Proposition 5

a) Let H be an increasing directed set of functions $\geqslant 0$ that are lower semi-continuous on every compact subset of T. Then

b) Let H be a decreasing directed set of functions $\geqslant 0$ that are upper semi-continuous on every compact subset of T. If there exists in H a function $h_0$ such that $\mu^\bullet(h_0) < +\infty$, then

For every compact set $K \subset T$, we have in case a)

and in case b)

by Prop. 2 of No. 2, and Prop. 8 of Ch. V, §1, No. 2. Case a) follows at once, by passage to the supremum with respect to K (No. 2, Prop. 2). To treat case b), denote by $\varepsilon$ a number $> 0$, and choose a compact set K such that $\mu^\bullet(h_0 \varphi_K) \geqslant \mu^\bullet(h_0) - \varepsilon$. We then have (No. 5, Prop. 4) $\mu^\bullet(h_0 \varphi_{C_K}) \leqslant \varepsilon$; for every function $h \in H$ that is $\leqslant h_0$, we therefore have $\mu^\bullet(h \varphi_{C_K}) \leqslant \varepsilon$, and finally $\mu^\bullet(h \varphi_K) \geqslant \mu^\bullet(h) - \varepsilon$ by Prop. 4 of No. 5. Therefore

Consequently the left side of (6) is $\geqslant$ the right side; the reverse inequality being obvious, the proposition is established.