ABSTRACT. We use ^-measures to give a proof of a convergence theorem of Ruelle. The method of proof is used to gain information about the ergodic properties of equilibrium states for subshifts of finite type.0. In 1968 D. Ruelle proved a convergence theorem (Theorem 3 of [8]) which he used to obtain a large class of interactions, for an infinite one-dimensional classical lattice gas, which have no phase transitions. The equilibrium state of such a system was shown to have strong ergodic properties. In 1973 R. Bowen remarked that Ruelle's proof could be extended to show the convergence of the powers of a certain operator acting on the space of all real-valued continuous functions on a one-sided shift space [2]. In the paper [2] he used this result and the theory of Markov partitions, due to Ya. G. Sinai and himself, to show that if J24 is a basic set of an Axiom A diffeomorphism / then, with respect to f\ns' each Holder continuous c4: J2S -> R has a unique equilibrium state and if f\cis is topologically mixing then with respect to the equilibrium state f\ns is a Bernoulli shift.This was done by using Markov partitions to move the problem to one about a two-sided subshift of finite type ([1], [10]). It can then be reduced to a problem about a one-sided subshift of finite type ([2], [11, p. 28]) and then Ruelle's theorem can be used.In this paper we connect these results with the idea of a g-measure, studied by M. Keane [5]. We shall give a proof of Ruelle's operator theorem using the notion of g-measure. The structure of this proof allows us to deduce elementary proofs of several results about equilibrium states. There is a dense subset V of CLX) whose members each have a unique equilibrium state, the natural extension of the one-sided shift with respect to these measures are Bernoulli shifts, and two members of V have the same equilibrium state if and only if they differ by a function of the form f° T-f+c where / G C(X) and cGR (here T denotes the one-sided shift). In §4 we extend these results to transformations more general than one-sided subshifts of finite type.