An integral on a complete metric measure space

We study a Henstock-Kurzweil type integral defined on a complete metric measure space XX endowed with a Radon measure μμ and with a family of “cells” FF that satisfies the Vitali covering theorem with respect to μμ. This integral encloses, in particular, the classical Henstock-Kurzweil integral on the real line, the dyadic Henstock-Kurzweil integral, the Mawhin’s integral [19], and the ss-HK integral [4]. The main result of this paper is the extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line, in terms of ACG∗ACG∗ functions (Main Theorem 1) and in terms of variational measures (Main Theorem 2).