Let $\sigma^n$ be the spherical measure on $S^n$ and $\lambda^n$ be the Lebesgue measure on $\mathbb{R}^n$.
Let us fix a point $a \in S^n$ and consider the stereographic projection $\Phi : S^n - \{a \} \to \mathbb{R}^n$.
By slight abuse of notation, let us use the same symbol $\sigma^n$ for restriction of the spherical measure on $S^n - \{a \}$.
Then, we can think of the pushforward measure $\Phi_*(\sigma^n)$ on $\mathbb{R}^n$.
Then my question is
What is the precise relation between $\Phi_*(\sigma^n)$ and $\lambda^n$? Apparently, the former is a finite measure while the latter is not, so they cannot coincide.
This ME post seems to deal with a similar situation, but it only focuses on $B(0,1) \subset \mathbb{R}^n$, so that $\phi$ in that post is different from my $\Phi$ above.
Could anyone pleaes clarfiy for me?
