Definition 2

— Given a positive measure $\mu$ on $X$, for every open set $G \subset X$ the upper integral $\mu^*(\varphi_G)$ is called the outer measure of $G$ and is denoted $\mu^*(G)$.

The outer measure of an open set $G$ is thus a number $\geq 0$, finite or equal to $+\infty$. Clearly $\mu^*(\varnothing) = 0$. Moreover, $\mu^*(X) = \| \mu \|,$ as is shown by formula (22) of Ch. III, §1, No. 8.