A boundary uniqueness theorem in $\mathbf C^n$

The classical boundary theorem of F. and M. Riesz asserts that, if the radial limits of a bounded holomorphic function $f(z)$ in the disk $|z|<1$ lie in a set of capacity zero for a set of positive measure on the circle $|z|=1$, then $f(z)\equiv\mathrm{constant}$. The main result of this paper is the proof of an analogous theorem for maps $F\colon D\to\mathbf C^n$, where $D$ is a domain in $\mathbf C^n$. We take as uniqueness set on the boundary any set of positive Lebesgue measure on a generating submanifold.
Bibliography: 12 titles.