Proper actions and proper invariant metrics

We show that if a (locally compact) group $G$ acts properly on a locally compact $σ$-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 $X$.

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