# existence of the essential supremum

We state the existence of the essential supremum for a set $\mathcal{S}$ of extended real valued functions on a $\sigma$-finite (http://planetmath.org/SigmaFinite) measure space $(\Omega,\mathcal{F},\mu)$.

###### Theorem.

Suppose that the measure space $(\Omega,\mathcal{F},\mu)$ is $\sigma$-finite. Then, the essential supremum of $\mathcal{S}$ exists. Furthermore, if $\mathcal{S}$ is nonempty then there exists a sequence $(f_{n})_{n=1,2,\ldots}$ in $\mathcal{S}$ such that

 $\operatorname{esssup}\mathcal{S}=\sup_{n}f_{n}.$ (1)

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

 $\operatorname{essinf}\mathcal{S}=\inf_{n}f_{n}.$
Title existence of the essential supremum ExistenceOfTheEssentialSupremum 2013-03-22 18:39:22 2013-03-22 18:39:22 gel (22282) gel (22282) 6 gel (22282) Theorem msc 28A20 EssentialSupremum