author | wenzelm |
Fri, 01 Jan 2016 10:49:00 +0100 | |
changeset 62020 | 5d208fd2507d |
parent 61973 | 0c7e865fa7cb |
child 62397 | 5ae24f33d343 |
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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section \<open>Projective Limit\<close> |
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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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||
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subsection \<open>Sequences of Finite Maps in Compact Sets\<close> |
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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)" |
22 |
assumes f_in_K: "\<And>n. K n \<noteq> {}" |
|
23 |
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)" |
50088 | 29 |
using proj_in_K f_in_K |
30 |
proof cases |
|
31 |
obtain k where "k \<in> K (Suc 0)" using f_in_K by auto |
|
32 |
assume "\<forall>n. t \<notin> domain (f n)" |
|
33 |
thus ?thesis |
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61808 | 34 |
by (auto intro!: exI[where x=1] image_eqI[OF _ \<open>k \<in> K (Suc 0)\<close>] |
35 |
simp: domain_K[OF \<open>k \<in> K (Suc 0)\<close>]) |
|
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qed blast |
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lemma proj_in_KE: |
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39 |
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 |
41 |
||
42 |
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) |
45 |
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46 |
end |
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47 |
||
48 |
lemma compactE': |
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50884
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differentiate (cover) compactness and sequential compactness
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49 |
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) \<longlongrightarrow> l) sequentially" |
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proof atomize_elim |
53 |
have "subseq (op + m)" by (simp add: subseq_def) |
|
54 |
have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto |
|
61808 | 55 |
from seq_compactE[OF \<open>compact S\<close>[unfolded compact_eq_seq_compact_metric] this] guess l r . |
61969 | 56 |
hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) \<longlonglongrightarrow> l" |
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using subseq_o[OF \<open>subseq (op + m)\<close> \<open>subseq r\<close>] by (auto simp: o_def) |
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thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l" by blast |
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qed |
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||
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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) \<longlonglongrightarrow> l)" |
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proof |
63 |
fix n s |
|
64 |
assume "subseq s" |
|
65 |
from proj_in_KE[of n] guess n0 . note n0 = this |
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66 |
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 |
68 |
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 " |
50088 | 72 |
by auto |
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qed |
|
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from compactE'[OF compact_projset this] guess ls rs . |
|
61969 | 75 |
thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^sub>F n) \<longlonglongrightarrow> l)" by (auto simp: o_def) |
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qed |
77 |
||
61969 | 78 |
lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) \<longlonglongrightarrow> l" |
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proof - |
61969 | 80 |
obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) \<longlonglongrightarrow> l" |
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81 |
proof (atomize_elim, rule diagseq_holds) |
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82 |
fix r s n |
8cc7f76b827a
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|
83 |
assume "subseq r" |
61969 | 84 |
assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) \<longlonglongrightarrow> l" |
85 |
then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) \<longlonglongrightarrow> l" |
|
52681
8cc7f76b827a
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86 |
by (auto simp: o_def) |
61969 | 87 |
hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) \<longlonglongrightarrow> l" using \<open>subseq r\<close> |
52681
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tuned definition of seqseq; clarified usage of diagseq via diagseq_holds
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|
88 |
by (rule LIMSEQ_subseq_LIMSEQ) |
61969 | 89 |
thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) \<longlonglongrightarrow> l" by (auto simp add: o_def) |
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qed |
61969 | 91 |
hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) \<longlonglongrightarrow> l" by (simp add: ac_simps) |
92 |
hence "(\<lambda>i. (f (diagseq i))\<^sub>F n) \<longlonglongrightarrow> l" by (rule LIMSEQ_offset) |
|
52681
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parents:
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93 |
thus ?thesis .. |
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qed |
95 |
||
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subsection \<open>Daniell-Kolmogorov Theorem\<close> |
50088 | 97 |
|
61808 | 98 |
text \<open>Existence of Projective Limit\<close> |
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|
100 |
locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure" |
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for I::"'i set" and P |
|
102 |
begin |
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103 |
||
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104 |
lemma emeasure_lim_emb: |
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assumes X: "J \<subseteq> I" "finite J" "X \<in> sets (\<Pi>\<^sub>M i\<in>J. borel)" |
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cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
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parents:
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shows "lim (emb I J X) = P J X" |
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107 |
proof (rule emeasure_lim) |
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108 |
write mu_G ("\<mu>G") |
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109 |
interpret generator: algebra "space (PiM I (\<lambda>i. borel))" generator |
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parents:
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110 |
by (rule algebra_generator) |
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hoelzl
parents:
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changeset
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111 |
|
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
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diff
changeset
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112 |
fix J and B :: "nat \<Rightarrow> ('i \<Rightarrow> 'a) set" |
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hoelzl
parents:
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113 |
assume J: "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)" "incseq J" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
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parents:
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and B: "decseq (\<lambda>n. emb I (J n) (B n))" |
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cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
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parents:
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and "0 < (INF i. P (J i) (B i))" (is "0 < ?a") |
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cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
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116 |
moreover have "?a \<le> 1" |
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cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
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changeset
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117 |
using J by (auto intro!: INF_lower2[of 0] prob_space_P[THEN prob_space.measure_le_1]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
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118 |
ultimately obtain r where r: "?a = ereal r" "0 < r" "r \<le> 1" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
119 |
by (cases "?a") auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
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120 |
def Z \<equiv> "\<lambda>n. emb I (J n) (B n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
121 |
have Z_mono: "n \<le> m \<Longrightarrow> Z m \<subseteq> Z n" for n m |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
122 |
unfolding Z_def using B[THEN antimonoD, of n m] . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
123 |
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
124 |
using \<open>incseq J\<close> by (force simp: incseq_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
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changeset
|
125 |
note [simp] = \<open>\<And>n. finite (J n)\<close> |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
126 |
interpret prob_space "P (J i)" for i using J prob_space_P by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
127 |
|
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
128 |
have P_eq[simp]: |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
129 |
"sets (P (J i)) = sets (\<Pi>\<^sub>M i\<in>J i. borel)" "space (P (J i)) = space (\<Pi>\<^sub>M i\<in>J i. borel)" for i |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
130 |
using J by (auto simp: sets_P space_P) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
131 |
|
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
132 |
have "Z i \<in> generator" for i |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
133 |
unfolding Z_def by (auto intro!: generator.intros J) |
50088 | 134 |
|
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
135 |
have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
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changeset
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136 |
def Utn \<equiv> "to_nat_on (\<Union>n. J n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
137 |
interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
138 |
by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
139 |
have inj_on_Utn: "inj_on Utn (\<Union>n. J n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
140 |
unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
141 |
hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
142 |
def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
143 |
interpret P': prob_space "P' n" for n |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
144 |
unfolding P'_def mapmeasure_def using J |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
145 |
by (auto intro!: prob_space_distr fm_measurable simp: measurable_cong_sets[OF sets_P]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
146 |
|
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
147 |
let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
148 |
{ fix n |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
149 |
have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
150 |
using J by (auto simp: P'_def mapmeasure_PiM space_P sets_P) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
151 |
also |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
152 |
have "\<dots> = ?SUP n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
153 |
proof (rule inner_regular) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
154 |
show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
155 |
next |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
156 |
show "fm n ` B n \<in> sets borel" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
157 |
unfolding borel_eq_PiF_borel by (auto simp: P'_def fm_image_measurable_finite sets_P J(3)) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
158 |
qed simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
159 |
finally have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
160 |
} note R = this |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
161 |
have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a \<and> compact K \<and> K \<subseteq> fm n ` B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
162 |
proof |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
163 |
fix n show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and> |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
164 |
compact K \<and> K \<subseteq> fm n ` B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
165 |
unfolding R[of n] |
50088 | 166 |
proof (rule ccontr) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
167 |
assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and> |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
168 |
compact K' \<and> K' \<subseteq> fm n ` B n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
169 |
have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
170 |
proof (intro SUP_least) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
171 |
fix K |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
172 |
assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
173 |
with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
174 |
by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
175 |
hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
176 |
unfolding not_less[symmetric] by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
177 |
hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K" |
61808 | 178 |
using \<open>0 < ?a\<close> by (auto simp add: ereal_less_minus_iff ac_simps) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
179 |
thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
180 |
qed |
61808 | 181 |
hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using \<open>0 < ?a\<close> by simp |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
182 |
hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
183 |
hence "0 \<le> - (2 powr (-n) * ?a)" |
61808 | 184 |
using \<open>?SUP n \<noteq> \<infinity>\<close> \<open>?SUP n \<noteq> - \<infinity>\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
185 |
by (subst (asm) ereal_add_le_add_iff) (auto simp:) |
61808 | 186 |
moreover have "ereal (2 powr - real n) * ?a > 0" using \<open>0 < ?a\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
187 |
by (auto simp: ereal_zero_less_0_iff) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
188 |
ultimately show False by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
189 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
190 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
191 |
then obtain K' where K': |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
192 |
"\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
193 |
"\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
194 |
unfolding choice_iff by blast |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
195 |
def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
196 |
have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
197 |
unfolding K_def |
61808 | 198 |
using compact_imp_closed[OF \<open>compact (K' _)\<close>] |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
199 |
by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
200 |
(auto simp: borel_eq_PiF_borel[symmetric]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
201 |
have K_B: "\<And>n. K n \<subseteq> B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
202 |
proof |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
203 |
fix x n assume "x \<in> K n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
204 |
then have fm_in: "fm n x \<in> fm n ` B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
205 |
using K' by (force simp: K_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
206 |
show "x \<in> B n" |
61808 | 207 |
using \<open>x \<in> K n\<close> K_sets sets.sets_into_space J(1,2,3)[of n] inj_on_image_mem_iff[OF inj_on_fm] |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61359
diff
changeset
|
208 |
by (metis (no_types) Int_iff K_def fm_in space_borel) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
209 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
210 |
def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
211 |
have Z': "\<And>n. Z' n \<subseteq> Z n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
212 |
unfolding Z'_def Z_def |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
213 |
proof (rule prod_emb_mono, safe) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
214 |
fix n x assume "x \<in> K n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
215 |
hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
216 |
by (simp_all add: K_def space_P) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
217 |
note this(1) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
218 |
also have "K' n \<subseteq> fm n ` B n" by (simp add: K') |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
219 |
finally have "fm n x \<in> fm n ` B n" . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
220 |
thus "x \<in> B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
221 |
proof safe |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
222 |
fix y assume y: "y \<in> B n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
223 |
hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
224 |
by (auto simp add: space_P sets_P) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
225 |
assume "fm n x = fm n y" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
226 |
note inj_onD[OF inj_on_fm[OF space_borel], |
61808 | 227 |
OF \<open>fm n x = fm n y\<close> \<open>x \<in> space _\<close> \<open>y \<in> space _\<close>] |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
228 |
with y show "x \<in> B n" by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
229 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
230 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
231 |
have "\<And>n. Z' n \<in> generator" using J K'(2) unfolding Z'_def |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
232 |
by (auto intro!: generator.intros measurable_sets[OF fm_measurable[of _ "Collect finite"]] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
233 |
simp: K_def borel_eq_PiF_borel[symmetric] compact_imp_closed) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
234 |
def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
235 |
hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
236 |
hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
237 |
have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
238 |
hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto |
50088 | 239 |
|
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
240 |
have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
241 |
proof - |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
242 |
fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
243 |
have "Y n = (\<Inter>i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
244 |
by (auto simp: Y_def Z'_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
245 |
also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. emb (J n) (J i) (K i))" |
61808 | 246 |
using \<open>n \<ge> 1\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
247 |
by (subst prod_emb_INT) auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
248 |
finally |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
249 |
have Y_emb: |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
250 |
"Y n = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" . |
61808 | 251 |
hence "Y n \<in> generator" using J J_mono K_sets \<open>n \<ge> 1\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
252 |
by (auto simp del: prod_emb_INT intro!: generator.intros) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
253 |
have *: "\<mu>G (Z n) = P (J n) (B n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
254 |
unfolding Z_def using J by (intro mu_G_spec) auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
255 |
then have "\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
256 |
note * |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
257 |
moreover have *: "\<mu>G (Y n) = P (J n) (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" |
61808 | 258 |
unfolding Y_emb using J J_mono K_sets \<open>n \<ge> 1\<close> by (subst mu_G_spec) auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
259 |
then have "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
260 |
note * |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
261 |
moreover have "\<mu>G (Z n - Y n) = |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
262 |
P (J n) (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))" |
61808 | 263 |
unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets \<open>n \<ge> 1\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
264 |
by (subst mu_G_spec) (auto intro!: sets.Diff) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
265 |
ultimately |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
266 |
have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)" |
61808 | 267 |
using J J_mono K_sets \<open>n \<ge> 1\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
268 |
by (simp only: emeasure_eq_measure Z_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
269 |
(auto dest!: bspec[where x=n] intro!: measure_Diff[symmetric] set_mp[OF K_B] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
270 |
simp: extensional_restrict emeasure_eq_measure prod_emb_iff sets_P space_P) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
271 |
also have subs: "Z n - Y n \<subseteq> (\<Union>i\<in>{1..n}. (Z i - Z' i))" |
61808 | 272 |
using \<open>n \<ge> 1\<close> unfolding Y_def UN_extend_simps(7) by (intro UN_mono Diff_mono Z_mono order_refl) auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
273 |
have "Z n - Y n \<in> generator" "(\<Union>i\<in>{1..n}. (Z i - Z' i)) \<in> generator" |
61808 | 274 |
using \<open>Z' _ \<in> generator\<close> \<open>Z _ \<in> generator\<close> \<open>Y _ \<in> generator\<close> by auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
275 |
hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union>i\<in>{1..n}. (Z i - Z' i))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
276 |
using subs generator.additive_increasing[OF positive_mu_G additive_mu_G] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
277 |
unfolding increasing_def by auto |
61808 | 278 |
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using \<open>Z _ \<in> generator\<close> \<open>Z' _ \<in> generator\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
279 |
by (intro generator.subadditive[OF positive_mu_G additive_mu_G]) auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
280 |
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
281 |
proof (rule setsum_mono) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
282 |
fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
283 |
have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
284 |
unfolding Z'_def Z_def by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
285 |
also have "\<dots> = P (J i) (B i - K i)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
286 |
using J K_sets by (subst mu_G_spec) auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
287 |
also have "\<dots> = P (J i) (B i) - P (J i) (K i)" |
61808 | 288 |
using K_sets J \<open>K _ \<subseteq> B _\<close> by (simp add: emeasure_Diff) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
289 |
also have "\<dots> = P (J i) (B i) - P' i (K' i)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
290 |
unfolding K_def P'_def |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
291 |
by (auto simp: mapmeasure_PiF borel_eq_PiF_borel[symmetric] |
61808 | 292 |
compact_imp_closed[OF \<open>compact (K' _)\<close>] space_PiM PiE_def) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
293 |
also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
294 |
finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
295 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
296 |
also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real_of_ereal i)) * real_of_ereal ?a)" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
297 |
by (simp add: setsum_left_distrib r) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
298 |
also have "\<dots> < ereal (1 * real_of_ereal ?a)" unfolding less_ereal.simps |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
299 |
proof (rule mult_strict_right_mono) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
300 |
have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
301 |
by (rule setsum.cong) (auto simp: powr_realpow powr_divide power_divide powr_minus_divide) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
302 |
also have "{1..<Suc n} = {..<Suc n} - {0}" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
303 |
also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) = |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
304 |
setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
305 |
also have "\<dots> < 1" by (subst geometric_sum) auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
306 |
finally show "(\<Sum>i = 1..n. 2 powr - real_of_ereal i) < 1" by simp |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
307 |
qed (auto simp: r ereal_less_real_iff zero_ereal_def[symmetric]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
308 |
also have "\<dots> = ?a" by (auto simp: r) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
309 |
also have "\<dots> \<le> \<mu>G (Z n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
310 |
using J by (auto intro: INF_lower simp: Z_def mu_G_spec) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
311 |
finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
312 |
hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)" |
61808 | 313 |
using \<open>\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>\<close> by (simp add: ereal_minus_less) |
314 |
have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by auto |
|
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
315 |
also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))" |
61808 | 316 |
apply (rule ereal_less_add[OF _ R]) using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
317 |
finally have "\<mu>G (Y n) > 0" |
61808 | 318 |
using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by (auto simp: ac_simps zero_ereal_def[symmetric]) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
319 |
thus "Y n \<noteq> {}" using positive_mu_G by (auto simp add: positive_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
320 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
321 |
hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
322 |
then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
323 |
{ |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
324 |
fix t and n m::nat |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
325 |
assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp |
61808 | 326 |
from Y_mono[OF \<open>m \<ge> n\<close>] y[OF \<open>1 \<le> m\<close>] have "y m \<in> Y n" by auto |
327 |
also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF \<open>1 \<le> n\<close>] . |
|
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
328 |
finally |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
329 |
have "fm n (restrict (y m) (J n)) \<in> K' n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
330 |
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
331 |
moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
332 |
using J by (simp add: fm_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
333 |
ultimately have "fm n (y m) \<in> K' n" by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
334 |
} note fm_in_K' = this |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
335 |
interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
336 |
proof |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
337 |
fix n show "compact (K' n)" by fact |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
338 |
next |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
339 |
fix n |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
340 |
from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
341 |
also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
342 |
finally |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
343 |
have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
344 |
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
345 |
thus "K' (Suc n) \<noteq> {}" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
346 |
fix k |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
347 |
assume "k \<in> K' (Suc n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
348 |
with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
349 |
then obtain b where "k = fm (Suc n) b" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
350 |
thus "domain k = domain (fm (Suc n) (y (Suc n)))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
351 |
by (simp_all add: fm_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
352 |
next |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
353 |
fix t and n m::nat |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
354 |
assume "n \<le> m" hence "Suc n \<le> Suc m" by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
355 |
assume "t \<in> domain (fm (Suc n) (y (Suc n)))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
356 |
then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto |
61808 | 357 |
hence "j \<in> J (Suc m)" using J_mono[OF \<open>Suc n \<le> Suc m\<close>] by auto |
358 |
have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using \<open>n \<le> m\<close> |
|
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
359 |
by (intro fm_in_K') simp_all |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
360 |
show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
361 |
apply (rule image_eqI[OF _ img]) |
61808 | 362 |
using \<open>j \<in> J (Suc n)\<close> \<open>j \<in> J (Suc m)\<close> |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
363 |
unfolding j by (subst proj_fm, auto)+ |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
364 |
qed |
61969 | 365 |
have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
366 |
using diagonal_tendsto .. |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
367 |
then obtain z where z: |
61969 | 368 |
"\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z t" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
369 |
unfolding choice_iff by blast |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
370 |
{ |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
371 |
fix n :: nat assume "n \<ge> 1" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
372 |
have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
373 |
by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
374 |
moreover |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
375 |
{ |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
376 |
fix t |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
377 |
assume t: "t \<in> domain (finmap_of (Utn ` J n) z)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
378 |
hence "t \<in> Utn ` J n" by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
379 |
then obtain j where j: "t = Utn j" "j \<in> J n" by auto |
61969 | 380 |
have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z t" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
381 |
apply (subst (2) tendsto_iff, subst eventually_sequentially) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
382 |
proof safe |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
383 |
fix e :: real assume "0 < e" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
384 |
{ fix i and x :: "'i \<Rightarrow> 'a" assume i: "i \<ge> n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
385 |
assume "t \<in> domain (fm n x)" |
61808 | 386 |
hence "t \<in> domain (fm i x)" using J_mono[OF \<open>i \<ge> n\<close>] by auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
387 |
with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
388 |
using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
389 |
} note index_shift = this |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
390 |
have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
391 |
apply (rule le_SucI) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
392 |
apply (rule order_trans) apply simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
393 |
apply (rule seq_suble[OF subseq_diagseq]) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
394 |
done |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
395 |
from z |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
396 |
have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" |
61808 | 397 |
unfolding tendsto_iff eventually_sequentially using \<open>0 < e\<close> by auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
398 |
then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow> |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
399 |
dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
400 |
show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e " |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
401 |
proof (rule exI[where x="max N n"], safe) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
402 |
fix na assume "max N n \<le> na" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
403 |
hence "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) = |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
404 |
dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
405 |
by (subst index_shift[OF I]) auto |
61808 | 406 |
also have "\<dots> < e" using \<open>max N n \<le> na\<close> by (intro N) simp |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
407 |
finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" . |
50088 | 408 |
qed |
409 |
qed |
|
61969 | 410 |
hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> (finmap_of (Utn ` J n) z)\<^sub>F t" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
411 |
by (simp add: tendsto_intros) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
412 |
} ultimately |
61969 | 413 |
have "(\<lambda>i. fm n (y (Suc (diagseq i)))) \<longlonglongrightarrow> finmap_of (Utn ` J n) z" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
414 |
by (rule tendsto_finmap) |
61969 | 415 |
hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) \<longlonglongrightarrow> finmap_of (Utn ` J n) z" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
416 |
by (intro lim_subseq) (simp add: subseq_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
417 |
moreover |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
418 |
have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)" |
61808 | 419 |
apply (auto simp add: o_def intro!: fm_in_K' \<open>1 \<le> n\<close> le_SucI) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
420 |
apply (rule le_trans) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
421 |
apply (rule le_add2) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
422 |
using seq_suble[OF subseq_diagseq] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
423 |
apply auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
424 |
done |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
425 |
moreover |
61808 | 426 |
from \<open>compact (K' n)\<close> have "closed (K' n)" by (rule compact_imp_closed) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
427 |
ultimately |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
428 |
have "finmap_of (Utn ` J n) z \<in> K' n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
429 |
unfolding closed_sequential_limits by blast |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
430 |
also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
431 |
unfolding finmap_eq_iff |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
432 |
proof clarsimp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
433 |
fix i assume i: "i \<in> J n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
434 |
hence "from_nat_into (\<Union>n. J n) (Utn i) = i" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
435 |
unfolding Utn_def |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
436 |
by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
437 |
with i show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
438 |
by (simp add: finmap_eq_iff fm_def compose_def) |
50088 | 439 |
qed |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
440 |
finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
441 |
moreover |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
442 |
let ?J = "\<Union>n. J n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
443 |
have "(?J \<inter> J n) = J n" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
444 |
ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
445 |
unfolding K_def by (auto simp: space_P space_PiM) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
446 |
hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
447 |
using J by (auto simp: prod_emb_def PiE_def extensional_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
448 |
also have "\<dots> \<subseteq> Z n" using Z' by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
449 |
finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" . |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
450 |
} note in_Z = this |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
451 |
hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
452 |
thus "(\<Inter>i. Z i) \<noteq> {}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
453 |
using INT_decseq_offset[OF antimonoI[OF Z_mono]] by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
454 |
qed fact+ |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
455 |
|
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
456 |
lemma measure_lim_emb: |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
457 |
"J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> X \<in> sets (\<Pi>\<^sub>M i\<in>J. borel) \<Longrightarrow> measure lim (emb I J X) = measure (P J) X" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
458 |
unfolding measure_def by (subst emeasure_lim_emb) auto |
50088 | 459 |
|
460 |
end |
|
461 |
||
50090 | 462 |
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
|
463 |
hide_const (open) Pi\<^sub>F |
50090 | 464 |
hide_const (open) Pi' |
465 |
hide_const (open) Abs_finmap |
|
466 |
hide_const (open) Rep_finmap |
|
467 |
hide_const (open) finmap_of |
|
468 |
hide_const (open) proj |
|
469 |
hide_const (open) domain |
|
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
470 |
hide_const (open) basis_finmap |
50090 | 471 |
|
61605 | 472 |
sublocale polish_projective \<subseteq> P: prob_space lim |
50088 | 473 |
proof |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
474 |
have *: "emb I {} {\<lambda>x. undefined} = space (\<Pi>\<^sub>M i\<in>I. borel)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
475 |
by (auto simp: prod_emb_def space_PiM) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
476 |
interpret prob_space "P {}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
477 |
using prob_space_P by simp |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
478 |
show "emeasure lim (space lim) = 1" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
479 |
using emeasure_lim_emb[of "{}" "{\<lambda>x. undefined}"] emeasure_space_1 |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
480 |
by (simp add: * PiM_empty space_P) |
50088 | 481 |
qed |
482 |
||
483 |
locale polish_product_prob_space = |
|
484 |
product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set" |
|
485 |
||
486 |
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)" |
|
487 |
proof qed |
|
488 |
||
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
489 |
lemma (in polish_product_prob_space) limP_eq_PiM: "lim = PiM I (\<lambda>_. borel)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
60809
diff
changeset
|
490 |
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_lim_emb) |
50088 | 491 |
|
492 |
end |