Abstract. Let T be a locally compact Hausdorff space and let Co(T ) = {f : T → C, f is continuous and vanishes at infinity} be provided with the supremum norm. Let Bc(T ) and Bo(T) be the σ-rings generated by the compact subsets and by the compact G δ subsets of T , respectively. The members of Bc(T ) are called σ-Borel sets of T since they are precisely the σ-bounded Borel sets of T . The members of Bo(T) are called the Baire sets of T . M (T ) denotes the dual of Co(T ). Let X be a quasicomplete locally convex Hausdorff space. Suppose u : Co(T ) → X is a continuous linear operator. Using the Baire and σ-Borel characterizations of weakly compact sets in M (T ) as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator u to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of σ-additive X-valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.
DEDICATED TO THE MEMORY OF PROFESSOR I. KLUVANEḰ
Ž .Let R R be a ring of subsets of a nonempty set ⍀ and ⌺ R R the Banach space of uniform limits of sequences of R R-simple functions in ⍀. Let X be a quasicom-Ž . plete locally convex Hausdorff space briefly, lcHs . Given a bounded X-valued Ž . vector measure m on R R, the concepts of m-integrability of functions in ⌺ R R and Ž . of representing measure of a continuous linear mapping u : ⌺ R R ª X are introduced. Based on these concepts and a theorem of Grothendieck on the range ofis shown that such a mapping u is weakly compact if and only if its representing measure is strongly additive. The result Ž . subsumes the range theorems of I. Tweddle Glasgow Math. J. 9, 1968, 123᎐127 Ž . and I. Kluvanek Math. Systems Theory 7, 1973, 44᎐54 . Also the theorem oń extension is deduced. The method of proof for all these results in vector measures is more natural than the known ones. ᮊ 1997 Academic Press Ž Let X be a quasicomplete locally convex Hausdorff space briefly, a . quasicomplete lcHs . Using James' criterion for the weak compactness of a w x set, Tweddle 12 showed that the closed convex hull of the range of a -additive X-valued vector measure defined on a -ring of sets is weakly compact. His proof is first given for the case of a -algebra, and then, by w x appealing to the Eberlein theorem 6, Theorem 8.12.7 , is extended to the
Abstract. Let T be a locally compact Hausdorff space and let M (T ) be the Banach space of all bounded complex Radon measures on T . Let Bo(T) and Bc(T) be the σ-rings generated by the compact G δ subsets and by the compact subsets of T , respectively. The members of Bo(T ) are called Baire sets of T and those of Bc(T ) are called σ-Borel sets of T (since they are precisely the σ-bounded Borel sets of T ). Identifying M (T ) with the Banach space of all Borel regular complex measures on T , in this note we characterize weakly compact subsets A of M (T ) in terms of the Baire and σ-Borel restrictions of the members of A. These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.
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