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.
