In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer. Math. Soc.353(1) (2001), 327–347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Öberg. Math. Res. Lett.10(5–6) (2003), 587–601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys.11 (1991), 279–307], under square summability of variations.