Luzin-C-property
A characteristic property of a measurable function that is finite almost-everywhere on its domain of definition. A function , finite almost-everywhere on
, has the
-property on
if for every
there is a perfect set
in
with measure
on which
is continuous if considered only on
. The
-property was introduced by N.N. Luzin [1], who also proved that for a function to have the
-property it is necessary and sufficient that it be measurable and finite almost-everywhere on
. This theorem of Luzin (the Luzin criterion) can be generalized to the case of functions of several variables (see [3], [4]) and is one of the main theorems in the metric theory of functions.
References
[1] | N.N. Luzin, Mat. Sb. , 28 (1912) pp. 266–294 |
[2] | N.N. Luzin, "Collected works" , 1 , Moscow (1953) (In Russian) MR0059845 |
[3] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05 |
[4] | E. Kamke, "Das Lebesgue–Stieltjes Integral" , Teubner (1960) MR0125193 Zbl 0071.05401 |
Comments
See Luzin criterion for additional references and comments.
Luzin-C-property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin-C-property&oldid=28870