Chapter 24 - Geometric Measure Theory: Selected Concepts, Results and Problems

Abstract

This chapter elaborates various aspects of geometric measure theory. One of the main themes of geometric measure theory is the detailed study of the geometric properties of general sets and Borel measures. In measure theory, usually a measure means a nonnegative countably additive set function defined on an σ-algebra of subsets. For any two positive integers, there are many distinct dimensional Borel regular measures that are invariant under the Euclidean group and coincide with the surface measure, when restricted to nice dimensional subset. It is found that using Hausdorff measures is the most natural way to measure lower dimensional objects in higher dimensional space. The structure theory for integral dimensional sets is elaborated in the chapter. A characterization of rectifiable sets in terms of their projection and tangential properties is presented in the chapter. Concerning tangential properties, a rectifiable 1-set has an approximate tangent at almost all its points. In contrast, at almost all points of a purely unrectifiable 1-set no approximate tangent exists. The densities of measures and rectifiability are also elaborated in the chapter.