 Wiener covering lemma

A finite collection of Euclidean open balls has a disjoint subcollection whose concentric triples cover the original union. Greedily retain a largest remaining ball and discard all intersecting ones. Each discarded ball has radius no greater than the retained one and lies in its triple. Lebesgue measure consequently bounds the original union by $3^d$ times the sum of the retained volumes. This covering lemma is distinct from inverse-closedness results for the Wiener algebra.