Definition 3

— A set $A \subset X$ is said to be locally negligible (for the measure $\mu$) if, for every $x \in X$, there exists a neighborhood $V$ of $x$ such that $V \cap A$ is negligible.

By the principle of localization, every locally negligible set is measurable. The properties of negligible sets (§2) show that every subset of a locally negligible set is locally negligible, and that every countable union of locally negligible sets is locally negligible.