  This paper investigates a variational principle for the Bowen metric mean
dimension of saturated sets $G_K$, where $K$ is a compact connected subset of
the convex combination of finite invariant measures for the systems with
g-almost product property. In fact, we prove the variational principle of a
saturated set with more information, that is $G_K\cap \{x\in X: C_f(X) \subset
\omega_f(x)\}$, which reveals that the limit point set of a saturated set
contains all structure of the orbits. As an application, we obtain a more
general version of multifractal analysis, which is derived independently and
can imply partial results of Backes and Rodrigues (2023 IEEE Trans. Inform.
Theory. 69 5485-5496).
