## On some descriptive characterizations of the $$N^{-1}$$ property.(English)Zbl 0891.26007

The author says that a function has $$N^{-1}$$ property iff whenever $$m(A)= 0$$, then $$m(f^{-1}(A))= 0$$. If $$X$$ is a space of continuous functions from a compact subset of $$\mathbb{R}^m$$ into $$\mathbb{R}^k$$, then the subset consisting of all mappings with the $$N^{-1}$$ property is the first category $$F_{\sigma\delta}$$ and if $$X$$ is a space of a.e. differentiable functions on a closed cube $$Q$$, then $$N^{-1}$$ is a $$G_\delta$$ subset of $$X$$.

### MSC:

 26B35 Special properties of functions of several variables, Hölder conditions, etc. 26B30 Absolutely continuous real functions of several variables, functions of bounded variation

### Keywords:

Luzin condition; $$N^{-1}$$ property
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