Vitali Covering Lemma - Vitali Covering Theorem

Vitali Covering Theorem

In the covering theorem, the aim is to cover, up to  a "negligible set", a given set E ⊆ Rd by a disjoint subcollection extracted from a Vitali covering for E : a Vitali class or Vitali covering for E  is a collection of sets such that, for every x ∈ E  and δ > 0, there is a set U  in the collection such that x ∈ U  and the diameter of U  is non-zero and less than δ.

In the classical setting of Vitali, the negligible set is a Lebesgue negligible set, but measures other than the Lebesgue measure, and spaces other than Rd have also been considered, see below.

The following observation is useful: if is a Vitali covering for E  and if E  is contained in an open set Ω ⊆ Rd, then the subcollection of sets U  in that are contained in Ω  is also a Vitali covering for E.