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Applications of a Theorem of Grothendieck to Vector Measures

✍ Scribed by T.V Panchapagesan

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 229 KB
Volume: 214
Category: Article
ISSN: 0022-247X
DOI: 10.1006/jmaa.1997.5589

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✦ Synopsis

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 of UU Ž Ž . . the biadjoint u of u g L L ⌺ R R , X , it is 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.

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