A Research on the Theory of Integration on Locally Compact Spaces: A Case Study of Generalized Riemann Integral

Exploring the Correspondence between Borel Gauges and Valuations on Locally Compact Spaces

Abstract

We extend the fundamental results on the hypothesis of the generalized Kiemann integral to the setting of limited or locally limited measures on locally compact second countable Hausdorff spaces. The correspondence between Borel gauges on X and persevering valuations on the upper space UX offers rise to a topological embeddings between the space of locally limited measures and locally limited reliable valuations, both contributed with the Scott topology. We assemble an approximating chain of basic valuations on the upper space of a locally compact space, whose least upper bound is the given locally limited measure. The generalized Kiemann integral is portrayed for limited capacities with respect to both limited and locally limited measures.