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

Karlsson and Margulis [A. Karlsson, G. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces. Commun. Math. Phys. 208 (1999), 107-123] 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.