Going beyond variation of sets. (April 2017) - Chlebík, Miroslav
Abstract: We study integralgeometric representations of variations of general sets A ⊂ R n without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function χ A is a signed Borel measure on R n with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss–Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a 'measure-theoretic boundary' plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of 'measure-theoretic boundary' and one can address the question to find notions of measure-theoretic boundary that are as fine as possible.