ON THE APPLICATION OF THE

We recall that the events Ek are said to be (mutually) independent (with respect to the probability measure P) if for any finite number of distinct subscripts kl, , k, we have P(Ek1 E.. ) = P(Ek1) . P(Ek). The celebrated Borel-Cantelli lemma asserts that (A) If ZP(Ek) < ?, then P (lim sup Ek) =0; (B) If the events Ek are independent and if ZP(Ek) = ?, then P(lim sup Ek)=l. In intuitive language P(lim stip Ek) is the probability that the events Ek occur "infinitely often" and will be denoted by P(Ek i.o.). This lemma is the basis of all theorems of the strong type in probability theory. Its application is made difficult by the assumption of independence