It is the purpose of this paper to characterize the complete spaces in the sense of [6] by measure-theoretic properties. Let (X, ) be a measurable space and let be a subpaving of satisfying certain closure properties, then X is -complete iff every 0,1-valued -regular measure on is a Dirac measure. In particular, we obtain Hewitt's well-known theorem that a completely regular space X is realcompact iff every 0,1-valued Baire measure on X is a Dirac measure. The main tool for our investigations is an extension theorem for measures due to Topsoe [10].