I was working through The **Borel-Cantelli lemma** from ***Real Analysis*** problem book and ran into the following comment:

Let $\{E\_k\}\_{k\geq1}$ is a countable family of measurable subsets of $\mathbb{R}^d$ and that $\sum \limits\_{k=1}^{\infty}m(E\_k)<\infty$.

Let $$E=\{x\in \mathbb{R}^d: x\in E\_k, \text{for infinitely many} \ k\}=\limsup\limits\_{k\to \infty}(E\_k).$$

I know the notion of $\limsup$ and $\liminf$ for sequences from $[-\infty,+\infty]$. But here $E\_k$ are sets (NOT numbers!). How did they get that $$\{x\in \mathbb{R}^d: x\in E\_k, \text{for infinitely many} \ k\}=\limsup\limits\_{k\to \infty}(E\_k)?$$

This equality seems to me quite weird.

Would be very grateful for explanation!

**EDIT:** Is there some essential difference between $\limsup$ of sequence of sets and reals?