L1(Ω,F,μ)的子集的弱紧性及其应用

Tke Weak Compactness of the Sets of L1 ( Ω,F,μ) and Applications

设（Ω，Ｆ，μ）为任一测度空间，μ可以为无限测度，本文给出了Ｌ１（Ω，Ｆ，μ）的任一子集为弱相对紧的充要条件。此外，当μ为 限测度时，我们还讨论了（Ω，Ｆ）上关于μ绝对连续的有限广义测度的集集收敛性，得到了较为满意的结果，它可看作是Ｖｉｔａｌｉ－Ｈａｈｎ－Ｓａｋｓ定理的推广。

Let ( Ω,F,μ) be any complete measurable space, may be infinite measure.In this paper, a sufficient and necessary condition about the weak compactness of thesets of L1 ( Ω,F,μ) is obtained. Furtherimore, when be infinite measure, we considerthe sets by sets convergence for finite generalized measures defined on the measurablespace ( Ω,F) which are absolutely continuous with and obtained some interestingresults which generalize the Vitali-Hahn-Saks theorem.