On the McShane-Dunford-Stieltjes Integral and McShane-Pettis-Stieltjes Integral

Abstract

This paper combines the McShane-Stieltjes integral and Pettis approaches by utilizing Pettis' definition, which coincides with the Dunford integral rather than the version applicable to weakly measurable functions with Lebesgue integrable images. In this way, another integration process without measure theoretic standpoint will be introduced. To this end, we will define the McShane-Dunford-Stieltjes integral and McShane-Pettis-Stieltjes integral in Banach Space and provide its simple properties such as the uniqueness, linearity property of both the integrand and integrator, additivity and formulate the Cauchy criterion of these integrals. In addition, the existence theorem of McShane-Dunford Stieltjes integral will also be presented.