  The concept of (stable) weak containment for measure-preserving actions of a
countable group $\Gamma$ is analogous to the classical notion of (stable) weak
containment of unitary representations. If $\Gamma$ is amenable then the
Rokhlin lemma shows that all essentially free actions are weakly equivalent.
However if $\Gamma$ is non-amenable then there can be many different weak and
stable weak equivalence classes. Our main result is that the set of stable weak
equivalence classes naturally admits the structure of a Choquet simplex. For
example, when $\Gamma=\mathbb{Z}$ this simplex has only a countable set of
extreme points but when $\Gamma$ is a nonamenable free group, this simplex is
the Poulsen simplex. We also show that when $\Gamma$ contains a nonabelian free
group, this simplex has uncountably many strongly ergodic essentially free
extreme points.
