A+ CATEGORY SCIENTIFIC UNIT

The joint modulus of variation of metric space valued functions and pointwise selection principles

Volume 238 / 2017

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova Studia Mathematica 238 (2017), 37-57 MSC: Primary 26A45, 28A20; Secondary 54C35, 54E50. DOI: 10.4064/sm8522-8-2016 Published online: 28 March 2017

Abstract

Given $T\subset\mathbb R$ and a metric space $M$, we introduce a nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^T$ (the set of all functions from $T$ into $M$), called the {joint modulus of variation}. We prove that {if two sequences $\{f_j\}$ and $\{g_j\}$ of functions from $M^T$ are such that $\{f_j\}$ is pointwise precompact, $\{g_j\}$ is pointwise convergent, and $\limsup_{j\to\infty}\nu_n(f_j,g_j) = o(n)$ as $n\to\infty$, then $\{f_j\}$ admits a pointwise convergent subsequence whose limit is a conditionally regulated function}. We illustrate the sharpness of this result by examples (in particular, the assumption on the $\limsup$ is necessary for uniformly convergent sequences $\{f_j\}$ and $\{g_j\}$, and ‘almost necessary’ when they converge pointwise) and show that most of the known Helly-type pointwise selection theorems are its particular cases.

Authors

  • Vyacheslav V. ChistyakovDepartment of Informatics, Mathematics and Computer Science
    National Research University Higher School of Economics
    Bol’shaya Pechërskaya St. 25/12
    Nizhny Novgorod 603155, Russia
    e-mail
    e-mail
  • Svetlana A. ChistyakovaDepartment of Informatics, Mathematics and Computer Science
    National Research University Higher School of Economics
    Bol’shaya Pechërskaya St. 25/12
    Nizhny Novgorod 603155, Russia
    e-mail

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