Fatou’s theorem (1906) states that a bounded analytic function in the unit disc of the complex plane has radial limits a.e. on the unit circleT. We prove the following.
Theorem 1.LetEbe a subset onT. There exists a bounded analytic function in the open unit disc which has no radial limit at each point ofEbut has unrestricted limit at each point ofT\Eif and only ifEis anFσset of measure zero.
The necessity of the condition thatEis of measure zero follows from Fatou’s theorem. The necessity of the condition thatEis anFσset is a known elementary (or just obvious) fact. Thus the sufficiency part of Theorem 1 is a converse to Fatou’s theorem. The converses to Fatou’s theorem proved by N. N. Lusin (1919) and by S. V. Kolesnikov (1994) are well-known. The method of the proof of Theorem 1 implies the first elementary and a few line proof for Lusin’s theorem, the original proof of which is very difficult. The same method also simplifies the proof of Kolesnikov’s theorem. Also, the sufficiency part of Theorem 1 immediately implies a well-known theorem of A. J. Lohwater and G. Piranian of 1957.