On the Sobczyk-Hammer decomposition of additive set functions

Abstract:

It is observed that continuity for charges is equivalent to the absence of two-valued minorants. This characterization forms the basis of a new short proof within a functional-analytic context of a decomposition theorem by A. Sobczyk and P. C. Hammer [5] for charges on a field $\mathfrak {A}$ into a continuous part and a part which can be written as a sum of at most two-valued charges on $\mathfrak {A}$. A counterexample shows that in general the decomposition of a charge into a nonatomic part and a part which has no nonnull nonatomic minorant is not unique.