[Math] Can someone explain the Borel-Cantelli Lemma

Question

I’m looking for an informal and intuitive explanation of the Borel-Cantelli Lemma. The symbolic version can be found here.

What is confusing me is what ‘probability of the limit superior equals $ 0 $’ means.

Thanks!

Answer 1

Let $\{E_n\}$ be a sequence of events. Each event $E_i$ is a collection of outcomes. The limit superior of the collection $\{E_n\}$ is the collection of all those outcomes that appear in infinitely many events. The Borel Cantelli Lemma says that if the sum of the probabilities of the $\{E_n\}$ are finite, then the collection of outcomes that occur infinitely often must have probability zero.

To give an example, suppose I randomly pick a real number $x \in [0,1]$ using an arbitrary probability measure $\mu$. I then challenge my (infinitely many) friends to guess a subset $E_n$ of $[0,1]$ containing $x$. If infinitely many friends guess correctly, I have to buy pizza for everyone. To make sure they don't just guess the whole interval, I make them pay for choosing large subsets -- the cost of choosing a set $E$ is $\mu(E)$. I check my bank account at the end of the guessing and find that I have only made a finite amount of money, i.e. $\sum \mu(E_n) < \infty$ (cheap friends!). Is it likely I will have to buy everyone pizza? No! Because the set $E_{\infty}$ of numbers that infinitely many friends agreed on has measure zero, so my random number $x$ has probability zero of being in there.