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Let g be a doubling gauge. We consider the packing measure ${}^g$ and the packing premeasure ${₀}^g$ in a metric space X. We first show that if ${₀}^g\left(X\right)$ is finite, then as a function of X, ${₀}^g$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $\lambda sup{₀}^g\left(F\right){\le }^g\left(B\right)\le sup{₀}^g\left(F\right)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite ${₀}^g$-premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge...