ON THE THEORY OF SET-VALUED MAPS OF BOUNDED VARIATION OF ONE REAL VARIABLE

(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 lin ear spaces are studied. A structure theorem (more general than the Jor dan decomposition) is proved for such maps; an analogue of Helly's sel ection principle is established. A compact set-valued map into a Banac h space that is a map of bounded variation (or a Lipschitz or an absol utely continuous map) is shown to have a continuous selection of bound ed variation (respectively, Lipschitz or absolutely continuous selecti on).