Hello everybody.


I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits\_{r \rightarrow 1-} f(re^{i\phi})=0$.


Does anyone know such an example.


Best
CJ



Holomorphic function with a.e. vanishing radial boundary limits
I am not sure if this would qualify as 'easy' but the first example of such a function was constructed by Lusin. It can be found in N. Lusin, J. Priwaloff, [Sur l'unicité et la multiplicité des fonctions analytiques](http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1925_3_42_/ASENS_1925_3_42__143_0/ASENS_1925_3_42__143_0.pdf), *Ann. Sci. École Norm. Sup.* (3), 1925, p. 143-191 (see p. 185).


