# Luzin examples

in the theory of functions of a complex variable

Examples that characterize boundary uniqueness properties of analytic functions (see [1], [2]).

1) For any set $E$ of measure zero on the unit circle $\Gamma = \{ {z } : {| z | = 1 } \}$, N.N. Luzin constructed (1919, see [1]) a function $f ( z)$ that is regular, analytic and bounded in the unit disc $D = \{ {z } : {| z | < 1 } \}$ and is such that $f ( z)$ does not have radial boundary values along each of the radii that end at points of $E$.

A similar example of Luzin and I.I. Privalov (1925, see [2], [3]) differs only by insignificant changes.

2) Luzin also constructed (1925, see [2]) regular analytic functions $f _ {1} ( z)$ and $f _ {2} ( z) \not\equiv 0$ in $D$ that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure $2 \pi$ on $\Gamma$. This set $E$ is of the first Baire category (cf. Baire classes) on $\Gamma$.

#### References

 [1] N.N. Luzin, , Collected works , 1 , Moscow (1953) pp. 267–269 (In Russian) [2] N.N. Luzin, , Collected works , 1 , Moscow (1953) pp. 280–318 (In Russian) [3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) [4] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauki i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)