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, L2-Betti
numbers,
We also propose various convergence results about L2-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.