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

Vyacheslav V. Chistyakov¹ ; Svetlana A. Chistyakova¹

¹ Department of Informatics, Mathematics and Computer Science National Research University Higher School of Economics Bol’shaya Pechërskaya St. 25/12 Nizhny Novgorod 603155, Russia

Studia Mathematica, Tome 238 (2017) no. 1, pp. 37-57

Résumé

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.

Détail

DOI : 10.4064/sm8522-8-2016

Keywords: given subset mathbb metric space introduce nondecreasing sequence pseudometrics set functions called joint modulus variation prove sequences functions pointwise precompact nbsp pointwise convergent limsup infty g infty admits pointwise convergent subsequence whose limit conditionally regulated function illustrate sharpness result examples particular assumption limsup necessary uniformly convergent sequences almost necessary converge pointwise known helly type pointwise selection theorems its particular cases