Exercise 10, § 2

With the notations of §1, No. 6, Prop. 6 let, for every $\alpha \in A$, $\mu_\alpha$ be a left Haar measure on $G_\alpha$, so that the inverse limit of the $\mu_\alpha$ is a left Haar measure $\mu$ on G. For every $\alpha \in A$, denote by $\lambda_\alpha$ the normalized Haar measure on $K_\alpha$.

a) Let $p$ be a finite number $\geqslant 1$ and let $f$ be a function in $\mathscr{L}_F^p(G, \mu)$ (F a Banach space or $F = \overline{\mathbf{R}}$). For every $\alpha \in A$, the function

defined almost everywhere for $\mu$, belongs to $\mathscr{L}_F^p(G, \mu)$ (Exer. 9). Show that, with respect to the directed set A, $f_\alpha$ tends in mean of order $p$ to $f$ (make use of Lemma 2 of §1, No. 6).

b) Assume that $A = \mathbf{N}$ and $p = 1$. Show that $f_n$ then tends to $f$ almost everywhere (for $\mu$) in G (use a method analogous to that of Ch. V, §8, Exer. 16).