Countable state shifts and uniqueness of g-measures

In this paper we present a new approach to studying g-measures which is based upon local absolute continuity. We extend the result in [11] that square summability of variations of g-functions ensures uniqueness of g-measures. The first extension is to the case of countably many symbols. The second extension is to some cases where $g\ge 0$, relaxing the earlier requirement in [11] that inf g>0.