Exercise 9, § 1

Adopt the notations of the proof of Th. 1 of No. 2. Let $\beta$ be the unique left Haar measure on $G$ such that $\beta(f_0) = 1$. Show that, for every $f \in \mathscr{K}(G)$, $I_g(f)$ tends to $\beta(f)$ with respect to $\mathscr{B}$. (Let $a$ be a cluster point of $g \mapsto I_g(f)$ with respect to $\mathscr{B}$. There exists an ultrafilter $\mathscr{U}$ finer than $\mathscr{B}$ such that $I_g(f)$ tends to $a$ with respect to $\mathscr{U}$. On the other hand, $I_g(f)$ tends to $\beta(f)$ with respect to $\mathscr{U}$.)