A Lusin type measurability property for vector-valued functions

We study the class of 'Riemann measurable' vector-valued functions based on a Lusin type property. This class contains all Riemann integrable functions and is closely related to the restricted versions of the McShane and Henstock integrals, the M- and H-integrals, defined by means of Lebesgue measurable gauges. Our developments are in the spirit of the Riemann type integral theory for real-valued functions. In particular, we prove that a bounded Riemann measurable vector-valued function is M-integrable.