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

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