Editing Boole's inequality (section)

{{Short description|Inequality applying to probability spaces}}
{{more footnotes needed|date=February 2012}}
{{Probability fundamentals}}

In [[probability theory]], '''Boole's inequality''', also known as the '''union bound''', says that for any [[finite set|finite]] or [[countable]] [[Set (mathematics)|set]] of [[Event (probability theory)|event]]s, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one of a countable number of events in terms of the individual probabilities of the events. Boole's inequality is named for its discoverer, [[George Boole]].<ref>{{Cite book|last=Boole|first=George|url=https://books.google.com/books?id=zv4YAQAAIAAJ&q=George+Boole|title=The Mathematical Analysis of Logic|date=1847|publisher=Philosophical Library|isbn=9780802201546|language=en}}</ref>

Formally, for a countable set of events ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, ..., we have

:<math>{\mathbb P}\left(\bigcup_{i=1}^{\infty} A_i \right) \le \sum_{i=1}^{\infty} {\mathbb P}(A_i).</math>

In [[measure theory|measure-theoretic]] terms, Boole's inequality follows from the fact that a measure (and certainly any [[probability measure]]) is ''σ''-[[Subadditivity|sub-additive]]. Thus Boole's inequality holds not only for probability measures <math>{\mathbb P}</math>, but more generally when <math>{\mathbb P}</math> is replaced by any finite measure.