Nets and sequences of $p$-Bochner, $p$-Dunford and $p$-Pettis integrable functions with values in a Banach space

Abstract

In this paper, we study convergence of nets and sequences of $p$-Bochner, $p$-Dunford and $p$-Pettis integrable functions, $p \in [1, \infty)$, defined on a finite measure space with values in a real Banach space. Applying Hölder's inequality, we study some properties of these functions and convergence of their nets and sequences. We introduce the idea of $\delta$-Cauchy nets in terms of which we establish convergence theorems for nets of above types of functions. We see that $\delta$-Cauchyness of a net plays the similar role as uniform integrability does in case of sequences.