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author | immler |

Fri, 16 Nov 2012 11:34:34 +0100 | |

changeset 50095 | 94d7dfa9f404 |

parent 50094 | 84ddcf5364b4 |

child 50096 | 7c9c5b1b6cd7 |

child 50102 | 5e01e32dadbe |

renamed to more appropriate lim_P for projective limit

--- a/src/HOL/Probability/Infinite_Product_Measure.thy Fri Nov 16 11:22:22 2012 +0100 +++ b/src/HOL/Probability/Infinite_Product_Measure.thy Fri Nov 16 11:34:34 2012 +0100 @@ -96,7 +96,7 @@ using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def) ultimately have "0 < ?a" by auto - have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = emeasure (PiP J M (\<lambda>J. (Pi\<^isub>M J M))) X" + have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = emeasure (limP J M (\<lambda>J. (Pi\<^isub>M J M))) X" using A by (intro allI generator_Ex) auto 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)" and A': "\<And>n. A n = emb I (J' n) (X' n)"

--- a/src/HOL/Probability/Projective_Family.thy Fri Nov 16 11:22:22 2012 +0100 +++ b/src/HOL/Probability/Projective_Family.thy Fri Nov 16 11:34:34 2012 +0100 @@ -57,22 +57,24 @@ (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X) definition - PiP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where - "PiP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) + limP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where + "limP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))" -lemma space_PiP[simp]: "space (PiP I M P) = space (PiM I M)" - by (auto simp add: PiP_def space_PiM prod_emb_def intro!: space_extend_measure) +abbreviation "lim\<^isub>P \<equiv> limP" + +lemma space_limP[simp]: "space (limP I M P) = space (PiM I M)" + by (auto simp add: limP_def space_PiM prod_emb_def intro!: space_extend_measure) -lemma sets_PiP[simp]: "sets (PiP I M P) = sets (PiM I M)" - by (auto simp add: PiP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure) +lemma sets_limP[simp]: "sets (limP I M P) = sets (PiM I M)" + by (auto simp add: limP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure) -lemma measurable_PiP1[simp]: "measurable (PiP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'" +lemma measurable_limP1[simp]: "measurable (limP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'" unfolding measurable_def by auto -lemma measurable_PiP2[simp]: "measurable M' (PiP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)" +lemma measurable_limP2[simp]: "measurable M' (limP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)" unfolding measurable_def by auto locale projective_family = @@ -84,11 +86,11 @@ assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)" begin -lemma emeasure_PiP: +lemma emeasure_limP: assumes "finite J" assumes "J \<subseteq> I" assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)" - shows "emeasure (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)" + shows "emeasure (limP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)" proof - have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" proof safe @@ -97,13 +99,13 @@ also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto finally show "x j \<in> space (M j)" . qed - hence "emeasure (PiP J M P) (Pi\<^isub>E J A) = - emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))" + hence "emeasure (limP J M P) (Pi\<^isub>E J A) = + emeasure (limP J M P) (prod_emb J M J (Pi\<^isub>E J A))" using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def) also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)" - proof (rule emeasure_extend_measure_Pair[OF PiP_def]) - show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto - show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def + proof (rule emeasure_extend_measure_Pair[OF limP_def]) + show "positive (sets (limP J M P)) (P J)" unfolding positive_def by auto + show "countably_additive (sets (limP J M P)) (P J)" unfolding countably_additive_def by (auto simp: suminf_emeasure proj_sets[OF `finite J`]) show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))" using assms by auto @@ -121,10 +123,10 @@ finally show ?thesis . qed -lemma PiP_finite: +lemma limP_finite: assumes "finite J" assumes "J \<subseteq> I" - shows "PiP J M P = P J" (is "?P = _") + shows "limP J M P = P J" (is "?P = _") proof (rule measure_eqI_generator_eq) let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}" let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)" @@ -132,26 +134,26 @@ show "Int_stable ?J" by (rule Int_stable_PiE) interpret prob_space "P J" using prob_space `finite J` by simp - show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_PiP) + show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP) show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space) - show "sets (PiP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J" + show "sets (limP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J" using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff) fix X assume "X \<in> ?J" then obtain E where X: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto - with `finite J` have "X \<in> sets (PiP J M P)" by simp + with `finite J` have "X \<in> sets (limP J M P)" by simp have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E" using E sets_into_space by (auto intro!: prod_emb_PiE_same_index) - show "emeasure (PiP J M P) X = emeasure (P J) X" + show "emeasure (limP J M P) X = emeasure (P J) X" unfolding X using E - by (intro emeasure_PiP assms) simp + by (intro emeasure_limP assms) simp qed (insert `finite J`, auto intro!: prod_algebraI_finite) lemma emeasure_fun_emb[simp]: assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)" - shows "emeasure (PiP L M P) (prod_emb L M J X) = emeasure (PiP J M P) X" + shows "emeasure (limP L M P) (prod_emb L M J X) = emeasure (limP J M P) X" using assms - by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective) + by (subst limP_finite) (auto simp: limP_finite finite_subset projective) lemma prod_emb_injective: 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)" @@ -235,30 +237,30 @@ definition "\<mu>G A = - (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (PiP J M P) X))" + (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (limP J M P) X))" lemma \<mu>G_spec: assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" - shows "\<mu>G A = emeasure (PiP J M P) X" + shows "\<mu>G A = emeasure (limP J M P) X" unfolding \<mu>G_def proof (intro the_equality allI impI ballI) 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)" - have "emeasure (PiP K M P) Y = emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) K Y)" + have "emeasure (limP K M P) Y = emeasure (limP (K \<union> J) M P) (emb (K \<union> J) K Y)" using K J by simp also have "emb (K \<union> J) K Y = emb (K \<union> J) J X" using K J by (simp add: prod_emb_injective[of "K \<union> J" I]) - also have "emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (PiP J M P) X" + also have "emeasure (limP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (limP J M P) X" using K J by simp - finally show "emeasure (PiP J M P) X = emeasure (PiP K M P) Y" .. + finally show "emeasure (limP J M P) X = emeasure (limP K M P) Y" .. qed (insert J, force) lemma \<mu>G_eq: - "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (PiP J M P) X" + "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (limP J M P) X" by (intro \<mu>G_spec) auto lemma generator_Ex: assumes *: "A \<in> generator" - shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (PiP J M P) X" + shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (limP J M P) X" proof - 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)" unfolding generator_def by auto @@ -267,10 +269,10 @@ lemma generatorE: assumes A: "A \<in> generator" - obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (PiP J M P) X" + obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (limP J M P) X" proof - 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" - "\<mu>G A = emeasure (PiP J M P) X" by auto + "\<mu>G A = emeasure (limP J M P) X" by auto then show thesis by (intro that) auto qed @@ -334,7 +336,7 @@ using J K by simp_all 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))" by simp - also have "\<dots> = emeasure (PiP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" + also have "\<dots> = emeasure (limP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un) also have "\<dots> = \<mu>G A + \<mu>G B" using J K JK_disj by (simp add: plus_emeasure[symmetric]) @@ -356,8 +358,8 @@ show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1) qed simp_all -lemma (in product_prob_space) PiP_PiM_finite[simp]: - assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "PiP J M (\<lambda>J. PiM J M) = PiM J M" - using assms by (simp add: PiP_finite) +lemma (in product_prob_space) limP_PiM_finite[simp]: + assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "limP J M (\<lambda>J. PiM J M) = PiM J M" + using assms by (simp add: limP_finite) end

--- a/src/HOL/Probability/Projective_Limit.thy Fri Nov 16 11:22:22 2012 +0100 +++ b/src/HOL/Probability/Projective_Limit.thy Fri Nov 16 11:34:34 2012 +0100 @@ -189,13 +189,13 @@ for I::"'i set" and P begin -abbreviation "PiB \<equiv> (\<lambda>J P. PiP J (\<lambda>_. borel) P)" +abbreviation "lim\<^isub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)" lemma - emeasure_PiB_emb_not_empty: + emeasure_limB_emb_not_empty: assumes "I \<noteq> {}" assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" - shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)" + shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)" proof - let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space borel" let ?G = generator @@ -208,7 +208,7 @@ fix A assume "A \<in> ?G" with generatorE guess J X . note JX = this interpret prob_space "P J" using prob_space[OF `finite J`] . - show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: PiP_finite) + show "\<mu>G A \<noteq> \<infinity>" using JX by (simp add: limP_finite) next fix Z assume Z: "range Z \<subseteq> ?G" "decseq Z" "(\<Inter>i. Z i) = {}" then have "decseq (\<lambda>i. \<mu>G (Z i))" @@ -222,7 +222,7 @@ ultimately have "0 < ?a" by auto hence "?a \<noteq> -\<infinity>" by auto have "\<forall>n. \<exists>J B. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> B \<in> sets (Pi\<^isub>M J (\<lambda>_. borel)) \<and> - Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (PiB J P) B" + Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^isub>B J P) B" using Z by (intro allI generator_Ex) auto then obtain J' B' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" "\<And>n. B' n \<in> sets (\<Pi>\<^isub>M i\<in>J' n. borel)" @@ -243,10 +243,10 @@ unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto) interpret prob_space "P (J i)" for i using prob_space by simp have "?a \<le> \<mu>G (Z 0)" by (auto intro: INF_lower) - also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq \<mu>G_eq PiP_finite proj_sets) + also have "\<dots> < \<infinity>" using J by (auto simp: Z_eq \<mu>G_eq limP_finite proj_sets) finally have "?a \<noteq> \<infinity>" by simp have "\<And>n. \<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" unfolding Z_eq using J J_mono - by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq) + by (subst \<mu>G_eq) (auto simp: limP_finite proj_sets \<mu>G_eq) interpret finite_set_sequence J by unfold_locales simp def Utn \<equiv> Un_to_nat @@ -380,20 +380,20 @@ (\<Inter> i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" . hence "Y n \<in> ?G" using J J_mono K_sets `n \<ge> 1` by (intro generatorI[OF _ _ _ _ Y_emb]) auto hence "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" unfolding Y_emb using J J_mono K_sets `n \<ge> 1` - by (subst \<mu>G_eq) (auto simp: PiP_finite proj_sets \<mu>G_eq) - interpret finite_measure "(PiP (J n) (\<lambda>_. borel) P)" + by (subst \<mu>G_eq) (auto simp: limP_finite proj_sets \<mu>G_eq) + interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)" proof - have "emeasure (PiP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>" - using J by (subst emeasure_PiP) auto - thus "emeasure (PiP (J n) (\<lambda>_. borel) P) (space (PiP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>" + have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^isub>E space borel) \<noteq> \<infinity>" + using J by (subst emeasure_limP) auto + thus "emeasure (limP (J n) (\<lambda>_. borel) P) (space (limP (J n) (\<lambda>_. borel) P)) \<noteq> \<infinity>" by (simp add: space_PiM) qed - have "\<mu>G (Z n) = PiP (J n) (\<lambda>_. borel) P (B n)" + have "\<mu>G (Z n) = limP (J n) (\<lambda>_. borel) P (B n)" unfolding Z_eq using J by (auto simp: \<mu>G_eq) moreover have "\<mu>G (Y n) = - PiP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" + limP (J n) (\<lambda>_. borel) P (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst \<mu>G_eq) auto - moreover have "\<mu>G (Z n - Y n) = PiP (J n) (\<lambda>_. borel) P + moreover have "\<mu>G (Z n - Y n) = limP (J n) (\<lambda>_. borel) P (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))" unfolding Z_eq Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1` by (subst \<mu>G_eq) (auto intro!: Diff) @@ -420,7 +420,7 @@ unfolding Z'_def Z_eq by simp also have "\<dots> = P (J i) (B i - K i)" apply (subst \<mu>G_eq) using J K_sets apply auto - apply (subst PiP_finite) apply auto + apply (subst limP_finite) apply auto done also have "\<dots> = P (J i) (B i) - P (J i) (K i)" apply (subst emeasure_Diff) using K_sets J `K _ \<subseteq> B _` apply (auto simp: proj_sets) @@ -593,10 +593,10 @@ qed then guess \<mu> .. note \<mu> = this def f \<equiv> "finmap_of J B" - show "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (PiB J P) (Pi\<^isub>E J B)" - proof (subst emeasure_extend_measure_Pair[OF PiP_def, of I "\<lambda>_. borel" \<mu>]) - show "positive (sets (PiB I P)) \<mu>" "countably_additive (sets (PiB I P)) \<mu>" - using \<mu> unfolding sets_PiP sets_PiM_generator by (auto simp: measure_space_def) + show "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (lim\<^isub>B J P) (Pi\<^isub>E J B)" + proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>]) + show "positive (sets (lim\<^isub>B I P)) \<mu>" "countably_additive (sets (lim\<^isub>B I P)) \<mu>" + using \<mu> unfolding sets_limP sets_PiM_generator by (auto simp: measure_space_def) next show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> B \<in> J \<rightarrow> sets borel" using assms by (auto simp: f_def) @@ -610,11 +610,11 @@ hence "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))" using \<mu> by simp also have "\<dots> = emeasure (P J) (Pi\<^isub>E J X)" using JX assms proj_sets - by (subst \<mu>G_eq) (auto simp: \<mu>G_eq PiP_finite intro: sets_PiM_I_finite) + by (subst \<mu>G_eq) (auto simp: \<mu>G_eq limP_finite intro: sets_PiM_I_finite) finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = emeasure (P J) (Pi\<^isub>E J X)" . next - show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (PiP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)" - using assms by (simp add: f_def PiP_finite Pi_def) + show "emeasure (P J) (Pi\<^isub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^isub>E J B)" + using assms by (simp add: f_def limP_finite Pi_def) qed qed @@ -631,56 +631,56 @@ hide_const (open) domain hide_const (open) enum_basis_finmap -sublocale polish_projective \<subseteq> P!: prob_space "(PiB I P)" +sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^isub>B I P)" proof - show "emeasure (PiB I P) (space (PiB I P)) = 1" + show "emeasure (lim\<^isub>B I P) (space (lim\<^isub>B I P)) = 1" proof cases assume "I = {}" interpret prob_space "P {}" using prob_space by simp show ?thesis - by (simp add: space_PiM_empty PiP_finite emeasure_space_1 `I = {}`) + by (simp add: space_PiM_empty limP_finite emeasure_space_1 `I = {}`) next assume "I \<noteq> {}" then obtain i where "i \<in> I" by auto interpret prob_space "P {i}" using prob_space by simp - have R: "(space (PiB I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))" + have R: "(space (lim\<^isub>B I P)) = (emb I {i} (Pi\<^isub>E {i} (\<lambda>_. space borel)))" by (auto simp: prod_emb_def space_PiM) moreover have "extensional {i} = space (P {i})" by (simp add: proj_space space_PiM) ultimately show ?thesis using `i \<in> I` apply (subst R) - apply (subst emeasure_PiB_emb_not_empty) - apply (auto simp: PiP_finite emeasure_space_1) + apply (subst emeasure_limB_emb_not_empty) + apply (auto simp: limP_finite emeasure_space_1) done qed qed context polish_projective begin -lemma emeasure_PiB_emb: +lemma emeasure_limB_emb: assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" - shows "emeasure (PiB I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)" + shows "emeasure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = emeasure (P J) (Pi\<^isub>E J B)" proof cases interpret prob_space "P {}" using prob_space by simp assume "J = {}" - moreover have "emb I {} {\<lambda>x. undefined} = space (PiB I P)" + moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^isub>B I P)" by (auto simp: space_PiM prod_emb_def) - moreover have "{\<lambda>x. undefined} = space (PiB {} P)" + moreover have "{\<lambda>x. undefined} = space (lim\<^isub>B {} P)" by (auto simp: space_PiM prod_emb_def) ultimately show ?thesis - by (simp add: P.emeasure_space_1 PiP_finite emeasure_space_1 del: space_PiP) + by (simp add: P.emeasure_space_1 limP_finite emeasure_space_1 del: space_limP) next assume "J \<noteq> {}" with X show ?thesis - by (subst emeasure_PiB_emb_not_empty) (auto simp: PiP_finite) + by (subst emeasure_limB_emb_not_empty) (auto simp: limP_finite) qed -lemma measure_PiB_emb: +lemma measure_limB_emb: assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel" - shows "measure (PiB I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)" + shows "measure (lim\<^isub>B I P) (emb I J (Pi\<^isub>E J B)) = measure (P J) (Pi\<^isub>E J B)" proof - interpret prob_space "P J" using prob_space assms by simp show ?thesis - using emeasure_PiB_emb[OF assms] - unfolding emeasure_eq_measure PiP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure + using emeasure_limB_emb[OF assms] + unfolding emeasure_eq_measure limP_finite[OF `finite J` `J \<subseteq> I`] P.emeasure_eq_measure by simp qed @@ -693,9 +693,9 @@ proof qed lemma (in polish_product_prob_space) - PiP_eq_PiM: - "I \<noteq> {} \<Longrightarrow> PiP I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) = + limP_eq_PiM: + "I \<noteq> {} \<Longrightarrow> lim\<^isub>P I (\<lambda>_. borel) (\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)) = PiM I (\<lambda>_. borel)" - by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_PiB_emb) + by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_limB_emb) end