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

Rihab Guedri, Najmeddine Attia

Abstract

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 $g$.

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

Abstract

Let and be two Borel probability measures on two separable metric spaces and respectively. For be two Hausdorff functions and , we introduce and investigate the generalized pseudo-packing measure and the weighted generalized packing measure to give some product inequalities : and for all and , where and is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant such that These appropriate inequalities are more refined than well know results since we do no assumptions on and .

Paper Structure

This paper contains 12 sections, 15 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Generalized fractal measures
  3. Generalized packing $h$-measures
  4. Example
  5. Weighted generalized packing $h$-measure
  6. Generalized Hausdorff $h$-measures
  7. Proof of Theorem A
  8. Proofs of Theorems B and C
  9. Densities
  10. Proof of Theorem B
  11. Proof of Theorem C
  12. appendix

Key Result

Corollary 1

Let $E\subseteq {\mathbb X}$, $F\subseteq {\mathbb Y}$, $\mu \in \mathcal{P}(\mathbb{X})$, $\nu \in \mathcal{P}(\mathbb{Y})$ and $h,g \in {\mathcal{F}}$. Assume that ${\mathbb X}$ and ${\mathbb Y}$ are amenable to packing, then there exist a constant $c>0$ such that provided that the product on the right-hand side of the first and the last inequalities is not of the form $0\times \infty$ or $\in

Theorems & Definitions (29)

  • Corollary 1
  • Remark 1
  • Definition 1
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Definition 2
  • ...and 19 more