Meaning of σ-algebra | Babel Free

Definitions

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.

Equivalents

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.”

“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).”

“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.”

CEFR level