Abstract
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.
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Communicated by G. Teschl.
G. A. Monteiro was supported by RVO: 67985840 and by the Academic Human Resource Program of the Czech Academy of Sciences. U. M. Hanung was supported by the Grant No. 573/E4.4/K/2011 of the Ministry of Research and Technology and the Directorate General of Higher Education, Indonesia.
M. Tvrdý was supported by RVO: 67985840 and by the Grant No. 14-06958S of the Grant Agency of the Czech Republic.
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Antunes Monteiro, G., Hanung, U.M. & Tvrdý, M. Bounded convergence theorem for abstract Kurzweil–Stieltjes integral. Monatsh Math 180, 409–434 (2016). https://doi.org/10.1007/s00605-015-0774-z
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DOI: https://doi.org/10.1007/s00605-015-0774-z