  We improve and subsume the conditions of Johansson and \"Oberg [18] and
Berbee [2] for uniqueness of a g-measure, i.e., a stationary distribution for
chains with complete connections. In addition, we prove that these unique
g-measures have Bernoulli natural extensions. In particular, we obtain a unique
g-measure that has the Bernoulli property for the full shift on finitely many
states under any one of the following additional assumptions. (1)
$$\sum_{n=1}^\infty (\var_n \log g)^2<\infty,$$ (2) For any fixed $\epsilon>0$,
$$\sum_{n=1}^\infty e^{-(\{1}{2}+\epsilon) (\var_1 \log g+...+\var_n \log
g)}=\infty,$$ (3) $$\var_n \log g=\ordo{\{1}{\sqrt{n}}}, \quad n\to \infty.$$
That the measure is Bernoulli in the case of (1) is new. In (2) we have an
improved version of Berbee's condition (concerning uniqueness and
Bernoullicity) [2], allowing the variations of log g to be essentially twice as
large. Finally, (3) is an example that our main result is new both for
uniqueness and for the Bernoulli property. We also conclude that we have
convergence in the Wasserstein metric of the iterates of the adjoint transfer
operator to the g-measure.
