# Luzin-C-property

A characteristic property of a measurable function that is finite almost-everywhere on its domain of definition. A function $f$, finite almost-everywhere on $[0,1]$, has the $\mathcal{C}$-property on $[0,1]$ if for every $\epsilon>0$ there is a perfect set $Q$ in $[0,1]$ with measure $>1-\epsilon$ on which $f$ is continuous if considered only on $Q$. The $\mathcal{C}$-property was introduced by N.N. Luzin [1], who also proved that for a function to have the $\mathcal{C}$-property it is necessary and sufficient that it be measurable and finite almost-everywhere on $[0,1]$. 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