The M _α and C-integrals

Abstract

In this paper, we define the ${M_\alpha }$-integral of real-valued functions defined on an interval [a, b] and investigate important properties of the ${M_\alpha }$-integral. In particular, we show that a function f: [a, b] → R is ${M_\alpha }$-integrable on [a, b] if and only if there exists an $AC{G_\alpha }$ function F such that F′ = f almost everywhere on [a, b]. It can be seen easily that every McShane integrable function on [a, b] is ${M_\alpha }$-integrable and every ${M_\alpha }$-integrable function on [a, b] is Henstock integrable. In addition, we show that the ${M_\alpha }$-integral is equivalent to the C-integral.