# Borel-Cantelli Lemma (Pictorially)

The Borel-Cantelli Lemma says that if $(X,\Sigma,\mu)$ is a measure space with $\mu(X) < \infty$ and if $\{E_n\}_{n=1}^\infty$ is a sequence of measurable sets such that $\sum_n\mu(E_n) < \infty$, then $$\mu\left(\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k\right)=\mu\left(\limsup_{n\to\infty} En \right)=0.$$
(For the record, I didn't understand this when I first saw it (or for a long time afterwards). My only thought was, "But what does that *mean? In English??*") To help our intuition, notice the conclusion is the same as saying $$\mu(\{x\in X: \text{there exists infinitely many $n$ such that $x\in E_n$}\})=0.$$ And *this* is another way of saying

*almost every $x\in X$ lives in at most finitely many $E_n.$*

So for a ("almost every") fixed $x\in X$ we have a picture like this:

Note! For each of the "almost every" $x$'s, we get a *different* finite collection of $E_n$. For an example of this picture in action, see this post.