Monatshefte für Mathematik

, Volume 180, Issue 3, pp 409–434 | Cite as

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

  • Giselle Antunes Monteiro
  • Umi Mahnuna Hanung
  • Milan TvrdýEmail author


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 Arzelà or Arzelà–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 \(\sigma \)-Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà 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.


Kurzweil–Stieltjes integral Bounded convergence theorem Integral over elementary set 

Mathematics Subject Classification

26A39 28B05 


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Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  • Giselle Antunes Monteiro
    • 1
  • Umi Mahnuna Hanung
    • 1
    • 2
    • 3
  • Milan Tvrdý
    • 1
    Email author
  1. 1.Institute of MathematicsCzech Academy of SciencesPraha 1Czech Republic
  2. 2.Department of MathematicsUniversitas Gadjah MadaYogyakartaIndonesia
  3. 3.KdV Institute for MathematicsUniversiteit van AmsterdamAmsterdamThe Netherlands

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