The Vitali Covering Theorem states that from any collection of balls covering a set, each ball small enough relative to its neighbors, a countable disjoint subcollection can be chosen that still covers almost all of the set, up to a set of measure zero. Named for Giuseppe Vitali, it is a basic tool of real analysis and measure theory, underlying the proof of the Lebesgue Differentiation Theorem and other covering-based arguments.
Facts
StatementIt is possible to cover, up to a Lebesgue-negligible set, a given subset E of R^d by a disjoint family extracted from a Vitali covering of E. 1 Classification
Statement Form Vitali Covering Theorem
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Source Vitali covering lemma, Wikipedia
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Source Giuseppe Vitali (Wikipedia)
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Source Vitali covering lemma, Wikipedia
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1. Vitali covering lemma, Wikipedia
Lead paragraph
The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
In Branch: Measure Theory, Lead sentence
mbinatorial and geometric result commonly used in measure theory of Euclidean spaces.
View the SourceGiuseppe Vitali (Wikipedia)
Principal Result Of: Giuseppe Vitali, Article textQuote, Principal Result Of: Giuseppe Vitali, Article text
(Title translation) "On groups of points and functions of real variables" is the paper containing the first proof of Vitali covering theorem.
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