σ-algebra
Contents
EnglishEdit
Alternative formsEdit
PronunciationEdit
- IPA^{(key)}: /ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/
NounEdit
σ-algebra (plural σ-algebras)
- (analysis) A collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements (with respect to the given set) and the collection is closed under countable unions.
- 2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4,
- Consider all the sigma-algebras that contain and take their intersection, which we call , i.e., a subset is in if and only if is in every sigma-algebra containing . It is easy to check that is indeed a sigma-algebra.
- 2003, Zoltán Ésik, Axiomatizing the Least Fixed Point Operation and Binary Supremum, Peter G. Clote, Helmut Schwichtenberg (editors), Computer Science Logic: 14th International Workshop, CSL 2000 Annual, Springer, page 305,
- A homomorphism of continuous Σ-algebras is a Σ-algebra homomorphism which is an ω-continuous function. It follows that any homomorphism of ω-continuous Σ-algebras is a homomorphism of the corresponding preiteration Σ-algebras.
- 2013, Alexandr A. Borovkov, Probability Theory, Springer, page 15,
- Consider all the σ-algebras on [0,1] containing all intervals from that segment (there is at least one such σ-algebra, for the collection of all the subsets of a given set clearly forms a σ-algebra).
- 2015, Arno Berger, Theodore P. Hill, An Introduction to Benford's Law, Princeton University Press, page 14,
- As detailed below, therefore, instead of the power set, the standard σ-algebra on used to [define] and analyze both continuous and discrete distributions is the so-called Borel σ-algebra , a proper subset (sub-σ-algebra) of the power set of .
- 2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4,
SynonymsEdit
- (collection of subsets that obeys certain conditions): σ-field
HypernymsEdit
- (collection of subsets that obeys certain conditions): field of sets, set algebra
HyponymsEdit
- (collection of subsets that obeys certain conditions): Borel σ-algebra, μ-completion
MeronymsEdit
HolonymsEdit
See alsoEdit
Further readingEdit
- Algebra of sets on Wikipedia.Wikipedia
- Sigma-ring on Wikipedia.Wikipedia
- sigma-algebra on nLab
- Algebra of sets on Encyclopedia of Mathematics
- Sigma-algebra (Computer Science) on Encyclopedia of Mathematics