6533b862fe1ef96bd12c6417

RESEARCH PRODUCT

VARIANTS OF A SELECTION PRINCIPLE FOR SEQUENCES OF REGULATED AND NON-REGULATED FUNCTIONS

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Let $T$ be a nonempty subset of $\RB$, $X$ a metric space with metric $d$ and $X^T$ the set of all functions mapping $T$ into $X$. Given $\vep>0$ and $f\in X^T$, we denote by $N(\vep,f,T)$ the least upper bound of those $n\in\NB$, for which there exist numbers $s_1,\dots,s_n,t_1,\dots,t_n$ from $T$ such that $s_1\vep$ for all $i=1,\dots,n$ ($N(\vep,f,T)=0$ if there are no such $n$'s). The following pointwise selection principle is proved: {\em If a sequence of functions\/ $\{f_j\}_{j=1}^\infty\subset X^T$ is such that the closure in $X$ of the sequence\/ $\{f_j(t)\}_{j=1}^\infty$ is compact for each $t\in T$ and\/ $\limsup_{j\to\infty}N(\vep,f_j,T)0$, then\/ $\{f_j\}_{j=1}^\infty$ contains a subsequence, converging pointwise on $T$ to a function $f\in X^T$, such that $N(\vep,f,T)0$}. We establish several variants of this result for functions with values in a metric semigroup and reflexive separable Banach space as well as for the weak pointwise and almost everywhere convergence of extracted subsequences, and comment on the necessity of conditions in the selection principles. We show that many Helly-type pointwise selection principles are consequences of our results, which can be applied to sequences of non-regulated functions, and compare them with recent results by Chistyakov [J. Math. Anal. Appl. 310 (2005) 609--625] and Chistyakov and Maniscalco [J. Math. Anal. Appl. 341 (2008) 613--625].