π-system
noun1
Mathematics, Measure theory, Probability theory, Sciences, Set theory A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections.
- To see this, first check that #92;sigma(X#95;0,X#95;1,#92;dots)#61;#92;sigma(#92;mathcalF#95;0), where #92;textstyle#92;mathcalF#95;0#58;#61;#92;bigcup#92;infty#95;#123;k#61;0#125;#92;sigma(X#95;0,#92;dots,X#95;k) is a field and, in particular, a #92;boldsymbol#92;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 #92;Delta of #92;Phi. Then we apply all possible elementary transformations and add the resulting #92;boldsymbol#92;pi-systems to the list. Of course, if #92;Gamma is a #92;boldsymbol#92;pi-system, and #92;Gamma' is a #92;boldsymbol#92;pi-system obtained from it by an elementary transformation and the diagrams of #92;Gamma and #92;Gamma' are the same, the root subsystems they span are the same, and therefore we do not add #92;Gamma'.2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups, Taylor & Francis (CRC Press), page 221:
1 more example
- Clearly the definitions for a #92;boldsymbol#92;pi-system and a #92;lambda-system are both satisfied by a #92;sigma-algebra.[…] Proposition 4.1.8 Let #92;Omega be a set and B be a collection of subsets of #92;Omega. The collection B is a #92;sigma-algebra if and only if B is a #92;lambda-system and a #92;boldsymbol#92;pi-system.2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis, Taylor & Francis (CRC Press):
- Related terms
- δ-ring, π bond, π system, π-calculus