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].
"A remark on the commensurability for inclusions of ergodic measured equivalence relations." Hokkaido Math. J. 37 (3) 545 - 560, August 2008. https://doi.org/10.14492/hokmj/1253539535