Let Θ be an interior function in the upper half-plane. Positive measures μ on the real line ℝ such that ∫_ℝ|f| ²dμ = ∥f∥₂ ² for all f ∈ K _Θ = H² ⊖ΘH² (i.e., the embedding K_Θ ⊂ L²(μ) is isometric) are studied. A similar problem for the unit disk was considered by A. B. Aleksandrov. In the case of the upper half-plane and special interior functions, such measures were described by De Branges. The goal of this paper is to explain why these results are essentially different. As is shown, the fact that the isomterty fails is connected with the existence of the finite angular derivative and can be expressed in terms of factorization parameters of Θ. Owing to this results, it is possible to formulate a criterion (in terms of zeros of the generating entire function E) for an orthogonal system of reproducing kernels to be a basis for the space H(E).