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

Abstract

Given $T\subset\mathbb{R}$ and a metric space $M$, we introduce a nondecreasing sequence of pseudometrics $\{ν_n\}$ on $M^T$ (the set of all functions from $T$ into $M$), called the \emph{joint modulus of variation}. We prove that if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^T$ are such that $\{f_j\}$ is pointwise precompact, $\{g_j\}$ is pointwise convergent, and the limit superior of $ν_n(f_j,g_j)$ 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 $\limsup$ 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.