# σ-algebra

## English

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### Pronunciation

• IPA(key): /ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/

### Noun

σ-algebra (plural σ-algebras)

1. (mathematical 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 ${\displaystyle {\mathcal {F}}}$  and take their intersection, which we call ${\displaystyle \Sigma }$ , i.e., a subset ${\displaystyle A\subset \Omega }$  is in ${\displaystyle \Sigma }$  if and only if ${\displaystyle A}$  is in every sigma-algebra containing ${\displaystyle {\mathcal {F}}}$ . It is easy to check that ${\displaystyle \Sigma }$  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 ${\displaystyle \mathbb {R} }$  used to [define] and analyze both continuous and discrete distributions is the so-called Borel σ-algebra ${\displaystyle {\mathcal {B}}}$ , a proper subset (sub-σ-algebra) of the power set of ${\displaystyle \mathbb {R} }$ .

#### Synonyms

• (collection of subsets that obeys certain conditions): σ-field