A Selection Principle for Mappings of Bounded Variation

E. Helly’s selection principle states that an infinite bounded family of real functions on the closed inter al, which is bounded in ariation, contains a pointwise con ergent sequence whose limit is a function of bounded ariation. We extend this theorem to metric space valued mappings of bounded variation. Then we apply the extended Helly selection principle to obtain the existence of regular selections of Žnon-convex.set-valued mappings: any set- alued mapping from an inter al of the real line into nonempty compact subsets of a metric space, which is of bounded ariation with respect to the Hausdorff metric, admits a selection of bounded ariation. Also, we show that a compact-valued set-valued mapping which is Lipschitzian, absolutely continuous, or of bounded Riesz -variation admits a selection which is Lipschitzian, absolutely continuous, or of bounded Riesz -variation, respectively