Sigmaadditiva

Sigmaadditiva, or sigma-additive, is a fundamental property of a set function in measure theory. A function μ defined on a sigma-algebra F of subsets of a given set Ω is sigma-additive if μ(∅) = 0 and, for any countable collection of pairwise disjoint sets {A1, A2, ...} in F, the equality μ(∪i Ai) = ∑i μ(Ai) holds. When μ takes only finite values, this condition is equivalent to the standard definition of a measure.

Most familiar measures are sigma-additive. Examples include the counting measure on any set, Lebesgue measure on the real line, and probability measures, where the total measure μ(Ω) equals 1. Dirac measures, which assign mass to a single point, are also sigma-additive. The concept is central to integration and probability theory, since sigma-additivity ensures consistency across countable unions and underpins many limit and convergence results.

The requirement of sigma-additivity distinguishes measures from merely finitely additive set functions. Finite additivity ensures additivity for finite unions only and does not guarantee the behavior over infinite or countable unions, leading to potential pathological examples in foundations.

Sigma-additivity has important theoretical consequences. It enables the Carathéodory extension theorem, which constructs a unique sigma-additive measure from a pre-measure on an algebra, and underlies constructions of probability spaces via Kolmogorov’s extension theorem. In analysis, sigma-additivity is essential for the Monotone Convergence and Dominated Convergence Theorems, which justify interchange of limits and integrals.