Abstract
We show the existence, for an arbitrary vector measureμ: Σ → x (where X is a Banach space and gs is aΣ-algebra of subsets of a set S) of a functional x′ ∋ X′ (X′ is the conjugate space of X) such thatμ is absolutely continuous with respect toμ x′,μ x′ (E)=<x′μ(E)>, E ∋ gs.
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R. G. Bartle, N. Dunford, and J. Schwartz, “Weak compactness and vector measures,” Canad. J. Math.,7, 289–305 (1955).
W. G. Bade, “On Boolean algebras of projections and algebras of operators,” Trans. Amer. Math. Soc.,80, 345–360 (1955).
N. Dunford and J. Schwartz, Linear Operators [Russian translation], Moscow (1962).
P. Halmos, Measure Theory [Russian translation], Moscow (1953).
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Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.
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Rybakov, V.I. Theorem of Bartle, Dunford, and Schwartz concerning vector measures. Mathematical Notes of the Academy of Sciences of the USSR 7, 147–151 (1970). https://doi.org/10.1007/BF01093500
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DOI: https://doi.org/10.1007/BF01093500