Definition 22.5.3 (Decomposable).label Let $(X, \cm, \mu)$ be a measure space and $\seqi{A}\subset \cm$, then $\seqi{A}$ is a decomposition of $(X, \cm, \mu)$ if:

1. (1)

   For each $i \in I$, $\mu(A_{i}) < \infty$.

2. (2)

   $X = \bigsqcup_{i \in I}X_{i}$.

3. (3)

   $\cm = \bracs{E \subset X|E \cap A_i \in \cm \forall i \in I}$.

4. (4)

   For each $E \in \cm$, $\mu(E) = \sum_{i \in I}\mu(E \cap A_{i})$.

If $(X, \cm, \mu)$ admits a decomposition, then it is decomposable.