The packing measure of rectifiable sets (Q1066294)

In a recent paper [Trans. Am. Math. Soc. 288, 679-699 (1985; Zbl 0537.28003)] the authors defined \(\phi\)-packing measure, by trying to maximize the \(\phi\) content of balls centered in E with radii less than \(\delta\) and allowing \(\delta \downarrow 0\). In general, \(0\leq \phi - m(E)\leq \phi -p(E)\leq +\infty\) where \(\phi -m\) and \(\phi -p\) denote the Hausdorff and packing measure respectively. The object of the present report is to use packing measure for geometric measure theory. A set E is called strongly regular with respect to \(\phi\) if \(0<\phi -m(E)=\phi - p(E)<\infty\). The authors show that for \(\phi (s)=s^ k\), k a positive integer, regularity in the Besicovitch-Federer sense is implied by strong regularity, but not conversely. Precise density inequalities are obtained for subsets of the plane and \(\phi (s)=s\), and there is an example to show that Hausdorff dimension may be quite different from packing dimension.