Vitali–Hahn–Saks theorem

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In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), states that given μn 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) = \mu(X)

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.

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