# Difference between revisions of "Luzin criterion"

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

− | For a function | + | For a function $f$ defined on the interval $[a,b]$ and almost-everywhere finite, to be [[Measurable function|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 | + | is less than $\epsilon$. 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==== | ====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> | + | <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 {{MR|}} {{ZBL|43.0484.04}} </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) {{MR|0640867}} {{MR|0354979}} {{MR|0148805}} {{MR|0067952}} {{MR|0039790}} {{ZBL|}} </TD></TR> | ||

+ | </table> | ||

====Comments==== | ====Comments==== | ||

− | In the West, Luzin's criterion is known as Luzin's theorem (in spite of an ambiguity — cf. [[ | + | 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|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 | + | 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. [[#References|[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 [[ | + | The Luzin criterion is intimately related to the [[Egorov theorem]] and to the notion of measurability according to Carathéodory (cf. [[Carathéodory measure]]). |

====References==== | ====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> | + | <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) {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR> | ||

+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR> | ||

+ | <TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 {{MR|0210528}} {{ZBL|0142.01701}} </TD></TR> | ||

+ | <TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR> | ||

+ | </table> | ||

+ | |||

+ | {{TEX|done}} |

## Latest revision as of 19:43, 28 December 2017

*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 |

**How to Cite This Entry:**

Luzin criterion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Luzin_criterion&oldid=15197