The Henstock-Kurzweil integral for functions defined on unbounded intervals and with values in Banach spaces

In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among which the fact that our integral contains the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.