Some properties of new general fractal measures

被引:15
作者
Achour, Rim [1 ]
Selmi, Bilel [1 ]
机构
[1] Univ Monastir, Fac Sci Monastir, Dept Math, Probabil & Fractals Lab LR18ES17, Monastir 5000, Tunisia
来源
MONATSHEFTE FUR MATHEMATIK | 2024年 / 204卷 / 04期
关键词
General Hausdorff measure; General packing measure; Doubling condition; Outer regular property; PACKING PREMEASURE;
D O I
10.1007/s00605-024-01979-7
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this research, we adopt a comprehensive approach to address the multifractal and fractal analysis problem. We introduce a novel definition for the general Hausdorff and packing measures by considering sums involving certain functions and variables. Specifically, we explore the sums of the form & sum;ih-1(qh(mu(B(xi,ri)))+tg(ri)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum \limits _i h<^>{-1}\Big (q h\big (\mu \bigl (B(x_i,r_i)\bigl )\big )+tg(r_i)\Big ), \end{aligned}$$\end{document}where mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} represents a Borel probability measure on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R<^>d$$\end{document}, and q and t are real numbers. The functions h and g are predetermined and play a significant role in our proposed intrinsic definition. Our observation reveals that estimating Hausdorff and packing pre-measures is significantly easier than estimating the exact Hausdorff and packing measures. Therefore, it is natural and essential to explore the relationships between the Hausdorff and packing pre-measures and their corresponding measures. This investigation constitutes the primary objective of this paper. Additionally, the secondary aim is to establish that, in the case of finite pre-measures, they possess a form of outer regularity in a metric space X that is not limited to a specific context or framework.
引用
收藏
页码:659 / 678
页数:20
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