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.
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References
J. Diestel and B. Faires: On vector measures. Trans. Amer. Math. Soc. 198, 253–271 (1974).
J. Diestel and J.J. Uhl jr.: Vector Measures. Providence, Rhode Island: American Mathematical Society 1977.
B. Faires and T.J. Morrison: The Jordan decomposition of vector-valued measures. Proc. Amer. Math. Soc. 60, 139–143 (1976).
H.H. Schaefer: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer 1974.
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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