Luzin criterion

for measurability of a function of a real variable

For a function $f$ defined on the interval $[a,b]$ and almost-everywhere finite, to be measurable it is necessary and sufficient that for any $\epsilon>0$ there is a function $\phi$, continuous on $[a,b]$, such that the measure of the set $$ \{ x \in [a,b] : f(x) \ne \phi(x) \} $$ is less than $\epsilon$. 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 Zbl 43.0484.04
[2]	I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian) MR0640867 MR0354979 MR0148805 MR0067952 MR0039790

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 $C$-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 $[a,b]$ is replaced by any completely-regular space and the (restriction of the) Lebesgue measure by any tight bounded measure on the Borel $\sigma$-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 $f$ 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) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[a2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[a3]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701
[a4]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202