Definition 16.1.3 (Essential Supremum).label 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
\[\norm{f}_{\mathcal{L}^\infty(X; E)}= \norm{f}_{\mathcal{L}^\infty(\mu; E)}= \norm{f}_{\mathcal{L}^\infty(X, \cm, \mu; E)}= \inf\bracs{\alpha \ge 0|\mu(\bracs{f > \alpha}) = 0}< \infty\]
In which case, $\norm{f}_{\mathcal{L}^\infty(X; E)}$ is the essential supremum of $f$.
The set $\mathcal{L}^{\infty}(X; E) = \mathcal{L}^{\infty}(\mu; E) = \mathcal{L}^{\infty}(X, \cm, \mu; E)$ is the space of all $E$-valued essentially bounded functions on $X$.
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