2023/07/11 by Waldemar Sieg, Sieg, Waldemar
Mathematics · #Advanced Topology and Set Theory #Bounded function #Combinatorics #Discrete mathematics #Mathematical analysis #Mathematical and Theoretical Analysis #Mathematics #Omega #Physics #Space (punctuation) #Uniform norm #math.CA
paper · pdf · doi:10.48550/arxiv.2307.05783
openalex publication_date 2023/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be a perfectly normal topological space, let A be a non-empty Gδ-subset of Ω and let B1(A) denote the space of all functions A→ℝ of Baire-one class on A. Let also ‖⋅‖_∞ be the supremum norm. The symbol χA stands for the characteristic function of A. We prove that for every bounded function f∈ B1(A) there is a sequence (Hn) of both Fσ- and Gδ-subsets of Ω such that the function f\colonΩ→ℝ given by the uniformly convergent series on Ω with the formula: f:=c∑n=0^∞((2)/(3))n+1((1)/(2)-χHn) extends f with f∈ B1(Ω) and the condition (\triangle) of the form: ‖f(A)‖_∞=‖f(Ω)‖_∞. We apply the above series to obtain an extension of f positive to f positive with the condition (\triangle). A similar technique allows us to obtain an extension of Baire-alpha function on A to Baire-alpha function on Ω.