Essential Properties of $L^{\infty }$-functions

Abstract

The paper deals with local characteristics of $L^{\infty }$-elements given as equivalence classes of measurable, essentially bounded functions $f:{\mathbb{R}}^m\to \mathbb{R}$. Besides of essential lower and upper limit functions we introduce a new set-valued map carrying the information on a class, the essential limit set at a point, and analyze their main properties. Criteria for qualifying the continuity of function representatives are appended. The results can be applied e.g. in control theory to intcrprete "almost everywhere" conditions.