Relations between packing premeasure and measure on metric space*

Let X be a metric space and μ a finite Borel measure on X. Let P¯μq,t and P¯μq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge q(μB(x,r))t(2r)(μB(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¯μq,t(E)=P¯μq,t(E) if q > 0 and μ is doubling on E then P¯μq,t(E) and P¯μq,t(E) are both zero or neither.