Hausdorff measure functions

Abstract

In many works on Hausdorff Measure Theory it has been the practice to place certain restrictions on the measure functions used. These restrictions usually ensure both the monotonloity and the continuity of the functions The aim of the first four chapters of this thesis is to find conditions under which the restrictions of continuity and monotonicity may be relaxed.In the first chapter we deal with the monotonicity condition with respect to both measures and pre-measures. The second and third chapters are concerned with an investigation of the continuity condition with regard to measures and pre-measures, respectively. Then, having found conditions under which these restrictions may or may not be relaxed, we are able, in the fourth chapter, to generalize some known results to the case of discontinuous and non-mono tonic functions.Seme of the results of the first four chapters prompted an Investigation of the properties of measures corresponding to sequences of measure functions, and this is incorporated in the fifth chapter.The main purpose of the final chapter is to determine whether or not some of the results of the earlier chapters may be extended to Hilbert space.