Baire one-Stieltjes integration

Abstract

This study introduces the Baire One-Stieltjes integral. The integral is shown to possess the desired elementary integral properties, including Cauchy Criterion and Henstock Lemma. It is known that every Riemann-Stieltjes integrable function is Baire One-Stieltjes integrable, while every Baire One-Stieltjes integrable function is Henstock-Stieltjes integrable. Between the Riemann-Stieltjes and the Baire One-Stieltjes, strict inclusion is exhibited in the space of regulated functions. We also formulate convergence theorems for this Stieltjes integral such as Uniform Convergence Theorem, Equi-integrability Convergence Theorem and Monotone Convergence Theorem. Also, some results leading to Riesz Representation Theorem are presented for this Stiletjes integral.