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Abstract

Let $\mu $ and $\lambda$ be bounded positive singular measures on the unit circle such that $\mu \perp \lambda$. It is proved that there exist positive measures $\mu_0$ and $\lambda_0$ such that $\mu_0 \sim \mu$, $\lambda_0 \sim \lambda$, and $\{|\psi_{\mu_0}| < 1\} \cap \{|\psi_{\lambda_0}| < 1\} = \emptyset$, where $\psi_\mu$ is the associated singular inner function of $\mu$. Let ${\cal Z}(\mu) = \bigcap_{\{\nu;\,\nu \sim \mu\}} Z(\psi_\nu)$ be the common zeros of equivalent singular inner functions of $\psi_\mu$. Then ${\cal Z}(\mu) \not= \emptyset$ and ${\cal Z}(\mu) \cap {\cal Z}(\lambda) = \emptyset$. It follows that $\mu \ll \lambda$ if and only if ${\cal Z}(\mu) \subset {\cal Z}(\lambda)$. Hence ${\cal Z}(\mu)$ is the set in the maximal ideal space of $H^\infty$ which relates naturally to the set of measures equivalent to $\mu$. Some basic properties of ${\cal Z}(\mu)$ are given.