Semicontinuity Properties of the Hausdorff Measure

Abstract

In this chapter, we first define a natural metric on the set of the closed sets of a metric space E: the Hausdorff distance. For this distance, we prove that the set of closed subsets of a compact set is compact. It would be extremely convenient to have the following continuity property: If a sequence of sets A_n converges for the Hausdorff distance towards a set A, then the Hausdorff measures of the A_n also converge to the Hausdorff measure of A. We give examples which show that this is in general not true. However, simple and useful conditions can be given on the sequence A_n in order that the Hausdorff measure is lower semicontinuous, that is,

$${{\cal H}^\alpha }(A)\, \le \,\mathop {\lim }\limits_{\,\,\,\,\,\,\,\,\,\,\,\,\,n} \inf \,\,\,\,{{\cal H}^\alpha }({A_n}).$$

(1)

We show that this last property is true when the sets A_n are “uniformly concentrated” (a property which we shall show to be true in Chapter 15 of this book for minimizing sequences of segmentations.