Nonstandard measure theory–Hausdorff measure
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- Proc. Amer. Math. Soc. 65 (1977), 326-331 Request permission
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, $f:X \to {\mathbf {R}}$, \[ \int {f\;d{\lambda ^t}} = {\text {St}}\left ( {\frac {1}{{{\nu ^t}}}\sum \limits _{x \in G} {*f(x)} } \right ).\]References
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Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 65 (1977), 326-331
- MSC: Primary 02H25; Secondary 28A75
- DOI: https://doi.org/10.1090/S0002-9939-1977-0444466-7
- MathSciNet review: 0444466