On Doob's conjecture about fine limit and Julia point

In the 1960, professor Doob proved that if f(z) is a meromorphic function on a disk D, then at almost every point of the perimeter of D either (a) f has both an angular limit and an equal fine limit, (b) f has every point of the plane as cluster value in every angle opening into D with vertex at the point, and f has a fine limit at the point, or (c) f has every point of the plane both as cluster value in every angle opening into D with vertex at the point and as fine cluster value. He then mentioned that it would be interesting to find an example to show that case (b) can really occur on a perimeter set of strictly positive measure. In this article, we prove this conjecture to be true.