Introduction of Inner Distributions and Their Approximation with Blaschke Distributions. Characterization of Blaschke Distributions

We define inner distributions on the unit disk as boundary values of classical inner functions. Also, we introduce Blaschke distributions on the unit disk. We give properties of the introduced inner and Blaschke distributions. We prove that an inner distribution on the unit disk can be uniformly approximated by a sequence of Blaschke distributions. We characterize the Blaschke distributions in the spirit of Fatou's theorem.