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 R^d 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 ${\displaystyle B_{j_{1}}}$ be the ball of largest radius. Inductively, assume that ${\displaystyle B_{j_{1}},\dots ,B_{j_{k}}}$ have been chosen. If there is some ball in ${\displaystyle B_{1},\dots ,B_{n}}$ that is disjoint from ${\displaystyle B_{j_{1}}\cup B_{j_{2}}\cup \cdots \cup B_{j_{k}}}$, let ${\displaystyle B_{j_{k+1}}}$ be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set m := k and terminate the inductive definition.

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