The modulus of a vector measure

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.