For every inner function Θ, we set Θ*(H²) =^def H²⊖ΘH² and Θ*(H^P) =^def clos_HP(H^P ∩ Θ*(H^p)) if p ≠ 2. Denote by C _P(Θ) the set of all (complex) measures μ, on the closed unit disk double-struck D̄ such that Θ*(H^p) ⊂ L ^P(|μ|). An inner function Θ is said to be one-component if the set (z ∈ double-struck D:|Θ(z)| < ε) is connected for some ε ∈ (0, 1). A series of criteria for an inner function to be one-component are obtained. For example, it is proved that Θ is one-component if and only if C_p(Θ) does not depend on p ∈ (0,+∞). Moreover, a criterion in terms of reproducing kernels of Θ *(H²) for an inner function Θ to be one-component is obtained. The set C_P(Θ) is described in the case where Θ is a Blaschke product of special form. This description implies that the set of all p such that a given measure μ belongs to C_p(Θ) can have any finite or infinite number of connected components. The following examples of interpolating Blaschke products Θ and positive measures μ are constructed: (1) Θ*(H¹) ⊂ L¹(μ) and Θ*(H²) ⊂ L²(μ) but Θ* (H^p) ⊄ L^P(μ) for any p ∈ (1, 2); (2) Θ*(H^P) ⊂ L^p(n) if and only if p = 1/n, where n is a positive integer; (3) Θ*(H^P) ⊂ L ^p(μ) if and only if p ≠ 1/n, where n is a positive integer.