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| 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]].
| | The name of the writer is Garland. Delaware is the only location I've been residing in. To perform badminton is something he truly enjoys doing. Managing people is what I do in my day job.<br><br>Here is my web site [http://www.clan-myreality.de/index.php?mod=users&action=view&id=9712 http://www.clan-myreality.de] |
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| 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.
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| ==Properties==
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| 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.
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| 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]].
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| ==External links== | |
| *[http://eom.springer.de/L/l061050.htm Springer Online]
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| [[Category:Real analysis]]
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| [[Category:Measure theory]]
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| {{mathanalysis-stub}}
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Latest revision as of 10:19, 24 May 2014
The name of the writer is Garland. Delaware is the only location I've been residing in. To perform badminton is something he truly enjoys doing. Managing people is what I do in my day job.
Here is my web site http://www.clan-myreality.de