--- a/src/HOL/Probability/Lebesgue_Integration.thy Wed Dec 08 16:15:14 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Wed Dec 08 19:32:11 2010 +0100
@@ -1176,8 +1176,6 @@
apply (rule positive_integral_mono)
using `f \<up> u` unfolding isoton_def le_fun_def by auto
next
- have "u \<in> borel_measurable M"
- using borel_measurable_SUP[of UNIV f] assms by (auto simp: isoton_def)
have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
show "(SUP i. positive_integral (f i)) = positive_integral u"
@@ -1198,8 +1196,6 @@
shows "(SUP i. positive_integral (f i)) = positive_integral (\<lambda>x. SUP i. f i x)"
(is "_ = positive_integral ?u")
proof -
- have "?u \<in> borel_measurable M"
- using borel_measurable_SUP[of _ f] assms by (simp add: SUPR_fun_expand)
show ?thesis
proof (rule antisym)
show "(SUP j. positive_integral (f j)) \<le> positive_integral ?u"
@@ -1209,9 +1205,9 @@
have "\<And>i. rf i \<in> borel_measurable M" unfolding rf_def
using assms by (simp cong: measurable_cong)
moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
- unfolding isoton_def SUPR_fun_expand le_fun_def fun_eq_iff
+ unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
using SUP_const[OF UNIV_not_empty]
- by (auto simp: restrict_def le_fun_def SUPR_fun_expand fun_eq_iff)
+ by (auto simp: restrict_def le_fun_def fun_eq_iff)
ultimately have "positive_integral ru \<le> (SUP i. positive_integral (rf i))"
unfolding positive_integral_alt[of ru]
by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
@@ -1354,24 +1350,16 @@
lemma (in measure_space) positive_integral_lim_INF:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes "\<And>i. u i \<in> borel_measurable M"
- shows "positive_integral (SUP n. INF m. u (m + n)) \<le>
+ shows "positive_integral (\<lambda>x. SUP n. INF m. u (m + n) x) \<le>
(SUP n. INF m. positive_integral (u (m + n)))"
proof -
- have "(SUP n. INF m. u (m + n)) \<in> borel_measurable M"
- by (auto intro!: borel_measurable_SUP borel_measurable_INF assms)
-
- have "(\<lambda>n. INF m. u (m + n)) \<up> (SUP n. INF m. u (m + n))"
- proof (unfold isoton_def, safe intro!: INF_mono bexI)
- fix i m show "u (Suc m + i) \<le> u (m + Suc i)" by simp
- qed simp
- from positive_integral_isoton[OF this] assms
- have "positive_integral (SUP n. INF m. u (m + n)) =
- (SUP n. positive_integral (INF m. u (m + n)))"
- unfolding isoton_def by (simp add: borel_measurable_INF)
+ have "positive_integral (\<lambda>x. SUP n. INF m. u (m + n) x)
+ = (SUP n. positive_integral (\<lambda>x. INF m. u (m + n) x))"
+ using assms
+ by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
+ (auto simp del: add_Suc simp add: add_Suc[symmetric])
also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
- apply (rule SUP_mono)
- apply (rule_tac x=n in bexI)
- by (auto intro!: positive_integral_mono INFI_bound INF_leI exI simp: INFI_fun_expand)
+ by (auto intro!: SUP_mono bexI le_INFI positive_integral_mono INF_leI)
finally show ?thesis .
qed
@@ -1960,10 +1948,10 @@
have borel_f: "\<And>i. (\<lambda>x. Real (f i x)) \<in> borel_measurable M"
using i unfolding integrable_def by auto
- hence "(SUP i. (\<lambda>x. Real (f i x))) \<in> borel_measurable M"
+ hence "(\<lambda>x. SUP i. Real (f i x)) \<in> borel_measurable M"
by auto
hence borel_u: "u \<in> borel_measurable M"
- using pos_u by (auto simp: borel_measurable_Real_eq SUPR_fun_expand SUP_F)
+ using pos_u by (auto simp: borel_measurable_Real_eq SUP_F)
have integral_eq: "\<And>n. positive_integral (\<lambda>x. Real (f n x)) = Real (integral (f n))"
using i unfolding integral_def integrable_def by (auto simp: Real_real)
@@ -2144,8 +2132,8 @@
thus ?thesis by simp
next
assume neq_0: "positive_integral (\<lambda>x. Real (2 * w x)) \<noteq> 0"
- have "positive_integral (\<lambda>x. Real (2 * w x)) = positive_integral (SUP n. INF m. (\<lambda>x. Real (?diff (m + n) x)))"
- proof (rule positive_integral_cong, unfold SUPR_fun_expand INFI_fun_expand, subst add_commute)
+ have "positive_integral (\<lambda>x. Real (2 * w x)) = positive_integral (\<lambda>x. SUP n. INF m. Real (?diff (m + n) x))"
+ proof (rule positive_integral_cong, subst add_commute)
fix x assume x: "x \<in> space M"
show "Real (2 * w x) = (SUP n. INF m. Real (?diff (n + m) x))"
proof (rule LIMSEQ_imp_lim_INF[symmetric])