TY - CHAP

T1 - Variants of a selection principle for sequences of regulated and non-regulated functions

AU - Maniscalco, Caterina

AU - Tretyachenko, Yuliya V.

AU - Chistyakov, Vyacheslav V.

PY - 2008

Y1 - 2008

N2 - Let $T$ be a nonempty subset of $\RB$, $X$ a metric space withmetric $d$ and $X^T$ the set of all functions mapping $T$ into$X$. Given $\vep>0$ and $f\in X^T$, we denote by $N(\vep,f,T)$ theleast upper bound of those $n\in\NB$, for which there existnumbers $s_1,\dots,s_n,t_1,\dots,t_n$ from $T$ such that$s_1\vep$ for all $i=1,\dots,n$ ($N(\vep,f,T)=0$ ifthere are no such $n$'s). The following pointwise selectionprinciple 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\/ $\limsup_{j\to\infty}N(\vep,f_j,T)<\infty$ for all$\vep>0$, then\/ $\{f_j\}_{j=1}^\infty$ contains a subsequence,converging pointwise on $T$ to a function $f\in X^T$, such that$N(\vep,f,T)<\infty$ for all $\vep>0$}. We establish severalvariants of this result for functions with values in a metricsemigroup and reflexive separable Banach space as well as for theweak pointwise and almost everywhere convergence of extractedsubsequences, and comment on the necessity of conditions in theselection principles. We show that many Helly-type pointwiseselection principles are consequences of our results, which can beapplied to sequences of non-regulated functions, and compare themwith recent results by Chistyakov [J. Math. Anal. Appl. 310(2005) 609--625] and Chistyakov and Maniscalco [J. Math. Anal.Appl. 341 (2008) 613--625].

AB - Let $T$ be a nonempty subset of $\RB$, $X$ a metric space withmetric $d$ and $X^T$ the set of all functions mapping $T$ into$X$. Given $\vep>0$ and $f\in X^T$, we denote by $N(\vep,f,T)$ theleast upper bound of those $n\in\NB$, for which there existnumbers $s_1,\dots,s_n,t_1,\dots,t_n$ from $T$ such that$s_1\vep$ for all $i=1,\dots,n$ ($N(\vep,f,T)=0$ ifthere are no such $n$'s). The following pointwise selectionprinciple 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\/ $\limsup_{j\to\infty}N(\vep,f_j,T)<\infty$ for all$\vep>0$, then\/ $\{f_j\}_{j=1}^\infty$ contains a subsequence,converging pointwise on $T$ to a function $f\in X^T$, such that$N(\vep,f,T)<\infty$ for all $\vep>0$}. We establish severalvariants of this result for functions with values in a metricsemigroup and reflexive separable Banach space as well as for theweak pointwise and almost everywhere convergence of extractedsubsequences, and comment on the necessity of conditions in theselection principles. We show that many Helly-type pointwiseselection principles are consequences of our results, which can beapplied to sequences of non-regulated functions, and compare themwith recent results by Chistyakov [J. Math. Anal. Appl. 310(2005) 609--625] and Chistyakov and Maniscalco [J. Math. Anal.Appl. 341 (2008) 613--625].

KW - Banach space

KW - Pointwise convergence

KW - almost everywhere convergence.

KW - double sequence

KW - generalized
variation

KW - metric semigroup

KW - metric space

KW - regulated function

KW - selection principle

KW - weak convergence

KW - Banach space

KW - Pointwise convergence

KW - almost everywhere convergence.

KW - double sequence

KW - generalized
variation

KW - metric semigroup

KW - metric space

KW - regulated function

KW - selection principle

KW - weak convergence

UR - http://hdl.handle.net/10447/40110

M3 - Chapter

SN - 978-981-283-443-0; 978-981-283-4

SP - 45

EP - 72

BT - Topics in Classical Analysis and Applications in Honor of Professor Dan Waterman on the occasion of his 80th birthday

ER -