A pointwise selection principle for metric semigoup valued functions

Caterina Maniscalco, Vyacheslav V. Chistyakov

Risultato della ricerca: Article

6 Citazioni (Scopus)

Abstract

Let $\emptyset\neq T \subset \RB, \hspace{.1in} (X,d,+)$ be an additive commutative semigroup with metric $d$ satisfying $d(x+z,y+z)=d(x,y)$ for all $x,y,z \in X,$ and $X^T$ the set of all functions from $T$ into $X$. If $n \in \NB$ and $f,g \in X^T$, we set $\nu (n,f,g,T) = \sup \sum _{i=1} ^{n} d(f(t_i)+g(s_i), g(t_i)+f(s_i))$, where the supremum is taken over all numbers $s_1,...,s_n,t_1,....,t_n$ from $T$ such that $s_1 \leq t_1 \leq s_2 \leq t_2 \leq ...\leq s_n \leq t_n.$ We prove the following pointwise selection theorem: \textit{If a sequence of functions $\{f_j\}_{j \in \NB} \subset X^T$ is such that the closure in $X$ of the set $\{f_j(t)\}_{j \in \NB}$ is compact for each t \in T, and $$\lim_{n\rightarrow \infty}(\frac{1}{n} \lim_{N\rightarrow \infty} \sup_{j,k\geq N, j \neq K} \nu (n,f_j,f_k,T)) = 0,$$ then it contains a subsequence which converges pointwise on $T$.} We show by examples that this result is sharp and present two of its variants.
Lingua originaleEnglish
pagine (da-a)613-625
RivistaJournal of Mathematical Analysis and Applications
Volume341
Stato di pubblicazionePublished - 2008

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Selection Principles
Metric
Set theory
Subset
Supremum
Subsequence
Closure
Semigroup
Converge
Theorem

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cita questo

A pointwise selection principle for metric semigoup valued functions. / Maniscalco, Caterina; Chistyakov, Vyacheslav V.

In: Journal of Mathematical Analysis and Applications, Vol. 341, 2008, pag. 613-625.

Risultato della ricerca: Article

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AB - Let $\emptyset\neq T \subset \RB, \hspace{.1in} (X,d,+)$ be an additive commutative semigroup with metric $d$ satisfying $d(x+z,y+z)=d(x,y)$ for all $x,y,z \in X,$ and $X^T$ the set of all functions from $T$ into $X$. If $n \in \NB$ and $f,g \in X^T$, we set $\nu (n,f,g,T) = \sup \sum _{i=1} ^{n} d(f(t_i)+g(s_i), g(t_i)+f(s_i))$, where the supremum is taken over all numbers $s_1,...,s_n,t_1,....,t_n$ from $T$ such that $s_1 \leq t_1 \leq s_2 \leq t_2 \leq ...\leq s_n \leq t_n.$ We prove the following pointwise selection theorem: \textit{If a sequence of functions $\{f_j\}_{j \in \NB} \subset X^T$ is such that the closure in $X$ of the set $\{f_j(t)\}_{j \in \NB}$ is compact for each t \in T, and $$\lim_{n\rightarrow \infty}(\frac{1}{n} \lim_{N\rightarrow \infty} \sup_{j,k\geq N, j \neq K} \nu (n,f_j,f_k,T)) = 0,$$ then it contains a subsequence which converges pointwise on $T$.} We show by examples that this result is sharp and present two of its variants.

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