Theorem 2.9.222C0F0197F3BF

Proof. We obtain $\mu $ by choosing, in a Borel way, a point of $E$ on each circle of radius $r\in [R/2,R]$ centered at $x$ in a complex line through $x$, and then we estimate its energy. For $\xi \in \mathbb P^{n-1}(\mathbb C)$ we first choose a unit vector $e_0(\xi )\in \xi $ in a Borel way: if $j$ is the first index with $(P_\xi )_{jj}>0$, put $e_0(\xi )=P_\xi e_ j/\sqrt{(P_\xi )_{jj}}$. Since $|P_\xi e_ j|^2=\langle P_\xi e_ j,e_ j\rangle =(P_\xi )_{jj}$, this is a unit vector in $\xi $. The set of $\xi $ with a given first index $j$ is Borel, and $e_0$ is continuous on it.

Let $r\in [R/2,R]$ and $\xi \in \mathbb P^{n-1}(\mathbb C)$. The circle $\theta \mapsto x+r\mathrm e^{\mathrm i\theta }e_0(\xi )$ lies in the closed disc $\{ x+\zeta e_0(\xi ):|\zeta |\leq r\} $, which is connected, contains $x$ and is contained in $D$; hence it lies in the connected component containing $x$ of the section of $D$ by the line $x+\xi $. On this component $u$ is subharmonic; in particular it is upper semicontinuous and bounded above on the circle, and by theorem 3.4.5(2) and the arc-length formula for circle means at the beginning of this section, it satisfies the submean inequality

Let

This set is closed by upper semicontinuity, and it has positive length. If $A=-\infty $, it is the whole interval $[0,2\pi ]$. If $A$ is finite and $F_{r,\xi }$ were a null set, then $A-u$ would be positive almost everywhere on the circle, so the circle mean of $u$ would be strictly less than $A$, contradicting the submean inequality.

The set of $(r,\xi ,\theta )$ with $\theta \in F_{r,\xi }$ is Borel, because $u$ is Borel and the map $(r,\xi ,\theta )\mapsto x+r\mathrm e^{\mathrm i\theta }e_0(\xi )$ into $\overline{B(x,R)}$ is Borel, being continuous on each of the finitely many Borel sets on which $e_0$ is continuous. Since $[R/2,R]\times \mathbb P^{n-1}(\mathbb C)$ is a compact, hence separable, metric space, lemma 2.9.21 gives a Borel map $(r,\xi )\mapsto \alpha (r,\xi )$ with $\alpha (r,\xi )\in F_{r,\xi }$. Put $e(r,\xi )=\mathrm e^{\mathrm i\alpha (r,\xi )}e_0(\xi )$ and $\Psi (r,\xi )=x+re(r,\xi )$. Then $\Psi $ is Borel, hence product-measurable (lemma 2.2.8), and it takes its values in $E$. Let $\mu $ be the image under $\Psi $ of the product of normalized Lebesgue measure on $[R/2,R]$ and $\sigma $. It is a Borel probability measure carried by $E$.

It remains to estimate the energy; the estimate does not depend on the choice of the phases $\alpha (r,\xi )$. For $r,r'\in [R/2,R]$ and unit vectors $e,e'$ spanning $\xi ,\xi '$,

where the identity follows by expanding both sides with $|e|=|e'|=1$ and $|e-e'|^2=2-2\operatorname{Re}\langle e,e'\rangle $, and the inequality holds since $rr'\geq R^2/4$ and $\operatorname{Re}\langle e,e'\rangle \leq |\langle e,e'\rangle |$. With $a=d(\xi ,\xi ')\in [0,\pi /2]$, we have $2-2|\langle e,e'\rangle |=4\sin ^2(a/2)\geq 4a^2/\pi ^2$, because $\sin y\geq 2y/\pi $ for $0\leq y\leq \pi /2$ by concavity of the sine on this interval. Hence $|re-r'e'|^2\geq (r-r')^2+R^2a^2/\pi ^2$.

Write $r=Rt$ with $t\in [1/2,1]$. Equip $[1/2,1]\times \mathbb P^{n-1}(\mathbb C)$ with the metric $\rho $ defined by $\rho ((t,\xi ),(t',\xi '))^2=(t-t')^2+d(\xi ,\xi ')^2$, and with the product probability $m$ of normalized Lebesgue measure and $\sigma $, which is defined on all Borel sets of this separable metric space by lemma 2.2.8. Then $\mu $ is the image of $m$ under $(t,\xi )\mapsto \Psi (Rt,\xi )$, and the lower bound for $|re-r'e'|^2$, together with $\pi >1$, gives

A $\rho $-ball of radius $s\leq 1$ is contained in the product of an interval of length $2s$, whose normalized measure is at most $4s$, and a ball of radius $s$ for the metric $d$. Since $s\leq 1<\pi /2$, lemma 2.9.19 shows that its $m$-measure is at most $4s^{2n-1}$. In particular $m$ has no atoms, so the diagonal, a closed and hence product-measurable set (lemma 2.2.8), is null for $m\times m$ by Tonelli's theorem. Fix $z\in [1/2,1]\times \mathbb P^{n-1}(\mathbb C)$. Splitting the integral into the set where $\rho (z,z')\geq 1$ and the shells where $2^{-k-1}\leq \rho (z,z')<2^{-k}$, $k\geq 0$, we obtain

Since $\mu $ is the image of $m$, the product $\mu \times \mu $ is the image of $m\times m$ under the product map, because both measures agree on rectangles (proposition 2.2.7). Therefore the lower bound for $|\Psi (Rt,\xi )-\Psi (Rt',\xi ')|$, valid off the $m\times m$-null diagonal, and an integration in $z$ give

This is the assertion, with $C_ n=\pi ^{2n-2}(1+2^{2n+1})$. $\square$