--- a/src/HOL/Probability/Projective_Limit.thy Wed Oct 07 15:31:59 2015 +0200
+++ b/src/HOL/Probability/Projective_Limit.thy Wed Oct 07 17:11:16 2015 +0200
@@ -101,440 +101,361 @@
for I::"'i set" and P
begin
-abbreviation "lim\<^sub>B \<equiv> (\<lambda>J P. limP J (\<lambda>_. borel) P)"
+lemma emeasure_lim_emb:
+ assumes X: "J \<subseteq> I" "finite J" "X \<in> sets (\<Pi>\<^sub>M i\<in>J. borel)"
+ shows "lim (emb I J X) = P J X"
+proof (rule emeasure_lim)
+ write mu_G ("\<mu>G")
+ interpret generator: algebra "space (PiM I (\<lambda>i. borel))" generator
+ by (rule algebra_generator)
+
+ fix J and B :: "nat \<Rightarrow> ('i \<Rightarrow> 'a) set"
+ assume J: "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)" "incseq J"
+ and B: "decseq (\<lambda>n. emb I (J n) (B n))"
+ and "0 < (INF i. P (J i) (B i))" (is "0 < ?a")
+ moreover have "?a \<le> 1"
+ using J by (auto intro!: INF_lower2[of 0] prob_space_P[THEN prob_space.measure_le_1])
+ ultimately obtain r where r: "?a = ereal r" "0 < r" "r \<le> 1"
+ by (cases "?a") auto
+ def Z \<equiv> "\<lambda>n. emb I (J n) (B n)"
+ have Z_mono: "n \<le> m \<Longrightarrow> Z m \<subseteq> Z n" for n m
+ unfolding Z_def using B[THEN antimonoD, of n m] .
+ have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
+ using \<open>incseq J\<close> by (force simp: incseq_def)
+ note [simp] = \<open>\<And>n. finite (J n)\<close>
+ interpret prob_space "P (J i)" for i using J prob_space_P by simp
+
+ have P_eq[simp]:
+ "sets (P (J i)) = sets (\<Pi>\<^sub>M i\<in>J i. borel)" "space (P (J i)) = space (\<Pi>\<^sub>M i\<in>J i. borel)" for i
+ using J by (auto simp: sets_P space_P)
+
+ have "Z i \<in> generator" for i
+ unfolding Z_def by (auto intro!: generator.intros J)
-lemma 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 (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (lim\<^sub>B J P) (Pi\<^sub>E J B)"
-proof -
- let ?\<Omega> = "\<Pi>\<^sub>E i\<in>I. space borel"
- let ?G = generator
- interpret G!: algebra ?\<Omega> generator by (intro algebra_generator) fact
- note mu_G_mono =
- G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`],
- THEN increasingD]
- write mu_G ("\<mu>G")
-
- have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
- proof (rule G.caratheodory_empty_continuous[OF positive_mu_G additive_mu_G,
- OF `I \<noteq> {}`, OF `I \<noteq> {}`])
- fix A assume "A \<in> ?G"
- with generatorE guess J X . note JX = this
- interpret prob_space "P J" using proj_prob_space[OF `finite J`] .
- 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))"
- by (auto intro!: mu_G_mono simp: decseq_def)
- moreover
- have "(INF i. \<mu>G (Z i)) = 0"
+ have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
+ def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
+ interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n
+ by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J])
+ have inj_on_Utn: "inj_on Utn (\<Union>n. J n)"
+ unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
+ hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
+ def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
+ interpret P': prob_space "P' n" for n
+ unfolding P'_def mapmeasure_def using J
+ by (auto intro!: prob_space_distr fm_measurable simp: measurable_cong_sets[OF sets_P])
+
+ let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
+ { fix n
+ have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
+ using J by (auto simp: P'_def mapmeasure_PiM space_P sets_P)
+ also
+ have "\<dots> = ?SUP n"
+ proof (rule inner_regular)
+ show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
+ next
+ show "fm n ` B n \<in> sets borel"
+ unfolding borel_eq_PiF_borel by (auto simp: P'_def fm_image_measurable_finite sets_P J(3))
+ qed simp
+ finally have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
+ } note R = this
+ have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a \<and> compact K \<and> K \<subseteq> fm n ` B n"
+ proof
+ fix n show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
+ compact K \<and> K \<subseteq> fm n ` B n"
+ unfolding R[of n]
proof (rule ccontr)
- assume "(INF i. \<mu>G (Z i)) \<noteq> 0" (is "?a \<noteq> 0")
- moreover have "0 \<le> ?a"
- using Z positive_mu_G[OF `I \<noteq> {}`] by (auto intro!: INF_greatest simp: positive_def)
- 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\<^sub>M J (\<lambda>_. borel)) \<and>
- Z n = emb I J B \<and> \<mu>G (Z n) = emeasure (lim\<^sub>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>\<^sub>M i\<in>J' n. borel)"
- and Z_emb: "\<And>n. Z n = emb I (J' n) (B' n)"
- unfolding choice_iff by blast
- moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
- moreover def B \<equiv> "\<lambda>n. emb (J n) (J' n) (B' n)"
- ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I"
- "\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)"
- by auto
- have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
- unfolding J_def by force
- have "\<forall>n. \<exists>j. j \<in> J n" using J by blast
- then obtain j where j: "\<And>n. j n \<in> J n"
- unfolding choice_iff by blast
- note [simp] = `\<And>n. finite (J n)`
- from J Z_emb have Z_eq: "\<And>n. Z n = emb I (J n) (B n)" "\<And>n. Z n \<in> ?G"
- unfolding J_def B_def by (subst prod_emb_trans) (insert Z, auto)
- interpret prob_space "P (J i)" for i using proj_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 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: limP_finite proj_sets mu_G_eq)
+ assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and>
+ compact K' \<and> K' \<subseteq> fm n ` B n)"
+ have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
+ proof (intro SUP_least)
+ fix K
+ assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
+ with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
+ by auto
+ hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
+ unfolding not_less[symmetric] by simp
+ hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
+ using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
+ thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
+ qed
+ hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
+ hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
+ hence "0 \<le> - (2 powr (-n) * ?a)"
+ using `?SUP n \<noteq> \<infinity>` `?SUP n \<noteq> - \<infinity>`
+ by (subst (asm) ereal_add_le_add_iff) (auto simp:)
+ moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
+ by (auto simp: ereal_zero_less_0_iff)
+ ultimately show False by simp
+ qed
+ qed
+ then obtain K' where K':
+ "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
+ "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
+ unfolding choice_iff by blast
+ def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
+ have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))"
+ unfolding K_def
+ using compact_imp_closed[OF `compact (K' _)`]
+ by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
+ (auto simp: borel_eq_PiF_borel[symmetric])
+ have K_B: "\<And>n. K n \<subseteq> B n"
+ proof
+ fix x n assume "x \<in> K n"
+ then have fm_in: "fm n x \<in> fm n ` B n"
+ using K' by (force simp: K_def)
+ show "x \<in> B n"
+ using `x \<in> K n` K_sets sets.sets_into_space J(1,2,3)[of n]
+ by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto
+ qed
+ def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
+ have Z': "\<And>n. Z' n \<subseteq> Z n"
+ unfolding Z'_def Z_def
+ proof (rule prod_emb_mono, safe)
+ fix n x assume "x \<in> K n"
+ hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
+ by (simp_all add: K_def space_P)
+ note this(1)
+ also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
+ finally have "fm n x \<in> fm n ` B n" .
+ thus "x \<in> B n"
+ proof safe
+ fix y assume y: "y \<in> B n"
+ hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
+ by (auto simp add: space_P sets_P)
+ assume "fm n x = fm n y"
+ note inj_onD[OF inj_on_fm[OF space_borel],
+ OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
+ with y show "x \<in> B n" by simp
+ qed
+ qed
+ have "\<And>n. Z' n \<in> generator" using J K'(2) unfolding Z'_def
+ by (auto intro!: generator.intros measurable_sets[OF fm_measurable[of _ "Collect finite"]]
+ simp: K_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
+ def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
+ hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
+ hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
+ have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
+ hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
- have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
- def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
- interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n
- by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J])
- have inj_on_Utn: "inj_on Utn (\<Union>n. J n)"
- unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
- hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
- def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
- let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
- {
- fix n
- interpret finite_measure "P (J n)" by unfold_locales
- have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
- using J by (auto simp: P'_def mapmeasure_PiM proj_space proj_sets)
- also
- have "\<dots> = ?SUP n"
- proof (rule inner_regular)
- show "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
- unfolding P'_def
- by (auto simp: P'_def mapmeasure_PiF fm_measurable proj_space proj_sets)
- show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
- next
- show "fm n ` B n \<in> sets borel"
- unfolding borel_eq_PiF_borel
- by (auto simp del: J(2) simp: P'_def fm_image_measurable_finite proj_sets J)
- qed
- finally
- have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
- } note R = this
- have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a
- \<and> compact K \<and> K \<subseteq> fm n ` B n"
- proof
- fix n
- have "emeasure (P' n) (space (P' n)) \<noteq> \<infinity>"
- by (simp add: mapmeasure_PiF P'_def proj_space proj_sets)
- then interpret finite_measure "P' n" ..
- show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
- compact K \<and> K \<subseteq> fm n ` B n"
- unfolding R
- proof (rule ccontr)
- assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n) * ?a \<and>
- compact K' \<and> K' \<subseteq> fm n ` B n)"
- have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
- proof (intro SUP_least)
- fix K
- assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
- with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
- by auto
- hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
- unfolding not_less[symmetric] by simp
- hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
- using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
- thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
- qed
- hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
- hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
- hence "0 \<le> - (2 powr (-n) * ?a)"
- using `?SUP _ \<noteq> \<infinity>` `?SUP _ \<noteq> - \<infinity>`
- by (subst (asm) ereal_add_le_add_iff) (auto simp:)
- moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
- by (auto simp: ereal_zero_less_0_iff)
- ultimately show False by simp
- qed
- qed
- then obtain K' where K':
- "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
- "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
- unfolding choice_iff by blast
- def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
- have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))"
- unfolding K_def
- using compact_imp_closed[OF `compact (K' _)`]
- by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
- (auto simp: borel_eq_PiF_borel[symmetric])
- have K_B: "\<And>n. K n \<subseteq> B n"
- proof
- fix x n
- assume "x \<in> K n" hence fm_in: "fm n x \<in> fm n ` B n"
- using K' by (force simp: K_def)
- show "x \<in> B n"
- using `x \<in> K n` K_sets sets.sets_into_space J[of n]
- by (intro inj_on_image_mem_iff[OF inj_on_fm _ fm_in, of "\<lambda>_. borel"]) auto
- qed
- def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
- have Z': "\<And>n. Z' n \<subseteq> Z n"
- unfolding Z_eq unfolding Z'_def
- proof (rule prod_emb_mono, safe)
- fix n x assume "x \<in> K n"
- hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
- by (simp_all add: K_def proj_space)
- note this(1)
- also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
- finally have "fm n x \<in> fm n ` B n" .
- thus "x \<in> B n"
- proof safe
- fix y assume y: "y \<in> B n"
- hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
- by (auto simp add: proj_space proj_sets)
- assume "fm n x = fm n y"
- note inj_onD[OF inj_on_fm[OF space_borel],
- OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
- with y show "x \<in> B n" by simp
+ have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
+ proof -
+ fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
+ have "Y n = (\<Inter>i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
+ by (auto simp: Y_def Z'_def)
+ also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. emb (J n) (J i) (K i))"
+ using `n \<ge> 1`
+ by (subst prod_emb_INT) auto
+ finally
+ have Y_emb:
+ "Y n = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
+ hence "Y n \<in> generator" using J J_mono K_sets `n \<ge> 1`
+ by (auto simp del: prod_emb_INT intro!: generator.intros)
+ have *: "\<mu>G (Z n) = P (J n) (B n)"
+ unfolding Z_def using J by (intro mu_G_spec) auto
+ then have "\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" by auto
+ note *
+ moreover have *: "\<mu>G (Y n) = P (J n) (\<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_spec) auto
+ then have "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" by auto
+ note *
+ moreover have "\<mu>G (Z n - Y n) =
+ P (J n) (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
+ unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
+ by (subst mu_G_spec) (auto intro!: sets.Diff)
+ ultimately
+ have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
+ using J J_mono K_sets `n \<ge> 1`
+ by (simp only: emeasure_eq_measure Z_def)
+ (auto dest!: bspec[where x=n] intro!: measure_Diff[symmetric] set_mp[OF K_B]
+ simp: extensional_restrict emeasure_eq_measure prod_emb_iff sets_P space_P)
+ also have subs: "Z n - Y n \<subseteq> (\<Union>i\<in>{1..n}. (Z i - Z' i))"
+ using `n \<ge> 1` unfolding Y_def UN_extend_simps(7) by (intro UN_mono Diff_mono Z_mono order_refl) auto
+ have "Z n - Y n \<in> generator" "(\<Union>i\<in>{1..n}. (Z i - Z' i)) \<in> generator"
+ using `Z' _ \<in> generator` `Z _ \<in> generator` `Y _ \<in> generator` by auto
+ hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union>i\<in>{1..n}. (Z i - Z' i))"
+ using subs generator.additive_increasing[OF positive_mu_G additive_mu_G]
+ unfolding increasing_def by auto
+ also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> generator` `Z' _ \<in> generator`
+ by (intro generator.subadditive[OF positive_mu_G additive_mu_G]) auto
+ also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
+ proof (rule setsum_mono)
+ fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
+ have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
+ unfolding Z'_def Z_def by simp
+ also have "\<dots> = P (J i) (B i - K i)"
+ using J K_sets by (subst mu_G_spec) auto
+ also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
+ using K_sets J `K _ \<subseteq> B _` by (simp add: emeasure_Diff)
+ also have "\<dots> = P (J i) (B i) - P' i (K' i)"
+ unfolding K_def P'_def
+ by (auto simp: mapmeasure_PiF borel_eq_PiF_borel[symmetric]
+ compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
+ also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
+ finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
+ qed
+ also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
+ by (simp add: setsum_left_distrib r)
+ also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
+ proof (rule mult_strict_right_mono)
+ have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)"
+ by (rule setsum.cong) (auto simp: powr_realpow powr_divide powr_minus_divide)
+ also have "{1..<Suc n} = {..<Suc n} - {0}" by auto
+ also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
+ setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1)
+ also have "\<dots> < 1" by (subst geometric_sum) auto
+ finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
+ qed (auto simp: r ereal_less_real_iff zero_ereal_def[symmetric])
+ also have "\<dots> = ?a" by (auto simp: r)
+ also have "\<dots> \<le> \<mu>G (Z n)"
+ using J by (auto intro: INF_lower simp: Z_def mu_G_spec)
+ finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
+ hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
+ using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
+ have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
+ also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
+ apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
+ finally have "\<mu>G (Y n) > 0"
+ using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
+ thus "Y n \<noteq> {}" using positive_mu_G by (auto simp add: positive_def)
+ qed
+ hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
+ then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
+ {
+ fix t and n m::nat
+ assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
+ from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
+ also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
+ finally
+ have "fm n (restrict (y m) (J n)) \<in> K' n"
+ unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
+ moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
+ using J by (simp add: fm_def)
+ ultimately have "fm n (y m) \<in> K' n" by simp
+ } note fm_in_K' = this
+ interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
+ proof
+ fix n show "compact (K' n)" by fact
+ next
+ fix n
+ from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
+ also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
+ finally
+ have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
+ unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
+ thus "K' (Suc n) \<noteq> {}" by auto
+ fix k
+ assume "k \<in> K' (Suc n)"
+ with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
+ then obtain b where "k = fm (Suc n) b" by auto
+ thus "domain k = domain (fm (Suc n) (y (Suc n)))"
+ by (simp_all add: fm_def)
+ next
+ fix t and n m::nat
+ assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
+ assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
+ then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
+ hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
+ have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
+ by (intro fm_in_K') simp_all
+ show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)"
+ apply (rule image_eqI[OF _ img])
+ using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
+ unfolding j by (subst proj_fm, auto)+
+ qed
+ have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z"
+ using diagonal_tendsto ..
+ then obtain z where z:
+ "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
+ unfolding choice_iff by blast
+ {
+ fix n :: nat assume "n \<ge> 1"
+ have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
+ by simp
+ moreover
+ {
+ fix t
+ assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
+ hence "t \<in> Utn ` J n" by simp
+ then obtain j where j: "t = Utn j" "j \<in> J n" by auto
+ have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
+ apply (subst (2) tendsto_iff, subst eventually_sequentially)
+ proof safe
+ fix e :: real assume "0 < e"
+ { fix i and x :: "'i \<Rightarrow> 'a" assume i: "i \<ge> n"
+ assume "t \<in> domain (fm n x)"
+ hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
+ with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t"
+ using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
+ } note index_shift = this
+ have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
+ apply (rule le_SucI)
+ apply (rule order_trans) apply simp
+ apply (rule seq_suble[OF subseq_diagseq])
+ done
+ from z
+ have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e"
+ unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
+ then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
+ dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
+ show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e "
+ proof (rule exI[where x="max N n"], safe)
+ fix na assume "max N n \<le> na"
+ hence "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) =
+ dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t
+ by (subst index_shift[OF I]) auto
+ also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
+ finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
qed
qed
- { fix n
- have "Z' n \<in> ?G" using K' unfolding Z'_def
- apply (intro generatorI'[OF J(1-3)])
- unfolding K_def proj_space
- apply (rule measurable_sets[OF fm_measurable[of _ "Collect finite"]])
- apply (auto simp add: P'_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
- done
- }
- def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
- hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
- hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
- have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
- hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
- have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
- proof -
- fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
- have "Y n = (\<Inter>i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
- by (auto simp: Y_def Z'_def)
- also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. emb (J n) (J i) (K i))"
- using `n \<ge> 1`
- by (subst prod_emb_INT) auto
- finally
- have Y_emb:
- "Y n = prod_emb I (\<lambda>_. borel) (J n)
- (\<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: limP_finite proj_sets mu_G_eq)
- interpret finite_measure "(limP (J n) (\<lambda>_. borel) P)"
- proof
- have "emeasure (limP (J n) (\<lambda>_. borel) P) (J n \<rightarrow>\<^sub>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) = 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) =
- 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) = 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!: sets.Diff)
- ultimately
- have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
- using J J_mono K_sets `n \<ge> 1`
- by (simp only: emeasure_eq_measure)
- (auto dest!: bspec[where x=n]
- simp: extensional_restrict emeasure_eq_measure prod_emb_iff simp del: limP_finite
- intro!: measure_Diff[symmetric] set_mp[OF K_B])
- also have subs: "Z n - Y n \<subseteq> (\<Union>i\<in>{1..n}. (Z i - Z' i))" using Z' Z `n \<ge> 1`
- unfolding Y_def by (force simp: decseq_def)
- have "Z n - Y n \<in> ?G" "(\<Union>i\<in>{1..n}. (Z i - Z' i)) \<in> ?G"
- using `Z' _ \<in> ?G` `Z _ \<in> ?G` `Y _ \<in> ?G` by auto
- hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union>i\<in>{1..n}. (Z i - Z' i))"
- using subs G.additive_increasing[OF positive_mu_G[OF `I \<noteq> {}`] additive_mu_G[OF `I \<noteq> {}`]]
- unfolding increasing_def by auto
- also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> ?G` `Z' _ \<in> ?G`
- by (intro G.subadditive[OF positive_mu_G additive_mu_G, OF `I \<noteq> {}` `I \<noteq> {}`]) auto
- also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
- proof (rule setsum_mono)
- fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
- have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
- unfolding Z'_def Z_eq by simp
- also have "\<dots> = P (J i) (B i - K i)"
- using J K_sets by (subst mu_G_eq) auto
- 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)
- done
- also have "\<dots> = P (J i) (B i) - P' i (K' i)"
- unfolding K_def P'_def
- by (auto simp: mapmeasure_PiF proj_space proj_sets borel_eq_PiF_borel[symmetric]
- compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
- also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
- finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
- qed
- also have "\<dots> = (\<Sum> i\<in>{1..n}. ereal (2 powr -real i) * ereal(real ?a))"
- using `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` by (subst ereal_real') auto
- also have "\<dots> = ereal (\<Sum> i\<in>{1..n}. (2 powr -real i) * (real ?a))" by simp
- also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
- by (simp add: setsum_left_distrib)
- also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
- proof (rule mult_strict_right_mono)
- have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)"
- by (rule setsum.cong) (auto simp: powr_realpow powr_divide powr_minus_divide)
- also have "{1..<Suc n} = {..<Suc n} - {0}" by auto
- also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
- setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1)
- also have "\<dots> < 1" by (subst geometric_sum) auto
- finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
- qed (auto simp:
- `0 < ?a` `?a \<noteq> \<infinity>` `?a \<noteq> - \<infinity>` ereal_less_real_iff zero_ereal_def[symmetric])
- also have "\<dots> = ?a" using `0 < ?a` `?a \<noteq> \<infinity>` by (auto simp: ereal_real')
- also have "\<dots> \<le> \<mu>G (Z n)" by (auto intro: INF_lower)
- finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
- hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
- using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
- have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
- also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
- apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
- finally have "\<mu>G (Y n) > 0"
- using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
- thus "Y n \<noteq> {}" using positive_mu_G `I \<noteq> {}` by (auto simp add: positive_def)
- qed
- hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
- then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
- {
- fix t and n m::nat
- assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
- from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
- also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
- finally
- have "fm n (restrict (y m) (J n)) \<in> K' n"
- unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
- moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
- using J by (simp add: fm_def)
- ultimately have "fm n (y m) \<in> K' n" by simp
- } note fm_in_K' = this
- interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
- proof
- fix n show "compact (K' n)" by fact
- next
- fix n
- from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
- also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
- finally
- have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
- unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
- thus "K' (Suc n) \<noteq> {}" by auto
- fix k
- assume "k \<in> K' (Suc n)"
- with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
- then obtain b where "k = fm (Suc n) b" by auto
- thus "domain k = domain (fm (Suc n) (y (Suc n)))"
- by (simp_all add: fm_def)
- next
- fix t and n m::nat
- assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
- assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
- then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
- hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
- have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
- by (intro fm_in_K') simp_all
- show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)"
- apply (rule image_eqI[OF _ img])
- using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
- unfolding j by (subst proj_fm, auto)+
- qed
- have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z"
- using diagonal_tendsto ..
- then obtain z where z:
- "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
- unfolding choice_iff by blast
- {
- fix n :: nat assume "n \<ge> 1"
- have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
- by simp
- moreover
- {
- fix t
- assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
- hence "t \<in> Utn ` J n" by simp
- then obtain j where j: "t = Utn j" "j \<in> J n" by auto
- have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
- apply (subst (2) tendsto_iff, subst eventually_sequentially)
- proof safe
- fix e :: real assume "0 < e"
- { fix i x
- assume i: "i \<ge> n"
- assume "t \<in> domain (fm n x)"
- hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
- with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t"
- using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
- } note index_shift = this
- have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
- apply (rule le_SucI)
- apply (rule order_trans) apply simp
- apply (rule seq_suble[OF subseq_diagseq])
- done
- from z
- have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e"
- unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
- then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
- dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
- show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e "
- proof (rule exI[where x="max N n"], safe)
- fix na assume "max N n \<le> na"
- hence "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) =
- dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t
- by (subst index_shift[OF I]) auto
- also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
- finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
- qed
- qed
- hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> (finmap_of (Utn ` J n) z)\<^sub>F t"
- by (simp add: tendsto_intros)
- } ultimately
- have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
- by (rule tendsto_finmap)
- hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
- by (intro lim_subseq) (simp add: subseq_def)
- moreover
- have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
- apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
- apply (rule le_trans)
- apply (rule le_add2)
- using seq_suble[OF subseq_diagseq]
- apply auto
- done
- moreover
- from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
- ultimately
- have "finmap_of (Utn ` J n) z \<in> K' n"
- unfolding closed_sequential_limits by blast
- also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))"
- unfolding finmap_eq_iff
- proof clarsimp
- fix i assume i: "i \<in> J n"
- hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
- unfolding Utn_def
- by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
- with i show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)"
- by (simp add: finmap_eq_iff fm_def compose_def)
- qed
- finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
- moreover
- let ?J = "\<Union>n. J n"
- have "(?J \<inter> J n) = J n" by auto
- ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
- unfolding K_def by (auto simp: proj_space space_PiM)
- hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
- using J by (auto simp: prod_emb_def PiE_def extensional_def)
- also have "\<dots> \<subseteq> Z n" using Z' by simp
- finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
- } note in_Z = this
- hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
- hence "(\<Inter>i. Z i) \<noteq> {}" using Z INT_decseq_offset[OF `decseq Z`] by simp
- thus False using Z by simp
+ hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> (finmap_of (Utn ` J n) z)\<^sub>F t"
+ by (simp add: tendsto_intros)
+ } ultimately
+ have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
+ by (rule tendsto_finmap)
+ hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
+ by (intro lim_subseq) (simp add: subseq_def)
+ moreover
+ have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
+ apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
+ apply (rule le_trans)
+ apply (rule le_add2)
+ using seq_suble[OF subseq_diagseq]
+ apply auto
+ done
+ moreover
+ from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
+ ultimately
+ have "finmap_of (Utn ` J n) z \<in> K' n"
+ unfolding closed_sequential_limits by blast
+ also have "finmap_of (Utn ` J n) z = fm n (\<lambda>i. z (Utn i))"
+ unfolding finmap_eq_iff
+ proof clarsimp
+ fix i assume i: "i \<in> J n"
+ hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
+ unfolding Utn_def
+ by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
+ with i show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)"
+ by (simp add: finmap_eq_iff fm_def compose_def)
qed
- ultimately show "(\<lambda>i. \<mu>G (Z i)) ----> 0"
- using LIMSEQ_INF[of "\<lambda>i. \<mu>G (Z i)"] by simp
- qed
- then guess \<mu> .. note \<mu> = this
- def f \<equiv> "finmap_of J B"
- show "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (lim\<^sub>B J P) (Pi\<^sub>E J B)"
- proof (subst emeasure_extend_measure_Pair[OF limP_def, of I "\<lambda>_. borel" \<mu>])
- show "positive (sets (lim\<^sub>B I P)) \<mu>" "countably_additive (sets (lim\<^sub>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)
- next
- fix J and X::"'i \<Rightarrow> 'a set"
- show "prod_emb I (\<lambda>_. borel) J (Pi\<^sub>E J X) \<in> Pow (I \<rightarrow>\<^sub>E space borel)"
- by (auto simp: prod_emb_def)
- assume JX: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> J \<rightarrow> sets borel"
- hence "emb I J (Pi\<^sub>E J X) \<in> generator" using assms
- by (intro generatorI[where J=J and X="Pi\<^sub>E J X"]) (auto intro: sets_PiM_I_finite)
- hence "\<mu> (emb I J (Pi\<^sub>E J X)) = \<mu>G (emb I J (Pi\<^sub>E J X))" using \<mu> by simp
- also have "\<dots> = emeasure (P J) (Pi\<^sub>E J X)"
- using JX assms proj_sets
- by (subst mu_G_eq) (auto simp: mu_G_eq limP_finite intro: sets_PiM_I_finite)
- finally show "\<mu> (emb I J (Pi\<^sub>E J X)) = emeasure (P J) (Pi\<^sub>E J X)" .
- next
- show "emeasure (P J) (Pi\<^sub>E J B) = emeasure (limP J (\<lambda>_. borel) P) (Pi\<^sub>E J B)"
- using assms by (simp add: f_def limP_finite Pi_def)
- qed
-qed
+ finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
+ moreover
+ let ?J = "\<Union>n. J n"
+ have "(?J \<inter> J n) = J n" by auto
+ ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
+ unfolding K_def by (auto simp: space_P space_PiM)
+ hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
+ using J by (auto simp: prod_emb_def PiE_def extensional_def)
+ also have "\<dots> \<subseteq> Z n" using Z' by simp
+ finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
+ } note in_Z = this
+ hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
+ thus "(\<Inter>i. Z i) \<noteq> {}"
+ using INT_decseq_offset[OF antimonoI[OF Z_mono]] by simp
+qed fact+
+
+lemma measure_lim_emb:
+ "J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> X \<in> sets (\<Pi>\<^sub>M i\<in>J. borel) \<Longrightarrow> measure lim (emb I J X) = measure (P J) X"
+ unfolding measure_def by (subst emeasure_lim_emb) auto
end
@@ -548,70 +469,24 @@
hide_const (open) domain
hide_const (open) basis_finmap
-sublocale polish_projective \<subseteq> P!: prob_space "(lim\<^sub>B I P)"
+sublocale polish_projective \<subseteq> P!: prob_space lim
proof
- show "emeasure (lim\<^sub>B I P) (space (lim\<^sub>B I P)) = 1"
- proof cases
- assume "I = {}"
- interpret prob_space "P {}" using proj_prob_space by simp
- show ?thesis
- 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 proj_prob_space by simp
- have R: "(space (lim\<^sub>B I P)) = (emb I {i} (Pi\<^sub>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 PiE_def)
- ultimately show ?thesis using `i \<in> I`
- apply (subst R)
- apply (subst emeasure_limB_emb_not_empty)
- apply (auto simp: limP_finite emeasure_space_1 PiE_def)
- done
- qed
+ have *: "emb I {} {\<lambda>x. undefined} = space (\<Pi>\<^sub>M i\<in>I. borel)"
+ by (auto simp: prod_emb_def space_PiM)
+ interpret prob_space "P {}"
+ using prob_space_P by simp
+ show "emeasure lim (space lim) = 1"
+ using emeasure_lim_emb[of "{}" "{\<lambda>x. undefined}"] emeasure_space_1
+ by (simp add: * PiM_empty space_P)
qed
-context polish_projective begin
-
-lemma emeasure_limB_emb:
- assumes X: "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
- shows "emeasure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = emeasure (P J) (Pi\<^sub>E J B)"
-proof cases
- interpret prob_space "P {}" using proj_prob_space by simp
- assume "J = {}"
- moreover have "emb I {} {\<lambda>x. undefined} = space (lim\<^sub>B I P)"
- by (auto simp: space_PiM prod_emb_def)
- moreover have "{\<lambda>x. undefined} = space (lim\<^sub>B {} P)"
- by (auto simp: space_PiM prod_emb_def simp del: limP_finite)
- ultimately show ?thesis
- 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_limB_emb_not_empty) (auto simp: limP_finite)
-qed
-
-lemma measure_limB_emb:
- assumes "J \<subseteq> I" "finite J" "\<forall>i\<in>J. B i \<in> sets borel"
- shows "measure (lim\<^sub>B I P) (emb I J (Pi\<^sub>E J B)) = measure (P J) (Pi\<^sub>E J B)"
-proof -
- interpret prob_space "P J" using proj_prob_space assms by simp
- show ?thesis
- 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
-
-end
-
locale polish_product_prob_space =
product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
proof qed
-lemma (in polish_product_prob_space) limP_eq_PiM:
- "I \<noteq> {} \<Longrightarrow> lim\<^sub>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_limB_emb)
+lemma (in polish_product_prob_space) limP_eq_PiM: "lim = PiM I (\<lambda>_. borel)"
+ by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_lim_emb)
end