  We give a new proof of a theorem of Giordano, Putnam and Skau characterizing
orbit equivalence of minimal homeomorphisms of the Cantor space in terms of
their sets of invariant Borel probability measures. The proof is based on a
strengtehning of a theorem of Krieger concerning minimal actions of certain
locally finite groups of homeomorphisms, and we also give a new proof of the
Giordano--Putnam--Skau characterization of orbit equivalence for these actions.
