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Zeitschrift für Analysis und ihre Anwendungen

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Volume 17, Issue 1, 1998, pp. 229–242
DOI: 10.4171/ZAA/818

Published online: 1998-03-31

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

U. Felgenhauer[1] and M. Wagner[2]

(1) Brandenburgische Technische Universität Cottbus, Germany
(2) Brandenburgische Technische Universität Cottbus, Germany

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.

Keywords: $L^{\infty}$ function space, upper and lower limit functions, set-valued maps, integrability and continuity criteria

Felgenhauer U., Wagner M.: Essential Properties of $L^{\infty}$-functions. Z. Anal. Anwend. 17 (1998), 229-242. doi: 10.4171/ZAA/818