Generalized densities on $\R^n$ and their applications

We examine some generalized densities (called $(\psi, n)$-densities) obtained as a result of strengthening the Lebesgue Density Theorem. It turnes out that these notions are the generalizations of superdensity, enhanced density and $m$-density and have some applications in the theory of sets of finite perimeter and in Sobolev spaces.