Let $\mu$ be a finite spherically symmetric measure over $\mathbb R^d$, so that $\mu(TB) = \mu(B)$ for all orthogonal transformations $T: \mathbb R^d \rightarrow \mathbb R^d$ and Borel set $B$. Let $g: \mathbb R \rightarrow \mathbb R $ be Borel-measurable function such that $\int| g(z \cdot x) |\mu(dz) < \infty$ for all $x \in \mathbb R^d$. Let $e_1 = (1,0,...,0) \in \mathbb R^d$. (The dot $\cdot$ denotes the usual inner product in $\mathbb R^d$ and $\| \cdot \|$ denotes the usual Euclidan norm.)
Prove that for any $x \in \mathbb R^d$ we have
$$ \int g(z \cdot x) \mu(dz) = \int g( \| x \|z \cdot e_1 ) \mu(dz). $$
This is a transformation that I got from a paper by Iosif Pinelis, but I don't know how it's done. It seems trivial but I couldn't see why (though it is intuitively clear).
