Some properties of Hausdorff measure theory

Abstract

CHAPTER I The definition of all the measure functions used in the thesis. CHAPTER II The condition for a measure function to to be a Hausdorff diametral dimension function in p-dimensional real Euclidean space is first established. Then the fact that an analytical set of infinite Hausdorff diametral measure is then proved and the necessary and sufficient conditions for a subset of a set with Hausdorff diametral dimension function h(x) to have dimension function g(x) are established. CHAPTER III Conditions on the dimension function of the cartesian product of two one-dimensional sets whose dimension functions are known, are established. CHAPTER IV The proof of the existence of a plane set S with Hausdorff diametral dimension function x2[alpha] [equation], such that if S log 3 is translated through any distance in the plane then the intersection of S with itself translated has zero Hausdorff diametral measure with dimension function x2[alpha]/ CHAPTER V The two area measures are considered in two dimensional real Euclidean space only. The necessary and sufficient condition for a measure function to be a non-metric-area dimension function is established and the metric area measure of sets which are the cartesian products of intervals with linear sets is found. These a re used to deduce that non-metric-area measure is in fact non-metric. The condition for x[alpha] to he a metric area measure is also established. CHAPTER VI This deals with sets on the frontier of the unit circle. First the connection between the area measures and the generalized affine length is established. Then the triangle of minimum area covering a given total arc length is found and finally the necessary and sufficient condition for a measure function to be a Hausdopff diametral dimension function for such sets is found.