Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc

Let $\phi$ be a holomorphic self-map of the unit disc $\mathbb{D}$. For every $\alpha \in \partial\mathbb{D}$, there is a measure $\tau_\alpha$ on $\partial\mathbb{D}$ (sometimes called Aleksandrov measure) defined by the Poisson representation $\operatorname{Re}(\alpha+\phi(z))/(\alpha-\phi(z)) = \int P(z,\zeta) \,d\tau_\alpha(\zeta)$. Its singular part $\sigma_\alpha$ measures in a natural way the ``affinity'' of $\phi$ for the boundary value $\alpha$. The affinity for values $w$ inside $\mathbb{D}$ is provided by the Nevanlinna counting function $N(w)$ of $\phi$. We introduce a natural measure-valued refinement $M_w$ of $N(w)$ and establish that the measures $\{\sigma_\alpha\}_{\alpha\in\partial\mathbb{D}}$are obtained as boundary values of the refined Nevanlinna counting function $M$. More precisely, we prove that $\sigma_\alpha$ is the weak$^*$ limit of $M_w$ whenever $w$ converges to $\alpha$non-tangentially outside a small exceptional set $E$. We obtain a sharp estimate for the size of $E$ in the sense of capacity.