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Topological Vector Space Valued Measures on Topological Spaces


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1 The University of Iowa, Department of Mathematics, Iowa City, IA, United States
     

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If X is a compact Hausdorff space space, E is a complete Hausdorff topological vector space and μ : (C(X),ll.ll) → E a linear continuous exhaustive mapping, we rst give a different proof that there is then a unique reqular, L∞-bounded, exhaustive E-valued Borel measure μ on X such that μ(f) = ∫ fdμ, ∀f ∈ C(X). Then we consider X to be a completely regular Hausdorff space and prove the extension of Alexanderov's theorem: X is a completely regular Hausdorff space and μ : Cb(X) → E a linear, continuos, exhaustive mapping and F is the algebra generated by zero-sets in X. Then there exist a unique nitely additive, exhaustive measure ν : F → E such that (i) ν is L∞-bounded i.e. the absolute convex hull of ν(F) (Γ(ν(F))) is bounded in E; (ii) ν is inner regular by zero-sets and outer regular by positive-sets; (iii) ∫ fdν = µ(f), ∀f ∈ Cb(X).

Keywords

Vector Measures, Measure Representation of Linear Operators, Alexandrov's Theorem.
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  • Topological Vector Space Valued Measures on Topological Spaces

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Authors

Surjit Singh Khurana
The University of Iowa, Department of Mathematics, Iowa City, IA, United States

Abstract


If X is a compact Hausdorff space space, E is a complete Hausdorff topological vector space and μ : (C(X),ll.ll) → E a linear continuous exhaustive mapping, we rst give a different proof that there is then a unique reqular, L∞-bounded, exhaustive E-valued Borel measure μ on X such that μ(f) = ∫ fdμ, ∀f ∈ C(X). Then we consider X to be a completely regular Hausdorff space and prove the extension of Alexanderov's theorem: X is a completely regular Hausdorff space and μ : Cb(X) → E a linear, continuos, exhaustive mapping and F is the algebra generated by zero-sets in X. Then there exist a unique nitely additive, exhaustive measure ν : F → E such that (i) ν is L∞-bounded i.e. the absolute convex hull of ν(F) (Γ(ν(F))) is bounded in E; (ii) ν is inner regular by zero-sets and outer regular by positive-sets; (iii) ∫ fdν = µ(f), ∀f ∈ Cb(X).

Keywords


Vector Measures, Measure Representation of Linear Operators, Alexandrov's Theorem.

References