volume preserving version of Moser's theorem

Question

There exists an well-known theorem of Moser :

Thereom(Moser) Let $M$ be a compact oriented smooth manifold and $\alpha,\beta$ be volume forms whose total volumes are the same. Then, there exists a diffeomorphism $\phi:M\rightarrow M$ such that $\phi^*\beta=\alpha$.

After that, I modify this theorem under the fixed volume form.

??? Let $M$ be a compact oriented smooth manifold and $\mu_0$ be a volume form. For positive $f,g\in C^\infty(M)$ with $\int_M f\mu_0=\int_M g\mu_0$, there exists a volume-preserving diffeomorphism $\phi\in \text{Diff}(M,\mu_0)$ such that $f\circ \phi = g$.

Is the theorem is true? How could I prove this? I try to prove this by using Moser's decomposition technique but there is some difficulty to verify the existence of solution of some PDE due to the volume preserving condition.

Answer 1

This is false. For example, if $f$ is constant, then $f\circ\phi$ will also be constant. If $g$ is nonconstant, you won't be able to make $f\circ\phi = g$.