Lemma 2.9.247D4E3784406E

Proof. We write the inner integral in polar coordinates on $H$ and show that the normalized surface measures of the unit spheres of the hyperplanes $H$ average to the normalized surface measure of $S^{2n-1}$. For $H\in \mathcal H$, let $S_ H$ be the unit sphere of $H$, and let $\tau _ H$ be the image under $T_ H$ of the normalized surface measure $\mathrm dS/s_{2n-3}$ of the unit sphere $S^{2n-3}$ of $\mathbb C^{n-1}=\mathbb R^{2n-2}$. We regard $\tau _ H$ as a Borel probability measure on $S^{2n-1}$ carried by $S_ H$; it does not depend on the frame, since surface measure is invariant under orthogonal transformations (subsection 3.1.1). Polar coordinates (3.1.2) on all of $\mathbb C^{n-1}$, transported to $H$ by $T_ H$, give

Measurability. Let $h$ be a nonnegative Borel function on $(0,\infty )\times S^{2n-1}$. By the definition of $\tau _ H$,

The map $(H,r,\omega )\mapsto (r,T_ H\omega )$ is Borel by the frame construction of subsection 2.9.17, since its two components are Borel. Hence $(H,r,\omega )\mapsto h(r,T_ H\omega )$ is a nonnegative Borel function on the product of the separable metric spaces $\mathcal H\times (0,\infty )$ and $S^{2n-3}$, and by lemma 2.2.8 and Tonelli's theorem the right side of (2.9.25) is a Borel function of $(H,r)$. In particular, taking $h(r,\theta )=g(\theta )$, we see that $H\mapsto \int g\, \mathrm d\tau _ H$ is Borel for every nonnegative Borel function $g$ on $S^{2n-1}$; for instance, $H\mapsto \tau _ H(B)$ is Borel for every Borel set $B\subset S^{2n-1}$.

Averaging. For a Borel set $B\subset S^{2n-1}$ put $\tau (B)=\int \tau _ H(B)\, \mathrm d\nu (H)$. By monotone convergence, $\tau $ is countably additive, so it is a Borel probability measure on $S^{2n-1}$. For a unitary matrix $U$, the map $UT_ H$ is a complex linear isometry from $\mathbb C^{n-1}$ onto $UH$, so $U$ maps $\tau _ H$ to $\tau _{UH}$, that is, $\tau _ H(U^{-1}B)=\tau _{UH}(B)$. Since $\nu $ is invariant, it follows that $\tau (U^{-1}B)=\int \tau _{UH}(B)\, \mathrm d\nu (H)=\tau (B)$, so $\tau $ is unitarily invariant. The normalized surface measure $\mathrm dS/s_{2n-1}$ of $S^{2n-1}$ is also unitarily invariant, because unitary matrices are orthogonal transformations of $\mathbb R^{2n}$. By the uniqueness in proposition 2.9.18, $\tau =\mathrm dS/s_{2n-1}$. By the definition of $\tau $ for indicator functions, by linearity for simple functions, and by monotone convergence in general, $\int _{\mathcal H}\int g\, \mathrm d\tau _ H\, \mathrm d\nu (H)=\int g\, \mathrm d\tau $ for every nonnegative Borel function $g$ on $S^{2n-1}$.

Conclusion. Integrate the polar-coordinate identity over $H$ with respect to $\nu $. The function $(r,\theta )\mapsto f(r\theta )$ is Borel on $(0,\infty )\times S^{2n-1}$, so by the measurability step $(H,r)\mapsto r^{2n-3}\int f(r\theta )\, \mathrm d\tau _ H(\theta )$ is a nonnegative Borel function on $\mathcal H\times (0,\infty )$, which is product-measurable by lemma 2.2.8. Tonelli's theorem therefore allows us to integrate first over $H$, and the averaging identity with $g(\theta )=f(r\theta )$ shows that the left side of the lemma equals

In polar coordinates $\xi =r\theta $ on all of $\mathbb C^ n=\mathbb R^{2n}$, (3.1.2) gives $|\xi |^{-2}\, \mathrm d\xi =r^{2n-3}\, \mathrm d r\, \mathrm dS(\theta )$, so this is the right side. All integrals may be infinite. $\square$