Definition 12.1.2 (Essential Supremum). Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $f: X \to E$ be strongly measurable, then $f$ is essentially bounded if
\[\norm{f}_{L^\infty(X; E)}= \norm{f}_{L^\infty(\mu; E)}= \norm{f}_{L^\infty(X, \cm, \mu; E)}= \inf\bracs{\alpha \ge 0|\mu(\bracs{f > \alpha}) = 0}< \infty\]
In which case, $\norm{f}_{L^\infty(X; E)}$ is the essential supremum of $f$.