Abstract
The notion of RNP can be localised to a subset A of E. We say that A has RNP if any E-valued measure μ such that μ(B)/P(B) is in A for any set B has representation f ° P. If we say that A has the martingale convergence property (MCP) if any A-valued martingale converges a.s. then the discussion of §3 shows that for bounded, closed, convex sets A, dentability, RNP and MCP are equivalent conditions.
Many other probabilistic aspects of RNP have been ignored in this article. We mention here at least the work on vector-valued amarts due to Bellow, Brunel, Chacon, Edgar, Sucheston and others and the work of Assouad, Enflo, James, Pisier and others on super-reflexivity, super-RNP and uniform convexity.
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Chatterji, S.D. (1979). The radon-nikodym property. In: Beck, A. (eds) Probability in Banach Spaces II. Lecture Notes in Mathematics, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071949
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DOI: https://doi.org/10.1007/BFb0071949
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