  We provide an operator algebraic interpretation of discrete measurable
groupoids in the course of re-proving (and slightly generalizing) a result on
treeability due to Adams and Spatzier. Then, we reconstruct Gaboriau's
beautiful work on costs of equivalence relations in operator algebra framework,
avoiding any measure theoretic argument, and clarify what kind of his results
can or cannot be generalized to the non-principal groupoid case.
