Definition 15.6.3 (Local $L^{p}$ Space*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded measure space, $E$ be a normed vector space over $K \in \RC$, and $p \in [1, \infty]$. For each $A \in \cf$ and $f \in \mathcal{L}^{p}_{\cf}(X, \cm, \mu; E)$, let $[f]_{L^p_A(X; \mu)}= \norm{\one_A f}_{L^p(X; \mu)}$, then the $[\cdot]_{L^p_A(X; \mu)}$ is a seminorm on $\mathcal{L}^{p}_{\cf}(X; E)$. The set

equipped with the seminorms $\bracsn{[\cdot]_{L^p_A(X;\mu)}|A \in \cf}$ is a separated locally convex space, and the local $L^{p}$ space of $(X, \cm, \mu)$.