Suppose $(M,g)$ is a oriented connected Riemannian manifold (but not necessarily compact). Let $\omega_g$ denote the volume form on $M$ determined by $g$, and let $m_g$ denote the probability measure on $M$ defined by
$ m_g(\phi) = \frac{1}{\mathrm{vol}(M,g)}\int_M \phi \omega_g, $
where $m_g$ is thought of as a positive linear functional on the space of continuous bounded functions on $M$.
If $n$ is another probability measure on $M$ then if $n$ is absolutely continuous with respect to $m_g$, the Radon-Nikodym derivative $a = \frac{dn}{dm_g} $ is a well defined function in $L^1(M,m_g)$.
My question is the following: Is there a nice way to describe the subset of the space $P(M)$ of probability measures on $M$ which are absolutely continuous with respect to my given $m_g$?
The book I am reading seems to imply that every $n \in P(M)$ has a well-defined Radon-Nikodym derivative $a = \frac{dn}{dm_g} $. But this seems false to me (at least, I think it's false for $M = \mathbb{R}$).
As I already pointed out in the comments, one nice way of expressing the set $P$ of probability measures that are absolutely continuous with respect to some $\sigma$-finite measure $\mu$ is
$$ P = \bigg\{f \, d\mu \, \mid \, f\in L^1(\mu) \text{ with } f \geq 0\text{ and } \int f \, d\mu =1 \bigg\}. $$
The inclusion "$\supset$" is immediate and the reverse inclusion is a consequence of the Radon Nikodym theorem.
Because manifolds are second countable and since the measure generated by the volume form is locally finite, the measure $m_g$ is $\sigma$-finite. Hence, the above identity also holds for $d\mu = dm_g$.