Vitali covering lemma

A theorem that extracts pairwise disjoint sets from a suitable family of sets such that fixed enlargements of the selected sets cover the family’s union.

Vitali covering theorem: A measure-theoretic theorem asserting that suitable fine coverings admit disjoint subfamilies covering a set up to measure zero. Unlike the lemma’s enlarged-set conclusion, this theorem gives almost-everywhere coverage without enlargement.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, identity, and the triangle inequality. Many formulations use metric balls as the sets being selected and enlarged.
Hardy–Littlewood maximal function: A function assigning each point the supremum of local averages of an integrable function over neighborhoods containing it. Covering lemmas establish weak-type bounds for this operator.
Lindelöf covering theorem: A theorem stating that every open cover of a second-countable space has a countable subcover. It extracts a countable cover, not a disjoint family whose enlargements cover.
Greedy algorithm: An algorithm that repeatedly makes a locally optimal choice among the remaining options. Selecting a largest available set and discarding intersecting sets captures the lemma’s basic construction.
Diameter: The supremum of distances between pairs of points in a set. Diameter controls the scale of sets in formulations for general families.
Lebesgue differentiation theorem: A theorem stating that local averages of an integrable function converge almost everywhere to its value. Covering arguments control the exceptional sets in differentiation proofs.
Heine–Borel theorem: A theorem characterizing compact subsets of Euclidean space by closedness and boundedness. Compactness produces finite subcovers, while the Vitali lemma targets disjoint selections.
Besicovitch covering theorem: A theorem bounding the overlap of selected balls in Euclidean space while covering a specified set. It controls bounded overlap rather than relying on enlarged disjoint selections.
Disjoint sets: Sets are disjoint when their intersection is empty. Pairwise disjointness is the key constraint on the selected subfamily.

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