关键词: Doubling condition, packing premeasure, packing measure

Abstract:

Let X be a metric space and [[mu]] a finite Borel measure on X. Let $\bar{\mathcal{P}}_{\mu}^{q,t}$ and ${\mathcal{P}}_{\mu}^{q,t}$ be
the packing premeasure and the packing measure on $X$, respectively, defined by the gauge $(\mu B(x,r))^q(2r)^t$, where $q,t\in\mathbb{R}$. For
any compact set $E$ of finite packing premeasure the authors prove: (1)
if $q\leq 0$ then $\bar{\mathcal{P}}_\mu^{q,t}(E)={\mathcal{P}}_\mu^{q,t}(E)$; (2) if $q>0$ and $\mu$ is doubling on $E$ then
$\bar{\mathcal{P}}_\mu^{q,t}(E)$ and ${\mathcal{P}}_\mu^{q,t}(E)$ are both zero or neither.

Key words: Doubling condition, packing premeasure, packing measure