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.