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A generalization of the Helly theorem for functions with values in a uniform space. (English. Russian original) Zbl 1210.54007

Russ. Math. 54, No. 5, 35-46 (2010); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2010, No. 5, 41-54 (2010).
Let \(T\) be a subset of the real line and \(Y\) a Hausdorff uniform space. In terms of generalized \(p\)-variation a sufficient condition for the existence of a pointwise convergent subsequence for a relatively sequentially compact sequence \(f_n: T\to Y\) is proved. This condition is also necessary for the uniform convergence of \(f_n\). A selection principle for the a.e. convergence is deduced.
The results may be useful e.g. for proving the existence of selections of bounded \(p\)-variation for multifunctions, in study of Niemytski superposition operator, stochastic processes, harmonic analysis etc.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54E15 Uniform structures and generalizations
26A45 Functions of bounded variation, generalizations
26A48 Monotonic functions, generalizations
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