Some properties of packing measure with doubling gauge
Volume 165 / 2004
Abstract
Let $g$ be a doubling gauge. We consider the packing measure ${\mathcal P}^g$ and the packing premeasure ${\mathcal P}_0^g$ in a metric space $X$. We first show that if ${\mathcal P}_0^g(X)$ is finite, then as a function of $X$, ${\mathcal P}_0^g$ has a kind of “outer regularity”. Then we prove that if $X$ is complete separable, then $\lambda \mathop {\rm sup}{\mathcal P}_0^g(F)\leq {\mathcal P}^g(B)\leq \mathop {\rm sup}{\mathcal P}_0^g(F)$ for every Borel subset $B$ of $X$, where the supremum is taken over all compact subsets of $B$ having finite ${\mathcal P}_0^g$-premeasure, and $\lambda $ is a positive number depending only on the doubling gauge $g$. As an application, we show that for every doubling gauge function, there is a compact metric space of finite positive packing measure.
