My problem is, if we take the $n$-dimensional Lebesgue measure $m_n$, we can calculate that, for the unit $(n-1)$-sphere, is
$$\sigma(S^n) = \frac{2\pi^{(n+1)/2}}{\Gamma(\frac{n+1}{2})}.$$
If we want to obtain the limit, we can do:
\begin{align}\lim_{n \to \infty} \sigma(S^{n-1}) &= \lim_{n \to \infty} \int_{\mathbb{R}^{n}} \chi_{\{x \ :\ x \in S^{n-1}\}}(x) \mathrm{d}x \\ & = \lim_{n \to \infty} \int_{\varphi_{n-1}=0}^{2\pi} \int_{\varphi_{n-2}=0}^{\pi} \cdots \int_{\varphi_1}^\pi \prod_{k=1}^{n-1} \sin(\varphi_k)^{n-k-1} \mathrm{d}\varphi_1 \mathrm{d}\varphi_2 \cdots\mathrm{d}\varphi_{n-1}.\end{align}
So we should be able to use Dominated Convergence since the integrand is bounded by $1$.
In this case the integrand becomes $0$ almost everywhere.
I have not treated infinite-dimensional spaces almost at all so I don't know if this is sound.