Vector measures: where are their integrals?

Let ν be a vector measure with values in a Banach space Z. The integration map Iν:L1(ν)→Z, given by f↦∫fdν for f ∈ L 1(ν), always has a formal extension to its bidual operator I∗∗ν:L1(ν)∗∗→Z∗∗. So, we may consider the "integral" of any element f ** of L 1(ν)** as I **ν(f **). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z **. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X ** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X′* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I **ν for the particular vector measure ν defined by ν(A) := T(χ A ).