See π-system on Wiktionary
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"ref": "2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49:",
"text": "To see this, first check that #x5C;sigma(X#x5F;0,X#x5F;1,#x5C;dots)#x3D;#x5C;sigma(#x5C;mathcalF#x5F;0), where #x5C;textstyle#x5C;mathcalF#x5F;0#x3A;#x3D;#x5C;bigcup#x5C;infty#x5F;#x7B;k#x3D;0#x7D;#x5C;sigma(X#x5F;0,#x5C;dots,X#x5F;k) is a field and, in particular, a #x5C;boldsymbol#x5C;pi-system.",
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"ref": "2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups, Taylor & Francis (CRC Press), page 221:",
"text": "We start with a basis of simple roots #x5C;Delta of #x5C;Phi. Then we apply all possible elementary transformations and add the resulting #x5C;boldsymbol#x5C;pi-systems to the list. Of course, if #x5C;Gamma is a #x5C;boldsymbol#x5C;pi-system, and #x5C;Gamma' is a #x5C;boldsymbol#x5C;pi-system obtained from it by an elementary transformation and the diagrams of #x5C;Gamma and #x5C;Gamma' are the same, the root subsystems they span are the same, and therefore we do not add #x5C;Gamma'.",
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"ref": "2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis, Taylor & Francis (CRC Press):",
"text": "Clearly the definitions for a #x5C;boldsymbol#x5C;pi-system and a #x5C;lambda-system are both satisfied by a #x5C;sigma-algebra.[…]\nProposition 4.1.8 Let #x5C;Omega be a set and B be a collection of subsets of #x5C;Omega. The collection B is a #x5C;sigma-algebra if and only if B is a #x5C;lambda-system and a #x5C;boldsymbol#x5C;pi-system.",
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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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