A comparison theorem for Clark-Aleksandrov measures on the torus

Given two Clark-Aleksandrov measures ${\sigma }^1$ and ${\sigma }^2$ on ${\text{bT}}^2$, we prove a theorem relating the property that ${\sigma }^1\ll {\sigma }^2$ to containment of a concrete function in a certain de Branges-Rovnyak space. We show that our theorem is applicable for all rational inner functions on ${\text{bD}}^2$, and we provide several examples of how the theorem can be applied to investigate Clark-Aleksandrov measures related to such functions.