The Mellin transform of a function is given by:
$$\mathcal{M}[f](s) = \int_0^{\infty}x^{s-1}f(x)dx$$
Supposedly, the magnitude of the Mellin transform is invariant to scaling, analogous to how the magnitude of the Fourier transform is invariant to translation. Yet, when I try to derive this I get:
$g(x) = f(kx)$
$|\mathcal{M}[g](s)| = |\int_0^{\infty}x^{s-1}f(kx)dx|=|\int_0^{\infty}(\frac{x'}{k})^{s-1}f(x')\frac{dx'}{k}|=|k^{-s}||\mathcal{M}[f](s)|$
which is not invariant, though it does have a simple relation between them. Where did I go wrong?