Definition 15.6.2 (Locally Bounded*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded measure space, $E$ be a normed vectorr space over $K \in \RC$, and $f: X \to E$ be strongly measurable, then $f$ is essentially bounded if for every $A \in \cf$, $\norm{\one_A f}_{L^\infty(A; E)}< \infty$.

The set $\mathcal{L}^{\infty}_{\cf}(X, \cm, \mu; E) = \mathcal{L}^{\infty}_{\cf}(X; E) = \mathcal{L}^{\infty}_{\cf}(\mu; E)$ is the space of all locally bounded $E$-valued functions on $X$.