I am reading Mattila's "Fourier analysis and Hausdorff dimension", the author leaves as an exercise to prove that the set $S_infty$ is a Borel set. I will now define the set $S_infty$:

**Theorem 5.1:** Let $Asubsetmathbb{R}^n$ be a Borel set with $text{dim}A=sleq1$. Then for all $tin$
$$ text{dim}{ ein S^{n-1} : text{dim}P_e(A)<t }leq n-2+t $$

Where $P_e: mathbb{R}^nto mathbb{R}$, is the proyection $P_e(x) = ecdot x$ for some $ein S^{n-1}$. For the proof the autor takes $sigma<tleq s$ and finds a Borel measure $mu$ with support on $A$ such that $0<mu(A)<infty$ and such that $I_sigma(mu)<infty$, where $I_sigma(mu)$ is the energy defined as

$$ I_sigma(mu) = iint |x-y|^{-sigma},dmu x,dmu y.$$

He then has to prove that

$$S_infty = {ein S^{n-1} : I_sigma(mu_e) =infty}$$

is a Borel set. Where $mu_e(B) = mu(P_e^{-1}(B))$ is the push-forward of $mu$ under $P_e$.

**This is what I tried doing:**

begin{align*}
I_sigma(mu_e) = int_{-infty}^inftyint_{-infty}^infty |x-y|^{-sigma},dmu_ex,dmu_ey &= int_{mathbb{R}^n}int_{mathbb{R}^n} |ecdotxi - ecdotzeta|^{-sigma},dmu xi,dmuzeta \
&= int_{mathbb{R}^n}int_{mathbb{R}^n} |e|^{-sigma}|xi - zeta|^{-sigma},dmuxi,dmuzeta\
&= int_{mathbb{R}^n}int_{mathbb{R}^n} |xi - zeta|^{sigma},dmuxi,dmuzeta\
&= I_sigma(mu),
end{align*}

where I'm using that because $ein S^{n-1}$ then $|e| =1$. But then I get stuck and I do not know how to continue with the calculations, because the set
$${ ein S^{n-1} : I_sigma(mu)=infty }$$
doesn't make sense to me, maybe my previous calculation is wrong.