50088
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(* Title: HOL/Probability/Projective_Family.thy
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Author: Fabian Immler, TU München
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*)
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header {* Projective Limit *}
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theory Projective_Limit
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imports
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Caratheodory
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Fin_Map
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Regularity
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Projective_Family
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Infinite_Product_Measure
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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 "PiB \<equiv> (\<lambda>J P. PiP J (\<lambda>_. borel) P)"
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lemma
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emeasure_PiB_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 (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB 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: PiP_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 (PiB 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 PiP_finite proj_sets)
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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: PiP_finite proj_sets \<mu>G_eq)
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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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291 |
proof (intro SUP_least)
|
|
292 |
fix K
|
|
293 |
assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
|
|
294 |
with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
|
|
295 |
by auto
|
|
296 |
hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
|
|
297 |
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`
|
|
383 |
by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq)
|
|
384 |
interpret finite_measure "(PiP (J n) (\<lambda>_. borel) P)"
|
|
385 |
proof
|
|
386 |
have "emeasure (PiP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>"
|
|
387 |
using J by (subst emeasure_PiP) auto
|
|
388 |
thus "emeasure (PiP (J n) (\<lambda>_. borel) P) (space (PiP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>"
|
|
389 |
by (simp add: space_PiM)
|
|
390 |
qed
|
|
391 |
have "\<mu>G (Z n) = PiP (J n) (\<lambda>_. borel) P (B n)"
|
|
392 |
unfolding Z_eq using J by (auto simp: \<mu>G_eq)
|
|
393 |
moreover have "\<mu>G (Y n) =
|
|
394 |
PiP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
|
|
395 |
unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst \<mu>G_eq) auto
|
|
396 |
moreover have "\<mu>G (Z n - Y n) = PiP (J n) (\<lambda>_. borel) P
|
|
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
|
|
423 |
apply (subst PiP_finite) apply auto
|
|
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"
|
|
596 |
show "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)"
|
|
597 |
proof (subst emeasure_extend_measure_Pair[OF PiP_def, of I "\<lambda>_. borel" \<mu>])
|
|
598 |
show "positive (sets (PiB I P)) \<mu>" "countably_additive (sets (PiB I P)) \<mu>"
|
|
599 |
using \<mu> unfolding sets_PiP sets_PiM_generator by (auto simp: measure_space_def)
|
|
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
|
|
613 |
by (subst \<mu>G_eq) (auto simp: \<mu>G_eq PiP_finite intro: sets_PiM_I_finite)
|
|
614 |
finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" .
|
|
615 |
next
|
|
616 |
show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (PiP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)"
|
|
617 |
using assms by (simp add: f_def PiP_finite Pi_def)
|
|
618 |
qed
|
|
619 |
qed
|
|
620 |
|
|
621 |
end
|
|
622 |
|
|
623 |
sublocale polish_projective \<subseteq> P!: prob_space "(PiB I P)"
|
|
624 |
proof
|
|
625 |
show "emeasure (PiB I P) (space (PiB I P)) = 1"
|
|
626 |
proof cases
|
|
627 |
assume "I = {}"
|
|
628 |
interpret prob_space "P {}" using prob_space by simp
|
|
629 |
show ?thesis
|
|
630 |
by (simp add: space_PiM_empty PiP_finite emeasure_space_1 `I = {}`)
|
|
631 |
next
|
|
632 |
assume "I \<noteq> {}"
|
|
633 |
then obtain i where "i \<in> I" by auto
|
|
634 |
interpret prob_space "P {i}" using prob_space by simp
|
|
635 |
have R: "(space (PiB I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))"
|
|
636 |
by (auto simp: prod_emb_def space_PiM)
|
|
637 |
moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM)
|
|
638 |
ultimately show ?thesis using `i \<in> I`
|
|
639 |
apply (subst R)
|
|
640 |
apply (subst emeasure_PiB_emb_not_empty)
|
|
641 |
apply (auto simp: PiP_finite emeasure_space_1)
|
|
642 |
done
|
|
643 |
qed
|
|
644 |
qed
|
|
645 |
|
|
646 |
context polish_projective begin
|
|
647 |
|
|
648 |
lemma emeasure_PiB_emb:
|
|
649 |
assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
|
|
650 |
shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)"
|
|
651 |
proof cases
|
|
652 |
interpret prob_space "P {}" using prob_space by simp
|
|
653 |
assume "J = {}"
|
|
654 |
moreover have "emb I {} {\<lambda>x. undefined} = space (PiB I P)"
|
|
655 |
by (auto simp: space_PiM prod_emb_def)
|
|
656 |
moreover have "{\<lambda>x. undefined} = space (PiB {} P)"
|
|
657 |
by (auto simp: space_PiM prod_emb_def)
|
|
658 |
ultimately show ?thesis
|
|
659 |
by (simp add: P.emeasure_space_1 PiP_finite emeasure_space_1 del: space_PiP)
|
|
660 |
next
|
|
661 |
assume "J \<noteq> {}" with X show ?thesis
|
|
662 |
by (subst emeasure_PiB_emb_not_empty) (auto simp: PiP_finite)
|
|
663 |
qed
|
|
664 |
|
|
665 |
lemma measure_PiB_emb:
|
|
666 |
assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
|
|
667 |
shows "measure (PiB I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)"
|
|
668 |
proof -
|
|
669 |
interpret prob_space "P J" using prob_space assms by simp
|
|
670 |
show ?thesis
|
|
671 |
using emeasure_PiB_emb[OF assms]
|
|
672 |
unfolding emeasure_eq_measure PiP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure
|
|
673 |
by simp
|
|
674 |
qed
|
|
675 |
|
|
676 |
end
|
|
677 |
|
|
678 |
locale polish_product_prob_space =
|
|
679 |
product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
|
|
680 |
|
|
681 |
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
|
|
682 |
proof qed
|
|
683 |
|
|
684 |
lemma (in polish_product_prob_space)
|
|
685 |
PiP_eq_PiM:
|
|
686 |
"I \<noteq> {} \<Longrightarrow> PiP I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) =
|
|
687 |
PiM I (\<lambda>_. borel)"
|
|
688 |
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_PiB_emb)
|
|
689 |
|
|
690 |
end
|