Let (E_{n}) be a sequence of events in some probability space. The BorelCantelli Lemma states that
if the sum of the probabilities of the E_{n} is finite, then the probability that infinitely many of them occur is 0.
Note that no assumption of independence is required.
For example, suppose (X_{n}) is a sequence of random variables, with P(X_{n} = 0) = 1/n^{2} for each n. The sum of the P(X_{n} = 0) is finite (in fact it is π^{2}/6  see Riemann zeta function), so the BorelCantelli Lemma says that the probability of X_{n} = 0 occurring for infinitely many n is 0. In other words, with probability 1, X_{n} is nonzero for all but finitely many n.
For general measure spaces, the BorelCantelli Lemma takes the following form:
Let μ be a measure on a set X, with σalgebra F, and let (A_{n}) be a sequence in F. Ifthen μ(lim sup A_{n}) = 0.
 <math>\sum_{n=1}^\infty\mu(A_n)<\infty,</math>
To see that this really is a generalization of the version given earlier, recall that lim sup A_{n} consists of those elements which are in A_{n} for infinitely many values of n.
A similar result, sometimes called one of two "BorelCantelli" lemmas, says that if the events E_{n} are independent and the sum of their probabilities diverges to infinity, then the probability that infinitely many of them occur is 1. (The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult.)
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