The $N^{-1}$-property of maps and Luzin's condition $(N)$

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.