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

G. Y. H. Chi
1977 Proceedings of the American Mathematical Society  
Let F be a quasi-complete locally convex space, (fi, 2, ¡i) a complete probability space, and L\¡i; F) the space of all strongly integrable functions /: Ü -> F with the Egoroff property. If F is a Banach space, then the Radon-Nikodym theorem was proved by Rieffel. This result extends to Freenet spaces. If F is dual nuclear, then the Lebesgue-Nikodym theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodym theorem is not
more » ... ble in general. It is shown in this article that the Radon-Nikodym 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 c 1^{F}, the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset M c F such that 2" ./»*(*,.) < 1, V(jc,) e B. In fact, these spaces have the Radon-Nikodym property, and they include the Montel (ßFJ-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-Nikodym theorem reduces to the Lebesgue-Nikodym theorem. An application to probability theory is considered.
doi:10.1090/s0002-9939-1977-0435338-2 fatcat:gwdk34pierbvfghrnolnvbjhgu