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

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.