The optimal form of selection principles for functions of a real variable |
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Authors: | Vyacheslav V. Chistyakov |
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Affiliation: | Department of Mathematics, State University Higher School of Economics, Bol'shaya Pechërskaya Street 25, Nizhny Novgorod 603600, Russia |
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Abstract: | Let T be a nonempty set of real numbers, X a metric space with metric d and XT the set of all functions from T into X. If fXT and n is a positive integer, we set , where the supremum is taken over all numbers a1,…,an,b1,…,bn from T such that a1b1a2b2anbn. The sequence is called the modulus of variation of f in the sense of Chanturiya. We prove the following pointwise selection principle: If a sequence of functions is such that the closure in X of the set is compact for each tT and then there exists a subsequence of , which converges in X pointwise on T to a function fXT satisfying limn→∞ν(n,f)/n=0. We show that condition (*) is optimal (the best possible) and that all known pointwise selection theorems follow from this result (including Helly's theorem). Also, we establish several variants of the above theorem for the almost everywhere convergence and weak pointwise convergence when X is a reflexive separable Banach space. |
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Keywords: | Modulus of variation Selection principle Pointwise convergence Proper function Generalized variation |
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