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Title: | On the Radon-Nikodym-Property, and related topics in locally convex spaces | Authors: | EGGHE, Leo | Issue Date: | 1978 | Publisher: | Springer Berlin Heidelberg | Source: | Aron, Richard M.; Dineen, Seán (Ed.). Vector Space Measures and Applications II, p. 77-90 | Series/Report: | Lecture Notes in Mathematics | Series/Report no.: | 645 | Abstract: | We introduce L X 1 (μ), the space of classes of X-valued μ-integrable functions used by Saab, which is an extension of the space of classes of Bochner-integrable functions, in Banach spaces. X denotes here a sequentially complete locally convex space. We give examples of spaces which are dentable, σ-dentable, having the Radon-Nikodym-Property, or having the Bishop-Phelps-Property, by proving some projective limit results. We also prove the following theorem : The following implications are valid : TeX 1.X has the Radon-Nikodym-Property. 2.Every uniformly bounded martingale is L X 1 -convergent. 3.Every uniformly bounded martingale is L X 1 -Cauchy. 4.Every uniformly bounded and finitely generated martingale is L X 1 -Cauchy. 5.X is σ-dentable. So we have the equivalency of (i) through (v) for quasi-complete (BM)-spaces. | Document URI: | http://hdl.handle.net/1942/17934 | ISBN: | 978-3-540-08669-7 | DOI: | 10.1007/BFb0069664 | Category: | B2 | Type: | Book Section |
Appears in Collections: | Research publications |
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