Luzin N property

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 $N\subset[a,b]$ such that $\lambda(N)=0$ , there holds: $\lambda(f(N))=0$ , where $\lambda$ stands for the Lebesgue 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. 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.

References

External links

Springer Online

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