Corollary 3

— Let $f$ be a bounded numerical function, upper semi-continuous on $X$ and with compact support. Then, the mapping $\mu \mapsto \int f\, d\mu$ is upper semi-continuous on $\mathscr{M}_+(X)$ for the vague topology.

If $h$ is a function in $\mathscr{K}_+(X)$ such that $|f| \leq h$ (Ch. III, §1, No. 2, Lemma 1) then $0 \leq f + h \leq 2h$, and since $f + h$ is upper semi-continuous, it follows from Cor. 1 that $f$ is $\mu$-integrable for every measure $\mu$ on $X$. Moreover, $\mu(f) = \mu(h) - \mu(h - f)$ and $h - f$ is a lower semi-continuous function $\geq 0$. Since the mapping $\mu \mapsto \mu(h - f)$ is lower semi-continuous on $\mathscr{M}_+(X)$ for the vague topology (§1, No. 1, Prop. 4), this proves the corollary.