Definition of $L^2(S^{n-1})$

Question

I'm puzzled with how we're supposed to define $L^2(S^{n-1})$ where $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$. How do we even define the inner product there? the only way that comes to mind currently is $(f,g) = \int_{S^{n-1}}fg\,d\mathcal{H}^{n-1}$. But this is not well defined since we need $f, g $ to be functions from $\mathbb{R}^n$ to $\mathbb{R}$ in order to use the Hausdorff measure. Any help or pointer to the right books would be appreciated.

Answer 1

There are many ways to define the desired measure on $S^{n-1}$.

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One way to do it is to note that $SO(n)$ acts transitively on $S^{n-1}$. Fix a point $x_0 \in S^{n-1}$, and let $H$ be the stabilizer of $x_0$. Note that $H \cong SO(n-1)$, so it is a compact subgroup of $SO(n)$. By the orbit-stabilizer theorem, we get a homeomorphism $SO(n)/H \to S^{n-1}$.

$SO(n)$ is a compact Hausdorff topological group, so it admits a canonical Haar probability measure. We push this measure forward to the quotient to obtain a probability measure on $S^{n-1}$. The fact that this measure is nonzero and invariant under the action of $SO(n)$ tells you that it is the one you want (perhaps up to scaling).

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Another way is to let $\nu$ be the vector field $\nu(x) = x$ on $\mathbb{R}^n$ (where we identify the tangent spaces of $\mathbb{R}^n$ with $\mathbb{R}^n$ in the standard way).

Then let $\omega$ be the standard volume form on $\mathbb{R}^n$, i.e. $\omega = dx_1 \wedge \dots \wedge dx_n$.

Let $\iota_\nu \omega$ be the contraction of $\omega$ along $\nu$, i.e. $\iota_\nu \omega$ is the $(n-1)$-form such that for all $x \in \mathbb{R}^n$ we have $(\iota_\nu \omega)_x(v_1, \dots, v_{n-1}) = \omega_x(v_1, \dots, v_{n-1}, \nu(x))$.

Finally, let $\alpha = i^*(\iota_\nu \omega)$, where $i : S^{n-1} \to \mathbb{R}^n$ is the inclusion. Then $\alpha$ is an $(n-1)$-form on $S^{n-1}$, and it is nonzero because $\omega$ and $\nu$ are nowhere-vanishing on $S^{n-1}$.

By integrating smooth functions against $\alpha$, we obtain a nonzero, positive linear functional on the space of smooth functions $S^{n-1} \to \mathbb{R}$. This extends (uniquely) to a nonzero, positive linear functional on the space of continuous functions $S^{n-1} \to \mathbb{R}$ by (e.g.) Stone-Weierstrass + dominated convergence. Then Riesz-Markov tells us that this comes from a unique nonzero Radon measure on $S^{n-1}$, which is the measure you want (perhaps up to scaling).