Corollary

— Let $\mathfrak{F}$ be a set, directed for the relation $\supset$, of integrable closed sets in $X$; then the closed set $B = \bigcap_{H \in \mathfrak{F}} H$ is integrable, and one has

For, let $H_0$ be a set in $\mathfrak{F}$; since $H_0$ is integrable, it is contained in an integrable open set $U$ (§ 1, No. 4, Prop. 19); the open sets $U \cap \mathbf{C}H$ form a set directed for the relation $\subset$, are contained in $U$, and have union $U \cap \mathbf{C}B$; we are thus reduced to Prop. 11.