Strong Barrelledness Properties in Certain l∞0(A) Spaces

It is known that the simple scalar function space l∞0(Ω, Σ) is barrelled of class ℵ0 whenever Σ is a σ-algebra of subsets of a set Ω. In this paper we extend this result by showing that there exist certain algebras of sets having property (L1), namely algebras with the strong Nikodym property, that fail to have property (VHS)-although every σ-algebra of sets has the strong Nikodým property-and such that if A belongs to this new class, then its corresponding space l∞0(A) is barrelled of class ℵ0.