Exercise 23, § 4

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