Vector measures: where . . .

Let ν be a vector measure with values in a Banach space Z. The integration map Iν: L 1(ν) → Z, given by f → ∫ f dν for f ∈ L1(ν), always has a formal extension to its bidual operator I∗∗ν: L 1(ν)∗ ∗ → Z∗∗. So, we may consider the “integral ” of any element f∗ ∗ of L1(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 (χA)