**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series A, Pt. 1, pp. 154--157

**SOME MEASURE THEORETICAL CHARACTERIZATIONS OF COMPACTNESS OF METRIC SPACES**

By

D. PLACHKY, *University of Münster*

SUMMARY. Let C_{b}(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 $\mathcal{B}^x$ denote the X-fold direct product of the Borel $\sigma$-algebra $\mathcal{B} of $\Re$. Then it is shown that X is compact if and only if for any closed, uncountable subset $\mathcal{F}$ of C_{b}(X) there exists some atomless probability measure P on $\mathcal{B}^x$ such that $P^*(\mathcal{F})= 1$ is valid, where P^{*} stands for the outer probability measure P, and the Borel $\sigma$-algebra $\mathcal{B}(C_b(X))$ of C_{b}(X) is countably generated and coincides with ${\mathcal{B^X} \cap C_b(X)$. Further more, it is proved that the metric space X is compact, if and only if any regular probability charge defined on the $\sigma$-algebra $\mathcal{B}(X)$ of Borel subsets of X is already $\sigma$-additive and $\mathcal{B}(X)$ is countably generated.

*AMS (1991) subject classification.* 60B05.

*Key words and phrases.* Continuous probability measure, regular probability charge.