Abstract
We introduce L 1X (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 :
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(i)
X has the Radon-Nikodym-Property.
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(ii)
Every uniformly bounded martingale is L 1X -convergent.
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(iii)
Every uniformly bounded martingale is L 1X -Cauchy.
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(iv)
Every uniformly bounded and finitely generated martingale is L 1X -Cauchy.
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(v)
X is σ-dentable.
So we have the equivalency of (i) through (v) for quasi-complete (BM)-spaces.
Keywords
- Banach Space
- Convex Space
- Factor Space
- Projective Limit
- Finite Partition
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References
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© 1978 Springer-Verlag
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Egghe, L. (1978). On the Radon-Nikodym-property, and related topics in locally convex spaces. In: Aron, R.M., Dineen, S. (eds) Vector Space Measures and Applications II. Lecture Notes in Mathematics, vol 645. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069664
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DOI: https://doi.org/10.1007/BFb0069664
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Publisher Name: Springer, Berlin, Heidelberg
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