View source for Luzin examples

''in the theory of functions of a complex variable''

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

1) For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610301.png" /> of measure zero on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610302.png" />, N.N. Luzin constructed (1919, see [[#References|[1]]]) a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610303.png" /> that is regular, analytic and bounded in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610304.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610305.png" /> does not have radial boundary values along each of the radii that end at points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610306.png" />.

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

2) Luzin also constructed (1925, see [[#References|[2]]]) regular analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610309.png" /> that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103011.png" />. This set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103012.png" /> is of the first Baire category (cf. [[Baire classes|Baire classes]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103013.png" />.

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

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>