The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified treatment

The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified... We present proofs, based on the Shapley–Folkman theorem, of the convexity of the range of a strongly continuous, finitely additive measure, as well as that of an atomless, countably additive measure. We also present proofs, based on diagonalization and separation arguments respectively, of the closure of the range of a purely atomic or purely nonatomic countably additive measure. A combination of these results yields Lyapunov’s celebrated theorem on the range of a countably additive measure. We also sketch, through a comprehensive bibliography, the pervasive diversity of the applications of the Shapley–Folkman theorem in mathematical economics. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified treatment

Positivity , Volume 17 (3) – Mar 20, 2012
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Publisher
Springer Basel
Copyright
Copyright © 2012 by Springer Basel AG
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0173-9
Publisher site
See Article on Publisher Site

Abstract

We present proofs, based on the Shapley–Folkman theorem, of the convexity of the range of a strongly continuous, finitely additive measure, as well as that of an atomless, countably additive measure. We also present proofs, based on diagonalization and separation arguments respectively, of the closure of the range of a purely atomic or purely nonatomic countably additive measure. A combination of these results yields Lyapunov’s celebrated theorem on the range of a countably additive measure. We also sketch, through a comprehensive bibliography, the pervasive diversity of the applications of the Shapley–Folkman theorem in mathematical economics.

Journal

PositivitySpringer Journals

Published: Mar 20, 2012

References

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