  In this paper, we introduce the mean $\Psi$-intermediate dimension which has
a value between the mean Hausdorff dimension and the metric mean dimension, and
prove the equivalent definition of the mean Hausdorff dimension and the metric
mean dimension. Furthermore, we delve into the core properties of the mean
$\Psi$-intermediate dimensions. Additionally, we establish the mass
distribution principle, a Frostman-type lemma, H\"older distortion, and derive
the corresponding product formula. Finally, we provide illustrative examples of
the mean $\Psi$-intermediate dimension, demonstrating its practical
applications.
