I am looking for a reference that states and proves Liouville's theorem regarding the existence of invariant measure(s) (i.e. Liouville measure(s)) on energy shells of some nice enough observables. To be precise, I am reading the lecture notes about semiclassical analysis by Zworski, and he essentially writes that the characterizing property of the Liouville measure $\mu$ is that if $p:T^*M\to\mathbb{R}$ is some nice-enough (e.g. smooth) real-valued function, which is invertible at least when we restrict the maximum and minimum values of $p$ to some closed real interval $[a, b]$, then
$$\int\int_{p^{-1}[a,b]}fdxd\xi = \int_a^b\int_{p^{-1}(c)}fd\mu dc$$
holds for any continuous observable $f:T^*M\to\mathbb{R}$. I have already tried to look for a proper proof of this theorem online, but I could not find anything.