Analytic functions in the unit disk with radial limit zero. In the following paper: 
http://www.ams.org/journals/proc/1955-006-01/S0002-9939-1955-0068625-6/S0002-9939-1955-0068625-6.pdf
there is the following claim. 

Examples have been given [5, p. 185] of functions $f(z)$, analytic in the
  unit-circle $K$: $|z| <1$, and not identically constant, for which the
  radial $\text{limit} f(e^{i\theta}) =\lim_{r\to 1}f(re^{i\theta})$ is zero for all $e^{i\theta}$  on $|z| =1$ except
  for a set of linear measure zero. In view of the Riesz-Nevanlinna
  theorem [6, p. 197], such functions cannot be bounded, or even of
  bounded characteristic, in $|z| < 1$.

This is not very intuitive as it seems at first contradictory to the maximum modulus principle applied in $\mathbb{D}$. Can someone explain what is actually happening here?
 A: When you say 

This is not very intuitive as it seems at first contradictory to the maximum modulus principle applied in $\mathbb{D}$

I think you are probably trying to interchange the two limits $r\uparrow 1$ and $\sup_\theta$, which is a big no-no.  Indeed, the $f$ in this example has
$$
\sup_\theta \lvert f(re^{i\theta})\rvert\to\infty\text{ as }r\uparrow 1.
$$
i.e., $f\notin H^\infty$.  In fact, it isn't in any $H^p$.
A toy example, to get a feeling of what could be happening, is to work with harmonic functions.  The Poisson kernel
$$
P_r(\theta)=\operatorname{Re}\frac{1+re^{i\theta}}{1-re^{i\theta}}=\frac{1-r^2}{1-2r\cos\theta+r^2}
$$
is harmonic on $\mathbb{D}$ and satisfies $\lim_{r\uparrow 1}P_r(\theta)\to 0$ for almost all $\theta$ (indeed, as long as $\cos\theta\neq 1$).  However, this isn't the full counterexample because the conjugate Poisson kernel
$$
Q_r(\theta)=\operatorname{Im}\frac{1+re^{i\theta}}{1-re^{i\theta}}=\frac{2r\sin\theta}{1-2r\cos\theta+r^2}
$$
does not have (almost everywhere) constant radial limit.
You can read Lusin's original paper here, although according to David Hansen's answer in this MO thread Hardy and Ramanujan might have a claim to be the first to construct such example.
