The relations of Hausdorff,*-Hausdorff and packing measures
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)].
scientific article; zbMATH DE number 4204724