Some properties of packing measure with doubling gauge

Let g be a doubling gauge. We consider the packing measure P-g and the packing premeasure P-0(g) in a metric space X. We first show that if P-0(g)(X) is finite, then as a function of X, P-0(g) has a kind of "outer regularity". Then we prove that if X is complete separable, then lambda sup P-0(g) (F) less than or equal to P-g (B) less than or equal to sup P-0(g) (F) for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite 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.