Abstract
An analog of the classical Michael theorem on continuous single-valued selections of lower semicontinuous maps whose values are closed and convex in a Fréchet space is proved for maps into metrizable (non-locally-convex) vector spaces. It turns out that, instead of the local convexity of the whole space containing these values, it is sufficient to require that the family of values of the map be uniformly locally convex. In contrast to the standard selection theorems, the proof bypasses the process of successively improving the approximations, and the desired selection is constructed as the result of pointwise integration with respect to a suitable probability distribution.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 2, pp. 89–94, 2008
Original Russian Text Copyright © by P. V. Semenov
The research was supported by the Russian Foundation for Basic Research (grant 08-01-00663).
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Semenov, P.V. Continuous selections as parametrically defined integrals. Funct Anal Its Appl 42, 155–159 (2008). https://doi.org/10.1007/s10688-008-0024-4
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DOI: https://doi.org/10.1007/s10688-008-0024-4