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

 

On the nonstandard representation of measures


Author: C. Ward Henson
Journal: Trans. Amer. Math. Soc. 172 (1972), 437-446
MSC: Primary 28A25; Secondary 02H25
MathSciNet review: 0315082
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Abstract: In this paper it is shown that every finitely additive probability measure $ \mu $ on $ S$ which assigns 0 to finite sets can be given a nonstandard representation using the counting measure for some $ ^ \ast $-finite subset $ F$ of $ ^ \ast S$. Moreover, if $ \mu $ is countably additive, then $ F$ can be chosen so that

$\displaystyle \int {fd\mu } = {\text{st( }}\frac{1}{{\vert\vert F\vert\vert}}\sum _{p \in F} ^\ast f(p))$

for every $ \mu $-integrable function $ f$. An application is given of such representations. Also, a simple nonstandard method for constructing invariant measures is presented.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0315082-2
PII: S 0002-9947(1972)0315082-2
Article copyright: © Copyright 1972 American Mathematical Society