Sigmafields

A sigma-field, or sigma-algebra, on a set Omega is a collection F of subsets of Omega that satisfies three conditions: Omega is in F; whenever A is in F, its complement Omega \ A is in F; and whenever A1, A2, … are in F, the union A1 ∪ A2 ∪ … is in F. From these it follows that the empty set is in F and F is closed under complements and countable unions, hence under countable intersections as well.

The sigma-field generated by a collection C of subsets of Omega, denoted σ(C), is the smallest sigma-field containing C. Equivalently, σ(C) is the intersection of all sigma-fields that contain C. If C partitions Omega into atoms, σ(C) consists of all unions of these atoms.

Examples: The power set P(Omega) is a sigma-field. The trivial sigma-field {∅, Omega} is always a sigma-field. On the real line R, the Borel sigma-field is generated by the open sets. If C is finite, σ(C) is finite and equals the finite-field generated by C, since only finitely many unions of sets can occur.

Properties: Sigma-fields form a lattice under inclusion; the intersection of any collection of sigma-fields is a sigma-field, and the sigma-field generated by a collection is the intersection of all sigma-fields containing that collection. Not every subset of Omega is measurable with respect to a given sigma-field; measurability depends on membership in F.

Relation to probability and analysis: A probability measure P is defined on a sigma-field F on Omega, assigning probabilities to events in F and respecting countable additivity. Measurable functions, such as random variables, are defined relative to (Omega, F) and a target measurable space.