On the theory of Hausdorff measures in metric spaces

In this work the main objective is to extend the theory of Hausdorff measures in general metric spaces. Throughout the thesis Hausdorff measures are defined using premeasures. A condition on premeasures of ‘finite order’ is introduced which enables the use of a Vitali type covering theorem. Weighted Hausdorff measures are shown to be an important tool when working with Hausdorff measures defined by a premeasure of finite order. The main result of this thesis is the existence of subsets of finite positive Hausdorff measure for compact metric spaces when the Hausdorff measure has been generated by a premeasure of finite order. This result then extends to analytic subsets of complete separable metric spaces by standard techniques in the case when the increasing sets lemma holds. The proof of this result uses techniques from functional analysis. In this respect the proof presented is quite different from those of the previous literature. A discussion on Hausdorff–Besicovitch dimension is also to be found. In particular the problem of whether dim (X) + dim (Y ) ≤ dim (X × Y ) is solved in complete generality. Generalised dimensions involving partitions of Hausdorff functions are also discussed for product spaces. These results follow from a study of the weighted Hausdorff measure on product spaces. An investigation is made of the sufficiency of some conditions for the increasing sets lemma to hold. Some counterexamples are given to show insufficiency of some of these conditions. The problem of finding a counterexample to the increasing sets lemma for Hausdorff measures generated by Hausdorff functions is also examined. It is also proved that for compact metric spaces we may also approximate the weighted Hausdorff measure by finite Borel measures that are ‘dominated’ by the premeasure generating the weighted Hausdorff measure. I am greatly indebted to Professor David Preiss for proposing this investigation and for much wise advice. In particular he suggested the study of the weighted Hausdorff measures. I should like to express my gratitude to Professor C.A. Rogers for his help and constructive criticism on the manuscript. I am also grateful to the Science and Engineering Research Council for the funding of my studies. J.D.H.