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On the theory of set-valued maps of bounded variation of one real variable

V. V. Chistyakov

N. I. Lobachevski State University of Nizhni Novgorod

Abstract: (Set-valued) maps of bounded variation in the sense of Jordan defined on a subset of the real line and taking values in metric or normed linear spaces are studied. A structure theorem (more general than the Jordan decomposition) is proved for such maps; an analogue of Helly's selection principle is established. A compact set-valued map into a Banach space that is a map of bounded variation (or a Lipschitz or an absolutely continuous map) is shown to have a continuous selection of bounded variation (respectively, Lipschitz or absolutely continuous selection).

UDC: 517.518.24+515.124

MSC: Primary 26A16, 26A45, 26A99; Secondary 46B99

Received: 09.09.1997

DOI: 10.4213/sm321

 English version: Sbornik: Mathematics, 1998, 189:5, 797–819

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