)])
unfolding K_def proj_space
apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]])
apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
done
}
def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
proof -
fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
have "Y n = (\<Inter> i\<in>{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 "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))"
using `n \<ge> 1`
by (subst prod_emb_INT) auto
finally
have Y_emb:
"Y n = prod_emb I (\<lambda>_. borel) (J n)
(\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto
hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1`
by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq)
interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)"
proof
have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^sub>E space borel) \<noteq> \<infinity>"
using J by (subst emeasure_limP) auto
thus "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
by (simp add: space_PiM)
qed
have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)"
unfolding Z_eq using J by (auto simp: mu_G_eq)
moreover have "\<mu>G (Y n) =
limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_eq) auto
moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P
(B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
by (subst mu_G_eq) (auto intro!: sets.Diff)
ultimately
have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
using J J_mono K_sets `n \<ge> 1`
by (simp only: emeasure_eq_measure)
(auto dest!: bspec[where x=n]
simp: extensional_restrict emeasure_eq_measure prod_emb_iff
intro!: measure_Diff[symmetric] set_mp[OF K_B])
also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1`
unfolding Y_def by (force simp: decseq_def)
have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G"
using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto
hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))"
using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]]
unfolding increasing_def by auto
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G`
by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
proof (rule setsum_mono)
fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
unfolding Z'_def Z_eq by simp
also have "\<dots> = P (J i) (B i - K i)"
apply (subst mu_G_eq) using J K_sets apply auto
apply (subst limP_finite) apply auto
done
also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets)
done
also have "\<dots> = P (J i) (B i) - P' i (K' i)"
unfolding K_def P'_def
by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric]
compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
qed
also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))"
using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto
also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp
also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
by (simp add: setsum_left_distrib)
also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
proof (rule mult_strict_right_mono)
have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)"
by (rule setsum_cong)
(auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide)
also have "{1..<Suc n} = {0..<Suc n} - {0}" by auto
also have "setsum (op ^ (1 / 2::real)) ({0..<Suc n} - {0}) =
setsum (op ^ (1 / 2)) ({0..<Suc n}) - 1" by (auto simp: setsum_diff1)
also have "\<dots> < 1" by (subst sumr_geometric) auto
finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
qed (auto simp:
`0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric])
also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real')
also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower)
finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
finally have "\<mu>G (Y n) > 0"
using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def)
qed
hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
{
fix t and n m::nat
assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
finally
have "fm n (restrict (y m) (J n)) \<in> 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) \<in> K' n" by simp
} note fm_in_K' = this
interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>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) \<in> Y (Suc n)" by auto
also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
finally
have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> 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) \<noteq> {}" by auto
fix k
assume "k \<in> K' (Suc n)"
with K'[of "Suc n"] sets.sets_into_space have "k \<in> 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 \<le> m" hence "Suc n \<le> Suc m" by simp
assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
by (intro fm_in_K') simp_all
show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)"
apply (rule image_eqI[OF _ img])
using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
unfolding j by (subst proj_fm, auto)+
qed
have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z"
using diagonal_tendsto ..
then obtain z where z:
"\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
unfolding choice_iff by blast
{
fix n :: nat assume "n \<ge> 1"
have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
by simp
moreover
{
fix t
assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
hence "t \<in> Utn ` J n" by simp
then obtain j where j: "t = Utn j" "j \<in> J n" by auto
have "(\<lambda>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 x assume "i \<ge> n" "t \<in> domain (fm n x)"
moreover
hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
ultimately 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: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
apply (rule le_SucI)
apply (rule order_trans) apply simp
apply (rule seq_suble[OF subseq_diagseq])
done
from z
have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e"
unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
show "\<exists>N. \<forall>na\<ge>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 \<le> 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 "\<dots> < e" using `max N n \<le> na` by (intro N) simp
finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
qed
qed
hence "(\<lambda>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 "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
by (rule tendsto_finmap)
hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
by (intro lim_subseq) (simp add: subseq_def)
moreover
have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> 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 \<in> K' n"
unfolding closed_sequential_limits by blast
also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))"
unfolding finmap_eq_iff
proof clarsimp
fix i assume "i \<in> J n"
moreover hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
unfolding Utn_def
by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
ultimately show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)"
by (simp add: finmap_eq_iff fm_def compose_def)
qed
finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
moreover
let ?J = "\<Union>n. J n"
have "(?J \<inter> J n) = J n" by auto
ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
unfolding K_def by (auto simp: proj_space space_PiM)
hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
using J by (auto simp: prod_emb_def PiE_def extensional_def)
also have "\<dots> \<subseteq> Z n" using Z' by simp
finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
} note in_Z = this
hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp
thus False using Z by simp
qed
ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0"
using LIMSEQ_INF[of "\<lambda>i. \<mu>G (Z i)"] by simp
qed
then guess \<mu> .. note \<mu> = this
def f \<equiv> "finmap_of J B"
show "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (lim\<^sub>B J P) (Pi\<^sub>E J B)"
proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>])
show "positive (sets (lim\<^sub>B I P)) \<mu>" "countably_additive (sets (lim\<^sub>B I P)) \<mu>"
using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def)
next
show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel"
using assms by (auto simp: f_def)
next
fix J and X::"'i \<Rightarrow> 'a set"
show "prod_emb I (\<lambda>_. borel) J (Pi\<^sub>E J X) \<in> Pow (I \<rightarrow>\<^sub>E space borel)"
by (auto simp: prod_emb_def)
assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel"
hence "emb I J (Pi\<^sub>E J X) \<in> generator" using assms
by (intro generatorI[where J=J and X="Pi\<^sub>E J X"]) (auto intro: sets_PiM_I_finite)
hence "\<mu> (emb I J (Pi\<^sub>E J X)) = \<mu>G (emb I J (Pi\<^sub>E J X))" using \<mu> by simp
also have "\<dots> = emeasure (P J) (Pi\<^sub>E J X)"
using JX assms proj_sets
by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite)
finally show "\<mu> (emb I J (Pi\<^sub>E J X)) = emeasure (P J) (Pi\<^sub>E J X)" .
next
show "emeasure (P J) (Pi\<^sub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^sub>E J B)"
using assms by (simp add: f_def limP_finite Pi_def)
qed
qed
end
hide_const (open) PiF
hide_const (open) Pi\<^sub>F
hide_const (open) Pi'
hide_const (open) Abs_finmap
hide_const (open) Rep_finmap
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\<^sub>B I P)"
proof
show "emeasure (lim\<^sub>B I P) (space (lim\<^sub>B I P)) = 1"
proof cases
assume "I = {}"
interpret prob_space "P {}" using proj_prob_space by simp
show ?thesis
by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`)
next
assume "I \<noteq> {}"
then obtain i where "i \<in> I" by auto
interpret prob_space "P {i}" using proj_prob_space by simp
have R: "(space (lim\<^sub>B I P)) = (emb I {i} (Pi\<^sub>E {i} (\<lambda>_. space borel)))"
by (auto simp: prod_emb_def space_PiM)
moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM PiE_def)
ultimately show ?thesis using `i \<in> I`
apply (subst R)
apply (subst emeasure_limB_emb_not_empty)
apply (auto simp: limP_finite emeasure_space_1 PiE_def)
done
qed
qed
context polish_projective begin
lemma emeasure_limB_emb:
assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
shows "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (P J) (Pi\<^sub>E J B)"
proof cases
interpret prob_space "P {}" using proj_prob_space by simp
assume "J = {}"
moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^sub>B I P)"
by (auto simp: space_PiM prod_emb_def)
moreover have "{\<lambda>x. undefined} = space (lim\<^sub>B {} P)"
by (auto simp: space_PiM prod_emb_def)
ultimately show ?thesis
by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP)
next
assume "J \<noteq> {}" with X show ?thesis
by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite)
qed
lemma measure_limB_emb:
assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
shows "measure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = measure (P J) (Pi\<^sub>E J B)"
proof -
interpret prob_space "P J" using proj_prob_space assms by simp
show ?thesis
using emeasure_limB_emb[OF assms]
unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
by simp
qed
end
locale polish_product_prob_space =
product_prob_space "\<lambda>_. 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)"
proof qed
lemma (in polish_product_prob_space) limP_eq_PiM:
"I \<noteq> {} \<Longrightarrow> lim\<^sub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
PiM I (\<lambda>_. borel)"
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb)
end