  Let $\mathcal{E}(X,d,\mu)$ be a Banach function space over a space of
homogeneous type $(X,d,\mu)$. We show that if the Hardy-Littlewood maximal
operator $M$ is bounded on the space $\mathcal{E}(X,d,\mu)$, then its
boundedness on the associate space $\mathcal{E}'(X,d,\mu)$ is equivalent to a
certain condition $\mathcal{A}_\infty$. This result extends a theorem by Andrei
Lerner from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of
homogeneous type.
