existence of the essential supremumExistenceOfTheEssentialSupremum2008-12-27 16:12:382009-02-01 14:39:51Theoremessential supremummeasure space$\sigma$-finitesupremuminfimum% almost certainly you want these
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We state the existence of the essential supremum for a set $\mathcal{S}$ of extended real valued functions on a \PMlinkname{$\sigma$-finite}{SigmaFinite} measure space $(\Omega,\mathcal{F},\mu)$.
\begin{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
\begin{equation}\label{eq:1}
\operatorname{esssup}\mathcal{S}=\sup_n f_n.
\end{equation}
\end{theorem*}
Note that, by reversing the inequalities, this result also applies to the essential infimum, except that equation (\ref{eq:1}) is replaced by
\begin{equation*}
\operatorname{essinf}\mathcal{S}=\inf_nf_n.
\end{equation*}