The first Borel-Cantelli lemma is the principle means by which information about expectations can be converted into almost sure information.

Let \((\Omega, {\cal F}, P)\) be a probability space and let \(A_k\), \(k=1,2,3,…\) by a sequence of events in \({\cal F}\) for which

\[

\sum_{k=1}^{\infty} P(A_k) < \infty

\]

Prove the (first) Borell-Cantelli lemma which says that

\[

P\Big\{\bigcap_{N=1}^{\infty}\bigcup_{k=N}^{\infty} A_k \Big\}=0

\]

Interpret this event whose probability is zero. (Say sometime like “it is the event in which all of the events \(A_k\) happen”. This of course not the right answer.)

Now let us use this result: Let \(X_k\) be a sequence of random variables and suppose that there is another random variable \(X\) such that if

\[

A_k =\{\omega\in\Omega ~|~ ~~ |X_k(\omega)-X(\omega)|>\delta \}

\]

then

\[

\sum_{k=1}^{\infty} P(A_k) < \infty

\]

for all \(\delta >0\). Show that \(X_n \to X\) with probability one. (That is the set of \(\omega\) so that \(X_n(\omega) \to X(\omega)\) has measure (probability) one.)