Vitali–Hahn–Saks theorem

In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), states that given μ_n for each integer n >0, a countably additive function defined on a fixed sigma-algebra Σ, with values in a given Banach space 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 vector measures is a vector measure.