Riemann integrability versus weak continuity

TL;DR

This paper explores the relationship between Riemann integrability and weak continuity in Banach spaces, introducing the weak Lebesgue property, its stability, and characterizations of operators related to integrability.

Contribution

It introduces the weak Lebesgue property for Banach spaces, studies its stability under sums, and characterizes Dunford-Pettis operators via Riemann integrability.

Findings

Weak Lebesgue property is stable under ℓ₁-sums.

New examples of Banach spaces with/without the property.

Characterization of Dunford-Pettis operators in terms of Riemann integrability.

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.