  The purpose of this article is to provide an alternative proof of the
weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for $m$-multilinear
Calder\'on-Zygmund operators on $\mathbb{R}^n$ first proved by Grafakos and
Torres. Subsequent proofs in the bilinear setting have been given by Maldonado
and Naibo and also by P\'erez and Torres. The proof given here is motivated by
the proof of the weak-type $(1,1)$ estimate for Calder\'on-Zygmund operators in
the nonhomogeneous setting by Nazarov, Treil, and Volberg.
