  In this paper, we study bimodules over a von Neumann algebra $M$ in two
related contexts. The first is an inclusion $M \subseteq M \rtimes_\alpha G$,
where $G$ is a discrete group acting on a factor $M$ by outer automorphisms.
The second is a regular inclusion $M \subseteq N$ of finite factors. In the
case of crossed products, we characterize the $M$-bimodules $X$ that lie
between $M$ and $M \rtimes_\alpha G$ and are closed in the Bures topology, in
terms of the subsets of $G$. We show that this characterization also holds for
$w^*$-closed bimodules when $G$ has the approximation property ($AP$), a class
of groups that includes all amenable and weakly amenable ones. As an
application, we prove a version of Mercer's extension theorem for certain
$w^*$-continuous isometric maps on $X$. We establish a similar theorem for
bimodules arising from regular inclusions of finite factors, which generalizes
the crossed product situation when $G$ acts on a finite factor. In the final
section we apply these ideas to provide new examples of singly generated finite
factors.
