The differential entropy is translation invariant but not scaling invariant: $h(X+c) = h(x)$ for some constant $c$,and $h(aX) = h(X) + \ln (|a|)$ .
I am interested in an extension of the scaling case, where $Y = m(X)$ for a general function $m$. From Wikipedia, it seems there is an inequality $h(Y)\leq h(x) + \int f_X(x)\log |\frac{\partial m}{\partial x}| dx$ assuming that $X$ and $Y$ have the same dimension. The equality holds when $m$ is a bijection.
I googled around and couldn't find any reference to the material for the inequality above. If directly going from the change of variable for a continuous r.v., it should be $h(Y) = h(X) + \int f_X \log |\frac{\partial m}{\partial x}| dx$ for a general function $m$.
Can someone suggest me any reference to the material in Wikipedia, or point out if I miss something when applying change of variables? Thanks very much!