JUCS - Journal of Universal Computer Science 14(6): 845-860, doi: 10.3217/jucs-014-06-0845

Computable Riesz Representation for Locally Compact Hausdorff Spaces

‡ Nanjing University, Nanjing, China§ FernUniversität Hagen, Hagen, Germany

Corresponding author: Hong Lu ( luhong@nju.edu.cn ) Citation:
Lu H, Weihrauch K (2008) Computable Riesz Representation for Locally Compact Hausdorff Spaces. JUCS - Journal of Universal Computer Science 14(6): 845-860. https://doi.org/10.3217/jucs-014-06-0845 |

Abstract

By the Riesz Representation Theorem for locally compact Hausdorff spaces, for every positive linear functional I on K(X) there is a measure μ such that I(f) =∫ f dμ where K(X) is the set of continuous real functions with compact support on the locally compact Hausdorff space X. In this article we prove a uniformly computable version of this theorem for computably locally compact computable Hausdorff spaces X. We introduce a representation of the positive linear functionals I on K(X) and a representation of the Borel measures on X and prove that for every such functional I a measure μ can be computed and vice versa such that I(f) = ∫ f dμ.

Keywords

computable analysis, computable topology, Hausdorff spaces, Riesz representation theorem