What is essential supremum

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The concept of essential supremums or essential supremums is required in mathematics when introducing the \ ({\ displaystyle L ^ {p}} \) spaces for the case \ ({\ displaystyle p = \ infty} \) 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 considered identical, one can only speak of function values ​​in individual points to a limited extent. The concept of restricted function must be adapted accordingly.

Table of Contents

definition


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

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

is, that is, there is a modification of \ ({\ displaystyle f} \) on a null set, so that the resulting function is limited in the classical sense. Every such \ ({\ displaystyle M} \) becomes one essential limit called. As essential supremum, in characters \ ({\ displaystyle \ mathrm {ess} \ sup \ | f \ | _ {X}} \), one denotes

\ ({\ displaystyle \ mathrm {ess} \ sup \ | f \ | _ {X} = \ inf \ {M \ geq 0 \ | \ M \ {\ text {is essential limit}} \}} \)

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 designate the essential supremum with \ ({\ displaystyle \ mathrm {vrai} \ max \ | f \ | _ {X}} \).

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

L.-Room


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

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

example


If one considers the Dirichlet jump function on \ ({\ displaystyle \ mathbb {R}} \) provided with the Lebesgue measure, the supremum is \ ({\ displaystyle 1} \). Since the set of rational numbers is a Lebesgue null set, the essential supremum is \ ({\ 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. Smirnov: Textbook of higher mathematics (= University books in mathematics. Vol. 6). Volume 5. 11th edition. Deutscher Verlag der Wissenschaften, Berlin 1991, ISBN 3-817-11303-X, p. 232, no. 6.









Categories:Functional analysis | Measure theory




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