Vol. 107, No. 2, 1983

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On a Radon-Nikodým problem for vector-valued measures

C. Debiève

Vol. 107 (1983), No. 2, 335–339

The purpose of this paper is to show that if m is a Banach space-valued measure with finite variation on a σ-algebra, then the variation |m| of m has a Radon-Nikodym derivative with respect to m.

This Radon-Nikodym derivative takes its values in the dual of the Banach space, is integrable in Dinculeanu’s sense and may be chosen of norm as close to one as we want.

From this theorem we deduce that if m and mare Banach space-valued measures on the same σ-algebra, such that m m, then m has a Radon-Nikodym derivative with respect to mand this derivative is m-integrable in Dinculeanu’s sense if we assume that the image space of m has Radon-Nikodym property.

Mathematical Subject Classification 2000
Primary: 46G10
Secondary: 28C05
Received: 21 October 1981
Published: 1 August 1983
C. Debiève