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 Ž . non-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 boundedariation 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.

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We introduce a concept of metric space valued mappings of two variables with finite total variation and define a counterpart of the Hardy space. Then we establish the following Helly type selection principle for mappings of two variables: Let X be a metric space and a commutative additive semigroup whose metric is translation invariant. Then an infinite pointwise precompact family of X-valued mappings on the closed rectangle of the plane, which is of uniformly bounded total variation, contains a pointwise convergent sequence whose limit is a mapping with finite total variation.

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