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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-2,
author = {V. V. Chistyakov and A. Rychlewicz},
title = {On the extension and generation of set-valued mappings of bounded variation},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {235-247},
zbl = {1019.26010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-2}
}
V. V. Chistyakov; A. Rychlewicz. On the extension and generation of set-valued mappings of bounded variation. Studia Mathematica, Tome 151 (2002) pp. 235-247. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-2/