If S is a locally compact
Hausdorff space, let βS be its Stone-Cech compactification and let M(S) be the
space of all finite complex valued regular Borel measures on S. In this paper we will
prove that whenever S is paracompact and {μn} is a sequence in M(βS) which
converges to zero in the weak star topology, then lim∫Sf dμn= 0 for every
continuous function f, and {μn} satisfies a certain uniformity condition on S. This
generalizes a result of R. S. Phillips on weak star sequential convergence
in the dual of l∞. Moreover, by using our theorem we can obtain many
previously known results whose proofs, though all similar, were apparently
independent.
