View source for Luzin criterion

''for measurability of a function of a real variable''

For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610101.png" />, defined on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610102.png" /> and almost-everywhere finite, to be measurable it is necessary and sufficient that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610103.png" /> there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610104.png" />, continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610105.png" />, such that the measure of the set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610106.png" /></td> </tr></table>

is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610107.png" />. It was proved by N.N. Luzin [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,   "Sur les propriétés des fonctions mesurables"  ''C.R. Acad. Sci. Paris'' , '''154'''  (1912)  pp. 1688–1690</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)</TD></TR></table>



====Comments====
In the West, Luzin's criterion is known as Luzin's theorem (in spite of an ambiguity — cf. [[Luzin theorem|Luzin theorem]]) and is generally stated a little bit differently, more like in [[Luzin-C-property|Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610108.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l0610109.png" /> is replaced by any [[Completely-regular space|completely-regular space]] and the (restriction of the) Lebesgue measure by any tight bounded measure on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l06101010.png" />-field. In this general setting the Luzin property may be used in order to give an alternative definition of the notion of measurability (cf. [[#References|[a1]]]) or, in recent works, a more adequate definition of this notion when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061010/l06101011.png" /> is no longer a real-valued function but, for example, a Banach-valued function.

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

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