Vector valued functions equivalent to measurable functions

Abstract:

Let X be a Banach space with dual ${X^\ast }$ and let $(\Omega ,\Sigma ,\mu )$ be a finite measure space. Suppose $f:\Omega \to X$ is weakly measurable. There exists a (norm) measurable $g:\Omega \to X$ such that $\langle {x^\ast },f\rangle = \langle {x^\ast },g\rangle$ a.e. for each ${x^\ast } \in {X^\ast }$ if and only if each set A of positive $\mu$-measure has a subset B of positive $\mu$-measure such that there is a weakly compact convex subset K of X with the property that \[ \langle {x^\ast },f\rangle \leqslant \sup \limits _{x \in K} \;\langle {x^\ast },x\rangle \] $\mu$-almost everywhere on B for each ${x^\ast } \in {X^\ast }$.