Geodesic
The $N^{-1}$-property of maps and Luzin's condition $(N)$
Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 411-418

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A function $f\colon G\to\mathbb R^n$, where $G$ is an open set in $\mathbb R^n$, has the $N^{-1}$-property if for all $E\subset\mathbb R^n$ we have $\bigl\{|E|=0\Rightarrow|f^{-1}(E)|=0\bigr\}$ ($|\cdot|$ is the Lebesgue measure). The article is concerned with the relations between the $N^{-1}$-property of functions, the maximal rank of derivatives, and the differentiability almost everywhere of composite functions. 
@article{MZM_1995_58_3_a8,
     author = {S. P. Ponomarev},
     title = {The $N^{-1}$-property of maps and {Luzin's} condition $(N)$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {411--418},
     publisher = {mathdoc},
     volume = {58},
     number = {3},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_3_a8/}
}
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S. P. Ponomarev. The $N^{-1}$-property of maps and Luzin's condition $(N)$. Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 411-418. http://geodesic.mathdoc.fr/item/MZM_1995_58_3_a8/