Corollary 3

— In a locally compact space X that is countable at infinity, every locally negligible set is negligible.

Since X is the union of a sequence $(K_n)$ of compact sets, every locally negligible set A is the union of the negligible sets $A \cap K_n$, hence is negligible.

One can give examples of locally compact spaces that are not countable at infinity, and of measures on such a space X such that there exist sets in X that are locally negligible but not negligible (§ 1, Exer. 5).