# 波莱尔－坎泰利引理

## 概率空间中的定理

$E_n$ 为某个概率空间中的一个事件序列。波莱尔－坎泰利引理说明： 如果所有的事件$E_n$ 发生的概率$\mathbb{P}$的总和是有限的，

$\sum_{n=1}^\infty \mathbb{P}(E_n)<\infty,$

$\mathbb{P} \left(\limsup_{n\to\infty} E_n\right) = 0\,$

$\limsup_{n\to\infty} E_n = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k.$

## 证明

$\sum_{n=1}^\infty \mathbb{P}(E_n)<\infty.$

$\inf_{N\geq 1} \sum_{n=N}^\infty \mathbb{P}(E_n) = 0. \,$

$\mathbb{P}\left(\limsup_{n\to\infty} E_n\right) = \mathbb{P} \left(\bigcap_{N=1}^\infty \bigcup_{n=N}^\infty E_n\right) \leq \inf_{N \geq 1} \mathbb{P} \left( \bigcup_{n=N}^\infty E_n\right) \leq \inf_{N\geq 1} \sum_{n=N}^\infty \mathbb{P} (E_n) = 0$[1]

## 推广

$\sum_{n=1}^\infty\mu(A_n)<\infty$

$\mu\left(\limsup_{n\to\infty} A_n\right) = 0\,$

## 参考来源

• Prokhorov, A.V., Borel–Cantelli lemma// (编) Hazewinkel, Michiel, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Feller William, An Introduction to Probability Theory and Its Application, John Wiley & Sons, 1961.
• Stein Elias, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.
• Bruss, F. Thomas, A counterpart of the Borel Cantelli Lemma, J. Appl. Prob., 1980, 17: 1094–1101.
• Durrett, Rick. "Probability: Theory and Examples." Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005.