Proposition 7

— Let T be a completely regular space. Let $\mu$ be a bounded positive measure on T, and let f be a bounded positive function on T, such that the set of points of T where f is not continuous is locally $\mu$-negligible. Then the mapping $\lambda \mapsto \lambda^*(f)$ of $\mathscr{M}_+^b(T)$ into $\mathbf{R}$ is continuous at the point $\mu$.

For every $t \in T$, set $f'(t) = \liminf_{s \to t} f(s)$, $f''(t) = \limsup_{s \to t} f(s)$. Obviously $f' \leqslant f \leqslant f''$, with equality at every point of T where f is continuous (hence $\mu$-almost everywhere). On the other hand, $f'$ is lower semi-continuous, $f''$ is upper semi-continuous and bounded (GT, IV, §6, No. 2, Prop. 4). We therefore have the following relations by Prop. 6,

One concludes by observing that $\mu^*(f') = \mu^*(f'')$, because $f'$ and $f''$ are equal locally $\mu$-almost everywhere.