# First Borel-Cantelli Lemma

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.)