Upper and lower bounds for the packing measure in relation to the Hausdorff measure
type: Item
instance of: scholarly article
title: Upper and lower bounds for the packing measure in relation to the Hausdorff measure
zbMATH Open document ID: 0729.28006
author: Sandra Meinershagen
published in: Real Analysis Exchange
publication date: 1991-01-01T00:00:00Z
review text: The packing measure \(p^ h(E)\) of a set E in an Euclidean space as considered here is defined by \[ p^ h(E)=\inf \{\sum_{i}P^ h(E_ i);\quad E\subseteq \cup_{i}E_ i\}, \] where h is a Hausdorff function and \(P^ h(E_ i)\) denotes the upper limit of all sums \(\sum_{j}h(2r_ j)\) of packings of \(E_ i\) by pairwise disjoint closed balls \(B(x_ j,r_ j)\) centred at \(E_ i\). If further \(m^ h\) denotes the Hausdorff measure and \(\underline d^ h_ m\) the lower Hausdorff density function defined as follows \[ \underline d^ h_ m(x)=\liminf_{r\to 0} [m^ h(E\cap B(x,r))/h(2r)] \] then for a partition of a \(p^ h\)-measurable set E into \(p^ h\)-measurable sets \(E_ n\) satisfying \(\underline d^ h_ m(x)<a_ n<1\forall x\in E\) a.e. \(m^ h\) the author obtains \[ p^ h(E)\geq \sum_{n}a_ n^{- 1}m^ h(E_ n). \] A similar estimate for an upper bound is found. In the second part symmetric Cantor sets of the real line are considered. Provided that Hausdorff and packing dimension coincide the author gives an upper bound for the lower Hausdorff density function. The estimations concerning endpoints of symmetric sets are convincing, but for limit points which are not endpoints the reviewer could only follow the proof under the additional assumption that the length of the removed middle interval is larger than the two which are left.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 4204723
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: packing measure
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: lower Hausdorff density function
zbMATH Keywords: symmetric Cantor sets
zbMATH Keywords: packing dimension
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4204723