A nonstandard characterization of weak convergence
Authors:
Robert M. Anderson and Salim Rashid
Journal:
Proc. Amer. Math. Soc. 69 (1978), 327332
MSC:
Primary 28A32; Secondary 02H25, 60B10
MathSciNet review:
0480925
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Abstract: Let X be any topological space, and the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on 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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 Robert M. Anderson, A nonstandard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), 1546. MR 0464380 (57:4311)
 [2]
 , Starfinite representations of measure spaces (to appear).
 [3]
 , Starfinite probability theory, Ph.D. Dissertation, Yale Univ., May, 1977.
 [4]
 Patrick Billingsley, Convergence of probability measures, Wiley, New York, 1968. MR 0233396 (38:1718)
 [5]
 Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113122. MR 0390154 (52:10980)
 [6]
 , Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25 (1976), 154187. MR 0457757 (56:15961)
 [7]
 W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability, Holt, Rinehart, and Winston, New York, 1969, pp. 1886. MR 0244931 (39:6244)
 [8]
 D. W. Müller, Nonstandard proofs of invariance principles in probability theory, Applications of Model Theory to Algebra, Analysis, and Probability, Holt, Rinehart, and Winston, New York, 1969, pp. 186194. MR 0239645 (39:1002)
 [9]
 Salim Rashid, Economies with infinitely many traders, Ph.D. Dissertation, Yale Univ., May, 1976.
 [10]
 K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York, 1976. MR 0491163 (58:10429)
 [11]
 V. S. Varadarajan, Measures on topological spaces, Amer. Math Soc. Transl. (2) 48 (1965), 161228.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919780480925X
PII:
S 00029939(1978)0480925X
Keywords:
Weak convergence,
tight,
relatively weakly compact,
topological measure theory,
nonstandard analysis
Article copyright:
© Copyright 1978
American Mathematical Society
