Representation of certain functions under Martin's axiom
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)].
representation of functions on the real line
scientific article; zbMATH DE number 4204332