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

• ${\displaystyle P}$ is non-empty.
• If ${\displaystyle A,B\in P}$ then ${\displaystyle A\cap B\in P.}$

That is, ${\displaystyle P}$ is a non-empty family of subsets of ${\displaystyle \Omega }$ that is closed under non-empty finite intersections.^{[nb 1]} 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.

• For any real numbers ${\displaystyle a}$ and ${\displaystyle b,}$ the intervals ${\displaystyle (-\infty ,a}$ form a π-system, and the intervals ${\displaystyle (a,b}$ form a π-system if the empty set is also included.
• The topology (collection of open subsets) of any topological space is a π-system.
• Every filter is a π-system. Every π-system that doesn't contain the empty set is a prefilter (also known as a filter base).
• For any measurable function ${\displaystyle f:\Omega \to \mathbb {R} ,}$ the set  ${\displaystyle {\mathcal {I}}_{f}=\left\{f^{-1}((-\infty ,x):x\in \mathbb {R} \right\}}$ defines a π-system, and is called the π-system generated by ${\displaystyle f.}$ (Alternatively, ${\displaystyle \left\{f^{-1}((a,b):a,b\in \mathbb {R} ,a<b\right\}\cup \{\varnothing \}}$ defines a π-system generated by ${\displaystyle f.}$)
• If ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ are π-systems for ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2},}$ respectively, then ${\displaystyle \{A_{1}\times A_{2}:A_{1}\in P_{1},A_{2}\in P_{2}\}}$ is a π-system for the Cartesian product ${\displaystyle \Omega _{1}\times \Omega _{2}.}$
• Every 𝜎-algebra is a π-system.

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

• ${\displaystyle \mathrm{\Omega <!--\; \Omega \; -->}\in <!--\; \in \; -->D,}$Zdroj:https://en.wikipedia.org?pojem=Pi-system
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