Weak $L^{\infty}$ and BMO in metric spaces

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.