  Let $G$ be a sofic group, and let $\Sigma = (\sigma_n)_{n\geq 1}$ be a sofic
approximation to it. For a probability-preserving $G$-system, a variant of the
sofic entropy relative to $\Sigma$ 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 $(G\times H)$-system obtained by co-induction from a
$G$-system, where $H$ is any other infinite sofic group.
