# volume preserving version of Moser's theorem

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.

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 $$f\circ\phi = g$$.