Covering Theorem (other)
In mathematics, covering theorem can refer to *Besicovitch covering theorem *Jensen's covering theorem *Vitali covering lemma In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The co ...

Besicovitch Covering Theorem
In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset ''E'' of the Euclidean space R''N'' by balls such that each point of ''E'' is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant ''c''N depending only on the dimension ''N'' with the following property: * Given any Besicovitch cover F of a bounded set ''E'', there are ''c''''N'' subcollections of balls ''A''1 = , …, ''A''''c''''N'' =  contained in F such that each collection ''A''i consists of disjoint balls, and : E \subseteq \bigcup_^ \bigcup_ B. Let G denote the subcollection of F consisting of all balls from the ''c''''N'' disjoint families ''A''1,...,''A''''c''''N''. The less precise following statement is clearly true: every point ''x'' ∈ R''N'' belongs to at most ''c''''N'' different balls from the subcollection G, and G remains a cover for ''E'' (every p ...
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Jensen's Covering Theorem
In set theory, Jensen's covering theorem states that if 0# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universe is close to the universe of all sets. The first proof appeared in . Silver later gave a fine-structure-free proof using his machines and finally gave an even simpler proof. The converse of Jensen's covering theorem is also true: if 0# exists then the countable set of all cardinals less than ℵω cannot be covered by a constructible set of cardinality less than ℵω. In his book ''Proper Forcing'', Shelah proved a strong form of Jensen's covering lemma. Hugh Woodin William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ... states it as:"In search of U ...
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