On $p$-Dunford integrable functions with values in Banach spaces

Abstract

Let $(\Omega ,\Sigma ,\mu )$ be a complete probability space, $X$ a Banach space and $1\le p<\infty$. In this paper we discuss several aspects of $p$-Dunford integrable functions $f:\Omega \to X$. Special attention is paid to the compactness of the Dunford operator of $f$. We also study the $p$-Bochner integrability of the composition $u\circ f:\Omega \to Y$, where $u$ is a $p$-summing operator from $X$ to another Banach space $Y$. Finally, we also provide some tests of $p$-Dunford integrability by using $w^{\ast }$-thick subsets of $X^{\ast }$.