Essential supremum and supremum of summable functions. (December 2019) - Phu, H. X.; Hoffmann, A.
Abstract : Let D ≌ RN, 0 < μ(D) < +∞ and f : D → R is an arbitary summable function. Then the function [image omitted] is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative [image omitted] . Further on, it holds ess sup ƒ = supα ↦ R : F(α) > 0, where ess sup ƒ denotes the essential supremum of ƒ. These properties can be used for computing ess sup ƒ. As example, two algorithms are stated. If the function ƒ is dense, or lower semicontinuous, or if -ƒ is robust, then sup ƒ = ess sup ƒ. In this case, the algorithms mentioned can be applied for determining the supremum of ƒ, i.e., also the global maximum of ƒ if it exists.