In the book title Mathematical Foundations of Statistical Mechanics, by Khinchin, it is stated that
if $M$ and $M_t$ are two regions in the phase space, where in fact $M_t$ is the image of the set $M$ under the time evolution, and both regions have the same measure, and if $f$ is any function defined on the phase space, then
$$\int_M f dV = \int_{M_t} f dV_t,$$ The author proves this by arguing that we can replace the volume element $dV_t$ by $dV$ in the last section of the proof.
However, even though both sets $M$ and $M_t$ have the same measure, what is the rationale behind the fact that we can interchange volume element $dV_t$ by $dV$ ?