A comparison theorem for Aleksandrov-Clark measures on the torus

Given two Aleksandrov-Clark measures σ1 and σ2 on T2, we prove a theorem relating the property that σ1≪σ2 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 D2, and we provide several examples of how the theorem can be applied to investigate Aleksandrov-Clark measures related to such functions.