On a class of Hausdorff measure of cartesian product sets

Abstract

In this paper, we study, in a separable metric space, the generalized Hausdorff measures H q,ξ µ defined using a measure µ and a premeasure ξ. We discuss the Hausdorff structure of product sets. Weighted generalized Hausdorff measures W q,ξ µ appeared as an important tool when studying the product sets. When ξ is defined as a Hausdorff function a necessary and sufficient condition is given to get H q,ξ µ = W q,ξ µ .