  Previous work introduced two measure-conjugacy invariants: the $f$-invariant
(for actions of free groups) and $\Sigma$-entropy (for actions of sofic
groups). The purpose of this paper is to show that the $f$-invariant is a
special case of $\Sigma$-entropy. There are two applications: the $f$-invariant
is invariant under group automorphisms and there is a uniform lower bound on
the $f$-invariant of a factor in terms of the original system.
