TITLE: Measures and differential forms on manifolds
QUESTION [13 upvotes]: 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.)

REPLY [5 votes]: Any smooth manifold has a canonical σ-ideal of negligible subsets,
and μ must vanish on these.
Apart from that, the Lie derivative of μ with respect to any smooth vector field
must exist.
This is how smooth measures are defined by Ramanan in Definition 1.9 of Chapter 3 of Global Calculus, for example.
Remark 2.8 in Chapter 8 there explains how this definition
is equivalent to the traditional definition of a smooth measure as a smooth section
of the line bundle of densities.