A McShane type integral on complete metric measure space

Abstract

In this article, a McShane type integral, the -McShane integral, is defned on a
complete measure metric space endowed with a non-atomic Radon measure \mu, and a family of \mu valued set functions that satisfy the Vitali covering theorem. The Saks-Henstock type lemma are investigated in terms of additive functions and some of the fundamental properties of such integrals are investigated, and a relationship between Lebesgue and McShane integrals is estab-lished. Finally, we have demonstrated that the extension of real-valued AC functions are the \mu McShane integral.