The approximate variation of univariate uniform space valued functions and pointwise selection principles

Abstract

Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the uniformity $\mathcal{U}$. Given $f\in X^T$ (=the family of all functions from $T$ into $X$), the approximate variation of $f$ is the two-parameter family $\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where $V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations $V_p(g)$ on $T$ with respect to $d_p$ of all functions $g\in X^T$ such that $d_p(f(t),g(t))\le\varepsilon$ for all $t\in T$. We establish the following pointwise selection principle: If a pointwise relatively sequentially compact sequence of functions $\{f_j\}_{j=1}^\infty\subset X^T$ is such that $\limsup_{j\to\infty}V_{\varepsilon,p}(f_j)<\infty$ for all $\varepsilon>0$ and $p\in\mathcal{P}$, then it contains a subsequence which converges pointwise on $T$ to a bounded regulated function $f\in X^T$. We illustrate this result by appropriate examples, and present a characterization of regulated functions $f\in X^T$ in terms of the approximate variation.