  We consider probability measure preserving discrete groupoids, group actions
and equivalence relations in the context of general probability spaces. We
study for these objects the notions of cost, $\beta$-invariant and some
higher-dimensional variants. We also propose various convergence results about
$\ell^2$-Betti numbers and rank gradient for sequences of actions, groupoids or
equivalence relations under weak finiteness assumptions. In particular we
connect the combinatorial cost with the cost of the ultralimit equivalence
relations. Finally a relative version of Stuck-Zimmer property is also
considered.
