Remark 1

In particular, let $X$ be a locally compact space on which a discrete group $D$ operates continuously and properly on the right; suppose $X/D$ is paracompact. Then, there exists a continuous function $h \geqslant 0$ on $X$ whose support has compact intersection with the saturation of every compact subset of $X$, and is such that $\sum_{d \in D} h(xd) = 1$ for every $x \in X$ (all terms of the sum being zero except for a finite number of them).