On the boundary behavior of bounded analytic functions

We prove a result which sheds new light on the fundamental theorems of complex analysis due to P. Fatou, F. and M. Riesz, N. N. Lusin, I. I. Privalov, and A. Beurling. Only classical tools available at the time of Fatou are used. The proofs are very simple and short. Also, we present the solution to an open problem proposed by M. von Renteln in 1980, which has been published in a well-known collection of open problems by D. M. Campbell, J. G. Clunie, and W. K. Hayman.