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

 

Nonstandard measure theory-Hausdorff measure


Author: Frank Wattenberg
Journal: Proc. Amer. Math. Soc. 65 (1977), 326-331
MSC: Primary 02H25; Secondary 28A75
MathSciNet review: 0444466
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Abstract: In this paper it is shown that the Hausdorff measures $ {\lambda ^t}$ for $ t \in [0,\infty )$ can be simultaneously represented as $ ^\ast$finite counting measures in an appropriate nonstandard model. That is, the following theorem is proved. Theorem. Suppose X is a metric space and $ \nu $ is an infinite positive $ ^\ast$integer. Then there is a $ ^\ast$finite set G such that for every standard $ t \in [0,\infty )$ and every $ {\lambda ^t}$-integrable Borel function, $ f:X \to {\mathbf{R}}$,

$\displaystyle \int {f\;d{\lambda ^t}} = {\text{St}}\left( {\frac{1}{{{\nu ^t}}}\sum\limits_{x \in G} {*f(x)} } \right).$


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0444466-7
PII: S 0002-9939(1977)0444466-7
Article copyright: © Copyright 1977 American Mathematical Society