A note on the generalized Hausdorff and packing measures of product sets in metric space

TL;DR

This paper introduces generalized measures for product sets in metric spaces, establishing new inequalities that extend classical results without restrictive assumptions on measures or functions.

Contribution

It develops and investigates generalized Hausdorff and packing measures, deriving refined product inequalities applicable under minimal conditions.

Findings

Established inequalities relating product measures and individual measures.

Proved existence of a constant c under geometric conditions for measure bounds.

Extended classical measure inequalities without assumptions on measures or functions.

Abstract

Let $\mu$ and $\nu$ be two Borel probability measures on two separable metric spaces $\X$ and $\Y$ respectively. For $h, g$ be two Hausdorff functions and $q\in \R$, we introduce and investigate the generalized pseudo-packing measure ${\RRR}_{\mu}^{q, h}$ and the weighted generalized packing measure ${\QQQ}_{\mu}^{q, h}$ to give some product inequalities : $${\HHH}_{\mu\times \nu}^{q, hg}(E\times F) \le {\HHH}_{\mu}^{q, h}(E) \; {\RRR}_{\nu}^{q, g}(F) \le {\RRR}_{\mu\times \nu}^{q, hg}(E\times F)$$ and $${\PPP}_{\mu\times \nu}^{q, hg}(E\times F) \le {\QQQ}_{\mu}^{q, h}(E) \; {\PPP}_{\nu}^{q, g}(F) $$ for all $E\subseteq \X$ and $F\subseteq \Y$, where ${\HHH}_{\mu}^{q, h}$ and ${\PPP}_{\mu}^{q, h}$ is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant $c$ such that $${\HHH}_{\mu\times \nu}^{q, hg}(E\times F) \le c\, {\HHH}_{\mu}^{q, h}(E) \; {\PPP}_{\nu}^{q, g}(F) $$ $${\HHH}_{\mu}^{q, h}(E) \; {\PPP}_{\nu}^{q, g}(F) \le c\, {\PPP}_{\mu}^{q, hg}(E \times F) $$ $${\PPP}_{\mu\times \nu}^{q, hg}(E\times F) \le c\, {\PPP}_{\mu}^{q, h}(E) \; {\PPP}_{\nu}^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $\mu, \nu, h$ and $g$.