This is the proof of Rado's theorem in Rudin's PMA.
The first claim of the proof is a nonzero set $\Omega$ is dense in $U$. This is okay for me.
However, the application of maximum modulus theorem for $\Omega$ seems strange for me.
Specifically, author says: given $\alpha \in \Omega$
$$ |f(\alpha)g(\alpha)^n| \leq \| fg^n\|_{\partial\Omega} = \| fg^n\|_T \leq \|f\|_T \|g\|^n_T$$, but it seems not always true. The first inequality is okay, by the maximum modulus theorem, but how can we say that $\| fg^n\|_{\partial\Omega} = \| fg^n\|_T$? Although we have shown the density of $\Omega$, saying $\bar\Omega = \bar U$, we can only deduce $T = \partial \bar U = \partial \bar \Omega \subseteq \partial \Omega$. This is because $\partial \bar A \subseteq \partial A$ in general and not vice versa.
This may be a very minor problem, but it seems to be an important part of the proof. Any help would be appreciated.
** $T$: Unit circle, $U$: Open unit ball in $\mathbb{C}$
You know that $fg^n=0$ everywhere on $\partial\Omega\cap U$. So, if you're computing $\|fg^n\|_{\partial\Omega}$, you can ignore the points in $U$; the value of $|fg^n|$ at those points (which is $0$) cannot be larger than the largest value it takes on $T$.