# 波莱尔－坎泰利引理

## 概率空间中的定理

${\displaystyle E_{n}}$ 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${\displaystyle E_{n}}$ 发生的概率${\displaystyle \mathbb {P} }$ 的总和是有限的，

${\displaystyle \sum _{n=1}^{\infty }\mathbb {P} (E_{n})<\infty ,}$

${\displaystyle \mathbb {P} \left(\limsup _{n\to \infty }E_{n}\right)=0\,}$

${\displaystyle \limsup _{n\to \infty }E_{n}=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }E_{k}}$

## 证明

${\displaystyle \sum _{n=1}^{\infty }\mathbb {P} (E_{n})<\infty }$

${\displaystyle \inf _{N\geq 1}\sum _{n=N}^{\infty }\mathbb {P} (E_{n})=0.\,}$

${\displaystyle \mathbb {P} \left(\limsup _{n\to \infty }E_{n}\right)=\mathbb {P} \left(\bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }E_{n}\right)\leq \inf _{N\geq 1}\mathbb {P} \left(\bigcup _{n=N}^{\infty }E_{n}\right)\leq \inf _{N\geq 1}\sum _{n=N}^{\infty }\mathbb {P} (E_{n})=0}$ [1]

## 推广

${\displaystyle \sum _{n=1}^{\infty }\mu (A_{n})<\infty }$

${\displaystyle \mu \left(\limsup _{n\to \infty }A_{n}\right)=0\,}$

## 参考来源

• Prokhorov, A.V., Borel–Cantelli lemma, Hazewinkel, Michiel (编), 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.