On sets of nonexistence of radial limits of bounded analytic functions

Let $f(z)$ be a function defined in the unit disc $D$: $|z|<1$; $\Gamma$ the unit circle $|z|=1$; $E(f)$ the set of points of $\Gamma$ at which $f(z)$ has no radial limits. In the paper a complete characterization is given of the sets $E(f)$ for bounded analytic functions $f$ in $D$. It is proved that for any $G_{\delta\sigma}$ set $E\subset \Gamma$ of linear measure zero there exists a function $f(z)$, bounded and analytic in $D$, such that $E(f)=E$.