Exercise 26, § 4

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$. For a subset $H$ of $L^p(G, \beta)$ ($1 \leq p < +\infty$) to be relatively compact (for the topology of convergence in mean of order $p$), it is necessary and sufficient that the following conditions be satisfied: $1^\circ$ $H$ is bounded in $L^p(G, \beta)$; $2^\circ$ for every $\varepsilon > 0$, there exists a compact subset $K$ of $G$ such that $\|f \varphi_{G-K}\|_p \leq \varepsilon$ for every function $f \in H$; $3^\circ$ for every $\varepsilon > 0$, there exists a neighborhood $V$ of $e$ in $G$ such that $\|\gamma(s)f - f\|_p \leq \varepsilon$ for all $f \in H$ and $s \in V$. (To prove that the conditions are sufficient, observe that if $g \in \mathscr{K}(G)$ and if $L$ is a compact subset of $G$, then the image, under the mapping $f \mapsto g * f$, of the set $\varphi_L \cdot H$, is an equicontinuous subset of $\mathscr{K}(G)$.)