Nets and sequences of Riemann and Riemann-type integrable functions with values in a Banach space

Abstract

In this article, we discuss several aspects of convergence theorems for nets and sequences of Riemann and Riemann-type integrable functions defined on a closed bounded interval in $\mathbb{R}$ with values in a Banach space. We introduce the notions of Riemann $\Delta$-Cauchy nets of functions with its analogous variants and derive some correlations between such kind of nets of functions and equi-Riemann integrability. Moreover, we establish equi-integrability of the pointwise closure of different types of equi-integrable collections of functions. Finally, several related results, e.g., relative compactness of equi-integrable collections of functions with respect to different topologies are studied.