  We show that if a (locally compact) group $G$ acts properly on a locally
compact $\sigma$-compact space $X$ then there is a family of $G$-invariant
proper continuous finite-valued pseudometrics which induces the topology of
$X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and only
if there exists a $G$-invariant proper compatible metric on $X$.
