Consider $(\Omega,\Sigma)$ a measurable space and let $P(\Sigma)$ denote the space of all probability measures on $(\Omega,\Sigma)$. I wonder if $P(\Sigma)$ can be identified with the dual space of some space $X$. If so, what is X?
It can be assumed that $\Omega$ is atmost countable.
As I noted in the comments, the "space" of probability measures is not even a vector space (not even closed under multiplication with scalars). Therefore, it cannot be a dual space of anything.