Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$. 


What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of compatibility of the topology or the differentiable structure of $M$ with the $\sigma$-algebra of $\mu$ would be required.


(Apologies if the question is too elementary for this forum. A pointer to the relevant result in the literature would suffice.)



Measures and differential forms on manifolds
I assume that $\mu$ is a measure defined on the $\sigma$-algebra of Borel sets. First, on any manifold the notion of negligible set is well defined. 


If $M$ is orientable and $\mu(N)=0$ for any negligible Borel set then the Radon-Nicodym theorem implies that, for any smooth volume form $\omega$ on $M$, there is a positive measurable function $\rho\_\omega\colon M\to\mathbb{R}$ such that


$$\mu(U)=\int\_U \rho\_\omega \omega, $$


for any open set $U\subset M$.


