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

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 $g$.