Boundedness conditions of Hausdorff h-measure in metric spaces

The fractal dimensions are very important characteristics of the fractal sets. A problem which arises in the study of the fractal sets is the determination of their dimensions. The Hausdorff dimension of this type of sets is difficult to be determined, even if the Box dimensions can be computed. In this article we present some boundedness conditions on the Hausdorff h-measure of a set, using their Box dimensions. Subject Classification: 28A78. 1 Background The calculus of the dimensions is fundamental in the study of fractals. The Hausdorff measures and the h-measures, the box dimensions, the packing dimensions are widely used and in many articles the relations between them are given ([5] [8]). In the papers [1] [4] we gave some boundedness conditions for a class of fractal sets, in R. This type of conditions is important in order to prove theorems concerning the module and the capacities and the relations between them ([10]). In this paper we work in metric spaces and we give some boundedness conditions of the Hausdorff h measures. Definition 1. Let (X, d) be a metric space. If r0 > 0 is a given number, then, a continuous function h(r), defined on [0, r0) , nondecreasing and such that lim r→0 h(r) = 0 is called a measure function.