existence of the essential supremum

We state the existence of the essential supremumMathworldPlanetmath for a set 𝒮 of extended real valued functions on a σ-finite (http://planetmath.org/SigmaFinite) measure spaceMathworldPlanetmath (Ω,,μ).


Suppose that the measure space (Ω,F,μ) is σ-finite. Then, the essential supremum of S exists. Furthermore, if S is nonempty then there exists a sequence (fn)n=1,2, in S such that

esssup𝒮=supnfn. (1)

Note that, by reversing the inequalitiesMathworldPlanetmath, this result also applies to the essential infimum, except that equation (1) is replaced by

Title existence of the essential supremum
Canonical name ExistenceOfTheEssentialSupremum
Date of creation 2013-03-22 18:39:22
Last modified on 2013-03-22 18:39:22
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 28A20
Related topic EssentialSupremum