pi-system
Sign in to saveAlso known as π-system, pi system, π system
In mathematics, a -system (or pi-system) on a set \Omega is a collection P of certain subsets of \Omega, such that
~11 min read
Article
13 sectionsContents
- Definitions
- Examples
- Relationship to {{lambda}}-systems
- The {{pi}}-{{lambda}} theorem
- Example
- {{pi}}-Systems in probability
- Equality in distribution
- Independent random variables
- Example
- See also
- Notes
- Citations
- References
In mathematics, a -system (or pi-system) on a set \Omega is a collection P of certain subsets of \Omega, such that P is non-empty. If A, B \in P then A \cap B \in P.
That is, P is a non-empty family of subsets of \Omega that is closed under non-empty finite intersections. The importance of -systems arises from the fact that if two probability measures agree on a -system, then they agree on the -algebra generated by that -system. Moreover, if other properties, such as equality of integrals, hold for the -system, then they hold for the generated -algebra as well. This is the case whenever the collection of subsets for which the property holds is a -system. -systems are also useful for checking independence of random variables.