The approximate variation to pointwise selection principles

Abstract

Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$, where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations $V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all $t\in T$. The notion of $\varepsilon$-variation $V_\varepsilon(f)$ was introduced by Fra\'nkov\'a [Math. Bohem. 116 (1991), 20-59] for intervals $T=[a,b]$ in $\mathbb{R}$ and $M=\mathbb{R}^N$ and extended to the general case by Chistyakov and Chistyakova [Studia Math. 238 (2017), 37-57]. We prove directly the following basic pointwise selection principle: If a sequence of functions $\{f_j\}_{j=1}^\infty$ from $M^T$ is such that the closure in $M$ of the set $\{f_j(t):j\in\mathbb{N}\}$ is compact for all $t\in T$ and $\limsup_{j\to\infty}V_\varepsilon(f_j)$ is finite for all $\varepsilon>0$, then it contains a subsequence, which converges pointwise on $T$ to a bounded regulated function $f\in M^T$. We establish several variants of this result for sequences of regulated and nonregulated functions, for functions with values in reflexive separable Banach spaces, for the almost everywhere convergence and weak pointwise convergence of extracted subsequences, and comment on the necessity of assumptions in the selection principles. The sharpness of all assertions is illustrated by examples.