sigmaadditive

Sigma-additive, also called sigma-additive or countably additive, describes a property of a set function defined on a sigma-algebra. Formally, a function μ: F → [0, ∞] on a sigma-algebra F over a set X is sigma-additive if for every countable collection {A_n} of pairwise disjoint sets in F, the equality μ(⋃_{n=1}^∞ A_n) = ∑_{n=1}^∞ μ(A_n) holds. This is equivalent to the notion of countable additivity.

Relation to measures: A measure is a nonnegative sigma-additive set function with the additional normalization μ(∅) = 0. If μ is a probability measure, then μ(X) = 1. Conversely, measures include common examples such as the Lebesgue measure on the real line, or Dirac measures concentrated at a point.

Key properties: Sigma-additivity implies several continuity properties. If A_n ↑ A (increasing to A), then μ(A) = lim μ(A_n) (continuity from below). If A_n ↓ A (decreasing to A) and μ(A_1) < ∞, then μ(A) = lim μ(A_n) (continuity from above). These forms of continuity underpin many limit arguments in integration and probability.

Examples and notes: The Lebesgue measure on R and the Dirac measure δ_x are sigma-additive. Not every finitely additive set function is sigma-additive; there exist finitely additive “charges” that fail countable additivity, illustrating the distinction between finite additivity and sigma-additivity.

Applications: Sigma-additivity is fundamental to probability theory, integration, and the rigorous development of measure and integration theories.