σ-algebra

plural σ-algebras · alternative sigma-algebra · alternative Σ-algebra

1. mathematical analysisA 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.

   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).

   2013, Alexandr A. Borovkov, Probability Theory, Springer, page 15

   2 more examples

   Consider all the sigma-algebras that contain #92;mathcal#123;F#125; and take their intersection, which we call #92;Sigma, i.e., a subset A#92;subset#92;Omega is in #92;Sigma if and only if A is in every sigma-algebra containing #92;mathcal#123;F#125;. It is easy to check that #92;Sigma is indeed a sigma-algebra.

   2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4

   Denote by #92;mathcal#123;F#125; the set of subsets of [the sample space] Ω which are considered events. #92;mathcal#123;F#125; is called the space of events. In rigorous probability theory, #92;mathcal#123;F#125; is required to be a sigma-algebra on Ω.

   2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics, 2nd edition, San Bernardino, CA, USA, →ISBN, §10.5.1, page 75

   Synonyms: σ-field