Definition 23.4.1 (Scaffold*).label Let $(X, \cm, \mu)$ be a measure space and $\cf \subset \cm$, then $\cf$ is a scaffold for $\mu$ if:

1. (S1)

   For each $A \in \cf$, $\mu(A) < \infty$.

2. (S2)

   For all $E \in \cm$, $\mu(E) = \sup\bracs{\mu(E \cap A)|A \in \cf}$.

3. (S3)

   For any $A, B \in \cf$, $A \cup B \in \cf$.

and the quadruple $(X, \cm, \cf, \mu)$ is a scaffolded measure space.

For any semifinite measure space $(X, \cm, \mu)$, $\cf = \bracs{A \in \cm|\mu(A) < \infty}$ is the canonical scaffold for $\mu$, and $(X, \cm, \mu)$ will be equipped with this scaffold unless specified otherwise.