Lemma 15.6.4.label Let $(X, \cm, \cf, \mu)$ be a scaffolded localisable measure space, $E$ be a normed vector space over $K \in \RC$, $p \in [1, \infty]$, and $\bracsn{f_A}_{A \in \cf}$ such that:

1. (a)

   For each $A \in \cf$, $f_{A} \in \mathcal{L}^{p}(A; E)$.

2. (b)

   For each $A, B \in \cf$, $f_{A}|_{A \cap B}= f_{B}|_{A \cap B}$ almost everywhere.

3. (c)

   $\bigcup_{A \in \cf}f_{A}(A)$ is a separable subset of $E$.

then there exists a unique $f \in L^{p}_{\cf}(X; E)$ such that $f|_{A} = f_{A}$ for all $A \in \cf$.