On Uniform Semigroup-Valued Additive Set Functions

Abstract

The main results of this paper are the following: (1) An extension theorem for a uniform semigroup-valued measure on a ring to the generated σ-ring. This result unifies the classieal Hahn-Carathéodory theorem, the extension theorem of Sion and a more recent result of Weber.(2) A theorem stating that every monocompact additive uniform semigroup-valued set function on a semiring is σ-additive. This result generalizes several earlier theorems of Alexandroff, Dinculeanu-Kluvanek, Glicksberg, Huneycutt, Mallory, Marczewski, Millington and Topsøe.

The support of a uniform semigroup valued measure

“…Let S = S; +, 0, U be a Hausdorff uniform semigroup and let τ be the topology generated by U. Following [12], a family (a i ) i∈I of elements of L is summable in S if the net ( i∈F a i ) F ∈ F (I);⊇ is τ -convergent and we write τ -lim F i∈F a i = i∈I a i . It is easy to prove that a family (a i ) i∈I in S is summable to an element s ∈ S if and only if, for every p ∈ P and every > 0, there exists F 0 ∈ F(I) such that F ∈ F(I) and F ⊇ F 0 imply p( i∈F a i , s) < .…”