Proper Action in Steps, with Application to Density Ratios of Maximal Invariants

Abstract

Let $G$ be a locally compact group acting continuously on the left of a locally compact space $\mathscr{X}$. It is assumed that $G = HK$ where $H$ and $K$ are closed subgroups. It is shown that if $K$ acts properly on $\mathscr{X}$ and $H$ acts properly on $\mathscr{X}/K$, then $G$ acts properly on $\mathscr{X}$. Under a mild additional condition the converse is also true. Several examples are given to show how these results can help decide the properness of composite actions. Proper action can be used to justify the representation of the density ratio of a maximal invariant as a ratio of integrals over the group.