Borel-Cantelli Lemma

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Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of $\Sigma$-measurable sets.


$\ds \sum_{n \mathop = 1}^\infty \map \mu {E_n} < \infty$


$\ds \map \mu {\limsup_{n \mathop \to \infty} E_n} = 0$

where $\limsup$ denotes limit superior of sets.


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 $i \in \N$.

By Measure is Subadditive:

$\ds \map \mu {\bigcup_{j \mathop = i}^\infty E_j} \le \sum_{j \mathop = i}^\infty \map \mu {E_j}$

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.


$\ds \map \mu {\limsup_{n \mathop \to \infty} E_n} = 0$


Borel-Cantelli Lemma in Probability

As each probability space $\struct {X, \Sigma, \Pr}$ is a measure space, the result carries over to probability theory.

Hence, given any countable sequence of events $E_n$, the sum of whose probabilities is finite, the probability that infinitely many of the events occur is zero.

Also see

Source of Name

This entry was named for Émile Borel and Francesco Paolo Cantelli.