Change of variables formula under minimal assumptions
In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need the a.e. existence of approximate partial derivatives. In Section 3 we consider the general form of the change of variables formula for Sobolev mappings. The author wishes to thank Professor Bogdan Bojarski for many stimulating conversations and suggestions.