Open AccessComputable Riesz Representation for Locally Compact Hausdorff Spaces
Author(s) -
Hong Lu,
Klaus Weihrauch
Publication year - 2008
Publication title -
electronic notes in theoretical computer science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.242
H-Index - 60
ISSN - 1571-0661
DOI - 10.1016/j.entcs.2008.03.002
Subject(s) - mathematics , hausdorff space , locally compact space , hausdorff measure , continuous functions on a compact hausdorff space , riesz–markov–kakutani representation theorem , riesz representation theorem , representation (politics) , borel measure , pure mathematics , compact space , discrete mathematics , measure (data warehouse) , normal space , computable analysis , space (punctuation) , locally compact group , computable function , hausdorff dimension , topological space , topological vector space , probability measure , computer science , database , politics , political science , law , operating system
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)=∫fdμ, 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)=∫fdμ
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