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Mat. Zametki, 2012, Volume 92, Issue 3, Pages 410–416 (Mi mz9072)  

A Generalization of the Set Averaging Theorem

G. Ivanov, E. S. Polovinkin

Moscow Institute of Physics and Technology

Abstract: We consider the possibility of generalizing the averaging theorem from the case of sets from $n$-dimensional Euclidean space to the case of sets from Banach spaces. The result is a cornerstone for constructing the theory of the Riemann integral for non-convex-valued multivalued mappings and for proving the convexity of this multivalued integral. We obtain a generalization of the averaging theorem to the case of sets from uniformly smooth Banach spaces as well as some corollaries.

Keywords: set averaging theorem, $n$-dimensional Euclidean space, Banach space, Riemann integral, non-convex-valued multivalued mapping, convex compact set, Hausdorff metric

DOI: https://doi.org/10.4213/mzm9072

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English version:
Mathematical Notes, 2012, 92:3, 369–374

Bibliographic databases:

UDC: 517.9
Received: 30.08.2011
Revised: 12.12.2011

Citation: G. Ivanov, E. S. Polovinkin, “A Generalization of the Set Averaging Theorem”, Mat. Zametki, 92:3 (2012), 410–416; Math. Notes, 92:3 (2012), 369–374

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