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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A nonstandard characterization of weak convergence


Authors: Robert M. Anderson and Salim Rashid
Journal: Proc. Amer. Math. Soc. 69 (1978), 327-332
MSC: Primary 28A32; Secondary 02H25, 60B10
MathSciNet review: 0480925
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Abstract: Let X be any topological space, and $ C(X)$ the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on $ C(X)$ to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0480925-X
PII: S 0002-9939(1978)0480925-X
Keywords: Weak convergence, tight, relatively weakly compact, topological measure theory, nonstandard analysis
Article copyright: © Copyright 1978 American Mathematical Society