author | hoelzl |
Tue, 18 Mar 2014 15:53:48 +0100 | |
changeset 56193 | c726ecfb22b6 |
parent 53374 | a14d2a854c02 |
child 57418 | 6ab1c7cb0b8d |
permissions | -rw-r--r-- |
50091 | 1 |
(* Title: HOL/Probability/Projective_Limit.thy |
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Author: Fabian Immler, TU München |
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*) |
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header {* Projective Limit *} |
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theory Projective_Limit |
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imports |
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Caratheodory |
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Fin_Map |
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Regularity |
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Projective_Family |
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Infinite_Product_Measure |
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"~~/src/HOL/Library/Diagonal_Subsequence" |
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begin |
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subsection {* Sequences of Finite Maps in Compact Sets *} |
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locale finmap_seqs_into_compact = |
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fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a)" and M |
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assumes compact: "\<And>n. compact (K n)" |
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assumes f_in_K: "\<And>n. K n \<noteq> {}" |
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assumes domain_K: "\<And>n. k \<in> K n \<Longrightarrow> domain k = domain (f n)" |
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assumes proj_in_K: |
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"\<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" |
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begin |
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lemma proj_in_K': "(\<exists>n. \<forall>m \<ge> n. (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n)" |
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using proj_in_K f_in_K |
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proof cases |
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obtain k where "k \<in> K (Suc 0)" using f_in_K by auto |
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assume "\<forall>n. t \<notin> domain (f n)" |
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thus ?thesis |
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by (auto intro!: exI[where x=1] image_eqI[OF _ `k \<in> K (Suc 0)`] |
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simp: domain_K[OF `k \<in> K (Suc 0)`]) |
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qed blast |
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lemma proj_in_KE: |
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obtains n where "\<And>m. m \<ge> n \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n" |
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using proj_in_K' by blast |
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lemma compact_projset: |
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shows "compact ((\<lambda>k. (k)\<^sub>F i) ` K n)" |
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using continuous_proj compact by (rule compact_continuous_image) |
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end |
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lemma compactE': |
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differentiate (cover) compactness and sequential compactness
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fixes S :: "'a :: metric_space set" |
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assumes "compact S" "\<forall>n\<ge>m. f n \<in> S" |
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obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially" |
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proof atomize_elim |
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have "subseq (op + m)" by (simp add: subseq_def) |
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have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto |
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from seq_compactE[OF `compact S`[unfolded compact_eq_seq_compact_metric] this] guess l r . |
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hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l" |
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using subseq_o[OF `subseq (op + m)` `subseq r`] by (auto simp: o_def) |
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thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) ----> l" by blast |
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qed |
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sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^sub>F n) ----> l)" |
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proof |
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fix n s |
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assume "subseq s" |
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from proj_in_KE[of n] guess n0 . note n0 = this |
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have "\<forall>i \<ge> n0. ((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0" |
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proof safe |
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fix i assume "n0 \<le> i" |
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also have "\<dots> \<le> s i" by (rule seq_suble) fact |
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finally have "n0 \<le> s i" . |
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with n0 show "((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0 " |
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by auto |
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qed |
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from compactE'[OF compact_projset this] guess ls rs . |
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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) |
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qed |
77 |
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lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l" |
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proof - |
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obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) ----> l" |
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proof (atomize_elim, rule diagseq_holds) |
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fix r s n |
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assume "subseq r" |
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assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) ----> l" |
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then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) ----> l" |
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by (auto simp: o_def) |
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hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) ----> l" using `subseq r` |
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by (rule LIMSEQ_subseq_LIMSEQ) |
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thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) ----> l" by (auto simp add: o_def) |
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qed |
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hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) ----> l" by (simp add: ac_simps) |
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hence "(\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l" by (rule LIMSEQ_offset) |
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thus ?thesis .. |
50088 | 94 |
qed |
95 |
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subsection {* Daniell-Kolmogorov Theorem *} |
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text {* Existence of Projective Limit *} |
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locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure" |
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for I::"'i set" and P |
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begin |
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103 |
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abbreviation "lim\<^sub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)" |
50088 | 105 |
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tuned: use induction rule sigma_sets_induct_disjoint
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lemma emeasure_limB_emb_not_empty: |
50088 | 107 |
assumes "I \<noteq> {}" |
108 |
assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" |
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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)" |
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proof - |
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let ?\<Omega> = "\<Pi>\<^sub>E i\<in>I. space borel" |
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let ?G = generator |
113 |
interpret G!: algebra ?\<Omega> generator by (intro algebra_generator) fact |
|
50252 | 114 |
note mu_G_mono = |
115 |
G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`], |
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116 |
THEN increasingD] |
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write mu_G ("\<mu>G") |
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||
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have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>" |
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proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G, |
50088 | 121 |
OF `I \<noteq> {}`, OF `I \<noteq> {}`]) |
122 |
fix A assume "A \<in> ?G" |
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with generatorE guess J X . note JX = this |
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interpret prob_space "P J" using proj_prob_space[OF `finite J`] . |
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show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite) |
50088 | 126 |
next |
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fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}" |
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then have "decseq (\<lambda>i. \<mu>G (Z i))" |
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50252 | 129 |
by (auto intro!: mu_G_mono simp: decseq_def) |
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moreover |
131 |
have "(INF i. \<mu>G (Z i)) = 0" |
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proof (rule ccontr) |
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assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0") |
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moreover have "0 \<le> ?a" |
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50252 | 135 |
using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def) |
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ultimately have "0 < ?a" by auto |
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hence "?a \<noteq> -\<infinity>" by auto |
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138 |
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> |
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Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^sub>B J P) B" |
50088 | 140 |
using Z by (intro allI generator_Ex) auto |
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then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" |
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"\<And>n. B' n \<in> sets (\<Pi>\<^sub>M i\<in>J' n. borel)" |
50088 | 143 |
and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)" |
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unfolding choice_iff by blast |
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moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)" |
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moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)" |
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ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" |
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148 |
"\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)" |
50088 | 149 |
by auto |
150 |
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" |
|
151 |
unfolding J_def by force |
|
152 |
have "\<forall>n. \<exists>j. j \<in> J n" using J by blast |
|
153 |
then obtain j where j: "\<And>n. j n \<in> J n" |
|
154 |
unfolding choice_iff by blast |
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155 |
note [simp] = `\<And>n. finite (J n)` |
|
156 |
from J Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G" |
|
157 |
unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto) |
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|
158 |
interpret prob_space "P (J i)" for i using proj_prob_space by simp |
50088 | 159 |
have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower) |
50252 | 160 |
also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq mu_G_eq limP_finite proj_sets) |
50088 | 161 |
finally have "?a \<noteq> \<infinity>" by simp |
162 |
have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono |
|
50252 | 163 |
by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq) |
50088 | 164 |
|
50243 | 165 |
have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite) |
166 |
def Utn \<equiv> "to_nat_on (\<Union>n. J n)" |
|
167 |
interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n |
|
168 |
by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J]) |
|
169 |
have inj_on_Utn: "inj_on Utn (\<Union>n. J n)" |
|
170 |
unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on) |
|
171 |
hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto |
|
50088 | 172 |
def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)" |
173 |
let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K" |
|
174 |
{ |
|
175 |
fix n |
|
176 |
interpret finite_measure "P (J n)" by unfold_locales |
|
177 |
have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))" |
|
178 |
using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets) |
|
179 |
also |
|
180 |
have "\<dots> = ?SUP n" |
|
181 |
proof (rule inner_regular) |
|
182 |
show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>" |
|
183 |
unfolding P'_def |
|
184 |
by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets) |
|
185 |
show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def) |
|
186 |
next |
|
187 |
show "fm n ` B n \<in> sets borel" |
|
188 |
unfolding borel_eq_PiF_borel |
|
189 |
by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J) |
|
190 |
qed |
|
191 |
finally |
|
192 |
have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto |
|
193 |
} note R = this |
|
194 |
have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a |
|
195 |
\<and> compact K \<and> K \<subseteq> fm n ` B n" |
|
196 |
proof |
|
197 |
fix n |
|
198 |
have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>" |
|
199 |
by (simp add: mapmeasure_PiF P'_def proj_space proj_sets) |
|
200 |
then interpret finite_measure "P' n" .. |
|
201 |
show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and> |
|
202 |
compact K \<and> K \<subseteq> fm n ` B n" |
|
203 |
unfolding R |
|
204 |
proof (rule ccontr) |
|
205 |
assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and> |
|
206 |
compact K' \<and> K' \<subseteq> fm n ` B n)" |
|
207 |
have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a" |
|
208 |
proof (intro SUP_least) |
|
209 |
fix K |
|
210 |
assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}" |
|
211 |
with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a" |
|
212 |
by auto |
|
213 |
hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a" |
|
214 |
unfolding not_less[symmetric] by simp |
|
215 |
hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K" |
|
216 |
using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps) |
|
217 |
thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp |
|
218 |
qed |
|
219 |
hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp |
|
220 |
hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def . |
|
221 |
hence "0 \<le> - (2 powr (-n) * ?a)" |
|
222 |
using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>` |
|
223 |
by (subst (asm) ereal_add_le_add_iff) (auto simp:) |
|
224 |
moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a` |
|
225 |
by (auto simp: ereal_zero_less_0_iff) |
|
226 |
ultimately show False by simp |
|
227 |
qed |
|
228 |
qed |
|
229 |
then obtain K' where K': |
|
230 |
"\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a" |
|
231 |
"\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n" |
|
232 |
unfolding choice_iff by blast |
|
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|
233 |
def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" |
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changeset
|
234 |
have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))" |
50088 | 235 |
unfolding K_def |
236 |
using compact_imp_closed[OF `compact (K' _)`] |
|
237 |
by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"]) |
|
238 |
(auto simp: borel_eq_PiF_borel[symmetric]) |
|
239 |
have K_B: "\<And>n. K n \<subseteq> B n" |
|
240 |
proof |
|
241 |
fix x n |
|
242 |
assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n" |
|
243 |
using K' by (force simp: K_def) |
|
244 |
show "x \<in> B n" |
|
50244
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immler
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50243
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changeset
|
245 |
using `x \<in> K n` K_sets sets.sets_into_space J[of n] |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
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changeset
|
246 |
by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto |
50088 | 247 |
qed |
248 |
def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)" |
|
249 |
have Z': "\<And>n. Z' n \<subseteq> Z n" |
|
250 |
unfolding Z_eq unfolding Z'_def |
|
251 |
proof (rule prod_emb_mono, safe) |
|
252 |
fix n x assume "x \<in> K n" |
|
53015
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|
253 |
hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" |
50088 | 254 |
by (simp_all add: K_def proj_space) |
255 |
note this(1) |
|
256 |
also have "K' n \<subseteq> fm n ` B n" by (simp add: K') |
|
257 |
finally have "fm n x \<in> fm n ` B n" . |
|
258 |
thus "x \<in> B n" |
|
259 |
proof safe |
|
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|
260 |
fix y assume y: "y \<in> B n" |
53015
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wenzelm
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changeset
|
261 |
hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"] |
50088 | 262 |
by (auto simp add: proj_space proj_sets) |
263 |
assume "fm n x = fm n y" |
|
264 |
note inj_onD[OF inj_on_fm[OF space_borel], |
|
265 |
OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`] |
|
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changeset
|
266 |
with y show "x \<in> B n" by simp |
50088 | 267 |
qed |
268 |
qed |
|
269 |
{ fix n |
|
270 |
have "Z' n \<in> ?G" using K' unfolding Z'_def |
|
271 |
apply (intro generatorI'[OF J(1-3)]) |
|
272 |
unfolding K_def proj_space |
|
273 |
apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]]) |
|
274 |
apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed) |
|
275 |
done |
|
276 |
} |
|
277 |
def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i" |
|
278 |
hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def) |
|
279 |
hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add) |
|
280 |
have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def) |
|
281 |
hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto |
|
282 |
have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}" |
|
283 |
proof - |
|
284 |
fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact |
|
285 |
have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono |
|
286 |
by (auto simp: Y_def Z'_def) |
|
287 |
also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))" |
|
288 |
using `n \<ge> 1` |
|
289 |
by (subst prod_emb_INT) auto |
|
290 |
finally |
|
291 |
have Y_emb: |
|
292 |
"Y n = prod_emb I (\<lambda>_. borel) (J n) |
|
293 |
(\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" . |
|
294 |
hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto |
|
295 |
hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1` |
|
50252 | 296 |
by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq) |
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immler
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changeset
|
297 |
interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)" |
50088 | 298 |
proof |
53015
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parents:
52681
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changeset
|
299 |
have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^sub>E space borel) \<noteq> \<infinity>" |
50095
94d7dfa9f404
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immler
parents:
50091
diff
changeset
|
300 |
using J by (subst emeasure_limP) auto |
94d7dfa9f404
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immler
parents:
50091
diff
changeset
|
301 |
thus "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>" |
50088 | 302 |
by (simp add: space_PiM) |
303 |
qed |
|
50095
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renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
304 |
have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)" |
50252 | 305 |
unfolding Z_eq using J by (auto simp: mu_G_eq) |
50088 | 306 |
moreover have "\<mu>G (Y n) = |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
307 |
limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" |
50252 | 308 |
unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_eq) auto |
50095
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renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
309 |
moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P |
50088 | 310 |
(B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))" |
311 |
unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1` |
|
50252 | 312 |
by (subst mu_G_eq) (auto intro!: sets.Diff) |
50088 | 313 |
ultimately |
314 |
have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)" |
|
315 |
using J J_mono K_sets `n \<ge> 1` |
|
316 |
by (simp only: emeasure_eq_measure) |
|
317 |
(auto dest!: bspec[where x=n] |
|
318 |
simp: extensional_restrict emeasure_eq_measure prod_emb_iff |
|
319 |
intro!: measure_Diff[symmetric] set_mp[OF K_B]) |
|
320 |
also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1` |
|
321 |
unfolding Y_def by (force simp: decseq_def) |
|
322 |
have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G" |
|
323 |
using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto |
|
324 |
hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))" |
|
50252 | 325 |
using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]] |
50088 | 326 |
unfolding increasing_def by auto |
327 |
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G` |
|
50252 | 328 |
by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto |
50088 | 329 |
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)" |
330 |
proof (rule setsum_mono) |
|
331 |
fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp |
|
332 |
have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))" |
|
333 |
unfolding Z'_def Z_eq by simp |
|
334 |
also have "\<dots> = P (J i) (B i - K i)" |
|
50252 | 335 |
apply (subst mu_G_eq) using J K_sets apply auto |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
336 |
apply (subst limP_finite) apply auto |
50088 | 337 |
done |
338 |
also have "\<dots> = P (J i) (B i) - P (J i) (K i)" |
|
339 |
apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets) |
|
340 |
done |
|
341 |
also have "\<dots> = P (J i) (B i) - P' i (K' i)" |
|
342 |
unfolding K_def P'_def |
|
343 |
by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric] |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset
|
344 |
compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def) |
50088 | 345 |
also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] . |
346 |
finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" . |
|
347 |
qed |
|
348 |
also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))" |
|
349 |
using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto |
|
350 |
also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp |
|
351 |
also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)" |
|
352 |
by (simp add: setsum_left_distrib) |
|
353 |
also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps |
|
354 |
proof (rule mult_strict_right_mono) |
|
355 |
have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)" |
|
356 |
by (rule setsum_cong) |
|
357 |
(auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide) |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
53374
diff
changeset
|
358 |
also have "{1..<Suc n} = {..<Suc n} - {0}" by auto |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
53374
diff
changeset
|
359 |
also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) = |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
53374
diff
changeset
|
360 |
setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1) |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
53374
diff
changeset
|
361 |
also have "\<dots> < 1" by (subst geometric_sum) auto |
50088 | 362 |
finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" . |
363 |
qed (auto simp: |
|
364 |
`0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric]) |
|
365 |
also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real') |
|
366 |
also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower) |
|
367 |
finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" . |
|
368 |
hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)" |
|
369 |
using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less) |
|
370 |
have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto |
|
371 |
also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))" |
|
372 |
apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto |
|
373 |
finally have "\<mu>G (Y n) > 0" |
|
374 |
using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric]) |
|
50252 | 375 |
thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def) |
50088 | 376 |
qed |
377 |
hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto |
|
378 |
then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force |
|
379 |
{ |
|
380 |
fix t and n m::nat |
|
381 |
assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp |
|
382 |
from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto |
|
383 |
also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] . |
|
384 |
finally |
|
385 |
have "fm n (restrict (y m) (J n)) \<in> K' n" |
|
386 |
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) |
|
387 |
moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)" |
|
388 |
using J by (simp add: fm_def) |
|
389 |
ultimately have "fm n (y m) \<in> K' n" by simp |
|
390 |
} note fm_in_K' = this |
|
391 |
interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel |
|
392 |
proof |
|
393 |
fix n show "compact (K' n)" by fact |
|
394 |
next |
|
395 |
fix n |
|
396 |
from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto |
|
397 |
also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto |
|
398 |
finally |
|
399 |
have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)" |
|
400 |
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) |
|
401 |
thus "K' (Suc n) \<noteq> {}" by auto |
|
402 |
fix k |
|
403 |
assume "k \<in> K' (Suc n)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50243
diff
changeset
|
404 |
with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto |
50088 | 405 |
then obtain b where "k = fm (Suc n) b" by auto |
406 |
thus "domain k = domain (fm (Suc n) (y (Suc n)))" |
|
407 |
by (simp_all add: fm_def) |
|
408 |
next |
|
409 |
fix t and n m::nat |
|
410 |
assume "n \<le> m" hence "Suc n \<le> Suc m" by simp |
|
411 |
assume "t \<in> domain (fm (Suc n) (y (Suc n)))" |
|
412 |
then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto |
|
413 |
hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto |
|
414 |
have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m` |
|
415 |
by (intro fm_in_K') simp_all |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
416 |
show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)" |
50088 | 417 |
apply (rule image_eqI[OF _ img]) |
418 |
using `j \<in> J (Suc n)` `j \<in> J (Suc m)` |
|
419 |
unfolding j by (subst proj_fm, auto)+ |
|
420 |
qed |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
421 |
have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z" |
50088 | 422 |
using diagonal_tendsto .. |
423 |
then obtain z where z: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
424 |
"\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z t" |
50088 | 425 |
unfolding choice_iff by blast |
426 |
{ |
|
427 |
fix n :: nat assume "n \<ge> 1" |
|
428 |
have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)" |
|
429 |
by simp |
|
430 |
moreover |
|
431 |
{ |
|
432 |
fix t |
|
433 |
assume t: "t \<in> domain (finmap_of (Utn ` J n) z)" |
|
434 |
hence "t \<in> Utn ` J n" by simp |
|
435 |
then obtain j where j: "t = Utn j" "j \<in> J n" by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
436 |
have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> z t" |
50088 | 437 |
apply (subst (2) tendsto_iff, subst eventually_sequentially) |
438 |
proof safe |
|
439 |
fix e :: real assume "0 < e" |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
440 |
{ fix i x |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
441 |
assume i: "i \<ge> n" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
442 |
assume "t \<in> domain (fm n x)" |
50088 | 443 |
hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
444 |
with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t" |
50243 | 445 |
using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn]) |
50088 | 446 |
} note index_shift = this |
447 |
have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n" |
|
448 |
apply (rule le_SucI) |
|
449 |
apply (rule order_trans) apply simp |
|
450 |
apply (rule seq_suble[OF subseq_diagseq]) |
|
451 |
done |
|
452 |
from z |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
453 |
have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" |
50088 | 454 |
unfolding tendsto_iff eventually_sequentially using `0 < e` by auto |
455 |
then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow> |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
456 |
dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
457 |
show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e " |
50088 | 458 |
proof (rule exI[where x="max N n"], safe) |
459 |
fix na assume "max N n \<le> na" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
460 |
hence "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
461 |
dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t |
50088 | 462 |
by (subst index_shift[OF I]) auto |
463 |
also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
464 |
finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" . |
50088 | 465 |
qed |
466 |
qed |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
467 |
hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> (finmap_of (Utn ` J n) z)\<^sub>F t" |
50088 | 468 |
by (simp add: tendsto_intros) |
469 |
} ultimately |
|
470 |
have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z" |
|
471 |
by (rule tendsto_finmap) |
|
472 |
hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z" |
|
473 |
by (intro lim_subseq) (simp add: subseq_def) |
|
474 |
moreover |
|
475 |
have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)" |
|
476 |
apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI) |
|
477 |
apply (rule le_trans) |
|
478 |
apply (rule le_add2) |
|
479 |
using seq_suble[OF subseq_diagseq] |
|
480 |
apply auto |
|
481 |
done |
|
482 |
moreover |
|
483 |
from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed) |
|
484 |
ultimately |
|
485 |
have "finmap_of (Utn ` J n) z \<in> K' n" |
|
486 |
unfolding closed_sequential_limits by blast |
|
487 |
also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))" |
|
50243 | 488 |
unfolding finmap_eq_iff |
489 |
proof clarsimp |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
490 |
fix i assume i: "i \<in> J n" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
491 |
hence "from_nat_into (\<Union>n. J n) (Utn i) = i" |
50243 | 492 |
unfolding Utn_def |
493 |
by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
494 |
with i show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)" |
50243 | 495 |
by (simp add: finmap_eq_iff fm_def compose_def) |
496 |
qed |
|
50088 | 497 |
finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" . |
498 |
moreover |
|
499 |
let ?J = "\<Union>n. J n" |
|
500 |
have "(?J \<inter> J n) = J n" by auto |
|
501 |
ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n" |
|
502 |
unfolding K_def by (auto simp: proj_space space_PiM) |
|
503 |
hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset
|
504 |
using J by (auto simp: prod_emb_def PiE_def extensional_def) |
50088 | 505 |
also have "\<dots> \<subseteq> Z n" using Z' by simp |
506 |
finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" . |
|
507 |
} note in_Z = this |
|
508 |
hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto |
|
509 |
hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp |
|
510 |
thus False using Z by simp |
|
511 |
qed |
|
512 |
ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0" |
|
51351 | 513 |
using LIMSEQ_INF[of "\<lambda>i. \<mu>G (Z i)"] by simp |
50088 | 514 |
qed |
515 |
then guess \<mu> .. note \<mu> = this |
|
516 |
def f \<equiv> "finmap_of J B" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
517 |
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)" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
518 |
proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
519 |
show "positive (sets (lim\<^sub>B I P)) \<mu>" "countably_additive (sets (lim\<^sub>B I P)) \<mu>" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
520 |
using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def) |
50088 | 521 |
next |
522 |
show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel" |
|
523 |
using assms by (auto simp: f_def) |
|
524 |
next |
|
525 |
fix J and X::"'i \<Rightarrow> 'a set" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
526 |
show "prod_emb I (\<lambda>_. borel) J (Pi\<^sub>E J X) \<in> Pow (I \<rightarrow>\<^sub>E space borel)" |
50088 | 527 |
by (auto simp: prod_emb_def) |
528 |
assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
529 |
hence "emb I J (Pi\<^sub>E J X) \<in> generator" using assms |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
530 |
by (intro generatorI[where J=J and X="Pi\<^sub>E J X"]) (auto intro: sets_PiM_I_finite) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
531 |
hence "\<mu> (emb I J (Pi\<^sub>E J X)) = \<mu>G (emb I J (Pi\<^sub>E J X))" using \<mu> by simp |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
532 |
also have "\<dots> = emeasure (P J) (Pi\<^sub>E J X)" |
50088 | 533 |
using JX assms proj_sets |
50252 | 534 |
by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
535 |
finally show "\<mu> (emb I J (Pi\<^sub>E J X)) = emeasure (P J) (Pi\<^sub>E J X)" . |
50088 | 536 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
537 |
show "emeasure (P J) (Pi\<^sub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^sub>E J B)" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
538 |
using assms by (simp add: f_def limP_finite Pi_def) |
50088 | 539 |
qed |
540 |
qed |
|
541 |
||
542 |
end |
|
543 |
||
50090 | 544 |
hide_const (open) PiF |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
545 |
hide_const (open) Pi\<^sub>F |
50090 | 546 |
hide_const (open) Pi' |
547 |
hide_const (open) Abs_finmap |
|
548 |
hide_const (open) Rep_finmap |
|
549 |
hide_const (open) finmap_of |
|
550 |
hide_const (open) proj |
|
551 |
hide_const (open) domain |
|
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
552 |
hide_const (open) basis_finmap |
50090 | 553 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
554 |
sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^sub>B I P)" |
50088 | 555 |
proof |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
556 |
show "emeasure (lim\<^sub>B I P) (space (lim\<^sub>B I P)) = 1" |
50088 | 557 |
proof cases |
558 |
assume "I = {}" |
|
50101
a3bede207a04
renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents:
50095
diff
changeset
|
559 |
interpret prob_space "P {}" using proj_prob_space by simp |
50088 | 560 |
show ?thesis |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
561 |
by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`) |
50088 | 562 |
next |
563 |
assume "I \<noteq> {}" |
|
564 |
then obtain i where "i \<in> I" by auto |
|
50101
a3bede207a04
renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents:
50095
diff
changeset
|
565 |
interpret prob_space "P {i}" using proj_prob_space by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
566 |
have R: "(space (lim\<^sub>B I P)) = (emb I {i} (Pi\<^sub>E {i} (\<lambda>_. space borel)))" |
50088 | 567 |
by (auto simp: prod_emb_def space_PiM) |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset
|
568 |
moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM PiE_def) |
50088 | 569 |
ultimately show ?thesis using `i \<in> I` |
570 |
apply (subst R) |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
571 |
apply (subst emeasure_limB_emb_not_empty) |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset
|
572 |
apply (auto simp: limP_finite emeasure_space_1 PiE_def) |
50088 | 573 |
done |
574 |
qed |
|
575 |
qed |
|
576 |
||
577 |
context polish_projective begin |
|
578 |
||
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
579 |
lemma emeasure_limB_emb: |
50088 | 580 |
assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
581 |
shows "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (P J) (Pi\<^sub>E J B)" |
50088 | 582 |
proof cases |
50101
a3bede207a04
renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents:
50095
diff
changeset
|
583 |
interpret prob_space "P {}" using proj_prob_space by simp |
50088 | 584 |
assume "J = {}" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
585 |
moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^sub>B I P)" |
50088 | 586 |
by (auto simp: space_PiM prod_emb_def) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
587 |
moreover have "{\<lambda>x. undefined} = space (lim\<^sub>B {} P)" |
50088 | 588 |
by (auto simp: space_PiM prod_emb_def) |
589 |
ultimately show ?thesis |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
590 |
by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP) |
50088 | 591 |
next |
592 |
assume "J \<noteq> {}" with X show ?thesis |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
593 |
by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite) |
50088 | 594 |
qed |
595 |
||
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
596 |
lemma measure_limB_emb: |
50088 | 597 |
assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
598 |
shows "measure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = measure (P J) (Pi\<^sub>E J B)" |
50088 | 599 |
proof - |
50101
a3bede207a04
renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents:
50095
diff
changeset
|
600 |
interpret prob_space "P J" using proj_prob_space assms by simp |
50088 | 601 |
show ?thesis |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
602 |
using emeasure_limB_emb[OF assms] |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
603 |
unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure |
50088 | 604 |
by simp |
605 |
qed |
|
606 |
||
607 |
end |
|
608 |
||
609 |
locale polish_product_prob_space = |
|
610 |
product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set" |
|
611 |
||
612 |
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)" |
|
613 |
proof qed |
|
614 |
||
50125
4319691be975
tuned: use induction rule sigma_sets_induct_disjoint
hoelzl
parents:
50124
diff
changeset
|
615 |
lemma (in polish_product_prob_space) limP_eq_PiM: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52681
diff
changeset
|
616 |
"I \<noteq> {} \<Longrightarrow> lim\<^sub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) = |
50088 | 617 |
PiM I (\<lambda>_. borel)" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
618 |
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb) |
50088 | 619 |
|
620 |
end |