  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 $g \geq 0$, relaxing the earlier requirement in [11]
that inf g>0.
