Let 𝒮 be a class of sets and let
U(𝒮) denote the class of all finite unions of sets from 𝒮. This paper concerns itself
with the question whether a vector valued function on 𝒮 has an additive
extension to U(𝒮). Several characterizations of functions with such an extension
property are presented. One of these characterizations establishes a relationship
between such extensions and integrals of certain types of simple functions.
The special cases when 𝒮 is the class of all convex polytopes or the class of
all compact convex subsets of a euclidean space are investigated in more
detail. Some examples are given to show that various extension problems
that have been solved previously by methods particularly designed for each
individual problem can also be solved by the application of these general
results.
