Let \((X,\mcm,\mu)\) be a measure space, let \((E_j)_{j\geq 1}\) be a sequence in \(\mcm\) with \(\sum_{j\geq 1}\mu(E_j) < \infty\), and let \(E\coloneqq\limsup_j E_j\) be its limit superior. Then \(E\in\mcm\) and \(\mu(E) = 0\).
We have \(E = \bigcap_{k\geq 1}\bigcup_{j\geq k} E_j\), a countable intersection of countable unions of sets in \(\mcm\), so \(E\in\mcm\). For every \(k\geq 1\), \(E\subseteq\bigcup_{j\geq k}E_j\), so by countable subadditivity \[ \mu(E)\leq\sum_{j\geq k}\mu(E_j). \] The right side is the tail of a convergent series, so it tends to \(0\) as \(k\to\infty\), and \(\mu(E) = 0\).