Infinite-dimensional vector function

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On vector measures

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 o-algebras be strongly bounded are presented, namely, X can contain no copy of /„. The next two sections treat the Jordan decomposition for measures with values in Z.|-spaces on c0(r) 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 c0. 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 normithenitsdualhas the Radon-Nikodym property. Consequently, a C(H) 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. The present paper contains results on various aspects of the general theory of vector-valued measures. It proceeds in four sections which are unrelated to each other except for their general relationship to the topic of the title. A brief outline of the results of each section is presented below—a more complete discussion of the sections is delayed (largely because of their disconnected nature) until the sections themselves. §1 is concerned with the theory of strongly bounded vector measures. The main result of this section (Theorem 1.1) provides criteria for a Banach space X to possess the property that every X-valued bounded additive map with values in X be strongly bounded. This theorem sharpens the classical Pettis theorem on weakly countably additive set functions and allows a sharpening of several other related results. §2 is concerned with the Jordan decomposition of vector measures with values in a Banach lattice. The results of this section are necessarily meager: not much is possible. Our most precise results are in case the range space is an abstract £space or c0. A few remarks are also made concerning the range of certain vector measures. §3 deals with the integrability of certain scalar functions with respect to a vector measure. Utilizing the series representation of a scalar function and its integral, a result of D. R. Lewis is generalized. Also, a criterion for integrability Received by the editors February 5, 1973 and, in revised form, May 25, 1973. AMS (MOS) subject classifications (1970). Primary 46B05.