It is the purpose of this paper to characterize the complete spaces in the sense of [6] by measure-theoretic properties. Let (X,0L) be a measurable space and let D6 be a subpaving of 0t satisfying certain closure properties, then X is ]}6-complete iff every 0,l-vaiued -regular measure on 0L is a Dirac measure. In particular,we obtain Hewitt's well-known theorem that a completely regular space X is realcompact iff every 0,l-valued Baire measure on X is a Dirac measure. The main tool for our investigations is an extension theorem for measures due to Topsoe [i0].

In previous papers, various notions of compact, T3, T4, and Tychonoff objects in a topological category were introduced and compared. The main objective of this paper is to characterize each of these classes of objects in the categories of filter and local filter convergence spaces as well as to examine how these various generalizations are related.

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Let Xx, X2 be lattices of subsets of a set X with X, c X2. The main result of this paper states that every semifinite tight set function on Xx can be extended to a semifinite tight set function on JT2. Furthermore, conditions assuring that such an extension is uniquely determined or o-smooth at are given. Since a semifinite tight set function defined on a lattice Jf"[and being o-smooth at ] can be identified with a semifinite ^regular content [measure] on the algebra generated by X, the general results are applied to various extension problems in abstract and topological measure theory.