Finitary measures for subshifts of finite type and sofic systems

Is there a class of measures which is natural for sofic systems in the same way that Markov measures are natural for subshifts of finite type? Motivated by this question, we identify and study a class of finitary measures on sofic systems. We aim to convince the reader that, in addition to answering the above question, those measures are related to Markov measures in the way that sofic systems are related to subshifts of finite type.