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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
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