A unifying Radon-Nikodym theorem through nonstandard hulls

We present a Radon-Nikodym theorem for vector measures of bounded variation that are absolutely continuous with respect to the Lebesgue measure on the unit interval. Traditional Radon-Nikod m derivatives are Banach space-valued Bochner integrable functions defined on the unit interval or some other measure space. The derivatives we construct are functions from *[0, 1], the nonstandard extension of the unit interval into a nonstandard hull of the Banach space E. For these generalized derivatives we have an integral that resembles the Bochner integral. Furthermore, we can standardize the generalized derivatives to produce the weak *-measurable E"-valued derivatives that Ionescu-Tulcea, Dinculeanu and others obtained in [8] and [5].