  In this paper, we study the separable and weak convergence of mixed-norm
Lebesgue spaces. Furthermore, we prove that the block space
$\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$ is the K\"othe dual of the
mixed Morrey space $\mathcal{M}_{\vec{p}}^{p_0}(\mathbb{R}^n)$ by the Fatou
property of these block spaces. The boundedness of the Hardy--Littlewood
maximal function is further obtained on the block space
$\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$. As applications, the
characterizations of $BMO(\mathbb{R}^n)$ via the commutators of the fractional
integral operator $I_{\alpha}$ on mixed Morrey spaces are proved as well as the
block space $\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$.
