A function $f(ax)$ that satisfies $$ f(ax)=a^\Delta f(x)\,\,\, (\Delta \in R) $$ is said to be scale invariant. The most general function $f(x)$ that satisfies the previous condition is of the form $$ f(x)=C x^{\Delta} $$ If we consider the set of distributions, namely the set of all $f(x)$ such that $\int f(x)dx=1$, is it possible to prove that the only distribution that is invariant under scale transformation is $$ f(x)=C x^{-1}. $$ This means that in general I can build a generic distribution as a power law that still satisfies the scale invariance properties $f(ax)=a^\Delta f(x)$, but this generic distribution will not be invariant under scale transformations. So, what is the physical meaning of the scale invariance property in a generic power law distribution?
First Edit: For a distribution $f$, if I perform a change of variables, I must preserve the probability, namely $$ |f(x)dx|=|g(y)dy|, $$ so that the new distribution $g$ will be given by $f(x)|\frac{dx}{dy}|=g(y)$. If a distribution is invariant under change of variables, I must have $$ g(y)=f(y). $$ Replacing $x$ with $ax$ and using the invariance of the distribution $f(x)$ under this transformation, I have $f(x)=C x^{-1}$.