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