Pettis integrability and the equality of the norms of the weak$^ \ast$ integral and the Dunford integral

Abstract:

If $(\Omega ,\sum ,\mu )$ is a perfect finite measure space and $X$ is a Banach space, then it is shown that ${X^ * }$ has the $\mu$-Pettis Integral Property if and only if \[ \left \| {({\text {wea}}{{\text {k}}^ * }) - \int \limits _\Omega {fd\mu } } \right \| = \left \| {({\text {Dunford}}) - \int \limits _\Omega {fd\mu } } \right \|\] for every bounded weakly measurable function $f:\Omega \to {X^ * }$.