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 of measure zero on the unit circle , N.N. Luzin constructed (1919, see [1]) a function that is regular, analytic and bounded in the unit disc and is such that does not have radial boundary values along each of the radii that end at points of .

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 and in that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure on . This set is of the first Baire category (cf. Baire classes) on .

See also Boundary properties of analytic functions; Luzin–Privalov theorems; Cluster set.

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)

References

[a1]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9