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(* Title: HOL/Probability/Infinite_Product_Measure.thy
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Author: Johannes Hölzl, TU München
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*)
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header {*Infinite Product Measure*}
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theory Infinite_Product_Measure
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imports Probability_Measure
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42147
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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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locale product_prob_space =
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fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set"
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assumes prob_spaces: "\<And>i. prob_space (M i)"
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and I_not_empty: "I \<noteq> {}"
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locale finite_product_prob_space = product_prob_space M I
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for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set" +
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assumes finite_index'[intro]: "finite I"
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sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
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by (rule prob_spaces)
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sublocale product_prob_space \<subseteq> product_sigma_finite
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by default
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sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite
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by default (fact finite_index')
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sublocale finite_product_prob_space \<subseteq> prob_space "Pi\<^isub>M I M"
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proof
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show "measure P (space P) = 1"
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by (simp add: measure_times measure_space_1 setprod_1)
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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 -
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interpret K: finite_product_prob_space M K
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by default (insert assms, auto)
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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 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"
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unfolding generator_def by auto
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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)
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and measure_fold_measurable:
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"(\<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)
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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)
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apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure
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intro!: I.positive_integral_cong)
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done
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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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lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
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unfolding measure_space_1[symmetric]
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using sets_into_space
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by (intro measure_mono) auto
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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:
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"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = 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"
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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"
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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
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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)"
|
|
274 |
shows "merge J x K -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
|
|
275 |
proof -
|
|
276 |
interpret J: finite_product_sigma_algebra M J by default fact
|
|
277 |
interpret K: finite_product_sigma_algebra M K by default fact
|
|
278 |
interpret JK: pair_sigma_algebra J.P K.P ..
|
|
279 |
|
|
280 |
from JK.measurable_cut_fst[OF
|
|
281 |
measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x
|
|
282 |
show ?thesis
|
|
283 |
by (simp add: space_pair_measure comp_def vimage_compose[symmetric])
|
|
284 |
qed
|
|
285 |
|
|
286 |
lemma (in product_prob_space) merge_emb:
|
|
287 |
assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
|
|
288 |
shows "(merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
|
|
289 |
emb I (K - J) (merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
|
|
290 |
proof -
|
|
291 |
have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x"
|
|
292 |
by (auto simp: restrict_def merge_def)
|
|
293 |
have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x"
|
|
294 |
by (auto simp: restrict_def merge_def)
|
|
295 |
have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
|
|
296 |
have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
|
|
297 |
have [simp]: "(K - J) \<inter> K = K - J" by auto
|
|
298 |
from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
|
|
299 |
by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto
|
|
300 |
qed
|
|
301 |
|
|
302 |
definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where
|
|
303 |
"infprod_algebra = sigma generator \<lparr> measure :=
|
|
304 |
(SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
|
|
305 |
measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>"
|
|
306 |
|
|
307 |
syntax
|
|
308 |
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3PIP _:_./ _)" 10)
|
|
309 |
|
|
310 |
syntax (xsymbols)
|
|
311 |
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10)
|
|
312 |
|
|
313 |
syntax (HTML output)
|
|
314 |
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10)
|
|
315 |
|
|
316 |
abbreviation
|
|
317 |
"Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I"
|
|
318 |
|
|
319 |
translations
|
|
320 |
"PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)"
|
|
321 |
|
|
322 |
sublocale product_prob_space \<subseteq> G: algebra generator
|
|
323 |
proof
|
|
324 |
let ?G = generator
|
|
325 |
show "sets ?G \<subseteq> Pow (space ?G)"
|
|
326 |
by (auto simp: generator_def emb_def)
|
|
327 |
from I_not_empty
|
|
328 |
obtain i where "i \<in> I" by auto
|
|
329 |
then show "{} \<in> sets ?G"
|
|
330 |
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
|
|
331 |
simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def)
|
|
332 |
from `i \<in> I` show "space ?G \<in> sets ?G"
|
|
333 |
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
|
|
334 |
simp: generator_def emb_def)
|
|
335 |
fix A assume "A \<in> sets ?G"
|
|
336 |
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"
|
|
337 |
by (auto simp: generator_def)
|
|
338 |
fix B assume "B \<in> sets ?G"
|
|
339 |
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"
|
|
340 |
by (auto simp: generator_def)
|
|
341 |
let ?RA = "emb (JA \<union> JB) JA XA"
|
|
342 |
let ?RB = "emb (JA \<union> JB) JB XB"
|
|
343 |
interpret JAB: finite_product_sigma_algebra M "JA \<union> JB"
|
|
344 |
by default (insert XA XB, auto)
|
|
345 |
have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
|
|
346 |
using XA A XB B by (auto simp: emb_simps)
|
|
347 |
then show "A - B \<in> sets ?G" "A \<union> B \<in> sets ?G"
|
|
348 |
using XA XB by (auto intro!: generatorI')
|
|
349 |
qed
|
|
350 |
|
|
351 |
lemma (in product_prob_space) positive_\<mu>G: "positive generator \<mu>G"
|
|
352 |
proof (intro positive_def[THEN iffD2] conjI ballI)
|
|
353 |
from generatorE[OF G.empty_sets] guess J X . note this[simp]
|
|
354 |
interpret J: finite_product_sigma_finite M J by default fact
|
|
355 |
have "X = {}"
|
|
356 |
by (rule emb_injective[of J I]) simp_all
|
|
357 |
then show "\<mu>G {} = 0" by simp
|
|
358 |
next
|
|
359 |
fix A assume "A \<in> sets generator"
|
|
360 |
from generatorE[OF this] guess J X . note this[simp]
|
|
361 |
interpret J: finite_product_sigma_finite M J by default fact
|
|
362 |
show "0 \<le> \<mu>G A" by simp
|
|
363 |
qed
|
|
364 |
|
|
365 |
lemma (in product_prob_space) additive_\<mu>G: "additive generator \<mu>G"
|
|
366 |
proof (intro additive_def[THEN iffD2] ballI impI)
|
|
367 |
fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this
|
|
368 |
fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this
|
|
369 |
assume "A \<inter> B = {}"
|
|
370 |
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
|
|
371 |
using J K by auto
|
|
372 |
interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact
|
|
373 |
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
|
|
374 |
apply (rule emb_injective[of "J \<union> K" I])
|
|
375 |
apply (insert `A \<inter> B = {}` JK J K)
|
|
376 |
apply (simp_all add: JK.Int emb_simps)
|
|
377 |
done
|
|
378 |
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)"
|
|
379 |
using J K by simp_all
|
|
380 |
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))"
|
|
381 |
by (simp add: emb_simps)
|
|
382 |
also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
|
|
383 |
using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un)
|
|
384 |
also have "\<dots> = \<mu>G A + \<mu>G B"
|
|
385 |
using J K JK_disj by (simp add: JK.measure_additive[symmetric])
|
|
386 |
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
|
|
387 |
qed
|
|
388 |
|
|
389 |
lemma (in product_prob_space) finite_index_eq_finite_product:
|
|
390 |
assumes "finite I"
|
|
391 |
shows "sets (sigma generator) = sets (Pi\<^isub>M I M)"
|
|
392 |
proof safe
|
|
393 |
interpret I: finite_product_sigma_algebra M I by default fact
|
|
394 |
have [simp]: "space generator = space (Pi\<^isub>M I M)"
|
|
395 |
by (simp add: generator_def product_algebra_def)
|
|
396 |
{ fix A assume "A \<in> sets (sigma generator)"
|
|
397 |
then show "A \<in> sets I.P" unfolding sets_sigma
|
|
398 |
proof induct
|
|
399 |
case (Basic A)
|
|
400 |
from generatorE[OF this] guess J X . note J = this
|
|
401 |
with `finite I` have "emb I J X \<in> sets I.P" by auto
|
|
402 |
with `emb I J X = A` show "A \<in> sets I.P" by simp
|
|
403 |
qed auto }
|
|
404 |
{ fix A assume "A \<in> sets I.P"
|
|
405 |
moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto
|
|
406 |
ultimately show "A \<in> sets (sigma generator)"
|
|
407 |
using `finite I` I_not_empty unfolding sets_sigma
|
|
408 |
by (intro sigma_sets.Basic generatorI[of I A]) auto }
|
|
409 |
qed
|
|
410 |
|
|
411 |
lemma (in product_prob_space) extend_\<mu>G:
|
|
412 |
"\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
|
|
413 |
measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
|
|
414 |
proof cases
|
|
415 |
assume "finite I"
|
|
416 |
interpret I: finite_product_sigma_finite M I by default fact
|
|
417 |
show ?thesis
|
|
418 |
proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI)
|
|
419 |
fix A assume "A \<in> sets generator"
|
|
420 |
from generatorE[OF this] guess J X . note J = this
|
|
421 |
from J(1-4) `finite I` show "measure I.P A = \<mu>G A"
|
|
422 |
unfolding J(6)
|
|
423 |
by (subst J(5)[symmetric]) (simp add: measure_emb)
|
|
424 |
next
|
|
425 |
have [simp]: "space generator = space (Pi\<^isub>M I M)"
|
|
426 |
by (simp add: generator_def product_algebra_def)
|
|
427 |
have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr>
|
|
428 |
= I.P" (is "?P = _")
|
|
429 |
by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`])
|
|
430 |
then show "measure_space ?P" by simp default
|
|
431 |
qed
|
|
432 |
next
|
|
433 |
let ?G = generator
|
|
434 |
assume "\<not> finite I"
|
|
435 |
note \<mu>G_mono =
|
|
436 |
G.additive_increasing[OF positive_\<mu>G additive_\<mu>G, THEN increasingD]
|
|
437 |
|
|
438 |
{ fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G"
|
|
439 |
|
|
440 |
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
|
|
441 |
by (metis rev_finite_subset subsetI)
|
|
442 |
moreover from Z guess K' X' by (rule generatorE)
|
|
443 |
moreover def K \<equiv> "insert k K'"
|
|
444 |
moreover def X \<equiv> "emb K K' X'"
|
|
445 |
ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
|
|
446 |
"K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X"
|
|
447 |
by (auto simp: subset_insertI)
|
|
448 |
|
|
449 |
let "?M y" = "merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
|
|
450 |
{ fix y assume y: "y \<in> space (Pi\<^isub>M J M)"
|
|
451 |
note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
|
|
452 |
moreover
|
|
453 |
have **: "?M y \<in> sets (Pi\<^isub>M (K - J) M)"
|
|
454 |
using J K y by (intro merge_sets) auto
|
|
455 |
ultimately
|
|
456 |
have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G"
|
|
457 |
using J K by (intro generatorI) auto
|
|
458 |
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)"
|
|
459 |
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto
|
|
460 |
note * ** *** this }
|
|
461 |
note merge_in_G = this
|
|
462 |
|
|
463 |
have "finite (K - J)" using K by auto
|
|
464 |
|
|
465 |
interpret J: finite_product_prob_space M J by default fact+
|
|
466 |
interpret KmJ: finite_product_prob_space M "K - J" by default fact+
|
|
467 |
|
|
468 |
have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
|
|
469 |
using K J by simp
|
|
470 |
also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
|
|
471 |
using K J by (subst measure_fold_integral) auto
|
|
472 |
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)"
|
|
473 |
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)")
|
|
474 |
proof (intro J.positive_integral_cong)
|
|
475 |
fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
|
|
476 |
with K merge_in_G(2)[OF this]
|
|
477 |
show "measure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
|
|
478 |
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto
|
|
479 |
qed
|
|
480 |
finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" .
|
|
481 |
|
|
482 |
{ fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
|
|
483 |
then have "\<mu>G (?MZ x) \<le> 1"
|
|
484 |
unfolding merge_in_G(4)[OF x] `Z = emb I K X`
|
|
485 |
by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) }
|
|
486 |
note le_1 = this
|
|
487 |
|
|
488 |
let "?q y" = "\<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M))"
|
|
489 |
have "?q \<in> borel_measurable (Pi\<^isub>M J M)"
|
|
490 |
unfolding `Z = emb I K X` using J K merge_in_G(3)
|
|
491 |
by (simp add: merge_in_G \<mu>G_eq measure_fold_measurable
|
|
492 |
del: space_product_algebra cong: measurable_cong)
|
|
493 |
note this fold le_1 merge_in_G(3) }
|
|
494 |
note fold = this
|
|
495 |
|
|
496 |
show ?thesis
|
|
497 |
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
|
|
498 |
fix A assume "A \<in> sets ?G"
|
|
499 |
with generatorE guess J X . note JX = this
|
|
500 |
interpret JK: finite_product_prob_space M J by default fact+
|
|
501 |
with JX show "\<mu>G A \<noteq> \<infinity>" by simp
|
|
502 |
next
|
|
503 |
fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}"
|
|
504 |
then have "decseq (\<lambda>i. \<mu>G (A i))"
|
|
505 |
by (auto intro!: \<mu>G_mono simp: decseq_def)
|
|
506 |
moreover
|
|
507 |
have "(INF i. \<mu>G (A i)) = 0"
|
|
508 |
proof (rule ccontr)
|
|
509 |
assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0")
|
|
510 |
moreover have "0 \<le> ?a"
|
|
511 |
using A positive_\<mu>G by (auto intro!: le_INFI simp: positive_def)
|
|
512 |
ultimately have "0 < ?a" by auto
|
|
513 |
|
|
514 |
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"
|
|
515 |
using A by (intro allI generator_Ex) auto
|
|
516 |
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)"
|
|
517 |
and A': "\<And>n. A n = emb I (J' n) (X' n)"
|
|
518 |
unfolding choice_iff by blast
|
|
519 |
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
|
|
520 |
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
|
|
521 |
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)"
|
|
522 |
by auto
|
|
523 |
with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G"
|
|
524 |
unfolding J_def X_def by (subst emb_trans) (insert A, auto)
|
|
525 |
|
|
526 |
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
|
|
527 |
unfolding J_def by force
|
|
528 |
|
|
529 |
interpret J: finite_product_prob_space M "J i" for i by default fact+
|
|
530 |
|
|
531 |
have a_le_1: "?a \<le> 1"
|
|
532 |
using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq
|
|
533 |
by (auto intro!: INF_leI2[of 0] J.measure_le_1)
|
|
534 |
|
|
535 |
let "?M K Z y" = "merge K y (I - K) -` Z \<inter> space (Pi\<^isub>M I M)"
|
|
536 |
|
|
537 |
{ fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
|
|
538 |
then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto
|
|
539 |
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
|
|
540 |
interpret J': finite_product_prob_space M J' by default fact+
|
|
541 |
|
|
542 |
let "?q n y" = "\<mu>G (?M J' (Z n) y)"
|
|
543 |
let "?Q n" = "?q n -` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)"
|
|
544 |
{ fix n
|
|
545 |
have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)"
|
|
546 |
using Z J' by (intro fold(1)) auto
|
|
547 |
then have "?Q n \<in> sets (Pi\<^isub>M J' M)"
|
|
548 |
by (rule measurable_sets) auto }
|
|
549 |
note Q_sets = this
|
|
550 |
|
|
551 |
have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))"
|
|
552 |
proof (intro le_INFI)
|
|
553 |
fix n
|
|
554 |
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
|
|
555 |
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)"
|
|
556 |
unfolding fold(2)[OF J' `Z n \<in> sets ?G`]
|
|
557 |
proof (intro J'.positive_integral_mono)
|
|
558 |
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
|
|
559 |
then have "?q n x \<le> 1 + 0"
|
|
560 |
using J' Z fold(3) Z_sets by auto
|
|
561 |
also have "\<dots> \<le> 1 + ?a / 2^(k+1)"
|
|
562 |
using `0 < ?a` by (intro add_mono) auto
|
|
563 |
finally have "?q n x \<le> 1 + ?a / 2^(k+1)" .
|
|
564 |
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
|
|
565 |
by (auto split: split_indicator simp del: power_Suc)
|
|
566 |
qed
|
|
567 |
also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
|
|
568 |
using `0 \<le> ?a` Q_sets J'.measure_space_1
|
|
569 |
by (subst J'.positive_integral_add) auto
|
|
570 |
finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
|
|
571 |
by (cases rule: extreal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"])
|
|
572 |
(auto simp: field_simps)
|
|
573 |
qed
|
|
574 |
also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
|
|
575 |
proof (intro J'.continuity_from_above)
|
|
576 |
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
|
|
577 |
show "decseq ?Q"
|
|
578 |
unfolding decseq_def
|
|
579 |
proof (safe intro!: vimageI[OF refl])
|
|
580 |
fix m n :: nat assume "m \<le> n"
|
|
581 |
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
|
|
582 |
assume "?a / 2^(k+1) \<le> ?q n x"
|
|
583 |
also have "?q n x \<le> ?q m x"
|
|
584 |
proof (rule \<mu>G_mono)
|
|
585 |
from fold(4)[OF J', OF Z_sets x]
|
|
586 |
show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto
|
|
587 |
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
|
|
588 |
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
|
|
589 |
qed
|
|
590 |
finally show "?a / 2^(k+1) \<le> ?q m x" .
|
|
591 |
qed
|
|
592 |
qed (intro J'.finite_measure Q_sets)
|
|
593 |
finally have "(\<Inter>n. ?Q n) \<noteq> {}"
|
|
594 |
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
|
|
595 |
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
|
|
596 |
note Ex_w = this
|
|
597 |
|
|
598 |
let "?q k n y" = "\<mu>G (?M (J k) (A n) y)"
|
|
599 |
|
|
600 |
have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_leI)
|
|
601 |
from Ex_w[OF A(1,2) this J(1-3), of 0] guess w0 .. note w0 = this
|
|
602 |
|
|
603 |
let "?P k wk w" =
|
|
604 |
"w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and> (\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)"
|
|
605 |
def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))"
|
|
606 |
|
|
607 |
{ fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and>
|
|
608 |
(\<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))"
|
|
609 |
proof (induct k)
|
|
610 |
case 0 with w0 show ?case
|
|
611 |
unfolding w_def nat_rec_0 by auto
|
|
612 |
next
|
|
613 |
case (Suc k)
|
|
614 |
then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
|
|
615 |
have "\<exists>w'. ?P k (w k) w'"
|
|
616 |
proof cases
|
|
617 |
assume [simp]: "J k = J (Suc k)"
|
|
618 |
show ?thesis
|
|
619 |
proof (intro exI[of _ "w k"] conjI allI)
|
|
620 |
fix n
|
|
621 |
have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)"
|
|
622 |
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps)
|
|
623 |
also have "\<dots> \<le> ?q k n (w k)" using Suc by auto
|
|
624 |
finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp
|
|
625 |
next
|
|
626 |
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)"
|
|
627 |
using Suc by simp
|
|
628 |
then show "restrict (w k) (J k) = w k"
|
|
629 |
by (simp add: extensional_restrict)
|
|
630 |
qed
|
|
631 |
next
|
|
632 |
assume "J k \<noteq> J (Suc k)"
|
|
633 |
with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
|
|
634 |
have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G"
|
|
635 |
"decseq (\<lambda>n. ?M (J k) (A n) (w k))"
|
|
636 |
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))"
|
|
637 |
using `decseq A` fold(4)[OF J(1-3) A_eq(2), of "w k" k] Suc
|
|
638 |
by (auto simp: decseq_def)
|
|
639 |
from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"]
|
|
640 |
obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)"
|
|
641 |
"\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto
|
|
642 |
let ?w = "merge (J k) (w k) ?D w'"
|
|
643 |
have [simp]: "\<And>x. merge (J k) (w k) (I - J k) (merge ?D w' (I - ?D) x) =
|
|
644 |
merge (J (Suc k)) ?w (I - (J (Suc k))) x"
|
|
645 |
using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"]
|
|
646 |
by (auto intro!: ext split: split_merge)
|
|
647 |
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w"
|
|
648 |
using w'(1) J(3)[of "Suc k"]
|
|
649 |
by (auto split: split_merge intro!: extensional_merge_sub) force+
|
|
650 |
show ?thesis
|
|
651 |
apply (rule exI[of _ ?w])
|
|
652 |
using w' J_mono[of k "Suc k"] wk unfolding *
|
|
653 |
apply (auto split: split_merge intro!: extensional_merge_sub ext)
|
|
654 |
apply (force simp: extensional_def)
|
|
655 |
done
|
|
656 |
qed
|
|
657 |
then have "?P k (w k) (w (Suc k))"
|
|
658 |
unfolding w_def nat_rec_Suc unfolding w_def[symmetric]
|
|
659 |
by (rule someI_ex)
|
|
660 |
then show ?case by auto
|
|
661 |
qed
|
|
662 |
moreover
|
|
663 |
then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
|
|
664 |
moreover
|
|
665 |
from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto
|
|
666 |
then have "?M (J k) (A k) (w k) \<noteq> {}"
|
|
667 |
using positive_\<mu>G[unfolded positive_def] `0 < ?a` `?a \<le> 1`
|
|
668 |
by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
|
|
669 |
then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto
|
|
670 |
then have "merge (J k) (w k) (I - J k) x \<in> A k" by auto
|
|
671 |
then have "\<exists>x\<in>A k. restrict x (J k) = w k"
|
|
672 |
using `w k \<in> space (Pi\<^isub>M (J k) M)`
|
|
673 |
by (intro rev_bexI) (auto intro!: ext simp: extensional_def)
|
|
674 |
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)"
|
|
675 |
"\<exists>x\<in>A k. restrict x (J k) = w k"
|
|
676 |
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1)"
|
|
677 |
by auto }
|
|
678 |
note w = this
|
|
679 |
|
|
680 |
{ fix k l i assume "k \<le> l" "i \<in> J k"
|
|
681 |
{ fix l have "w k i = w (k + l) i"
|
|
682 |
proof (induct l)
|
|
683 |
case (Suc l)
|
|
684 |
from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto
|
|
685 |
with w(3)[of "k + Suc l"]
|
|
686 |
have "w (k + l) i = w (k + Suc l) i"
|
|
687 |
by (auto simp: restrict_def fun_eq_iff split: split_if_asm)
|
|
688 |
with Suc show ?case by simp
|
|
689 |
qed simp }
|
|
690 |
from this[of "l - k"] `k \<le> l` have "w l i = w k i" by simp }
|
|
691 |
note w_mono = this
|
|
692 |
|
|
693 |
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"
|
|
694 |
{ fix i k assume k: "i \<in> J k"
|
|
695 |
have "w k i = w (LEAST k. i \<in> J k) i"
|
|
696 |
by (intro w_mono Least_le k LeastI[of _ k])
|
|
697 |
then have "w' i = w k i"
|
|
698 |
unfolding w'_def using k by auto }
|
|
699 |
note w'_eq = this
|
|
700 |
have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined"
|
|
701 |
using J by (auto simp: w'_def)
|
|
702 |
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
|
|
703 |
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
|
|
704 |
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
|
|
705 |
using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ }
|
|
706 |
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
|
|
707 |
|
|
708 |
have w': "w' \<in> space (Pi\<^isub>M I M)"
|
|
709 |
using w(1) by (auto simp add: Pi_iff extensional_def)
|
|
710 |
|
|
711 |
{ fix n
|
|
712 |
have "restrict w' (J n) = w n" using w(1)
|
|
713 |
by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def)
|
|
714 |
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto
|
|
715 |
then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) }
|
|
716 |
then have "w' \<in> (\<Inter>i. A i)" by auto
|
|
717 |
with `(\<Inter>i. A i) = {}` show False by auto
|
|
718 |
qed
|
|
719 |
ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
|
|
720 |
using LIMSEQ_extreal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp
|
|
721 |
qed
|
|
722 |
qed
|
|
723 |
|
|
724 |
lemma (in product_prob_space) infprod_spec:
|
|
725 |
shows "(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> measure_space (Pi\<^isub>P I M)"
|
|
726 |
proof -
|
|
727 |
let ?P = "\<lambda>\<mu>. (\<forall>A\<in>sets generator. \<mu> A = \<mu>G A) \<and>
|
|
728 |
measure_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
|
|
729 |
have **: "measure infprod_algebra = (SOME \<mu>. ?P \<mu>)"
|
|
730 |
unfolding infprod_algebra_def by simp
|
|
731 |
have *: "Pi\<^isub>P I M = \<lparr>space = space generator, sets = sets (sigma generator), measure = measure (Pi\<^isub>P I M)\<rparr>"
|
|
732 |
unfolding infprod_algebra_def by auto
|
|
733 |
show ?thesis
|
|
734 |
apply (subst (2) *)
|
|
735 |
apply (unfold **)
|
|
736 |
apply (rule someI_ex[where P="?P"])
|
|
737 |
apply (rule extend_\<mu>G)
|
|
738 |
done
|
|
739 |
qed
|
|
740 |
|
|
741 |
sublocale product_prob_space \<subseteq> measure_space "Pi\<^isub>P I M"
|
|
742 |
using infprod_spec by auto
|
|
743 |
|
|
744 |
lemma (in product_prob_space) measure_infprod_emb:
|
|
745 |
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
|
|
746 |
shows "measure (Pi\<^isub>P I M) (emb I J X) = measure (Pi\<^isub>M J M) X"
|
|
747 |
proof -
|
|
748 |
have "emb I J X \<in> sets generator"
|
|
749 |
using assms by (rule generatorI')
|
|
750 |
with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto
|
|
751 |
qed
|
|
752 |
|
|
753 |
end |