Borel-Cantelli Lemma
Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of $\Sigma$-measurable sets.
Let:
- $\ds \sum_{n \mathop = 1}^\infty \map \mu {E_n} < \infty$
Then:
- $\ds \map \mu {\limsup_{n \mathop \to \infty} E_n} = 0$
where $\limsup$ denotes limit superior of sets.
Proof
By definition of limit superior:
- $\ds \limsup_{n \mathop \to \infty} E_n = \bigcap_{i \mathop = 1}^\infty \bigcup_{j \mathop = i}^\infty E_j$
Thus, by Measure is Monotone and Intersection is Subset:
- $(1): \quad \ds \map \mu {\limsup_{n \mathop \to \infty} E_n} = \map \mu {\bigcap_{i \mathop = 1}^\infty \bigcup_{j \mathop = i}^\infty E_j} \le \map \mu {\bigcup_{j \mathop = i}^\infty E_j}$
for all $i \in \N$.
- $\ds \map \mu {\bigcup_{j \mathop = i}^\infty E_j} \le \sum_{j \mathop = i}^\infty \map \mu {E_j}$
However, by assumption $\ds \sum_{n \mathop = 1}^\infty \map \mu {E_n}$ converges.
By Tail of Convergent Series tends to Zero this implies:
- $\ds \lim_{i \mathop \to \infty} \sum_{n \mathop = i}^\infty \map \mu {E_n} = 0$
Now $(1)$ implies, together with Lower and Upper Bounds for Sequences, that:
- $\ds \map \mu {\limsup_{n \mathop \to \infty} E_n} \le 0$
But as $\mu$ is a measure, the converse inequality also holds.
Hence:
- $\ds \map \mu {\limsup_{n \mathop \to \infty} E_n} = 0$
$\blacksquare$
Borel-Cantelli Lemma in Probability
As each probability space $\struct {X, \Sigma, \Pr}$ is a measure space, the result carries over to probability theory, as follows:
Let $\sequence {E_n}$ be an arbitrary countable sequence of events.
Let the sum of the probabilities of $\sequence {E_n}$ be finite.
Then the probability that infinitely many of the events occur is zero.
Also known as
The is also known as the first Borel-Cantelli lemma, in order to distinguish it from the second Borel-Cantelli lemma.
Also see
Source of Name
This entry was named for Émile Borel and Francesco Paolo Cantelli.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Borel-Cantelli lemma
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 6$: Problem $9$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Borel-Cantelli Lemma
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Borel-Cantelli Lemma