On uniform boundedness properties in exhausting additive set function spaces

Abstract

SynopsisValdivia (1978) introduced the class of suprabarrelled spaces, and (1979) deduced some uniform boundedness properties for scalar valued exhausting additive set functions on a σ-algebra from the suprabarrelledness of certain spaces. In this paper, it is shown that those uniform boundedness properties hold for G-valued exhausting additive set functions, G being a commutative topological group, on a larger class of Boolean algebras. Such properties are proved in Valdivia (1979) by means of duality theory arguments and ‘sliding hump’ methods, whereas here they are derived from the Baire category theorem. This generalization enables us to find a wide class of compact topological spaces K such that the subspaces of C(K) which satisfy a mild property are suprabarrelled.