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A UNIFORM CONVERGENCE THEOREM FOR APPROXIMATE HENSTOCK-STIELTJES INTEGRAL

  • Im, Sung-Mo (Department of Mathematics, Chungbuk National University) ;
  • Kim, Yung-Jinn (Department of Mathematics, Chungbuk National University) ;
  • Rim, Dong-Il (Department of Mathematics, Chungbuk National University)
  • Published : 2004.05.01

Abstract

In this paper, we introduce, for each approximate distribution $\~{T}$ of [a, b], the approximate Henstock-Stieltjes integral with value in Banach spaces. The Henstock integral is a special case of this where $\~{T}\;=\;\{(\tau,\;[a,\;b])\;:\;{\tau}\;{\in}\;[a,\;b]\}$. This new concept generalizes Henstock integral and abstract Perron-Stieltjes integral. We establish a uniform convergence theorem for approximate Henstock-Stieltjes integral, which is an improvement of the uniform convergence theorem for Perron-Stieltjes integral by Schwabik [3].

References

  1. Lectures on the theory of integration R.Henstock
  2. Math. Bohem. v.121 no.4 Abstract Perron-Stieltjes integral S.Schwabik
  3. Amer. Math. Soc., Providence The integrals of Lebesgue, Denjoy, Perron and Henstock R.A.Gordon