Essential Supremum

The concept of the essential supremum or essential supremum is needed in mathematics when introducing the spaces for the case as an extension of the supremum concept. Since, in the construction of these function spaces, functions that differ from one another only in terms of zero sets are regarded as identical, one can only speak of function values ​​in individual points to a limited extent. The concept of limited function must be adapted accordingly. ${\displaystyle L^{p}}$ ${\displaystyle p=\infty }$

definition

Let be a measure space and a Banach space . A measurable function is called essentially constrained if there is a number such that ${\displaystyle (\Omega ,{\mathcal {L}},\mu )}$ ${\displaystyle X}$ ${\displaystyle f\colon \Omega \rightarrow X}$ ${\displaystyle M\in \mathbb {R} }$

${\displaystyle \mu (\{x\in \Omega \ |\ \|f(x)\|_{X}>M\})=0}$

is, that is, there is a modification of on a zero set, so that the resulting function is restricted in the classical sense. Each such is called an essential limit . The essential supremum , in signs , is called ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle \mathrm {ess} \sup \|f\|_{X}}$

${\displaystyle \mathrm {ess} \sup \|f\|_{X}=\inf\{M\geq 0\ |\ M\ {\text{ist wesentliche Schranke}}\}}$

or also (for ) ${\displaystyle A\subset \Omega }$

${\displaystyle \mathrm {ess} \sup _{x\in A}\|f(x)\|_{X}=\inf _{N\subset A,\mu (N)=0}\sup _{x\in A\setminus N}\|f(x)\|_{X}}$.

Some authors also refer to the essential supremum as . ${\displaystyle \mathrm {vrai} \max \|f\|_{X}}$

For a continuous or section-wise continuous function , the identity to the classical supremum results if the Lebesgue measure is. ${\displaystyle \mu }$

L ^∞ space

The set of all essentially restricted functions is denoted by. Let us denote the set of essentially restricted functions with bound 0. Then is the set of equivalence classes of functions that differ only on a null set. ${\displaystyle {\mathcal {L}}^{\infty }(\Omega ,X)}$ ${\displaystyle {\mathcal {N}}\subset {\mathcal {L}}^{\infty }(\Omega ,X)}$ ${\displaystyle L^{\infty }(\Omega ,X):={\mathcal {L}}^{\infty }(\Omega ,X)/{\mathcal {N}}}$

${\displaystyle L^{\infty }(\Omega ,X)}$is a linear space with norm

${\displaystyle \|[f]\|_{L^{\infty }}=\mathrm {ess} \sup \|f\|_{X},\ f\in [f]}$.

This norm is independent of the choice of representative in the equivalence class . With this norm becomes a Banach space . In the mathematical literature, the square brackets that stand for the equivalence class of are dispensed with . As a rule, one simply writes and points out to the reader that the equations that arise can only be understood down to zero quantities. ${\displaystyle f}$ ${\displaystyle [f]}$ ${\displaystyle L^{\infty }(\Omega ,X)}$ ${\displaystyle f}$ ${\displaystyle f}$

example

Considering the Dirichlet step function to bear the Lebesgue measure , as is the supremum . Since the set of rational numbers is a Lebesgue null set, the essential supremum is . ${\displaystyle \mathbb {R} }$ ${\displaystyle 1}$ ${\displaystyle 0}$

literature

• Jürgen Elstrodt : Measure and integration theory. 6th, corrected edition. Springer, Berlin et al. 2009, ISBN 978-3-540-89727-9 , p. 223.
• Vladimir I. Smirnow : Textbook of higher mathematics (= university books for mathematics . Vol. 6). Volume 5. 11th edition. Deutscher Verlag der Wissenschaften, Berlin 1991, ISBN 3-817-11303-X , p. 232, no. 6.