Exercise 8, § 2

Let G be a locally compact abelian group, $\mu$ a Haar measure on G, $f$ a $\mu$-measurable finite numerical function on G. Assume that for every $s \in G$, the numerical function

is continuous on G. Show that $f$ is then continuous. (Argue by contradiction; assuming that the function $f$ is not continuous at a point $x_0 \in G$, show first that there exists on G a filter $\mathfrak{F}$ with limit $e$ such that and deduce from this that also $\lim_{\mathfrak{F}, s} |f(sx)| = +\infty$ for every $x \in G$. If $g = |f|/(1 + |f|)$, deduce from the last result a contradiction to the fact that for every compact subset $K \subset G$,