sigmaadditivity

Sigmaadditivity, also called countable additivity, is a central property of measures in measure theory. A set function μ defined on a sigma-algebra 𝔽 over a set Ω is sigma-additive if μ(∅) = 0 and for every sequence (A_n) of pairwise disjoint sets in 𝔽, μ(∪_{n=1}^∞ A_n) = ∑_{n=1}^∞ μ(A_n), with the sum taken in the extended nonnegative reals [0, ∞]. Equivalently, μ is countably additive.

This property ensures a measure behaves well with respect to countable unions. It implies monotonicity and yields important continuity results: for an increasing sequence A_1 ⊆ A_2 ⊆ … in 𝔽, μ(∪_{n=1}^∞ A_n) = lim_{n→∞} μ(A_n); and, if μ(A_1) < ∞ for a decreasing sequence A_1 ⊇ A_2 ⊇ …, then μ(∩_{n=1}^∞ A_n) = lim_{n→∞} μ(A_n).

Examples of sigma-additive set functions include the Lebesgue measure on the real line, the counting measure on any set, and probability measures, where μ(Ω) = 1. Sigma-additivity underpins many core results and constructions, such as the Carathéodory extension theorem, which extends a pre-measure defined on an algebra to a measure on the generated sigma-algebra.

By contrast, finite additivity alone does not guarantee this countable property; measures must be sigma-additive to fit the standard axioms of measure theory.