Exercise 9, § 2

Let G be a locally compact group, $\mu$ a left Haar measure on G, $f$ a $\mu$-integrable function. Let $\mathscr{B}$ be a filter base formed of $\mu$-integrable sets of measure $> 0$, having $e$ as limit. For every $B \in \mathscr{B}$, set $f_B(t) = \frac{1}{\mu(B)} \int_B f(st)\, d\mu(s)$. Show that for every integrable subset A of G, $\lim_{\mathscr{B}} \int_A f_B(t)\, d\mu(t) = \int_A f(t)\, d\mu(t)$.