Let's say I've a probability distribution $f(x)$ for the variable $x$. With a change of variable to $y$, I can find the probability distribution for $y$: $g(y) = f(x)\left|\frac{dx}{dy}\right|$.
The entropy should be: $$S = \int f(x)\log(f(x))dx=\int g(y)\log(g(y))dy$$
But now, in this paper [NIHMS480577], eq 20. It is said that the entropy is coordinate dependent:
$$S' = \int f(x)\log(f(x))dx = \int f(x(y))\left|\frac{dx}{dy}\right|\log(f(x(y)))dy = \int g(y)\log(f(x(y)))dy$$
And of course $S'\neq S$. Something is wrong right? Care must be taken when we average a probability distribution?