Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

Abstract

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 LX1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX1 has the weak Lebesgue property whenever X has the Radon-Nikodým 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 L1[0, 1] has the weak Lebesgue property.