The assertion in question comes from the short final section in Theory of capacities of Choquet (1953/54), in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this operation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and similarly for superadditivity. His treatment of this point was more like an outline, and his proof limited to a rather narrow special case. Thus the proper context and scope of the assertion has remained open. In this paper we present a counterexample which shows that the initial context has to be modified, and then in a new context we prove a comprehensive theorem which fulfils all the needs that have turned up so far.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-2-4,
author = {Heinz K\"onig},
title = {The (sub/super)additivity assertion of Choquet},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {171-197},
zbl = {1021.28003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-2-4}
}
Heinz König. The (sub/super)additivity assertion of Choquet. Studia Mathematica, Tome 157 (2003) pp. 171-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-2-4/