π-system

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pi-system

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• pi-system

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

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

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

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

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

• 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.

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

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

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

• stymes