Theorem

Let {An}n=0∞\{A_n\}_{n=0}^\infty be sequence of events occurring with certain probability distribution (in a sample space Ω\Omega), AA be the event consisting of the occurrence of finite number of events AnA_n for n=1,2,...n= 1,2,..., then probability of an infinite number of the AnA_n occurring is zero if

∑n=1∞P(An)<∞\sum_{n=1}^\infty P(A_n) < \infty

Equivalently, if P(An)=0P(A_n) = 0 for all nn, probability that none of them occurs is 11, and in particular probability of AA that a finite number occur is also 11.

If the events {An}n=0∞\{A_n\}_{n=0}^\infty are independent, then probability of an infinite number of the AnA_n occurring is one if

∑n=1∞P(An)=∞\sum_{n=1}^\infty P(A_n) = \infty

Theorem (stated in terms of measure spaces)

Consider measure space (X,Σ,μ)(X,\Sigma,\mu), sequence of Σ\Sigma-measurable sets ⟨En⟩n∈ℕ\langle E_n \rangle_{n \in \mathbb{N}}, if

∑n=1∞μ(En)<∞\sum_{n=1}^\infty \mu(E_n) < \infty

then

μ(lim supn→∞⁡En)=0\mu \left({\limsup_{n \to \infty} E_n} \right)= 0

(limit superior of set)

References

  1. https://mathworld.wolfram.com/Borel-CantelliLemma.html
  2. https://proofwiki.org/wiki/Borel-Cantelli_Lemma
  3. https://www.math.mcgill.ca/dstephens/OldCourses/556-2006/Math556-BorelCantelli.pdf
  4. https://en.wikipedia.org/wiki/Borel–Cantelli_lemma