An asymptotic equipartition property for measures on model spaces

Let G be a sofic group, and let Σ “ pσnqně1 be a sofic approximation to it. For a probability-preserving G-system, a variant of the sofic entropy relative to Σ has recently been defined in terms of sequences of measures on its model spaces that ‘converge’ to the system in a certain sense. Here we prove that, in order to study this notion, one may restrict attention to those sequences that have the asymptotic equipartition property. This may be seen as a relative of the Shannon– McMillan theorem in the sofic setting. We also give some first applications of this result, including a new formula for the sofic entropy of a pGˆHq-system obtained by co-induction from a G-system, whereH is any other infinite sofic group.