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Semigroup Forum

, Volume 45, Issue 1, pp 120–128 | Cite as

Extensions of semigroup valued, finitely additive measures

  • K. P. S. Bhaskara Rao
  • R. M. Shortt
Research Article
  • 22 Downloads

Abstract

LetC be a field of subsets of a non-empty setX and let μ:CE be a finitely additive measure (a “charge”) taking values in a commutative semigroupE. We consider the problem of extending μ to a charge\(\bar \mu :{\cal P}(X) \to E\) defined on the power set\({\cal P}(X)\) and we say thatE has the charge extension property (CEP) if such extensions always exist. Los and Marczewski proved [4] that the semigroup of non-negative reals has CEP, and Carlson and Prikry [2] have shown that everygroup has CEP. We prove that every compact semigroup has CEP and show that CEP follows from certain completeness and distributivity conditions. Specializing to the case of lattices considered as semigroups under the operation of supremum, we characterize the class of lattices with CEP. An application to closure operators in general topology is also discussed.

Keywords

Boolean Algebra Commutative Semigroup Outer Measure Finite Semigroup Compact Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Bhaskara Rao, K.P.S. and Bhaskara Rao, M., “Theory of Charges”, Academic Press, London-New York, 1983.MATHGoogle Scholar
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    Carlson, T. and Prikry, K.,Ranges of Signed Measures, Periodica Math. Hungarica13 (1982), 151–155.MATHCrossRefMathSciNetGoogle Scholar
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    Chang, C.C., and Keisler, H.J., “Model Theory”, North Holland/Elsevier, Amsterdam-New York, 1973.MATHGoogle Scholar
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    Łos, J. and Marczewski, E.,Extension of measures, Fund. Math36 (1949), 267–276.MATHMathSciNetGoogle Scholar
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    Wehrung, F.,Embedding into injective ordered monoids, Rapport de recherche, Université de Caen (4) 1990.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • K. P. S. Bhaskara Rao
    • 1
    • 2
  • R. M. Shortt
    • 1
    • 2
  1. 1.Bangalore Centre Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of MathematicsWesleyan UniversityMiddletown

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