On the mean $\Psi$-intermediate dimensions

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.