The generalized riemann integral on locally compact spaces

摘要

We extend the basic results on the theory of the generalized Riemann integral to the setting of locally compact second countable Hausdorff spaces with bounded or locally finite measures, the latter being measures which are finite on compact subsets of the space. We show that the correspondence between Borel measures on X and continuous valuations on the uppre 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. We define the generalized Riemann integral for bounded functions with respect to bounded and unbounded measures. Also in this setting generalized R-integrability for a bounded function is proved to be equivalent to the conditions that the set of its discontinuities has measure zero. Finally we prove that if a bounded function is R-integrable then it is also Lebesgue integrable and the two integrals coincide.