## Illinois Journal of Mathematics

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

### 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}.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1258138224

**Mathematical Reviews number (MathSciNet)**

MR2210264

**Zentralblatt MATH identifier**

1106.46027

**Subjects**

Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

Secondary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 28E05: Nonstandard measure theory [See also 03H05, 26E35]

#### 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