Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and ${|ψ_{μ₀}| < 1} ∩ {|ψ_{λ₀}| < 1} = ∅$, where $ψ_{μ}$ is the associated singular inner function of μ. Let $𝓩(μ) = ⋂_{ν;ν∼ μ} Z(ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$. Then 𝓩(μ) ≠ ∅ and 𝓩(μ) ∩ 𝓩(λ) = ∅. It follows that μ ≪ λ if and only if 𝓩(μ) ⊂ 𝓩(λ). Hence 𝓩(μ) is the set in the maximal ideal space of $H^{∞}$ which relates naturally to the set of measures equivalent to μ. Some basic properties of 𝓩(μ) are given.
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We study connected components of a common zero set of equivalent singular inner functions in the maximal ideal space of the Banach algebra of bounded analytic functions on the open unit disk. To study topological properties of zero sets of inner functions, we give a new type of factorization theorem for inner functions.