Theorem 2.9.27AF10D8C2B9D9

Proof. We parametrize the affine complex lines that can meet $K$ by a compact set carrying a finite measure $\lambda $, and bound from below, independently of $j$, the $\lambda $-measure of a set of lines meeting $E_ j$. Choose $M>0$ with $K\subset B(0,M)$. Every affine complex line can be written as $c+H^\perp $ with $H\in \mathcal H$ and $c\in H$, and $c=P_ Hy$ for every point $y$ of the line. Indeed, if the line is $a+\mathbb Cb$, let $H$ be the hyperplane orthogonal to $b$, so that $H^\perp =\mathbb Cb$, and put $c=P_ Ha$; then $P_ H(a+\zeta b)=c$ for every $\zeta \in \mathbb C$. If the line meets $K$ at a point $y$, then $|c|=|P_ Hy|\leq |y|<M$. Hence the lines that can meet $K$ are parametrized by the set

which is compact, being closed in $\mathcal H\times \overline{B(0,M)}$ because $c\in H$ means $P_ Hc=c$. We equip $\mathcal L_ M$ with the finite measure $\lambda $ defined by

for nonnegative Borel functions $h$. In the frames of subsection 2.9.17, $\lambda $ is the image of the product of $\nu $ and Lebesgue measure on the closed ball of radius $M$ in $\mathbb C^{n-1}$. Thus it is a Borel measure, and its total mass is the volume of a ball of radius $M$ in $\mathbb R^{2n-2}$.

By inner regularity (proposition 2.2.2), choose compact sets $F_ j\subset E_ j$ with $\mu _ j(F_ j)\geq 1/2$. The set

parametrizes the lines meeting $F_ j$. It is compact, as the image of $\mathcal H\times F_ j$ under the continuous map $(H,y)\mapsto (H,P_ Hy)$. Its section over $H$ is $P_ HF_ j$, so $\lambda (J_ j)=\int m_ H(P_ HF_ j)\, \mathrm d\nu (H)$. Let $g_{j,H}$ be the densities given by theorem 2.9.26 for $\mu _ j$. Since $F_ j\subset P_ H^{-1}(P_ HF_ j)$, the Cauchy–Schwarz inequality gives, for $\nu $-almost every $H$,

Integrate this inequality over $H$ with respect to the probability measure $\nu $, and apply the Cauchy–Schwarz inequality once more. Squaring the result and using theorem 2.9.26, we obtain

In particular $C>0$, and $\lambda (J_ j)\geq \delta =1/(4\beta _ n C)$ for every $j$. The sets $\bigcup _{j\geq m}J_ j$ decrease in $m$ and have measure at least $\delta $. Since $\lambda $ is finite, continuity from above gives

so this set is nonempty. If $(H,c)$ belongs to it, then $c\in P_ HF_ j$ for infinitely many $j$. For each such $j$ there is $y\in F_ j$ with $c=P_ Hy$, so $y\in c+H^\perp $. Hence the line $c+H^\perp $ meets $F_ j$, and therefore $E_ j$, for infinitely many $j$. $\square$