# Density Properties

• Jean Michel Morel
• Sergio Solimini
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 14)

## Abstract

In this chapter and the following, we shall be concerned with properties which are true for all points of a given set A, except a set of points with zero Hausdorff measure. In this situation, we say that the considered property is true almost everywhere (a.e.) in K. We call α-set any $${{\cal H}^\alpha }(K)\, < \, + \infty .$$- measurable subset of a metric space E such that $$\lim {\inf _{r \to 0}}{{{{\cal H}^1}(K \cap B(x,r))} \over {2r}}$$. We shall begin to develop the Besicovitch theory of sets with finite Hausdorff measure.

In order to explain the relevance of the results proved below, let us take as an (important) example, the case a = 1. Then obvious examples of 1-sets are the differentiable Jordan curves with finite length. It is easily seen that for such sets A, the length contained in a ball B(x,r) centered at a point x of A is equivalent to 2r as r tends to zero. This relation is true for all points of A except for the endpoints of the curve. Let us take an example of quite different structure: the “planar Cantor set” defined in Section 4 of Chapter 4. Let us denote this set by K. It is easily seen (and we prove at the end of the present chapter) that the “spherical lower density” lim inf $${{\cal H}^\alpha } -$$ is not 1. So we observe a quite different spatial repartition for K and for smooth curves. It is expected that a general 1-set will have points where “curve-like” behaviour occurs (the spherical density is 1) and points where “Cantor- like” behaviour occurs. The first kind of point will be called “regular” and the other one “irregular”. One of the most remarkable results of the Besicovitch theory is that this classication is stable when one cuts a set into measurable pieces: The set of the regular points of an α-set K which are contained in a measurable subset J is essentially the same as the set of regular points of J!

The same thing happens for irregular points. So the properties of “regularity” and “irregularity” are intrinsic, local, stable properties. Every set will be divided by the Besicovitch theory into its regular part (which we will later prove to be contained in a bunch of rectifiable curves) and its irregular part (which will be, under many aspects, similar to the planar Cantor set). Those parts clearly behave like two nonmiscible fluids, and the Besicovitch theory yields clear computational criteria (the local spherical densities) to decide whether a point is regular or irregular. From the Image Processing viewpoint, we can say that the Besicovitch theory yields an absolute definition of which points in an “edge map” are really points of an edge and which are not (the irregular points).