  We introduce a natural pseudometric on the space of actions of $d$-generated
groups, such that the zero classes are exactly the weak equivalence classes and
the metric identification with respect to this pseudometric is compact.
  We analyze convergence in this space and prove that every class contains an
action that properly satisfies every combinatorial type condition that it
satisfies with arbitrarily small error. We also show that the class of every
free non-amenable action contains an action that satisfies the measurable von
Neumann problem.
  The results have analogues in the realm of unitary representations as well.
