# σ-algebra

## English

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### Pronunciation

• IPA(key): /ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/

### Noun

σ-algebra (plural σ-algebras)

1. 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.
• 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).
• 2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics[1], 2nd edition, San Bernardino, CA, USA, →ISBN, §10.5.1, page 75:
Denote by ${\displaystyle {\mathcal {F}}}$  the set of subsets of [the sample space] Ω which are considered events. ${\displaystyle {\mathcal {F}}}$  is called the space of events. In rigorous probability theory, ${\displaystyle {\mathcal {F}}}$  is required to be a sigma-algebra on Ω.

#### Synonyms

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