Abstract. For every doubling gauge g, we prove that there is a Cantor set of positive finite H g -measure, P g -measure, and P g 0 -premeasure. Also, we show that every compact metric space of infinite P g 0 -premeasure has a compact countable subset of infinite P g 0 -premeasure. In addition, we obtain a class of uniform Cantor sets and prove that, for every set E in this class, there exists a countable set F , with F = E ∪ F , and a doubling gauge g such that E ∪ F has different positive finite P g -measure and P g 0 -premeasure.