Radon–Nikodým derivatives for vector measures belonging to Köthe function spaces

Let m, n be a couple of vector measures with values on a Banach space. We develop a separation argument which provides a characterization of when the Radon–Nikodým derivative of n with respect to m—in the sense of the Bartle–Dunford–Schwartz integral—exists and belongs to a particular sublattice Z(μ) of the space of integrable functions L1(m). We show that this theorem is in fact a particular feature of our separation argument, which can be applied to prove other results in both the vector measure and the function space settings.