Corollary 2

— Every locally negligible open set is negligible (and is therefore contained in the complement of the support of $\mu$).

For, the outer measure of an open set G is the supremum of the measures $|\mu|(K)$ of the compact sets $K \subset G$ (§ 4, No. 6, Cor. 4 of Th. 4); if G is locally negligible then $|\mu|(K) = 0$ for every compact set K contained in G, therefore $|\mu|^*(G) = 0$.