2014/12/02 by Giselle Antunes Monteiro, Monteiro, Giselle Antunes, Umi Mahnuna Hanung +4
Mathematics · #26A39 (Primary) #26A42 (Secondary) #28B05 #Advanced Banach Space Theory #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #math.CA #msc:26A39 #msc:26A42 #msc:28B05
paper · pdf · doi:10.48550/arxiv.1412.0993
openalex publication_date 2014/12/02 · arxiv created 2014/12/12 · arxiv updated 2014/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann-Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzela or Arzela-Osgood or Osgood Theorem. In the setting of the Kurzweil-Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the Young-Stieltjes integral by Hildebrandt in his monograph from 1963. However, it is clear that the proof by Hildebrandt cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzela Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil-Stieltjes integral in a setting elementary as much as possible.