A Selection Principle for Mappings of Bounded Variation

AbstractE. Helly's selection principle states that an infinite bounded family of real functions on the closed interval, which is bounded in variation, contains a pointwise convergent sequence whose limit is a function of bounded variation. 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 (non-convex) set-valued mappings: any set-valued mapping from an interval of the real line into nonempty compact subsets of a metric space, which is of bounded variation with respect to the Hausdorff metric, admits a selection of bounded variation. Also, we show that a compact-valued set-valued mapping which is Lipschitzian, absolutely continuous, or of bounded Riesz Φ-variation admits a selection which is Lipschitzian, absolutely continuous, or of bounded Riesz Φ-variation, respectively