sigmafinitas

Sigmafinitas, or sigma-finiteness, is a property of a measure space in measure theory. A measure space is sigma-finite if its underlying set can be written as a countable union of measurable subsets, each having finite measure. More precisely, a measure space (X, Σ, μ) is sigma-finite if there exist sets A1, A2, ... in Σ with μ(An) < ∞ for all n and X = ∪n An (the sets need not be disjoint).

Examples and basic cases include: Lebesgue measure on the real line is sigma-finite because R = ∪n=1∞ [−n, n] with μ([−n, n]) = 2n < ∞. Any probability measure is finite, hence sigma-finite. The counting measure on a countable set is sigma-finite, since the space can be written as a union of singletons, each of measure 1; on an uncountable set, the counting measure is not sigma-finite because a countable union of finite sets is countable, not the whole space.

Properties and implications: If μ is sigma-finite and E is a measurable subset with μ(E) < ∞, then the restricted measure μ|E is also sigma-finite. The property is preserved under common constructions, such as product measures: if μ and ν are sigma-finite, then their product μ×ν is sigma-finite. Sigma-finiteness is a standard hypothesis in many fundamental theorems of integration and probability, including the Radon–Nikodym theorem, Fubini’s theorem in general form, and disintegration results.

See also: σ-finite, measure theory, locally finite measure, Radon–Nikodym theorem.