Exercise 2, § 1

Show that, for every function $f \geqslant 0$ defined on $T$, the mapping $\mu \mapsto \mu^\bullet(f)$ of $\mathscr{M}_+(T)$ into $\overline{\mathbf{R}}$ is lower semi-continuous for the quasi-strong topology (Ch. III, §1, Exer. 8; cf. Ch. IV, §1, Exer. 6 b)). Under what condition is this mapping continuous?