SOME MEASURE THEORETICAL CHARACTERIZATIONS OF COMPACTNESS OF METRIC SPACES

SUMMARY. Let Cb(X) denote the Banach space consisting of all real-valued, bounded, and continuous functions on some metric space X together with the supremum norm and let BX denote the X-fold direct product of the Borel σ-algebra B of <. Then it is shown that X is compact if and only if for any closed, uncountable subset F of Cb(X) there exists some atomless probability measure P on BX such that P ∗F = 1 is valid, where P ∗ stands for the outer probability measure of P, and the Borel σ-algebra BCb(X)) of Cb(X) is countably generated and coincides with BX ∩Cb(X). Furthermore, it is proved that a metric space X is compact, if and only if any regular probability charge defined on the σ-algebra B(X) of Borel subsets of X is already σ-additive and B(X) is countably generated. Motivation and starting point for this note is the following characterization of separability of metric spaces: Lemma 1. Let (X, d) denote some metric space with metric d. Then X is separable if and only if the Borel σ-algebra of any closed, uncountable subset Y of X such that (Y, d) is complete, admits some continuous probability measure. Proof. Cf. Plachky (1992). Clearly, any probability measure P on some σ-algebra A of subsets of some set Ω, which is atomless, i.e. there does not exist some set A ∈ A, satisfyin