In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Vitali–Hahn–Saks theorem (named for Giuseppe Vitali

Giuseppe Vitali

Giuseppe Vitali was an Italian mathematician who worked in several branches of mathematical analysis.- Mathematical contributions :...

, Hans Hahn

Hans Hahn

Hans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.-Biography:...

, and Stanisław Saks) states that given μ_n for each integer n >0, a countably additive function defined on a fixed sigma-algebra

Sigma-algebra

In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

 Σ, with values in a given Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

 B, such that


exists for every set X in Σ, then μ is also countably additive. In other words, the limit of a sequence of spectral measures is a spectral measure.