Maximal arc: Difference between revisions
en>FrescoBot m Bot: fixing section wikilinks |
en>Wcherowi m →Properties: clarification |
||
| Line 1: | Line 1: | ||
In [[mathematics]], a function ''f'' on the interval [''a'', ''b''] has the '''Luzin N property''', named after [[Nikolai Luzin]] (also called Luzin property or N property) if for all <math>N\subset[a,b]</math> such that <math>\lambda(N)=0</math>, there holds: <math>\lambda(f(N))=0</math>, where <math>\lambda</math> stands for the [[Lebesgue measure]]. | |||
Note that the image of such a set ''N'' is not necessarily [[measurable set|measurable]], but since the Lebesgue measure is [[Complete measure|complete]], it follows that if the Lebesgue [[outer measure]] of that set is zero, then it is measurable and its Lebesgue measure is zero as well. | |||
==Properties== | |||
Every [[absolutely continuous]] function has the Luzin N property. The [[Cantor function]] on the other hand does not: the Lebesgue measure of the [[Cantor set]] is zero, however its image is the complete [0,1] interval. | |||
Also, if a function ''f'' on the interval [''a'',''b''] is [[continuous function|continuous]], is of [[bounded variation]] and has the Luzin N property, then it is [[absolutely continuous]]. | |||
==External links== | |||
*[http://eom.springer.de/L/l061050.htm Springer Online] | |||
[[Category:Real analysis]] | |||
[[Category:Measure theory]] | |||
{{mathanalysis-stub}} | |||
Revision as of 19:23, 25 November 2012
In mathematics, a function f on the interval [a, b] has the Luzin N property, named after Nikolai Luzin (also called Luzin property or N property) if for all such that , there holds: , where stands for the Lebesgue measure.
![{\displaystyle N\subset [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33194ec123afd1a8ffdd697e1464f0a3b321608a)
Note that the image of such a set N is not necessarily measurable, but since the Lebesgue measure is complete, it follows that if the Lebesgue outer measure of that set is zero, then it is measurable and its Lebesgue measure is zero as well.
Properties
Every absolutely continuous function has the Luzin N property. The Cantor function on the other hand does not: the Lebesgue measure of the Cantor set is zero, however its image is the complete [0,1] interval.
Also, if a function f on the interval [a,b] is continuous, is of bounded variation and has the Luzin N property, then it is absolutely continuous.