If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as:
\begin{equation}
\nu(A) = \int\_M I\_A \mu. 
\end{equation}


My question is does the converse hold also? Is not, then when would it.  

That is, if $\nu$ is a measure on $M$ then when does there exist a $1$-density $\mu$ such that $\nu$ and $\mu$ are related as above. 


In short is there a version of the Radon–Nikodym theorem relating densities and measures on manifolds? 



Riemannian Measures, Densities and Radon–Nikodym Theorem
This follows from the usual Radon-Nikodym theorem. Observe that, given two metrics $g\_0,g\_1$ on $M$ with volume densities $dV\_{g\_0}$, $dV\_{g\_1}$ then there exists a *positive* smooth function $\rho\_{10}: M\to (0,\infty)$ such that


$$ dV\_{g\_1}= \rho\_{10} dV\_{g\_0}. $$


Hence, if $B\subset M$ is a Borel subset, then 


$$ V\_{g\_0}(B)=0 \Longleftrightarrow V\_{g\_1}(B) =0.$$


In other words, the concept of negligible set has an *intrinsic* meaning on a manifold.


You can characterize a density as being a Borel measure $\mu$ on $M$ such that $\mu(S)=0$ for any any negligible Borel subset $S\subset M$.


