Exercise 23, § 3

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$, $f$ and $g$ two positive locally $\beta$-integrable functions. Suppose that $f$ and $g$ are convolable and that one of these functions is zero in the complement of a countable union of compact sets. Now prove that $f * g$ is equal locally almost everywhere to a lower semi-continuous function (use Prop. 15 of No. 5).