Additive functions of intervals and Hausdorff measure

Abstract

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define