Limit superior of sets
= Limit superior of sets
{title2=$\limsup_n E_n=\bigcap_N\bigcup_{n\geq N}E_n$}
Membership means that for every index $N$ there is a later $n\geq N$ with $x\in E_n$, equivalently membership for infinitely many indices. If $\sum_n\mu(E_n)<\infty$, <countable subadditivity> gives $\mu(\limsup E_n)\leq\sum_{n\geq N}\mu(E_n)\to0$. This is the measure form of the first <Borel-Cantelli lemma>, and requires no independence assumption.