In mathematics, a π-system (or pi-system) on a set Ω {\displaystyle \Omega } is a collection P {\displaystyle P} of certain subsets of Ω , {\displaystyle \Omega ,} such that
P {\displaystyle P} is non-empty. If A , B ∈ P {\displaystyle A,B\in P} then A ∩ B ∈ P . {\displaystyle A\cap B\in P.}
That is, P {\displaystyle P} is a non-empty family of subsets of Ω {\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 σ ( E 1 , E 2 , … ) . {\displaystyle \sigma (E_{1},E_{2},\ldots ).} So instead we may examine the union of all 𝜎-algebras generated by finitely many sets ⋃ n σ ( E 1 , … , E n ) . {\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.
Definitions A π-system is a non-empty collection of sets P {\displaystyle P} that is closed under non-empty finite intersections, which is equivalent to P {\displaystyle P} containing the intersection of any two of its elements. If every set in this π-system is a subset of Ω {\displaystyle \Omega } then it is called a π-system on Ω . {\displaystyle \Omega .} For any non-empty family Σ {\displaystyle \Sigma } of subsets of Ω , {\displaystyle \Omega ,} there exists a π-system I Σ , {\displaystyle {\mathcal {I}}_{\Sigma },} called the π-system generated by Σ {\displaystyle {\boldsymbol {\varSigma }}} , that is the unique smallest π-system of Ω {\displaystyle \Omega } containing every element of Σ . {\displaystyle \Sigma .} It is equal to the intersection of all π-systems containing Σ , {\displaystyle \Sigma ,} and can be explicitly described as the set of all possible non-empty finite intersections of elements of Σ : {\displaystyle \Sigma :}
{ E 1 ∩ ⋯ ∩ E n : 1 ≤ n ∈ N and E 1 , … , E n ∈ Σ } . {\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.
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