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Khurana, Surjit Singh
- Weak Integration of Vector-Valued Functions
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1 Department of Mathematics, 101 MacLean Hall, The University of Iowa, Iowa City, Iowa 52240, US
1 Department of Mathematics, 101 MacLean Hall, The University of Iowa, Iowa City, Iowa 52240, US
Source
The Journal of the Indian Mathematical Society, Vol 39, No 1-4 (1975), Pagination: 155-166Abstract
In this paper, X is a Hausdorff topological space, (μ a finite, positive, and countably additive set function on a σ-algebra β of subsets of X which includes the Borel sets, and μ* the outer measure associated with μ (μ*(A)=Inf{(μ(B):⊃A, B ∈ μ}).
- Topological Vector Space Valued Measures on Topological Spaces
Abstract Views :357 |
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Authors
Affiliations
1 The University of Iowa, Department of Mathematics, Iowa City, IA, US
1 The University of Iowa, Department of Mathematics, Iowa City, IA, US
Source
The Journal of the Indian Mathematical Society, Vol 86, No 3-4 (2019), Pagination: 259-271Abstract
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
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