author  wenzelm 
Tue, 28 Feb 2012 21:53:36 +0100  
changeset 46731  5302e932d1e5 
parent 45777  c36637603821 
child 46898  1570b30ee040 
permissions  rwrr 
42147  1 
(* 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 
42147  9 
begin 
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lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B" 

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unfolding restrict_def extensional_def by auto 

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lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)" 

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unfolding restrict_def by (simp add: fun_eq_iff) 

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lemma split_merge: "P (merge I x J 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)" 

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unfolding merge_def by auto 

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lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I x J y \<in> extensional K" 

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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)" 

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have "x \<in> A \<longleftrightarrow> merge J x (I  J) y \<in> (\<lambda>x. restrict x J) ` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" 

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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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lemma (in product_prob_space) measure_preserving_restrict: 

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assumes "J \<noteq> {}" "J \<subseteq> K" "finite K" 

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shows "(\<lambda>f. restrict f J) \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" (is "?R \<in> _") 

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proof  

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

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interpret K: finite_product_prob_space M K by default fact 
42147  49 
have J: "J \<noteq> {}" "finite J" using assms by (auto simp add: finite_subset) 
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interpret J: finite_product_prob_space M J 

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by default (insert J, auto) 

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from J.sigma_finite_pairs guess F .. note F = this 

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then have [simp,intro]: "\<And>k i. k \<in> J \<Longrightarrow> F k i \<in> sets (M k)" 

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by auto 

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let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. F k i" 
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let ?J = "product_algebra_generator J M \<lparr> measure := measure (Pi\<^isub>M J M) \<rparr>" 
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have "?R \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (sigma ?J)" 

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proof (rule K.measure_preserving_Int_stable) 

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show "Int_stable ?J" 

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by (auto simp: Int_stable_def product_algebra_generator_def PiE_Int) 

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show "range ?F \<subseteq> sets ?J" "incseq ?F" "(\<Union>i. ?F i) = space ?J" 

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using F by auto 

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show "\<And>i. measure ?J (?F i) \<noteq> \<infinity>" 

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using F by (simp add: J.measure_times setprod_PInf) 

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have "measure_space (Pi\<^isub>M J M)" by default 

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then show "measure_space (sigma ?J)" 

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by (simp add: product_algebra_def sigma_def) 

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show "?R \<in> measure_preserving (Pi\<^isub>M K M) ?J" 

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proof (simp add: measure_preserving_def measurable_def product_algebra_generator_def del: vimage_Int, 

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safe intro!: restrict_extensional) 

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fix x k assume "k \<in> J" "x \<in> (\<Pi> i\<in>K. space (M i))" 

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then show "x k \<in> space (M k)" using `J \<subseteq> K` by auto 

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next 

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fix E assume "E \<in> (\<Pi> i\<in>J. sets (M i))" 

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then have E: "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" by auto 

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then have *: "?R ` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i)) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))" 

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(is "?X = Pi\<^isub>E K ?M") 

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using `J \<subseteq> K` sets_into_space by (auto simp: Pi_iff split: split_if_asm) blast+ 

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with E show "?X \<in> sets (Pi\<^isub>M K M)" 

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by (auto intro!: product_algebra_generatorI) 

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have "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = (\<Prod>i\<in>J. measure (M i) (?M i))" 

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using E by (simp add: J.measure_times) 

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also have "\<dots> = measure (Pi\<^isub>M K M) ?X" 

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unfolding * using E `finite K` `J \<subseteq> K` 

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by (auto simp: K.measure_times M.measure_space_1 

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cong del: setprod_cong 

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intro!: setprod_mono_one_left) 

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finally show "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = measure (Pi\<^isub>M K M) ?X" . 

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qed 

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qed 

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then show ?thesis 

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by (simp add: product_algebra_def sigma_def) 

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qed 

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lemma (in product_prob_space) measurable_restrict: 

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assumes *: "J \<noteq> {}" "J \<subseteq> K" "finite K" 

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shows "(\<lambda>f. restrict f J) \<in> measurable (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" 

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using measure_preserving_restrict[OF *] 

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by (rule measure_preservingD2) 

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definition (in product_prob_space) 

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"emb J K X = (\<lambda>x. restrict x K) ` X \<inter> space (Pi\<^isub>M J M)" 

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lemma (in product_prob_space) emb_trans[simp]: 

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"J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> emb L K (emb K J X) = emb L J X" 

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by (auto simp add: Int_absorb1 emb_def) 

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lemma (in product_prob_space) emb_empty[simp]: 

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"emb K J {} = {}" 

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by (simp add: emb_def) 

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lemma (in product_prob_space) emb_Pi: 

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assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K" 

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shows "emb K J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))" 

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using assms space_closed 

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by (auto simp: emb_def Pi_iff split: split_if_asm) blast+ 

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lemma (in product_prob_space) emb_injective: 

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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 "emb L J X = emb L J Y" 

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shows "X = Y" 

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proof  

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interpret J: finite_product_sigma_finite M J by default fact 

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show "X = Y" 

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proof (rule injective_vimage_restrict) 

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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 J.sets_into_space sets by auto 

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

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using M.not_empty by auto 

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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 `emb L J X = emb L J Y` by (simp add: emb_def) 

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qed fact 

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qed 

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lemma (in product_prob_space) emb_id: 

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"B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> emb L L B = B" 

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by (auto simp: emb_def Pi_iff subset_eq extensional_restrict) 

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lemma (in product_prob_space) emb_simps: 

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shows "emb L K (A \<union> B) = emb L K A \<union> emb L K B" 

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and "emb L K (A \<inter> B) = emb L K A \<inter> emb L K B" 

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and "emb L K (A  B) = emb L K A  emb L K B" 

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by (auto simp: emb_def) 

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lemma (in product_prob_space) measurable_emb[intro,simp]: 

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assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)" 

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shows "emb L J X \<in> sets (Pi\<^isub>M L M)" 

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using measurable_restrict[THEN measurable_sets, OF *] by (simp add: emb_def) 

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lemma (in product_prob_space) measure_emb[intro,simp]: 

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assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)" 

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shows "measure (Pi\<^isub>M L M) (emb L J X) = measure (Pi\<^isub>M J M) X" 

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using measure_preserving_restrict[THEN measure_preservingD, OF *] 

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by (simp add: emb_def) 

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definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) measure_space" where 

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"generator = \<lparr> 

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space = (\<Pi>\<^isub>E i\<in>I. space (M i)), 

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sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)), 

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measure = undefined 

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\<rparr>" 

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lemma (in product_prob_space) generatorI: 

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"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> sets generator" 

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unfolding generator_def by auto 

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lemma (in product_prob_space) generatorI': 

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"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> sets generator" 

171 
unfolding generator_def by auto 

172 

173 
lemma (in product_sigma_finite) 

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assumes "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" 

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shows measure_fold_integral: 

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"measure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. measure (Pi\<^isub>M J M) (merge I x J ` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I) 

177 
and measure_fold_measurable: 

178 
"(\<lambda>x. measure (Pi\<^isub>M J M) (merge I x J ` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B) 

179 
proof  

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interpret I: finite_product_sigma_finite M I by default fact 

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interpret J: finite_product_sigma_finite M J by default fact 

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interpret IJ: pair_sigma_finite I.P J.P .. 

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show ?I 

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unfolding measure_fold[OF assms] 

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apply (subst IJ.pair_measure_alt) 

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apply (intro measurable_sets[OF _ A] measurable_merge assms) 

187 
apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure 

188 
intro!: I.positive_integral_cong) 

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done 

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191 
have "(\<lambda>(x, y). merge I x J y) ` A \<inter> space (I.P \<Otimes>\<^isub>M J.P) \<in> sets (I.P \<Otimes>\<^isub>M J.P)" 

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by (intro measurable_sets[OF _ A] measurable_merge assms) 

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from IJ.measure_cut_measurable_fst[OF this] 

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show ?B 

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apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure) 

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apply (subst (asm) measurable_cong) 

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apply auto 

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done 

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qed 

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definition (in product_prob_space) 

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"\<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 = measure (Pi\<^isub>M J M) X))" 

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lemma (in product_prob_space) \<mu>G_spec: 

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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 = measure (Pi\<^isub>M J M) X" 

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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 "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)" 

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using K J by simp 

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also have "emb (K \<union> J) K Y = emb (K \<union> J) J X" 

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using K J by (simp add: emb_injective[of "K \<union> J" I]) 

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also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X" 

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using K J by simp 

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finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" .. 

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qed (insert J, force) 

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lemma (in product_prob_space) \<mu>G_eq: 

221 
"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) = measure (Pi\<^isub>M J M) X" 

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by (intro \<mu>G_spec) auto 

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lemma (in product_prob_space) generator_Ex: 

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assumes *: "A \<in> sets 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 = measure (Pi\<^isub>M J M) X" 

227 
proof  

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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)" 

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unfolding generator_def by auto 

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

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qed 

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lemma (in product_prob_space) generatorE: 

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assumes A: "A \<in> sets 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 = measure (Pi\<^isub>M J M) X" 

236 
proof  

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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 = measure (Pi\<^isub>M J M) X" by auto 

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then show thesis by (intro that) auto 

240 
qed 

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lemma (in product_prob_space) merge_sets: 

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assumes "finite J" "finite K" "J \<inter> K = {}" and A: "A \<in> sets (Pi\<^isub>M (J \<union> K) M)" and x: "x \<in> space (Pi\<^isub>M J M)" 

244 
shows "merge J x K ` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)" 

245 
proof  

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interpret J: finite_product_sigma_algebra M J by default fact 

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interpret K: finite_product_sigma_algebra M K by default fact 

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interpret JK: pair_sigma_algebra J.P K.P .. 

249 

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from JK.measurable_cut_fst[OF 

251 
measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x 

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show ?thesis 

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by (simp add: space_pair_measure comp_def vimage_compose[symmetric]) 

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qed 

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256 
lemma (in product_prob_space) merge_emb: 

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assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)" 

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shows "(merge J y (I  J) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = 

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emb I (K  J) (merge J y (K  J) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M))" 

260 
proof  

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have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x" 

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by (auto simp: restrict_def merge_def) 

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have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x" 

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by (auto simp: restrict_def merge_def) 

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have [simp]: "(I  J) \<inter> K = K  J" using `K \<subseteq> I` `J \<subseteq> I` by auto 

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

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

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

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by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto 

270 
qed 

271 

272 
definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where 

273 
"infprod_algebra = sigma generator \<lparr> measure := 

274 
(SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and> 

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

275 
prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>" 
42147  276 

277 
syntax 

278 
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3PIP _:_./ _)" 10) 

279 

280 
syntax (xsymbols) 

281 
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10) 

282 

283 
syntax (HTML output) 

284 
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10) 

285 

286 
abbreviation 

287 
"Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I" 

288 

289 
translations 

290 
"PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)" 

291 

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

292 
lemma (in product_prob_space) algebra_generator: 
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

293 
assumes "I \<noteq> {}" shows "algebra generator" 
42147  294 
proof 
295 
let ?G = generator 

296 
show "sets ?G \<subseteq> Pow (space ?G)" 

297 
by (auto simp: generator_def emb_def) 

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

298 
from `I \<noteq> {}` obtain i where "i \<in> I" by auto 
42147  299 
then show "{} \<in> sets ?G" 
300 
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"] 

301 
simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def) 

302 
from `i \<in> I` show "space ?G \<in> sets ?G" 

303 
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"] 

304 
simp: generator_def emb_def) 

305 
fix A assume "A \<in> sets ?G" 

306 
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" 

307 
by (auto simp: generator_def) 

308 
fix B assume "B \<in> sets ?G" 

309 
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" 

310 
by (auto simp: generator_def) 

311 
let ?RA = "emb (JA \<union> JB) JA XA" 

312 
let ?RB = "emb (JA \<union> JB) JB XB" 

313 
interpret JAB: finite_product_sigma_algebra M "JA \<union> JB" 

314 
by default (insert XA XB, auto) 

315 
have *: "A  B = emb I (JA \<union> JB) (?RA  ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)" 

316 
using XA A XB B by (auto simp: emb_simps) 

317 
then show "A  B \<in> sets ?G" "A \<union> B \<in> sets ?G" 

318 
using XA XB by (auto intro!: generatorI') 

319 
qed 

320 

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

321 
lemma (in product_prob_space) positive_\<mu>G: 
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322 
assumes "I \<noteq> {}" 
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323 
shows "positive generator \<mu>G" 
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324 
proof  
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325 
interpret G!: algebra generator by (rule algebra_generator) fact 
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326 
show ?thesis 
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327 
proof (intro positive_def[THEN iffD2] conjI ballI) 
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328 
from generatorE[OF G.empty_sets] guess J X . note this[simp] 
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329 
interpret J: finite_product_sigma_finite M J by default fact 
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330 
have "X = {}" 
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331 
by (rule emb_injective[of J I]) simp_all 
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332 
then show "\<mu>G {} = 0" by simp 
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333 
next 
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334 
fix A assume "A \<in> sets generator" 
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335 
from generatorE[OF this] guess J X . note this[simp] 
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336 
interpret J: finite_product_sigma_finite M J by default fact 
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337 
show "0 \<le> \<mu>G A" by simp 
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338 
qed 
42147  339 
qed 
340 

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341 
lemma (in product_prob_space) additive_\<mu>G: 
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342 
assumes "I \<noteq> {}" 
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343 
shows "additive generator \<mu>G" 
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344 
proof  
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345 
interpret G!: algebra generator by (rule algebra_generator) fact 
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346 
show ?thesis 
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347 
proof (intro additive_def[THEN iffD2] ballI impI) 
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348 
fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this 
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349 
fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this 
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350 
assume "A \<inter> B = {}" 
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351 
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)" 
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352 
using J K by auto 
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353 
interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact 
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354 
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}" 
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355 
apply (rule emb_injective[of "J \<union> K" I]) 
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356 
apply (insert `A \<inter> B = {}` JK J K) 
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357 
apply (simp_all add: JK.Int emb_simps) 
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358 
done 
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359 
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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360 
using J K by simp_all 
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361 
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))" 
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362 
by (simp add: emb_simps) 
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363 
also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" 
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364 
using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un) 
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365 
also have "\<dots> = \<mu>G A + \<mu>G B" 
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366 
using J K JK_disj by (simp add: JK.measure_additive[symmetric]) 
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367 
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" . 
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368 
qed 
42147  369 
qed 
370 

371 
lemma (in product_prob_space) finite_index_eq_finite_product: 

372 
assumes "finite I" 

373 
shows "sets (sigma generator) = sets (Pi\<^isub>M I M)" 

374 
proof safe 

375 
interpret I: finite_product_sigma_algebra M I by default fact 

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376 
have space_generator[simp]: "space generator = space (Pi\<^isub>M I M)" 
42147  377 
by (simp add: generator_def product_algebra_def) 
378 
{ fix A assume "A \<in> sets (sigma generator)" 

379 
then show "A \<in> sets I.P" unfolding sets_sigma 

380 
proof induct 

381 
case (Basic A) 

382 
from generatorE[OF this] guess J X . note J = this 

383 
with `finite I` have "emb I J X \<in> sets I.P" by auto 

384 
with `emb I J X = A` show "A \<in> sets I.P" by simp 

385 
qed auto } 

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386 
{ fix A assume A: "A \<in> sets I.P" 
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387 
show "A \<in> sets (sigma generator)" 
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388 
proof cases 
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389 
assume "I = {}" 
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390 
with I.P_empty[OF this] A 
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391 
have "A = space generator \<or> A = {}" 
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392 
unfolding space_generator by auto 
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393 
then show ?thesis 
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394 
by (auto simp: sets_sigma simp del: space_generator 
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395 
intro: sigma_sets.Empty sigma_sets_top) 
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396 
next 
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397 
assume "I \<noteq> {}" 
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398 
note A this 
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399 
moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto 
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400 
ultimately show "A \<in> sets (sigma generator)" 
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401 
using `finite I` unfolding sets_sigma 
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402 
by (intro sigma_sets.Basic generatorI[of I A]) auto 
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403 
qed } 
42147  404 
qed 
405 

406 
lemma (in product_prob_space) extend_\<mu>G: 

407 
"\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and> 

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408 
prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>" 
42147  409 
proof cases 
410 
assume "finite I" 

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411 
interpret I: finite_product_prob_space M I by default fact 
42147  412 
show ?thesis 
413 
proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI) 

414 
fix A assume "A \<in> sets generator" 

415 
from generatorE[OF this] guess J X . note J = this 

416 
from J(14) `finite I` show "measure I.P A = \<mu>G A" 

417 
unfolding J(6) 

418 
by (subst J(5)[symmetric]) (simp add: measure_emb) 

419 
next 

420 
have [simp]: "space generator = space (Pi\<^isub>M I M)" 

421 
by (simp add: generator_def product_algebra_def) 

422 
have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr> 

423 
= I.P" (is "?P = _") 

424 
by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`]) 

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425 
show "prob_space ?P" 
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426 
proof 
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427 
show "measure_space ?P" using `?P = I.P` by simp default 
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428 
show "measure ?P (space ?P) = 1" 
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429 
using I.measure_space_1 by simp 
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430 
qed 
42147  431 
qed 
432 
next 

433 
let ?G = generator 

434 
assume "\<not> finite I" 

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435 
then have I_not_empty: "I \<noteq> {}" by auto 
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436 
interpret G!: algebra generator by (rule algebra_generator) fact 
42147  437 
note \<mu>G_mono = 
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438 
G.additive_increasing[OF positive_\<mu>G[OF I_not_empty] additive_\<mu>G[OF I_not_empty], THEN increasingD] 
42147  439 

440 
{ fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G" 

441 

442 
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J" 

443 
by (metis rev_finite_subset subsetI) 

444 
moreover from Z guess K' X' by (rule generatorE) 

445 
moreover def K \<equiv> "insert k K'" 

446 
moreover def X \<equiv> "emb K K' X'" 

447 
ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X" 

448 
"K  J \<noteq> {}" "K  J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X" 

449 
by (auto simp: subset_insertI) 

450 

46731  451 
let ?M = "\<lambda>y. merge J y (K  J) ` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K  J) M)" 
42147  452 
{ fix y assume y: "y \<in> space (Pi\<^isub>M J M)" 
453 
note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X] 

454 
moreover 

455 
have **: "?M y \<in> sets (Pi\<^isub>M (K  J) M)" 

456 
using J K y by (intro merge_sets) auto 

457 
ultimately 

458 
have ***: "(merge J y (I  J) ` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G" 

459 
using J K by (intro generatorI) auto 

460 
have "\<mu>G (merge J y (I  J) ` emb I K X \<inter> space (Pi\<^isub>M I M)) = measure (Pi\<^isub>M (K  J) M) (?M y)" 

461 
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto 

462 
note * ** *** this } 

463 
note merge_in_G = this 

464 

465 
have "finite (K  J)" using K by auto 

466 

467 
interpret J: finite_product_prob_space M J by default fact+ 

468 
interpret KmJ: finite_product_prob_space M "K  J" by default fact+ 

469 

470 
have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K  J)) M) (emb (J \<union> (K  J)) K X)" 

471 
using K J by simp 

472 
also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K  J) M) (?M x) \<partial>Pi\<^isub>M J M)" 

473 
using K J by (subst measure_fold_integral) auto 

474 
also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G (merge J y (I  J) ` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)" 

475 
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)") 

476 
proof (intro J.positive_integral_cong) 

477 
fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 

478 
with K merge_in_G(2)[OF this] 

479 
show "measure (Pi\<^isub>M (K  J) M) (?M x) = \<mu>G (?MZ x)" 

480 
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto 

481 
qed 

482 
finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" . 

483 

484 
{ fix x assume x: "x \<in> space (Pi\<^isub>M J M)" 

485 
then have "\<mu>G (?MZ x) \<le> 1" 

486 
unfolding merge_in_G(4)[OF x] `Z = emb I K X` 

487 
by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) } 

488 
note le_1 = this 

489 

46731  490 
let ?q = "\<lambda>y. \<mu>G (merge J y (I  J) ` Z \<inter> space (Pi\<^isub>M I M))" 
42147  491 
have "?q \<in> borel_measurable (Pi\<^isub>M J M)" 
492 
unfolding `Z = emb I K X` using J K merge_in_G(3) 

493 
by (simp add: merge_in_G \<mu>G_eq measure_fold_measurable 

494 
del: space_product_algebra cong: measurable_cong) 

495 
note this fold le_1 merge_in_G(3) } 

496 
note fold = this 

497 

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

498 
have "\<exists>\<mu>. (\<forall>s\<in>sets ?G. \<mu> s = \<mu>G s) \<and> 
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

499 
measure_space \<lparr>space = space ?G, sets = sets (sigma ?G), measure = \<mu>\<rparr>" 
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

500 
(is "\<exists>\<mu>. _ \<and> measure_space (?ms \<mu>)") 
42147  501 
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G]) 
502 
fix A assume "A \<in> sets ?G" 

503 
with generatorE guess J X . note JX = this 

504 
interpret JK: finite_product_prob_space M J by default fact+ 

505 
with JX show "\<mu>G A \<noteq> \<infinity>" by simp 

506 
next 

507 
fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}" 

508 
then have "decseq (\<lambda>i. \<mu>G (A i))" 

509 
by (auto intro!: \<mu>G_mono simp: decseq_def) 

510 
moreover 

511 
have "(INF i. \<mu>G (A i)) = 0" 

512 
proof (rule ccontr) 

513 
assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0") 

514 
moreover have "0 \<le> ?a" 

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

515 
using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def) 
42147  516 
ultimately have "0 < ?a" by auto 
517 

518 
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) = measure (Pi\<^isub>M J M) X" 

519 
using A by (intro allI generator_Ex) auto 

520 
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)" 

521 
and A': "\<And>n. A n = emb I (J' n) (X' n)" 

522 
unfolding choice_iff by blast 

523 
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)" 

524 
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)" 

525 
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)" 

526 
by auto 

527 
with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G" 

528 
unfolding J_def X_def by (subst emb_trans) (insert A, auto) 

529 

530 
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m" 

531 
unfolding J_def by force 

532 

533 
interpret J: finite_product_prob_space M "J i" for i by default fact+ 

534 

535 
have a_le_1: "?a \<le> 1" 

536 
using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq 

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

537 
by (auto intro!: INF_lower2[of 0] J.measure_le_1) 
42147  538 

46731  539 
let ?M = "\<lambda>K Z y. merge K y (I  K) ` Z \<inter> space (Pi\<^isub>M I M)" 
42147  540 

541 
{ fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)" 

542 
then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto 

543 
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I" 

544 
interpret J': finite_product_prob_space M J' by default fact+ 

545 

46731  546 
let ?q = "\<lambda>n y. \<mu>G (?M J' (Z n) y)" 
547 
let ?Q = "\<lambda>n. ?q n ` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)" 

42147  548 
{ fix n 
549 
have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)" 

550 
using Z J' by (intro fold(1)) auto 

551 
then have "?Q n \<in> sets (Pi\<^isub>M J' M)" 

552 
by (rule measurable_sets) auto } 

553 
note Q_sets = this 

554 

555 
have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))" 

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

556 
proof (intro INF_greatest) 
42147  557 
fix n 
558 
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto 

559 
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)" 

560 
unfolding fold(2)[OF J' `Z n \<in> sets ?G`] 

561 
proof (intro J'.positive_integral_mono) 

562 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 

563 
then have "?q n x \<le> 1 + 0" 

564 
using J' Z fold(3) Z_sets by auto 

565 
also have "\<dots> \<le> 1 + ?a / 2^(k+1)" 

566 
using `0 < ?a` by (intro add_mono) auto 

567 
finally have "?q n x \<le> 1 + ?a / 2^(k+1)" . 

568 
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)" 

569 
by (auto split: split_indicator simp del: power_Suc) 

570 
qed 

571 
also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)" 

572 
using `0 \<le> ?a` Q_sets J'.measure_space_1 

573 
by (subst J'.positive_integral_add) auto 

574 
finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1` 

43920  575 
by (cases rule: ereal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"]) 
42147  576 
(auto simp: field_simps) 
577 
qed 

578 
also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)" 

579 
proof (intro J'.continuity_from_above) 

580 
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto 

581 
show "decseq ?Q" 

582 
unfolding decseq_def 

583 
proof (safe intro!: vimageI[OF refl]) 

584 
fix m n :: nat assume "m \<le> n" 

585 
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)" 

586 
assume "?a / 2^(k+1) \<le> ?q n x" 

587 
also have "?q n x \<le> ?q m x" 

588 
proof (rule \<mu>G_mono) 

589 
from fold(4)[OF J', OF Z_sets x] 

590 
show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto 

591 
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x" 

592 
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto 

593 
qed 

594 
finally show "?a / 2^(k+1) \<le> ?q m x" . 

595 
qed 

596 
qed (intro J'.finite_measure Q_sets) 

597 
finally have "(\<Inter>n. ?Q n) \<noteq> {}" 

598 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 

599 
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto } 

600 
note Ex_w = this 

601 

46731  602 
let ?q = "\<lambda>k n y. \<mu>G (?M (J k) (A n) y)" 
42147  603 

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

604 
have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_lower) 
42147  605 
from Ex_w[OF A(1,2) this J(13), of 0] guess w0 .. note w0 = this 
606 

46731  607 
let ?P = 
608 
"\<lambda>k wk w. w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and> 

609 
(\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)" 

42147  610 
def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))" 
611 

612 
{ fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and> 

613 
(\<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))" 

614 
proof (induct k) 

615 
case 0 with w0 show ?case 

616 
unfolding w_def nat_rec_0 by auto 

617 
next 

618 
case (Suc k) 

619 
then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

620 
have "\<exists>w'. ?P k (w k) w'" 

621 
proof cases 

622 
assume [simp]: "J k = J (Suc k)" 

623 
show ?thesis 

624 
proof (intro exI[of _ "w k"] conjI allI) 

625 
fix n 

626 
have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)" 

627 
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps) 

628 
also have "\<dots> \<le> ?q k n (w k)" using Suc by auto 

629 
finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp 

630 
next 

631 
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)" 

632 
using Suc by simp 

633 
then show "restrict (w k) (J k) = w k" 

634 
by (simp add: extensional_restrict) 

635 
qed 

636 
next 

637 
assume "J k \<noteq> J (Suc k)" 

638 
with J_mono[of k "Suc k"] have "J (Suc k)  J k \<noteq> {}" (is "?D \<noteq> {}") by auto 

639 
have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G" 

640 
"decseq (\<lambda>n. ?M (J k) (A n) (w k))" 

641 
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))" 

642 
using `decseq A` fold(4)[OF J(13) A_eq(2), of "w k" k] Suc 

643 
by (auto simp: decseq_def) 

644 
from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"] 

645 
obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)" 

646 
"\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto 

647 
let ?w = "merge (J k) (w k) ?D w'" 

648 
have [simp]: "\<And>x. merge (J k) (w k) (I  J k) (merge ?D w' (I  ?D) x) = 

649 
merge (J (Suc k)) ?w (I  (J (Suc k))) x" 

650 
using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"] 

651 
by (auto intro!: ext split: split_merge) 

652 
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w" 

653 
using w'(1) J(3)[of "Suc k"] 

654 
by (auto split: split_merge intro!: extensional_merge_sub) force+ 

655 
show ?thesis 

656 
apply (rule exI[of _ ?w]) 

657 
using w' J_mono[of k "Suc k"] wk unfolding * 

658 
apply (auto split: split_merge intro!: extensional_merge_sub ext) 

659 
apply (force simp: extensional_def) 

660 
done 

661 
qed 

662 
then have "?P k (w k) (w (Suc k))" 

663 
unfolding w_def nat_rec_Suc unfolding w_def[symmetric] 

664 
by (rule someI_ex) 

665 
then show ?case by auto 

666 
qed 

667 
moreover 

668 
then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto 

669 
moreover 

670 
from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto 

671 
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

672 
using positive_\<mu>G[OF I_not_empty, unfolded positive_def] `0 < ?a` `?a \<le> 1` 
42147  673 
by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq) 
674 
then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto 

675 
then have "merge (J k) (w k) (I  J k) x \<in> A k" by auto 

676 
then have "\<exists>x\<in>A k. restrict x (J k) = w k" 

677 
using `w k \<in> space (Pi\<^isub>M (J k) M)` 

678 
by (intro rev_bexI) (auto intro!: ext simp: extensional_def) 

679 
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)" 

680 
"\<exists>x\<in>A k. restrict x (J k) = w k" 

681 
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k  1)) = w (k  1)" 

682 
by auto } 

683 
note w = this 

684 

685 
{ fix k l i assume "k \<le> l" "i \<in> J k" 

686 
{ fix l have "w k i = w (k + l) i" 

687 
proof (induct l) 

688 
case (Suc l) 

689 
from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto 

690 
with w(3)[of "k + Suc l"] 

691 
have "w (k + l) i = w (k + Suc l) i" 

692 
by (auto simp: restrict_def fun_eq_iff split: split_if_asm) 

693 
with Suc show ?case by simp 

694 
qed simp } 

695 
from this[of "l  k"] `k \<le> l` have "w l i = w k i" by simp } 

696 
note w_mono = this 

697 

698 
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" 

699 
{ fix i k assume k: "i \<in> J k" 

700 
have "w k i = w (LEAST k. i \<in> J k) i" 

701 
by (intro w_mono Least_le k LeastI[of _ k]) 

702 
then have "w' i = w k i" 

703 
unfolding w'_def using k by auto } 

704 
note w'_eq = this 

705 
have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined" 

706 
using J by (auto simp: w'_def) 

707 
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)" 

708 
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]]) 

709 
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)" 

710 
using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ } 

711 
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this 

712 

713 
have w': "w' \<in> space (Pi\<^isub>M I M)" 

714 
using w(1) by (auto simp add: Pi_iff extensional_def) 

715 

716 
{ fix n 

717 
have "restrict w' (J n) = w n" using w(1) 

718 
by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def) 

719 
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto 

720 
then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) } 

721 
then have "w' \<in> (\<Inter>i. A i)" by auto 

722 
with `(\<Inter>i. A i) = {}` show False by auto 

723 
qed 

724 
ultimately show "(\<lambda>i. \<mu>G (A i)) > 0" 

43920  725 
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

726 
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

727 
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

728 
show ?thesis 
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

729 
proof (intro exI[of _ \<mu>] conjI) 
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

730 
show "\<forall>S\<in>sets ?G. \<mu> S = \<mu>G S" using \<mu> by simp 
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

731 
show "prob_space (?ms \<mu>)" 
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

732 
proof 
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

733 
show "measure_space (?ms \<mu>)" using \<mu> by simp 
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

734 
obtain i where "i \<in> I" using I_not_empty by auto 
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

735 
interpret i: finite_product_sigma_finite M "{i}" by default auto 
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

736 
let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)" 
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

737 
have X: "?X \<in> sets (Pi\<^isub>M {i} M)" 
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

738 
by auto 
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

739 
with `i \<in> I` have "emb I {i} ?X \<in> sets generator" 
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

740 
by (intro generatorI') auto 
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

741 
with \<mu> have "\<mu> (emb I {i} ?X) = \<mu>G (emb I {i} ?X)" by auto 
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

742 
with \<mu>G_eq[OF _ _ _ X] `i \<in> I` 
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

743 
have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))" 
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

744 
by (simp add: i.measure_times) 
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

745 
also have "emb I {i} ?X = space (Pi\<^isub>P I M)" 
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

746 
using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def) 
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

747 
finally show "measure (?ms \<mu>) (space (?ms \<mu>)) = 1" 
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

748 
using M.measure_space_1 by (simp add: infprod_algebra_def) 
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

749 
qed 
42147  750 
qed 
751 
qed 

752 

753 
lemma (in product_prob_space) infprod_spec: 

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

754 
"(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> prob_space (Pi\<^isub>P I M)" 
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

755 
(is "?Q infprod_algebra") 
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

756 
unfolding infprod_algebra_def 
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

757 
by (rule someI2_ex[OF extend_\<mu>G]) 
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

758 
(auto simp: sigma_def generator_def) 
42147  759 

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

760 
sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M" 
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

761 
using infprod_spec by simp 
42147  762 

763 
lemma (in product_prob_space) measure_infprod_emb: 

764 
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" 

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

765 
shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X" 
42147  766 
proof  
767 
have "emb I J X \<in> sets generator" 

768 
using assms by (rule generatorI') 

769 
with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto 

770 
qed 

771 

42166
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

772 
lemma (in product_prob_space) measurable_component: 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

773 
assumes "i \<in> I" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

774 
shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

775 
proof (unfold measurable_def, safe) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

776 
fix x assume "x \<in> space (Pi\<^isub>P I M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

777 
then show "x i \<in> space (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

778 
using `i \<in> I` by (auto simp: infprod_algebra_def generator_def) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

779 
next 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

780 
fix A assume "A \<in> sets (M i)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

781 
with `i \<in> I` have 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

782 
"(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

783 
"(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

784 
by (auto simp: infprod_algebra_def generator_def emb_def) 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

785 
from generatorI[OF _ _ _ this] `i \<in> I` 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

786 
show "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

787 
unfolding infprod_algebra_def by auto 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

788 
qed 
efd229daeb2c
products of probability measures are probability measures
hoelzl
parents:
42148
diff
changeset

789 

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

790 
lemma (in product_prob_space) emb_in_infprod_algebra[intro]: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

791 
fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

792 
shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

793 
proof cases 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

794 
assume "J = {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

795 
with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

796 
by (auto simp: emb_def infprod_algebra_def generator_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

797 
product_algebra_def product_algebra_generator_def image_constant sigma_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

798 
then show ?thesis by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

799 
next 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

800 
assume "J \<noteq> {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

801 
show ?thesis unfolding infprod_algebra_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

802 
by simp (intro in_sigma generatorI' `J \<noteq> {}` J X) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

803 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

804 

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

805 
lemma (in product_prob_space) finite_measure_infprod_emb: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

806 
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

807 
shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

808 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

809 
interpret J: finite_product_prob_space M J by default fact+ 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

810 
from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

811 
with assms show "\<mu>' (emb I J X) = J.\<mu>' X" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

812 
unfolding \<mu>'_def J.\<mu>'_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

813 
unfolding measure_infprod_emb[OF assms] 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

814 
by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

815 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

816 

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

817 
lemma (in finite_product_prob_space) finite_measure_times: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

818 
assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

819 
shows "\<mu>' (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu>' i (A i))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

820 
using assms 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

821 
unfolding \<mu>'_def M.\<mu>'_def 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

822 
by (subst measure_times[OF assms]) 
43920  823 
(auto simp: finite_measure_eq M.finite_measure_eq setprod_ereal) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

824 

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

825 
lemma (in product_prob_space) finite_measure_infprod_emb_Pi: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

826 
assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

827 
shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

828 
proof cases 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

829 
assume "J = {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

830 
then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

831 
by (auto simp: infprod_algebra_def generator_def sigma_def emb_def) 
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

832 
then show ?thesis using `J = {}` P.prob_space 
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

833 
by simp 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

834 
next 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

835 
assume "J \<noteq> {}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

836 
interpret J: finite_product_prob_space M J by default fact+ 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

837 
have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

838 
using J `J \<noteq> {}` by (subst J.finite_measure_times) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

839 
also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

840 
using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

841 
finally show ?thesis by simp 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

842 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

843 

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

844 
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

845 
proof 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

846 
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

847 
by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

848 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

849 

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

850 
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

851 
proof 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

852 
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

853 
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

854 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

855 

43679
052eaf7509cf
rename lemma Infinite_Product_Measure.sigma_sets_subseteq, it hides Sigma_Algebra.sigma_sets_subseteq
hoelzl
parents:
42950
diff
changeset

856 
lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

857 
by (auto intro: sigma_sets.Basic) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

858 

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

859 
lemma (in product_prob_space) infprod_algebra_alt: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

860 
"Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

861 
sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i))), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

862 
measure = measure (Pi\<^isub>P I M) \<rparr>" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

863 
(is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>") 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

864 
proof (rule measure_space.equality) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

865 
let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

866 
have "sigma_sets ?O ?M = sigma_sets ?O ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

867 
proof (intro equalityI sigma_sets_mono UN_least) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

868 
fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

869 
have "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> emb I J ` sets (Pi\<^isub>M J M)" by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

870 
also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper) 
43679
052eaf7509cf
rename lemma Infinite_Product_Measure.sigma_sets_subseteq, it hides Sigma_Algebra.sigma_sets_subseteq
hoelzl
parents:
42950
diff
changeset

871 
also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_superset_generator) 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

872 
finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" . 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

873 
have "emb I J ` sets (Pi\<^isub>M J M) = emb I J ` sigma_sets (space (Pi\<^isub>M J M)) (Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

874 
by (simp add: sets_sigma product_algebra_generator_def product_algebra_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

875 
also have "\<dots> = sigma_sets (space (Pi\<^isub>M I M)) (emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

876 
using J M.sets_into_space 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

877 
by (auto simp: emb_def_raw intro!: sigma_sets_vimage[symmetric]) blast 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

878 
also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

879 
using J by (intro sigma_sets_mono') auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

880 
finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

881 
by (simp add: infprod_algebra_def generator_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

882 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

883 
then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

884 
by (simp_all add: infprod_algebra_def generator_def sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

885 
qed simp_all 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

886 

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

887 
lemma (in product_prob_space) infprod_algebra_alt2: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

888 
"Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

889 
sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)), 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

890 
measure = measure (Pi\<^isub>P I M) \<rparr>" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

891 
(is "_ = ?S") 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

892 
proof (rule measure_space.equality) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

893 
let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

894 
let ?G = "(\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

895 
have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

896 
by (subst infprod_algebra_alt) (simp add: sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

897 
also have "\<dots> = sigma_sets ?O ?A" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

898 
proof (intro equalityI sigma_sets_mono subsetI) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

899 
interpret A: sigma_algebra ?S 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

900 
by (rule sigma_algebra_sigma) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

901 
fix A assume "A \<in> ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

902 
then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

903 
and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

904 
by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

905 
then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) ` (B j) \<inter> space (Pi\<^isub>P I M))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

906 
by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

907 
{ fix j assume "j\<in>J" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

908 
with `J \<subseteq> I` have "j \<in> I" by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

909 
with `j \<in> J` B have "(\<lambda>x. x j) ` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

910 
by (auto simp: sets_sigma intro: sigma_sets.Basic) } 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

911 
with `finite J` `J \<noteq> {}` have "A \<in> sets ?S" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

912 
unfolding A by (intro A.finite_INT) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

913 
then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

914 
next 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

915 
fix A assume "A \<in> ?A" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

916 
then obtain i B where i: "i \<in> I" "B \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

917 
and "A = (\<lambda>x. x i) ` B \<inter> space (Pi\<^isub>P I M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

918 
by auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

919 
then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

920 
by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

921 
with i show "A \<in> sigma_sets ?O ?G" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

922 
by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

923 
qed 
42866  924 
also have "\<dots> = sets ?S" 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

925 
by (simp add: sets_sigma) 
42866  926 
finally show "sets (Pi\<^isub>P I M) = sets ?S" . 
42257
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

927 
qed simp_all 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

928 

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

929 
lemma (in product_prob_space) measurable_into_infprod_algebra: 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

930 
assumes "sigma_algebra N" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

931 
assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

932 
assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

933 
shows "f \<in> measurable N (Pi\<^isub>P I M)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

934 
proof  
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

935 
interpret N: sigma_algebra N by fact 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

936 
have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

937 
using f by (auto simp: measurable_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

938 
{ fix i A assume i: "i \<in> I" "A \<in> sets (M i)" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

939 
then have "f ` (\<lambda>x. x i) ` A \<inter> f ` space infprod_algebra \<inter> space N = (\<lambda>x. f x i) ` A \<inter> space N" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

940 
using f_in ext by (auto simp: infprod_algebra_def generator_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

941 
also have "\<dots> \<in> sets N" 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

942 
by (rule measurable_sets f i)+ 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

943 
finally have "f ` (\<lambda>x. x i) ` A \<inter> f ` space infprod_algebra \<inter> space N \<in> sets N" . } 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

944 
with f_in ext show ?thesis 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

945 
by (subst infprod_algebra_alt2) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

946 
(auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def) 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

947 
qed 
08d717c82828
prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
42166
diff
changeset

948 

42865  949 
lemma (in product_prob_space) measurable_singleton_infprod: 
950 
assumes "i \<in> I" 

951 
shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)" 

952 
proof (unfold measurable_def, intro CollectI conjI ballI) 

953 
show "(\<lambda>x. x i) \<in> space (Pi\<^isub>P I M) \<rightarrow> space (M i)" 

954 
using M.sets_into_space `i \<in> I` 

955 
by (auto simp: infprod_algebra_def generator_def) 

956 
fix A assume "A \<in> sets (M i)" 

957 
have "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E _\<in>{i}. A)" 

958 
by (auto simp: infprod_algebra_def generator_def emb_def) 

959 
also have "\<dots> \<in> sets (Pi\<^isub>P I M)" 

960 
using `i \<in> I` `A \<in> sets (M i)` 

961 
by (intro emb_in_infprod_algebra product_algebraI) auto 

962 
finally show "(\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" . 

963 
qed 

964 

965 
lemma (in product_prob_space) sigma_product_algebra_sigma_eq: 

966 
assumes M: "\<And>i. i \<in> I \<Longrightarrow> M i = sigma (E i)" 

967 
shows "sets (Pi\<^isub>P I M) = sigma_sets (space (Pi\<^isub>P I M)) (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))" 

968 
proof  

969 
let ?E = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))" 

970 
let ?M = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i))" 

971 
{ fix i A assume "i\<in>I" "A \<in> sets (E i)" 

972 
then have "A \<in> sets (M i)" using M by auto 

973 
then have "A \<in> Pow (space (M i))" using M.sets_into_space by auto 

974 
then have "A \<in> Pow (space (E i))" using M[OF `i \<in> I`] by auto } 

975 
moreover 

976 
have "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) \<in> space infprod_algebra \<rightarrow> space (E i)" 

977 
by (auto simp: M infprod_algebra_def generator_def Pi_iff) 

978 
ultimately have "sigma_sets (space (Pi\<^isub>P I M)) ?M \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?E" 

979 
apply (intro sigma_sets_mono UN_least) 

980 
apply (simp add: sets_sigma M) 

981 
apply (subst sigma_sets_vimage[symmetric]) 

982 
apply (auto intro!: sigma_sets_mono') 

983 
done 

984 
moreover have "sigma_sets (space (Pi\<^isub>P I M)) ?E \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?M" 

985 
by (intro sigma_sets_mono') (auto simp: M) 

986 
ultimately show ?thesis 

987 
by (subst infprod_algebra_alt2) (auto simp: sets_sigma) 

988 
qed 

989 

990 
lemma (in product_prob_space) Int_proj_eq_emb: 

991 
assumes "J \<noteq> {}" "J \<subseteq> I" 

992 
shows "(\<Inter>i\<in>J. (\<lambda>x. x i) ` A i \<inter> space (Pi\<^isub>P I M)) = emb I J (Pi\<^isub>E J A)" 

993 
using assms by (auto simp: infprod_algebra_def generator_def emb_def Pi_iff) 

994 

995 
lemma (in product_prob_space) emb_insert: 

996 
"i \<notin> J \<Longrightarrow> emb I J (Pi\<^isub>E J f) \<inter> ((\<lambda>x. x i) ` A \<inter> space (Pi\<^isub>P I M)) = 

997 
emb I (insert i J) (Pi\<^isub>E (insert i J) (f(i := A)))" 

998 
by (auto simp: emb_def Pi_iff infprod_algebra_def generator_def split: split_if_asm) 

999 

42257
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prove measurable_into_infprod_algebra and measure_infprod
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diff
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1000 
subsection {* Sequence space *} 
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parents:
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diff
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1001 

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parents:
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diff
changeset

1002 
locale sequence_space = product_prob_space M "UNIV :: nat set" for M 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1003 

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prove measurable_into_infprod_algebra and measure_infprod
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diff
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1004 
lemma (in sequence_space) infprod_in_sets[intro]: 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1005 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)" 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1006 
shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)" 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1007 
proof  
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prove measurable_into_infprod_algebra and measure_infprod
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diff
changeset

1008 
have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))" 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1009 
using E E[THEN M.sets_into_space] 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1010 
by (auto simp: emb_def Pi_iff extensional_def) blast 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1011 
with E show ?thesis 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1012 
by (auto intro: emb_in_infprod_algebra) 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1013 
qed 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1014 

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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1015 
lemma (in sequence_space) measure_infprod: 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1016 
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)" 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1017 
shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) > \<mu>' (Pi UNIV E)" 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1018 
proof  
46731  1019 
let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^isub>E {.. n} E)" 
42257
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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changeset

1020 
{ fix n :: nat 
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prove measurable_into_infprod_algebra and measure_infprod
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diff
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1021 
interpret n: finite_product_prob_space M "{..n}" by default auto 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1022 
have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)" 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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1023 
using E by (subst n.finite_measure_times) auto 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
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1024 
also have "\<dots> = \<mu>' (?E n)" 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1025 
using E by (intro finite_measure_infprod_emb[symmetric]) auto 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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1026 
finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . } 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1027 
moreover have "Pi UNIV E = (\<Inter>n. ?E n)" 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
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diff
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1028 
using E E[THEN M.sets_into_space] 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1029 
by (auto simp: emb_def extensional_def Pi_iff) blast 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1030 
moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)" 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1031 
using E by auto 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1032 
moreover have "decseq ?E" 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
changeset

1033 
by (auto simp: emb_def Pi_iff decseq_def) 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1034 
ultimately show ?thesis 
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prove measurable_into_infprod_algebra and measure_infprod
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parents:
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diff
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1035 
by (simp add: finite_continuity_from_above) 
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prove measurable_into_infprod_algebra and measure_infprod
hoelzl
parents:
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diff
changeset

1036 
qed 
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hoelzl
parents:
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diff
changeset

1037 

42147  1038 
end 