Definition 7

*A set $\mathfrak{A}$ of subsets of a topological space $T$ is said to be locally countable if, for every $t \in T$, there exists a neighborhood $V$ of $t$ such that the set of $A \in \mathfrak{A}$ that intersect $V$ is countable.*

If the set $\mathfrak{A}$ of subsets of $T$ is locally countable then, for every compact subset $K$ of $T$, the set of $A \in \mathfrak{A}$ that intersect $K$ is countable, since $K$ can be covered by a finite number of open neighborhoods of points of $K$, each of which intersects only a countable set of subsets belonging to $\mathfrak{A}$.

Def. 7 shows that the union of a locally countable set of $\mu$-measurable (resp. locally $\mu$-negligible) subsets of a locally compact space is $\mu$-measurable (resp. locally $\mu$-negligible) (No. 1, Prop. 3 and No. 2, Prop. 5).