In the proof of theorem 2.12 in Thomas-Fermi and related theories (Lieb) it is stated that: if $$\int_{\mathbb{R}^{3}}\rho\leq Z $$ then $$f(x)=Z \vert x\vert^{-1}- \int_{\mathbb{R}^{3}}\frac{\rho^{*}(y)}{\vert x-y\vert}dy$$ is symmetric decreasing function, where $\rho^{*}$ is the symmetric , decreasing rearrangement of $\rho$ and $Z>0$. I can not show the part of the decreasing monotony. Any help would be appreciated, thank you. Note:
Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$.
The symmetric-decreasing rearrangement of a measurable function $f:\mathbb{R}^n \to \mathbb{R}$ is then defined by
$$f^*(x):=\int_0^{\infty} \chi_{\{|f|>t\}^*}(x)dt,$$
by comparison to the "layercake" representation of $f$, namely $$f(x)=\int_0^{\infty} \chi_{\{f>t\}}(x)dt.$$