On the Application of the Borel-cantelli Lemma

The celebrated Borel-Cantelli lemma asserts that (A) If ~P(E,,) < 00, then P (lim sup Ek) =O; (B) If the events Es are independent and if xP(Ek) = m, then P(lim sup Eh) = 1. In intuitive language P(lim. sup &) is the probability that the events Eh occur “infinitely often” and will be denoted by P(Ek i.0.). 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 in part (B). As Bore1 already noticed [l, p. 48 ff.], this assumption can be removed if we assume that(2)