Let θ be an inner function, let θ*(H²) = H² Θ θH², and let p be a finite Borel measure on the unit circle double-struck T sign. Our main purpose is to prove that, if every function f ∈ θz.ast;(H²) can be defined μ-almost everywhere on double-struck T sign in a certain (weak) natural sense, then every function f ∈ θ*(H²) has finite angular boundary values μ-almost everywhere on double-struck T sign. A similar result is true for the L^p-analog of θ*(H²) (p > 0).