Let $(X,M,\mu)$ be a measure space and let $E_{n}\in M$. Under the assumption that $\sum\mu(E_{n})<\infty$, we can get $\mu(\limsup E_{n})=0$. Now I am trying to prove or disprove that the conclusion follows if the assumption is replaced by $\sum\mu(E_{n})^{2}<\infty$.
In my attempt to disprove it, I wanted $\sum\mu(E_{n})^{2}<\infty$ but $\sum\mu(E_{n})=\infty$ so I tried with intervals of length $\frac{1}{n}$ (either taking $E_{n}=(0,\frac{1}{n})$ or the corresponding closed interval, or picking the $E_{n}$'s to be disjoint) but I end up with the $\limsup$ being either empty or a singleton. On the other hand, I don't know what I can use to prove it.