ESSENTIAL SUPREMUM AND SUPREMUM OF SUMMABLE FUNCTIONS

摘要

Let D C IRN, 0 < fJ.(D) < +oo and : D -￥ JR is an arbitrary summable function. Then the function F(a) := SixeD-f(x)>a)(f(x) ~ a^d^ (a e ffi) '1B continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative F'{a) = —fj{f > a). Further on, it holds ess sup/ = sup{o; 6 JR : F(a) > 0}, where ess sup denotes the essential supremum of f. These properties can be used for computing ess sup . As example, two algorithms are stated. If the function f is dense, or lower semicontinuous, or if — f is robust, then sup/ = ess sup/. In this case, the algorithms mentioned can be applied for determining the supremum of f, i.e., also the global maximum of if it exists.