A remark on the commensurability for inclusions of ergodic measured equivalence relations

Abstract

It is shown that, for each inclusion of ergodic discrete measured equivalence relations, the commensurability can be characterized in terms of measure theoretical arguments. As an application, we also include a measure theoretical proof concerning a property of the commensurability groupoid which determines the commensurability in terms of operator algebras. It is proven that a family of typical elements in the commensurability groupoid is closed under the product operation. This proof supplements a gap in the proof of [2, Lemma 7.5].