π-system

English

Alternative forms

• pi-system

Noun

π-system (plural π-systems)

1. (set theory, measure theory, probability theory) A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections.
   • 2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49:

     To see this, first check that $\sigma (X_{0},X_{1},\dots )=\sigma ({\mathcal {F}}_{0})$, where $\textstyle {\mathcal {F}}_{0}:=\bigcup _{k=0}^{\infty }\sigma (X_{0},\dots ,X_{k})$ is a field and, in particular, a ${\boldsymbol {\pi }}$-system.

   • 2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups‎^[1], Taylor & Francis (CRC Press), page 221:

     We start with a basis of simple roots $\Delta$ of $\Phi$. Then we apply all possible elementary transformations and add the resulting ${\boldsymbol {\pi }}$-systems to the list. Of course, if $\Gamma$ is a ${\boldsymbol {\pi }}$-system, and $\Gamma '$ is a ${\boldsymbol {\pi }}$-system obtained from it by an elementary transformation and the diagrams of $\Gamma$ and $\Gamma '$ are the same, the root subsystems they span are the same, and therefore we do not add $\Gamma '$.

   • 2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis‎^[2], Taylor & Francis (CRC Press):

     Clearly the definitions for a ${\boldsymbol {\pi }}$-system and a $\lambda$-system are both satisfied by a $\sigma$-algebra. […]
     Proposition 4.1.8 Let $\Omega$ be a set and $B$ be a collection of subsets of $\Omega$. The collection $B$ is a $\sigma$-algebra if and only if $B$ is a $\lambda$-system and a ${\boldsymbol {\pi }}$-system.

Usage notes

• By convention, the empty intersection (aka nullary intersection: the "intersection of no sets") is taken to be Ώ itself: its explicit exclusion means that Ώ need not be a member of any arbitrary π-system (i.e., of every π-system).
• The system is said to be a π-system on Ώ.
• For any family Σ of subsets of Ώ, there exists a unique smallest π-system that contains every element of Σ: it is called the π-system generated by Σ.

Hyponyms

• (collection of subsets): filter, topology, σ-algebra

Translations

collection of subsets closed under non-empty finite intersections

• French: π-système m, pi-système m
• German: π-System n
• Italian: π-sistema m, sistema pi m

See also

• δ-ring
• π bond (type of chemical bond)
• π system (system involving π bonds)
• π-calculus

Further reading

• Finite intersection property on Wikipedia.Wikipedia
• Delta-ring on Wikipedia.Wikipedia

Anagrams

• stymes