sigmafiniteness

Sigma-finiteness is a property of a measure space that ensures the space can be decomposed into a countable union of sets of finite measure. Formally, a measure space (X, Σ, μ) is sigma-finite if there exists a sequence (E_n) of measurable subsets of X such that X = ⋃_{n=1}^∞ E_n and μ(E_n) < ∞ for every n.

Equivalently, the space is covered by a countable family of finite-measure sets. This condition is automatically satisfied by finite measure spaces, and it is preserved under many standard constructions. For example, if μ is sigma-finite and ν is absolutely continuous with respect to μ, then ν is also sigma-finite, since the Radon-Nikodym derivative f ∈ L^1(μ) yields ν(E_n) = ∫_{E_n} f dμ < ∞ for each finite μ(E_n).

Examples include the Lebesgue measure on R^n, which is sigma-finite because R^n is the union of the finite intervals [-k, k]^n, the counting measure on a countable set (each singleton has finite measure, and the entire space is a countable union of singletons), and Haar measure on any σ-compact locally compact group.

Non-examples include the counting measure on an uncountable set, since a countable union of finite sets cannot exhaust an uncountable space. Some singular measures that assign infinite measure to every nonempty set also fail sigma-finiteness.

Sigma-finiteness is important in several fundamental results, such as the Radon-Nikodym theorem and Fubini/Tonelli theorems, where sigma-finiteness of the measures involved ensures the validity of interchange and density arguments.