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 $U\subset M$.

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.