Vitali Covering Lemma

In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali (Vitali 1908). The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E  of Rd by a disjoint family extracted from a Vitali covering of E.