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

Non-standard limits of graphs and some orbit equivalence invariants