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 ${\mathcal W}_\mu^{q, \xi}$ appear as an important tool when studying the product sets. When $\mu$ and $\xi$ satisfy the doubling condition, we prove that ${\mathcal H}_\mu^{q, \xi} = {\mathcal W}_\mu^{q, \xi}$. As an application, the case where $\xi$ is defined as the Hausdorff function is considered.
Citation
Najmeddine Attia. Hajer Jebali. Rihab Guedri. "On a class of Hausdorff measure of cartesian product sets in metric spaces." Topol. Methods Nonlinear Anal. 62 (2) 601 - 623, 2023. https://doi.org/10.12775/TMNA.2023.016
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