## Illinois Journal of Mathematics

### A unifying Radon-Nikodým theorem through nonstandard hulls

G. Beate Zimmer

#### 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}.

#### Article information

Source
Illinois J. Math., Volume 49, Number 3 (2005), 873-883.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138224

Digital Object Identifier
doi:10.1215/ijm/1258138224

Mathematical Reviews number (MathSciNet)
MR2210264

Zentralblatt MATH identifier
1106.46027

#### Citation

Zimmer, G. Beate. A unifying Radon-Nikodým theorem through nonstandard hulls. Illinois J. Math. 49 (2005), no. 3, 873--883. doi:10.1215/ijm/1258138224. https://projecteuclid.org/euclid.ijm/1258138224