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Hausdorff measures in topological spaces Willmott, Richard C.

Abstract

Given a non-negative set function τ on a family ɑ of subsets of a metric space X, an outer measure ν can be generated on X as follows: for B⊂X and δ > 0, [Equations omitted] The Hausdorff s-dimensional and h-measures are special cases of this measure. A number of processes have been suggested for generating a measure on an arbitrary topological space, which generalize this Hausdorff measure process in a metric space. In this thesis we introduce and study a process for generating a measure on.an arbitrary space, which abstracts the essential idea behind all the Hausdorff measures and their generalizations, and contains them as special cases. In chapter I the concept of a measure generated on a space by a gauge and a filterbase is introduced. We show that with any such filterbase is automatically associated a topology for the space, the filterbase topology. We then impose different conditions on the filterbase and deduce resulting properties of the filterbase topology and of the measure. Measurability and approximation properties of the measure are obtained for sets defined in terms of the filterbase, and then for sets defined in terms of the filterbase topology, such as closed, compact, etc. In chapter II we consider measures generated on a topological space. We show that previous measures are special cases of our measure and that known measurability and approximation results can be obtained for them from our general theory. The relationship between the given topology and the topologies of the filterbases used to generate the various measures is examined. A number of additional processes for generating a measure on a topological space are investigated and relations among the various measures are studied. In chapter III we consider several processes for generating measures on a quasi-uniform space, showing that a number of the previously studied measures are included. In particular, we study the measure generated on a uniform space, and obtain some measurability properties by applying our general theory. In chapter IV we work in a compact Hausdorff space and generate a measure using the uniformity for the space and the process of the previous chapter. For the first time, restrictions are placed on the generating set function τ. We examine some consequences of this restriction and then introduce a partial ordering on the family of such functions which generalizes the usual ordering on the h-functions in Hausdorff h-measure theory. This ordering has been used in connection with studies of non-σ- finiteness. We show here that its interest is essentially limited to the metric case.

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