Vitali covering theorem

A measure-theoretic theorem that selects pairwise disjoint balls from a fine family so their fixed enlargements cover almost all of the set.

Vitali covering lemma: A selection result that extracts disjoint balls from a family whose fixed dilations cover the original family. This formulation makes the theorem’s disjoint-ball selection explicit.
Lebesgue measure: The standard measure assigning lengths, areas, and volumes to broad classes of subsets of Euclidean space. The classical theorem identifies the uncovered remainder as Lebesgue-null.
Lebesgue differentiation theorem: A theorem stating that local averages of an integrable function converge almost everywhere to its point values. Disjoint-ball estimates control the exceptional points where differentiation fails.
Besicovitch covering theorem: A Euclidean covering theorem that selects a covering with bounded overlap from families of arbitrarily small balls. Its bounded-overlap conclusion differs from disjoint balls whose enlargements cover almost all points.
Hardy–Littlewood maximal function: A function assigning each point the supremum of averages of an absolute-valued function over neighborhoods containing it. The covering theorem bounds the measure of points where this maximal function is large.
Null set: A set that can be covered by measurable sets of arbitrarily small total measure. The theorem permits a leftover set only when it has measure zero.
Weak-type inequality: An operator bound that controls the measure of an output’s superlevel set by an input norm. Covering selections yield the weak-type estimate for the maximal operator.
Five-covering lemma: A metric covering lemma that selects disjoint balls whose fivefold dilations cover the original centers. It gives a full covering by enlarged balls under hypotheses different from the almost-everywhere Vitali conclusion.
Outer regularity: A property of a measure that approximates measurable sets from outside by open sets. Open-set approximation supplies convenient ball families in many covering arguments.
Open ball: The set of points at distance less than a fixed radius from a center in a metric space. Balls are the basic covering sets selected by the theorem.

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Denjoy–Luzin theoremRelated: Interval coverings support measure estimates used in classical arguments about derivatives and exceptional points.