The relations of Hausdorff,*-Hausdorff and packing measures
type: Item
instance of: scholarly article
title: The relations of Hausdorff,*-Hausdorff and packing measures
zbMATH Open document ID: 0729.28007
author: Hung Hwan Lee
author: In-Soo Baek
published in: Real Analysis Exchange
publication date: 1991-01-01T00:00:00Z
review text: The authors investigate the size of thin sets in \({\mathbb{R}}^ d\) by Hausdorff measure, the so-called centered covering measure and the packing measure. The definition of centered covering measure differs from that of Hausdorff measure one in that only centered balls are allowed for covering. The packing measure has a two stage definition involving first the introduction of a packing premeasure P which is defined as follows \(\phi -P(E)=\limsup_{\delta \to 0} \{\sum^{\infty}_{n=1}\phi (d(B_ n))\); \(B_ n\) are disjoint closed balls centered in E with \(d(B_ n)\leq \delta \}\), where \(\phi\) is a Hausdorff function. Unfortunately, the authors have not consequently used closed balls in their definition, but this is necessary to apply Vitali's theorem in density inequalities for the considered measures cited in section 3 of the paper. The second stage to define the final packing measure is \[ \phi -p(E)=\inf \{\sum^{\infty}_{n=1}\phi -P(E_ n);\quad E\subseteq \cup_{n}E_ n\}. \] In section 4 two notions of strong \(\phi\)- regularity are considered, that packing and Hausdorff measure (or centered covering measure) coincide for a measurable set of finite packing measure. Most obtained results are at least implicitely known and due to \textit{X. Saint Raymond} and \textit{C. Tricot} [Math. Proc. Camb. Philos. Soc. 103, No.1, 133-145 (1988; Zbl 0639.28005)].
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 4204724
zbMATH Keywords: strong regularity
zbMATH Keywords: size of thin sets
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: centered covering measure
zbMATH Keywords: packing measure
zbMATH Keywords: density inequalities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4204724