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

DOI:
https://doi.org/10.1090/S0002-9939-1978-0480925-X

MathSciNet review:
0480925

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**Robert M. Anderson,*A nonstandard representation for Brownian motion and Itô integration*, Israel J. Math.**25**(1976), 15-46. MR**0464380 (57:4311)****[2]**-,*Star-finite representations of measure spaces*(to appear).**[3]**-,*Star-finite 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), 113-122. MR**0390154 (52:10980)****[6]**-,*Applications of nonstandard analysis to ideal boundaries in potential theory*, Israel J. Math.**25**(1976), 154-187. 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. 18-86. 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. 186-194. 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), 161-228.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0480925-X

Keywords:
Weak convergence,
tight,
relatively weakly compact,
topological measure theory,
nonstandard analysis

Article copyright:
© Copyright 1978
American Mathematical Society