Definition of "π-system"

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

2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49

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

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

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

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