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

TL;DR

This paper introduces a new joint modulus of variation for functions into metric spaces and establishes pointwise selection principles, generalizing many classical Helly-type theorems with applications to convergence and regulation of functions.

Contribution

It defines the joint modulus of variation and proves a new pointwise selection principle linking it to convergence and regulation of functions in metric spaces.

Findings

The joint modulus of variation generalizes classical variation concepts.

Sequences with bounded joint modulus admit pointwise convergent subsequences.

The results encompass and extend known Helly-type theorems.

Abstract

Given $T\subset\mathbb{R}$ and a metric space $M$, we introduce a nondecreasing sequence of pseudometrics $\{\nu_n\}$ 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 $\nu_n(f_j,g_j)$ 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 $\limsup$ 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.