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