Given a measurable space, the vector space of signed measures is a Banach space. Does it have the Radon-Nikodym property? What if the space is of a special type, such as a nice topological space with the Borel $\sigma$-algebra?


If there is any related information, such as the Radon-Nikodym property relative to some special measure spaces, I would also be interested.


I assume the answer is well known, but I have not been able to find information in standard sources such as Diestel and Uhl's book.



Radon-Nikodym property for space of signed measures
The spaces you are interested in are abstractly AL-spaces and by [Kakutani's representation theorem](https://www.jstor.org/stable/pdf/1968915.pdf?casa_token=x9kvIzsjxHUAAAAA:btJzFUxpBLfAZcTE-3USjRKjmUMMNF-uGO24KQdIaD6F0fTqt3pijagbQptR5XaVx3uv1SD2NfUb0WE5GEW5eQHfXawLS61OLV0Nyv2NnW2teq3Plg_P), they can be represented as $L\_1(\mu)$ for some measure. In particular, they have the RNP if and only if the measure $\mu$ is purely discrete, in which case $L\_1(\mu)$ is isometric to $\ell\_1({\rm supp}\, \mu)$.


Thus, for compact spaces $K$, the space of measures $M(K)$ has RNP if and only if $K$ is [scattered](https://encyclopediaofmath.org/wiki/Scattered_space).


