On a class of Hausdorff measure of cartesian product sets

Abstract

In this paper, we study, in a separable metric space, a class of Hausdorff measures $\mathcal{H}_\mu^{q, \xi}$ defined using a measure $\mu$ and a premeasure $\xi$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures $\mathcal{W}_\mu^{q, \xi}$ appeared as an important tool when studying the product sets. When $\mu$ and $\xi$ are blanketed, we prove that $\mathcal{H}_\mu^{q, \xi} = \mathcal{W}_\mu^{q, \xi}$. As an application, the case when $\xi$ is defined as the Hausdorff function is considered.