TITLE: On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?
QUESTION [10 upvotes]: I could not answer or find references of this question, even for the following special case: 
On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function with integral $1$ the Jacobian of some diffeomorphism?
An equivalent formulation of the question is: On $S^2$, is every positive smooth probability measure the translate of the standard one by some diffeomorphism?

REPLY [17 votes]: Here is an "answer-version" of my comment:
Yes, this is true in general.  The reference I know is Moser's 1965 paper "On the volume elements on a manifold" (http://www.jstor.org/stable/1994022).
Specifically, let $M$ be a compact connected orientable manifold, and let $\sigma$ and $\tau$ be smooth volume forms on $M$ both with integral 1.  Then there exists a diffeomorphism $\varphi:M\to M$ such that $\varphi^*\tau=\sigma$.
The orientability hypothesis isn't really necessary (just use densities rather than volume forms; see Moser's footnote (2)).