Proposition 8

a) Let H be a set of functions $\geq 0$, lower semi-continuous, directed for the relation $\leq$; then

b) Let H be a set of functions $\geq 0$, upper semi-continuous, directed for the relation $\geq$; if there exists in H a function $h_0$ such that $\mu^\bullet(h_0) < +\infty$, then

The assertion a) is, in view of Prop. 4, a repetition of Theorem 1 of Ch. IV, §1, No. 1. To establish b), set $\eta = \inf_{h \in H} h$, and let $a$ be a number $> 0$. There exists a compact set K such that (Ch. IV, §4, No. 4, Cor. 1 of Prop. 5):

The functions $h\varphi_K$, where $h$ runs over $H$, form a set of upper semi-continuous functions, directed for the relation $\geqslant$, which contains an integrable function. Therefore (Ch. IV, §4, No. 4, Cor. 2 of Prop. 5):

But (Ch. IV, §4, No. 4, Cor. 1 of Prop. 5) $\mu^\bullet(h_0\varphi_{C_K}) \leqslant a$, whence $\mu^\bullet(h\varphi_{C_K}) \leqslant a$ for every function $h \in H$ such that $h \leqslant h_0$. Therefore, finally:

The inequality $\mu^\bullet(\eta) \leqslant \inf_{h \in H} \mu^\bullet(h)$ being obvious, and $a$ being arbitrary, the proposition is established.