Essential Supremum and Supremum of Summable Functions

Let D lR N , 0 < (D) < +1 and f : D ! lR is an arbitrary summable function. Then the function F() := R fx2D:f(x)g (f(x) ?) dd (2 lR) is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative F 0 () = ?f ]. Further on, it holds ess supf = supf 2 lR : F() > 0g, where ess supf denotes the essential supremum of f. These properties can be used for computing esssup f. As example, two algorithms are stated. If the function f is dense, or lower semicontinuous, or if ?f is robust, then supf = ess supf. In this case, the algorithms mentioned can be applied for determining the supremum of f, i.e., also the global maximum of f if it exists.