On the existence of nontangential boundary values of pseudocontinuable functions

Let $\theta$ be an inner functions, let $\theta^*(H^2)=H^2\ominus\theta H^2$, and let $\mu$ be a finite Borel measure on the unit circle $\mathbb T$. Our main purpose is to prove that, if every function $f\in\theta^*(H^2)$ can be defined $\mu$-almost everywhere on $\mathbb T$ in a certain (weak) natural sense, then every function $f\in\theta^*(H^2)$ has finite nontangential boundary values $\mu$-almost everywhere on $\mathbb T$. A similar result is true for the $\mathcal L^p$-analog of $\theta^*(H^2)$ ($p>0$). Bibliography: 17 titles.