Density measures and applications

Abstract

The paper, that continuous some previous work of Sch\"onherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive measures. We study their basic properties and investigate related integrals. Measures taking only the values 0 and 1 are considered as special case. The results are first applied to weak convergence in $\mathcal{L}^\infty(\Omega)$. Then we derive integral representations by means of such measures for several notions of differentiability for integrable functions and we show a kind of mean value theorem for some class of Sobolev functions. Finally we provide a new approach to the generalized Jacobians in the sense of Clarke.