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Measures in topological spaces

Let X be a completely-regular topological space and let C*(X) denote the space of all bounded, real-valued continuous functions on X. For a positive linear functional [...] on C*(X), consider the following two continuity conditions. [...] is said to be a B-integral if whenever [...] and [...] for all [...], then [...]. [...] is said to be B-normal if whenever [...] is a directed system with [...] for all [...], then [...]. It is obvious that a B-normal functional is always a B-integral. The main concern of this paper is what can be said in the converse direction.

Methods are developed for discussing this question. Of particular importance is the representation of C*(X) as a space [...] of finitely-additive set functions on a certain algebra of subsets of X. This result was first announced by A. D. Alexandrov, but his proof was obscure. Since there seem to be no proofs readily available in the literature, a complete proof is given here. Supports of functionals are discussed and a relatively simple proof is given of the fact that every B-integral is B-normal if and only if every B-integral has a support.

The space X is said to be B-compact if every B-integral is B-normal. It is shown that B-compactness is a topological invariant and various topological properties of B-compact spaces are investigated. For instance, it is shown that B-compactness is permanent on the closed sets and the co-zero sets of a B-compact space. In the case that the spaces involved are locally-compact, it is shown that countable products and finite intersections of B-compact spaces are B-compact.

Also B-compactness is studied with reference to the classical compactness conditions. For instance, it is shown that if X is B-compact, then X is realcompact. Or that if X is paracompact and if the continuum hypothesis holds, then X is B-compact if and only if X is realcompact.

Finally, the methods and results developed in the paper are applied to formulate and prove a very general version of the classical Kolmogorov consistency theorem of probability theory. The result is as follows. If X is a locally-compact, B-compact space and if S is an abstract set, then a necessary and sufficient condition that a finitely-additive set function defined on the Baire (or the Borel) cylinder sets of X[superscript S] be a measure is that its projection on each of the finite coordinate spaces be Baire (or regular Borel) measures.

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Measures in topological spaces