author | immler |
Fri, 16 Nov 2012 11:34:34 +0100 | |
changeset 50095 | 94d7dfa9f404 |
parent 50091 | b3b5dc2350b7 |
child 50101 | a3bede207a04 |
permissions | -rw-r--r-- |
50091 | 1 |
(* Title: HOL/Probability/Projective_Limit.thy |
50088 | 2 |
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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begin |
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subsection {* Enumeration of Countable Union of Finite Sets *} |
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locale finite_set_sequence = |
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fixes Js::"nat \<Rightarrow> 'a set" |
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assumes finite_seq[simp]: "finite (Js n)" |
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begin |
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text {* Enumerate finite set *} |
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definition "enum_finite_max J = (SOME n. \<exists> f. J = f ` {i. i < n} \<and> inj_on f {i. i < n})" |
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definition enum_finite where |
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"enum_finite J = |
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(SOME f. J = f ` {i::nat. i < enum_finite_max J} \<and> inj_on f {i. i < enum_finite_max J})" |
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lemma enum_finite_max: |
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assumes "finite J" |
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shows "\<exists>f::nat\<Rightarrow>_. J = f ` {i. i < enum_finite_max J} \<and> inj_on f {i. i < enum_finite_max J}" |
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unfolding enum_finite_max_def |
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by (rule someI_ex) (rule finite_imp_nat_seg_image_inj_on[OF `finite J`]) |
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lemma enum_finite: |
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assumes "finite J" |
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shows "J = enum_finite J ` {i::nat. i < enum_finite_max J} \<and> |
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inj_on (enum_finite J) {i::nat. i < enum_finite_max J}" |
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unfolding enum_finite_def |
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by (rule someI_ex[of "\<lambda>f. J = f ` {i::nat. i < enum_finite_max J} \<and> |
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inj_on f {i. i < enum_finite_max J}"]) (rule enum_finite_max[OF `finite J`]) |
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lemma in_set_enum_exist: |
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assumes "finite A" |
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assumes "y \<in> A" |
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shows "\<exists>i. y = enum_finite A i" |
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using assms enum_finite by auto |
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definition set_of_Un where "set_of_Un j = (LEAST n. j \<in> Js n)" |
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definition index_in_set where "index_in_set J j = (SOME n. j = enum_finite J n)" |
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definition Un_to_nat where |
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"Un_to_nat j = to_nat (set_of_Un j, index_in_set (Js (set_of_Un j)) j)" |
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lemma inj_on_Un_to_nat: |
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shows "inj_on Un_to_nat (\<Union>n::nat. Js n)" |
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proof (rule inj_onI) |
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fix x y |
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assume "x \<in> (\<Union>n. Js n)" "y \<in> (\<Union>n. Js n)" |
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then obtain ix iy where ix: "x \<in> Js ix" and iy: "y \<in> Js iy" by blast |
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assume "Un_to_nat x = Un_to_nat y" |
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hence "set_of_Un x = set_of_Un y" |
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"index_in_set (Js (set_of_Un y)) y = index_in_set (Js (set_of_Un x)) x" |
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by (auto simp: Un_to_nat_def) |
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moreover |
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{ |
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fix x assume "x \<in> Js (set_of_Un x)" |
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have "x = enum_finite (Js (set_of_Un x)) (index_in_set (Js (set_of_Un x)) x)" |
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unfolding index_in_set_def |
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apply (rule someI_ex) |
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using `x \<in> Js (set_of_Un x)` finite_seq |
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apply (auto intro!: in_set_enum_exist) |
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done |
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} note H = this |
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moreover |
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have "y \<in> Js (set_of_Un y)" unfolding set_of_Un_def using iy by (rule LeastI) |
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note H[OF this] |
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moreover |
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have "x \<in> Js (set_of_Un x)" unfolding set_of_Un_def using ix by (rule LeastI) |
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note H[OF this] |
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ultimately show "x = y" by simp |
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qed |
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lemma inj_Un[simp]: |
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shows "inj_on (Un_to_nat) (Js n)" |
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by (intro subset_inj_on[OF inj_on_Un_to_nat]) (auto simp: assms) |
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lemma Un_to_nat_injectiveD: |
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assumes "Un_to_nat x = Un_to_nat y" |
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assumes "x \<in> Js i" "y \<in> Js j" |
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shows "x = y" |
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using assms |
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by (intro inj_onD[OF inj_on_Un_to_nat]) auto |
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end |
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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>\<^isub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^isub>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)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>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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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 compactE[OF `compact S` 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)\<^isub>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)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>F n \<in> (\<lambda>k. (k)\<^isub>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)\<^isub>F n) ----> l)" by (auto simp: o_def) |
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qed |
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lemma (in finmap_seqs_into_compact) |
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diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^isub>F n) ----> l" |
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proof - |
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have "\<And>i n0. (f o seqseq i) i = f (diagseq i)" unfolding diagseq_def by simp |
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from reducer_reduces obtain l where l: "(\<lambda>i. ((f \<circ> seqseq (Suc n)) i)\<^isub>F n) ----> l" |
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unfolding seqseq_reducer |
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by auto |
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have "(\<lambda>i. (f (diagseq (i + Suc n)))\<^isub>F n) = |
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(\<lambda>i. ((f o (diagseq o (op + (Suc n)))) i)\<^isub>F n)" by (simp add: add_commute) |
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also have "\<dots> = |
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(\<lambda>i. ((f o ((seqseq (Suc n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))))) i)\<^isub>F n)" |
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unfolding diagseq_seqseq by simp |
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also have "\<dots> = (\<lambda>i. ((f o ((seqseq (Suc n)))) i)\<^isub>F n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))" |
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by (simp add: o_def) |
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also have "\<dots> ----> l" |
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proof (rule LIMSEQ_subseq_LIMSEQ[OF _ subseq_diagonal_rest], rule tendstoI) |
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fix e::real assume "0 < e" |
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from tendstoD[OF l `0 < e`] |
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show "eventually (\<lambda>x. dist (((f \<circ> seqseq (Suc n)) x)\<^isub>F n) l < e) |
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sequentially" . |
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qed |
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finally show ?thesis by (intro exI) (rule LIMSEQ_offset) |
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qed |
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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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abbreviation "lim\<^isub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)" |
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lemma |
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emeasure_limB_emb_not_empty: |
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assumes "I \<noteq> {}" |
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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\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)" |
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proof - |
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let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel" |
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let ?G = generator |
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interpret G!: algebra ?\<Omega> generator by (intro algebra_generator) fact |
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note \<mu>G_mono = |
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G.additive_increasing[OF positive_\<mu>G[OF `I \<noteq> {}`] additive_\<mu>G[OF `I \<noteq> {}`], THEN increasingD] |
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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, |
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OF `I \<noteq> {}`, OF `I \<noteq> {}`]) |
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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 prob_space[OF `finite J`] . |
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show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite) |
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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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by (auto intro!: \<mu>G_mono simp: decseq_def) |
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moreover |
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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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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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have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^isub>M J (\<lambda>_. borel)) \<and> |
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Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^isub>B J P) B" |
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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>\<^isub>M i\<in>J' n. borel)" |
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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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"\<And>n. B n \<in> sets (\<Pi>\<^isub>M i\<in>J n. borel)" |
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by auto |
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have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" |
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unfolding J_def by force |
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have "\<forall>n. \<exists>j. j \<in> J n" using J by blast |
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then obtain j where j: "\<And>n. j n \<in> J n" |
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unfolding choice_iff by blast |
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note [simp] = `\<And>n. finite (J n)` |
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from J Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G" |
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unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto) |
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interpret prob_space "P (J i)" for i using prob_space by simp |
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have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower) |
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also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq \<mu>G_eq limP_finite proj_sets) |
50088 | 247 |
finally have "?a \<noteq> \<infinity>" by simp |
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have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono |
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by (subst \<mu>G_eq) (auto simp: limP_finite proj_sets \<mu>G_eq) |
50088 | 250 |
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interpret finite_set_sequence J by unfold_locales simp |
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def Utn \<equiv> Un_to_nat |
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interpret function_to_finmap "J n" Utn "inv_into (J n) Utn" for n |
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by unfold_locales (auto simp: Utn_def) |
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def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)" |
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let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K" |
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{ |
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fix n |
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interpret finite_measure "P (J n)" by unfold_locales |
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have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))" |
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using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets) |
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also |
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have "\<dots> = ?SUP n" |
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proof (rule inner_regular) |
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show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>" |
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unfolding P'_def |
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by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets) |
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show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def) |
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next |
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show "fm n ` B n \<in> sets borel" |
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unfolding borel_eq_PiF_borel |
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by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J) |
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qed |
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finally |
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have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto |
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} note R = this |
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have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a |
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\<and> compact K \<and> K \<subseteq> fm n ` B n" |
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proof |
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fix n |
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have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>" |
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by (simp add: mapmeasure_PiF P'_def proj_space proj_sets) |
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then interpret finite_measure "P' n" .. |
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show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and> |
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compact K \<and> K \<subseteq> fm n ` B n" |
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unfolding R |
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proof (rule ccontr) |
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assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and> |
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compact K' \<and> K' \<subseteq> fm n ` B n)" |
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have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a" |
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proof (intro SUP_least) |
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fix K |
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assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}" |
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with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a" |
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by auto |
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hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a" |
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unfolding not_less[symmetric] by simp |
|
298 |
hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K" |
|
299 |
using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps) |
|
300 |
thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp |
|
301 |
qed |
|
302 |
hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp |
|
303 |
hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def . |
|
304 |
hence "0 \<le> - (2 powr (-n) * ?a)" |
|
305 |
using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>` |
|
306 |
by (subst (asm) ereal_add_le_add_iff) (auto simp:) |
|
307 |
moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a` |
|
308 |
by (auto simp: ereal_zero_less_0_iff) |
|
309 |
ultimately show False by simp |
|
310 |
qed |
|
311 |
qed |
|
312 |
then obtain K' where K': |
|
313 |
"\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a" |
|
314 |
"\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n" |
|
315 |
unfolding choice_iff by blast |
|
316 |
def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" |
|
317 |
have K_sets: "\<And>n. K n \<in> sets (Pi\<^isub>M (J n) (\<lambda>_. borel))" |
|
318 |
unfolding K_def |
|
319 |
using compact_imp_closed[OF `compact (K' _)`] |
|
320 |
by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"]) |
|
321 |
(auto simp: borel_eq_PiF_borel[symmetric]) |
|
322 |
have K_B: "\<And>n. K n \<subseteq> B n" |
|
323 |
proof |
|
324 |
fix x n |
|
325 |
assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n" |
|
326 |
using K' by (force simp: K_def) |
|
327 |
show "x \<in> B n" |
|
328 |
apply (rule inj_on_image_mem_iff[OF inj_on_fm _ fm_in]) |
|
329 |
using `x \<in> K n` K_sets J[of n] sets_into_space |
|
330 |
apply (auto simp: proj_space) |
|
331 |
using J[of n] sets_into_space apply auto |
|
332 |
done |
|
333 |
qed |
|
334 |
def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)" |
|
335 |
have Z': "\<And>n. Z' n \<subseteq> Z n" |
|
336 |
unfolding Z_eq unfolding Z'_def |
|
337 |
proof (rule prod_emb_mono, safe) |
|
338 |
fix n x assume "x \<in> K n" |
|
339 |
hence "fm n x \<in> K' n" "x \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" |
|
340 |
by (simp_all add: K_def proj_space) |
|
341 |
note this(1) |
|
342 |
also have "K' n \<subseteq> fm n ` B n" by (simp add: K') |
|
343 |
finally have "fm n x \<in> fm n ` B n" . |
|
344 |
thus "x \<in> B n" |
|
345 |
proof safe |
|
346 |
fix y assume "y \<in> B n" |
|
347 |
moreover |
|
348 |
hence "y \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" using J sets_into_space[of "B n" "P (J n)"] |
|
349 |
by (auto simp add: proj_space proj_sets) |
|
350 |
assume "fm n x = fm n y" |
|
351 |
note inj_onD[OF inj_on_fm[OF space_borel], |
|
352 |
OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`] |
|
353 |
ultimately show "x \<in> B n" by simp |
|
354 |
qed |
|
355 |
qed |
|
356 |
{ fix n |
|
357 |
have "Z' n \<in> ?G" using K' unfolding Z'_def |
|
358 |
apply (intro generatorI'[OF J(1-3)]) |
|
359 |
unfolding K_def proj_space |
|
360 |
apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]]) |
|
361 |
apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed) |
|
362 |
done |
|
363 |
} |
|
364 |
def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i" |
|
365 |
hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def) |
|
366 |
hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add) |
|
367 |
have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def) |
|
368 |
hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto |
|
369 |
have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}" |
|
370 |
proof - |
|
371 |
fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact |
|
372 |
have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono |
|
373 |
by (auto simp: Y_def Z'_def) |
|
374 |
also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))" |
|
375 |
using `n \<ge> 1` |
|
376 |
by (subst prod_emb_INT) auto |
|
377 |
finally |
|
378 |
have Y_emb: |
|
379 |
"Y n = prod_emb I (\<lambda>_. borel) (J n) |
|
380 |
(\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" . |
|
381 |
hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto |
|
382 |
hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1` |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
383 |
by (subst \<mu>G_eq) (auto simp: limP_finite proj_sets \<mu>G_eq) |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
384 |
interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)" |
50088 | 385 |
proof |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
386 |
have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>" |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
387 |
using J by (subst emeasure_limP) auto |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
388 |
thus "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>" |
50088 | 389 |
by (simp add: space_PiM) |
390 |
qed |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
391 |
have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)" |
50088 | 392 |
unfolding Z_eq using J by (auto simp: \<mu>G_eq) |
393 |
moreover have "\<mu>G (Y n) = |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
394 |
limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" |
50088 | 395 |
unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst \<mu>G_eq) auto |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
396 |
moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P |
50088 | 397 |
(B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))" |
398 |
unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1` |
|
399 |
by (subst \<mu>G_eq) (auto intro!: Diff) |
|
400 |
ultimately |
|
401 |
have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)" |
|
402 |
using J J_mono K_sets `n \<ge> 1` |
|
403 |
by (simp only: emeasure_eq_measure) |
|
404 |
(auto dest!: bspec[where x=n] |
|
405 |
simp: extensional_restrict emeasure_eq_measure prod_emb_iff |
|
406 |
intro!: measure_Diff[symmetric] set_mp[OF K_B]) |
|
407 |
also have subs: "Z n - Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1` |
|
408 |
unfolding Y_def by (force simp: decseq_def) |
|
409 |
have "Z n - Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i - Z' i)) \<in> ?G" |
|
410 |
using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto |
|
411 |
hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i - Z' i))" |
|
412 |
using subs G.additive_increasing[OF positive_\<mu>G[OF `I \<noteq> {}`] additive_\<mu>G[OF `I \<noteq> {}`]] |
|
413 |
unfolding increasing_def by auto |
|
414 |
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G` |
|
415 |
by (intro G.subadditive[OF positive_\<mu>G additive_\<mu>G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto |
|
416 |
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)" |
|
417 |
proof (rule setsum_mono) |
|
418 |
fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp |
|
419 |
have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))" |
|
420 |
unfolding Z'_def Z_eq by simp |
|
421 |
also have "\<dots> = P (J i) (B i - K i)" |
|
422 |
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
|
423 |
apply (subst limP_finite) apply auto |
50088 | 424 |
done |
425 |
also have "\<dots> = P (J i) (B i) - P (J i) (K i)" |
|
426 |
apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets) |
|
427 |
done |
|
428 |
also have "\<dots> = P (J i) (B i) - P' i (K' i)" |
|
429 |
unfolding K_def P'_def |
|
430 |
by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric] |
|
431 |
compact_imp_closed[OF `compact (K' _)`] space_PiM) |
|
432 |
also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] . |
|
433 |
finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" . |
|
434 |
qed |
|
435 |
also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))" |
|
436 |
using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto |
|
437 |
also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp |
|
438 |
also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)" |
|
439 |
by (simp add: setsum_left_distrib) |
|
440 |
also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps |
|
441 |
proof (rule mult_strict_right_mono) |
|
442 |
have "(\<Sum>i\<in>{1..n}. 2 powr - real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)" |
|
443 |
by (rule setsum_cong) |
|
444 |
(auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide) |
|
445 |
also have "{1..<Suc n} = {0..<Suc n} - {0}" by auto |
|
446 |
also have "setsum (op ^ (1 / 2::real)) ({0..<Suc n} - {0}) = |
|
447 |
setsum (op ^ (1 / 2)) ({0..<Suc n}) - 1" by (auto simp: setsum_diff1) |
|
448 |
also have "\<dots> < 1" by (subst sumr_geometric) auto |
|
449 |
finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" . |
|
450 |
qed (auto simp: |
|
451 |
`0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric]) |
|
452 |
also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real') |
|
453 |
also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower) |
|
454 |
finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" . |
|
455 |
hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)" |
|
456 |
using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less) |
|
457 |
have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto |
|
458 |
also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))" |
|
459 |
apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto |
|
460 |
finally have "\<mu>G (Y n) > 0" |
|
461 |
using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric]) |
|
462 |
thus "Y n \<noteq> {}" using positive_\<mu>G `I \<noteq> {}` by (auto simp add: positive_def) |
|
463 |
qed |
|
464 |
hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto |
|
465 |
then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force |
|
466 |
{ |
|
467 |
fix t and n m::nat |
|
468 |
assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp |
|
469 |
from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto |
|
470 |
also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] . |
|
471 |
finally |
|
472 |
have "fm n (restrict (y m) (J n)) \<in> K' n" |
|
473 |
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) |
|
474 |
moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)" |
|
475 |
using J by (simp add: fm_def) |
|
476 |
ultimately have "fm n (y m) \<in> K' n" by simp |
|
477 |
} note fm_in_K' = this |
|
478 |
interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel |
|
479 |
proof |
|
480 |
fix n show "compact (K' n)" by fact |
|
481 |
next |
|
482 |
fix n |
|
483 |
from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto |
|
484 |
also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto |
|
485 |
finally |
|
486 |
have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)" |
|
487 |
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) |
|
488 |
thus "K' (Suc n) \<noteq> {}" by auto |
|
489 |
fix k |
|
490 |
assume "k \<in> K' (Suc n)" |
|
491 |
with K'[of "Suc n"] sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto |
|
492 |
then obtain b where "k = fm (Suc n) b" by auto |
|
493 |
thus "domain k = domain (fm (Suc n) (y (Suc n)))" |
|
494 |
by (simp_all add: fm_def) |
|
495 |
next |
|
496 |
fix t and n m::nat |
|
497 |
assume "n \<le> m" hence "Suc n \<le> Suc m" by simp |
|
498 |
assume "t \<in> domain (fm (Suc n) (y (Suc n)))" |
|
499 |
then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto |
|
500 |
hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto |
|
501 |
have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m` |
|
502 |
by (intro fm_in_K') simp_all |
|
503 |
show "(fm (Suc m) (y (Suc m)))\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K' (Suc n)" |
|
504 |
apply (rule image_eqI[OF _ img]) |
|
505 |
using `j \<in> J (Suc n)` `j \<in> J (Suc m)` |
|
506 |
unfolding j by (subst proj_fm, auto)+ |
|
507 |
qed |
|
508 |
have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z" |
|
509 |
using diagonal_tendsto .. |
|
510 |
then obtain z where z: |
|
511 |
"\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) ----> z t" |
|
512 |
unfolding choice_iff by blast |
|
513 |
{ |
|
514 |
fix n :: nat assume "n \<ge> 1" |
|
515 |
have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)" |
|
516 |
by simp |
|
517 |
moreover |
|
518 |
{ |
|
519 |
fix t |
|
520 |
assume t: "t \<in> domain (finmap_of (Utn ` J n) z)" |
|
521 |
hence "t \<in> Utn ` J n" by simp |
|
522 |
then obtain j where j: "t = Utn j" "j \<in> J n" by auto |
|
523 |
have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> z t" |
|
524 |
apply (subst (2) tendsto_iff, subst eventually_sequentially) |
|
525 |
proof safe |
|
526 |
fix e :: real assume "0 < e" |
|
527 |
{ fix i x assume "i \<ge> n" "t \<in> domain (fm n x)" |
|
528 |
moreover |
|
529 |
hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto |
|
530 |
ultimately have "(fm i x)\<^isub>F t = (fm n x)\<^isub>F t" |
|
531 |
using j by (auto simp: proj_fm dest!: |
|
532 |
Un_to_nat_injectiveD[simplified Utn_def[symmetric]]) |
|
533 |
} note index_shift = this |
|
534 |
have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n" |
|
535 |
apply (rule le_SucI) |
|
536 |
apply (rule order_trans) apply simp |
|
537 |
apply (rule seq_suble[OF subseq_diagseq]) |
|
538 |
done |
|
539 |
from z |
|
540 |
have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" |
|
541 |
unfolding tendsto_iff eventually_sequentially using `0 < e` by auto |
|
542 |
then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow> |
|
543 |
dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" by auto |
|
544 |
show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e " |
|
545 |
proof (rule exI[where x="max N n"], safe) |
|
546 |
fix na assume "max N n \<le> na" |
|
547 |
hence "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) = |
|
548 |
dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^isub>F t) (z t)" using t |
|
549 |
by (subst index_shift[OF I]) auto |
|
550 |
also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp |
|
551 |
finally show "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e" . |
|
552 |
qed |
|
553 |
qed |
|
554 |
hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) ----> (finmap_of (Utn ` J n) z)\<^isub>F t" |
|
555 |
by (simp add: tendsto_intros) |
|
556 |
} ultimately |
|
557 |
have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z" |
|
558 |
by (rule tendsto_finmap) |
|
559 |
hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z" |
|
560 |
by (intro lim_subseq) (simp add: subseq_def) |
|
561 |
moreover |
|
562 |
have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)" |
|
563 |
apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI) |
|
564 |
apply (rule le_trans) |
|
565 |
apply (rule le_add2) |
|
566 |
using seq_suble[OF subseq_diagseq] |
|
567 |
apply auto |
|
568 |
done |
|
569 |
moreover |
|
570 |
from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed) |
|
571 |
ultimately |
|
572 |
have "finmap_of (Utn ` J n) z \<in> K' n" |
|
573 |
unfolding closed_sequential_limits by blast |
|
574 |
also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))" |
|
575 |
by (auto simp: finmap_eq_iff fm_def compose_def f_inv_into_f) |
|
576 |
finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" . |
|
577 |
moreover |
|
578 |
let ?J = "\<Union>n. J n" |
|
579 |
have "(?J \<inter> J n) = J n" by auto |
|
580 |
ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n" |
|
581 |
unfolding K_def by (auto simp: proj_space space_PiM) |
|
582 |
hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def |
|
583 |
using J by (auto simp: prod_emb_def extensional_def) |
|
584 |
also have "\<dots> \<subseteq> Z n" using Z' by simp |
|
585 |
finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" . |
|
586 |
} note in_Z = this |
|
587 |
hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto |
|
588 |
hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp |
|
589 |
thus False using Z by simp |
|
590 |
qed |
|
591 |
ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0" |
|
592 |
using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (Z i)"] by simp |
|
593 |
qed |
|
594 |
then guess \<mu> .. note \<mu> = this |
|
595 |
def f \<equiv> "finmap_of J B" |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
596 |
show "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)" |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
597 |
proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>]) |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
598 |
show "positive (sets (lim\<^isub>B I P)) \<mu>" "countably_additive (sets (lim\<^isub>B I P)) \<mu>" |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
599 |
using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def) |
50088 | 600 |
next |
601 |
show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel" |
|
602 |
using assms by (auto simp: f_def) |
|
603 |
next |
|
604 |
fix J and X::"'i \<Rightarrow> 'a set" |
|
605 |
show "prod_emb I (\<lambda>_. borel) J (Pi\<^isub>E J X) \<in> Pow ((I \<rightarrow> space borel) \<inter> extensional I)" |
|
606 |
by (auto simp: prod_emb_def) |
|
607 |
assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel" |
|
608 |
hence "emb I J (Pi\<^isub>E J X) \<in> generator" using assms |
|
609 |
by (intro generatorI[where J=J and X="Pi\<^isub>E J X"]) (auto intro: sets_PiM_I_finite) |
|
610 |
hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp |
|
611 |
also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)" |
|
612 |
using JX assms proj_sets |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
613 |
by (subst \<mu>G_eq) (auto simp: \<mu>G_eq limP_finite intro: sets_PiM_I_finite) |
50088 | 614 |
finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" . |
615 |
next |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
616 |
show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)" |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
617 |
using assms by (simp add: f_def limP_finite Pi_def) |
50088 | 618 |
qed |
619 |
qed |
|
620 |
||
621 |
end |
|
622 |
||
50090 | 623 |
hide_const (open) PiF |
624 |
hide_const (open) Pi\<^isub>F |
|
625 |
hide_const (open) Pi' |
|
626 |
hide_const (open) Abs_finmap |
|
627 |
hide_const (open) Rep_finmap |
|
628 |
hide_const (open) finmap_of |
|
629 |
hide_const (open) finmapset |
|
630 |
hide_const (open) proj |
|
631 |
hide_const (open) domain |
|
632 |
hide_const (open) enum_basis_finmap |
|
633 |
||
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
634 |
sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^isub>B I P)" |
50088 | 635 |
proof |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
636 |
show "emeasure (lim\<^isub>B I P) (space (lim\<^isub>B I P)) = 1" |
50088 | 637 |
proof cases |
638 |
assume "I = {}" |
|
639 |
interpret prob_space "P {}" using prob_space by simp |
|
640 |
show ?thesis |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
641 |
by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`) |
50088 | 642 |
next |
643 |
assume "I \<noteq> {}" |
|
644 |
then obtain i where "i \<in> I" by auto |
|
645 |
interpret prob_space "P {i}" using prob_space by simp |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
646 |
have R: "(space (lim\<^isub>B I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))" |
50088 | 647 |
by (auto simp: prod_emb_def space_PiM) |
648 |
moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM) |
|
649 |
ultimately show ?thesis using `i \<in> I` |
|
650 |
apply (subst R) |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
651 |
apply (subst emeasure_limB_emb_not_empty) |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
652 |
apply (auto simp: limP_finite emeasure_space_1) |
50088 | 653 |
done |
654 |
qed |
|
655 |
qed |
|
656 |
||
657 |
context polish_projective begin |
|
658 |
||
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
659 |
lemma emeasure_limB_emb: |
50088 | 660 |
assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
661 |
shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)" |
50088 | 662 |
proof cases |
663 |
interpret prob_space "P {}" using prob_space by simp |
|
664 |
assume "J = {}" |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
665 |
moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^isub>B I P)" |
50088 | 666 |
by (auto simp: space_PiM prod_emb_def) |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
667 |
moreover have "{\<lambda>x. undefined} = space (lim\<^isub>B {} P)" |
50088 | 668 |
by (auto simp: space_PiM prod_emb_def) |
669 |
ultimately show ?thesis |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
670 |
by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP) |
50088 | 671 |
next |
672 |
assume "J \<noteq> {}" with X show ?thesis |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
673 |
by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite) |
50088 | 674 |
qed |
675 |
||
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
676 |
lemma measure_limB_emb: |
50088 | 677 |
assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
678 |
shows "measure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)" |
50088 | 679 |
proof - |
680 |
interpret prob_space "P J" using prob_space assms by simp |
|
681 |
show ?thesis |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
682 |
using emeasure_limB_emb[OF assms] |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
683 |
unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure |
50088 | 684 |
by simp |
685 |
qed |
|
686 |
||
687 |
end |
|
688 |
||
689 |
locale polish_product_prob_space = |
|
690 |
product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set" |
|
691 |
||
692 |
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)" |
|
693 |
proof qed |
|
694 |
||
695 |
lemma (in polish_product_prob_space) |
|
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
696 |
limP_eq_PiM: |
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
697 |
"I \<noteq> {} \<Longrightarrow> lim\<^isub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) = |
50088 | 698 |
PiM I (\<lambda>_. borel)" |
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset
|
699 |
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb) |
50088 | 700 |
|
701 |
end |