Luzin criterion

for measurability of a function of a real variable

For a function , defined on the interval and almost-everywhere finite, to be measurable it is necessary and sufficient that for any there is a function , continuous on , such that the measure of the set

is less than . It was proved by N.N. Luzin [1]. In other words, an almost-everywhere finite function is measurable if and only if it becomes continuous if one neglects a set of arbitrary small measure.

References

[1]	N.N. [N.N. Luzin] Lusin, "Sur les propriétés des fonctions mesurables" C.R. Acad. Sci. Paris , 154 (1912) pp. 1688–1690
[2]	I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian)

Comments

In the West, Luzin's criterion is known as Luzin's theorem (in spite of an ambiguity — cf. Luzin theorem) and is generally stated a little bit differently, more like in Luzin -property (but with a compact set instead of a perfect set). The tightness of the measure and the normality of the space makes all these formulations equivalent.

The Luzin criterion remains true if the interval is replaced by any completely-regular space and the (restriction of the) Lebesgue measure by any tight bounded measure on the Borel -field. In this general setting the Luzin property may be used in order to give an alternative definition of the notion of measurability (cf. [a1]) or, in recent works, a more adequate definition of this notion when is no longer a real-valued function but, for example, a Banach-valued function.

The Luzin criterion is intimately related to the Egorov theorem and to the notion of measurability according to Carathéodory (cf. Carathéodory measure).

References

[a1]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[a2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[a3]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98
[a4]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)