On a class of Hausdorff measure of cartesian product sets in metric spaces - Attia, Najmeddine; Jebali, Hajer; Guedri, Rihab (Topological Methods in Nonlinear Analysis, 2023)
In this paper we study, in a separable metric space, a class of Hausdorff measures ${\mathcal H}_μ^{q, ξ}$ defined using a measure $μ$and a premeasure $ξ$.We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures ${\mathcal W}_μ^{q, ξ}$ appeare as an important tool when studying the product sets. When $μ$ and $ξ$ satisfy the doublingcondition, we prove that ${\mathcal H}_μ^{q, ξ} = {\mathcal W}_μ^{q, ξ}$.As an application, the case where $ξ$ is defined as the Hausdorff function is considered.