Everyone knows and loves the Borel-Cantelli lemma: it is widely used, intuitive and has a cute proof. Let’s recall it.

**Borel-Cantelli Lemma.** Let be a measure space, and a sequence of measurable sets such that . Then the points of that lie in infinitely many form a set of measure zero.

*Proof.* We present the usual slick proof of the lemma, and postpone intuition until the development of a quantitative statement. Define to be the set of points lying in infinitely many , and notice that . It follows that is measurable, and contained in each cofinite union . Now, , and the latter goes to zero as , because converges. Thus , as claimed.

Alright, so the set of points lying in “very many” is “very small”. A natural quantitative question to ask at this point is: how small is the set of points lying in “moderately many” ? More precisely, if is the set of points lying in at least of the , and the set of points lying in exactly of the , can we get bounds on and , depending on ?