Bounded convergence theorem for abstract Kurzweil–Stieltjes integral
- 221 Downloads
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.
KeywordsKurzweil–Stieltjes integral Bounded convergence theorem Integral over elementary set
Mathematics Subject Classification26A39 28B05
- 6.Gordon, R.A.: The integrals of Lebesgue, Denjoy, Perron, and Henstock. In: Graduate Studies in Mathematics. AMS, Providence (1994)Google Scholar
- 8.Hönig, C.S.: The abstract Riemann–Stieltjes integral and its applications to linear differential equations with generalized boundary conditions. Notas do Instituto de Matemática e Estatística da Universidade de S. Paulo, Série de Matemática no. 1 (1973)Google Scholar
- 9.Hönig, C.S.: Volterra Stieltjes-Integral Equations. In: Mathematics Studies, vol. 16. North Holland and American Elsevier, Amsterdam, New York (1975)Google Scholar