On a global property of approximately differentiable functions (Q1093751)

Let a jet of order \(r\) from a Borel set \(B\) into \(R^m\) denote the set of measurable functions \(\{f^a\}_{| a| \leq r}: B\to R^m,\) where \(a=(a_1,\ldots,a_n)\) is a multi-index. The main result of the paper says that if \(m(B)<\infty\), then the jet \(\{f^a\} \in \operatorname{ap} T^r(B,R^m)\) if and only if for each \(\varepsilon >0\) there exist a compact set \(K\subset B\) and a function \(F\in C^r(R^n,R^m)\) such that \(m(K)>m(B)-\varepsilon\) and \(f^a| K=D^aF| K\). This is an answer to the question formulated by \textit{H. Federer} in ``Geometric measure theory'' [Berlin: Springer Verlag (1969; Zbl 0176.00801), p. 229], and a generalization (for higher orders) of Th. 3.1.16 in Federer's book.