An Henstock-Kurzweil type integral on a meausure metric space

We consider an Henstock-Kurzweil type integral defined on a complete measure metric space $X=(X, d)$ endowed with a Radon measure $\mu$ and with a family $\F$ of ``intervals" that satisfies, besides usual conditions, the Vitali covering theorem. In particular, for such integral, we obtain extensions of the descriptive characterization of the classical Henstock-Kurzweil integral on the real line, in terms of $ACG_*$ functions and in terms of variational measures. Moreover we show that, besides the usual Henstock-Kurzweil integral on the real line, such integral includes also the dyadic Henstock-Kurzweil integral, the $GP$-integral and the $s$-HK integral. For this last integral we prove a better version of the Fundamental Theorem of Calculus since the classical one is not true in this setting.