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On the Radon-Nikodým theorem and locally convex spaces with the Radon-Nikodým property


Author: G. Y. H. Chi
Journal: Proc. Amer. Math. Soc. 62 (1977), 245-253
MSC: Primary 28A45; Secondary 60G99, 46G10
MathSciNet review: 0435338
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Abstract: Let F be a quasi-complete locally convex space, $ (\Omega ,\Sigma ,\mu )$ a complete probability space, and $ {L^1}(\mu ;F)$ the space of all strongly integrable functions $ f:\Omega \to F$ with the Egoroff property. If F is a Banach space, then the Radon-Nikodým theorem was proved by Rieffel. This result extends to Fréchet spaces. If F is dual nuclear, then the Lebesgue-Nikodým theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodým theorem is not available in general. It is shown in this article that the Radon-Nikodým theorem for the strong integral can be established for quasi-complete locally convex spaces F having the following property:

(CM) For every bounded subset $ B \subset l_N^1\{ F\} $, the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset $ M \subset F$ such that $ \Sigma _{i = 1}^\infty {p_M}({x_i}) < 1,\forall ({x_i}) \in B$.

In fact, these spaces have the Radon-Nikodým property, and they include the Montel (DF)-spaces, the strong duals of metrizable Montel spaces, the strong duals of metrizable Schwartz spaces, and the precompact duals of separable metrizable spaces. When F is dual nuclear, the Radon-Nikodým theorem reduces to the Lebesgue-Nikodým theorem. An application to probability theory is considered.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0435338-2
Keywords: Radon-Nikodým theorem, vector measures, nuclear spaces, Montel (DF)-spaces, Schwartz spaces, metrizable spaces
Article copyright: © Copyright 1977 American Mathematical Society