Some Relations Between Packing Premeasure And Packing Measure

Packing measure and Packing dimension were introduced by Tricot in1982. Like Hausdorff dimension, we may use the strong measure theoretical skills in the study of Packing dimension. In some degree, the Packing-measure in many ways can be dual to Hausdorff measure, so the introduction of Packing-measure has led to a further understanding of fractal geometry. In1999, D.J.Feng, S. Hua and Z.Y. Wen proved that if K is a compact set in Rn with Ps0(K)＜∞, then Ps(K)=Ps0(K). This result can be extended to include Packing measures induced by gauge functions g(t)=tsL(t), where L(t) is a slowly varying function. In2001, MariannaCsornyei proved that there is a compact set Kof R and a doubling gauge function g, such that Pg0(K)＜∞, and Pg(K)≠Pg0(K).Let K(?)Rn be a compact set and g a gauge function, g(t)=tsL(t), where L(t) is slowly varying, i.e., L(t) is increasing on (0+∞) and satisfies lim t→0L(ct)/L(t)=1for some c＞1. We proved, in this paper, that if Pg0(K)＜∞, then Pg(K)=Pg0(K). In this proof, we adopt Feng-Hua-Wen’s idea. On the other hand, we give a careful proof for Marianna Csornyei’s result.