Luzin N property

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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 $N\subset[a,b]$ that $λ(N) = 0$ , it holds that $λ(f (N)) = 0$ , where $λ$ stands for the Lebesgue measure.

Note that the image of such an N set 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. The definition asserts that the image of such a set N is actually measurable.

[edit] Properties

Every absolutely continuous function has the Luzin N property. The Cantor function on the other hand has not: the Lebesuge measure of the Cantor set is zero, however its image is the complete [0,1] interval.

Also, if an f function on the interval [a,b] is continuous, is of bounded variation and has the Luzin N property, then it is absolutely continuous.