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- W3208987044 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 $qin R$, 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 : $${HHH}_{mutimes nu}^{q, hg}(Etimes F) le {HHH}_{mu}^{q, h}(E) ; {RRR}_{nu}^{q, g}(F) le {RRR}_{mutimes nu}^{q, hg}(Etimes F)$$ and $${PPP}_{mutimes nu}^{q, hg}(Etimes F) le {QQQ}_{mu}^{q, h}(E) ; {PPP}_{nu}^{q, g}(F) $$ for all $Esubseteq X$ and $Fsubseteq Y$, 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 $${HHH}_{mutimes nu}^{q, hg}(Etimes F) le c, {HHH}_{mu}^{q, h}(E) ; {PPP}_{nu}^{q, g}(F) $$ $${HHH}_{mu}^{q, h}(E) ; {PPP}_{nu}^{q, g}(F) le c, {PPP}_{mu}^{q, hg}(E times F) $$ $${PPP}_{mutimes nu}^{q, hg}(Etimes F) le c, {PPP}_{mu}^{q, h}(E) ; {PPP}_{nu}^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $mu, nu, h$ and $g$." @default.
- W3208987044 created "2021-11-08" @default.
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- W3208987044 date "2022-01-01" @default.
- W3208987044 modified "2023-10-15" @default.
- W3208987044 title "A note on the generalized Hausdorff and packing measures of product sets in metric space" @default.
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- W3208987044 doi "https://doi.org/10.7153/mia-2022-25-20" @default.
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