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.

Concrete Representation of Abstract (L)-Spaces and the Mean Ergodic Theorem

On vector measures

Semantics of probabilistic programs

Soit K un espace compact, C(K) l'espace des fonctions complexes continues sur K, muni de la norme uniforme, son dual (espace des mesures de Radon sur K). Cet article est consacre essentiellement a l'etude des applications lineaires faiblement compactes de C(K) dans des espaces localement convexes F quelconques i.e. les applications lineaires qui transforment la boule unite de C(K) en une partie faiblement relativement compacte de F.

Sur Les Applications Lineaires Faiblement Compactes D'Espaces Du Type C(K)

0. Introduction. The present paper is concerned with real-valued measures which enjoy the property of finite additivity but not necessarily the property of countable additivity. Our interest in such measures arose from two sources. First, the junior author has been concerned with the space of all finitely additive complex measures on a certain family of sets, which under certain conditions can be made into an algebra over the complex numbers. Second, both of us were informed by S. Kakutani of a result similar to our Theorem 1.23, the proof given by Kakutani being different from ours. The problem of characterizing finitely additive measures in some specific way occurred to us as being quite natural, and this problem we have succeeded in solving under fairly general conditions (Theorem 1.22). The body of the paper is divided into four sections. In §1, we consider finitely additive measures in a reasonably general context, obtaining first a characterization of such measures in terms of a countably additive part and a purely finitely additive part. Purely finitely additive measures are then characterized explicitly. In §2, we extend the theorem of Fichtenholz and Kantorovic [i](l) and thereby characterize the general bounded linear functional on the Banach space of bounded measurable functions on a general measurable space. In §3, we consider a number of phenomena which appear in the special case of the real number system. Here we exhibit a number of finitely additive measures which have undeniably curious properties. In §4, we describe connections between our finitely additive measures and certain countably additive Borel measures defined on a special class of compact Hausdorff spaces. We are indebted to Professor S. Kakutani for comments on and improvements in the results obtained. Throughout the present paper, the symbol R designates the real number system, and points are denoted by lower-case Latin letters, sets by capital Latin letters, families of sets by capital script letters. Sets of functions are denoted by capital German letters. For any set X and any A CZX, the characteristic function of A is denoted by \\a1. General finitely additive measures. 1.1 Definition. Let X be an abstract set, and let M be a family of subsets of X closed under the formation of finite unions and of complements. Let \"M be the smallest family of sets containing Vît and closed under the formation of countable unions and of complements.

https://www.ams.org/tran/1952-072-01/S0002-9947-1952-0045194-X/S0002-9947-1952-0045194-X.pdf

Finitely additive measures

Overview Preliminaries The Cauchy transform as a function The Cauchy transform as an operator Topologies on the space of Cauchy transforms Which functions are Cauchy integrals? Multipliers and divisors The distribution function for Cauchy transforms The backward shift on $H^2$ Clark measures The normalized Cauchy transform Other operators on the Cauchy transforms List of symbols Bibliography Index.

Concrete Representation of Abstract (L)-Spaces and the Mean Ergodic Theorem

Citations

On vector measures

Semantics of probabilistic programs

Sur Les Applications Lineaires Faiblement Compactes D'Espaces Du Type C(K)

Finitely additive measures

The Cauchy Transform

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