On convergence of McShane type integral in regards with radon measur

Abstract

McShane integrals are generalized Riemann type integrals. In this paper, monotone convergence of $\mu$-McShane integrable function is discussed. Further we introduce $\mu$-equi-integrability, and $\mu$-uniformly absolutely continuity on a complete metric space, endowed with a Radon measure $\mu$ with a family of cells that satisfies the Vitali covering theorem with respect to $\mu$. We find several convergence theorems based on $\mu$-equi-integrability, and $\mu$-uniformly absolutely continuity. Finally, we establish a sufficient and necessary condition for the sequence of $\mu$-McShane integrable functions to be $\mu$-equi-integrable with regard to $\mu$-uniformly absolutely continuity.