Covering Lemmas in a Metric Space

Abstract

Of course, we only have access to the Hausdorff measure of a given set A by looking at its coverings. Clearly, the definition (6.1) is very abstract and gives no indication about how a covering of A should be to give an accurate account of the Hausdorff measure. This section is devoted to several criteria which will tell us when a covering is a “good” covering, correctly approximating the Hausdorff measure. As we shall see, and this is rather surprising, the first criterion simply is that the covering is made of sets with diameter small enough. We shall also focus on additional properties that we would like to require from a covering. The main property is that the sets of the covering should be disjoint. Indeed, if they are, we may add inequalities concerning A obtained on each one of the sets of the covering and obtain global estimates on A. Thus, we may pass from local estimates on a covering of A to global estimates on the set A itself. The Vitali Covering Lemma is an essential tool to do that and we shall use it constantly in the next chapters. This chapter ends with a classical application of the Vitali Covering Lemma: Lebesgue measure and the N- dimensional Hausdorff measure agree on ℝ^N.