Vitali–Hahn–Saks theorem

In mathematics, the Vitali–Hahn–Saks theorem (named for Giuseppe Vitali, Hans Hahn, and Stanisław Saks) 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

$\lim_{n \rightarrow \infty} \mu_n(X) \rightarrow \mu(X)$

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.