Exercise 20, § 3

Let $G$ be a locally compact group, $\beta$ a left Haar measure on G.

a) Let $f$ be a bounded numerical function in G, uniformly continuous for the left uniform structure. Prove 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 $f_v * f$ following the filter of sections of $\mathscr{B}$.

b) Take $G = \mathbf{T}$; give an example of a continuous function $h$ in $\mathbf{T}$ which is not of the form $f * g$, where $f$ and $g$ belong to $L^2(\mathbf{T}, \beta)$. (Use exercises 19 b) and 20 of Chapter IV, § 6.)