Let \((X,\mcm,\mu)\) be a measure space and let \(f\colon X\to\RR\) be measurable.
An essential lower bound of \(f\) is a real number \(b\) such that \(S_{b} \coloneqq \theset{x\in X\suchthat f(x) < b } = \inverseof{f}((-\infty, b))\) has measure zero. The essential infimum of \(f\) is the supremum of its essential lower bounds: \[ \ess\inf f \coloneqq \sup \theset{b\in\RR\suchthat \mu (S_b) = 0}. \]
An essential upper bound of \(f\) is a real number \(c\) such that \(T_c \coloneqq \theset{x\in X\suchthat f(x) > c} = \inverseof{f}((c, \infty))\) has measure zero. The essential supremum of \(f\) is the infimum of its essential upper bounds: \[ \ess\sup f \coloneqq \inf \theset{c\in\RR\suchthat \mu (T_c) = 0}. \]
A measurable function \(g\colon X\to\CC\) is essentially bounded if \(\norm{g}_\infty \coloneqq \ess\sup \abs{g} < \infty\).
The essential infimum is the largest \(b\) with \(f\geq b\) almost everywhere, and the essential supremum is the smallest \(c\) with \(f\leq c\) almost everywhere, when these are finite. An essentially bounded function is bounded outside a set of measure zero.