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 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 $\mathcal{B}^x$ 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 Cb(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 Cb(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.

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This article in mathematical reviews.