Lemma 6.3.25898ED42C26F

Proof. Fix $z_0$ and $a$, and put $g(\theta )=v(z_0+\mathrm e^{\mathrm i\theta }a)$, an upper semicontinuous function on the circle. By lemma 3.4.4 there are continuous real functions $g_ m$ decreasing to $g$ whose integrals decrease to the integral of $g$, which may be $-\infty $. Fix $m$ and $\delta >0$. We approximate $g_ m$ uniformly by real trigonometric polynomials, using the Fejér kernel

The two expressions agree because the product $\sum _{k,l=0}^{N-1}\mathrm e^{\mathrm i(k-l)\theta }$ contains exactly $N-|h|$ pairs $(k,l)$ with $k-l=h$, for each integer $h$ with $|h|<N$. By the first expression $F_ N\ge 0$, and summing the geometric series gives $F_ N(\theta )=\sin ^2(N\theta /2)/(N\sin ^2(\theta /2))\le 1/(N\sin ^2(\theta /2))$ for $0<|\theta |\le \pi $; by the second, $F_ N$ has integral one with respect to the normalized measure $\mathrm d\theta /(2\pi )$ on $[-\pi ,\pi ]$.

Let $q_ N=g_ m*F_ N$ be the convolution with respect to $\mathrm d\theta /(2\pi )$, that is, $q_ N(\theta )=\frac1{2\pi }\int _{-\pi }^{\pi }g_ m(\theta -\tau )F_ N(\tau )\, \mathrm d\tau $. For $0<\delta '\le \pi $ let $\omega (\delta ')$ be the largest oscillation of $g_ m$ over arcs of length at most $\delta '$, which tends to zero with $\delta '$ by uniform continuity. Splitting the integral at $|\tau |=\delta '$, we obtain

so $q_ N\to g_ m$ uniformly as $N\to \infty $. Inserting the second expression for $F_ N$ and substituting $\sigma =\theta -\tau $, we see that $q_ N(\theta )=\sum _{|k|<N}c_ k\mathrm e^{\mathrm ik\theta }$ with $c_ k=(1-|k|/N)\frac1{2\pi }\int _{-\pi }^{\pi }g_ m(\sigma )\mathrm e^{-\mathrm ik\sigma }\, \mathrm d\sigma $. Since $g_ m$ is real, $c_{-k}=\bar c_ k$, so $q_ N(\theta )=\operatorname{Re}P(\mathrm e^{\mathrm i\theta })$ for the polynomial $P(t)=c_0+2\sum _{0<k<N}c_ kt^ k$, and $\operatorname{Re}P(0)=c_0$ is the mean of $g_ m$. Choose $N$ with $|q_ N-g_ m|<\delta $. Then $\operatorname{Re}(P(t)+\delta )\ge g_ m\ge g$ on the circle, and the hypothesis gives

Letting $\delta \to 0$ and then $m\to \infty $, we obtain the asserted inequality. $\square$