Borel–Cantelli lemma

The Borel–Cantelli lemma is a result in measure theory. It is often stated in the context of probability theory, where it is used to study whether, in a given sequence of events, a finite or infinite number of these events occur. The statement of the lemma is often split into two parts:

  • The first Borel–Cantelli lemma, which states that if the sum of the probabilities of the events is finite, then the probability that infinitely many of them occur is 0. This result holds for any sequence of events, without additional assumptions;
  • The second Borel–Cantelli lemma, which states that if the events are independent and the sum of their probabilities is infinite, then the probability that infinitely many of them occur is 1.

It follows that the probability of the limit superior of a sequence of independent events is always either zero or one. For this reason, the Borel–Cantelli lemma is often referred to as a zero-one law. Other examples or similar results include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law.

The Borel–Cantelli lemma is named after Émile Borel and Francesco Paolo Cantelli, who stated it in the first decades of the 20th century.

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