Some properties of packing measure with doubling gauge

Sheng-You Wen¹ ; Zhi-Ying Wen²

¹ Department of Mathematics Hubei University Hubei, 430062, P.R. China
² Department of Mathematics Tsinghua University Beijing, 100084, P.R. China

Studia Mathematica, Tome 165 (2004) no. 2, pp. 125-134

Résumé

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.

Détail

DOI : 10.4064/sm165-2-3

Keywords: doubling gauge consider packing measure mathcal packing premeasure mathcal metric space first mathcal finite function mathcal has kind outer regularity prove complete separable lambda mathop sup mathcal leq mathcal leq mathop sup mathcal every borel subset where supremum taken compact subsets having finite mathcal g premeasure lambda positive number depending only doubling gauge application every doubling gauge function there compact metric space finite positive packing measure