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

Studia Mathematica. 2017. Vol. 238. No. 1. P. 37–57.

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova

Given a subset $T$ of the reals $R$ 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 the limit superior of $\nu_n(f_j,g_j)$ as $j\to\infty$ is $o(n)$ 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 lim sup 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.

Keywords: metric space, метрическое пространство, selection principle, pointwise convergence, принцип выбора, поточечная сходимость, joint modulus of variation, regulated function, generalized variation, совместный модуль вариации, обобщенная вариация, регулярная функция

Publication based on the results of: Модулярные пространства, являющиеся модулярно полными (Modular spaces that are modular complete) (2017)