Covering theorems and applications

Abstract

Geometric Measure Theory is a highly active field of research in mathematics in large part due to its applications to several fields ranging from calculus of variations to differential geometry. One area of interest in Geometric Measure Theory is that of coverings. The notion of a set covering another is highly convenient when performing analysis. Through coverings we may construct, for example, the Hausdorff Measure. This measure is most widely used due to its ability to obtain the Hausdorff Dimension. This value is highly studied in various contexts such as in dynamical systems and it is used to generalise the usual notion of dimensionality by allowing for fractional dimensions which are in turn useful when analyzing fractal sets. Other widely known types of coverings are the Vitali, Besicovitch and Whitney coverings. Through these one may derive important results such as the Lebesgue Differentiation Theorem as well as the existence and construction of a highly useful extension operator. In this dissertation, we shall first discuss different ways of constructing measures and use these to derive the well-known Lebesgue measure. The Hasudorff measure is then subsequently derived and discussed and finally the coverings mentioned will also be elaborated upon. Applications to derive other results in abstract mathematics shall also be given.