Limit superior of sets (${\mathrm{limsup}}_n{E}_n={\bigcap }_N{\bigcup }_{n\ge N}{E}_n$)

Membership means that for every index $N$ there is a later $n\ge N$ with $x\in E_n$, equivalently membership for infinitely many indices. If ${\sum }_n\mu (E_n)<\infty$, countable subadditivity gives $\mu (\mathrm{limsup}{E}_n)\le {\sum }_{n\ge N}\mu (E_n)\to 0$. This is the measure form of the first Borel-Cantelli lemma, and requires no independence assumption.