A unifying Radon-Nikodým theorem through nonstandard hulls

Abstract

We present a Radon-Nikodým theorem for vector measures of bounded variation that are absolutely continuous with respect to the Lebesgue measure on the unit interval. Traditional Radon-Nikodým 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 $\ster[0,1]$, 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 \cite{8} and \cite{5}.