Hausdorff measures and packing premeasures

Abstract

We estimate the HausdorR measures and the packing premeasures of symmetric generalized Cantor sets in the d-dimensional Euclidean space R^d. Two simple estimations will be obtained. Let φ_1 and φ_2 be two measure functions. Suppose limt_(t→0) φ_2(t)/φ_1(t) = 0, limt_(t→0) φ_2(t)/t^d = ∞, and φ_1(t)/t^d is strictly decreasing as t increases. Then we can construct a compact set K in R^d such that 0 < Λ_(φ1) (K) < ∞ and 0 < φ_2 - P(K) < ∞ with the aid of the above estimations.