On the extension and generation of set-valued mappings of bounded variation

Tom 153 / 2002

V. V. Chistyakov, A. Rychlewicz Studia Mathematica 153 (2002), 235-247 MSC: Primary 26A45, 26A51, 54C65; Secondary 26A16, 54C60, 54E50. DOI: 10.4064/sm153-3-2


We study set-valued mappings of bounded variation of one real variable. First we prove the existence of an extension of a metric space valued mapping from a subset of the reals to the whole set of reals with preservation of properties of the initial mapping: total variation, Lipschitz constant or absolute continuity. Then we show that a set-valued mapping of bounded variation defined on an arbitrary subset of the reals admits a regular selection of bounded variation. We introduce a notion of generated set-valued mappings and show that, under suitable assumptions, set-valued mappings (with arbitrary domains) which are Lipschitzian, of bounded variation or absolutely continuous are generated by certain families of mappings with nice properties. Finally, we prove a Castaing type representation theorem for set-valued mappings of bounded variation.


  • V. V. ChistyakovDepartment of Mathematics
    University of Nizhny Novgorod
    23 Gagarin Avenue
    Nizhny Novgorod 603950, Russia
  • A. RychlewiczFaculty of Mathematics
    Łódź University
    Stefana Banacha 22
    90-238 Łódź, Poland

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