Proposition 11

— Let $\mathscr{G}$ be a set, directed for the relation $\subset$, of integrable open sets in $X$; for $A = \bigcup_{G \in \mathscr{G}} G$ to be integrable, it is necessary and sufficient that $\sup_{G \in \mathscr{G}} |\mu|(G) < +\infty$, in which case $\mu(A) = \lim_\mathscr{G} \mu(G)$ and $|\mu|(A) = \sup_{G \in \mathscr{G}} |\mu|(G)$.

For, one knows that $|\mu|^*(A) = \sup_{G \in \mathscr{G}} |\mu|(G)$ (\§1, No. 2, Prop. 7); the proposition therefore follows from Prop. 10.