On the application of the Borel-Cantelli lemma

Abstract

Consider a probability space (£2, Q, P) and a sequence of events ((^-measurable sets in £2) {Ek}, A = 1, 2, • • • .The upper (or outer) limiting set of the sequence {Ek} is defined by oo oo lim sup Ek = fi U Ek.n=l k=nWe 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 Ai, • • • , As we have P(Eki ■ ■ ■ Ekt) = P(Ekl) ■ ■ ■ P(EK).