A note on the generalized Hausdorff and packing measures of product sets in metric space
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929200054665216 |
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| author | Guedri, Rihab Attia, Najmeddine |
| author_facet | Guedri, Rihab Attia, Najmeddine |
| contents | Let $μ$ and $ν$ 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 ${\RRR}_μ^{q, h}$ and the weighted generalized packing measure ${\QQQ}_μ^{q, h}$ to give some product inequalities :
$${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le {\HHH}_μ^{q, h}(E) \; {\RRR}_ν^{q, g}(F) \le {\RRR}_{μ\times ν}^{q, hg}(E\times F)$$ and
$${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le {\QQQ}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ for all $E\subseteq \X$ and $F\subseteq \Y$, where ${\HHH}_μ^{q, h}$ and ${\PPP}_μ^{q, h}$ 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
$${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$
$${\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) \le c\, {\PPP}_μ^{q, hg}(E \times F) $$
$${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\PPP}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $μ, ν, h$ and $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_15774 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A note on the generalized Hausdorff and packing measures of product sets in metric space Guedri, Rihab Attia, Najmeddine Classical Analysis and ODEs Let $μ$ and $ν$ 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 ${\RRR}_μ^{q, h}$ and the weighted generalized packing measure ${\QQQ}_μ^{q, h}$ to give some product inequalities : $${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le {\HHH}_μ^{q, h}(E) \; {\RRR}_ν^{q, g}(F) \le {\RRR}_{μ\times ν}^{q, hg}(E\times F)$$ and $${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le {\QQQ}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ for all $E\subseteq \X$ and $F\subseteq \Y$, where ${\HHH}_μ^{q, h}$ and ${\PPP}_μ^{q, h}$ 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 $${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ $${\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) \le c\, {\PPP}_μ^{q, hg}(E \times F) $$ $${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\PPP}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $μ, ν, h$ and $g$. |
| title | A note on the generalized Hausdorff and packing measures of product sets in metric space |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2110.15774 |