This is an announcement for the paper "Riemann integrability versus weak continuity" by Gonzalo Martinez-Cervantes.
Abstract: In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space $X$ is said to have the weak Lebesgue property if every Riemann integrable function from $[0,1]$ into $X$ is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under $\ell_1$-sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford-Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of $\tau$-continuous non Riemann integrable functions, with $\tau$ a locally convex topology weaker than the norm topology.
Archive classification: math.FA
Mathematics Subject Classification: 46G10, 28B05, 03E10
Submitted from: gonzalo.martinez2@um.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1510.08801
or