Nonstandard measure theory–Hausdorff measure

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, $f:X \to {\mathbf {R}}$, \[ \int {f\;d{\lambda ^t}} = {\text {St}}\left ( {\frac {1}{{{\nu ^t}}}\sum \limits _{x \in G} {*f(x)} } \right ).\]