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On the jordan decomposition for vector measures

Part of the Lecture Notes in Mathematics book series (LNM,volume 990)

Abstract

It is shown that a vector measure μ on an algebra of sets with values in an order complete Banach lattice G is the difference of two positive vector measures if either μ is bounded and G is an order complete AM-space with unit, or μ has bounded variation and there exists a positive contractive projection G" → G. This result is a complete counterpart to the corresponding one on the regularity of a bounded linear operator.

Keywords

  • Linear Operator
  • Vector Lattice
  • Bounded Linear Operator
  • Bounded Variation
  • Banach Lattice

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References

  1. J. Diestel and B. Faires: On vector measures. Trans. Amer. Math. Soc. 198, 253–271 (1974).

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  2. J. Diestel and J.J. Uhl jr.: Vector Measures. Providence, Rhode Island: American Mathematical Society 1977.

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  3. B. Faires and T.J. Morrison: The Jordan decomposition of vector-valued measures. Proc. Amer. Math. Soc. 60, 139–143 (1976).

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  4. H.H. Schaefer: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer 1974.

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  5. K.D. Schmidt: A general Jordan decomposition. Arch. Math. 38 556–564 (1982).

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© 1983 Springer-Verlag

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Schmidt, K.D. (1983). On the jordan decomposition for vector measures. In: Beck, A., Jacobs, K. (eds) Probability in Banach Spaces IV. Lecture Notes in Mathematics, vol 990. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064272

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  • DOI: https://doi.org/10.1007/BFb0064272

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12295-1

  • Online ISBN: 978-3-540-39870-7

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