Let τ be a premeasure on a complete separable metric space and let τ＊ be the Method I measure constructed from τ. We give conditions on T such that τ＊ has a regularity as follows： Every τ＊-measurable set has measure equivalent to the supremum of premeasures of its compact subsets. Then we prove that the packing measure has this regularity if and only if the corresponding packing premeasure is locally finite.
Let X be a metric space and u a finite Borel measure on X. Let P^- u^q,t and P u^q,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge （uB（x, r））^q（2r）^t, where q, t ∈ R. For any compact set E of finite packing premeasure the authors prove： （1） if q ≤ 0 then P^- u^q,t（E） =P u^q,t （E）; （2） if q 〉 0 and u is doubling on E then P^- u^q,t （E） and P u^q,t （E） are both zero or neither.
