Essential supremum and essential infimum

The concept of essential supremum and essential supremum is required in mathematics at the launch of the rooms for the event as an extension of the supremum term. As will be considered in the design of this function rooms functions that differ only on null sets as identical, one can only speak limited function values ​​in individual points. The term of the limited function must be adjusted accordingly.

Definition

Be a measure space and a Banach space. A measurable function is called essentially bounded if there is a number such that

, that is, there is a modification of on a set of measure zero, so that the resulting function in the classical sense is limited. Each such is called an essential barrier. As an essential supremum, in characters, is referred to

Or ( for )

Some authors refer to the essential supremum with also.

For a continuous or sectionally continuous function gives the identity to the classical supremum, if the Lebesgue measure is.

L ∞ - space

With the set of all essentially bounded functions is called. It is denoted by the set of essentially bounded functions with limit 0. Then the set of the equivalence classes.

Is a linear space with norm

This standard is independent of the choice of the representative in the equivalence class. With this norm becomes a Banach space. In the mathematical literature one renounces the square brackets stand for the equivalence class of. In general, you simply writes and instructs the reader then that the equations that occur are to be understood only up to null sets.

Example

If we consider the Dirichlet jump function provided with the Lebesgue measure, then the supremum. Since the set of rational numbers but is a Lebesgue -null set, is the essential supremum.

Swell

  • Jürgen Elstrodt: measure and integration theory. 6, corrected edition. Springer, Berlin and others 2009, ISBN 978-3-540-89727-9, p 223
  • Vladimir I. Smirnov: Textbook of higher mathematics ( = high math textbooks Vol 6. ). Volume 5 11th edition. German Academic Publishers, Berlin, 1991, ISBN 3- 817-11303 -X, pp. 232, No. 6
  • Functional Analysis
  • Measure theory
316579
de