existence of the essential supremum

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

Theorem.

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

$$\mathrm{esssup}𝒮=\underset{n}{sup}{f}_n.$$(1)

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

$$\mathrm{essinf}𝒮=\underset{n}{inf}{f}_n.$$