Representation of certain functions under Martin's axiom
type: Item
instance of: scholarly article
title: Representation of certain functions under Martin's axiom
zbMATH Open document ID: 0729.03028
DOI: 10.1007/BF01137560
published in: Mathematical Notes
publication date: 1991-01-01T00:00:00Z
review text: The author studies sequences of real functions \(f_ n\) converging quasi- normally on a subset A of the real line R of power less than continuum. More precisely, a sequence \(f_ n\) converges quasi-normally to a function f on a set A of real numbers if there is a sequence \(\epsilon_ n\) of positive real numbers with lim \(\epsilon\) \({}_ n=0\) such that for every real number \(a\in A\), there is a natural number \(k_ a\) such that \(| f(a)-f_ n(a)| <\epsilon_ n\) holds for every \(n\geq k_ a\). The concept of quasi-normal convergence was introduced and studied by Z. Bukovská and L. Bukovský quite recently. Assuming Martin's Axiom (MA), the author shows that for every function f defined on a subset A of R of power less than continuum, there is a sequence of continuous functions on R that converges quasi-normally to f on A. This result and some of its consequences following from the Booth lemma (a weaker form of MA) strengthen results due to \textit{J. Shinoda} [Nagoya Math. J. 49, 117-125 (1973; Zbl 0266.04004)].
reviewed by: Petr Stepanek
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 4204332
zbMATH Keywords: representation of functions on the real line
zbMATH Keywords: quasi-normal convergence
zbMATH Keywords: Martin's Axiom
zbMATH Keywords: Booth lemma
author: Natalia Kholshchevnikova
Wikidata QID: Q114694050
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4204332