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.