author  immler@in.tum.de 
Wed, 07 Nov 2012 14:41:49 +0100  
changeset 50040  5da32dc55cd8 
parent 50039  bfd5198cbe40 
child 50041  afe886a04198 
permissions  rwrr 
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(* Title: HOL/Probability/Infinite_Product_Measure.thy 
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Author: Johannes Hölzl, TU München 

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*) 

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header {*Infinite Product Measure*} 

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theory Infinite_Product_Measure 

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imports Probability_Measure Caratheodory Projective_Family 
42147  9 
begin 
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49780  11 
lemma split_merge: "P (merge I J (x,y) i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J  I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)" 
42147  12 
unfolding merge_def by auto 
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49780  14 
lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K" 
42147  15 
unfolding merge_def extensional_def by auto 
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lemma injective_vimage_restrict: 

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assumes J: "J \<subseteq> I" 

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and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}" 

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and eq: "(\<lambda>x. restrict x J) ` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) ` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" 

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shows "A = B" 

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proof (intro set_eqI) 

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fix x 

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from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto 

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have "J \<inter> (I  J) = {}" by auto 

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show "x \<in> A \<longleftrightarrow> x \<in> B" 

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proof cases 

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assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)" 

49780  29 
have "x \<in> A \<longleftrightarrow> merge J (I  J) (x,y) \<in> (\<lambda>x. restrict x J) ` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" 
42147  30 
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub split: split_merge) 
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then show "x \<in> A \<longleftrightarrow> x \<in> B" 

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using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub eq split: split_merge) 

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next 

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assume "x \<notin> (\<Pi>\<^isub>E i\<in>J. S i)" with sets show "x \<in> A \<longleftrightarrow> x \<in> B" by auto 

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qed 

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qed 

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47694  38 
lemma (in product_prob_space) distr_restrict: 
42147  39 
assumes "J \<noteq> {}" "J \<subseteq> K" "finite K" 
47694  40 
shows "(\<Pi>\<^isub>M i\<in>J. M i) = distr (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D") 
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proof (rule measure_eqI_generator_eq) 

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have "finite J" using `J \<subseteq> K` `finite K` by (auto simp add: finite_subset) 

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interpret J: finite_product_prob_space M J proof qed fact 

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interpret K: finite_product_prob_space M K proof qed fact 

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let ?J = "{Pi\<^isub>E J E  E. \<forall>i\<in>J. E i \<in> sets (M i)}" 

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let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)" 

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let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))" 

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show "Int_stable ?J" 

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by (rule Int_stable_PiE) 

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show "range ?F \<subseteq> ?J" "(\<Union>i. ?F i) = ?\<Omega>" 
47694  52 
using `finite J` by (auto intro!: prod_algebraI_finite) 
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{ fix i show "emeasure ?P (?F i) \<noteq> \<infinity>" by simp } 

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show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space) 

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show "sets (\<Pi>\<^isub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J" 

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using `finite J` by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff) 

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fix X assume "X \<in> ?J" 

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then obtain E where [simp]: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto 

50003  60 
with `finite J` have X: "X \<in> sets (Pi\<^isub>M J M)" 
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by simp 

47694  62 

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have "emeasure ?P X = (\<Prod> i\<in>J. emeasure (M i) (E i))" 

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using E by (simp add: J.measure_times) 

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also have "\<dots> = (\<Prod> i\<in>J. emeasure (M i) (if i \<in> J then E i else space (M i)))" 

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by simp 

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also have "\<dots> = (\<Prod> i\<in>K. emeasure (M i) (if i \<in> J then E i else space (M i)))" 

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using `finite K` `J \<subseteq> K` 

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by (intro setprod_mono_one_left) (auto simp: M.emeasure_space_1) 

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also have "\<dots> = emeasure (Pi\<^isub>M K M) (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))" 

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using E by (simp add: K.measure_times) 

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also have "(\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) ` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i))" 

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using `J \<subseteq> K` sets_into_space E by (force simp: Pi_iff split: split_if_asm) 

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finally show "emeasure (Pi\<^isub>M J M) X = emeasure ?D X" 

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using X `J \<subseteq> K` apply (subst emeasure_distr) 

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by (auto intro!: measurable_restrict_subset simp: space_PiM) 

42147  77 
qed 
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47694  79 
lemma (in product_prob_space) emeasure_prod_emb[simp]: 
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assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)" 

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shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X" 
47694  82 
by (subst distr_restrict[OF L]) 
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(simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X) 

42147  84 

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sublocale product_prob_space \<subseteq> projective_family I "\<lambda>J. PiM J M" M 
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proof 
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fix J::"'i set" assume "finite J" 
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interpret f: finite_product_prob_space M J proof qed fact 
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show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) \<noteq> \<infinity>" by simp 
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show "\<exists>A. range A \<subseteq> sets (Pi\<^isub>M J M) \<and> 
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(\<Union>i. A i) = space (Pi\<^isub>M J M) \<and> 
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(\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`] 
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by (auto simp add: sigma_finite_measure_def) 
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show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1) 
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qed simp_all 
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lemma (in projective_family) prod_emb_injective: 
47694  98 
assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)" 
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assumes "prod_emb L M J X = prod_emb L M J Y" 

100 
shows "X = Y" 

101 
proof (rule injective_vimage_restrict) 

102 
show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" 

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using sets[THEN sets_into_space] by (auto simp: space_PiM) 

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have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)" 

49780  105 
using M.not_empty by auto 
47694  106 
from bchoice[OF this] 
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show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto 

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show "(\<lambda>x. restrict x J) ` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) ` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))" 

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using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def) 

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qed fact 

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abbreviation (in projective_family) 
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"emb L K X \<equiv> prod_emb L M K X" 
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definition (in projective_family) generator :: "('i \<Rightarrow> 'a) set set" where 
47694  116 
"generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))" 
42147  117 

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lemma (in projective_family) generatorI': 
47694  119 
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> generator" 
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unfolding generator_def by auto 

42147  121 

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lemma (in projective_family) algebra_generator: 
47694  123 
assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G") 
47762  124 
unfolding algebra_def algebra_axioms_def ring_of_sets_iff 
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proof (intro conjI ballI) 

47694  126 
let ?G = generator 
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show "?G \<subseteq> Pow ?\<Omega>" 

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by (auto simp: generator_def prod_emb_def) 

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from `I \<noteq> {}` obtain i where "i \<in> I" by auto 

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then show "{} \<in> ?G" 

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by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"] 

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simp: sigma_sets.Empty generator_def prod_emb_def) 

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from `i \<in> I` show "?\<Omega> \<in> ?G" 

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by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"] 

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simp: generator_def prod_emb_def) 

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fix A assume "A \<in> ?G" 

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then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA" 

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by (auto simp: generator_def) 

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fix B assume "B \<in> ?G" 

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then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB" 

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by (auto simp: generator_def) 

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let ?RA = "emb (JA \<union> JB) JA XA" 

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let ?RB = "emb (JA \<union> JB) JB XB" 

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have *: "A  B = emb I (JA \<union> JB) (?RA  ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)" 

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using XA A XB B by auto 

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show "A  B \<in> ?G" "A \<union> B \<in> ?G" 

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unfolding * using XA XB by (safe intro!: generatorI') auto 

42147  148 
qed 
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lemma (in projective_family) sets_PiM_generator: 
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"sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" 
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proof cases 

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assume "I = {}" then show ?thesis 

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unfolding generator_def 

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by (auto simp: sets_PiM_empty sigma_sets_empty_eq cong: conj_cong) 

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next 

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assume "I \<noteq> {}" 

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show ?thesis 

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proof 

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show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" 

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unfolding sets_PiM 

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proof (safe intro!: sigma_sets_subseteq) 

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fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator" 

50003  164 
by (auto intro!: generatorI' sets_PiM_I_finite elim!: prod_algebraE) 
49804  165 
qed 
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qed (auto simp: generator_def space_PiM[symmetric] intro!: sigma_sets_subset) 

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qed 

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lemma (in projective_family) generatorI: 
47694  170 
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> generator" 
42147  171 
unfolding generator_def by auto 
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definition (in projective_family) 
42147  174 
"\<mu>G A = 
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(THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (PiP J M P) X))" 
42147  176 

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lemma (in projective_family) \<mu>G_spec: 
42147  178 
assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" 
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shows "\<mu>G A = emeasure (PiP J M P) X" 
42147  180 
unfolding \<mu>G_def 
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proof (intro the_equality allI impI ballI) 

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fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)" 

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have "emeasure (PiP K M P) Y = emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) K Y)" 
42147  184 
using K J by simp 
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also have "emb (K \<union> J) K Y = emb (K \<union> J) J X" 

47694  186 
using K J by (simp add: prod_emb_injective[of "K \<union> J" I]) 
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also have "emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (PiP J M P) X" 
42147  188 
using K J by simp 
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finally show "emeasure (PiP J M P) X = emeasure (PiP K M P) Y" .. 
42147  190 
qed (insert J, force) 
191 

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lemma (in projective_family) \<mu>G_eq: 
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"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (PiP J M P) X" 
42147  194 
by (intro \<mu>G_spec) auto 
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lemma (in projective_family) generator_Ex: 
47694  197 
assumes *: "A \<in> generator" 
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shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (PiP J M P) X" 
42147  199 
proof  
200 
from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" 

201 
unfolding generator_def by auto 

202 
with \<mu>G_spec[OF this] show ?thesis by auto 

203 
qed 

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lemma (in projective_family) generatorE: 
47694  206 
assumes A: "A \<in> generator" 
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obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (PiP J M P) X" 
42147  208 
proof  
209 
from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" 

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"\<mu>G A = emeasure (PiP J M P) X" by auto 
42147  211 
then show thesis by (intro that) auto 
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qed 

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lemma (in projective_family) merge_sets: 
50003  215 
"J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^isub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^isub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) ` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)" 
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by simp 

42147  217 

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lemma (in projective_family) merge_emb: 
42147  219 
assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)" 
49780  220 
shows "((\<lambda>x. merge J (I  J) (y, x)) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = 
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emb I (K  J) ((\<lambda>x. merge J (K  J) (y, x)) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M))" 

42147  222 
proof  
49780  223 
have [simp]: "\<And>x J K L. merge J K (y, restrict x L) = merge J (K \<inter> L) (y, x)" 
42147  224 
by (auto simp: restrict_def merge_def) 
49780  225 
have [simp]: "\<And>x J K L. restrict (merge J K (y, x)) L = merge (J \<inter> L) (K \<inter> L) (y, x)" 
42147  226 
by (auto simp: restrict_def merge_def) 
227 
have [simp]: "(I  J) \<inter> K = K  J" using `K \<subseteq> I` `J \<subseteq> I` by auto 

228 
have [simp]: "(K  J) \<inter> (K \<union> J) = K  J" by auto 

229 
have [simp]: "(K  J) \<inter> K = K  J" by auto 

230 
from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis 

47694  231 
by (simp split: split_merge add: prod_emb_def Pi_iff extensional_merge_sub set_eq_iff space_PiM) 
232 
auto 

42147  233 
qed 
234 

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235 
lemma (in projective_family) positive_\<mu>G: 
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236 
assumes "I \<noteq> {}" 
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237 
shows "positive generator \<mu>G" 
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238 
proof  
47694  239 
interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact 
45777
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240 
show ?thesis 
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241 
proof (intro positive_def[THEN iffD2] conjI ballI) 
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242 
from generatorE[OF G.empty_sets] guess J X . note this[simp] 
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243 
interpret J: finite_product_sigma_finite M J by default fact 
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244 
have "X = {}" 
47694  245 
by (rule prod_emb_injective[of J I]) simp_all 
45777
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246 
then show "\<mu>G {} = 0" by simp 
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247 
next 
47694  248 
fix A assume "A \<in> generator" 
45777
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249 
from generatorE[OF this] guess J X . note this[simp] 
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250 
interpret J: finite_product_sigma_finite M J by default fact 
47694  251 
show "0 \<le> \<mu>G A" by (simp add: emeasure_nonneg) 
45777
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252 
qed 
42147  253 
qed 
254 

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255 
lemma (in projective_family) additive_\<mu>G: 
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256 
assumes "I \<noteq> {}" 
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257 
shows "additive generator \<mu>G" 
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258 
proof  
47694  259 
interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact 
45777
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260 
show ?thesis 
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261 
proof (intro additive_def[THEN iffD2] ballI impI) 
47694  262 
fix A assume "A \<in> generator" with generatorE guess J X . note J = this 
263 
fix B assume "B \<in> generator" with generatorE guess K Y . note K = this 

45777
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264 
assume "A \<inter> B = {}" 
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265 
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)" 
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266 
using J K by auto 
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267 
interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact 
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268 
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}" 
47694  269 
apply (rule prod_emb_injective[of "J \<union> K" I]) 
45777
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270 
apply (insert `A \<inter> B = {}` JK J K) 
47694  271 
apply (simp_all add: Int prod_emb_Int) 
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272 
done 
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273 
have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)" 
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274 
using J K by simp_all 
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275 
then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))" 
47694  276 
by simp 
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277 
also have "\<dots> = emeasure (PiP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" 
47694  278 
using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un) 
45777
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279 
also have "\<dots> = \<mu>G A + \<mu>G B" 
47694  280 
using J K JK_disj by (simp add: plus_emeasure[symmetric]) 
45777
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281 
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" . 
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282 
qed 
42147  283 
qed 
284 

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285 
lemma (in product_prob_space) PiP_PiM_finite[simp]: 
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286 
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "PiP J M (\<lambda>J. PiM J M) = PiM J M" 
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287 
using assms by (simp add: PiP_finite) 
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288 

47694  289 
lemma (in product_prob_space) emeasure_PiM_emb_not_empty: 
290 
assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. X i \<in> sets (M i)" 

291 
shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)" 

42147  292 
proof cases 
47694  293 
assume "finite I" with X show ?thesis by simp 
42147  294 
next 
47694  295 
let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space (M i)" 
42147  296 
let ?G = generator 
297 
assume "\<not> finite I" 

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298 
then have I_not_empty: "I \<noteq> {}" by auto 
47694  299 
interpret G!: algebra ?\<Omega> generator by (rule algebra_generator) fact 
42147  300 
note \<mu>G_mono = 
45777
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301 
G.additive_increasing[OF positive_\<mu>G[OF I_not_empty] additive_\<mu>G[OF I_not_empty], THEN increasingD] 
42147  302 

47694  303 
{ fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> ?G" 
42147  304 

305 
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J" 

306 
by (metis rev_finite_subset subsetI) 

307 
moreover from Z guess K' X' by (rule generatorE) 

308 
moreover def K \<equiv> "insert k K'" 

309 
moreover def X \<equiv> "emb K K' X'" 

310 
ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X" 

47694  311 
"K  J \<noteq> {}" "K  J \<subseteq> I" "\<mu>G Z = emeasure (Pi\<^isub>M K M) X" 
42147  312 
by (auto simp: subset_insertI) 
49780  313 
let ?M = "\<lambda>y. (\<lambda>x. merge J (K  J) (y, x)) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M)" 
42147  314 
{ fix y assume y: "y \<in> space (Pi\<^isub>M J M)" 
315 
note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X] 

316 
moreover 

317 
have **: "?M y \<in> sets (Pi\<^isub>M (K  J) M)" 

318 
using J K y by (intro merge_sets) auto 

319 
ultimately 

49780  320 
have ***: "((\<lambda>x. merge J (I  J) (y, x)) ` Z \<inter> space (Pi\<^isub>M I M)) \<in> ?G" 
42147  321 
using J K by (intro generatorI) auto 
49780  322 
have "\<mu>G ((\<lambda>x. merge J (I  J) (y, x)) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = emeasure (Pi\<^isub>M (K  J) M) (?M y)" 
42147  323 
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto 
324 
note * ** *** this } 

325 
note merge_in_G = this 

326 

327 
have "finite (K  J)" using K by auto 

328 

329 
interpret J: finite_product_prob_space M J by default fact+ 

330 
interpret KmJ: finite_product_prob_space M "K  J" by default fact+ 

331 

47694  332 
have "\<mu>G Z = emeasure (Pi\<^isub>M (J \<union> (K  J)) M) (emb (J \<union> (K  J)) K X)" 
42147  333 
using K J by simp 
47694  334 
also have "\<dots> = (\<integral>\<^isup>+ x. emeasure (Pi\<^isub>M (K  J) M) (?M x) \<partial>Pi\<^isub>M J M)" 
335 
using K J by (subst emeasure_fold_integral) auto 

49780  336 
also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G ((\<lambda>x. merge J (I  J) (y, x)) ` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)" 
42147  337 
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)") 
47694  338 
proof (intro positive_integral_cong) 
42147  339 
fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 
340 
with K merge_in_G(2)[OF this] 

47694  341 
show "emeasure (Pi\<^isub>M (K  J) M) (?M x) = \<mu>G (?MZ x)" 
42147  342 
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto 
343 
qed 

344 
finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" . 

345 

346 
{ fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 

347 
then have "\<mu>G (?MZ x) \<le> 1" 

348 
unfolding merge_in_G(4)[OF x] `Z = emb I K X` 

349 
by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) } 

350 
note le_1 = this 

351 

49780  352 
let ?q = "\<lambda>y. \<mu>G ((\<lambda>x. merge J (I  J) (y,x)) ` Z \<inter> space (Pi\<^isub>M I M))" 
42147  353 
have "?q \<in> borel_measurable (Pi\<^isub>M J M)" 
354 
unfolding `Z = emb I K X` using J K merge_in_G(3) 

47694  355 
by (simp add: merge_in_G \<mu>G_eq emeasure_fold_measurable cong: measurable_cong) 
42147  356 
note this fold le_1 merge_in_G(3) } 
357 
note fold = this 

358 

47694  359 
have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>" 
42147  360 
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G]) 
47694  361 
fix A assume "A \<in> ?G" 
42147  362 
with generatorE guess J X . note JX = this 
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363 
interpret JK: finite_product_prob_space M J by default fact+ 
46898
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364 
from JX show "\<mu>G A \<noteq> \<infinity>" by simp 
42147  365 
next 
47694  366 
fix A assume A: "range A \<subseteq> ?G" "decseq A" "(\<Inter>i. A i) = {}" 
42147  367 
then have "decseq (\<lambda>i. \<mu>G (A i))" 
368 
by (auto intro!: \<mu>G_mono simp: decseq_def) 

369 
moreover 

370 
have "(INF i. \<mu>G (A i)) = 0" 

371 
proof (rule ccontr) 

372 
assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0") 

373 
moreover have "0 \<le> ?a" 

45777
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374 
using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def) 
42147  375 
ultimately have "0 < ?a" by auto 
376 

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377 
have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = emeasure (PiP J M (\<lambda>J. (Pi\<^isub>M J M))) X" 
42147  378 
using A by (intro allI generator_Ex) auto 
379 
then obtain J' X' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" "\<And>n. X' n \<in> sets (Pi\<^isub>M (J' n) M)" 

380 
and A': "\<And>n. A n = emb I (J' n) (X' n)" 

381 
unfolding choice_iff by blast 

382 
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)" 

383 
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)" 

384 
ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. X n \<in> sets (Pi\<^isub>M (J n) M)" 

385 
by auto 

47694  386 
with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> ?G" 
387 
unfolding J_def X_def by (subst prod_emb_trans) (insert A, auto) 

42147  388 

389 
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" 

390 
unfolding J_def by force 

391 

392 
interpret J: finite_product_prob_space M "J i" for i by default fact+ 

393 

394 
have a_le_1: "?a \<le> 1" 

395 
using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq 

44928
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hoelzl
parents:
43920
diff
changeset

396 
by (auto intro!: INF_lower2[of 0] J.measure_le_1) 
42147  397 

49780  398 
let ?M = "\<lambda>K Z y. (\<lambda>x. merge K (I  K) (y, x)) ` Z \<inter> space (Pi\<^isub>M I M)" 
42147  399 

47694  400 
{ fix Z k assume Z: "range Z \<subseteq> ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)" 
401 
then have Z_sets: "\<And>n. Z n \<in> ?G" by auto 

42147  402 
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I" 
403 
interpret J': finite_product_prob_space M J' by default fact+ 

404 

46731  405 
let ?q = "\<lambda>n y. \<mu>G (?M J' (Z n) y)" 
406 
let ?Q = "\<lambda>n. ?q n ` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)" 

42147  407 
{ fix n 
408 
have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)" 

409 
using Z J' by (intro fold(1)) auto 

410 
then have "?Q n \<in> sets (Pi\<^isub>M J' M)" 

411 
by (rule measurable_sets) auto } 

412 
note Q_sets = this 

413 

47694  414 
have "?a / 2^(k+1) \<le> (INF n. emeasure (Pi\<^isub>M J' M) (?Q n))" 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
43920
diff
changeset

415 
proof (intro INF_greatest) 
42147  416 
fix n 
417 
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto 

418 
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)" 

47694  419 
unfolding fold(2)[OF J' `Z n \<in> ?G`] 
420 
proof (intro positive_integral_mono) 

42147  421 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 
422 
then have "?q n x \<le> 1 + 0" 

423 
using J' Z fold(3) Z_sets by auto 

424 
also have "\<dots> \<le> 1 + ?a / 2^(k+1)" 

425 
using `0 < ?a` by (intro add_mono) auto 

426 
finally have "?q n x \<le> 1 + ?a / 2^(k+1)" . 

427 
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)" 

428 
by (auto split: split_indicator simp del: power_Suc) 

429 
qed 

47694  430 
also have "\<dots> = emeasure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)" 
431 
using `0 \<le> ?a` Q_sets J'.emeasure_space_1 

432 
by (subst positive_integral_add) auto 

433 
finally show "?a / 2^(k+1) \<le> emeasure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1` 

434 
by (cases rule: ereal2_cases[of ?a "emeasure (Pi\<^isub>M J' M) (?Q n)"]) 

42147  435 
(auto simp: field_simps) 
436 
qed 

47694  437 
also have "\<dots> = emeasure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)" 
438 
proof (intro INF_emeasure_decseq) 

42147  439 
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto 
440 
show "decseq ?Q" 

441 
unfolding decseq_def 

442 
proof (safe intro!: vimageI[OF refl]) 

443 
fix m n :: nat assume "m \<le> n" 

444 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 

445 
assume "?a / 2^(k+1) \<le> ?q n x" 

446 
also have "?q n x \<le> ?q m x" 

447 
proof (rule \<mu>G_mono) 

448 
from fold(4)[OF J', OF Z_sets x] 

47694  449 
show "?M J' (Z n) x \<in> ?G" "?M J' (Z m) x \<in> ?G" by auto 
42147  450 
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x" 
451 
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto 

452 
qed 

453 
finally show "?a / 2^(k+1) \<le> ?q m x" . 

454 
qed 

47694  455 
qed simp 
42147  456 
finally have "(\<Inter>n. ?Q n) \<noteq> {}" 
457 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 

458 
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto } 

459 
note Ex_w = this 

460 

46731  461 
let ?q = "\<lambda>k n y. \<mu>G (?M (J k) (A n) y)" 
42147  462 

44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
43920
diff
changeset

463 
have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_lower) 
42147  464 
from Ex_w[OF A(1,2) this J(13), of 0] guess w0 .. note w0 = this 
465 

46731  466 
let ?P = 
467 
"\<lambda>k wk w. w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and> 

468 
(\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)" 

42147  469 
def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))" 
470 

471 
{ fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and> 

472 
(\<forall>n. ?a / 2 ^ (k + 1) \<le> ?q k n (w k)) \<and> (k \<noteq> 0 \<longrightarrow> restrict (w k) (J (k  1)) = w (k  1))" 

473 
proof (induct k) 

474 
case 0 with w0 show ?case 

475 
unfolding w_def nat_rec_0 by auto 

476 
next 

477 
case (Suc k) 

478 
then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

479 
have "\<exists>w'. ?P k (w k) w'" 

480 
proof cases 

481 
assume [simp]: "J k = J (Suc k)" 

482 
show ?thesis 

483 
proof (intro exI[of _ "w k"] conjI allI) 

484 
fix n 

485 
have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)" 

486 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps) 

487 
also have "\<dots> \<le> ?q k n (w k)" using Suc by auto 

488 
finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp 

489 
next 

490 
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)" 

491 
using Suc by simp 

492 
then show "restrict (w k) (J k) = w k" 

47694  493 
by (simp add: extensional_restrict space_PiM) 
42147  494 
qed 
495 
next 

496 
assume "J k \<noteq> J (Suc k)" 

497 
with J_mono[of k "Suc k"] have "J (Suc k)  J k \<noteq> {}" (is "?D \<noteq> {}") by auto 

47694  498 
have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> ?G" 
42147  499 
"decseq (\<lambda>n. ?M (J k) (A n) (w k))" 
500 
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))" 

501 
using `decseq A` fold(4)[OF J(13) A_eq(2), of "w k" k] Suc 

502 
by (auto simp: decseq_def) 

503 
from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"] 

504 
obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)" 

505 
"\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto 

49780  506 
let ?w = "merge (J k) ?D (w k, w')" 
507 
have [simp]: "\<And>x. merge (J k) (I  J k) (w k, merge ?D (I  ?D) (w', x)) = 

508 
merge (J (Suc k)) (I  (J (Suc k))) (?w, x)" 

42147  509 
using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"] 
510 
by (auto intro!: ext split: split_merge) 

511 
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w" 

512 
using w'(1) J(3)[of "Suc k"] 

47694  513 
by (auto simp: space_PiM split: split_merge intro!: extensional_merge_sub) force+ 
42147  514 
show ?thesis 
515 
apply (rule exI[of _ ?w]) 

516 
using w' J_mono[of k "Suc k"] wk unfolding * 

47694  517 
apply (auto split: split_merge intro!: extensional_merge_sub ext simp: space_PiM) 
42147  518 
apply (force simp: extensional_def) 
519 
done 

520 
qed 

521 
then have "?P k (w k) (w (Suc k))" 

522 
unfolding w_def nat_rec_Suc unfolding w_def[symmetric] 

523 
by (rule someI_ex) 

524 
then show ?case by auto 

525 
qed 

526 
moreover 

527 
then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

528 
moreover 

529 
from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto 

530 
then have "?M (J k) (A k) (w k) \<noteq> {}" 

45777
c36637603821
remove unnecessary sublocale instantiations in HOLProbability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
44928
diff
changeset

531 
using positive_\<mu>G[OF I_not_empty, unfolded positive_def] `0 < ?a` `?a \<le> 1` 
42147  532 
by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 
533 
then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto 

49780  534 
then have "merge (J k) (I  J k) (w k, x) \<in> A k" by auto 
42147  535 
then have "\<exists>x\<in>A k. restrict x (J k) = w k" 
536 
using `w k \<in> space (Pi\<^isub>M (J k) M)` 

47694  537 
by (intro rev_bexI) (auto intro!: ext simp: extensional_def space_PiM) 
42147  538 
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)" 
539 
"\<exists>x\<in>A k. restrict x (J k) = w k" 

540 
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k  1)) = w (k  1)" 

541 
by auto } 

542 
note w = this 

543 

544 
{ fix k l i assume "k \<le> l" "i \<in> J k" 

545 
{ fix l have "w k i = w (k + l) i" 

546 
proof (induct l) 

547 
case (Suc l) 

548 
from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto 

549 
with w(3)[of "k + Suc l"] 

550 
have "w (k + l) i = w (k + Suc l) i" 

551 
by (auto simp: restrict_def fun_eq_iff split: split_if_asm) 

552 
with Suc show ?case by simp 

553 
qed simp } 

554 
from this[of "l  k"] `k \<le> l` have "w l i = w k i" by simp } 

555 
note w_mono = this 

556 

557 
def w' \<equiv> "\<lambda>i. if i \<in> (\<Union>k. J k) then w (LEAST k. i \<in> J k) i else if i \<in> I then (SOME x. x \<in> space (M i)) else undefined" 

558 
{ fix i k assume k: "i \<in> J k" 

559 
have "w k i = w (LEAST k. i \<in> J k) i" 

560 
by (intro w_mono Least_le k LeastI[of _ k]) 

561 
then have "w' i = w k i" 

562 
unfolding w'_def using k by auto } 

563 
note w'_eq = this 

564 
have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined" 

565 
using J by (auto simp: w'_def) 

566 
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)" 

567 
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]]) 

568 
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)" 

47694  569 
using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq space_PiM)+ } 
42147  570 
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this 
571 

572 
have w': "w' \<in> space (Pi\<^isub>M I M)" 

47694  573 
using w(1) by (auto simp add: Pi_iff extensional_def space_PiM) 
42147  574 

575 
{ fix n 

576 
have "restrict w' (J n) = w n" using w(1) 

47694  577 
by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def space_PiM) 
42147  578 
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto 
47694  579 
then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: prod_emb_def space_PiM) } 
42147  580 
then have "w' \<in> (\<Inter>i. A i)" by auto 
581 
with `(\<Inter>i. A i) = {}` show False by auto 

582 
qed 

583 
ultimately show "(\<lambda>i. \<mu>G (A i)) > 0" 

43920  584 
using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp 
45777
c36637603821
remove unnecessary sublocale instantiations in HOLProbability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
44928
diff
changeset

585 
qed fact+ 
c36637603821
remove unnecessary sublocale instantiations in HOLProbability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
44928
diff
changeset

586 
then guess \<mu> .. note \<mu> = this 
c36637603821
remove unnecessary sublocale instantiations in HOLProbability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
44928
diff
changeset

587 
show ?thesis 
47694  588 
proof (subst emeasure_extend_measure_Pair[OF PiM_def, of I M \<mu> J X]) 
589 
from assms show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))" 

590 
by (simp add: Pi_iff) 

591 
next 

592 
fix J X assume J: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))" 

593 
then show "emb I J (Pi\<^isub>E J X) \<in> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))" 

594 
by (auto simp: Pi_iff prod_emb_def dest: sets_into_space) 

595 
have "emb I J (Pi\<^isub>E J X) \<in> generator" 

50003  596 
using J `I \<noteq> {}` by (intro generatorI') (auto simp: Pi_iff) 
47694  597 
then have "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" 
598 
using \<mu> by simp 

599 
also have "\<dots> = (\<Prod> j\<in>J. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" 

600 
using J `I \<noteq> {}` by (subst \<mu>G_spec[OF _ _ _ refl]) (auto simp: emeasure_PiM Pi_iff) 

601 
also have "\<dots> = (\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. 

602 
if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" 

603 
using J `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1) 

604 
finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = \<dots>" . 

605 
next 

606 
let ?F = "\<lambda>j. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j))" 

607 
have "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = (\<Prod>j\<in>J. ?F j)" 

608 
using X `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1) 

609 
then show "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = 

610 
emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)" 

611 
using X by (auto simp add: emeasure_PiM) 

612 
next 

613 
show "positive (sets (Pi\<^isub>M I M)) \<mu>" "countably_additive (sets (Pi\<^isub>M I M)) \<mu>" 

49804  614 
using \<mu> unfolding sets_PiM_generator by (auto simp: measure_space_def) 
42147  615 
qed 
616 
qed 

617 

47694  618 
sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>M I M" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

619 
proof 
47694  620 
show "emeasure (Pi\<^isub>M I M) (space (Pi\<^isub>M I M)) = 1" 
621 
proof cases 

622 
assume "I = {}" then show ?thesis by (simp add: space_PiM_empty) 

623 
next 

624 
assume "I \<noteq> {}" 

625 
then obtain i where "i \<in> I" by auto 

626 
moreover then have "emb I {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i)) = (space (Pi\<^isub>M I M))" 

627 
by (auto simp: prod_emb_def space_PiM) 

628 
ultimately show ?thesis 

629 
using emeasure_PiM_emb_not_empty[of "{i}" "\<lambda>i. space (M i)"] 

630 
by (simp add: emeasure_PiM emeasure_space_1) 

631 
qed 

42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

632 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

633 

47694  634 
lemma (in product_prob_space) emeasure_PiM_emb: 
635 
assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" 

636 
shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. emeasure (M i) (X i))" 

637 
proof cases 

638 
assume "J = {}" 

639 
moreover have "emb I {} {\<lambda>x. undefined} = space (Pi\<^isub>M I M)" 

640 
by (auto simp: space_PiM prod_emb_def) 

641 
ultimately show ?thesis 

642 
by (simp add: space_PiM_empty P.emeasure_space_1) 

643 
next 

644 
assume "J \<noteq> {}" with X show ?thesis 

645 
by (subst emeasure_PiM_emb_not_empty) (auto simp: emeasure_PiM) 

42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

646 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

647 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

648 
lemma (in product_prob_space) emeasure_PiM_Collect: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

649 
assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

650 
shows "emeasure (Pi\<^isub>M I M) {x\<in>space (Pi\<^isub>M I M). \<forall>i\<in>J. x i \<in> X i} = (\<Prod> i\<in>J. emeasure (M i) (X i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

651 
proof  
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

652 
have "{x\<in>space (Pi\<^isub>M I M). \<forall>i\<in>J. x i \<in> X i} = emb I J (Pi\<^isub>E J X)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

653 
unfolding prod_emb_def using assms by (auto simp: space_PiM Pi_iff) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

654 
with emeasure_PiM_emb[OF assms] show ?thesis by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

655 
qed 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

656 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

657 
lemma (in product_prob_space) emeasure_PiM_Collect_single: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

658 
assumes X: "i \<in> I" "A \<in> sets (M i)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

659 
shows "emeasure (Pi\<^isub>M I M) {x\<in>space (Pi\<^isub>M I M). x i \<in> A} = emeasure (M i) A" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

660 
using emeasure_PiM_Collect[of "{i}" "\<lambda>i. A"] assms 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

661 
by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

662 

47694  663 
lemma (in product_prob_space) measure_PiM_emb: 
664 
assumes "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" 

665 
shows "measure (PiM I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. measure (M i) (X i))" 

666 
using emeasure_PiM_emb[OF assms] 

667 
unfolding emeasure_eq_measure M.emeasure_eq_measure by (simp add: setprod_ereal) 

42865  668 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

669 
lemma sets_Collect_single': 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

670 
"i \<in> I \<Longrightarrow> {x\<in>space (M i). P x} \<in> sets (M i) \<Longrightarrow> {x\<in>space (PiM I M). P (x i)} \<in> sets (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

671 
using sets_Collect_single[of i I "{x\<in>space (M i). P x}" M] 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

672 
by (simp add: space_PiM Pi_iff cong: conj_cong) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

673 

47694  674 
lemma (in finite_product_prob_space) finite_measure_PiM_emb: 
675 
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> measure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))" 

676 
using measure_PiM_emb[of I A] finite_index prod_emb_PiE_same_index[OF sets_into_space, of I A M] 

677 
by auto 

42865  678 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

679 
lemma (in product_prob_space) PiM_component: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

680 
assumes "i \<in> I" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

681 
shows "distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

682 
proof (rule measure_eqI[symmetric]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

683 
fix A assume "A \<in> sets (M i)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

684 
moreover have "((\<lambda>\<omega>. \<omega> i) ` A \<inter> space (PiM I M)) = {x\<in>space (PiM I M). x i \<in> A}" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

685 
by auto 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

686 
ultimately show "emeasure (M i) A = emeasure (distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i)) A" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

687 
by (auto simp: `i\<in>I` emeasure_distr measurable_component_singleton emeasure_PiM_Collect_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

688 
qed simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

689 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

690 
lemma (in product_prob_space) PiM_eq: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

691 
assumes "I \<noteq> {}" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

692 
assumes "sets M' = sets (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

693 
assumes eq: "\<And>J F. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>j. j \<in> J \<Longrightarrow> F j \<in> sets (M j)) \<Longrightarrow> 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

694 
emeasure M' (prod_emb I M J (\<Pi>\<^isub>E j\<in>J. F j)) = (\<Prod>j\<in>J. emeasure (M j) (F j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

695 
shows "M' = (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

696 
proof (rule measure_eqI_generator_eq[symmetric, OF Int_stable_prod_algebra prod_algebra_sets_into_space]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

697 
show "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

698 
by (rule sets_PiM) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

699 
then show "sets M' = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

700 
unfolding `sets M' = sets (PiM I M)` by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

701 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

702 
def i \<equiv> "SOME i. i \<in> I" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

703 
with `I \<noteq> {}` have i: "i \<in> I" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

704 
by (auto intro: someI_ex) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

705 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

706 
def A \<equiv> "\<lambda>n::nat. prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. space (M i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

707 
then show "range A \<subseteq> prod_algebra I M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

708 
by (auto intro!: prod_algebraI i) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

709 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

710 
have A_eq: "\<And>i. A i = space (PiM I M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

711 
by (auto simp: prod_emb_def space_PiM Pi_iff A_def i) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

712 
show "(\<Union>i. A i) = (\<Pi>\<^isub>E i\<in>I. space (M i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

713 
unfolding A_eq by (auto simp: space_PiM) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

714 
show "\<And>i. emeasure (PiM I M) (A i) \<noteq> \<infinity>" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

715 
unfolding A_eq P.emeasure_space_1 by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

716 
next 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

717 
fix X assume X: "X \<in> prod_algebra I M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

718 
then obtain J E where X: "X = prod_emb I M J (PIE j:J. E j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

719 
and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

720 
by (force elim!: prod_algebraE) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

721 
from eq[OF J] have "emeasure M' X = (\<Prod>j\<in>J. emeasure (M j) (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

722 
by (simp add: X) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

723 
also have "\<dots> = emeasure (PiM I M) X" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

724 
unfolding X using J by (intro emeasure_PiM_emb[symmetric]) auto 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

725 
finally show "emeasure (PiM I M) X = emeasure M' X" .. 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

726 
qed 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

727 

42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

728 
subsection {* Sequence space *} 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

729 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

730 
lemma measurable_nat_case: "(\<lambda>(x, \<omega>). nat_case x \<omega>) \<in> measurable (M \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)) (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

731 
proof (rule measurable_PiM_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

732 
show "(\<lambda>(x, \<omega>). nat_case x \<omega>) \<in> space (M \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)) \<rightarrow> (UNIV \<rightarrow>\<^isub>E space M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

733 
by (auto simp: space_pair_measure space_PiM Pi_iff split: nat.split) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

734 
fix i :: nat and A assume A: "A \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

735 
then have *: "{\<omega> \<in> space (M \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)). prod_case nat_case \<omega> i \<in> A} = 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

736 
(case i of 0 \<Rightarrow> A \<times> space (\<Pi>\<^isub>M i\<in>UNIV. M)  Suc n \<Rightarrow> space M \<times> {\<omega>\<in>space (\<Pi>\<^isub>M i\<in>UNIV. M). \<omega> n \<in> A})" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

737 
by (auto simp: space_PiM space_pair_measure split: nat.split dest: sets_into_space) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

738 
show "{\<omega> \<in> space (M \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)). prod_case nat_case \<omega> i \<in> A} \<in> sets (M \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

739 
unfolding * by (auto simp: A split: nat.split intro!: sets_Collect_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

740 
qed 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

741 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

742 
lemma measurable_nat_case': 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

743 
assumes f: "f \<in> measurable N M" and g: "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

744 
shows "(\<lambda>x. nat_case (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

745 
using measurable_compose[OF measurable_Pair[OF f g] measurable_nat_case] by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

746 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

747 
definition comb_seq :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a)" where 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

748 
"comb_seq i \<omega> \<omega>' j = (if j < i then \<omega> j else \<omega>' (j  i))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

749 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

750 
lemma split_comb_seq: "P (comb_seq i \<omega> \<omega>' j) \<longleftrightarrow> (j < i \<longrightarrow> P (\<omega> j)) \<and> (\<forall>k. j = i + k \<longrightarrow> P (\<omega>' k))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

751 
by (auto simp: comb_seq_def not_less) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

752 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

753 
lemma split_comb_seq_asm: "P (comb_seq i \<omega> \<omega>' j) \<longleftrightarrow> \<not> ((j < i \<and> \<not> P (\<omega> j)) \<or> (\<exists>k. j = i + k \<and> \<not> P (\<omega>' k)))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

754 
by (auto simp: comb_seq_def) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

755 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

756 
lemma measurable_comb_seq: "(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> measurable ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)) (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

757 
proof (rule measurable_PiM_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

758 
show "(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> space ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)) \<rightarrow> (UNIV \<rightarrow>\<^isub>E space M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

759 
by (auto simp: space_pair_measure space_PiM Pi_iff split: split_comb_seq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

760 
fix j :: nat and A assume A: "A \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

761 
then have *: "{\<omega> \<in> space ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)). prod_case (comb_seq i) \<omega> j \<in> A} = 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

762 
(if j < i then {\<omega> \<in> space (\<Pi>\<^isub>M i\<in>UNIV. M). \<omega> j \<in> A} \<times> space (\<Pi>\<^isub>M i\<in>UNIV. M) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

763 
else space (\<Pi>\<^isub>M i\<in>UNIV. M) \<times> {\<omega> \<in> space (\<Pi>\<^isub>M i\<in>UNIV. M). \<omega> (j  i) \<in> A})" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

764 
by (auto simp: space_PiM space_pair_measure comb_seq_def dest: sets_into_space) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

765 
show "{\<omega> \<in> space ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M)). prod_case (comb_seq i) \<omega> j \<in> A} \<in> sets ((\<Pi>\<^isub>M i\<in>UNIV. M) \<Otimes>\<^isub>M (\<Pi>\<^isub>M i\<in>UNIV. M))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

766 
unfolding * by (auto simp: A intro!: sets_Collect_single) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

767 
qed 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

768 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

769 
lemma measurable_comb_seq': 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

770 
assumes f: "f \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" and g: "g \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

771 
shows "(\<lambda>x. comb_seq i (f x) (g x)) \<in> measurable N (\<Pi>\<^isub>M i\<in>UNIV. M)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

772 
using measurable_compose[OF measurable_Pair[OF f g] measurable_comb_seq] by simp 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

773 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

774 
locale sequence_space = product_prob_space "\<lambda>i. M" "UNIV :: nat set" for M 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

775 
begin 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

776 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

777 
abbreviation "S \<equiv> \<Pi>\<^isub>M i\<in>UNIV::nat set. M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

778 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

779 
lemma infprod_in_sets[intro]: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

780 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

781 
shows "Pi UNIV E \<in> sets S" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

782 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

783 
have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))" 
47694  784 
using E E[THEN sets_into_space] 
785 
by (auto simp: prod_emb_def Pi_iff extensional_def) blast 

786 
with E show ?thesis by auto 

42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

787 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

788 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

789 
lemma measure_PiM_countable: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

790 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

791 
shows "(\<lambda>n. \<Prod>i\<le>n. measure M (E i)) > measure S (Pi UNIV E)" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

792 
proof  
46731  793 
let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^isub>E {.. n} E)" 
50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

794 
have "\<And>n. (\<Prod>i\<le>n. measure M (E i)) = measure S (?E n)" 
47694  795 
using E by (simp add: measure_PiM_emb) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

796 
moreover have "Pi UNIV E = (\<Inter>n. ?E n)" 
47694  797 
using E E[THEN sets_into_space] 
798 
by (auto simp: prod_emb_def extensional_def Pi_iff) blast 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

799 
moreover have "range ?E \<subseteq> sets S" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

800 
using E by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

801 
moreover have "decseq ?E" 
47694  802 
by (auto simp: prod_emb_def Pi_iff decseq_def) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

803 
ultimately show ?thesis 
47694  804 
by (simp add: finite_Lim_measure_decseq) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

805 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

806 

50000
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

807 
lemma nat_eq_diff_eq: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

808 
fixes a b c :: nat 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

809 
shows "c \<le> b \<Longrightarrow> a = b  c \<longleftrightarrow> a + c = b" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

810 
by auto 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

811 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

812 
lemma PiM_comb_seq: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

813 
"distr (S \<Otimes>\<^isub>M S) S (\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') = S" (is "?D = _") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

814 
proof (rule PiM_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

815 
let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

816 
let "distr _ _ ?f" = "?D" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

817 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

818 
fix J E assume J: "finite J" "J \<subseteq> ?I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

819 
let ?X = "prod_emb ?I ?M J (\<Pi>\<^isub>E j\<in>J. E j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

820 
have "\<And>j x. j \<in> J \<Longrightarrow> x \<in> E j \<Longrightarrow> x \<in> space M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

821 
using J(3)[THEN sets_into_space] by (auto simp: space_PiM Pi_iff subset_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

822 
with J have "?f ` ?X \<inter> space (S \<Otimes>\<^isub>M S) = 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

823 
(prod_emb ?I ?M (J \<inter> {..<i}) (PIE j:J \<inter> {..<i}. E j)) \<times> 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

824 
(prod_emb ?I ?M ((op + i) ` J) (PIE j:(op + i) ` J. E (i + j)))" (is "_ = ?E \<times> ?F") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

825 
by (auto simp: space_pair_measure space_PiM prod_emb_def all_conj_distrib Pi_iff 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

826 
split: split_comb_seq split_comb_seq_asm) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

827 
then have "emeasure ?D ?X = emeasure (S \<Otimes>\<^isub>M S) (?E \<times> ?F)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

828 
by (subst emeasure_distr[OF measurable_comb_seq]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

829 
(auto intro!: sets_PiM_I simp: split_beta' J) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

830 
also have "\<dots> = emeasure S ?E * emeasure S ?F" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

831 
using J by (intro P.emeasure_pair_measure_Times) (auto intro!: sets_PiM_I finite_vimageI simp: inj_on_def) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

832 
also have "emeasure S ?F = (\<Prod>j\<in>(op + i) ` J. emeasure M (E (i + j)))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

833 
using J by (intro emeasure_PiM_emb) (simp_all add: finite_vimageI inj_on_def) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

834 
also have "\<dots> = (\<Prod>j\<in>J  (J \<inter> {..<i}). emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

835 
by (rule strong_setprod_reindex_cong[where f="\<lambda>x. x  i"]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

836 
(auto simp: image_iff Bex_def not_less nat_eq_diff_eq ac_simps cong: conj_cong intro!: inj_onI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

837 
also have "emeasure S ?E = (\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

838 
using J by (intro emeasure_PiM_emb) simp_all 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

839 
also have "(\<Prod>j\<in>J \<inter> {..<i}. emeasure M (E j)) * (\<Prod>j\<in>J  (J \<inter> {..<i}). emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

840 
by (subst mult_commute) (auto simp: J setprod_subset_diff[symmetric]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

841 
finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" . 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

842 
qed simp_all 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

843 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

844 
lemma PiM_iter: 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

845 
"distr (M \<Otimes>\<^isub>M S) S (\<lambda>(s, \<omega>). nat_case s \<omega>) = S" (is "?D = _") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

846 
proof (rule PiM_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

847 
let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

848 
let "distr _ _ ?f" = "?D" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

849 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

850 
fix J E assume J: "finite J" "J \<subseteq> ?I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

851 
let ?X = "prod_emb ?I ?M J (PIE j:J. E j)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

852 
have "\<And>j x. j \<in> J \<Longrightarrow> x \<in> E j \<Longrightarrow> x \<in> space M" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

853 
using J(3)[THEN sets_into_space] by (auto simp: space_PiM Pi_iff subset_eq) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

854 
with J have "?f ` ?X \<inter> space (M \<Otimes>\<^isub>M S) = (if 0 \<in> J then E 0 else space M) \<times> 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

855 
(prod_emb ?I ?M (Suc ` J) (PIE j:Suc ` J. E (Suc j)))" (is "_ = ?E \<times> ?F") 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

856 
by (auto simp: space_pair_measure space_PiM Pi_iff prod_emb_def all_conj_distrib 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

857 
split: nat.split nat.split_asm) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

858 
then have "emeasure ?D ?X = emeasure (M \<Otimes>\<^isub>M S) (?E \<times> ?F)" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

859 
by (subst emeasure_distr[OF measurable_nat_case]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

860 
(auto intro!: sets_PiM_I simp: split_beta' J) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

861 
also have "\<dots> = emeasure M ?E * emeasure S ?F" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

862 
using J by (intro P.emeasure_pair_measure_Times) (auto intro!: sets_PiM_I finite_vimageI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

863 
also have "emeasure S ?F = (\<Prod>j\<in>Suc ` J. emeasure M (E (Suc j)))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

864 
using J by (intro emeasure_PiM_emb) (simp_all add: finite_vimageI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

865 
also have "\<dots> = (\<Prod>j\<in>J  {0}. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

866 
by (rule strong_setprod_reindex_cong[where f="\<lambda>x. x  1"]) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

867 
(auto simp: image_iff Bex_def not_less nat_eq_diff_eq ac_simps cong: conj_cong intro!: inj_onI) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

868 
also have "emeasure M ?E * (\<Prod>j\<in>J  {0}. emeasure M (E j)) = (\<Prod>j\<in>J. emeasure M (E j))" 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

869 
by (auto simp: M.emeasure_space_1 setprod.remove J) 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

870 
finally show "emeasure ?D ?X = (\<Prod>j\<in>J. emeasure M (E j))" . 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

871 
qed simp_all 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

872 

cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

873 
end 
cfe8ee8a1371
infinite product measure is invariant under adding prefixes
hoelzl
parents:
49804
diff
changeset

874 

42147  875 
end 