An essentially bounded $f$ agrees almost everywhere with a bounded measurable $g$ with $\sup\abs{g}\le\norm{f}_\infty$.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space and let \(f\colon X\to\CC\) be measurable with \(\norm{f}_\infty<\infty\), that is, \(f\) is essentially bounded. Then there is a bounded measurable function \(g\colon X\to\CC\) with \(\sup_{x\in X}\abs{g(x)}\leq\norm{f}_\infty\) and \(f=g\) \(\mu\)-almost everywhere.