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

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