Hausdorff Measure

Abstract

From now on, we abandon the heuristic style which was used in Part I. We shall give a rigorous definition of one of the terms used previously: the length. In this chapter, we define the Hausdorff outer measure of dimension a in a metric space E. We shall see that when E = ℝN and a is an integer, this definition is a generalization of length (α = 1), area (α = 2), volume (α = 3). Those quantities can be easily defined for smooth curves, surfaces, etc. The main difficulty of a more general definition, suitable for objects like the “edges” in an image, is maintaining the basic properties which are expected from such a measure, particularly the additivity. It must be emphasized that, in the very general framework of outer measures, the additivity is not ensured. In the following, we shall define the main formal properties of outer measures and give a characterization of the sets on which an outer measure becomes additive (measurable sets) when it is defined on a topological space (Section 2). Then, we shall apply this theory to the Hausdorff outer measure. Sections 3 and 4 are devoted to two basic examples of sets with finite Hausdorff length: the path connected sets on the one side which will be a paradigm for “edge sets”. The second example, somehow orthogonal to the first, is the classical two-dimensional Cantor set. This set shows all characteristics of what we shall later call “fully irregular” or “fully unrectifiable” sets.