The Hausdorff Dimension Distribution of Finite Measures in Euclidean Space

Abstract

Let E be a Borel set of R^N. The α-outer Hausdorff measure of E has been defined to be where and each B_i is a closed ball. d(B_i) denotes the diameter of B_i.It is easily seen that the same value H^α(E) is obtained if we consider coverings of E by open balls or by balls which may be either open or closed.By dim(E) we will mean the usual Hausdorff-Besicovitch dimension of E, where