I am a physics student. I am studying the Maxwell-Boltzmann velocity distribution. Initially I began working with the distribution expressed in terms of momentum $p$, lets coll it $f_p(p)$, which is such that $$ n = f_p(p) \ dp $$ is the number of particles with momentum $ p \in (p, p + dp) $. In one problem I was asked to perform a change of variable from momentum to velocity using $ p = mv $. Initially I just replaced $ f_v (v) = f_p (mv) $ but I found this is wrong and the correct answer was $$ f_v(v) = f_p(mv) \frac{dp}{dv} .$$ After some investigation I understood why this is like this using physical arguments.
My question is if this is a characteristic of distributions in general. I mean, let $ f_x(x) $ to be a distribution and let $ x = g(y) $ to be a change of variable. Then $$ f_y (y) = f_x (g(y)) \frac{dg}{dy} $$ Is this a general property of distributions? If I have some object $ f_x(x) $ which I don't know if it is a distribution, can I assert that it is a distribution if I find it transforms like that? (similar to what happens with tensors, they are recognized by the way they transform).
The reason for this transform property is that if $\Omega_p$ is a region in $p$-space then the number of particles with $p \in \Omega_p$ is given by $$ n(\Omega_p) = \int_{\Omega_p} f_p(p) \, dp = \int_{\Omega_v} f_p(p) \, \frac{dp}{dv} dv = \int_{\Omega_v} f_v(v) \, dv, $$ where $\Omega_v = \{ v \mid p(v) \in \Omega_p \}$ is the set of $v$ corresponding to a $p \in \Omega_p.$