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 covering theorem is credited to the Italian mathematician Giuseppe Vitali. 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 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 covering theorem is credited to the Italian mathematician Giuseppe Vitali. 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. Comments. Without loss of generality, we assume that the collection of balls is not empty; that is, n > 0. Let B j 1 {displaystyle B_{j_{1}}} be the ball of largest radius. Inductively, assume that B j 1 , … , B j k {displaystyle B_{j_{1}},dots ,B_{j_{k}}} have been chosen. If there is some ball in B 1 , … , B n {displaystyle B_{1},dots ,B_{n}} that is disjoint from B j 1 ∪ B j 2 ∪ ⋯ ∪ B j k {displaystyle B_{j_{1}}cup B_{j_{2}}cup cdots cup B_{j_{k}}} , let B j k + 1 {displaystyle B_{j_{k+1}}} be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set m := k and terminate the inductive definition. Now set X := ⋃ k = 1 m 3 B j k {displaystyle X:=igcup _{k=1}^{m}3,B_{j_{k}}} . It remains to show that B i ⊂ X {displaystyle B_{i}subset X} for every i = 1 , 2 , … , n {displaystyle i=1,2,dots ,n} . This is clear if i ∈ { j 1 , … , j m } {displaystyle iin {j_{1},dots ,j_{m}}} . Otherwise, there necessarily is some k ∈ { 1 , … , m } {displaystyle kin {1,dots ,m}} such that Bi intersects B j k {displaystyle B_{j_{k}}} and the radius of B j k {displaystyle B_{j_{k}}} is at least as large as that of Bi. The triangle inequality then easily implies that B i ⊂ 3 B j k ⊂ X {displaystyle B_{i}subset 3,B_{j_{k}}subset X} , as needed. This completes the proof of the finite version. Let F denote the collection of all balls Bj, j ∈ J, that are given in the statement of the covering lemma. The following result provides a certain disjoint subcollection G of F. If this subcollection G is described as { B j , j ∈ J ′ } {displaystyle {B_{j},jin J'}} , the property of G, stated below, readily proves that Precise form of the covering lemma. Let  F be a collection of (nondegenerate) balls in a metric space, with bounded radii. There exists a disjoint subcollection  G of  F with the following property: (Degenerate balls only contain the center; they are excluded from this discussion.)Let R  be the supremum of the radii of balls in F. Consider the partition of F into subcollections Fn, n ≥ 0, consisting of balls B  whose radius is in (2−n−1R, 2−nR]. A sequence Gn, with Gn ⊂ Fn, is defined inductively as follows. First, set H0 = F0 and let G0 be a maximal disjoint subcollection of H0. Assuming that G0,...,Gn have been selected, let and let Gn+1 be a maximal disjoint subcollection of Hn+1. The subcollection