Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space. Suppose that \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) satisfies \(\lim _{n ightarrow \infty} \mu\left(A_{n}ight)=0\). Show that \(\lim _{n ightarrow \infty} \int_{A_{n}} u d \mu=0\) for all \(u \in \mathcal{L}^{1}(\mu)\).

[ use Vitali's convergence theorem.]