  Bennett, DeVore and Sharpley introduced the space weak $L^{\infty}$ in 1981
and studied its relationship with functions of bounded mean oscillation. Here
we characterize weak $L^{\infty}$ in measure spaces without using the
decreasing rearrangement of a function. Instead, we obtain exponential
estimates for the distribution function. In addition, we consider a localized
version of the characterization that leads to a new characterization of BMO.
