The joint modulus of variation of metric space valued functions and pointwise selection principles
Volume 238 / 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.
