A Characterization of Strongly Measurable Pettis Integrable Functions

Abstract

Let $\mathfrak {X}$ be a Banach space and $(\Omega ,\Sigma ,\mu )$ be a finite measure space. A strongly measurable $f:\Omega \to \mathfrak {X}$ is Pettis integrable if and only if there exists a Young’s function $\Phi$ with ${\lim _{t \to \infty }}\Phi (t)/t = \infty$ such that ${x^ \ast }f \in {L^\Phi }(\mu )$ for all ${x^ \ast } \in {\mathfrak {X}^ \ast }$.