Proper actions and proper invariant metrics

Abstract

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 $X$.Comment: The paper has been completely rewritten and differs essentially from "Constructing invariant Heine-Borel metrics for proper G-spaces". The main result extended to the more general case when $G$ is a topological group which acts properly on a locally compact $\sigma$-compact Hausdorff space $X$. Note that there is a gap in the proof of Theorem 2.4 of the old versio