On a problem of Doob about the fine topology and normal functions

Abstract:

Early in the sixties, Joseph L. Doob proved that if $f(z)$ is a normal function in a disk, then every angular cluster value at a boundary point is also a fine cluster value at that point. He then asked whether or not the converse of this theorem is true. It was pointed out by the referee that Choquet was the first one to show that the converse of the above theorem is false. His example was published by Brelot and Doob (see [2, p. 404]). Recently, the referee suggests the question as to whether a bounded analytic function can actually have an angular limit but not a fine limit. We construct a Blaschke product which has the fine cluster value 0 and the angular limit 1 at the same boundary point. This answers both questions of Doob and the referee.