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• W1985613118 abstract "The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space $X$ to have the property that bounded additive $X$-valued maps on $sigma$-algebras be strongly bounded are presented, namely, $X$ can contain no copy of ${l_infty }$. The next two sections treat the Jordan decomposition for measures with values in ${L_1}$-spaces on ${c_0}(Gamma )$ spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of ${c_0}$. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space $X$ has an equivalent very smooth norm (in particular, a Fréchet differentiable norm) then its dual has the Radon-Nikodym property. Consequently, a $C(Omega )$ space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper." @default.
• W1985613118 created "2016-06-24" @default.
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• W1985613118 date "1974-01-01" @default.
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• W1985613118 title "On vector measures" @default.
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• W1985613118 doi "https://doi.org/10.1090/s0002-9947-1974-0350420-8" @default.
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