If $\sum_k m(E_k)<\infty$, then almost every point lies in only finitely many of the sets $E_k$.Suggest a slogan
Let \((E_k)_{k\geq1}\) be a sequence of Lebesgue measurable subsets of \(\RR^n\) with \[ \sum_{k=1}^\infty m(E_k) < \infty . \] Then \(m\qty{\limsup_{k\to\infty} E_k} = 0\), where \(\limsup_k E_k\) is the limit superior of the sets \(E_k\); equivalently, almost every \(x\in\RR^n\) lies in only finitely many of the sets \(E_k\).
The same statement holds for a sequence of measurable sets in any measure space. For a probability measure \(P\) and events \(E_k\) with \(\sum_k P(E_k)<\infty\), it says that with probability \(0\) infinitely many of the events \(E_k\) occur.