Weak Continuity of Riemann Integrable Functions in Lebesgue-Bochner Spaces

作     者：J.M.CALABUIG J.RODRíGUEZ E.A.SNCHEZ-PREZ 

作者机构：Instituto Universitario de Matemlica Pura y Aplicada Universidad Politcnica de Valencia 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2010年第26卷第2期

页      面：241-248页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported by MEC and FEDER (Project MTM2005-08350-C03-03) Generalitat Valenciana (Project GV/2007/191) supported by MEC and FEDER (Project MTM2005-08379) Fundacion Seneca (Project 00690/PI/04) the "Juan de la Cierva" Programme (MEC and FSE) supported by MEC and FEDER (Project MTM2006-11690-C02-01) 

主　　题：Riemann integral Bochner integral Lebesgue-Bochner space weak Lebesgue property 

摘      要：In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.