Let \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) be a sequence of mutually disjoint sets. Show that

\[u \mathbb{1}_{\cup_{n} A_{n}} \in \mathcal{L}^{1}(\mu) \Longleftrightarrow u \mathbb{1}_{A_{n}} \in \mathcal{L}^{1}(\mu) \quad \text { and } \quad \sum_{n=1}^{\infty} \int_{A_{n}}|u| d \mu<\infty\]