  We characterize the set of all measurable functions on $\RR^n$ possessing an
$A_1$ majorant, denoted as $\cM_{A_1}(\RR^n)$, by certain Banach function
spaces. We prove that a function has an $A_1$ majorant if and only if it
belongs to some Banach function space for which the Hardy-Littlewood maximal
operator is bounded. This answers the question posted by G. Knese, J.
M$^{c}$Carthy, and K. Moen.
