# 几乎必然

## 定义

${\displaystyle (\Omega ,{\mathcal {F}},P)}$为一概率空间。若${\displaystyle {\mathcal {F}}}$中一个事件${\displaystyle E}$满足${\displaystyle P(E)=1}$，则称其几乎必然发生。等价地，若${\displaystyle E}$不发生的概率是0，即${\displaystyle P(E^{C})=0}$，则称${\displaystyle E}$几乎必然发生。更一般地，若对于一个事件${\displaystyle E\subseteq \Omega }$（不一定要属于${\displaystyle {\mathcal {F}}}$），存在一个零测集${\displaystyle N}$，满足${\displaystyle E^{C}\subset N}$${\displaystyle P(N)=0}$，则称${\displaystyle E}$几乎必然发生。[5]

## 参考资料

1. The Definitive Glossary of Higher Mathematical Jargon — Almost. Math Vault. 2019-08-01 [2019-11-16]. （原始内容存档于2020-02-28） （美国英语）.
2. ^ Weisstein, Eric W. Almost Surely. mathworld.wolfram.com. [2019-11-16]. （原始内容存档于2021-08-13） （英语）.
3. ^ Almost surely - Math Central. mathcentral.uregina.ca. [2019-11-16]. （原始内容存档于2021-08-13）.
4. ^ Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott. Finite Model Theory and Its Applications. Springer. 2007: 232. ISBN 978-3-540-00428-8.
5. ^ Jacod, Jean; Protter. Probability Essentials. Springer. 2004: 37. ISBN 978-3-540-438717.
6. ^ Williamson, Timothy. How probable is an infinite sequence of heads?. Analysis. 2007-07-01, 67 (3): 173–180 [2021-08-13]. ISSN 0003-2638. doi:10.1093/analys/67.3.173. （原始内容存档于2021-03-08） （英语）.
7. ^ Friedgut, Ehud; Rödl, Vojtech; Rucinski, Andrzej; Tetali, Prasad. A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring. Memoirs of the American Mathematical Society (AMS Bookstore). January 2006, 179 (845): 3–4. ISSN 0065-9266. S2CID 9143933. doi:10.1090/memo/0845.
8. ^ Spencer, Joel H. 0. Two Starting Examples. The Strange Logic of Random Graphs. Algorithms and Combinatorics 22. Springer. 2001: 4 [2021-08-13]. ISBN 978-3540416548. （原始内容存档于2021-08-13）.

## 参考书目

• Rogers, L. C. G.; Williams, David. Diffusions, Markov Processes, and Martingales. 1: Foundations. Cambridge University Press. 2000. ISBN 978-0521775946.
• Williams, David. Probability with Martingales. Cambridge Mathematical Textbooks. Cambridge University Press. 1991. ISBN 978-0521406055.