Some remarks on Radon-Nikodym compact spaces

The class of quasi Radon-Nikodym compact spaces is introduced. We prove that this class is closed under countable products and continuous images. It includes the Radon-Nikodym compact spaces. Adapting Alster's proof we show that every quasi Radon-Nikodym and Corson compact space is Eberlein. This generalizes earlier results by J. Orihuela, W. Schachermayer, M. Valdivia and C. Stegall. Further the class of almost totally disconnected spaces is defined and it is shown that every quasi Radon-Nikodym space which is almost totally disconnected is actually a Radon-Nikodym compact space embeddable in the space of probability measures on a scattered compact space.