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

 

Egoroff's theorem and the distribution of standard points in a nonstandard model


Authors: C. Ward Henson and Frank Wattenberg
Journal: Proc. Amer. Math. Soc. 81 (1981), 455-461
MSC: Primary 03H05; Secondary 26E35, 28A12
MathSciNet review: 597662
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Abstract: We study the relationship between the Loeb measure $ {}^0({}^*\mu)$ of a set $ E$ and the $ \mu$-measure of the set $ S(E) = \{ x \vert {}^* x \in E \}$ of standard points in $ E$. If $ E$ is in the $ \sigma $-algebra generated by the standard sets, then $ {}^0({}^ * \mu )(E) = \mu (S(E))$. This is used to give a short nonstandard proof of Egoroff's Theorem. If $ E$ is an internal, * measurable set, then in general there is no relationship between the measures of $ S(E)$ and $ E$. However, if $ {}^ * X$ is an ultrapower constructed using a minimal ultrafilter on $ \omega $, then $ {}^ * \mu (E) \approx 0$ implies that $ S(E)$ is a $ \mu $-null set. If, in addition, $ \mu $ is a Borel measure on a compact metric space and $ E$ is a Loeb measurable set, then

$\displaystyle \underline \mu (S(E)) \leqslant {}^0({}^ * \mu )(E) \leqslant \overline \mu (S(E))$

where $ \underline \mu $ and $ \overline \mu $ are the inner and outer measures for $ \mu $.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0597662-3
PII: S 0002-9939(1981)0597662-3
Article copyright: © Copyright 1981 American Mathematical Society