On a class of Hausdorff measure of cartesian product sets

TL;DR

This paper investigates a class of Hausdorff measures in separable metric spaces, explores their structure on product sets, and establishes conditions under which weighted measures coincide with standard ones, with applications to Hausdorff functions.

Contribution

It introduces and analyzes a new class of Hausdorff measures based on measures and premeasures, proving their equivalence with weighted measures under certain conditions.

Findings

Proved equality of Hausdorff and weighted measures when measures are blanketed.

Analyzed Hausdorff structure of product sets.

Applied results to cases involving Hausdorff functions.

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.