Families of Sets of Positive Measure

We present a combinatorial description of those families 5a of sets, for which there is a finite measure p. such that inf{ß{P) : P 6 a"} > 0 . This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions. The main part of this paper deals with the following problem. Given a family ¿P of subsets of a certain set X, under what conditions does there exist a finite measure p. defined on some o-algebra containing £P such that M{p(P) : P e 3°} > 0! The solution to the "finitely additive" version of this problem has been known since Kelley [11] introduced the notion of intersection numbers in order to characterize Boolean algebras having strictly positive (finitely additive) measures. Kelley's idea was subsequently used to describe compact topological spaces that support Radon measures (cf. [6, Chapter 6]). Casteren [5] applied intersection numbers to a problem on the existence of certain Radon measures in topological spaces. In [ 18] topological spaces having strictly positive Baire measures are characterized with the help of intersection numbers. This notion appears also in Mägerl-Namioka [14] in the context of separability of spaces of measures. In §2 of this paper we introduce a combinatorial condition involving intersection numbers which provides an answer to the question mentioned above. The subject is investigated in the context of measures being regular with respect to a given ¿-lattice. This setting is suitable for some applications of the main result. Section 3 contains a characterization of measure-compact and Borel measure-compact topological spaces which seems to answer the problems posed by Wheeler [19] and Gardner-Pfeffer [8]. Next we discuss a problem on the existence of regular measure extensions.