# Hopefully Interesting

## March 14, 2008

### Quantitative Borel-Cantelli Lemma

Filed under: math.CA,motivated theorems — Pietro @ 12:00 pm
Tags: ,

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 $(X,\mathcal{M},\mu)$ be a measure space, and $(E_n)_{n\in\mathbb{N}}$ a sequence of measurable sets such that $\sum_{n\in\mathbb{N}} \mu(E_n) < \infty$. Then the points of $X$ that lie in infinitely many $E_n$ 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 $F_\infty$ to be the set of points lying in infinitely many $E_n$, and notice that $F_\infty = \bigcap_{n\in\mathbb{N}} \bigcup_{k\geq n} E_k$. It follows that $F_\infty$ is measurable, and contained in each cofinite union $\bigcup_{k\geq n} E_k$. Now, $\mu(\bigcup_{k\geq n}E_k) \leq \sum_{k\geq n} \mu(E_k)$, and the latter goes to zero as $n\to\infty$, because $\sum_{n\in\mathbb{N}} \mu(E_n)$ converges. Thus $\mu(F_\infty)=0$, as claimed. $\Box$

Alright, so the set of points lying in “very many” $E_n$ is “very small”. A natural quantitative question to ask at this point is: how small is the set of points lying in “moderately many” $E_n$? More precisely, if $F_k$ is the set of points lying in at least $k$ of the $E_n$, and $G_k = F_k \backslash F_{k+1}$ the set of points lying in exactly $k$ of the $E_n$, can we get bounds on $\mu(F_k)$ and $\mu(G_k)$, depending on $k$?