Exercise 20, § 4

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$. a) Let $f$ be a bounded numerical function on $G$, uniformly continuous for the left uniform structure. Show that for every bounded measure $\mu$ on $G$, $\mu * f$ is uniformly continuous for the left uniform structure; moreover, with the notations of No. 7, Prop. 19, $f$ is the limit, for the topology of uniform convergence in $G$, of the functions $f_V * f$ with respect to the section filter of $\mathscr{B}$. b) Take $G = T$; give an example of a continuous function $h$ on $T$ that is not of the form $f * g$, where $f$ and $g$ belong to $L^2(T, \beta)$. (Use Exers. 15 b) and 16 of Ch. IV, §6.)