sigmafield

A sigma-field, or sigma-algebra, on a set X is a nonempty collection F of subsets of X that satisfies three properties: X is in F and the empty set is in F; if A is in F, then the complement of A relative to X is in F; and if A1, A2, A3, ... are in F, then the union of all A_n is in F. From these conditions, F is closed under countable unions and, by De Morgan’s laws, under countable intersections. Every member of F is a subset of X, and F is a subset of the power set P(X).

Examples include the smallest sigma-field {∅, X}, the largest P(X), and, in the finite case, sigma-fields corresponding to partitions of X (the sigma-field consists of all unions of the atoms of the partition). The Borel sigma-field on the real line is the smallest sigma-field containing all open intervals. Given a collection C of subsets of X, the sigma-field generated by C is the smallest sigma-field containing C, equivalently the intersection of all sigma-fields that contain C.

Measures are defined on sigma-fields. A measure μ on a sigma-field F is a function μ: F → [0, ∞] with μ(∅) = 0 and countable additivity: for disjoint A_n in F, μ(∪ A_n) = ∑ μ(A_n). A probability measure is a measure with μ(X) = 1. Sigma-fields provide the natural domains for defining probability and integration, ensuring consistency under countable operations. In many texts, sigma-field and sigma-algebra are used synonymously, though terminology varies slightly by author.