The generalized Riemann integral on locally compact spaces

We extend the basic results on the theory of the generalized Riemann integral to the setting of bounded or locally finite measures on locally compact second countable Hausdorff spaces. The correspondence between Borel measures on X and continuous valuations on the upper space UX gives rise to a topological embedding between the space of locally finite measures and locally finite continuous valuations, both endowed with the Scott topology. We construct an approximating chain of simple valuations on the upper space of a locally compact space, whose least upper bound is the given locally finite measure. The generalized Riemann integral is defined for bounded functions with respect to both bounded and locally finite measures. Also in this setting, generalized R-integrability for a bounded function is proved to be equivalent to the condition that the set of its discontinuities has measure zero. Furthermore, if a bounded function is R-integrable then it is also Lebesgue integrable and the two integrals coincide. Finally, we extend R-integration to an open set and provide a sufficient condition for the computability of the integral of a bounded almost everywhere continuous function.