A note on an ergodic theorem in weakly uniformly convex geodesic spaces

Karlsson and Margulis (Commun. Math. Phys. 208:107–123, 1999) proved in the setting of uniformly convex geodesic spaces, which additionally satisfy a nonpositive curvature condition, an ergodic theorem that focuses on the asymptotic behavior of integrable cocycles of nonexpansive mappings over an ergodic measure-preserving transformation. In this note we show that this result holds true when assuming a weaker notion of uniform convexity.