Pettis integration via the Stonian transform

Let (, , ) be a finite measure space, and let / be a bounded weakly measurable function from into a Banach space X. Let S be the Stone space of the measure algebra /~\). Then/induces a continuous map /: S -* (A r **,weak*) in a natural way. Criteria for Pettis integrability of / are investigated in the context of this "Stonian transform" / of /. In particular, some insight is achieved as to how / can be Pettis integrable without being weakly equivalent to any strongly measurable function. The fine structure of (X**, weak*) is also examined in this setting.