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Dinamatematika,aljabar σ (atawawidang σ)X pikeun sasétS hartina anggotasubsétS nu katutup ku sét operasi-operasi nu bisa diitung; aljabar σ utamana dipaké pikeun nangtukeunukuranS. Ieu konsép penting dinaanalisis matematika jeungtéori probabilitas.
Sacara formal,X kaasup aljabar σ mun jeung ukur mun (jika dan hanya jika,if and only if) miboga pasipatan di handap ieu:
From 1 and 2 it follows thatS is inX; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections (viaDe Morgan's laws).
An ordered pair (S,X), whereS is a set andX is a σ-algebra overS, is called améasurable space.
MunS mangrupa sét naon baé, then the family consisting only of the empty set andS is a σ-algebra overS, the so-calledtrivial σ-algebra. Another σ-algebra overS is given by the fullpower set ofS.
If {Xa} is a family of σ-algebras overS, then the intersection of allXa is also a σ-algebra overS.
IfU is an arbitrary family of subsets ofS then we can form a special σ-algebra fromU, called theσ-algebra generated by U. We denote it by σ(U) and define it as follows.First note that there is a σ-algebra overS that containsU, namely the power set ofS.Let Φ be the family of all σ-algebras overS that containU (that is, a σ-algebraX overS is in Φ if and only ifU is a subset ofX.)Then we define σ(U) to be the intersection of all σ-algebras in Φ. σ(U) is then the smallest σ-algebra overS that containsU.
This léads to the most important example: theBorel algebra over anytopological space is the σ-algebra generated by theopen sets (or, equivalently, by theclosed sets).Note that this σ-algebra is not, in general, the whole power set.For a non-trivial example, see theVitali set.
On theEuclidean spaceRn, another σ-algebra is of importance: that of allLebesgue measurable sets. This σ-algebra contains more sets than the Borel algebra onRn and is preferred inintegration théory.
See alsomeasurable function.