Research Article

The modulus of a vector measure

Published in: Quaestiones Mathematicae
Volume 47 , issue sup1 , 2024 , pages: 121–136

DOI: 10.2989/16073606.2023.2287823

Author(s): Ben de Pagter Delft University of Technology, The Netherlands , Werner J. Ricker Katholische Universität Eichstätt-Ingolstadt, Germany

Keywords: Primary: 46A40 , 46G10 , Secondary: 06F20 , Vector measure , finitely additive , order bounded , modulus

Download full text Get new issue alerts

Abstract

It is known that if L is a Dedekind complete Riesz space and (Ω, Σ) is a measurable space, then the partially ordered linear space of all L-valued, finitely additive and order bounded vector measures m on Σ is also a Dedekind complete Riesz space (for the natural operations). In particular, the modulus |m| _o of m exists in this space of measures and |m| _o is given by a well known formula. Some 20 years ago L. Drewnowski and W. Wnuk asked the question (for L not Dedekind complete) if there is an m for which |m|_o exists but, |m| _o is not given by the usual formula? We show that such a measure m does indeed exist.

« The order bidual of C(X) for a realcompact space

On the diagonal of Riesz operators on Banach lattices »