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.
Read more about this topic: Vitali Covering Lemma
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