Some questions of the theory of approximatively differentiable functions (Q1101648)

Let \(\Omega\) be a Borel subset in \(R^ n\) and \(\mu\) is a regular Borel measure on \(\Omega\) such that \(\mu (\Omega)<\infty\). Let f be a mapping from \(\Omega\) into \(R^ m\) such that \(| f| \in L_{\infty}(\Omega,\mu)\). Then for any positive real number \(\epsilon\) there exist \(g\in C(R^ n,R^ m)\) and a compact subset K of \(\Omega\) such that \(\mu(K)>\mu (\Omega)-\epsilon\) and \(f|_ K=g|_ K\). The author considered the case in which we can choose \(g\in C^ r(R^ n,R^ m).\) Using the Whitney extension theorem he reduced the problem to consider some special classes of functions. The author also obtained a right inverse mapping theorem for these classes and extended the results to the case in which \(R^ m\) is replaced by a normed space.