Title:

Packing measures, packing dimensions, and the existence of sets of positive finite measure

A number of definitions of packing measures have been proposed at one time or another, differing from each other both in the notion of packing they employ, and in whether the radii or the diameters of the balls of the packing are used. In Chapter 1 various definitions of packing measures are considered and relationships between these definitions established. Chapter 2 presents work which was done jointly with Professor D. Preiss, and which has been published as such. It is shown here that, with one of the possible radiusbased definitions of packing measure, every analytic metric space of infinite packing measure contains a compact subset of positive finite measure. It is also indicated how this result carries over to other radiusbased packing measures in the case of Hausdorff functions satisfying a doubling condition. In Chapter 3 a construction is described which provides, for every Hausdorff function h, a compact metric space of infinite diameterbased h packing measure, with no subsets of positive finite measure. It is then indicated how such a construction may be modified to deal with certain Hausdorff functions which do not satisfy a doubling condition, and a radiusbased definition of packing measure. In Chapter 4 we consider topological and packing dimensions, and show that if X is a separable metric space, then dimT(X) = min {dimQ(X') : X' is homeomorphic to X} , where dimQ denotes the packing dimension associated with any one of the packing measures considered in this work, and dimT denotes topological dimension. Chapter 5 answers the question, for which Hausdorff functions h may the Hausdorff and packing measures, and HhA and PhA, agree and be positive and finite for some A ⊆ Rn. We show that the assumption that the two measures agree and are positive and finite on some subset of Rn implies that the function h is a regular density function (in the sense of Preiss). The converse result is also provided, that for each regular density function h, there is a subset A of Rn such that HhA = PhA and this common measure is positive and finite.
