# 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 |

#### Comments

See Luzin criterion for additional references and comments.

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Luzin-C-property.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Luzin-C-property&oldid=28880