A Probabilistic Proof of the Vitali Covering Lemma

The Vitali Covering Lemma states that, given a finite collection of balls in R^d, there exists a disjoint subcollection that fills at least 3^{−d} of the measure of the union of the original collection. We present classical proofs of this lemma due to Banach and Garnett. Subsequently, we provide a new proof of this lemma that utilizes probabilistic “Erdo ̈s” type techniques and Padovan numbers.