--- a/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy Tue Jun 30 13:29:30 2015 +0200
+++ b/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy Tue Jun 30 13:30:04 2015 +0200
@@ -1547,25 +1547,25 @@
lemma sup_continuous_nn_integral:
assumes f: "\<And>y. sup_continuous (f y)"
- assumes [measurable]: "\<And>F x. (\<lambda>y. f y F x) \<in> borel_measurable (M x)"
- shows "sup_continuous (\<lambda>F x. (\<integral>\<^sup>+y. f y F x \<partial>M x))"
+ assumes [measurable]: "\<And>x. (\<lambda>y. f y x) \<in> borel_measurable M"
+ shows "sup_continuous (\<lambda>x. (\<integral>\<^sup>+y. f y x \<partial>M))"
unfolding sup_continuous_def
proof safe
- fix C :: "nat \<Rightarrow> 'b \<Rightarrow> ereal" assume C: "incseq C"
- then show "(\<lambda>x. \<integral>\<^sup>+ y. f y (SUPREMUM UNIV C) x \<partial>M x) = (SUP i. (\<lambda>x. \<integral>\<^sup>+ y. f y (C i) x \<partial>M x))"
- using sup_continuous_mono[OF f]
- by (simp add: sup_continuousD[OF f C] fun_eq_iff nn_integral_monotone_convergence_SUP mono_def le_fun_def)
+ fix C :: "nat \<Rightarrow> 'b" assume C: "incseq C"
+ with sup_continuous_mono[OF f] show "(\<integral>\<^sup>+ y. f y (SUPREMUM UNIV C) \<partial>M) = (SUP i. \<integral>\<^sup>+ y. f y (C i) \<partial>M)"
+ unfolding sup_continuousD[OF f C]
+ by (subst nn_integral_monotone_convergence_SUP) (auto simp: mono_def le_fun_def)
qed
lemma inf_continuous_nn_integral:
assumes f: "\<And>y. inf_continuous (f y)"
- assumes [measurable]: "\<And>F x. (\<lambda>y. f y F x) \<in> borel_measurable (M x)"
- assumes bnd: "\<And>x F. (\<integral>\<^sup>+ y. f y F x \<partial>M x) \<noteq> \<infinity>"
- shows "inf_continuous (\<lambda>F x. (\<integral>\<^sup>+y. f y F x \<partial>M x))"
+ assumes [measurable]: "\<And>x. (\<lambda>y. f y x) \<in> borel_measurable M"
+ assumes bnd: "\<And>x. (\<integral>\<^sup>+ y. f y x \<partial>M) \<noteq> \<infinity>"
+ shows "inf_continuous (\<lambda>x. (\<integral>\<^sup>+y. f y x \<partial>M))"
unfolding inf_continuous_def
proof safe
- fix C :: "nat \<Rightarrow> 'b \<Rightarrow> ereal" assume C: "decseq C"
- then show "(\<lambda>x. \<integral>\<^sup>+ y. f y (INFIMUM UNIV C) x \<partial>M x) = (INF i. (\<lambda>x. \<integral>\<^sup>+ y. f y (C i) x \<partial>M x))"
+ fix C :: "nat \<Rightarrow> 'b" assume C: "decseq C"
+ then show "(\<integral>\<^sup>+ y. f y (INFIMUM UNIV C) \<partial>M) = (INF i. \<integral>\<^sup>+ y. f y (C i) \<partial>M)"
using inf_continuous_mono[OF f]
by (auto simp add: inf_continuousD[OF f C] fun_eq_iff antimono_def mono_def le_fun_def bnd
intro!: nn_integral_monotone_convergence_INF)