I've been studying Liouville's theorem lately. The statistical mechanics textbook by Kardar proves the theorem by showing that $dq\cdot dp$ is unchanged for each coordinate for an infinitessimal rectangle of length and width $dq$ and $dp$.
My thought is:
this proof shows that the $2n$-volume of a set in an $2n$ dimensional phase space is unchanged over time.
by taking a subset with fixed energy level, and recognizing that it stays within the subspace of that energy level and projecting onto that subspace, the $2n-1$ volume of that subset also stays the same.
however, the area of a set as projected on arbitrary coordinates $q_i,p_i$ won't necessarily stay constant, because even though this projection is preserved by a first order approximation, it is not once we take interaction with other variables into account. Hence, we can have that the projection of the phase space onto some variables $q_i,p_i$ increases or decreases over time.
Is this correct?