Vol. 75, No. 2, 1978

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On the extension of additive functionals on classes of convex sets

H. Groemer

Vol. 75 (1978), No. 2, 397–410

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.

Mathematical Subject Classification 2000
Primary: 52A05
Received: 2 February 1977
Published: 1 April 1978
H. Groemer