Definition 15.6.1 (Locally Integrable*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded measure space, $E$ be a normed vector space over $K \in \RC$, $f: X \to E$ be strongly measurable, and $p \in [1, \infty]$, then $f$ is locally $p$-integrable if for every $A \in \cf$, $\int_{A} \norm{f}_{A}^{p} d\mu < \infty$.

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