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The Riesz theorem is not true for sets of measure zero since it is known that given any set PC C of measure zero (in particular, P may be of second category) there exists a nonconstant function f(z), holomorphic and bounded in D, with radial limit zero at each point of P [4, p. 214]. However, the following theorem shows that something very definitely can be said about the rate at which a bounded holomorphic function approaches zero along radii terminating at a subset of C of second category.

A asymptotic analog of the F. and M. Riesz radial uniqueness theorem.