Proposition 5

— For a set A to be locally negligible, it is necessary and sufficient that, for every compact set K, $A \cap K$ be negligible.

The condition is obviously sufficient since every point of X has a compact neighborhood. It is necessary, because if, for every $x \in K$, there exists a neighborhood $V_x$ of $x$ such that $A \cap V_x$ is negligible, then there exists a finite number of points $x_i \in K$ such that the $V_{x_i}$ form a covering of K, and $A \cap K$ is contained in the union of the negligible sets $A \cap V_{x_i}$.