- Luzin N property
In
mathematics , a function "f" on the interval ["a", "b"] has the Luzin N property, named afterNikolai Luzin (also called Luzin property or N property) if for all such that , there holds: , where stands for theLebesgue measure .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. TheCantor function on the other hand does not: the Lebesgue measure of theCantor 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 ofbounded variation and has the Luzin N property, then it isabsolutely continuous .External links
* [http://eom.springer.de/L/l061050.htm Springer Online]
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