Vitali covering lemma

E451527

mathematical theorem result in measure theory

The Vitali covering lemma is a fundamental result in measure theory that provides conditions under which a collection of sets can be reduced to a disjoint subcollection that still covers almost all of the original set, and it underpins many key theorems in real analysis and differentiation.

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Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in measure theory ⓘ
appearsIn	advanced real analysis textbooks ⓘ harmonic analysis textbooks ⓘ measure theory textbooks ⓘ
appliesTo	collections of balls in Euclidean space ⓘ collections of intervals in the real line ⓘ metric spaces with a Vitali covering property ⓘ
asserts	from a suitable covering one can extract a disjoint subcollection covering almost all of the set ⓘ the uncovered part of the set has measure zero under suitable hypotheses ⓘ
assumes	a Vitali covering of a measurable set ⓘ boundedness or local finiteness conditions on the covering ⓘ
concerns	Lebesgue measure NERFINISHED ⓘ covering of sets by families of sets ⓘ outer measure ⓘ selection of disjoint subcollections ⓘ
field	measure theory ⓘ real analysis ⓘ
generalizes	elementary interval selection arguments in real analysis ⓘ
hasVersion	metric space version ⓘ n-dimensional version for balls in R^n ⓘ one-dimensional version for intervals in R ⓘ outer measure version ⓘ
historicalPeriod	early 20th century mathematics ⓘ
implies	existence of a countable disjoint subcollection ⓘ the union of the disjoint subcollection covers the set up to a null set ⓘ
isToolIn	differentiation theory of integrals ⓘ geometric measure theory ⓘ real-variable harmonic analysis ⓘ singular integral theory ⓘ
namedAfter	Giuseppe Vitali NERFINISHED ⓘ
relatedTo	Besicovitch covering theorem NERFINISHED ⓘ Hardy–Littlewood maximal operator NERFINISHED ⓘ Lebesgue differentiation theorem NERFINISHED ⓘ Vitali covering theorem NERFINISHED ⓘ Vitali set ⓘ
requires	basic properties of Lebesgue measure ⓘ countable subadditivity of outer measure ⓘ
usedFor	proof of Hardy–Littlewood maximal inequality ⓘ proof of Lebesgue differentiation theorem ⓘ proof of Lebesgue differentiation theorem for integrable functions ⓘ proof of Lebesgue differentiation theorem for signed and vector measures ⓘ proof of weak-type (1,1) estimates for maximal functions ⓘ proofs in differentiation of measures ⓘ proofs in geometric measure theory ⓘ proofs of Calderón–Zygmund decomposition variants ⓘ proofs of density theorems ⓘ

Referenced by (1)

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Hardy–Littlewood maximal function → relatedTo → Vitali covering lemma ⓘ