In mathematics, a π-system (or pi-system) on a set ${\displaystyle \Omega }$ is a collection ${\displaystyle P}$ of certain subsets of ${\displaystyle \Omega ,}$ such that

That is, ${\displaystyle P}$ is a non-empty family of subsets of ${\displaystyle \Omega }$ that is closed under non-empty finite intersections. The importance of π-systems arises from the fact that if two probability measures agree on a π-system, then they agree on the 𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra as well. This is the case whenever the collection of subsets for which the property holds is a 𝜆-system. π-systems are also useful for checking independence of random variables.

This is desirable because in practice, π-systems are often simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many sets ${\displaystyle \sigma (E_{1},E_{2},\ldots ).}$ So instead we may examine the union of all 𝜎-algebras generated by finitely many sets ${\textstyle \bigcup _{n}\sigma (E_{1},\ldots ,E_{n}).}$ This forms a π-system that generates the desired 𝜎-algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a π-system that generates the very important Borel 𝜎-algebra of subsets of the real line.

A π-system is a non-empty collection of sets ${\displaystyle P}$ that is closed under non-empty finite intersections, which is equivalent to ${\displaystyle P}$ containing the intersection of any two of its elements. If every set in this π-system is a subset of ${\displaystyle \Omega }$ then it is called a π-system on ${\displaystyle \Omega .}$

For any non-empty family ${\displaystyle \Sigma }$ of subsets of ${\displaystyle \Omega ,}$ there exists a π-system ${\displaystyle {\mathcal {I}}_{\Sigma },}$ called the π-system generated by ${\displaystyle {\boldsymbol {\varSigma }}}$, that is the unique smallest π-system of ${\displaystyle \Omega }$ containing every element of ${\displaystyle \Sigma .}$ It is equal to the intersection of all π-systems containing ${\displaystyle \Sigma ,}$ and can be explicitly described as the set of all possible non-empty finite intersections of elements of ${\displaystyle \Sigma :}$ $${\displaystyle \left\{E_{1}\cap \cdots \cap E_{n}~:~1\leq n\in \mathbb {N} {\text{ and }}E_{1},\ldots ,E_{n}\in \Sigma \right\}.}$$

A non-empty family of sets has the finite intersection property if and only if the π-system it generates does not contain the empty set as an element.

A 𝜆-system on ${\displaystyle \Omega }$ is a set ${\displaystyle D}$ of subsets of ${\displaystyle \Omega ,}$ satisfying

Whilst it is true that any 𝜎-algebra satisfies the properties of being both a π-system and a 𝜆-system, it is not true that any π-system is a 𝜆-system, and moreover it is not true that any π-system is a 𝜎-algebra. However, a useful classification is that any set system which is both a 𝜆-system and a π-system is a 𝜎-algebra. This is used as a step in proving the π-𝜆 theorem.