Multidimensional measures on Cantor sets

Abstract

Cantor sets in R are common examples of sets on which Hausdorff measures can be positive and finite. However, there exists Cantor sets on which no Hausdorff measure is supported and finite. The purpose of this thesis is to try to resolve this problem by studying an extension of the Hausdorff measures h on R by allowing test functions to depend on the midpoint of the covering intervals instead of only on the diameter. As a partial result a theorem about the Hausdorff measure of any regular enough Cantor set, with respect to a chosen test function, is obtained.