AbstractIt is shown that for the inclusion of factors (B⊆A):=(W∗(S,ω)⊆W∗(R,ω)) corresponding to an inclusion of ergodic discrete measured equivalence relations S⊆R, S is normal in R in the sense of Feldman–Sutherland–Zimmer [J. Feldman, C.E. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989) 239–269] if and only if A is generated by the normalizing groupoid of B. Moreover, we show that there exists the largest intermediate equivalence subrelation NR(S) which contains S as a normal subrelation. We further give a definition of “commensurability groupoid” as a generalization of normality. We show that the commensurability groupoid of B in A generates A if and only if the inclusion B⊆A...