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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Loeb-measure approach to theorems by Prohorov, Sazonov and Gross


Author: Tom L. Lindstrøm
Journal: Trans. Amer. Math. Soc. 269 (1982), 521-534
MSC: Primary 60B11; Secondary 03H05, 26E35, 28C20
MathSciNet review: 637706
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Abstract: We use the Loeb-measure of nonstandard analysis to prove three classical results on limit measures: Let $ {\{ {\mu _i}\} _{i \in I}}$ be a projective system of Radon measures, we use the Loeb-measure $ L({\tilde \mu _E})$ for an infinite $ E \in {}^{\ast}I$ and a standard part map to construct a Radon limit measure on the projective limit (Prohorov's Theorem). Using the Loeb-measures on hyperfinite dimensional linear spaces, we characterize the Fourier-transforms of measures on Hilbert spaces (Sazonov's Theorem), and extend cylindrical measures on Hilbert spaces to $ \sigma $-additive measures on Banach spaces (Gross' Theorem).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637706-9
PII: S 0002-9947(1982)0637706-9
Keywords: Nonstandard analysis, Loeb-measure, cylindrical measures, projective limits, Fourier-transforms, measurable norms
Article copyright: © Copyright 1982 American Mathematical Society