Corollary

— a) Let $(U_\alpha)_{\alpha \in I}$ be an increasing directed family of open subsets of $T$, with union $U$. Then $\mu^\bullet(U) = \sup_{\alpha \in I} \mu^\bullet(U_\alpha)$.

b) Let $(F_\alpha)_{\alpha \in I}$ be a decreasing directed family of closed subsets of $T$, with intersection $F$. If there exists an $\alpha \in I$ such that $\mu^\bullet(F_\alpha)$ is finite, then $\mu^\bullet(F) = \inf_{\alpha \in I} \mu^\bullet(F_\alpha)$.

By the preceding corollary, there exists a largest locally negligible open set; this justifies the following definition: