Essential supremum and supremum of summable functions

Abstract

Let D ≌ R^N , 0 < μ(D) < +∞ and f : D → R is an arbitary summable function. Then the function is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative . Further on, it holds ess sup ƒ = sup{α ↦ 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.

Keywords:

• Essential supremum
• supremum
• global maximum
• summable function
• convex function