19.6.3.1 Convergence (in measure, almost everywhere)

Problem: Let \( \left(X, A, \mu\right. \) ) be a measurable space, \( A_{n} \in A, n \in \mathbb{N} \) is the sequence of such measurable sets that \[ \sum_{n=1}^{\infty} \mu\left(A_{n}\right)<\infty . \] Prove that \( \lim \sup _{n \rightarrow \infty} A_{n} \) is measurable and its measure is equal to 0 .