Vol. 28, No. 1, 1969

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A theorem on sequential convergence of measures and some applications

John Bligh Conway

Vol. 28 (1969), No. 1, 53–60

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.

Mathematical Subject Classification
Primary: 46.25
Secondary: 28.00
Received: 11 December 1967
Published: 1 January 1969
John Bligh Conway