author  wenzelm 
Wed, 28 Nov 2012 15:59:18 +0100  
changeset 50252  4aa34bd43228 
parent 50245  dea9363887a6 
child 50884  2b21b4e2d7cb 
permissions  rwrr 
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(* Title: HOL/Probability/Projective_Limit.thy 
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Author: Fabian Immler, TU München 
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*) 

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header {* Projective Limit *} 

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theory Projective_Limit 

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imports 

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Caratheodory 

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Fin_Map 

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Regularity 

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Projective_Family 

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Infinite_Product_Measure 

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"~~/src/HOL/Library/Countable_Set" 
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begin 
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subsection {* Sequences of Finite Maps in Compact Sets *} 

18 

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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) 

45 

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end 

47 

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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 

58 
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) 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" . 

96 
qed 

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finally show ?thesis by (intro exI) (rule LIMSEQ_offset) 

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qed 

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subsection {* DaniellKolmogorov Theorem *} 

101 

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text {* Existence of Projective Limit *} 

103 

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locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure" 

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for I::"'i set" and P 

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begin 

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abbreviation "lim\<^isub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)" 
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lemma emeasure_limB_emb_not_empty: 
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assumes "I \<noteq> {}" 
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assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" 

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shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)" 
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proof  
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let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel" 

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let ?G = generator 

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interpret G!: algebra ?\<Omega> generator by (intro algebra_generator) fact 

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note mu_G_mono = 
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G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`], 

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THEN increasingD] 

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write mu_G ("\<mu>G") 

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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 proj_prob_space[OF `finite J`] . 
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show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite) 
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next 
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fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}" 

132 
then have "decseq (\<lambda>i. \<mu>G (Z i))" 

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by (auto intro!: mu_G_mono simp: decseq_def) 
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moreover 
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have "(INF i. \<mu>G (Z i)) = 0" 

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proof (rule ccontr) 

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assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0") 

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moreover have "0 \<le> ?a" 

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using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def) 
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ultimately have "0 < ?a" by auto 
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hence "?a \<noteq> \<infinity>" by auto 

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have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^isub>M J (\<lambda>_. borel)) \<and> 

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Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^isub>B J P) B" 
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using Z by (intro allI generator_Ex) auto 
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then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" 

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"\<And>n. B' n \<in> sets (\<Pi>\<^isub>M i\<in>J' n. borel)" 

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and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)" 

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unfolding choice_iff by blast 

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moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)" 

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moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)" 

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ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" 

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"\<And>n. B n \<in> sets (\<Pi>\<^isub>M i\<in>J n. borel)" 

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by auto 

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have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" 

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unfolding J_def by force 

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have "\<forall>n. \<exists>j. j \<in> J n" using J by blast 

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then obtain j where j: "\<And>n. j n \<in> J n" 

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unfolding choice_iff by blast 

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note [simp] = `\<And>n. finite (J n)` 

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from J Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G" 

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unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto) 

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interpret prob_space "P (J i)" for i using proj_prob_space by simp 
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have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower) 
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also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq mu_G_eq limP_finite proj_sets) 
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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: limP_finite proj_sets mu_G_eq) 
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have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite) 
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def Utn \<equiv> "to_nat_on (\<Union>n. J n)" 

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interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n 

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by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J]) 

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have inj_on_Utn: "inj_on Utn (\<Union>n. J n)" 

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unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on) 

175 
hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto 

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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) 

183 
also 

184 
have "\<dots> = ?SUP n" 

185 
proof (rule inner_regular) 

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show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>" 

187 
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) 

190 
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) 

194 
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" 

212 
proof (intro SUP_least) 

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fix K 

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assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}" 

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with H have "\<not> ?SUP n  emeasure (P' n) K \<le> 2 powr (n) * ?a" 

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by auto 

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hence "?SUP n  emeasure (P' n) K > 2 powr (n) * ?a" 

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unfolding not_less[symmetric] by simp 

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hence "?SUP n  2 powr (n) * ?a > emeasure (P' n) K" 

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using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps) 

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thus "?SUP n  2 powr (n) * ?a \<ge> emeasure (P' n) K" by simp 

222 
qed 

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hence "?SUP n + 0 \<le> ?SUP n  (2 powr (n) * ?a)" using `0 < ?a` by simp 

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hence "?SUP n + 0 \<le> ?SUP n +  (2 powr (n) * ?a)" unfolding minus_ereal_def . 

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hence "0 \<le>  (2 powr (n) * ?a)" 

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using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq>  \<infinity>` 

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by (subst (asm) ereal_add_le_add_iff) (auto simp:) 

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moreover have "ereal (2 powr  real n) * ?a > 0" using `0 < ?a` 

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by (auto simp: ereal_zero_less_0_iff) 

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ultimately show False by simp 

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qed 

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qed 

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then obtain K' where K': 

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"\<And>n. emeasure (P (J n)) (B n)  emeasure (P' n) (K' n) \<le> ereal (2 powr  real n) * ?a" 

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"\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n" 

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unfolding choice_iff by blast 

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def K \<equiv> "\<lambda>n. fm n ` K' n \<inter> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" 

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have K_sets: "\<And>n. K n \<in> sets (Pi\<^isub>M (J n) (\<lambda>_. borel))" 

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unfolding K_def 

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using compact_imp_closed[OF `compact (K' _)`] 

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by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"]) 

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(auto simp: borel_eq_PiF_borel[symmetric]) 

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have K_B: "\<And>n. K n \<subseteq> B n" 

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proof 

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fix x n 

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assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n" 

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using K' by (force simp: K_def) 

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show "x \<in> B n" 

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using `x \<in> K n` K_sets sets.sets_into_space J[of n] 
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by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto 
50088  251 
qed 
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def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)" 

253 
have Z': "\<And>n. Z' n \<subseteq> Z n" 

254 
unfolding Z_eq unfolding Z'_def 

255 
proof (rule prod_emb_mono, safe) 

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fix n x assume "x \<in> K n" 

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hence "fm n x \<in> K' n" "x \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" 

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by (simp_all add: K_def proj_space) 

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note this(1) 

260 
also have "K' n \<subseteq> fm n ` B n" by (simp add: K') 

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finally have "fm n x \<in> fm n ` B n" . 

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thus "x \<in> B n" 

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proof safe 

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fix y assume "y \<in> B n" 

265 
moreover 

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hence "y \<in> space (Pi\<^isub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"] 
50088  267 
by (auto simp add: proj_space proj_sets) 
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assume "fm n x = fm n y" 

269 
note inj_onD[OF inj_on_fm[OF space_borel], 

270 
OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`] 

271 
ultimately show "x \<in> B n" by simp 

272 
qed 

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qed 

274 
{ fix n 

275 
have "Z' n \<in> ?G" using K' unfolding Z'_def 

276 
apply (intro generatorI'[OF J(13)]) 

277 
unfolding K_def proj_space 

278 
apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]]) 

279 
apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed) 

280 
done 

281 
} 

282 
def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i" 

283 
hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def) 

284 
hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add) 

285 
have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def) 

286 
hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto 

287 
have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}" 

288 
proof  

289 
fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact 

290 
have "Y n = (\<Inter> i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono 

291 
by (auto simp: Y_def Z'_def) 

292 
also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter> i\<in>{1..n}. emb (J n) (J i) (K i))" 

293 
using `n \<ge> 1` 

294 
by (subst prod_emb_INT) auto 

295 
finally 

296 
have Y_emb: 

297 
"Y n = prod_emb I (\<lambda>_. borel) (J n) 

298 
(\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" . 

299 
hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto 

300 
hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1` 

50252  301 
by (subst mu_G_eq) (auto simp: limP_finite proj_sets mu_G_eq) 
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interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)" 
50088  303 
proof 
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have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>" 
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using J by (subst emeasure_limP) auto 
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thus "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>" 
50088  307 
by (simp add: space_PiM) 
308 
qed 

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have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)" 
50252  310 
unfolding Z_eq using J by (auto simp: mu_G_eq) 
50088  311 
moreover have "\<mu>G (Y n) = 
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312 
limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" 
50252  313 
unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_eq) auto 
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314 
moreover have "\<mu>G (Z n  Y n) = limP (J n) (\<lambda>_. borel) P 
50088  315 
(B n  (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))" 
316 
unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1` 

50252  317 
by (subst mu_G_eq) (auto intro!: sets.Diff) 
50088  318 
ultimately 
319 
have "\<mu>G (Z n)  \<mu>G (Y n) = \<mu>G (Z n  Y n)" 

320 
using J J_mono K_sets `n \<ge> 1` 

321 
by (simp only: emeasure_eq_measure) 

322 
(auto dest!: bspec[where x=n] 

323 
simp: extensional_restrict emeasure_eq_measure prod_emb_iff 

324 
intro!: measure_Diff[symmetric] set_mp[OF K_B]) 

325 
also have subs: "Z n  Y n \<subseteq> (\<Union> i\<in>{1..n}. (Z i  Z' i))" using Z' Z `n \<ge> 1` 

326 
unfolding Y_def by (force simp: decseq_def) 

327 
have "Z n  Y n \<in> ?G" "(\<Union> i\<in>{1..n}. (Z i  Z' i)) \<in> ?G" 

328 
using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto 

329 
hence "\<mu>G (Z n  Y n) \<le> \<mu>G (\<Union> i\<in>{1..n}. (Z i  Z' i))" 

50252  330 
using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]] 
50088  331 
unfolding increasing_def by auto 
332 
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i  Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G` 

50252  333 
by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto 
50088  334 
also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr real i * ?a)" 
335 
proof (rule setsum_mono) 

336 
fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp 

337 
have "\<mu>G (Z i  Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i  K i))" 

338 
unfolding Z'_def Z_eq by simp 

339 
also have "\<dots> = P (J i) (B i  K i)" 

50252  340 
apply (subst mu_G_eq) using J K_sets apply auto 
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341 
apply (subst limP_finite) apply auto 
50088  342 
done 
343 
also have "\<dots> = P (J i) (B i)  P (J i) (K i)" 

344 
apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets) 

345 
done 

346 
also have "\<dots> = P (J i) (B i)  P' i (K' i)" 

347 
unfolding K_def P'_def 

348 
by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric] 

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349 
compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def) 
50088  350 
also have "\<dots> \<le> ereal (2 powr  real i) * ?a" using K'(1)[of i] . 
351 
finally show "\<mu>G (Z i  Z' i) \<le> (2 powr  real i) * ?a" . 

352 
qed 

353 
also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr real i) * ereal(real ?a))" 

354 
using `?a \<noteq> \<infinity>` `?a \<noteq>  \<infinity>` by (subst ereal_real') auto 

355 
also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr real i) * (real ?a))" by simp 

356 
also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr real i)) * real ?a)" 

357 
by (simp add: setsum_left_distrib) 

358 
also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps 

359 
proof (rule mult_strict_right_mono) 

360 
have "(\<Sum>i\<in>{1..n}. 2 powr  real i) = (\<Sum>i\<in>{1..<Suc n}. (1/2) ^ i)" 

361 
by (rule setsum_cong) 

362 
(auto simp: powr_realpow[symmetric] powr_minus powr_divide inverse_eq_divide) 

363 
also have "{1..<Suc n} = {0..<Suc n}  {0}" by auto 

364 
also have "setsum (op ^ (1 / 2::real)) ({0..<Suc n}  {0}) = 

365 
setsum (op ^ (1 / 2)) ({0..<Suc n})  1" by (auto simp: setsum_diff1) 

366 
also have "\<dots> < 1" by (subst sumr_geometric) auto 

367 
finally show "(\<Sum>i = 1..n. 2 powr  real i) < 1" . 

368 
qed (auto simp: 

369 
`0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq>  \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric]) 

370 
also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real') 

371 
also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower) 

372 
finally have "\<mu>G (Z n)  \<mu>G (Y n) < \<mu>G (Z n)" . 

373 
hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)" 

374 
using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less) 

375 
have "0 \<le> ( \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto 

376 
also have "\<dots> < ( \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))" 

377 
apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto 

378 
finally have "\<mu>G (Y n) > 0" 

379 
using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric]) 

50252  380 
thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def) 
50088  381 
qed 
382 
hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto 

383 
then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force 

384 
{ 

385 
fix t and n m::nat 

386 
assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp 

387 
from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto 

388 
also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] . 

389 
finally 

390 
have "fm n (restrict (y m) (J n)) \<in> K' n" 

391 
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) 

392 
moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)" 

393 
using J by (simp add: fm_def) 

394 
ultimately have "fm n (y m) \<in> K' n" by simp 

395 
} note fm_in_K' = this 

396 
interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel 

397 
proof 

398 
fix n show "compact (K' n)" by fact 

399 
next 

400 
fix n 

401 
from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto 

402 
also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto 

403 
finally 

404 
have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)" 

405 
unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff) 

406 
thus "K' (Suc n) \<noteq> {}" by auto 

407 
fix k 

408 
assume "k \<in> K' (Suc n)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
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diff
changeset

409 
with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto 
50088  410 
then obtain b where "k = fm (Suc n) b" by auto 
411 
thus "domain k = domain (fm (Suc n) (y (Suc n)))" 

412 
by (simp_all add: fm_def) 

413 
next 

414 
fix t and n m::nat 

415 
assume "n \<le> m" hence "Suc n \<le> Suc m" by simp 

416 
assume "t \<in> domain (fm (Suc n) (y (Suc n)))" 

417 
then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto 

418 
hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto 

419 
have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m` 

420 
by (intro fm_in_K') simp_all 

421 
show "(fm (Suc m) (y (Suc m)))\<^isub>F t \<in> (\<lambda>k. (k)\<^isub>F t) ` K' (Suc n)" 

422 
apply (rule image_eqI[OF _ img]) 

423 
using `j \<in> J (Suc n)` `j \<in> J (Suc m)` 

424 
unfolding j by (subst proj_fm, auto)+ 

425 
qed 

426 
have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) > z" 

427 
using diagonal_tendsto .. 

428 
then obtain z where z: 

429 
"\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) > z t" 

430 
unfolding choice_iff by blast 

431 
{ 

432 
fix n :: nat assume "n \<ge> 1" 

433 
have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)" 

434 
by simp 

435 
moreover 

436 
{ 

437 
fix t 

438 
assume t: "t \<in> domain (finmap_of (Utn ` J n) z)" 

439 
hence "t \<in> Utn ` J n" by simp 

440 
then obtain j where j: "t = Utn j" "j \<in> J n" by auto 

441 
have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) > z t" 

442 
apply (subst (2) tendsto_iff, subst eventually_sequentially) 

443 
proof safe 

444 
fix e :: real assume "0 < e" 

445 
{ fix i x assume "i \<ge> n" "t \<in> domain (fm n x)" 

446 
moreover 

447 
hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto 

448 
ultimately have "(fm i x)\<^isub>F t = (fm n x)\<^isub>F t" 

50243  449 
using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn]) 
50088  450 
} note index_shift = this 
451 
have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n" 

452 
apply (rule le_SucI) 

453 
apply (rule order_trans) apply simp 

454 
apply (rule seq_suble[OF subseq_diagseq]) 

455 
done 

456 
from z 

457 
have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" 

458 
unfolding tendsto_iff eventually_sequentially using `0 < e` by auto 

459 
then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow> 

460 
dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^isub>F t) (z t) < e" by auto 

461 
show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e " 

462 
proof (rule exI[where x="max N n"], safe) 

463 
fix na assume "max N n \<le> na" 

464 
hence "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) = 

465 
dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^isub>F t) (z t)" using t 

466 
by (subst index_shift[OF I]) auto 

467 
also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp 

468 
finally show "dist ((fm n (y (Suc (diagseq na))))\<^isub>F t) (z t) < e" . 

469 
qed 

470 
qed 

471 
hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^isub>F t) > (finmap_of (Utn ` J n) z)\<^isub>F t" 

472 
by (simp add: tendsto_intros) 

473 
} ultimately 

474 
have "(\<lambda>i. fm n (y (Suc (diagseq i)))) > finmap_of (Utn ` J n) z" 

475 
by (rule tendsto_finmap) 

476 
hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) > finmap_of (Utn ` J n) z" 

477 
by (intro lim_subseq) (simp add: subseq_def) 

478 
moreover 

479 
have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)" 

480 
apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI) 

481 
apply (rule le_trans) 

482 
apply (rule le_add2) 

483 
using seq_suble[OF subseq_diagseq] 

484 
apply auto 

485 
done 

486 
moreover 

487 
from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed) 

488 
ultimately 

489 
have "finmap_of (Utn ` J n) z \<in> K' n" 

490 
unfolding closed_sequential_limits by blast 

491 
also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))" 

50243  492 
unfolding finmap_eq_iff 
493 
proof clarsimp 

494 
fix i assume "i \<in> J n" 

495 
moreover hence "from_nat_into (\<Union>n. J n) (Utn i) = i" 

496 
unfolding Utn_def 

497 
by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto 

498 
ultimately show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^isub>F (Utn i)" 

499 
by (simp add: finmap_eq_iff fm_def compose_def) 

500 
qed 

50088  501 
finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" . 
502 
moreover 

503 
let ?J = "\<Union>n. J n" 

504 
have "(?J \<inter> J n) = J n" by auto 

505 
ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n" 

506 
unfolding K_def by (auto simp: proj_space space_PiM) 

507 
hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def 

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hoelzl
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50101
diff
changeset

508 
using J by (auto simp: prod_emb_def PiE_def extensional_def) 
50088  509 
also have "\<dots> \<subseteq> Z n" using Z' by simp 
510 
finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" . 

511 
} note in_Z = this 

512 
hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto 

513 
hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp 

514 
thus False using Z by simp 

515 
qed 

516 
ultimately show "(\<lambda>i. \<mu>G (Z i)) > 0" 

517 
using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (Z i)"] by simp 

518 
qed 

519 
then guess \<mu> .. note \<mu> = this 

520 
def f \<equiv> "finmap_of J B" 

50095
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immler
parents:
50091
diff
changeset

521 
show "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)" 
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

522 
proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>]) 
94d7dfa9f404
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immler
parents:
50091
diff
changeset

523 
show "positive (sets (lim\<^isub>B I P)) \<mu>" "countably_additive (sets (lim\<^isub>B I P)) \<mu>" 
94d7dfa9f404
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immler
parents:
50091
diff
changeset

524 
using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def) 
50088  525 
next 
526 
show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel" 

527 
using assms by (auto simp: f_def) 

528 
next 

529 
fix J and X::"'i \<Rightarrow> 'a set" 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset

530 
show "prod_emb I (\<lambda>_. borel) J (Pi\<^isub>E J X) \<in> Pow (I \<rightarrow>\<^isub>E space borel)" 
50088  531 
by (auto simp: prod_emb_def) 
532 
assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel" 

533 
hence "emb I J (Pi\<^isub>E J X) \<in> generator" using assms 

534 
by (intro generatorI[where J=J and X="Pi\<^isub>E J X"]) (auto intro: sets_PiM_I_finite) 

535 
hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp 

536 
also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)" 

537 
using JX assms proj_sets 

50252  538 
by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite) 
50088  539 
finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" . 
540 
next 

50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

541 
show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)" 
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

542 
using assms by (simp add: f_def limP_finite Pi_def) 
50088  543 
qed 
544 
qed 

545 

546 
end 

547 

50090  548 
hide_const (open) PiF 
549 
hide_const (open) Pi\<^isub>F 

550 
hide_const (open) Pi' 

551 
hide_const (open) Abs_finmap 

552 
hide_const (open) Rep_finmap 

553 
hide_const (open) finmap_of 

554 
hide_const (open) proj 

555 
hide_const (open) domain 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

556 
hide_const (open) basis_finmap 
50090  557 

50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

558 
sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^isub>B I P)" 
50088  559 
proof 
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

560 
show "emeasure (lim\<^isub>B I P) (space (lim\<^isub>B I P)) = 1" 
50088  561 
proof cases 
562 
assume "I = {}" 

50101
a3bede207a04
renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents:
50095
diff
changeset

563 
interpret prob_space "P {}" using proj_prob_space by simp 
50088  564 
show ?thesis 
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

565 
by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`) 
50088  566 
next 
567 
assume "I \<noteq> {}" 

568 
then obtain i where "i \<in> I" by auto 

50101
a3bede207a04
renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents:
50095
diff
changeset

569 
interpret prob_space "P {i}" using proj_prob_space by simp 
50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

570 
have R: "(space (lim\<^isub>B I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))" 
50088  571 
by (auto simp: prod_emb_def space_PiM) 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset

572 
moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM PiE_def) 
50088  573 
ultimately show ?thesis using `i \<in> I` 
574 
apply (subst R) 

50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

575 
apply (subst emeasure_limB_emb_not_empty) 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50101
diff
changeset

576 
apply (auto simp: limP_finite emeasure_space_1 PiE_def) 
50088  577 
done 
578 
qed 

579 
qed 

580 

581 
context polish_projective begin 

582 

50095
94d7dfa9f404
renamed to more appropriate lim_P for projective limit
immler
parents:
50091
diff
changeset

583 
lemma emeasure_limB_emb: 
50088  584 
assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" 
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585 
shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)" 
50088  586 
proof cases 
50101
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587 
interpret prob_space "P {}" using proj_prob_space by simp 
50088  588 
assume "J = {}" 
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589 
moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^isub>B I P)" 
50088  590 
by (auto simp: space_PiM prod_emb_def) 
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591 
moreover have "{\<lambda>x. undefined} = space (lim\<^isub>B {} P)" 
50088  592 
by (auto simp: space_PiM prod_emb_def) 
593 
ultimately show ?thesis 

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594 
by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP) 
50088  595 
next 
596 
assume "J \<noteq> {}" with X show ?thesis 

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597 
by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite) 
50088  598 
qed 
599 

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600 
lemma measure_limB_emb: 
50088  601 
assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" 
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602 
shows "measure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)" 
50088  603 
proof  
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604 
interpret prob_space "P J" using proj_prob_space assms by simp 
50088  605 
show ?thesis 
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606 
using emeasure_limB_emb[OF assms] 
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607 
unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure 
50088  608 
by simp 
609 
qed 

610 

611 
end 

612 

613 
locale polish_product_prob_space = 

614 
product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set" 

615 

616 
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)" 

617 
proof qed 

618 

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619 
lemma (in polish_product_prob_space) limP_eq_PiM: 
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620 
"I \<noteq> {} \<Longrightarrow> lim\<^isub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) = 
50088  621 
PiM I (\<lambda>_. borel)" 
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622 
by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb) 
50088  623 

624 
end 